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+{"text":"\\section{Introduction. Dimension increase and splitting}\n\nFinite dimensional integrable systems have been a rarity for a\nvery long time; up to few decades ago, they could be easily\nenumerated: harmonic oscillators, the Kepler system, a few\nspinning tops.\n\nApart from harmonic oscillators -- which give linear equations and are the prototype of integrability -- these are integrated by exploiting their symmetries and the\nassociated integrals of motion to reduce the problem to a lower\ndimensional one; thus, we have {\\it symmetry reduction}.\n\nThe situation changed radically in the seventies, when entire\nclasses of new integrable systems appeared, integrable by the Lax\npair construction \\cite{Lax}. Among these a\nprominent role -- also due to priority -- is taken by the Calogero\nsystem \\cite{Cal}. In this case, one has an arbitrary number $N$\nof points on the line, interacting via a certain  pair potential;\nthus the system is described by a natural hamiltonian $H = (1\/2)\n\\sum_i (\\dot{x}_i )^2 + V (x)$. Calogero showed that there is an\ninvertible map from ${\\bf R}^N$ to $GL(N)$, mapping the vector $x$ to a\nmatrix $L(x)$, such that the evolution of $x (t)$ results in an\nevolution of $L (t) := L [x(t)]$ governed by Lax equations; thus\none integrates $L(t)$ and obtains  $x(t)$.\n\nAs pointed out by Kazhdan, Kostant and Sternberg \\cite{KKS}, this\nalso initiated a new way of integrating systems (which they\nstudied geometrically): indeed, rather than trying to reduce the\ndimension of the system, Calogero passed to consider a system of\n{\\it higher dimension}\\footnote{Another classical\ninstance of simplification by dimension increase is the trick to\nlinearize the matrix Riccati equation. See also \\cite{CGM,ShW} in\nthis context.}. In this note we want to show that this approach is fruitful also beyond the realm of integrable hamiltonian systems.\n\nThe key observation made by Kazhdan, Kostant and Sternberg\n\\cite{KKS}, see also \\cite{Mar,MSSV}, in the context of\nHamiltonian systems is that given a differential equation $\\Delta$\nin dimension $n$, in some cases it may be helpful to increase its\ndimension: this is in particular the case when it is possible to\ndescribe $\\Delta$ as originating from either the symmetry\nreduction or the projection to an $n$-dimensional manifold of a\nsimpler (e.g. linear) equations $\\Delta^*$ in dimension $n+m$.\n\nIt was remarked in \\cite{con} that this approach does apply also\nto non-hamiltonian systems, and in particular to systems in\nPoincar\\'e-Dulac normal form \\cite{Arn,CiGa,enf} (a remark to this effect was already contained in \\cite{enf}). Essentially, a system\nin normal form with only a finite number of resonances can\nalways be mapped to a linear system, and is equivalent to the\nlatter on a certain invariant manifold. As observed there, such a\nprocedure is geometrically interesting, but equivalent\nto a classical technique already known to Dulac \\cite{Dul}, and\nabscribed by him to Horn and Lyapounov. In more general terms,\ni.e. outside the scope of normal forms theory, this corresponds to\nthe situation of a finite dimensional centralizer for a certain\nalgebra associated to the vector field \\cite{cag}.\n\nIn the present note, we show how ``dimension increasing''\nnicely combines with a ``splitting'' (in a sense to be described\nbelow) approach \\cite{split} in the case of an infinite\nnumber of resonances\\footnote{For a different use of the splitting approach in normal forms type problems, see \\cite{YP}.}; this case cannot be tackled by the Horn--Lyapounov--Dulac approach.\n\nRoughly speaking, the procedure presented in this note combines\nthe two approaches mentioned above, i.e. describing a nonlinear\nsystem as projection of a linear one and symmetry reduction in the\nsense described in \\cite{split}.\n\nA more abstract (and general) treatment is given elsewhere \\cite{seb}; in the present note we adopt an approach and notation aimed at\napplications, and discuss a number of concrete applications. We\nfocus on systems in normal form (sect.2) and in particular on the\nembedding and splitting of systems with infinitely many resonances (sect.3). Some simple examples are discussed in sect.4, while sect.5 is devoted to (in general non-hamiltonian) nonlinearly perturbed oscillators, and sect.6 to bifurcation problems.\n\n\\section{Equations in normal form; resonances}\n\nWe consider an ODE in $R^n$ with a fixed point in the origin and\nexpanded around this in a power series; we write this in the form\n$$ {{\\rm d} x \\over {\\rm d} t } \\ = \\ f (x) \\ = \\ A x \\, + \\, \\sum_{k=1}^\\infty f_k (x) \\eqno(1) $$\nwhere $f_k$ is polynomial with $f_k (a x) = a^{k+1} (x)$, and we have singled out the\nlinear part $f_0 (x) = A x$. We also write $F$ for the nonlinear\npart of $f$, i.e. $f^i (x) = A^i_j x^j + F^i (x)$.\n\nAs well known, the matrix $A$ can be uniquely decomposed into its\nsemisimple and nilpotent parts, which commute with each other and\nhence with $A$:\n$$ A = A_s + A_n \\ ; \\ [A_s , A_n ] = 0 . $$\nIn the following, we will use the following vector fields ($\\partial_i\n:= \\partial \/ \\partial x^i$):\n$$ X_A := (Ax)^i \\partial_i \\ , \\ X_0 = (A_s)^i \\partial_i \\ , \\ X_F = F^i (x) \\partial_i \\ , \\ X_f = f^i (x) \\partial_i \\ . $$\n\n\\subsection{Resonances}\n\nLet us denote by $\\{ \\lambda_1 , ... , \\lambda_n \\}$ the eigenvalues of\n$A$; take a basis $\\{ {\\bf e}_1 , ... , {\\bf e}_n \\}$ in $R^n$ consisting\nof generalized eigenvectors of $A$, i.e. eigenvectors of $A_s$: $A_s {\\bf e}_j = \\lambda_j {\\bf e}_j$. We will use $x$ coordinates in this basis, and the multiindex notation\n$$ x^\\mu \\ := \\ x_1^{\\mu_1} ... x_n^{\\mu_n} \\ . $$\n\nWe say that the vector monomial ${\\bf v}_{\\mu,\\alpha} := x^\\mu {\\bf e}_\\alpha$ is {\\it\nresonant} with $A$ if\n$$ (\\mu \\cdot \\lambda) \\ := \\sum_{i=1}^n \\mu_i \\lambda_i \\ = \\ \\lambda_\\alpha \\ \\\n{\\rm with} \\ \\ \\mu_i \\ge 0 \\ , \\ |\\mu| := \\sum_{i=1}^n \\mu_i \\ge 1 \\ . \\eqno(2) $$\n\nThe relation $(\\mu \\cdot \\lambda) = \\lambda_\\alpha$ is said to be a {\\it\nresonance relation} related to the eigenvalue $\\lambda_\\alpha$, and the\ninteger $|\\mu|$ is said to be the {\\it order} of the resonance.\nIn our context it is useful to include order one resonances in the definition (albeit the {\\em trivial} order one resonances given by $\\lambda_\\alpha = \\lambda_\\alpha$ are of little interest).\nNote that here one could as well consider $A_s$ rather than $A$.\n\nThe space of vectors resonant with (the semisimple part of) $A$ is\ndefined as the linear span of the vectors ${\\bf v}_{\\mu,\\alpha}$ defined\nabove. \n\n\n\\subsection{Normal forms}\n\nWe say (see e.g. \\cite{Arn,CiGa,enf}) that (1) is in\nPoincar\\'e-Dulac normal form\\footnote{The reader should be warned that some different definition is also in use \\cite{Elp}.} if its nonlinear part $F(x)$ is resonant with $A$. This implies that\n$$ \\[ \\, X_0 \\, , \\, X_F \\, \\] \\ = \\ 0 \\ . \\eqno(3) $$\n\nIt should be mentioned that the presence of a nilpotent part $A_n$\nin $A$ introduces some subtleties. If (3) holds, then both $X_A$ and $X_F$ commute with $X_0$, and therefore\n$$ \\[ \\, X_0 \\, , \\, X_f \\, \\] \\ = \\ 0 \\ , \\eqno(4) $$\ni.e. the system has a symmetry described by a semisimple matrix.\n\nAs well known, starting from any dynamical system (or vector\nfield)  of the form (1), we can arrive at a dynamical system (or\nvector field) in Poincar\\'e-Dulac normal form by means of a\nsequence (in general, infinite) of near-identity transformations \nobtained by means of the Poincar\\'e algorithm; these combine into\na near-identity transformation $H$ defined by a series which is in\ngeneral only formal. \n\n\\medskip\\noindent\n{\\bf Remark 1.} We may reformulate the definition of systems in\nnormal form by saying that $f$ is is normal form if and only if\n$X_F$ is in the centralizer of $X_0$. $\\odot$\n\n\\medskip\\noindent\n{\\bf Remark 2.} If the system is required to have some symmetry,\nsay $[X_f , X_g ] = 0$ with (in an obvious notation) $g^i (x) =\nB^i_j x^j + G^i$, then $[X_f , X_B ] = 0$ as well, i.e. $X_F$ is\nin the centralizers of both $X_0$ and $X_B$. More generally, if\n$X_f$ is an element of some Lie algebra ${\\cal G}$, then under suitable conditions, see \\cite{CiGa}, it can be put in joint normal form, i.e. (again with obvious notations) $[X_f , X_{B_i} ] = 0$.  $\\odot$\n\n\\subsection{Sporadic resonances and invariance relations}\n\nLet us consider again the resonance equation (2). It is clear that\nif there are non-negative integers $\\sigma_i$ (some of them nonzero) such that \n$$ \\sum_{i=1}^n \\, \\sigma_i \\lambda_i \\ = \\ 0 \\ , \\eqno(5) $$\nthen we always have infinitely many resonances. The monomial $\\phi\n= x^\\sigma$ will be called a {\\it resonant scalar monomial}. It is an invariant of $X_0$, and any multiindex $\\mu$\nwith $\\mu_i = k \\sigma_i + \\delta_{i \\alpha}$ provides a resonance relation\n$(\\mu \\cdot \\lambda ) = \\lambda_\\alpha$ related to the eigenvalue $\\lambda_\\alpha$; in\nother words, any monomial $x^{k \\sigma} x^\\alpha = \\phi x^\\alpha$ is resonant,\nand so is any vector ${\\bf v}_{k \\sigma + e_\\alpha , \\alpha}$.\n\nTherefore, we say that (5) identifies a {\\it invariance relation}.\nThe presence of invariance relations is the only way to have\ninfinitely many resonances in a finite dimensional system (see \\cite{enf}).\n\nAny nontrivial resonance (2) such that there is no $\\sigma$ with $\\sigma_i \\le \\mu_i$ (for all $i=1,...,n$) providing an invariance relation, is said to be a {\\it sporadic resonance}. Sporadic resonances are always in finite number (if any) in a finite dimensional system \\cite{enf}.\n\nAny invariance relation (5) such that there is no $\\nu$ with\n$\\nu_i \\le \\sigma_i$ (and of course $\\nu \\not= \\sigma$) providing another\ninvariance relation, is said to be an {\\it elementary invariance\nrelation}. Every invariance relation is a linear combination (with nonnegative integer coefficients) of elementary ones. Elementary invariance relations are always in finite number (if any) in a finite dimensional system \\cite{enf}.\n\n\n\n\n\\section{Embedding systems with invariance relations in quasi-linear systems}\n\nIn this section we will discuss how the procedure described in\n\\cite{con} generalizes, in connection with a ``splitting'' of the\nsystem described by $Y$ \\cite{split}, in the presence of\ninvariance relations.\n\nWe should preliminarily identify all sporadic resonances $(\\mu\n\\cdot \\lambda ) = \\lambda_\\alpha$ and elementary invariance relations $(\\sigma\n\\cdot \\lambda ) = 0$. We associate resonant monomials $x^\\mu$ and\nresonant vectors ${\\bf v}_{\\mu,\\alpha}$ to the former ones, and invariant\nmonomials $x^\\sigma$ to the latter ones.\n\nWe then introduce two set of new coordinates: these will be the\ncoordinates $w^1 , ... , w^r$ in correspondence with sporadic\nresonances (as in \\cite{con}), and other new coordinates $\\phi^1,\n... , \\phi^m$ in correspondence with elementary invariance\nrelations.\n\nWe should also assign evolution equations for the $w$ and $\\phi$ coordinates; these will be given in agreement with (1) itself. That is, the equations for the $w$ will be\n$$ {{\\rm d} w^j \\over {\\rm d} t} \\ = \\ {\\partial w^j \\over \\partial x^i} \\, {{\\rm d} x^i \\over {\\rm d} t} \\ := \\ h^j (x,w,\\phi) \\ ; \\eqno(6) $$\nand as for the $\\phi$'s we assign\n$$ {{\\rm d} \\phi^a \\over {\\rm d} t} \\ = \\ {\\partial \\phi^a \\over \\partial x^i} \\, {{\\rm d} x^i \\over {\\rm d} t} \\ := \\ z^a (x,w,\\phi) \\ . \\eqno(7) $$\n\nWe will thus consider the enlarged space $W = (x,w,\\phi) =\n{\\bf R}^{n+r+m}$, and in this the vector field\n$$ Y \\ = \\ f^i (x,w,\\phi) \\, {\\partial \\over \\partial x^i} \\ + \\ h^j (x,w,\\phi) \\, {\\partial \\over \\partial w^j} \\ + \\ z^a (x,w,\\phi) \\, {\\partial \\over \\partial \\phi^a} \\ . \\eqno(8) $$\n\nNote that some ambiguity is present here, in that we can write the\ncoefficients of this vector field in different ways as a function\nof the $x,w,\\phi$. Indeed, the vector field $Y$ is uniquely\ndefined only on the manifold identified by $\\psi^i := w^i -\nx^{\\mu_{(i)}} = 0$, $\\phi_a - \\zeta_a (x) = 0$.\n\n\\medskip\\noindent\n{\\bf Lemma 1.} {\\it The $(n+m)$-dimensional manifold $M \\subset W$ identified by $\\psi^i := w^i - x^{\\mu^{(i)}} = 0$ is invariant under the flow of $Y$.}\n\n\\medskip\\noindent\n{\\bf Proof.} Obvious by construction. $\\triangle$\n\n\\medskip\\noindent\n{\\bf Lemma 2.} {\\it The functions $z^a$ defined in (7) can be written in terms of the $\\phi$ variables alone, i.e. $\\partial z^a \/ \\partial x^i = \\partial z^a \/ \\partial w^j = 0$.}\n\n\\medskip\\noindent\n{\\bf Proof.} Every analytic invariant of $X_0$ can be represented as a convergent series in the $z^a$ (see \\cite{enf}); their evolution is also invariant under the $X_0$ action, hence can also be written in terms of invariants, hence of the $z^a$ themselves. $\\triangle$\n\n\\medskip\\noindent\n{\\bf Corollary 1.} {\\it The evolution of the $\\phi$ variables is described by a (nonlinear) equation in the $\\phi$ variables only.}\n\n\\medskip\\noindent\n{\\bf Proof.} This is merely a restatement of lemma 2 above. Note that the equations for $x$ and $w$ depend on $\\phi$ and are therefore nonautonomous. $\\triangle$\n\n\n\\medskip\\noindent\n{\\bf Proposition 3.} {\\it The analytic functions $f^i$ and $h^j$\ndefined above can be written as linear in the $x$ and $w$\nvariables, the coefficients being functions of the $\\phi$ variables.}\n\n\\medskip\\noindent\n{\\bf Proof.} Recall each $w^j$ is a monomial $w^j =\nx_1^{p_1} ... x_n^{p_n}$ with $(p , \\lambda) = \\lambda_s$ for some\n$s=1,...,n$; with reference to this integer $s$, we add a label to\n$w_j$, i.e. write $w_j^{(s)}$. \nBy construction and by the results above, each $f^m$ can be written in \nthe form $f^m = a^m (\\phi) \\cdot x_m + \\sum_k c_k^m (\\phi) w_k^{(m)}$, with analytic $a_m$ and $c_k^m$. So the assertion for the $f^m$ is obvious.\n\nThe time derivative of $w_j^{(s)}$ under the flow of (1) will be ${\\dot w}_j^{(s)} = (\\partial w_j^{(s)} \/ \\partial x^m ) f^m$. \nTherefore it is sufficient to show that $(\\partial w_j^{(s)}\/\\partial x^m ) w_k^{(m)}$ is zero or a multiple of $x^m$ or of some $w_\\ell^{(s)}$ with a suitable resonant scalar monomial as a factor. But the above operation just means to replace one factor $x^m$ by $w_k^{(m)}$, and the resulting linear combination of the eigenvalues still yields a resonance relation (2) with $\\lambda_s$ on the right hand side. See \\cite{seb} for a different approach to the proof. $\\triangle$\n\n\n\\medskip\\noindent\n{\\bf Corollary 2.} {\\it The evolution of the $x$ and $w$ variables is described by nonautonomous linear equations, obtained by inserting the solution $\\phi = \\phi (t)$ of the equations for $\\phi$ in the general equations ${\\dot x} = f (x,w,\\phi)$, ${\\dot w} = h (x,w,\\phi)$.}\n\\medskip\n\n\\medskip\\noindent\n{\\bf Proof.} Obvious. $\\triangle$\n\\medskip\n\n\\medskip\\noindent\n{\\bf Remark 3.} We will also say that the vector field $Y$ is quasi-linear, meaning by this that it is linear in the $x$ and $w$ variables. In this way we recover -- as a special case -- the situation discussed in \\cite{Lie} as well as the terminology used there. $\\odot$\n\n\\medskip\\noindent\n{\\bf Remark 4.} The results obtained here extend and unify those given in \\cite{split,con}; see also \\cite{enf}. As the $\\phi$ identify group orbits for the group $G$ generated by the Lie algebra, we interpret $\\dot\\phi = z (\\phi)$ as an equation in orbit space, and the equation for $(x,w)$ as an equation on the Lie group $G$. Methods for the solution of the latter are discussed in \\cite{WN}, see also \\cite{CGM}. $\\odot$\n\n\\medskip\\noindent\n{\\bf Remark 5.} If no invariance relations are present, hence no\n$\\phi$ variables are introduced, then the system describing the\ntime evolution of the $x,w$ variables is linear; this is the\nsituation studied in \\cite{con}. Note that in this case we have\nexactly the interpretation of normal forms as projection of a\nlinear system to an invariant manifold, without symmetry\nreduction. $\\odot$\n\n\\medskip\\noindent\n{\\bf Remark 6.} If there are no sporadic resonances of order greater than one then Proposition 3 yields a linear system for the $f^i$, with functions of the $\\phi$ variables as coefficients. Therefore, upon solving the reduced equation for the $\\phi$ variables one obtains a non-autonomous linear system. Moreover, if all eigenvalues are distinct then we have a product system of one-dimensional equations. $\\odot$\n\n\n\\medskip\\noindent\n{\\bf Remark 7.} Finally, we note that if $\\phi (t)$ converges to some $\\phi_0$ (this is always the case if the $\\phi$ space is one-dimensional and $|\\phi (t)|$ does not escape to infinity), the asymptotic evolution of the system is governed by a linear autonomous equation for $x$ and $w$ (see \\cite{Thi} for the behavior of \nasymptotically autonomous equations). Similarly, if there is a periodic solution $\\bar\\phi (t)$ with $\\phi (t) \\to \\bar\\phi (t)$, the asymptotic evolution of the system is governed by a linear equation with periodic coefficients for $x$ and $w$. $\\odot$\n\n\\section{Examples}\n\n\n{\\bf Example 1.} (See \\cite{con}). For $A = {\\rm diag} (1,k)$, $k \\in {\\bf N}$, the only resonant vector is ${\\bf v} = x^k {\\bf e}_2$, corresponding to a sporadic resonance, and there is no invariance relation. Systems in normal form correspond to\n$$ X = x \\partial_x + (k y + c x^k) \\partial_y $$\nwith $c$ a real constant. According to our procedure, we define $w = x^k$, and obtain\n$$ Y \\ = \\ x \\partial_x + (k y + c w) \\partial_y + k w \\partial_w \\ ; $$\nthe invariant manifold $M$ is given by $\\psi := w - x^k = 0$. The\nsolution to the system in $W$ for initial data $(x_0,y_0,w_0)$ is\n$ x(t) = x_0 e^t$, $y(t) = y_0 e^{kt} + (c_1 k w_0) t e^{kt}$,\n$w(t) = w_0 e^{kt}$; for initial data on $M$, i.e. $w_0 = x_0^k$,\nthe solution remains on $M$ and its projection to ${\\bf R}^2 = (x,y)$\nis $ x(t)= x_0 e^t$, $y (t) = [ y_0 + (c_1 k x_0^k) t ] e^{kt} $.\n\n\n\\medskip\\noindent\n{\\bf Example 2.} Consider the matrix\n$$ A \\ = \\ \\pmatrix{ 0 & -1 & 0 \\cr 1 & 0 & 0 \\cr 0 & 0 & 1 \\cr} $$\nwith eigenvalues $(-i,i,1)$. There is one elementary invariance\nrelation, $\\lambda_1 + \\lambda_2 = 0$, and no sporadic resonance. The linear centralizer is spanned by $A$ itself and by matrices $ D_1 = {\\rm diag} (1,1,0)$ and $D_2 = {\\rm diag} (0,0,1)$. The ring of\ninvariants is generated by $r^2 := x^2 + y^2$. Systems in normal\nforms are written as\n$$ \\begin{array}{l}\n{\\dot x} = \\alpha (r^2) x - \\beta (r^2) y \\\\\n{\\dot y} = \\beta (r^2) x + \\alpha (r^2) y \\\\\n{\\dot z} = \\gamma (r^2) z \\end{array} $$\nwhere $\\alpha , \\beta , \\gamma $ are arbitrary power series.\n\nFollowing our procedure, we introduce one further variable $\\phi = r^2$; for this we have $d \\phi \/ d t = 2 (x {\\dot x} + y {\\dot y} ) = 2 r^2 \\alpha (r^2 ) $. Hence the system in $W = {\\bf R}^4$ is written as\n$$ \\begin{array}{l}\n{\\dot x} \\ = \\ \\alpha (\\phi) \\, x \\ - \\ \\beta (\\phi) \\, y \\\\\n{\\dot y} \\ = \\ \\beta (\\phi) \\, x \\ + \\ \\alpha (\\phi) \\, y \\\\\n{\\dot z} \\ = \\ \\gamma (\\phi) \\, z \\\\\n{\\dot \\phi} = \\ 2 \\, \\phi \\, \\alpha (\\phi) \\ . \\end{array} $$\n\n\\medskip\\noindent\n{\\bf Example 3.} Consider the matrix\n$$ A \\ = \\ \\pmatrix{ 0 & -1 & 0 & 0 \\cr 1 & 0 & 0 & 0 \\cr  0 & 0 & 1 & 0 \\cr 0 & 0 & 0 & k \\cr} $$\n($k \\in {\\bf N}$, $k > 1$) with eigenvalues $(-i,+i,1,k)$. There\nis one sporadic resonance, $k \\lambda_3 = \\lambda_4$, and one elementary\ninvariance relation, $\\lambda_1 + \\lambda_2 = 0$. The linear centralizer\nis spanned the  matrices $D_1 = {\\rm diag} (1,1,0,0)$, $D_2 = {\\rm\ndiag} (0,0,1,0)$, $D_3 = {\\rm diag} (0,0,0,1)$ together with\n$$ M_1 = \\pmatrix{0&-1&0&0\\cr 1&0&0&0\\cr 0&0&0&0\\cr 0&0&0&0\\cr} \\ \\ {\\rm and} \\ \\ M_2 = \\pmatrix{0&0&0&0\\cr 0&0&0&0\\cr 0&0&0&0\\cr 0&0&k&0\\cr} \\ . $$\nSystems in normal forms are written as\n$$ \\begin{array}{l}\n{\\dot x}_1 = \\alpha (r^2) x_1 - \\beta (r^2) x_2 \\\\\n{\\dot x}_2 = \\beta (r^2) x_1 + \\alpha (r^2) x_2 \\\\\n{\\dot x}_3 = \\gamma (r^2) x_3 \\\\\n{\\dot x}_4 = \\eta (r^2) x_4 + \\theta (r^2) x_3^k \\end{array} $$\nwhere $r^2 := x^2 + y^2$ and $\\alpha , \\beta , \\gamma , \\eta , \\theta $ are arbitrary power series.\n\nFollowing our procedure we introduce $\\phi = r^2$, for which $d \\phi \/ d t = 2 (x {\\dot x} + y {\\dot y} ) = 2 r^2 \\alpha (r^2 ) $, and $w = x_3^k$ for which $d w \/ d t = k x_3^{k-1} {\\dot x}_3 = k \\gamma (r^2) x_3^k$. Hence the system in $W = {\\bf R}^5$ is written as\n$$ \\begin{array}{l}\n{\\dot x}_1 = \\alpha (\\phi) x_1 - \\beta (\\phi) x_2 \\\\\n{\\dot x}_2 = \\beta (\\phi) x_1 + \\alpha (\\phi) x_2 \\\\\n{\\dot x}_3 = \\gamma (\\phi) x_3 \\\\\n{\\dot w} = k \\gamma (\\phi) w \\\\\n{\\dot \\phi} = 2 \\phi \\alpha (\\phi) \\ . \\end{array} $$\n\n\\section{Perturbed oscillators}\n\n{\\bf Example 4. Perturbation of oscillators in 1:1 resonance.}\nConsider the matrix\n$$ A \\ = \\ \\pmatrix{ 0 & -1 & 0 & 0 \\cr 1 & 0 & 0 & 0 \\cr  0 & 0 & 0 & -1 \\cr 0 & 0 & 1 & 0 \\cr} \\ , $$\nwhich we also write in block form as\n$$ A \\ = \\ \\pmatrix{ J & 0 \\cr 0 & J \\cr} \\ , \\ {\\rm where} \\ J \\ = \\ \\pmatrix{0&-1\\cr 1&0\\cr} \\ , $$\nwith eigenvalues $(-i,+i,-i,i)$. There are no sporadic resonances of order greater than one, and four elementary invariance relations:\n$$ \\lambda_1 + \\lambda_2 = 0 \\ , \\ \\lambda_3 + \\lambda_4 = 0 \\ , \\\n\\lambda_1 + \\lambda_4 = 0 \\ , \\ \\lambda_2 + \\lambda_3 = 0 \\ ; $$\nall other resonances can be described in terms of these. \nWe stress that the equations describing these invariance relations are\nlinearly dependent; however they should be considered, according\nto our definition, as different elementary ones. Corresponding to\nthis, the associated invariant quantities will not be functionally\nindependent (obviously, we cannot have more than three independent\ninvariants for a flow in ${\\bf R}^4$).\n\nThe linear centralizer of $A$ is an eight-dimensional algebra, spanned by the following matrices (in two by two block notation):\n$$\\begin{array}{l}\nB_1 = \\pmatrix{I&0\\cr 0&0\\cr} \\ , \\\nB_2 = \\pmatrix{0&0\\cr 0&I\\cr} \\ , \\\nB_3 = \\pmatrix{0&I\\cr I&0\\cr} \\ , \\\nB_4 = \\pmatrix{0&J\\cr -J&0\\cr} \\ , \\\\\n~ \\\\\nS_1 = \\pmatrix{J&0\\cr 0&0\\cr} \\ , \\\nS_2 = \\pmatrix{0&0\\cr 0&J\\cr} \\ , \\\nS_3 = \\pmatrix{0&I\\cr -I&0\\cr} \\ , \\\nS_4 = \\pmatrix{0&J\\cr J&0\\cr} \\ .\n\\end{array} \\eqno(9) $$\nNote here we have chosen a basis with $B_i = B_i^+$, $S_i = - S_i^+$.\n\nThe linear system $\\.\\xi = A \\xi$ describes two oscillators in 1:1 resonance; the normal form will correspond to a perturbation of these, generically breaking the exchange symmetry among the two oscillators.\nSystems in normal form are compactly written as\n$$\n\\pmatrix{ {\\dot x}\\cr {\\dot y}\\cr {\\dot z}\\cr {\\dot w}\\cr} \\ = \\\n\\pmatrix{\n\\alpha & - \\beta & \\gamma & - \\eta \\cr\n\\beta & \\alpha & \\eta & \\gamma \\cr\n\\mu & - \\nu & \\sigma & - \\tau \\cr\n\\nu & \\mu & \\tau & \\sigma \\cr}\n\\ \\pmatrix{ x\\cr y\\cr z\\cr w\\cr} $$\nwhere $\\alpha , \\beta , ... , \\tau$ are arbitrary power series in the elementary invariants\n$$ \\phi_1 = x^2 + y^2 \\ , \\ \\phi_2 = z^2 + w^2 \\ , \\ \\phi_3 = xz + yw \\ , \\ \\phi_4 = xw - yz \\ ; $$\nnote that $\\phi_a = (\\xi , B_a \\xi)$, with $(.,.)$ the scalar product.\nWe abbreviate the above evolution equation as\n$$ {\\dot \\xi} \\ = \\ K (\\phi) \\ \\xi \\ . $$\nThe evolution equations for the $\\phi$, as required by our procedure, are simply (only the selfadjoint matrices defined in (9) appear)\n$$ {\\dot \\phi}_a = \\( \\xi , (B_a K + K^+ B_a ) \\xi \\) \\ . $$\n\n\n\n\\medskip\\noindent\n{\\bf Example 5. Perturbation of oscillators in $1:k$ resonance.}\nConsider the matrix\n$$ A \\ = \\ \\pmatrix{ 0 & -1 & 0 & 0 \\cr 1 & 0 & 0 & 0 \\cr  0 & 0 & 0 & -k \\cr 0 & 0 & k & 0 \\cr} $$\n(with $k \\in {\\bf N}$, $k > 1$) with eigenvalues $(-i,+i,-i\nk,ik)$. This is put in diagonal form passing to\nvariables\n$$ \\xi_1 = (x_1 - i x_2)\/2 \\ , \\ \\xi_2 = (x_1 + i x_2)\/2 \\ , \\ \n\\xi_3 = (x_3 - i x_4)\/2 \\ , \\ \\xi_4 = (x_3 + i x_4)\/2 \\ , $$\nwhich we use in intermediate computations below.\nWe also write $ \\xi = \\Lambda x$, $x = \\Lambda^{-1} \\xi$, with\n$$ \\Lambda = {1 \\over 2} \\ \\pmatrix{1&-i&0&0\\cr 1&1&0&0\\cr 0&0&1&-i\\cr 0&0&1&i\\cr} \\ ; \\ \\Lambda^{-1} = \\pmatrix{1&1&0&0\\cr i&-i&0&0\\cr 0&0&1&1\\cr 0&0&i&-i\\cr} \\ . $$\n\nThere are four sporadic resonances:\n$$ k \\lambda_1 = \\lambda_3 \\ ,  \\ k \\lambda_2 = \\lambda_4 \\ , \\\n\\lambda_3 + (k-1) \\lambda_2 = \\lambda_1 \\ ,  \\ \\lambda_4 + (k-1) \\lambda_1 = \\lambda_2 \\ . $$\nMoreover, there are four elementary invariance relations:\n$$ \\lambda_1 + \\lambda_2 = 0 \\ , \\ \\lambda_3 + \\lambda_4 = 0 \\ , \\\nk \\lambda_1 + \\lambda_4 = 0 \\ , \\ k \\lambda_2 + \\lambda_3 = 0 \\ . $$\nAll other resonances can be described in terms of these. \n\nSystems in normal forms are written as\n$$ \\begin{array}{l}\n\\.\\xi_1 = \\alpha_1 \\xi_1 + \\vartheta_1 \\xi_2^{k-1} \\xi_3 \\\\\n\\.\\xi_2 = \\alpha_2 \\xi_2 + \\vartheta_2 \\xi_1^{k-1} \\xi_4 \\\\\n\\.\\xi_3 = \\alpha_3 \\xi_3 + \\vartheta_3 \\xi_1^k \\\\\n\\.\\xi_4 = \\alpha_4 \\xi_4 + \\vartheta_4 \\xi_2^k \\ ,\n\\end{array} \\eqno(10) $$\nwhere $\\alpha_i , \\vartheta_i $ are arbitrary power series in the invariants of the linear flow.\n\nFollowing our procedure, we introduce variables\n$$ w_1 = \\xi_1^k \\ , \\ w_2 = \\xi_2^k \\ , \\\nw_3 = \\xi_2^{k-1} \\xi_3 \\ , \\ w_4 = \\xi_1^{k-1} \\xi_4 \\ ,  $$\nrelated to sporadic resonances. We also introduce variables\nrelated to elementary invariance relations, given by\n$$ \\phi_1 = \\xi_1 \\xi_2 \\ , \\ \\phi_2 = \\xi_3 \\xi_4 \\ , \\\n\\phi_3 = \\xi_1^k \\xi_4 \\ , \\ \\phi_4 = \\xi_2^k \\xi_3 \\ . $$\n\nWe must then introduce evolution equations for the $w$ and $\\phi$ variables according to our procedure, i.e. according to (6) and (7) above.\nAs for the $w$, we get\n$$ \\begin{array}{l}\n\\.w_1 = k \\alpha_1 w_1 + \\vartheta_1 \\phi_1^{k-1} \\xi_3 \\\\\n\\.w_2 = k \\alpha_2 w_2 + \\vartheta_2 \\phi_1^{k-1} \\xi_4 \\\\\n\\.w_3 = [\\alpha_3 + (k-1) \\alpha_2 ] w_3 + [ (k-1) \\vartheta_2 \\phi_1^{k-2} \\phi_2 + \\vartheta_3 \\phi_1^{k-1} ] \\xi_1 \\\\\n\\.w_4 = [\\alpha_4 + (k-1) \\alpha_2 ] w_4 + [ (k-1) \\vartheta_1 \\phi_1^{k-2} \\phi_2 + \\vartheta_4 \\phi_1^{k-1} ] \\xi_2 \\ .\n\\end{array}$$\n\nLet us now consider the equations for the $\\phi$; we easily get\n$$ \\begin{array}{l}\n\\.\\phi_1 = (\\alpha_1 + \\alpha_2 ) \\phi_1 + \\vartheta_2 \\phi_3 + \\vartheta_1 \\phi_4 \\\\\n\\.\\phi_2 = (\\alpha_3 + \\alpha_4 ) \\phi_2 + \\vartheta_3 \\phi_3 + \\vartheta_4 \\phi_4 \\\\\n\\.\\phi_3 = (k \\alpha_1 + \\alpha_4 ) \\phi_3 + k \\vartheta_1 \\phi_1^{k-1} \\phi_2 + \\vartheta_4 \\phi_1^k \\\\\n\\.\\phi_4 = (k \\alpha_2 + \\alpha_3 ) \\phi_4 + k \\vartheta_2 \\phi_1^{k-1} \\phi_2 + \\vartheta_3 \\phi_1^k \\ .\n\\end{array} \\eqno(11) $$\n\nSummarizing, all systems of the form (10) -- i.e. in normal form with respect to the linear part $\\.\\xi = A \\xi$ -- are written as the autonomous system (11) for the $\\phi$ variables, plus a linear nonautonomous system, which introducing the notation $\\eta = (\\xi ; w)$ can be written as\n$$ \\.\\eta \\ = \\ M \\ \\eta $$\nwhere $M = M (\\phi)$ is a matrix which we write as $M = D + L$, where\n$$ D \\ = \\ {\\rm diag} (\\alpha_1 , \\alpha_2 , \\alpha_3 , \\alpha_4 ; k \\alpha_1 , k \\alpha_2 , \\alpha_3 + (k-1) \\alpha_2 , \\alpha_4 + (k-1) \\alpha_1 ) $$\nand $L$ is an off-diagonal sparse matrix with nonzero terms\n$$ \\begin{array}{c}\nL_{17} = \\vartheta_1 \\ , \\ L_{28} = \\vartheta_2 \\ , \\ L_{35} = \\vartheta_3 \\ , \\ L_{46} = \\vartheta_4 \\ ; \\ \nL_{53} = \\vartheta_1 \\phi_1^{k-1} \\ , \\ L_{64} = \\vartheta_2 \\phi_1^{k-1} \\ , \\\\\nL_{71} = [ (k-1) \\vartheta_2 \\phi_1^{k-2} \\phi_2 + \\vartheta_3 \\phi_1^{k-1} ] \\ , \\ \nL_{82} = [ (k-1) \\vartheta_1 \\phi_1^{k-2} \\phi_2 + \\vartheta_4 \\phi_1^{k-1} ] \\ .\n\\end{array}\n$$\n\n\n\\medskip\\noindent\n{\\bf Example 6. Perturbation of two oscillators with no resonance.} Consider the matrix \n$$ A \\ = \\ \\pmatrix{ 0 & -1 & 0 & 0 \\cr 1 & 0 & 0 & 0 \\cr  0 & 0 & 0 & -\\pi \\cr 0 & 0 & \\pi & 0 \\cr} $$\nwith eigenvalues $(-i,+i,-\\pi i ,\\pi i)$.\n\nNow there are no sporadic resonances of order greater than one, and two elementary\ninvariance relations:\n$$ \\lambda_1 + \\lambda_2 = 0 \\ , \\ \\lambda_3 + \\lambda_4 = 0 \\ . $$\nNote that now we have only two invariants, $r_1^2 = x_1^2 + x_2^2$ and $r_2^2 = x_3^2 + x_4^2$: this is easily understood as we have irrational flow on the two-torus ${\\bf T}^2 \\subset {\\bf R}^4$.\nThe centralizer of $A$ corresponds to matrices\n$$ M \\ = \\ \\pmatrix{ a & - b & 0 & 0 \\cr b & a & 0 & 0 \\cr 0&0& c & - d \\cr 0&0& d & c \\cr} $$\nThus systems in normal form will be written as\n$$ \\pmatrix{{\\dot x}_1 \\cr {\\dot x}_2 \\cr {\\dot x}_3 \\cr {\\dot x}_4 \\cr} \\ = \\ \\pmatrix{ \\alpha & - \\beta & 0 & 0 \\cr \\beta & \\alpha & 0 & 0 \\cr 0&0& \\gamma & - \\eta \\cr 0&0& \\eta & \\gamma \\cr} \\ \\pmatrix{x_1 \\cr x_2 \\cr x_3 \\cr x_4 \\cr} $$\nwith $\\alpha , \\beta , \\gamma , \\eta$ being power series in $r_1^2 , r_2^2$.\n\nFollowing our procedure we introduce variables $ \\phi_1 = x_1^2 + x_2^2$ and $\\phi_2 = x_3^2 + x_4^2 $, whose evolution is given by\n$$ {\\dot \\phi}_1 \\ = \\ 2 \\, \\alpha \\, \\phi_1 \\ , \\ {\\dot \\phi}_2 \\ = \\ 2 \\, \\gamma \\, \\phi_2 \\ . $$\n\n\\medskip\\noindent\n{\\bf Example 7. Perturbation of oscillators in 1:1:1 resonance.}\nConsider the six-dimensional matrix written in block form as\n$$ A \\ = \\ \\pmatrix{\\omega J & 0 & 0 \\cr 0 & \\omega J & 0 \\cr 0 & 0 & \\omega J \\cr}  \\ \\ {\\rm where} \\ \\ J \\ = \\ \\pmatrix{0 & - 1 \\cr 1 & 0 \\cr} \\ . $$\nPassing to coordinates $(\\xi_j,\\eta_j) = (p_j+i q_j, p_j- i q_j)$,\nthis reads\n$$ \\^A \\ = \\ {\\rm diag} (i \\omega , - i \\omega , i \\omega , - i \\omega , i \\omega , - i \\omega ) \\ . $$\nThe eigenvalues are $\\lambda_k \\ = \\ (-1)^k i \\omega $, hence there is no sporadic resonance of order greater than one\\footnote{Note these are present in the case of $1:k:\\ell$ resonance (with integers $k,\\ell > 1$).} and nine invariance relations:\n$$ \\begin{array}{lll}\n\\lambda_1 + \\lambda_2 = 0 & \\lambda_3 + \\lambda_4 = 0 & \\lambda_5 + \\lambda_6 = 0 \\\\\n\\lambda_1 + \\lambda_4 = 0 & \\lambda_1 + \\lambda_6 = 0 & \\lambda_3 + \\lambda_6 = 0 \\\\\n\\lambda_2 + \\lambda_3 = 0 & \\lambda_2 + \\lambda_5 = 0 & \\lambda_4 + \\lambda_5 = 0\n\\end{array} $$ The invariants corresponding to these are of course\n$$ \\begin{array}{lll}\n\\^\\phi_1 = \\xi_1 \\eta_1 , & \\^\\phi_2 = \\xi_2 \\eta_2 , & \\^\\phi_3 = \\xi_3 \\eta_3 , \\\\\n\\^\\phi_4 = \\xi_1 \\eta_2 , & \\^\\phi_5 = \\xi_1 \\eta_3 , & \\^\\phi_6 = \\xi_2 \\eta_3 , \\\\\n\\^\\phi_7 = \\xi_2 \\eta_1 , & \\^\\phi_8 = \\xi_3 \\eta_2 , & \\^\\phi_9 =\n\\xi_3 \\eta_2 .\n\\end{array} $$\nGoing back to the original coordinates, expressions for these are\nrecovered from (no sum on repeated indices)\n$$ \\xi_j \\eta_j \\ = \\ (p_j^2 + q_j^2) \\ \\ ; \\ \\ \\xi_j \\eta_k \\ = \\ (p_j p_k + q_j q_k) + i (q_j p_k - p_j q_k) $$\nand is thus more convenient to pass to a different basis for\ninvariant functions, i.e.\n$$ \\begin{array}{l}\n\\phi_1 := \\^\\phi_1 = (p_1^2 + q_1^2) \\ \\ , \\ \\ \\phi_2 := \\^\\phi_2\n= (p_2^2 + q_2^2) \\ \\ , \\ \\\n\\phi_3 := \\^\\phi_3 =  (p_3^2 + q_3^2) \\ ; \\\\\n\\phi_4 := (\\^\\phi_4 + \\^\\phi_7)\/2 = (p_1 p_2 + q_1 q_2) \\ \\ , \\ \\\n\\phi_5 := (\\^\\phi_4 - \\^\\phi_7)\/(2i) = (q_1 p_2 - p_1 q_2) \\ ; \\\\\n\\phi_6 := (\\^\\phi_5 + \\^\\phi_9)\/2 = (p_1 p_3 + q_1 q_3) \\ \\ , \\ \\\n\\phi_7 := (\\^\\phi_5 - \\^\\phi_9)\/(2i) = (q_1 p_3 - p_1 q_3) \\ ; \\\\\n\\phi_8 := (\\^\\phi_6 + \\^\\phi_8)\/2 = (p_2 p_3 + q_2 q_3) \\ \\ , \\ \\\n\\phi_9 := (\\^\\phi_6 - \\^\\phi_8)\/(2i) = (q_2 p_3 - p_2 q_3) \\ .\n\\end{array} $$\n\nThe centralizer $C (A)$ of $A$ in ${\\rm Mat} (6,R)$ is an algebra\nspanned by eighteen matrices; these are written in $2 \\times 2$\nblock form as\n$$ \\pmatrix{C_{11} & C_{12} & C_{13} \\cr C_{21} & C_{22} & C_{23} \\cr\nC_{31} & C_{32} & C_{33} \\cr} $$ where the $C_{ij}$ are \ntwo-dimensional matrices written as ($\\alpha_{ij}$ and $\\beta_{ij}$ real\nconstants)\n$$ C_{ij} \\ = \\ \\alpha_{ij} \\, I \\ + \\ \\beta_{ij} \\, J \\ . $$\nIt is easy to extract from these a basis made of nine selfadjoint\nmatrices $B_k = B_k^+$ and nine antiselfadjoint ones $S_k = -\nS_k^+$. With ${\\bf x} = (p_1,q_1,p_2,q_2,p_3,q_3)$, and $(.,.)$ the\nstandard scalar product in ${\\bf R}^6$ these can be chosen so that $\n\\phi_a = ({\\bf x} , B_a {\\bf x}) $.\n\nIn this compact notation the $1:1:1$ resonant three-dimensional\noscillator is described by ${\\dot {\\bf x}} = A {\\bf x}$. We write the\nnormal form of evolution equations for perturbation of this as\n$$ \\begin{array}{ll}\n{\\dot {\\bf x}} \\ &= \\ K (\\phi ) \\ {\\bf x} \\\\\n{\\dot \\phi}_a \\ &= \\( {\\bf x} \\, , \\, (B_a K + K^+ B_a ) {\\bf x} \\right)\n\\end{array} $$ with $K$ an arbitrary matrix in $C(A)$.\n\n\\section{Bifurcations}\n\n{\\bf Example 8. Hopf bifurcation.} Consider the matrix\n$$ A \\ = \\ \\pmatrix{0&-\\omega_0\\cr \\omega_0&0\\cr} $$\nwith eigenvalues $\\lambda_1 = - i \\omega_0$, $\\lambda_2 = i \\omega_0$. There is\nno sporadic resonance, and one elementary invariance relation, $\\lambda_1 + \\lambda_2 = 0$, with associated invariant $\\phi = x^2 + y^2$.\nThe most general system in normal form is therefore\n$ {\\dot x} = \\alpha (\\phi ) x - \\beta (\\phi ) y$, ${\\dot y} = \\beta (\\phi ) x + \\alpha (\\phi ) y $. According to our procedure, the evolution equation for the new coordinate $\\phi$ will be $ \\.\\phi = 2 \\phi \\alpha (\\phi)$. \nAs the linear part of the system is given by $A$, we must require $\\alpha (0) = 0 $, $\\beta (0) = \\omega_0$.\n\nIn applications, one is interested in the case where the system does also depend on an external (``control'') parameter $\\mu$, which usually does not evolve in time\\footnote{A different framework is provided by dynamic bifurcations \\cite{Ben,Nei}.}, the linear part being given by $A$ at the critical value. In this case the normal form and the $\\phi$ evolution equation read\n$$\\begin{array}{l}\n{\\dot x} \\ = \\ \\alpha (\\phi , \\mu ) \\, x \\ - \\ \\beta (\\phi , \\mu) \\, y \\\\\n{\\dot y} \\ = \\ \\beta (\\phi , \\mu ) \\, x \\ + \\ \\alpha (\\phi , \\mu) \\, y \\\\\n\\.\\phi \\ = \\ 2 \\, \\phi \\, \\alpha (\\phi , \\mu) \\ .\n\\end{array} $$\n\nIn the standard model of Hopf bifurcation, $\\alpha (\\phi , \\mu ) = \\mu - c \\phi $, and we write $\\beta (\\phi , \\mu ) = \\omega_0 + b (\\phi , \\mu)$ with $b(0,0) = 0$. This corresponds to the normal form\n$$ \\cases{\n{\\dot x} = \\mu x - \\omega_0 y - b (x^2+y^2 , \\mu) y - c (x^2 + y^2 ) x & \\cr\n{\\dot y} = \\omega_0 x + \\mu y + b (x^2+y^2 , \\mu) x - c (x^2 + y^2 ) x & \\cr} $$\nwhich in our approach reads\n$$ \\cases{\n{\\dot x} = \\mu x - \\omega_0 y - b (\\phi , \\mu) y - c (\\phi ) x & \\cr\n{\\dot y} = \\omega_0 x + \\mu y + b (\\phi , \\mu) x - c (\\phi ) x & \\cr \n\\.\\phi  =  2 \\, \\phi \\, \\alpha (\\phi , \\mu) \\ . & \\cr}\n$$\nNote that the space of invariants is one-dimensional (with the additional constraint $\\phi \\ge 0$); thus, either $\\phi (t)$ is unbounded for $t > 0$, or it approaches one of the zeroes of the function $\\alpha (\\phi , \\mu)$, say $\\phi_0$. In this case, the system reduces asymptotically to a linear one: \n$$ \n\\pmatrix{{\\dot x} \\cr {\\dot y} \\cr} \\ = \\ \\pmatrix{\n0 & - \\omega_0 - b (\\phi_0 , \\mu ) \\cr \n\\omega_0 + b (\\phi_0 , \\mu) & 0 \\cr} \\ \\pmatrix{ x \\cr y \\cr} \\ . $$\nThe standard analysis of Hopf bifurcation is readily recovered in this way. \n\\bigskip\n\nNote that we can also look at Hopf bifurcation in a slightly\ndifferent way, i.e. include the $\\mu$ variable from the beginning.\nIn this case the matrix $A$ is given by\n$$ A = \\pmatrix{0&-1&0\\cr 1&0&0\\cr 0&0&0\\cr} $$\nwith eigenvalues $(-i,i,0)$ and invariance relations $\\lambda_1 +\n\\lambda_2 = 0 $ and $\\lambda_3 = 0$; the associated invariants are $\\phi$\nand $\\mu$.\nThe linear centralizer of $A$ is spanned by matrices\n$$ M = \\pmatrix{a&-b&0\\cr b&a&0\\cr 0&0&c\\cr} $$\nand correspondingly the normal form will be\n$$ \\begin{array}{l}\n{\\dot x} = \\alpha (\\phi , \\mu) x - \\beta (\\phi , \\mu) y \\\\\n{\\dot y} = \\beta (\\phi , \\mu) x + \\alpha (\\phi , \\mu) y \\\\\n\\.\\mu = \\gamma (\\phi , \\mu)\n\\end{array} $$\n\nThe equation for $\\phi$ is just the one given above, and we are\nled to the same system; interpreting $\\mu$ as an external control\nparameter forces $\\gamma (\\phi , \\mu ) \\equiv 0$ (if not, there is a feedback of the system on the control parameter).\n\n\n\\medskip\\noindent\n{\\bf Example 9. Hamiltonian Hopf bifurcation.} Consider the matrix\n$$ A \\ = \\ \\pmatrix{\\mu & - \\omega & 0 & 0 \\cr \\omega & \\mu & 0 & 0 \\cr 0&0&- \\mu &-\\omega \\cr 0&0&\\omega & -\\mu \\cr} \\ = \\ \\pmatrix{\\mu I + \\omega J & 0 \\cr 0 & - \\mu I + \\omega J \\cr} $$\nwith eigenvalues $\\lambda_1 = -\\mu - i \\omega$, $\\lambda_2 = -\\mu + i \\omega$, $\\lambda_3 = \\mu -i \\omega$, $\\lambda_4 = \\mu + i \\omega$. We assume $\\omega \\not= 0$ (so it could be rescaled to $\\omega = 1$) and $\\mu \\not=0$; the case $\\mu = 0$ corresponds to example 4. In applications, one considers the case where $\\mu$ is an external control parameter and studies the changes as this is varied; $\\mu = 0$ is a critical value.\n\nThe matrix $A$ is diagonalized, for all $\\mu$, passing to variables $\\xi^i$ as in example 5 above. The evolution ${\\dot x} = A x$ preserves the symplectic structure $\\kappa = {\\rm d} x^1 \\wedge {\\rm d} x^3 + {\\rm d} x^2 \\wedge {\\rm d} x^4$.\n\nIt is easy to check, for generic $\\mu$, that there is no sporadic resonance and that there are two elementary invariance relations, given by $\\lambda_1 + \\lambda_4 = 0$ and $\\lambda_2 + \\lambda_3 = 0$. The corresponding invariants will be $ \\varphi_1 = \\xi^1 \\xi^4$, $\\varphi_2 = \\xi^2 \\xi^3$; they are complex conjugate, and correspond to real invariants \n$ \\phi_1 = x_1 x_3 + x_2 x_4$ and $\\phi_2 = x_1 x_4 - x_2 x_3$. \nThese are also written for later reference as $\\phi_a = (1\/2) (\\xi , B_a \\xi)$, with $B_a$ obvious four-dimensional symmetric matrices. \n\nThe linear centralizer of $A$ (for $\\mu \\not= 0$) is given by matrices written in block form, with $\\alpha_k$ and $\\beta_k$ real constants, as\n$$ M \\ = \\ \\pmatrix{\n\\alpha_1 I + \\beta_1 J & 0 \\cr\n0 & \\alpha_2 I + \\beta_2 J \\cr} \\ . $$\n\nCorrespondingly, systems in normal form will be given by\n$$ {\\dot x} \\ = \\ M \\, x $$\nwhere now $\\alpha_k , \\beta_k $ will be functions of the\ninvariants $\\phi_1 , \\phi_2$. Note that such systems in\ngeneral do not preserve the symplectic form $\\kappa$, unless $\\alpha_2 = - \\alpha_1 $ and $\\beta_2 = \\beta_1$. \nThe evolution of the $\\phi$ is given by\n$ {\\dot \\phi}_a = (1\/2) \\( x , (M^+ B_a + B_a M ) x \\)$. \n\nIt is convenient to write the system in terms of the two-dimensional vectors $\\eta_1 = (x^1,x^2)$, $\\eta_2 = (x^3,x^4)$ and $\\phi = (\\phi_1,\\phi_2)$. Moreover, we single out the linear part in the functions $\\alpha_k$ and $\\beta_k$, writing $\\alpha_k (\\phi) = (-1)^{k+1} \\mu + a (\\phi)$, $\\beta_k (\\phi) = \\omega + b_k (\\phi)$ with $a_k (0) = b_k (0) = 0$; the smooth functions are $a_k$, $b_k$ are arbitrary apart from this constraint, and can also depend on the parameters $\\mu$ and $\\omega$.\n\n\nThe system is hence described in our approach by the following equations: \n$$ \\begin{array}{ll}\n{\\dot \\eta}_1 =& (\\mu I + \\omega J) \\, \\eta_1 \\ + \\ [a_1 (\\phi) \\cdot I \\, - \\, b_1 (\\phi) \\cdot J ] \\, \\eta_1 \\ , \\\\\n{\\dot \\eta}_2 =& (- \\mu I + \\omega J) \\, \\eta_2 \\ + \\ [a_2 (\\phi) \\cdot I \\, - \\, b_2 (\\phi) \\cdot J ] \\, \\eta_2 \\ , \\\\ \n{\\dot \\phi} =& \\[ \\( a_1 (\\phi) + a_2 (\\phi) \\) \\cdot I \\, + \\, \\( b_2 (\\phi) - b_1 (\\phi) \\) \\cdot J \\] \\ \\phi \\ . \\end{array} $$ \n\nIf $a_k , b_k$ are such that the system is hamiltonian, the $\\phi$ are always strictly invariant, and we are reduced to a linear system on each level set of $\\phi$; if the $a_k,b_k$ are such that the system is not hamiltonian but the (two-dimensional) equation for the $\\phi$ satisfies the condition for the existence of a limit cycle, as is often the case in bifurcation problems, then remark 7 applies.\n\nLet us look more closely to the perturbation of the case $\\mu = 0$; as recalled above, the analysis of this case can be recovered from example 4. More precisely, we can rescale time so that $\\omega = 1$, and allow the arbitrary functions appearing in the analysis of example 4 to also depend on the parameter $\\mu$.\n\nAlternatively, we can proceed as at the end of the previous example, and include the $\\mu$ variable from the beginning. With a $2 \\oplus 2 \\oplus 1$ block notation, we have now\n$$ A \\ = \\ \\pmatrix{\\mu I + \\omega J & 0 & 0 \\cr 0 & - \\mu I + \\omega J & 0 \\cr 0 & 0 & 0 \\cr} $$\nwith linear centralizer \n$$ M \\ = \\ \\pmatrix{\n\\alpha_1 I + \\beta_1 J & 0 & 0 \\cr\n0 & \\alpha_2 I + \\beta_2 J & 0 \\cr\n0 & 0 & c \\cr} \\ . $$\nIn the normal form, going back to the original time parametrization,  we have \n$$ \\begin{array}{ll}\n{\\dot \\eta}_1 =& (\\mu I + \\omega J) \\, \\eta_1 \\ + \\ [\\alpha_1 (\\phi,\\mu) \\cdot I \\, - \\, \\beta_1 (\\phi,\\mu) \\cdot J ] \\, \\eta_1 \\ , \\\\\n{\\dot \\eta}_2 =& (- \\mu I + \\omega J) \\, \\eta_2 \\ + \\ [\\alpha_2 (\\phi,\\mu) \\cdot I \\, - \\, \\beta_2 (\\phi,\\mu) \\cdot J ] \\, \\eta_2 \\ , \\\\ \n{\\dot \\phi} =& \\[ \\( \\alpha_1 (\\phi,\\mu) + \\alpha_2 (\\phi,\\mu) \\) \\cdot I \\, + \\, \\( \\beta_2 (\\phi,\\mu) - \\beta_1 (\\phi,\\mu) \\) \\cdot J \\] \\ \\phi \\ , \\\\\n{\\dot \\mu} =& \\gamma (\\phi,\\mu) \\ . \\end{array} $$ \nAgain, interpreting $\\mu$ as an external control parameter requires $\\gamma (\\phi , \\mu ) \\equiv 0$.\n\n\n\\section*{Acknowledgements}\n\nThis work was started by discussions during a visit by SW in the\nDepartment of Mathematics of Universit\\`a di Milano, and while GG\nwas a guest at the DFG-Graduierten\\-kol\\-leg ``Hierarchie und\nSymmetrie in Mathema\\-ti\\-schen Mo\\-del\\-len'' at RWTH Aachen. We\nwould like to thank these Institutions for support to our work. GG\nalso acknowledges partial support by Fondazione CARIPLO and by\nGNFM-INdAM.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{sec:intro}}\nSkyrmions are particle-like magnetization swirls in magnetic materials,\\cite{Pf_MnSi_Science_09,Tokura_review_skyrmion_Natnano_13,Nanjing_Theory_artificial_skyrmion_PRL_13,Jiang_blowing_bubble_Science_15,CNRS_[CoFeB\/Pt]n_STXM_Natphys_15,White_GaV4S8_Natmater_15,CNRS_Ta\/Pt\/Co\/MgO\/Ta_PEEM_Natnano_16} which are promising candidates for advanced magnetic memory applications.\\cite{Tokura_review_skyrmion_Natnano_13}\nIn these memory schemes, individual skyrmions are used to encode the binary information `1' and `0', e.g., via their presence\\cite{Hamburg_Ir\/FePd_write_Science_13,VA,MIT_[Ta\/Pt\/Co]n_STXM_Natmater_16} or their internal structures.\\cite{Cu2OSeO3_multidomain_Nanoletters_16,Diaz_helicity_theory_IOP_16} Skyrmions are topological objects, which makes them robust against superparamagnetism.\nConsequently, it should be possible to reduce the bit size beyond the limits of conventional ferromagnetic memory. Further, the energy required to manipulate skyrmions is several orders of magnitude less than domain wall-based ferromagnetic memory.\\cite{Pf_MnSi_STT_rotation_Science,Rosch_MnSi_emergent_NatPhys,Tokura_CuOSeO-MnSi_ratchet_Natmater_14}\n\nProminent skyrmion-carrying materials are noncentrosymmetric helimagnets, such as MnSi,\\cite{Pf_MnSi_Science_09} Fe$_{1-x}$Co$_x$Si,\\cite{Pf_FeCoSi_SANS_PRB_10} FeGe,\\cite{Tokura_FeGe_STT_NatComm_12} Cu$_2$OSeO$_3$,\\cite{Tokura_CuOSeO_LTEM_Science_12,Tokura_CuOSeO_REXS_PRL_14} and $\\beta$-type Co$_8$Zn$_8$Mn$_4$.\\cite{CoMnZn} In such materials, the broken crystalline inversion symmetry induces the Dzyaloshinskii-Moriya interaction (DMI), leading to periodic, incommensurate, modulated spiral spin structures.\nAssisted by thermal fluctuations at finite temperature and in an external field, the topologically protected skyrmion lattice phase forms, consisting of chiral skyrmions.\\cite{Pf_MnSi_Science_09}\nThe advantages of chiral skyrmions for memory devices are twofold. First, compared to magnetic bubbles with similar topological spin configurations,\\cite{Bubble_review_Scinece_80} skyrmions are more robust and can be manipulated with ease.\nThe size of an individual skyrmion in these materials is usually between 3-100~nm,\\cite{Pf_MnSi_Science_09,2010:Yu:Nature,YuFeGe2011,2011:Kanazawa} i.e., smaller than other types of skyrmions.\\cite{Bubble_review_Scinece_80,Tokura_BaFeScMgO_reversal_PNAS_12,Nanjing_[Co\/Pt]n_MOKE_PRB_14,Jiang_blowing_bubble_Science_15,CNRS_[CoFeB\/Pt]n_STXM_Natphys_15,Berkeley_[Fe\/Ni]n_SPLEEM_APL_15,CNRS_Ta\/Pt\/Co\/MgO\/Ta_PEEM_Natnano_16,MIT_[Ta\/Pt\/Co]n_STXM_Natmater_16}\nSecond, the skyrmion phase is a rigid, hexagonally-ordered periodic lattice, resulting in an equal spacing between neighboring skyrmions across the entire sample.\\cite{Pf_MnSi_long_range,Tokura_review_skyrmion_Natnano_13}\nAs a consequence, in a racetrack-like memory scheme,\\cite{Parkin_racetrack_Science_08} no extra effort is needed to assure that they keep their distance, as the control of the skyrmion-skyrmion distance is otherwise experimentally rather challenging.\\cite{Hamburg_Ir\/FePd_write_Science_13,Jiang_blowing_bubble_Science_15,MIT_[Ta\/Pt\/Co]n_STXM_Natmater_16}\n\n\n\nThe skyrmion lattice phase in noncentrosymmetric helimagnets is usually a long-range-ordered state, with the correlation length reaching hundreds of micrometers.\\cite{Pf_MnSi_long_range}\nThis largely limits the applicability of this type of skyrmion order for device applications. Therefore, the formation and manipulation of skyrmion lattice domains, which break the long-range order, is an indispensable step towards skyrmion-based racetrack memories.\\cite{SkyRace_Tomasello}\nWe have recently reported the observation of a multidomain skyrmion lattice state on the surface of Cu$_2$OSeO$_3$ in a magnetic diffraction experiment, created by tilting the magnetic field away from the major crystalline axis.\\cite{Cu2OSeO3_multidomain_Nanoletters_16,2016:Zhang-PRB} \nHowever, several key questions remained unanswered.\nMost importantly, the size, shape, and distribution of the domains remained unknown, which are crucial pieces of information needed for designing skyrmion devices.  \n\n\nIn this Letter, we use an imaging technique based on resonant elastic x-ray scattering (REXS) to map out the lateral distribution of skyrmion lattice domains. Moreover, we demonstrate that by tuning the tilt angle of the applied field, the size of the domains can be efficiently manipulated, which is an important step towards future applications.\n\n\n\n\n\n\n\\begin{figure*}[ht!]\n\t\\begin{center}\n\t\t\\includegraphics[width=16cm]{Fig1_Setup-PD.pdf}\n\t\t\\caption{(a) Schematic of the REXS setup, showing the field tilt angle $\\gamma$, defined with respect to the surface normal.\n\t\t\t(b) Photograph of the sample and its orientation. The blue square marks the real-space region ($1 \\times 1$~mm$^2$) mapped using magnetic diffraction imaging, shown in Fig.\\ \\ref{fig:domains}.\n\t\t\t(c) Sketch of the magnetic phase diagram of Cu$_2$OSeO$_3$, as obtained by REXS.\n\t\t\t(d) Magnetization patterns and (e) corresponding REXS diffraction patterns in reciprocal space $(hk1)$ plane (experimental data) for the conical (50~K, 32~mT, $\\gamma=90^\\circ$), skyrmion (57~K, 32~mT, $\\gamma=0^\\circ$), and helical (15~K, 0~mT) phase, respectively. The sampling area is $500\\times500~\\mu\\mathrm{m}^2$.}\n\t\t\\vspace*{-0.5cm}\n\t\t\\label{fig:setup}\n\t\\end{center}\n\\end{figure*}\n\nFigure \\ref{fig:setup}(a) shows a schematic of the REXS setup used for the characterization of the magnetically ordered phases of Cu$_2$OSeO$_3$ as a function of applied magnetic field and temperature. A $\\omega$-$2\\theta$ geometry was used for the experiments, where $\\omega$ is the angle of incidence of the x-rays and $2 \\theta$ the scattering angle. The diffracted x-rays are detected with a charge-coupled device (CCD) camera in the ultrahigh vacuum scattering chamber RASOR on beamline I10 at the Diamond Light Source.\\cite{RASOR_review} A magnetic field was applied to the sample whose strength and tilt angle, $\\gamma$, with respect to the surface normal can be varied.\nThe incident, $\\sigma$-polarized x-rays were at the resonance with the $L_3$ edge of Cu (931.25~eV), resulting in a wavelength of 13.3~\\AA.\nFor Cu$_2$OSeO$_3$ with its relatively large lattice constant of 8.925~\\AA, the (001) Bragg peak can be reached at $2 \\theta \\approx 96.5^\\circ$.\n\nIn noncentrosymmetric $P2_13$ helimagnets, a `universal' magnetic phase diagram is observed that shows helical, conical, or ferrimagnetic order below the transition temperature $T_\\mathrm{c}$ as a function of increasing field.\\cite{Rosch_3D_Monte_Carlo_PRB_13}\nBelow $T_\\mathrm{c}$ of $\\sim$57.5~K for Cu$_2$OSeO$_3$, at finite fields, the skyrmion phase can be found. \nFigure \\ref{fig:setup}(c) shows a schematic of the magnetic phase diagram for the Cu$_2$OSeO$_3$ bulk crystal as observed by REXS.\\cite{Cu2OSeO3_multidomain_Nanoletters_16} \nThe magnetization patterns corresponding to the helical, skyrmion, and conical phase are shown in Fig.\\ \\ref{fig:setup}(d), and the experimental REXS results in (e), respectively (from top to bottom).\nThe modulation wavevector for all three phases has a length of 0.0158-0.0162~reciprocal lattice units (r.l.u.), corresponding to a real-space modulation pitch of $\\sim$56~nm, in agreement with the values reported in the literature.\\cite{Pf_CuOSeO_PRL_12, Tokura_CuOSeO_LTEM_Science_12, Tokura_CuOSeO_rotation_PRB_12, White_Cu2OSeO3_E_rotation_IOP_12, Tokura_CuOSeO_REXS_PRL_14, White_Cu2OSeO3_LTEM_PNAS_15}\nIn the helical state, the weak cubic anisotropy locks the propagation wave vector along the $\\langle$100$\\rangle$ direction.\nAt 57~K and in an applied magnetic field of 32~mT, $\\gamma=0^\\circ$, the sharp six-fold-symmetric diffraction pattern emerges, which is a signature of the single-domain, long-range-ordered skyrmion lattice state.\nOne of the skyrmion wave vectors is along [010] due to the cubic anisotropy.\\cite{Cu2OSeO3_multidomain_Nanoletters_16}\n\n\nBy titling the magnetic field to $\\gamma=17^\\circ$, the skyrmion lattice breaks up into domains, resulting in a necklace-like diffraction pattern, as shown in Fig.\\ \\ref{fig:focus}(a).\nThe formation of domains is the result of the competing magnetic anisotropies.\\cite{Cu2OSeO3_multidomain_Nanoletters_16}\nNote that the diffraction pattern was obtained with an incident x-ray beam focused to an area of 300 $\\times$ 300~$\\mu$m$^2$ on the sample. Once the beam is skimmed down to an area of 20~$\\mu$m in diameter (using a pinhole), the single domain state with its six-fold-symmetric pattern is recovered [cf.\\ Fig.\\ \\ref{fig:focus}(b)]. \nThis means that the domains are $>$$100\\pi~\\mu$m$^2$ in area, and that this beam spot can be used to map out the real-space domain pattern.\n\n\\begin{figure}[ht!]\n\t\\begin{center}\n\t\t\\includegraphics[width=8.5cm]{Fig2_BeamSize.pdf}\n\t\t\\caption{Reciprocal space map of the skyrmion lattice plane that is perpendicular to the external field, reached by field cooling down to 57~K in a field of 32~mT with $\\gamma=15^\\circ$. Area sampled by the beam: (a) $300\\times300~\\mu\\text{m}^2$ and (b) $20~\\mu\\text{m}$ in diameter.\n\t\tThe scattering intensity is in arbitrary units.}\n\t\t\\vspace*{-1.0cm}\n\t\t\\label{fig:focus}\n\t\\end{center}\n\\end{figure}\n\n\n\n\\begin{figure*}[ht!]\n\t\\begin{center}\n\t\t\\includegraphics[width=16cm]{Fig3_Domains.pdf}\n\t\t\\caption{Scanning REXS images. (a) Definition of the lattice rotation angle $\\Psi$ in the skyrmion diffraction pattern ($\\Psi \\in [0^\\circ,60^\\circ]$ as indicated by the color bar). The case of $\\Psi=30^\\circ$ is shown.\n\t\t\t(b-h) 1~mm $\\times$ 1~mm raster scans showing in-plane rotational domain maps for the indicated field angle $\\gamma$.}\n\t\t\\vspace*{-0.9cm}\n\t\t\\label{fig:domains}\n\t\\end{center}\n\\end{figure*}\n\n\nThe sample shape and geometry is shown Fig.\\ \\ref{fig:setup}(b), in which the raster-scanned area ($1\\times1$~mm$^2$) is indicated by the blue square.\nThe skyrmion lattice state is reached by field-cooling from 65~K (paramagnetic phase) down to 57~K in a field of 32~mT with the $\\gamma$ angle as indicated in Fig.\\ \\ref{fig:domains}(b-h), followed by a 15~min waiting period. \nEach area scan image is composed of $50\\times50$ pixels. \nFor most of the pixels, a six-fold-symmetric, single domain diffraction pattern is observed. Its rotational state is characterized by the in-plane rotation angle $\\Psi$, as shown Fig.\\ \\ref{fig:domains}(a). An angle of $\\gamma =0^\\circ$ corresponds to two of the six diffraction spots being aligned along the $q_x$-axis.\nFor some pixels, especially for larger field tilt angles $\\gamma$, a multidomain state is found at the domain boundaries.\nThe evolution of the domain pattern as a function of field tilt angle is shown in Fig.\\ \\ref{fig:domains}(b-h).\nFor $\\gamma=0^\\circ$, a perfect long-range-ordered single skyrmion lattice domain is observed [cf.\\ Fig.\\ \\ref{fig:domains}(b)], locked in one direction determined by the anisotropies.\nWhen the field is slightly tilted ($\\gamma = 5^\\circ$), two domains start to emerge, which do not differ very much in $\\Psi$.\nAs the field is further tilted, the domains become randomly oriented and the average size decreases (most strongly for $\\gamma = 15^\\circ \\rightarrow 19^\\circ$).\nFor $\\gamma = 22^\\circ$, a mosaic pattern is observed at the bottom-left corner of Fig.\\ \\ref{fig:domains}(g). This results from a lack of resolution, as determined by the x-ray beam size.\nAs the domain size decreases to less than the beam size, a multidomain diffraction pattern is observed across each pixel.\nNote that the pinhole diameter can not be reduced much below 20~$\\mu$m in our experimental configuration, as this will result in Fresnel diffraction (near-field condition giving cylindrical wave fronts), i.e., the plane wave approximation can no longer be applied for modeling the scattering process.\n\n\nThe shape of the domains is generally rather irregular, and their distribution random, suggesting that the domain formation is spontaneous and not governed by defect-pinning.\nThe domain pattern obtained for each $\\gamma$ is stable over time, as confirmed by multiple scans of the same area.\nNote, however, that for the same $\\gamma$, if the temperature is increased above $T_\\mathrm{c}$ and then lowered down to the skyrmion phase again, the domain image has no resemblance to the previous pattern.\nThe orientation of neighboring domains is similar and the system generally prefers the orientation of neighboring domains to differ by less than $10^\\circ$. At higher tilt angles ($\\gamma \\geq 15^\\circ$), the domain boundaries are sharp, however, for smaller angles the transition between domains becomes almost continuous, as can be seen in Fig.\\ \\ref{fig:domains}(d).\n\n\nThis `polycrystalline' appearance of a skyrmion lattice reflects the delicate balance of the system's competing interactions.\nThe isotropic exchange interaction, the anisotropic exchange (DMI) term, and the Zeeman energy compete\nabout the principal magnetic order, while anisotropy, demagnetization, and possibly the ferroelectric effect in the multiferroic material Cu$_2$OSeO$_3$ serve as perturbations which are responsible for the fine adjustment of the system's structure.\nSkyrmions behave like quasi-particles and prefer to keep their close-packed order, i.e., less ferromagnetic-like space in-between them is preferred. However, if defects are introduced into the system, some region would have to become ferromagnetic, unless the skyrmions adapt their shape and size. \nAs a result, the skyrmions in the defect zone can become elliptically distorted, as observed using Lorentz transmission electron microscopy (LTEM) on thinned-down FeGe$_{1-x}$Si$_x$ bulk samples.\\cite{Tokyo_FeGeSi_LTEM_Scienceadv_16} \nMoreover, the close-packed ordering of the skyrmion lattice can have defects at the domain boundaries, e.g., `5-7 defects' where five or seven skyrmions are surrounding a reference skyrmion, instead of six.\\cite{White_Cu2OSeO3_LTEM_PNAS_15} In this case, a line defect can appear, forming the domain boundary. Depending on its configuration, the neighboring domains can change their orientations and take an arbitrary angle, depending on the width of the defect-containing boundary. The thicker this boundary, the larger the relative rotation. If we relate this to our domain observations, it points towards the fact that for small $\\gamma$, the domain boundary contains thin defect lines, leading to small relative rotations. With increasing $\\gamma$, the number of defects increases across the domain boundary, giving rise to sharp rotational transitions.\nNote that LTEM imaging, despite being immensely useful for studying the local defects of the skyrmions, is not able to map out the large-scale skyrmion structures like skyrmion lattice domains. Moreover, for systems with such a delicate energy balance as Cu$_2$OSeO$_3$, LTEM sample preparation-induced defects may affect the intrinsic surface domain structure (essentially a property of the bulk crystal). \nTherefore, the type and density of defects, and their influence on the observed domains, can be significantly different between bulk and thinned-down LTEM samples.\n\n\nIn summary, we have presented a study of the domain structure in the skyrmion lattice phase of Cu$_2$OSeO$_3$ using REXS imaging.\nThis method provides important information on the domain distribution, shape, size, and formation.\nBy tuning the field tilt angle, in-plane rotational domains in the hexagonally ordered skyrmion lattice phase can be generated. The lateral dimensions of the domains are for small tilt angles $>$20~$\\mu$m. The domain transition is abrupt, expect for very small tilt angles, where the change is almost gradual.\nThe study of skyrmion domains may enable potential device applications making use of the skyrmion lattice state. \nIn skyrmion lattice-based memory, the single-domain state must be broken up into domains, which each encode information in a spatially separated manner.\nThis was demonstrated using the field tilt angle as an additional handle.\n\n\n\n\n\n\nThe REXS experiments were carried out on beamline I10 at the Diamond Light Source, UK, under proposals SI-11784 and SI-12958. \nWe thank the EPSRC for support under grant EP\/N032128\/1.\nS.L.Z.\\ and T.H.\\ acknowledge financial support by the Semiconductor Research Corporation. A.B.\\ and C.P.\\ acknowledge financial support through DFG TRR80 and ERC AdG (291079, TOPFIT).\n\n\n\n\n \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\large Introduction}}\n\\label{secIntr}\n\n    The main part of the paper is a a review of our results obtained\nin~\\cite{GribPavlov2012}--\\cite{GribPavlov2013a}.\n    In the end we discuss some new features of particles with negative energies\nconcerning their origination and collisions with ordinary particles.\n\n    Recently many papers were\npublished~\\cite{BanadosSilkWest09}--\\cite{Zaslavskii12b} in which some\nspecific properties of collisions of particles close to the horizon of\nthe rotating black holes were discussed.\n    In~\\cite{BanadosSilkWest09} the effect of unlimited growing energy of\ncolliding two particles in the centre of mass frame for critical Kerr\nrotating black holes (call it BSW effect) was discovered.\n    In our papers~\\cite{GribPavlov2010}--\\cite{GribPavlov2011b}\nthe same effect was found for noncritical black holes when multiple\ncollisions took place.\n\n    All this shows that close to the horizon there is natural supercollider\nof particles with the energies up to Planck's scale.\n    However its location close to the horizon makes difficult due to the large\nred shift get some observed effects outside of the ergosphere when particles\ngo from the ``black hole supercollider'' outside.\n    Here we shall discuss some other effect valid at any point of\nthe ergosphere.\n    The energy in the centre of mass frame can take large values for large\nnegative values of the angular momentum projection on the rotation axis of\nthe black hole.\n    The problem is how to get this large (in absolute value) negative values.\n    That is why first we shall obtain limitations on the values of\nthe projection of angular momentum outside ergosphere for particles falling\ninto the black hole and inside the ergosphere.\n    It occurs that inside the ergosphere of black hole there is no limit for\nnegative values of the momentum and arbitrary large energy in the centre\nof mass frame is possible for large angular momentum of the two colliding\nparticles.\n    In a sense the effect is similar to the BSW-effect.\n    One of the particle with large negative angular momentum can be called\n``critical'', the other particle ``ordinary''.\n\n\n    Also in this paper we analyse geodesics for particles with negative energies\nfor rotating black holes.\n    Kerr's metric~\\cite{Kerr63}, predicts differently from the Schwarzschild\nmetric  existence of the special region outside the horizon of the black hole\ncalled ergosphere.\n    In the egosphere elementary particles must rotate together with the black\nhole.\n    The new feature of ergosphere is the existence of geodesics with negative\nrelative to infinity energy.\n    Existence of such geodesics leads to the possibility of extraction of\nenergy from the rotating black holes due to the\nPenrose process~\\cite{Penrose69}\n    so one can call these geodesics the Penrose geodesics.\n    However in spite of the more than 40 years passing after the discovery of\nthe Penrose effect there is still no information about the full picture of the\nPenrose geodesics especially about their origin.\n    Here we shall investigate the problem of properties of such geodesics.\n    Particle can arrive on such trajectory in the result of collisions or\ndecays in the ergosphere.\n    But the world line of the geodesic in geodesically complete space-time\nmust originate or terminate either in singularity or in\nthe infinity~\\cite{HawkingEllis}.\n    Note that here one means infinity in space-time so that it can be infinity\nin time for finite value of the space distances.\n    The problem is to find where originate and terminate Penrose geodesics and\nthis is the subject of the paper.\n    It will be shown that the length of the Penrose geodesics is always finite\ninside the ergosphere so that the only possibility of their origination and\ntermination is outside of the ergosphere.\n    But the geodesics for particles with negative energy don't exist in the\nexternal space out of ergosphere so that one comes to the conclusion that they\noriginate and terminate inside the gravitational radius of the black hole.\n    The fact that they terminate inside the horizon is well\nknown~\\cite{Contopoulos84}.\n    So our interest in this paper is to find the place of their origination.\n    These geodesics occur to be ``white hole'' geodesics originating at the\nsingularity arriving to ergosphere and then returning back to the horizon\nand singularity.\n    A new feature is that if particle with negative energy collides with\nordinary particle the energy in the centre of mass frame is growing without\nlimit on the horizon.\n\n    The system of units $G=c=1$ is used in the paper.\n\n\n\n\\vspace{9mm}\n{\\centering \\section{General formulas for the energy of particles\nclose to the black hole}\n\\label{sec2}}\n\n    The Kerr's metric of the rotating black hole~\\cite{Kerr63} in\nBoyer--Lindquist coordinates~\\cite{BoyerLindquist67} has the form:\n    \\begin{equation}\nd s^2 = d t^2 -\n\\frac{2 M r}{\\rho^2} \\, ( d t - a \\sin^2 \\! \\theta\\, d \\varphi )^2\n-\\, \\rho^2 \\left( \\frac{d r^2}{\\Delta} + d \\theta^2 \\right)\n- (r^2 + a^2) \\sin^2 \\! \\theta\\, d \\varphi^2,\n\\label{Kerr}\n\\end{equation}\n    where\n    \\begin{equation} \\label{Delta}\n\\rho^2 = r^2 + a^2 \\cos^2 \\! \\theta, \\ \\ \\ \\ \\\n\\Delta = r^2 - 2 M r + a^2,\n\\end{equation}\n    $M$ is the mass of the black hole, $ aM $ its angular momentum.\n    The rotation axis direction corresponds to $\\theta =0$, i.e. $a \\ge 0$.\n    The event horizon of the Kerr's black hole corresponds to\n    \\begin{equation}\nr = r_H \\equiv M + \\sqrt{M^2 - a^2} .\n\\label{Hor}\n\\end{equation}\n    For Cauchy horizon one has\n    \\begin{equation}\nr = r_C \\equiv M - \\sqrt{M^2 - a^2} .\n\\label{HorC}\n\\end{equation}\n    The surface of the static limit is defined by\n    \\begin{equation}\nr = r_1(\\theta) \\equiv M + \\sqrt{M^2 - a^2 \\cos^2 \\theta} .\n\\label{Lst}\n\\end{equation}\n    In case $ a \\le M $ the region of space-time between the static limit\nand event horizon is called ergosphere.\n\n    For geodesics in Kerr's metric~(\\ref{Kerr}) one obtains\n(see \\cite{Chandrasekhar}, Sec.~62 or \\cite{NovikovFrolov}, Sec.~3.4.1)\n    \\begin{equation} \\label{geodKerr1}\n\\rho^2 \\frac{d t}{d \\lambda } = -a \\left( a E \\sin^2 \\! \\theta - J \\right)\n+ \\frac{r^2 + a^2}{\\Delta}\\, P,\n\\end{equation}\n    \\begin{equation}\n\\rho^2 \\frac{d \\varphi}{d \\lambda } =\n- \\left( a E - \\frac{J}{\\sin^2 \\! \\theta} \\right) + \\frac{a P}{\\Delta} ,\n\\label{geodKerr2}\n\\end{equation}\n    \\begin{equation} \\label{geodKerr3}\n\\rho^2 \\frac{d r}{d \\lambda} = \\sigma_r \\sqrt{R}, \\ \\ \\ \\\n\\rho^2 \\frac{d \\theta}{d \\lambda} = \\sigma_\\theta \\sqrt{\\Theta},\n\\end{equation}\n    \\begin{equation} \\label{geodP}\nP = \\left( r^2 + a^2 \\right) E - a J,\n\\end{equation}\n    \\begin{equation} \\label{geodR}\nR = P^2 - \\Delta [ m^2 r^2 + (J- a E)^2 + Q],\n\\end{equation}\n    \\begin{equation} \\label{geodTh}\n\\Theta = Q - \\cos^2 \\! \\theta \\left[ a^2 ( m^2 - E^2) +\n\\frac{J^2}{\\sin^2 \\! \\theta} \\right].\n\\end{equation}\n    Here $E$ is conserved energy (relative to infinity)\nof the probe particle,\n$J$ is conserved angular momentum projection on the rotation axis\nof the black hole,\n$m$ is the rest mass of the probe particle, for particles with nonzero\nrest mass $\\lambda = \\tau \/m $,\nwhere $\\tau$ is the proper time for massive particle,\n$Q$ is the Carter's constant.\n    The constants $\\sigma_{r}, \\sigma_{\\theta}$ in formulas~(\\ref{geodKerr3})\nare equal to $\\pm 1$ and are defined by the direction\nof particle movement in coordinates $r$, $\\theta$.\n    For massless particles one must take $m = 0$\nin~(\\ref{geodR}), (\\ref{geodTh}).\n\n    The permitted region for particle movement outside the event horizon\nis defined by conditions\n    \\begin{equation} \\label{ThB0}\nR \\ge 0, \\ \\ \\ \\ \\ \\Theta \\ge 0, \\ \\ \\ \\ \\\n\\frac{d t}{d \\lambda} \\ge 0 .\n\\end{equation}\n    The last inequality forbids movement ``back in time''~\\cite{Wald}.\n    Let us find limitations for the particle angular momentum from the\nconditions~(\\ref{ThB0}) at the point $(r, \\theta)$,\ntaking the fixed values of $\\Theta$~\\cite{GribPavlov2013}.\n\n    Outside the ergosphere $ r^2 -2 r M +a^2 \\cos^2 \\! \\theta >0 $ one obtains\n    \\begin{equation} \\label{EvErg}\nE \\ge \\frac{1}{\\rho^2} \\sqrt{(m^2 \\rho^2 + \\Theta)\n(r^2 -2 r M +a^2 \\cos^2 \\! \\theta)}, \\ \\ \\ \\ \\\n   \nJ \\in \\left[ J_{-}, \\ J_{+} \\right],\n\\end{equation}\n    \\begin{equation}\nJ_{\\pm} = \\frac{\\sin \\theta}{r^2 -2 r M +a^2 \\cos^2 \\! \\theta}\n\\biggl[ - 2 r M a E \\sin \\theta\n\\pm \\sqrt{ \\Delta \\left( \\rho^4 E^2 \\!-\\! (m^2 \\rho^2 \\!+\\! \\Theta)\n(r^2 \\!-\\! 2 r M \\!+\\! a^2 \\cos^2 \\! \\theta) \\right)} \\biggr] .\n\\label{Jpm}\n\\end{equation}\n\n    On the boundary of ergosphere\n    \\begin{equation} \\label{rEgErg}\nr = r_1(\\theta) \\ \\ \\ \\Rightarrow \\ \\ \\ E \\ge 0, \\ \\ \\ \\\nJ \\le E \\left[ \\frac{M r_1(\\theta) }{a} + a \\sin^2 \\! \\theta \\left(\\!\n1 - \\frac{m^2}{2 E^2} - \\frac{\\Theta}{4 M r_1(\\theta) E^2} \\!\\right)\n\\! \\right]\\!.\n\\end{equation}\n\n    Inside ergosphere\n    \\begin{equation} \\label{lHmdd}\nr_H < r < r_1(\\theta) \\ \\ \\ \\Rightarrow \\ \\ \\\n(r^2 -2 r M +a^2 \\cos^2 \\! \\theta) <0 ,\n\\end{equation}\n    \\begin{equation}\nJ \\le J_{-}(r,\\theta) = \\frac{- \\sin \\theta}{\nr^2 \\!-\\! 2 r M \\!+\\! a^2 \\cos^2 \\! \\theta} \\biggl[ 2 r M a E \\sin \\theta\n- \\sqrt{ \\Delta \\left( \\rho^4 E^2 \\!-\\! (m^2 \\rho^2 + \\Theta)\n(r^2 \\!-\\! 2 r M \\!+\\! a^2 \\cos^2 \\! \\theta) \\right)} \\biggr].\n\\label{JmErg}\n\\end{equation}\n    So it is only inside the ergosphere that the energy $E$ of the particle\nrelative to infinity can be negative.\n    From~(\\ref{JmErg}) one can see that for negative energy~$E$ of the particle\nin ergosphere its angular momentum projection on the rotation axis of the\nblack hole must be also negative.\n\n    For negative values of the energy $E$ the function $ J_{-}(r,\\theta) $\nis decreasing with growing $r$ in ergosphere, so that\n    \\begin{equation} \\label{JgEHf}\n\\theta \\ne 0 , \\pi , \\ \\ \\ r \\to r_1(\\theta) \\ \\ \\Rightarrow \\ \\\nJ_{-}(r,\\theta) \\to - \\infty .\n\\end{equation}\n    So in order to come to the upper frontier of the ergosphere particle\nwith negative energy one must have infinitely large in absolute value negative\nangular momentum.\n\n    In the limit $r \\to r_H$ from~(\\ref{JmErg}) (for $\\theta \\ne 0, \\pi$)\none obtains\n    \\begin{equation} \\label{JgEH}\nJ \\le J_H = \\frac{ 2 M r_H E}{a} .\n\\end{equation}\n    So $J_H$ is the maximal value of the angular momentum of the particle with\nthe energy~$E$ close to the gravitational radius.\n\n\n\\vspace{9mm}\n{\\centering \\section{The energy of particles collision close to the black hole}\n\\label{secCollision}}\n\n\n    One can find the energy in the centre of mass frame of two colliding\nparticles $E_{\\rm c.m.}$ with rest masses~$m_1$, $m_2$ taking the square of\n    \\begin{equation} \\label{SCM}\n\\left( E_{\\rm c.m.}, 0\\,,0\\,,0\\, \\right) = p^{\\,i}_{(1)} + p^{\\,i}_{(2)},\n\\end{equation}\n    where $p^{\\,i}_{(n)}$ are 4-momenta of particles $(n=1,2)$.\n    Due to $p^{\\,i}_{(n)} p_{(n)i}= m_n^2$ one has\n    \\begin{equation} \\label{SCM2af}\nE_{\\rm c.m.}^{\\,2} = m_1^2 + m_2^2 + 2 p^{\\,i}_{(1)} p_{(2)i} .\n\\end{equation}\n    Note that the energy of collisions of particles in the centre of mass\nframe is always positive (while the energy of one particle due to Penrose\neffect~\\cite{Penrose69} can be negative!) and satisfies the condition\n    \\begin{equation} \\label{Eb0}\nE_{\\rm c.m.} \\ge m_1 + m_2.\n\\end{equation}\n    This follows from the fact that the colliding particles move one\ntowards another with some velocities.\n\n    It is important to note that $E_{\\rm c.m.}$ for two colliding particles\nis not a conserved value differently from energies of particles (relative\nto infinity) $E_1$,  $E_2$.\n\n    For the free falling particles with energies $E_1$,  $E_2$ and angular\nmomentum projections $J_1, J_2$ from~(\\ref{geodKerr1})--(\\ref{geodR})\none obtains~\\cite{HaradaKimura11}:\n    \\begin{eqnarray}\nE_{\\rm c.m.}^{\\,2} = m_1^2 + m_2^2 + \\hspace{57mm}\n\\nonumber \\\\[7pt]\n+\\, \\frac{2}{\\rho^2} \\biggl[ \\, \\frac{P_1 P_2 -\n\\sigma_{1 r} \\sqrt{R_1} \\, \\sigma_{2 r} \\sqrt{R_2}}{\\Delta}\n- \\frac{ (J_1 - a  E_1 \\sin^2 \\! \\theta) (J_2 - a  E_2 \\sin^2 \\! \\theta)}\n{\\sin^2 \\! \\theta}\n - \\sigma_{1 \\theta} \\sqrt{\\Theta_1} \\,\n\\sigma_{2 \\theta} \\sqrt{\\Theta_2} \\, \\biggr].\n\\label{KerrL1L2}\n\\end{eqnarray}\n    The big values of the collision energy can occur near the event horizon\nif one of the particles has the ``critical'' angular moment $J_H$:\n    \\begin{eqnarray}\nE_{\\rm c.m.}^{\\,2}(r \\to r_H) = \\frac{ (J_{1H} J_2 - J_{2H} J_1)^2}\n{4 M^2 (J_{1H} - J_1) (J_{2H} - J_2)}\n+ m_1^2 \\left[1+ \\frac{J_{2H} \\!- J_2}{J_{1H} \\!- J_1}\\right] +\nm_2^2 \\left[ 1+ \\frac{J_{1H} \\!- J_1}{J_{2H} \\!- J_2}\\right]\n\\label{GrPvPi}\n\\end{eqnarray}\n(see Eq.~16 in~\\cite{GribPavlovPiattella2012}).\n    This is the BSW-effect.\n\n    The energy in the centre of mass frame can be written\nthrough the relative velocity~$ v_{\\rm rel}$ of particles at the moment\nof collision~\\cite{Zaslavskii11}, \\cite{GribPavlovPiattella2012}:\n    \\begin{equation} \\label{Relsk03}\nE_{\\rm c.m.}^{\\,2} = m_1^2 + m_2^2 +\n\\frac{2 m_1 m_2}{\\sqrt{1 \\!- v_{\\rm rel}^2}}\n\\end{equation}\n    and the nonlimited growth of the collision energy in the centre of mass\nframe occurs due to growth of the relative velocity to the velocity of\nlight~\\cite{Zaslavskii11}.\n\n    Let's consider another opportunity for big values of collision energy.\n    As we can see from~(\\ref{JmErg}),\non the boundary and inside ergosphere there exist geodesics on\nwhich particle with fixed energy can have arbitrary large in absolute value\nnegative angular momentum projection.\n    Let us find the asymptotic of~(\\ref{KerrL1L2}) for $J_2 \\to -\\infty$ and\nsome fixed value $r$ in ergosphere supposing the value of Carter's\nconstant $Q_2$ to be such that~(\\ref{ThB0}) is valid and $\\Theta_2 \\ll J_2^2$.\n    Then from~(\\ref{KerrL1L2}) one obtains\n    \\begin{eqnarray}\nE_{\\rm c.m.}^{\\,2} \\approx \\frac{- 2 J_2}{\\rho^2 \\Delta } \\,\n\\biggl[ \\frac{J_1}{\\sin^2 \\! \\theta}\n\\left( r^2 \\!-\\! 2 r M \\!+\\! a^2 \\cos^2 \\! \\theta \\right)\n+ 2 r M a E_1\n- \\frac{\\sigma_{1r} \\sigma_{2r} \\sqrt{R_1}}{\\sin \\theta}\n\\sqrt{-(r^2 \\!-\\! 2 r M \\!+\\! a^2 \\cos^2 \\! \\theta) } \\biggr] .\n\\label{KerrJB}\n\\end{eqnarray}\n    This asymptotic formula is valid for all possible $E_1$, $J_1$\n(see~(\\ref{JmErg})) for $r_H < r < r_{1}(\\theta)$ and for\n$E_1>0$ and $J_1$ satisfying~(\\ref{rEgErg}) for $r=r_{1}(\\theta)$.\n    The poles $\\theta = 0, \\pi$ are not considered here because the points\non surface of static limit are on the event horizon.\n\n    Note that expression in brackets in~(\\ref{KerrJB}) is positive\nin ergosphere.\n    This is evident for $r=r_{1}(\\theta)$ and follows from\nlimitations~(\\ref{JmErg}) for $r_H < r < r_{1}(\\theta)$,\nand inside ergosphere~(\\ref{KerrJB}) can be written as\n    \\begin{eqnarray}\nE_{\\rm c.m.}^{\\,2} \\approx J_2 \\frac{r^2 -2 r M +a^2 \\cos^2 \\! \\theta}\n{\\rho^2 \\Delta \\sin^2 \\! \\theta}\n\\left( \\sigma_{1r} \\sqrt{J_{1 +}- J_1} -\n\\sigma_{2r} \\sqrt{J_{1 -}- J_1} \\right)^2.\n\\label{KerrJBner}\n\\end{eqnarray}\n\n    So from~(\\ref{KerrJB}) one comes to the conclusion that\n{\\it when particles fall on the rotating black hole collisions with arbitrarily\nhigh energy in the centre of mass frame are possible at any point of\nthe ergosphere if $J_2 \\to -\\infty$ and the energies $E_1, E_2$ are fixed}.\n    The energy of collision in the centre of mass frame is growing\nproportionally to $\\sqrt{|J_2|}$.\n\n    Note that large negative values of the angular momentum projection are\nforbidden for fixed values of energy of particle out of the ergosphere.\n    So particle which is nonrelativistic on space infinity ($E=m$)\ncan arrive to the horizon of the black hole if its angular momentum projection\nis located in the interval\n    \\begin{equation}\n-2 m M \\left[ 1 + \\sqrt{1+ \\frac{a}{M}}\\, \\right] \\le J \\le 2 m M\n\\left[ 1 + \\sqrt{1 - \\frac{a}{M}}\\, \\right].\n\\label{KerrEM}\n\\end{equation}\n    The left boundary is a minimal value of the angular momentum of particles\nwith $E=m$ capable to achieve ergosphere falling from infinity.\n    That is why collisions with $J_2 \\to -\\infty$ do not occur for particles\nfollowing from infinity.\n    But if the particle came to ergosphere and there in the result of\ninteractions with other particles is getting large negative values of the\nangular momentum projection (no need for getting high energies!)\nthen its subsequent collision with the particle falling on the black hole\nleads to high energy in the centre of mass frame.\n\n    Getting superhigh energies for collision of usual particles (i.e. protons)\nin such mechanism occur however physically nonrealistic.\n    Really from~(\\ref{KerrJB}) the value of angular momentum necessary for\ngetting the collision energy $E_{\\rm c.m.}$ has the order\n    \\begin{equation}\nJ_2 \\approx - \\frac{a E_{\\rm c.m.}^{\\,2}}{2 E_1}.\n\\label{KerrEMR}\n\\end{equation}\n    So from~(\\ref{KerrEM}) absolute value of the angular momentum $J_2$\nmust acquire the order $ E_{\\rm c.m.}^{\\,2} \/ (m_1 m_2)$ relative to the\nmaximal value of the angular momentum of the particle incoming to ergosphere\nfrom infinity.\n    For example if $E_1=E_2 = m_p$ (the proton mass) then $|J_2|$ must\nincrease with a factor $10^{18}$ for $ E_{\\rm c.m.} = 10^9 m_p$.\n    To get this one must have very large number of collisions with getting\nadditional negative angular momentum in each collision.\n\n    However the situation is different for supermassive particles.\n    In~\\cite{GribPavlov2002(IJMPD)}--\\cite{GribPavlov2008c} we discussed\nthe hypothesis that dark matter contains stable superheavy neutral particles\nwith mass of the Grand Unification scale created by gravitation in the end\nof the inflation era.\n    These particles are nonstable for energies of interaction of the order\nof Grand Unification and decay on particles of visual matter but are\nstable at low energies.\n    But in ergosphere of the rotating black holes such particles due to\ngetting large relative velocities can increase their energy from $2m$ to\nvalues of $3m$ and larger so that the mechanism considered in our paper\ncan lead to their decays as it was in the early universe.\n    The number of intermediate collisions for them is not very large\n(of the order of 10).\n\n\n\\vspace{9mm}\n{\\centering \\section{Properties of movement of particles with negative energy\nin ergosphere}\n\\label{secErgo}}\n\n\n    From~(\\ref{geodKerr3}), (\\ref{geodTh}), (\\ref{JmErg}) one can see that the\nCarter's constant $Q \\ge 0$ for particles with negative energy in ergosphere.\n    One can have $Q=0$ for $E \\le0$ only in case of movement in\nequatorial plane.\n    The value $Q=0$ is necessary for equatorial movement, but for the positive\nenergy $E$ with $Q=0$ the non-equatorial movement can take place.\n\n    Let's consider other specifics for movements of the negative\nenergy particles.\n    Define the effective potential by the formula\n    \\begin{equation} \\label{Leff}\nV_{\\rm eff} = - \\frac{R}{2 \\rho^4}.\n\\end{equation}\n    Then due to~(\\ref{geodKerr3}),\n    \\begin{equation} \\label{LeffUR}\n\\frac{1}{2} \\left( \\frac{d r}{d \\lambda} \\right)^{\\!2} + V_{\\rm eff}=0\n\\end{equation}\n    and so\n    \\begin{equation} \\label{LeffUR2}\n\\frac{d^2 r}{d \\lambda^2} = - \\frac{d V_{\\rm eff}}{d r} .\n\\end{equation}\n\n    Inside the event horizon up to Cauchy horizon\nfrom~(\\ref{geodKerr3}), (\\ref{geodR}), (\\ref{geodTh}), (\\ref{JgEH})\none has $V_{\\rm eff} <0$ for any falling particles.\n    So any particle intersecting the event horizon must achieve the\nCauchy horizon.\n    After going through the Cauchy horizon the particle can achieve\nsingularity.\n    The necessary condition for this is that Carter constant $Q \\le 0$\n(see~(\\ref{geodKerr3})--(\\ref{geodTh})).\n    For particles with negative energy in ergosphere this is true only for\nmovement in equatorial plane, i.e. $Q=0$.\n    Further, one consider the case of equatorial movement $(\\theta = \\pi\/2)$.\n\n    Let us show that for particles with nonzero and zero masses (photons) with\nnegative relative to infinity energy there are no orbits inside ergosphere\nin equatorial plane with constant $r$ or with $r$ changing for all geodesic inside\nthe interval $r_1 \\ge r \\ge r_H$~\\cite{GribPavlov2013a}.\n\n    The necessary condition of existence of orbits with constant $r$ is\n    \\begin{equation} \\label{LeffCucl}\nV_{\\rm eff}=0, \\ \\ \\ \\ \\frac{d V_{\\rm eff}}{d r} =0\\,.\n\\end{equation}\n    To prove our statement it is sufficient to show that for\n$dt\/d \\lambda >0$ and\n    \\begin{equation} \\label{LeffdVefg0}\nE<0, \\ \\ \\ r > r_H, \\ \\ \\ V_{\\rm eff}(r)=0 \\ \\ \\Rightarrow\n\\ \\ V^{\\, \\prime}_{\\rm eff}(r) > 0 .\n\\end{equation}\n\n    For $V_{\\rm eff}(r)=0$ the derivative of the effective potential\ncan be written as\n    \\begin{equation} \\label{Leffder}\nV_{\\rm eff}^{\\, \\prime}(r) = \\frac{1}{2 \\rho^4}\n\\left( 2 (r -M) \\frac{P^2}{\\Delta} + 2 m^2 r \\Delta - 4 r E P \\right).\n\\end{equation}\n    So in order to prove our statement it is sufficient to prove that\nfor the negative energy of the particle in ergosphere one has $P >0$.\n\n    From the condition of movement ``forward in time'' and~(\\ref{geodKerr1})\none obtains\n    \\begin{equation} \\label{geodKett}\nP \\ge  \\frac{ a \\left( a E \\sin^2 \\! \\theta - J \\right) \\Delta}\n{r^2 + a^2}.\n\\end{equation}\n    For particles in ergosphere with $E<0$ and $\\theta=\\pi\/2$, $ r<2$\nfrom~(\\ref{lHmdd}), (\\ref{JmErg}) one has\n    \\begin{equation}\na E  - J \\ge a E  - J_{-} \\ge \\frac{a E }{ r^2 -2 r M} >0.\n\\label{LPerg}\n\\end{equation}\n    That is why $P>0$ and our statement is proved.\n\n    So there are no circular orbits for Penrose trajectories\nin Kerr's black holes.\n    The permitted zone for such particles in ergosphere can have only\nupper boundary.\n\n    Note that one gets absence of orbits with constant $r=r_H$ on the horizon\nof the nonextremal black holes $a<M$ from the fact that for $V_{\\rm eff}=0$\nand $\\Theta \\ge 0$ one  has $J=2 r_H \/M$ and\n    \\begin{eqnarray}\nV^{\\, \\prime}_{\\rm eff}(r_H)=\n\\frac{(r_H - M) (m^2 r_H^2 + (J-a E)^2 + Q}{\\rho^4} \\ge\n\\nonumber \\\\\n\\frac{(r_H-M)}{\\rho^4 a^2} ( m^2 \\rho^2 a^2 +\n(r_H^4 - a^4 \\cos^2 \\! \\theta ) E^2 +\n4 r^2_H M^2 \\cot^2 \\! \\theta ) > 0 .\n\\label{VderH}\n\\end{eqnarray}\n    For extremal black holes $a=M$ one can see from~(\\ref{VderH})\n$V^{\\, \\prime}_{\\rm eff}(r_H)=0$ for $\\theta= \\pi\/2$.\n    However circular orbits for $E\\ne 0$ are also absent in this\ncase as it is shown in~\\cite{HaradaKimura10}.\n\n\n\n\n\\vspace{9mm}\n{\\centering \\section{The time of movement of particles with negative\nenergy in the ergosphere}\n\\label{secErttau}}\n\n\n    Let us analyze the problem of the time of movement for particles with\nnegative energy in ergosphere.\n    As it was shown in the previous section the geodesic with negative energy\nin ergosphere begins from $r=r_H$, then achieves the upper point of the\ntrajectory $r_b$ and after it falls to horizon.\n    So the proper time interval of movement of the particle along all geodesic\nin ergosphere is defined by the integral\n    \\begin{equation} \\label{VsIntdlbO}\n\\Delta \\lambda = 2 \\int \\limits_{r_H}^{r_b} \\frac{ d r }{|d r \/d \\lambda |}\n= 2 \\int \\limits_{r_H}^{r_b} \\frac{d r}{\\sqrt{- 2 V_{\\rm eff}(r)}} .\n\\end{equation}\n    The factor 2 before the integral is due to taking into account the fact\nthat the proper time of movement along geodesic up from some value of\nthe radial coordinate $r$ is equal to the time of falling down to the same\nvalue of $r$.\n\n    In the vicinity of the upper point $r_b$\non the trajectory of the particle with negative energy one has\nfrom (\\ref{LeffUR}) and $V_{\\rm eff}(r_b)=0$ that\n    \\begin{equation} \\label{VsdrdlrH}\n\\left| \\frac{d r}{d \\lambda} \\right| = \\sqrt{- 2 V_{\\rm eff}} \\approx\n\\sqrt{2 (r_b - r) V^{\\, \\prime}_{\\rm eff}(r_b)}.\n\\end{equation}\n    As it was shown in the preceding part for the boundary point of the\npermitted zone $V^{\\, \\prime}_{\\rm eff}(r_b)>0$, so the integral\n    \\begin{equation} \\label{VsIntdlb}\n\\int \\frac{ d r }{|d r \/d \\lambda |} \\sim\n\\int \\frac{d r}{\\sqrt{2 (r_b - r) V^{\\, \\prime}_{\\rm eff}(r_b)}}\n\\end{equation}\n    is convergent and the proper time of the lifting to the upper point\n(falling from the upper point) of the trajectory in the vicinity of this\npoint is finite.\n\n    Due to the fact that permitted zones for particles with negative energies\nin ergosphere can have only upper boundary there are no zeros\nfor $dr \/ d \\lambda$ in the other points of the trajectory.\n    So the integral~(\\ref{VsIntdlbO}) is convergent and the proper time of\nmovement along geodesic in the ergosphere for the particle with the negative\nenergy is finite.\n\n    Note that the coordinate time of movement to the horizon is infinite.\n    For equatorial movement the evaluations of the divergence of the\nintegral for coordinate time were given in~\\cite{GribPavlov2013a}.\n\n    Finiteness of the proper time for movement of particles with negative\nenergy in ergosphere of the black hole leads to the problem of the\norigination and termination of such trajectories.\n    As we said in the Introduction these lines cannot arrive to ergosphere\nfrom the region outside of the ergosphere.\n    So they must originate and terminate inside the gravitational radius.\n    This means that they originate as ``white hole'' geodesics originating\ninside the horizon.\n\n    Note that similar situation takes place for some geodesics of particles\nwith positive energy in Schwarzschild metric (see the text book~\\cite{LL_II}).\n    The geodesic completeness leads to the necessity of taking into account\n``white hole'' geodesics originating in the past singularity of the eternal\nblack hole for radial geodesics with specific energy $E\/m<1$\narising from the region inside the gravitational radius.\n    However for Penrose geodesics we show that all such geodesics\nin ergosphere of the Kerr's black hole have such behaviour.\n\n\n    From~(\\ref{geodKerr3})--(\\ref{ThB0}), (\\ref{JgEH}) one can see that all\nlight like particles (photons) with negative energy falling in equatorial\nplane from ergosphere achieve singularity.\n    Massive particles also achieve singularity for example if $E=-m$\nor the angular momentum is such that\n    \\begin{equation} \\label{FalSin}\nJ \\le J_H \\biggl( 1 + \\frac{a}{2M} \\sqrt{1 + \\frac{m^2}{E^2} } \\, \\biggr).\n\\end{equation}\n    The proof of all these results is the same for\n$d r \/ d \\lambda >0$ and $d r \/ d \\lambda < 0$.\n    That is why the same conditions are valid for ``white hole''geodesics\noriginating in Kerr's singularity, arriving to ergosphere and then going back\ninside the gravitational radius.\n    Particles moving in equatorial plane do not achieve singularity\nif for example\n    \\begin{equation} \\label{FalNS}\n\\frac{|E|}{m} \\ll \\frac{r_C}{M} , \\ \\ \\frac{|J|}{m M} \\ll \\frac{r_C}{M} .\n\\end{equation}\n    Then after achieving some minimal values of the radial coordinate\nthe particle can turn it's movement in the direction of larger $r$ and come\nback to ergosphere along the white hole geodesics.\n\n    Let's come back to the question on the energy of collisions for\nparticle falling onto black hole and  the particle with negative energy\nmoving in ergosphere.\n    As it was shown previously (Eqs.~(\\ref{JmErg}), (\\ref{JgEHf}))\nparticles with large in absolute value negative angular momentum can achieve\nthe region close to the upper frontier of ergosphere.\n    The energy of collision of such particle with the ordinary particle\nwill be very large due to~(\\ref{KerrJB}) at any point of the ergosphere.\n\n    The existence of particles moving from the gravitational radius in the\ndirection of larger $r$ along white hole geodesic can give us\nthe new opportunity for collisions independent of angular momentum\nwith non-limited energy near black holes.\n    As we can see from~(\\ref{KerrL1L2}) the difference between energy\nof collisions with particle moving to increasing $r$  $ (\\sigma_{2r} =1)$\nand decreasing $r$  $ (\\sigma_{2r} =-1)$ is\n    \\begin{equation} \\label{DDD}\n\\Delta E_{c.m}^2 = \\frac{4 \\sqrt{R_1} \\sqrt{R_2}} {\\rho^2 \\Delta} .\n\\end{equation}\n    This difference is equal to zero for top point of trajectory,\nwhen $R_2 =0$.\n    But for non-critical particles $(J \\ne J_H)$ the difference is infinite\nlarge for collision on the horizon $(r \\to r_H)$.\n    So from~(\\ref{KerrL1L2}) one has for such collisions\n$(\\sigma_{1r} \\sigma_{2r} =-1)$ for $r\\to r_H$: \\,\n$P_1 P_2 - \\sigma_{1r} \\sqrt{R_1} \\sigma_{2r} \\sqrt{R_2} > 0$, \\, $\\Delta \\to 0$,\n    \\begin{equation} \\label{ss12m}\nE_{c.m} \\sim \\frac{ \\sqrt{ 2 (P_1 P_2 - \\sigma_{1r} \\sqrt{R_1} \\sigma_{2r} \\sqrt{R_2})}}\n{\\rho \\sqrt{\\Delta}} \\approx\n\\frac{2 a}{ \\sqrt{r-r_H} } \\sqrt{\n\\frac{(J_{1H} - J_1 ) (J_{2H} - J_2 ) }\n{(r_H^2 + a^2 \\cos^2 \\theta )(r_H-r_C)}} \\to +\\infty .\n\\end{equation}\n    The analogue of the result~(\\ref{ss12m}) is valid not only for Kerr's black\nholes but also for Schwarzschild black holes\n    \\begin{equation} \\label{SHw}\na=0, \\ \\ \\sigma_1 \\sigma_2 =-1, \\ \\ \\\nE_{c.m} \\sim 2 \\sqrt{ \\frac{E_1 E_2}{ 1 - (r_H\/ r)}} \\to +\\infty, \\ \\\nr \\to r_H=2M.\n\\end{equation}\n    As it is known from the textbook~\\cite{HawkingEllis}\nfor the Schwarzschild case geodesic completeness leads to existence of\n``white hole'' geodesics.\n    Note that the energies of particles $E_1, E_2$ can not be negative\noutside event horizon of Schwarzschild black holes~\\cite{GribPavlov2010NE}.\n    One can see from~(\\ref{Relsk03}) that the effect of the large energy\nis explained by the large Lorentz factor, so that $v_{\\rm rel} \\sim 1$\non the horizon.\n\n    The formulas~(\\ref{ss12m}), (\\ref{SHw}) can be interpreted as nonstability\nof the configuration with ``white hole'' geodesics and therefore instability of\neternal black holes.\n    Evaluation of the role of the gravitational wave radiation for large\nvalues of the energy in the centre of mass frame is needed to get the final\nanswer about the physical sense of the obtained results.\n\n\n\\vspace{7mm}\n{\\centering \\section{Conclusion}\n\\label{Conclusion}}\n\n\n1) Geodesics with negative energy in equatorial plane\noriginate inside the gravitational radius of the rotating black hole\nare ``white hole'' geodesics!\n    This is similar to the case of Schwarzschild eternal black hole for which\nit was shown in~\\cite{GribPavlov2010NE} that negative energy\ntrajectories arise in the white hole past singularity.\n    However differently from the eternal Schwarzschild black hole when there\nare two different space like singular surfaces --- the black hole and white\nhole --- here we have one Kerr's time like singular surface on which some\ngeodesics originate and some terminate.\n\n2)\nOne can get information about the interior of the gravitational radius if\nsome particles move along these geodesics!\n    So there is no cosmic censorship for such rotating black holes if one\nunderstands cosmic censorship as impossibility to get information from\nthe region inside the gravitational radius.\n    The cosmonaut can get direct information about the interior of the\ngravitational radius only inside the ergosphere.\nHowever if one considers interaction of negative energy particles radiated by\nthe ``black-white'' hole to ergosphere with ordinary positive energy particles\nescaping the ergosphere this information can be obtained by any external observer.\n    The electromagnetic interaction of the negative energy photons with usual\nmatter can lead to some new physical process of ``annihilation'' of this\nmatter inside ergosphere.\n    The energy in the centre of mass frame of the collision of ordinary\npositive energy particle with the particle with negative energy is growing\nwithout limit on the horizon.\n\n3)\nThe mass of the ``black-white'' hole can grow due to the radiation of negative\nenergy particles to ergosphere where these particles interact with\npositive energy particles.\n\n    Surely all results in this paper correspond to exact Kerr's solution.\n    For physically realized rotating black holes as the result of the star\ncollapse due to nonstability of interior solution inside the gravitational\nradius especially inside the Cauchy horizon (see~\\cite{GribPavlov2008UFN})\nthe results can be different and the special research is needed.\n\n\n\\vspace{0mm}\n {\\bf Acknowledgments.}\n    The research is supported by the grant from The John Templeton Foundation.\n\n\n\\vspace{5mm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Light quark confinement\\footnote{This is the third lecture on quark \nconfinement given by V.N.Gribov in 1992 in Orsay. An extensive discussion of the\nconsequences of all this for the structure of the Green function can be found in\n[5,6] - in the two last papers concluding his 20 years long study of the problem\nof quark confinement in QCD.}\n\\footnote{The text was prepared for\npublication by Yu. Dokshitzer, B. Metsch and J. Nyiri on the basis of a tape    \nrecording and notes taken during the lecture}}\nWe have described the confinement of heavy quarks in an analogy with \nthe theory of the supercharged nucleus [1,2]. Let us now suppose again\nthat $\\alpha$ is behaving like\n\\[ \\picbox{O1.pstex_t}  \\]\nMaking this assumption, we are neglecting gluon-gluon interactions and the\nexistence of gluons as real particles. Our aim is to see, what can arise\nfrom the discussion of light quarks only. We introduce $\\lambda$\ncorresponding to $\\alpha_c$ and consider quark masses $m_0 \\ll \\lambda$.\nThe interactions of light quarks (for which $m_0 \\ll \\lambda$) will be\ndiscussed in a rather simplified way. We will take into account all\npossible interactions \n\\[ \\picbox{O2.pstex_t} \\]\nwhere the gluon propagator (considered as an effective photon),\ncorresponding to the dotted line is \n\\begin{equation}\n\\label{1}\nD_{\\mu\\nu} = \\frac{\\alpha}{q^2}\\delta_{\\mu\\nu} .\n\\end{equation}\nFurther, we look for a model which enables us to see, what happens to\nthe fermions if there is an interaction \nbetween them, as indicated above. The\nquestion is, how the bound states or the Green function behave in such\na case.\n\nLet us consider the energy of two quarks, $u$ and $\\overline{d}$, for\nexample. Without interaction there will be positive energy states with\n$E > 2m$\nand negative energy states with $E<-2m$:\n\\[  \\picbox{O3.pstex_t} \\]\nIntroducing the interaction, for small $\\alpha$ we will\nfind that there are some bound states near $2m$ and $-2m$.\n\\[  \\picbox{O4.pstex_t}  \\]\nSo far, we consider the usual Dirac vacuum: the negative \nenergy\nstates are\noccupied, and the positive ones are empty. Increasing the coupling, i.e.\nincreasing $\\alpha$, we could expect that the \nmagnitude of the energy is decreasing\nand the levels corresponding to the bound states will come closer and\ncloser to zero.\n\\[  \\picbox{O5.pstex_t}  \\]\nWith a further increase of the coupling up to a critical value, one\npossibility for the levels will be just to approach the zero line and\nnever cross. There is, however, also a possibility of crossing.\nWe will see that the first case corresponds to normal spontaneous\nsymmetry breaking. But, if the levels cross, and especially in the most\nclear case, when they pass the lines $2m$ and $-2m$, respectively,\neverything will change and we arrive at\nvery different phenomena:\n\\[ \\picbox{O6.pstex_t}  \\]\nIndeed, now the \noriginal \nvacuum which corresponds to the case when level $2$ is\nempty and level $1$ is occupied, is absolutely unstable. We have to\nfill the new negative energy state and leave the positive energy\nlevel empty. But by filling this new state, we get an excitation, a meson-type\nstate with a mass $\\mu$. For free quarks this would mean that the quark\nwith negative energy decays into a negative energy meson (filling the\nnegative energy levels) and creates a positive energy quark. As a result,\nthe Dirac picture in which all negative energy levels are filled up and\nall positive energy levels are empty, is destroyed. But if so, a positive\nenergy quark also decays into a positive energy meson and a quark with\nnegative energy. This means that both decays\n\\begin{eqnarray*}\n  q^- & \\rightarrow & \\mu_- + q^+ \\\\\n  q^+ & \\rightarrow & \\mu_+ + q^-\n\\end{eqnarray*}\nare possible, and both $q^-$ and $q^+$ are unstable.\n\nThe question is now, how to deal with the bound state\nproblem. Of course, we could\njust start to calculate the bound states, considering the interactions\nwithout corrections to the Green function. However, one has to take\ninto account \nthat the fermion-fermion interaction changes the effective mass of the\nquarks and this in its turn will change the bound states considerably,\n\\[  \\picbox{O7.pstex_t}  \\]\nwhich makes the problem more complicated. We \nthus \nwill have to consider\nbound states and the Green functions\non equal footing. \n\nUp to now, the only approach which deals with this problem and is\nself-consistent is the Nambu - Jona-Lasinio model [3]. It considers the\nfermion Green function\ncorrections \ndue to a four-fermion interaction:\n\\[  \\picbox{O8.pstex_t}  \\]\nIn spite of the strong dependence on the cut-off, the model preserves\nall symmetries in the Green function and in two-particle interactions.\nLet us present the result of Nambu and Jona-Lasinio in a way somewhat\ndifferent from what is given in [3]. We express it as the dependence\nof the renormalized mass \n$m$ on the bare mass\n$m_0$. \nThey found that if the\neffective coupling (it depends on the definition in their case)\n$\\frac{g^2}{2\\pi}$ is less\nthan unity, the curve will be just the usual one:\n\\[  \\picbox{O9.pstex_t}  \\]\nIf, however, \nthe coupling\n$\\frac{g^2}{2\\pi}$ is larger than unity, the dependence\nwill be like \n\\footnote{This result is not always quoted, but it is\npresent in their paper.}:\n\\[  \\picbox{O10.pstex_t}  \\]\nAccording to the interpretation of Nambu and Jona-Lasinio, the upper part\nof the curve, which at $m_0=0$ reaches a finite point, corresponds to the\nspontaneous symmetry breaking. But there are three solutions at $m_0=0$\nand at sufficiently small $m_0$ values. What Nambu and Jona-Lasinio claim\nis that the lower part of the curve is unstable, and there is a real\nvacuum. I agree with this, if $m_0>m_{crit}$.\nWe can ask: what is the source of instability\nof this curve? The general argumentation is the following. Talking\nabout spontaneous symmetry breaking, $m$ is like a magnetic field in a\nferromagnetic; we just choose a definite direction. But $m_0$ is like an\nexternal field, and the system is like a compass. If the external field\nand the induced field are pointing at the same direction, the situation\nis stable. If they point to opposite directions, the compass will change.\n\nI am, however, not sure that the instability of the almost perturbative\nsolution which contains no condensate at all can be explained in such\na way. The explanation may be right for the part $b$ of the curve in\nFig.2 which corresponds to a big spontaneous magnetic moment and the\nopposite direction of $m_0$. It does not work for the part $a$ close to\nperturbation theory which has no spontaneous magnetic moment. And,\nlooking more carefully at the curve, we see that the part $b$ corresponds\nto pseudoscalar states inside the Dirac see, while on the piece $a$ both\nthe pseudoscalar and scalar states are inside, both levels passed.\nRecognizing this, one can conclude that indeed, the mentioned state is\nunstable, but for a trivial reason: the corresponding level is inside\nthe Dirac see and it is not filled up. The problem is, what happens if\nwe fill up this level. It remains an open question, what can be\nconsidered as a ground state under these conditions. And\nit is a problem how to get these results in a more self-consistent\nway, not depending on the cut-off so strongly.\n\nThe Nambu - Jona-Lasinio model can be reproduced in our picture. For this\npurpose, just as a theoretical exercise, let us use $\\alpha$  not going\nto unity, i.e. draw\n\\[ \\picbox{O11.pstex_t} \\]\ninstead of the curve in Fig.1. In this case there will be a second scale\n$\\lambda_2$, and outside this scale we will have just point-like\ninteraction. This reminds of \nthe Nambu - Jona-Lasinio picture, which,\napparently, can be reached somehow in our approach. The problem is, how\nto write constructively the corresponding equation, and whether this\ncan be done at all. Of course, this constructive part can be only\napproximate. But if we recognize that it can be written, then we will\nbe able to develop a theory in which we put the main ingredient of our\ndiscussion as an input into our solution and try to find the real\nconstruction. The main difference will appear in the analytic properties\nof the Green function. The Green function of a fermion for such a case\nwould be quite different in \nits\nanalytic structure compared to the usual one.\n\nI am afraid I will not have the time to come to this point today, but\nI would like to explain just the physics.\n\nHow to write the equation? What happens in the real case and how to\ndeal with it? Let us start with with the Green function. What do we\nknow, what are we supposed to know about this Green function as a\nfunction of $q^2$ ?\n\\[ \\picbox{unnamed.pstex_t} \\]\nBeyond a certain\n$\\lambda$ in the region where the coupling is small, it\nis asymptotically free; here the Green function has to satisfy the\nrenorm group equation. But if as a result of the interaction a mass is\nacquired, this mass would be somewhere at smaller $q^2$; \nhere the equation\nbecomes essentially very complicated and we are not able to extract\na reasonable structure.\n\nThe idea to write an equation which is correct in both regions, near\nthe threshold and at large $q^2$, and to match these two solutions,\ncomes from the following consideration. Suppose that $\\alpha_{crit}\/\\pi$\nis small:\n\\[   \\frac{\\alpha_{crit}}{\\pi} \\approx 1-\\sqrt{\\frac{2}{3}} = 0.2 .\\]\nNow, however, we may ask: how could new masses, new solutions etc.\nappear at all at such a small $\\alpha$. Obviously, this $\\alpha$ has\nto be multiplied by something large. What happens, for example, at large\n$q$? We know, that there is always a logarithm of $q^2\/\\lambda^2$ and the\nreal parameter becomes\n\\[ \\alpha_0 \\ln \\frac{q^2}{\\lambda^2} \\]\nwhich is, in spite of the smallness of $\\alpha$, big enough to change\nthe Green function essentially. But near the threshold there is also a\nlogarithm:\n\\[  \\ln \\frac{q^2-m^2}{\\tilde{\\lambda}^2}\\,,\\qquad \n        \\mbox{ with some scale }\\tilde{\\lambda}= \\lambda \\mbox{ or } m  \\]\nwhich is always present. In other words, in this region there could be\nalso a quantity which changes seriously in spite of the relative\nsmallness of $\\alpha$. Hence, we want to write the equation which is\ncorrect near the threshold, taking into account correctly the singularity\nof a supposed mass, and after that compare this with the renorm group\nequation; we shall see whether it is possible to write an equation which\nis correct in both regions, and if yes, we will try to solve it.\nIn order to get the singularity correctly, we take the second derivative\nof $G^{-1}$ with respect to the momenta.\n\\begin{equation}\n\\label{2}\n \\picbox{O13.pstex_t}\n\\end{equation}\nThe contribution of the first term is trivial, the second derivative of\n$q+m$ gives zero. Taking the second derivative of the first graph, it can\nbe easily seen that\n\\begin{equation}\n\\label{3}\n\\partial^2 \\frac{1}{(q-q')^2-i\\varepsilon} = -4\\pi i\\delta^{(4)}(q-q')  .\n\\end{equation}\nThis gives for the first diagram $\\gamma_{\\mu}G(q)\\gamma_{\\mu}\n\\frac{\\alpha}{\\pi}$ -- just by direct calculation. In other words, we\nmake it local.  From this diagram we take the contribution where $k\n\\equiv (q-q')$ -- the momentum of the photon -- essentially equals zero.\n\nLet us now look at the second diagram. We have here the choice of taking\nthe second derivative at one of the photon lines, or to differentiate\nonce at one line and once at the second line. Having in mind that all\nthe integrations would have a structure which need some logarithmic\nenhancement, it would mean that the most important regions of integration\nin this integral would be those where $k_1 \\ll k_2$ and $k_2 \\ll k_1$.\nWe take the derivative at $k_1$ and then set it to be zero, but for $k_2$ the\nintegration will give the same as before. If this integration gives us\ntwo logarithms, we kill one and recover it after the integration of our\ndifferential equation; but we still have the first one. But if we\ndifferentiate once one line and once the other, we will always sit on\nthe region $k_1 \\sim k_2$, because they have to be of the same order.\nAnd, in this case, there is no logarithm at all; after the integration,\nwe will recover one, but one order will be lost. Clearly, a possible\napproach is to try not to choose different diagrams, but to use the small\n$k$ region of integration. Ordering the\nintegration inside the diagram in such a way that one momentum is much\nsmaller than the others, and differentiating this diagram, we will find\na relatively simple answer. Indeed, suppose that we have any diagrams\nwith any loops. If we differentiate some lines twice (it can be any line)\nand neglect all first derivatives, we get an amplitude of the following\nstructure:\n\\[ \\picbox{O14.pstex_t}  \\]\nThis is just the Compton scattering of a zero momentum photon $k=0$, and\nfor this quantity the most singular contribution is obviously\n\\[  \\Gamma_{\\mu}(0,q)G(q)\\Gamma_{\\mu}(0,q)  \\]\nwhich corresponds to the diagram\n\\[ \\picbox{O15.pstex_t} .\\]\n\nBut \nthe vertex $\\Gamma$ is at zero momentum \nand hence $\\Gamma_{\\mu}(0,q) = \\partial_{\\mu}G^{-1}(q)$. \nIn this approximation we\ncan write a very simple equation:\n\\begin{equation}\n\\label{4}\n\\partial^2 G^{-1}(q) = \\frac{\\alpha(q)}{\\pi}\\partial_{\\mu}G^{-1}(q)G(q)\n \\partial_{\\mu}G^{-1}(q)\n\\end{equation}\nwhich differs essentially from any Bethe-Salpeter type equation. Indeed,\nusing a Bethe-Salpeter type equation, we do not change the vertex part\nand end up with rather bad properties. Equation (\\ref{4}) is scale\ninvariant, it is $\\gamma_5$-invariant, it has many nice symmetry\nproperties and, what is most important, it has a correct behaviour near\nthe threshold.\n\nThe gauge is fixed, because we used\n\\[  D_{\\mu\\nu} = \\frac{\\delta_{\\mu\\nu}}{q^2}.  \\]\nIt is an important question, what we would get in different gauges.\nIn Feynman gauge we are very lucky: we find an expression which does\nnot depend explicitly on the expression for the Green function.\nUsing a different gauge, we would find the infrared behaviour of this\ndiagram to be more complicated, and we would not be able to extract\nuniversally the region of small $k$. We would have integrals over $q$\nwhich are also possible to use, but with the necessity to think about the\nbehaviour near the threshold.\n\nWe, however, have chosen this gauge; we did not destroy the general\nfeatures and used the current conservation which just corresponds to\n$\\Gamma_{\\mu} = \\partial_{\\mu}G^{-1} $. Accepting this, we can now ask,\nwhat is the relation to the renorm group equation.\n\nSuppose that we would like to write the renorm group equation in the\nsame spirit. Let us take again the second derivatives. In this case\nwe would be definitely correct, because we know that it is a logarithmic\napproximation.\n\nIn our logarithmic approximation we will do exactly the same with the\nonly difference that $\\alpha$ would be $\\alpha(q^2)$ .\nIn the renorm group equation $\\alpha$ is a function of $q^2$. But, of\ncourse, $\\alpha$ in general depends on two momenta: $k^2$ and $q^2$.\nAnd in the renorm group equation at large \nmomenta, in the ultraviolet\nregion, $\\alpha$ depends on the variable which is the largest.\nSince we are close to $k=0$, this means that here we will have\n$\\alpha(q^2)$, \nand we will recover the renorm\ngroup equation at large $q^2$. If we solve this equation with a\nslowly varying $\\alpha$, we will be correct in both the threshold region\nand the ultraviolet region.\n\nWe also have to formulate an equation for the bound states under the same\nassumption. Looking for bound states, we consider scalar and pseudoscalar\nvertices. This vertex\n\\[ \\picbox{O16.pstex_t}   \\]\nhas to be equal to \n\\[ \\picbox{O17.pstex_t}   \\]\n\nHere $\\varphi(q,p)$ depends on $p$, the total momentum of a pair, and\n  $q$, the quark momenta being given by $q+p\/2$ and\n$q-p\/2$. \nWith the same procedure as in obtaining the equation for the Green\nfunction we find for the vertex (see [4] for some details):\n\\begin{equation}\n\\label{5}\n\\partial^2 \\varphi(q,p) = \\frac{\\alpha}{\\pi}\\left[ A_{\\mu}(q)\\partial_{\\mu}\n \\varphi(q,p) + \\partial_{\\mu}\\varphi\\tilde{A_{\\mu}}(q) - A_{\\mu}(q)\\varphi(q,p) \\tilde{A_{\\mu}}(q)\\right]\\,,\n\\end{equation}\nwhere $A_{\\mu}=\\partial_{\\mu}G^{-1}G$, $\\tilde A_{\\mu}= G\n\\partial_{\\mu}G^{-1}$\\,.\nIt means that we have two equations in this approximation. We used this\napproximation just to be constructive and to study what will result if we\nmake this approximation. In principle, solving both equations \nwe will get everything\nwhat is necessary: we know $G$ and $A_{\\mu}$, we have a linear equation\nfor bound states, we can see what is the type of the energy etc. The\nequation for the bound states has very nice features.  It is beautiful\nfrom the point of view of the Goldstone theorem in the following\nsense.\n\nSuppose I have some symmetry in my equation,\ne.g. $\\gamma_5$-invariance. Since there is no $\\gamma_5$ in\nequation (5), it is  $\\gamma_5$-invariant. But of course the boundary\ncondition for $G^{-1}$ at $q \\to \\infty$ is just $G_0^{-1}=(\\hat q -\nm_0)$, and thus destroys the symmetry. But suppose that $m_0=0$. In\nthis case there would be symmetry here, which means that the Green\nfunction will not be unique, since it can be \n\\begin{displaymath}\n  G^{-1} + \\delta G^{-1}, \n\\end{displaymath}\nwhere $G^{-1}$ is some solution and $ \\delta G^{-1} \\propto \\gamma_5\nG^{-1}$\\,. This means that the variation  $\\delta G^{-1}$ also is\nimportant. What would be the equation for the variation? If we\ncalculate the variation on both sides of equation (4) we obtain\n\\begin{displaymath}\n  \\partial_\\mu\\,(\\delta G^{-1}) = \\frac{\\alpha}{\\pi}\\left(\n    \\partial_\\mu(\\delta G^{-1})G\\partial_\\mu G^{-1} +\n        \\partial_\\mu G^{-1} G \\partial_\\mu (\\delta G^{-1}) -\n        \\partial_\\mu G^{-1} G (\\delta G^{-1}) G \\partial_\\mu G^{-1}\n    \\right)   \\,,\n\\end{displaymath}\nso we find that $\\varphi=\\delta G^{-1}$ fulfils equation (5) at\n$p=0$. It means that if some symmetry is broken, i.e. if there are\nmultiple solutions of the equation for the Green function, we always\nwill have some solution of the equation for the vertex at $p=0$,\nwhich is the Goldstone.\n\nIt is clear, that in the present model we can discuss many questions,\nuse a running coupling $\\alpha$ as in Fig. 1 and reproduce the\nNJL-features without any essentials depending on a cutoff.\nBefore discussing this point further, we will first look for the\nsolution of (4) and discuss the result. \n\nAbove, we introduced $A_\\mu = (\\partial_\\mu G^{-1}) G$, which is a very\nuseful quantity. Since $G^{-1}=a\\frac{\\hat q}{q}+b$ is essentially a\n$2 \\times 2$ matrix, $A_\\mu$ is a $U(2)$ gauge potential: \n\\begin{equation}\n\\label{6}\n  \\partial^2G^{-1} = \\partial_\\mu((\\partial_\\mu G^{-1})\\,G\\,G^{-1}) \n  = (\\partial_\\mu A_\\mu)G^{-1} + A_\\mu (\\partial_\\mu G^{-1}) =\n  \\frac{\\alpha}{\\pi}A_\\mu\\partial_\\mu G^{-1}\n\\end{equation}\nwhere in the last step we used Eq. (4). Multiplying from the right by\n$G$ we thus find\n\\begin{equation}\n\\label{7}\n\\partial_{\\mu}A_{\\mu} + A_{\\mu}A_{\\mu} = \\frac{\\alpha}{\\pi}A_{\\mu}A_{\\mu}.\n\\end{equation}\nThis means that\n\\[\\partial_{\\mu}A_{\\mu}=-\\beta A_{\\mu}A_{\\mu};\\]\nand $A_\\mu$ is a pure $U(2)$-gauge potential with a condition $\\beta = 1-\\frac{\\alpha}{\\pi}$.\nOf course, this is just a useful trick. Important is to write down the real\nequation for the Green function. The most natural thing is to express\n$G^{-1}$ in the form\n\\[   G^{-1} = \\rho\\, e^{\\frac{\\varphi}{2}\\hat{n}} ,\\]\nwhere $\\hat{n}$ is a $2 \\times 2$-matrix\n\\[  \\hat{n} = \\frac{\\hat{q}}{q} .\\]\nIt is just easier to use this form for our purpose: we can find an\nequation for $\\rho$ and an equation for $\\varphi$. Both are functions of\n$q^2$: $\\rho(q^2)$, $\\varphi(q^2)$. There is, however, no scale in the\nequation; it contains only a derivative of $q$. If we introduce\n\\[  \\xi = \\ln q ,\\]\nwe will find an equation in which $\\xi$ can be considered as ''time'', and\nwhich is an oscillator equation. In fact there are two oscillators, one\nfor $\\rho$, the other for $\\varphi$, and they will satisfy non-linear\nequations. For $\\varphi$ we find\n\\begin{equation}\n\\label{8}\n\\ddot{\\varphi} + 2\\left(1+\\beta\\frac{\\dot{\\rho}}{\\rho}\\right)\\dot{\\varphi}\n - 3\\sinh\\varphi = 0 .\n\\end{equation}\nThis is just an oscillator with damping; a similar equation can be\nwritten for $\\rho$. Important is that that there has to be ''energy''\nconservation in this equation. Indeed, we said that $\\xi$ plays the\nrole of time; it, however, did not enter the equation explicitly.\nThus there has to be a conservation law which, as it is easy to show,\nleads to\n\\begin{equation}\n\\label{9}\n\\left(1+\\beta \\frac{\\dot{\\rho}}{\\rho}\\right)^2 =  1 + \n \\beta^2\\left(\\frac{\\dot{\\varphi}^2}{4}-3\\sinh^2 \\frac{\\varphi}{2}\\right) .\n\\end{equation}\nWe thus can eliminate $\\rho$ altogether, and find the equation for\n$\\varphi$\n\\begin{displaymath}\n\\ddot{\\varphi} +\n 2\\sqrt{1-\\beta^2\\left(3\\sinh^2\\!\\frac{\\varphi}{2}-\\frac{\\dot\\varphi^2}{4}\\right)}\\dot{\\varphi}\n - 3\\sinh\\varphi = 0\\,,\n\\end{displaymath}\nwhich is an oscillator with damping. Having this in mind is sufficient\nto understand the structure of the solution. Indeed, what is this\n$\\varphi$ ? We have \n\\begin{equation}\n\\label{10}\nG^{-1} = \\rho\\cosh\\frac{\\varphi}{2} + \\frac{\\hat{q}}{q}\\rho\\sinh\n \\frac{\\varphi}{2} .\n\\end{equation}\nThe perturbative solution is $\\varphi$ close to $i\\pi$. In this case\nthe first term is zero, the other is proportional to $\\hat{q}\/q$ -\nthis corresponds to the massless solution. Since $m_0$ is small, we\nhave to have solutions like this at $q \\rightarrow \\infty$.\n\nNow we have to find the solution everywhere. Let us first investigate\nthe equation without damping; we get\n\\[ \\ddot{\\varphi} - 3\\sinh\\varphi = 0 .\\]\nIf we go to the Euclidean space, $\\varphi = i\\psi$, the potential becomes\na periodical potential:\n\\[ \\ddot{\\psi} - 3\\sin\\psi = 0 .\\]\n\\[ \\picbox{O18.pstex_t}    \\]\nWe have to look for a possible solution for this\nstructure with damping. What does this mean? For the oscillator\nwith damping any solution at $\\xi \\rightarrow \\infty$ has to be in a\nminimum, because the energy is decreasing. But if $\\xi$ is going\nto $-\\infty$, the energy is growing. What could be in this case a normal,\nreasonable solution? It is almost clear that the only possibility is\nto put at $\\xi \\rightarrow -\\infty$ the ''particles'' in this\noscillator at the maximum,\nand start to move them slowly; eventually, they will appear inside the\nwell.\n\n\n\n\n\nThere is a most important question, namely: what is the critical\ncoupling in this case? What do we know about an oscillator with\ndamping? If the damping is large enough, all the trajectories will go\nmonotonically to the minimum. If the damping is sufficiently small,\nthe solution will start to oscillate. In order to see when this\ntransition happens, we have to look for the equation just near the\nminimum $\\psi=\\pi$. With $\\phi \\equiv \\pi - \\psi$ we have for small\n$\\phi$\n\\begin{displaymath}\n  \\ddot \\phi + 2 \\sqrt{1+3\\beta^2}\\dot\\phi + 3 \\phi = 0\\,,\n\\end{displaymath}\nwith fundamental solutions\n\\begin{displaymath}\n  \\phi_{1,2} = \\mbox{e}^{\\nu_{1,2}\\xi} \\mbox{ where } \\nu_{1,2} =\n  -\\sqrt{1+3\\beta^2}\\pm\\sqrt{3\\beta^2-2}\\,.\n\\end{displaymath}\nSo we have monotonic behaviour for $3\\beta^2-2>0$. On the other hand\nif $\\beta^2<\\frac{2}{3}$, i.e.\n\\begin{displaymath}\n  \\frac{\\alpha_{crit}}{\\pi} = 1 - \\sqrt{\\frac{2}{3}} <\n  \\frac{\\alpha}{\\pi} <  1 + \\sqrt{\\frac{2}{3}}\\,,\n\\end{displaymath}\nwe will have oscillations before reaching the minimum. The critical angle $\\psi_c$, which\nseparates the regions where the solution is monotonic and where it\noscillates can be\nshown (see e.g. (4)) to be given by \n\\begin{displaymath}\n  \\sin^2 \\frac{\\psi}{2} = \\left(\\frac{2}{3}-\\beta^2\\right)\n  \\sqrt{\\frac{1+3\\beta^2}{1-\\beta^2}}\n  \\frac{1}{1 + \\sqrt{(1+3\\beta^2)(1-\\beta^2)}}\\,.\n\\end{displaymath}\n\\[  \\picbox{O19.pstex_t}  \\]\n\nUp to now we have considered a constant coupling $\\alpha$. We know that\nfor $q>\\lambda$ the Green function is determined by perturbation\ntheory, which has to match the solutions in the region of smaller\n$q$. If $\\beta^2>\\frac{2}{3}$ for all $q^2$, the solution which goes\nas $\\psi \\approx \\frac{q}{m_c}$ for $q\\to 0$ matches the solution\n$\\frac{i}{2}(\\psi-\\pi) \\approx \\frac{m_0}{q} + \\frac{\\nu_1^2}{q^3}$ for\n$q\\to \\infty$ monotonically. This determines $m_0$ as a function of $m_c$ in a\nunique way.\nLet $\\lambda$ be the value of $q$ where\n$\\beta^2(\\lambda^2)=\\frac{2}{3}$.\nIf, however, $\\beta^2<\\frac{2}{3}$ below $q=\\lambda$, the solutions can\noscillate and $m_0(m_{c_i})=0$ for some $m_{c_i}$ as indicated in\nFig.6. \n\\[  \\picbox{O21.pstex_t}  \\]\n  \\[   Fig.6 \\]\n This then is a\nsolution corresponding to broken chiral symmetry.\n\n\\section*{References}\n\\begin{description}\n\\item[1] V. N. Gribov, Orsay lectures on confinement (I), \npreprint  LPTHE\n Orsay 92-60 (1993); hep-ph\/9403218\n\\item[2] V. N. Gribov, Orsay lectures on confinement (II), preprint LPTHE\n Orsay 94-20 (1994); hep-ph\/9407269\n\\item[3] Y. Nambu, G. Jona-Lasinio, Phys. Rev. \\underline{122} (1965), 345\n\\item[4] V. N. Gribov, Lund preprint LU 91-7 (1991)\n\\item[5] V. N. Gribov, QCD at large and short distances, \n  Bonn preprint TK 97-08 (1997), hep-ph\/9807224.\n\\item[6] V. N. Gribov, The theory of quark confinement, Bonn preprint\n TK 98-09 (1998), hep-ph\/9902279\n\\end{description}\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe following bound was introduced by Lov{\\'a}sz in his celebrated paper on Shannon capacity. In particular it implies famous Hoffman bound~\\cite{haemers2021hoffman}. We provide the proof in the interest of completeness.\n\n\\begin{theorem}[Lov{\\'a}sz,~\\cite{lovasz1979shannon}]\nLet $G = (V,E)$ be a simple graph. Consider a symmetric real matrix $A$ such that $A_{ij} = 1$ for every pair $\\{i,j\\} \\notin E(G)$.\nThen\n\\[\n\\alpha(G) \\leq \\lambda_{max}(A),\n\\]\nwhere $\\alpha(G)$ is the size of a maximal independent set of $G$ and $\\lambda_{max}(A)$ is the maximal eigenvalue of $A$.\n\\label{lovasz}\n\\end{theorem}\n\n\\begin{proof} Let $I$ be an independent set, denote by $\\chi_I$ its characteristic vector. Then\n\\[\n(A\\chi_I, \\chi_I) = |I|^2.\n\\]\nFrom the other hand, one has\n\\[\n(A\\chi_I, \\chi_I) = \\sum c_i^2\\lambda_i \\leq \\left(\\sum c_i^2 \\right) \\lambda_{max} = |I| \\cdot \\lambda_{max},\n\\]\nwhere $\\chi_I = \\sum c_i v_i$ is the decomposition of $\\chi_I$ via orthonormal eigenbasis $\\{v_i\\}$ of $A$. \n(We use that symmetric matrix has real spectrum and the length of $\\chi_I$ is the same in the standard basis and in $\\{v_i\\}$, i.e. $\\sum c_i^2 = |I|$.)\n\\end{proof}\n\nThe minimum of $\\lambda_{max}(A)$ over the appropriate $A$ is called \\textit{Lov{\\'a}sz number} or \\textit{Lov{\\'a}sz theta function} of a graph.\n\nAlso we need the following corollaries. \nSuppose that $A$ and $G$ satisfy the conditions of Theorem~\\ref{lovasz}.\nLet $c$, $\\lambda_{max}$ and $\\spp$ stand for the minimal entry, the maximal eigenvalue and the spectral radius of $A$, respectively.\n\n\\begin{corollary}\nLet $I$ be a set with at most $\\varepsilon |I|^2\/2$ edges inside. \nSuppose that $(1-c)\\varepsilon < 1$. Then\n\\[\n|I| \\leq \\frac{\\lambda_{max}}{1-(1-c)\\varepsilon}.\n\\]\n\\label{supersaturation}\n\\end{corollary}\n\n\n\\begin{corollary}\nLet $I$ and $J$ be subsets of $V(G)$ with at most $\\varepsilon |I| \\cdot |J|$ edges between $I$ and $J$ (edges in $I\\cap J$ are counted twice here).\nSuppose that $(1-c)\\varepsilon < 1$. Then\n\\[\n|I| \\cdot |J| \\leq \\left( \\frac{\\spp}{1-(1-c)\\varepsilon} \\right)^2.\n\\]\n\\label{cross}\n\\end{corollary}\n\nFor the bounds on \\textit{disjoint} $I$ and $J$ one can make the class of appropriate matrices slightly wider, i.e.\nnot demand $A_{ii} = 1$. Then one may combine the proof of Corollary~\\ref{cross} with Proposition 4.1 in the paper of Haemers~\\cite{haemers2001bicliques}.\n\n\n\n\\subsection{A straightforward application to eventown problem}\n\nLet $F$ be a family of subsets of $[n]$ is \\textit{eventown} if the intersection of any two members is even (in particular all sets have even size).\nBerlekamp~\\cite{berlekamp1969subsets} and Graver~\\cite{graver1975boolean} independently proved $F$ has at most $2^{\\lf n\/2\\right\\rfloor}$ members, which is also best possible.\nThe proof is very short up to general linear algebra. Note that every maximal eventown $F$ is a linear subspace of $\\mathbb{F}_2^n$; otherwise one can replace $F$ with $\\sppan F$.\nSince $F$ lies in the orthogonal complement $F^\\perp$ and $\\dim F + \\dim F^\\perp = n$, $|F|$ has the dimension at most $\\lf n\/2 \\right\\rfloor$.\n\nConsider the following Hadamard matrix\n\\[\nA = \n\\begin{pmatrix}\n1 & 1 \\\\\n1 & -1\n\\end{pmatrix}.\n\\]\nIts spectrum is $\\{\\pm \\sqrt{2}\\}$. \nThen the spectrum of $M=A^{\\otimes n}$ is $\\{\\pm 2^\\frac{n}{2} \\}$. \nLet us consider $G=(2^{[n]},E)$ and $(X,Y)\\in E$ iff $|X\\cap Y|$ is odd. Then we identify elements of $2^{[n]}$ with $\\{0,1\\}^n$ by usual way as well as indices for rows and columns of matrix $M$ (where we mean that  $A=(a_{ij})_{i,j\\in \\{0,1\\}}$). Let $M=(m_{st})_{s,t\\in \\{0,1\\}^n}$. Then for each $X,Y\\in 2^{[n]}$ we have\n\\[\n m_{\\chi(X),\\chi(Y)}=\\prod_{r=1}^n a_{(\\chi(X))_r, (\\chi(Y))_r}=(-1)^{|\\{ r\\in \\{1,\\dots,n\\}| a_{(\\chi(X))_r, (\\chi(Y))_r}=-1 \\}|}=(-1)^{|X\\cap Y|}.\n\\]\nThus we see that graph $G$ and matrix $M$ satisfy the conditions of Theorem~\\ref{lovasz}.\n\nApplying Theorem~\\ref{lovasz} one has $|F| \\leq 2^{n\/2}$. For even $n$ we already get another proof of eventown theorem.\nFor even $n$ we should also recall that $F$ is a linear subspace, so $|F| \\leq 2^{\\lf n\/2 \\right\\rfloor}$.\n\n\nLet $op(F)$ denote the number of distinct pairs $f_1, f_2 \\in F$ for which $|f_1 \\cap f_2|$ is odd.\nO'Neill~\\cite{oNeill2021towards} showed that for $1 \\leq s \\leq 2^{\\lf n\/2 \\right\\rfloor} - 2^{\\lf n\/4 \\right\\rfloor}$ there is a family $F$ with $|F| = 2^{\\lf n\/2 \\right\\rfloor} + s$\nand $op(F) = s \\cdot 2^{\\lf n\/2 \\right\\rfloor - 1}$.\nAlso he conjectured that this example is tight and proved the conjecture for $s = 1,2$.\nThe application of Corollary~\\ref{supersaturation} gives twice weaker bound for even $n$ (and much weaker bound for odd $n$).\n\n\\begin{theorem}\n\\label{theoremop}\nLet $|F| = 2^{n\/2} + s$ for some integer $s$. Then\n\\[\nop(F) \\geq s \\cdot 2^{\\lf \\frac{n}{2} \\right\\rfloor - 2}.\n\\]\n\\end{theorem}\n\n\n\n\\subsection{An application to $k$-town problem}\n\n\nLet $F$ be a family of vectors from $\\{0,\\dots,k-1\\}^n$, such that $(f_1,f_2) = 0 \\pmod k$ for any $f_1,f_2 \\in F$ (in particular for $f_1 = f_2$);\nsuch $F$ is further called a \\textit{$k$-town family}.\n\nFirst, for prime $k$ the classical argument gives the tight upper bound $k^{\\lf n \/2\\right\\rfloor}$.\n\nIf $k$ is square free we can obtain the same bound from the prime $k$ case in the following way. Let us see that the case $k=pq$ is a formal consequence of the cases $k=p$, $k=q$ if $p,q$ are coprime. Indeed let $F\\subset (\\mathbb{Z}\/pq\\mathbb{Z})^n$ such that $(f_1,f_2) = 0$ for each $f_1,f_2 \\in F$.  Then by assumption the number of residues modulo $p$ for elements $F$ does not exceed $p^{\\frac{n}{2}}$ and number of residues modulo $q$ for elements $F$ does not exceed $q^{\\frac{n}{2}}$. Then by the Chinese remainder theorem  $|F|\\leq p^{\\frac{n}{2}}q^{\\frac{n}{2}}$.\n\n\nThe observations above should be folklore, meanwhile we do not know how to prove the first inequality in the following theorem without spectral graph theory for an arbitrary $k$. \n\\begin{theorem}\nIf $F$ is a $k$-town family then \n\\[\n|F|\\leq k^{\\frac{n}{2}}.\n\\]\nMoreover suppose that $k$ is prime, and $(f_1,f_2) \\neq 0 \\pmod k$ for at most $\\varepsilon|F|^2$ pairs $f_1,f_2 \\in F$.\nIf $\\varepsilon < \\frac{k-1}{k}$ then\n\\[\n|F| \\leq \\frac{k^{\\frac{n}{2}}} {1 - \\frac{k}{k-1} \\varepsilon}.\n\\]\n\\label{ktown}\n\\end{theorem}\n\n\nObtaining an example of $k$-town $F$ with $|F| = k^{\\frac{n}{2}}$ in the case of prime $k$ and even $n$ is equivalent to finding a set of $\\frac{n}{2}$ pairwise (and-self) orthogonal linear independent vectors in $(\\mathbb{Z}\/k\\mathbb{Z})^n$. There are a lots of such sets; for example for a prime $k=4t+1$ one can consider vectors of the form $v_j=e_{2j-1}+\\varepsilon^t e_{2j}$ for $1\\leq j\\leq \\frac{n}{2}$ (here $\\{e_j\\}$ --- is the standard basis in $(\\mathbb{Z}\/k\\mathbb{Z})^n$, $\\varepsilon$ is a primitive root in $\\mathbb{Z}\/k\\mathbb{Z}$). \n\nFor a general prime $k$ one can choose $v_1,v_2,\\dots$ inductively and almost arbitrarily such that $v_j\\in \\langle v_1,\\dots,v_{j-1}\\rangle^{\\perp}\\setminus \\langle v_1,\\dots,v_{j-1}\\rangle$  and $(v_j,v_j)=0$. This can be done: indeed, for $j<\\frac{n}{2}-2$ we can choose arbitrary $4<\\dim \\langle v_1,\\dots,v_{j-1}\\rangle^{\\perp} - \\dim \\langle v_1,\\dots,v_{j-1}\\rangle$ linearly independent vectors and find $v_j$ in their span (as any quadratic form with $\\geq 3$ variables over finite field has an isotropic vector). When $j=[\\frac{n}{2}]-2$, we can choose $4$-dimensional \n$V\\subset (\\langle v_1,\\dots,v_{j-1}\\rangle^{\\perp}\\setminus \\langle v_1,\\dots,v_{j-1}\\rangle)\\cup{0}$ as well, \nand it is also well-known that there exists $v_{\\frac{n}{2}-1}, v_{\\frac{n}{2}}\\in V$ such that $(v_{\\frac{n}{2}-1},v_{\\frac{n}{2}-1})=(v_{\\frac{n}{2}-1},v_{\\frac{n}{2}})=\n(v_{\\frac{n}{2}},v_{\\frac{n}{2}}) = 0$.\n\nIn some cases were $k$ is non-prime  we can obtain examples of different nature. For example, when $k=m^{2}$ for some integer $m$ one can consider the set of vectors of the form $(mx_1,\\dots, mx_n)$. This example shows that in the case of $k$ being a perfect square the first inequality of Theorem~\\ref{ktown} is also tight for an odd $n$.   \n\nSuppose that we are interested in the scalar product $t$ instead of $0$. Then the statement of Theorem~\\ref{ktown} can be slightly improved.\n\n\\begin{corollary}\nLet $F$ be a family of vectors from $\\{0,\\dots,k-1\\}^n$, and $(f_1,f_2) = t \\pmod k$ for every $f_1,f_2 \\in F$. Then\n\\[\n|F| \\leq c(t,k) \\cdot k^{\\frac{n}{2}}\n\\]\nfor some $\\frac{1}{\\sqrt{2}} < c(t,k) \\leq 1$. Moreover if $\\frac{k}{\\gcd (k,t)}$ tends to infinity $c(t,k)$ tends to $\\frac{1}{\\sqrt{2}}$ ($\\gcd$ stands for the greatest common divisor).\n\nAssume also that $k$ is prime and $(f_1,f_2) \\neq t \\pmod k$ for at most $\\varepsilon|F|^2$ pairs $f_1,f_2 \\in F$ for some $t \\neq 0$.\nIf $\\varepsilon < \\frac{k-1}{k}$ then\n\\[\n|F| \\leq c(k) \\frac{k^{\\frac{n}{2}}} {1 - \\frac{k}{k-1} \\varepsilon},\n\\]\nwhere $c(k) < 1$ and $c(k) \\to \\frac{2\\sqrt{2}}{\\pi}$ with $k \\to \\infty$.\n\\label{slightlybetter}\n\\end{corollary}\n\n\n\\section{Proofs}\n\n\\begin{proof}[Proof of Corollary~\\ref{supersaturation}]\nDenote by $\\chi_I$ the characteristic vector of $I$. Then\n\\[\n(A\\chi_I, \\chi_I) \\geq |I|^2 \\cdot ( 1-(1-c)\\varepsilon ).\n\\]\nFrom the other hand, one has\n\\[\n(A\\chi_I, \\chi_I) = \\sum c_i^2\\lambda_i \\leq \\left(\\sum c_i^2 \\right) \\lambda_{max} = |I| \\cdot \\lambda_{max},\n\\]\nwhere $\\chi_I = \\sum c_i v_i$ is the decomposition of $\\chi_I$ via orthonormal eigenbasis $\\{v_i\\}$ of $A$. \n\\end{proof}\n\n\n\\begin{proof}[Proof of Corollary~\\ref{cross}]\nDenote by $\\chi_I$ and $\\chi_J$ the characteristic vectors of $I$ and $J$ respectively. Then\n\\[\n(A\\chi_I, \\chi_J) \\geq |I| \\cdot |J| \\cdot ( 1-(1-c)\\varepsilon ).\n\\]\nFrom the other hand, one has\n\\[\n(A\\chi_I, \\chi_I) = \\sum c_i d_i \\lambda_i \\leq \\left (\\sum |c_i| \\cdot |d_i| \\right ) \\spp \\leq \\sqrt{\\left(\\sum c_i^2 \\right)  \\left(\\sum d_i^2 \\right) } \\cdot \\spp = \n\\sqrt{|I| \\cdot |J|} \\cdot \\spp,\n\\]\nwhere $\\chi_I = \\sum c_i v_i$ and $\\chi_J = \\sum d_i v_i$ are the decompositions of $\\chi_I$ and $\\chi_J$ via orthonormal eigenbasis $\\{v_i\\}$ of $A$. \n\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{ktown}]\n\nConsider the following $k\\times k$ matrix:\n\\[\na_{jl} = \\phi^{jl},\n\\]\nwhere $\\phi$ is a primitive $k$-th root of unity and $0 \\leq j,l \\leq k-1$ and the matrix $M=A^{\\otimes n}$. Then we have $A^4=k^{2}E$, therefore $|\\lambda|=(k^{\\frac {1}{2}})^n=k^\\frac{n}{2}$ for each eigenvalue $\\lambda$ of $M$, i.e., $\\spp(M)=k^\\frac{n}{2}$. \n\nLet us see that $A$ (and, as a consequence, $M$) has an eigenbasis in $\\mathbb{R}^n$. Let $\\{e_i\\}$ be a standard basis. Then moving to the basis\n$\\{e_1, \\frac{1}{\\sqrt{2}}(e_j\\pm e_{k+2-j})\\}$ or to $\\{e_1, e_{\\frac{k}{2}}, \\frac{1}{\\sqrt{2}}(e_j\\pm e_{k+2-j})\\}$ for even $k$ (note that it is a unitary transformation) we obtain a block matrix with two blocks of the form $A_1, iA_2$, where $A_1,A_2$ are the real symmetric matrices (they have the sizes $\\frac{k+1}{2}$ \u0438 $\\frac{k-1}{2}$ respectively for odd $k$ and $\\frac{k}{2}+1$, $\\frac{k}{2}+1$ for even $k$). As a matter of fact, for each $2\\leq j\\leq k$ we have\n\\[\nA(e_j+e_{k+2-j})=\\sum_{l=1}^k \\phi^{(j-1)(l-1)} e_l +\n\\sum_{l=1}^k \\phi^{(k+1-j)(l-1)} e_l = 2+ \n\\sum_{l=2}^k (\\phi^{(j-1)(l-1)}  +\n  \\overline{\\phi^{(j-1)(l-1)}  })e_l = 2+\n  \\sum_{l=2}^{[\\frac{k}{2}]}2\\cdot \\Rre (\\phi^{(j-1)(l-1)})(e_l+e_{k+2-l})\n\\]\nand for each $2\\leq j\\leq k$, $j\\neq \\frac{k}{2}$\n\\[\nA(e_j-e_{k+2-j})=\\sum_{l=1}^k \\phi^{(j-1)(l-1)} e_l -\n\\sum_{l=1}^k \\phi^{(k+1-j)(l-1)} e_l =  \n\\sum_{l=2}^k (\\phi^{(j-1)(l-1)}  -\n  \\overline{\\phi^{(j-1)(l-1)}  })e_l =\n  i\\cdot \\sum_{l=2}^{[\\frac{k}{2}]}2\\cdot \\Iim(\\phi^{(j-1)(l-1)})(e_l-e_{k+2-l}).\n\\]\n\nThus we see that this change of the base (over $\\mathbb{R}$) leads to real symmetric matrix and to pure imaginary symmetric matrix, which both have real eigenbasis, therefore $A$ has real eigenbasis as well. Hence $N := \\Rre M$ has the same real eigenbasis with $M$. \nObviously, all eigenvalues of $N$ lie in $\\{\\pm k^{n\/2},0\\}$. \n\n\nLet us consider $G=((\\mathbb{Z}\/k\\mathbb{Z})^n,E)$, where  $(X,Y)=((x_1,\\dots,x_n), (y_1, \\dots, y_n))\\in E$ iff $\\sum_{ r=1}^n x_ry_r\\neq 0$. Then we also identify $(\\mathbb{Z}\/k\\mathbb{Z})^n$ as indices for rows and columns of matrix $M$ (taking all indices in $A$ modulo $k$). Let $M=(m_{st})_{s,t\\in \\{0,\\dots, k-1\\}^n}$. Now for each $X,Y\\in ((\\mathbb{Z}\/k\\mathbb{Z})^n, (X,Y)\\notin E$ we have\n\\[\n m_{X,Y}=\\prod_{r=1}^n a_{x_r, y_r}=\\phi^{\\sum_{r=1}^n x_ry_r}=\\phi^{0}=1.\n\\]\nThus graph $G$ and matrix $M$ (and also $N$) satisfy the conditions of Theorem~\\ref{lovasz}, so the first statement of the theorem is proved.\n\n\nNow let $k$ be a prime number. An immediate application of Corollary~\\ref{supersaturation} gives\n\\[\n|F| \\leq \\frac{k^{\\frac{n}{2}}} {1 - t \\varepsilon},\n\\]\nwhere $t = 1 - \\cos \\left ( \\lf \\frac{k}{2} \\right\\rfloor \\frac{2 \\pi}{k} \\right )$, which tends to 2 with $k \\to \\infty$.\nSo we modify the proof of Corollary~\\ref{supersaturation} in the following way.\n\nFor every root $\\phi$ the matrix $N = N(\\phi)$ satisfies\n\\[\n(N(\\phi) \\chi_F, \\chi_F) = \\sum_{i,j \\in F} N_{ij}(\\phi) \n\\]\nand\n\\[\n(N(\\phi) \\chi_F, \\chi_F) \\leq |F| \\cdot k^{\\frac{n}{2}}.\n\\]\nSumming up these inequalities for all $k$-th roots except $1$ one has\n\\begin{equation}\n (k-1) \\cdot |F|^2 \\cdot (1-\\varepsilon) - \\varepsilon \\cdot |F|^2  \\leq (k-1) \\cdot |F| \\cdot k^{\\frac{n}{2}}\n\\label{13231}    \n\\end{equation}\nsince for every $i,j$ that corresponds to sets with nonzero scalar product\n\\[\n\\sum_{\\phi \\neq 1} N_{ij}(\\phi) = -1,\n\\]\nand for $i,j$ that corresponds to sets with zero scalar product\n\\[\n\\sum_{\\phi \\neq 1} N_{ij}(\\phi) = k-1.\n\\]\nRewriting~\\eqref{13231} finishes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Corollary~\\ref{slightlybetter}]\nFix a primitive $k$-th root of unity $\\phi$ and consider the same matrices $A$ and $M$ as in the previous proof. \nLet us consider $G=((\\mathbb{Z}\/k\\mathbb{Z})^n,E)$, where  $(X,Y)=((x_1,\\dots,x_n), (y_1, \\dots, y_n))\\in E$ iff $\\sum_{ r=1}^n x_ry_r\\neq t$.\nThen $G$ and $N := \\Rre (\\phi^t M)$ satisfy the conditions of Theorem~\\ref{lovasz}.\nNote that $N$ and $M$ shares a real eigenbasis, so all eigenvalues of $N$ lie in $\\Rre \\{ \\pm \\phi^t k^{n\/2}, \\pm \\phi^t k^{n\/2}i \\}$. \nHence the spectral radius of $N$ lies between $\\frac{1}{\\sqrt{2}} k^{n\/2}$ and $k^{n\/2}$. \n\nLets, check the second part of first proposition. Note that for any coprime  $x,y$ we have $\\frac{xy}{\\gcd (xy,t)}=\\frac{x}{\\gcd (x,t)}\\cdot \\frac{y}{\\gcd (y,t)}$ and $c(t,xy)\\leq c(t,x)c(t,y)$ --- this follows from remarks before Theorem~\\ref{ktown}. Therefore we need only treat the case $k=p^s$ for prime $p$.\n\nNote that any upper bound for $|F|$ with some $t=t_0$ is also an upper bound for $t=t_0r^2, r\\in (\\mathbb{Z}\/k\\mathbb{Z})^*$. Indeed, if $(f_1,f_2)=t_0r^2$ for each $f_1,f_2\\in F$, then $(\\frac{1}{r}f_1, \\frac{1}{r}f_2)=t_0$ and we can apply a bound for $t=t_0$. Now we are to show that we can choose $r=r(k,t)$ such that $|\\Rre (\\phi^{tr^2})|$ tends to $\\frac{1}{\\sqrt{2}}$ if $\\frac{k}{\\gcd (k,t)}$ tends to infinity --- in this case $|\\Rre (\\phi^{tr^2}i^l)|$ tends to $\\frac{1}{\\sqrt{2}}$ as well. \nTo obtain this, it's sufficient to choose $r$ such that $|\\frac{tr^2}{k\/8}|$ tends to $1$ (with suitable choice of  representative of $tr^2$ modulo $k$).  \n\n\nAny $t\\in \\mathbb{Z}\/p^s \\mathbb{Z}$ we can consider as $t=p^mt', r\\in\\mathbb{Z}\/p^{s-m}\\mathbb{Z}$ and replacing $(t,p^s)$ with $(r, p^{s-m})$ (this doesn't change both ratios in question) we can assume that $t\\in (\\mathbb{Z}\/p^s \\mathbb{Z})^*$ (and $\\frac{k}{\\gcd (k,t)}=k$).\n\n\nLet $p>2$. Let's prove that we can choose $l$ such that $|l|<k^{\\frac{1}{2}}$ and $t=lw^2$ modulo $k$ for some $w$. Suppose that $t$ is non-square modulo $k$ (else we can take $l=1$). If $p=2$, then all non-squares are known to be of the form\n$3w^2,5w^2$ or $7w^2$ for some $w$, so we can take $l\\in\\{1,3,5,7\\}$ (when $s>5$). If $p>2$ we can take $l_1,l_2$ with $|l_i|<\\sqrt{p}$ such that $l_1=l_2t$ modulo $p$ by Thues lemma. Then for some $i$ $l_i$ is a non-square modulo $m$ and therefore modulo $k=p^s$. As $t$ is non square as well we can write $t=l_iw^2$ and take $l=l_i$.\n\n\nNow take $x \\in \\mathbb{Z}$ such that $x^2\\in \\left (\\frac{k}{8|l|} - \\left( \\frac{k}{|l|} \\right)^{\\frac{1}{2}}, \\frac{k}{8|l|}+ \\left (\\frac{k}{|l|} \\right)^{\\frac{1}{2}} \\right)$, then\n$||lx^2|-\\frac{k}{8}| \\leq (kl)^{\\frac{1}{2}}<k^{\\frac{3}{4}}$\nand $|\\frac{lx^2}{\\frac{k}{8}}|$ tends to $1$ when $k$ tends to infinity, as desired (we put $r=\\frac{x}{w}$ modulo $k$).\n\n\n\n\nNow we prove the second part of Corollary. Repeat the proof of the corresponding part of Theorem~\\ref{ktown}.\nThen the right part of~\\eqref{13231} is replaced with\n\\[\n\\sum_{a=1}^{k-1} \\spp  \\Rre (\\phi^{ta} M) .\n\\]\nNote that for every $a$ \n\\[\n\\spp  \\Rre (\\phi^{ta} M) = \\max ( \\Rre (\\phi^{ta}) , \\Rre(\\phi^{ta}\\cdot i) , -\\Rre(\\phi^{ta}) , -\\Rre(\\phi^{ta}\\cdot i) ) \\cdot k^{n\/2} =\n\\]\n\\[\\max \\left ( \\left | \\cos \\frac{2\\pi t a }{k} \\right |,\\left | \\cos  \\left ( \\frac{2\\pi t a }{k} + \\frac{\\pi}{2} \\right ) \\right |, \\left | \\cos  \\left ( \\frac{2\\pi t a }{k} + \\pi \\right )\\right |, \\left | \\cos  \\left ( \\frac{2\\pi t a }{k} + \\frac{3\\pi}{2} \\right ) \\right | \\right ) k^{n\/2}.\n\\]\nAfter summation over $1 \\leq a \\leq k-1$ and taking $k^{n\/2}$ out of the brackets and dividing by $k-1$, one has a Riemann sum for the following integral\n\\[\n\\frac{\\int_{0}^{\\pi\/4} \\cos(x) d x }{\\pi\/4} = \\frac{2\\sqrt{2}}{\\pi}.\n\\]\n\n\n\\end{proof}\n\n\n    \n\\section{Discussion}\n\n\\subsection{Related subjects}\n\nIt turns out that Theorem~\\ref{ktown} is closely related with several well-studied themes.\n\n\\paragraph{Erd{\\H o}s distance problem over a finite field.}\nLet $\\mathbb{F}_q$ be a finite field. Hart and Iosevich~\\cite{hart2008sums} proved that if $t \\neq 0$ and $E \\subset \\mathbb{F}^n_q$, then\n\\[\n|\\{(x, y) \\in E \\times E :  x \\cdot y = t\\}| = \\frac{|E|^2}{q} + R(E),\n\\]\nwhere\n\\[\n|R(E)| \\leq q^\\frac{n-1}{2} |E|.\n\\]\nIn the notation of Theorem~\\ref{ktown} (for this part we assume $k$ to be prime) it implies inequality\n\\[\n|F| \\leq \\frac{k^\\frac{n-1}{2}}{\\frac{k-1}{k} - \\varepsilon}.\n\\]\nThis bound is better than Theorem~\\ref{ktown} for a large prime $k$, but it essentially requires the condition $t \\neq 0$. The bound for $t = 0$ is obtained in~\\cite{hart2011averages} and is slightly weaker than Theorem~\\ref{ktown}.\n\n\nDefine the distance in an $n$-dimensional vector space over a finite field as\n\\[\n\\dist(\\bar x, \\bar y) = (x_1 - y_1)^2 + \\dots (x_n - y_n)^2,\n\\]\ni.e. we omit the square root in the standard formula.\nLet $\\Delta(E)$ be the set of all distances between points in $E$.\nThe Erd{\\H o}s distance problem is to determine the smallest possible cardinality of the set $\\Delta(E)$ over all sets $E$ of a given size.\n\n\nIn the case of prime $k > 2$ Theorem~\\ref{ktown} gives the following elementary bound which is far from the Iosevich--Rudnev estimate~\\cite{iosevich2007erdos}.\nLet $E$ be a set with $|\\Delta(E)| = s$. Without loss of generality $(0,\\dots,0) \\in E$; otherwise one may shift $E$. \nThen $E$ lies on $s$ spheres centered at $(0,\\dots,0)$. \nBy a pigeon-hole principle there is a sphere containing a subset $E'\\subset E$ of size at least $(|E|-1)\/s$. \nSince $k$ is odd and $E'$ lie on a sphere, there are at most $s$ different scalar products between vectors of $E'$.\nLet $t$ be a most popular scalar product in $E' \\times E'$. Then one may apply Theorem~\\ref{ktown} with $\\varepsilon = \\frac{s-1}{s}$ and receive that\n\\[\n|E'| \\leq \\frac{k^{n\/2}}{1 - \\frac{k}{k-1}\\frac{s-1}{s}} = \\frac{s(k-1)k^{n\/2}}{k-s}.\n\\]\nHence\n\\[\n|E| \\leq \\frac{s^2(k-1)k^{n\/2}}{k-s} + 1.\n\\]\n\nFor instance, $s = k-1$ means that we evaluate the size of a set without a single distance $r$ by $k^{{\\frac{n+6}{2}}}$, meanwhile Iosevich--Rudnev estimate is\n$Ck^{\\frac{n+1}{2}}$ with an absolute constant $C > 0$.  \n\n\n\\paragraph{A bound via singular numbers.}\nLet $G = (V,E)$ be a simple graph. Suppose that $A$ is a complex matrix such that $A_{ij} = 1$ for every pair $\\{i,j\\} \\notin E(G)$.\nThen\n\\[\n\\alpha(G) \\leq \\sigma_{max}(A),\n\\]\nwhere $\\alpha(G)$ is the size of a maximal independent set of $G$ and $\\sigma_{max}(A)$ is the maximal singular value of $A$ (i.e. the square root of a\nmaximal eigenvalue of a self-adjoint operator $A^*A$, where $A^*$ denotes the adjoint of $A$).\nThe proof immediately follows from the main theorem of~\\cite{danciger2006min}.\n\nFor matrix $A$ in the proof of Theorem~\\ref{ktown} one has\n\\[\nA^*A = kE,\n\\]\nso every singular number of matrix $A$ is equal to $\\sqrt{k}$.\nHence every singular number of $M = A^{\\otimes n}$ is equal to $k^{n\/2}$.\nThis implies the first inequality of Theorem~\\ref{ktown}.\n\n\\paragraph{Hypergraph discrepancy and asymptotic precision of Theorem~\\ref{theoremop} when $\\varepsilon$ is close to $1\/2$.}\nA hypergraph is a pair $(V, E)$, where $V$ is a finite set whose elements are called vertices and $E$ is a family of subsets of $V$, called edges. \nA vertex $2$-coloring of a hypergraph $(V, E)$ is a map $\\pi : V \\rightarrow \\{\\pm 1\\}$. \nA \\textit{discrepancy} of a coloring $\\pi$ is the largest value of $|\\sum_{v\\in e} \\pi (v) |$ over $e \\in E$.\n\n\nConsider an explicit hypergraph $H = (V,E)$, where $V = [N] \\times [N]$ and edges have form $I \\times J$ for $I,J \\subset [N]$. \nNote that every $\\{\\pm 1\\}$ matrix of size $N$ produces a 2-coloring of $H$.\nLet $N = 2^n$ and $\\chi$ be the coloring from $A^{\\otimes n}$ (recall that $A$ is $2\\times2$ Hadamard matrix).\nLet $I \\times J$ be an edge of $H$ providing the discrepancy $\\disc$ of $\\chi$; without loss of generality let $\\disc$ be positive. Then\n\\[\n\\disc = (A^{\\otimes n} \\chi_I, \\chi_J) =  (1-2\\varepsilon) \\cdot |I| \\cdot |J|,\n\\]\nwhere $\\varepsilon$ satisfies the conditions of Theorem~\\ref{theoremop}. Theorem~\\ref{theoremop} implies \n\\begin{equation}\n    \\disc^2 = |I|^2 \\cdot |J|^2 (1-2\\varepsilon)^2  \\leq 2^n |I|\\cdot |J| \\leq 2^{3n} = N^3.\n\\label{precise}\n\\end{equation}\n\nAstashkin proved~\\cite{astashkin2010rademacher} that $H$ has the discrepancy at least $cN^{3\/2}$ for \\textit{every} coloring $\\chi$ and some absolute constant $c > 0$, i.e. inequality~(\\ref{precise}) is precise up to an absolute constant. It means that Theorem~\\ref{theoremop} is precise up to an absolute constant in the case when $|I|$ and $|J|$ are close to $N$ and $\\frac{1}{2} - \\varepsilon$ is of order $2^{-n\/2}$.  \n\n\n\n\n\\subsection{Further questions}\n\n\\paragraph{Generalization on $k$-eventown.}\nWe say that $F \\subset 2^{[n]}$ is $k$-eventown if the size of the intersection of any (not necessarily different) $f_1, f_2 \\in F$ is zero modulo $k$.\nThe problem of determine the maximal size of $k$-eventown was studied by Frankl and Odlyzko~\\cite{frankl1983subsets}.\nThey found a nice construction based on Hadamard matrices of $k$-eventowns of size at least $(ck)^{\\lf n\/(4k) \\right\\rfloor}$, where $c > 0$ is an absolute constant. \nIn addition, they showed that any $k$-eventown has size at most $2^{O(\\log k\/k)n}$ as $n$ tends to $\\infty$. \n\nIn particular, for $k = 3$ the best known lower and upper bounds are $24^{\\lf n\/12 \\right\\rfloor}$ and $2^{\\lf n\/2 \\right\\rfloor}$ respectively.\n\n\n\n\\paragraph{Generalization on $t$-wise $k$-eventown.}\nWe say that $F \\subset 2^{[n]}$ is $t$-wise $k$-eventown if the size of the intersection of any different $f_1,\\dots, f_t \\in F$ is zero modulo $k$.\nNote that a 2-wise $k$-eventown is not the same as an $k$-eventown, since in the former we do not require that the sets themselves\nhave size zero modulo $k$. \n\nSudakov and Vieira~\\cite{sudakov2018two} show that a $t$-wise eventown has for $t \\geq 3$ a unique extremal configuration and obtain a stability result for this problem. Gishboliner, Sudakov and Tomon~\\cite{gishboliner2021small} show that for every $k$ there is $t = t(k)$ that\nthe size of any $t$-wise $k$-eventown is bounded by\n\\[\n|F| \\leq 2^{[n\/k]} + const(k,t).\n\\]\n\nA generalization of the methods of the paper, if exists, will have a deal with some tensor analysis.\n\n\n\n\n\n\n\\paragraph{Acknowledgments.} The authors are grateful to Fedor Petrov, Andrey Kupavskii and Pavel Prozorov for useful discussions. \nThe reviewers remarks were extremely helpful. The work of Danila Cherkashin was supported by the Russian Science Foundation grant 21-11-00040.\n\n\n\\bibliographystyle{plain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{intro}\nMassive stars contribute to the chemical composition of matter as we know it in the universe, and their deaths are accompanied by energetic core-collapse supernovae (SNe) that seed our universe with black holes (BHs) and neutron stars (NSs) -- the most exotic objects of the stellar graveyard. Large time-domain surveys of the sky \\citep[e.g.,][]{York2000,Drake2009,Law2009,Kaiser2010,Shappee2014,Abbott2016,Tonry2018,Bellm2019}, paired with targeted follow-up efforts, have greatly enriched our view on the final stages of massive star evolution. Yet, a lot remains to be understood about the diverse paths that bring massive stars toward\ntheir violent deaths \\citep{Langer2012}.  \n\nCore-collapse SNe can occur in stars with a hydrogen envelope\n(Type II) or in stars where hydrogen is almost or completely missing \\citep[Type Ib\/c, also referred to as stripped-envelope SNe;][]{Filippenko1997,Matheson2001,Li2011,Modjaz2014,Perley2020b,Frohmaier2021}. Type Ib\/c SNe constitute approximately 25\\% of all massive star explosions \\citep{Smith2011}, and their pre-SN progenitors are thought to be either massive ($M\\gtrsim 20-25$\\,M$_{\\odot}$) and metal-rich single stars that have been stripped through stellar mass loss; or the mass donors in close binary systems (at any metallicity) that have initial masses $\\gtrsim 8$\\,M$_{\\odot}$ \\citep[e.g.,][and references therein]{Langer2012}. \n\nA small fraction of Type Ib\/c SNe show velocities in their optical spectra that are systematically higher than those measured in ordinary SNe Ic at similar epochs. Hence, these explosions are referred to as SNe of Type Ic with broad lines  \\citep[hereafter, Ic-BL; e.g.,][]{Filippenko1997,Modjaz2016,GalYam2017}. Compared to Type Ib\/c SNe, broad-lined events are found to prefer environments with lower metallicity (in a single star scenario, mass loss mechanisms also remove angular momentum and are enhanced by higher metallicities), and in galaxies with higher star-formation rate density. Thus, it has been suggested that SN Ic-BL progenitors may be stars younger and more massive than those of normal Type Ic (more massive progenitors can lose their He-rich layers to winds at lower metallicity due to the higher luminosities driving the winds), and\/or tight massive binary systems that can form efficiently in dense stellar clusters \\citep[e.g.,][]{Kelly2014,Japelj2018,Modjaz2020}. \n\nThe spectroscopic and photometric properties used to classify core-collapse SNe are largely determined by the stars' outer envelopes \\citep[envelope mass, radius, and chemical composition;][]{Young2004}. On the other hand, quantities such as explosion energies, nickel masses, and ejecta geometries can be inferred via extensive multi-wavelength and multi-band observations. These quantities, in turn, can help constrain the properties of the stellar cores  \\citep[such as mass, density structure, spin, and magnetic fields; see e.g. ][and references therein]{Woosley2002,Burrows2007,Jerkstrand2015} that are key to determine the nature of the explosion. For example, based on nickel masses and ejecta masses derived from bolometric light curve analyses, \\citet{Taddia2019} found that $\\gtrsim 21\\%$ of Ic-BL  progenitors are compatible with massive ($\\gtrsim 28$\\,M$_{\\odot}$), possibly single stars, whereas $\\gtrsim 64\\%$ could be associated with less massive stars in close binary systems.  \n\nUnderstanding the progenitor scenario of SNe Ic-BL is particularly important as these SNe challenge greatly the standard explosion mechanism of massive stars \\citep[e.g.,][and references therein]{Mezzacappa1998,MacFadyen1999,Heger2003,WoosleyHeger2006,Janka2007,Janka2012,Smith2014,Foglizzo2015,Muller2020,Schneider2021}. The energies inferred from optical spectroscopic modeling of Ic-BL events are of order $\\approx 10^{52}\\,{\\rm erg}$, in excess of the $\\approx 10^{51}\\,{\\rm erg}$ inferred in typical SNe Ib\/c, while ejecta masses are comparable or somewhat higher \\citep{Taddia2019}. In the traditional core-collapse scenario, neutrino irradiation from the proto-NS revives the core-bounce shock, making the star explode. However, the neutrino mechanism cannot explain the more energetic SNe Ic-BL. Unveiling the nature of an engine powerful enough to account for the extreme energetics of SNe Ic-BL is key to understanding the physics behind massive stellar deaths, and remains as of today an open question. \n\nA compelling scenario invokes  the existence of a jet or a newly-born magnetar as the extra source of energy needed to explain SNe Ic-BL \\citep[e.g.,][]{Burrows2007,Papish2011,Gilkis2014,Mazzali2014,Lazzati2012,Gilkis2016,Soker2017,Barnes2018,Shankar2021}. The rapid rotation of a millisecond proto-NS formed in the collapse of a rotating massive star can amplify the NS magnetic field to $\\gtrsim 10^{15}$\\,G, creating a magnetar. The magnetar spins down quickly via magnetic braking, and in some cases magneto-rotational instabilities can launch a collimated jet that drills through the outer layers of the star producing a gamma-ray burst \\citep[GRB; e.g.,][]{Heger2003,Izzard2004,WoosleyHeger2006,Burrows2007,Bugli2020,Bugli2021}.  These jets can transfer sufficient energy to the surrounding stellar material to explode it into a SN.\n\nThe above scenario is particularly interesting in light of the fact that SNe Ic-BL are also the only type of core-collapse events that, observationally, have been unambiguously linked to GRBs \\citep[e.g.,][and references therein]{Woosley2006,Cano2017}. GRBs are characterized by bright afterglows that emit radiation from radio to X-rays, and are unique laboratories for studying relativistic particle acceleration and magnetic field amplification processes \\citep{Piran2004,Meszaros2006}. In between ordinary SNe Ic-BL and cosmological GRBs is a variety of transients that we still have to fully characterize. Among those are low-luminosity GRBs, of which the most famous example is GRB\\,980425, associated with the radio-bright Type  Ic-BL SN\\,1998bw \\citep{Galama1998,Kulkarni1998}.  \n\nRecently, \\citet{Shankar2021} used the jetted outflow model produced from a consistently formed proto-magnetar in a 3D core-collapse SN to extract a range of central engine parameters  (energy $E_{eng}$ and  opening  angle $\\theta_{eng}$)  that were then used as inputs to hydrodynamic models of  jet-driven explosions. The output of these models, in turn, were used to derive predicted SN light curves and spectra from different viewing angles, and found to be in agreement with SN Ic-BL optical observables \\citep[see also][]{Barnes2018}. It was also shown that additional energy from the engine can escape through the tunnel drilled in the star as an ultra-relativistic jet (GRB) with energy $\\approx 10^{51}$\\,erg. On the other hand, a SN Ic-BL can be triggered even if the jet engine fails to produce a successful GRB jet. The duration of the central engine, $t_{eng}$, together with $E_{eng}$ and $\\theta_{eng}$, are critical to determining the fate of the jet \\citep{Lazzati2012}. \n\nA more general scenario where the high velocity ejecta found in SNe Ic-BL originate from a cocoon driven by a relativistic jet (regardless of the nature of the central engine) is also receiving attention. In this scenario, cosmological long GRBs are explosions where the relativistic jet breaks out successfully from the stellar envelope, while low-luminosity GRBs and SNe Ic-BL that are not associated with GRBs represent cases where the jet is choked \\citep[see e.g. ][and references therein]{Piran2019,Eisenberg2022,Gottlieb2022,Pais2022}. \n\nOverall, the dividing line between successful GRB jets and failed ones is yet to be fully explored observationally, and observed jet outcomes in SNe Ic-BL have not yet been systematically compared to model outputs. While we know that SNe discovered by means of a GRB are all of Type Ic-BL, the question that remains open is whether all SNe Ic-BL make a GRB (jet), at least from some viewing angle, or if instead the jet-powered SNe Ic-BL are intrinsically different and rarer than ordinary SNe Ic-BL. Indeed, due to the collimation of GRB jets, it is challenging to understand whether all SNe Ic-BL are linked to  successful GRBs: a non-detection in $\\gamma$- or X-rays could simply be due to the explosion being directed away from us. \n Radio follow-up observations are needed to probe  the explosions' fastest-moving ejecta ($\\gtrsim 0.2c$) largely free of geometry and viewing angle constraints.  Determining observationally what is the fraction of Type Ic-BL explosions that output jets which successfully break out of the star (as mildly-relativistic or ultra-relativistic ejecta), and measuring their kinetic energy via radio calorimetry, can provide jet-engine  explosion models a direct test of their predictions. \n \n Using one of the largest sample of SNe Ic-BL with deep radio follow-up observations \\citep[which included 15 SNe Ic-BL discovered by the Palomar Transient Factory, PTF\/iPTF;][]{Law2009}, \\citet{Corsi2016}  already established that $<41\\%$ of SNe Ic-BL harbor relativistic ejecta similar to that of SN\\,1998bw. Here, we present the results of a systematic radio follow-up campaign of an additional 16 SNe Ic-BL (at $z\\lesssim 0.05$) detected independently of $\\gamma$-rays by the Zwicky Transient Facility \\citep[ZTF;][]{Bellm2019,Graham2019}. This study greatly expands our previous works on the subject \\citep[][]{Corsi2017,Corsi2016,Corsi2014}. Before the advent of PTF and ZTF, the comparison between jet-engine model outcomes and radio observables was severely limited by the rarity of SN Ic-BL discoveries \\citep[e.g.,][]{Berger2003,Soderberg2006} and\/or by selection effects \\citep[e.g.,][]{Woosley2006}---out of the thousands of jets identified, nearly  all were discovered at large distances via their high-energy emission (implying aligned jet geometry and ultra-relativistic speeds). In this work, we aim to provide a study free of these biases. \n \n Our paper is organized as follows. In Section \\ref{sec:discovery} we describe our multi-wavelength observations; in Section \\ref{section:sample} we describe in more details the SNe Ic-BL included in our sample; in Section \\ref{sec:modeling} we model the optical, X-ray, and radio properties of the SNe presented here and derive constraints on their progenitor and ejecta properties. Finally, in Section \\ref{sec:conclusion} we summarize and conclude. Hereafter we assume cosmological parameters  $H_0 = 69.6$\\,km\\,s$^{-1}$\\,Mpc$^{-1}$, $\\Omega_{\\rm M }= 0.286$, $\\Omega_{\\rm vac} = 0.714$ \\citep{Bennett2014}. All times are given in UT unless otherwise stated. \n\n\n\\begin{table*}\n\\begin{center}\n\\caption{The sample of 16 SNe Ic-BL  analyzed in this work. For each SN we provide the IAU name, the ZTF name, the position, redshift, and luminosity distance. \\label{tab:sample}}\n\\begin{tabular}{lccc}\n\\hline\n\\hline\nSN (ZTF name)  & RA, Dec (J2000) &  $z$& $d_L$  \\\\\n & (hh:mm:ss~~dd:mm:ss) & &(Mpc)\\\\\n\\hline\n2018etk (18abklarx) & 15:17:02.53 +03:56:38.7 & 0.044& 196 \\\\\n2018hom (18acbwxcc)& 22:59:22.96 +08:45:04.6 & 0.030 & 132   \\\\\n2018hxo (18acaimrb) &  21:09:05.80 +14:32:27.8 & 0.048 & 214  \\\\\n2018jex (18acpeekw) & 11:54:13.87 +20:44:02.4 & 0.094 & 434  \\\\\n2019hsx (19aawqcgy) & 18:12:56.22 +68:21:45.2 & 0.021 & 92  \\\\\n2019xcc (19adaiomg) & 11:01:12.39 +16:43:29.1 & 0.029 & 128  \\\\\n2020jqm (20aazkjfv) & 13:49:18.57 $-$03:46:10.4 & 0.037 & 164  \\\\\n2020lao (20abbplei) & 17:06:54.61 +30:16:17.3 & 0.031 & 137  \\\\\n2020rph (20abswdbg) & 03:15:17.83 +37:00:50.8 & 0.042 & 187\\\\\n2020tkx (20abzoeiw) &  18:40:09.01 +34:06:59.5 & 0.027 & 119 \\\\\n2021xv (21aadatfg) & 16:07:32.82 +36:46:46.2 & 0.041 & 182 \\\\\n2021aug (21aafnunh) & 01:14:04.81 +19:25:04.7 & 0.041 & 182 \\\\\n2021epp (21aaocrlm) & 08:10:55.27 $-$06:02:49.3 & 0.038 & 168\\\\\n2021htb (21aardvol) & 07:45:31.19 +46:40:01.3  & 0.035& 155 \\\\\n2021hyz (21aartgiv) &  \n09:27:36.51 +04:27:11.0 &  0.046& 205\\\\\n2021ywf (21acbnfos) & 05:14:10.99 +01:52:52.4 & 0.028 & 123 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{CorsiFig1_lcs_mag.png}\n    \\caption{P48 $r$- (top) and $g$-band (middle) light curves for the SNe Ic-BL in our sample, compared with the $R$- and $B$-band light curves of SN\\,1998bw, respectively. The bottom panel shows the corresponding color evolution. Observed AB magnitudes are corrected for Milky Way extinction. The archetypal SN\\,1998bw is shown in black solid points, and its Gaussian Process interpolation in black dashed lines. See also \\citet{Anand2022} and \\citet{Gokul2022}.}\n    \\label{fig:opt-lc-mag}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{CorsiFig2_lcs_flux.png}\n    \\caption{Top and middle panels: same as Figure \\ref{fig:opt-lc-mag} but in flux space and with fluxes  normalized to their Gaussian Process maximum. Bottom panel: bolometric light curves. We converted $g-r$ to bolometric magnitudes with the empirical relations by \\cite{Lyman2014,Lyman2016}. See also \\citet{Anand2022} and \\citet{Gokul2022}.\n    \\label{fig:opt-lc-flux}}\n\\end{figure*}\n\n\\section{Multi-wavelength observations}\n\\label{sec:discovery}\nWe have collected a sample of 16 SNe Ic-BL observed with the ZTF and with follow-up observations in the radio. The SNe Ic-BL included in our sample are listed in Table \\ref{tab:sample}. We selected these SNe largely based on the opportunistic availability of follow-up observing time on the Karl G. Jansky Very Large Array (VLA).  The sample of SNe presented here doubles the sample of SNe Ic-BL with deep VLA observations presented in \\citet{Corsi2016}. \n\nThe SNe considered in this work are generally closer than the PTF\/iPTF sample of SNe Ic-BL presented in \\citet{Taddia2019}. In fact, their median redshift ($\\approx 0.037$) is about twice as small as the median redshift of the PTF\/iPTF SN Ic-BL sample \\citep[$\\approx 0.076$;][]{Taddia2019}. However, the median redshift of the ZTF SNe in our sample is compatible with the median redshift ($\\approx 0.042$) of the full ZTF SN Ic-BL population presented in \\citet{Gokul2022}. A subset of the SNe Ic-BL presented here is also analyzed in a separate paper and in a different context \\citep[r-process nucleosynthesis;][]{Anand2022}. In this work, we report for the first time the results of our radio follow-up campaign of these events. We note that the ``Asteroid Terrestrial-impact Last Alert System''\\citep[ATLAS;][]{Tonry2018} has contributed to several of the SN detections considered here (see Section \\ref{section:sample}). Three of the Ic-BL in our sample were also reported in the recently released bright SN catalog by the All-Sky Automated Survey for Supernovae \\citep[ASAS-SN;][]{Neumann2022}.\n\n\nIn what follows, we describe the observations we carried for this work. In Section \\ref{section:sample} we give more details on each of the SNe Ic-BL in our sample. \n\n\\subsection{ZTF photometry}\nAll photometric observations presented in this work were conducted with the Palomar Schmidt 48-inch (P48) Samuel Oschin telescope as part of the ZTF survey \\citep{Bellm2019,Graham2019}, using the ZTF camera \\citep{dekany2020}. \nIn default observing mode, ZTF uses 30\\,s exposures, and survey observations are carried out in $r$ and $g$ band, down to a typical limiting magnitude of $\\approx 20.5$\\,mag. P48 light curves were derived using the ZTF pipeline \\citep{Masci2019}, and forced photometry \\cite[see][]{Yao2019}. Hereafter, all reported magnitudes are in the AB system. \nThe P48 light curves are shown in Figures~\\ref{fig:opt-lc-mag}-\\ref{fig:opt-lc-flux}. All the light curves presented in this work will be made public on the Weizmann Interactive Supernova Data Repository (WISeREP\\footnote{\\url{https:\/\/www.wiserep.org\/}}).\n\n\n\n\\begin{table}\n \\begin{center}\n\\caption{\\label{tab:xrt_summary}\\textit{Swift}\/XRT observations of 9 of the 16 SNe Ic-BL in our sample. We provide the MJD of the \\textit{Swift} observations, the XRT exposure time, and the 0.3-10\\,keV unabsorbed flux measurements (or $3\\sigma$ upper-limits for non detections).  \\label{tab:x-ray}}\n\\begin{tabular}{llccc}\n\\hline\n\\hline\nSN  & T$_{\\rm XRT}$  & Exp. & $F_{\\rm 0.3-10\\,keV}$ \\\\\n & (MJD)  & (ks) & ($10^{-14}$\\,erg\\,cm$^{-2}$\\,s$^{-1}$)\\\\\n\\hline\n2018etk & 58377.85 & 4.8 & $< 4.2$ \\\\\n2018hom & 58426.02 & 4.3 & $< 6.4$ \\\\\n2019hsx & 58684.15 & 15 & $6.2_{-1.8}^{+2.3}$ \\\\\n2020jqm & 59002.09 & 7.4 & $< 3.3 $ \\\\\n2020lao & 59007.40 & 14 & $< 2.9 $ \\\\\n2020rph & 59088.89 & 7.5 & $< 3.6 $ \\\\\n2020tkx & 59125.38 & 8.1 & $< 3.3 $ \\\\\n2021hyz & 59373.09 & 4.7 & $< 3.5 $ \\\\\n2021ywf & 59487.60 & 7.2 & $5.3_{-3.3}^{+4.9} $ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Optical Spectroscopy}\nPreliminary spectral type classifications of several of the SNe in our sample were obtained with the Spectral Energy Distribution Machine (SEDM) mounted on the Palomar 60-inch telescope (P60), and quickly reported to the Transient Name Server (TNS). The SEDM is a very low resolution ($R\\sim 100$) integral field unit spectrograph optimized for transient classification with high observing efficiency \\citep{Blagorodnova2018,Rigault2019}. \n\nAfter initial classification, typically further spectroscopic observations are carried out as part of the ZTF transient follow-up programs to confirm and\/or improve classification, and to characterize the time-evolving spectral properties of interesting events. Amongst the series of spectra obtained for each of the SNe presented in this work, we select one good quality photospheric phase spectrum  (Figures~\\ref{fig:spectra1}-\\ref{fig:spectra2}; grey) on which we run SNID \\citep{Blondin2007} to obtain the best match to a SN Ic-BL template (black), after clipping the host emission lines and fixing the redshift to that derived either from the SDSS host galaxy or from spectral line fitting (H$\\alpha$; see Section \\ref{section:sample} for further details).\nHence, in addition to the SEDM, in this work we also made use of the following instruments: the Double Spectrograph \\citep[DBSP;][]{Oke1995}, a low-to-medium resolution grating instrument for the Palomar 200-inch telescope Cassegrain focus that uses a dichroic to split light into separate red and blue channels observed simultaneously; the Low Resolution Imaging Spectrometer \\citep[LRIS;][]{Oke1982,Oke1995}, a visible-wavelength imaging and spectroscopy instrument operating at the Cassegrain focus of Keck-I; the Alhambra Faint Object Spectrograph and Camera (ALFOSC), a CCD camera and spectrograph installed at the Nordic Optical Telescope \\citep[NOT;][]{Djupvik2010}. All spectra presented in this work will be made public on WISeREP.\n\n\\subsection{X-ray follow up with \\textit{Swift}}\nFor 9 of the 16 SNe presented in this work we carried out follow-up observations in the X-rays using the X-Ray Telescope  \\citep[XRT;][]{Burrows+2005} on the \\textit{Neil Gehrels Swift Observatory} \\citep{Gehrels+2004}. \n\nWe analyzed these observations using the online XRT tools\\footnote{See \\url{https:\/\/www.swift.ac.uk\/user_objects\/}.}, as described in \\citet{Evans+2009}. \nWe correct for Galactic absorption, and adopt a power-law spectrum with photon index $\\Gamma = 2$  for count rate to flux conversion for non-detections, and for detections (two out of ninw events) where the number of photons collected is too small to enable a meaningful spectral fit (one out of two detections).  Table~\\ref{tab:xrt_summary} presents the results from co-adding all observations of each source.\n\n\n\\subsection{Radio follow up with the VLA}\n\\label{sec:radioobs}\nWe observed the fields of the SNe Ic-BL in our sample with the VLA via several of our programs using various array configurations and receiver settings (Table \\ref{tab:data}). \n\nThe VLA raw data were calibrated in \\texttt{CASA} \\citep{McMullin2007} using the automated VLA calibration pipeline. After manual inspection for further flagging, the calibrated data were imaged using the \\texttt{tclean} task. Peak fluxes were measured from the cleaned images using \\texttt{imstat} and circular regions centered on the optical positions of the SNe, with radius equal to the nominal width (FWHM) of the VLA synthesized beam (see Table \\ref{tab:data}). The RMS noise values were estimated with \\texttt{imstat} from the residual images. Errors on the measured peak flux densities in Table \\ref{tab:data} are calculated adding  a 5\\% error in quadrature to measured RMS values. This accounts for systematic uncertainties on the absolute flux calibration. \n\nWe checked all of our detections (peak flux density above $3\\sigma$) for the source morphology (extended versus point-like), and ratio between integrated and peak fluxes using the CASA task \\texttt{imfit}. All sources for which these checks provided evidence for extended or marginally resolved emission are marked accordingly in Table \\ref{tab:data}. For non detections, upper-limits on the radio flux densities are given at the $3\\sigma$ level unless otherwise noted.\n\n\\renewcommand\\arraystretch{1.5}\n\\setlength\\LTcapwidth{2\\linewidth}\n\\begin{longtable*}{ccclccc}\n\\caption{VLA follow-up observations of the 16 SN\\lowercase{e} I\\lowercase{c}-BL in our sample. For all of the observations of the SNe in our sample we report: the mid MJD of the VLA observation; the central observing frequency; the measured flux density (all flux density upper-limits are calculated at $3\\sigma$ unless otherwise noted); the VLA array configuration; the FWHM of the VLA nominal synthesized beam; the VLA project code under which the observations were conducted. See Sections \\ref{sec:radioobs} and \\ref{sec:radioanalysis} for discussion.} \\label{tab:data}\\\\\n\\toprule\n\\toprule\nSN & ${\\rm T_{VLA}}$$^{a}$    & $\\nu$  & $F_{\\nu}$ & Conf. & Nom.Syn.Beam  & Project \\\\\n & (MJD)  & (GHz) & ($\\mu$Jy) &  & (FWHM; arcsec) &  \\\\\n\\midrule\n2018etk & 58363.08  & 6.3&  $90.1\\pm8.7$$^{b}$   & D & 12&  VLA\/18A-176$^{d}$\\\\\n & 58374.09   &14 & $41\\pm11$ & D  & 4.6&  VLA\/18A-176$^{d}$ \\\\\n  & 58375.03   & 6.3& $89.7\\pm8.8$$^{b}$  & D  &$12$&  VLA\/18A-176$^{d}$ \\\\\n &  59362.27  &6.2 & $78.5\\pm6.3$$^{b}$  & D  &12 &  VLA\/20B-149$^{d}$\\\\\n\\midrule\n2018hom &  58428.04  & 6.6 &  $133\\pm11$  & D & 12&  VLA\/18A-176$^{d}$\\\\\n\\midrule\n2018hxo  & 58484.73  & 6.4 & $\\lesssim 234$$^{c}$  & C & 3.5 &  VLA\/18A-176$^{d}$\\\\\n\\midrule\n2018jex &  58479.38 &6.4 & $\\lesssim 28$ & C & 3.5&  VLA\/18A-176$^{d}$\\\\\n\\midrule\n2019hsx   & 58671.14 & 6.2 & $\\lesssim 19$ & BnA & 1.0&  VLA\/19B-230$^{d}$\\\\\n\\midrule\n2019xcc &  58841.43 &6.3 &$62.7\\pm 8.7$$^{b}$& D  & 12 &  VLA\/19B-230$^{d}$\\\\\n   & 58876.28  & 6.3  &$60.1\\pm8.5$$^{b}$ & D  & 12&  VLA\/19B-230$^{d}$  \\\\\n  &59363.00 & 6.3 & $50.4\\pm8.1$$^{b}$& D & 12&  VLA\/20B-149$^{d}$ \\\\\n\\midrule\n2020jqm &  58997.03  & 5.6& $175\\pm13$& C  &  3.5&SG0117$^{d}$\\\\\n  &59004.03 & 5.6 & $310\\pm19$&C  & 3.5 &SG0117$^{d}$ \\\\\n &59028.48  & 5.5& $223\\pm18$&B & 1.0& VLA\/20A-568$^{d}$\\\\\n &59042.95  &5.7 & $202\\pm15$&B &1.0& VLA\/20A-568$^{d}$ \\\\\n  &59066.09  &5.5  &$136\\pm13$ &B  & 1.0& VLA\/20A-568$^{d}$ \\\\\n &59088.03  & 5.5  &$168\\pm13$ &B  & 1.0& VLA\/20A-568$^{d}$ \\\\\n &59114.74 & 5.5 & $620\\pm33$ &B  & 1.0& VLA\/20A-568$^{d}$ \\\\\n  &59240.37 & 5.5 & $720\\pm37$ &A &0.33& VLA\/20B-149$^{d}$\\\\\n\\midrule\n2020lao &  59006.21 & 5.2& $\\lesssim 33$ &C & 3.5& SG0117$^{d}$\\\\\n  & 59138.83 &5.5& $\\lesssim 21$&  B & 1.0& SG0117$^{d}$  \\\\\n\\midrule\n2020rph & 59089.59 &5.5 & $42.7\\pm7.4$&   B & 1.0& SG0117$^{d}$\\\\\n  & 59201.28 &  5.5 & $43.9\\pm7.0$&  A &0.33 & SG0117$^{d}$ \\\\\n\\midrule\n2020tkx &   59117.89 & 10 & $272\\pm 16$& B &0.6 & VLA\/20A-374$^{e}$\\\\\n & 59136.11  & 10  & $564\\pm29$& B  &0.6 & VLA\/20A-374$^{e}$ \\\\\n & 59206.92 & 5.5&$86.6\\pm7.3$& A & 0.33 & VLA\/20B-149$^{d}$\\\\\n\\midrule\n2021xv &  59242.42  & 5.5 & $\\lesssim 23$& A & 0.33& VLA\/20B-149$^{d}$\\\\\n & 59303.24 & 5.2& $\\lesssim 29$& D &12 & VLA\/20B-149$^{d}$ \\\\\n &59353.11 & 5.2 & $34.3\\pm8.1$$^{b}$& D & 12 & VLA\/20B-149$^{d}$ \\\\\n\\midrule\n2021aug &  59254.75  & 5.2& $\\lesssim 22$& A & 0.33& VLA\/20B-149$^{d}$\\\\\n  & 59303.62 &5.4 & $\\lesssim 45$& D & 12& VLA\/20B-149$^{d}$\\\\\n & 59353.48 & 5.4 & $\\lesssim 30$& D &12 & VLA\/20B-149$^{d}$\\\\\n\\midrule\n2021epp &  59297.06 &5.3 & $(2.62\\pm0.13)\\times10^3$$^{b}$ & D&12 & VLA\/20B-149$^{d}$\\\\\n &  59302.99 & 5.1 &$(2.82\\pm0.18)\\times10^3$$^{b}$ & D & 12& VLA\/20B-149$^{d}$ \\\\\n & 59352.83& 5.3  &$(2.75\\pm0.20)\\times10^3$$^{b}$ & D &12 & VLA\/20B-149$^{d}$ \\\\\n\\midrule\n2021htb  &59324.94 & 5.2& $50\\pm10$$^{b}$& D&12 & VLA\/20B-149$^{d}$\\\\\n  & 59352.87 & 5.2 &$59.4\\pm9.5$$^{b}$ &D  & 12& VLA\/20B-149$^{d}$ \\\\\n\\midrule\n2021hyz & 59326.08  & 5.2& $38\\pm11$&D &12 & VLA\/20B-149$^{d}$\\\\\n &  59352.99& 5.5 & $\\lesssim 30$& D & 12& VLA\/20B-149$^{d}$ \\\\\n\\midrule\n2021ywf &  59487.57  & 5.0  & $83\\pm10$ & B &1.0 &SH0105$^{d}$\\\\\n &  59646.12 & 5.4 &$19.8\\pm 6.3$ & A & 0.33& SH0105$^{d}$\\\\\n\\bottomrule\n\\multicolumn{7}{l}{$^{a}$ Mid MJD time of VLA observation (total time including calibration).}\\\\\n\\multicolumn{7}{l}{$^{b}$ Resolved or marginally resolved with emission likely dominated by the host galaxy.}\\\\\n\\multicolumn{7}{l}{$^{c}$ Image is dynamic range limited due to the presence of a bright source in the field.}\\\\\n\\multicolumn{7}{l}{$^{d}$ PI: Corsi.}\\\\\n\\multicolumn{7}{l}{$^{e}$ PI: Ho.}\n\\end{longtable*}\n\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=15cm]{CorsiFig3_IcBL_snid_spectra_part1.pdf}\n    \\caption{Photospheric phase spectra (grey) plotted along with their SNID best match templates (black) for the first half of the SNe Ic-BL in our sample. Spectra are labeled with their IAU name and spectroscopic phase (since $r$-band maximum; see Table \\ref{tab:opt_data}). We note that the spectra used to estimate the photospheric velocities of SN\\,2019xcc, SN\\,2020lao, and SN\\,2020jqm presented in Table \\ref{tab:opt_data} are different from the ones shown here for classification purposes. This is because for spectral classification we prefer later-time but higher-resolution spectra, while for velocity measurements we prefer earlier-time spectra even if taken with the lower-resolution SEDM. All spectra presented in this work will be made public on WISeREP. }\n    \\label{fig:spectra1}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=15cm]{CorsiFig4_IcBL_snid_spectra_part2.pdf}\n    \\caption{Same as Figure \\ref{fig:spectra1} but for the second half of the SNe Ic-BL in our sample. All spectra presented in this work will be made public on WISeREP. }\n    \\label{fig:spectra2}\n\\end{figure*}\n\n\n\\section{Sample description}\n\\label{section:sample} \n\\subsection{SN 2018etk}\nOur first ZTF photometry of SN\\,2018etk (ZTF18abklarx) was\nobtained on 2018 August 1 (MJD 58331.16) with the P48. This first ZTF detection was in the $r$ band, with a host-subtracted magnitude of $19.21\\pm0.12$\\,mag (Figure~\\ref{fig:opt-lc-mag}), at $\\alpha$=15$^{\\rm h}$17$^{\\rm m}$02.53$^{\\rm s}$,\n$\\delta=+03^{\\circ}56'38\\farcs7$ (J2000). The object was reported to the TNS by ATLAS on 2018 August 8, who discovered it on 2018 August 6  \\citep{Tonry2018_ZTF18abklarx}.\nThe last ZTF non-detection prior to ZTF discovery was on 2018 July 16, and the last shallow ATLAS non-detection was on 2018 August 2, at $18.75\\,$mag.\nThe transient was classified as a Type Ic SN by \\citet{Fremling2018_ZTF18abklarx} based on a spectrum obtained on 2018 August 13 with the SEDM.\nWe re-classify this transient as a SN Type  Ic-BL most similar to SN\\,2006aj based on a P200 DBSP spectrum obtained on 2018 August 21 (see Figure~\\ref{fig:spectra1}).\nSN\\,2018etk exploded in a star-forming galaxy with a known\nredshift of $z= 0.044$ derived from SDSS data.\n\n\\subsection{SN 2018hom}\nOur first ZTF photometry of SN\\,2018hom (ZTF18acbwxcc) was\nobtained on 2018 November 1  (MJD 58423.54) with\nthe P48. This first ZTF detection was in the $r$ band, with a host-subtracted magnitude of $16.60\\pm0.04$\\,mag (Figure \\ref{fig:opt-lc-mag}), at $\\alpha$=22$^{\\rm h}$59$^{\\rm m}$22.96$^{\\rm s}$,\n$\\delta=+08^{\\circ}45'04\\farcs6$ (J2000). The object was reported to the TNS by ATLAS on 2018 October 26, and discovered by ATLAS on 2018 October 24 at $o\\approx 17.3\\,$mag \\citep{Tonry2018_ZTF18acbwxcc}.\nThe last ZTF non-detection prior to ZTF discovery was on 2018 October 9 at $g>20.35\\,$mag, and the last ATLAS non-detection was on 2018 October 22 at $o> 18.25\\,$mag.\nThe transient was classified as a SN Type Ic-BL by \\citet{Fremling2018_ZTF18acbwxcc} based on a spectrum obtained on 2018 November 2 with the SEDM.\nSN\\,2018etk exploded in a galaxy with unknown redshift. We measure a redshift of $z = 0.030$ from star-forming emission lines in a Keck-I LRIS spectrum obtained on 2018 November 30. We plot this spectrum in Figure~\\ref{fig:spectra1}, along with its SNID template match to the Type Ic-BL SN\\,1997ef. We note that this SN was also reported in the recently released ASAS-SN bright SN catalog \\citep[][]{Neumann2022}.\n\n\\subsection{SN 2018hxo}\nOur first ZTF photometry of SN\\,2018hxo (ZTF18acaimrb) was\nobtained on 2018 October 9  (MJD 58400.14) with the P48. This first detection was in the $g$ band, with a host-subtracted magnitude of $18.89\\pm0.09$\\,mag (Figure~\\ref{fig:opt-lc-mag}), at $\\alpha$=21$^{\\rm h}$09$^{\\rm m}$05.80$^{\\rm s}$,\n$\\delta=+14^{\\circ}32'27\\farcs8$ (J2000). The object was first reported to the TNS by ATLAS on 2018 November 6, and first detected by ATLAS on 2018 September 25 at $o=18.36\\,$mag \\citep{Tonry2018_ZTF18acaimrb}.\nThe last ZTF non-detection prior to discovery was on 2018 September 27 at $r>20.12\\,$mag, and the last ATLAS non-detection was on 2018 September 24 at $o> 18.52\\,$mag.\nThe transient was classified as a SN Type Ic-BL by \\citet{Dahiwale2020_ZTF18acaimrb} based on a spectrum obtained on 2018 December 1 with Keck-I LRIS. In Figure \\ref{fig:spectra1} we plot this spectrum along with its SNID match to the Type Ic-BL SN\\,2002ap.\nSN\\,2018etk exploded in a galaxy with unknown redshift. We measure a redshift of $z=0.048$ from star-forming emission lines in the Keck spectrum. \n\n\\subsection{SN 2018jex}\nOur first ZTF photometry of SN\\,2018jex (ZTF18acpeekw) was\nobtained on 2018 November 16 (MJD 58438.56) with\nthe P48. This first detection was in the $r$ band, with a host-subtracted magnitude of $20.07\\pm0.29$\\,mag, at $\\alpha$=11$^{\\rm h}$54$^{\\rm m}$13.87$^{\\rm s}$,\n$\\delta=+20^{\\circ}44'02\\farcs4$ (J2000). The object was reported to the TNS by AMPEL on November 28 \\citep{Nordin2018_ZTF18acpeekw}.\nThe last ZTF last non-detection prior to ZTF discovery was on 2018 November 16 at $r>19.85\\,$mag.\nThe transient was classified as a SN Type Ic-BL based on a spectrum obtained on 2018 December 4 with Keck-I LRIS. In Figure~\\ref{fig:spectra1} we show this spectrum plotted against the SNID template of the Type Ic-BL SN\\,1997ef.\nAT2018jex exploded in a galaxy with unknown redshift. We measure a redshift of $z=0.094$ from star-forming emission lines in the Keck spectrum.\n\n\\subsection{SN 2019hsx}\nWe refer the reader to \\citet{Anand2022} for details about this SN Ic-BL.  Its P48 light curves and the spectrum used for classification are shown in Figures \\ref{fig:opt-lc-mag} and \\ref{fig:spectra1}. We note that this SN was also reported in the recently released ASAS-SN bright SN catalog  \\citep{Neumann2022}.\n\n\\subsection{SN 2019xcc}\nWe refer the reader to \\citet{Anand2022} for details about this SN Ic-BL.  Its P48 light curves and the spectrum used for classification are shown in Figures \\ref{fig:opt-lc-mag} and \\ref{fig:spectra1}.\n\n\\subsection{SN 2020jqm}\nOur first ZTF photometry of SN\\,2020jqm (ZTF20aazkjfv) was\nobtained on 2020 May 11 (MJD 58980.27) with\nthe P48. This first detection was in the $r$ band, with a host-subtracted magnitude of $19.42\\pm0.13$\\,mag, at $\\alpha$=13$^{\\rm h}$49$^{\\rm m}$18.57$^{\\rm s}$,\n$\\delta=-03^{\\circ}46'10\\farcs4$ (J2000). The object was reported to the TNS by ALeRCE on May 11 \\citep{Forster2020_ZTF20aazkjfv}.\nThe last ZTF  non-detection prior to ZTF discovery was on 2020 May 08 at $g>17.63\\,$mag.\nThe transient was classified as a SN Type  Ic-BL based on a spectrum obtained on 2020 May 26 with the SEDM \\citep{Dahiwale2020_ZTF20aazkjfv}.\nSN\\,2020jqm exploded in a galaxy with unknown redshift. We measure a redshift of $z = 0.037$ from host-galaxy emission lines in a NOT ALFOSC spectrum obtained on 2020 June 6. We plot the ALFOSC spectrum along with its SNID match to the Type Ic-BL SN\\,1998bw in Figure~\\ref{fig:spectra1}.\n\n\\subsection{SN 2020lao}\nWe refer the reader to \\citet{Anand2022} for details about this SN Ic-BL.  Its P48 light curves and the spectrum used for classification are shown in Figures \\ref{fig:opt-lc-mag} and \\ref{fig:spectra1}. We note that this SN was also reported in the recently released ASAS-SN bright SN catalog  \\citep{Neumann2022}.\n\n\\subsection{SN 2020rph}\nWe refer the reader to \\citet{Anand2022} for details about this SN Ic-BL.  Its P48 light curves and the spectrum used for classification are shown in Figures \\ref{fig:opt-lc-mag} and \\ref{fig:spectra2}.\n\n\\subsection{SN 2020tkx}\nWe refer the reader to \\citet{Anand2022} for details about this SN Ic-BL.  Its P48 light curves and the spectrum used for classification are shown in Figures \\ref{fig:opt-lc-mag} and \\ref{fig:spectra2}.\n\n\\subsection{SN 2021xv}\nWe refer the reader to \\citet{Anand2022} for details about this SN Ic-BL.  Its P48 light curves and the spectrum used for classification are shown in Figures \\ref{fig:opt-lc-mag} and \\ref{fig:spectra2}.\n\n\\subsection{SN 2021aug}\nOur first ZTF photometry of SN\\,2021aug (ZTF21aafnunh) was\nobtained on 2021 January 18  (MJD 59232.11) with the P48. This first detection was in the $g$ band, with a host-subtracted magnitude of $18.73\\pm0.08$\\,mag, at $\\alpha$=01$^{\\rm h}$14$^{\\rm m}$04.81$^{\\rm s}$,\n$\\delta=+19^{\\circ}25'04\\farcs7$ (J2000).\nThe last ZTF non-detection prior to ZTF discovery was on 2021 January 16 at $g>20.12\\,$mag. The transient was publicly reported to the TNS by ALeRCE on 2021 January 18 \\citep{MunozArancibia2021}, and classified as a SN Type Ic-BL\nbased on a spectrum obtained on 2021 February 09 with the SEDM \\citep{Dahiwale2021_ZTF21aafnunh}.\nSN\\,2020jqm exploded in a galaxy with unknown redshift. We measure a redshift of $z= 0.041$  from star-forming emission lines in a P200 DBSP spectrum obtained on 2021 February 08. This spectrum is shown in Figure~\\ref{fig:spectra2} along with its template match to the Type Ic-BL SN\\,1997ef.\n\n\\subsection{SN 2021epp}\nOur first ZTF photometry of SN\\,2021epp (ZTF21aaocrlm) was\nobtained on 2021 March 5 (MJD 59278.19) with the P48. This first ZTF detection was in the $r$ band, with a host-subtracted magnitude of $19.61\\pm0.15$\\,mag (Figure~\\ref{fig:opt-lc-mag}), at $\\alpha$=08$^{\\rm h}$10$^{\\rm m}$55.27$^{\\rm s}$, $\\delta=-06^{\\circ}02'49\\farcs3$ (J2000). \nThe transient was publicly reported to the TNS by ALeRCE on 2021 March 5 \\citep{MunozArancibia20210305}, and classified as a SN Type Ic-BL based on a spectrum obtained on 2021 March 13 by ePESSTO+ with the ESO Faint Object Spectrograph and Camera \\citep{Kankare2021}.\nThe last ZTF non-detection prior to discovery was on 2021 March 2 at $r>19.72\\,$mag.\nIn Figure~\\ref{fig:spectra2} we show the classification spectrum \nplotted against the SNID template of the Type Ic-BL SN\\,2002ap.\nSN\\,2021epp exploded in a galaxy with known redshift of $z = 0.038$.\n\n\\subsection{SN 2021htb}\nOur first ZTF photometry of SN\\,2021htb (ZTF21aardvol) was\nobtained on 2021 March 31 (MJD 59304.164) with the P48. This first ZTF detection was in the $r$ band, with a host-subtracted magnitude of $20.13\\pm0.21$\\,mag (Figure~\\ref{fig:opt-lc-mag}), at $\\alpha$=07$^{\\rm h}$45$^{\\rm m}$31.19$^{\\rm s}$,\n$\\delta=46^{\\circ}40'01\\farcs4$ (J2000).\nThe transient was publicly reported to the TNS by SGLF on 2021 April 2 \\citep{Poidevin_2021}.\nThe last ZTF non-detection prior to ZTF discovery was on 2021 March 2, at $r>19.88\\,$mag.\nIn Figure~\\ref{fig:spectra2} we show a P200 DBSP spectrum taken on 2021 April 09   plotted against the SNID template of the Type Ic-BL SN\\,2002ap. SN\\, 2021htb exploded in a SDSS galaxy with redshift $z= 0.035$.\n\n\\subsection{SN 2021hyz}\nOur first ZTF photometry of SN\\,2021hyz (ZTF21aartgiv) was\nobtained on 2021 April 03  (MJD 59307.155) with the P48. This first ZTF detection was in the $g$ band, with a host-subtracted magnitude of $20.29\\pm0.17$\\,mag (Figure~\\ref{fig:opt-lc-mag}), at $\\alpha$=09$^{\\rm h}$27$^{\\rm m}$36.51$^{\\rm s}$,\n$\\delta=04^{\\circ}27'11\\farcs$ (J2000).\nThe transient was publicly reported to the TNS by ALeRCE on 2021 April 3 \\citep{Forster_2021}.\nThe last ZTF non-detection prior to ZTF discovery was on 2021 April 1, at $g>19.15\\,$mag. In Figure~\\ref{fig:spectra2} we show a P60 SEDM spectrum taken on 2021 April 30  plotted against the SNID template of the Type Ic-BL SN\\,1997ef.\nSN\\,2021hyz exploded in a galaxy with redshift $z= 0.046$.\n\n\\subsection{SN 2021ywf}\nWe refer the reader to \\citet{Anand2022} for details about this SN Ic-BL.  Its P48 light curves and the spectrum used for classification are shown in Figures \\ref{fig:opt-lc-mag} and \\ref{fig:spectra2}.\n\n\n\\begin{footnotesize}\n\\begin{longtable*}{lcccllllll}\n\\caption{Optical properties of the 16 SNe Ic-BL in our sample. We list the SN name; the MJD of maximum light in $r$ band; the absolute magnitude at $r$-band peak; the absolute magnitude at $g$-band peak; the explosion time estimated as days since $r$-band maximum; the estimated nickel mass; the characteristic timescale of the bolometric light curve; the photospheric velocity; the ejecta mass; and the kinetic energy of the explosion.  See Sections \\ref{sec:vphot} and \\ref{sec:optical_properties} for discussion. \\label{tab:opt_data}}\\\\\n\\toprule\n\\toprule\nSN & T$_{r,\\rm max}$ & M$^{\\rm peak}_{r}$ & M$^{\\rm peak}_{g}$  & ${\\rm T}_{\\rm exp}$-T$_{r,\\rm max}$  & $M_{\\rm Ni}$   & $\\tau_{m}$  & v$_{\\rm ph}$($^{a}$) & $M_{\\rm ej}$ & $E_{\\rm k}$ \\\\\n &  (MJD) & (AB mag)  & (AB mag) & (d) & (M$_{\\odot}$) & (d) & ($10^4$\\,km\/s) & (M$_{\\odot}$) & ($10^{51}$erg) \\\\\n\\midrule[0.5pt]\n2018etk & 58337.40 & $-18.31\\pm0.03$ & $-18.30\\pm0.02$ & $-9\\pm1$ & $0.13_{-0.02}^{+0.01}$ & $5.0_{-2}^{+2}$ &  $2.6\\pm0.2$ (5)\n& $0.7\\pm0.5$ & $3\\pm2$ \\\\\n2018hom & 58426.31 & $-19.30\\pm0.11$ & $-18.91\\pm0.01$ & $-9.3_{-0.4}^{+0.7}$ & $0.4\\pm0.1$ & $6.9\\pm0.2$ & $1.7\\pm0.2$ (27) & $> 0.7$ & $>1$ \\\\\n2018hxo & 58403.76 & $-18.68\\pm0.06$ & $-18.4\\pm0.1$ & $-28.6_{-0.3}^{+0.2}$ & $0.4\\pm0.2$ & $6\\pm2$ & $0.6\\pm0.1$ (48)\n& $>0.1$ & $>0.02$ \\\\\n2018jex & 58457.01 & $-19.06\\pm0.02$ & $-18.61\\pm0.04$ & $-18.49\\pm0.04$ & $0.53_{-0.06}^{+0.07}$ & $13_{-3}^{+2}$ & $1.8\\pm0.3$ (8)\n& $3\\pm1$ & $7\\pm3$ \\\\\n2019hsx & 58647.07 & $-17.08\\pm0.02$ & $-16.14\\pm0.04$ & $-15.6_{-0.5}^{+0.4}$ & $0.07_{-0.01}^{+0.01}$ & $12\\pm1$ & $1.0\\pm0.2$ (-0.2)\n& $1.6\\pm0.4$ & $1.0\\pm0.5$ \\\\\n2019xcc & 58844.59 & $-16.58\\pm0.06$ & $-15.6\\pm0.2$ & $-11\\pm2$ & $0.04\\pm0.01$ & $5.0_{-0.9}^{+1.4}$ & $2.4\\pm0.2$ (6)\n& $0.7\\pm0.3$ & $2\\pm1$ \\\\\n2020jqm & 58996.21 & $-18.26\\pm0.02$ & $-17.39\\pm0.04$ & $-17\\pm1$ & $0.29_{-0.04}^{+0.05}$ & $18\\pm2$ & $1.3\\pm0.3$ (-0.5) \n& $5\\pm1$ & $5\\pm3$ \\\\\n2020lao & 59003.92 & $-18.66\\pm0.02$ & $-18.55\\pm0.02$ & $-11\\pm1$ & $0.23\\pm0.01$ & $7.7\\pm0.2$ & $1.8\\pm0.2$ (9)\n& $1.2\\pm0.2$ & $2.5\\pm0.7$ \\\\\n2020rph & 59092.34 & $-17.48\\pm0.02$ & $-16.94\\pm0.03$ & $-19.88\\pm0.02$  & $0.07\\pm0.01$ & $17.23_{-0.9}^{+1.2}$ & $1.2\\pm0.5$ (-1)\n& $4\\pm2$ & $3\\pm3$ \\\\\n2020tkx & 59116.50 & $-18.49\\pm0.05$ & $-18.19\\pm0.03$ & $-13\\pm4$  & $0.22\\pm0.01$ & $10.9_{-0.8}^{+0.7}$ & $1.32\\pm0.09$ (53)\n& $> 1.5$ & $> 1.5$ \\\\\n2021xv & 59235.56 & $-18.92\\pm0.07$ & $-18.99\\pm0.05$ & $-12.8_{-0.3}^{+0.2}$ & $0.30_{-0.02}^{+0.01}$ & $7.7_{-0.5}^{+0.7}$ & $1.3\\pm0.1$ (3)\n& $0.9\\pm0.1$ & $1.0\\pm0.2$ \\\\\n2021aug & 59251.98 & $-19.42\\pm0.01$ & $-19.32\\pm0.06$ & $-24\\pm2$ & $0.7\\pm0.1$ & $17\\pm7$ & $0.8\\pm0.3$ (1) \n& $3\\pm2$ & $1\\pm1$ \\\\\n2021epp & 59291.83 & $-17.49\\pm0.03$ & $-17.12\\pm0.09$ & $-15\\pm1$ & $0.12\\pm0.02$ & $17_{-3}^{+4}$ & $1.4\\pm0.5$ (-4)\n& $5\\pm2$ & $6\\pm5$ \\\\\n2021htb & 59321.56 & $-16.55\\pm0.03$ & $-15.66\\pm0.07$ & $-19.38\\pm0.02$  & $0.04\\pm0.01$ & $13\\pm2$ & $1.0\\pm0.5$ (-6)\n& $1.8\\pm0.9$ & $1\\pm1$ \\\\\n2021hyz & 59319.10 & $-18.83\\pm0.05$ & $-18.81\\pm0.01$ & $-12.9\\pm0.9$  & $0.29_{-0.02}^{+0.01}$ & $7.7_{-0.4}^{+0.5}$ & $2.3\\pm0.3$ (16) & $> 1.3$ & $>4$ \\\\\n2021ywf & 59478.64 & $-17.10\\pm0.05$ & $-16.5\\pm0.1$ & $-10.7\\pm0.5$  & $0.06\\pm0.01$ & $8.9\\pm0.8$ & $1.2\\pm0.1$ (0.5) \n& $1.1\\pm0.2$ & $0.9\\pm0.3$ \\\\\n\\bottomrule\n\\multicolumn{9}{l}{$^{a}$ Rest-frame phase days of the spectrum that was used to measure the velocity.}\\\\\n\\end{longtable*}\n\\end{footnotesize}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[height=8cm]{CorsiFig5_speed.png}\n    \\caption{Photospheric velocities of the ZTF SNe in our sample (black) plotted as a function of (rest frame) time since explosion (see Table \\ref{tab:opt_data}).  Velocities are measured using Fe II (5169 $\\AA$); velocities quoted refer to 84\\% confidence and are measured relative to the Ic template velocity using the open source software \\texttt{SESNspectraLib} \\citep{Liu2016,Modjaz2016}. We compare our results with photospheric velocities derived from spectroscopic modeling for a number of Ib\/c SNe. Red symbols represent GRB-SNe \\citep{Iwamoto1998,Mazzali2003,Mazzali2006b}; magenta is used for XRF\/X-ray transients-SNe \\citep{Mazzali2006a,Pian2006,Modjaz2009}; blue represents SNe Ic-BL \\citep{Mazzali2000,Mazzali2002}; green is used for the ``normal'' Type Ic SN 1994I \\citep{Sauer2006}. Finally, for comparison we also plot the photospheric velocities for the SNe Ic-BL in the \\citet{Corsi2016} sample as measured by \\citet{Taddia2019} (see their Tables 2 and A1; yellow crosses). Errors on the times since explosion account for the uncertainties on T$_{\\rm exp}$ as reported in Table \\ref{tab:opt_data}. \\label{fig:velocities}}\n\\end{figure*}\n\n\n\\section{Multi-wavelength analysis}\n\\label{sec:modeling}\n\\subsection{Photospheric velocities}\n\\label{sec:vphot}\nWe confirm the SN Type Ic-BL classification of each object in our sample by measuring the photospheric velocities (${\\rm v_{ph}}$).\nSNe Ic-BL are characterized by high expansion velocities evident in the broadness of their spectral lines. A good proxy for the photospheric velocity is that derived from the maximum absorption position of the Fe II \\citep[$\\lambda 5169$; e.g.,][]{Modjaz2016}. We caution, however, that estimating this velocity is not easy given the strong line blending. We first pre-processed one high-quality spectrum per object using the IDL routine \\texttt{WOMBAT}, then smoothed the spectrum using the IDL routine \\texttt{SNspecFFTsmooth} \\citep{Liu2016}, and finally ran \\texttt{SESNSpectraLib} \\citep{Liu2016,Modjaz2016} to obtain final velocity estimates. \n\nIn  Figure~\\ref{fig:velocities} we show a comparison of the photospheric velocities estimated for the SNe in our sample with those derived from spectroscopic modeling for a number of SNe Ib\/c. The velocities measured for our sample are compatible, within the measurement errors, with what was observed for the PTF\/iPTF samples.\nMeasured values for the photospheric velocities with the corresponding rest-frame phase in days since maximum $r$-band light of the spectra that were used to measure them are also reported in  Table~\\ref{tab:opt_data}.  \n\nWe note that the spectra used to estimate the photospheric velocities of SN\\,2019xcc, SN\\,2020lao, and SN\\,2020jqm are different from those used for the classification of those events as SNe Ic-BL (see Section \\ref{section:sample} and Figure \\ref{fig:spectra1}). This is because for spectral classification we prefer later-time but higher-resolution spectra, while for velocity measurements we prefer earlier-time spectra even if taken with the lower-resolution SEDM. \n\n\\subsection{Bolometric light curve analysis}\n\\label{sec:optical_properties}\nIn our analysis we correct all ZTF photometry for Galactic extinction, using the Milky Way (MW) color excess $E(B-V)_{\\mathrm{MW}}$ toward the position of the SNe.\nThese are all obtained from \\cite{Schlafly2011}. All reddening corrections are applied using the \\cite{Cardelli1989} extinction law with $R_V=3.1$. \nAfter correcting for Milky Way extinction, we interpolate our P48 forced-photometry light curves \nusing a Gaussian Process via the {\\tt GEORGE}\\footnote{\\href{https:\/\/george.readthedocs.io}{https:\/\/george.readthedocs.io}} package with a stationary Matern32 kernel and the analytic functions of \\citet{Bazin2009} as mean for the flux form.\nAs shown in Figure~\\ref{fig:opt-lc-mag}, the colour evolution of the SNe  in our sample are not too dissimilar with one another, which implies that the amount of additional host extinction is small. Hence, we set the host extinction to zero.\nNext, we derive bolometric light curves calculating bolometric corrections from the $g$- and $r$-band data following the empirical relations by \\cite{Lyman2014,Lyman2016}. For SN\\,2018hxo, since there is only one $g$-band detection, we assume a constant bolometric correction to estimate its bolometric light curve. These bolometric light curves are shown in the bottom panel of Figure \\ref{fig:opt-lc-flux}. \n\nWe estimate the explosion time ${\\rm T}_{\\rm exp}$ of the SNe in our sample as follows.  For SN\\,2021aug, we fit the early ZTF light curve data following the method presented in \\citet{Miller2020}, where we fix the power-law index of the rising early-time temporal evolution to $\\alpha=2$, and derive an estimate of the explosion time from the fit. For most of the other SNe in our sample, the ZTF $r$- and $g$-band light curves lack enough early-time data to determine an estimate of the explosion time following the formulation of \\citet{Miller2020}. For all these SNe we instead set the explosion time to the mid-point between the last non-detection prior to discovery, and the first detection. \nResults on ${\\rm T}_{\\rm exp}$ are reported in Table  \\ref{tab:opt_data}.\n\nWe fit the bolometric light curves around peak ($-20$ to 60 rest-frame days relative to peak) to a model using the Arnett formalism \\citep{Arnett1982}, with the nickel mass ($M_{\\rm Ni}$) and characteristic time scale $\\tau_m$ as free parameters \\citep[see e.g. Equation A1 in][]{Valenti2008}. The derived values of $M_{\\rm Ni}$ (Table \\ref{tab:opt_data}) have a median of $\\approx 0.22$\\,M$_{\\odot}$, compatible with the median value found for SNe Ic-BL in the PTF sample by \\citet{Taddia2019}, somewhat lower than for SN\\,1998bw for which the estimated nickel mass values are in the range $(0.4-0.7)$\\,M$_{\\odot}$, but comparable to the $M_{\\rm Ni}\\approx 0.19-0.25$\\,M$_{\\odot}$ estimated for\nSN\\,2009bb \\citep[see e.g.,][]{Lyman2016,Afsariardchi2021}. We note that events such as SN\\,2019xcc and SN\\,2021htb have relatively low values of $M_{\\rm Ni}$, which are however compatible with the range of $0.02-0.05$\\,M$_{\\odot }$ expected for the nickel mass of magnetar-powered SNe Ic-BL \\citep{Nishimura2015,Chen2017,Suwa2015}. \n\nNext, from the measured  characteristic timescale $\\tau_m$ of the bolometric light curve, and the photospheric velocities estimated via spectral fitting  (see previous Section) we derive the ejecta mass ($M_{\\rm ej}$) and the kinetic energy ($E_{\\rm k}$) via the following relations \\citep[see e.g. Equations 1 and 2 in][]{Lyman2016}:\n\\begin{eqnarray}\n    \\tau^2_m {\\rm v_{ph, max}}=\\frac{2\\kappa}{13.8 c} M_{\\rm ej}~~~ & \n    ~~~{\\rm v}^2_{\\rm ph, max}=\\frac{5}{3}\\frac{2 E_{\\rm k}}{M_{\\rm ej}},\\label{eq:ejecta}\n\\end{eqnarray}\nwhere we assume a constant opacity of $\\kappa = 0.07$\\,g\\,cm$^{-2}$. \n\nWe note that to derive $M_{\\rm ej}$ and $E_{\\rm k}$ as described above we assume the photospheric velocity evolution is negligible within 15 days relative to peak epoch, and use the spectral velocities measured within this time frame to estimate ${\n\\rm v}_{\\rm ph, max}$ in Equation \\ref{eq:ejecta}. However, there are four objects in our sample (SN\\,2018hom, SN\\,2018hxo, SN\\,2020tkx, and SN\\,2021hyz) for which the spectroscopic analysis constrained the photospheric velocity only after day 15 relative to peak epoch. For these events, we only provide lower limits on the ejecta mass and kinetic energy (see Table \\ref{tab:opt_data}). \n \nConsidering only the SNe in our sample for which we are able to measure the photospheric velocity within 15\\,d since peak epoch, we derive median values for the ejecta masses and kinetic energies of  $1.7$\\,M$_{\\odot}$ and $2.2\\times10^{51}$\\,erg, respectively. These are both a factor of $\\approx 2$ smaller than the median values derived for the PTF\/iPTF sample of SNe Ic-BL \\citep{Taddia2019}. This could be due to either an intrinsic effect, or to uncertainties on the measured photospheric velocities. In fact, we note that the photospehric velocity is expected to decrease very quickly after maximum light (see e.g. Figure \\ref{fig:velocities}). Since the photospheric velocity in Equation (\\ref{eq:ejecta}) of the Arnett formulation is the one at peak, our estimates of ${\\rm v}_{\\rm ph,max}$ could easily underestimate that velocity by a factor of $\\approx 2$ for many of the SNe in our sample. This would in turn yield an underestimate of $M_{\\rm ej}$ by a factor of $\\approx 2$ (though the kinetic energy would be reduced by a larger factor). A more in-depth analysis of these trends and uncertainties will be presented in \\citet{Gokul2022}. \n \n\\subsection{Search for gamma-rays}\nBased on the explosion dates derived for each object in Section \\ref{sec:optical_properties} (Table \\ref{tab:opt_data}), we searched for potential GRB coincidences in several online archives. No potential counterparts were identified in both spatial and temporal coincidence with either the Burst Alert Telescope (BAT; \\citealt{Barthelmy2005}) on the \\textit{Neil Gehrels\nSwift Observatory}\\footnote{See \\url{https:\/\/swift.gsfc.nasa.gov\/results\/batgrbcat}.} or the Gamma-ray Burst Monitor  \\citep[GBM;][]{Meegan2009} on \\textit{Fermi}\\footnote{See \\url{https:\/\/heasarc.gsfc.nasa.gov\/W3Browse\/fermi\/fermigbrst.html}.}.\n\nSeveral candidate counterparts were found with \\textit{temporal} coincidence in the online catalog from the KONUS instrument on the \\textit{Wind} satellite (SN\\,2018etk, SN\\,2018hom, SN\\,2019xcc, SN\\,2020jqm, SN\\,2020lao, SN\\,2020tkx, SN\\,2021aug). However, given the relatively imprecise explosion date constraints for several of the events in our sample (see Table \\ref{tab:opt_data}), and the coarse localization information from the KONUS instrument, we cannot firmly associate any of these GRBs with the SNe Ic-BL. In fact, given the rate of GRB detections by KONUS ($\\sim 0.42$\\,d$^{-1}$) and the time window over which we searched for counterparts ($30$\\,d in total; derived from the explosion date constraints), the observed number of coincidences (13) is consistent with random fluctuations. Finally, none of the possible coincidences were identified in events with explosion date constraints more precise than 1\\,d. \n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=18cm]{CorsiFig6_GokulXray.png}\n    \\caption{\\textit{Swift}\/XRT upper-limits and detections (downward pointing triangles and filled circles with error bars, respectively) obtained for 9 of the 16 SNe Ic-BL in our sample. We plot the observed X-ray luminosity as a function of time since explosion. We compare these observations with the X-ray light curves of the low-luminosity GRB\\,980425 \\citep[red stars;][]{Kouveliotou2004}, GRB\\,060218 \\citep[red squares;][]{Campana2006},  GRB\\,100316D \\citep[red crosses;][]{Margutti2013}, and with the relativistic iPTF17cw \\citep[blue cross;][]{Corsi2017}. Dotted red lines connect the observed data points (some of which at early and late times are not shown in the plot) for these three low-luminosity GRBs. We also plot the observed  X-ray luminosity predicted by off-axis GRB models \\citep[black, green, and orange lines;][]{vanEerten2011, VanEerten2012}. We assume  $\\epsilon_B=\\epsilon_e=0.1$, a constant density ISM in the range $n = 1-10\\,{\\rm cm}^{-3}$, a top-hat jet of opening angle $\\theta_j=0.2$, and various observer's angles $\\theta_{\\rm obs}=(2.5-3)\\theta_j$.}\n    \\label{fig:X-raymodel}\n\\end{figure*}\n\n\\subsection{X-ray constraints}\n\\label{sec:X-raymodel}\nSeven of the 9 SNe Ic-BL observed with \\textit{Swift}-XRT did not result in a significant detection. In Table \\ref{tab:x-ray} we report the derived 90\\% confidence flux upper limits in the 0.3--10\\,keV band after correcting for Galactic absorption \\citep{Willingale+2013}. \n\nWhile observations of SN\\,2021ywf resulted in a  $\\approx 3.2\\sigma$  detection significance (Gaussian equivalent), the limited number of photons (8) precluded a meaningful spectral fit. Thus, a $\\Gamma = 2$ power-law spectrum was adopted for the flux conversion for this source as well. We note that because of the relatively poor spatial resolution of the \\textit{Swift}-XRT (estimated positional uncertainty of 11.7\\arcsec~radius  at 90\\% confidence), we cannot entirely rule out unrelated emission from the host galaxy of SN 2021ywf (e.g., AGN, X-ray binaries, diffuse host emission; see Figure \\ref{fig:hosts-det} for the host galaxy).\n\nFor SN\\,2019hsx we detected enough photons to perform a spectral fit for count rate to flux conversion. The spectrum is found to be relatively soft, with a best-fit power-law index of $\\Gamma = 3.9^{+3.0}_{-2.1}$. \nOur \\textit{Swift} observations of SN\\,2019hsx do not show significant evidence for variability of the source X-ray flux over the timescales of our follow up. While the lack of temporal variability is not particularly constraining given the low signal-to-noise ratio in individual epochs, we caution that also in this case the relatively poor spatial resolution of the \\textit{Swift}-XRT (7.4\\arcsec~radius position uncertainty at 90\\% confidence) implies that unrelated emission from the host galaxy cannot be excluded.\n\nThe constraints derived from \nthe \\textit{Swift}-XRT observations can be compared with the\nX-ray light curves of low-luminosity GRBs, or models of GRB afterglows \nobserved slightly off-axis. For the latter, we use\nthe numerical model by \\citet{vanEerten2011, VanEerten2012}. We assume equal energies in the electrons and magnetic fields ($\\epsilon_B= \\epsilon_e=0.1$), and an interstellar medium (ISM) of density $n = 1-10$\\,cm$^{-3}$. We note that a constant density ISM (rather than a wind profile) has been shown to fit the majority of GRB afterglow light curves, implying that most GRB progenitors might have relatively small wind termination-shock radii \\citep{Schulze2011}. We generate the model light curves for a nominal redshift of $z=0.05$ and then convert the predicted flux densities into X-ray luminosities by integrating over the 0.3-10\\,keV energy range and  neglecting the small redshift corrections. We plot the model light curves in Figure \\ref{fig:X-raymodel}, for various energies, different power-law indices $p$ of the electron energy distribution, and various off-axis angles (relative to a jet opening angle, set to $\\theta_j=0.2$). In the same Figure we also plot  the X-ray light curves of low-luminosity GRBs for comparison (neglecting redshift corrections). \nAs evident from this Figure, our \\textit{Swift}\/XRT upper limits (downward-pointing triangles) \nexclude X-ray afterglows associated with higher-energy GRBs observed slightly off-axis.  However, X-ray emission as faint as the afterglow of\nthe low-luminosity GRB\\,980425 cannot be excluded. As we discuss in the next Section, radio data collected with the VLA enable us to exclude GRB\\,980425\/SN\\,1998bw-like emission for most of the SNe in our sample. \n\nWe note that our two X-ray detections of SN\\,2019hsx and SN\\,2021ywf are consistent with several GRB off-axis light curve models and, in the case of SN\\,2021ywf, also with GRB\\,980425-like emission within the large errors. However, for this interpretation of their X-ray emission to be compatible with our radio observations (see Table \\ref{tab:data}), one needs to invoke a flattening of the radio-to-X-ray spectrum, similar to what has been invoked for other stripped-envelope SNe in the context of cosmic-ray dominated shocks \\citep{Ellison2000, Chevalier2006}.\n\n\\subsection{Radio constraints}\n\\label{sec:radioanalysis}\nAs evident from Table \\ref{tab:data}, we have obtained at least one radio detection for 11 of the 16 SNe in our sample.  None of these 11 radio sources were found to be coincident with known radio sources in the VLA FIRST \\citep{Becker1995} catalog (using a search radius of 30\\arcsec~around the optical SN positions). This is not surprising since the FIRST survey had a typical RMS sensitivity of $\\approx 0.15$\\,mJy at 1.4\\,GHz, much shallower than the deep VLA follow-up observations carried out within this follow-up program. We also checked the quick look images from the VLA Sky Survey (VLASS) which reach a typical RMS sensitivity of $\\approx 0.12$\\,mJy at 3\\,GHz \\citep{Hernandez2018,Law2018}. We could find images for all but one (SN\\,2021epp) of the fields containing the 16 SNe BL-Ic in our sample. The VLASS images did not provide any radio detection at the locations of the SNe in our sample.\n\nFive of the 11 SNe Ic-BL with radio detections are associated with extended or marginally resolved radio emission. Two other radio-detected events (SN\\,2020rph and SN\\,2021hyz) appear point-like in our images, but show no evidence for significant variability of the detected radio flux densities over the timescales of our observations. Thus, for a total of 7 out of 11 SNe Ic-BL with radio detections, we consider the measured flux densities as upper-limits corresponding to the brightness of their host galaxies, similarly to what was done in e.g. \\citet{Soderberg2006} and \\citet{Corsi2016}. The remaining 4 SNe Ic-BL with radio detections are compatible with point sources (SN\\,2018hom, SN\\,2020jqm, SN\\,2020tkx, and SN\\,2021ywf), and all but one (SN\\,2018hom) had more than one observation in the radio via which we were able to establish the presence of substantial variability of the radio flux density. Hereafter we consider these 4 detections as genuine radio SN counterparts, though we stress that with only one observation of SN\\,2018hom we cannot rule out a contribution from host galaxy emission, especially given that the radio follow up of this event was carried out with the VLA in its most compact (D) configuration with poorer angular resolution.\n\nIn summary, our radio follow-up campaign of 16 SNe Ic-BL resulted in 4 radio counterpart detections, and 12 deep upper-limits on associated radio counterparts.\n\n\\begin{figure*}\n    \\begin{center}\n    \\includegraphics[width=6cm]{CorsiFig7_ZTF18abklarx_host_contour.png}\n    \\includegraphics[width=6cm]{CorsiFig7_ZTF19adaiomg_host_contour.png}\n    \\includegraphics[width=6.cm,height=5.6cm]{CorsiFig7_ZTF20abswdbg_host_contour.png}\n    \\includegraphics[width=6cm]{CorsiFig7_ZTF21aadatfg_host_contour.png}\n    \\includegraphics[width=6cm,height=5.6cm]{CorsiFig7_ZTF21aaocrlm_host_contour.png}\n    \\includegraphics[width=6cm]{CorsiFig7_ZTF21aardvol_host_contour.png}\n    \\includegraphics[width=6cm]{CorsiFig7_ZTF21aartgiv_host_contour.png}\n    \\caption{PanSTARRS-1 \\citep{Flewelling2020} reference $r$-band images of the fields of the SNe in our sample for which host galaxy light dominates the radio emission. Contours in magenta are 30\\%, 50\\%, and 90\\% of the radio peak flux reported in Table \\ref{tab:data} for the first radio detection of each field. The blue circles centered on the optical SN positions (not shown in the images) have sizes of 2\\arcsec  \\citep[comparable to the ZTF PSF at average seeing;][]{Bellm2019}. The red-dotted circles enclose the region in which we search for radio counterparts (radii equal to the nominal FWHM of the VLA synthesized beams;  Table \\ref{tab:data}). The sizes of the actual VLA synthesized beams  are shown as filled magenta ellipses. The red dots mark the locations of the radio peak fluxes measured in the radio search areas. \\label{fig:hosts}}\n    \\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n    \\begin{center}\n    \\hbox{\n    \\includegraphics[height=8.3cm]{CorsiFig8_ZTF18acbwxcc_contour.png}\n    \\hspace{0.2cm}\n    \\includegraphics[height=8.3cm]{CorsiFig8_ZTF20aazkjfv_contour.png}}\n    \\hbox{\n    \\includegraphics[height=7.8cm]{CorsiFig8_ZTF20abzoeiw_contour.png}\n    \\includegraphics[height=7.8cm]{CorsiFig8_ZTF21acbnfos_contour.png}}\n    \\caption{Same as Figure \\ref{fig:hosts} but for the fields containing the SNe in our sample for which we detected a SN radio counterpart.  We stress that with only one observation of SN\\,2018hom we cannot rule out a contribution from host galaxy emission, especially given that the radio follow up of this event was carried out with the VLA in its D configuration. \\label{fig:hosts-det}}\n    \\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=18cm]{CorsiFig9_RadioLight.png}\n    \\caption{Radio ($\\approx 6$\\,GHz) observations of the 16 SNe Ic-BL in our sample (filled circles and downward pointing triangles in shades of pink, purple, and blue). Upper-limits associated with non-detections ($3\\sigma$ or brightness of the host galaxy at the optical location of the SN) are plotted with downward-pointing triangles; detections are plotted with filled circles. We compare these observations with the radio light curves of GRB-SNe (red); of relativistic-to-mildly relativistic SNe Ic-BL discovered independently of a $\\gamma$-ray trigger (cyan); and with PTF11qcj \\citep{Corsi2014}, an example of a radio-loud non-relativistic and CSM-interacting SN Ic-BL (yellow). As evident from this Figure, our observations exclude SN\\,1998bw-like radio emission for all but one (SN\\,2021epp) of the events in our sample. This doubles the sample of SNe Ic-BL for which radio emission observationally similar to SN\\,1998bw was previously excluded \\citep{Corsi2016}, bringing the upper limit on the fraction of SNe compatible with SN\\,1998bw down to $< 19\\%$ \\citep[compared to $< 41\\%$ previously reported in][]{Corsi2016}.\n    For 10 of the 16 SNe presented here we also exclude relativistic ejecta with radio luminosity densities in between $\\approx 5\\times10^{27}$\\,erg\\,s$^{-1}$\\,Hz$^{-1}$ and  $\\approx 10^{29}$\\,erg\\,s$^{-1}$\\,Hz$^{-1}$ at $t\\gtrsim 20$\\,d, similar to SNe associated with low-luminosity GRBs such as SN\\,1998bw \\citep{Kulkarni1998}, SN\\,2003lw \\citep{Soderberg2004}, SN\\,2010dh \\citep{Margutti2013}, or to the relativistic SN\\,2009bb \\citep{Soderberg2010} and iPTF17cw \\citep{Corsi2017}.  None of our observations exclude radio emission similar to that of SN\\,2006aj. }\n    \\label{fig:radiolight}\n\\end{figure*}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=8.6cm]{CorsiFig10_RadioSpeed.png}\n    \\caption{Properties of the radio-emitting ejecta of the SNe in our sample for which we detect a radio counterpart (magenta dots), compared with those of GRB-SNe (red stars) and of relativistic-to-mildly\nrelativistic SNe Ic-BL discovered independently of a $\\gamma$-ray trigger (cyan stars). As evident from this Figure, only SN\\,2018hom is compatible with an ejecta speed $\\gtrsim 0.3c$, though with the caveat that we only have one radio observation for this SN. None of the other ZTF SNe Ic-BL in our sample shows evidence for ejecta faster than $0.3c$. We also note that SN\\,2020jqm lies in the region of the parameter space occupied by radio-loud CSM-interacting SNe similar to PTF11qcj. See Section \\ref{sec:radio_properties} for discussion.}\n    \\label{fig:radiospeed}\n\\end{figure}\n\n\\subsubsection{Fraction of SN\\,1998bw-like SNe Ic-BL}\nThe local rate of SNe Ic-BL is estimated to be $\\approx 5\\%$ of the core-collapse SN rate \\citep{Li2011,Shivvers2017,Perley2020b} or $\\approx 5\\times10^3$\\,Gpc$^{-3}$\\,yr$^{-1}$ assuming a core-collapse SN rate of $\\approx 10^5$\\,Gpc$^{-3}$\\,yr$^{-1}$ \\citep{Perley2020b}.  Observationally, we know that cosmological long GRBs are characterized by ultra-relativistic jets observed on-axis, and have an \\textit{intrinsic} (corrected for beaming angle) local volumetric rate of $79^{+57}_{-33}$\\,Gpc$^{-3}$\\,yr$^{-1}$ \\citep[e.g.,][and references therein]{Ghirlanda2022}. Hence, only ${\\cal O}(1)\\%$ of SNe Ic-BL can make long GRBs. For low-luminosity GRBs, the \\textit{observed} local rate is affected by large errors, $230^{+490}_{-190}$\\,Gpc$^{-3}$\\,yr$^{-1}$ \\citep[see][and references therein]{Bromberg2011}, and their typical beaming angles are largely unconstrained. Hence, the question of what fraction of SNe Ic-BL can make low-luminosity GRBs remains to be answered. \n\nRadio observations of SNe Ic-BL are a powerful way to constrain this fraction independently of relativistic beaming effects that preclude observations of jets in X-rays and $\\gamma$-rays for off-axis observers. However, observational efforts aimed at constraining the fraction of SNe Ic-BL harboring low-luminosity GRBs independently of $\\gamma$-ray observations have long been challenged by the rarity of the SN Ic-BL optical detections (compared to other core-collapse events), coupled with the small number of these rare SNe for which the community has been able to collect deep radio follow-up observations within 1\\,yr since explosion \\citep[see e.g.,][]{Soderberg2006catalog}. Progress in this respect has been made since the advent of the PTF, and more generally with synoptic optical surveys that have greatly boosted the rate of stripped-envelope core-collapse SN discoveries \\citep[e.g.,][]{Shappee2014,Sand2018,Tonry2018}. \n\nIn our previous work \\citep{Corsi2016}, we presented one of the most extensive samples of SNe Ic-BL with deep VLA observations, largely composed of events detected by the PTF\/iPTF. Combining our sample with the SN Ic-BL\\,2002ap \\citep{GalYam2002,Mazzali2002} and SN\\,2002bl \\citep{Armstrong2002,Berger2003}, and the CSM-interacting SN Ic-BL\\,2007bg \\citep{Salas2013}, we had overall 16 SNe Ic-BL for which radio emission observationally similar to SN\\,1998bw was excluded, constraining the rate of SNe Ic-BL observationally similar to SN\\,1998bw to $< 6.61\/16\\approx 41\\%$, where we have used the fact that the Poisson 99.865\\% confidence (or $3\\sigma$ Gaussian equivalent for a single-sided distribution) upper-limit on zero SNe compatible with SN\\,1998bw is $\\approx 6.61$.\n\nWith the addition of the 16 ZTF SNe Ic-BL presented in this work, we now have doubled the sample of SNe Ic-BL with deep VLA observations presented in \\citet{Corsi2016}, providing evidence for additional 15 SNe Ic-BL (all but SN\\,2021epp; see Figure \\ref{fig:radiolight}) that are observationally different from SN\\,1998bw in the radio. Adding to our sample also SN\\,2018bvw \\citep{Ho2020_ZTF18aaqjovh}, AT\\,2018gep \\citep{Ho2019_ZTF18abukavn}, and SN\\,2020bvc \\citep{Ho2020_ZTF20aalxlis}, whose radio observations exclude SN\\,1998bw-like emission, we are now at 34 SNe Ic-BL that are observationally different from SN\\,1998bw. Hence, we can tighten our constraint on the fraction of 1998bw-like SNe Ic-BL to $< 6.61\/34\\approx 19\\%$ (99.865\\% confidence). This upper-limit implies that the \\textit{intrinsic} rate of 1998bw-like GRBs is $\\lesssim 950$\\,Gpc$^{-3}$\\,yr$^{-1}$. Combining this constraint with the rate of low-luminosity GRBs derived from their high-energy emission, we conclude that low-luminosity GRBs have inverse beaming factors $2\/\\theta^2\\lesssim 4^{+20}_{-3}$, corresponding to  jet half-opening angles $\\theta \\gtrsim 40^{+40}_{-24}$\\,deg. \n\nWe note that for 10 of the SNe in the sample presented here we also exclude relativistic ejecta with radio luminosity densities in between $\\approx 5\\times10^{27}$\\,erg\\,s$^{-1}$\\,Hz$^{-1}$ and  $\\approx 10^{29}$\\,erg\\,s$^{-1}$\\,Hz$^{-1}$ at $t\\gtrsim 20$\\,d, pointing to the fact that SNe Ic-BL similar to those associated with low-luminosity GRBs, such as SN\\,1998bw \\citep{Kulkarni1998}, SN\\,2003lw \\citep{Soderberg2004}, SN\\,2010dh \\citep{Margutti2013}, or to the relativistic SN\\,2009bb \\citep{Soderberg2010} and iPTF17cw \\citep{Corsi2017}, are intrinsically rare.  However, none of our observations exclude radio emission similar to that of SN\\,2006aj. This is not surprising since the afterglow of this low-luminosity GRB faded on timescales much faster than the $20-30$ days since explosion that our VLA monitoring campaign allowed us to target. To enable progress, obtaining prompt ($\\lesssim 5$\\,d since explosion) and accurate spectral classification paired with deep radio follow-up observations of SNe Ic-BL should be a major focus of future studies. At the same time, as discussed in \\citet{Ho2020_ZTF20aalxlis}, high-cadence optical surveys can provide an alternative way to measure the rate of SNe Ic-BL that are similar to SN\\,2006aj independently of $\\gamma$-ray and radio observations, by catching potential optical signatures of shock-cooling emission at early times.  Based on an analysis of ZTF SNe with early high-cadence light curves, \\citet{Ho2020_ZTF20aalxlis} concluded that it appears that SN\\,2006aj-like events are uncommon, but more events will be needed to measure a robust rate.\n\n\\subsubsection{Properties of the radio-emitting ejecta}\n\\label{sec:radio_properties}\nGiven that none of the SNe in our sample shows evidence for relativistic ejecta, hereafter we consider their radio properties within the synchrotron self-absorption (SSA) model for radio SNe \\citep{Chevalier1998}. Within this model, constraining the radio peak frequency and peak flux can provide information on the size of the radio emitting material. We start from Equations (11) and (13) of \\citet{Chevalier1998}:\n\\begin{eqnarray}\n\\nonumber R_{p}\\approx 8.8\\times10^{15}\\,{\\rm cm}\\left(\\frac{\\eta}{2\\alpha}\\right)^{1\/(2p+13)} \\left(\\frac{F_p}{\\rm Jy}\\right)^{(p+6)\/(2p+13)}\\times\\\\\\left(\\frac{d_L}{\\rm Mpc}\\right)^{(2p+12)\/(2p+13)}\\left(\\frac{\\nu_p}{5\\,\\rm GHz}\\right)^{-1},~~~\n\\label{eq:ejspeed}\n\\end{eqnarray}\nwhere $\\alpha \\approx 1$ is the ratio of relativistic electron energy density to magnetic energy density, $F_{p}$  is the flux density at the time of SSA peak, $\\nu_{p}$ is the SSA frequency, and where $R\/\\eta$ is the thickness of the radiating electron shell. The normalization of the above Equation has a small dependence on $p$ and in the above we assume $p\\approx 3$ for the power-law index of the electron energy distribution. Setting $R_p \\approx {\\rm v_s} t_p$ in Equation (\\ref{eq:ejspeed}), and considering that $L_p \\approx 4\\pi d^2_L F_p$ (neglecting redshift effects), we get:\n\\begin{eqnarray}\n \\nonumber  \\left(\\frac{L_p}{\\rm erg\\,s^{-1}\\,Hz^{-1}}\\right) \\approx 1.2\\times10^{27} \\left(\\frac{\\beta_s}{3.4}\\right)^{(2p+13)\/(p+6)} \\\\\\times\\left(\\frac{\\eta}{2\\alpha}\\right)^{-1\/(p+6)} \\left(\\frac{\\nu_p}{5\\,\\rm GHz}\\frac{t_p}{\\rm 1\\,d}\\right)^{(2p+13)\/(p+6)}\n\\end{eqnarray}\nwhere we have set $\\beta_s = {\\rm v_s}\/c$. We plot in Figure \\ref{fig:radiospeed} with blue-dotted lines the relationship above for various values of $\\beta_s$ (and for $p=3$, $\\eta=2$, $\\alpha=1$). As evident from this Figure, relativistic events such as SN\\,1998bw (for which the non-relativistic approximation used in the above Equations breaks down) are located at $\\beta_s\\gtrsim 1$. None of the ZTF SNe Ic-BL in our sample for which we obtained a radio counterpart detection shows evidence for ejecta faster than $0.3c$, except possibly for SN\\,2018hom. However, for this event we only have one radio observation and hence contamination from the host galaxy cannot be excluded. We also note that SN\\,2020jqm lies in the region of the parameter space occupied by radio-loud CSM interacting SNe similar to PTF\\,11qcj. \n\nThe magnetic field can be expressed as \\citep[see Equations (12) and (14) in][]{Chevalier1998}:\n\\begin{eqnarray}\n\\nonumber B_p \\approx 0.58\\rm\\,G \\left(\\frac{\\eta}{2\\alpha}\\right)^{4\/(2p+13)}\\left(\\frac{F_{p}}{\\rm Jy}\\right)^{-2\/(2p+13)}\\times\\\\\\left(\\frac{d_{L}}{\\rm Mpc}\\right)^{-4\/(2p+13)}\\left(\\frac{\\nu_p}{5\\,\\rm GHz}\\right).\n\\label{eq2}\n\\end{eqnarray} \nConsider a SN shock expanding in a circumstellar medium (CSM) of density:\n\\begin{equation}\n\\rho\\approx 5 \\times10^{11} \\,{\\rm g\\,cm}^{-1}A_{*}R^{-2}\n\\end{equation}\nwhere:\n\\begin{equation}\n  A_{*}=  \\frac{\\dot{M}\/(10^{-5}M_{\\odot}\/{\\rm yr})}{4\\pi {\\rm v}_w\/(10^3{\\rm km\/s})}.\n\\label{eq:rho}\n\\end{equation}\nAssuming that a fraction $\\epsilon_B$ of the energy density $\\rho{\\rm v}^2_s$  goes into magnetic fields:\n\\begin{equation}\n \\frac{B_p^2}{8\\pi} = \\epsilon_B \\rho {\\rm v_s}^2 = \\epsilon_B \\rho R_p^2 t_p^{-2},\n \\label{eq3}\n\\end{equation}\none can write:\n\\begin{eqnarray}\n\\nonumber \\left(\\frac{L_p}{\\rm erg\\,s^{-1}\\,Hz^{-1}}\\right) \\approx 1.2\\times10^{27} \\left(\\frac{\\eta}{2\\alpha}\\right)^{2}\\left(\\frac{\\nu_p}{5\\,\\rm GHz}\\frac{t_p}{1\\,\\rm d}\\right)^{(2p+13)\/2}\\\\\\times \\left(5\\times10^3\\epsilon_B A_*\\right)^{-(2p+13)\/4},~~~~~~\n\\end{eqnarray}\nwhere we have used Equations  (\\ref{eq2}), (\\ref{eq:rho}), and (\\ref{eq3}). We plot in Figure \\ref{fig:radiospeed} with yellow-dashed lines the relationship above for various values of $\\dot{M}$ (and for $p=3$, $\\eta=2$, $\\alpha=1$, $\\epsilon_B =0.33$, $v_w=1000$\\,km\\,s$^{-1}$).  As evident from this Figure, relativistic events such as SN\\,1998bw show a preference for smaller mass-loss rates. We note that while the above relationship depends strongly on the assumed values of $\\eta$, $\\epsilon_B$, and $v_w$, this trend for $\\dot{M}$ remains true regardless of the specific values of these (uncertain) parameters. We also note that the above analysis assumes mass-loss in the form of a steady wind. While this is generally considered to be the case for relativistic SNe Ic-BL, binary interaction or eruptive mass loss in core-collapse SNe can produce denser CSM with more complex profiles \\citep[e.g.][]{Montes1998,Soderberg2006,Salas2013,Corsi2014,Margutti2017,Balasubramanian2021,Maeda2021,Stroh2021}.\n\nFinally, the total energy coupled to the fastest (radio emitting) ejecta can be expressed as \\citep[e.g.,][]{Soderberg2006}:\n\\begin{equation}\nE_r\\approx \\frac{4\\pi R_p^3}{\\eta}\\frac{B_p^2}{8\\pi\\epsilon_B}=\\frac{R_p^3}{\\eta}\\frac{B_p^2}{2\\epsilon_B}.\n\\label{eq4}\n\\end{equation}\n In Table \\ref{tab:radioproperties} we summarize the properties of the radio ejecta derived for the four SNe for which we detect a radio counterpart. These values can be compared with  $\\dot{M}\\approx2.5\\times10^{-7}M_{\\odot}{\\rm yr}^{-1}$ and $E_r\\approx (1-10)\\times10^{49}$\\,erg estimated for SN\\,1998bw by \\citet{Li1999}, with $\\dot{M}\\approx2\\times10^{-6}M_{\\odot}{\\rm yr}^{-1}$ and $E_r\\approx 1.3\\times10^{49}$\\,erg estimated for SN\\,2009bb by \\citet{Soderberg2010}, and with  with $\\dot{M}\\approx(0.4-1)\\times10^{-5}M_{\\odot}{\\rm yr}^{-1}$ and $E_r\\approx (0.3-4)\\times10^{49}$\\,erg estimated for GRB\\,100316D by \\citet{Margutti2013}. \n\n\\begin{center}\n\\begin{table}\n\\caption{Properties of the radio ejecta of the SNe in our sample for which we detect a radio counterpart. We report the SN name, the estimated SN shock speed normalized to the speed of light ($\\beta_s$), the mass-loss rate of the pre-SN progenitor ($\\dot{M}$), the energy coupled to the fastest (radio-emitting) ejecta ($E_r$), and the ratio between the last and the kinetic energy of the explosion (estimated from the optical light curve modeling, $E_k$). See Section \\ref{sec:radioanalysis} for discussion. \\label{tab:radioproperties}}\n\\begin{tabular}{lcccc}\n\\hline\n\\hline\nSN  & $\\beta_s$ &  $\\dot{M}$ & $E_r$ & $E_r\/E_k$ \\\\\n & & (M$_{\\odot}$\\,yr$^{-1}$) & (erg) & \\\\\n\\hline\n2018hom &  0.35 & $1.1\\times10^{-6}$ & $3.6\\times10^{47}$ & $<0.04$\\%\\\\\n2020jqm &  0.048 & $2.7\\times10^{-4}$ & $5.7\\times10^{48}$ & 0.1\\% \\\\\n2020tkx &  0.14 & $1.7\\times10^{-5}$ & $1.1\\times10^{48}$ & $<0.07$\\% \\\\\n2021ywf & 0.19 &  $2.2\\times10^{-6}$ & $2.3\\times10^{47}$ & 0.02\\%\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\end{center}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=16cm]{CorsiFig11_OffAxisRadioLight.png}\n    \\caption{Approximate radio luminosity density for GRBs observed largely off-axis during the sub-relativistic phase (black solid, dotted, dashed, and dash-dotted lines) compared with the radio detections and upper-limits of the SNe Ic-BL in our sample. Most of our observations exclude fireballs with energies $E\\gtrsim 10^{50}$\\,erg expanding in ISM media with densities $\\gtrsim 1$\\,cm$^{-3}$. However, our observations become less constraining for smaller energy and ISM density values. For example, most of our radio data cannot exclude off-axis jets with energies $E\\sim 10^{49}$\\,erg and $n\\sim 0.1$\\,cm$^{-3}$. See Section \\ref{sec:off-axis-radio} for discussion.}\n    \\label{fig:off-axis-radio}\n\\end{figure*}\n\n\\subsubsection{Off-axis GRB radio afterglow constraints}\n\\label{sec:off-axis-radio}\nWe finally consider what type of constraints our radio observations put on a scenario where the SNe Ic-BL in our sample could be accompanied by relativistic ejecta from \na largely (close to 90\\,deg) off-axis GRB afterglow that would become visible in the radio band when the relativistic fireball enters the sub-relativistic phase and approaches spherical symmetry.\nBecause our radio observations do not extend past 100-200 days since explosion, we can put only limited constraints on this scenario. Hence, hereafter we present some general order-of-magnitude considerations rather than a detailed event-by-event modeling. \n\nFollowing \\citet{Corsi2016}, we can model approximately the late-time radio emission from an off-axis GRB based on the results by \\citet{Livio2000}, \\citet{Waxman2004b}, \\citet{Zhang2009}, and \\citet{VanEerten2012}. For fireballs expanding in an interstellar medium (ISM) of constant\ndensity $n$ (in units of cm$^{-3}$), at timescales $t$ such that:\n\\begin{equation}\n    t \\gtrsim (1+z) \\times t_{\\rm SNT}\/2\n\\end{equation}\nwhere the transition time to the spherical Sedov\u2013Neumann\u2013Taylor (SNT) blast wave, $t_{\\rm SNT}$,  reads:\n\\begin{equation}\n    t_{\\rm SNT}\\approx 92\\,{\\rm d}\\left(E_{51}\/n\\right)^{1\/3},\n\\end{equation}\nthe luminosity density can be approximated analytically via the following formula \\citep[see Equation (23) in][ where we neglect redshift corrections and assume $p=2$]{Zhang2009}:\n\\begin{eqnarray}\n\\nonumber L_{\\nu}({\\rm t}) \\approx 4\\pi d^{2}_{\\rm L} F_{\\nu}({\\rm t}) \\approx 2\\times10^{30}\\left(\\frac{\\epsilon_e}{0.1}\\right)\\left(\\frac{\\epsilon_B}{0.1}\\right)^{3\/4}n^{9\/20}\\\\\\times E^{13\/10}_{51}\\left(\\frac{\\nu}{\\rm 1\\,GHz}\\right)^{-1\/2}\\left(\\frac{t}{92\\,{\\rm d}}\\right)^{-9\/10}\\,{\\rm erg\\,s^{-1}\\,{\\rm Hz}^{-1}}.~~~~~~~\n\\end{eqnarray}\nIn the above Equations, $E_{51}$ is the beaming-corrected ejecta energy in units of $10^{51}$\\,erg. We note that here we assume again a constant density ISM in agreement with the majority of GRB afterglow observtions \\citep[e.g.,][]{Schulze2011}. \n\nWe plot the above luminosity in Figure \\ref{fig:off-axis-radio} together with our radio observations and upper-limits, assuming $\\epsilon_e=0.1$, $\\epsilon_B=0.1$, and for representative values of low-luminosity GRB energies and typical values of long GRB ISM densities $n$. As evident from this Figure, our observations exclude fireballs with energies $E\\gtrsim 10^{50}$\\,erg expanding in ISM media with densities $\\gtrsim 1$\\,cm$^{-3}$. However, our observations become less constraining for smaller energy and ISM density values. \n\n\\section{Summary and Conclusion}\\label{sec:conclusion}\nWe have presented deep radio follow-up observations of 16 SNe Ic-BL that are part of the ZTF sample.  \nOur campaign resulted in 4 radio counterpart detections and 12 deep radio upper-limits. For 9 of these 16 events we have also carried out X-ray observations with \\textit{Swift}\/XRT. All together, these results constrain the fraction of SN\\,1998bw-like explosions to $< 19\\%$ (3$\\sigma$ Gaussian equivalent), tightening previous constraints by a factor of $\\approx 2$. Moreover, our results exclude relativistic ejecta with radio luminosities densities in between $\\approx 5\\times10^{27}$\\,erg\\,s$^{-1}$\\,Hz$^{-1}$ and  $\\approx 10^{29}$\\,erg\\,s$^{-1}$\\,Hz$^{-1}$ at $t\\gtrsim 20$\\,d since explosion for $\\approx 60\\%$ of the events in our sample, pointing to the fact that SNe Ic-BL similar to low-luminosity-GRB-SN such as SN\\,1998bw, SN\\,2003lw, SN\\,2010dh, or to the relativistic SN\\,2009bb and iPTF17cw, are intrinsically rare. This result is in line with numerical simulations that suggest that a SN Ic-BL can be triggered even if a jet\nengine fails to produce a successful GRB jet.\n\nWe showed that our radio observations exclude an association of the SNe Ic-BL in our sample with largely off-axis GRB afterglows with energies $E\\gtrsim 10^{50}$\\,erg expanding in ISM media with densities $\\gtrsim 1$\\,cm$^{-3}$. On the other hand, our radio observations are less constraining for smaller energy and ISM density values, and cannot exclude off-axis jets with energies $E\\sim 10^{49}$\\,erg.\n\nWe noted that the main conclusion of our work is subject to the caveat that the parameter space of SN\\,2006aj-like explosions (with faint radio emission peaking only a few days after explosion) is left largely unconstrained by current systematic radio follow-up efforts like the one presented here. In other words, we cannot exclude that a larger fraction of SNe Ic-BL harbors GRB\\,060218\/SN\\,2006aj-like emission. In the future, obtaining fast and accurate spectral classification of SNe Ic-BL paired with deep radio follow-up observations executed within $5$\\,d since explosion would overcome this limitation. While high-cadence optical surveys can provide an alternative way to measure the rate of SNe Ic-BL that are similar to SN\\,2006aj via shock-cooling emission at early times, more optical detections are also needed to measure a robust rate. \n\nThe Legacy Survey of Space and Time on the Vera C. Rubin Observatory \\citep[LSST;][]{Rubin2019} promises to provide numerous discoveries of even the rarest type of explosive transients, such as the SNe Ic-BL discussed here. The challenge will be to recognize and classify these explosions promptly \\citep[e.g.,][]{Villar2019,Villar2020}, so that they can be followed up in the radio with current and next generation radio facilities. Indeed, Rubin, paired with the increased sensitivity of the next generation VLA \\citep[ngVLA;][]{ngVLA}, could provide a unique opportunity for building a large statistical sample of SNe Ic-BL with deep radio observations that may be used to guide theoretical modeling in a more systematic fashion, beyond what has been achievable over the last $\\approx 25$ years (i.e., since the discovery of GRB-SN\\,1998bw).  In addition, the Square Kilometer Array (SKA) will enable discoveries of radio SNe and other transients in an untargeted and optically-unbiased way \\citep{Lien2011}. Hence, one can envision that the Rubin-LSST+ngVLA and SKA samples will, together, provide crucial information on\nmassive star evolution, as well as SNe Ic-BL physics and CSM properties. \n\nWe conclude by noting that understanding the evolution of single and stripped binary stars up to core collapse is of special interest in the new era of time-domain multi-messenger (gravitational-wave and neutrino) astronomy \\citep[see e.g., ][for recent reviews]{Murase2018,Scholber2012,Ernazar2020,Guepin2022}. Gravitational waves from nearby core-collapse SNe, in particular, represent an exciting prospect for expanding multi-messenger studies beyond the current realm of compact binary coalescences. While they may come into reach with the current LIGO \\citep{Aasi2015} and Virgo \\citep{Acernese2015} detectors, it is more likely that next generation gravitational-wave observatories, such as the Einstein Telescope \\citep{Maggiore2020} and the Cosmic Explorer \\citep{Evans2021}, will enable painting the first detailed multi-messenger picture of a core-collapse explosion. The physics behind massive stars' evolution and deaths also impacts the estimated rates and mass distribution of compact object mergers \\citep[e.g.,][]{Schneider2021} which, in turn, are current primary sources for LIGO and Virgo, and will be detected in much large numbers by next generation gravitational-wave detectors. Hence, continued and coordinated efforts dedicated to understanding massive stars' deaths and the link between pre-SN progenitors and properties of SN explosions, using multiple messengers, undoubtedly represent an exciting path forward.\n\n\\begin{acknowledgements}\n\\small\nA.C. and A.B. acknowledge support from NASA \\textit{Swift} Guest investigator programs (Cycles 16 and 17 via grants \\#80NSSC20K1482 and \\#80NSSC22K0203). S.A. gratefully acknowledges support from the National Science Foundation GROWTH PIRE grant No. 1545949. S.Y. has been supported by the research project grant ``Understanding the Dynamic Universe'' funded by the Knut and Alice Wallenberg Foundation under Dnr KAW 2018.0067, and the G.R.E.A.T research environment, funded by {\\em Vetenskapsr\\aa det}, the Swedish Research Council, project number 2016-06012.\nBased on observations obtained with the Samuel Oschin Telescope 48-inch and the 60-inch Telescope at the Palomar\nObservatory as part of the Zwicky Transient Facility project. ZTF is supported by the National Science Foundation under Grants\nNo. AST-1440341 and No. AST-2034437, and a collaboration including Caltech, IPAC, the Weizmann Institute for Science, the Oskar Klein Center at\nStockholm University, the University of Maryland, Deutsches Elektronen-Synchrotron and Humboldt University, Los Alamos\nNational Laboratories, the TANGO\nConsortium of Taiwan, the University of Wisconsin at Milwaukee, Trinity College Dublin, Lawrence Berkeley\nNational Laboratories, Lawrence Livermore National\nLaboratories, and IN2P3, France. Operations are conducted by COO, IPAC, and UW. The SED Machine is based upon work supported by the National Science Foundation under Grant No. 1106171. The ZTF forced-photometry service was funded under the Heising-Simons Foundation grant \\#12540303 (PI: Graham).\nThe National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.\nThe Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation Grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation. Based in part on observations made with the Nordic Optical Telescope, owned in collaboration by the University of Turku and Aarhus University, and operated jointly by Aarhus University, the University of Turku and the University of Oslo, representing Denmark, Finland and Norway, the University of Iceland and Stockholm University at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias.\n\\end{acknowledgements}\n\n\\bibliographystyle{aasjournal}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMelanoma is the most dangerous type of skin cancer which cause almost 60,000 deaths annually. In order to improve the efficiency, effectiveness, and accuracy of melanoma diagnosis, International Skin Imaging Collaboration (ISIC) provides over 2,000 dermoscopic images of various skin problems for lesion segmentation, disease classification and other relative research. \n\nLesion segmentation aims to extract the lesion segmentation boundaries from dermoscopic images to assist exports in diagnosis. In recent years, U-Net, FCN and other deep learning methods are widely used for medical image segmentation. However, the existing algorithms have the restriction in lesion segmentation because of various appearance of lesion caused by the diversity of persons and different collection environment. FCN, U-Net and other one-stage segmentation methods are sensitive to the size of the lesion. Too large or too small size of lesion decreasing the accuracy of these one-stage methods. Two-stage method could reduce the negetive influence of diverse size of lesion. Mask R-CNN can be viewed as a two-stage method which has an outstanding performance in COCO. However, Mask R-CNN still has some brawbacks in lesion segmentation. Unlike clear boundary between different objects in COCO data, the boundary between lesion and healthy skin is vague in this challenge. The vague boundary reduces the accuracy of RPN part in Mask R-CNN which may cause the negative influence in following segmentation part. Furthermore, the low resolution of input image in the segmentation of Mask R-CNN also reduce the accuracy of segmentation.\n\nIn this report, we propose a method for lesion segmentation. Our method is a two-stage process including detection and segmentation. Detection part is used to detect the location of the lesion and crop the lesion from images. Following the detection, segmentation part segments the cropped image and predict the region of lesion. Furthermore, we also propose an optimised processed for cropping image. Instead of cropping the image by bounding box exactly, in training, we crop the image with a random expansion and contraction to increase robustness of neural networks. Finally, image augmentation and other ensemble methods are used in our method. Our method is based on deep convolutional networks and trained by the dataset provided by ISIC  \\cite{Tschandl2018_HAM10000} \\cite{DBLP:journals\/corr\/abs-1710-05006}. \n\n\\section{Materials and Methods}\n\n\\begin{figure*}\n \\centering\n  \\includegraphics[height=6cm,width=0.75\\textwidth]{process.png}\n  \\caption{Process}\n  \\label{fig:Process}\n\\end{figure*}\n\n\\subsection{Database and Metrics}\nFor Lesion Segmentation, ISIC 2018 provides 2594 images with corresponding ground truth for training. There are 100 images for validation and 1000 images for testing without ground truth. The number of testing images in this year is 400 more than which in 2017. The size and aspect ratio of images are various. The lesion in images has different appearances and appear in different parts of people. There are three criterions for ground truth which are a fully-automated algorithm, a semi-automated flood-fill algorithm and manual polygon tracing. The ground truth labelled under different criterions has different shape of boundary which is a challenge in this task. We split the whole training set into two sets with ratio 10:1.\n\nThe Evaluation of this challenge is Jaccard index which is shown in Equation \\ref{eq:jaccard}. In 2018, the organiser adds a penalty when the Jaccard index is below 0.65.\n\\begin{equation}\nJ(A,B) = \n\\begin{cases}\n\\frac{|A \\bigcap B|}{|A \\bigcup B|}  & J(A,B) \\geq 0.65 \\\\\n0 & \\text{otherwise}\\\\\n\\end{cases}\n\\label{eq:jaccard}\n\\end{equation}\n\n\n\n\n\\subsection{Methods}\n\nWe design a two-stage process combining detection and segmentation. Figure \\ref{fig:Process} shows our process. Firstly, the detection part detects the location of the lesion with the highest probability. According to the bounding box, the lesion is cropped from the original image and the cropped image is normalised to 512*512 which is the size of the input image of segmentation part. A fine segmentation boundray of the cropped image is provided by the segmentation part.\n\n\\subsubsection{Detection Part}\nIn the detection process, we use the detection part in MaskR-CNN \\cite{DBLP:journals\/corr\/HeGDG17}. We also use the segmentation branch of Mask R-CNN to supervise the training of neural networks.\n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[width=0.4\\textwidth]{detection.png}\n  \\caption{Detection}\n  \\label{fig:Detection}\n\\end{figure}\n\n\\subsubsection{Segmentation Part}\nIn the segmentation part, we design an encode-decode architecture of network inspired by DeepLab \\cite{DBLP:journals\/corr\/ChenPK0Y16}, PSPNet \\cite{DBLP:journals\/corr\/ZhaoSQWJ16}, DenseASPP \\cite{Yang_2018_CVPR} and Context Contrasted Local \\cite{Ding_2018_CVPR}. Our architecture is shown in Figure \\ref{fig:Segmentation}. The features are extracted by extended ResNet 101 with three cascading blocks. After ResNet, a modified ASPP is used to compose various scale features. We also use a skip connection to transfer detailed information.\n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{segmentation.png}\n  \\caption{Segmentation}\n  \\label{fig:Segmentation}\n\\end{figure}\n\nFigure \\ref{fig:ASPP} shows the structure of the modified ASPP block. A 1x1 convolutional layer is used to compose the feature extracted by ResNet and reduce the number of feature maps. After that, we use a modified ASPP block to extract information in various scales. The modified ASPP has three parts which are dense ASPP, standard convolution layers and pooling layers. Dense ASPP is proposed by \\cite{Yang_2018_CVPR} which reduce the influence of margin in ASPP. Considering the vague boundary and low contrast appearance of the lesion, we add standard convolution layers to enhance the ability of neural networks in distinguishing the boundary. The aim of pooling layers is to let the networks consider the surrounding area of the low contrast lesion. The modified ASPP includes three dilated convolutions with rate 3, 6, 12 respectively, three standard convolution layers with size 3, 5, 7 respectively and four pooling layers with size 5, 9, 13, 17 respectively.\n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[height=13cm,width=0.4\\textwidth]{ASPP.png}\n  \\caption{Modified ASPP}\n  \\label{fig:ASPP}\n\\end{figure}\n\n\n\n\\subsection{Pre-processing}\nInstead of only using RBG channels, we combine the SV channels in Hue-Saturation-Value colour space and lab channels in CIELAB space with the RGB channels. These 8 channels are the input of segmentation part. Figure \\ref{fig:hsvlab} shows different channels of HSV and CIELAB colour space.\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[width=0.4\\textwidth]{hsvlab.png}\n  \\caption{Single channel in HSV and CIELAB colour space}\n  \\label{fig:hsvlab}\n\\end{figure}\n\n\\subsection{Post-processing}\nThe ensemble is used as the post-processing to increase the performance of segmentation. The input image of the segmentation part is rotated 90, 180 degrees and flipped to generate the other three images. Each image has a result predicted by the segmentation part. The results of the rotated and flipped image need to rotate and flip back to the original image. The final mask is the average of four results of these images.\n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{ensemable.png}\n  \\caption{Ensamble}\n  \\label{fig:Ensamble}\n\\end{figure}\n\n\\subsection{Training}\nImage augmentation is used in training of detection and segmentation. Rotation, colour jitter, flip, crop and shear is operated in each image. Each channel of images is scaled to 0 to 1. In segmentation part the size of input images is set to 512x512. Some examples are shown in Figure \\ref{fig:aug1}. \n\n\\begin{figure}[H]\n\\centering\n\\begin{subfigure}{.25\\textwidth}\n  \\centering\n  \\includegraphics[width=0.4\\linewidth]{fliplr.png}\n  \\caption{Flip Left to Right}\n  \\label{fig:aug1-1}\n\\end{subfigure}%\n\\begin{subfigure}{.25\\textwidth}\n  \\centering\n  \\includegraphics[width=0.4\\linewidth]{rotate45.png}\n  \\caption{Roatate 45$^\\circ$}\n  \\label{fig:aug1-2}\n\\end{subfigure}\n\\caption{Examples for image augmentation}\n\\label{fig:aug1}\n\\end{figure}\n\nWe use the Adam optimiser and set the learning rate to 0.001. The learning rate will be set at 92\\% of the previous after each epoch. The batch size is 8. We early stop the training when the net start overfitting. We use the dice loss which is shown in Equation \\ref{eq:diceloss}, where $p_{i,j}$ is the prediction in pixel $(i,j)$ and $g_{i,j}$ is the ground truth in pixel $(i,j)$. \n\n\\begin{equation}\n\\label{eq:diceloss}\nL =  -\\frac{\\sum_{i,j}(p_{i,j} g_{i,j})}{\\sum_{i,j}p_{i,j} + \\sum_{i,j}g_{i,j}-\\sum_{i,j}(p_{i,j} g_{i,j})}\n\\end{equation}\n\nThe input image of the segmentation part is crop randomly in a range near the bounding box predicted by detection part. In order to improve the diversity of input images and provide more information about background context. In training, the input image of the segmentation part is cropped from 81\\% to 121\\% of the bounding box randomly which is shown in Figure \\ref{fig:crop}.\n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[width=0.3\\textwidth]{crop.png}\n  \\caption{Crop image}\n  \\label{fig:crop}\n\\end{figure}\n\n\\begin{figure*}\n \\centering\n  \\includegraphics[width=0.9\\textwidth]{results.png}\n  \\caption{Predicted masks of different segmentation methods}\n  \\label{fig:result}\n\\end{figure*}\n\n\\subsection{Implementation}\nThe detection part of our method is implemented by using Pytorch 0.4 in Ubuntu 14.04. The framework is from \\url{https:\/\/github.com\/roytseng-tw\/Detectron.pytorch}. The segmentation part is implemented by using Pytorch 0.3.1 in Ubuntu 14.04. The neural networks are trained by two Nvidia 1080 Ti with 60 GB of RAM.\n\n\\section{Results}\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{llll}\n\\hline\nMethod & Jaccard & Jaccard(>0.65)\\\\\n\\hline\nMask R-CNN & 0.825 & 0.787\\\\\nOne-stage Segmentation & 0.820 & 0.783\\\\\nTwo-stage Segmentation & 0.846 & 0.816\\\\\n\\hline\n\\end{tabular}\n\\label{table:results}\n\\caption{Evaluation metrics of different segmentation methods}\n\\end{table}\n\n\nThe evaluation metrics on 257 images of our two-stage segmentation method with MaskR-CNN and our one-stage segmentation method is shown in Table 1. The thresholded Jaccard of our two-stage method on the official testing set in 0.802. Figure \\ref{fig:result} shows the outputs of different segmentation methods. Compared with other methods, The results of our two-stage method has a better location and more smooth edge.\n\n\n\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:I}\n\nNeutron stars, which are produced through core-collapse supernovae, are one of the most suitable environments for probing physics under extreme conditions, which are quite difficult to realize on the Earth. The density inside the star significantly exceeds the standard nuclear density, while the gravitational and magnetic fields become much stronger than those observed in our solar system. So, by observing the neutron star itself and\/or the phenomena associated with neutron stars, inversely one would be able to extract the information for the extreme conditions. For example, the discovery of the $2M_\\odot$ neutron stars excludes some of the soft equations of state (EOSs) \\cite{D10, A13, C20}. Namely, if the maximum mass of the neutron star constructed with an EOS does not reach the observed mass, such an EOS is not good for the EOS describing neutron stars. The light bending due to the strong gravitational field, which is one of the relativistic effects, may also tell us the neutron star properties. That is, owing to the strong gravitational field induced by the neutron star, the light curve from the neutron star would be modulated. Thus, by carefully observing the pulsar light curve, one could mainly constrain the stellar compactness, i.e., the ratio of the stellar mass to the radius (e.g., \\cite{PFC83, LL95, PG03, PO14, SM18, Sotani20a}). In practice, the mass and radius of the neutron stars, i.e., PSR J0030+0451 \\cite{Riley19, Miller19} and PSR J0740+6620 \\cite{Riley21, Miller21}, are successfully constrained through the observations with the Neutron star Interior Composition Explorer (NICER) on the International Space Station. The current uncertainty in the mass and radius constraints is still large, but this type of constraint must help us to know the true EOS when the uncertainty is reduced through future observations. \n\n\nIn addition to the direct observations of the stellar mass and radius, the oscillation frequency from the neutron star is another important observable. Since the oscillation frequency strongly depends on the interior properties of the object, as an inverse problem, one can extract the stellar properties by observing the frequency. This technique is known as asteroseismology, which is similar to seismology on the Earth and helioseismology on the Sun. In fact, by identifying the quasi-periodic oscillations observed in the afterglow following the giant flares with the neutron star crustal oscillations, one could constrain the crust properties (e.g., Refs. \\cite{GNHL2011, SNIO2012, SIO2016}). Similarly, one may know the neutron star mass, radius, and EOS by observing the gravitational wave frequencies from neutron stars and by identifying them with specific oscillation modes  (e.g., Refs. \\cite{AK1996,AK1998,STM2001,SH2003,SYMT2011,PA2012,DGKK2013,Sotani20b,Sotani20c,Sotani21,SD22}). Furthermore, this technique is recently adopted for understanding the gravitational wave signals appearing in the numerical simulations for core-collapse supernova (e.g., Refs. \\cite{FMP2003,FKAO2015,ST2016,SKTK2017,MRBV2018,SKTK2019,TCPOF19,SS2019,ST2020,STT2021}).\n\n\nIn general, the gravitational wave frequencies from the neutron stars are complex numbers, where the real and imaginary parts respectively correspond to an oscillation frequency and damping rate, and the corresponding eigenmodes are called quasi-normal modes. So, to determine the quasi-normal modes of the neutron stars, one has to somehow solve the eigenvalue problem with respect to the eigenvalue in two-dimensional parameter space with the real and imaginary parts. Since this solution may be a bother, one sometimes adopts the approximation to estimate the frequency of gravitational waves. The simple approximation is the relativistic Cowling approximation, where the metric perturbations are neglected during the fluid oscillations, i.e., the frequencies of fluid oscillations are determined with the fixed metric. One can qualitatively discuss the behavior of frequencies with the Cowling approximation, even though the accuracy of the determined frequencies is within $\\sim 20\\%$ \\cite{ST2020, YK97}.  Another approximation adopted so far is the zero-damping approximation, where one takes into account the metric perturbations as well as the fluid perturbations, but the imaginary part of the eigenvalue is assumed to be zero. This is because the damping rate, i.e, the imaginary part of the eigenvalue, for the gravitational waves induced by the fluid oscillations is generally much smaller than the oscillation frequency, i.e., the real part of the eigenvalue. This approximation is considered to estimate the frequency of the gravitational waves well, but it is not discussed how well this approximation works. So, in this study, we first discuss how well the zero-damping approximation works by comparing the frequencies determined with the approximation to those determined through the proper eigenvalue problem. Anyway, with either the Cowling approximation or the zero-damping approximation, one can discuss only the frequency of the gravitational waves induced by the fluid oscillations, such as the fundamental, pressure, and gravity modes, but one cannot discuss the gravitational waves associated with the oscillations of the spacetime itself, i.e., the $w$-modes. \n\n\nIn order to estimate the $w_1$-mode frequency, in this study, we find the empirical relation for the ratio of the damping rate of the $w_1$-mode gravitational wave, i.e., the imaginary part of the quasi-normal mode, to its frequency, i.e., the corresponding real part, as a function of the stellar compactness almost independently of the adopted EOSs. With this empirical relation, we newly propose a one-dimensional approximation for estimating the $w_1$-mode frequency and show how well this approximation works. Unless otherwise mentioned, we adopt geometric units in the following, $c=G=1$, where $c$ and $G$ denote the speed of light and the gravitational constant, respectively, and the metric signature is $(-,+,+,+)$.\n\n\n\\section{Neutron star models}\n\\label{sec:EOS}\n\nIn this study, we simply consider the static, spherically symmetric stars, where the metric describing the system is given by \n\\begin{equation} \n  ds^2 = -e^{2\\Phi}dt^2 + e^{2\\Lambda}dr^2 + r^2\\left(d\\theta^2 + \\sin^2\\theta d\\phi^2\\right).  \\label{eq:metric}\n\\end{equation}\nThe metric functions $\\Phi$ and $\\Lambda$ in Eq. (\\ref{eq:metric}) are functions only of $r$, while $e^{2\\Lambda}$ is directly related to the mass function, $m(r)$, as $e^{-2\\Lambda}=1-2m\/r$. The stellar structure is determined by integrating the Tolman-Oppenheimer-Volkoff equation with an appropriate EOS for neutron star matter. In this study, we adopt the same EOSs as in Refs. \\cite{Sotani20c, Sotani21}, which are listed in Table \\ref{tab:EOS} together with the EOS parameters and maximum mass of the neutron star. Here, $K_0$ and $L$ are the incompressibility for symmetric nuclear matter and the density dependence of the nuclear symmetry energy. We note that any EOSs can be characterized by these nuclear saturation parameters, which are constrained via terrestrial nuclear experiments, such as $K_0= 230\\pm 40$ MeV \\cite{KM13} and $L\\simeq 58.9\\pm 16$ MeV \\cite{Li19}. Comparing these fiducial values of $K_0$ and $L$ to those for the adopted EOSs, some of the EOSs listed in Table \\ref{tab:EOS} seem to be excluded, but we consider even such EOSs in this study to examine the EOS dependence in the wide parameter space. Additionally, $\\eta$ in Table \\ref{tab:EOS} is the specific combination of $K_0$ and $L$ as $\\eta\\equiv \\left(K_0 L^2\\right)^{1\/3}$ \\cite{SIOO14}, which is a suitable parameter not only for expressing the properties of low-mass neutron stars but also for discussing the maximum mass of neutron stars \\cite{SSB16,Sotani17,SK17}. The mass and radius relations for the neutron star models constructed with the EOS listed in Table \\ref{tab:EOS} are shown in Fig. \\ref{fig:MR}, where the filled and open marks respectively correspond to the EOSs constructed with the relativistic framework and with the Skyrme-type effective interaction, while the double-square is the EOS constructed with a variational method. As in Ref. \\cite{Sotani20c}, we consider the stellar models denoted with marks in this figure. \n\n\n\\begin{table}\n\\caption{EOS parameters adopted in this study, $K_0$, $L$, and $\\eta$, and the maximum mass, $M_{\\rm max}$, of the neutron star constructed with each EOS.} \n\\label{tab:EOS}\n\\begin {center}\n\\begin{tabular}{ccccc}\n\\hline\\hline\nEOS & $K_0$ (MeV) & $L$ (MeV) & $\\eta$ (MeV) & $M_{\\rm max}\/M_\\odot$   \\\\\n\\hline\nDD2\n & 243 & 55.0  & 90.2  & 2.41  \\\\ \nMiyatsu\n & 274 &  77.1 & 118  & 1.95  \\\\\nShen\n\\ & 281 & 111  & 151  &  2.17  \\\\  \nFPS\n & 261 & 34.9 & 68.2  & 1.80  \\\\  \nSKa\n & 263 & 74.6 & 114 & 2.22  \\\\ \nSLy4\n & 230 & 45.9 &  78.5 & 2.05  \\\\ \nSLy9\n & 230 & 54.9 &  88.4 & 2.16  \\\\  \nTogashi\n & 245  & 38.7  & 71.6 & 2.21  \\\\ \n\\hline \\hline\n\\end{tabular}\n\\end {center}\n\\end{table}\n\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[scale=0.5]{MR1}\n\\vspace{0.5cm}\n\\caption\nFor the neutron star models constructed with various EOSs, the mass is shown as a function of the radius. Figure is taken from Ref. \\cite{Sotani20c}.\n\n\\label{fig:MR}\n\\end{figure}\n\n\n\n\n\\section{Quasi-normal modes}\n\\label{sec:QNMs}\n\nIn order to determine the quasi-normal modes of gravitational waves from compact objects, one has to solve the eigenvalue problem. The perturbation equations are derived from the linearized Einstein equation by adding the metric and fluid perturbations on the background stellar models. By imposing the appropriate boundary conditions at the stellar center, surface, and spacial infinity, the problem to solve becomes the eigenvalue problem with respect to the eigenvalue, $\\omega$. As a practical matter, how to numerically deal with the boundary condition at the spatial infinity may become a problem. In this study, we especially adopt the continued fraction method, the so-called Leaver's method \\cite{Leaver85,Leaver90}. That is, the perturbation variable outside the star is expressed as a power series around the stellar surface, which also satisfies the boundary condition at the infinity. By substituting this expansion into the Regge-Wheeler equation, one can get a three-term recurrence relation including $\\omega$, which is rewritten in the form of a continued fraction. So, one has to find the eigenvalue, $\\omega$, which satisfies the continued fraction. Here, we symbolically express this condition as $f(\\omega)=0$. The resulatant $\\omega$ is generally a complex value, where the real and imaginary parts correspond to the oscillation frequency,  Re($\\omega$)\/$2\\pi$, and damping rate of the corresponding gravitational waves. The concrete perturbation equations, boundary conditions, and functional form of $f(\\omega)$ are shown in Refs. \\cite{STM2001,ST2020}. \n\nIn practice, since $f(\\omega)$ is also a complex value, we try to find the value of $\\omega$ numerically, with which the absolute value of $f(\\omega)$ becomes the local minimum. In this study, we especially focus on the fundamental ($f$-), 1st pressure ($p_1$-), and 1st spacetime ($w_1$-) modes for the various neutron star models constructed with the EOSs listed in Table \\ref{tab:EOS}. We note that the $f$- and $p_1$-modes are the quasi-normal modes induced by the fluid oscillations, while the $w_1$-mode is the quasi-normal mode associated with the oscillations of spacetime itself. \n\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[scale=0.5]{ratio-ffDf}\n\\vspace{0.5cm}\n\\caption\nIn the top panel, the ratio of Im($\\omega$) to Re($\\omega$) for the $f$-mode is shown as a function of the stellar compactness for various neutron star models, where the solid line denotes the fitting formula given by Eq. (\\ref{eq:fitting_ff}). In the bottom pane, the relative deviation of the value of Im($\\omega$)\/Re($\\omega$) estimated with the empirical formula from that calculated via the eigenvalue problem is shown. \n\n\\label{fig:ratio-ff}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[scale=0.5]{ratio-p1}\n\\vspace{0.5cm}\n\\caption\nThe ratio of Im($\\omega$) to Re($\\omega$) for the $p_1$-mode is shown as a function of the stellar compactness for various neutron star models.\n\n\\label{fig:ratio-p1}\n\\end{figure}\n\nWith the resultant $f$-mode frequency, we show the ratio of Im$(\\omega)$ to Re$(\\omega)$ in the top panel of Fig. \\ref{fig:ratio-ff}. From this figure, one can observe that the values of Im$(\\omega)$\/Re$(\\omega)$ are well expressed as a function of the stellar compactness, $M\/R$, almost independently of the EOS. In fact, we can derive the empirical relation for Im$(\\omega)$\/Re$(\\omega)\\, \\raisebox{-0.8ex}{$\\stackrel{\\textstyle >}{\\sim}$ } 10^{-5}$, i.e., except for the quite low-mass neutron stars, given by \n\\begin{equation}\n  \\frac{{\\rm Im}(\\omega_f)}{{\\rm Re}(\\omega_f)} = \\left[0.13193 -4.4754\\left(\\frac{M}{R}\\right)\n        +290.9\\left(\\frac{M}{R}\\right)^2 -756.14 \\left(\\frac{M}{R}\\right)^3\\right]\\times10^{-4},\\label{eq:fitting_ff}\n\\end{equation}\nwith which the expected values are shown with the thick-solid line in the top panel. In the bottom panel of Fig. \\ref{fig:ratio-ff}, we also show the relative deviation calculated with\n\\begin{equation}\n  \\Delta = \\frac{{\\rm abs}[{\\cal R}-{\\cal R}_{\\rm fit}]}{{\\cal R}}, \\label{eq:Delta}\n\\end{equation}\nwhere ${\\cal R}$ and ${\\cal R_{\\rm fit}}$ denote the ratio of Im$(\\omega)$ to Re$(\\omega)$ detemined through the eigenvalue problem and estimated with Eq. (\\ref{eq:fitting_ff}) for each stellar model, respectively. From this figure, we find that the values of Im$(\\omega)$\/Re$(\\omega)$ can be usually estimated within $\\sim10\\%$ accuracy by using the empirical relation. In a similar way, we show Im$(\\omega)$\/Re$(\\omega)$ for the $p_1$-modes in Fig. \\ref{fig:ratio-p1}. But, for the case of the $p_1$-mode we can not derive the empirical relation as a function of $M\/R$, unlike the case of the $f$-mode. \n\n\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[scale=0.5]{ReIm-MR1}\n\\vspace{0.5cm}\n\\caption\nThe ratio of Im$(\\omega)$ to Re$(\\omega)$ of the $w_1$-mode for various EOSs is shown as a function of the corresponding stellar compactness (top panel), where the solid line denotes the fitting line given by Eq. (\\ref{eq:fitting_w1}). The bottom panel denotes the relative deviation of the estimation with the fitting formula from the values of Im$(\\omega)$\/Re$(\\omega)$ calculated via the eigenvalue problem.\n\n\\label{fig:ratio}\n\\end{figure}\n\nOn the other hand, we also find that one can express the value of Im(${\\omega}$)\/Re($\\omega$) for the $w_1$-mode as a function of $M\/R$ almost independently of the adopted EOS. In the top panel of Fig. \\ref{fig:ratio}, we show the ratio of Im$(\\omega)$ to Re$(\\omega)$ for the $w_1$-mode determined through the eivenvlue problem for each stellar model as a function of the corresponding stellar compactness, where the solid line denotes the fitting formula given by\n\\begin{equation}\n  \\frac{{\\rm Im}(\\omega_{w_1})}{{\\rm Re}(\\omega_{w_1})} = 1.0659 -4.1598\\left(\\frac{M}{R}\\right)\n        +16.4565\\left(\\frac{M}{R}\\right)^2 -39.5369\\left(\\frac{M}{R}\\right)^3.  \\label{eq:fitting_w1}\n\\end{equation}\nIn the bottom panel, we show the relative deviation calculated with Eq. (\\ref{eq:Delta}) for the $w_1$-mode, where ${\\cal R}_{\\rm fit}$ is estimated with Eq. (\\ref{eq:fitting_w1}). Considering to the fact that $M\/R=0.172$ (0.207) for the canonical neutron star model with $M=1.4M_\\odot$ and $R=12$ km (10 km), one can observe that the fitting formula given by Eq. (\\ref{eq:fitting_w1}) works better for the low-mass neturon star models. \n\n\n\n\n\\section{One-dimensional approximation}\n\\label{sec:Approximation0}\n\nIn order to determine the quasi-normal modes, one has to somehow search the solution of $\\omega$ in two-dimensional parameter space with the real and imaginary parts, and this procedure may be trouble. One may sometimes adopt a suitable approximation to get out of this trouble, even if one can estimate only the frequency of quasi-normal modes. In this study, we especially consider the approximation, where the eigenvalue belongs to one-dimensional parameter space, depending only on the real part of the eigenvalue. We refer to this approximation as the one-dimensional approximation in this study. For example, since the imaginary part of the quasi-normal modes induced by the fluid oscillations is much smaller than the real part of them, as shown in Figs. \\ref{fig:ratio-ff} and \\ref{fig:ratio-p1}, one may assume that Im$(\\omega)=0$. This approximation, referred to as zero-damping approximation, is a special case of one-dimensional approximation. In fact, this approximation has been adopted in some previous studies, but it has not been discussed how well this approximation works. So, in the next subsections, we will see the accuracy of the zero-damping approximation for the $f$- and $p_1$-mode frequencies, and then we also propose the one-dimensional approximation for estimating the $w_1$-mode frequency.\n\n\n\\subsection{Zero-damping approximation}\n\\label{sec:Approximation1}\n\nThe zero-damping approximation, neglecting the imaginary part of the eigenvalue, is the simplest one-dimensional approximation, i.e., the eigenvalue, $\\omega$, is assumed to be \n\\begin{equation}\n  \\omega_{{\\rm 1D},f} = \\omega_{r}, \\label{eq:omega_1D0}\n\\end{equation}\nwhere $\\omega_r$ is some real number and $\\omega_{{\\rm 1D},f}$ denotes the eigenvalue with the approximation for the gravitational wave induced by the fluid oscillations. With the zero-damping approximation, one can estimate the frequency, with which the absolute value of $f(\\omega_{{\\rm 1D},f})$ becomes the local minimum. Once the value of $\\omega_r$ would be determined, the frequency of a gravitational wave is given by $\\omega_r\/2\\pi$. As an example, we show the absolute value of $f(\\omega_{{\\rm 1D},f})$ as a function of the frequency for the neutron star model with $1.46M_\\odot$ constructed with SLy4 EOS in Fig. \\ref{fig:SLy4-13}, where the vertical dashed lines denote the frequencies of the $f$- and $p_1$-modes determined through the proper eigenvalue problem without the approximation and the inserted panel is an enlarged drawing in the vicinity of the $p_1$-mode frequency. \n\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[scale=0.5]{SLy4-13fp1}\n\\vspace{0.5cm}\n\\caption\nFor the neutron star model with $1.46M_\\odot$ constructed with SLy4 EOS, we show the absolute value of $f(\\omega_{{\\rm 1D},f})$ as a function of frequency. The vertical dashed lines denote the Re$(\\omega)\/2\\pi$ for the $f$- and $p_1$-modes determined through the proper eigenvalue problem without the approximation. The panel inserted in the figure is an enlarged drawing in the vicinity of the $p_1$-mode frequency. \n\n\\label{fig:SLy4-13}\n\\end{figure}\n\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[scale=0.5]{Dfp1}\n\\vspace{0.5cm}\n\\caption\nThe relative deviation of the value of $\\omega_r$ determined with the zero-damping approximation from the value of Re$(\\omega)$ determined through the proper eigenvalue problem without the approximation, which is calculated with Eq. (\\ref{eq:DRef}), is shown as a function of $M\/R$ for various neutron star models. The top and bottom panels correspond to the results for the $f$- and $p_1$-modes, respectively. \n\n\\label{fig:Dfp1}\n\\end{figure}\n\nIn order to check the accuracy of the frequencies of the $f$- and $p_1$-modes with the zero-damping approximation, we estimate the relative deviation of them from the corresponding frequencies determined through the proper eigenvalue problem, which is calculated by\n\\begin{equation}\n  \\Delta {\\rm Re}(\\omega_i) = \\frac{{\\rm abs}[{\\rm Re}(\\omega_i)-\\omega_{r,i}]}{{\\rm Re}(\\omega_i)}, \n      \\label{eq:DRef}\n\\end{equation}\nwhere ${\\rm Re}(\\omega_i)$ and $\\omega_{r,i}$ respectively denote the real part of $\\omega$ determined through the proper eivenvalue problem without the approximation and the value of $\\omega_r$ determined with the zero-damping approximation for the $f$- ($i=f$) and $p_1$-modes ($i=p_1$). For various neutron star models, we show the values of $\\Delta {\\rm Re}(\\omega_i)$ for the $f$-mode ($p_1$-mode) in the top (bottom) panel of Fig. \\ref{fig:Dfp1}. From this figure, one can observe that the zero-damping approximation works significantly well.\n\n\n\n\n\\subsection{One-dimensional approximation for the $w_1$-mode}\n\\label{sec:Approximation2}\n\nUnlike the gravitational waves induced by fluid oscillations, such as the $f$- and $p_1$-modes, the imaginary part of the spacetime modes ($w$-modes) becomes generally comparable to the real part of them, as shown in Fig. \\ref{fig:ratio}. So, one cannot estimate the $w_1$-mode frequency with the zero-damping approximation. However, owing to the finding of the emprical relation for Im($\\omega$)\/Re($\\omega$) as shown in Fig. \\ref{fig:ratio}, we propose the one-dimensional approximation for the $w_1$-mode, i.e., the eigenvalue with the one-dimensional approximation is assumed to be\n\\begin{equation}\n   \\omega_{{\\rm 1D},w}=\\omega_r\\left(1+{\\rm i}{\\cal R}_{\\rm fit}\\right), \\label{eq:omega_1D1}\n\\end{equation}\nwhere $\\omega_r$ is some real value, while ${\\cal R}_{\\rm fit}$ denotes the ratio of Im$(\\omega_{w_1})$ to Re$(\\omega_{w_1})$ estimated with Eq. (\\ref{eq:fitting_w1}). Then, one should find the suitable value of $\\omega_r$, with which the absolute value of $f(\\omega_{{\\rm 1D},w})$ becomes local minimum. As an example, in Fig. \\ref{fig:SLy4-08} we show the absolute value of $f(\\omega_{{\\rm 1D},w})$ as a function of the frequency given by $\\omega_r\/2\\pi$ for the neutron star model with $1.46M_\\odot$ constructed with SLy4 EOS. In this figure, for reference we also show the $w_1$-mode frequency determined through the proper eigenvalue problem with the dashed vartical line. \n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[scale=0.5]{SLy4-08w1}\n\\vspace{0.5cm}\n\\caption\nThe absolute value of $f(\\omega_{{\\rm 1D},w})$ is shown as a function of the frequency for the neutron star model with $1.46M_\\odot$ constructed with SLy4 EOS, where the vertical dashed line denotes the $w_1$-mode frequency determined though the proper eigenvalue problem without the approximation. \n\n\\label{fig:SLy4-08}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[scale=0.5]{Dw1Im}\n\\end{center}\n\\vspace{0.5cm}\n\\caption\nThe relative deviation of the value of $\\omega_r$ determined with the one-dimensional approximation from that of Re$(\\omega)$ determined through the proper eigenvalue problem is shown as a function of stellar compactness for various neutron star models in the top panel. In the bottom panel, we show the relative deviation of the imaginary part of the eigenvalue estimated with the one-dimensional approximation from that of Im$(\\omega)$ determined through the proper eigenvalue problem, calculated with Eq. (\\ref{eq:DIm})\n\n\\label{fig:Dw1}\n\\end{figure}\n\nTo check the accuracy of the one-dimensional approximation, in the top panel of Fig. \\ref{fig:Dw1} we show the relative deviation of the value of $\\omega_r$ determined with the one-dimensional approximation from that of Re$(\\omega)$ determined through the proper eigenvalue problem, calculated with Eq. (\\ref{eq:DRef}) for the $w_1$-mode, as a function of $M\/R$ for various neutron star models. In the bottom panel of Fig. \\ref{fig:Dw1}, we also show the relative deviation, $\\Delta {\\rm Im}(\\omega_{w_1})$, of the imaginary part of the eigenvalue estimated with the one-dimensional approximation, i.e., $\\omega_r{\\cal R}_{\\rm fit}$, from Im$(\\omega)$ determined through the proper eigenvalue problem, which is calculated with\n\\begin{equation}\n  \\Delta {\\rm Im}(\\omega_{w_1}) \n    = \\frac{{\\rm abs}[{\\rm Im}(\\omega_{w_1})- \\omega_r{\\cal R}_{\\rm fit}]}{{\\rm Im}(\\omega_{w_1})}.\n    \\label{eq:DIm}\n\\end{equation}\nFrom this figure, one can observe that the $w_1$-mode frequency can be estimated with the one-dimensional approximation within $\\sim 1\\%$ accuracy independently of the adopted EOSs. On the other hand, the damping rate of the $w_1$-mode can be estimated $\\sim 30\\%$ accuracy, which seems to strongly depend on the accuracy of the empirical relation for Im$(\\omega_{w_1})$\/Re$(\\omega_{w_1})$ given by Eq. (\\ref{eq:fitting_w1}).\n\n\n\n\n\\section{Conclusion}\n\\label{sec:Conclusion}\n\nQuasi-normal modes are one of the important properties characterizing compact objects. In this study, first, we show that the ratio of the imaginary part to the real part of the quasi-normal mode for the $f$- and $w_1$-modes can be expressed as a function of the stellar compactness almost independently of the adopted EOSs, and we derive the corresponding empirical relations. Then, focusing on the $f$- and $p_1$-modes, which are gravitational wave frequencies induced by the stellar fluid oscillations, we examine the accuracy of the zero-damping approximation to estimate the corresponding frequencies of gravitational waves, where the damping rate, i.e., the imaginary part of the quasi-normal mode, is neglected. As a result, we show that one can estimate the frequencies of the $f$- and $p_1$-mode with the zero-damping approximation with considerable accuracy. In addition, we newly propose the one-dimensional approximation for estimating the $w_1$-mode frequency by adopting the empirical relation (that we find in this study) for the ratio of the imaginary part to the real part of the $w_1$-modes, and show that this approximation works well, where one can estimate the frequency within $\\sim 1\\%$ accuracy. \n\n\n\n\n\\bmhead{Acknowledgments}\nThis work is supported in part by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Numbers \nJP19KK0354, \nJP20H04753,  and\nJP21H01088, \nand by Pioneering Program of RIKEN for Evolution of Matter in the Universe (r-EMU).\n\n\n\n\\section*{Declarations}\n\nThe author has no relevant financial or non-financial interests to disclose.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\setcounter{equation}{0} \\ALTsect{\\setcounter{equation}{0} \\ALTsect}\n\\def{\\arabic{section}}.{\\arabic{equation}}{{\\arabic{section}}.{\\arabic{equation}}}\n\\begin{document}\n\\sloppy\n\\begin{center}\n{\\Large \\bf\nBody Fixed Frame, Rigid Gauge Rotations \\\\\nand  Large N Random Fields in QCD}\\\\\n\\vskip .5cm\n{\\large{\\bf Shimon Levit}\\\\}\n\\vskip .5cm\n{\\it Department of Physics}\\\\\n{\\it Weizmann Institute of Science}\\\\\n{\\it Rehovot 76100 Israel\\ \\footnote{ Permanent address. Supported in\npart by\nUS -- Israel Binational Science Foundation grant no. 89--00393}}\\\\\n{\\it and}\\\\\n{\\it Max-Planck-Institut f\\\"ur Kernphysik}\\\\\n{\\it D-6900 Heidelberg, Germany\\ \\footnote{ Supported by Humboldt Award.}}\\\\\n\\end{center}\n\\vskip .5cm\n\\centerline{\\Large{\\bf Abstract}}\n\n\n The \"body fixed frame\" with respect to  local gauge transformations is\n introduced.  Rigid gauge \"rotations\" in QCD and their Schr\\\"odinger  equation\n are studied for static and dynamic quarks.\nPossible choices of the rigid gauge field configuration\n corresponding to a nonvanishing\nstatic colormagnetic field in the \"body fixed\" frame are discussed.\nA gauge invariant variational equation is derived in this frame.\nFor large number  N of colors\n the rigid gauge field configuration is regarded as random\nwith maximally random probability distribution under constraints on\nmacroscopic--like quantities. For the uniform magnetic field\nthe joint probability distribution of the field components\nis determined  by maximizing the\nappropriate entropy under the area law constraint for the Wilson loop.\nIn the quark sector the gauge invariance requires\nthe  rigid gauge field configuration to appear\nnot only as a background  but  also  as inducing\nan instantaneous quark-quark interaction.  Both are random in the large N\nlimit.\n\\vskip .5cm\n\\setcounter{equation}{0} \\ALTsect{Introduction}\n\nStudies of  non perturbative aspects of dynamics\nof non abelian gauge fields will continue to remain one of\nthe focuses of  theoretical activities. These fields appear\nat all levels of the \"elementary\" interactions and even begin to\nenter at a more phenomenological macroscopic level in condensed matter\nsystems. Quantum Chromodynamics represents a prime example of a\nstrongly coupled theory with non abelian gauge fields.  Despite many\nefforts, e.g. instantons \\cite{ins}, large N expansion\n\\cite{lar,fra,wit,mgm,egk,ind},\nlattice gauge theory and  strong coupling\nexpansion \\cite{lat,str}, topological considerations \\cite{pol,top},\n QCD sum rules \\cite{svz}, \"spaghetti\" vacuum \\cite{cop},  light cone\napproach \\cite{bro}, explicit color projection \\cite{jon}, and others\n\\cite{adl,yaf,dua}, the quantitative understanding\nof the basic QCD features is still far from satisfactory. A sustained\neffort with different angles of attack is clearly in order with the hope\nthat accumulated qualitative experience will finally lead to the\ndevelopment of quantitative calculational tools. This paper is a\ncontribution to this effort.\n\n Invariance under local gauge transformations is the most important\nfeature of a non abelian gauge theory. In the framework of the Hamiltonian\nformulation of QCD I wish to explore the consequences of this\ninvariance using some\ngeneral methods common to molecular and nuclear physics. I wish to\ndefine an appropriate generalization of the body-fixed\n (intrinsic, rotating) frame formalism in the context of\nthe local gauge transformations. After doing so one can attempt to\nseparately investigate the\ndynamics of the gauge \"rotations\" of the frame and the intrinsic frame\ndynamics. These would be the analogs of rigid rotations and\nintrinsic vibrations in molecular and nuclear physics. Most of my\ninterest in this paper will concentrate on the study of the \"rigid\ngauge rotations\".\nThe study of the couplings between the \"rotations\" and\nthe \"vibrations\" of the gauge field is deferred to future work.\nTo avoid misunderstanding I wish to stress that by \"gauge rotations\" or\n\"rotations of the gauge field\"\nin this paper I will always mean Eq.(\\ref{rgd}) below, which  includes\nthe proper SU(N) rotation  as well as the inhomogeneous \"shift\" term.\n\nPerhaps the most important conceptual advantage of using the body-fixed\nframe associated with a given symmetry\nlies in the fact that one can freely approximate the dynamics in this\nframe without fears to violate the symmetry.\nIn particular the use of this  formalism appears to be\nfruitful provided there exist such a body-fixed,  intrinsic\nframe in which the \"rotational - vibrational\" coupling can be considered\nas small. This in turn generically\nhappens when the \"rotational inertia\" is much\nlarger than the \"inertia\" associated with the intrinsic motion\nso that a variant of the Born-Oppenheimer approximation is valid.  A typical\nsituation is when the system's ground state is strongly \"deformed\" away\nfrom a symmetric state.\nBy deformation I mean absence of symmetry with respect to\n\"intrinsic transformations\",\ni.e. transformations in the body-fixed frame  and not\nwith respect to the transformations in \"laboratory\". Absence of symmetry\nin \"laboratory\" would correspond to the symmetry breakdown\nwhich can not occur for a local gauge symmetry.\nQuantum mechanical examples of deformed bodies are e.g. non spherical\nmolecules, deformed nuclei, etc.\n\nI do not have a priori arguments that the QCD vacuum\nis strongly \"deformed\" in the above sense. Appearance of various QCD\ncondensates, Ref. \\cite{svz}, suggests  that this may be true. The\ncondensate wavefunction should then play a role of the strongly deformed\nconfiguration. Another positive indication is the large N master field concept,\nRef. \\cite{wit}, according to which a special gauge field  configuration\nshould exist which dominates the vacuum wave function or the\ncorresponding functional integral. It is expected,  however, that the\nmaster  field is not\nsimply a fixed classical configuration. It should rather be regarded as\na statistical distribution allowing to calculate quantities which are\nanalogous to macroscopic\nthermodynamic quantities in statistical physics, i.e. such that their\nfluctuations are suppressed\nin the large N limit, Refs. \\cite{mat,ran}. Glimpses of the meaning of\nthese vague notions were found in various matrix models,\ncf. Refs. \\cite{mgm,egk,mat,ran},\nand, e.g. in 1+1 dimensional QCD, Ref. \\cite{bor}.  If this view point\nis correct then suitably chosen rigidly rotating \"deformed\" gauge field\ncould play a role of the master field provided one understands\nin which sense it should also be statistical.\nThe following formalism will clarify some of these issues and\nprovide a general framework in which they could be further discussed.\n\nWorks in the spirit of our study have already appeared in the\npast,cf. Refs. \\cite{lee,jac,sim,dos} and the analogy with various\ntypes of rotational motion  is frequently used in QCD.\nIn this sense the present study is a continuation of these works.\n\nThis paper is organized as following. In Section 2\nI introduce the transformation to the body-fixed frame in the context\nof  the simplest model of the gauge rotational motion  --\nthe rigid gauge rotor.             Giving a  natural definition\nof the rigid gauge \"rotations\" I  proceed to determine\nthe appropriate generalization of the standard space rigid rotor\n results --  the moment of inertia tensor,\nthe \"body-fixed\"\nframe, the generators of the \"body-fixed\" gauge group\nin terms of which the character of \"deformation\" can be classified, etc.\nDespite severe limitation on the set of the allowed gauge field\nconfigurations the model is gauge invariant.\n I work out in Section 3\nthe quantum mechanics of the  model. As with the\nspace rotor the generators of the \"laboratory\" and \"body-fixed\"\ngauge transformations provide a complete set of quantum numbers\nfor the wave functionals of the model. The vacuum has zero energy and\nis the most disordered state. Higher states correspond to the presence\nof very heavy, i.e. static quarks and antiquarks in the system.\nAs an important example I consider the wave function and the\ncorresponding Schr\\\"odinger   equation  for  a pair of static quark and antiquark.\nThis and other similar equations in the model are\nsimple matrix equation with the inverse \"moment of inertia\" determining\nthe interaction between the color sources and depending on the assumed\nrigid gauge configuration which plays the role of the \"free parameter\".\n\nIn Section 4 I discuss the meaning of the results obtained so far and\npossible choices of the rigid gauge field which physically represents\na non vanishing  colormagnetic field\nin the body-fixed  frame. In the ground state this frame\ndoes not \"rotate\" but has random orientations in local\ncolor spaces at every space point. Introduction of static quarks forces\nthe frame to \"rotate\" quantum mechanically\nat the points where the quarks are situated.  The energy eigenvalues\nof these \"rotations\" are the energies of the quantum states of the\ncolorelectric field generated by the  quarks. The propagator of\n this field\nis the moment of inertia of the model and depends explicitly on the\nassumed configuration of the rigid static colormagnetic field.\nFor zero field the propagator is a simple Coulomb while for\na uniform field diagonal in color the\npropagator  behaves asymptotically as a decaying Gaussian.\n  The so called dual Meissner effect picture of the confinement,\n Ref. \\cite{man}, could be implemented  if a configuration\nof the rigid colormagnetic field is found which \"channels\" the\ncolorelectric field and makes its propagator effectively one dimensional.\nIt turns out that the creation of such a magnetic \"wave-guide\"\nis connected with existance of a zero eigenvalue of a certain\noperator in the model.\n\nSince  the quark color degrees of freedom are treated\n quantum mechanically\nthe model allows for a possibility that confinement of fundamental\nrepresentations  does not automatically mean confinement of higher\nrepresentations.   I discuss this\npossibility and derive a variational equation for the rigid field.\nThis equation is fully gauge invariant.\n\nIn Section 5\nexpecting that  rigid gauge rotations should be relevant for the\nmaster field concept I study the model in the large N limit.\nAny candidate for the master field must be allowed to undergo free\n\"gauge rotations\" which can not be frozen by this limit\nand should induce an interaction between the quarks.\nGoing to the body fixed frame of these rotations I regard the\nrigid gauge field configuration as random and introduce a natural\nrequirement that it is least biased under  constraints that it\nshould reproduce\ngauge invariant quantities which can be regarded macroscopic-like in the\nlarge N limit. This means that it should be maximally random under these\nconstraints. In order to make these ideas explicit I discuss in some\ndetail the case of the uniform\ncolormagnetic field. Such configuration in QCD was already discussed in the\npast, Refs. \\cite{sav,cop} but it seems that its appearance in the interaction\nis a novel feature\nof the model. The detailed form of  this interaction depends on the\ndifferences of the color components of the magnetic field. It is not\nconfining for any finite number of colors. For $N \\rightarrow \\infty$\n I assume  that the form of the density of the color components of\n  the field is known.\nIn 2+1 dimensions I choose  it such that it gives area\nlaw for space oriented Wilson loops.\nI treat then the entire distribution of these components as\n a joint distribution of their probabilities and regard\n the adopted \"single component\" density as an analog of a\nmacroscopic quantity that must be reproduced by this\njoint distribution. I postulate that it must otherwise be maximally random,\n i.e.  must have the maximum entropy (minimum\ninformation content) under suitable constraints. In this way I\nderive the maximally random distribution for this model. I discuss its relation\nto the large N limit of the Schr\\\"odinger   equation for the static quark--antiquark\nsystem. I also give possible generalizations of this development\nto  3+1 dimensions.\n\nIn Section 6 I include dynamical quarks\nand show that the rigid gauge rotor limit corresponds exactly to the\nlimit in QED in which only the instantaneous Coulomb interaction\nbetween the charges is retained. The major difference in QCD is that\nin this limit   the quarks not only interact instantaneously via a more\ncomplicated interaction  controlled by the rigid gauge field configuration,\nbut at the same time are also found  in a static  colormagnetic field\ninduced by this configuration. This dual appearance of the rigid field is\n a consequence of the gauge invariance and in the large N limit\nis apparently the way the\nmaster field should enter the quark sector of the theory. According to the\nideology developed in Section 5 both the field in which the quarks\nmove and their interaction should be considered as random in the large N\nlimit. The random interaction between the quarks  opens interesting\npossibilities to discuss the relationship between confinement and\nlocalization.\n\nThe body fixed  Hamiltonian with dynamical quarks is gauge invariant.\nIts invariance  with  respect to global symmetries however is not\nguaranteed for an arbitrary choice of the rigid gauge configuration.\nI discuss possible variational approaches to determine this configuration\nand derive an analogue of the Hartree-Fock equations for the model.\n\nIn the rest of the Introduction I will establish my notations, cf.,\n Ref.\\cite{bjo}.  I\nconsider the QCD Hamiltonian in d=3 space dimensions in the $A^{0} = 0$\ngauge,\n\\begin{equation}\nH=\\frac{1}{2}\\int d^{3}x [(E_{a}^{i}(x))^{2}+(B_{a}^{i}(x))^{2}]\n+ \\int d^{3}x q_{\\gamma}^{+}(x)[\\alpha^{i} \\left(p^{i}-g\nA_{a}^{i}(x) \\frac{ \\lambda_{\\gamma \\delta}^{a}}{2} \\right)\n+\\beta m]q_{\\delta}(x).\n\\label{ham}\\end{equation}\nwith\n\\begin{equation}\nB_{a}^{i}(x)=\\epsilon_{ijk}(\\partial_{j}A_{a}^{k}+gf_{abc}A_{b}^{j}A_{c}^{k}),\n\\end{equation}\n$i,j,k=1,...,3$;  $\\gamma,\\delta$=1,...,N;  $a=1,...,N^2 - 1$ for SU(N)\ngauge group and  $f_{abc}$ -- the structure constants of the SU(N).\nDirac and flavor indices are omitted and the summation\nconvention for all repeated indices is employed here and in the following.\nThe gluon vector potential  $A_{a}^{i}(x)$ and  minus the electric\nfield  $-E_{a}^i (x)$ are canonically conjugate variables,\n\\begin{equation}\n[E_{a}^{i}(x),A_{b}^{j}(y)]=i\\delta_{ab}\\delta_{ij}\\delta (x-y),\n\\end{equation}\nand the quark fields obey the standard anticommutation relations.\n\nThe Hamiltonian (\\ref{ham}) is invariant under\nthe time independent gauge transformations. Using the matrix valued\nhermitian fields\n\\begin{equation}\nA_{\\alpha\\beta}^{i}(x)=A_{a}^{i}(x) \\frac{ \\lambda_{\\alpha\\beta}^{a}}{2},\n\\; \\; \\; \\; \\;\nE_{\\alpha\\beta}^{i}(x)=E_{a}^{i}(x) \\frac{\\lambda_{\\alpha\\beta}^{a}}{2},\n\\end{equation}\nwhere $\\lambda^{a}$ are the SU(N) generators with the properties\n\\begin{eqnarray}\n[\\lambda^{a},\\lambda^{b}]&=&2if_{abc}\\lambda^{c};\\; \\; \\;\n\\{\\lambda^a,\\lambda^b\\} = \\frac{4}{N}\n\\delta_{ab} + 2d_{abc} \\lambda^c;\\nonumber \\\\\nTr\\lambda^{a}\\lambda^{b}&=&2\\delta_{ab};\\; \\; \\; \\; \\;\n\\lambda_{\\alpha \\beta}^{a} \\lambda_{\\gamma\\delta}^{a} = 2[\\delta_{\\alpha\\delta}\n\\delta_{\\beta\\gamma}-\\frac{1}{N}\\delta_{\\alpha\\beta}\\delta_{\\gamma\\delta}]\n\\end{eqnarray}\none can write the gauge transformation as\n\\begin{equation}\n A^i \\rightarrow SA^{i}S^{+}+\\frac{i}{g}S\\partial^{i}S^{+};\\;\\; \\;\nE^i\\rightarrow SE^{i}S^{+};\\; \\; \\;  q\\rightarrow Sq  \\end{equation}\nwhere S(x) are time independent but x - dependent unitary $N\\times N$\nmatrices, elements of the SU(N) group. The generators of this\n transformation\n\\begin{equation}\n G_{a}(y) \\equiv G_a^A (y) + G_a^q (y),  \\nonumber \\end{equation}\n\\begin{equation}\nG_{a}^A (y) =  \\partial_{i}E_{a}^{i}(y)+gf_{abc}A_{b}^{i}(y)E_{c}^{i}(y),\n\\; \\; G_{a}^q (y) = -gq^{+}(y)\n\\frac{\\lambda^{a}}{2} q(y) \\label{gen} \\end{equation}\nare conserved,\n\\begin{equation}\n\\frac{\\partial G_{a}(x)}{\\partial t}=i[H,G_{a}(x)]=0\\end{equation}\nand it is consistent to impose the Gauss law constraints\n\\begin{equation}\nG_{a}(x)|\\Psi>=0 \\label{gl} \\end{equation}\nfor all physical states.\nAlthough $G_{a}(x)$ do not commute, their commutators\n\\begin{equation}\n[G_{a}(x),G_{b}(y)]=gf_{abc}\\delta (x-y)G_{c}(x)\\end{equation}\nallow to set  them all simultaneously zero.\n\n\n\\setcounter{equation}{0} \\ALTsect{Rigid Gauge Rotor.}\n\nIn this section I will discuss the rigid gauge \"rotations\". Classically\nI define them as gauge field configurations of the type\n\\begin{equation}\n A^i(x,t) = U(x,t)a^{i}(x)U^{+}(x,t)+\\frac{i}{g}U(x,t)\\partial^{i}U^{+}(x,t)\n\\label{rgd}\\end{equation}\nwhere $a^i(x)$ are t-independent, fixed as far as their x-dependence is\nconcerned, \"rigid\" fields which I do not\nspecify and leave them arbitrary for the moment. Eq.(\\ref{rgd}) is\nthe simplest example of the transformation to the \"body fixed\" frame\nof the local gauge symmetry in which I have assumed that the dynamics\nof the field in this frame is  very stiff so that     the field\ncan  be approximately replaced by\nits static  average. In general $a^i$ is of course dynamical but should\nbe viewed as constrained since $U(x,t)$ already contains a third of the\ndegrees of freedom. For non rigid  $a^i$  there is no obvious choice of\nthe body-fixed frame and it can be constrained in a variety of ways,\nsay, $a^3 = 0$ (axial gauge), $\\partial_i a^i = 0$ (Coulomb gauge),\netc. In our language\nthese different gauge fixings correspond to different \"rotating\" frames.\nSince they are \"non inertial\" the dynamics will look very differently\ndepending on the choice of the frame and  different fictitious\nforces, the analogue of  Coriolis and centrifugal forces,\nwill be present.  I am planning to discuss these issues elsewhere.\n\nWith the anzatz (\\ref{rgd}) the covariant derivatives are\n\\begin{equation}\nD^i \\equiv \\partial ^i - igA^i = U(x,t) d^i(x) U^{+}(x,t),\\end{equation}\nwith fixed, rigid\n\\begin{equation}\nd^i(x) \\equiv \\partial ^i - iga^i(x). \\end{equation}\nInserting (\\ref{rgd}) in the Hamiltonian (\\ref{ham})\none finds that the gauge invariant potential term\n$\\sum_{i,a}(B_{a}^{i}(x))^{2} \\sim \\sum_{i,j}Tr[D^i,D^j]^2 = \\sum_{i,j}\nTr[d^i,d^j]^2$ is\nindependent of the $U$'s, i.e. it is fixed, nondynamical in this model.\nThe dynamics of the gauge field\nis governed by the kinetic energy, i.e. the term with\nthe electric field  in (\\ref{ham}).  Using\n$\\partial_0(U\\partial_iU^{+}) = (i\/2)U(\\partial_i\\omega)U^{+}$ with\n$\\omega = 2i U^{+}\\partial_0 U$ one finds\n\\begin{eqnarray}\n-E^{i} = \\partial_0 A^{i} =  \\frac{1}{2g}U [\\omega,d^i] U^{+},\n\\label{dai}\\end{eqnarray}\nand therefore the kinetic energy in (\\ref{ham}) is\n\\begin{equation}\n\\frac{1}{4}\\int d^3 x Tr(\\partial_0 A^i)^2 =\n - \\frac{1}{16g^2}\\int d^3 x Tr\\left( \\omega[d^i,[d^i,\\omega]]\\right)\n \\label{ke} \\end{equation}\nwhere I have disregarded surface terms, ignoring for the moment\npossible non vanishing fields at infinity, non trivial topologies and\nother global issues (cf.  below).\n\nThe double commutator in (\\ref{ke}) with the summation over all indices,\n$x,i$ and the color is the straightforward generalization of the\nfamiliar double vector product summed over all particles indices\n in the moment of inertia tensor\nappearing in the kinetic energy of rigid space rotations of system of\nparticles with fixed relative positions.\n Following this analogy  the energy (\\ref{ke}) of the rigid gauge\n rotations can be written\n\\begin{equation}\nE_{rot} = \\frac{1}{4}\\int d^3 x Tr(\\omega I \\omega) \\label{en} \\end{equation}\nwhere the moment of inertia is defined as a differential matrix operator\nsuch that\n\\begin{eqnarray}\nI\\omega &\\equiv& - \\frac{1}{4g^2} \\left[ d^i,[d^i,\\omega]\\right] = \\nonumber \\\\\n&=& - \\frac{1}{4g^2}\\left(\\partial_i^2 - ig[\\partial_i a^i,\\omega] -\n2ig[a^i,\\partial_i\\omega] - g^2 \\left[ a^i,[a^i,\\omega]\\right]\\right) .\n \\label{mom}\n\\end{eqnarray}\n To obtain the corresponding\nHamiltonian one can use the gauge field part $G^A$ of the generators\n(\\ref{gen}). Using  Eqs. (\\ref{rgd}), (\\ref{dai}) and the\ndefinition (\\ref{mom}) one finds\n\\begin{equation}\nG^A = [\\partial^i - ig A^i,E^i] =\n\\frac{1}{2g}U \\left[d^i,[d^i,\\omega]\\right] U^{+} = -2g U (I\\omega) U^{+}.\n\\label{omg} \\end{equation}\nDefining the gauge generators in the rotating frame\n$\\hat{G} = U^{+} G^A U$, expressing $\\omega = - I^{-1} \\hat{G} \/2g$ from (\\ref{omg}) and\nsubstituting in (\\ref{en}) one finds the Hamiltonian of the rigid gauge\nrotor\n\\begin{equation}\nH_{rot}^A = \\frac{1}{16g^2}\\int d^3 x d^3 y  Tr \\hat{G} (x) I^{-1}(x,y)\n\\hat{G} (y) = \\frac{1}{4g^2}\\int d^3 x d^3 y \\hat{G}_a(x) I_{ab}^{-1}(x,y)\n\\hat{G}_b(y).\n\\label{hrot} \\end{equation}\nwhere $I_{ab}^{-1}(x,y) = (1\/4) Tr(\\lambda^a I^{-1}(x,y)\n\\lambda^b)$ is proportional to the inverse of the operator $-d^i_{ac}\nd^i_{cb}$ with $d^i_{ab} = \\partial_i \\delta_{ab} - g f_{abc} a^i_{c}$\nand I assumed  that this operator  does not have zero eigenvalues.\nIn a more careful way of handling fields at infinity one should avoid\nthe integration by parts in (\\ref{ke}). The  inverse \"moment\nof inertia\" operator is then replaced by a less transparent\n\\begin{equation}\nK_{aa'}(x,x') = \\int d^3 y \\left[ d^i_{bc}(y) I^{-1}_{ca}(y,x)\\right]\n\\left[ d^i_{bc'}(y) I^{-1}_{c'a'}(y,x')\\right].\\end{equation}\nMost of the following results remain valid for both forms of this\noperator.\n\nThe meaning of the preceeding expressions is quite obvious. They are the\nfield-theoretic generalization of the standard rigid rotor results.\nThe unbroken\nlocal gauge symmetry of QCD means that there are free SU(N) color\ngauge \"rotations\" at\nevery space point. Expression (\\ref{hrot}) shows that the \"rotations\"\nat different  points as well as around different color axes are coupled\nvia the non diagonal\nelements of the moment of inertia \"tensor\" $I_{ab}(x,y)$  in the manner\nsimilar to the coupling between rigid rotations around different space axes in\nsystems of particles.\n\nIt does not seem to be useful to diagonalize the operator $I^{-1}$ in\n(\\ref{hrot}).\nThe standard diagonal form of the rigid rotor Hamiltonian , i.e., $H = \\frac{1}{2}\\sum_a\nL_a^2\/I_a$  can be usefully achieved only in the case of\nrotations corresponding to a single SU(2) group to which the familiar\nrigid space rotations  belong. Diagonalizing the moment of inertia\nin the case of higher groups will introduce combinations\nof the generators multiplied by  matrices of orthogonal rotations.\nThese in general will not have\nthe group commutation relations.  Already for a single\nSU(3) the group O(8) of\nrotations in the adjoint space needed in order to diagonalize the\nmoment of inertia is much larger than SU(3).\n\n The actual values of the moment of inertia depend on the rigid\nconfiguration $a^i (x)$ of the gauge field via the\nexpression (\\ref{mom}). This comprises the \"free parameter\" of the rigid\ngauge rotor model. For abelian theory or alternatively\nin the limit $g \\rightarrow 0$ the inverse of $I(x,y)$\nappearing in Eq.(\\ref{hrot})\nis just the Coulomb propagator.\n In the opposite large $g$ or long wavelength limit\n$I(x,y)$ becomes a local tensor  given by the last term in (\\ref{mom})\nwhich is obviously the SU(N)\ngeneralization of the moment of inertia expression.\n\n An important feature of\nthe Hamiltonian  (\\ref{hrot}) is that despite the severe limitation of the\nallowed gauge field configurations imposed by  (\\ref{rgd}) it remaines\ngauge invariant. This is because (\\ref{hrot}) depends on $\\hat{G}$\nrather than $G^A$. Under a gauge transformation $U \\rightarrow SU$,\n$G^A$ transforms as $SG^A S^{+}$ so that  $\\hat{G}(x)$ and therefore\n$H_{rot}^A$ stay invariant. The gauge invariance of (\\ref{hrot}) is the\nsimplest illustration of the usefulness of the introduction of the\nbody fixed frame. One can freely approximate the dynamics in this frame\nwithout fears of violating the symmetry with respect to which the frame\n has been defined, i.e. the local gauge symmetry in the present case.\n\nConsider another transformation, $U \\rightarrow US$. Referring to\nEq.(\\ref{rgd}) one can interpret this transformation\neither as the change of $U$ i.e. the transformation of the intrinsic\nframe with respect to the rigid \"shape\" $a^i$ or\nas the change of $a^i$ ,\n $a^i \\rightarrow Sa^{i}S^{+}+\\frac{i}{g}S\\partial^{i}S^{+}$, i.e.\nthe  transformation of the intrinsic \"shape\" with respect\nto the intrinsic frame.\nSuch transformations obviously form a group of local SU(N) gauge\ntransformations  which I will call\n the intrinsic or \"body fixed\"\n  gauge group to distinguish it from the \"laboratory\" gauge\ngroup of the ordinary gauge transformations. According to two different\ninterpretations of the intrinsic gauge transformations\ngiven above one has two options. One option is to\n regard  $\\hat{G}$'s as the generators of the intrinsic group.\nThey act on  the dynamical variables $U$ but they have a disadvantage in\nthat  the \"laboratory\"\ngroup is not completely independent of such an  intrinsic group, e.g.\nthey both have identical Casimir operators. Another option is to formally\nintroduce operators which gauge transform  the intrinsic variables $a^i$.\nThey will have the same form as $G^A$'s but with $a^i$ replacing $A^i$.\nDefined in this way the intrinsic group will be completely independent\nof the \"laboratory\" gauge group but will act on nondynamical variables\nwhich do not appear in the wavefunctions. Convenience should dictate\nwhich one to use.\n\nThe above introduction of the intrinsic vs \"laboratory\" gauge groups\nis obviously quite general with e.g. the definition of $\\hat G(x)$ being\nindependent of the rigid rotor restrictions set by fixing $a^i$\nto be nondynamical in (\\ref{rgd}).\nUnlike the local gauge symmetry in \"laboratory\", the symmetry in the\n\"body fixed\" frame can be broken.\nE.g., the gauge invariant Hamiltonian  (\\ref{hrot})\n is in general not invariant under the transformations of the intrinsic\ngauge group. This is a simple example of the situation to which I\nreferred earlier as a possible existence of \"deformation\"\nvs impossibility of the symmetry breakdown in the context of non\nabelian local\ngauge theory. The character of the deformation can be classified\nusing the intrinsic gauge group, e.g. in classification of\npossible \"deformed shapes\" of the rigid gauge rotor (\\ref{hrot})\nby the  transformation properties of the moment of inertia\n$I_{ab}(x,y)$ under this group. Here I obviously adopt the second\ninterpretation of the intrinsic group. The invariance of\n$I_{ab}(x,y)$ under all\nintrinsic transformations would be analogous\nto the spherical rotor limit  in the space rotation case.\nThe invariance under a continuous subgroup of the intrinsic group is the\nanalog of the axial symmetric rotor, etc.  Discrete intrinsic\nsubgroups should also be considered.\n\nConsider a rigid gauge configuration $a^{i'}$ which is a gauge\ntransform of $a^i$, $a^{i'} = S(a^i +(i\/g)\\partial_i)S^{+}$.\n  The Hamiltonian\n(\\ref{hrot}) will have the same form with the same moment of inertia\nbut with $\\hat{G}$ replaced by $S^{+}\\hat{G} S$. The eigenvalues of this\ntransformed Hamiltonian will not change and will therefore depend only\non gauge invariant combinations of the rigid $a^i$, i.e. on the Wilson\nloop variables $Tr P exp (ig\\oint a^i dx_i)$.\n\\setcounter{equation}{0} \\ALTsect{Static Quarks}\n\nSo far I have discussed the rigid gauge rotor limit\nof only the first term in (\\ref{ham}). The resulting $H_{rot}^A$ is relevant\nfor the discussion of very heavy quarks.\nThey can be considered static as far as their translational motion is\nconcerned. They still have a wavefunction describing the motion of their color\ndegrees of freedom. Because of (\\ref{gl}) this motion is coupled to\nthe \"rotations\" of the gauge field which I will treat\nusing (\\ref{hrot}).\n\nIn the limit of $m\\rightarrow \\infty$  the quark kinetic energy\nterm $q^{+}\\vec{\\alpha}\\vec{p}q$  and the quark color current coupling\n$q^{+}\\vec{\\alpha}\\lambda^{a}q$ in Eq.(\\ref{ham}) can be neglected\nand the resulting Hamiltonian decouples into a part containing the gauge field\nand another  containing the quarks, $H=H_{A}+H_{q}$, where\n$H_{A}$ is the first term in (\\ref{ham})\nand $H_q = m\\int d^3x q^{+} (x)\\beta q(x)$.\nThe coupling  appears only via the Gauss law constraint, Eq.(\\ref{gl}).\nThe wave function can not be taken\nas a product $\\Psi=\\Psi(q)\\Psi(A)$ but should be a local color singlet.\nIn the representation  in which\n\\[ \\beta=\\left( \\begin{array}{cc}  1 & 0 \\\\  0 & -1\\end{array}\\right) \\]\n\\[ q_{\\alpha}(x)=a_{\\alpha}(x) \\left( \\begin{array}{cc} 1 \\\\ 0\\end{array}\n\\right)   + b_{\\alpha}^{+}(x) \\left( \\begin{array}{cc} 0 \\\\ 1\\end{array}\n\\right)  \\]\nwith\n\\begin{eqnarray}\n\\{a_{\\alpha}(x),a_{\\beta}^{+}(y)\\}&=&\\delta_{\\alpha\\beta}\\delta (x-y)\n\\nonumber \\\\\n\\{b_{\\alpha}(x),b_{\\beta}^{+}(y)\\}&=&\\delta_{\\alpha\\beta}\n\\delta (x-y), etc...\\nonumber\n\\end{eqnarray}\nthe eigenfunctions of $H_{q}$ are trivially written down.\nConsider, e.g.,\n\\begin{eqnarray}\n|\\Psi(q)>\\equiv |vac(q)> &=& |0>, \\label{vac} \\\\\n|\\Psi(q)>\\equiv |x_{0},\\alpha> &=& a_{\\alpha}^{+}(x_{0})|0> \\label{1qu} \\\\\n|\\Psi(q)>\\equiv |x_{0},\\alpha ;y_{0},\\beta > &=& a_{\\alpha}^{+}(x_{0})b_{\\beta}\n^{+}(y_{0})|0> \\label{2qu} \\end{eqnarray}\nThese wave functions describe respectively  zero quarks, one\nstatic quark at $x_{0}$ with color component $\\alpha$ and a static\nquark - antiquark pair at $x_0$ and $y_0$. It is easy to form local\ncolor singlets with these wave functions.\n For e.g. the quark-antiquark pair it is\n\\begin{equation}\n |\\Psi>=\\sum_{\\alpha\\beta} \\Psi_{x_{0},\\alpha ;y_{0},\\beta}(A)|x_{0},\n\\alpha ;y_{0},\\beta> \\end{equation}\nwith the wave functional\n$\\Psi_{x_{0},\\alpha ;y_{0},\\beta}(A)$ satisfying\n\\begin{eqnarray}G_{a}^{A}(x)\\Psi_{x_{0},\\alpha ;y_{0},\\beta }(A) =\ng\\delta(x-x_{0})\\frac{\\lambda_{\\alpha\\alpha^{'}}^{a}}{2}\n\\Psi_{x_{0},\\alpha^{'} ;y_{0},\\beta}(A)+g\\delta(x-y_{0})\n\\frac{\\bar{\\lambda}_{\\beta\n\\beta^{'}}^{a}}{2}\\Psi_{x_{0},\\alpha ;y_{0},\\beta^{'}}(A)\\label{cnd}\n\\end{eqnarray}\nwhere $\\bar{\\lambda}_{\\alpha\\beta}^{a}=-\\lambda_{\\alpha\\beta}^{a*}$.\nThe wave functional of the\ngauge field should be a singlet at every point in space except\nat the position of the quarks where it should transform as  N and $\\bar{N}$\nmultiplets of SU(N).  This constraint together with the\nSchr\\\"odinger   equation\n\\begin{equation}\nH_{A}\\Psi_{x_{0},\\alpha;y_{0},\\beta}(A)=E\\Psi_{x_{0},\\alpha;y_{0},\\beta}(A)\n\\end{equation}\ncompletely defines the problem for the gauge field.\n\n In the rigid gauge rotor limit $H_A$ is given by Eq.(\\ref{hrot}).\nThe wavefunctions of this Hamiltonian  are general functionals\n $\\Psi [U(x)]$ of the SU(N) matrices $U_{\\alpha \\beta} (x)$.\nTheir scalar product is determined by functional integration over the\n$U$'s with the corresponding group invariant measure.\nThe vacuum wave functional must obey $G_a^A (x)\\Psi_{vac}[U] = 0$.\nThis means that it is a constant independent of $U_{\\alpha \\beta}(x)$.\nSince also $\\hat{G}_a (x)\\Psi_{vac}[U] = 0$ the vacuum energy is zero according\nto (\\ref{hrot}).\nRegarding the parametrization of the $U$'s in terms of the appropriate Euler\nangles of the SU(N) rotations at every space point, the constant\n$\\Psi_{vac}[U(x)]$ means that all the \"orientations\" of $U(x)$ at all points\nare equally\nprobable, i.e. there are no correlations between the \"orientations\"\nof the rigid gauge rotor at different points. This is as \"random\" as\nthe distribution of the $U$'s can get. The absence of correlations\nis the property only of the vacuum. For other states the \"orientations\"\nof the gauge fields at different space points are correlated via the\n\"moment of inertia\" operator.\n\nIn order to discuss\nthe  wave functions with non zero number of quarks it is sufficient\nto know some simple\nproperties of the gauge generators $G_a^A(x)$ and $\\hat{G}_a(x)$. Since\n$\\hat{G}_a(x)$ are gauge scalars, they commute with $G_a^A (x)$,\n\\begin{equation}\n[G_a^A(x),\\hat{G}_b(y)] = 0 \\end{equation}\nwhich means that together the generators of the \"laboratory\" and the\nintrinsic gauge groups provide a complete set of commuting\nquantum numbers for the wave functionals $\\Psi [U_{\\alpha \\beta }(x)]$.\nIndeed, since the Casimir operators for $G^A$'s and $\\hat{G}$'s coincide one has\ne.g., for the SU(2) the $(G_a^A(x))^2$, $G_3^A(x)$ and $\\hat{G}_3(x)$,\ni.e. three local commuting operator fields  for the three fields of the\nEuler angles needed to specify the $U(x)$.\nIn the SU(3) one has eight fields of the \"Euler angles\" and eight local\ncommuting generators made off $G^A$'s and $\\hat{G}$'s --\n the two group Casimir operators, one Casimir operator\nof an SU(2) subgroup for $G^A$'s, say $\\sum_{a=1}^{3} (G_a^A)^2(x)$ and the\ncorresponding one for the $\\hat{G}$'s and respectively two\npairs of the Cartan generators -- $G_3^A(x)$, $G_8^A(x)$ and $\\hat{G}_3(x)$ and\n $\\hat{G}_8(x)$. This counting continues correctly for any N, i.e.\n$N-1$ for the SU(N) Casimir operators, $2((N-1)+(N-2)+... +1)$\nfor the  Casimir operators of pairs of SU$(N-1)$...SU(2) subgroups and $2(N-1)$\nfor the Cartan generators. Altogether there are $N^2 - 1$ local commuting\noperators as needed. An eigenfunction\nof this complete set of operators is  the Wigner function\n$D_{K K^{\\prime}}^L(U(x))$ of U at a certain space point. $K$ and $K^{\\prime}$\nare the quantum numbers of the \"laboratory\" and the intrinsic groups and $L$\ndetermines the representation.\n\nSince under an infinitesimal gauge transformations $U\\rightarrow\n(1+i\\epsilon_a(x) \\frac{\\lambda_a}{2})U$  one can easily verify that\n\\begin{eqnarray}\n\\left[ G_{a} (x),U_{\\alpha \\beta}(y) \\right] &=& g \\delta (x - y)\n\\frac{\\lambda^a_{\\alpha \\gamma}}{2} U_{\\gamma \\beta}(x) \\nonumber \\\\\n\\left[ G_{a} (x),U_{\\alpha \\beta}^{+}(y) \\right] &=& - g \\delta (x -y)\nU_{\\alpha \\gamma}^{+}(x)\\frac{\\lambda^a_{\\gamma \\beta}}{2} \\label{com} \\\\\n\\left[ \\hat{G}_{a} (x),U_{\\alpha \\beta}(y) \\right] &=& g \\delta (x - y)\nU_{\\alpha \\gamma}(x) \\frac{\\lambda^a_{\\gamma \\beta}}{2} \\nonumber \\\\\n\\left[ \\hat{G}_{a} (x),U_{\\alpha \\beta}^{+}(y) \\right] &=& - g \\delta (x -y)\n\\frac{\\lambda^a_{\\alpha \\gamma}}{2} U_{\\gamma \\beta}^{+}(x)\n \\nonumber \\end{eqnarray}\n All the operators in the rigid gauge rotor model are functions of the\n$G$'s and $U$'s. E.g. consider the electric field operator. According\nto Eq.(\\ref{dai}) it is\n\\begin{equation}\nE^i =  -\\frac{1}{4g^2} U [d^i,I^{-1} \\hat{G}] U^{+} \\label{elf} \\end{equation}\nwhere I have expressed $\\omega$ in terms of $\\hat{G}$ using (\\ref{omg}).\n\nUsing the relations (\\ref{com}) it is easy to write the general form of\nthe wave functions for a single quark and for a quark--antiquark pair,\n\\begin{eqnarray}\n\\Psi_{x_0,\\alpha}[U] = U_{\\alpha \\gamma}(x_0)c_{\\gamma} \\nonumber \\\\\n\\Psi_{x_0,\\alpha;y_0,\\beta}[U] = U_{\\alpha \\gamma}(x_0)U_{\\delta\\beta}\n^{+}(y_0)c_{\\gamma \\delta} \\label{wfn}\\end{eqnarray}\nThey satisfy the conditions (\\ref{cnd}) following from the\nGauss law with constant coefficients\n $c_{\\gamma}$ and $c_{\\gamma \\delta}$\nwhich give  the probability amplitudes of the intrinsic quantum\nnumbers $\\gamma$ and $\\delta$. They should be normalized,\n$\\sum |c_{\\gamma}|^2 = 1 ; \\sum |c_{\\gamma \\delta}|^2 = 1$ to assure\nthe normalization $\\int d[U(x)] |\\Psi [U]|^2 = 1$\n\nThese amplitudes\n must be found by solving the corresponding Schr\\\"odinger  equations but before\ndescribing this I wish to remark that the above form of\nthe wave functions is valid also when the limitation of\nthe rigid gauge rotations is relaxed and the most general gauge\nconfigurations are allowed. The parametrization (\\ref{rgd}) is still very\nuseful but now with fully dynamical fields $a^i$\n the variation of which should be limited only by a \"gauge fixing\"\ncondition to avoid overcounting as described above.\n  The dynamics will of course be\nthat of the full QCD but  the wave functions\nof the static quark and the quark--antiquark pair will have the same form\n(\\ref{wfn}).  The difference will be that amplitudes $c_{\\gamma}$ and\n$c_{\\gamma \\delta}$ will be functionals of $a^i(x)$  describing the\nspace and\ncolor fluctuations of the \"string\" attached to the quark or between the\nquark and the antiquark. In the rigid gauge rotation case there are only\ncolor fluctuations described by constant amplitudes.\n\nFor quarks in higher representations the wave functions have the same\nform with $U$ replaced by the appropriate Wigner D-function. E.g. in\nthe adjoint representation\n\\begin{eqnarray}\n\\Psi_{x_0,a}[U] = Tr(U(x_0)\\lambda^aU^{+}(x_0)\\lambda^b)c_b,\n\\nonumber \\end{eqnarray}\netc.\n\nI will now derive the Schr\\\"odinger   equation for the string amplitudes\n$c_{\\gamma}$ and $c_{\\gamma\\delta}$.\n Acting with the Hamiltonian  (\\ref{hrot}) on (\\ref{wfn}), using (\\ref{com})\nand the orthogonality of $U$'s  with respect to the integration over the\ngroup, $\\int dU U_{\\alpha \\beta}^{*} U_{\\mu \\nu} = \\delta_{\\alpha \\mu}\n\\delta_{\\beta \\nu}$,  I find\n\\begin{eqnarray}\nQ_{\\alpha \\gamma}(x_0)c_{\\gamma} =  E c_{\\alpha}, \\nonumber \\\\\nQ_{\\alpha \\gamma}(x_0)c_{\\gamma \\beta} + Q^{*}_{\\beta \\mu}(y_0)c_{\\alpha \\mu}\n- P_{\\alpha \\beta , \\gamma \\mu }(x_0,y_0)c_{\\gamma \\mu} =\nE c_{\\alpha \\beta}, \\label{sch1} \\end{eqnarray}\nwhere I denoted\n\\begin{eqnarray}\nQ_{\\alpha \\gamma}(x_0) = \\frac{1}{4} I_{ab}^{-1}(x_0,x_0)(\\lambda^a\n\\lambda^b)_{\\alpha \\gamma}, \\\\\nP_{\\alpha \\beta , \\gamma \\mu }(x_0,y_0) =  \\frac{1}{2} I_{ab}^{-1}(x_0,y_0)\n\\lambda^a_{\\alpha \\gamma} \\lambda^b_{\\mu \\beta}.\\end{eqnarray}\nIn SU(2) $Q_{\\alpha \\gamma}$ takes a particularly simple diagonal form,\n$Q_{\\alpha \\gamma} = \\delta_{\\alpha \\gamma}(1\/4)I_{aa}^{-1}(x_0,x_0)$.\nand is the eigenvalue for a single quark. For quarks in e.g. adjoint\nrepresentation the lambda matrices in the expressions above are replaced\nby the corresponding group generators $if_{abc}$.\nThe first two terms in the second line of\n(\\ref{sch1}) are the quark and the antiquark self\nenergies whereas the last term is their interaction. In QCD one expects\nthat terms like $Q$ are inflicted by the long and short distance\ndivergences and should be properly regularized which I will assume for the\nrest of the paper. I will further assume the translational invariance of\n$Q$, i.e. its independence of $x_0$. One can then  rewrite Eq.(\\ref{sch1})\nby transforming it to the basis in\nwhich $Q$ is diagonal. Defining its eigenvectors\n$Qb^{(n)} = \\epsilon_n b^{(n)}$ and expanding\n$c_{\\gamma \\beta} = d_{mn} b_{\\gamma}^{(m)} b_{\\beta}^{(n)*}$ one finds\n\\begin{equation}\n<kl| P (x_0,y_0) |mn> d_{mn} = (E - \\epsilon^k - \\epsilon^l) d_{kl}\n\\;\\;\\; {\\rm(no\\;\\;sum\\;\\;over\\;\\;k\\;\\;and\\;\\;l)} \\label{peq} \\end{equation}\nwhere $<kl| P |mn> = P_{\\alpha \\beta , \\gamma \\mu } b_\\gamma^{(m)}\nb_\\mu^{(n)*} b_\\alpha^{(k)} b_\\beta^{(l)*}$. The Schr\\\"odinger  equation (\\ref{peq})\nis $N^2 \\times N^2$ matrix equation and the most interesting question of course\nconcerns the dependence of its eigenvalues on the distance $|x_0 - y_0|$\nfor various possible choices of the rigid gauge field configuration\n$a^i(x)$ on which the matrix $P$ depends. I will address this question\nin the next section.\n\n\\setcounter{equation}{0} \\ALTsect{Choices of The Rigid Field. Mean Field Equations.}\n\nThe rigid configuration if it exists in QCD must reflect the\nproperties of the gluon condensate of the  vacuum. One of the more\naccepted views of the QCD vacuum  is that this is a condensate of non\ntrivial topological configurations -- the Z(N) vortices,\nc.f.,\\cite{top}.  Although\nsuch configurations are easily incorporated in the above formalism\nI was not able to overcome\ntechnical difficulties in working out a theory of their condensation.\n\nOn a heuristic level each Z(N) vortex carries\n a unit of flux of the colormagnetic\nfield. Condensation of the vortices presumably\nmeans that there is a non zero average of this field in  the vacuum.\n Of course due to unbroken local gauge symmetry  it\nmust undergo free \"gauge rotations\" at each space point. In the ground\nstate this means that there are\nequal probabilities of all the \"orientations\"   yielding  zero\naverage value in the laboratory. The finite   average\nvalue of the condensate field\ncan only be \"seen\" in the \"body fixed\" frame and should appear in this\npicture in the manner similar to\n $a^i$ in the expression (\\ref{rgd}) for our rigid gauge\nrotations. The field strength\n\\begin{equation}\nB^i(x) = U(x)b^i(x)U^{+}(x),\\,\\,\\, \\  b^i = \\frac{i}{g} \\epsilon_{ijk}\n [d^j,d^k], \\end{equation}\nalso averages to zero in the ground state but has a non zero value $b^i$\nin the \"body fixed\" frame.\n\nVia the dynamics of $U(x)$ the anzatz (\\ref{rgd})  leads to colorelectric\nfield  (\\ref{elf}) which propagates away from points\nwhere $\\hat{G}(x)$ is non zero, i.e. from  the location of static quarks.\nThe propagator of this field is controlled by the condensate\nfield $a^i$ which enters the expressions for $I^{-1}$ and $d^i$.\nThis propagator is a long range Coulomb potential for zero $a^i$ and\n  is a Gaussian for\n$a^i$ corresponding to a uniform colormagnetic $b^i$ (cf., below).\nThe screening of the propagation range of the colorelectric field\nin the presence of the colormagnetic \"condensate\"  $b^i$ is\nreminiscent of the dual to the Meissner effect of screening of\na magnetic field by the electric condensate of a superconductor.\nThis possibility of the dual Meissner effect is  of course a\nstandard scenario for confinement in QCD. It is expected that\ntubes of flux of the colorelectric field are formed which\nconnect quarks and make their energy depend linearly on the distance.\n\nIn the present formalism a way to attempt to model the formation\nof a confining string is to look for  such a configuration of\nthe rigid field $a^i(x)$ for which the propagator $I^{-1}$ behaves\nroughly speaking  as\none dimensional for large separations along some  given  line in space\nat the end of which quarks can be placed. This means that\na sort of magnetic \"wave guide\" should be constructed so that\nthe Green's function of the operator $-d^2_{ab} =\n-d^i_{ac} d^i_{cb} =\n -(\\partial_i \\delta_{ac} - g f_{acc'} a^i_{c'})(\\partial_i - g f_{cbb'}\n a^i_{b'})$ is\nasymptotically  $\\propto |x-x'|$ along, say, one of the coordinate\naxes. In order to see the difficulties in finding such a configuration\nconsider for simplicity 2 space dimensions and choose\n$a^1 = c(y)$ and $a^2 = 0$ with an arbitrary $c(y)$. This choice\ncorresponds to the colormagnetic field $b(y) = \\partial_y c(y)$\ndepending only on one coordinate $y$. The operator to invert is then\n\\begin{equation}\n-(\\partial_x -ig c(y) \\cdot F)^2- \\partial_y^2   \\end{equation}\nwhere I denoted the color spin matrices $F^a_{bc} = if_{bac}$\nand $c(y) \\cdot F = c_a (y) F^a$. The propagator  is then\n\\begin{equation}\n\\int_{-\\infty}^{\\infty}\n dk e^{ik (x - x')} \\sum_{n} \\frac{\\chi_n(k,y) \\chi_n (k,y')}\n{\\epsilon_n (k)}, \\label{prop} \\end{equation}\nwhere $\\chi_n (k,x)$ and $\\epsilon_n (k)$ are solutions of\n\\begin{equation}\n [-\\partial_y^2 + (k +c(y) \\cdot F)^2 ] \\chi_n (k,y) = \\epsilon_n (k)\n\\chi_n (k,y). \\label{lan} \\end{equation}\nIn order to achieve the desired confining behaviour of the propagator\nthe sum in (\\ref{prop}) must be\n$\\sim k^2$ for $k \\rightarrow 0$. The simplest\nis to assume that the lowest eigenvalue of (\\ref{lan}),\n$\\epsilon_0 (k)$ should vanish  as $k^2$ for small $k$. However\nthe operator in (\\ref{lan}) is a sum of squares and does not have zero\neigenvalues for non trivial regular $c(y)$. It is also not symmetric\nin $k$ for small values of $k$ but this seems to be less of  a problem.\nThe same conclusions seem to hold in 3 space dimensions. It is quite\npossible that perhaps a singular configuration $a^i$ exists\nwhich leads to  zero eigenvalue in (\\ref{lan}) at zero $k$\nbut I was not able to find it.\n\nThe strong coupling limit of lattice QCD suggests\n that  quarks in the fundamental reperesentation are confined\nwhereas they are only\nscreened if put  in the adjoint representation. This\ncrucial difference  comes from fairly simple quantum mechanics\nof color degrees of freedom related to  matching of group\nrepresentations in neighboring lattice points.\n In our rigid gauge rotor model a similar\nsimple quantum mechanics of  colors is retained. As a result the\neigenenergies of a systems of static quarks will be determined by\ndifferent\ncombinations of the color components of $I^{-1}$ depending on the\nrepresentation of the quarks.  E.g.,  as already mentioned\nin Section 3  when the quarks are taken in\nthe adjoint representation the $\\lambda$ matrices in the expressions\nfor $P$ and $Q$ in the Schr\\\"odinger  equations (\\ref{sch1})\nare replaced by their adjoint counterpartners $F$.\n\nIn order to find the optimal $a^i$ in a systematic way  one can\nfollow a variational approach and  minimize the\nground state energy of the rigid gauge rotations for  fixed positions\nof static quarks.\n This energy is given by a sum\nof the lowest eigenenergy of $H^A_{rot}$, Eq.(\\ref{hrot}) and the\ncolormagnetic energy given by the second term in (\\ref{ham}) with\nrigidly \"rotating\" $A(x)$, Eq.(\\ref{rgd}), i.e.\n\\begin{equation}\nE[a^i] = E_{rot}[a^i]  - \\frac{1}{2g^2}\\int d^3x Tr [d^i,d^j]^2 \\end{equation}\nVariation of this expression gives\n\\begin{equation}\n\\partial_i f^{ij} - ig [a^i,f^{ij}] = \\frac{1}{2}\\frac{\\delta E_{rot}}\n{\\delta a^i} , \\label{mfld} \\end{equation}\nwhere $f^{ij} = (i\/g) [d^i,d^j]$ .\n Eq.(\\ref{mfld}) is obviously gauge invariant.\n In the vacuum $E_{rot}$ is zero and the minimization of the second\nterm  simply gives the classical equation for $a^i$ in the vacuum.\nFor a quark- antiquark system $E_{rot}$ is non trivial and\ndepends on the distance between the quarks.  I plan to discuss\nthe solutions of the equation (\\ref{mfld}) and their relation\nto confinement elsewhere.\n\nIn the rest of this Section  as an illustration of a simple\n choice for the rigid field $a^i$ which allows\nto obtain some analytic results I consider it  to be diagonal,\n$a^i_{\\alpha \\beta} (x) = \\delta_{\\alpha \\beta} a^i_\\alpha (x)$.\nThe moment of inertia operator with such $a^i$ is\n\\begin{equation}\n-\\frac{1}{4g^2}\\left[d^i ,[d^i ,\\omega ] \\right] _{\\alpha \\beta} =\n-\\frac{1}{4g^2} \\left[ \\partial ^i - ig(a^i_\\alpha - a^i_\\beta )\\right] ^{2}\n\\omega_{\\alpha \\beta} . \\label{dmi} \\end{equation}\nUsing Green's function satisfying\n\\begin{equation}\n\\left(\\partial ^i - ig(a^i_\\alpha - a^i_\\beta )\\right) ^{2}\n J_{\\alpha \\beta }(x,y) = - \\delta (x-y),\\label{grf} \\end{equation}\nand following the procedure leading to Eq. (\\ref{hrot}) one finds the rigid\ngauge rotor Hamiltonian in this case\n\\begin{equation}\nH = \\frac{1}{4} \\int d^2 x d^2 y \\hat{G}_{\\alpha \\beta}(x) J_{\\alpha \\beta}(x,y)\n\\hat{G}_{\\beta \\alpha}(y). \\end{equation}\nThe Schr\\\"odinger   equation for the static quark--antiquark wave function\n(\\ref{wfn}) has the form (\\ref{peq}) with\n\\begin{equation}\nQ_{\\alpha \\gamma}(x_0) = \\frac{1}{4} \\delta_{\\alpha \\gamma}\\left[\n\\sum_\\beta J_{\\alpha \\beta}(x_0,x_0) - \\frac{1}{N}\n\\left(2 J_{\\alpha \\alpha}(x_0,x_0) -\n\\frac{1}{N}\\sum_\\beta J_{\\beta \\beta}(x_0,x_0)\\right)\\right] \\end{equation}\n\\begin{eqnarray}\nP_{\\alpha \\beta , \\gamma \\mu }(x_0,y_0) &=&- \\frac{1}{2}\n \\delta_{\\gamma \\mu} \\delta_{\\alpha \\beta} J_{\\alpha \\gamma}(x_0,y_0) + \\\\\n&+& \\frac{1}{2N}\\delta_{\\alpha \\gamma} \\delta_{\\mu \\beta}\\left[J_{\\beta \\beta}\n(x_0,y_0) + J_{\\gamma \\gamma}(x_0,y_0)\n - \\frac{1}{N} \\sum_\\nu J_{\\nu \\nu}(x_0,y_0)\\right]. \\nonumber \\end{eqnarray}\nThe diagonal components of $J$ are simple Coulomb propagators independent\nof the color so that the expressions for $Q$ and $P$ can be simplified\n further but I will not go into the details of this. Instead\nI will now consider the choice of $a^i$\nwhich corresponds to a much discussed in the\nliterature situation of a uniform colormagnetic field.\nI emphasize that in the present model  this field is uniform in the\nintrinsic, \"body fixed\" frame.   For simplicity I will\nfirst work in 2+1 dimensions  and will  try to extend to 3+1 in the\nnext section. I set\n\\begin{equation}\n a^i_\\alpha (x) =\n \\frac{1}{2}b_{\\alpha} \\epsilon_{ij}x^j\n\\label{cb1} \\end{equation}\nwhere the space indices $i,j$ presently run over the values 1 and 2.\nIn two space dimensions one can take $b$ diagonal in  color\nsince the transformation diagonalizing it is a part of $U$'s in\n (\\ref{rgd}).\nExplicit expression for $J$ is easily obtained\nin this case from the known  Green's function\nof a Schr\\\"odinger   equation in a constant magnetic field, cf. Ref.\\cite{fey},\n\\begin{equation}\nJ_{\\alpha \\beta}(x,y) = \\frac{1}{4\\pi} e^{i(gb_{\\alpha \\beta}\/2)\n\\epsilon_{ij} x^i y^j} \\int_0^{\\infty} \\frac{ds}{\\sinh s}\ne^{-(|g b_{\\alpha \\beta}|\/4)(x-y)^2 \\coth s} \\end{equation}\nwhere $b_{\\alpha \\beta} = b_\\alpha - b_\\beta$. For $x = y$ this expression\nis independent of color indices. It must be regulated to prevent the\ndivergence, e.g.\n\\begin{equation}\n \\frac{g^2 N}{2\\pi} \\int_{s_0}^{\\infty} \\frac{ds}{\\sinh s},\\end{equation}\nwhere $s_0$ is a regularization\ncutoff. Although $J(x,y)$ does not depend only on\nthe distance $|x-y|$ the Schr\\\"odinger  equation\n(\\ref{peq}) with $P$ and $Q$ based on such $J$\nis translationally invariant. Shifting  the coordinates by say a vector\n$h$ and simultaneously performing a gauge transformation of the wave function\n$c_{\\alpha \\alpha} \\rightarrow c_{\\alpha \\alpha}\\exp\\left[i(gb_{\\alpha}\/2)\n\\epsilon_{ij}h^i (x_0^j - y_0^j)\\right]$ leaves  Eq.(\\ref{peq})\ninvariant.  The integral in  the expression for $J(x,y)$\ncan be expressed in terms of the Bessel function\n$K_0(|g(b_{\\alpha} - b_{\\beta}|z^2\/4)$  with $z = x - y$ and for\n $|g (b_{\\alpha} - b_{\\beta})| z^2 \\rightarrow \\infty$\nit has the following asymptotic form\n\\begin{equation}\n\\frac{1}{4\\sqrt{\\pi}}(2|g(b_\\alpha - b_\\beta)|)^{-1\/4} \\exp\\left[ -\n |g(b_\\alpha - b_\\beta)| z^2\/4\\right]. \\end{equation}\nFor finite values of  $g|b_{\\alpha} - b_{\\beta}|$\nit decreases as a Gaussian at large separations\n$z$. This should lead to a similar decrease of\nthe eigenvalues of (\\ref{peq})  -- an entirely unsatisfactory behavior\nas far as the confinement is concerned.\nIn the next Section it will be seen that the situation may be\ndifferent in the large N limit.\n\n\\setcounter{equation}{0} \\ALTsect{ Large N Random Colormagnetic Fields.}\n\nAs mentioned in the Introduction\nrigid gauge field rotations should be relevant\nfor QCD in the large N limit  where it is expected that a\nmaster field configuration dominates the vacuum, \\cite{wit}.\nAs in the case of a condensate\nsuch a configuration can not be just some fixed\ngauge field potential $A^i_a(x)$.\nIt must be allowed to undergo free gauge \"rotations\"\nexactly as $a^i$ in\nEq.(\\ref{rgd}) since the gauge invariance is not expected to be  broken\nin the large N limit. The dynamics of these rotations can not be \"frozen\"\nand must be described by the gauge rotor Hamiltonian considered in\n Section 2. These \"rotations\" induce an interaction\nbetween quarks as was shown in Section 3 for static quarks and will be\ndemonstrated for dynamical quarks in Section 6 below where it will\n also be shown that in addition\n$a^i$ appears as a background field in the Dirac operator.\n\nAnother  important consideration is that for large N there is a large\nnumber of degrees of freedom operating at each space point which\nintroduces statistical elements in the theory, cf. Refs.\n\\cite{mat,ran,bor}.\nExperience with this limit for simple systems indicates that\ntwo types of gauge invariant physical operators  should exist, analogoes to\nmacroscopic and microscopic observables in thermodynamics. The former\ndepend on finite (relative to N) number of dynamical variables\nand involve sums over all labels of the degrees of freedom,\ni.e. the color indices. A simple example is\n$a^i_{\\alpha\\beta}a^i_{\\beta\\alpha}$, etc. Operators without such\nsummations, e.g., $a^i_{\\alpha\\beta}$ with fixed $\\alpha$ and $\\beta$\nbelong to the second  type which must be regarded as\nmicroscopic observables\n like ,e.g., a coordinate of a particle or a single\nspin variable in thermodynamic systems. The fluctuations of the macroscopic\noperators are suppressed and expectations of their products factorize at\n$N = \\infty$. This is not so for microscopic observables.\n\nOn the basis of these considerations one can adopt the following point of view.\nAfter allowing for free gauge rotations according to (\\ref{rgd}), i.e.\nafter transformation to the  body-fixed frame, one should\n consider $a^i(x)$ as  static\n random matrix functions described by a probability\ndistribution $P[a^i(x)]$. This distribution can be determined following\nthe ideas of the random matrix theory, cf.\n Ref.\\cite{ent}. To this end one should introduce the amount  of information\n (negative entropy)\n\\begin{equation}\nI\\left\\{P\\left[ a \\right]\\right\\} =\n \\int D\\mu[a^i(x)] P[a^i(x)] ln P[a^i(x)]  \\label{inf}\n\\end{equation}\nassociated with the $P[a^i(x)]$.\nMinimizing $I\\{P[a]\\}$ subject to suitably chosen constraints on\nmacroscopic-like variables should determine the least biased distribution\n$P[a^i(x)]$. As in statistical mechanics\nthe large N factorization should   then simply appear as a consequence of\nthe central limit theorem.\n\nThere are two crucial questions which need to be answered\nin following this procedure - what is the appropriate measure in\nthe integral (\\ref{inf}) and  what are the variables which should\nbe constrained. I hope to address the general answer to\nthese questions in the future work. Presently I will illustrate\nhow the procedure can be put to work for a uniform colormagnetic field,\nEq.(\\ref{cb1}).\n\n   In the limit of  large N\nonly the first terms in the expressions for $P$ and $Q$ above should\nbe retained and the Schr\\\"odinger   equation (\\ref{peq}) for diagonal components\nof the string amplitude becomes\n\\begin{equation}\n-2g^2\\sum_\\beta J_{\\alpha \\beta} (x_0,y_0) c_{\\beta \\beta} =\n(E - E^0_\\alpha) c_{\\alpha \\alpha}, \\label{meq} \\end{equation}\nwhere $E^0_\\alpha = 2g^2 \\sum_\\mu J_{\\alpha \\mu}(x_0,x_0)$.\nThe non diagonal string amplitudes decouple and satisfy a trivial equation\n$(E^0_\\mu + E^0_\\nu)c_{\\mu \\nu} = 2 E c_{\\mu \\nu}$ the eigenvalues of which are\nsimply the sums of the selfenergies. Without careful treatment of long and\nshort distance regularization in the large N limit one can not reliably\ndiscuss these eigenvalues taken separately and I will concentrate\non Eq.(\\ref{meq}).  Using translational invariance and\nwriting this equation for $x_0 = 0$ and $y_0 = z$ one obtains\n\\begin{equation}\n\\sum_{\\beta=1}^N \\int_0^{\\infty} \\frac{ds}{4\\pi \\sinh s}\ne^{-(|g (b_{\\alpha} - b_{\\beta})|\/4) z^2 \\coth s} c_{\\beta \\beta} =\n- \\frac{E}{2g^2} c_{\\alpha \\alpha}.\\label{teq}  \\end{equation}\nas the large N limit of the Schr\\\"odinger  equation for a static quark --\nantiquark pair in the rigid  gauge\nconfiguration corresponding to a uniform colormagnetic field in 2+1\ndimensions. One must still specify the large N scaling of various quantities\nwhich enter this equation. Provided each term in the sum on the left hand\nside is of the same order of magnitude  I get the standard\nscaling of the coupling constant  requiring that $g^2 N$ is held fixed.\nIn the exponential of the integrand one can then extract the finite combination\n$\\bar g = g\\sqrt{N}$. The problem is then to determine the\nscaling and in general the entire distribution of\nthe field components  $b_{\\alpha}\/\\sqrt{N}$. Regarding the behavior at\nlarge separations $z$ one notes that if\nthe limit of $N \\rightarrow \\infty$ is taken first in such a way that\nthe differences $|b_{\\alpha} - b_{\\beta}|\/\\sqrt{N}$ decrease then the Gaussian\ndecay can possibly be prevented.\n\nI use this example to  demonstrate how the ideas about the\nstatistical nature of the large N limit can be used to determine the\ndistribution of the components $b_\\alpha\/\\sqrt{N}$.\nI consider what happens with the Wilson loop\n$W(C) = \\frac{1}{N}<Tr P \\exp(ig\\oint_C A^idx_i)>$ in the present theory.\nChoosing the loop perpendicular to the time axis, inserting (\\ref{rgd})\nin $W(C)$, using its  gauge invariance and the explicit form (\\ref{cb1}) of\n$a^i$  one finds\n\\begin{equation}\n W(C) = \\frac{1}{N}\\sum_\\alpha e^{ig b_\\alpha S} = \\int_{-\\infty}^\\infty\n db \\rho (b) e^{igbS}\n\\end{equation}\nwhere $S$ is the area of the loop and $\\rho (b) =\n(1\/N)\\sum_{\\alpha=1}^N \\delta (b - b_\\alpha)$ is the density of the field\ncomponents. In the large N limit $\\rho(b)$ can be approximated\nby a smooth function provided the range of variations of b does\nnot grow with N. Assuming this one easily finds simple expressions for\n$\\rho (b)$ , e.g. Lorenzian\n\\begin{equation}\n\\rho (b) = \\frac{b_0 \\sqrt{N}}{\\pi (b^2 + Nb_0^2)} \\label{lor} \\end{equation}\nwhich lead to the area law dependence of the Wilson loop,\n$W(C) = \\exp (-\\bar{g}b_0 S)$. The combination $\\bar{g} b_0$ plays the role of\nthe string constant. The placing of $N$'s in (\\ref{lor}) was chosen in such\na way as to have this constant finite for $N \\rightarrow \\infty$.\n\nThe choice (\\ref{lor}) is the simplest possible.\nAny meromorfic function $\\rho(b)$ with poles in the upper plane\nwill give the area law with the string tension controlled by the\nposition of the pole closest to the real axis. One can also take functions\nwith other type of singularities in the upper complex plane, etc. The simple\nchoice (\\ref{lor}) gives area law for any S,\nmissing entirely the asymptotic freedom behavior\nat small S. One can attempt to correct this by choosing more\ninvolved expressions for $\\rho$. A  much more serious problem\nis that the space oriented Wilson loop may not be a good measure\nof confining properties in the model where one has\nnever worried about the  Lorenz invariance.\n\nAdopting any  form of the \"single component\"  density $\\rho(b)$\nstill leaves the distribution of the values of $b_\\alpha$ needed\nin, e.g., Eq. (\\ref{teq}) largely undetermined. Using\n statistical concepts described in the beginning of this\nSection one should  view $b_\\alpha$'s as random quantities\nand introduce their joint probability distribution\n$P(b_1, b_2,...,b_N)$ which should be such\nthat $\\rho(b)$ is reproduced but is\notherwise  maximally random, i.e. contains least amount of  information.\nThe question immediately arises as to whether $\\rho(b)$ is the only\nquantity which should constrain $P(b_1,...,b_N)$ and what is the complete\nset of such constraints. In the absence of  general answers\nI take  $\\rho(b)$ controlling $W(C)$ as an example\nand determine the distribution $P(b_1,...,b_N)$ by\nminimizing the appropriate negative entropy (information)\nwith this constraint.\n\nThe quantities $b_\\alpha$'s are eigenvalues of a\nhermitian, in general complex matrix. The information\ncontent of a probability distribution $P(b_1,...,b_N)$ of such eigenvalues\nis a well studied question, cf. Ref.\\cite{ent}. It is\n\\begin{equation} \\int d\\mu[b] P(b_1,...,b_N) ln P(b_1,...,b_N)\\label{ent}\\end{equation}\nwhere the measure is $d\\mu[b] = const \\prod_{\\alpha > \\beta} |b_\\alpha\n- b_\\beta|^2\ndb_1 db_2...db_N$,  reflecting the repulsion of the eigenvalues. Minimizing\n(\\ref{ent}) under the condition of a given $\\rho (b) =\n(1\/N)\\sum_\\alpha (b - b_\\alpha)$ one finds\n\\begin{equation}\nP(b_1,...,b_N) = const \\;\\exp \\left(\\sum_{\\alpha\\neq\\beta}^N ln|b_\\alpha -\nb_\\beta| - 2N\\sum_\\alpha^N \\int_{-\\infty}^\\infty ln|b_\\alpha - b^\\prime|\n\\rho (b^\\prime ) db^\\prime\\right).\\label{dis} \\end{equation}\n Using, e.g. Eq. (\\ref{lor}) for $\\rho (b)$ this expression becomes explicitly\n\\begin{equation}\nP(b_1,...,b_N) = const\\; \\exp \\left(\\sum_{\\alpha\\neq\\beta}^N ln|b_\\alpha -\nb_\\beta| - N\\sum_\\alpha^N ln (b_\\alpha^2 +Nb_0^2) \\right).\\label{pbb} \\end{equation}\nThe constant in front of this expression must assure the normalization of\n$P$ and can be calculated by the methods described in Ref.\\cite{meh}.\nUsing the standard  interpretation  of $P(b_1,...,b_N)$  as a partition\nfunction of a fictitious Coulomb gas\n one can say that the \"particles\" $b_\\alpha$\nare \"repelled\" from each other by the first\nterm in its exponential but are kept within the interval  $b_0$ by the second\nterm representing the interaction with the background \"charge\" distributed\naccording to $\\rho(b)$. The average distance $|b_\\alpha - b_\\beta| \\sim b_0\/N$\nbecomes very small in the large N limit. The Schr\\\"odinger   equation (\\ref{teq}) is now\na random matrix equation with the probability distribution of its elements\ncontrolled by the $P(b_1,...,b_N)$ above. The actual numerical\nsolution of this equation is now in progress.\n\nIn a similar way one can consider rigid gauge configuration representing\nuniform colormagnetic field in $3 + 1$ dimensions. This field in the intrinsic\nframe corresponds to the choice\n\\begin{equation}\na^i(x) = \\frac{1}{2} \\epsilon_{ijk} b^j x^k \\label{cb2} \\end{equation}\nwhere  now however  the three color matrices $b^i$ in general cannot be\nassumed diagonal. For such non-diagonal colormagnetic field the inversion\nof the moment of inertia operator  $-(1\/4g^2)[d^i,[d^i,\\omega]]$\nrequires a solution of a matrix differential equation. This equation\nsimplifies considerably if $b^i$'s are nonetheless restricted to be diagonal,\n$b^i_{\\alpha \\beta} = \\delta_{\\alpha \\beta} b^i_\\alpha$. Then\nthe equation (\\ref{dmi}) is still valid with the index $i$ now running from\n1 to 3. In the following equation (\\ref{grf}) for the Green's function  one\nshould just replace $(b_\\alpha - b_\\beta)\\epsilon_{ij}x^j$ by\n$\\epsilon_{ijk}(b^j_\\alpha -\n b^j_\\beta)x^k$. The expression for this Green's function is known\n and one can repeat all the\nsteps leading to the  static quark--antiquark\nSchr\\\"odinger  equation which is the analog of Eq.(\\ref{teq}) in 3+1 dimensions.\n\nTurning again to the Wilson loop one finds in this case\n $\\oint_C a^i dx_i =  b^j S_j$ with $S_j = (1\/2) \\epsilon_{jki}\n\\oint_C x^k dx_i$ so that $\\left( \\oint_c a^i dx_i \\right)_\\alpha =\nb_\\alpha S \\cos \\theta _\\alpha$ (no sum over $\\alpha$) where\n$S = \\sqrt{\\sum_i (S^i)^2},  b_\\alpha = \\sqrt {\\sum_i (b^i_\\alpha)^2}$\nand $\\theta_\\alpha$ -- the angle between the vectors $b^i_\\alpha$ and $S^i$\nat a given $\\alpha$. S is the area of the loop when it is planar and\nis related to the minimal area in general.\nThe Wilson loop is\n\\begin{eqnarray}\nW(C) & = & \\frac{1}{N}<\\sum_\\alpha e^{igSb_\\alpha \\cos \\theta_\\alpha}>\n = \\nonumber \\\\\n   & = & \\frac{1}{N}\\sum_\\alpha 2\\pi \\int _{-1}^{1} d(\\cos \\theta_{\\alpha})\ne^{igSb_\\alpha \\cos \\theta_\\alpha} = \\label{wl3} \\\\\n    & = & \\frac{4\\pi}{gS} \\int_0^{\\infty} \\frac{db}{b} \\rho(b) \\sin (gbS),\n \\nonumber \\end{eqnarray}\nwhere $\\rho (b) = (1\/N) \\sum_\\alpha \\delta (b - b_\\alpha)$ is the density\nof the positive lengths of the color components of the vector\n$b^i$. In (\\ref{wl3}) I have performed the angle averaging which must be\npresent in the vacuum wavefunction.\nOne can easily choose $\\rho (b)$, e.g. the square of Lorentzian\n\\begin{equation}\n\\rho(b) = \\frac{4 b_0 b^2\\sqrt{N}}{\\pi (b^2 + Nb_0^2)^2} \\label{rh2} \\end{equation}\nwhich gives  the area law $W(C) = 4\\pi\\exp (-\\bar{g}b_0S)$. This choice is\nagain not unique  and gives the area law for any $S$. It has a powerlike\ntale as opposed to the perturbative Gaussian.\n\nThe statistical arguments for finding the entire distribution\nof $b^i_\\alpha$  can be used in $3+1$ dimensional\ncase as well with the difference that in this case the density of the lengths\nof the vectors $b^i_\\alpha$ is fixed by, e.g. Eq.(\\ref{rh2}) and their directions\nare distributed isotropically.\n\n\\setcounter{equation}{0} \\ALTsect{Dynamic Quarks}\n\nDynamic quarks can be easily included in the rigid gauge rotation model.\nFor this I define quark fields in the \"rotating frame\", $q = U\\hat{q}$,\nuse Eq. (\\ref{rgd}) in the second term of the QCD Hamiltonian  (\\ref{ham})\nand replace the first term in it by (\\ref{hrot}). Using moreover\n the Gauss law constraint (\\ref{gl}) I can write the original\nQCD Hamiltonian (\\ref{ham}) in the rigid gauge rotor limit as\nexpressed in terms of the quark fields only,\n\\begin{equation}\nH_{rot} = \\frac{1}{2}\\int d^3 x d^3 y\n\\hat{\\rho}_a (x) I_{ab}^{-1}(x,y,[a^i]) \\hat{\\rho}_b (y)\n+\\int d^3 x \\hat{q}^{+}(x)[\\alpha^{i}(p^{i}-g\na^{i}(x))+\\beta m]\\hat{q}(x), \\label{hrt1} \\end{equation}\nwhere $\\hat{\\rho}_a = \\hat{q}^{+}\\lambda ^a\\hat{q}$\nare the color quark densities in the rotating frame.\nThe Hamiltonian  $H_{rot}$  describes quarks with gauge strings attached to them,\ni.e. $q(x)$  are multiplied by $U^{+}(x)$. They  move in an external\ncolormagnetic field described by the vector potential $a^i(x)$ and\n interact\nvia an instantaneous  interaction  $I^{-1}_{ab}(x,y,[a^i])$ also\ndepending on $a^i$ via Eq.(\\ref{mom}). The simultaneous appearance of\nthe rigid gauge field configuration both as a background and as \"inducing\"\nthe quark-quark interaction is ultimately a consequence of the\ngauge invariance which requires that non dynamical rigid gauge fields\nappear only in the form (\\ref{rgd}).\n\n$H_{rot}$ is gauge invariant since the operators $\\hat{q}$ are. Moreover\nthis Hamiltonian should only be used in the color singlet sector of the\ntheory since I have used the Gauss law to derive it.\n\nFor the vanishing $a^i$ the Hamiltonian  $H_{rot}$ describes free quarks interacting\nCoulombically. Also for a general non zero $a^i$\n$H_{rot}$ should be regarded as the QCD analogue\nof the QED Hamiltonian  in which only the instantaneous Coulomb\ninteraction between the charges has been retained.\nIndeed the analogue of the rigid gauge \"rotations\"  (\\ref{rgd}) in QED\nis $A^i(x,t) = a^i (x) + \\partial ^i \\chi(x,t)$\nwith abelian $U = \\exp (ig\\chi(x,t))$, fixed rigid $a^i (x)$ and dynamical\n$\\chi (x,t)$. Repeating the steps leading to (\\ref{hrt1}) one will\nderive in the QED case the Coulomb interaction between the charge densities.\n\nRegarding the possible role of $a^i(x)$ as the master field in the large N\nlimit\none has in $H_{rot}$ a way in which this field should enter the quark\nsector of the theory, i.e. serving both as a background field and\nperhaps somewhat surprisingly also  controlling\nthe quark interaction. Following the developments of Section 5 this\nfield should be regarded as random. The appearance of a random interaction\nbetween the quarks means that a possible mechanism for confinement\nof dynamical quarks in the large N limit could be\nrelated to the localization of their relative distances.\nThe possible connection between\n confinement and localization has already been mentioned\nin the past but usually in the context of a random background field and not\nwith random interactions as appear in the present model.\n\nThe Hamiltonian  (\\ref{hrt1}) takes exact account of the gauge symmetry.\nOne must however also worry about global symmetries.\nFor any N the Hamiltonian  $H_{rot}$ may serve as a possible basis for various\nphenomenological\ndevelopments. Both for this matter and conceptually  one must face the\nissue that allowing for an arbitrary x-dependence of various color components\nof $a^i$  in (\\ref{hrt1}) leads to breaking\nof important symmetries such as translational, rotational, Lorenz,\ntime reversal, and various discrete symmetries. Of course the breaking\nof continuous space symmetries is not uncommon in phenomenology,\ne.g. the bag model, the quark potential model, the Skyrme model, etc.\n Symmetries can be restored by\nconsidering all configurations translated by the symmetry and integrating\nover them using collective coordinates. This of course\napplies to both continuous and discrete symmetries.\nIn the absence of the guidance from the\nsymmetries a more dynamical criterion for fixing\n$a^i$ seems to be the condition of lowest  energy.\nThis leads naturally to a  generalization of the variational approach\nof Section 4 in which the  variational energy\nshould be replaced by the ground state energy  $E_0[a^i(x)]$\nof (the suitably regularized)  $H_{rot}$  found for a given $a^i (x)$.\nShould the solution $a^i$\n break a global symmetry, the symmetry \"images\" of this $a^i$\nwill also be solutions and one should \"sum\" over all of them in a standard\nway thereby restoring the symmetry.\nThis variational approach may be combined with the Hartree-Fock\nmethod which should allow to calculate $E_0 [a^i(x)]$ approximately. The\nHartree-Fock approximation for fermions was shown to be consistent with\nthe large N approximation, c.f., Ref.\\cite{sal}.\nIn a combined approach one should form for fixed $a^i$\nan expectation value of\n$H_{rot}$ with respect to a trial state of a chosen\ncolor singlet  configuration of quarks\n(e.g., vacuum, baryon, etc) which must be a product state, i.e. such that\nthe expectation values with respect to this state have a non interacting\nfactorized form,i.e.,\n\\begin{eqnarray}\n <\\hat{\\rho}_a (x) \\hat{\\rho}_b (y)> &=& \\lambda^a_{\\alpha \\beta}\n\\lambda^b_{\\gamma \\delta}(\n<\\hat{q}_\\alpha^{+}(x)\\hat{q}_\\beta(x)><\\hat{q}_\\gamma^{+}(y)\\hat{q}_\\delta(y)> -\\nonumber\n\\\\\n&-&<\\hat{q}_\\alpha^{+}(x)\\hat{q}_\\delta(y)><\\hat{q}_\\gamma^{+}(y)\\hat{q}_\\beta(x)>).\n\\label{hf}  \\end{eqnarray}\nFor  a global color singlet state\n$<\\hat{q}_\\alpha^{+}(x)\\hat{q}_\\beta(y)> = \\delta_{\\alpha \\beta}\\rho (y,x)$, with\n$\\rho(x,y) = \\frac {1}{N}\\sum_{\\gamma}<\\hat{q}_\\gamma^{+}(y)\\hat{q}_\\gamma(x)>$\nand therefore\n\\begin{equation}\n<\\hat{\\rho}_a (x) \\hat{\\rho}_b (y)> = -2\\delta_{ab}\\rho(x,y)\\rho(y,x)\n\\label{fck} \\end{equation}\n\nFollowing the standard Hartree-Fock routine, cf., Ref.\\cite{hfr},\nthe single quark density matrix can be expanded\nin terms of a complete set of functions, i.e.,  $\\rho (x,y) =\n\\sum_n f_n \\psi_n (x) \\psi_n^* (y)$. At this stage it is customary\nto add the so called Slater determinant condition\nwhich means that the single quark states $\\psi_n$ have sharp occupations\n $f_n = 0$ or $1$.  This condition and its\ncompatibility with the large N limit and Lorenz invariance were discussed in\nRef.\\cite{bor}. Adopting this condition, using (\\ref{fck}) in forming the\n expectation\n $E_{HF}[\\rho(x,y),a^i(x)] = <H_{rot}>$ and varying with the respect to\n$\\psi_n(x)$'s with constraints on their\nnormalization one obtains the Hartree-Fock equation\n\\begin{equation}\n[\\alpha^{i}p^{i}+\\beta m]\\psi_n(x)\n- 2\\int d^3 y I_{aa}^{-1}(x,y,[a^i]) \\rho (x,y)\\psi_n(y)\n = \\epsilon_n \\psi_n(x), \\label{feq} \\end{equation}\nwhich appears here as a selfconsistent\nDirac equation for the quark wave functions $\\psi_n(x)$.\nNote that  in this equation $a^i$ disappeared from the\nDirac operator and enters only the interaction $I_{aa}^{-1}(x,y)$.\n Solutions of (\\ref{feq}) determine the optimum $\\rho$ and thus\n$E_{HF}$ for a given $a^i$.\n\n Hartree-Fock equation similar to\n(\\ref{feq}) has  been investigated in the  1+1 dimensional QCD, \\cite{bor}\n although\nthere for obvious reasons the field $a^i$ was absent and $I^{-1}(x,y)$ was\nsimply $|x - y|$.   Both\nthe t'Hooft meson spectrum and  the baryon soliton solutions\nwere found in this approach\nin 1+1 dimensions. For small quark masses the baryon was the realization of the\nskyrmion described in the quark language. If\nsuccessful the approach based on Eq.(\\ref{feq}) can possibly lead to similar\nresults in 2+1 and 3+1 dimensions.\n\n\n\\setcounter{equation}{0} \\ALTsect*{Acknowledgement}\nUseful discussions with S. Elizur, S. Finkelstein, J. Goldstone, K. Johnson,\nA. Kerman,  F. Lenz, M. Milgrom, J. Negele, J. Polonyi, E. Rabinovici,\nS. Solomon and H. Weidenm\\\"uller are acknowledged gratefully. Special\nacknowledgement is to A. Schwimmer for patient teaching and\nanswering my questions and for critical reading of the manuscript. Parts of\nthis work were done during visits in Max-Planck-Institut fur Kernphysik\nat Heidelberg and I wish to thank H. Weidenm\\\"uller for warm hospitality\n there.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe hadronic decays of heavy quarkonia below the threshold for heavy \nmeson pair production are understood to proceed predominantly via\nthree intermediate gluons. One of the gluons can be replaced by a \nphoton with a penalty of order the ratio of coupling constants, \n$\\alpha \/ \\alpha_s$. Such exclusive radiative decays of the heavy vector\nmesons $J\/\\psi$\\ and $\\Upsilon$\\ have been the subject of many experimental and\ntheoretical studies. For the experimenter, the final states from\nradiative decays are relatively easy to identify as they have a high\nenergy photon, a low multiplicity of other particles, and low\nbackground. Theoretically, the radiative decays of heavy quarkonia\ninto a single light hadron provide a particularly clean environment to \nstudy the conversion of gluons into hadrons, and thus their study is a\ndirect test of QCD. $\\Uos \\to \\gamma \\eta^{\\prime}$\\ is one such candidate channel. This\ndecay channel has been observed to be produced in the $J\/\\psi$\\\ncharmonium system (the $1^3\\text{S}_1$ state of $c \\bar{c}$) with\n$\\mathcal{B}(J\/\\psi\\to\\gamma \\eta^{\\prime}) = (4.71\\pm0.27)\\times10^{-3}$~\\cite{PDG}.  \nNaive scaling predicts that decay rates for radiative $\\Upsilon(1\\text{S})$ decays\nare suppressed by the factor \n$(q_{b}m_{c}\/q_{c}m_{b})^{2}$ $\\approx 1\/40$ \nwith respect to the corresponding $J\/\\psi$\\ radiative decays.\nThis factor arises because the quark-photon coupling is proportional\nto the electric charge, and the quark propagator is roughly\nproportional to $1\/m$ for low momentum quarks. Taking into account the\ntotal widths~\\cite{PDG} of $J\/\\psi$\\ and $\\Upsilon(1\\text{S})$, the branching fraction\nof a particular $\\Upsilon(1\\text{S})$ radiative decay mode is expected to be around\n0.04 of the corresponding $J\/\\psi$\\ branching fraction. However, the CLEO \nsearch~\\cite{CleoEtaPrimeStudy} for $\\Uos \\to \\gamma \\eta^{\\prime}$\\ in $61.3\\,\\rm pb^{-1}$ of data\ncollected with the CLEO~II detector found no signal in this mode, and\nresulted in a 90\\% confidence level \nupper limit of $1.6\\times10^{-5}$ for the branching\nfraction $\\Uos \\to \\gamma \\eta^{\\prime}$, an order of magnitude smaller than this\nexpectation.\n\nThe two-body decay \\UpsToGammaFTwo\\ has been\nobserved~\\cite{CleoF2_1270} in the older CLEO~II $\\Upsilon(1\\text{S})$ analysis,\nand this observation has been confirmed~\\cite{LuisAnalysis,Holger}, with much\ngreater statistics, in CLEO~III data. \nThe measurement $\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma f_2(1270)) = (10.2\\pm1.0) \\times10^{-5}$, \nfrom the combination of the two CLEO~III measurements, \nis $0.074\\pm0.010$ times the corresponding $J\/\\psi$\\\ndecay mode, showing a deviation of roughly a factor of two from the\nnaive scaling estimates.\nIn radiative $J\/\\psi$\\ decays the\nratio of $\\eta^{\\prime}$\\ to \\fTwo\\ production is $3.4\\pm0.4$. If the same ratio\nheld in $\\Upsilon(1\\text{S})$, the $\\eta^{\\prime}$\\ channel would be clearly visible.\nThe channel $\\Uos \\to \\gamma \\eta$\\ has\nreceived significant theoretical attention. \nThis channel has been observed in $J\/\\psi$\\ decays~\\cite{PDG}\nwith the branching fraction of $(9.8\\pm1.0)\\times 10^{-4}$, a value\nsmaller by a factor of \nfive\nthan $\\mathcal{B}(J\/\\psi\\to\\gamma \\eta^{\\prime})$. \nThe previous CLEO search of $\\Upsilon(1\\text{S})$ decays produced an upper limit of\n$2.1\\times10^{-5}$ at the 90\\% confidence level \nfor this mode~\\cite{CleoEtaStudy}.\n\n\nSeveral authors have tried to explain the lack of signals in radiative \n$\\Upsilon(1\\text{S})$ decays into pseudoscalar mesons, using a variety of models\nwhich produce branching fraction predictions of \n$10^{-6}\\ \\text{to}\\ 10^{-4}$.\nEmploying the Vector \nMeson Dominance Model (VDM), Intemann~\\cite{Intemann} predicts the\nbranching fractions for the heavy vector meson radiative decay into\nlight pseudoscalar mesons. Using  the mixing mechanism of $\\eta$,\n$\\eta^{\\prime}$\\ with the as-yet-unobserved pseudoscalar resonance $\\eta_{b}$,\nChao~\\cite{KTChao} first calculated the mixing angle\n$\\lambda_{\\eta\\eta_{b}}$ in order to estimate the radiative branching\nfractions. Baier and Grozin~\\cite{BaierGrozin} showed that for light\nvector mesons (such as $J\/\\psi$) there might be an additional ``anomaly''\ndiagram that contributes significantly to the radiative decays. Noting\nthat VDM has no direct relation to QCD as the fundamental theory of\nstrong interactions, and referring to~\\cite{Intemann},\nMa tries to address the problem by using factorization at tree level\nwith NRQCD matrix elements to describe the heavy vector meson portion\nmultiplied by a set of twist-2 and twist-3 gluonic distribution\namplitudes~\\cite{JPMa}.\n\n\n\\section{Detector and Data Sample}\nThis study is based upon data collected by the CLEO III detector\nat the Cornell Electron Storage Ring (CESR). CLEO III is\na versatile multi-purpose particle detector described fully\nelsewhere~\\cite{cleoiii-detector}. Centered on the $e^+e^-$ interaction\nregion of CESR, the inner detector consists of a silicon strip vertex\ndetector and a wire drift chamber measuring the momentum vectors and\nthe ionization energy losses ($dE\/dx$) of charged tracks based on their\ntrajectories in the presence of a 1.5T solenoidal magnetic field. The\nsilicon vertex detector and the drift chamber tracking system together\nachieve a charged particle momentum resolution of 0.35\\% (1\\%) at\n1\\,\\rm GeV\/$c$\\ (5\\,\\rm GeV\/$c$)\nand a fractional $dE\/dx$\\ resolution of 6\\% for hadrons and 5\\% for electrons. \nBeyond the drift chamber is a Ring Imaging Cherenkov Detector, RICH,\nwhich covers 80\\% of the solid angle \nand is used to further identify charged particles by giving for each\nmass hypothesis the fit likelihood to the measured Cherenkov radiation\npattern. After the RICH is a CsI crystal calorimeter that covers 93\\%\nof the solid angle, allowing both photon detection and electron\nsuppression. The calorimeter provides an energy \nresolution of 2.2\\% (1.5\\%) for 1\\ \\rm GeV\\ (5\\ \\rm GeV) photons. Beyond the calorimeter \nis a superconducting solenoidal coil providing the magnetic field,\nfollowed by iron flux return plates with wire chambers interspersed \nat 3, 5, and 7 hadronic interaction lengths (at normal \nincidence) to provide\nmuon identification.\n\nThe data sample has an integrated luminosity of $1.13\\,\\rm fb^{-1}$ taken at the\n$\\Upsilon(1\\text{S})$ energy $\\sqrt{s} = 9.46\\ \\rm GeV$, which corresponds to \n$N_{\\Upsilon(1\\text{S})} = 21.2\\pm0.2$ million $\\Upsilon(1\\text{S})$ decays~\\cite{CLEO-III-NUPS}. \nThe efficiencies for decay chain reconstruction were obtained from\nMonte \nCarlo simulated radiative events generated with the \n($1+\\cos^{2}\\theta$) angular distribution expected for decays \n$\\Upsilon(1\\text{S}) \\to \\gamma+\\text{pseudoscalar}$. The Monte Carlo simulation of\nthe detector response was based upon GEANT~\\cite{GEANT}, and\nsimulation  events were processed in an identical fashion to data.\n\n\n\n\\newcommand{$\\sigma_{\\pi,i}$}{$\\sigma_{\\pi,i}$}\n\\newcommand{$\\sigma^{2}_{\\pi}$}{$\\sigma^{2}_{\\pi}$}\n\\newcommand{$\\sum_{i}^{3}\\sigma^{2}_{\\pi,i}$}{$\\sum_{i}^{3}\\sigma^{2}_{\\pi,i}$}\n\n\\newcommand{$S_{\\pi,i}$}{$S_{\\pi,i}$}\n\\newcommand{$S^{2}_{\\pi}$}{$S^{2}_{\\pi}$}\n\\newcommand{$\\sum_{i}^{3}S^{2}_{\\pi,i}$}{$\\sum_{i}^{3}S^{2}_{\\pi,i}$}\n\n\\newcommand{$(m_{\\gamma\\gamma} - m_{\\pi^0})\/\\sigma_{\\gamma\\gamma}$}{$(m_{\\gamma\\gamma} - m_{\\pi^0})\/\\sigma_{\\gamma\\gamma}$}\n\\newcommand{$((M_{reconstructed}-M_{\\pi^0})\/resolution)$}{$((M_{reconstructed}-M_{\\pi^0})\/resolution)$}\n\\newcommand{\\SDeDxDefn}{\\text{S}_{dE\/dx} \\equiv\n(dE\/dx(\\text{measured}) - dE\/dx(\\text{expected}))\/\\sigma_{dE\/dx}}\n\\newcommand{\\text{S}_{dE\/dx}}{\\text{S}_{dE\/dx}}\n\\newcommand{$\\text{S}_{dE\/dx}^{2}$}{$\\text{S}_{dE\/dx}^{2}$}\n\\section{Event Selection and Results}\\label{sec:event-selection}\nIn our search for $\\Uos \\to \\gamma \\eta$\\ and $\\Uos \\to \\gamma \\eta^{\\prime}$, we reconstruct $\\eta$\nmesons in the modes $\\eta \\to \\gamma \\gamma$, $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$, and $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$; the\nlatter two will collectively be referred to as $\\eta\\to3\\pi$. We\nreconstruct the\n$\\eta^{\\prime}$\\ meson in the mode $\\eta \\pi^+ \\pi^-$ with\n$\\eta$ decaying in any of the above modes, and in addition, \nthe mode $\\eta^{\\prime} \\to \\gamma \\rho^0$, where $\\rho^0 \\to \\pi^+\\pi^-$. \nFrom the CLEO~II studies~\\cite{CleoEtaPrimeStudy,CleoEtaStudy} we\nexpected five out of the seven modes under investigation to be relatively\nbackground free and so we employ minimal selection\ncriteria to maximize sensitivity and minimize possible systematic\nbiases. \nThe other two, $\\eta \\to \\gamma \\gamma$ and $\\eta^{\\prime} \\to \\gamma \\rho^0$, have large branching\nfractions, but also large backgrounds, and so our event selection for these\nmodes aims to decrease the background with a corresponding loss of\nefficiency.  \n\nOur general analysis strategy is to reconstruct the complete decay\nchain ensuring that none of the constituent tracks or showers have\nbeen used more than once, then kinematically constrain the intermediate\n$\\pi^{0}$\\ and $\\eta$ meson candidates  to their nominal masses~\\cite{PDG},\nand finally require the event to be consistent with having the 4-momentum of the\ninitial $e^+e^-$ system. Multiply-reconstructed $\\Upsilon(1\\text{S})$ candidates in an\nevent, a problem of varying severity from mode to mode, is dealt with \nby selecting the combination with lowest $\\chi^{2}_{\\mathrm{Total}}$, the sum of\nchi-squared of the 4-momentum constraint ($\\chi^{2}_{\\text{P}4}$) and chi-squared of\nall the mass-constraints involved in a particular decay chain. For\nexample, there are four mass-constraints involved in the decay chain\n$\\Uos \\to \\gamma \\eta^{\\prime};\\EtaThreePZ$, three $\\pi^{0}$\\ mass-constraints and one $\\eta$\nmass-constraint. \nThe mode $ \\Uos \\to \\gamma \\eta; \\EtaThreePZ $ is an exception in which we\npreferred to accept the $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$ candidate having the lowest\n$S^{2}_{\\pi}$\\ $\\equiv$ $\\sum_{i}^{3}S^{2}_{\\pi,i}$, with $S_{\\pi,i}$\\ $\\equiv$\n$(m_{\\gamma\\gamma} - m_{\\pi^0})\/\\sigma_{\\gamma\\gamma}$\\ of the \\textit{i}th $\\pi^{0}$ candidate.  \nThe yield is obtained by counting the number of final state $\\eta$ or\n$\\eta^{\\prime}$\\ candidates within our acceptance mass window defined as the\ninvariant mass region centered around the mean value and providing\n98\\% signal acceptance as determined from signal Monte Carlo. Whenever\npossible, an event vertex is calculated using the information from\nthe charged tracks, and the 4-momentum of the photon candidates is then\nrecalculated, assuming that the showers originate from the event\nvertex rather than the origin of the CLEO coordinate system. This produces\nan improvement in the $\\eta$ and $\\eta^{\\prime}$\\\ncandidates' invariant mass resolution of roughly 10\\%, leading to a slight\nincrease in the sensitivity of the measurement.\n\nThe CLEO~III trigger~\\cite{cleoiii-trigger} relies upon two components:\n(1) the tracking-based ``axial'' and ``stereo'' triggers derived from\nthe signals on the 16 axial layers of the drift chamber, and  the signals\nregistered on the chamber's 31 stereo layers, and (2) the calorimeter-based\ntrigger derived from the energy deposition in the CsI crystal\ncalorimeter. The events for the ``all neutral'' modes $\\Uos \\to \\gamma \\eta; \\EtaGG$\\ and\n$ \\Uos \\to \\gamma \\eta; \\EtaThreePZ $ are collected by the calorimeter-based trigger\ncondition requiring two high energy back-to-back showers.\nWe demand that triggered events meet the\nfollowing analysis requirements: (a) a high energy calorimeter shower not\nassociated with a charged track, having a lateral profile consistent\nwith being a photon, and  having a measured energy greater than\n4.0\\ \\rm GeV\\ must be present; \n(b) there must be the correct\nnumber of pairs of oppositely charged, \ngood quality tracks with usable $dE\/dx$\\ information. \nThe efficiency of these requirements is more than 60\\% in\nmodes involving charged tracks and approximately 54\\% and 45\\% \nfor cases where $\\eta \\to \\gamma \\gamma$ and $\\eta\\to 3\\pi^{0}$, respectively.\n\n\nThe photon candidates we use in forming $\\pi^{0}$\\ and $\\eta \\to \\gamma \\gamma$ candidates\nhave minimum energy depositions of 30\\ \\rm MeV\\ and 50\\ \\rm MeV,\nrespectively. All photon candidates are required to be not associated\nto charged tracks, and at least one of the photon candidates of \neach pair must have\na lateral profile consistent with that expected for a photon. The\nphoton  candidates we use in reconstructing the $\\eta$ meson in the\n$\\gamma \\gamma$\\ mode must be detected either in the fiducial barrel or\nthe fiducial endcap\\footnote{The fiducial regions of the barrel and \nendcap are defined by \n$|\\cos(\\theta)|<0.78$ and $0.85<|\\cos(\\theta)|<0.95$,\nrespectively; the region between the barrel fiducial region and the\nendcap fiducial region is not used due to its relatively poor\nresolution. \n} calorimeter region only. \nThese candidates are then kinematically\nconstrained to the nominal meson mass, the exception being $\\Uos \\to \\gamma \\eta; \\EtaGG$,\nwhere no mass-constraining was done to the $\\eta$ candidate, because\nwe examine $m_{\\gamma\\gamma}$ in this mode to determine our yield.\n\nThe $\\eta$ candidates in the mode $\\pi^+\\pi^-\\pi^0$ are built by first\nforcing pairs of oppositely charged quality tracks to originate from a\ncommon vertex. The $\\pi^{0}$\\ candidate having invariant mass within\n$7\\sigma_{\\gamma\\gamma}$ is then added to complete the reconstruction\nof $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$ candidates. The charged tracks are required to be\nconsistent with being pions by adding the pion hypothesis $\\SDeDxDefn$\nin quadrature for two tracks and requiring the sum of $\\text{S}_{dE\/dx}^{2}$\\ to be\nless than 16. \n\nIn the case of $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$, the $\\eta$ candidate is simply built by\nadding three different $\\pi^{0}$\\ candidates, where no constituent photon\ncandidate contributes more than once in a candidate $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$\nreconstruction. The $\\pi^{0}$\\ candidates are selected by requiring\n$S_{\\pi}<10.0$. In order to increase the efficiency in this mode, \nan exception was made to the fiducial region requirement, and\nphotons in the gap between the barrel and endcap fiducial regions\nwere allowed.\n\n\n\\subsection{The Decay $\\Upsilon\\to\\gamma\\eta ,\\eta\\to 3\\pi$}\nThe $\\Upsilon$ candidate in the mode $\\gamma \\eta$ is\nformed by combining a high-energy photon ($E>4\\ \\rm GeV$) with the $\\eta$\ncandidate, requiring that this photon is not a daughter of the $\\eta$\ncandidate. The $\\Upsilon$ candidate is then subjected to the\n4-momentum constraint of the initial $e^+e^-$ system. In the case of\n$\\eta\\to3\\pi$, multiply reconstructed $\\Upsilon$ candidates were\nrestricted by selecting only one candidate. \nFor $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$, we select the candidate with the lowest $\\chi^{2}_{\\mathrm{Total}}$,\nthe sum of chi-squared of the 4-momentum constraint\nand chi-squared of the mass-constraint to the $\\pi^{0}$\\ candidate.\nFor $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$, we select the candidate with the smallest $S^{2}_{\\pi}$.\nThe selected $\\Upsilon$ candidate is further required\nto satisfy the 4-momentum consistency criterion, restricting \n$\\chi^{2}_{\\text{P}4}$\\ $<100$ for $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$ and a less stringent cut of 200\nfor $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$ measurements.\nIn addition, we limit the number of reconstructed calorimeter showers \nfor the mode $ \\Uos \\to \\gamma \\eta; \\EtaThreePZ $ to minimize backgrounds such as\n$e^{+} e^{-} \\to \\gamma \\phi$\\ where \\phiKsKl\\ without jeopardizing the signal efficiency.\n\nFrom Monte Carlo simulations, the overall reconstruction efficiencies,\n$\\epsilon_i$, for each channel are determined to be $(28.5\\pm4.3)\\%$\nand $(11.8\\pm1.9)\\%$ for the decay chains $\\Upsilon\\to\\gamma\\eta,\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$ and\n$\\Upsilon\\to\\gamma\\eta, \\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$, respectively. \nThe uncertainties in the efficiency\ninclude the Monte Carlo samples' statistical uncertainty and our\nestimate of possible systematic biases, which are discussed further\nin Section~\\ref{sec:sys-lim}.\n\nWe find no candidate events within our acceptance invariant mass\nwindow for the search $\\Uos \\to \\gamma \\eta$, $\\eta\\to3\\pi$. The invariant mass\ndistributions for candidate $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$ and $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$, after\nimposing all the selection criteria are shown in Figure~\\ref{fig:eta-3pi}.\n\\begin{figure}\n\\includegraphics*[width=3.25in]{0990307-001.eps}\n\\caption{Candidate $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$ (top) and $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$ (bottom)\ninvariant mass distributions from $\\Upsilon(1\\text{S})$ data.\nThe large number of events near 780\\,\\rm MeV\/$c^2$\\ (top) is due to the \nabundant process $e^{+} e^{-} \\to \\gamma \\omega$. No events are observed in our acceptance\nregion, bounded by the arrows.}  \n\\label{fig:eta-3pi}\n\\end{figure}\n\n\\subsection{The Decay $\\Upsilon\\to\\gamma\\eta ,\\eta\\to\\gamma\\gamma$}\nThe 3-photon final state resulting from $\\Uos \\to \\gamma \\eta; \\EtaGG$\\ is dominated\nby the QED process $e^{+} e^{-} \\to \\gamma \\gamma \\gamma$. Our selection criteria of loosely\nreconstructing an $\\eta \\to \\gamma \\gamma$ meson and requiring the \\chisq\\ of\n4-momentum constraint on the $\\Upsilon(1\\text{S})$ meson formed by adding a\nhard-photon to be $<200$ are not sufficient to suppress this\nbackground. The QED background, however has a distinct feature - the\ntwo photons having energies \n$E_{hi}$ and $E_{lo}$ used in reconstructing the $\\eta$ candidate have\na large energy asymmetry, where asymmetry is defined as\n$(E_{hi}-E_{lo})\/(E_{hi}+E_{lo})$.  \nReal $\\eta$ mesons are expected to have an approximately\nuniform distribution of\nasymmetry in the range (0,1). We require the asymmetry to be less than\n0.8. To further discriminate between the signal and the background, we\nused a neural net approach. \n\nThe input to the neural net is a vector of six variables, namely the\nmeasured energy and the polar angle $\\theta$ of each of the three calorimeter\nshowers used in the reconstruction chain. The training sample is\ncomprised of 20,000 simulated signal and background events in equal\nproportion. The simulated $e^{+} e^{-} \\to \\gamma \\gamma \\gamma$\\ background events have a\nhigh-energy photon ($E>4\\ \\rm GeV$), $\\gamma \\gamma$\\ invariant mass for the two\nlower-energy photons in the range 0.4-0.7\\,\\rm GeV\/$c^2$, and energy asymmetry\nless than 0.8.\n\n\nFor our final selection, we choose neural-net output with $51\\%$\nefficiency while rejecting $86\\%$ of the background.\nThe combined efficiency of our\nselection criteria for this mode is $(23.8\\pm2.4)\\%$, which includes\npossible systematic biases and statistical uncertainties from the\nsimulation. The resulting $\\gamma \\gamma$\\ invariant mass distribution from\n$\\Upsilon(1\\text{S})$ data is fit, as shown in Figure~\\ref{fig:eta-gg},\nto a double Gaussian function, whose mass and widths are fixed to values\nfound from signal Monte Carlo data, along with a second order polynomial\nbackground function. From this likelihood fit, we obtain $-2.3\\pm8.7$\nevents; consistent with zero. We then perform the same likelihood fit\nmultiple times fixing the signal area to different values, assigning\neach of the fits a probability proportional to  $e^{-{\\chi^2}\/2}$,\nwhere \\chisq\\ is obtained from the likelihood fit.\nThe resulting probability distribution is normalized and numerically\nintegrated up to 90\\% of the area to obtain the yield at 90\\%\nconfidence level. Our limit thus obtained is 14.5 events at 90\\%\nconfidence level. \n\\begin{figure}\n\\includegraphics*[width=3.25in]{0990307-002.eps}\n\\caption{Invariant mass distribution of $\\gamma \\gamma$\\ candidates in $\\Upsilon(1\\text{S})$\n  data for the mode $\\Uos \\to \\gamma \\eta; \\EtaGG$, overlaid with fits using a) floating area\n  (solid red) yielding $-2.3\\pm8.7$ events, and b) area fixed to 14.5\n  events (dashed blue), the upper limit corresponding to 90\\% C.L.}\n\\label{fig:eta-gg}\n\\end{figure}\n\n\\subsection{The Decay  $\\Upsilon\\to\\gamma\\eta^{\\prime} ,\\eta^{\\prime}\\to\\eta\\pi^+\\pi^-$}\nReconstruction of the decay chains $\\Uos \\to \\gamma \\eta^{\\prime}$, where \n$\\eta^{\\prime} \\to \\eta \\pi^+ \\pi^-$, builds on the search $\\Uos \\to \\gamma \\eta$\\. \nThe reconstructed $\\eta$ candidate is\nconstrained to the nominal $\\eta$ mass. The mass-constrained $\\eta$\ncandidate is further combined with a pair of oppositely charged\nquality tracks by forcing the tracks and the $\\eta$ candidate to\noriginate from a common vertex. In reconstruction of $\\eta^{\\prime}; \\EtaPMZ$,\ncare is exercised to ensure that no track is used more than once in the\ndecay chain. The high energy photon is combined with the $\\eta^{\\prime}$\\\ncandidate to build an $\\Upsilon$ candidate which is further\nconstrained to the 4-momentum of the initial $e^+e^-$ system. \nIn the reconstruction chain $\\eta^{\\prime}; \\EtaGG$, the $\\Upsilon$ candidate with\nthe lowest sum of chi-squared to the 4-momentum constraint ($\\chi^{2}_{\\text{P}4}$)\ncombined with the chi-squared of the mass-constraint to the $\\eta$\ncandidate (\\EtaMassFitChisq) is accepted as\nthe representative $\\Upsilon$ candidate in the reconstructed event.\nIn the modes where $\\eta\\to3\\pi$, \nthe $\\pi^{0}$\\ mass-constraint chi-squared, $\\chi^2_{\\pi^0}$, also contributes to the \n$\\chi^{2}_{\\mathrm{Total}}$ . \n\n\nTo ensure that only good quality $\\eta$ candidates participate in the decay\nchain, the \\EtaMassFitChisq\\ values of ``$\\eta\\to \\text{all neutral}$''\ncandidates are required to be less than 200. Owing to the better\nmeasurements of charged track momenta, this criterion is more\nstringent (\\EtaMassFitChisq $<100$) in the case of $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$. The \ntargeted efficiency (around 99\\%) of this requirement is achieved in\nall three cases.\n \nThe charged tracks used in reconstructing $\\eta^{\\prime}$\\ candidates have to be\nconsistent with the pion hypothesis.\nWe again require the sum of squared $\\text{S}_{dE\/dx}$ added in quadrature to\nbe less than 16 for both the two track and four track cases. The\nefficiency of this requirement alone is around 99\\%.\n\nThe selected $\\Upsilon$ candidate is further required to satisfy the\n4-momentum consistency criterion, restricting $\\chi^{2}_{\\text{P}4}$\\ $<100$\nin the $\\eta \\to \\gamma \\gamma$ case and a less stringent value of 200 for\n$\\eta\\to3\\pi$. The overall reconstruction efficiencies of our\nselection criteria as determined from signal Monte Carlo simulations are\n$(35.3\\pm5.2)\\%$, $(24.5\\pm2.2)\\%$ and $(14.4\\pm2.9)\\%$ for $\\eta$\ndecays to $\\gamma \\gamma$, $\\pi^+\\pi^-\\pi^0$ and $3\\pi^0$, respectively.\n\nAfter these selection criteria, we find\nno candidate events in the modes \\ModeEtapGG\\ and $\\Uos \\to \\gamma \\eta^{\\prime};\\EtaThreePZ$, as\nshown in Figure~\\ref{fig:etap-eta-pi-pi}. However, in the mode\n\\ModeEtapPMZ, we find two good candidate events passing our selection\ncriteria as shown in Figure~\\ref{fig:etap-eta-pi-pi}.\nThese two events have been looked at in detail and appear to be\ngood signal events. However, they are insufficient to allow\nus to claim a positive signal, as no candidate events are observed \nin the modes \\ModeEtapGG\\ and $\\Uos \\to \\gamma \\eta^{\\prime};\\EtaThreePZ$, each providing higher\nsensitivity than the decay chain \\ModeEtapPMZ.\n\\begin{figure}\n\\includegraphics*[width=3.44in]{0990307-003.eps}\n\\caption{Invariant mass distributions of $\\eta\\pi^+\\pi^-$ candidates from $\\Upsilon(1\\text{S})$\ndata. The $\\eta$ candidate is constrained to the nominal $\\eta$ meson\nmass. No events are observed in the signal box for $\\eta \\to \\gamma \\gamma$ (top) and\n$\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$ (bottom); two signal events are observed for\n$\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$ (middle).}\n\\label{fig:etap-eta-pi-pi}\n\\end{figure}\n\n\\subsection{The Decay $\\Upsilon\\to\\gamma\\eta^{\\prime} ,\\eta^{\\prime}\\to\\gamma\\rho^0$}\nThe reconstruction scheme for the decay chain \\ModeEtapGR\\ is slightly\ndifferent from those previously described. We first build $\\rho^0$\ncandidates by forcing pairs of oppositely charged tracks to originate\nfrom a common vertex. Next, we add a photon candidate \n(which we refer to as the ``soft shower'' having energy $E_s$ in\ncontrast with the high energy radiative photon)  \nnot associated with charged tracks, and having a lateral profile\nconsistent with being a photon, to build $\\eta^{\\prime}$\\ candidates. To obtain\nthe maximum yield, we neither restrict the energy $E_s$ of the\nphoton nor the invariant mass of the $\\rho^0$ candidate at this stage. \nA high energy photon is then added, ensuring that the soft shower and\nhigh energy photon are distinct, to build the $\\Upsilon$ candidate. The\n$\\Upsilon$ candidate is then constrained to the 4-momentum of the\ninitial $e^+e^-$ system and the candidate with the lowest\n$\\chi^{2}_{\\text{P}4}$\\ value is selected.\n\nThe candidate $\\eta^{\\prime}$\\ invariant mass resolution is vastly improved due\nto the mass-constraints on the candidate $\\pi^{0}$\\ and $\\eta$ mesons in\n$\\eta^{\\prime} \\to \\eta \\pi^+ \\pi^-$ decays. In reconstruction of\n$\\eta^{\\prime} \\to \\gamma \\rho^0$, a significant improvement in candidate $\\eta^{\\prime}$\\ invariant mass\nresolution ($\\approx 30\\%$) as well as the energy resolution of the\nsoft shower is achieved by performing the 4-momentum constraint on\nthe $\\Upsilon$ candidate.\n\nParticle identification in the channel $\\eta^{\\prime} \\to \\gamma \\rho^0$\\ is achieved by\ndemanding the combined RICH and $dE\/dx$\\ likelihood for the pion\nhypothesis be greater than the combined likelihood for each of the\nelectron, kaon and proton hypotheses. Copiously produced QED processes\nsuch as $e^+e^- \\to \\gamma \\gamma e^+e^-$ are\nsuppressed by imposing an electron veto, requiring that\n$|E\/p-1.0|>0.05$, where $p$ is the measured momentum and $E$ is the\nassociated calorimeter energy of the charged track. QED events of the type \n$e^+e^- \\to \\gamma \\gamma \\mu^+\\mu^-$ are suppressed by requiring that\nneither track registers a hit five hadronic interaction lengths deep\ninto the muon detector system. Continuum background of the type \n$e^+e^- \\to \\gamma \\gamma \\rho^0$ is suppressed by demanding\n$E_s>100$\\ \\rm MeV. Finally, the event is ensured to be complete by\ndemanding $\\chi^{2}_{\\text{P}4}$\\ $< 100$. \nThe overall efficiency of the selection criteria for this mode is\n$(40.1\\pm2.1)\\%$, including possible systematic uncertainties and\nthe statistical uncertainty of the Monte Carlo sample.\n\n\\begin{figure}\n\\includegraphics*[width=3.25in]{0990307-004.eps}\n\\caption{Invariant mass distribution of $\\gamma\\rho^0$ candidates in $\\Upsilon(1\\text{S})$\n  data for the mode \\ModeEtapGR\\ overlaid with fits using a) floating area\n  (solid red) yielding $-3.1\\pm5.3$ events, and b) area fixed to\n  8.6 events (dashed blue), corresponding to the upper limit at 90\\% C.L.}\n\\label{fig:etap-gr}\n\\end{figure}\n\nAlthough highly efficient, our selection criteria are not sufficient to\nsuppress the smooth continuum background from the reaction\n$e^+e^- \\to \\gamma \\gamma \\rho^0$.\nThe candidate $\\eta^{\\prime} \\to \\gamma \\rho^0$\\ invariant mass distribution after our selection\ncriteria, shown in Figure~\\ref{fig:etap-gr}, is fit to\na double Gaussian function over a floating polynomial background\nfunction of order one. The parameters of the double Gaussian function\nare fixed to the values obtained from a fit to signal Monte\nCarlo and the area is left to float. The likelihood fit yields\n$-3.1\\pm5.3$ events, which is consistent with zero. In the absence of a\nclear signal, we determine the upper limit yield\nas we do in the case of $\\Uos \\to \\gamma \\eta; \\EtaGG$, and find an upper limit at 90\\%\nconfidence level of 8.6\nevents. \n\n\n\n\n\\section{Systematic Uncertainties and Combined Upper Limits}\\label{sec:sys-lim}\nSince we do not have a signal in any of the modes, and since the\nkinematic efficiency is near-maximal, statistical uncertainties\ndominate over systematic uncertainties. \nBy comparison of the expected yield of the QED process $e^{+} e^{-} \\to \\gamma \\gamma \\gamma$\\\nwith the calculated cross-section for this process, we estimate the\nuncertainty on the trigger simulation for ``all neutral'' modes to be\n4.5\\%. For modes with only two charged tracks, we have studied the \nQED processes $e^+e^- \\to \\gamma \\rho^0$ and $e^+e^- \\to \\gamma \\phi$, \nand assign a 13\\% uncertainty\non the efficiency due to possible trigger mismodeling. For events with\nmany charged tracks, we assign a systematic\nuncertainty of 1\\% as the relevant trigger lines are very well understood,\nredundant, and\nvery efficient.\nWe assign 1\\% uncertainty per track in charged track\nreconstruction based upon CLEO studies~\\cite{CLEOSystematics}\nof low-multiplicity events, and\n2.5\\% systematic uncertainty per photon from mismodeling of\ncalorimeter response which translates to 5\\% uncertainty per meson\n($\\pi^{0}$\\ and $\\eta$) decaying into $\\gamma \\gamma$, \nagain based upon CLEO studies~\\cite{CLEOSystematics}.\nThe systematic uncertainty in $\\text{S}_{dE\/dx}$ for two\ntracks added in quadrature (as in $ \\Uos \\to \\gamma \\eta; \\EtaPMZ $) was evaluated to be 4\\%\nby considering the efficiency difference of this requirement in\nMonte Carlo and data samples of\n$e^{+} e^{-} \\to \\gamma \\omega$. Consequently, we assign 4\\% and 5.7\\% uncertainty to the\nreconstruction efficiencies of modes involving two and four charged\ntracks, respectively, excepting $\\eta^{\\prime} \\to \\gamma \\rho^0$\\ where this requirement was not\nimposed. For the mode $\\eta^{\\prime} \\to \\gamma \\rho^0$, the systematic uncertainty in the\nefficiency of analysis cuts, found to be 3.9\\%, was evaluated by\ncomparing the efficiency difference in Monte Carlo and data by\nstudying the $\\rho^0$ signal due to the QED processes. \nFor the neural-net cut in the mode $\\Uos \\to \\gamma \\eta; \\EtaGG$, we studied the\nefficiency in QED $e^{+} e^{-} \\to \\gamma \\gamma \\gamma$\\ simulated events and the real data\ndominated by the same QED process for a wide range of neural-net\noutput values. We find a maximum difference of 7\\% in these two\nnumbers, which we take as a conservative estimate of the\nassociated systematic uncertainty.\nThe systematic uncertainties for various $\\eta$ and $\\eta^{\\prime}$\\ decay modes\nare listed in Table~\\ref{tab:sys-errs}. \nThese uncertainties were added in quadrature, along with the\nstatistical error due to the limited size of Monte Carlo samples, to\nobtain the overall systematic uncertainties in the efficiencies.\n\n\\begin{small}\n\\begingroup\n\\squeezetable\n\\begin{table*}[!]\n\\begin{center}\n\\caption{\\label{tab:sys-errs}Contributions to systematic\nuncertainties in the efficiencies for $\\Uos \\to \\gamma \\eta^{\\prime}$\\ (upper half) and\n$\\Uos \\to \\gamma \\eta$\\ (lower half). The uncertainties are expressed as relative\npercentages and combined in quadrature.}\n\\begin{ruledtabular}\n\\begin{tabular}{lcccc}\nUncertainty source & ~~$\\eta^{\\prime}; \\EtaGG$ & ~~$\\eta^{\\prime}; \\EtaPMZ$ & ~~$\\eta^{\\prime}; \\EtaThreePZ$\n& $\\eta^{\\prime} \\to \\gamma \\rho^0$ \\\\  \n\n\\hline\nTrigger mismodeling    & 13   & 1   & 13    & 1   \\\\\nTrack reconstruction   & 2   & 4   & 2    & 2   \\\\\nCalorimeter response   & 5   & 5   & 15   & 2.5 \\\\\nAnalysis cuts          & 4   & 5.7 & 4    & 3.9 \\\\\nMonte Carlo statistics & 1.0 & 1.6 & 2.4  & 1.0 \\\\\n\\hline\nCombined uncertainty   & 14.7 & 8.8 & 20.4 & 5.2 \\\\\n\\hline\n& & & & \\\\\nUncertainty source &   $\\eta \\to \\gamma \\gamma$ & $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$ & $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$ & \\\\\n\\hline\nTrigger mismodeling    & 4.5   & 13   & 4.5    & \\\\\nTrack reconstruction   & -   & 2   & -    & \\\\\nCalorimeter response   & 5   & 5   & 15   & \\\\\nAnalysis cuts          & 7   & 4   & -    & \\\\\nMonte Carlo statistics & 1.3 & 1.2 & 1.7  & \\\\\n\\hline\nCombined uncertainty   & 9.8 & 15.2 & 16.0 & \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{center}\n\\end{table*}\n\\endgroup\n\\end{small}\n\n\n\\begin{small}\n\\begingroup\n\\squeezetable\n\\begin{table*}[ht]\n\\begin{center}\n\\caption{Results of the search for $\\Uos \\to \\gamma \\eta^{\\prime}$\\ and $\\Uos \\to \\gamma \\eta$. Results include\nstatistical and systematic uncertainties, as described in the\ntext. The combined limit is obtained after including the systematic\nuncertainties.}\n\\label{tab:limits}\n\\begin{ruledtabular}\n\\begin{tabular}{lcccc}\n& ~~$\\eta^{\\prime}; \\EtaGG$ & ~~$\\eta^{\\prime}; \\EtaPMZ$ & ~~$\\eta^{\\prime}; \\EtaThreePZ$ & ~~$\\eta^{\\prime} \\to \\gamma \\rho^0$ \\\\\n\\hline\n\nObserved events & 0 & 2 & 0 & $-3.1\\pm5.3$ \\\\\n\n$\\mathcal{B}_{\\eta^{\\prime},i}\\%$ & \n$17.5\\pm0.6$ & $10.0\\pm0.4$ & $14.4\\pm0.5$ & $29.5\\pm1.0$ \\\\\n\nReconstruction efficiency (\\%) & \n$35.2\\pm5.2$ & $24.5\\pm2.2$ & $14.4\\pm2.9$ & $40.1\\pm2.1$ \\\\ \n\n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta^{\\prime})(90\\%~\\text{C.L.})$\\footnotemark[1] &\n\\ulbr{1.8} & \\ulbr{10.3} & \\ulbr{5.2} & \\ulbr{3.4} \\\\ \n\n\n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta^{\\prime})(90\\%~\\text{C.L.})$\\footnotemark[2] &\n\\ulbr{1.9} & \\ulbr{10.4} & \\ulbr{5.8} & \\ulbr{3.4} \\\\ \n\n\\hline\n\n\\multicolumn{2}{l}{Combined limit on \n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta^{\\prime})$} &\n\\multicolumn{2}{c}{\\ulbr{1.9}} \\\\\n\n\\hline\n\n& & & & \\\\\n\n& $\\eta \\to \\gamma \\gamma$ & $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$ & $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$ & \\\\\n\n\\hline\n\nObserved events & $-2.3\\pm8.7$ & 0 & 0 & \\\\\n\n$\\mathcal{B}_{\\eta,i}\\%$ & $39.4\\pm0.3$ & $22.6\\pm0.4$ & $32.5\\pm0.3$ & \\\\\n\nReconstruction efficiency (\\%) & $23.8\\pm2.4$ & $28.5\\pm2.9$ &\n$11.8\\pm1.9$ & \\\\\n\n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta)(90\\%~\\text{C.L.})$\\footnotemark[1]\n& \\ulbr{7.3} & \\ulbr{1.7} & \\ulbr{2.8} & \\\\\n\n\n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta)(90\\%~\\text{C.L.})$\\footnotemark[2] & \n\\ulbr{7.4} & \\ulbr{1.8} & \\ulbr{2.9} & \\\\\n\\hline\n\nCombined limit on $\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta)$ &\n\\multicolumn{3}{c}{\\ulbr{1.0}} &\\\\\n\n\n\\end{tabular}\n\\end{ruledtabular}\n\\footnotetext[1]{excluding systematic uncertainties}\n\\footnotetext[2]{including systematic uncertainties}\n\\end{center}\n\\end{table*}\n\\endgroup\n\\end{small}\n\n\n\\begin{figure*}[!]\n\\mbox{\\includegraphics*[width=6.3in]{0990307-005.eps}}\n\\caption{Likelihood distributions as a function of branching fraction\nfor the decay mode $\\Uos \\to \\gamma \\eta$\\ (left) and $\\Uos \\to \\gamma \\eta^{\\prime}$\\ (right). All\ndistributions are smeared by respective systematic uncertainties and\nnormalized to the same area. The solid black curve denotes the combined\nlikelihood distribution.} \n\\label{fig:limit-plots}\n\\end{figure*}\nThe systematic uncertainties in efficiencies, uncertainties in the\nproduct branching ratios, and the statistical uncertainty in the number\nof $\\Upsilon(1\\text{S})$ decays, $N_{\\Uos}$, are incorporated~\\cite{Cousins}\nby a ``toy'' Monte Carlo procedure to obtain smeared likelihood\ndistributions for the branching fraction in each mode,\n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\text{P}) = N_{\\text{P}}\/(\\epsilon_{i}\\cdot\\mathcal{B}_{\\text{P},i}\\cdot N_{\\Uos})$, \nwhere $\\text{P} = \\eta , \\eta^{\\prime}$, and $\\epsilon_i$ and\n$\\mathcal{B}_{\\text{P},i}$ denote the efficiency and branching\nfractions of the $i$th mode. \nTo obtain the smeared likelihood distribution $\\mathcal{L}_{\\text{P},i}$, the\nexperiment is performed multiple times, randomly selecting\n$N_{\\text{P}}$  from the likelihood function appropriate for each\nmode\\footnote{For modes with zero or few observed events, the\n  appropriate likelihood function is generated from Poisson statistics.\n  For the background limited modes $\\eta \\to \\gamma \\gamma$ and $\\eta^{\\prime} \\to \\gamma \\rho^0$, we \n  already have the likelihood function which we used in calculating\n  the upper limit of the observed number of events at 90\\% CL.} and then\ndividing by the sensitivity factor $\\epsilon_{i}\\cdot\\mathcal{B}_{\\text{P},i}\\cdot N_{\\Uos}$, where each term is picked\nfrom a Gaussian distribution about their mean values with the\nappropriate standard deviation. \n\nThe combined likelihood distribution for\n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\text{P})$ is derived as\n$\\mathcal{L}_{\\text{P}} = \\prod_{i}{\\mathcal{L}_{\\text{P},i}}$ which\nis summed up to 90\\% of the area in the physically allowed region to obtain\nthe upper limit branching fraction for $\\Upsilon(1\\text{S})\\to\\gamma\\text{P}$.\nFrom the constituent $\\mathcal{L}_{\\text{P},i}$ and the combined\n$\\mathcal{L}_{\\text{P}}$ as shown in \nFigure~\\ref{fig:limit-plots}, \nwe obtain upper limits on\n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta)$ of \\br{7.4}, \\br{1.8}, \\br{2.9}, \nand \\br{1.0} for $\\eta$ decaying into $\\gamma \\gamma$,\n$\\pi^+\\pi^-\\pi^0$, $\\pi^0\\pi^0\\pi^0$, and all three combined,\nrespectively. We obtain upper limits for\n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta^{\\prime})$ \nof \\br{1.9}, \\br{10.4}, \\br{5.8}, and \\br{3.4} for $\\eta$\ndecaying into $\\gamma \\gamma$, $\\pi^+\\pi^-\\pi^0$, $\\pi^0\\pi^0\\pi^0$,\nand $\\eta^{\\prime} \\to \\gamma \\rho^0$, respectively. The combined upper limit for $\\mathcal{B}$(\\ModeEtap)\\ is\n$1.9\\times10^{-6}$, a value larger than one of the sub-modes\n(\\ModeEtapGG), due to the two candidate events in \\ModeEtapPMZ.\nThe numbers of observed events, detection efficiencies and upper\nlimits are listed in Table~\\ref{tab:limits}. \n\n\n\\section{Summary and Conclusion}\\label{sec: summary}\nWe report on a new search for the radiative decay of $\\Upsilon(1\\text{S})$ to the \npseudoscalar mesons $\\eta$ and $\\eta^{\\prime}$\\ in $21.2\\times10^{6}$ $\\Upsilon(1\\text{S})$ \ndecays collected with the CLEO~III detector. The $\\eta$ meson was reconstructed\nin the three modes $\\eta \\to \\gamma \\gamma$, $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$ or $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$.\nThe $\\eta^{\\prime}$\\ meson was reconstructed either in the mode $\\eta^{\\prime} \\to \\gamma \\rho^0$\\ or \n$\\eta^{\\prime} \\to \\pi^{+} \\pi^{-} \\eta$ with $\\eta$ decaying through any\nof the above three modes. All these modes except for $\\eta^{\\prime} \\to \\gamma \\rho^0$\\ had earlier\nbeen investigated in CLEO~II data amounting to \n$N_{\\Uos}$ $ = 1.45\\times 10^6$ $\\Upsilon(1\\text{S})$ mesons and resulted in\nprevious upper limits $\\mathcal{B}$(\\ModeEtap)\\ $< 1.6 \\times 10^{-5}$ and \n$\\mathcal{B}$(\\ModeEta)\\ $< 2.1 \\times 10^{-5}$  \nat 90\\% C.L. These limits were already smaller than \nthe naive predictions based upon the scaling of the decay rate for the\ncorresponding $J\/\\psi$\\ radiative decay mode by the factor \n$(q_{b}m_{c}\/q_{c}m_{b})^{2}$,\nand also the model of K\\\"orner {\\it et al.,}~\\cite{KKKS}, \nwhose perturbative QCD approach predictions\nfor $\\mathcal{B}(J\/\\psi\\to\\gamma X)$ where $X = \\eta,\n\\eta^{\\prime}, f_2$ as well as $\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma f_2)$ agree\nwith experimental results.\n\nWith a CLEO~III data sample 14.6 times as large as the CLEO~II data\nsample, we find no convincing signal in any of the modes. Based purely\nupon the luminosities, we would expect the new upper limits to be scaled \ndown by a factor of between 14.6 (in background-free modes) and\n$\\sqrt{14.6}$ in background dominated modes if the two CLEO detectors \n(CLEO~II and CLEO~III) offered similar particle detection efficiencies. \nIn the search for $\\Uos \\to \\gamma \\eta$\\ we find no hint of \na signal, and manage to reduce the limit by an even larger factor. In\nthe search for $\\Uos \\to \\gamma \\eta^{\\prime}$, however, we find two clean candidate events\nin the channel \\ModeEtapPMZ, which, though we\ncannot claim them as signal, do indicate the possibility that we are\nclose to the sensitivity necessary to obtain a positive result. \nBecause of these two events, our combined limit for $\\Uos \\to \\gamma \\eta^{\\prime}$\\ is not \nreduced by as large a factor as the luminosity ratio, and in fact is \nlooser than that which would be obtained if we analyzed \none sub-mode (\\ModeEtapGG) alone.\nIn this analysis we found upper limits which we\nreport at 90\\% confidence level as\n\\begin{displaymath}\n\\mathcal{B}(\\Upsilon(1\\text{S}) \\to \\gamma \\eta) < 1.0 \\times 10^{-6},\n\\end{displaymath}\n\\begin{displaymath}\n\\mathcal{B}(\\Upsilon(1\\text{S}) \\to \\gamma \\eta^{\\prime}) < 1.9 \\times 10^{-6}.\n\\end{displaymath}\n\nOur results are sensitive enough to test the appropriateness of the\npseudoscalar mixing approach as pursued by Chao~\\cite{KTChao}, where\nmixing angles among various pseudoscalars including $\\eta_{b}$\\ are\ncalculated. Then, using a calculation for the M1 transition\n$\\Upsilon\\to\\gamma\\eta_{b}$, he predicts \n$\\mathcal{B}(\\Upsilon(1\\text{S}) \\to \\gamma \\eta) = 1\\times 10^{-6}$ and\n$\\mathcal{B}(\\Upsilon(1\\text{S}) \\to \\gamma \\eta^{\\prime}) = 6\\times 10^{-5}$. Our\nlimit for $\\Uos \\to \\gamma \\eta^{\\prime}$\\ is significantly smaller than Chao's prediction\nand does not support his approach.\n\nThe sensitivity challenge posed by both the extended vector dominance\nmodel and the higher twist approach of Ma are beyond our reach. \nIn extended VDM, Intemann predicts \n$1.3\\times 10^{-7} < \\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta) < 6.3\\times 10^{-7}$ \nand \n$5.3\\times 10^{-7} < \\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta^{\\prime}) <\n2.5\\times 10^{-6}$, where the two limits are determined by having\neither destructive or constructive interference, respectively, between\nthe terms involving $\\Upsilon(1\\text{S})$ and $\\Upsilon(2\\text{S})$. Even if it is determined that \nthe amplitudes are added constructively, our limit remains higher than\nthe VDM prediction for $\\Upsilon(1\\text{S})\\to\\gamma\\eta$. \n\nMa's prediction of \n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta^{\\prime}) \\approx 1.7\\times 10^{-6}$ is\nconsistent with our result. However, his prediction for\n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta) \\approx 3.3\\times 10^{-7}$ is a factor of\n$\\sim 3$ smaller than our limit.\n\n\n\n\n\n\\begin{comment}\n\\begin{figure}\n\\includegraphics*[bb = 35 150 535 610, width=3.25in]{\\etalimitplotps}\n\\caption{Likelihood distribution as function of branching\nratio for the decay mode $\\Uos \\to \\gamma \\eta$. Black curve denotes the combined\ndistribution.}\n\\label{fig:eta log limits}\n\\end{figure}\n\\begin{figure}\n\\includegraphics*[bb = 35 150 535 610, width=3.25in]{\\etaplimitplotps}\n\\caption{Likelihood distribution as function of branching\nratio for the decay mode $\\Uos \\to \\gamma \\eta^{\\prime}$. Black curve denotes the combined\ndistribution.}\n\\label{fig:etap log limits}\n\\end{figure}\n\\end{comment}\n\\begin{comment}\n\\begin{figure*}\n\\mbox{\\includegraphics*[bb = 55 150 535 610, width=3.3in]{\\etalimitplotps}}\n\\mbox{\\includegraphics*[bb = 55 150 535 610, width=3.3in]{\\etaplimitplotps}}\n\\caption{Likelihood distributions as function of branching fraction\nfor the decay mode $\\Uos \\to \\gamma \\eta$\\ (left) and $\\Uos \\to \\gamma \\eta^{\\prime}$\\ (right). All\ndistributions are smeared by respective systematic uncertainties and\nnormalized to unit area. Black curve denotes the combined likelihood\ndistribution.}\n\\label{fig:limit-plots}\n\\end{figure*}\n\\end{comment}\n\n\nWe gratefully acknowledge the effort of the CESR staff \nin providing us with excellent luminosity and running conditions. \nD.~Cronin-Hennessy and A.~Ryd thank the A.P.~Sloan Foundation. \nThis work was supported by the National Science Foundation,\nthe U.S. Department of Energy, and \nthe Natural Sciences and Engineering Research Council of Canada.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Basic computational properties of $\\mathsf{LEM}$}\n\\label{sec: basic computational properties of IMLLshpos}\nThe observations in the previous sections lead us to set the reduction rules on terms, which allow duplication and erasure for values only, as in Figure~\\ref{fig: term reduction rules}.\nThose reductions are more restrictive than the cut-elimination\nsteps that we could perform on $\\mathsf{LEM}$ if we look at\nit as it was a pure logical system, i.e.~not a type-assignment. Since the cut-elimination of $\\mathsf{LEM}$\nworks as in $\\mathsf{LL}$, once replaced \\enquote{!} for \\enquote{$ \\shpos $}, we can observe the effect of moving the $ cut $ in:\n\\begin{equation}\\label{eqn: non linear duplication}\n\\AxiomC{\\vdots}\n\\noLine\n\\UnaryInfC{$y: \\shpos A, z: \\shpos \\mathbf{1} \\vdash yz:  \\mathbf{1}$}\n\\RightLabel{$p$}\n\\UnaryInfC{$y: \\shpos A, z: \\shpos \\mathbf{1} \\vdash yz: \\shpos  \\mathbf{1}$}\n\\AxiomC{$\\vdots$}\n\\noLine\n\\UnaryInfC{$\\Gamma, x_1: \\shpos \\mathbf{1}, x_2: \\shpos \\mathbf{1} \\vdash M:\\ \\tau$}\n\\AxiomC{\\vdots}\n\\noLine\n\\UnaryInfC{$\\vdash I: \\mathbf{1}$}\n\\RightLabel{$c$}\n\\BinaryInfC{$\\Gamma, x: \\shpos \\mathbf{1} \\vdash \\mathtt{ copy} ^{I}_{\\mathbf{1}}\\, x \\mathtt{ \\   as \\ } y,z \\mathtt{ \\   in \\ }M: \\tau$}\n\\RightLabel{$cut$}\n\\BinaryInfC{$\\Gamma, y: \\shpos A, z: \\shpos \\mathbf{1} \\vdash \\mathtt{ copy} ^{I}_{\\mathbf{1}}\\, yz \\mathtt{ \\   as \\ } y,z \\mathtt{ \\   in \\ }M: \\tau$}\n\\DisplayProof\n\\end{equation}\nupward, in order to eventually eliminate it.\nThe move would require to duplicate the \\emph{open} term $yz$, erroneously \nyielding a non linear term. So, at the proof-theoretical level, moves \nof the cut rule exist that cannot correspond to any reduction on terms. \nIn order to circumvent the here above misalignment, we proceed as follows:\n\\begin{itemize}\n\t\\item \n\tWe define the \\emph{lazy cut-elimination steps}. Their introduction rules out any attempt to eliminate instances \n\tof cuts like~\\eqref{eqn: non linear duplication}.\n\tThe apparent drawback is to transform cuts like~\\eqref{eqn: non linear duplication} into \\emph{deadlocks}, i.e. into instances of $ cut $\n\tthat we cannot eliminate.\n\t\n\t\\item\n\tDeadlocks are not a problem. Once defined \\emph{lazy types},\n\twe can show that a \\emph{lazy cut-elimination strategy} exists\n\t\tsuch that it eliminates all the cut rules that may\n\t\toccur in a derivation of a lazy type. \n\t\tThe cost of the elimination is cubical\n\t\t(Theorem~\\ref{thm: cut elimination for downarrow IMLL2}).\n\\end{itemize}\nLast, we show that the reduction on terms in Figure~\\ref{fig: term reduction rules} and Figure~\\ref{fig: term commuting conversions} enjoy Theorem~\\ref{thm:Subject Reduction for IMLL2shpos}, i.e.~Subject reduction.\n\n\\subsection{Cut-elimination and  its cubical complexity}\n\\label{Cut elimination, complexity, and subject reduction for IMLL2^shpos}\n\\begin{defn}[The cuts of $\\mathsf{LEM} $]\n\\label{defn: definitions for cut} \nLet $ (X,Y) $ identify an instance:\n\\begin{equation}\\label{eqn: cut elimination format}\n\\AxiomC{\\vdots}\n\\RightLabel{$X$}\n\\UnaryInfC{$\\Gamma \\vdash N:\\sigma$}\n\\AxiomC{\\vdots}\n\\RightLabel{$Y$}\n\\UnaryInfC{$\\Delta, x:\\sigma \\vdash M: \\tau$}\n\\RightLabel{$ cut $}\n\\BinaryInfC{$\\Gamma,\\Delta \\vdash M[N\/x]: \\tau$}\n\\DisplayProof\n\\end{equation}\nof the rule $ cut $ that occurs in a given derivation \n$ \\mathcal{D} $ of \n$\\mathsf{LEM}$, where $X$ and $Y$ are two of the rules in Figure~\\ref{fig: the system shpos add IMLL2}. \n\\emph{Axiom cuts} involve $ ax $, and are of the form $(X,ax)$ or $(ax,Y)$, for some $ X$ and $ Y $. \n\\emph{Exponential cuts} are ($p$,$d$), ($p$,$w$), and ($p$,$c$). \n\\emph{Principal cuts} are $(\\multimap$R$,\\multimap$L$)$,\n$(\\forall$R$,\\forall$L$)$ and every exponential cut.\n\\emph{Symmetric cuts} contain axiom and principal cuts.\nEvery symmetric cut that is not exponential is \\emph{multiplicative}. \n\\emph{Commuting cuts} are all the remaining instances of $ cut $, not mentioned here above, $(p, p)$ included, for example.\n\\par\nA \\textit{lazy cut} is every instance of the cut~\\eqref{eqn: cut elimination format} which is both exponential and such that \n$N$ is a value.\n\\par\nA \\emph{deadlock} is every instance of the cut~\\eqref{eqn: cut elimination format} which is both exponential and such that $\\Gamma\\neq \\emptyset$. Otherwise, it is \\emph{safe}.\n\\qed\n\\end{defn}\n\n\n\\begin{figure}[t]\n\\begin{equation*}\n\\scalebox{0.6}{$\n\\begin{aligned}\n\\AxiomC{$\\mathcal{D}$}\n\t\\noLine\n\t\\UnaryInfC{$\\Gamma, x:\\sigma \\vdash M: A$}\n\t\\RightLabel{$\\multimap R$}\n\t\\UnaryInfC{$\\Gamma \\vdash \\lambda x.M: \\sigma\\multimap A$}\n\t\\AxiomC{$\\mathcal{D}'$}\n\t\\noLine\n\t\\UnaryInfC{$\\Delta\\vdash N:\\sigma$}\n\t\\AxiomC{$\\mathcal{D}''$}\n\t\\noLine\n\t\\UnaryInfC{$\\Theta, z:A  \\vdash P:\\tau$}\n\t\\RightLabel{$\\multimap L$}\n\t\\BinaryInfC{$\\Delta,\\Theta, y:\\sigma\\multimap A \\vdash P[yN\/z]:\\tau$}\n\t\\RightLabel{$cut$}\n\t\\BinaryInfC{$\\Gamma, \\Delta, \\Theta \\vdash P[(\\lambda x.M)N\/z]:\\tau$}\n\t\\DisplayProof\n&\\rightsquigarrow \\ \n\\AxiomC{$\\mathcal{D}'$}\n\t\\noLine\n\t\\UnaryInfC{$\\Delta\\vdash N:\\sigma$}\n\t\\AxiomC{$\\mathcal{D}$}\n\t\\noLine\n\t\\UnaryInfC{$\\Gamma, x:\\sigma \\vdash M:A$}\n\t\\RightLabel{$cut$}\n\t\\BinaryInfC{$\\Gamma, \\Delta \\vdash M[N\/x]:A$}\n\t\\AxiomC{$\\mathcal{D}''$}\n\t\\noLine\n\t\\UnaryInfC{$\\Theta, z:A \\vdash P:\\tau$}\n\t\\RightLabel{$cut$}\n\t\\BinaryInfC{$\\Gamma, \\Delta, \\Theta \\vdash P[M[N\/x]\/z]:\\tau$}\n\t\\DisplayProof\\\\ \\\\ \n\t\t\\AxiomC{$\\mathcal{D}$}\n\t\\noLine\n\t\\UnaryInfC{$\\Gamma \\vdash M: A\\langle\\gamma\/\\alpha\\rangle$}\n\t\\RightLabel{$\\forall R$}\n\t\\UnaryInfC{$\\Gamma \\vdash M: \\forall\\alpha. A$}\n\t\\AxiomC{$\\mathcal{D}'$}\n\t\\noLine\n\t\\UnaryInfC{$\\Delta, x: A\\langle B\/\\alpha\\rangle \\vdash N:\\tau$}\n\t\\RightLabel{$\\forall L$}\n\t\\UnaryInfC{$\\Delta, x: \\forall\\alpha. A \\vdash N:\\tau$}\n\t\\RightLabel{$cut$}\n\t\\BinaryInfC{$\\Gamma,\\Delta \\vdash N[M\/x]:\\tau$}\n\t\\DisplayProof\n& \\rightsquigarrow \\ \n\t\\AxiomC{$\\mathcal{D} \\langle B \/\\alpha\\rangle$}\n\t\\noLine\n\t\\UnaryInfC{$\\Gamma\\vdash M:A\\langle B\/\\alpha\\rangle$}\n\t\\AxiomC{$\\mathcal{D}'$}\n\t\\noLine\n\t\\UnaryInfC{$\\Delta, x:A\\langle B\/\\alpha\\rangle \\vdash N:\\tau$}\n\t\\RightLabel{$cut$}\n\t\\BinaryInfC{$\\Gamma, \\Delta \\vdash N[M\/x]:\\tau$}\n\t\\DisplayProof\\\\ \\\\ \n\t\t\\AxiomC{$\\mathcal{D}$}\n\t\\noLine\n\t\\UnaryInfC{$\\vdash V: \\sigma$}\n\t\\RightLabel{$p$}\n\t\\UnaryInfC{$\\vdash V: \\shpos \\sigma$}\n\t\\AxiomC{$\\mathcal{D}'$}\n\t\\noLine\n\t\\UnaryInfC{$\\Gamma, x: \\sigma \\vdash M:\\tau$}\n\t\\RightLabel{$d$}\n\t\\UnaryInfC{$\\Gamma, y: \\shpos \\sigma \\vdash M[y\/x]:\\tau$}\n\t\\RightLabel{$cut$}\n\t\\BinaryInfC{$\\Gamma \\vdash M[ V\/x]:\\tau$}\n\t\\DisplayProof\n& \\rightsquigarrow \\ \n\t\\AxiomC{$\\mathcal{D}$}\n\t\\noLine\n\t\\UnaryInfC{$\\vdash V:\\sigma$}\n\t\\AxiomC{$\\mathcal{D}'$}\n\t\\noLine\n\t\\UnaryInfC{$\\Gamma, x:\\sigma \\vdash M:\\tau$}\n\t\\RightLabel{$cut$}\n\t\\BinaryInfC{$\\Gamma \\vdash M[V\/x]:\\tau$}\n\t\\DisplayProof\\\\\\\\ \n\\AxiomC{$\\mathcal{D}$}\n\\noLine\n\\UnaryInfC{$\\vdash V:\\sigma$}\n\\RightLabel{$p$}\n\\UnaryInfC{$\\vdash  V:  \\shpos \\sigma$}\n\\AxiomC{$\\mathcal{D}'$}\n\\noLine\n\\UnaryInfC{$\\Delta \\vdash M:\\tau$}\n\\RightLabel{$w$}\n\\UnaryInfC{$\\Delta, x: \\shpos  \\sigma \\vdash \\mathtt{discard}_{\\sigma}\\, x \\mathtt{\\ in\\ }M:\\tau$}\n\\RightLabel{$cut$}\n\\BinaryInfC{$\\Delta \\vdash \\mathtt{discard}_{\\sigma}\\, V \\mathtt{\\ in\\ }M:\\tau$}\n\\DisplayProof\n& \\rightsquigarrow \\ \n\\AxiomC{$\\mathcal{D}'$}\n\\noLine\n\\UnaryInfC{$\\Delta \\vdash M:\\tau$}\n\\DisplayProof\\\\ \\\\\n\\AxiomC{$\\mathcal{D}$}\n\\noLine\n\\UnaryInfC{$\\vdash V: \\sigma$}\n\\RightLabel{$p$}\n\\UnaryInfC{$\\vdash  V: \\shpos \\sigma$}\n\\AxiomC{$\\mathcal{D}_1$}\n\\noLine\n\\UnaryInfC{$\\Delta, y: \\shpos \\sigma, z:\\shpos \\sigma \\vdash M: \\tau$}\n\\AxiomC{$\\mathcal{D}_2$}\n\\noLine\n\\UnaryInfC{$\\vdash U :\\sigma$}\n\\RightLabel{$c$}\n\\BinaryInfC{$\\Delta, x:\\shpos \\sigma \\vdash \\mathtt{copy}^{U }_{\\sigma}\\,  x \\mathtt{\\ as\\ } y, z \\mathtt{\\ in\\ }M:\\tau$}\n\\RightLabel{$cut$}\n\\BinaryInfC{$\\Delta \\vdash \\mathtt{copy}^{U} _{\\sigma}\\, \nV \\mathtt{\\ as\\ } y, z \\mathtt{\\ in \\ }M:\\tau$}\n\\DisplayProof\n&\\rightsquigarrow \\ \n\\AxiomC{$\\mathcal{D}$}\n\\noLine\n\\UnaryInfC{$\\vdash V : \\sigma$}\n\\RightLabel{$p$}\n\\UnaryInfC{$\\vdash  V  : \\shpos \\sigma$}\n\\AxiomC{$\\mathcal{D}$}\n\\noLine\n\\UnaryInfC{$\\vdash V  :\\sigma$}\n\\RightLabel{$p$}\n\\UnaryInfC{$\\vdash  V : \\shpos \\sigma$}\n\\AxiomC{$\\mathcal{D}_1$}\n\\noLine\n\\UnaryInfC{$\\Delta, y: \\shpos \\sigma, z: \\shpos \\sigma \\vdash M:\\tau$}\n\\RightLabel{$cut$}\n\\BinaryInfC{$\\Delta, z: \\shpos \\sigma \\vdash M[V\/y]:\\tau$}\n\\RightLabel{$cut$}\n\\BinaryInfC{$\\Delta \\vdash M[V\/y, V\/z]:\\tau$}\n\\DisplayProof\n\\end{aligned}\n$}\n\\end{equation*}\n\\caption{\\emph{Lazy} cut-elimination rules for the principal cuts $(\\multimap$R$,\\multimap$L$)$, $(\\forall$L$,\\forall$R$)$, \n$(p, d)$, $(p, w)$, and $(p, c)$.}\n\\label{fig: cut elimination exponential}\n\\end{figure}\n\n\\noindent \nThe lazy cut-elimination rules that we introduce here below \nare the standard ones, but restricted to avoid the elimination of non lazy instances of the\nexponential cuts $(p, d)$, $(p, w)$ and $(p, c)$. \n\n\\begin{defn}[Lazy cut-elimination rules]\n\\label{defn:Lazy cut-elimination rules}\nFigure~\\ref{fig: cut elimination exponential} \nintroduces the \\emph{lazy cut-elimination rules} for the principal \ncuts.\nThe elimination rules for commuting and axiom cuts are standard, so we omit\nthem all from Figure~\\ref{fig: cut elimination exponential};\nthe (possibly) less obvious commuting ones can be recovered \nfrom the reductions on terms in Figure~\\ref{fig: term commuting conversions}.\nWe remark that the elimination of the principal cuts\n$(\\forall$R$,\\forall$L$)$ and $(p, d)$ does not modify the subject \nof their concluding judgment. So, we call them \\textit{insignificant} as every other cut-elimination rule non influencing their concluding subject.\nGiven a derivation $ \\mathcal{D} $, we write $\\mathcal{D}\\rightsquigarrow \\mathcal{D}'$ if $\\mathcal{D}$ rewrites  to some $\\mathcal{D}$ by \none of the above rules.\n\\qed\n\\end{defn}\n\\noindent\nLazy cut-elimination is a way of preventing the erasure and the duplication of terms that are not values, and hence to restore a correspondence between cut-elimination and term reduction.  However, one can run into derivations containing exponential cuts that will never turn into lazy cuts, like the deadlock in~\\eqref{eqn: non linear duplication}. The solution we adopt is to identify a set of judgments whose derivations can be rewritten into cut-free ones by a sequence of lazy cut-elimination steps.\n\n\\begin{defn}[Lazy types, lazy judgments and lazy derivations]\n\\label{defn: lazyness} \nWe say that $\\sigma$ is a \\textit{lazy type} if it contains no \\emph{negative} occurrences of $\\forall$. \nAlso, we say that $x_1:\\sigma_1, \\ldots, x_n:\\sigma_n  \\vdash M:\\tau$ \nis a \\textit{lazy judgment} if $\\tau$ is a lazy type and \n$\\sigma_1$, \\ldots, $\\sigma_n$ contain no \\emph{positive}  \noccurrences of  $\\forall$. \nLast, $\\mathcal{D}\\triangleleft \\Gamma \\vdash \nM: \\tau$ is called a \\textit{lazy derivation}  if $\\Gamma \\vdash M: \\tau$ \nis a lazy judgment.\n\\qed\n\\end{defn}\n\\noindent\nLemma~\\ref{fact: quantification propagation} and \n\\ref{lem: lazy derivation properties} here below, as well as \nDefinition~\\ref{defn: bound mio} and\n\\ref{defn:Height of an inference rule},  are the last preliminaries\nto show the relevance of lazy cuts that occur in lazy derivations.\n\n\\begin{lem} {\\ }\n\\label{fact: quantification propagation}\n\\begin{enumerate}[(1)]\n\\item \n\\label{enum: shpos is lazy} \nEvery type of the form $\\shpos \\sigma$ is closed and lazy.\n\\item \n\\label{enum: closed implies positive forall} \nEvery closed type has at least a positive quantification. \n\\item \n\\label{enum: rules c,d,w,forallL and some p are not lazy} \nLet $ \\rho $ be any instance of \n$\\forall \\mathrm{L}$, $d$, $w$, $c$, and $ p $, the latter with a\nnon empty context. The conclusion of $ \\rho $ is not lazy.\n\\item \n\\label{enum: quantification propagation} \nLet $ \\rho $ be any instance of $ ax $, $ \\multimap\\mathrm{R}$, \n$ \\multimap\\mathrm{L}$, $ \\forall\\mathrm{R}$, and $ p $, the latter with an empty context. If the conclusion of $ \\rho $ is lazy, then, every  premise of $ \\rho $ is lazy.\n\\item \n\\label{enum: cut-free and lazy implies all judgements lazy}  \nIf $\\mathcal{D}$ is a cut-free and lazy derivation of $\\mathsf{LEM}$, \nthen  all its judgments are lazy and no occurrences of $\\forall \\mathrm{L}$, $d$, $w$, $c$, and $ p $, the latter with a\nnon empty context, can exist in $ \\mathcal{D} $.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nPoint~\\ref{enum: shpos is lazy} holds by Definition~\\ref{defn:Types for IMLL2shpos}. \nPoint~\\ref{enum: closed implies positive forall} is by a structural induction on types. \nConcerning Point~\\ref{enum: rules c,d,w,forallL and some p are not lazy}, the conclusions of  $d$, $w$, $c$, and $ p $ contain $ \\shpos\\sigma $. This is a closed type,  hence,  by Point~\\ref{enum: closed implies positive forall}, such conclusions are not lazy judgments. Moreover, \n$\\forall \\mathrm{L}$   introduces  a positive occurrence of\n$ \\forall $ in the context of its conclusion, so that this latter cannot be a lazy judgment. \nPoint~\\ref{enum: quantification propagation} is a case analysis on every listed inference rule.\nAs for Point~\\ref{enum: cut-free and lazy implies all judgements lazy},\nwe can proceed by structural induction on $ \\mathcal{D} $. By definition,\nthe conclusion of $ \\mathcal{D} $ is a lazy judgment. \nPoint~\\ref{enum: rules c,d,w,forallL and some p are not lazy} excludes\nthat one among $\\forall \\mathrm{L}$, $d$, $w$, $c$, and $ p $ (with a\nnon empty context) may be the last rule of $ \\mathcal{D} $. So, \nonly one among $ ax $, $ \\multimap\\mathrm{R}$, \n$ \\multimap\\mathrm{L}$, $ \\forall\\mathrm{R}$, and $ p $ (with an empty context) can be the concluding rule, say $ r $, of $ \\mathcal{D} $.\nPoint~\\ref{enum: quantification propagation} implies that \nall the premises of $ r $ are lazy. Hence, we can apply the inductive hypothesis to the derivations of the premises of $ r $  and conclude.\n\\end{proof}\n\n\\begin{defn}[Size of a derivation]\n\t\\label{defn: bound mio}\n\tThe \\emph{size} $\\vert\\mathcal{D}\\vert$ of a derivation $\\mathcal{D}$ in $\\mathsf{LEM}$ is defined by induction:\n\t\\begin{enumerate}\n\t\t\\item \n\t\tIf $\\mathcal{D}$ is $ax$ then $\\vert\\mathcal{D}\\vert=1$.\n\t\t\\item \n\t\tIf $\\mathcal{D}$ is a derivation $\\mathcal{D}'$ that concludes by a\n\t\trule with a single premise, then $\\vert\\mathcal{D}\\vert= \\vert\\mathcal{D}'\\vert+1$.\n\t\t\\item \n\t\tIf $\\mathcal{D}$ composes two derivations $\\mathcal{D}'$ and $\\mathcal{D}'' $ by a rule with two premises, but different from $c$, \n\t\tthen $\\vert\\mathcal{D}\\vert= \\vert\\mathcal{D}'\\vert+ \\vert\\mathcal{D}''\\vert+1$. \n\t\t\\item \n\t\t\\label{enum:on standard dimension of c}\n\t\tIf $\\mathcal{D}$ composes two derivations $\\mathcal{D}'$ and $\\mathcal{D}'' $ by the rule $c$,  \n\t\tthen $\\vert\\mathcal{D}\\vert= \\vert\\mathcal{D}'\\vert+ \\vert\\mathcal{D}''\\vert+3$. \n\t\t\\qed\n\t\\end{enumerate}\n\\end{defn}\n\n\\begin{rem}\nAdding ``3'' instead of the possibly expected ``1'' in the clause~\\eqref{enum:on standard dimension of c} of Definition~\\ref{defn: bound mio} highlights the non linearity that the instances of $ c $ \nintroduce in the course of the lazy cut-elimination on \\textsf{LEM} of Definition~\\ref{defn:Lazy cut-elimination strategy} below.\n\\qed\n\\end{rem}\n\n\\begin{lem} \n\\label{lem: lazy derivation properties}\nLet $\\mathcal{D}\\triangleleft x_1: \\sigma_1, \\ldots, x_n: \\sigma_n \\vdash M: \\sigma$ be a cut-free and lazy derivation.\n\\begin{enumerate}[(1)]\n\\item \n\\label{enum: cut free and lazy implies linear normal form} \n$M$ is a linear $\\lambda$-term in normal form.\n\\item  \n\\label{enum: size term bounded size types LEML} \n$\\vert M \\vert \\leq \\sum_{i=1}^n \\vert \\sigma_i \\vert +  \\vert \\sigma \\vert $.\n\\item \n\\label{enum: relation lazy cut free derivation and term} \n$\\vert \\mathcal{D} \\vert= \\vert M \\vert +k$, where $k$  is the number of $\\forall$ and $\\shpos$ occurring in   $\\sigma_1, \\ldots, \\sigma_n, \\sigma$.\n\\item \n\\label{enum: lazy cut free derivations are smaller if terms are smaller} \nIf $\\mathcal{D}'\\triangleleft x_1: \\sigma_1, \\ldots, x_n: \\sigma_n \\vdash N: \\sigma$ is  a lazy and cut-free derivation, then $\\vert N \\vert \\leq \\vert M \\vert$ implies $\\vert \\mathcal{D}'\\vert \\leq \\vert \\mathcal{D} \\vert $.\n\\item \n\\label{enum: finite number of normal forms} \nThe set of values with lazy type $ \\sigma $ is finite.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nThe assumptions on $ \\mathcal{D}$ allow to apply Lemma~\\ref{fact: quantification propagation}.\\ref{enum: cut-free and lazy implies all judgements lazy}\nwhich implies that every judgment in $\\mathcal{D}$ is lazy and\nfree of $\\forall$L, $d$, $w$, $c$ and $ p $ (with a non empty context). \nHence, we can prove Points~\\ref{enum: cut free and lazy implies linear normal form}-\\ref{enum: relation lazy cut free derivation and term} by induction on the structure of $\\mathcal{D}$. Point~\\ref{enum: lazy cut free derivations are smaller if terms are smaller} is a corollary of\nPoint~\\ref{enum: relation lazy cut free derivation and term}. Point~\\ref{enum: finite number of normal forms} is a corollary of \nPoint~\\ref{enum: size term bounded size types LEML}.\n\\end{proof}\n\n\n\n\\begin{defn}[Height of an inference rule]\n\\label{defn:Height of an inference rule}\nLet $\\mathcal{D}\\triangleleft \\Gamma\\vdash M:\\sigma$ be a derivation and \n$ r $ a rule instance in it. The \\emph{height of $r$}, written $ h(r) $,  is the number of rule instances from the conclusion \n$ \\Gamma\\vdash M:\\sigma $ of $ \\mathcal{D} $ upward to the conclusion of $ r $. The \\textit{height of $\\mathcal{D}$}, written $h(\\mathcal{D})$, is the largest $h(r)$ among its rule instances.\n\\qed\n\\end{defn}\n\n\n\\noindent\nLemma~\\ref{lem: existence of the safe exponential cut} \nand~\\ref{lem: above the cut} assure that we can eliminate exponential lazy cuts from a lazy derivation.\n\n\\begin{lem}[Existence of a lazy $ cut $] \n\\label{lem: existence of the safe exponential cut} \nLet $\\mathcal{D}$ be a lazy derivation with only exponential cuts in it.\nAt least one of those cuts is \\emph{safe}.\n\\end{lem}\n\\begin{proof}\nLet $\\Gamma \\vdash M: \\tau$ be the conclusion of $ \\mathcal{D} $.\nBy contradiction, let us suppose that every occurrence of (exponential)\n$ cut $ in $ \\mathcal{D} $ is a deadlock.\nAt least one of them, say $ c_m $, has minimal height $ h(c_m) $, i.e.~no other $ cut $ occurs in the sequence of rule instances, say $ r_1,\\ldots r_n $, from the conclusion of $ c_m $ down to the\none of $ \\mathcal{D} $.\nSince $ c_m $ is a deadlock, its leftmost premise has form $ \\Delta \\vdash N:\\shpos\\sigma $, \nwhere $ \\Delta\\neq \\emptyset $. \nBy Proposition~\\ref{prop: exponential context}, $\\Delta$ is strictly exponential and the whole $ \\Delta \\vdash N:\\shpos\\sigma $ is a non lazy\njudgment by Lemma~\\ref{fact: quantification propagation}.\\ref{enum: shpos is lazy} and Lemma~\\ref{fact: quantification propagation}.\\ref{enum: closed implies positive forall}. \nThe contraposition of\nLemma~\\ref{fact: quantification propagation}.\\ref{enum: quantification propagation} \nimplies that the non lazy judgment in $ c_m $ can only be transformed to \na non lazy judgment by every $ r_i $, with $ 1\\leq i\\leq n $, letting the \nconclusion of $ \\mathcal{D} $ non lazy, so\ncontradicting the assumption.\nHence, $ c_m $ must be safe.\n\\end{proof}\n\n\\begin{lem}[Eliminating a lazy $ cut $]\n\\label{lem: above the cut}\nLet $\\mathcal{D}$ be a lazy derivation with only exponential cuts in it.\nA lazy derivation $\\mathcal{D}^*$ exists such that \nboth $\\mathcal{D} \\rightsquigarrow \\mathcal{D}^*$, by reducing a lazy cut, and \n$\\vert\\mathcal{D}^*\\vert < \\vert\\mathcal{D}\\vert$.\n\\end{lem}\n\\begin{proof}\nLemma~\\ref{lem: existence of the safe exponential cut}\nimplies that $ \\mathcal{D} $ contains at least an exponential cut which is  safe. Let us take $(p, X)$ with maximal height $ h((p, X)) $ among those safe  instances of $ cut $. \nSo, if $ (p, X) $ has form:\n\\begin{equation}\n\\label{eqn: above the cut}\n\\AxiomC{$\\mathcal{D}'$}\n\\noLine\n\\UnaryInfC{$\\vdash N: \\sigma$}\n\\RightLabel{$p$}\n\\UnaryInfC{$\\vdash  N: \\shpos \\sigma$}\n\\AxiomC{\\vdots}\n\\RightLabel{$X$}\n\\UnaryInfC{$\\Delta, x:\\shpos \\sigma \\vdash M:\\tau$}\n\\RightLabel{$cut$ \\enspace ,}\n\\BinaryInfC{$ \\Delta \\vdash M[N\/x]:\\tau$}\n\\DisplayProof\n\\end{equation}\n\\noindent\nthen $\\mathcal{D}'$ is a lazy derivation because $\\shpos \\sigma$ is a lazy type by  Lemma~\\ref{fact: quantification propagation}.\\ref{enum: shpos is lazy}.  Since $\\mathcal{D}'$ is lazy and can only contain exponential cuts, by Lemma~\\ref{lem: existence of the safe exponential cut} and by maximality of  $ h((p, X)) $,   it is forcefully cut-free.\nSo, by Lemma~\\ref{lem: lazy derivation properties}.\\ref{enum: cut free and lazy implies linear normal form}, we have that $ N $ is a value, \ni.e.~$(p, X)$ is lazy and we can reduce it to obtain $ \\mathcal{D}^*$. \nIf $ X $ in~\\eqref{eqn: above the cut} is $d$ or $w$, then it is \nsimple to show\nthat $\\vert\\mathcal{D}^*\\vert<\\vert\\mathcal{D}\\vert$. \nLet $ X $ be $ c $. Then, \\eqref{eqn: above the cut} is:\n\\begin{equation}\n\\label{eqn:above the cut (p,c)}\n\\AxiomC{$\\mathcal{D}'$}\n\\noLine\n\\UnaryInfC{$\\vdash V: \\sigma$}\n\\RightLabel{$p$}\n\\UnaryInfC{$\\vdash  V: \\shpos \\sigma$}\n\\AxiomC{$\\mathcal{D}''$}\n\\noLine\n\\UnaryInfC{$\\Delta, y: \\shpos \\sigma, z:\\shpos \\sigma \\vdash M': \\tau$}\n\\AxiomC{$\\mathcal{D}'''$}\n\\noLine\n\\UnaryInfC{$\\vdash U: \\sigma$}\n\\RightLabel{$c$}\n\\BinaryInfC{$\\Delta, x:\\shpos \\sigma \\vdash \\mathtt{copy}^{U}_{\\sigma}\\,  x \\mathtt{\\ as\\ }y,z \\mathtt{\\ in\\ }M':\\tau$}\n\\RightLabel{$cut$ \\enspace ,}\n\\BinaryInfC{$\\Delta \\vdash \\mathtt{copy}^{U} _{\\sigma}\\,   V \\mathtt{\\ as\\ }y,z \\mathtt{\\ in \\ }M':\\tau$}\n\\DisplayProof\n\\end{equation}\nwith $\\mathcal{D}'''$ lazy and cut-free for the same reasons as $ \\mathcal{D}' $ is. So, \\eqref{eqn:above the cut (p,c)} can reduce to:\n\\begin{equation}\n\\label{eqn:reducing above the cut (p,c)}\n\\AxiomC{$\\mathcal{D}'$}\n\\noLine\n\\UnaryInfC{$\\vdash V : \\sigma$}\n\\RightLabel{$p$}\n\\UnaryInfC{$\\vdash  V  : \\shpos \\sigma$}\n\\AxiomC{$\\mathcal{D}'$}\n\\noLine\n\\UnaryInfC{$\\vdash V  :\\sigma$}\n\\RightLabel{$p$}\n\\UnaryInfC{$\\vdash  V : \\shpos \\sigma$}\n\\AxiomC{$\\mathcal{D}''$}\n\\noLine\n\\UnaryInfC{$\\Delta, y: \\shpos \\sigma, z: \\shpos \\sigma \\vdash M':\\tau$}\n\\RightLabel{$cut$ \\enspace .}\n\\BinaryInfC{$\\Delta, z: \\shpos \\sigma \\vdash M'[V\/y]:\\tau$}\n\\RightLabel{$cut$}\n\\BinaryInfC{$\\Delta \\vdash M'[V\/y, V\/z]:\\tau$}\n\\DisplayProof\n\\end{equation}\n\\noindent\nBy Lemma~\\ref{lem: lazy derivation properties}.\\ref{enum: finite number of normal forms}, we can safely  assume that $U$ is a value with largest size among values of type $\\sigma$.\nSo, Lemma~\\ref{lem: lazy derivation properties}.\\ref{enum: size term bounded size types LEML} implies $\\vert V \\vert \\leq \\vert U \\vert$, \nfrom which $\\vert \\mathcal{D}' \\vert \\leq \\vert \\mathcal{D}''' \\vert$,\nby Lemma~\\ref{lem: lazy derivation properties}.\\ref{enum: lazy cut free derivations are smaller if terms are smaller}.\nBy applying Definition~\\ref{defn: bound mio}.\\ref{enum:on standard dimension of c} to \n\\eqref{eqn:above the cut (p,c)} and\n\\eqref{eqn:reducing above the cut (p,c)}, we have $\\vert \\mathcal{D}^* \\vert < \\vert \\mathcal{D} \\vert$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\begin{defn}[Lazy cut-elimination strategy]\n\\label{defn:Lazy cut-elimination strategy}\nLet $ \\mathcal{D} $ be a lazy derivation of $ \\textsf{LEM} $.\nLet a \\emph{round} on $ \\mathcal{D} $ be defined as follows:\n\\begin{enumerate}[\\{1\\}]\n\t\\item \\label{enum: round comm}  \n\tLet eliminate all the commuting instances of $ cut $.\n\t\\item \\label{enum: round sym} \n\tIf any instance of $ cut $ remains, it is necessarily symmetric.\n\tLet reduce a multiplicative cut, if any.\n\tOtherwise, let reduce a lazy exponential cut, if any.\n\\end{enumerate}\nThe \\emph{lazy cut-elimination strategy} iterates \nrounds, starting from $ \\mathcal{D} $, until instances of $ cut $\nexist in the obtained derivation.\n\\qed\n\\end{defn}\n\n\t\n\n\n\\begin{thm}[Lazy cut-elimination has a cubic bound]\n\\label{thm: cut elimination for downarrow IMLL2}\nLet $\\mathcal{D}$ be a lazy derivation. \nThe lazy cut-elimination can reduce $\\mathcal{D}$ to a cut-free $\\mathcal{D}^*$ in $\\mathcal{O}(\\vert \\mathcal{D} \\vert^3)$ steps.\n\\end{thm}\n\\begin{proof}\nLet $H(\\mathcal{D})$ be the sum of the heights $h(\\mathcal{D}')$ \nof all  sub-derivations $\\mathcal{D}'$ of \n$\\mathcal{D}$ whose conclusion is an instance of $ cut $.\nWe proceed by induction on the lexicographically order of the pairs \n$\\langle\\vert \\mathcal{D}\\vert , H(\\mathcal{D}) \\rangle$. \nTo show that the lazy cut-elimination strategy in Definition~\\ref{defn:Lazy cut-elimination strategy} terminates, we\nstart by applying a round to $ \\mathcal{D} $, using \nstep~\\ref{enum: round comm}.\nEvery commuting cut-elimination step  just moves an instance of $ cut $ upward, strictly decreasing $H(\\mathcal{D})$ and leaving\n$ \\vert \\mathcal{D}\\vert $ unaltered. \nLet us continue by applying\n\tstep~\\ref{enum: round sym} of the round.\nAs usual, $ \\vert \\mathcal{D}\\vert $ shrinks when eliminating a multiplicative cut.\nIf only exponential instances of $ cut $ remain,  by Lemma~\\ref{lem: above the cut} we can rewrite \n$ \\mathcal{D} $ to $ \\mathcal{D}' $ by reducing a lazy exponential cut in such a way that \n$ \\vert \\mathcal{D}'\\vert  < \\vert \\mathcal{D} \\vert $. Therefore,  the lazy cut-elimination strategy terminates with a cut-free derivation $ \\mathcal{D}^* $.\n\\par\nWe now exhibit a bound on the number of \ncut-elimination steps from $ \\mathcal{D} $ to $ \\mathcal{D}^* $.\nGenerally speaking, we can represent a lazy strategy as:\n\\begin{align}\n\\label{align:cut-elimination bound diagram}\n\\mathcal{D}\n= \\mathcal{D}_0\n\\underbrace{\\longrightarrow}_{cc_0}\n\\mathcal{D}'_0\n\\rightsquigarrow\n\\mathcal{D}_1\n\\,\\cdots \\underbrace{\\longrightarrow}_{cc_i}\n\\mathcal{D}'_{i}\n\\rightsquigarrow\n\\mathcal{D}_{i+1}\n\\underbrace{\\longrightarrow}_{cc_{i+1}}\\cdots\\,\n\\mathcal{D}'_{n-1}\n\\rightsquigarrow\n\\mathcal{D}_n\n\\underbrace{\\longrightarrow}_{cc_{n}}\n\\mathcal{D}'_{n}\n= \\mathcal{D}^*\n\\enspace ,\n\\end{align}\nwhere every $ cc_j $ denotes the number of \ncommuting cuts applied from derivation $ \\mathcal{D}_{j} $ to derivation\n$ \\mathcal{D}'_{j} $, for every $ 0\\leq j\\leq n $.\nA bound on every $ cc_j $ is $ \\vert\\mathcal{D}_{j}\\vert^2 $ because every\ninstance of rule in $ \\mathcal{D}_{j} $ can, in principle, be commuted with every other. \nThe first part of the proof implies\n$ \\vert\\mathcal{D}_{j}\\vert = \\vert\\mathcal{D}'_{j}\\vert $,\nfor every $ 0\\leq j\\leq n $.\nLemma~\\ref{lem: above the cut} implies\n$ \\vert\\mathcal{D}'_{j}\\vert > \\vert\\mathcal{D}_{j+1}\\vert $,\nfor every $ 0\\leq j\\leq n-1 $.\nSo, $ n\\leq \\vert\\mathcal{D}\\vert $ and the total number of cut-elimination steps in \n\\eqref{align:cut-elimination bound diagram} is \n$ O(\\vert\\mathcal{D}\\vert\\cdot\\vert\\mathcal{D}\\vert^2) $.\n\\end{proof}\n\n\\begin{rem}\nThe cubic bound on the lazy strategy keeps holding also in the case we apply the \\emph{lazy} cut-elimination to \\emph{non lazy} derivations. Of course, in that case, deadlocks may remain in the final derivation where no instance of $cut$ can be further eliminated.\n\\qed\n\\end{rem}\n\n\n\n\\subsection{Subject reduction theorem}\n\\label{sec: subject reduction}\nThe proof of the Subject reduction requires some typical preliminaries.\n\\begin{lem}[Substitution]\n\\label{lem: substitution type variables} \nIf $\\Gamma \\vdash M: \\tau$ then $\\Gamma[A\/ \\alpha] \\vdash M: \\tau[A\/ \\alpha]$, for every \\emph{linear} type $ A $.\n\\end{lem}\n\\begin{lem}[Generation] \\label{lem: generation}  {\\ }  \n\\begin{enumerate}[(1)]\n\\item \\label{lem:generation 1} \nIf $\\mathcal{D} \\triangleleft \\Gamma \\vdash \\lambda x. M:\\tau$, then \n$\\tau= \\shpos^n   \\forall \\vec{ \\alpha}. (\\sigma \\multimap A)$, where \n$\\shpos^n\\triangleq \\shpos \\overset{n}{\\ldots}\\shpos$ and \n$\\vec{\\alpha}= \\alpha_1,\\ldots, \\alpha_m$, for some $n\\geq0$ and $m\\geq 0$.\n\\item  \\label{lem:generation 5} \nIf $\\mathcal{D} \\triangleleft \\Delta, x: \\forall \\alpha.A \\vdash P:\\tau$, then an instance $ r $ of $\\forall\\mathrm{L}$ exists in $\\mathcal{D}$ \nwith \nconclusion $\\Delta', x: \\forall \\alpha.A \\vdash P':\\tau'$,\nfor some $\\Delta', P'$ and $\\tau'$. \nI.e., $ r $ introduces $x:\\forall \\alpha.A$.\n\\item  \\label{lem:generation 6}\nIf $\\mathcal{D} \\triangleleft \\Delta, x: \\sigma \\multimap B \n    \\vdash P[xN\/y]: \\tau$, \nthen an instance $ r $ of $\\multimap\\mathrm{L}$ \nexists in $\\mathcal{D}$ with\nconclusion $\\Delta', x: \\sigma \\multimap B \\vdash P'[xN'\/y] : \\tau'$,\nfor some $\\Delta', P', N'$ and $\\tau'$.\nI.e., $ r $ introduces $x:\\sigma \\multimap B$.\n\\item  \\label{lem:generation 2} \nIf $\\mathcal{D} \\triangleleft \\Gamma \\vdash \\lambda x. M : \\forall  \\alpha. A$, then a derivation $\\mathcal{D}'$ exists\nwhich is $\\mathcal{D}$ with some rule permuted in order to obtain an instance of $\\forall\\mathrm{R}$ as last rule of $\\mathcal{D}'$.\n\\item  \\label{lem:generation 3} \nIf $\\mathcal{D}\\triangleleft \\Gamma \\vdash \\lambda x. P: \\sigma \\multimap B$, then a derivation $\\mathcal{D}'$ exists\nwhich is $\\mathcal{D}$ with some rule permuted in order to obtain an instance of $\\multimap\\mathrm{R}$ as last rule of $\\mathcal{D}'$.\n\\item  \\label{lem:generation 7}\nIf $\\mathcal{D} \\triangleleft \\Delta, x: \\shpos \\sigma \\vdash P[xN\/y] : \n\\tau$, then an instance $ r $ of $ d $ exists in $\\mathcal{D}$ \nwith \nconclusion  $\\Delta', x: \\shpos \\sigma \\vdash P'[xN'\/y]:\\tau'$,\nfor some $\\Delta', P', N'$ and $\\tau'$.\nI.e.,  $d$ introduces $x:\\shpos  \\sigma$.\n\\item \\label{lem:generation 4} \nIf $\\mathcal{D}  \\triangleleft \\Gamma \\vdash M: \\shpos \\sigma$, then a derivation $\\mathcal{D}'$ exists\nwhich is $\\mathcal{D}$ with some rule permuted in order to get an instance of $p$ as last rule of $\\mathcal{D}'$.\n\\item \\label{lem:generation 8}\nIf $\\mathcal{D} \\triangleleft \\Delta, x: \\shpos  \\sigma \\vdash \n\\mathtt{discard}_{\\sigma}\\, x \\mathtt{\\ in \\ }P: \\tau$, then\nan instance $ r $ of $ w $ exists in $\\mathcal{D}$ with \nconclusion \n$ \\Delta', x: \\shpos\\sigma \n  \\vdash \n  \\mathtt{discard}_{\\sigma}\\, x \\mathtt{\\ in \\ }P': \\tau' $,\n  for some $ \\Delta', P' $ and $ \\tau' $.\nI.e., $r$ introduces $x:\\shpos \\sigma$.\n\\item \\label{lem:generation 9}\nIf $\\mathcal{D} \\triangleleft \\Delta, x: \\shpos \\sigma \\vdash \n\t\\mathtt{copy}_{\\sigma}^{U} \\, x  \\mathtt{\\ as \\ } x_1, x_2 \\mathtt{\\ in \\ }P: \\tau$, then\n\tan instance $ r $ of $ c $ exists in $\\mathcal{D}$ with \n\tconclusion \n\t$\\Delta', x: \\shpos \\sigma \\vdash \n\t\\mathtt{copy}_{\\sigma}^{U} \\, x  \\mathtt{\\ as \\ } \n\tx_1, x_2 \\mathtt{\\ in \\ }P': \\tau'$,\n\tfor some $ \\Delta', P' $ and $ \\tau' $.\n\tI.e., $r$ introduces $x:\\shpos \\sigma$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nWe can adapt the proof by \nGaboardi\\&Ronchi in~\\cite{gaboardi2009light} to $\\mathsf{LEM}$\nbecause the types in Definition~\\ref{defn:Types for IMLL2shpos}\nare a sub-set of Gaboardi\\&Ronchi's \\emph{essential types}.\nIn particular, Point\\!~\\textit{\\ref{lem:generation 4}} relies on\nProposition~\\ref{prop: exponential context}.\n\\end{proof}\n\n\\begin{thm}[Subject reduction] \n\\label{thm:Subject Reduction for IMLL2shpos} \n If $\\Gamma \n\\vdash M: \\tau$  and $M \\rightarrow M^\\prime$, then $\\Gamma \n\\vdash M^\\prime: \\tau$.\n\\end{thm}\n\\begin{proof}\nWe proceed by structural induction on $ \\mathcal{D} $.\nThe crucial case of $M \\rightarrow M^\\prime$\n(Figure~\\ref{fig: term reduction rules}) is when\n$(\\lambda x. P)Q$ exists in $M$ and $\\mathcal{D}$ contains:\n\\begin{prooftree}\n\\AxiomC{$\\mathcal{D}' \\triangleleft \\Delta \\vdash \\lambda x. P: \\sigma$}\n\\AxiomC{$\\mathcal{D}'' \\triangleleft \\Sigma, y: \\sigma \\vdash N[ yQ\/z ]:\\tau$}\n\\RightLabel{$ cut $ \\enspace .}\n\\BinaryInfC{$\\mathcal{D} \\triangleleft \\Delta, \\Sigma \\vdash N[(\\lambda x.P)Q\/z]:\\tau$}\n\\end{prooftree}\nLemma~\\ref{lem: generation}.\\textit{\\ref{lem:generation 1}} implies that  $\\sigma= \\shpos^n \\forall \\vec{ \\alpha}. (\\sigma_1  \\multimap C)$, where $\\shpos^n\\triangleq \\shpos \\overset{n}{\\ldots}\\shpos$ and $\\vec{\\alpha}= \\alpha_1,\\ldots, \\alpha_m$, for some $n\\geq0$ and $m\\geq 0$. \nLemma~\\ref{lem: generation}.\\textit{\\ref{lem:generation 5}},  \n\\ref{lem: generation}.\\textit{\\ref{lem:generation 6}}, \nand \\ref{lem: generation}.\\textit{\\ref{lem:generation 7}}, \nimply that $\\mathcal{D}''$ has form: \n\\begin{equation}\n\\label{eqn:conclusion of derivation}\n\\AxiomC{\\vdots}\n\\noLine\n\\UnaryInfC{$ \\Sigma_1'' \\vdash Q'':\\sigma_1'$}\n\\AxiomC{$\\vdots$}\n\\noLine\n\\UnaryInfC{$\\Sigma_2'', z: C' \\vdash N'': \\tau''$}\n\\RightLabel{$\\multimap$L}\n\\BinaryInfC{$\\Sigma''_1, \\Sigma''_2, y''':\\sigma_1' \\multimap C' \\vdash N''[y'''Q''\/z]: \\tau''$}\n\\noLine\n\\UnaryInfC{$\\vdots $}\n\\noLine\n\\UnaryInfC{$\\Sigma', y'': \\forall \\vec{\\alpha}. (\\sigma_1 \\multimap C )\\vdash N'[y''Q'\/z]: \\tau'$}\n\\RightLabel{$d$}\n\\UnaryInfC{$\\Sigma', y': \\shpos \\forall \\vec{\\alpha}.( \\sigma_1 \\multimap C) \\vdash N'[y'Q'\/z]: \\tau'$}\n\\noLine\n\\UnaryInfC{$\\vdots $}\n\\noLine\n\\UnaryInfC{$\\Sigma, y:\\shpos ^n \\forall \\vec{\\alpha}. (\\sigma_1 \\multimap C) \\vdash N[yQ\/z]: \\tau$}\n\\DisplayProof\n\\end{equation}\nwhere $\\sigma_1'= \\sigma_1[A_1\/\\alpha_1, \\ldots, A_m \/\\alpha_m]$ and $C'=C[A_1\/\\alpha_1, \\ldots, A_m \/\\alpha_m]$, for some $\\Sigma', \\Sigma''_1, \\Sigma''_2,y', y'',y''', N', N'', Q',Q'',  \\tau',\\tau'',  A_1, \\ldots, A_m$.\nLemma~\\ref{lem: generation}.\\textit{\\ref{lem:generation 2}},  \n\\ref{lem: generation}.\\textit{\\ref{lem:generation 3}} and \n\\ref{lem: generation}.\\textit{\\ref{lem:generation 4}} imply that, permuting some of its rules,\n$\\mathcal{D}'$ can be reorganized as:\n\\begin{prooftree}\n\\AxiomC{$\\vdots$}\n\\noLine\n\\UnaryInfC{$\\Delta, x:\\sigma_1 \\vdash P: C$}\n\\RightLabel{$\\multimap$R}\n\\UnaryInfC{$\\Delta \\vdash \\lambda x. P:\\sigma_1 \\multimap C$}\n\\RightLabel{$\\forall$R  \\enspace ,}\n\\doubleLine\n\\UnaryInfC{$\\Delta \\vdash \\lambda x.P: \\forall \\vec{\\alpha}. (\\sigma_1 \\multimap C)$}\n\\doubleLine\n\\RightLabel{$p$}\n\\UnaryInfC{$\\Delta \\vdash \\lambda x.P: \\shpos^n \\forall \\vec{\\alpha}. (\\sigma_1 \\multimap C)$}\n\\end{prooftree}\nwhere the concluding instances of $ p $ are necessary if $n>0$ and\nare legally introduced because $\\Delta$ is strictly exponential as consequence of Proposition~\\ref{prop: exponential context}\nthat we can apply to the judgment beacause $ \\shpos\\sigma $ is\nstrictly exponential as well.\nMoreover, Lemma~\\ref{lem: substitution type variables} assures that a derivation of $\\Delta, x:\\sigma_1' \\vdash P: C'$ exists\nbecause $\\alpha_1, \\ldots, \\alpha_m$ are not free in $\\Delta$. Therefore:\n\\begin{prooftree}\n\\AxiomC{$\\vdots$}\n\\noLine\n\\UnaryInfC{$\\Sigma''_1 \\vdash Q'': \\sigma'_1$}\n\\AxiomC{$\\vdots$}\n\\noLine\n\\UnaryInfC{$\\Delta, x:\\sigma'_1 \\vdash P: C'$}\n\\RightLabel{$cut$}\n\\BinaryInfC{$\\Delta, \\Sigma''_1 \\vdash P[Q''\/x]: C'$}\n\\AxiomC{$\\vdots$}\n\\noLine\n\\UnaryInfC{$\\Sigma''_2, z:C' \\vdash N'': \\tau''$}\n\\RightLabel{$cut$ \\enspace .}\n\\BinaryInfC{$\\Delta, \\Sigma''_1, \\Sigma''_2 \\vdash N''[P[Q''\/x]\/z]: \\tau''$}\n\\noLine\n\\UnaryInfC{$\\vdots$}\n\\noLine\n\\UnaryInfC{$\\Delta, \\Sigma \\vdash N[P[Q\/x]\/z]: \\tau$}\n\\end{prooftree}\n\\noindent\nwhich concludes with the same rules as in~\\eqref{eqn:conclusion of derivation}. A similar proof exists, which relies on\nLemma~\\ref{lem: generation}.\\textit{\\ref{lem:generation 8}}, \nor Lemma~\\ref{lem: generation}.\\textit{\\ref{lem:generation 9}},\nwhen reducing\n$\\mathtt{discard}_{\\sigma}$ $V\\mathtt{\\ in\\ }$ $M$, or \n$\\mathtt{copy}^{U}_{\\sigma}\\, V \\mathtt{\\ as\\ }y,z \\mathtt{\\ in\\ }M$. All the remaining cases are straightforward. \n\\end{proof}\n\\section{Conclusions}\n\\label{section:Conclusions}\\noindent\nWe introduce $\\mathsf{LEM}$. It is a type-assignment for the \nlinear $ \\lambda $-calculus extended with new constructs that can \nduplicate or erase values, i.e.~closed and normal linear \n$ \\lambda $-terms. $\\mathsf{LEM}$ enjoys a mildly weakened \ncut-elimination, and the Subject reduction. The internalization of the \nmechanism of linear weakening  and contraction by means of modal rules \nallows to exponentially compress derivations of $\\mathsf{IMLL}_2$.  \nOn one side, this enables to represent boolean circuits more compactly, as compared to previous ones, based on the multiplicative fragment of Linear Logic. On the other, $\\mathsf{LEM}$ can represent Church-like encoding of the natural numbers, with successor and the addition for them.\n\nWe conclude by briefly discussing possible future works.\n\nIn Section~\\ref{sec: duplication and erasure in a linear setting}, we \nconjecture that a version of the general separation property holds in \nthe linear $\\lambda$-calculus (Conjecture~\\ref{conj: linear separation}.)\nWere it true, we could show that a\n duplicator exists for all the finite sets of closed terms in \n$\\beta\\eta$-normal form, and connect\nlinear duplication with the standard notion of separation.\n\nIn Section~\\ref{sec: the system shpos, with, oplus IMLL2 cut elimination and bound},  we design $\\mathsf{LEM}$ to express linear weakening and \ncontraction in the same spirit as of the exponential rules of  \n$\\mathsf{LL}$. We are working to push the analogy further,\nby formulating a type-assignment that extends\n$\\mathsf{IMLL}_2$ with \\enquote{linear additives}. A candidate rule is:\n\\begin{prooftree}\n\\AxiomC{$x_1:A \\vdash M_1: A_1 $}\n\\AxiomC{$x_2: A \\vdash M_2: A_2$}\n\\AxiomC{$ \\vdash V:A$}\n\\RightLabel{$\\with$R}\n\\TrinaryInfC{$x: A \\vdash \\mathtt{copy}^V_A \\, x \\mathtt{\\ as \\ }x_1,x_2 \\mathtt{\\ in \\ } \\langle M_1, M_2\\rangle : A_1 \\with A_2$}\n\\end{prooftree}\nwhere $V$ is a value and $A, A_1, A_2$ are closed and without negative occurrences of $\\forall$. The intuition behind this rule is the one \nwe discuss in Section~\\ref{sec: the system shpos, with, oplus IMLL2 cut elimination and bound} for the contraction rule of $\\mathsf{LEM}$. \nWe also claim that the new system  would keep the normalization cost  linear, unlike standard additives (see~\\cite{mairson2003computational}). \n\nSection~\\ref{sec: encoding boolean circuits}  presents an encoding of the  boolean circuits not preserving their depth. Moving to unbounded fan-in proof nets for $\\mathsf{LEM}$ would improve the correspondence,  where the rules $ p, w, c $ and $ d $ would be expressed  by nodes and boxes. \nOperations on them would compactly perform duplication and get rid of garbage, possibly improving~\\cite{DBLP:conf\/lics\/Terui04,DBLP:conf\/lfcs\/MogbilR07,DBLP:journals\/corr\/abs-1201-1120}.  \nA reasonable question would then be whether the use of alternative and weaker exponential rules in $\\mathsf{LEM}$ could be the right approach to capture circuit complexity classes like $\\textsf{NC}, \\textsf{AC}$, and $\\textsf{P}\/_{\\operatorname{poly}}$ in analogy with the implicit characterizations of the Polynomial and Elementary-time computational complexity classes by means of Light Logics \\cite{lafont2004soft,danos2003linear}.\n\nSection~\\ref{section:Church numerals} contributes to the problem of defining numeral systems in linear settings. \nIn~\\cite{Mackie2018}, Mackie has recently introduced linear variants of numeral systems.\nHe shows that successor, addition, predecessor, and subtraction have representatives in the linear $\\lambda$-calculus. We could not find how giving type in $\\mathsf{LEM}$ to  some of  the terms of Mackie's numeral systems.\nHowever, by merging Mackie's encoding and Scott numerals \\cite{CF:58}, numeral systems seem to exist which  $ \\mathsf{LEM} $ can give a type to. The cost would be to extend $\\mathsf{LEM}$ with recursive types, following \nRoversi\\&Vercelli \\cite{Roversi+Vercelli:2010-DICE10}.\n\n\n\\section{Duplication and erasure for the linear $\\lambda$-calculus}\n\\label{sec: duplication and erasure in a linear setting}\nAs a motivational background we discuss  erasure and duplication in the linear $\\lambda$-calculus both in an untyped and in a type-assignment setting.\n\n\\subsection{The untyped setting}\nThe linear $\\lambda$-calculus forbids any form of \\emph{direct} duplication \nof $ \\lambda $-terms, by means of multiple occurrences of the same variable, \nor of erasure, by omitting occurrences of bound variables in a $ \\lambda $-term. \nNevertheless, erasure and duplication can be simulated. \nConcerning the former, a first approach has been developed by Klop \\cite{klop1980combinatory}, and can be called   \\enquote{erasure by garbage collection}.  It consists on   accumulating unwanted data during computation in place of erasing it. \nFor example, $K'=\\lambda xy.\\langle x,y \\rangle$ represents the classical \n$K= \\lambda xy.x$, the second component of $\\langle x,y \\rangle$ \nbeing garbage. Another approach is by Mackie,  and can be called \n\\enquote{erasure by data consumption} \\cite{Mackie2018}. It involves a step-wise erasure \nprocess that proceeds by $\\beta$-reduction, according to the following definition: \n\n\\begin{defn}[Erasability]\n\\label{defn: erasure} \nA linear $\\lambda$-term $M$ is \\emph{erasable} \nif $\\mathcal{C}[M]\\rightarrow_\\beta^* I$, for some  context $\\mathcal{C}[]$ \nsuch that $ \\mathcal{C}[M] $ is linear.\n\\qed \n\\end{defn}\n\\begin{exmp}\nThe context $\\mathcal{C}[]=(\\lambda z.[])III$ erases \n$\\lambda xy.zxy$ because, filling \n$ [] $ by $\\lambda xy.zxy$, we obtain a closed linear $\\lambda$-term that reduces to $I$.\n\\qed\n\\end{exmp}\n\\noindent\nIn~\\cite{mairsonlinear}, Mackie proves that all closed linear $\\lambda$-terms can be erased by means of very simple contexts.\n\\begin{lem}[\\cite{mairsonlinear}]\n\\label{lem: linear terms are solvable} \nLet $ M $ be any closed linear $\\lambda$-term. Then there exists $n \\geq 0$ such that $M I\\overset{n }{\\ldots}I\\rightarrow_\\beta^* I$.\n\\end{lem}\n\\noindent\nThe above result is closely related to solvability (see~\\cite{barendregt1984lambda}): \n``\\textit{A $\\lambda$-term $M$ in the standard $\\lambda$-calculus is said \\textit{solvable} if, for some $n$, there exist $\\lambda$-terms \n$N_1, \\ldots, N_n$ such that $M N_1 \\ldots N_n =_\\beta I$.}''\nLemma~\\ref{lem: linear terms are solvable} states that every closed linear $\\lambda$-term is solvable by linear contexts. \n\\par \nIn fact, the notion of erasability can be addressed in a more general setting.\n\\begin{defn}[Erasable sets]\\label{defn: erasable sets} Let $X$ be a set of  linear $\\lambda$-terms. We say that $X$ is an \\textit{erasable set} if a linear $\\lambda$-term $\\mathtt{E}_X$ exists such that $\\mathtt{E}_X \\, M \\rightarrow^*_{\\beta \\eta } I$, for all $M \\in X$. We call $\\mathtt{E}_X$ \\textit{eraser} of $X$.\n\\end{defn}\n\\begin{prop} \\label{prop: erasable if and only if closed} A finite set  $X$ of  linear $\\lambda$-terms is  erasable  if and only if all terms in $X$ are closed. \n\\end{prop}\n\\begin{proof}\nLet $X$ be a finite set of linear $\\lambda$-terms. To prove the left-to-right direction,  suppose  $X$ is erasable.   By definition, there exists a linear $\\lambda$-term $\\mathtt{E}_X$ such that $\\mathtt{E}_X \\, M \\rightarrow_{\\beta\\eta}^* I$, for all $M \\in X$. Since $ I $ is closed, by Fact~\\ref{fact:Stability} each $ M\\in X $ must be closed too. Let us now suppose that all terms in $X$ are closed, and let $M_1, \\ldots, M_n$ be such terms.  By Lemma~\\ref{lem: linear terms are solvable}, for every $i \\leq n$ there exists a $k_i\\geq 0$ such that $M_i I\\overset{k_i }{\\ldots}I\\rightarrow_\\beta^* I$.  It suffices to  set $\\mathtt{E}_{X}\\triangleq \\lambda x. x I\\overset{k }{\\ldots}I $, where $k= \\max_{i=1}^n k_i$. \n\\end{proof}\n\\noindent\nRecall from Definition~\\ref{eqn:datatype unity and product} that  $\\langle M, N \\rangle\\triangleq \\lambda z.z MN$. In the same spirit of Definition~\\ref{defn: erasable sets}, we now investigate duplicability in the linear $\\lambda$-calculus. \n\\begin{defn}[Duplicable sets]\\label{defn: duplicable sets} Let $X$ be a  set of   linear $\\lambda$-terms. We say that $X$ is a \\textit{duplicable set} if a linear $\\lambda$-term $\\mathtt{D}_X$ exists such that $\\mathtt{D}_X \\, M \\rightarrow^*_{\\beta \\eta} \\langle M,  M \\rangle$ and $FV(\\mathtt{D}_X)\\cap FV(M) =\\emptyset$, for all $M \\in X$. We call $\\mathtt{D}_X$ \\textit{duplicator} of $X$.\n\\end{defn}\n\\begin{prop}\\label{prop: duplicable implies closed} If a finite set  $X$ of  linear $\\lambda$-terms is  duplicable  then all terms in $X$ are closed. \n\\end{prop}\n\\begin{proof}\nLet $X$ be a finite set of  linear $\\lambda$-terms, and suppose  $X$ is  duplicable.  By definition,  there exists a linear $\\lambda$-term $\\mathtt{D}_X$ such that $\\mathtt{D}_X \\, M \\rightarrow_{\\beta\\eta}^* \\langle M,  M  \\rangle$, for all $M \\in X$. Since both $M$ and  $\\mathtt{D}_{X}$ are linear $\\lambda$-terms, and $FV(\\mathtt{D}_X)\\cap FV(M) =\\emptyset$, we have that $\\mathtt{D}_X\\, M$ is linear, for all  $M \\in X$. If there were a variable occurring free in a term $M\\in X$, then it would occur twice in $\\langle M, M \\rangle$, contradicting  Fact~\\ref{fact:Stability}.\n\\end{proof}\n\\noindent\nWe conjecture that the converse holds as well, as long as we restrict to sets of distincts $\\beta\\eta$-normal forms. Indeed, duplication in a linear setting ultimately relies on the following linear version of the  general separation theorem for the standard $\\lambda$-calculus proved by Coppo et al.~\\cite{coppo1978semi}:\n\\begin{conj}[General separation] \\label{conj: linear separation} Let $X=\\lbrace M_1, \\ldots, M_n \\rbrace$ be a set of distinct closed linear $\\lambda$-terms in  $\\beta\\eta$-normal form. Then, for all  $N_1, \\ldots, N_n$  closed linear $\\lambda$-terms, there exists a closed linear $\\lambda$-term $F$ such that $F\\, M_i =_{\\beta \\eta}N_i$, $\\forall i \\leq n$.\n\\end{conj}\n\\noindent\nNow, let $X= \\lbrace M_1, \\ldots, M_n \\rbrace$ be a finite set of distinct closed  linear $\\lambda$-terms in $\\beta\\eta$-normal form.  If Conjecture~\\ref{conj: linear separation} were true,  by fixing  $N_i\\triangleq \\langle M_i, M_i \\rangle$ for all $i \\leq  n$, there would exists a closed  linear $\\lambda$-term  $\\mathtt{D}_X$ such that $\\mathtt{D}_X \\, M_i \\rightarrow_{\\beta \\eta}^* \\langle M_i, M_i\\rangle$. \nSo, we could connect linear erasure and duplication to standard  $\\lambda$-calculus notions:\n\\begin{equation*}\n\\begin{split}\n\\textit{solvability} & \\textrm{ implies } \\textit{linear erasability}\\\\\n\\textit{separation} & \\textrm{ implies } \\textit{linear duplication} \n\\enspace .\n\\end{split}\n\\end{equation*}\nThis topic is left to future work (see Section~\\ref{section:Conclusions}).\n\n\\subsection{The typed setting}\nErasure and duplication are less direct and liberal in \n$ \\textsf{IMLL}_2 $ which assigns types to linear $ \\lambda $-terms. \nSpecifically, it is possible to erase or duplicate all values (Definition~\\ref{defn:Values among linear lambda-terms}) of what we call  \\enquote{ground type}, i.e.~Mairson\\&Terui's  notion of closed $\\Pi_1$-type~\\cite{mairson2003computational},  whose formal definition will be recalled shortly. A typical example of  ground type in $\\mathsf{IMLL}_2$  is the one representing  booleans.\nThe standard second-order intuitionistic formulation of booleans (i.e.~$\\forall \\alpha .\\alpha \\multimap \\alpha \\multimap \\alpha$)   is \nmeaningless for $\\mathsf{IMLL}_2$ due to the lack of free weakening. \nMairson\\&Terui \\citep{mairson2003computational} define them as:\n\\begin{align}\n&\\mathbf{B}\\triangleq\\forall \\alpha. \\alpha \\multimap \\alpha \\multimap \\alpha \\otimes \\alpha \n\\label{eqn: boolean data type}\n&&\\mathtt{tt}\\triangleq\t\\lambda x. \\lambda y. \\langle x,y \\rangle \n&&\\mathtt{ff}\\triangleq \\lambda x. \\lambda y. \\langle y,x \\rangle \n\\end{align} \nwhere the values \\enquote{truth} \\texttt{tt}\nand the \\enquote{falsity} \\texttt{ff} implement the\n\\enquote{erasure by garbage collection}: the first element of the pair is the \\enquote{real} output, while the second one is garbage.\nStarting from~(\\ref{eqn: boolean data type}), Mairson shows in~\\cite{mairsonlinear} that $\\mathsf{IMLL}$ is expressive enough to encode boolean functions. Mairson and Terui  reformulate that encoding \nin $\\mathsf{IMLL}_2$ in order to prove \nresults about the complexity of cut-elimination \n\\cite{mairson2003computational}. The advantage of $\\mathsf{IMLL}_2$ is to assign uniform types to the $\\lambda$-terms representing boolean functions. An \\emph{eraser} $\\mathtt{E}_{\\mathbf{B}}$ and a \\emph{duplicator} $\\mathtt{D}_{\\mathbf{B}}$ are the keys to obtain the encoding:\n\\begin{align}\n\\label{eqn: erasure booleans}\n\\mathtt{E}_{\\mathbf{B}}& \\triangleq\n\\lambda z. \\mathtt{let\\ }zII \\mathtt{\\ be \\ } x,y \\mathtt{ \\ in \\ }(\\mathtt{let \\ }y \\mathtt{ \\ be \\ } I \\mathtt{ \\ in \\ }x) :\\mathbf{B}\\multimap \\mathbf{1}\n\\\\\n\\label{eqn: duplication booleans}\n\\mathtt{D}_{\\mathbf{B}}& \\triangleq\n\\lambda z. \\pi_1(z\\langle \\mathtt{tt}, \\mathtt{tt}\\rangle\\langle \\mathtt{ff}, \\mathtt{ff}\\rangle) : \\mathbf{B}\\multimap \\mathbf{B}\\otimes \\mathbf{B}\n\\\\\n\\label{eqn: boolean projection}\n\\pi_1 &\\triangleq\n\\lambda z. \\mathtt{let\\ }z \\mathtt{\\ be \\ }x,y \\mathtt{\\ in\\ }(\\mathtt{let \\ }\\mathtt{E}_{\\mathbf{B}} \\, y \\mathtt{\\ be \\ }I \\mathtt{\\ in \\ }x) :\n(\\mathbf{B}\\otimes \\mathbf{B}) \\multimap\\mathbf{B}\n\\end{align}\nwith $ \\mathbf{B} $ as in \\eqref{eqn: boolean data type} and $\\pi_1$ the linear $\\lambda$-term projecting the first element of a pair.\\\\\n Switching to type-assignment setting we get uniform copying and erasing mechanisms of the whole \\textit{class} of values of a given ground type. Note that the type-theoretical constraints let the erasure of a typed linear $\\lambda$-term make use of something more than mere stacks of identities as in Lemma~\\ref{lem: linear terms are solvable}. Also, note that \\textit{both} the possible results $\\langle \\mathtt{tt}, \\mathtt{tt} \\rangle$ \nand $\\langle \\mathtt{ff}, \\mathtt{ff} \\rangle$ of duplications are built-in \ncomponents of  $ \\texttt{D}_{\\textbf{B}} $.\nIn accordance with the given input, $\\mathtt{D}_{\\mathbf{B}}$  selects \nthe right pair representing the result by \\textit{erasing} the \nunwanted one. Such a \\enquote{linear} form of \\emph{duplication by selection and erasure} is a step-by-step elimination of useless data until the desired result shows up. \n\\par\nThe analysis of~(\\ref{eqn: erasure booleans}) and~(\\ref{eqn: duplication booleans}) leads to the following formal notions: \n\\begin{defn} [Duplicable and erasable types in $\\mathsf{IMLL}_2$]  \n\\label{defn: duplicable and erasable types} \nLet $A$  be a type in $\\mathsf{IMLL}_2$. It is a \\textit{duplicable type}  \nif a linear $\\lambda$-term $\\mathtt{D}_A: A \\multimap A \\otimes A$ exists \nsuch that $\\mathtt{D}_{A} \\,V \\rightarrow_{\\beta \\eta}^* \n\\langle V, V \\rangle$, for every value $V$ of $A$.\n\nMoreover, $A$ is an \\textit{erasable type} if a linear $\\lambda$-term $\\mathtt{E}_A:A \\multimap \\mathbf{1}$ exists such that\n$\\mathtt{E}_{A} \\, V \\rightarrow_{\\beta \\eta}^* I$,\nfor every value  $V$ of $A$.\n\nWe call $\\mathtt{D}_A$ \\emph{duplicator} of $A$ and $\\mathtt{E}_{A}$ its \\emph{eraser}. \n\\qed\n\\end{defn}\n\\noindent\nDuplicators and erasers in Definition~\\ref{defn: duplicable and erasable types} apply to values of a given type, i.e.~\\textit{closed} and normal inhabitants. This is not a loss of generality because Proposition~\\ref{prop: erasable if and only if closed} and Proposition~\\ref{prop: duplicable implies closed} say that  only closed terms can be duplicated or erased linearly.\n\n\\begin{defn}[$\\Pi_1$, $\\Sigma_1$-types \\cite{mairson2003computational} and Ground types]\n\\label{defn:Pi and Sigma types} \nThe following mutually defined grammars generate $\\Pi_1$ and  $\\Sigma_1$-types:\n\\begin{align*}\n\\Pi_1&:= \\alpha  \\ \\vert \\ \\Sigma_1      \n\\multimap \\Pi_1 \\ \\vert \\ \\forall \\alpha. \\Pi_1  \\\\\n\\Sigma_1&:= \\alpha \\ \\vert \\ \\Pi_1 \n\\multimap \\Sigma_1 \n\\end{align*} \nWe call \\emph{ground types} the \\emph{closed} $\\Pi_1$-types.\n\\qed\n\\end{defn}\n\\noindent\nWe note that the universal quantifier $\\forall$ occurs only positively \nin a $\\Pi_1$-type, hence in ground types.\n\\par\nThe booleans $\\mathbf{B}$ in~\\eqref{eqn: boolean data type}, \nthe unit $\\mathbf{1}$ and the tensor $A\\otimes B$ as in Definition~\\ref{eqn:datatype unity and product} are ground types, if $A$ and $B$ are.\nIn fact, following \\cite{mairson2003computational}, tensors and units can \noccur \\emph{also} to the left-hand side of a linear implication \n\\enquote{$ \\multimap $}, even in negative positions. The reason is that \nwe can ignore them in practice, thanks to the isomorphisms:\n\\begin{align*}\n((A \\otimes B) \\multimap C) \\multimapboth (A \\multimap B \\multimap C) \n&&\n(\\mathbf{1}\\multimap C) \\multimapboth C\n\\enspace .\n\\end{align*} \nGround types represent \\emph{finite} data types, while the values with a ground type represent their data.\n\\begin{fact}\\label{rem: every normal term has Pi1 type} \nEvery closed linear $\\lambda$-term $ M $ has a ground type.\n\\qed\n\\end{fact}\n\\begin{proof}\nEvery closed  linear $\\lambda$-term $M$  is typable  in $\\mathsf{IMLL}$ (see~\\cite{hindley1989bck}). Types in $\\mathsf{IMLL}$ are  quantifier-free instances\nof  $\\Pi_1$-types. Hence, $M$ has also a $\\Pi_1$-type $A$ in $\\mathsf{IMLL}_2$. \nLet $FV(A)= \\lbrace\\alpha_1, \\ldots, \\alpha_n\\rbrace $. Since $M$ inhabits $A$, it also inhabits \n$\\overline{A} =\\forall \\alpha_1.\\cdots.\\forall \\alpha_n. A$, which  is a \nclosed $\\Pi_1$-type, i.e.~a ground type in $\\mathsf{IMLL}_2$.\n\\end{proof}\n\\noindent\nThe class of ground types is  a subset of both the classes of duplicable and erasable types. \n\\begin{thm}[\\cite{mairson2003computational}]\n\\label{thm: pi1 types are erasable}\nEvery ground type is erasable.\n\\end{thm}\n\\begin{proof} \nThe proof follows from proving two statements\nby simultaneous induction: (i)  For every $\\Pi_1$-type $A$ with free type variables $\\alpha_1, \\ldots, \\alpha_n$ there exists \na linear $ \\lambda $-term $\\mathtt{E}_A$  such \nthat $ \\vdash \\mathtt{E}_A: A[\\mathbf{1}\/\\alpha_1, \\ldots,\\mathbf{1} \/\\alpha_n] $ $\\multimap \\mathbf{1}$, and (ii) for every $\\Sigma_1$-type $A$ with free type variables $\\alpha_1, \\ldots, \\alpha_n$ there exists a linear $ \\lambda $-term $\\mathtt{H}_A$ such that \n$ \\vdash \\mathtt{H}_A:A[\\mathbf{1}\/\\alpha_1, \\ldots,\\mathbf{1}\/ \\alpha_n]$.\n\\end{proof}\n\n\\begin{thm}\n\\label{thm: pi1 are duplicable}\nEvery inhabited ground type is duplicable.\n\\end{thm}\n\\noindent\nMairson and Terui sketch the proof of Theorem~\\ref{thm: pi1 are duplicable} in\n\\cite{mairson2003computational}.\n\\ref{sec: the d-soundness theorem DICE}, which we see as an integral and relevant part of this work, develops it in every detail. \n\n\\section{Introduction}\n\\label{section:Introduction}\nGirard introduces \\textit{Linear Logic} ($\\mathsf{LL}$) in \n\\cite{Girard:TCS87} as a refinement of both classical and intuitionistic \nlogic. \\textsf{LL} decomposes the intuitionistic implication  \n\\enquote{$\\Rightarrow$} into the more primitive linear implication \n\\enquote{$\\multimap$} and modality \\enquote{$\\oc$} (of course), \nthe latter giving a logical status to weakening and contraction by\nmeans of the so-called  \\textit{exponential rules}. \nAccording to the \\emph{Curry-Howard correspondence},\n this  decomposition allows to identify a strictly linear \ncomponent of the functional computations that interacts with the non-linear one, in which duplication and erasure are allowed.\n\nThis work focuses on $\\mathsf{IMLL}_2$, i.e.~second-order intuitionistic \\textit{multiplicative} Linear Logic which, we recall, is free of any kind of exponential rules. \nThe Curry-Howard correspondence tightly relates $\\mathsf{IMLL}_2$ and \nthe linear $\\lambda$-calculus, a sub-language of the standard $\\lambda$-calculus without explicit erasure and duplication.\n\nInteresting works exist on the expressiveness of both the untyped and the typed linear $ \\lambda $-calculus.\n\nAlves et al.~\\cite{DBLP:conf\/csl\/AlvesFFM06} recover the full computational power of G\\\"odel System $T$ by adding booleans, natural numbers, and a linear iterator to the linear $\\lambda$-calculus, the non-linear features coming specifically from the iterator and the numerals. \n\nMatsuoka investigates the discriminating power of linear $ \\lambda $-terms  with types in $ \\textsf{IMLL}$, i.e.~intuitionistic multiplicative Linear Logic, proving typed variants of B\\\"ohm Theorem~\\cite{MATSUOKA200737}. We remark that, in this setting, discriminating among linear $ \\lambda $-terms relies on \\emph{a specific form of weakening} already inside $ \\textsf{IMLL} $. \n\nAnother work that exploits the built-in erasure and copying mechanisms of the linear $\\lambda$-calculus is by Mairson~\\cite{mairsonlinear}. \nWith no new constructors, Mairson encodes boolean circuits in the linear \n$ \\lambda $-calculus. Moreover, Mairson\\&Terui reformulate Mairson's \nresults inside $\\mathsf{IMLL}_2$ and prove bounds on the complexity of the cut-elimination in sub-systems of \\textsf{LL}~\\cite{mairson2003computational}.\n\n\\paragraph{Contributions}\nStarting from Mairson\\&Terui's \\cite{mairson2003computational}, this work investigates a structural proof-theory, and the related Curry-Howard correspondence,  of $\\mathsf{IMLL}_2$  extended with inference rules  for contraction and  weakening. \n\\begin{enumerate}\n\\item \nWe introduce the \\emph{Linearly Exponential and Multiplicative}  system $\\mathsf{LEM}$, giving a logical status to\nthe erasure and the duplication that~\\cite{mairson2003computational} identifies inside the linear $ \\lambda $-calculus. \n$\\mathsf{LEM}$ is a type-assignment for a \\emph{linear} \n$\\lambda$-calculus endowed with constructs for weakening and \ncontraction, and it is obtained by extending $\\mathsf{IMLL}_2$ with \nrules on modal formulas ``$ \\shpos A$''.  $\\mathsf{LEM}$ can be seen  as a sub-system of $\\mathsf{LL}$ \nwith a restricted form ``$ \\shpos $'' of \\enquote{$\\oc$}.\n\n\\item\nWe consider a mildly weakened cut-elimination, called ``lazy'',  that  faithfully represents the mechanism of linear erasure and  duplication discussed in~\\cite{mairson2003computational}, and we  identify a set of derivations in $\\mathsf{LEM}$\nthat rewrite to cut-free \nones under that lazy cut-elimination in a cubic number of steps \n(Section~\\ref{Cut elimination, complexity, and subject reduction for IMLL2^shpos}).  \nMoreover, we show the Subject reduction of $\\mathsf{LEM}$ \n(Section~\\ref{sec: subject reduction}).\n\n\\item \nWe prove that the cut-elimination of $\\mathsf{IMLL}_2$ can simulate the \none of $\\mathsf{LEM}$ at a cost which can be exponential in the size of \nthe given derivation of $\\mathsf{LEM}$ \n(Section~\\ref{sec: the expressiveness of the system}).\nSo,  $\\mathsf{LEM}$ can speed up the cut-elimination of $\\mathsf{IMLL}_2$,  meaning that it compresses in smaller derivations what can be algorithmically expressed in $\\mathsf{IMLL}_2$.\n\n\\item Hence, we explore the algorithmic expressiveness of $\\mathsf{LEM}$\n(Section~\\ref{section:The expressiveness of IMLLshpos2 and applications}):\n\\begin{enumerate}\n\t\\item \n\tBoth $\\mathsf{LEM}$ and $\\mathsf{IMLL}_2$ can represent boolean circuits.\t\n\tHowever, the copying mechanism, directly available in $\\mathsf{LEM}$, makes the encoding of the fan-out of the nodes of the circuit essentially natural, facilitating the modularity and the readability of the encoding itself.\n\tMoreover, the erasure in $\\mathsf{LEM}$ avoids to accumulate garbage when evaluating a circuit represented by a derivation of $\\mathsf{LEM}$, unlike in other proposals.\n\t\\item \n\tWe show that numerals, structurally related to Church ones, \n\texist in $\\mathsf{LEM}$. Their type is\n\t$ (\\shpos \\forall\\alpha.(\\alpha\\multimap\\alpha) ) \\multimap\\forall\\alpha.(\\alpha\\multimap\\alpha) $ that forbids\n\titerations longer than the complexity of the lazy cut-elimination.  Remarkably, the numerals in $\\mathsf{LEM}$ admit successor and addition that work as expected, thanks to the Subject reduction. \n\n    \\item \n\tFinally, we show that Hereditarily Finite Permutations, which form a group inside the  linear\n\t$ \\lambda $-calculus, inhabit a simple generalization \n\tof the here above type of numerals, so possibly connecting \n\t$\\mathsf{LEM}$ with reversible computations.\n\\end{enumerate}\nThe above contributions follow from a fully detailed, and not\nat all obvious, technical reworking of Mairson\\&Terui's \\cite{mairson2003computational} work. \nWe propose it as a solid base to further investigations concerning duplication and erasure   in a purely linear setting.\n\\end{enumerate}\nSection~\\ref{sec: background} is about (formal) preliminaries. Section~\\ref{sec: duplication and erasure in a linear setting} introduces the motivating background and \nSection~\\ref{sec:  the system shpos, with, oplus  IMLL2 cut elimination and bound} formally defines $\\mathsf{LEM}$.\n\n\\paragraph{Acknowledgments}\nWe are indebted to the anonymous reviewers for their patience and  constructive attitude with which they red and commented on previous versions of this work.\n\\section*{\\refname}\n\n\n\\section{Preliminaries} \\label{sec: background}\n\\subsection{The linear $\\lambda$-calculus}\n\nWe assume the reader to be familiar with standard $\\lambda$-calculus and related concepts like:\n(i) the set $ FV(M) $ of the free variables of the $ \\lambda $-term $ M $, \n(ii) the meta-level substitution $M[N\/x]$ that replaces \nthe $ \\lambda $-term $ N $ for every free occurrence of the variable $ x $\nin $ M $, \n(iii) the contexts $ \\mathcal{C}[] $, i.e.~$ \\lambda $-terms with a place-holder (hole) $ [] $ \n that may capture free variables of a $ \\lambda $-term plugged into $ [] $,\n (iv) the $\\alpha$-equivalence ($=_\\alpha$),\n(v) the $\\beta$-reduction $(\\lambda x.M)N\\rightarrow_\\beta M[N\/x]$, \n(vi) the $\\eta$-reduction $\\lambda x.Mx\\rightarrow_\\eta M$ that can apply \nif $ x $ is not free in $ M $. Both $\\rightarrow_\\beta$ and $\\rightarrow_\\eta$ are considered contextually closed.\n\n\n\nBy $\\rightarrow_\\beta^*$ we denote the reflexive and transitive closure of the $ \\beta $-reduction, and by  $=_{\\beta}$  its reflexive, symmetric and transitive closure.\n\nAlso, \nby $\\rightarrow_\\eta^*$ we denote the reflexive and transitive closure of the $ \\eta $-reduction, and by  $=_{\\eta}$  its reflexive, symmetric and transitive closure.\n\nFinally, by $\\rightarrow_{\\beta \\eta}$ we denote $\\rightarrow_\\beta \\cup \\rightarrow_{\\eta}$, and by $\\rightarrow^*_{\\beta \\eta}$ we denote its reflexive and transitive closure.\n\nA $\\lambda$-term is in $\\beta$-\\textit{normal form}, or simply \n($ \\beta $-)\\emph{normal},\nwhenever no $ \\beta $-reduction applies to it. \nA $\\lambda$-term is in $\\eta$-\\textit{normal form}, or simply \\emph{$ \\eta $-normal} if no $\\eta$-reduction applies to it. Finally, a $\\lambda$-term is in $\\beta\\eta$-\\textit{normal form}, or simply  $ \\beta \\eta $-\\emph{normal}, whenever no $ \\beta\\eta $-reduction applies to it.\n\nA $\\lambda$-term is \\textit{closed} if $FV(M)=\\emptyset$.  \n\nThe \\textit{size} $\\vert M \\vert $ of $M$ is the number of nodes in its syntax tree. \n\nThe \\emph{linear} $\\lambda$-calculus is the $\\lambda$-calculus restricted to \\emph{linear} $\\lambda$-terms: \n\n\\begin{defn}[Linear $\\lambda$-terms] \n\\label{defn:Linear lambda-terms}\nA $\\lambda$-term $M$ is \\textit{linear} if all of its free variables occur once in it and every proper sub-term $ \\lambda x.M' $ of $M$ is such that  $ x $ occurs in  $ M' $ and $M' $ is linear.\n\\qed\n\\end{defn}\n\\noindent\nFor example, \n$I\\triangleq \\lambda x.x$ and \n$C\\triangleq\\lambda x. \\lambda y. \\lambda z. xzy$ are linear, \nwhile $K \\triangleq \\lambda x. \\lambda y. x$ and \n$S\\triangleq\\lambda x. \\lambda y. \\lambda z. xz(yz) $ are not.\n\n\\vspace{\\baselineskip}\n\\noindent\nTo our purposes, we shall adopt the following notion of value:\n\\begin{defn}[Values]\n\\label{defn:Values among linear lambda-terms}\nA \\emph{value} is every linear $ \\lambda $-term which is both \n($ \\beta $-)\\emph{normal} and \\emph{closed}. \n\\qed\n\\end{defn}\n\\noindent\nWe shall generally use $ V $ and $ U $ to range over values.\n\n\\begin{fact}[Stability]\n\\label{fact:Stability}\nLinear $\\lambda$-terms are stable under $\\beta$-reduction, \ni.e.~$M$ linear and $M \\rightarrow_\\beta N $ imply $N$ is linear. \nAnalogously, linear $\\lambda$-terms are stable under $\\eta$-red\\-uct\\-ion, i.e.~$M$ linear and $M \\rightarrow_\\eta N$ imply $N$ is linear. In both cases,   $FV(N)= FV(M)$.\n\\qed\n\\end{fact}\n\\noindent\nFinally, we shall write $M \\circ N$ in place of $\\lambda z. M(Nz)$.\n\n\\begin{figure}[ht]\n\t\\begin{mathpar}\n\t\t\\inferrule*[Right= $ax$]\n\t\t{\\\\}\n\t\t{x: A \\vdash x:A} \n\t\t\\and\n\t\t\\inferrule*[Right= $cut$]\n\t\t{\\Gamma \\vdash N: A \\\\ \\Delta, x:A \\vdash M:C}\n\t\t{\\Gamma, \\Delta \\vdash M[N\/x]:C}\n\t\t\\\\\n\t\t\\inferrule*[Right= $\\multimap$R]\n\t\t{\\Gamma, x:A \\vdash M:B}\n\t\t{\\Gamma \\vdash \\lambda x. M : A \\multimap B} \n\t\t\\and\n\t\t\\inferrule*[Right= $\\multimap$L]\n\t\t{\\Gamma \\vdash N: A \\\\ \\Delta, x:B \\vdash M: C}\n\t\t{\\Gamma, \\Delta, y: A \\multimap B \\vdash M[yN\/x]: C}\n\t\t\\\\\n\t\t\\inferrule*[Right=$\\forall$R]\n\t\t{\\Gamma \\vdash M: A\\langle \\gamma\/\\alpha \\rangle \n\t\t \\\\ \\gamma \\not \\in FV(\\operatorname{rng}(\\Gamma))}\n\t\t{\\Gamma \\vdash M: \\forall \\alpha.A} \n\t\t\\and\n\t\t\\inferrule*[Right= $\\forall$L]\n\t\t{\\Gamma, x: A\\langle B\/ \\alpha \\rangle \\vdash M:C}\n\t\t{\\Gamma, x: \\forall \\alpha. A \\vdash M:C}\n\t\\end{mathpar}\n\t\\caption{$\\mathsf{IMLL}_2$ as a type-assignment system.}\n\t\\label{figure:IMLL2}\n\\end{figure}\n\n\\subsection{The systems $\\mathsf{IMLL}_2$ and $\\mathsf{IMLL}$} \\label{subsec: IMLL2}\nWe assume familiarity with basic proof-theoretical notions and with Linear Logic (see~\\cite{girard1987linear,troelstra2000basic}.)\n\\emph{Second-order Intuitionistic Multiplicative Linear Logic} ($\\mathsf{IMLL}_2$), seen as a type-assignment for the linear $\n \\lambda $-calculus, is in Figure~\\ref{figure:IMLL2}, where, we remark, the only logical operators are the universal quantifier ``$ \\forall$'' and the\n\tlinear implication ``$ \\multimap $''. $\\mathsf{IMLL}_2$ derives \\emph{judgments} $\\Gamma \\vdash M: A$, i.e.~a \\emph{type} $ A $ for the linear $ \\lambda $-term $ M $ from the \\emph{context} $\\Gamma$.\nA \\textit{type} is a (type) variable $\\alpha$, or an \\emph{implication} $A \\multimap B$, or a \\emph{universal quantification} $\\forall \\alpha. A$, where\n$A$ and $B$ are types. \nThe set of free type variables of $ A $ is $ FV(A) $. If $FV(A)= \\emptyset$, then $ A $ is \\textit{closed}. If $FV(A)= \\lbrace \\alpha_1, \\ldots, \\alpha_n \\rbrace$, then  \\textit{a closure} $\\overline{A}$ of $A$ is $\\forall \\alpha_1. \\cdots. \\forall \\alpha_n. A$, not necessarily linked to a specific order of $ \\alpha_1,\\ldots,\\alpha_n $.\nThe standard meta-level substitution of a type $ B $ for every free occurrence of $ \\alpha $ in $ A $ is $A \\langle B\/ \\alpha \\rangle$. \nThe \\textit{size} $\\vert A \\vert$ of the type $A$ is the number of nodes in its syntax tree. \nA \\textit{context} $\\Gamma$ has form $x_1: A_1, \\ldots, x_n:A_n$, with $ n\\geq 0 $, i.e.~it is a finite multiset of \\textit{assumptions} $x: A$, where $ x $ is a\n $ \\lambda $-variable. \nThe \\emph{domain} $\\operatorname{dom}(\\Gamma)$ of $ \\Gamma $ is \n$\\{x_1, \\ldots, x_n\\}$ and its range $ \\operatorname{rng}(\\Gamma) $\nis $\\{A_1, \\ldots, A_n\\}$.\nThe size $\\vert \\Gamma \\vert$ of $ \\Gamma $ is $\\sum^n_{i=1}\\vert A_i \\vert$.  \nTypically, names for contexts are $\\Gamma, \\Delta$ or $\\Sigma$.\n\nSince $ \\textsf{IMLL}_2 $ gives types to linear $ \\lambda $-terms,  \n$\\multimap$L is necessarily subject to the \\emph{linearity constraint} \n$\\operatorname{dom}(\\Gamma) \\cap \\operatorname{dom}(\\Delta)= \\emptyset$. \nWe range over the derivations of $\\mathsf{IMLL}_2$ by $\\mathcal{D}$. \nThe \\textit{size} $\\vert \\mathcal{D} \\vert$ of $\\mathcal{D}$ is the number of the rule instances that $\\mathcal{D}$ contains. \nWe say that $\\Gamma \\vdash M: B$ is \\textit{derivable} if a derivation $\\mathcal{D}$ exists that concludes with the judgment $ \\Gamma \\vdash M:B$, and we also say that $\\mathcal{D}$ is a derivation of $ \\Gamma \\vdash M:B$. In that case we write $\\mathcal{D}\\triangleleft \\Gamma \\vdash M: B$ saying that \n$M$ is an \\textit{inhabitant} of $B$ or that $B$ is \\textit{inhabited} by $M$ \nfrom $\\Gamma$. \nThe cut-elimination steps for $\\mathsf{IMLL}_2$ are standard and both\ncut-elimination and confluence hold for it \\cite{troelstra2000basic}. \n\n\n\\textit{Propositional Intuitionistic Multiplicative Linear Logic} ($\\mathsf{IMLL}$) is $\\mathsf{IMLL}_2$ without $\\forall$R and $\\forall$L. \nFrom Hindley \\cite{hindley1989bck}, we recall that $\\mathsf{IMLL}$, \nthus $\\mathsf{IMLL}_2$, gives a type to every linear $\\lambda$-term. \nThe converse holds as well, due to the above \\textit{linearity constraint} \non $\\multimap$L, so the class of linear $\\lambda$-terms is exactly the one \nof all typable $ \\lambda $-terms in $\\mathsf{IMLL}_2$. \nIt follows that second-order does not allow to type more terms but it is nevertheless useful to assign uniform types to structurally related \n$ \\lambda $-terms.\n\nWe conclude by recalling standard definitions of types in $\\mathsf{IMLL}_2$:\n\n\\begin{defn}[Basic datatypes]\n\t\\label{eqn:datatype unity and product}\nThe \\emph{unity} type is $ \\textbf{1} \\triangleq \\forall \\alpha. (\\alpha \\multimap \\alpha) $ with constructor $ I \\triangleq \\lambda x.x $, i.e.~the identity,\n\tand destructor $ \\mathtt{let}\\ M \\mathtt{\\ be \\ }I \\mathtt{\\ in \\ }N \\triangleq MN $;\n\nThe \\emph{tensor product} type\n\t$ A \\otimes B \\triangleq \n\t\\forall \\alpha. (A \\multimap B \\multimap \\alpha) \\multimap \\alpha $ \n\twith constructor \n\t$  \\langle M, N \\rangle \\triangleq \\lambda z. z\\,M\\,N $\n\tand destructor $ \\mathtt{let}\\ M \\mathtt{\\ be \\ }x,y \\mathtt{\\ in \\ }N\n\t\\triangleq M(\\lambda x. \\lambda y. N) $.\n\t\nBoth binary tensor product and pair extend to their obvious $n$-ary versions\n$A^n= \\underbrace{A \\otimes \\ldots\\otimes A}_{n}$ and $M^n \\triangleq  \\langle \\underbrace{ M, \\ldots, M}_{n}\\rangle$.\n\\qed\n\\end{defn}\n\\begin{rem}\n\tEvery occurrence of unity, ($ n $-ary) tensor and $ n $-tuple in the coming sections will be taken from Definition~\\ref{eqn:datatype unity and product}.\n\\end{rem}\n\\noindent\t\nFinally, Definition~\\ref{eqn:datatype unity and product} talks about \\emph{datatypes} because, by introducing a specific syntax for constructors and destructors, we implicitly adopt a pattern matching mechanism\nto operate on $ \\lambda $-terms typed with those types.\n\\section{The expressiveness of $\\mathsf{LEM}$ and applications}\n\\label{section:The expressiveness of IMLLshpos2 and applications}\nTheorem~\\ref{thm: translations for LAML} says that $\\mathsf{LEM}$  is not  \\enquote{algorithmically} more expressive than  $ \\mathsf{IMLL}_2 $. Nonetheless, terms with type in  $\\mathsf{LEM}$,  and their evaluation mechanisms, exponentially compress the corresponding linear $ \\lambda $-terms and evaluations in $ \\mathsf{IMLL}_2 $ (Theorem~\\ref{thm: exponential compression}).  The goal of this section is to explore  the benefits of this compression.\n\n\n\n\\begin{figure}[ht]\n\\centering\n \\subfigure[\\scriptsize{From left,  input,  internal,  fan-out, and  output nodes.}\t\\label{figure:node of boolean circuits} ]{\n\\scalebox{.65}{\n \\begin{tikzpicture}[\n node\/.style={circle,minimum size=1pt,fill=white},\n node distance=0.25cm\n ]\n\n \\node[circle, inner sep =1pt, draw](op){\\scriptsize{$\\textsl{op}^n$}};\n \\node[circle]  at  ($(op)+(0, +1)$)(ab){\\scriptsize{\\ldots $n\\geq 0$\\ldots}} ;\n   \\coordinate[right = of ab](abri){} ;\n   \\coordinate[left = of ab](able){} ;\n   \\draw[->](able)to[out=270, in=135](op);\n      \\draw[->](abri)to[out=270, in=45](op);\n       \\draw[->](op)--($(op)+(0, -1)$) ;\n \n      \n  \n \\node[node,  draw, left=3cm  of op](in){\\scriptsize{$\\textrm{x}$}};\n   \\draw[->](in)--($(in)+(0, -1)$) ;  \n  \n   \\node[circle, inner sep= 1pt, draw, right=3cm of op, fill=black](fan){};\n  \\node[node]  at ($(fan)+(0, -1)$)(befan){\\scriptsize{\\ldots $n\\geq 0$\\ldots}} ;\n   \\node[circle] at  ($(fan)+(0, +1)$)(abfan){} ;\n   \\coordinate[right = of befan](berifan){} ;\n   \\coordinate[left = of befan](belefan){} ;\n     \\draw[->, thick](fan)to[out=225, in=90](belefan);\n      \\draw[->,thick](fan)to[out=315, in=90](berifan);\n         \\draw[->](abfan)to[out=270, in=90](fan);        \n        \n   \\node[node,  draw, right=4cm  of belefan](out){\\scriptsize{$\\mathrm{y}$}};\n   \\draw[<-](out)--($(out)+(0, 1)$) ;     \n \\end{tikzpicture}\n }} \n \n\n\\subfigure[\\scriptsize{The 2-bits full-adder boolean circuit}\\label{figure:2-bits full adder}]{\n\\scalebox{.65}{\n \\begin{tikzpicture}[\n auto=left,\n empty\/.style = {minimum size=0cm,fill=white},\n el\/.style = {inner sep=2pt, align=left, sloped},\n node\/.style={circle,minimum size=.9cm,fill=white},\n sqnode\/.style={rectangle,inner sep=2pt,fill=white},\n node distance=1cm\n ]\n \\draw(0, 1)node(q){};\n \\node[node,draw            ] (b1)  {$ \\mathrm{x}_1 $};\n \\node[node,draw,below=of b1] (b2)  {$ \\mathrm{x}_2 $};\n \\node[node,draw,below=of b2] (cin) {$ \\mathrm{x}_{\\textrm{in}} $};\n \n \\node[circle, fill=black, inner sep=1pt, right=of b1, draw ](fop1)  {};\n \\node[circle, fill=black, inner sep=1pt, right=of b2, draw ](fop2){};\n\n\n \\node[node,draw,right=2cm of b1] (xor1)  {$\\bigoplus$};\n \\node[circle, fill=black, inner sep=1pt,  right=of xor1] (foxor1) {};\n \\node[node,draw,right=2cm of b2] (and1)  {$\\bigwedge$};\n \n \\node[node,draw,right= of foxor1] (xor2) {$\\bigoplus$};\n \\draw ($ (xor2)+(0,-3.825) $) node[node,draw] (and2) {$\\bigwedge$};\n  \\node[circle, fill=black, inner sep=1pt, draw, left= of and2] (focin) {};\n  \n  \\node[node, below=of xor2](preor){};\n \\node [node,draw, right= of preor] (or) {$\\bigvee$ };\n \n \n   \n \\node[node, right=of xor2, draw] (output1){$ \\mathrm{y}_1 $};\n \\node[node, right=of or, draw]  (output2){$ \\mathrm{y}_2 $};\n \\draw (xor2) to[out=0, in=180]node[above]{\\scriptsize{$s$}} (output1);\n  \\draw (or) to[out=0, in=180]node[above]{\\scriptsize{$y_{\\text{out}}$}} (output2);\n \\draw ($ (output1)+(1.75,0) $) node[sqnode] (s) {\\scriptsize{$(x'_1\\oplus x'_2)\\oplus y'$}};\n \\draw ($ (output2)+(2.50,0) $) \n   node[sqnode] (s) {\\scriptsize{$(x''_1\\wedge x''_2)\\vee((x'_1\\oplus x'_2)\\wedge y'')$}};\n\n \n \\draw (b1)   to[out=0,in=180]node[above]{\\scriptsize{$x_1$}} (fop1);\n \\draw (b2)   to[out=0,in=180]node[above]{\\scriptsize{$x_2$}} (fop2);\n \\draw (cin)  to[out=0,in=180]node [above]{\\scriptsize{$y_{\\text{in}}$}}(focin);\n \n \\draw[thick](fop1)  to[out=0,in=180] node[above  ] {\\scriptsize{$ x'_1  $}} (xor1);\n \\draw[thick](fop1)  to[out=0,in=135] node[below,pos=0.2] {\\scriptsize{$ x''_1\\ \\ $}} (and1);\n \n \\draw[ thick] (fop2)  to[out=0,in=225] node[above,pos=0.2] {\\scriptsize{$ x'_2\\ \\ $}} (xor1);\n \\draw[ thick] (fop2)  to[out=0,in=180] node[below        ] {\\scriptsize{$ x''_2 $}} (and1);\n \n \\draw[ thick] (focin) to[out=0,in=225] node[above, pos=0.1] {\\scriptsize{$ y'\\ \\ $}} (xor2);\n \\draw[ thick] (focin) to[out=0,in=180] node[below] {\\scriptsize{$ y''$}} (and2);\n\n \\draw (xor1)   to[out=  0,in=180]node[above]{\\scriptsize{$z_1$}}                                     (foxor1);\n \\draw[ thick] (foxor1) to[out=  0,in=180] node[above]       {\\scriptsize{$ z_1' $}}  (xor2);\n \\draw[ thick] (foxor1) to[out=0,in=135] node[below,pos=0.1] {\\scriptsize{$z''_1 \\ \\ $}} (and2);\n\n \\draw (and1) to[out=0,in=180] node[above,pos=0.1] {\\scriptsize{$ z_2 $}} (or);\n \\draw (and2) to[out=0,in=225] node[below, right] {\\scriptsize{$z_3 $}} (or);\n \\end{tikzpicture}\n}\n}\n\\subfigure[\\scriptsize{The 3-bits majority boolean circuit}\n\\label{figure:maj3}]{\n\\scalebox{.65}{\n\\begin{tikzpicture}[\nauto=left,\nempty\/.style = {minimum size=0cm,fill=white},\nel\/.style = {inner sep=2pt, align=left, sloped},\nnode\/.style={circle,minimum size=.9cm,fill=white},\nsqnode\/.style={rectangle,inner sep=2pt,fill=white},\nnode distance=1cm\n]\n \\draw(0, 1)node(q){};\n\\node[node,draw            ] (b1)  {$ \\mathrm{x}_1 $};\n\\node[node,draw,below=of b1] (b2)  {$ \\mathrm{x}_2 $};\n\n\\node[circle, fill=black, inner sep=1pt, right=of b1, draw] (fob1){}; \n\\node[circle, fill=black, inner sep=1pt, right=of b2, draw ] (fob2){};\n\n\\node[node,draw,right=of fob1] (o1)  {$\\bigvee$};\n\\node[node,draw,right=of fob2] (a1)  {$\\bigwedge$};\n\\node[node,draw,below=of b2  ] (b3)  {$ \\mathrm{x}_3 $};\n\n\n\n\\coordinate[right=of a1] (foa1);\n\n\\node[node,draw,right=of foa1] (o2)  {$\\bigvee$};\n\\node[node,draw,below=of o2] (a2)  {$\\bigwedge$};\n\n\\node[circle, fill=black, inner sep=1pt, right=of a1, draw ] (foa1){};\n\\node[node, below= of a1](beforefob3){};\n\\node[circle, fill=black, inner sep=1pt, right= of beforefob3, draw] (fob3){};\n\\node[circle, fill=black, inner sep=1pt, right=of o2, draw ]   (foo2){};\n\n\\node[node,draw,right=of foo2] (a3)  {$\\bigwedge$};\n\\node[node,draw,above=of a3] (o3)  {$\\bigvee$};\n\\node[circle, fill=black, inner sep=1pt, left= of o3, draw] (foo1){};\n\\draw ($ (o3)+(2,-1) $) node[node,draw] (a) {$\\bigwedge$};\n\n\\node[node, right = of a,draw ]  (output1){$\\mathrm{y_1}$};\n\\node[node, right=of a2, draw] (output2){$\\mathrm{y_2}$};\n\\draw (a) to[out=0, in=180]node[above]{\\scriptsize{$m$}} (output1);\n \\draw (a2) to[out=0, in=180]node[above]{\\scriptsize{$g$}} (output2);\n\\draw ($(output1)+(2,0)$)node[sqnode] (grabage) {\\scriptsize{$ \\textsl{maj}_3(x_1',x_2',x_3')$}};\n\\draw ($(output2)+(2,0)$)node[sqnode] (out) {\\scriptsize{$ \\textsl{min}_3\\{x_1'',x_2'',x_3''\\} $}};\n\n\\draw (b1)   to[out=  0,in=180]node[above]{\\scriptsize{$x_1$}} (fob1);\n\\draw[thick] (fob1) to[out=  0,in=180] node[above] { \\scriptsize{$x_1'$}} (o1);\n\\draw[thick] (fob1) to[out=  0,in=135] node[below, pos=0.2] { \\scriptsize{$x_1''\\ \\ $}} (a1);\n\\draw (b2)  to[out=  0,in=180] node[above]{\\scriptsize{$x_2$}}(fob2);\n\\draw[thick] (fob2)  to[out= 0,in=225] node[above, pos=0.2] {\\scriptsize{ $x_2'\\ \\ \\ $}} (o1);\n\\draw[thick] (fob2)  to[out= 0,in=180] node[below] { \\scriptsize{$x_2''$}}(a1);\n\n\\draw (o1)  to[out=  0,in=180] (foo1);\n\\draw (a1)  to[out=  0,in=180]node[above]{\\scriptsize{$y_2$}} (foa1);\n\\draw[thick] (foa1)  to[out=0,in=135] node[below, pos=0.2] {\\scriptsize{$ y_2''\\ \\ $}} (a2);\n\\draw [thick](foa1)  to[out=0,in=180] node[above] {\\scriptsize{$ y_2'$}} (o2);\n\\draw (b3)  to[out= 0,in=180]node[above]{\\scriptsize{$x_3$}} (fob3);\n\\draw[thick] (fob3)  to[out=  0,in=225] node[above, pos=0.2] {\\scriptsize{$ x_3'\\ \\ \\ $}}  (o2);\n\\draw[thick] (fob3)  to[out=  0,in=180] node[below] {\\scriptsize{$ x_3''$}} (a2);\n\n\\draw (o1)  to[out= 0,in=180]node[above]{\\scriptsize{$y_1$}} (foo1);\n\\draw[thick] (foo1)  to[out= 0,in=180] node[above] {\\scriptsize{$ y_1'$}} (o3);\n\\draw[thick] (foo1)  to[out= 0,in=135] node[below, pos=0.2] {\\scriptsize{$ y_1''\\ \\ $}} (a3);\n\\draw (o2)  to[out=  0,in=180]node[above]{\\scriptsize{$y_3$}} (foo2);\n\\draw[thick] (foo2)  to[out=  0,in=225] node[above, pos=0.2] {\\scriptsize{ $y_3'\\ \\ \\  $}}  (o3);\n\\draw[thick] (foo2)  to[out=  0,in=180] node[below] {\\scriptsize{$ y_3''$}} (a3);\n\n\\draw (o3)  to[out=  0,in=135]node[above, right]{\\scriptsize{$\\ z_1$}} (a);\n\\draw (a3)  to[out=  0,in=225]node[below, right]{\\scriptsize{$\\ z_2$}} (a);\n\\end{tikzpicture}\n}\n}\n\\caption{Nodes of boolean circuits and some examples. \nWriting, for example,\n$ \\textsl{maj}_3\\{x_1',x_2',x_3'\\} $ in place of \n$ \\textsl{maj}_3\\{x_1,x_2,x_3\\} $ would be equivalent. \nThe current notation just highlights which is the component of the fan-out nodes that an output depends on.}\n\\label{fig: some examples of boolean circuits}\n \\end{figure}\n \n\\begin{figure}\n\\scalebox{0.85}{\n\\bgroup\n\\def\\arraystretch{1.5}%\n\\begin{tabular}{l|ll} \n $\\neg$ & $\\mathtt{not}  \\triangleq\\lambda b. \\lambda x. \\lambda y. b yx $&$:\\mathbf{B} \\multimap\\mathbf{B}$\n \\\\ \\hline\n$\\wedge^0$   &$\\mathtt{and}^0 \\triangleq \\lambda x.\\lambda y. \\langle x,y \\rangle$ &$:\\mathbf{B}$\n\\\\\n$\\wedge^1$   &$\\mathtt{and}^1\\triangleq I$ &$:\\mathbf{B}\\multimap \\mathbf{B}$\n\\\\\n$\\wedge^2$&  $ \\mathtt{and}^2  \\triangleq \n\\lambda x_1. \\lambda x_2.   \\pi_1(x_1 x_2\\, \\mathtt{ff})$ &$:\n\\mathbf{B}\\multimap  \\mathbf{B} \\multimap \\mathbf{B}$\n\\\\\n$\\wedge^{n+2} $ &  $\\mathtt{and}^{n+2}\\triangleq \\lambda x_1\\ldots  x_{n+1}x_{n+2}. \\mathtt{and}^2\\, (\\mathtt{and}^{n+1}\\, x_1\\,   \\ldots\\, x_{n+1})\\, x_{n+2}$ &$:\\mathbf{B}\\multimap \\overset{n+2}{\\ldots}  \\multimap \\mathbf{B}\\multimap\\mathbf{B}$\n\\\\ \\hline\n$\\vee^0$ &$ \\mathtt{or}^0 \\triangleq \\lambda x.\\lambda y. \\langle y,x \\rangle$ &$:\\mathbf{B}$\\\\\n$\\vee^1$   &$\\mathtt{or}^1 \\triangleq  I $ &$:\\mathbf{B}\\multimap \\mathbf{B}$\n\\\\\n$\\vee^2$ & $\\mathtt{or}^2  \\triangleq \\lambda x_1. \\lambda x_2. \\pi_1(x_1 \\mathtt{tt}\\, x_2)$ &$:\n\\mathbf{B}\\multimap \\mathbf{B} \\multimap \\mathbf{B}$\n\\\\\n$\\vee^{n+2} $ &  $\\mathtt{or}^{n+2}\\triangleq \\lambda x_1\\ldots  x_{n+1}x_{n+2}. \\mathtt{or}^2\\, (\\mathtt{or}^{n+1}\\, x_1\\,   \\ldots\\, x_{n+1})\\, x_{n+2}$ &$: \\mathbf{B}\\multimap \\overset{n+2}{\\ldots}  \\multimap \\mathbf{B}\\multimap\\mathbf{B}$\\\\ \\hline\n\\textsl{fo}$^0$& $\\mathtt{out}^0\\triangleq \\lambda x. \\mathtt{discard}_{\\mathbf{B}}\\, x \\mathtt{ \\ in \\ } I$ &$: \\shpos \\mathbf{B}\\multimap \\mathbf{1}$\\\\\n\\textsl{fo}$^1$& $\\mathtt{out}^1\\triangleq  I$ &$: \\shpos \\mathbf{B}\\multimap \\mathbf{B}$ \\\\\n\\textsl{fo}$^2$& $\\mathtt{out}^2\\triangleq \\lambda x. \n\\texttt{copy}_{\\mathbf{B}}^{\\mathtt{tt}}\\, x\\ \\texttt{as}\\ x_1,x_2\\ \\texttt{in}\\ \\langle x_1, x_2\\rangle$ &$:\\shpos \\mathbf{B}\\multimap \\shpos \\mathbf{B}\\otimes  \\shpos \\mathbf{B}$ \\\\\n\\textsl{fo}$^{n+2}$& $\\mathtt{out}^{n+2}\\triangleq \\lambda x. \n\\texttt{copy}_{\\mathbf{B}}^{\\mathtt{tt}}\\, x\\ \\texttt{as}\\ x_1,x_2\\ \\texttt{in}\\ \n\\langle \n\\mathtt{out}^{n+1}\\, x_1,\nx_2\n\\rangle $ &$: \\shpos \\mathbf{B}\\multimap \\shpos \\mathbf{B} \\otimes \\overset{n+2}{\\ldots} \\otimes \\shpos \\mathbf{B}$ \\\\\n\\end{tabular}\n\\egroup \n}\n\\caption{Encoding of boolean functions and fan-out.}\n\\label{fig: enc boolean functions and fanout}\n\\end{figure}\n\n\\subsection{Boolean circuits in $\\mathsf{LEM}$}\n\\label{sec: encoding boolean circuits}\nWe encode boolean circuits as terms of $\\mathsf{LEM}$\n(Definition~\\ref{defn: from boolean circuits to terms}) and \nwe prove a simulation result \n(Proposition~\\ref{prop: simulation for boolean circuits}).\n\nThe encoding is inspired by Mairson\\&Terui~\\cite{mairsonlinear}. Other encodings of the boolean circuits have been shown in  Terui~\\cite{DBLP:conf\/lics\/Terui04}, \nMogbil\\&Rahli~\\cite{DBLP:conf\/lfcs\/MogbilR07} and \nAubert~\\cite{DBLP:journals\/corr\/abs-1201-1120} by considering  the  \\textit{unbounded} proof-nets for the multiplicative fragment \\textsf{MLL} of Linear Logic. Unbounded proof-nets are an efficient language able to express $n$-ary tensor products  by single nodes and to characterize parallel computational complexity classes such as \\textsf{NC}, \\textsf{AC}, and $\\textsf{P}\/_{\\operatorname{poly}}$. The contribution of this work to these encodings is in the use of   \\texttt{copy} and \\texttt{discard}  to directly express the fan-out nodes that  allow a more compact and  modular representation of circuits. In particular, as compared to~\\cite{DBLP:conf\/lics\/Terui04,DBLP:conf\/lfcs\/MogbilR07,DBLP:journals\/corr\/abs-1201-1120}, our encoding is able to  get rid of the garbage  that accumulates in the course of the simulation. \n\n\n\n\n\nWe start by briefly recalling the basics of boolean circuits from \nVollmer~\\cite{Vollmer:1999:ICC:520668}.\n\n\\begin{defn}[Boolean circuits] \nA \\textit{boolean circuit} $ C $ is a finite, directed and acyclic graph with $n$ \\emph{input nodes}, $m$ \\emph{output  nodes}, \\emph{internal nodes} and \\textit{fan-out nodes} as in Figure~\\ref{figure:node of boolean circuits}. \nThe incoming (resp.~outgoing) edges of a node are \\textit{premises} \n(resp.~\\textit{conclusions}). The \\textit{fan-in} of an internal node is the number of its premises. \nLabels for the $n$ input nodes of $ C $ are  $\\mathrm{x}_1, \\ldots, \\mathrm{x}_n$ and those ones for the $m$ outputs are $\\mathrm{y}_1, \\ldots, \\mathrm{y}_m$. \nEach internal node with fan-in $n \\geq 0$ has an $n$-ary boolean function $\\textsl{op}^n$ as its label, provided that if $n=0$, then $\\textsl{op}^n$ is a boolean constant in $\\lbrace 0,1 \\rbrace$. \nThe fan-out nodes have no label. \nInput and internal nodes are \\textit{logical nodes} and their conclusions are \\textit{logical edges}. If $\\nu$ and $\\nu'$ are logical nodes, then $\\nu'$ is a \\textit{successor} of $\\nu$ if a directed path from $\\nu$ to $\\nu'$ exists which crosses no logical node.\nThe \\emph{size} $ |C| $ of $C$ is the number of nodes. Its \\emph{depth} $ \\delta(C) $ is the length of the longest path from an input node to a output node. \nA \\textit{basis} $\\mathcal{B}$ is a set of boolean functions. A boolean circuit $C$ is \\textit{over a basis} $\\mathcal{B}$  if the label of every of its internal nodes belong to $\\mathcal{B}$ only. The standard unbounded fan-in basis is $\\mathcal{B}_1= \\lbrace \\neg, (\\wedge^n)_{n \\in \\mathbb{N}},  (\\vee^n)_{n \\in \\mathbb{N}}\\rbrace$. \n\\qed\n\\end{defn}\n\\noindent\nWhen representing boolean circuits as terms we label edges by $\\lambda$-variables, we omit their orientation, we assume that every fan-out always has a logical edge as its premise and we draw non-logical edges, i.e.~conclusions of fan-out nodes, as thick lines.\nFigures~\\ref{figure:2-bits full adder} and~\\ref{figure:maj3} are examples.\nA 2-bits full-adder is in the first one. \nIt takes two bits $ x_1, x_2 $ and a carrier $ y_{\\textrm{in}} $ as inputs.\nIts outputs are the sum $ s = (x'_1\\oplus x'_2)\\oplus y' $ and the \ncarrier $ y_{\\text{out}} = (x''_1\\wedge x''_2)\\vee((x'_1\\oplus x'_2)\\wedge y'')$, where $ \\oplus $ is the exclusive or that we can\nobtain by the functionally complete functions in $ \\mathcal{B}_1 $.\nFigure~\\ref{figure:maj3} is the 3-bits majority function $ \\textsl{maj}_3(x_1,x_2,x_3)$. \nIt serially composes three occurrences of the boolean circuit that switches two inputs $ x_1 $ and $ x_2 $ in order to put the greatest on the topmost output and the smallest on the bottommost one, under the convention that $ 0$ is smaller than $1$.\nSo, the 3-bits majority circuit first sorts its input bits and then checks if the topmost two, i.e. the majority, are both set to $1$.\nThe lowermost output is garbage. \n\n\nTranslating boolean circuits as terms of $\\mathsf{LEM}$ requires to encode the boolean functions in $\\mathcal{B}_1$ and the fan-out nodes. \nFigure~\\ref{fig: enc boolean functions and fanout} reports them, where $\\mathtt{tt}$ and $\\mathtt{ff}$ encode the boolean values in~\\eqref{eqn: boolean data type}, and $\\pi_1$ is the projection in~\\eqref{eqn: boolean projection}. \nAs a typographical convention, $\\mathtt{i}\\in \\lbrace \\mathtt{tt}, \\mathtt{ff} \\rbrace$ will code the boolean constant $i \\in \\lbrace 0,1 \\rbrace$, \nand $\\mathtt{op}^n$ the $n$-ary boolean function $\\textsl{op}^n$, according to Figure~\\ref{fig: enc boolean functions and fanout}. \nWe shorten $\\mathtt{and}^0$, $\\mathtt{or}^0$,  $\\mathtt{and}^2$, $\\mathtt{or}^2$, and $\\mathtt{out}^2$ as $\\mathtt{tt}$, $\\mathtt{ff}$, $\\mathtt{and}$, $\\mathtt{or}$, and $\\mathtt{out}$, respectively. The encoding of  the binary exclusive or $\\oplus$ is $\\mathtt{xor}$.\n\nWe recall that boolean circuits are a model of parallel computation, while the $\\lambda$-calculus models sequential computations. Mapping the former into the latter requires some technicalities. The notion of \\emph{level} allows to topologically sort the structure of the boolean circuits in order to preserve the node dependencies:\n\\begin{defn}[Level] The \\textit{level} $l$ of a logical node $\\nu$ in a boolean circuit $C$ is:\n\\begin{enumerate}\n\\item $ 0 $ if $\\nu$ has no  successors, and\n\\item $\\max \\lbrace l_1, \\ldots, l_k  \\rbrace +1$ if $\\nu$ has  successors $\\nu_1, \\ldots, \\nu_k$ with levels $l_1, \\ldots, l_k$.\n\\end{enumerate}\nThe \\textit{level} of a logical edge is the level of the logical node it is the conclusion of. \nThe \\textit{level} of a boolean circuit is the greatest level of its logical nodes.\n\\qed\n\\end{defn}\nWe define a level-by-level translation of unbounded fan-in boolean circuits over $\\mathcal{B}_1$ into  terms typable in $\\mathsf{LEM}$ taking inspiration from Schubert~\\cite{schubert2001complexity}:\n\\begin{defn}[From boolean circuits to terms]\\label{defn: from boolean circuits to terms}\nLet $C$ be a boolean circuit with $n$ inputs and $m$ outputs.  We define the term $\\mathtt{level}^l_C$ by induction on $l-1$:\n\\begin{enumerate}\n\\item $\\mathtt{level}_{C}^{-1} \\triangleq \\langle x_1, \\ldots, x_n  \\rangle$, where $x_1, \\ldots, x_n$ are  the variables labelling the logical edges of level $0$.\n\\item $\\mathtt{level}_{C}^{l} \\triangleq (\\lambda x_1\\ldots x_n x_{n+1}\\ldots x_{m}. \\mathtt{let}\\, (\\mathtt{out}^{k_{1}}\\, x_{1}) \\ \\mathtt{\\ be \\ }y^1_1, \\ldots, y^1_{k_1} \\mathtt{ \\ in \\ } \\ldots$\\\\\n$\\mathtt{let}\\, (\\mathtt{out}^{k_{n}}\\, x_{n})  \\mathtt{\\ be \\ }y^n_1, \\ldots, y^n_{k_n} \\mathtt{ \\ in \\ }  \\mathtt{level}_{C}^{l-1})\\, B_1 \\ldots B_m$, where:\n\\begin{enumerate}\n\\item $x_1,\\ldots ,x_n,x_{n+1},\\ldots, x_{m}$ are the variables labelling the logical edges of level  $ l $, \n\\item  for all $1 \\leq j \\leq n$,   $x_j$ is the premise of a fan-out node with conclusions labelled with   $y^j_1, \\ldots, y^j_{k_j}$ (see Figure~\\ref{fig: level fan out e internal nodes}).\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[node\/.style={circle,minimum size=1pt,fill=white},\n node distance=0.25cm]\n  \\node[circle, inner sep= 1pt, draw, fill=black](fan){};\n  \\node[node]  at ($(fan)+(0, -0.50)$)(befan){\\ldots} ;\n   \\node[circle] at  ($(fan)+(0, +0.50)$)(abfan){} ;\n   \\coordinate[right = of befan](berifan){} ;\n   \\coordinate[left = of befan](belefan){} ;\n     \\draw[-, thick](fan)to[out=225, in=90]node[left]{\\scriptsize{$y^j_1$}\\ \\ }(belefan);\n      \\draw[-,thick](fan)to[out=315, in=90]node[right]{\\ \\scriptsize{$y^j_{k_j}$}}(berifan);\n         \\draw[-](abfan)to[out=270, in=90]node[right]{\\scriptsize{$x_j$}}(fan);   \n\n \\node[circle, inner sep =1pt, draw, right=3cm of fan](op){\\scalebox{0.5}{$\\textsl{op}^h$}};\n \\node[circle]  at  ($(op)+(0, +0.50)$)(ab){\\scriptsize{\\ldots}} ;\n   \\coordinate[right = of ab](abri){} ;\n   \\coordinate[left = of ab](able){} ;\n   \\draw[-](able)to[out=270, in=135]node[left]{\\scriptsize{$z_1$}}(op);\n      \\draw[-](abri)to[out=270, in=45]node[right]{\\scriptsize{$\\, z_h$}}(op);\n       \\draw[-](op)--node[right]{\\scriptsize{$x_i$}} ($(op)+(0, -0.50)$) ;\n\\end{tikzpicture}\n\\caption{From left, a fan-out node and an internal node.}\n\\label{fig: level fan out e internal nodes}\n\\end{figure}\n\\item for all $1 \\leq i \\leq m$,  if $x_i$ is the variable labeling the conclusion of an internal node $\\textsl{op}^h$  with premises labeled  by $z_1,  \\ldots,  z_h$, respectively (see Figure~\\ref{fig: level fan out e internal nodes}), then $B_i\\triangleq \\mathtt{op}^h\\, z_1\\,  \\ldots\\,  z_h$. If $x_i$ is the variable labelling the conclusion of an input node  then $B_i \\triangleq x_i$.\n\\end{enumerate}\n\\end{enumerate}\nLast, if the input nodes have conclusions labeled by \n$x_1, \\ldots, x_{n}$, respectively, and if $C$ has level $l$,\nthen we define  \n$\\lambda(C) \\triangleq  \\lambda x. \\mathtt{let}\\ x \\mathtt{  \\ be \\ }x_1, \\ldots, x_{n} \\mathtt{ \\ in \\ }  \\mathtt{level}_C^l$.\n\\qed\n\\end{defn}\n\\begin{exmp}[$2$-bits full adder] The level-by-level translation of the boolean circuit $C$  in Figure~\\ref{figure:2-bits full adder} is the following:\n\\label{exmp:2-bits full-adder}\n\\allowdisplaybreaks\n\\begin{align*}\n\\mathtt{level}_{C}^{-1}&\\triangleq \\langle s, y_{\\text{out}} \\rangle\\\\\n\\mathtt{level}_{C}^0&\\triangleq (\\lambda s. \\lambda y_{\\text{out}}. \\mathtt{level}^{-1}_C)(\\mathtt{xor}\\, z'_1\\, y')(\\mathtt{or}\\, z_2\\, z_3)\\\\\n\\mathtt{level}_{C}^1&\\triangleq (\\lambda z_2. \\lambda z_3. \\mathtt{level}_{C}^0)(\\mathtt{and}\\, x''_1\\, x''_2)(\\mathtt{and}\\, z''_1\\, y'') \\\\\n\\mathtt{level}_{C}^2&\\triangleq(\\lambda z_1.\\lambda y_{\\text{in}}. \\mathtt{let}\\,(\\mathtt{out}\\, z_1)\\  \\mathtt{be}\\ z'_1, z''_1 \\ \\mathtt{ \\ in \\ }\\\\\n&\n\\phantom{\\ \\triangleq\\ \\qquad }\n\\mathtt{let}\\,(\\mathtt{out}\\, y_{\\text{in}}) \\mathtt{ \\ be\\  }y', y'' \\mathtt{ \\ in \\ }\\mathtt{level}_C ^1     )(\\mathtt{xor}\\, x'_1\\, x'_2)\\, y_{\\text{in}}\\\\\n\\mathtt{level}_{C}^3&\\triangleq (\\lambda x_1. \\lambda x_2.  \\mathtt{let}\\,(\\mathtt{out}\\, x_1) \\mathtt{ \\ be\\  }x_1', x_1'' \\mathtt{ \\ in \\ } \\\\\n&\n\\phantom{\\ \\triangleq\\ \\qquad }\n\\mathtt{let}\\,(\\mathtt{out}\\, x_2) \\mathtt{ \\ be\\  }x_2', x_2'' \\mathtt{ \\ in \\ } \\mathtt{level}_C^2)\\, x_1\\, x_2 \n\\end{align*}\nwhere we set $\\lambda(C)\\triangleq \\lambda x. \\mathtt{let }\\, x \\mathtt{\\ be \\ }  x_1,  x_2,  y_{\\text{in}} \\mathtt{ \\ in \\ } \\mathtt{level}_C ^3$ which reduces to:\n\\begin{align*}\n&\\lambda x. \\mathtt{let }\\, x \\mathtt{\\ be \\ }  x_1,  x_2,  y_{\\text{in}} \\mathtt{ \\ in \\ } (\\mathtt{let}\\, (\\mathtt{out}\\, x_1) \\mathtt{\\ be \\ }x'_1, x''_1 \\mathtt{ \\ in \\ }\\\\\n&\\mathtt{let}\\, (\\mathtt{out} \\, x_2) \\mathtt{\\ be \\ }x'_2, x''_2 \\mathtt{ \\ in \\ }( \\mathtt{let}\\, (\\mathtt{out}\\, y_{\\text{in}})\\mathtt{\\ be \\ }y', y'' \\mathtt{ \\ in \\ }\\\\\n&\\mathtt{let}\\,(\\mathtt{out}\\, (\\mathtt{xor}\\, x'_1\\, x'_2)) \\mathtt{ \\ be \\ } z'_1, z''_2  \\mathtt{ \\ in\\  } \\langle  \\mathtt{xor}\\, z'_1\\, y', \\mathtt{or}\\, (\\mathtt{and}\\, x''_1\\, x''_2)(\\mathtt{and}\\, z''_1\\, y'') \\rangle \\  ))\n\\enspace .\n\\qed\n\\end{align*}\n\\end{exmp}\n\\begin{exmp}[$3$-bits majority]  The level-by-level translation of the boolean circuit  $C$ in Figure~\\ref{figure:maj3} is the following:\n\\label{exmp:maj3}\n\\allowdisplaybreaks\n\\begin{align*}\n\\mathtt{level}_C^{-1}&\\triangleq \\langle m, g \\rangle           \n\\\\\n\\mathtt{level}_C^{0}&\\triangleq  (\\lambda m. \\lambda g. \\mathtt{level}_C^{-1})(\\mathtt{and}\\, z_1\\, z_2)(\\mathtt{and}\\, y''_2\\, x''_3)\\\\\n\\mathtt{level}_C^{1}&\\triangleq  (\\lambda z_1. \\lambda z_2. \\mathtt{level}_C^0)(\\mathtt{or}\\, y'_1\\, y'_3)(\\mathtt{and}\\, y''_1\\, y''_3)\\\\\n\\mathtt{level}_C^{2}&\\triangleq (\\lambda y_1. \\lambda y_3. \\mathtt{let}\\, (\\mathtt{out}\\, y_1)\\mathtt{\\ be \\   } y'_1, y''_1 \\mathtt{ \\ in \\ }\\\\\n& \n\\phantom{\\ \\triangleq \\quad}\n\\mathtt{let}\\, (\\mathtt{out} \\, y_3)\\mathtt{ \\ be \\ }y'_3, y''_3 \\mathtt{ \\ in \\ }\\mathtt{level}_C^1          )(\\mathtt{or}\\, x'_1\\, x'_2 )(\\mathtt{or}\\, y'_2\\, x'_3)\\\\\n\\mathtt{level}_C^3&\\triangleq (\\lambda y_2.\\lambda x_3. \\mathtt{let}\\, (\\mathtt{out}\\, y_2)\\mathtt{\\ be \\ }y'_2, y''_2 \\mathtt{ \\ in \\  }\\\\\n&  \n\\phantom{\\ \\triangleq\\quad}\n\\mathtt{let}(\\mathtt{out} \\, x_3)\\mathtt{\\ be \\ }x'_3, x''_3 \\mathtt{ \\ in \\ }\\mathtt{level}_C^{2}                  )(\\mathtt{and}\\, x''_1\\, x''_2 )\\, x_3\\\\\n\\mathtt{level}_C^4&\\triangleq (\\lambda x_1. \\lambda x_2.  \\mathtt{let}\\, (\\mathtt{out}\\, x_1)\\mathtt{\\ be \\ }x'_1, x''_1 \\mathtt{ \\ in \\  }\\\\\n& \n\\phantom{\\ \\triangleq \\quad}\n\\mathtt{let}\\, (\\mathtt{out}\\, x_2)\\mathtt{\\ be \\ }x'_2, x''_2 \\mathtt{ \\ in \\  }\\mathtt{level}_C^{3}      )\\, x_1\\, x_2\n\\end{align*}\nwhere we set $\\lambda(C)\\triangleq \\lambda x.  \\mathtt{let \\ }x \\mathtt{\\ be \\ }x_1, x_2, x_3 \\mathtt{ \\ in \\ }\\mathtt{level}_C^4$  which reduces to:\n\\begin{align*}\n&\\lambda x.  \\mathtt{let \\ }x \\mathtt{\\ be \\ }x_1, x_2, x_3 \\mathtt{ \\ in \\ } \\mathtt{let}\\, (\\mathtt{out}\\, x_1)\\mathtt{\\ be \\ }x'_1,x''_1 \\mathtt{ \\ in \\ }\\\\\n&\\mathtt{let}\\,(\\mathtt{out} \\, x_2 )\\mathtt{\\ be \\ }x'_2, x''_2 \\mathtt{\\ in\\ } ( \\mathtt{let}\\,(\\mathtt{out}\\, x_3)\\mathtt{\\ be \\ }x'_3, x''_3 \\mathtt{ \\ in \\ }\\\\\n&\\mathtt{let}\\,(\\mathtt{out}\\,(\\mathtt{or}\\, x'_1\\, x'_2))\\mathtt{\\ be \\ }y'_1, y''_1 \\mathtt{ \\ in \\ } (\n \\mathtt{let}\\,(\\mathtt{out}\\,(\\mathtt{and}\\, x''_1\\, x''_2)) \\mathtt{\\ be \\ }y'_2, y''_2 \\mathtt{ \\ in \\ }\\\\\n& \\mathtt{let }\\,(\\mathtt{out}\\,(\\mathtt{or}\\, y'_2\\, x'_3))\\mathtt{\\ be \\ }y'_3, y''_3 \\mathtt{ \\ in \\ }\n\\langle \\mathtt{ and }\\,(\\mathtt{or}\\, y'_1\\, y'_3)(\\mathtt{and}\\, \\, y''_1\\, y''_3) , \\mathtt{and}\\,y''_2\\, x''_3 \\rangle \\ ))\n\\enspace . \\qed\n\\end{align*}\n\\end{exmp}\n\n\\noindent\nThe size of the term coding an internal node depends on its fan-in.\nLikewise, the size of the term coding a fan-out node depends on the number of conclusions. The size of the circuit bounds both values.  Moreover, by Theorem~\\ref{thm:Subject Reduction for IMLL2shpos}, reducing a typable term yields a typable term. These observations imply:\n\\begin{prop}[Simulation of circuit evaluation]\\label{prop: simulation for boolean circuits}\nIf $ C$ is an unbounded fan-in boolean circuit over $\\mathcal{B}_1$ with $n$ inputs and $m$ outputs then $\\lambda(C)$  is such that:\n\\begin{enumerate}\n\\item  its size  is $ O(|C|) $, \n\\item it has type $(\\shpos \\mathbf{B}\\otimes \\overset{n}{\\ldots}\\otimes \\shpos \\mathbf{B})\\multimap ( \\mathbf{B}\\otimes \\overset{m}{\\ldots}\\otimes  \\mathbf{B})$ in $\\mathsf{LEM}$, and\n\\item for all $(i_1, \\ldots, i_n) \\in \\lbrace 0, 1 \\rbrace^n$, the evaluation of  $C$  on input $(i_1, \\ldots, i_n)$ outputs the tuple $(i'_1, \\ldots, i'_{m})\\in \\lbrace 0, 1 \\rbrace^m$ iff  $\\lambda(C) \\, \\langle \\mathtt{i}_1, \\dots, \\mathtt{i}_n \\rangle  \\rightarrow^*\\langle \\mathtt{i}'_1, \\ldots, \\mathtt{i}'_m\\rangle$.\n\\end{enumerate}\n\\end{prop}\n\\noindent\nIt should not be surprising that the translation cannot preserve the depth of a given circuit,  since $ \\mathsf{LEM}$ has only \\emph{binary} logical operators. That is why we use nested instances \n\\enquote{\\texttt{let}}\n(Definition~\\eqref{eqn:datatype unity and product}) to access single elements of\n$A_1\\otimes\\ldots\\otimes A_n $.  We could preserve the depth by extending \\textsf{LEM} with unbounded tensor products as done, for example, in~\\cite{DBLP:conf\/lics\/Terui04} for the multiplicative fragment of linear logic $\\mathsf{MLL}$. \n\n\n\\subsection{Numerals in $ \\mathsf{LEM} $}\n\\label{section:Church numerals}\nWe introduce  a class $ \\mathcal{N} $ of terms in $\\mathsf{LEM}$,   called  \\emph{numerals}, that represent natural numbers.\nWe give a successor \\texttt{S} and an addition \\texttt{A} on numerals, both typable in $\\mathsf{LEM}$,  and we show  that they  \nbehave as expected using  Subject reduction. Moreover, the numerals can operate as\niterators on a class of terms in $\\mathsf{LEM}$ that form a group with respect to the application.\n\n\\begin{defn}[Terms and types for $ \\mathcal{N} $]\n\\label{defn:Terms and types for mathcalN}\nLet us recall that $\\mathbf{1}$ is $\\forall \\alpha. \\alpha \\multimap \\alpha$ and $I$ is $\\lambda x.x$ (Section~\\ref{sec: background}).\nThe numerals of $ \\mathcal{N} $ have form:\n\\begin{align}\n\\nonumber\n\\overline{0} &\\triangleq \n \\lambda fx. \\texttt{discard}_{\\mathbf{1}}\\, f\\, \\texttt{in}\\, x: \\mathbb{N} \n\\\\\n\\nonumber\n\\overline{1} &\\triangleq\n \\lambda fx. fx: \\mathbb{N}\n\\\\\n\\overline{n+2} &\\triangleq\n \\lambda fx. \\texttt{copy}_{\\mathbf{1}}^{I}\\, f\\, \\texttt{as}\\,\n             f_1\\ldots f_n\\,\\texttt{in}\\, f_1(\\ldots(f_n\\, x)\\ldots): \\mathbb{N}\n\\enspace ,\n\\label{eqn: infinito}\n\\end{align}\n\\noindent\nwhere, for any $M$,\n$ \\texttt{copy}_{\\mathbf{1}}^{I}\\, f_0\\, \\texttt{as}\\, f_1\\ldots f_n\\,\\texttt{in}\\ M $ in~\\eqref{eqn: infinito} stands for:\n\\begin{align*}\n\\texttt{copy}_{\\mathbf{1}}^{I}\\, f_0\\, \\texttt{as}\\, f_1, f'_2 \\,\\texttt{in}\\,\n (\\texttt{copy}_{\\mathbf{1}}^{I}\\, f'_2\\, \\texttt{as}\\, f_2, f'_3 \\,\\texttt{in}\\, \\ldots\n (\\texttt{copy}_{\\mathbf{1}}^{I}\\, f'_{n-1}\\, \\texttt{as}\\, f_{n-1}, f_n \\,\\texttt{in}\\, M)\\ldots)\n \\enspace ,\n\\end{align*}\n\\noindent\nand $ \\mathbb{N} \\triangleq \\mathbf{N}[\\mathbf{1}\/\\alpha] $ with\n$ \\mathbf{N} \\triangleq\n(\\shpos \\alpha) \\multimap \\alpha $.\nIn order to identify terms that represent the same natural number, \nwe take numerals up to the following equivalences:\n\\begin{align*}\n&\\texttt{copy}_{\\boldmath{1}}^{I}\\, f\\, \\texttt{as}\\, f_1, f_2 \\,\\texttt{in}\\, \n(\\texttt{copy}_{\\boldmath{1}}^{I}\\, f_2\\, \\texttt{as}\\, f_3, f_4 \\,\\texttt{in}\\, M)\n\\\\\n&\n\\ \\,\n\\qquad\n\\qquad\n\\qquad\n\\qquad\n\\qquad\n= \\texttt{copy}_{\\boldmath{1}}^{I}\\, f\\, \\texttt{as}\\, f_2, f_4 \\,\\texttt{in}\\, (\n \\texttt{copy}_{\\boldmath{1}}^{I}\\, f_2\\, \\texttt{as}\\, f_1, f_3 \\,\\texttt{in}\\, M)\n\\\\\n&f (\\texttt{copy}_{\\boldmath{1}}^{I}\\, f'\\, \\texttt{as}\\, g, h \\ \\texttt{in}\\, M)\n=\n\\texttt{copy}_{\\boldmath{1}}^{I}\\, f'\\, \\texttt{as}\\, g, h \\,\\texttt{in}\\, f\\,M\n\\enspace .\n\\qquad \n\\qquad \n\\qquad \n\\qquad \n\\qed\n\\end{align*}\n\\end{defn}\nThe elements of $ \\mathcal{N} $ are the analogue of the Church numerals. Let us compare $\\mathbb{N}$ to the type $\\mathbf{int}$ of the Church numerals in Linear Logic: \n\\begin{equation*}\n\\begin{split}\n\\mathbf{int}&\\triangleq \\forall \\alpha .(\\oc (\\alpha \\multimap \\alpha)\\multimap (\\alpha \\multimap \\alpha))\\\\\n\\mathbb{N} &\\triangleq (\\shpos \\forall \\alpha .(\\alpha \\multimap \\alpha))\\multimap \\forall \\alpha . (\\alpha \\multimap \\alpha) \\enspace .\n\\end{split}\n\\end{equation*}\nIn the former the universal quantification is in positive position, while in the latter it  occurs on both sides of the main implication. This is because we can apply the modality $\\shpos$  only to  ground types, which are closed. Also, observe that the lack of an external quantifier in $\\mathbb{N}$ limits  the use of numerals as iterators.\n\nThe analogy with the Church numerals can be pushed further by defining a successor $\\texttt{S}$ and an addition $\\texttt{A}$:\n\\begin{align*}\n\\texttt{S} & \\triangleq\n\\lambda nfx.\n\\texttt{copy}_{\\boldmath{1}}^{I}\\,f\\,\\texttt{as}\\, f_1, f_2 \\,\\texttt{in}\\, f_1(n f_2 x): \\mathbb{N}\\multimap\\mathbb{N}\n\\\\\n\\texttt{A} & \\triangleq\n\\lambda mnfx.\n\\texttt{copy}_{\\boldmath{1}}^{I}\\,f\\,\\texttt{as}\\, f_1, f_2 \\,\\texttt{in}\\, \nm f_1(n f_2 x): \\mathbb{N}\\multimap \\mathbb{N}\\multimap\\mathbb{N}\n\\enspace .\n\\end{align*}\n\\noindent\nSticking to the computational behaviour of the terms\n(Figures~\\ref{fig: term reduction rules} \nand~\\ref{fig: term commuting conversions}), not of the underlying derivations, \nthe Subject reduction \n(Theorem~\\ref{thm:Subject Reduction for IMLL2shpos}) implies:\n\n\\begin{prop} \\label{prop: succ add}For all $n , m\\geq 0$, $ \\mathtt{S}\\,\\overline{n} \\rightarrow^* \\overline{n+1} $ and $ \\mathtt{A}\\,\\overline{m}\\,\\overline{n}\\rightarrow^* \\overline{m+n} $.\n\\end{prop}\n\\noindent\nFor example, the following reduction is legal:\n\\begin{align*}\n\\texttt{S} \\,\\overline{2} & \\triangleq\n(\\lambda nfx.\n\\texttt{copy}_{\\boldmath{1}}^{I}\\,f\\,\\texttt{as}\\, f_1, f_2 \\,\\texttt{in}\\, f_1(n f_2 x))\n( \\lambda gy. \\texttt{copy}_{\\mathbf{1}}^{I}\\,g\\, \\texttt{as}\\,\n             g_1,  g_2\\,\\texttt{in}\\, g_1(g_2\\, y))\n\\\\\n& \\rightarrow\n\\lambda fx.\n\\texttt{copy}_{\\boldmath{1}}^{I}\\,f\\,\\texttt{as}\\, f_1, f_2 \\,\\texttt{in}\\, f_1(( \\lambda gy. \\texttt{copy}_{\\mathbf{1}}^{I}\\,g\\, \\texttt{as}\\,\n             g_1,  g_2\\,\\texttt{in}\\, g_1(g_2\\, y)) f_2 x)\n\\\\\n& \\rightarrow\n\\lambda fx.\n\\texttt{copy}_{\\boldmath{1}}^{I}\\,f\\,\\texttt{as}\\, f_1, f_2 \\,\\texttt{in}\\, f_1(( \\lambda y. \\texttt{copy}_{\\mathbf{1}}^{I}\\,f_2\\, \\texttt{as}\\,\n             g_1,  g_2\\,\\texttt{in}\\, g_1(g_2\\, y)) x)\n\\\\\n& \\rightarrow\n\\lambda fx.\n\\texttt{copy}_{\\boldmath{1}}^{I}\\,f\\,\\texttt{as}\\, f_1, f_2 \\,\\texttt{in}\\, f_1( \\texttt{copy}_{\\mathbf{1}}^{I}\\,f_2\\, \\texttt{as}\\,\n             g_1,  g_2\\,\\texttt{in}\\, g_1(g_2\\, x)) \n      \\\\\n& =\n\\lambda fx.\n\\texttt{copy}_{\\boldmath{1}}^{I}\\,f\\,\\texttt{as}\\, f_1, f_2 \\,\\texttt{in}\\, ( \\texttt{copy}_{\\mathbf{1}}^{I}\\,f_2\\, \\texttt{as}\\,\n             g_1,  g_2\\,\\texttt{in}\\, f_1(g_1(g_2\\, x)))        \n              \\triangleq \\overline{3}\n\\end{align*}\n\\noindent\nObserve that Proposition~\\ref{prop: succ add} considers typable terms and term reductions  by exploiting Theorem~\\ref{thm:Subject Reduction for IMLL2shpos}. A similar result cannot be restated for the related  derivations and the  lazy-cut elimination  (Definition~\\ref{defn:Lazy cut-elimination strategy}).   For example, the here above term $\\texttt{S} \\,\\overline{2}$ has  type $\\mathbb{N}$, that is not lazy (Definition~\\ref{defn: lazyness}), due to the presence of a universal quantification in negative position.  Indeed, the  lazy cut-elimination strategy of a derivation of $\\texttt{S} \\,\\overline{2}$  runs  into  deadlocks before producing a cut-free derivation of $\\overline{3}$. \n\n\nAs far as we could see, the  \\enquote{zero-test}, the predecessor  and the subtraction on numerals cannot have type in $\\mathsf{LEM}$.  \nThe problem is the position of the universal quantifiers of $ \\mathbb{N} $. Consider, for example, the following predecessor:\n\\begin{align*}\n\\texttt{P} & \\triangleq\n\\lambda nsz.n\\, S[s]\\, B[z]\n&&\n(\\textrm{Predecessor})\n\\\\\nS[M] & \\triangleq\n\\lambda p.\\texttt{let}\\ p\\ \\texttt{be}\\ l, r\\ \\texttt{in}\\ \n\\langle M, lr \\rangle\n&&\n(\\textrm{Step function})\n\\\\\nB[N] & \\triangleq \\langle I, N\\rangle\n&&\n(\\textrm{Base function})\n\\end{align*}\nintroduced by Roversi \\cite{Roversi:1999-CSL}. Giving a type to \\texttt{P} would require to substitute $ (\\alpha\\multimap\\alpha)\\otimes\\alpha$ for $ \\alpha $ in $ \\mathbb{N} $, as suggested by the application of $ n:\\mathbb{N} $ to $ S[s] $. The position of the universal quantifiers in $\\mathbb{N}$ forbids it. Were such an instance legal, we could iterate functions, contradicting the cubic bound on the cut-elimination\n(Theorem~\\ref{thm: cut elimination for downarrow IMLL2}.)\n\nFurther, we can generalize $\\mathbb{N}$ to\n$\\mathbf{N}[\\overline{(A \\multimap A)} \/\\gamma]$,\nwhere $\\overline{(A \\multimap A)}$ is the closure of a \nquantifier-free type $A \\multimap A$,\nand find that \\emph{Hereditarily finite permutations} (\\textsf{HFP})\nby Dezani\n\\cite{DBLP:journals\/tcs\/Dezani-Ciancaglini76} \ninhabit $\\overline{A \\multimap A}$. \nAn  \\textsf{HFP}  is a $\\lambda$-term of the form $ P \\triangleq \\lambda z x_1 \\ldots x_n. z(P_1 x_{\\rho(1)})\\ldots (P_n x_{\\rho(n)}) $,\nfor some $ n\\geq 0 $,  where $\\rho \\in S_n$ (the symmetric group of $\\lbrace 1, \\ldots, n\\rbrace$)  and $ P_1, \\ldots, P_n$ are \\textsf{HFP}. The class $\\mathcal{H}_{\\text{lin}}$ of linear $\\lambda$-terms which are \\textsf{HFP}  (considered modulo $\\beta \\eta$-equivalence) forms a group:\n\\begin{enumerate}\n\\item The binary operation is $\\lambda fgx.f(g\\,x)$;\n\\item The identity is $I$;\n\\item If $P=\\lambda z x_1 \\ldots x_n. z(P_1 x_{\\rho(1)})\\ldots (P_n x_{\\rho(n)})$  is in  $\\mathcal{H}_{\\text{lin}}$, the inductively defined inverse is:\n \\[P^{-1} \\triangleq \\lambda z' x_1 \\ldots x_n. z'(P^{-1}_{\\rho^{-1}(1)} x_{\\rho^{-1}(1)})\\ldots (P^{-1}_{\\rho^{-1}(n)} x_{\\rho^{-1}(n)})\\] where, for all $1 \\leq i \\leq n$,  $\\rho^{-1}_i$ is the inverse permutation of $\\rho_i$  and $P^{-1}_{i}$ is the inverse of $P_{i}$.\n\\end{enumerate}\nFor example, let $ P =\\lambda wabc.w(\\lambda xy. ayx)(\\lambda xy. bxy)c $ which belongs to \\textsf{HFP} since\n$ \\lambda xy. ayx =_{\\beta } (\\lambda zxy. zyx) a $,\n$ \\lambda xy. bxy =_{\\beta } (\\lambda zxy. zxy) b $,\n$ c =_{\\beta } Ic $, where $ (\\lambda zxy. zyx)$, $ (\\lambda zxy. zxy) $ and $ I $ are in \\textsf{HFP}. Then, $P$ has type\n$\\forall \\alpha_x \\alpha_y\\alpha_a\\alpha_b\\alpha_c\\alpha. \nA \\multimap A$, \nwhich is a ground type\nwhere $A$ is \n$(\\alpha_x\\multimap\\alpha_y\\multimap\\alpha_a )\n \\multimap \n (\\alpha_x\\multimap\\alpha_y\\multimap\\alpha_b ) \n \\multimap \\alpha_c \n \\multimap \\alpha$.\nThese observations show an unexpected link between $\\mathsf{LEM}$ and reversible computation (see Perumalla \\cite{perumalla2013chc} for a thorough introduction) that can well be expressed in terms of monoidal structures where permutations play a central role \\cite{DBLP:conf\/types\/PaoliniPR15, paolini16ictcs, paolini2018ngc}.\n\n\\section{The proof of Theorem~\\ref{thm: pi1 are duplicable}} \n\\label{sec: the d-soundness theorem DICE}\nIn this section we give a detailed proof of Theorem~\\ref{thm: pi1 are duplicable} for $\\mathsf{IMLL}_2$, which states that if $A$ is an inhabited ground type, i.e~an inhabited  closed $\\Pi_1$-type, then $A$ is also a duplicable type. By Definition~\\ref{defn: duplicable and erasable types}, this amounts to show that a linear $\\lambda$-term $\\mathtt{D}_{A}: A\\multimap A \\otimes A$ exists such that  $\\mathtt{D}_{A}\\, V \\rightarrow^*_{\\beta \\eta} \\langle V, V \\rangle$ holds for every value  $V$ of $A$. We shall construct $ \\mathtt{D}_{A} $ as the composition of three linear $\\lambda$-terms, as diagrammatically displayed in Figure~\\ref{fig:DA diagram}. For each such component we dedicate a specific subsection. For the sake of presentation, in this section we  focus on  terms of $\\mathsf{IMLL}_2$ rather than on derivations, so that when we say that a term $M$ has type $A$ with context $\\Gamma$, we clearly mean that a derivation $\\mathcal{D}$ exists such that $\\mathcal{D}\\triangleleft \\Gamma \\vdash M: A$. Moreover, as assumed in Section~\\ref{sec: background}, terms are considered modulo $\\alpha$-equivalence. \n\\begin{figure}\n    \\begin{tikzpicture}[baseline= (a).base]\n    \\node[scale=.725] (a) at (0,0){\n        \\begin{tikzcd}[column sep=scriptsize, row sep=tiny]\n        A \\ar[rr, \"\\mathtt{sub}^s_{A}\"] \n        && \n        A^-[\\mathbf{B}^s] \\ar[rr, \"\\mathtt{enc}^s _{A}\"] \n        && \n        \\mathbf{B}^s \n        \\ar[rr, \"\\mathtt{dec}^s _{A} \"] \n        &&\n        A \\otimes A\n        \\\\\n        V \\ar[rr, mapsto]  \n        &&  \n        V_{A} \\ar[rr, mapsto] \n        && \n        \\lceil V_{A} \\rceil \\ar[rr, mapsto]\n        &&\n        \\langle V_{A}, V_{A}  \\rangle\n        \\end{tikzcd}\n    };\n    \\end{tikzpicture}\n    \\caption{The diagrammatic representation of $ \\mathtt{D}_{A}$.}\n    \\label{fig:DA diagram}\n\\end{figure}\n\\subsection{The linear $ \\lambda $-term $ \\mathtt{sub}^s_{A} $}\nRoughly, the $\\lambda$-term $\\mathtt{sub}^s_{A}$, when applied to a value  $V$ of ground type $A$, produces its $\\eta$-long normal form $V_{A}$ whose type is obtained from $A$ as follows: we strip away every occurrence of $\\forall$ and we substitute each type variable  with the $s$-ary tensor of boolean datatypes $\\mathbf{B}^s=\\mathbf{B}\\otimes \\overset{s}{\\ldots}\\otimes \\mathbf{B}$, for some $s>0$.\\\\\nBefore introducing the $\\lambda$-term $\\mathtt{sub}^s_{A}$, we need the definition of $\\eta$-long normal form:\n\\begin{defn}[$\\eta$-long normal forms]\\label{defn: eta long nf}\nLet $\\mathcal{D}\\triangleleft \\Gamma \\vdash M: B$ be cut-free.\nWe define the $\\eta$-\\textit{expansion} of $\\mathcal{D}$, denoted  $\\mathcal{D}^\\Gamma_B$,  as the derivation obtained from $\\mathcal{D}$ by substituting every occurrence of:\n\\begin{prooftree}\n\\AxiomC{}\n\\RightLabel{$ax$}\n\\UnaryInfC{$x: A \\vdash x: A$}\n\\end{prooftree}\n with a derivation of $ x: A \\vdash M':A $, for some $ M' $, whose axioms have form $y: \\alpha \\vdash y: \\alpha$. The  $\\eta$-expansion is unique and transforms the $\\lambda$-term $M$ to   its \\emph{$\\eta$-long normal form}, denoted by $M^{\\Gamma}_B$ and such that $M^\\Gamma_B\\rightarrow^*_\\eta M$.\nIf the context $\\Gamma$ of an $ \\eta $-expanded $ \\mathcal{D} $ is \n$x_1: A_1, \\ldots, x_n:A$ we may write \n$\\mathcal{D}^{A_1, \\ldots, A_n}_B$ and $M^{A_1, \\ldots, A_n}_B$.\nIf $\\Gamma$ is empty, we feel free to write $\\mathcal{D}_B$ and $M_B$.\n\\end{defn}\n\\begin{lem}\\label{lem: canonical definition for eta long normal forms} Let $\\mathcal{D}\\triangleleft \\Gamma \\vdash M: A$ be a cut-free derivation in $\\mathsf{IMLL}_2$, and let  $M^\\Gamma_A$ denote the $\\eta$-long normal form obtained by $\\eta$-expanding $\\mathcal{D}$. Then:\n\\begin{enumerate}[1]\n\\item  \\label{eqn: 1 canonical definition for eta long normal forms} If $M=x$, $A= \\alpha$, and  $\\Gamma= x: \\alpha$ then  $x^\\alpha_\\alpha=x$.\n\\item \\label{eqn: 2 canonical definition for eta long normal forms} If $M=x$, $A= \\forall \\alpha. B$, and  $\\Gamma= x: \\forall \\alpha. B$ then  $x^{\\forall \\alpha. B}_{\\forall \\alpha. B}= x^{B}_B$.\n\\item  \\label{eqn: 3 canonical definition for eta long normal forms} If $M=x$, $A= B \\multimap C$, and  $\\Gamma= x: B \\multimap C$ then  $x^{B \\multimap C}_{B \\multimap C}=\\lambda y.(xy^{B}_B)^C_C$.\n\\item \\label{eqn: 4 canonical definition for eta long normal forms} If $A= \\forall \\alpha. B$ then $M^{\\Gamma}_{\\forall \\alpha. B}= M^\\Gamma_{B\\langle \\gamma\/\\alpha \\rangle}$, for some $\\gamma$. \n\\item \\label{eqn: 5 canonical definition for eta long normal forms} If $M= \\lambda x. N$ and $A= B \\multimap C$ then $(\\lambda x. N)^{\\Gamma}_{B \\multimap C}= \\lambda x. N^{\\Gamma, x: B}_C$.\n\\item \\label{eqn: 7 canonical definition for eta long normal forms} If $M=P[yN\/x]$ and $\\Gamma=\\Delta, \\Sigma, y: B \\multimap C$, where $P$ has type $A$ with context $\\Delta, x:C$  and   $N$ has type $B$ with context $\\Sigma$,  then $(P[yN\/x])^{\\Gamma}_A=P^{\\Delta, x: C}_A[yN^\\Sigma _B\/x]$.\n\\item \\label{eqn: 6 canonical definition for eta long normal forms} If $M=P[yN\/x]$ and  $\\Gamma= \\Gamma', y: \\forall \\alpha. B$ then we have $(P[yN\/x])^{\\Gamma', y: \\forall \\alpha.B}_A=(P[yN\/x])^{\\Gamma', y: B\\langle D\/\\alpha \\rangle}_A$, for some type $D$. \n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nJust follow the definition of $\\eta$-long normal form.\n\\end{proof}\n\n\\begin{defn} \\label{defn: stripping forall} Let $A$ be a type in $\\mathsf{IMLL}_2$. We define $A^-$ by induction on the complexity of the type:\n\\begin{align*}\n\\alpha^- &\\triangleq \\alpha \\\\\n(A \\multimap B)^- &\\triangleq  A^- \\multimap B^-\\\\\n(\\forall \\alpha. A)^-&\\triangleq  A^-\\langle \\gamma\/\\alpha \\rangle \\enspace, \n\\end{align*}\nwhere $\\gamma$ is taken from the head of an infinite list of fresh type variables. \nThe notation $A[B]$ denotes the type obtained by replacing $B$ for every free type variable of $A$. Moreover, if $\\Gamma= x_1:A_1, \\ldots, x_n:A_n$, then $\\Gamma^-$ stands for $x_1:A_1^-, \\ldots, x_n:A^-_n$, and $\\Gamma[B]$  stands for $x_1:A_1[B], \\ldots, x_n:A_n[B]$.\\qed\n\\end{defn}\n\\begin{defn}[The linear $\\lambda$-term $\\mathtt{sub}^s_{A}$]\\label{defn: sub} Let $s>0$. We define the linear $\\lambda$-terms $\\mathtt{sub}^s_{A}: A[\\mathbf{B}^s]\\multimap A^-[\\mathbf{B}^s]$, where $A $ is a $ \\Pi_1$-type, and $\\overline{\\mathtt{sub}}^s_{A}: A^-[\\mathbf{B}^s] \\multimap A[\\mathbf{B}^s]$, where $A $ is a  $\\Sigma_1$-type, by simultaneous induction on the size of $A$:\n\\begin{align*}\n\t&\\mathtt{sub}^s_{\\alpha}\\triangleq  \\lambda x.x \n\t&&\\overline{\\mathtt{sub}}^s_{\\alpha} \\triangleq  \\lambda x.x \\\\\n\t&\\mathtt{sub}^s_{\\forall \\alpha. B}  \\triangleq  \\mathtt{sub}^s_{B} \n&&  \\\\\n\t&\\mathtt{sub}^s_{B \\multimap C}  \\triangleq  \\lambda x.\\lambda y.  \\mathtt{sub}^s_{C}(x\\, (\\overline{\\mathtt{sub}}^s_{B}\\,y)) \n\t&&  \\overline{\\mathtt{sub}}^s_{B \\multimap C}\n\t      \\triangleq  \n\t       \\lambda x.\\lambda \n\t       y.\\overline{\\mathtt{sub}}^s_C(x\\,(\\mathtt{sub}^s_{B}\\,y)).\t\n           \\qed\n\\end{align*}\n\\end{defn}\nThe following will be used to compact the proof of some of the coming lemmas.\n\\begin{defn}\\label{defn: sub substitution}\nLet $s>0$. Let $A$ be a $\\Pi_1$-type. Let $\\Gamma=x_1:A_1, \\ldots, x_n:A_n$ be a context of $\\Sigma_1$-types. Let $M$ be an inhabitant of \n$A[\\mathbf{B}^s]$ with context $\\Gamma[\\mathbf{B}^s]$.\nThen $M[\\Gamma]$ denotes the substitution:\n\\begin{align*}\n&M[\\overline{\\mathtt{sub}}^s_{A_1}\\,x'_1\/x_1, \\ldots,\\overline{\\mathtt{sub}}^s_{A_n}\\,x'_n\/x_n ]\n\\end{align*} \nfor some $x'_1, \\ldots, x'_n$.\n\\qed\n\\end{defn}\n\n\\begin{lem}\\label{lem: sub variable to eta long normal form} Let $s>0$ and $z$ be of type $A[\\mathbf{B}^s]$.\n\\begin{enumerate}[(1)]\n\\item \\label{eqn: 1 sub variable to eta long normal form} If $A$ is a $\\Pi_1$-type, then $\\mathtt{sub}^s_{A}\\, z \\rightarrow_\\beta^* z_{A}^A$.\n\\item \\label{eqn: 2 sub variable to eta long normal form}  If $A$ is a  $\\Sigma_1$-type, then $\\overline{\\mathtt{sub}}^s_A\\, z \\rightarrow_\\beta^* z^A_A$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nWe prove both points by simultaneous induction on $\\vert A \\vert$:\n\\begin{enumerate}\n\\item Case $A= \\alpha$. Both the statements are straightforward since we have  $z^\\alpha _\\alpha=z$ by Lemma~\\ref{lem: canonical definition for eta long normal forms}.\\ref{eqn:  1 canonical definition for eta long normal forms}.\n\\item Case $A= \\forall \\alpha. B$. This case applies to point~\\ref{eqn: 1 sub variable to eta long normal form} only. By induction hypothesis, for every variable $x$ of type $B[\\mathbf{B}^s]$, $\\mathtt{sub}^s_{B}\\,x \\rightarrow_\\beta^* x^B_B$. The $\\lambda$-term $\\mathtt{sub}^s_B$ has type  $B[\\mathbf{B}^s]\\multimap B^-[\\mathbf{B}^s]$, which is equal to $(B\\langle \\mathbf{B}^s\/\\alpha\\rangle )[\\mathbf{B}^s]\\multimap (\\forall \\alpha. B)^-[\\mathbf{B}^s]$. Hence, $\\mathtt{sub}^s_B$ has also type $(\\forall \\alpha. B)[\\mathbf{B}^s]\\multimap (\\forall \\alpha. B)^-[\\mathbf{B}^s]$. Moreover,  by Definition~\\ref{defn: sub} we have $\\mathtt{sub}^s_{B}= \\mathtt{sub}^s_{\\forall \\alpha. B}$. Therefore, for every variable $z$ of type $(\\forall \\alpha. B)[\\mathbf{B}^s]$ we have  $\\mathtt{sub}^s_{\\forall \\alpha. B}\\, z= \\mathtt{sub}^s_B\\, z \\rightarrow_\\beta^* z^{B}_B$. But $z^{B}_B=z^{\\forall \\alpha. B}_{\\forall \\alpha. B}$ by Lemma~\\ref{lem: canonical definition for eta long normal forms}.\\ref{eqn:  2 canonical definition for eta long normal forms}.\n\\item Case $A= B \\multimap C$. We prove point~\\ref{eqn: 1 sub variable to eta long normal form}   only (point~\\ref{eqn: 2 sub variable to eta long normal form} is similar). Let $z$ be of type $(B\\multimap C)[\\mathbf{B}^s]=B[\\mathbf{B}^s]\\multimap C[\\mathbf{B}^s]$. Then we have\n\\allowdisplaybreaks\n\\begin{align*}\n\\mathtt{sub}^s_{B \\multimap C}\\, z &=(\\lambda x.  \\lambda y.  \\mathtt{sub}^s_{C}(x(\\overline{\\mathtt{sub}}^s_{B}\\, y)))z &&\\text{Definition}~\\ref{defn: sub} \\\\\n&\\rightarrow_\\beta \\lambda y.  \\mathtt{sub}^s_{C}(z(\\overline{\\mathtt{sub}}^s_{B}\\, y))&&\\\\ \n&=\\lambda y. ( \\mathtt{sub}^s_{C}\\, w)[z(\\overline{\\mathtt{sub}}^s_{B}\\, y)\/w] &&\\\\\n& \\rightarrow_\\beta^* \\lambda y. w^C_C [z (\\overline{\\mathtt{sub}}^s_{B}\\, y)\/w]&& \\text{induction hyp., point}~\\ref{eqn: 1 sub variable to eta long normal form}\\\\\n& \\rightarrow_\\beta^* \\lambda y. w^C_C [z y^B_B\/w]&& \\text{induction hyp., point}~\\ref{eqn: 2 sub variable to eta long normal form}\\\\\n&= \\lambda y. (zy^B_B)^C_C &&\\\\\n&= z^{B \\multimap C}_{B\\multimap C}&& \\text{Lemma}~\\ref{lem: canonical definition for eta long normal forms}.\\ref{eqn: 3 canonical definition for eta long normal forms}.\n\\end{align*}\n\\end{enumerate}\n\\end{proof}\n\\begin{lem}\\label{lem: sub preservation of eta long normal form} Let $s>0$. \nIf $z:A[\\mathbf{B}^s]$, where $A$ is a  $\\Pi_1$-type, then $\\mathtt{sub}^s_A\\, z^A_A \\rightarrow_\\beta^* z^A_A $.\n\\end{lem}\n\\begin{proof}\n We prove it by induction on $\\vert A \\vert$:\n\\begin{enumerate}\n\\item Case $A= \\alpha$. The statement is straightforward since we have $z^\\alpha _\\alpha=z$ by Lemma~\\ref{lem: canonical definition for eta long normal forms}.\\ref{eqn: 1 canonical definition for eta long normal forms}\n\\item Case $A= \\forall \\alpha. B$.  By Definition~\\ref{defn: sub}, $\\mathtt{sub}^s_{\\forall \\alpha. B}= \\mathtt{sub}^s_{B}$ and we use the induction hypothesis.\n\\item Case $A= B \\multimap C$. Then we have\n\\allowdisplaybreaks\n\\begin{align*}\n \\mathtt{sub}^s_{B \\multimap C}\\, z^{B \\multimap C}_{B\\multimap C}&= (\\lambda x.  \\lambda y.  \\mathtt{sub}^s_{C}(x(\\overline{\\mathtt{sub}}^s_{B}\\, y)))z^{B \\multimap C}_{B\\multimap C}&& \\text{Definition}~\\ref{defn: sub}\\\\\n &= (\\lambda x.  \\lambda y.  \\mathtt{sub}^s_{C}(x(\\overline{\\mathtt{sub}}^s_{B}\\, y)))( \\lambda w. (zw^{B}_B)^C_C)&&\\text{Lemma}~\\ref{lem: canonical definition for eta long normal forms}.\\ref{eqn: 3 canonical definition for eta long normal forms}\\\\\n &\\rightarrow_\\beta  \\lambda y.  \\mathtt{sub}^s_{C}((\\lambda w. (zw^{B}_B)^C_C)(\\overline{\\mathtt{sub}}^s_{B}\\, y))\\\\\n  &\\rightarrow _\\beta \\lambda y.  \\mathtt{sub}^s_{C} (z(\\overline{\\mathtt{sub}}^s_{B}\\, y)^{B}_B)^C_C \\\\\n   &\\rightarrow_\\beta^*  \\lambda y.  \\mathtt{sub}^s_{C} (z y^{B}_B)^C_C&&\\text{Lemma}~\\ref{lem: sub variable to eta long normal form}.\\ref{eqn: 2 sub variable to eta long normal form}\\\\\n    &=  \\lambda y.  (\\mathtt{sub}^s_{C}\\, w^{C}_C)[ z y^{B}_B\/w] \\\\ \n       &\\rightarrow_\\beta^*  \\lambda y.  w^C_C[ z y^{B}_B\/w] &&\\text{induction hyp.} \\\\ \n          &=  \\lambda y. (zy^B_B)^C_C \\\\ \n            &=_\\alpha z^{B\\multimap C}_{B \\multimap C}&&\\text{Lemma}~\\ref{lem: canonical definition for eta long normal forms}.\\ref{eqn: 3 canonical definition for eta long normal forms}\n\\end{align*} \n\\end{enumerate}\n\\end{proof}\n\\begin{lem}\\label{lem: sub} Let $s>0$. Let $A$ be a $\\Pi_1$-type, and let  \n$\\Gamma=x_1:A_1, \\ldots, x_n:A_n$ be a context of $\\Sigma_1$-types. \nIf $\\Gamma[\\mathbf{B}^s]\\vdash M:A[\\mathbf{B}^s]$, with $M$ normal, then:\n\\begin{equation*}\n\\mathtt{sub}^s_{A}\\, M[\\Gamma]\\rightarrow_\\beta^* M^\\Gamma _A  \\enspace .\n\\end{equation*}\n\\end{lem}\n\\begin{proof}\nLet $Q_{\\Gamma, A}$ be the number of universal quantifications in $A_1, \\ldots, A_n, A$. We prove the result by induction on $\\vert M \\vert+ Q_{\\Gamma, A}$. If $M=z$ then $\\Gamma= z:A$ and $\\mathtt{sub}^s_A\\, M[\\Gamma]= \\mathtt{sub}^s_{A}(\\overline{\\mathtt{sub}}^s_{A}\\, z)$. By point~\\ref{eqn: 2 sub variable to eta long normal form} of Lemma~\\ref{lem: sub variable to eta long normal form} and by Lemma~\\ref{lem: sub preservation of eta long normal form} we have $\\mathtt{sub}^s_A( \\overline{\\mathtt{sub}}^s_A \\, z) \\rightarrow_\\beta^*  \\mathtt{sub}^s_A \\, z^A_A \\rightarrow_\\beta^* z^A_A$. If $M= \\lambda z.N$ then we have two cases depending on the type of $M$:\n\\begin{enumerate}\n\\item Case $A= \\forall \\alpha. B$.   The $\\lambda$-term $\\mathtt{sub}^s_B$ has type $B[\\mathbf{B}^s]\\multimap B^-[\\mathbf{B}^s]$, which is equal to $(B\\langle \\mathbf{B}^s\/\\alpha\\rangle )[\\mathbf{B}^s] \\multimap (\\forall \\alpha. B)^-[\\mathbf{B}^s]$, so that $\\mathtt{sub}^s_B$ has also type $(\\forall \\alpha. B)[\\mathbf{B}^s]\\multimap (\\forall \\alpha. B)^-[\\mathbf{B}^s]$. By Definition~\\ref{defn: sub} we have $\\mathtt{sub}^s_{B}= \\mathtt{sub}^s_{\\forall \\alpha. B}$. By using the induction hypothesis, for every $M$ of type $(\\forall \\alpha. B)[\\mathbf{B}^s]$ with context $\\Gamma[\\mathbf{B}^s]$, we have $\\mathtt{sub}^s_{\\forall \\alpha. B}\\, M[\\Gamma]= \\mathtt{sub}^s_B\\, M[\\Gamma]  \\rightarrow_\\beta^* M^{\\Gamma}_{B}$. Moreover, by Lemma~\\ref{lem: canonical definition for eta long normal forms}.\\ref{eqn: 4 canonical definition for eta long normal forms}, $M^{\\Gamma}_{B}= M^\\Gamma_{\\forall \\alpha. A}$.\n\\item Case $A= B \\multimap C$. Then we have:\n\\allowdisplaybreaks\n\\begin{align*}\n\\mathtt{sub}^s_{B \\multimap C}\\, M[\\Gamma]&=(\\lambda x.\\lambda y.  \\mathtt{sub}^s_{C}(x\\, (\\overline{\\mathtt{sub}}^s_{B}\\,y)))(\\lambda z. N)[\\Gamma]&&\\text{Definition}~\\ref{defn: sub}\\\\\n&\\rightarrow_\\beta \\lambda y.  \\mathtt{sub}^s_{C}((\\lambda z.N)[\\Gamma] \\, (\\overline{\\mathtt{sub}}^s_{B}\\,y))\\\\\n&\\rightarrow_\\beta  \\lambda y.  \\mathtt{sub}^s_{C}((N[\\Gamma])[\\overline{\\mathtt{sub}}^s_{B}\\,y\/z])\\\\\n&=  \\lambda y.  \\mathtt{sub}^s_{C}(N[\\Gamma, y:B] )&&\\text{Definition}~\\ref{defn: sub substitution}\\\\\n&\\rightarrow_\\beta^*  \\lambda y.N^{\\Gamma,y: B}_C&&\\text{induction hyp.}\\\\\n&=(\\lambda y.N)^{\\Gamma}_{B \\multimap C}&&\\text{Lemma}~\\ref{lem: canonical definition for eta long normal forms}.\\ref{eqn: 5 canonical definition for eta long normal forms}\\\\\n&=_{\\alpha} M^\\Gamma_{B \\multimap C}.\n\\end{align*}\n\\end{enumerate}\nIf $M=P[zN\/w]$ then the type of $z$ cannot have an outermost universal quantification, because $\\Gamma$ is a context of $\\Sigma_1$-types. So $z$ has type of the form $B \\multimap C$ in $\\Gamma$.  Let $\\Gamma'$ and $\\Gamma''$ be contexts  such that $\\Gamma= \\Gamma', \\Gamma'', z: B \\multimap C$, $\\operatorname{dom}(\\Gamma')=FV(P)$, and $\\operatorname{dom}(\\Gamma'')=FV(N)$. Then we have:\n\\allowdisplaybreaks\n\\begin{align*}\n\\mathtt{sub}^s _{A}\\, M[\\Gamma]&= \\mathtt{sub}^s_{A}(P[zN\/w])[\\Gamma]\\\\\n&= \\mathtt{sub}^s_{A}(P[\\Gamma']  [(zN)[\\Gamma'', z: B \\multimap C]\/w])\\\\\n&=\\mathtt{sub}^s_A (P[\\Gamma'][(\\overline{\\mathtt{sub}}^s_{B \\multimap C}\\, z)(N[\\Gamma''])\/w])\\\\\t    \n&\\rightarrow_\\beta^* \\mathtt{sub}^s_A (P[\\Gamma'][ \\overline{\\mathtt{sub}}^s_C(z \\,(\\mathtt{sub}^s_{B}\\,(N[\\Gamma''])))\/w])&&\\text{Definition}~\\ref{defn: sub}\\\\\t       \n&\\rightarrow_\\beta^* \\mathtt{sub}^s_A (P[\\Gamma'][ \\overline{\\mathtt{sub}}^s_C(z N^{\\Gamma''}_B)\/w] )&&\\text{induction hyp.}\\\\\t\n&= (\\mathtt{sub}^s_A \\, P[\\Gamma'][\\overline{\\mathtt{sub}}^s_C\\, w\/w] ) [z N^{\\Gamma''}_B\/w]\\\\\t\n&= (\\mathtt{sub}^s_A \\, P[\\Gamma', w: C])[z N^{\\Gamma''}_B\/w]&&\\text{Definition}~\\ref{defn: sub substitution}\\\\\n&\\rightarrow_\\beta^* P^{\\Gamma', w: C}_A [z N^{\\Gamma''}_B\/w]&&\\text{induction hyp.}\\\\\t\n&=(P[zN\/w])^{\\Gamma}_A &&\\text{Lemma}~\\ref{lem: canonical definition for eta long normal forms}.\\ref{eqn: 7 canonical definition for eta long normal forms}\n\\end{align*}\n\\end{proof}\n\n\n\\subsection{The linear $\\lambda$-term $ \\mathtt{enc}^s_{A} $}\nOne missing ingredient  in the previous subsection is the value of $ s $, which is fixed to some strictly positive integer.  To determine $s$ we need the following property:\n\\begin{lem} \\label{lem: pi1type bound} For every cut-free derivation $\\mathcal{D}\\triangleleft \\Gamma \\vdash M: B$ in $\\mathsf{IMLL}_2$ which does not contain applications of $\\forall$L, the following inequations hold:\n\\begin{equation}\\label{eqn: 1 inequation}\n\\vert M \\vert  \\leq  \\vert M^{\\Gamma}_B \\vert \\leq \\vert \\Gamma^-  \\vert + \\vert B^- \\vert \\leq 2 \\cdot \\vert M_B^{\\Gamma} \\vert \\enspace ,\n\\end{equation}\nwhere $(\\_ )^-$ is as in Definition~\\ref{defn: stripping forall}, and $M^\\Gamma_A$ is as in Definition~\\ref{defn: eta long nf}.\n\\end{lem}\n\\begin{proof}\nThe inequation $\\vert M \\vert  \\leq  \\vert M^{\\Gamma}_B \\vert$ is by definition of $\\eta$-long normal form. Now, let $\\mathcal{D}^{\\Gamma}_B$ be the $\\eta$-expansion of $\\mathcal{D}$, so that $\\mathcal{D}^{\\Gamma}_B \\triangleleft \\Gamma \\vdash M^{\\Gamma}_B: B$. We prove the remaining two inequations by induction on $\\mathcal{D}^{\\Gamma}_B$. If it is an axiom then, by definition of $\\eta$-expansion, it must be of the form $x: \\alpha \\vdash x:\\alpha$, where   $M^{\\Gamma}_B=x$. Hence, $\\vert x \\vert \\leq 2 \\cdot \\vert \\alpha \\vert \\leq  2 \\cdot \\vert x \\vert$. Both the rules $\\multimap$R and $\\multimap$L,  increase by one the overall size of the types in a judgment and of the corresponding term, so the inequalities still hold. Last, the rules for $\\forall$ do not affect the size of both  $\\Gamma^-, B^-$ and  $M^{\\Gamma}_B$.\n\\end{proof}\n\\noindent\nNotice that Lemma \\ref{lem: pi1type bound} does not hold in general whenever $\\mathcal{D}$ contains instances of the inference rule $\\forall$L,   since one can exploit the inference rule $\\forall$L to \\enquote{compress} the size of a type. \\\\\nNow, consider a cut-free derivation $\t\\mathcal{D}\\triangleleft \\vdash M:A$, where $A$ is  a ground type. Since negative occurrences of $\\forall$ are not allowed in $A$, $\\mathcal{D}$ contains no application of $\\forall$L and,  by Lemma~\\ref{lem: pi1type bound}, this implies  that $\\vert M \\vert \\leq \\vert A^-\\vert$. This limits the number of variables a generic inhabitant of $A$ has, so that we can safely say that the variables of $ M $ must certainly belong to a fixed set $\\lbrace \\mathrm{x}_1, \\ldots, \\mathrm{x}_{\\vert A^- \\vert} \\rbrace$. The next step is to show that we can encode every normal form as a tuple of booleans, i.e.~as elements in $ \\mathbf{B}^s $ with a sufficiently large $ s $. Actually, we are interested in $\\eta$-long normal forms only, due to the way the linear $\\lambda$-term $\\mathtt{sub}^s_{A}$ acts on inhabitants of $A$ as shown in the previous subsection. So, given a ground type $ A$, we can represent the $\\eta$-long normal forms of type $A $ with tuples of type $ \\mathbf{B}^{\\mathcal{O}(\\vert A^-\\vert \\, \\cdot \\, \\log \\vert A^-\\vert)} $, since each such linear $\\lambda$-term has at most $\\vert A^- \\vert$ symbols, each one encoded using around $\\log \\vert A^- \\vert$ bits. By setting $s=c \\cdot (\\vert A^-\\vert \\, \\cdot \\,  \\log \\vert A^- \\vert)$  for some $c>0$ large enough, there must exist a coding function $\\lceil \\_ \\rceil: \\Lambda_s \\longrightarrow \\mathbf{B}^s$, where  $\\Lambda_{s} $ is the set of all normal linear $\\lambda$-terms having size bounded by $s$. The role of the $\\lambda$-term $\\mathtt{enc}^s_{A}$ is to internalize the coding function   $\\lceil \\_ \\rceil$ in $\\mathsf{IMLL}_2$ as far as the $\\eta$-long normal forms of a fixed type $A$ are concerned. \n\nThe coming Lemma~\\ref{lem: existence abs app}   relies on an \niterated selection mechanism, i.e.~a nested  \\texttt{if}-\\texttt{then}-\\texttt{else} construction. In order to define selection, we first we need to extend the projection in~\\eqref{eqn: boolean projection} (Section~\\ref{sec: duplication and erasure in a linear setting}). \n\n\n\\begin{defn}[Generalized projection]\\label{defn: generalized projection}\nLet  $A$  be a ground type. For all $k \\geq 0$ and $\\vec{m}=m_1, \\ldots, m_ k \\geq 0$,  the  linear $\\lambda$-term $\\pi^{\\vec{m}}_1$  is defined below:\n\\begin{equation*}\n\\pi_1^{\\vec{m} } \\triangleq \\begin{cases}    \n\\lambda z. \\mathtt{let\\ }z \\mathtt{\\ be\\ }x,y \\mathtt{\\ in\\ \n}(\\mathtt{let\\ } \\mathtt{E}_{A}\\,y \\mathtt{\\ be\\ }I \\mathtt{\\ in\\ }x) &\\text{if }k=0\\\\ \n \\lambda z. \\mathtt{let\\ }z \\mathtt{\\ be\\ }x,y \\mathtt{\\ in\\ }(\\mathtt{let\\ } \\mathtt{E}_{A}\\,(y\\, \\mathtt{tt}^{m_1}  \\ldots \\mathtt{tt}^{m_k}) \\mathtt{\\ be\\ }I \\mathtt{\\ in\\ }x) &\\text{if }k>0\n \\end{cases} \n\\end{equation*}\nwith type $B \\otimes B \\multimap B$, where $B \\triangleq \\mathbf{B}^{m_1} \\multimap \\ldots \\multimap  \\mathbf{B}^{m_k} \\multimap A$. When $k=0$ we simply  write $\\pi_1$ in place of $\\pi_1^{\\vec{m}}$, whose type is $A \\otimes A \\multimap A$.\n\\end{defn}\n\n\\begin{defn}[Generalized selection] \\label{defn: generalized selection} Let  $A$ be a ground type and let $M_{\\mathtt{tt}^n}$, $ M_{\\langle \\mathtt{tt}^{n-1},\\mathtt{ff}\\rangle}$, \\ldots, $M_{\\langle \\mathtt{tt},\\mathtt{ff}^{n-1}\\rangle}$, $M_{\\mathtt{ff}^n}$ be (not necessarily distinct) normal inhabitants of $\\mathbf{B}^{m_1} \\multimap \\ldots \\multimap   \\mathbf{B}^{m_k} \\multimap A$, for some   $n \\geq 1$,  $k \\geq 0 $, and  $\\vec{m}=m_1, \\ldots, m_k \\geq 0$.  We define the linear $\\lambda$-term:\n\\begin{equation}\\label{eqn: generalized selection}\n\\mathtt{if \\ }x \\mathtt{\\ then \\ }[M_{\\mathtt{tt}^n},  M_{\\langle \\mathtt{tt}^{n-1}, \\mathtt{ff}\\rangle}, \\ldots, M_{\\langle \\mathtt{tt},\\mathtt{ff}^{n-1}\\rangle},M_{\\mathtt{ff}^n} ]^{\\vec{m} }\n\\end{equation}\nwith type $ \\mathbf{B}^{m_1} \\multimap \\ldots \\multimap \\mathbf{B}^{m_k} \\multimap A$ and context $x:\\mathbf{B}^n$ by induction on $n$:\n\\begin{itemize}\n\\item $n=1$: $ \\mathtt{if \\ }x \\mathtt{\\ then \\ }[M_{\\mathtt{tt}},M_{\\mathtt{ff}} ]^{\\vec{m} } \\triangleq  \\pi_1^{\\vec{m}} (x\\, M_{\\mathtt{tt}} \\, M_{\\mathtt{ff}})$.\n\\item $n>1$:  $ \\mathtt{if \\ }x \\mathtt{\\ then \\ }[M_{\\mathtt{tt}^n},  M_{\\langle \\mathtt{tt}^{n-1}, \\mathtt{ff}\\rangle}, \\ldots, M_{\\langle \\mathtt{tt},\\mathtt{ff}^{n-1}\\rangle},M_{\\mathtt{ff}^n} ]^{\\vec{m}} \\triangleq $\n\\begin{align*}\n&\\mathtt{ let \\ }x \\mathtt{ \\ be \\ } x_1, x_2  \\mathtt{\\ in\\ } (\\mathtt{if\\ }x_2 \\mathtt{\\ then \\ } \\\\\n& \\big[ (\\lambda y_1. \\mathtt{if \\ }y_1 \\mathtt{\\ then \\ } [P_{\\mathtt{tt}^{n-1}},  P_{\\langle \\mathtt{tt}^{n-2}, \\mathtt{ff}\\rangle}, \\ldots, P_{\\langle \\mathtt{tt},\\mathtt{ff}^{n-2}\\rangle},P_{\\mathtt{ff}^{n-1}}]^{\\vec{m}}),\\\\\n&\\phantom{\\big[}(\\lambda y_2. \\mathtt{if \\ }y_2 \\mathtt{\\ then \\ } [Q_{\\mathtt{tt}^{n-1}},  Q_{\\langle \\mathtt{tt}^{n-2}, \\mathtt{ff}\\rangle}, \\ldots, Q_{\\langle \\mathtt{tt},\\mathtt{ff}^{n-2 }\\rangle},Q_{\\mathtt{ff}^{n-1}}]^{\\vec{m} })  \\big]^{n-1, \\vec{m}}) \\, x_1 \n\\end{align*}\nwhere, $\\pi_1^{\\vec{m}}$ is as in Definition~\\ref{defn: generalized projection} and, for every $n$-tuple  $\\langle \\mathtt{b}_1, \\ldots, \\mathtt{b}_n \\rangle$ of booleans, \n$ P_{\\langle \\mathtt{b}_1, \\ldots, \\mathtt{b}_n \\rangle}\\triangleq M_{\\langle \\langle \\mathtt{b}_1, \\ldots, \\mathtt{b}_n \\rangle, \\mathtt{tt} \\rangle}$, $\nQ_{\\langle \\mathtt{b}_1, \\ldots, \\mathtt{b}_n \\rangle}\\triangleq M_{\\langle  \\langle \\mathtt{b}_1, \\ldots, \\mathtt{b}_n \\rangle,\\mathtt{ff} \\rangle}$.\n\\end{itemize}\nwhen $k=0$ we feel free of ruling out the apex $\\vec{m}$ in~\\eqref{eqn: generalized selection}.\n\\end{defn}\n\\begin{lem} Let  $A$ be a ground type and let $M_{\\mathtt{tt}^n}$, $ M_{\\langle \\mathtt{tt}^{n-1},\\mathtt{ff}\\rangle}$, \\ldots, $M_{\\langle \\mathtt{tt},\\mathtt{ff}^{n-1}\\rangle}$, $M_{\\mathtt{ff}^n}$ be (not necessarily distinct) normal inhabitants of $\\mathbf{B}^{m_1} \\multimap \\ldots \\multimap   \\mathbf{B}^{m_k} \\multimap A$, for some   $n \\geq 1$,  $k \\geq 0 $, and  $\\vec{m}=m_1, \\ldots, m_k \\geq 0$.  \nFor every $n$-tuple of booleans  $\\langle \\mathtt{b}_1, \\ldots, \\mathtt{b}_n \\rangle$ it holds that:\n\\begin{equation*}\n\\mathtt{if \\ } \\langle \\mathtt{b}_1, \\ldots, \\mathtt{b}_n \\rangle \\mathtt{\\ then \\ }(M_{\\mathtt{tt}^n},  M_{\\langle \\mathtt{tt}^{n-1}, \\mathtt{ff}\\rangle}, \\ldots, M_{\\langle \\mathtt{tt},\\mathtt{ff}^{n-1}\\rangle},M_{\\mathtt{ff}^n} ) \\rightarrow_\\beta^* M_{\\langle \\mathtt{b}_1, \\ldots, \\mathtt{b}_n \\rangle}\\enspace .\n\\end{equation*}\n\\end{lem}\n\\begin{proof}\nStraightforward.\n\\end{proof}\nNotice that, if $n=1$ and $k=0$  in Definition~\\eqref{defn: generalized selection}, we get  the usual  \\texttt{if}-\\texttt{then}-\\texttt{else} construction defined in~\\cite{gaboardi2009light} as:\n\\begin{equation}\\label{eqn: pi1m}\n\\mathtt{if}\\ x \\ \\mathtt{then}\\ M_1\\ \\mathtt{else}\\ M_2\n\\triangleq  \\pi_1(x\\,M_1\\,M_2)\n\\end{equation}\nwith type $A$ and context $x: \\mathbf{B}$, where $\\pi_1: A \\otimes A \\multimap A$ is as in Definition~\\ref{defn: generalized projection}. Clearly, if  $\\mathtt{b}_1\\triangleq\\mathtt{tt}$ and $\\mathtt{b}_2\\triangleq \\mathtt{ff}$, then $\\mathtt{if}\\ \\mathtt{b}_i\\ \\mathtt{then}\\ M_1\\ \\mathtt{else}\\ M_2 \n\\rightarrow_\\beta^* M_i$ for $i=1,2$.\n\nBefore defining the linear $\\lambda$-term $\\mathtt{enc}_A^s$ we need to encode the $ \\lambda $-abstractions and the applications in $\\mathsf{IMLL}_2$.\n\n\\begin{lem} \\label{lem: existence abs app} Let $s>0$. The following statements hold:\n\\begin{enumerate}[(1)]\n\\item \\label{eqn: abs}  A linear $\\lambda$-term $\\mathtt{abs}^s: \\mathbf{B}^s \\multimap \\mathbf{B}^{s}\\multimap \\mathbf{B}^s$ exists such that $\\mathtt{abs}\\lceil x \\rceil \\lceil M \\rceil \\rightarrow^*_\\beta \\lceil \\lambda x. M \\rceil$, if $\\vert \\lambda x .M \\vert \\leq s $ and $ x \\in \\lbrace \\mathrm{x}_1, \\ldots, \\mathrm{x}_{s} \\rbrace$.\n\\item\\label{eqn: app}  A linear $\\lambda$-term $\\mathtt{app}^s: \\mathbf{B}^s \\multimap \\mathbf{B}^{s}\\multimap \\mathbf{B}^s$ exists such that $\\mathtt{app} \\lceil M \\rceil \\lceil N \\rceil \\rightarrow^* _\\beta \\lceil MN \n\\rceil $, if $\\vert MN  \\vert \\leq s$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nWe sketch the proof of Point~\\ref{eqn: abs} only, since Point~\\ref{eqn: app} is similar. Recall the notation in Definition~\\ref{defn: generalized selection}. We let boolean values range over $b_1, b_2, \\ldots$ and with $\\mathtt{b}$ we denote the corresponding encoding of the boolean value $b$ in $\\mathsf{IMLL}_2$.   The linear $\\lambda$-term $\\mathtt{abs}$ is  of the form:\n\\begin{equation*}\n\\lambda x. \\lambda y. (\\mathtt{if \\ }x \\mathtt{\\ then\\ } [ P_{\\mathtt{tt}^s}, P_{\\langle \\mathtt{tt}^{s-1}, \\mathtt{ff}\\rangle}, \\ldots, P_{\\langle \\mathtt{ff}, \\mathtt{tt}^{s-1}\\rangle},P_{\\mathtt{ff}^s}]^s)\\, y\n\\end{equation*}\nwhere, for all $s$-tuple of booleans $T=\\langle \\mathtt{b}_1, \\ldots, \\mathtt{b}_s \\rangle$, the linear $\\lambda$-term  $P_{T}$ with type $\\mathbf{B}^s \\multimap \\mathbf{B}^s$ is as follows:\n\\begin{equation*}\n\\lambda y.\\mathtt{if \\ }y \\mathtt{\\ then\\ } [Q^T_{\\mathtt{tt}^s}, Q^T_{\\langle \\mathtt{tt}^{s-1}, \\mathtt{ff}\\rangle}, \\ldots, Q^T_{\\langle \\mathtt{ff}, \\mathtt{tt}^{s-1}\\rangle},Q^T_{\\mathtt{ff}^s}]\\enspace .\n\\end{equation*}\nFor all $T=\\langle \\mathtt{b}_1, \\ldots, \\mathtt{b}_s \\rangle$ and for all $T'=\\langle \\mathtt{b}'_1, \\ldots, \\mathtt{b}'_s \\rangle$ we define:\n\\begin{equation*}\n Q^T_{T'}= \\begin{cases} \\lceil \\lambda x. M \\rceil  &\\text{if } \\langle \\mathtt{b}_1, \\ldots, \\mathtt{b}_s \\rangle= \\lceil x\\rceil, \\  \\langle \\mathtt{b}'_1, \\ldots, \\mathtt{b}'_s \\rangle= \\lceil M \\rceil, \\\\ &  \\text{and }\\vert \\lambda x. M \\vert \\leq  s\\\\ \\\\\n\\langle \\mathtt{tt},\\overset{s}{ \\ldots}, \\mathtt{tt} \\rangle &\\text{otherwise}.\n\\end{cases}\n\\end{equation*}\n\\end{proof}\nThe $\\lambda$-term $\\mathtt{enc}^s_{A}$, given a value  $V_A$  in $\\eta$-long normal form and of type $A$, combines the $\\lambda$-terms $\\mathtt{abs}^s$ and $\\mathtt{app}^s$ to construct its encoding.\n\\begin{defn}[The linear $\\lambda$-term $\\mathtt{enc}^s_{A}$] \\label{defn: enc} Let $s>0$. We define the linear $\\lambda$-terms $\\mathtt{enc}^s_{A}: A^-[\\mathbf{B}^s]\\multimap \\mathbf{B}^s$, where $A$ is a  $\\Pi_1$-type, and $\\overline{\\mathtt{enc}}^s_A: \\mathbf{B}^s \\multimap A^-[\\mathbf{B}^s]$, where $A$ is a $\\Sigma_1$-type, by simultaneous induction on the size of $A$:\n    \\begin{align*}\n& \\mathtt{enc}^s_{\\alpha}  \\triangleq  \\lambda z. z &&     \\mathtt{enc}^s_{B \\multimap C}\n       \\triangleq \\lambda z. \\mathtt{abs}^s \\lceil x \\rceil\\,   (\\mathtt{enc}^s_{C}\\,   (z\\, (\\overline{\\mathtt{enc}}^s_{B}\\,   \\lceil x \\rceil))) \\\\\n       &    \\overline{\\mathtt{enc}}^s_{\\alpha} \\triangleq   \\lambda z. z\n       &&  \\overline{\\mathtt{enc}}^s_{B \\multimap C}\\triangleq \\lambda z. \\lambda x.   \\overline{\\mathtt{enc}}^s_{C}\\, (\\mathtt{app}^s    z\\, (\\mathtt{enc}^s_{B}\\,x))\n    \\end{align*}\n    \\noindent\n    with $x$  chosen fresh in $\\lbrace \\mathrm{x}_1, \\ldots, \\mathrm{x}_{s} \\rbrace$.\\qed\n    \\end{defn}\n    The following will be used to compact the proof of some of the coming lemmas.\n    \\begin{defn} \nLet $s>0$, and let $A$ be a $\\Pi_1$-type and $\\Gamma=x_1:A_1, \\ldots, x_n:A_n$ be a context of $\\Sigma_1$-types. If  $M$ is an inhabitant of type $A^-[\\mathbf{B}^s]$   with context $\\Gamma^-[\\mathbf{B}^s]$ then  $M[\\Gamma]$ denotes the substitution:\n\\begin{equation*}\n  M[\\overline{\\mathtt{enc}}^s_{A_1}\\,x'_1 \/x_1 ,\\ldots, \\overline{\\mathtt{enc}}^s_{A_n}\\, x'_n\/x_n] \n\\end{equation*}    \nfor some $x'_1, \\ldots, x'_n$. \\qed\n\\end{defn} \nTo prove that $\\mathtt{enc}^s_{A}$ is able to encode a value $V_A$ of type $A$ we need an intermediate step. We first prove that $\\mathtt{enc}^s_{A}$   substitutes every $\\lambda$-abstraction in $V_{A}$ with an instance of  $\\mathtt{abs}^s$, and every application with an instance of  $\\mathtt{app}^s$, thus producing a \\enquote{precode}. Then we  prove that, when every free variable in it has been substituted with its respective encoding,  the precode reduces to $\\lceil V_{A}\\rceil$.\n\\begin{defn} Let $s>0$. If $M$ is a linear $\\lambda$-term in normal form such that $\\vert M \\vert \\leq s$, we define $M^s$ by induction on $\\vert M \\vert$:\n\\begin{enumerate}\n\\item $M=x$ if and only if $M^s=x$,\n\\item $M= \\lambda x. N$ if and only if $M^s= \\mathtt{abs}^s \\, \\lceil x' \\rceil \\, N^s[\\lceil x' \\rceil\/x] $,\n\\item $M=PQ$ if and only if $M^s= \\mathtt{app}^s \\, P^s \\, Q^s$,\n\\end{enumerate} \nwhere $x'$ is fresh, chosen in $\\lbrace \\mathrm{x}_1, \\ldots, \\mathrm{x}_{s} \\rbrace$. \\qed\n\\end{defn}\n\\begin{lem} \\label{lem: substitution lemma for s} Let $s>0$. If $M$ and $N$ are linear $\\lambda$-terms, then $M^s[N^s\/x]=(M[N\/x])^s$.\n\\end{lem}\n\\begin{proof}\nBy induction on $\\vert M \\vert$. If $M=x$ then $x^s[N^s\/x]= x[N^s\/x]= N^s= (x[N\/x])^s$. If $M=PQ$ then either $x$ occurs in $P$ or it occurs in $Q$, and let us consider the case  $x \\in FV(P)$, the other case being similar: by using the induction hypothesis we have $(PQ)^s [N^s\/x]= \\mathtt{app}^s \\, P^s[N^s\/x] \\, Q^s= \\mathtt{app}^s (P[N\/x])^s \\, Q^s= (P[N\/x]Q)^s=((PQ)[N\/x])^s$. If $M= \\lambda y. P$ then we have that $(\\lambda y. P)^s [N^s \/x]$ $= \\mathtt{abs}^s \\, \\lceil y' \\rceil \\, P^s[N^s \/x][\\lceil y' \\rceil\/y]$ $= \\mathtt{abs}^s \\, \\lceil y' \\rceil   \\, (P[N \/x])^s [\\lceil y' \\rceil \/y] $ $= (\\lambda y. P[N\/x])^s$ $= ((\\lambda y. P)[N\/x])^s$.\n\\end{proof}\n\\begin{lem}\\label{lem: precoding lemma} Let $s>0$. If $M$ is a linear $\\lambda$-term in normal form such that $\\vert M \\vert \\leq s$ with free variables $x_1, \\ldots, x_n$ then \n\\[M^s [ \\vec{\\lceil x' \\rceil}]\\rightarrow_\\beta^* \\lceil M[x'_1\/x_1, \\ldots, x'_n\/x_n] \\rceil\\]\nwhere $ \\vec{\\lceil x' \\rceil}=[\\lceil x'_1 \\rceil\/x_1, \\ldots, \\lceil x'_n \\rceil\/x_n]$ and  $x'_1, \\ldots, x'_n$  are distinct and fresh in $\\lbrace \\mathrm{x}_1, \\ldots, \\mathrm{x}_{s} \\rbrace$.\n\\end{lem}\n\\begin{proof}\nBy induction on $\\vert M \\vert$. If $M=x$ then $\\exists i \\leq n$ $x_i=x$, so that  \n$x^s [\\lceil x'_i \\rceil\/x]= x [\\lceil x'_i \\rceil\/x]=\\lceil x'_i\\rceil= \\lceil  x[x'_i\/x] \\rceil$. If $M= \\lambda y. N$ then, using the induction hypothesis, we have: \n\\allowdisplaybreaks\n\\begin{align*}\n(\\lambda y. N)^s  [ \\vec{\\lceil x' \\rceil}]&= (\\mathtt{abs}^s \\, \\lceil y' \\rceil \\, N^s [\\lceil y' \\rceil\/y]) [ \\vec{\\lceil x' \\rceil} ]\\\\\n&=\\mathtt{abs}^s \\, \\lceil y' \\rceil \\, N^s[ \\vec{\\lceil x' \\rceil}, \\lceil y' \\rceil \/y]\\\\\n&\\rightarrow_\\beta^* \\mathtt{abs}^s \\, \\lceil y' \\rceil \\, \\lceil N[x_1'\/x_1, \\ldots, x'_n\/x_n, y'\/y] \\rceil\\\\\n&\\rightarrow_\\beta^* \\lceil \\lambda y'. N[x_1'\/x_1, \\ldots, x'_n\/x_n, y'\/y] \\rceil&& \\text{Lemma~\\ref{lem: existence abs app}}\\\\\n&=_{\\alpha}\\lceil  (\\lambda y. N)[x_1'\/x_1, \\ldots, x'_n\/x_n] \\rceil.\n\\end{align*}\nIf $M=PQ$ then let $y_1, \\ldots, y_m$ (resp. $z_1, \\ldots, z_k$) be the free variables of $P$ (resp. $Q$), and let $\\vec{x'}=y'_1, \\ldots, y'_m, z'_1, \\ldots, z'_k$. Then we have: \n\\allowdisplaybreaks\n\\begin{align*}\n&(PQ)^s [ \\vec{\\lceil x' \\rceil}]=\\\\\n&= \\mathtt{app}^s \\, P^s  [ \\vec{\\lceil y' \\rceil}] \\, Q^s [ \\vec{\\lceil z' \\rceil}] \\\\\n&\\rightarrow_\\beta^* \\mathtt{app}^s \\,  \\lceil P[y_1'\/y_1, \\ldots, y'_m\/y_m] \\rceil \\, \\lceil Q[z_1'\/z_1, \\ldots, z'_k\/z_k] \\rceil \\\\\n&\\rightarrow_\\beta^*  \\lceil P[y_1'\/y_1, \\ldots, y'_m\/y_m] Q[z_1'\/z_1, \\ldots, z'_k\/z_k]  \\rceil && \\text{Lemma~\\ref{lem: existence abs app}}\\\\\n&= \\lceil (PQ)[x_1'\/x_1, \\ldots, x'_n\/x_n] \\rceil.  \n\\end{align*}\n\\end{proof}\nIt is easy to check that  if $M$ is an inhabitant of a $\\Pi_1$-type $A$ with context $\\Gamma=x_1:A_1, \\ldots, x_n:A_n$ of $\\Sigma_1$-types, then $M$ has also type $A^-[\\mathbf{B}^s]$   with context $\\Gamma^-[\\mathbf{B}^s]$.\n\\begin{lem}\\label{lem: enc} Let $M$ be a $\\eta$-long normal form of type $A$  with context $\\Gamma=x_1:A_1, \\ldots, x_n:A_n$, where $A$ is a $\\Pi_1$-type and $\\Gamma$ is a context of $\\Sigma_1$-types, and let $\\sum_{i=1}^m\\vert A_i^- \\vert+ \\vert A^-  \\vert=k$ and $s=c \\cdot (k\\cdot \\log k)$, for some $c$ large enough. Then:\n\\begin{equation*}\n (\\mathtt{enc}^s_A\\, M[\\Gamma]) [\\vec{\\lceil x' \\rceil}] \\rightarrow_\\beta^*\\lceil M[x'_1\/x_1, \\ldots, x'_n\/x_n]  \\rceil \\enspace , \n\\end{equation*}\n where $ \\vec{\\lceil x' \\rceil}=[\\lceil x'_1 \\rceil\/x_1, \\ldots, \\lceil x'_n \\rceil\/x_n]$, with $x'_1, \\ldots, x'_n$  distinct and chosen fresh in $\\lbrace \\mathrm{x}_1, \\ldots, \\mathrm{x}_{s} \\rbrace$.\n\\end{lem}\n\\begin{proof} By Lemma~\\ref{lem: precoding lemma} it suffices to prove by induction on $\\vert M \\vert $ that the reduction  $\\mathtt{enc}^s_A\\, M[\\Gamma]  \\rightarrow_\\beta^* M^s$ holds.  If $M=x$ then $A= \\alpha$ and $\\Gamma= x:\\alpha$, because   $M$ is in $\\eta$-long normal form,  so that we have $\\mathtt{enc}_{\\alpha}^s \\, x[x: \\alpha]= \\mathtt{enc}^s_{\\alpha}(\\overline{\\mathtt{enc}}_\\alpha^s \\, x) \\rightarrow_\\beta^* x= x^s $.  If $M= \\lambda y.N$ then $A= B \\multimap C$, so that:\n\\allowdisplaybreaks\n\\begin{align*}\n&\\mathtt{enc}_{B \\multimap C}^s ((\\lambda y.N)[\\Gamma]) \\\\\n&\\rightarrow_\\beta \\mathtt{abs}^s\\, \\lceil x' \\rceil (\\mathtt{enc}^s_C((\\lambda y. N[\\Gamma])(\\overline{\\mathtt{enc}}^s_B \\, \\lceil y' \\rceil)))  &&\\text{Definition~\\ref{defn: enc}}\\\\\n&\\rightarrow_\\beta \\mathtt{abs}^s\\, \\lceil y' \\rceil (\\mathtt{enc}^s_C(N[\\Gamma][\\overline{\\mathtt{enc}}^s_B \\, \\lceil y' \\rceil\/y])) \\\\\n&= \\mathtt{abs}^s\\, \\lceil y' \\rceil  (\\mathtt{enc}^s_C(N[\\Gamma][\\overline{\\mathtt{enc}}^s_B \\, x\/x])) [ \\lceil y' \\rceil \/y]\\\\\n&= \\mathtt{abs}^s\\, \\lceil y' \\rceil (\\mathtt{enc}^s_C(N[\\Gamma,  y:B]))[ \\lceil y'\\rceil\/y]&&\\text{Definition~\\ref{defn: enc}}\\\\\n&\\rightarrow_\\beta^* \\mathtt{abs}^s\\, \\lceil y ' \\rceil (N^s [\\lceil y'\\rceil \/y]) && \\text{induction hyp.}\\\\\n&=  (\\lambda y. N)^s .\n\\end{align*}\nLast, suppose $M=P[yN\/x]$, and let $\\Sigma$, $\\Delta$ be contexts such that  $\\Gamma=\\Sigma, \\Delta, y: B \\multimap C$, $\\operatorname{dom}(\\Sigma)=FV(P)$, and $\\operatorname{dom}(\\Delta)=FV(N)$.   Then we have:\n\\allowdisplaybreaks\n\\begin{align*}\n&\\mathtt{enc}^s_{A} (P[yN\/x])[\\Gamma]\\\\\n&= \\mathtt{enc}^s_{A} (P[\\Sigma][(yN)[\\Delta, y:B \\multimap C]   \/x]) \\\\\n&= \\mathtt{enc}^s_{A} (P[\\Sigma][(\\overline{\\mathtt{enc}}^s_{B \\multimap C}\\,  y  ) N[\\Delta]   \/x])  \\\\\n&\\rightarrow_\\beta^* \\mathtt{enc}^s_{A} (P[\\Sigma][ \\overline{\\mathtt{enc}}^s_{C}(\\mathtt{app}^s \\,   y  (\\mathtt{enc}^s_B \\, N[\\Delta])) \/x]) &&\\text{Definition~\\ref{defn: enc}}\\\\\n& \\rightarrow_\\beta^* \\mathtt{enc}^s_{A} (P[\\Sigma][ \\overline{\\mathtt{enc}}^s_{C}(\\mathtt{app}^s \\,    y \\, N^s) \/x])&&\\text{induction hyp.}\\\\\n&= \\mathtt{enc}^s_{A} (P[\\Sigma][ \\overline{\\mathtt{enc}}^s_{C}(yN)^s  \/x])\\\\\n&=\\mathtt{enc}^s_{A} (P[\\Sigma, x: C])  [ (yN)^s   \/x]&&\\text{Definition~\\ref{defn: enc}}\\\\\n&\\rightarrow_\\beta^* P^s  [ (yN)^s    \/x] &&\\text{induction hyp.}\\\\\n&=(P[yN\/x])^s  && \\text{Lemma~\\ref{lem: substitution lemma for s}}. \n\\end{align*}\n\\end{proof}\n\n \n\n\\subsection{The linear $\\lambda$-term $ \\mathtt{dec}^s_{A}$} \\label{subsection: dec and DBn}\nThe linear $ \\lambda $-term $ \\mathtt{dec}^s_{A} $ is the component of $\\mathtt{D}_{A}$ requiring the type inhabitation. Roughly, it takes in input a tuple of boolean values encoding the $\\eta$-long normal form $V_{A}$ of a ground type $A$, and it produces the pair $\\langle V_{A}, V_{A}\\rangle$. To ensure that $ \\mathtt{dec}^s_{A} $ is defined on all possible inputs, it is built in such a way that it returns a default inhabitant of $A$ whenever the tuple of booleans in input does not encode any $\\lambda$-term. \n\\begin{defn}[The linear $\\lambda$-term $\\mathtt{dec}^s_{A}$]\\label{defn: dec} Let $A$ be a ground type  and let  $U$ be a value of type $A$. If  for some $c$ large enough $s=c \\cdot (\\vert A^- \\vert \\cdot \\log \\vert A^- \\vert )$,  then we  define the linear $\\lambda$-term $\\mathtt{dec}^s_{A}: \\mathbf{B}^s \\multimap A\\otimes A$ as follows:\n\\begin{equation*}\n\\lambda x. \\mathtt{if \\ }x \\mathtt{\\ then\\ } [P_{\\mathtt{tt}^s}, P_{\\langle \\mathtt{tt}^{s-1}, \\mathtt{ff}\\rangle}, \\ldots, P_{\\langle \\mathtt{ff}, \\mathtt{tt}^{s-1}\\rangle},P_{\\mathtt{ff}^s}]\n\\end{equation*}\nwhere, for all $T=\\langle \\mathtt{b}_1, \\ldots, \\mathtt{b}_s \\rangle$ of type $\\mathbf{B}^s$:\n\\begin{equation*}\n P_{T}= \\begin{cases} \\langle V_{A}, V_{A} \\rangle  &\\text{if }\\langle \\mathtt{b}_1, \\ldots, \\mathtt{b}_s \\rangle= \\lceil V_{A} \\rceil \\\\\n\\langle U, U \\rangle &\\text{otherwise}.\n\\end{cases}\n\\end{equation*}\n\\end{defn}\nWe are now able to prove the fundamental result of this section: \n\\begin{thm}[Duplication~\\cite{mairson2003computational}]\nEvery inhabited ground type is duplicable.\n\\end{thm}\n\\begin{proof}\n The duplicator $\\mathtt{D}_{A}$ of a inhabited ground type is defined as follows: we fix $s=c \\cdot (\\vert A^- \\vert  \\cdot \\log \\vert A^- \\vert )$,  we fix a default value  $U$ of $A$ (see Definition~\\ref{defn: dec}), and we set:\n\\begin{equation*}\n\\mathtt{D}_{A}\\triangleq \\mathtt{dec}^s_{A}\\circ \\mathtt{enc}^s_{A}\\circ \\mathtt{sub}^s_{A}\n\\end{equation*}\nwhich has type $A \\multimap A \\otimes A$. By Lemma~\\ref{lem: sub}, Lemma~\\ref{lem: enc}, and Definition~\\ref{defn: dec} the conclusion follows. Moreover, for all values $V$ of type $A$, we have:\n\\begin{equation*}\n\\mathtt{D}_A\\, V \\rightarrow_{\\beta}^* \\langle V_A, V_A\\rangle \\rightarrow_{\\eta}^* \\langle V, V\\rangle \\enspace .\n\\end{equation*}\n\\end{proof}\n\\begin{rem}\\label{rem: duplicator} If $A$ is a ground type inhabited by the value $U$, we shall  write $\\mathtt{D}_{A}^{U}$ to stress that the default inhabitant of $A$ used in  constructing the duplicator $\\mathtt{D}_{A}$ of $A$ is $U$.\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The system $\\mathsf{LEM}$}\\label{sec:  the system shpos, with, oplus  IMLL2 cut elimination and bound}\nTheorems~\\ref{thm: pi1 types are erasable} and~\\ref{thm: pi1 are duplicable}  say that the ground types can be weakened and contracted in $\\mathsf{IMLL}_2$. We here logically internalize those kinds of weakening a contraction in the deductive system $\\mathsf{LEM}$ (Linearly Exponential and Multiplicative). It extends $\\mathsf{IMLL}_2$ \nwith inference rules for the modality \n``$\\shpos$''  that closely recall the exponential rules in Linear Logic.\n\n\\begin{defn}[Types of $\\mathsf{LEM}$]\n\\label{defn:Types for IMLL2shpos}\nLet $\\mathcal{X}$ be a denumerable set of type variables. The following grammar~\\eqref{eqn: grammar mathcalAe} generates the \\textit{exponential types}, while the grammar~\\eqref{eqn: grammar A} generates the \\textit{linear types}:\n\\begin{align}\n\\sigma := &\\ A \\ \\vert \\  \\shpos \\sigma \\label{eqn: grammar mathcalAe}\\\\\nA:= &\\ \\alpha \\ \\vert \\ \\sigma \\multimap A \\ \\vert \\ \\forall \\alpha.A  \\label{eqn: grammar A} \n\\end{align}\nwhere $\\alpha \\in \\mathcal{X}$ and,  in the last clause of the grammar~\\eqref{eqn: grammar mathcalAe}, i.e.~the one introducing $ \\shpos \\sigma $, \\emph{the type $ \\sigma$ must be closed and without negative occurrences of $\\forall$}. The set of all types generated by the grammar~(\\ref{eqn: grammar mathcalAe})   will be denoted $\\Theta_\\shpos$.   A type is \\textit{strictly exponential} if it is of the form $\\shpos \\sigma$. A \\textit{strictly exponential context} is a context containing only  strictly exponential types and, similarly, a  \\textit{linear context} contains only linear types. Finally, $A\\langle B\/\\alpha \\rangle$ is the standard meta-level substitution of $B$, for every occurrence of $ \\alpha$ in $A$. \n\\qed\n\n\\end{defn}\n\\noindent\n\\begin{rem}\nThe modality ``$\\shpos$'' identifies where the ground types  (Definition~\\ref{defn:Pi and Sigma types}) occur in the \ngrammars~(\\ref{eqn: grammar mathcalAe}) and~(\\ref{eqn: grammar A})\n because it applies to closed types that are free from negative occurrences  of $\\forall$. \nSo, the occurrences of $\\shpos \\sigma$ identify where contraction and weakening rules can apply in the derivations of $\\mathsf{LEM}$.\n\\qed\n\\end{rem}\n\\noindent\nAlso, we observe that syntactically replacing the Linear Logic modality \\enquote{$\\oc$} for \n\\enquote{$\\shpos$} in~(\\ref{eqn: grammar mathcalAe}) and~(\\ref{eqn: grammar A}) yields  a subset of Gaboardi\\&Ronchi's \\textit{essential types}~\\cite{gaboardi2009light}, introduced to prove Subject reduction in a variant of Soft Linear Logic. Essential types forbid the occurrences of modalities in the right-hand side of an implication, such as in $A \\multimap \\oc B$.\n\n\n\n\n\n\n$\\mathsf{LEM}$ will be defined as  a type-assignment for the term calculus $\\Lambda_\\shpos$, that is essentially the standard linear $\\lambda$-calculus with  explicit and type-dependent constructs for erasure and  duplication, i.e.~$\\mathtt{discard}_\\sigma$ and $\\mathtt{copy}_\\sigma ^V$, the latter being also decorated with a value $V$. These new constructs are able to copy and discard values only, i.e.~closed and normal linear $\\lambda$-terms. \n\n\\begin{defn}[Terms and reduction of $\\mathsf{LEM} $] \n\\label{defn:lambda-terms and values of IMLLshpos2} \nLet $\\mathcal{V}$ be a denumerable set of variables. The \\textit{terms} of $\\mathsf{LEM}$ are given by the grammar:\n\\begin{align}\nM, N := \\ & x\\ \n     \\vert\\ \\lambda x.M\\ \n     \\vert\\ MN\\ \n     \\vert\\ \\mathtt{discard}_{\\sigma}\\, M \\mathtt{\\ in\\ } N\\ \n     \\vert\\ \\mathtt{copy}^{V}_{\\sigma}\\, M \\mathtt{\\ as\\ }x,y \n            \\mathtt{\\ in\\ } N\n&  \\label{eqn: term assignment shposIMLL2}\n\\end{align} \nwhere $x, y\\in \\mathcal{V}$, $ V $ is a value (Definition~\\ref{defn:Values among linear lambda-terms}, Section~\\ref{sec: background}), and $ \\sigma \\in \\Theta_\\shpos$. The set of all terms of $\\mathsf{LEM}$  will be denoted $\\Lambda_{\\shpos}$.\nThe set of the free variables of a term, and the notion of size are standard for variables, abstractions, and applications. \nThe extension to the new constructors are:\n\\begin{align}\n\\nonumber\nFV(\\mathtt{  discard }_{\\sigma}\\, M\\mathtt{ \\   in \\ }N)= \n& \\ FV(M)\\cup FV(N)\n\\\\\n\\nonumber\nFV(\\mathtt{  copy}^{V}_{\\sigma}\\, M \\mathtt{ \\   as \\ }x, y \\mathtt{ \\   in \\ }N)= & \\ FV(M) \\cup (FV(N) \\setminus \\lbrace x, y \\rbrace)\n\\\\\n\\nonumber\n\\vert \\mathtt{  discard }_{\\sigma}\\, M\\mathtt{ \\   in \\ }N\\vert = & \\ \\vert M\\vert + \\vert N\\vert +1 \n\\\\\n\\label{align: size of copy}\n\\vert \\mathtt{  copy}^{V}_{\\sigma}\\, M \\mathtt{ \\   as \\ }x, y \\mathtt{ \\   in \\ }N\\vert = & \\ \\vert M\\vert  + \\vert N \\vert + \\vert V \\vert + 1 \\enspace .\n\\end{align}\nA term $M$ in~\\eqref{eqn: term assignment shposIMLL2} is \\textit{linear} \nif all of its free variables occur once in it and every proper sub-term \nof $M$ with form $ \\lambda x.N $  (resp.~$ \\mathtt{copy}^{V}_{\\sigma}\\, M' \\mathtt{\\ as\\ }y,z \\mathtt{\\ in\\ } N $) is such that $ x$ (resp.~$y,z$)  must occur in $ N $ and $ N $ is linear. \\textit{Henceforth, we use linear terms only.}\n \nThe notions of meta-level substitution and context are as usual.\n\n The \\textit{one-step reduction relation} $\\rightarrow$ is a binary relation on terms. It is defined by the reduction rules in Figure~\\ref{fig: term reduction rules} and by the commuting conversions in Figure~\\ref{fig: term commuting conversions}. It applies in any context. Its reflexive and transitive closure is denoted $\\rightarrow^*$. A term is said a (or is in) \\textit{normal form} if no reduction step applies to it. \n\\qed\n\\end{defn}\n\\begin{figure}[t]\n\\centering\n\\begin{equation}\\nonumber\n\\begin{aligned}\n\\scalebox{0.7}{$  (\\lambda x. M)N  $}&\\scalebox{0.7}{$ \\  \\rightarrow  \\ M[N\/x] $}\\\\\n\\scalebox{0.7}{$  \\mathtt{discard}_{\\sigma}\\, V \\mathtt{\\ in\\ }M $} & \\scalebox{0.7}{$ \\  \\rightarrow \\ M $}\\\\\n\\scalebox{0.7}{$  \\mathtt{copy}^{U}_{\\sigma}\\, V \\mathtt{\\ as\\ } y, z \\mathtt{\\ in\\ }M $} & \\scalebox{0.7}{$ \\  \n\t\\rightarrow  \\ M[V\/y, V\/z] $}\n\\end{aligned}\n\\end{equation}\n\\caption{Reduction on terms.}\n\\label{fig: term reduction rules}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\begin{equation}\\nonumber\n\t\\begin{aligned}\n\t\\scalebox{0.7}{$(\\mathtt{discard}_{\\sigma}\\, M \\mathtt{\\ in\\ } N)P $}&\\scalebox{0.7}{$ \\ \\rightarrow\\ \\mathtt{discard}_{\\sigma}\\, M  \\mathtt{\\ in\\ }(NP) $}\\\\\n    \\scalebox{0.7}{$\\mathtt{discard}_{\\sigma}\\, (\\mathtt{discard}_{\\tau}\\, M \\mathtt{\\ in\\ } N) \\mathtt{\\ in \\ }P $}&\\scalebox{0.7}{$\\  \\rightarrow \\  \\mathtt{discard}_{\\tau}\\, M  \\mathtt{\\ in\\ }(\\mathtt{discard}_{\\sigma}\\, N \\mathtt{\\ in\\ }P)$}\\\\\n\t\\scalebox{0.7}{$\\mathtt{copy}^{V}_{\\sigma}\\, (\\mathtt{discard}_{\\tau}\\, M \\mathtt{\\ in\\ } N)\\mathtt{\\ as\\ }y,z \\mathtt{\\ in\\ }P $}& \\scalebox{0.7}{$\\ \\rightarrow \\  \\mathtt{discard}_{\\tau}\\, M  \\mathtt{\\ in\\ }(\\mathtt{copy}^{V}_{\\sigma}\\, N \\mathtt{\\ as\\ }y,z \\mathtt{\\ in\\ }P)$}\n\t\\\\\n\t\\scalebox{0.7}{$(\\mathtt{copy}^{V}_{\\sigma}\\, M \\mathtt{\\ as\\ }y,z \\mathtt{\\ in\\ }N)P $}&\\scalebox{0.7}{$\\  \\rightarrow \\  \\mathtt{copy}^{V}_{\\sigma}\\, M \\mathtt{\\ as\\ }y,z  \\mathtt{\\ in\\ }(NP) $}\\\\\n\t\\scalebox{0.7}{$\\mathtt{discard}_{\\sigma}\\, (\\mathtt{copy}^{V}_{\\tau}\\, M \\mathtt{\\ as\\ }y,z \\mathtt{\\ in\\ }N) \\mathtt{\\ in \\ }P $}&\\scalebox{0.7}{$\\ \\rightarrow  \\ \\mathtt{copy}^{V}_{\\tau}\\, M \\mathtt{\\ as\\ }y,z  \\mathtt{\\ in\\ }(\\mathtt{discard}_{\\sigma}\\, N \\mathtt{\\ in\\ }P)$}\\\\\n\t\\scalebox{0.7}{$\\mathtt{copy}^{U}_{\\sigma}\\, (\\mathtt{copy}^{V}_{\\tau}\\, M \\mathtt{\\ as\\ }y,z \\mathtt{\\ in\\ }N)\\mathtt{\\ as\\ }y',z' \\mathtt{\\ in\\ }P $}& \\scalebox{0.7}{$\\ \\rightarrow \\  \\mathtt{copy}^{V}_{\\tau}\\, M \\mathtt{\\ as\\ } y,z  \\mathtt{\\ in\\ }(\\mathtt{copy}^{U}_{\\sigma}\\, N \\mathtt{\\ as\\ }y',z' \\mathtt{\\ in\\ }P)$}\n\t\\end{aligned}\n\t\\end{equation}\n\t\\caption{Commuting conversions on terms.}\n\t\\label{fig: term commuting conversions}\n\\end{figure}\n\\noindent\n\n\\begin{figure}[htbp]\n\\begin{mathpar}\n\\scalebox{0.8}{$\n\\inferrule*[Right= $ax$]{\\\\}{x: A \\vdash x:A}$} \\and\n\\scalebox{0.8}{$\\inferrule*[Right= $cut$]{\\Gamma \\vdash N: \\sigma \\\\ \\Delta, x: \\sigma \\vdash M:\\tau}{\\Gamma, \\Delta \\vdash M[N\/x]: \\tau}$} \\\\\n\\scalebox{0.8}{$\\inferrule*[Right= $\\multimap$R]{\\Gamma, x:\\sigma \\vdash M:B}{\\Gamma \\vdash \\lambda x. M : \\sigma \\multimap B }$} \\and\n\\scalebox{0.8}{$\\inferrule*[Right= $\\multimap$L]{\\Gamma \\vdash N: \\sigma \\\\ \\Delta, x:B  \\vdash M: \\tau}{\\Gamma, \\Delta, y: \\sigma \\multimap B \\vdash M[yN\/x]: \\tau}$}\\\\\n\\scalebox{0.8}{$\\inferrule*[Right=$\\forall$R]{\\Gamma \\vdash M: A \\langle \\gamma \/ \\alpha \n\\rangle\\\\ \\gamma \\not \\in FV(\\operatorname{rng}(\\Gamma))}{\\Gamma \\vdash M: \\forall \\alpha.A}$} \\and\n\\scalebox{0.8}{$\\inferrule*[Right= $\\forall$L]{ \\Gamma, x: A\\langle B\/ \\alpha \\rangle  \\vdash M:\\tau}{\\Gamma, x: \\forall \\alpha. A \\vdash M:\\tau}$}\\\\\n\\scalebox{0.8}{$\\inferrule*[Right=$p$]{x_1:\\shpos \\sigma_1, \\ldots, x_n:\\shpos \\sigma_n \\vdash M:\\sigma }{x_1: \\shpos \\sigma_1, \\ldots, x_n: \\shpos \\sigma_n \\vdash  M: \\shpos \\sigma }$} \\and\n\\scalebox{0.8}{$\\inferrule*[Right=$d$]{\\Gamma, x: \\sigma \\vdash M:\\tau}{\\Gamma, y: \\shpos  \\sigma  \\vdash M[y\/x]:\\tau}$}\\\\\n\\scalebox{0.8}{$\\inferrule*[Right=$w$]{\\Gamma \\vdash M:\\tau}{\\Gamma, x: \\shpos  \\sigma \\vdash \\mathtt{ discard}_{\\sigma}\\,   x\\mathtt{ \\   in \\ }M:\\tau}$} \\and \n\\scalebox{0.8}{$\\inferrule*[Right=$c$]{\\Gamma, y: \\shpos  \\sigma, z: \\shpos \\sigma \\vdash M:\\tau \\\\ \\vdash V: \\sigma}{\\Gamma, x: \\shpos \\sigma \\vdash \\mathtt{ copy} ^{V}_{\\sigma}\\, x \\mathtt{ \\   as \\ } y,z \\mathtt{ \\   in \\ }M:\\tau}$}\n\\end{mathpar}\n\\caption{The system $\\mathsf{LEM}$.}\n\\label{fig: the system shpos add IMLL2} \n\\end{figure}\n\\noindent\nBoth the type and the term annotations in the constructs  $\\mathtt{discard}_\\sigma$  and $\\mathtt{copy}^V_\\sigma$ will become meaningful once we introduce the type-assignment system. The value $V$ will be an inhabitant of $\\sigma$, a necessary condition in order to faithfully express the mechanism of linear duplication.\n\nThe structure of the types in Definition~\\ref{defn:Types for IMLL2shpos} drives the definition of $ \\mathsf{LEM} $.\n\\begin{defn}[The system $\\mathsf{LEM} $]\n\\label{defn:The type assigment IMLLshpos2}\nIt is the type-assignment system for the term calculus $\\Lambda_\\shpos$ (Definition~\\ref{defn:lambda-terms and values of IMLLshpos2}) in Figure~\\ref{fig: the system shpos add IMLL2}. \nIt extends $\\mathsf{IMLL}_2$ with the rules \\emph{promotion} $p$, \\emph{dereliction} $d$, \\emph{weakening} $w$ and \\emph{contraction} $c$.\nAs usual, $\\multimap$R, $\\forall$R, and $p$ are right rules\nwhile\n$\\multimap$L, $\\forall$L, $d$, $w$, and $c$ are left ones.\n\\qed\n\\end{defn}\n\\noindent\nFirst, we observe that $ax$ cannot introduce exponential types, like\nin the \\emph{essential types} of the type systems in \\cite{gaboardi2009light}. This is the base for proving:\n\\begin{prop}[Exponential context from exponential conclusion]\n\\label{prop: exponential context} \nIf $\\mathcal{D} \\triangleleft \\Gamma \\vdash M: \\shpos \\sigma$ is a derivation in \n$\\mathsf{LEM}$, then $\\Gamma$ is a strictly exponential context.\n\\end{prop}\n\\begin{proof}\n\tBy structural induction on the derivation $\\mathcal{D}$.\n\\end{proof}\n\\noindent\nAlso, we observe that $p$, $d$, $w$, $c$ of $\\mathsf{LEM}$ are reminiscent of the namesake Linear Logic exponential rules, but they only apply to types $ \\shpos\\sigma $ that~\\eqref{eqn: grammar mathcalAe}  (Definition~\\ref{defn:Types for IMLL2shpos}) generates, i.e.~closed types with no negative occurrences of universal quantifiers.  \nThe rule $c$ has one premise more than the contraction rule in Linear Logic. This premise \\enquote{witnesses} that the type $\\sigma$ we want to contract is inhabited by at least one value $V$. This is because  $ c $ expresses the mechanism of linear contraction discussed in the previous section for $\\mathsf{IMLL}_2$.\nAs we shall see in Theorem~\\ref{thm: translations for LAML}, the term $\\mathtt{copy}^V_\\sigma$ that $c$ introduces is a (very) compact notation for duplicators of ground types, \nwhose detailed description is in~\\ref{sec: the d-soundness theorem DICE}. Roughly, the  duplicator of a ground type $A$ is a  linear $\\lambda$-term that, taking a value $U$ of type $A$ as argument,  implements the following three main operations:\n\\begin{enumerate}\n\t\\item \\label{enumerate: duplication step 1}\n\texpand $ U $ to its $ \\eta $-long normal form \n\t$ U_{A} $ \n\twhose dimension is bounded by the dimension of the type $A$. \n\tThis is why using a modality to identify types for which \n\tthis can be done is so important;\n\t\n\t\\item \\label{enumerate: duplication step 2}\n    compile $ U_{A} $ to a linear $ \\lambda $-term\n    $ \\lceil U_{A} \\rceil $ which encodes $ U_{A} $ as \n    a boolean tuple;\n\t\n\t\\item \\label{enumerate: duplication step 3}\n    copy and decode $ \\lceil U_{A} \\rceil $, obtaining the duplication \n    $ \\langle U_{A}, U_{A} \\rangle $ of $ U_{A} $ as final result.  This duplication \n    is by means of the term $\\mathtt{dec}^s_{A}$ in~\\ref{subsection: dec and DBn}.   It nests a series of  \\texttt{if}-\\texttt{then}-\\texttt{else} constructs which is a look-up table, possibly quite big, that stores all the pairs of normal inhabitants of $A$.\n    Each of them represents a possible outcome of the duplication.\n    Given a boolean tuple $ \\lceil U_{A} \\rceil $  in input, the nested \\texttt{if}-\\texttt{then}-\\texttt{else} select the corresponding pair \n    $ \\langle U_{A}, U_{A} \\rangle $, erasing all the remaining \\enquote{candidates}. The inhabitation condition for $A$ stated in Theorem~\\ref{thm: pi1 are duplicable} ensures that the default pair $ \\langle V, V \\rangle $ exists as a sort of \\enquote{exception}. We ``throw'' it each time the boolean tuple that $\\mathtt{dec}^s_{A}$ receives as input does not encode any term.\n\\end{enumerate}\n\\noindent\nPoint~\\ref{enumerate: duplication step 3} is the one implementing Mairson\\&Terui's \\enquote{\\emph{duplication by selection and erasure}} discussed in Section~\\ref{sec: duplication and erasure in a linear setting}. It involves the component of the duplicator which is exponential in the size of $A$.  Therefore, as we shall see in Theorem~\\ref{thm: exponential compression},  the construct $\\mathtt{copy}_\\sigma^V$ exponentially compresses the linear duplication mechanism encoded in a duplicator.\n\nWe conclude this section by commenting about how \\enquote{$ \\shpos $} and \\textsf{LL}'s \\enquote{$\\oc$} differ. Intuitively, the latter allows to duplicate or erase logical structure, or terms, \\emph{at once}, which is the standard way to computationally interpret contraction and weakening of a logical system.\nThe modality \\enquote{$ \\shpos $} identifies duplication and erasure processes with a more \\emph{constructive} nature. The duplication proceeds step-by-step among a whole set of possible choices in order to identify those ones that cannot contain the copies of the term we are interested to duplicate, until it eventually reaches what it searches. Then, it exploits erasure. Erasing means eroding step-by-step a derivation or a term, according to the type that drives its construction.\n\n\n\n\n\\section{Translation of $\\mathsf{LEM}$ into $\\mathsf{IMLL}_2$ and exponential compression}\\label{sec: the expressiveness of the system}\nThe system $\\mathsf{LEM}$ provides a logical status to copying \nand erasing operations that exist in $\\mathsf{IMLL}_2$. In what follows, we \nshow that a translation $(\\_)^\\bullet$ from $\\mathsf{LEM}$ into $\\mathsf{IMLL}_2$ exists \nthat \\enquote{unpacks} both the constructs $\\mathtt{discard}_{\\sigma}$ and  \n$\\mathtt{copy}_{\\sigma}^{V}$ by turning them into, respectively, an  eraser \nand a duplicator of ground types. Then,  we show that the reduction steps in \nFigure~\\ref{fig: term reduction rules} and the commuting conversions in \nFigure~\\ref{fig: term commuting conversions} can be simulated by the \n$\\beta \\eta$-reduction of the linear $\\lambda$-calculus, as long as we \nrestrict to terms of $\\Lambda_\\shpos$ typable in $\\mathsf{LEM}$. \nLast, we discuss the complexity of the translation, and we prove that \nevery term typable in $\\mathsf{LEM}$ is mapped to a linear \n$\\lambda$-term whose size can be is exponential in the one of the original \nterm.\n\n We start with defining the translation from $\\mathsf{LEM}$ to $\\mathsf{IMLL}_2$.\n\\begin{defn} [From $ \\mathsf{LEM} $ to $ \\mathsf{IMLL}_2 $]\n\\label{defn: translation IMLL2 shpos with oplus into IMLL2} \nThe map  \n$(\\_)^\\bullet: \\mathsf{LEM} \\longrightarrow\\mathsf{IMLL}_2$ \ntranslates a derivation \n$\\mathcal{D} \\triangleleft\\Gamma \\vdash_{\\mathsf{LEM}}  M: \\tau $ into a derivation $\\mathcal{D}^\\bullet \\triangleleft \\Gamma^\\bullet \\vdash_{\\mathsf{IMLL}_2} M^\\bullet :\\tau^\\bullet$:\n\\begin{enumerate}\n\\item For all types $\\sigma \\in \\Theta_\\shpos$:\n\\begin{align*}\n\\alpha^\\bullet &\\triangleq \\alpha\\\\\n(\\tau \\multimap A)^\\bullet &\\triangleq \\tau^\\bullet \\multimap A^\\bullet \\\\\n(\\forall \\alpha . A)^\\bullet &\\triangleq \\forall \\alpha. A^\\bullet \\\\\n(\\shpos \\tau)^\\bullet & \\triangleq \\tau^\\bullet \\enspace .\n\\end{align*}\n\\item For all contexts $\\Gamma= x_1: \\sigma_1, \\ldots, x_n: \\sigma_n$,  we set $\\Gamma^\\bullet\\triangleq  x_1: \\sigma_1 ^\\bullet, \\ldots, x_n: \\sigma_n^\\bullet$;\n\\item\n\\label{enum: translation on terms} \nFor all typable terms $ M\\in \\Lambda_\\shpos$:\n\\begin{align*}\nx^\\bullet  &\\triangleq x\n\\\\\n(\\lambda x. P )^\\bullet &\\triangleq \\lambda x. P^\\bullet \n\\\\\n(PQ)^\\bullet &\\triangleq P^\\bullet Q^\\bullet\n\\\\\n(\\mathtt{discard}_{\\sigma}\\, P \\mathtt{\\ in\\ }Q)^\\bullet\n&\\triangleq\n  \\mathtt{let\\ } \\mathtt{E}_{\\sigma^\\bullet}\\, P^\\bullet \\mathtt{\\ be\\ }I \\mathtt{\\ in\\ }Q^\\bullet \n\\\\\n(\\mathtt{copy}^{V}_{\\sigma}\\, P \\mathtt{\\ as\\ }x_1,x_2 \\mathtt{\\ in\\ \n}Q)^\\bullet &\\triangleq \\mathtt{let\\ }\\mathtt{D}^{V^\\bullet} _{\\sigma^\\bullet}\\, P^\\bullet \n\\mathtt{\\ be\\ } x_1,x_2 \\mathtt{\\ in \\ }Q^\\bullet \\enspace ,\n\\end{align*}\nwhere $\\mathtt{E}_{\\sigma^\\bullet}$ and $\\mathtt{D}^{V^\\bullet} _{\\sigma^\\bullet}$ (see Remark~\\ref{rem: duplicator} in~\\ref{sec: the d-soundness theorem DICE}) are the eraser and the duplicator of $\\sigma^\\bullet$ which is both\nground, because $\\sigma$ is closed and with no negative occurrences of $\\forall$, \nand inhabited by $V^\\bullet$. \n\n\\item \nThe definition of $ (\\_)^\\bullet $ extends to any derivation \n$\\mathcal{D}\\triangleleft\\Gamma\\vdash M:\\sigma$ of $\\mathsf{LEM}$\nin the obvious way, following\nthe structure of $ M^{\\bullet} $. \nFigure \\ref{fig: translation inference rules} collects the most interesting cases.\n\\qed\n\\end{enumerate}   \n\\end{defn}\n\n\\begin{rem}\nBoth $\\mathtt{E}_{\\sigma^\\bullet}$ and \n$\\mathtt{D}^{V^\\bullet} _{\\sigma^\\bullet}$ in \npoint~\\ref{enum: translation on terms} of \nDefinition~\\ref{defn: translation IMLL2 shpos with oplus into IMLL2} here above exist by Theorem~\\ref{thm: pi1 types are erasable} and Theorem~\\ref{thm: pi1 are duplicable}.\n\\qed\n\\end{rem}\n\n\\begin{figure}[t]\n\\centering\n\\begin{equation}\n\\begin{aligned}\\nonumber\n\\left(\n\\scalebox{0.6}{\n\\AxiomC{$\\mathcal{D}$}\n\\noLine\n\\UnaryInfC{$x_1: \\shpos \\sigma_1, \\ldots, x_n: \\shpos \\sigma_n \\vdash M: \\sigma$}\n\\RightLabel{$p$}\n\\UnaryInfC{$x_1: \\shpos \\sigma_1, \\ldots, x_n: \\shpos \\sigma_n \\vdash M: \\shpos \\sigma$}\n\\DisplayProof } \n\\right)^\\bullet\n\\triangleq &\n\\left(\n\\scalebox{0.6}{\n\\AxiomC{$\\mathcal{D}$}\n\\noLine\n\\UnaryInfC{$x_1: \\shpos \\sigma_1, \\ldots, x_n: \\shpos \\sigma_n \\vdash M :\\sigma$}\n\\DisplayProof } \n\\right)^\\bullet\n\\\\\n\\\\\n\\left(\n\\scalebox{0.6}{\n\\AxiomC{$\\mathcal{D}$}\n\\noLine\n\\UnaryInfC{$\\Gamma, x: \\sigma \\vdash M:\\tau$}\n\\RightLabel{$d$}\n\\UnaryInfC{$\\Gamma, y: \\shpos \\sigma \\vdash M[y\/x]: \\tau$}\n\\DisplayProof} \n\\right)^\\bullet\n\\triangleq &\n\\scalebox{0.6}{\n\\AxiomC{}\n\\RightLabel{ax}\n\\UnaryInfC{$  y:\\sigma^\\bullet \\vdash y: \\sigma^\\bullet$}\n\\def3pt{0pt}\n\\AxiomC{$\\left( \\begin{aligned}\\begin{gathered} \\mathcal{D} \\\\ \\Gamma, x: \\sigma \\vdash M: \\tau \\end{gathered}\\end{aligned}\\right)^\\bullet$}\n\\noLine\n\\UnaryInfC{\\hspace{2.5cm}}\n\\def3pt{3pt}\n\\RightLabel{$cut$}\n\\BinaryInfC{$\\Gamma^\\bullet, y: \\sigma^\\bullet \\vdash M^\\bullet  [y\/x]:\\tau^\\bullet$}\n\\DisplayProof} \n\\\\\n\\\\\n\\left(\n\\scalebox{0.6}{\n\\AxiomC{$\\mathcal{D}$}\n\\noLine\n\\UnaryInfC{$\\Gamma \\vdash M:\\tau$}\n\\RightLabel{$w$}\n\\UnaryInfC{$\\Gamma , x: \\shpos \\sigma \\vdash \\mathtt{discard }_{\\sigma}\\, x \\mathtt{\\  in\\  }M:\\tau$}\n\\DisplayProof}\n\\right)^\\bullet\n\\triangleq&\n\\scalebox{0.6}{\n\\AxiomC{$\\vdots$}\n\\noLine\n\\UnaryInfC{$x: \\sigma^\\bullet \\vdash \\mathtt{E}_{\\sigma^\\bullet}\\, x: \\mathbf{1}$}\n\\AxiomC{$ \\left( \\begin{aligned}\\begin{gathered}  \\mathcal{D} \\\\ \\Gamma \\vdash M :\\tau \\end{gathered}\\end{aligned} \\right)^\\bullet$}\n\\noLine\n\\UnaryInfC{\\vdots}\n\\noLine\n\\UnaryInfC{$\\Gamma^\\bullet, y: \\mathbf{1} \\vdash \\mathtt{let\\ }y \\mathtt{\\ be\\ }I \\mathtt{\\ in\\ }M^\\bullet :\\tau^\\bullet$}\n\\RightLabel{$cut$}\n\\BinaryInfC{$\\Gamma^\\bullet, x:\\sigma^\\bullet  \\vdash \\mathtt{let\\ }\\mathtt{E}_{\\sigma^\\bullet}\\, x \\mathtt{\\ be\\ }I \\mathtt{\\ in\\ }M^\\bullet :\\tau^\\bullet$}\n\\DisplayProof}\n\\\\\n\\\\\n\\left( \n\\scalebox{0.6}{\n\\AxiomC{$\\mathcal{D}_1$}\n\\noLine\n\\UnaryInfC{$\\Gamma, x_1: \\shpos \\sigma, x_2: \\shpos \\sigma \\vdash M: \\tau$}\n\\AxiomC{$\\mathcal{D}_2$}\n\\noLine\n\\UnaryInfC{$\\vdash V: \\sigma$}\n\\RightLabel{$c$}\n\\BinaryInfC{$\\Gamma, x: \\shpos \\sigma \\vdash \\mathtt{copy}^{V}_{\\sigma} \\, x \\mathtt{\\ as \\ } x_1, x_2 \\mathtt{ \\ in\\ } M: \\tau$}\n\\DisplayProof}\n\\right)^\\bullet\n\\triangleq & \n\\scalebox{0.6}{\n\\def\\hskip .1cm{\\hskip 0.4cm}\n\\def1pt{1pt}\n\\AxiomC{$\\mathcal{D}_2^\\bullet$}\n\\noLine\n\\UnaryInfC{$\\vdash V^\\bullet : \\sigma^\\bullet$}\n\\noLine\n\\UnaryInfC{$\\vdots$}\n\\noLine\n\\UnaryInfC{$x:\\sigma^\\bullet \\vdash \\mathtt{D}^{V^\\bullet } _{\\sigma^\\bullet} \\, x: \\sigma^\\bullet \\otimes \\sigma^\\bullet$}\n\\AxiomC{$\\left( \\begin{aligned} \\begin{gathered}\\mathcal{D}_1 \\\\   \\Gamma, x_1: \\shpos \\sigma, x_2: \\shpos \\sigma  \\vdash M :  \\tau \\end{gathered}\\end{aligned} \\right)^\\bullet$}\n\\noLine\n\\UnaryInfC{\\vdots}\n\\noLine\n\\UnaryInfC{$\\Gamma^\\bullet, y :\\sigma^\\bullet \\otimes \\sigma^\\bullet \\vdash  \\mathtt{let\\ }y \\mathtt{\\ be\\ }x_1,x_2 \\mathtt{\\ in\\ }M^\\bullet: \\tau^\\bullet$}\n\\RightLabel{$cut$}\n\\BinaryInfC{$\\Gamma^\\bullet, x: \\sigma^\\bullet \\vdash \\mathtt{let\\ }\\mathtt{D}_{\\sigma^\\bullet}^{V^\\bullet} \\, x \\mathtt{\\ be\\ }x_1,x_2 \\mathtt{\\ in\\ }M^\\bullet :\\tau^\\bullet$}\n\\DisplayProof}\n\\end{aligned}\n\\end{equation}\n\\caption{The translation of the rules $p$, $d$,  $w$ and $c$.}\n\\label{fig: translation inference rules}\n\\end{figure}\n\nThe simulation theorem requires some preliminaries.\n\n\\begin{lem} \\label{lem: translation on values}For every typable value $V$:\n\\begin{enumerate}[(1)]\n\\item \\label{enum: 1 translation on values} $V^\\bullet =V$.\n\\item \\label{enum: 2 translation on values} $V$ has type $\\sigma$ if and only if $V^\\bullet$ has type $\\sigma^\\bullet$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nStraightforward consequence of Definition~\\ref{defn: translation IMLL2 shpos with oplus into IMLL2}.\n\\end{proof}\n\n\\begin{lem} \\label{lem: bullet commutes well with substitution in LAML} For all terms $M, N \\in \\Lambda_{\\shpos}$ typable in $\\mathsf{LEM}$, $M^\\bullet [N^\\bullet\/x]= (M[N\/x])^\\bullet$.\n\\end{lem}\n\\begin{proof}\nWe can proceed by a standard structural induction on $ M $.\n\\end{proof}\n\\noindent\nWe now show that every reduction on terms typable in $\\mathsf{LEM}$ can be simulated in the linear $\\lambda$-calculus by means of the $\\beta \\eta$-reduction relation.  We recall that\nSubject reduction holds on every typable $M \\in \\Lambda_\\shpos$\n(Theorem~\\ref{thm:Subject Reduction for IMLL2shpos}). \nMoreover, every linear $\\lambda$-term that has a type in $\\mathsf{IMLL}$, has one in $\\mathsf{IMLL}_2$ (see~\\cite{hindley1989bck}). So, we state the simulation theorem \nfor terms, rather than the related derivations.\n\\begin{thm}[Simulation]\n\\label{thm: translations for LAML} Let $\\mathcal{D}\\triangleleft \\Gamma \\vdash M: \\sigma$ be a derivation in  $\\mathsf{LEM}$. If $M_1 \\rightarrow M_2$ then $M_1^\\bullet\\rightarrow^*_{\\beta \\eta}M_2^\\bullet$:\n\\begin{center}\n\\begin{tikzcd}\nM_1\\arrow[d, dashed] \\arrow[r]& M_2 \\arrow[d,dashed] \\\\\nM_1^\\bullet \\arrow[r, \"*\"  pos=1, \"\\beta \\eta \" ' pos=1.05] & M_2^{  \\bullet}\n\\end{tikzcd}\n\\enspace.\n\\end{center}\n\\end{thm}\n\\begin{proof} We can proceed by structural induction on $M_1$.\nOne of the most interesting cases is when\n$M_1$ is $ (\\lambda x. P)Q$ and $M_2= P[Q\/x]$. \nLemma~\\ref{lem: bullet commutes well with substitution in LAML}\nimplies $((\\lambda y. P)Q)^\\bullet= (\\lambda y. P^\\bullet)Q^\\bullet \\rightarrow_\\beta P^\\bullet[Q^\\bullet\/x] =(P[Q\/x])^\\bullet$. \nIf $M_1$ is $\\mathtt{discard}_{\\sigma}\\, V \\mathtt{\\ in\\ }N$ and\n$M_2$ is $N$, then $V$ is a value of type $\\sigma$. By Lemma~\\ref{lem: translation on values}.\\ref{enum: 2 translation on values}, $V^\\bullet$ is a value of $\\sigma^\\bullet$.  Hence:\n\\allowdisplaybreaks\n\\begin{align*}\n(\\mathtt{discard}_{\\sigma}\\, V \\mathtt{\\ in\\ }N)^\\bullet &\n\\triangleq \\mathtt{let\\ } \\mathtt{E}_{\\sigma^\\bullet}\\, V^\\bullet \\mathtt{\\ be\\ }I \\mathtt{\\ in\\ }N^\\bullet\n\\rightarrow^*_\\beta  N^\\bullet\n\\end{align*}\n\\noindent\nby Theorem~\\ref{thm: pi1 types are erasable}.\nIf $M_1$ is $\\mathtt{copy}^{U}_{\\sigma}\\, V \\mathtt{\\ as\\ }x_1,x_2 \\mathtt{\\ in\\ }N$ and $M_2$ is $N[V\/x_1, V\/x_2]$, \nthen $U$ and $V$ are both values of type $\\sigma$. By Lemma~\\ref{lem: translation on values}.\\ref{enum: 2 translation on values}, $U^\\bullet$ and $V^\\bullet$ are both values of type $\\sigma^\\bullet$. \nHence:\n\\allowdisplaybreaks\n\\begin{align*}\n(\\mathtt{copy}^{U}_{\\sigma}\\, V \\mathtt{\\ as\\ }x_1,x_2 \\mathtt{\\ in\\ \n}N)^\\bullet&\\triangleq  \\mathtt{let\\ }\\mathtt{D}^{U^\\bullet } _{\\sigma^\\bullet}\\, V^\\bullet\n\\mathtt{\\ be\\ } x_1,x_2 \\mathtt{\\ in \\ }N^\\bullet \\\\\n&\\rightarrow^*_{\\beta \\eta}  \\mathtt{let\\ }\\langle V^\\bullet, V^\\bullet \\rangle \\mathtt{\\ be\\ } x_1,x_2 \\mathtt{\\ in \\ }N^\\bullet  &&\\text{Theorem}~\\ref{thm: pi1 are duplicable} \\\\\n&\\rightarrow^*_\\beta N^\\bullet[V^\\bullet\/x_1, V^\\bullet\/x_2]\\\\\n&\\triangleq (N^\\bullet[V^\\bullet\/x_1]) [ V^\\bullet\/x_2]\\\\\n&=    ((N[V\/x_1]) [ V\/x_2])^\\bullet &&\\text{Lemma}~\\ref{lem: bullet commutes well with substitution in LAML}\\\\\n&\\triangleq (N[V\/x_1, V\/x_2])^\\bullet   \\enspace .\n\\end{align*}\n\\end{proof}\n\\noindent\nWe conclude by estimating the impact of the translation  on the size of terms  produced by  \\enquote{unpacking} the constructs $\\mathtt{discard}_{\\sigma}$ and $\\mathtt{copy}^V_{\\sigma}$. \nWe need to bound the dimension of erasers and duplicators with ground type $A$. We rely on the map $(\\_ )^-$ \n(Definition~\\ref{defn: stripping forall} \nin~\\ref{sec: the d-soundness theorem DICE})\nthat, intuitively, strips every occurrence of $\\forall$ away from a\ngiven type (Remark~\\ref{rem: duplicator} in~\\ref{sec: the d-soundness theorem DICE}.)\n\n\\begin{lem}[Size of duplicators and erasers] \n\\label{lem: size of duplicator}  For every ground type $A$:\n\\begin{enumerate}[(1)]\n\\item  \\label{enum: size of eraser}$\\vert \\mathtt{E}_{A} \\vert \\in \\mathcal{O}(\\vert A^- \\vert)$.\n\\item\\label{enum: size of duplicator} If $V$ is a value of $A$, then  $\\vert \\mathtt{D}_{A}^{V} \\vert \\in \\mathcal{O}(  2^{\\vert A^- \\vert^2})$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nPoint~\\ref{enum: size of eraser} is straightforward by looking at the proof of Theorem~\\ref{thm: pi1 types are erasable}. Concerning Point~\\ref{enum: size of duplicator}, from \n\\ref{sec: the d-soundness theorem DICE} we know that $\\mathtt{D}_{A}^{V}$ is\n$ \\mathtt{dec}^s_{A} \\circ  \n\\mathtt{enc}_{A}^s  \\circ\n\\mathtt{sub}^s_{A} $,\nwhere  $s= \\mathcal{O}(\\vert  A^- \\vert \\, \\cdot \\, \\log \\vert A^- \\vert )$. The components of $\\mathtt{D}_{A}^{V}$ with a\nsize not linear in $\\vert A^- \\vert $ are $\\mathtt{dec}^s_{A}$ and  $\\mathtt{enc}^s_{A}$. The $ \\lambda $-term $\\mathtt{dec}^s_{A}$ (see point~\\ref{enumerate: duplication step 3} in Section~\\ref{sec:  the system shpos, with, oplus  IMLL2 cut elimination and bound}) nests occurrences of\n\\texttt{if}-\\texttt{then}-\\texttt{else} each containing $2^{s}$ pairs of normal inhabitants of $A$, every of which,\nby Lemma~\\ref{lem: pi1type bound}, has size bounded by \n$\\vert A^- \\vert$. \nSimilarly, $\\mathtt{enc}^s_{A}$ alternates instances of\n$\\lambda$-terms $\\mathtt{abs}^s$ and $\\mathtt{app}^s$ which, \nagain, nest occurrences of \\texttt{if}-\\texttt{then}-\\texttt{else} \nevery one with $2^{s}$ instances of boolean strings of size $s$. \nThe overall complexity of $\\mathtt{D}_{A}^{V}$ is $\\mathcal{O}(s \\cdot 2^{s})= \\mathcal{O}(2^{\\vert A^-  \\vert^2})$. \n\\end{proof}\n\\noindent\n\n\\begin{thm}[Exponential Compression]\n\\label{thm: exponential compression} \nLet  $\\mathcal{D} \\triangleleft \\Gamma \\vdash M: \\sigma$ be a derivation in \n$\\mathsf{LEM}$. Then,  \n$\\vert M^\\bullet \\vert = \\mathcal{O}(2^{\\vert M \\vert^k})$, \nfor some $k \\geq 1$.\n\\end{thm}\n\\begin{proof} The proof is by structural induction on $M$. The  interesting case is when $M$ is \n$\\mathtt{copy}^{V}_{\\sigma}\\, P \\mathtt{\\ as\\ }x_1,x_2 \\mathtt{\\ in\\ }Q$. \nSince $M$ is typable,  $V$ has type $\\sigma$, that is  closed and free from negative occurrences of $\\forall$, hence lazy. \nBy Lemma~\\ref{lem: lazy derivation properties}.\\ref{enum: finite number of normal forms}, it is safe to assume that $V$ is a value with largest size among values of type $\\sigma$.\nBy Lemma~\\ref{lem: translation on values}, $V=V^\\bullet$ is also the largest value of type $\\sigma^\\bullet$ in $\\mathsf{IMLL_2}$. Finally, by Lemma~\\ref{lem: pi1type bound}, this implies that $V$  is a $\\eta$-long normal form of type $\\sigma^\\bullet$.  Now, by using  Definition~\\ref{eqn:datatype unity and product}, we have   $\\vert M^\\bullet \\vert= \\vert \\mathtt{let\\ }\\mathtt{D}^{V} _{\\sigma^\\bullet}\\, P^\\bullet \\mathtt{\\ be\\ } x_1,x_2 \\mathtt{\\ in \\ }Q^\\bullet\\vert = \\vert \\mathtt{D}^{V} _{\\sigma^\\bullet}\\vert + \\vert  P^\\bullet \\vert+ \\vert Q^\\bullet \\vert +4$.  On the one hand, by induction hypothesis, we obtain  $\\vert   P^\\bullet \\vert = \\mathcal{O}(2^{\\vert P \\vert^{k'}})$ and  $\\vert   Q^\\bullet \\vert = \\mathcal{O}(2^{\\vert Q \\vert^{k''}})$, for some $k', k'' \\geq 1$.  On the other hand, by applying both Lemma~\\ref{lem: size of duplicator} and  Lemma~\\ref{lem: pi1type bound},  we have $\\vert \\mathtt{D}^{V} _{\\sigma^\\bullet}\\vert=\\mathcal{O}(2^{\\vert A^- \\vert^2}) =\\mathcal{O}(2^{2 \\cdot \\vert V \\vert^2})$. \nTherefore, there exists $k\\geq 1$ such that $\\vert M^\\bullet \\vert = \\mathcal{O}(2^{(\\vert V \\vert + \\vert P \\vert + \\vert Q \\vert +1)^k})=\\mathcal{O}(2^{\\vert M \\vert^k })$.\n\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Vanishing topos and\nthe semi-continuity of the Swan conductor}\n\\label{svan}\n\\subsection{Calculus on vanishing topos}\\label{ssvan}\n\nLet $f\\colon X\\to S$\nbe a morphism of schemes.\nBy abuse of notation,\nlet $X, S$ also denote\nthe associated \\'etale toposes.\nFor the definition of\nthe {\\em vanishing topos}\n$X\\overset \\gets\\times_SS$\nand the morphisms \n$$\\begin{CD}\nX@>{\\Psi_f}>> X\\overset \\gets\\times_SS\n@>{p_2}>>S\\\\\n@.@V{p_1}VV\\\\\n@.X\\end{CD}$$\nof toposes,\nwe refer to \\cite[1.1, 4.1, 4.3]{Il} and\n\\cite[1.1]{TS}.\nFor a geometric point $x$ of \na scheme $X$,\nwe assume in this article\nthat the residue field\nof $x$ is a separable closure\nof the residue field at the\nimage of $x$ in $X$,\nif we do not say otherwise explicitly.\n\nFor a geometric point $x$ of $X$,\nthe fiber \n$x\\times_X(X\\overset \\gets\\times_SS)$\nof $p_1\\colon\nX\\overset \\gets\\times_SS\\to X$\nat $x$ is the vanishing topos\n$x\\overset \\gets\\times_SS$\nand is canonically\nidentified with the strict localization\n$S_{(s)}$\nat the geometric point $s=f(x)$\nof $S$ defined by the image of $x$\n(cf.\\  \\cite[(1.8.2)]{TS}).\n\nA point on the topos $X\\overset \\gets\\times_SS$\nis defined by a triple denoted $x\\gets t$\nconsisting of a geometric point $x$ of $X$,\na geometric point $t$ of $S$\nand a specialization $s=f(x)\\gets t$\nnamely a geometric point $S_{(s)}\\gets t$\nof the strict localization\nlifting $S\\gets t$.\nThe fiber \n$(X\\overset \\gets\\times_SS)\n\\times_{\nS\\overset \\gets\\times_SS}\n(s\\gets t)$\nof the canonical morphism\n$X\\overset \\gets\\times_SS\\to \nS\\overset \\gets\\times_SS$\nat a point $s\\gets t$\nis canonically identified with\nthe geometric fiber $X_{s}$.\nThe fiber products\n$X_{(x)}\\times_{S_{(s)}}S_{(t)}$\nand\n$X_{(x)}\\times_{S_{(s)}}t$\nare called the {\\em Milnor tube}\nand the {\\em Milnor fiber}\nrespectively.\n\nFor a commutative diagram\n$$\\xymatrix{\nX\\ar[rr]^f\\ar[rd]_p&&\nY\\ar[ld]^g\\\\\n&S}$$\nof morphisms of schemes,\nthe morphism\n$\\overset{\\gets} g\\colon X\\overset \\gets\\times_YY\\to\nX\\overset \\gets\\times_SS$\nis defined by functoriality and\nwe have a canonical isomorphism\n$\\Psi_p\\to \\overset{\\gets} g\n\\circ \\Psi_f$.\nOn the fibers of a geometric point $x$\nof $X$, the morphism\n$\\overset{\\gets} g$\ninduces a morphism\n\\begin{equation}\ng_{(x)}\\colon Y_{(y)}\n=x\\overset\\gets\\times_YY\\to S_{(s)}\n=x\\overset\\gets\\times_SS\n\\label{eqpx}\n\\end{equation}\non the strict localizations\nat $y=f(x)$ and $s=p(x)$ \\cite[(1.7.3)]{TS}.\nIn particular for $Y=X$,\nwe have a canonical isomorphism\n$\\Psi_p\\to \\overset{\\gets} g\n\\circ \\Psi_{\\rm id}$.\n\nLet $\\Lambda$ be a finite local\nring with residue\nfield $\\Lambda_0$ of characteristic \n$\\ell$ invertible on $S$.\nLet $D^+(-)$ denote\nthe derived category of \ncomplexes of $\\Lambda$-modules\nbounded below\nand let $D^b(-)$ denote\nthe subcategory consisting\nof complexes with bounded cohomology.\nIn the following,\nwe assume that $S$ and $X$ are quasi-compact\nand quasi-separated.\nWe say that an object of\n$D^b(X\\overset \\gets\\times_SS)$\nis constructible\nif there exist finite partitions\n$X=\\coprod_\\alpha X_\\alpha$\nand\n$S=\\coprod_\\beta S_\\beta$ \nby locally closed\nconstructible subschemes such that\nthe restrictions to $X_\\alpha\n\\overset \\gets\\times_SS_\\beta$\nof cohomology sheaves are locally constant\nand constructible \\cite[1.3]{TS}.\n\nLet $D^b_c(-)$ denote\nthe subcategory of $D^b(-)$ consisting\nof constructible objects\nand\nlet $D_{\\rm ctf}(-)\n\\subset D^b_c(-)$ denote\nits subcategory consisting\nof objects of finite tor-dimension.\nIf $\\Lambda$ is a field,\nwe have\n$D_{\\rm ctf}(-)\n=D^b_c(-)$.\n\nWe canonically identify\na function on the underlying set\nof a scheme $X$ with \nthe function on the set\nof isomorphism classes of\ngeometric points $x$ of $X$.\nSimilarly, we call a function on the set\nof isomorphism classes of\npoints $x\\gets t$ of $X\\overset \\gets\\times_SS$\na function on $X\\overset \\gets\\times_SS$.\nWe say that a function on \n$X\\overset \\gets\\times_SS$\nis a {\\em constructible function}\nif there exist finite partitions\n$X=\\coprod_\\alpha X_\\alpha$\nand\n$S=\\coprod_\\beta S_\\beta$ as above\nsuch that the restrictions\nto $X_\\alpha\n\\overset \\gets\\times_SS_\\beta$\nare locally constant.\n\nFor an object ${\\cal K}$ of\n$D_{\\rm ctf}(X\\overset \\gets\\times_SS)$,\nthe rank function \n$\\dim {\\cal K}_{x\\gets t}$\nis defined as a constructible function\non $X\\overset \\gets\\times_SS$.\nIf $\\Lambda=\\Lambda_0$ is a field, we have\n$\\dim {\\cal K}_{x\\gets t}=\n\\sum_q(-1)^q\\dim {\\cal H}^q{\\cal K}_{x\\gets t}$.\nIn general, \nwe have\n$\\dim_\\Lambda {\\cal K}\n=\n\\dim_{\\Lambda_0} \n{\\cal K}\\otimes_\\Lambda^L\\Lambda_0$.\n\n\n\n\\begin{df}\\label{df11}\nLet $Z$ be a quasi-finite scheme \nover $S$ \nand let $\\varphi\\colon Z\\to {\\mathbf Q}$\nbe a function.\nWe define the {\\em derivative} $\\delta(\\varphi)$\nof $\\varphi$ as a function\non $Z\\overset \\gets\\times_SS$\nby \n\\begin{equation}\n\\delta(\\varphi)(x\\gets t)\n=\\varphi(x)\n-\\sum_{z\\in Z_{(x)}\\times\n_{S_{(s)}}t}\\varphi(z)\n\\label{eqdel}\n\\end{equation}\nwhere $s=f(x)$.\nIf the derivative $\\delta(\\varphi)$\nis $0$ (resp.\\ $\\delta(\\varphi)\\geqq 0$),\nwe say that the function $\\varphi$\nis {\\em flat} (resp.\\ {\\em increasing})\nover $S$.\nIf the morphism $f\\colon Z\\to S$ is finite,\nwe define a function $f_*\\varphi$ on $S$\nby \n\\begin{equation}\nf_*\\varphi(s)=\n\\sum_{x\\in Z_s}\\varphi(x).\n\\label{eqf*}\n\\end{equation}\n\\end{df} \n\n\n\\begin{lm}\\label{lmscZ}\nLet $S$ be a noetherian scheme,\n$Z$ be a quasi-finite scheme \nover $S$ and \n$\\varphi\\colon Z\\to {\\mathbf Q}$ be a function.\n\n{\\rm 1.}\nAssume that\n$Z$ is \\'etale over $S$.\nThen $\\varphi$ is locally constant\n(resp.\\ upper semi-continuous)\nif and only if\nit is flat over $S$\n(resp.\\ constructible\nand increasing over $S$).\n\n\n{\\rm 2.} \nThe function $\\varphi$ is\nconstructible if and only if its\n{\\em derivative} $\\delta(\\varphi)\n\\colon Z\\overset \\gets\\times_SS\n\\to {\\mathbf Q}$\ndefined in {\\rm (\\ref{eqdel})}\nis constructible.\nConsequently, if $\\varphi$ is flat over $S$,\nthen $\\varphi$ is constructible.\n\n{\\rm 3.} \nAssume that\n$\\varphi$ is {\\em flat}\nover $S$.\nThen, $\\varphi=0$\nif and only if\n$\\varphi(x)=0$ for\nthe generic point $x$ of\nevery irreducible component of $Z$.\n\n{\\rm 4.}\nAssume that \nthe morphism $f\\colon Z\\to S$ is finite.\nIf $\\varphi$ is constructible,\nthe function $f_*\\varphi$\nis also constructible.\nAssume that $\\varphi$ is constructible\nand is increasing over $S$.\nThen the function $f_*\\varphi$ on $S$ is \nupper semi-continuous.\nFurther, the function $\\varphi$ is flat over $S$ if \nand only if\n$f_*\\varphi$ is locally constant.\n\\end{lm}\n\n\\proof{\n1. \nSince the question is \\'etale local on $Z$,\nwe may assume that $Z\\to S$ \nis an isomorphism.\nThen the assertion is clear.\n\n\n2.\nAssume $\\delta(\\varphi)$ is constructible.\nBy noetherian induction,\nit suffices to show the following:\nFor every geometric point $t$ of $S$\nand the closure $T\\subset S$ of its image,\nthere exists a dense open subset\n$V\\subset T$ such that\n$\\varphi$ is locally constant on\n$Z\\times_SV$.\nReplacing $S$ by $T$,\nit suffices to consider the case where \n$t$ dominates\nthe generic point of an irreducible scheme\n$S$.\nFor a geometric point $x$ of $Z$\nabove $t$,\nwe have $\\delta(\\varphi)(x\\gets t)=0$.\nBy further replacing $X$ by\nan \\'etale neighborhood of $x$,\nit suffices to consider the case\nwhere $Z$ is \\'etale over $S$\nand $\\delta(\\varphi)=0$.\nThen, by 1,\n$\\varphi$ is locally constant and hence\nconstructible.\n\nAssume $\\varphi$ is constructible.\nFor a closed subset $T$ of $S$,\nthe subtopos\n$(Z\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z\\times_ST)\\overset\\gets\\times_ST$\nis empty.\nHence, \nby noetherian induction,\nit suffices to show the following:\nFor every geometric point $t$ of $S$\nand the closure $T\\subset S$ of its image,\nthere exists a dense open subset\n$V\\subset T$ such that\n$\\delta(\\varphi)$ is locally constant on\n$(Z\\times_SV)\\overset\\gets\\times_SV$.\nSimilarly as above,\nit suffices to consider the case where \n$Z$ is \\'etale over $S$\nand $\\varphi$ is locally constant.\nThen, by 1.,\n$\\delta(\\varphi)=0$ and is constructible.\n\n\n3.\nBy (\\ref{eqdel}),\na function flat over $S$ is uniquely determined\nby the values at the generic points\nof irreducible components.\n\n\n4.\nAssume that $f\\colon Z\\to S$ is finite.\nIf $\\varphi$ is constructible,\nthere exists a dense open subscheme\n$U$ of $S$ such that\n$\\varphi$ is locally constant\non $Z\\times_SU$.\nHence \n$f_*\\varphi$ is constructible by\nnoetherian induction.\nFor a specialization $s\\gets t$,\nwe have\n$\\sum_{x\\in Z_s}\\delta(\\varphi)(x\\gets t)\n=f_*\\varphi(s)-\nf_*\\varphi(t)\n=\\delta(f_*\\varphi)(s\\gets t)$.\nHence we may assume\n$Z=S$ and then the assertion is clear.\n\\qed}\n\\medskip\n\nWe give an example\nof flat function.\nLet $S$ be a noetherian\nscheme, $X$ be a\nscheme of finite type over $S$\nand $Z\\subset X$ be a closed\nsubscheme quasi-finite over $S$.\nLet $A$ be a complex of \n${\\cal O}_X$-modules\nsuch that the cohomology sheaves\n${\\cal H}^q(A)$ are coherent\n${\\cal O}_X$-modules supported on $Z$\nfor all $q$ and\nthat $A$ is of finite tor-dimension \nas a complex of ${\\cal O}_S$-modules.\nFor a geometric point $z$ of $Z$ \nand its image $s$ in $S$,\nlet ${\\cal O}_{S,s}$ denote the {\\em strict}\nlocalization\nand $k(s)$ the separably closed\nresidue field of ${\\cal O}_{S,s}$.\nThen, the $k(s)$-vector spaces\n$Tor^{{\\cal O}_{S,s}}_q\n(A_z,k(s))$ are of finite dimension\nand are $0$ except for finitely many $q$.\nWe define a function\n$\\varphi_A\\colon Z\\to {\\mathbf Z}$\nby\n\\begin{equation}\n\\varphi_A(z)\n=\n\\sum_q\n(-1)^q\n\\dim_{k(s)}\nTor^{{\\cal O}_{S,s}}_q\n(A_z,k(s)).\n\\label{eqA}\n\\end{equation}\n\n\\begin{lm}\\label{lmA}\nLet noetherian schemes\n$Z\\subset X\\to S$ \nand a complex $A$ be as above.\n\n{\\rm 1.}\nThe function \n$\\varphi_A\\colon Z\\to {\\mathbf Z}$\ndefined by {\\rm (\\ref{eqA})} is\nconstructible and flat over $S$.\n\n{\\rm 2.}\nSuppose that $S$ and $Z$ are integral\nand that the image of the generic point\n$\\xi$ of $Z$ is the generic point\n$\\eta$ of $S$.\nIf $A={\\cal O}_Z$,\nthe value of the function $\\varphi_A$\nat a geometric point of $Z$ above $\\xi$ is\nthe inseparable degree $[k(\\xi):k(\\eta)]\n_{\\rm insep}$.\n\\end{lm}\n\nThe condition that\n$A$ is of finite tor-dimension \nas a complex of ${\\cal O}_S$-modules\nis satisfied if $S$ is regular.\n\n\\proof{1.\nSince the assertion is\n\\'etale local on $Z$,\nwe may assume that $Z$ is\nfinite over $S$, \nthat $X$ and $S$ are affine and that\n$z$ is the unique point in the geometric fiber $Z\n\\times_S{\\rm Spec}\\ k(s)$.\nThen, the complex $Rf_*A$\nis a perfect complex of\n${\\cal O}_S$-modules\nand $\\varphi_A(z)$ equal the rank of\n$Rf_*A$.\nHence, the assertion follows.\n\n2.\nWe may assume that\n$S={\\rm Spec}\\ k(\\eta)$\nand $Z={\\rm Spec}\\ k(\\xi)$\nand the assertion follows.\n\\qed}\n\\medskip\n\nWe generalize the definition\nof derivative\nto functions on vanishing topos.\n\n\\begin{df}\\label{df14}\nLet \n\\begin{equation}\n\\xymatrix{\nZ\\ar[rr]^f\\ar[rd]_p&&\nY\\ar[ld]^g\\\\\n&S}\n\\label{eqfgp}\n\\end{equation} \nbe a commutative diagram of\nmorphisms of schemes\nsuch that $Z$ is quasi-finite over $S$.\nLet $\\psi\\colon \nZ\\overset\\gets\\times_YY\\to {\\mathbf Q}$\nbe a function such that\n$\\psi(x\\gets w)=0$\nunless $w$ is not supported\non the image of $f_{(x)}\n\\colon Z_{(x)}\\to Y_{(y)}$\nwhere $y=f(x)$.\nWe define the derivative\n$\\delta(\\psi)$ as \na function on \n$Z\\overset\\gets\\times_SS\\to {\\mathbf Z}$\nby\n\\begin{equation}\n\\delta(\\psi)(x\\gets t)\n=\\psi(x\\gets y)\n-\\sum_{w\\in Y_{(y)}\\times_{S_{(s)}}t}\n\\psi(x\\gets w)\n\\label{eqdelY}\n\\end{equation}\nwhere $y=f(x)$ and $s=p(x)$.\nWe say that $\\psi$ is {\\em flat} over $S$\nif $\\delta(\\psi)=0$.\n\\end{df}\n\nThe sum in the right hand side\nof (\\ref{eqdelY})\nis a finite sum by the assumption\nthat $Z$ is quasi-finite over $S$\nand the assumption on the support of $\\psi$.\nIf $Z=Y$, we recover the\ndefinition (\\ref{eqdel})\nby applying (\\ref{eqdelY})\nto the pull-back $p_2^*\\varphi\\colon\nZ\\overset\\gets\\times_ZZ\\to {\\mathbf Z}$\nby $p_2\\colon\nZ\\overset\\gets\\times_ZZ\\to Z$.\n\nThe following elementary Lemma will be used\nin the proof of a generalization\nof the continuity of the Swan conductor.\n\n\\begin{lm}\\label{lmdelp}\nLet the assumption on the diagram\n{\\rm (\\ref{eqfgp})} be as in\nDefinition {\\rm \\ref{df14}} and let\n$\\varphi$ be a function on $Z$.\nWe define a function\n$\\psi$ on $Z\\overset\\gets\\times_YY$\nby\n\\begin{equation}\n\\psi(x\\gets w)=\n\\sum_{z\\in Z_{(x)}\\times_{Y_{(y)}}w}\n\\varphi(z)\n\\label{eqpsi}\n\\end{equation}\nwhere $y=f(x)$.\nThen the derivative $\\delta(\\varphi)$\non $Z\\overset\\gets\\times_SS$\ndefined by {\\rm (\\ref{eqdel})}\nequals \n$\\delta(\\psi)$\ndefined by {\\rm (\\ref{eqdelY})}.\n\\end{lm}\n\n\\proof{\nIt follows from\n$\\psi(x\\gets y)=\\varphi(x)$\nand \n$Z_{(x)}\\times_{S_{(s)}}t\n=\n\\coprod_{w\\in Y_{(y)}\\times_{S_{(s)}}t}\n(Z_{(x)}\\times_{Y_{(y)}}w)$.\nNote that except for finitely many\ngeometric points $w$ of\n$Y_{(y)}\\times_{S_{(s)}}t$\nthose supported on the image of\n$Z_{(x)}$, the fiber\n$Z_{(x)}\\times_{Y_{(y)}}w$\nis empty.\n\\qed}\n\n\\subsection{Nearby cycles and the local acyclicity}\\label{ssla}\n\nFor a morphism $f\\colon X\\to S$,\nthe morphism $\\Psi_f\\colon X\\to \nX\\overset \\gets\\times_SS$\ndefines the nearby cycles functor\n$R\\Psi_f:D^+(X)\\to D^+(X\\overset \\gets\\times_SS)$.\nThe canonical morphism\n$p_1^*\\to R\\Psi_f$\nof functors is defined by adjunction\nand by the isomorphism ${\\rm id}\\to p_1\\circ \\Psi_f.$\nThe cone of the morphism\n$p_1^*\\to R\\Psi_f$\ndefines the vanishing cycles functor\n$R\\Phi_f:D^+(X)\\to D^+(X\\overset \\gets\\times_SS)$.\nIf $S$ is the spectrum of \na henselian discrete valuation ring\nand if $s,\\eta$ denote\nits closed and generic points,\nwe recover the classical construction\nof complexes $\\psi, \\phi$ of nearby cycles\nand vanishing cycles as \nthe restrictions to\n$X_s\\overset \\gets\\times_S\\eta$\nof $R\\Psi_f$ and $R\\Phi_f$\nrespectively.\n\nRecall that $f\\colon X\\to S$\nis said to be locally acyclic relatively\nto a complex ${\\cal F}$ \nof $\\Lambda$-modules\non $X$\n{\\rm \\cite[Definition 2.12]{TF}}\nif the canonical morphism\n\\begin{equation}\n\\begin{CD}\n{\\cal F}_x\n@>>>\nR\\Gamma(X_{(x)}\\times_{S_{(s)}}t,\n{\\cal F}|_{X_{(x)}\\times_{S_{(s)}}t})\n\\end{CD}\n\\label{eqla}\n\\end{equation}\nis an isomorphism\nfor every $x\\gets t$.\nRecall that $f\\colon X\\to S$\nis said to be universally locally acyclic relatively\nto ${\\cal F}$,\nif for every morphism $S'\\to S$,\nits base change is locally acyclic relatively\nto the pull-back of ${\\cal F}$.\n\n\n\\begin{lm}[{\\rm cf.\\ \\cite[Proposition 7.6.2]{Fu}}]\\label{lmctf}\nLet $f\\colon X\\to S$\nbe a morphism of finite type and\nlet ${\\cal F}\\in D_{\\rm ctf}(X)$\nbe a complex of finite tor-dimension.\n\n{\\rm 1.}\nSuppose that ${\\cal F}$ is of tor-amplitude $[a,b]$ and that $f\\colon X\\to S$ is of relative dimension $d$. Then, \nfor points $x\\gets t$\nof $X\\overset{\\gets}\\times_SS$,\nthe complex\n$R\\Gamma(X_{(x)}\\times_{S_{(s)}}t,\n{\\cal F}|_{X_{(x)}\\times_{S_{(s)}}t})$\nof $\\Lambda$-modules\nis of\ntor-amplitude $[a,b+d]$ \nand,\nfor a $\\Lambda$-module $M$,\nthe canonical morphism\n\\begin{equation}\nR\\Gamma(X_{(x)}\\times_{S_{(s)}}t,\n{\\cal F}|_{X_{(x)}\\times_{S_{(s)}}t})\n\\otimes_\\Lambda^LM\n\\to  \nR\\Gamma(X_{(x)}\\times_{S_{(s)}}t,\n{\\cal F}|_{X_{(x)}\\times_{S_{(s)}}t}\n\\otimes_\\Lambda^LM)\n\\label{eqFu}\n\\end{equation}\nis an isomorphism.\n\n{\\rm 2.}\nLet $\\Lambda_0$\nbe the residue field of $\\Lambda$.\nThen,\n$f\\colon X\\to S$ is locally acyclic\n(resp.\\ universally locally acyclic)\nrelatively to \n${\\cal F}$ if and only if\nit is so relatively to\n${\\cal F}_0=\n{\\cal F}\\otimes_\\Lambda^L\\Lambda_0$.\n\\end{lm}\n\n\\proof{\n1.\nBy the assumption that\n$f\\colon X\\to S$\nis of finite type and of\nrelative dimension $d$,\nthe functor \n$R\\Gamma(X_{(x)}\\times_{S_{(s)}}t,-)$\nis of cohomological dimension $\\leqq d$\nby \\cite[Corollaire 3.2]{XIV}.\nHence, similarly as \\cite[(4.9.1)]{Rapport},\nthe canonical morphism (\\ref{eqFu})\nis an isomorphism.\nSince the complex\n${\\cal F}|_{X_{(x)}\\times_{S_{(s)}}t}\n\\otimes_\\Lambda^LM$\nis acyclic outside $[a,b]$,\nthe complex\n$R\\Gamma(X_{(x)}\\times_{S_{(s)}}t,$\n${\\cal F}|_{X_{(x)}\\times_{S_{(s)}}t}\n\\otimes_\\Lambda^LM)$\nis acyclic outside $[a,b+d]$.\nThus, the complex\n$R\\Gamma(X_{(x)}\\times_{S_{(s)}}t,\n{\\cal F}|_{X_{(x)}\\times_{S_{(s)}}t})$ is \nof tor-amplitude $[a,b+d]$.\n\n2.\nIt suffices to show the \nassertion for local acyclicity.\nIf the canonical morphism \n(\\ref{eqla})\nis an isomorphism for $x\\gets t$,\nthen (\\ref{eqla})\nfor ${\\cal F}_0$ is an isomorphism\nby the isomorphism (\\ref{eqFu}) \nfor $M=\\Lambda_0$.\n\nTo show the converse,\nlet $I^\\bullet$ be a filtration\nby ideals of $\\Lambda$\nsuch that ${\\rm Gr}\\Lambda$\nis a $\\Lambda_0$-vector space.\nThen, $I^\\bullet$ defines a filtration\non ${\\cal F}={\\cal F}\\otimes^L\\Lambda$ and\na canonical isomorphism\n${\\rm Gr}{\\cal F}\\to\n{\\cal F}_0\\otimes_{\\Lambda_0}\n{\\rm Gr}\\Lambda$.\nHence if (\\ref{eqla})\nis an isomorphism for ${\\cal F}_0$ then\n(\\ref{eqla})\nfor ${\\cal F}$ is an isomorphism.\n\\qed}\n\\medskip\n\nWe consider a commutative diagram\n\\begin{equation}\n\\xymatrix{\nX\\ar[rr]^f\\ar[rd]_p&&\nY\\ar[ld]^g\\\\\n&S}\n\\label{eqXYS}\n\\end{equation} \nof schemes.\nThe canonical isomorphism\n$\\overset \\gets g\\circ \\Psi_f\n\\to \\Psi_p$\ninduces an isomorphism of functors\n\\begin{equation}\nR\\overset \\gets g_{*}\\circ R\\Psi_f\n\\to R\\Psi_p\n\\label{eqPsPs}\n\\end{equation}\n\n\nFor an object ${\\cal K}$\nof $D^+(X\\overset \\gets\\times_YY)$\nand a geometric point $x$ of $X$,\nthe restriction of $R\\overset{\\gets} g_*{\\cal K}$\non $x\\overset \\gets\\times_SS=S_{(s)}$\nfor $s=f(x)$\nis canonically identified\nwith $Rg_{(x)*}({\\cal K}|_{Y_{(y)}})$\nfor $y=f(x)$\nin the notation of (\\ref{eqpx})\nby \\cite[(1.9.2)]{TS}.\nFor the stalk at a point $x\\gets t$ of\n$X\\overset \\gets\\times_SS$,\nthis identification gives a canonical isomorphism\n\\begin{equation}\nR\\overset{\\gets} g_*{\\cal K}_{x\\gets t}\n\\to\nR\\Gamma(Y_{(y)}\\times_{S_{(s)}}\nS_{(t)},{\\cal K}|_{Y_{(y)}\\times_{S_{(s)}}\nS_{(t)}}).\n\\label{eqMt}\n\\end{equation}\nFor an object ${\\cal F}$ of $D^+(X)$,\n(\\ref{eqMt}) applied to $Y=X$ \ngives a canonical identification\n\\begin{equation}\nR\\Psi_p{\\cal F}_{x\\gets t}\n\\to\nR\\Gamma(X_{(x)}\\times_{S_{(s)}}\nS_{(t)},{\\cal F}|_{X_{(x)}\\times_{S_{(s)}}\nS_{(t)}})\\label{eqMt2}\n\\end{equation}\nwith the cohomology\nof the Milnor tube \\cite[(1.1.15)]{TS}.\n\nA cartesian diagram\n$$\\begin{CD}\nX@<<< X_T\\\\\n@VfVV @VV{f_T}V\\\\\nS@<i<< T\n\\end{CD}$$\nof schemes defines\na 2-commutative diagram\n$$\n\\begin{CD}\nX_T@<{p_1}<<\nX_T\\overset \\gets\\times_TT\n@<{\\Psi_{f_T}}<< X_T\\\\\n@ViVV@V{\\overset \\gets i}VV@VViV\\\\\nX@<{p_1}<<\nX\\overset \\gets\\times_SS\n@<{\\Psi_f}<< X\n\\end{CD}$$\nand\nthe base change morphisms define\na morphism\n\\begin{equation}\n\\begin{CD}\n@>>> \\overset{\\gets*} ip_1^*\n@>>> \\overset{\\gets*} iR\\Psi_f\n@>>> \\overset{\\gets*} iR\\Phi_f@>>>\\\\\n@.@V{\\simeq}VV @VVV @VVV\\\\\n@>>> p_1^*i^*\n@>>> R\\Psi_{f_T}i^*\n@>>> R\\Phi_{f_T}i^*@>>>\\\\\n\\end{CD}\n\\label{eqbc}\n\\end{equation}\nof distinguished triangles of functors.\nFor an object ${\\cal F}$ of $D^+(X)$,\nwe say that the formation of\n$R\\Psi_f{\\cal F}$ commutes with\nthe base change $T\\to S$ if\nthe middle vertical arrow defines\nan isomorphism\n$\\overset{\\gets*} iR\\Psi_f{\\cal F}\n\\to R\\Psi_{f_T}i^*{\\cal F}$.\n\n\nFor a point $x\\gets t$\nof $X\\overset\\gets\\times_SS$,\nif $T\\subset S$ denotes\nthe closure of the image of $t$ in $S$,\nthe left square of (\\ref{eqbc})\ninduces a commutative diagram\n\\begin{equation}\n\\xymatrix{\n(p_1^*{\\cal F})_{x\\gets t}\n={\\cal F}_x\\ar[rr]\\ar[rrd]\n&&\nR\\Psi_f{\\cal F}_{x\\gets t}\n=\nR\\Gamma(X_{(x)}\\times_{S_{(s)}}\nS_{(t)},{\\cal F}|_{X_{(x)}\\times_{S_{(s)}}\nS_{(t)}})\\ar[d]\\\\\n&&\nR\\Psi_{f_T}({\\cal F}|_{X_T})_{x\\gets t}\n=\nR\\Gamma(X_{(x)}\\times_{S_{(s)}}\nt,{\\cal F}|_{X_{(x)}\\times_{S_{(s)}}\nt}).}\n\\label{eqMf}\n\\end{equation}\nThe vertical arrow\nis the canonical morphism from the cohomology \nof the Milnor tube\nto that of the Milnor fiber\nand the slant arrow\nis the canonical morphism (\\ref{eqla}).\nRecall that we assume\nthat the residue field of $t$\nis a separable closure of\nthe residue field at\nthe image in $S_{(s)}$.\n\nWe interpret the local acyclicity\nin terms of vanishing topos.\n\n\\begin{pr}\\label{lmapp}\nLet $f\\colon X\\to S$\nbe a morphism of schemes.\nThen, for an object ${\\cal F}$ of $D^+(X)$, \nthe conditions\nin {\\rm 1.{}\\!} and {\\rm 2.{}\\!} \nbelow are equivalent to each other respectively.\n\n\n{\\rm 1.}\n{\\rm (1)}\nFor every finite morphism $g\\colon T\\to S$,\nfor every geometric point $x$ of $X$\nand \nfor every specialization\n$t\\gets u$ of\ngeometric points of $T$\nsuch that \n$s=f(x)=g(t)$\nas geometric points of $S$,\nthe canonical morphism\n\\begin{equation}\nR\\Gamma(X_{(x)}\\times_{S_{(s)}}\nT_{(u)},{\\cal F}|_{X_{(x)}\\times_{S_{(s)}}\nT_{(u)}})\n\\to\nR\\Gamma(X_{(x)}\\times_{S_{(s)}}\nu,{\\cal F}|_{X_{(x)}\\times_{S_{(s)}}\nu})\n\\label{eqSTt}\n\\end{equation}\nthat is a vertical arrow in {\\rm (\\ref{eqMf})}\nfor the base change $f_T\\colon X_T\n=X\\times_ST\\to T$\nis an isomorphism.\n\n{\\rm (2)}\nThe formation of $R\\Psi_f{\\cal F}$\ncommutes with finite base change $T\\to S$.\n\n{\\rm 2. ({\\cite[Corollaire 2.6]{app}})}\n{\\rm (1)}\nThe morphism $f\\colon X\\to S$ is \n(resp.\\ universally) locally \nacyclic relatively to ${\\cal F}$.\n\n{\\rm (2)}\nThe canonical morphism\n$p_1^*{\\cal F}\\to\nR\\Psi_f{\\cal F}$\nis an isomorphism\nand the formation of $R\\Psi_f{\\cal F}$\ncommutes with finite (resp.\\ arbitrary) base change $T\\to S$.\n\n{\\rm (3)}\nThe canonical morphism\n$p_1^*{\\cal F}_T\\to\nR\\Psi_{f_T}{\\cal F}_T$\nis an isomorphism\nfor every finite \n(resp.\\ every) morphism $T\\to S$\nand the pull-back ${\\cal F}_T$\nof ${\\cal F}$ on $X_T=X\\times_ST$.\n\\end{pr}\n\n\\proof{\n1.\nLet $T\\to S$ be a finite morphism\nand $x\\mapsto s$ and $t\\gets u$ be as in\nthe condition (1).\nThen, \nfor the geometric point $x'$ of\n$X_T$ defined by a unique point of\n$x\\times_st$,\nthe Milnor tube\n$X_{T (x')}\\times_{T_{(t)}}\nT_{(u)}$\nis canonically isomorphic to\n$X_{(x)}\\times_{S_{(s)}}\nT_{(u)}$ since $T\\to S$ is finite.\nThus, the morphism\n(\\ref{eqSTt}) is identified with\nthe morphism\n$R\\Gamma(X_{(x)}\\times_{S_{(s)}}\nT_{(u)},{\\cal F}|_{X_{(x)}\\times_{S_{(s)}}\nT_{(u)}})\n\\to\nR\\Gamma(X_{(x)}\\times_{S_{(s)}}\nu,{\\cal F}|_{X_{(x)}\\times_{S_{(s)}}\nu})$\nthat is the vertical arrow in {\\rm (\\ref{eqMf})}\nfor $f_T\\colon X_T\\to T$\nat $x'\\gets u$.\n\nLet $T'\\subset T$\nbe the closure of the image of $u$.\nWe consider the\nbase change morphisms\n\\begin{equation}\n\\begin{CD}\nR\\Psi_f{\\cal F}_{x\\gets u}\n=&\nR\\Gamma(X_{(x)}\\times_{S_{(s)}}\nS_{(u)},{\\cal F}|_{X_{(x)}\\times_{S_{(s)}}\nS_{(u)}})\\\\\n&@VVV\\\\\nR\\Psi_{f_T}({\\cal F}|_{X_T})_{x\\gets u}\n=&\nR\\Gamma(X_{(x)}\\times_{S_{(s)}}\nT_{(u)},{\\cal F}|_{X_{(x)}\\times_{S_{(s)}}\nT_{(u)}})\\\\\n&@VVV\\\\\nR\\Psi_{f_{T'}}({\\cal F}|_{X_{T'}})_{x\\gets u}\n=&\nR\\Gamma(X_{(x)}\\times_{S_{(s)}}\nu,{\\cal F}|_{X_{(x)}\\times_{S_{(s)}}\nu}).\n\\end{CD}\n\\label{eqSTft}\n\\end{equation}\nThe condition (1) implies\nthat the lower arrow and\nthe composition are isomorphisms.\nHence, the upper arrow\nis an isomorphism\nand we have\n(1)$\\Rightarrow$(2).\n\nConversely, \nthe condition (2) implies\nthat the upper arrow and\nthe composition are isomorphisms.\nHence, the lower arrow\nthat is the same as\n(\\ref{eqSTt}) is an isomorphism.\nThus, we have\n(2)$\\Rightarrow$(1).\n\n\n2. \nFirst, we show the\nequivalence of (1) and (2)\nin the cases without resp.\nThe condition (1) is equivalent to\nthat the slant arrow in (\\ref{eqMf})\nis an isomorphism\nfor every point $x\\gets t$\nof $X\\overset \\gets\\times_SS$.\nHence the condition (2) implies the condition (1)\nby (2)$\\Rightarrow$(1) in 1.\\\nand the commutativity of the diagram (\\ref{eqMf}).\n\nConversely, by \\cite[Corollaire 2.6]{app},\nif the condition (1) is satisfied,\nthe formation of\n$Rf_{(x)*}({\\cal F}|_{X_{(x)}})$\ncommutes with finite base change\nfor every geometric point $x$ of $X$\nwhere $f_{(x)}\\colon X_{(x)}\\to S_{(s)}$\nis the morphism on the strict localizations\ninduced by $f$.\nThus, for every finite morphism\n$T\\to S$ and every point $x\\gets u$\nof $X_T\\overset\\gets\\times_TT$,\nthe upper arrow in (\\ref{eqSTft})\nis an isomorphism.\nHence\nthe formation of $R\\Psi_f{\\cal F}$\ncommutes with finite base change $T\\to S$.\nFurther the vertical arrow in (\\ref{eqMf})\nis an isomorphism. Thus\nthe canonical morphism\n$p_1^*{\\cal F}\\to\nR\\Psi_f{\\cal F}$\nis an isomorphism\nfurther by \nthe commutativity of the diagram (\\ref{eqMf}).\n\nIf $p_1^*{\\cal F}\\to\nR\\Psi_f{\\cal F}$\nis an isomorphism,\nthe formation of\n$R\\Psi_f{\\cal F}$ commutes\nwith base change $T\\to S$\nif and only if\n$p_1^*{\\cal F}_{X_T}\\to\nR\\Psi_{f_T}{\\cal F}_{X_T}$\nis an isomorphism\nby the left square of (\\ref{eqbc}).\nHence we have\nan equivalence (2)$\\Leftrightarrow$(3).\nThe equivalence (1)$\\Leftrightarrow$(3)\nin the cases with resp.\\ follows\nimmediately from that without resp.\n\\qed}\n\n\n\n\\begin{pr}\\label{prM}\nLet $f\\colon X\\to S$\nbe a morphism of finite type of noetherian schemes\nand let $Z\\subset X$ be a closed\nsubscheme quasi-finite over $S$.\nLet ${\\cal F}$\nbe an object of\n$D^b_c(X)$\nsuch that the restriction of $f\\colon X\\to S$\nto the complement $X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z$ is \n(resp.\\ universally) locally acyclic\nrelatively to the restriction of ${\\cal F}$.\n\n{\\rm 1. (cf.\\ {\\cite[Proposition 6.1]{Or}})}\n$R\\Psi_f{\\cal F}$ and $R\\Phi_f{\\cal F}$\nare constructible\nand their formations commute\nwith finite (resp.\\ arbitrary)\nbase change.\nThe constructible object \n$R\\Phi_f{\\cal F}$ is supported\non $Z\\overset \\gets\\times_SS$.\n\n{\\rm 2. }\nLet $x$ be a geometric point of $X$\nand $s=f(x)$ be the geometric point\nof $S$ defined by the image of $x$ by $f$.\nLet $t$ and $u$ be geometric points\nof $S_{(s)}$\nand  $t\\gets u$ be a specialization. \nThen, there exists a distinguished triangle\n\\begin{equation}\n\\begin{CD}\n@>>>\nR\\Psi_f{\\cal F}_{x\\gets t}\n@>>>\nR\\Psi_f{\\cal F}_{x\\gets u}\n@>>>\n{\\displaystyle\n\\bigoplus_{z\\in (Z\\times_XX_{(x)})\n\\times_{S_{(s)}}t}}\nR\\Phi_f{\\cal F}_{z\\gets u}\n@>>>\n\\end{CD}\n\\label{eqDel}\n\\end{equation}\nwhere \n$R\\Psi_f{\\cal F}_{x\\gets t}\n\\to\nR\\Psi_f{\\cal F}_{x\\gets u}$\nis the cospecialization.\n\\end{pr}\n\nThe commutativity of the formation of $R\\Psi_f{\\cal F}$ with any base change implies\nits constructibility by \\cite[8.1, 10.5]{Or}\nas noted after \\cite[Theorem 1.3.1]{TS}.\n\n\\proof{\n1.\nThe constructibility \nis proved by\ntaking a compactification\nin \\cite[Proposition 6.1]{Or}.\nThe commutativity with base change\nis proved similarly by\ntaking a compactification\nand applying the proper base change\ntheorem.\n\nThe assertion on the support of $R\\Phi_f{\\cal F}$\nfollows from Proposition \\ref{lmapp}.2 (1)$\\Rightarrow$(2).\n\n2.\nLet $t$ and $u$\nbe geometric points of\n$S_{(s)}$ and $t\\gets u$ be a specialization.\nBy replacing $S$ by the strict localization\n$S_{(s)}$\nand shrinking $X$, \nwe may assume that $S=S_{(s)}$, that $X$ is affine\nand that $Z=Z\\times_XX_{(x)}$ is finite over $S$.\n\nWe consider the diagram\n$$\\begin{CD}\ns@. t\\\\\n@V{i_{s}}VV @VV{i_t}V\\\\\nS@<j<< S_{(t)}@<k<< u\n\\end{CD}$$\nand let the morphisms obtained by\nthe base change $X\\to S$\ndenoted by the same letters,\nby abuse of notation.\nSimilarly as the sliced vanishing cycles\nin the proof of \\cite[Proposition 6.1]{Or},\nwe consider an object $\\Phi_{t\\gets u}{\\cal F}$ on \n$X\\times_SS_{(t)}$ fitting in the distinguished\ntriangle\n$\\to j^*{\\cal F}\\to Rk_*(j\\circ k)^*{\\cal F}\\to \n\\Phi_{t\\gets u}{\\cal F}\\to $.\nSince the formation of $R\\Psi_f{\\cal F}$ commutes\nwith finite base change by 1.,\nwe have a distinguished\ntriangle (\\ref{eqDel})\nwith the third term replaced by\n$\\Delta_x=(Rj_*\\Phi_{t\\gets u}{\\cal F})_x$.\nFurther,\nthe third term itself is canonically isomorphic\nto the direct sum of\n$\\Delta_z=(\\Phi_{t\\gets u}{\\cal F})_z$\nfor $z\\in Z_t$.\n\nSince  $R\\Phi_f{\\cal F}$ is acyclic\noutside $Z\\overset\\gets\\times_SS$ by 1.,\nthe canonical morphisms\n$i_s^*{\\cal F}\\to \ni_s^*Rj_*j^*{\\cal F}$\nand\n$i_s^*{\\cal F}\\to \ni_s^*R(j\\circ k)_*(j\\circ k)^*{\\cal F}$\nare isomorphisms on $X_s\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z_s$.\nHence, \nthe restriction $i_s^*Rj_*\\Phi_{t\\gets u}{\\cal F}$\nis acyclic on  $X_s\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z_s$.\nSimilalry, \nthe restriction $i_t^*\\Phi_{t\\gets u}{\\cal F}$\nis acyclic on  $X_t\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z_t$.\n\nWe take a compactification $\\bar X$ of $X$\nand an extension $\\bar {\\cal F}$ of ${\\cal F}$ to $\\bar X$.\nDefine $\\Phi_{t\\gets u}\\bar {\\cal F}$ on \n$\\bar X\\times_SS_{(t)}$ similarly as\n$\\Phi_{t\\gets u}{\\cal F}$\nand set $Y=\\bar X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} X$.\nBy the proper base change theorem,\nthe canonical morphisms\n$R\\Gamma(\\bar X_s,\ni_s^*Rj_*\\Phi_{t\\gets u}\\bar {\\cal F})\n\\gets\nR\\Gamma(\\bar X,\nRj_*\\Phi_{t\\gets u}\\bar {\\cal F})\n\\to\nR\\Gamma(\\bar X\\times_SS_{(t)},\n\\Phi_{t\\gets u}\\bar {\\cal F})\n\\to\nR\\Gamma(\\bar X_t,\ni_t^*\\Phi_{t\\gets u}\\bar {\\cal F})$\nare isomorphisms\nand similarly for the restrictions to $Y$.\nHence, we obtain a commutative diagram\n$$\\begin{CD}\n\\Delta_{x}=&R\\Gamma(Z_s,\ni_s^*Rj_*\\Phi_{t\\gets u}{\\cal F})\n@>>>\nR\\Gamma_{c}(X_s,\ni_s^*Rj_*\\Phi_{t\\gets u}{\\cal F})\n\\\\\n&@VVV@VVV\\\\\n\\bigoplus_{z\\in Z_t}\n\\Delta_z=\n&R\\Gamma(Z_t,\ni_t^*\\Phi_{t\\gets u} {\\cal F})\n@>>>\nR\\Gamma_c(X_t,\ni_t^*\\Phi_{t\\gets u} {\\cal F})\n\\end{CD}$$\nof isomorphisms\nand the assertion follows.\n\\qed}\n\n\\begin{cor}\\label{corsl}\nWe keep the assumptions in Proposition {\\rm \\ref{prM}}\nand \nlet $x$ be a geometric point of $X$\nand $s=f(x)$ be the geometric point\nof $S$ defined by the image of $x$ by $f$\nas in Proposition {\\rm \\ref{prM}.2}.\nThen, the restriction of\nthe constructible sheaf\n$R^q\\Psi_f{\\cal F}$\non $x\\overset\\gets\\times_SS\n=S_{(s)}$\nis locally constant \noutside the image of\nthe finite scheme $Z\\times_XX_{(x)}$\nfor every $q$.\n\\end{cor}\n\n\\proof{\nLet $t$ and $u$\nbe geometric points of $S_{(s)}$\nnot in the image of $Z\\times_XX_{(x)}$\nand $t\\gets u$ be a specialization.\nSince $R\\Psi_f{\\cal F}$\nis constructible,\nit suffices to show that\nthe cospecialization morphism\n$R\\Psi_f{\\cal F}_{x\\gets t}\n\\to\nR\\Psi_f{\\cal F}_{x\\gets u}$ is an isomorphism.\nThen by the assumption on\nthe local acyclicity,\nthe complex $\\Phi_{t\\gets u}{\\cal F}$ \nin the proof of Proposition \\ref{prM}.2 is acyclic.\nHence the assertion follows from (\\ref{eqDel})\nand the isomorphism\n$R\\Phi_f{\\cal F}_{z\\gets u}\\to \n(\\Phi_{t\\gets u}{\\cal F})_z$.\n\\qed}\n\n\\begin{cor}\\label{corsc}\nWe keep the assumptions in Proposition {\\rm \\ref{prM}}.\nWe further assume that \n${\\cal F}$ is of finite tor-dimension.\n\n{\\rm 1.}\nThe complexes\n$R\\Psi_f{\\cal F}$\nand \n$R\\Phi_f{\\cal F}$\nare of finite tor-dimension.\nConsequently,\nthe functions\n$\\dim R\\Psi_f{\\cal F}$\nand \n$\\dim R\\Phi_f{\\cal F}$\nare defined and constructible.\n\n{\\rm 2.}\nDefine a constructible\nfunction $\\delta_{\\cal F}$ \non $X\\overset\\gets\\times_SS$\nsupported on $Z\\overset \\gets\\times_SS$ \nby $\\delta_{\\cal F}(x\\gets t)=\n\\dim R\\Phi_f{\\cal F}_{x\\gets t}$.\nLet $\\Lambda_0$\ndenote the residue field\nof $\\Lambda$ and assume that\n$R\\Phi_f{\\cal F}\n\\otimes^L_{\\Lambda_0}\\Lambda$ \nis acyclic except at degree $0$.\nThen, we have $\\delta_{\\cal F}\\geqq 0$\nand the equality $\\delta_{\\cal F}=0$\nis equivalent to the condition that\nthe morphism $f$ is (resp.\\ universally) locally acyclic\nrelatively to ${\\cal F}$.\n\\end{cor}\n\n\\proof{\n1. By Proposition \\ref{prM}.1,\nProposition \\ref{lmapp}.1\nand Lemma \\ref{lmctf}.1,\nthe complex\n$R\\Psi_f{\\cal F}$\nis of finite tor-dimension\nand hence\n$R\\Phi_f{\\cal F}$\nis also of finite tor-dimension.\nSince they are constructible \nby Proposition \\ref{prM}.1,\nthe functions\n$\\dim R\\Psi_f{\\cal F}$\nand \n$\\dim R\\Phi_f{\\cal F}$\nare defined and constructible.\n\n\n2.\nThe positivity $\\delta_{\\cal F}\\geqq 0$\nfollows from \nthe assumption that\n$R\\Phi_f{\\cal F}$ is acyclic except at degree $0$.\nFurther the equality $\\delta_{\\cal F}=0$\nis equivalent to \n$R\\Phi_f{\\cal F}=0$.\nSince the formation of \n$R\\Psi_f{\\cal F}$\ncommutes with finite (resp.\\ arbitrary) \nbase change by Proposition \\ref{prM}.1,\nit is further\nequivalent to the condition that\nthe morphism $f$ is (resp.\\ universally) locally acyclic\nrelatively to ${\\cal F}$\nby Proposition \\ref{lmapp}.2.\n\\qed}\n\n\\begin{lm}\\label{lmP}\nAssume that\n$\\Lambda=\\Lambda_0$\nis a field.\nThen, the assumption \nthat $R\\Phi_f{\\cal F}$ \nis acyclic except at degree $0$\nin Corollary {\\rm \\ref{corsc}.2}\nis satisfied if the following conditions\nare satisfied:\nThe scheme $S$ is noetherian,\nthe restriction of $f\\colon X\\to S$\nto $X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z$ is \nuniversally locally acyclic\nrelatively to the restriction of ${\\cal F}$\nand the following condition {\\rm (P)}\nis satisfied.\n\n{\\rm (P)}\nFor every morphism\n$T\\to S$ from the spectrum $T$ of\na discrete valuation ring,\nthe pull-back of ${\\cal F}[1]$\nto $X_T$ is perverse.\n\\end{lm}\n\n\\proof{\nLet $x\\gets t$ be a point of\n$X\\overset \\gets\\times_SS$ \nand let $T\\to S$ be a morphism\nfrom the spectrum $T$ of\na discrete valuation ring\nsuch that the image of $T\\to S$\nis the same as that of $\\{f(x),t\\}$.\nSince the formation of $R\\Phi_f{\\cal F}$ commutes\nwith arbitrary base change\nby Proposition \\ref{prM}.1,\nthe base change morphism\n$R\\Phi_f{\\cal F}_{x\\gets t}\n\\to R\\Phi_{f_T}({\\cal F}|_{X_T})_{x\\gets t}$\nis an isomorphism.\nThe complex\n$R\\Phi_{f_T}({\\cal F}|_{X_T})$ \nis a perverse sheaf \nby the assumption (P)\nand by the theorem of Gabber\n\\cite[Corollaire 4.6]{au}.\nSince \n$R\\Phi_{f_T}({\\cal F}|_{X_T})$ \nvanishes outside the closed fiber $Z_s$,\nthis implies that\nthe complex $R\\Phi_f{\\cal F}$\nis acyclic except at degree $0$.\n\\qed}\n\\medskip\n\nThe condition {\\rm (P)}\nis satisfied\nif $f\\colon X\\to S$ is smooth of relative dimension $d$\nand\n${\\cal F}=j_!{\\cal G}[d]$\nfor the open immersion $j\\colon U\\to X$\nof the complement $U=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} D$\nof a Cartier divisor $D$\nand a locally constant constructible \nsheaf ${\\cal G}$\nof free $\\Lambda$-modules on $U$.\n\n\\subsection{Semi-continuity of the Swan conductor}\\label{sssc}\n\nIn this subsection,\nwe assume that $\\Lambda=\\Lambda_0$ is a field\nfor simplicity.\nIf $\\Lambda$ is not a field,\nthe same results hold without modifications\nfor constructible complexes of\n$\\Lambda$-modules\nof finite tor-dimension,\nby taking \n$\\otimes^L_{\\Lambda}\\Lambda_0$.\n\n\nWe reformulate the main result of\nDeligne-Laumon in \\cite{DL}\nin Proposition \\ref{prDL} below.\nLet $f\\colon X\\to S$\nbe a flat morphism of relative dimension $1$\nand let $Z\\subset X$ be a closed subscheme.\nAssume that \n$X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z$ is smooth over $S$\nand that $Z$ is quasi-finite over $S$.\nLet ${\\cal F}$ be a constructible\ncomplex of $\\Lambda$-modules on $X$ such that\nthe restrictions of the cohomology sheaves\non $X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z$ are locally constant.\n\nLet $s\\to S$ be a geometric point\nsuch that the residue\nfield is an {\\em algebraic closure}\nof the residue field of the image of $s$\nin $S$.\nFor a geometric point \n$x$ of $Z$ above $s$,\nthe normalization of\nthe strict localization\n$X_{s,(x)}$ is the finite disjoint\nunion $\\amalg_i{\\rm Spec}\\ {\\cal O}_{K_i}$ \nwhere ${\\cal O}_{K_i}$ are strictly local discrete valuation\nrings with algebraically\nclosed residue field $k(s)$.\nLet $\\bar\\eta_i$ denote\nthe geometric point\ndefined by a separable closure $\\bar K_i$ of\nthe fraction field $K_i$.\nFor a $\\Lambda$-representation $V$\nof the absolute Galois group $G_{K_i}\n={\\rm Gal}(\\bar K_i\/K_i)$,\nthe Swan conductor\n${\\rm Sw}_{K_i}V\\in {\\mathbf N}$\nis defined \\cite{DL}\nand the total dimension is\ndefined as the sum\n$\\dim{\\rm tot}_{K_i}V\n=\\dim V\n+{\\rm Sw}_{K_i}V$.\n\nThe stalk\n${\\cal H}^q({\\cal F})_{{\\bar \\eta}_i}$\nfor each integer $q$\ndefines a $\\Lambda$-representation\nof the absolute Galois group $G_{K_i}$\nand hence the total dimension\n$\\dim{\\rm tot}_{K_i}{\\cal F}_{{\\bar \\eta}_i}$\nis defined as the alternating sum\n$\\sum_q(-1)^q\n\\dim{\\rm tot}_{K_i}{\\cal H}^q({\\cal F})_{{\\bar \\eta}_i}$.\nWe define the Artin conductor by\n\\begin{equation}\na_x({\\cal F}|_{X_s})\n=\n\\sum_i\n\\dim{\\rm tot}_{K_i}{\\cal F}_{{\\bar \\eta}_i}\n-\n\\dim{\\cal F}_x.\n\\label{eqax}\n\\end{equation}\nWe define \na function $\\varphi_{\\cal F}$\non $X$ supported on $Z$ by\n\\begin{equation}\n\\varphi_{\\cal F}(x)=\na_x({\\cal F}|_{X_s})\n\\label{eqphis}\n\\end{equation}\nfor $s=f(x)$.\nThe derivative $\\delta(\\varphi_{\\cal F})$\non $X\\overset\\gets\\times_SS$\nis defined by (\\ref{eqdel}).\n\n\n\\begin{pr}[{\\cite[Th\\'eor\\`eme 2.1.1]{DL}}]\\label{prDL}\nLet $S$ be a noetherian scheme\nand \n$f\\colon X\\to S$\nbe a flat morphism of relative dimension $1$.\nLet $Z\\subset X$\nbe a closed subscheme quasi-finite\nover $S$\nsuch that $U=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z$ is smooth over $S$.\nLet ${\\cal F}$ be a constructible\ncomplex of $\\Lambda$-modules \non $X$ such that\nthe restrictions of cohomology sheaves\non $X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z$ are locally constant.\n\n{\\rm 1.}\nThe objects $R\\Psi_f{\\cal F}$\nand $R\\Phi_f{\\cal F}$\nare constructible\nand their formations commutes\nwith any base change.\nThe function\n$\\varphi_{\\cal F}$ {\\rm (\\ref{eqphis})}\nsatisfies\n\\begin{equation}\n\\dim R\\Phi_f{\\cal F}_{x\\gets t}=\n\\delta(\\varphi_{\\cal F})(x\\gets t)\n\\label{eqDL}\n\\end{equation}\nand is constructible.\n\n{\\rm 2.}\nAssume ${\\cal F}=j_!{\\cal G}[1]$\nfor the open immersion \n$j\\colon U=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z\\to X$\nand a locally constant constructible \nsheaf ${\\cal G}$\non $U$\nand that $Z$ is flat over $S$.\nThen, we have $\\delta(\\varphi_{\\cal F})\\geqq 0$.\nThe function $\\varphi_{\\cal F}$\nis {\\em flat} over $S$\nif and only if\n$f\\colon X\\to S$ is universally locally acyclic\nrelatively to ${\\cal F}=j_!{\\cal G}[1]$.\n\\end{pr}\n\n\\proof{We sketch and\/or recall an outline of\nproof with some modifications.\n\n{\\rm 1.}\nThe constructibility of $R\\Psi_f{\\cal F}$\nand $R\\Phi_f{\\cal F}$\nand the commutativity with base\nchange follow\nfrom Proposition \\ref{prM}.1\nand the local acyclicity of smooth morphism.\n\nBy devissage, the proof of \n(\\ref{eqDL}) is reduced to \nthe case where ${\\cal F}=j_!{\\cal G}[1]$\nfor the open immersion \n$j\\colon U=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z\\to X$\nand a locally constant sheaf ${\\cal G}$ on $U$.\nBy the commutativity with base\nchange,\nthe equality (\\ref{eqDL})\nis reduced to the case where $S$\nis the spectrum of a complete\ndiscrete valuation ring\nwith algebraically closed residue field.\nFurther by base change\nand the normalization, we may assume \nthat $X$ is normal\nand that its generic fiber is smooth.\nBy devissage, we may assume\nthat $Z$ is flat over $S$.\n\nIn this case, (\\ref{eqDL})\nwas first proved in \\cite{DL}, under an\nextra assumption that $X$ is smooth,\nby constructing a\ngood compactification using a deformation argument.\nLater it was reproved together with a generalization\n in \\cite[Remark (4.6)]{Kato}\nusing the semi-stable reduction theorem\nof curves\nwithout using the deformation argument.\n\nSince $R\\Phi_f{\\cal F}$ is constructible,\nthe equality (\\ref{eqDL}) implies that\nthe function $\\delta(\\varphi_{\\cal F})$\non $Z\\overset\\gets\\times_SS$\nis constructible.\nHence $\\varphi_{\\cal F}$\non $Z$ is constructible\nby Lemma \\ref{lmscZ}.2.\n\n2.\nThe complex $R\\Phi_f{\\cal F}$\nis acyclic except at degree $0$\nby Lemma \\ref{lmP}.\nHence the assertions follow\nfrom the equality (\\ref{eqDL}) \nand Corollary \\ref{corsc}.2.\n\\qed}\n\n\\begin{cor}\\label{corDL}\nAssume further that $Z$ is finite \nand flat over $S$\nand that ${\\cal F}=j_!{\\cal G}$\nfor a locally constant sheaf ${\\cal G}$\non $U$.\nThen, \nthe function $f_*\\varphi_{\\cal F}$ \n{\\rm (\\ref{eqf*})} on $S$\nis lower semi-continuous.\nThe function $f_*\\varphi_{\\cal F}$\nis locally constant if and only if\n$f\\colon X\\to S$ is universally locally acyclic\nrelatively to ${\\cal F}=j_!{\\cal G}$.\n\\end{cor}\n\n\\proof{\nIt follows from Proposition \\ref{prDL},\nLemma \\ref{lmscZ}.4 and Corollary \\ref{corsc}.2.\nThe lower semi-continuity\nreplaces the upper semi-continuity\nbecause of the shift $[1]$\nin Proposition \\ref{prDL}.2.\n\\qed}\n\\medskip\n\nWe give a slight generalization\nof Proposition \\ref{prDL}\nusing vanishing topos.\nLet \n\\begin{equation}\\xymatrix{\nZ\\ar[r]^{\\subset}&\nX\\ar[rr]^f\\ar[rd]_p&&\nY\\ar[ld]^g\\\\\n&&S}\n\\label{eqfpg}\n\\end{equation}\nbe a commutative diagram of morphisms\nof finite type of noetherian schemes\nsuch that\n$g\\colon Y\\to S$\nis flat of relative dimension $1$\nand that $Z\\subset X$ \nis a closed subscheme\nquasi-finite over $S$.\nFor geometric point\n$x\\to X$,\nwe set $y=f(x)$ and $s=p(x)$\nand define \n$T_{(x)}\\subset Y_{(y)}$ to be the image\nof the finite scheme\n$Z\\times_XX_{(x)}$ over $S_{(s)}$\nby $f_{(x)}\\colon X_{(x)}\\to Y_{(y)}$.\nAssume that, for every geometric point\n$x\\to X$,\nthe complement\n$Y_{(y)}\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} T_{(x)}$\nis essentially smooth over $S_{(s)}$.\n\nLet ${\\cal K}$\nbe an object of\n$D^b_c(X\\overset \\gets\\times_YY)$\nsuch that, \nfor every geometric point\n$x\\to X$, the restrictions of\ncohomology sheaves on\n$Y_{(y)}\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} T_{(x)}\n\\subset Y_{(y)}=x\\overset \\gets\\times_YY$ \nare locally constant.\nThen, similarly as (\\ref{eqphis}),\nwe define \na function $\\psi_{\\cal K}$\non $X\\overset\\gets\\times_YY$ by\n\\begin{equation}\n\\psi_{\\cal K}(x\\gets w)=\na_w({\\cal K}|_{Y_{(y)}\\times_{S_{(s)}t}})\n\\label{eqphi2}\n\\end{equation}\nwhere $y=f(x),s=p(x)$ and $t=g(w)$\nwith {\\em algebraically closed} residue field $k(t)$.\nWe also define a function $\\delta(\\psi_{\\cal K})$\non $X\\overset\\gets\\times_SS$ by\n(\\ref{eqdelY}).\n\n\\begin{pr}\\label{prc}\nLet the notation be as above.\nLet ${\\cal K}$\nbe an object of\n$D_c^b(X\\overset \\gets\\times_YY)$\nand $x\\gets t$ be a point of\n$X\\overset \\gets\\times_SS$.\nSet $y=f(x)$ and $s=p(x)$\nand assume that the restriction of\ncohomology sheaf ${\\cal H}^q{\\cal K}$\non $Y_{(y)}\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} T_{(x)}$ \nis locally constant for every $q$.\nThen, we have\n\\begin{equation}\n\\dim R\\overset\\gets g_*{\\cal K}_{x\\gets t}-\n\\dim R\\overset\\gets g_*{\\cal K}_{x\\gets s}=\n\\delta(\\psi_{\\cal K})(x\\gets t).\n\\label{eqDL2}\n\\end{equation}\n\\end{pr}\n\n\\proof{\nBy the  canonical isomorphisms\n$R\\overset\\gets g_*{\\cal K}_{x\\gets t}\n\\to\nR\\Gamma(Y_{(y)}\\times_{S_{(s)}}S_{(t)},\n{\\cal K}|_{Y_{(y)}\\times_{S_{(s)}}S_{(t)}})$\nand\n$R\\overset\\gets g_*{\\cal K}_{x\\gets s}\n\\to\nR\\Gamma(Y_{(y)},\n{\\cal K}|_{Y_{(y)}})={\\cal K}_y$\n(\\ref{eqMt}),\nwe obtain a distinguished triangle\n$\\to\nR\\overset\\gets g_*{\\cal K}_{x\\gets s}\n\\to\nR\\overset\\gets g_*{\\cal K}_{x\\gets t}\n\\to\nR\\Phi_{g_{(y)}}({\\cal K}|_{Y_{(y)}})_{y\\gets t}\n\\to $.\nHence it follows\nfrom Proposition \\ref{prDL}.1.\n\\qed}\n\\medskip\n\n\nIn fact, (\\ref{eqDL})\nis a special case of (\\ref{eqDL3}) below\nwhere $X=Y$.\n\n\\begin{cor}\\label{corc}\nWe keep the notation in Proposition {\\rm \\ref{prc}}.\nLet ${\\cal F}$ be an object of\n$D^b_c(X)$ such that ${\\cal K}=\nR\\Psi_f{\\cal F}$ is an object of\n$D^b_c(X\\overset\\gets\\times_YY)$.\nAssume that ${\\cal K}$\nand a point $x\\gets t$ of\n$X\\overset \\gets\\times_SS$\nsatisfies the condition in Proposition \n{\\rm \\ref{prc}}.\nThen, we have\n\\begin{equation}\n\\dim R\\Phi_p{\\cal F}_{x\\gets t}\n=\n\\delta(\\psi_{\\cal K})(x\\gets t).\n\\label{eqDL3}\n\\end{equation}\n\\end{cor}\n\n\\proof{\nBy the isomorphisms\n$R\\Psi_p{\\cal F}\n\\to R\\overset\\gets g_*{\\cal K}$\nand $R\\overset\\gets g_*{\\cal K}_{x\\gets s}\n\\to {\\cal K}_y\\to {\\cal F}_x$,\nwe obtain a distinguished triangle\n$\\to R\\overset\\gets g_*{\\cal K}_{x\\gets s}\n\\to R\\overset\\gets g_*{\\cal K}_{x\\gets t}\n\\to R\\Phi_p{\\cal F}_{x\\gets t}\\to$.\nHence it follows from\n(\\ref{eqDL2}).\n\\qed}\n\\medskip\n\nWe consider the diagram (\\ref{eqfpg})\nsatisfying the condition there\nand assume further that $g\\colon Y\\to S$ \nis smooth.\nLet ${\\cal F}$\nbe an object of\n$D^b_c(X)$ and \nassume that $p\\colon X\\to S$\nis locally acyclic\nrelatively to ${\\cal F}$\nand that the restriction of $f\\colon X\\to Y$\nto the complement $X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z$ is\nlocally acyclic\nrelatively to the restriction of ${\\cal F}$.\n\nWe define a function \n$\\varphi_{{\\cal F},f}$\non $Z$ as follows.\nFor a geometric point\n$x$ of $Z$,\nset $y=f(x)$ and $s=p(x)$.\nWe regard $s$ as a geometric point\nof $S$ such that the residue\nfield is an {\\em algebraic closure}\nof the residue field of the image of $s$\nin $S$.\nThe base change \n$f_s\\colon X_s\\to Y_s$\nof $f\\colon X\\to Y$\nis a morphism to a smooth curve\nover the algebraically closed field $k(s)$.\nThe strict localization\n$Y_{s,(y)}$ is ${\\rm Spec}\\ {\\cal O}_{K_y}$\nfor a strictly local\ndiscrete valuation ring ${\\cal O}_{K_y}$\nwith an algebraically closed\nresidue field $k(y)=k(s)$\nsince $Y_s$ is a smooth curve over $s$.\nThe cohomology of the \nstalk of the vanishing cycles complex\n$\\phi_x({\\cal F}|_{X_s},f_s)$\ndefine $\\Lambda$-representations\nof the absolute Galois group\n$G_{K_y}$ and hence the total dimension\n$\\dim{\\rm tot}_y\n\\phi_x({\\cal F}|_{X_s},f_s)$\nis defined as the alternating sum.\nSimilarly as (\\ref{eqphis}),\nwe define a function \n$\\varphi_{{\\cal F},f}$\non $Z$ by\n\\begin{equation}\n\\varphi_{{\\cal F},f}(x)\n=\n\\dim{\\rm tot}_y\n\\phi_x({\\cal F}|_{X_s},f_s)\n\\label{eqphKf}\n\\end{equation}\n\n\n\\begin{pr}\\label{prMsc}\nLet \n\\begin{equation*}\\xymatrix{\nZ\\ar[r]^{\\subset}&\nX\\ar[rr]^f\\ar[rd]_p&&\nY\\ar[ld]^g\\\\\n&&S}\n\\leqno{\\rm (\\ref{eqfpg})}\n\\end{equation*}\nbe a commutative diagram of morphisms\nof finite type of noetherian schemes\nsuch that\n$g\\colon Y\\to S$\nis {\\em smooth} of relative dimension $1$\nand that $Z\\subset X$ \nis a closed subscheme\nquasi-finite over $S$.\n\nLet ${\\cal F}$\nbe an object of\n$D^b_c(X)$ and \nassume that $p\\colon X\\to S$ is\nlocally acyclic\nrelatively to  ${\\cal F}$\nand that the restriction of $f\\colon X\\to Y$\nto the complement $X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z$ is\nlocally acyclic\nrelatively to the restriction of ${\\cal F}$.\nThen, the function \n$\\varphi_{{\\cal F},f}$ \n{\\rm (\\ref{eqphKf})} on $Z$\nis constructible and {\\em flat} over $S$.\nIf $Z$ is \\'etale over $S$,\nit is locally constant.\n\\end{pr}\n\n\n\\proof{\nLet $x$ be a geometric point of\n$Z$ and let $y=f(x)$ and\n$s=p(x)$ be its images.\nWe regard $s$ as a geometric point\nof $S$ such that the residue\nfield is an {\\em algebraic closure}\nof the residue field of the image of $s$\nin $S$.\nLet $Y_{s,{(y)}}={\\rm Spec}\\ {\\cal O}_{K_y}$\nbe the strict localization\nof the geometric fiber and let\n$x\\gets u$ be the point\nof $X\\overset\\gets \\times_YY$\ndefined by a separable closure\nof $K_y$.\nThe complex $R\\Phi_f{\\cal F}$\nis constructible\nand its construction commutes\nwith base change\nby Proposition \\ref{prM}.1.\nHence, we have a canonical\nisomorphism\n$\\phi_x({\\cal F}|_{X_s},f_s)\n\\to\nR\\Phi_f{\\cal F}_{x\\gets u}$\nand\n\\begin{equation}\n\\varphi_{{\\cal F},f}(x)\n=\n\\dim{\\rm tot}_y\n\\phi_x({\\cal F}|_{X_s},f_s)\n=\n\\dim{\\rm tot}_yR\\Phi_f{\\cal F}_{x\\gets u}.\n\\label{eqphw}\n\\end{equation}\n\nWe apply Proposition \\ref{prc} to\n${\\cal K}=R\\Psi_f{\\cal F}$.\nThe assumption in Proposition \\ref{prc} that\n${\\cal H}^q{\\cal K}\n=R^q\\Psi_f{\\cal F}$\non\n$Y_{(y)}\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} T_{(x)}\\subset\nY_{(y)}=x\\overset \\gets\\times_YY$\nis locally constant for every $q$\nis satisfied\nfor every geometric point $x$ of $X$\nby Corollary \\ref{corsl}.\nHence the function \n$\\psi_{\\cal K}$  (\\ref{eqphi2})\nfor ${\\cal K}=R\\Psi_f{\\cal F}$\nis defined as a function\non $X\\overset\\gets\\times_YY$.\n\nIn order to apply Lemma \\ref{lmdelp}, \nwe show\n$$\\psi_{\\cal K}(x\\gets w)\n=\\sum_{z\\in Z_{(x)}\\times_{Y_{(y)}}w}\n\\varphi_{{\\cal F},f}(z)$$\nfor a point $x\\gets w$\nof $Z\\overset\\gets\\times_YY$\nsuch that $w$ is supported on \nthe image $T_{(x)}\\subset Y_{(y)}$\nof $Z_{(x)}$.\nBy the assumption that $Y\\to S$ is\nsmooth, the fiber\n$Y_{(w)}\\times_{S_{(t)}}t$\nfor $t=p(w)$\nis the spectrum of a discrete valuation ring.\nLet $v$ be its geometric generic point\nregarded as a geometric point of\n$Y_{(w)}$.\n\nBy (\\ref{eqphi2}) and (\\ref{eqax}), we have\n$$\\psi_{\\cal K}(x\\gets w)\n=\n\\dim{\\rm tot}_w(R\\Psi_f{\\cal F}_{x\\gets v})\n-\n\\dim (R\\Psi_f{\\cal F}_{x\\gets w}).$$\nWe apply Proposition \\ref{prM}.2\nto $f\\colon X\\to Y$\nand specializations $y\\gets w\\gets v$\nto compute the right hand side.\nThen, the distinguished triangle (\\ref{eqDel})\nimplies that the right hand side equals\n$\\sum_{z\\in Z_{(x)}\\times_{Y_{(y)}}w}\n\\dim{\\rm tot}_w(R\\Phi_f{\\cal F}_{z\\gets v})$.\nBy (\\ref{eqphw})\napplied for $z$,\nit is further equal to $\n\\sum_{z\\in Z_{(x)}\\times_{Y_{(y)}}w}\n\\varphi_{{\\cal F},f}(z)\n$ as required.\n\nTherefore, by applying Lemma \\ref{lmdelp}, \nwe obtain\n$\\delta(\\psi_{\\cal K})=\n\\delta(\\varphi_{{\\cal F},f})$\nas functions on $Z\\overset\\gets\\times_SS$.\nSince $R\\Phi_p{\\cal F}=0$,\nthe function $\\psi_{\\cal K}$ is flat over $S$\nby (\\ref{eqDL3}).\nHence, \nthe function $\\varphi_{{\\cal F},f}$ is also flat\nover $S$.\nSince it is flat over $S$, \nthe function $\\varphi_{{\\cal F},f}$ is \nconstructible by Lemma \\ref{lmscZ}.2.\n\n\nIf $Z$ is \\'etale over $S$, the function\n$\\varphi_{{\\cal F},f}$ is locally constant by\nLemma \\ref{lmscZ}.1.\n\\qed}\n\n\\section{Closed conical subsets\non the cotangent bundle}\\label{sCc}\n\nIn this preliminary section,\nwe recall and study some notions\nintroduced in \\cite{Be}\nrelated to closed conical subsets\nof the cotangent bundles.\n\n\n\\subsection{$C$-transversality}\\label{ssCt}\n\n\n\nWe say that a closed subset \n$C\\subset E$ of a vector bundle\n$E$ over $X$ is {\\em conical} if it is stable\nunder the action of \nthe multiplicative group ${\\mathbf G}_m$.\nEquivalently,\nit is defined by a graded ideal ${\\cal I}$\nof the graded algebra $S^\\bullet_{{\\cal O}_X} \n{\\cal E}^\\vee$,\nif the vector bundle $E={\\mathbf V}({\\cal E})$ \nis associated to a locally free \n${\\cal O}_X$-module ${\\cal E}$.\nFor a closed conical subset\n$C\\subset E$,\nwe call its intersection\n$B$ with the $0$-section\nregarded as a closed\nsubset of $X$ the {\\em base} of $C$.\n\nLet \n${\\mathbf P}(C)={\\rm Proj}_X\n(S^\\bullet {\\cal E}^\\vee\/{\\cal I})\n\\subset {\\mathbf P}(E)={\\rm Proj}_X\nS^\\bullet {\\cal E}^\\vee$\ndenote the projectivization.\nThe projectivization\n${\\mathbf P}(C)$ is empty if and\nonly if $C$ is a subset of the\n$0$-section.\nThe projectivization \n${\\mathbf P}(C)$ itself does not\ndetermine $C$ \nbut the pair with the base\n$B$ determines $C$ uniquely.\n\nWe study the intersection of \na closed conical subset \nwith the inverse image of the 0-section\nby a morphism of vector bundles.\nFor a morphism $X\\to Y$ \nof finite type of locally noetherian schemes\nand a closed subset $Z\\subset X$,\nwe say that the restriction of $X\\to Y$ on $Z$\nis {\\em finite} (resp.\\ {\\em proper}) \nif for every closed subscheme\nstructure or equivalently\nfor the reduced closed subscheme\nstructure on $Z$,\nthe induced morphism\n$Z\\to Y$ is finite (resp.\\ proper).\n\n\n\\begin{lm}[{\\rm \\cite[Lemma 1.2 (ii)]{Be}}]\\label{lmnc}\nLet $E\\to F$ be \na morphism of vector bundles\nover a locally noetherian scheme $X$.\nFor a closed conical subset $C\n\\subset E$,\nthe following\nconditions are equivalent:\n\n\n{\\rm (1)}\nThe intersection of $C$\nwith the inverse image $K$ of\nthe $0$-section of $F$ by\n$E\\to F$ is a subset of\nthe $0$-section of $E$.\n\n{\\rm (2)}\nThe restriction of\n$E\\to F$ on $C$ is finite.\n\\end{lm}\n\n\\proof{\n(1)$\\Rightarrow$(2):\nSince the question is local on \n$X$, we may assume $X$ is\naffine.\nThen the assertion\nfollows from the elementary Lemma below.\n\n(2)$\\Rightarrow$(1):\nThe intersection $C\\cap K\\subset E$ is \na closed conical subset\nfinite over $X$.\nHence, it is a subset of the\n0-section.\n\\qed}\n\n\\begin{lm}\\label{lmRI}\nLet $R=\\bigoplus_{n\\geqq 0}\nR_n$ be a graded ring\nand $I=\\bigoplus_{n> 0}\nR_n\\subset R$ be the graded ideal.\nLet $M=\\bigoplus_{n\\geqq 0}\nM_n$ be a graded $R$-module.\nIf the $R\/I$-module\n$M\/IM$ is finitely generated,\nthen the $R$-module\n$M$ is finitely generated.\n\\end{lm}\n\n\\proof{\nLet $x_1,\\ldots,x_m\\in M$\nbe a lifting of a system of generators \nof $M\/IM$ consisting\nof homogeneous elements\nand let $N\\subset M$ be \nthe sub $R$-module\ngenerated by $x_1,\\ldots,x_m$.\nBy induction on $n$,\nthe morphism\n$N\/I^nN\\to M\/I^nM$ is \nsurjective for every $n\\geqq 0$.\nThus, we have $M_n\\subset N$\nfor every $n\\geqq 0$\nand the assertion follows.\n\\qed}\n\\medskip\n\n\n\nIn the rest of this article,\n$k$ denotes a field of characteristic \n$p\\geqq 0$\nand $X$ denotes a smooth scheme over $k$,\nunless otherwise stated.\nThe cotangent bundle\n$T^*X$ is the covariant\nvector bundle over $X$\nassociated to the locally free ${\\cal O}_X$-module\n$\\Omega^1_{X\/k}$.\nThe $0$-section of $T^*X$ is \nidentified with the conormal bundle\n$T^*_XX$ of $X\\subset X$.\n\nFor a closed conical subset\n$C\\subset T^*X$, \nwe define the condition for\na morphism coming into $X$\nto be $C$-transversal.\n\n\\begin{df}[{\\rm \\cite[1.2]{Be}}]\\label{dfCh}\nLet $X$ be a smooth scheme \nover a field $k$\nand let $C\\subset T^*X$ be a closed conical subset of \nthe cotangent bundle.\nLet $h\\colon W\\to X$ be a \nmorphism of smooth schemes over $k$.\nDefine  \n\\begin{equation}\nh^*C=W\\times_XC\\subset W\\times_XT^*X\n\\label{eqhC}\n\\end{equation}\nto be the pull-back of $C$ and\nlet $K\\subset W\\times_XT^*X$ \nbe the inverse image of the $0$-section \nby the canonical morphism\n$dh\\colon W\\times_XT^*X\\to T^*W$.\n\n{\\rm 1.}\nFor a point $w\\in W$,\nwe say that $h\\colon W\\to X$\nis {\\em $C$-transversal} at $w$\nif the fiber $(h^*C\\cap K)\\times_Ww$\nof the intersection \nis a subset of the $0$-section\n$W\\times_XT^*_XX\\subset W\\times_XT^*X$.\n\nWe say that $h\\colon W\\to X$\nis {\\em $C$-transversal}\nif the intersection $h^*C\\cap K$\nis a subset of the $0$-section\n$W\\times_XT^*_XX\\subset W\\times_XT^*X$.\n\n{\\rm 2}.\nIf $h\\colon W\\to X$\nis $C$-transversal,\nwe define a closed conical subset\n\\begin{equation}\nh^{\\circ}C\\subset T^*W\n\\label{eqhoC}\n\\end{equation}\nto be the image of\n$h^*C$ by $W\\times_XT^*X\\to T^*W$.\n\\end{df}\n\nBy Lemma {\\rm \\ref{lmnc}},\nif $h\\colon W\\to X$ is $C$-transversal,\nthen $h^{\\circ}C$ is a closed conical subset\nof $T^*W$.\n\n\\begin{lm}\\label{lmCh}\nLet $X$ be a smooth scheme \nover a field $k$\nand let $C\\subset T^*X$ be a closed conical subset of \nthe cotangent bundle.\nLet $h\\colon W\\to X$ be a morphism \nof smooth schemes over $k$.\n\n\n{\\rm 1.}\nIf $h$ is smooth,\nthen $h$ is $C$-transversal\nand the canonical morphism\n$h^{*}C\\to h^{o}C$ is an isomorphism.\n\n{\\rm 2.}\nIf $C\\subset T^{*}_{X}X$ is \na subset of the $0$-section,\nthen $h$ is $C$-transversal.\n\n{\\rm 3.}\n{\\rm (cf.\\ \\cite[Lemma 2.2 (i)]{Be})}\nFor a morphism $g\\colon V\\to W$\nof smooth schemes over $k$,\nthe following conditions are equivalent:\n\n{\\rm (1)}\n$h$ is $C$-transversal\non a neighborhood of\n$g(V)\\subset W$ and \n$g\\colon V\\to W$ is\n$h^{\\circ}C$-transversal.\n\n{\\rm (2)}\nThe composition $h\\circ g\\colon V\\to X$\nis $C$-transversal.\n\n{\\rm 4.} {\\rm (\\cite[Lemma 1.2 (i)]{Be})}\nThe subset of $W$ consisting of\npoints $w\\in W$ where\n$h\\colon W\\to X$ is $C$-transversal\nis an open subset of $W$.\n\n{\\rm 5.}\nLet $D=\\bigcup_{i=1}^mD_i$\nbe a divisor with simple normal crossings\nof $X$ relatively to $X\\to {\\rm Spec}\\ k$\nand let \n\\begin{equation}\nC_D=\\bigcup_{I\\subset\\{1,\\ldots\\,m\\}}\nT^*_{D_I}X\\subset T^*X\n\\label{eqCD}\n\\end{equation}\nbe the union of the conormal bundles\nof the intersections\n$D_I=\\bigcap_{i\\in I}D_i$\nof irreducible components\nfor all subsets $I\\subset\\{1,\\ldots,m\\}$\nof indices, including $D_\\varnothing=X$.\nThen, $h\\colon W\\to X$ is $C$-transversal\nif and only if\n$h^*D=D\\times_XW$\nis a divisor with simple normal crossings\nrelatively to $W\\to {\\rm Spec}\\ k$\nand \n$h^*D_i\\subset W$ are smooth divisors\nfor $i=1,\\ldots,m$.\n\\end{lm}\n\nThe assertion 3 implies that\n$C$-transversal morphisms\nhave similar properties as\n\\'etale morphisms.\nFor ${\\cal F}$-transversal morphisms \nintroduced in Definition \\ref{dfprpg},\nproperties corresponding to 1-3\nwill be proved \nin Lemma \\ref{lmprpg}.\n\n\\proof{\n1. \nSince\nthe canonical morphism\n$dh\\colon W\\times_XT^*X\\to T^*W$\nis a closed immersion,\nthe intersection $h^*C\\cap K$\nis a subset of the $0$-section \n$K\\subset W\\times_XT^*X$ and\nthe morphism $h^*C\\to h^{\\circ}C$ is\nan isomorphism.\n\n\n2. \nSince $h^*C\\subset\nW\\times_XT^*X$ is a subset of \nthe $0$-section,\nthe assertion follows.\n\n3.\nSince \n$V\\times_XT^*X\\to T^*V$\nis the composition\n$V\\times_XT^*X\\to V\\times_WT^*W\\to T^*V$,\nthe condition (2) is equivalent to\nthat both \nthe intersection of\n$(hg)^*C=g^*(h^*C)\n\\subset V\\times_XT^*X$ with the\ninverse image of the $0$-section by\n$V\\times_XT^*X\\to V\\times_WT^*W$\nand the intersection of the image of\n$(hg)^*C$ in\n$V\\times_WT^*W$\nwith the\ninverse image of the $0$-section by\n$V\\times_WT^*W\\to T^*V$\nare subsets of the $0$-sections.\nBy 4 below,\nthe first condition means that\n$h$ is $C$-transversal\non a neighborhood of\n$g(V)$ and then the second means that\n$g\\colon V\\to W$ is\n$h^{\\circ}C$-transversal.\n\n4.\nThe complement of the subset is\nthe image of the closed subset\n${\\mathbf P}(h^*C\\cap K)\\subset\n{\\mathbf P}(W\\times_XT^*X)$\nof a projective space bundle over $W$.\n\n5. The $C$-transversality is equivalent to\nthe injectivity of\n$W\\times_XT^*_{D_I}X\\to\nT^*W$ for all $I\\subset \\{1,\\ldots, m\\}$.\nHence the assertion follows.\n\\qed}\n\\medskip\n\n\nFor a closed conical subset\n$C\\subset T^*X$, \nwe define the condition for\na morphism going out of $X$\nto be $C$-transversal.\n\n\\begin{df}[{\\rm \\cite[1.2]{Be}}]\\label{dfCf}\nLet $X$ be a smooth scheme \nover a field $k$\nand let $C\\subset T^*X$ be a closed conical subset \nof the cotangent bundle.\nLet $f\\colon X\\to Y$ \nbe a morphism of\nsmooth schemes over $k$.\n\n\n{\\rm 1.}\nFor a point $x\\in X$,\nwe say that $f\\colon X\\to Y$\nis {\\em $C$-transversal} at $x$\nif the fiber $df^{-1}(C)\\times_Xx$\nof the inverse image\nof $C$ by the canonical morphism\n$df\\colon X\\times_YT^*Y\\to T^*X$\nis a subset of the $0$-section\n$X\\times_YT^*_YY\\subset X\\times_YT^*Y$.\n\nWe say that $f\\colon X\\to Y$\nis {\\em $C$-transversal}\nif the inverse image $df^{-1}(C)$\nof $C$ by the canonical morphism\n$df\\colon X\\times_YT^*Y\\to T^*X$\nis a subset of the $0$-section\n$X\\times_YT^*_YY\\subset X\\times_YT^*Y$.\n\n{\\rm 2.}\nWe say that a pair\nof morphisms $h\\colon W\\to X$ and\n$f\\colon W\\to Y$ of smooth schemes over $k$\nis {\\em $C$-transversal}\nif $h\\colon W\\to X$ \nis $C$-transversal\nand if\n$f\\colon W\\to Y$ \nis $h^{\\circ}C$-transversal.\n\\end{df}\n\n\n\n\\begin{lm}\\label{lmCf}\nLet $X$ be a smooth scheme \nover a field $k$\nand let $C\\subset T^*X$ be a closed conical subset \nof the cotangent bundle.\nLet $f\\colon X\\to Y$ \nbe a morphism of\nsmooth schemes over $k$.\n\n{\\rm 1.}\nFor $Y={\\rm Spec}\\ k$,\nthe canonical\nmorphism $f\\colon X\\to {\\rm Spec}\\ k$ \nis $C$-transversal.\n\n{\\rm 2.}\nAssume that $C$ is the $0$-section\n$T^*_XX\\subset T^*X$.\nThen, $f$ is $C$-transversal\nif and only if $f$ is smooth.\n\n{\\rm 3.}\nAssume that $f$ is \\'etale.\nThen, $f$ is $C$-transversal\nif and only if \n$C$ is a subset of the $0$-section\n$T^*_XX\\subset T^*X$.\n\n{\\rm 4.}\nAssume that $f\\colon X\\to Y$ \nis $C$-transversal and let\n$g\\colon Y\\to Z$ be a smooth morphism.\nThen, the composition\n$g\\circ f\\colon X\\to Z$ \nis $C$-transversal.\n\n\n{\\rm 5.} {\\rm (\\cite[Lemma 1.2 (i)]{Be})}\nThe subset of $X$ consisting of\npoints $x\\in X$ where\n$f\\colon X\\to Y$ is $C$-transversal\nis an open subset of $Y$.\n\n{\\rm 6.}\nAssume that\n$f\\colon X\\to Y$ \nis $C$-transversal.\nThen, the morphism \n$f\\colon X\\to Y$ is smooth\non a neighborhood of the\nbase $B$ of $C$.\n\n{\\rm 7.}\nAssume that $Y$ is a curve\nand let $x\\in X$ be a point.\nThen, $f\\colon X\\to Y$ is \n{\\em not} $C$-transversal at\n$x$ if and only if\nthe image of the fiber\n$(X\\times_YT^*Y)\\times_Xx$\nby the canonical morphism\n$df\\colon X\\times_YT^*Y\\to T^*X$\nis a subset of the fiber $C\\times_Xx$.\n\n\n{\\rm 8.}\nLet $D=\\bigcup_{i=1}^mD_i$\nbe a divisor with simple normal crossings\nof $X$ relatively to $X\\to {\\rm Spec}\\ k$\nand let \n$C_D=\\bigcup_{I\\subset\\{1,\\ldots\\,m\\}}\nT^*_{D_I}X\\subset T^*X$\n{\\rm (\\ref{eqCD})}\nbe the union of the conormal bundles\nof the intersections\n$D_I=\\bigcap_{i\\in I}D_i$\nof irreducible components\nfor all subsets $I\\subset\\{1,\\ldots,m\\}$\nof indices.\nThen, $f\\colon X\\to Y$ is $C$-transversal\nif and only if\n$D\\subset X$\nhas simple normal crossings\nrelatively to $f\\colon X\\to Y$.\n\n{\\rm 9}.\nLet $h\\colon W\\to X$ and $f\\colon W\\to Y$\nbe morphisms of smooth schemes\nover $k$. Then, the following conditions\nare equivalent:\n\n{\\rm (1)}\nThe pair $(h,f)$ is $C$-transversal.\n\n{\\rm (2)}\nThe morphism $(h,f)\\colon W\\to X\\times Y$ \nis $C\\times T^*Y$-transversal.\n\\end{lm}\n\n\\medskip\nIn next subsection,\nwe will see that\nthe property 1 is related to\nthe generic local acyclicity\n\\cite[Corollaire 2.16]{TF}.\nThe property 2 is related to\nthe local acyclicity of smooth morphism\n(see also Lemma \\ref{lmlcst}.1).\nThe property 3 is related to\nthe characterization of\nlocally constant sheaves\n\\cite[Proposition 2.11]{cst}\n(see also Lemma \\ref{lmlcst}.3).\nThe property 4 is related to\n\\cite[Corollaire 2.7]{app}.\n\n\\proof{\n1.\nSince $T^*Y=0$ for $Y={\\rm Spec}\\ k$,\nthe assertion follows.\n\n2.\nAssume that\n$C$ is the $0$-section\n$T^*_XX\\subset T^*X$.\nThen, $f\\colon X\\to Y$\nis $C$-transversal\nif and only if the canonical morphism\n$X\\times_YT^*Y\\to T^*X$\nis an injection.\nHence, this is equivalent to that\n$f$ is smooth.\n\n\n3.\nAssume that $f$ is \\'etale.\nThen, since\n$X\\times_YT^*Y\\to T^*X$\nis an isomorphism,\nthe morphism\n$f$ is $C$-transversal if and only if\n$C$ is a subset of the $0$-section\n$T^*_XX\\subset T^*X$.\n\n\n4.\nAssume that \n$g\\colon Y\\to Z$ is smooth.\nThen, since $X\\times_ZT^*Z\\to \nX\\times_YT^*Y$ is an injection,\nthe $C$-transversality for $f$ \nimplies that for $g\\circ f$.\n\n5.\nThe complement of the subset is\nthe image of the closed subset\n${\\mathbf P}(df^{-1}C)\\subset\n{\\mathbf P}(X\\times_YT^*Y)$\nof a projective space bundle over $X$.\n\n6.\nIf $f\\colon X\\to Y$ \nis $C$-transversal,\nthen $df\\colon X\\times_YT^*Y\\to T^*X$\nis an injection on a neighborhood of $B$\nand hence\n$f\\colon X\\to Y$ is smooth\non a neighborhood of $B$.\n\n7.\nSince the fiber\n$(X\\times_YT^*Y)\\times_Xx$\nis a line and the inverse image\nof the fiber $C\\times_Xx$\nis its conical subset,\nthe inverse image of $C\\times_Xx$\nis not the subset of the $0$-section\nif and only if it is equal to\nthe line\n$(X\\times_YT^*Y)\\times_Xx$\nitself.\n\n8.\nThe $C$-transversality \nis equivalent to the injectivity of\n$D_I\\times_YT^*Y\\to T^*D_I$\nfor all subsets $I\\subset \\{1,\\ldots,m\\}$.\nHence, it is \nequivalent to the smoothness of\n$D_I\\to Y$\nfor $I\\subset \\{1,\\ldots,m\\}$.\nThus the assertion follows.\n\n9.\nThe conditions (1) and (2)\nare rephrased as conditions\non elements\n$(\\alpha,\\beta)\n\\in {\\rm Ker}(W\\times_XT^*X\\oplus\nW\\times_YT^*Y\n\\to T^*W)_w$\nin the fiber at $w\\in W$ \nof the kernel as follows:\n\n(1) If $\\alpha$ is contained in the inverse\nimage of $C$ and $\\beta=0$,\nthen $\\alpha=0$.\nFurther, if\n$\\alpha$ is contained in the inverse\nimage of $C$ then $\\beta=0$.\n\n(2) \nIf $\\alpha$ is contained in the inverse\nimage of $C$ then $(\\alpha,\\beta)=(0,0)$.\n\n\\noindent\nThus, the conditions are equivalent.\n\\qed}\n\n\n\\begin{df}[{\\rm \\cite[1.2]{Be}}]\\label{dff!C}\nLet $f\\colon X\\to Y$ be a\nmorphism of smooth schemes over $k$\nand $C\\subset T^*X$ be a \nclosed conical subset.\nAssume that $f$ is {\\em proper}\non the base $B$ of $C$.\nWe define a closed conical subset \n\\begin{equation}\nf_{\\circ}C \\subset T^*Y\n\\label{eqf!C}\n\\end{equation}\nto be the closure of the image\nby the first arrow\n$$\\begin{CD}T^*Y\n@<<< \nX\\times_YT^*Y@>>> T^*X\n\\end{CD}$$\nof the inverse image of $C$\nby the second arrow.\n\\end{df}\n\n\\begin{lm}\\label{lmncf}\nLet $f\\colon X\\to Y$\nbe a morphism\nof smooth schemes over $k$.\nLet $C\\subset T^*X$ be a closed conical \nsubset.\nAssume that $f$ is {\\em proper}\non the base $B$ of $C$.\nThen, for a morphism\n$g\\colon Y\\to Z$ of smooth schemes over $k$,\nthe following conditions\nare equivalent:\n\n{\\rm (1)}\nThe morphism $g$ is $f_\\circ C$-transversal.\n\n{\\rm (2)}\nThe composition $gf\\colon X\\to Z$ \nis $C$-transversal.\n\\end{lm}\n\n\\proof{\nWe consider the commutative diagram\n$$\\begin{CD}\n@.\nX\\times_ZT^*Z@>>>\nX\\times_YT^*Y@>>> T^*X\\\\\n@.@VVV @VVV\\\\\nT^*Z@<<<\nY\\times_ZT^*Z@>>> T^*Y.\n\\end{CD}$$\nThe condition (1) \n(resp.\\ (2)) means that\nthe subset in \n$T^*Z$ obtained by taking\ninverse images and images\nin the diagram\nstarting from\n$C\\subset T^*X$\nvia $T^*Y$ \n(resp.\\ via $X\\times_ZT^*Z$) is a subset of\nthe $0$-section.\nThey are equivalent\nsince the square is cartesian.\n\\qed\n\n\n\n\n\\begin{lm}\\label{lmChf}\nLet $C\\subset T^*X$ be a closed conical subset\nand let \n\\begin{equation}\n\\begin{CD}\nX@<h<<W\\\\\n@VfVV @VVgV\\\\\nY@<i<<Z\n\\end{CD}\n\\label{eqfghi}\n\\end{equation}\nbe a cartesian diagram of\nsmooth schemes over $k$.\n\n{\\rm 1.}\nAssume that the horizontal arrows\nare regular immersions of the same codimension.\nThen, \nthe following conditions are equivalent:\n\n{\\rm (1)}\nThe morphism \n$f\\colon X\\to Y$ \nis $C$-transversal on a neighborhood of $W$.\n\n{\\rm (2)}\nThe pair of the immersion\n$h\\colon W\\to X$\nand the morphism \n$g\\colon W\\to Z$ \nis $C$-transversal.\n\n{\\rm 2.}\nAssume that $f\\colon X\\to Y$\nis smooth and\n$C$-transversal.\nThen, \nthe pair of $h\\colon W\\to X$ and \n$g\\colon W\\to Z$ is $C$-transversal.\n\n{\\rm 3.}\nAssume that $f$ is proper on\nthe base $B$ of $C$\nand is flat on a neighborhood\nof $B$.\nAssume that the horizontal arrows\nare immersions.\nThen, the following conditions\nare equivalent:\n\n{\\rm (1)}\n$i$ is $f_{\\circ}C$-transversal,\n\n{\\rm (2)}\n$h$ is $C$-transversal.\n\nFurther, if these equivalent conditions\nare satisfied, \nwe have $i^{\\circ}f_{\\circ}C\n=g_{\\circ}h^{\\circ}C$.\n\\end{lm}\n\n\n\\proof{\n1.\nWe have a commutative diagram\n\\begin{equation}\n\\begin{CD}\n0@>>> T^*_WX@>>>\nW\\times_XT^*X@>>> T^*W@>>>0\\\\\n@.@AAA @AAA @AAA@.\\\\\n0@>>> W\\times_ZT^*_ZY@>>>\nW\\times_YT^*Y@>>> W\\times_ZT^*Z@>>>0\n\\end{CD}\n\\label{eqTWZ}\n\\end{equation}\nof exact sequences\nof vector bundles on $W$.\nThe condition (1) is equivalent to\nthat the inverse image in \n$W\\times_YT^*Y$ of \n$h^*C\\subset W\\times_XT^*X$\nby the middle vertical arrow is\na subset of the $0$-section,\nby Lemma \\ref{lmCh}.4.\nThe condition (2) is equivalent to\nthat the inverse image in \n$T^*_WX$ of \n$h^*C\\subset W\\times_XT^*X$\nby the upper left horizontal arrow \nand the inverse image in \n$W\\times_ZT^*Z$ of \n$h^{\\circ}C\\subset T^*W$\nby the right vertical arrow \nare subsets of the $0$-sections.\nSince the left vertical arrow \nis an isomorphism,\nthe conditions (1) and (2) \nare equivalent.\n\n\n2.\nFirst, we assume $Z\\to Y$ is smooth.\nThen $h$ is smooth and $C$-transversal.\nFurther \nthe arrows in the right square\nof (\\ref{eqTWZ})\nare injections\nand the square is cartesian.\nThus the inverse image of\n$h^{\\circ}C\\subset \nW\\times_XT^*X\n\\subset T^*W$\nin $W\\times_ZT^*Z$\nis the same as the inverse image\nin $W\\times_YT^*Y$\nand hence $g$ is $h^{\\circ}C$-transversal.\n\nIf $Z\\to Y$ is an immersion,\nthe assertion follows from 1.\nIn general,\nit suffices to decompose \n$Z\\to Y$ as the composition\n$Z\\to Z\\times Y\\to Y$\nof the graph and the projection\nby Lemma \\ref{lmCh}.3.\n\n3. \nBy replacing $X$ by a neighborhood\nof $B$, we may assume\nthat $f$ is flat.\nThe condition (1) (reap.\\ (2))\nmeans that the inverse image of\n$C\\subset T^*X$\nin $T^*_WX$ \n(resp.\\ its image in $T^*_ZY$)\nis a subset of the $0$-section.\nSince the left vertical arrow\nin (\\ref{eqTWZ}) is an isomorphism,\nthese conditions are equivalent.\n\nThe subset\n$i^{\\circ}f_{\\circ}C$\n(resp.\\ $g_{\\circ}h^{\\circ}C$)\nof $T^*Z$\nis the image of\nthe subset of\n$W\\times_ZT^*Z$\nobtained from\n$h^*C\\subset\nW\\times_XT^*X$\nby taking inverse images\nand images\nin the right square of (\\ref{eqTWZ})\nvia $W\\times_YT^*Y$\n(resp.\\ via $T^*W$).\nSince the square is cartesian,\nthey are equal.\n\\qed}\n\n\n\n\n\\subsection{The universal family of\nhyperplane sections}\\label{ssRn}\n\nAssume that $X$ \nsmooth over a field $k$\nis quasi-projective\nand let ${\\cal L}$ be an ample invertible\n${\\cal O}_X$-module.\nLet $E$ be a  $k$-vector space of finite\ndimension and\n$E\\to \\Gamma(X,{\\cal L})$ \nbe a $k$-linear mapping inducing\na surjection\n$E\\otimes_k{\\cal O}_X\\to {\\cal L}$\nand an immersion \n\\begin{equation}\ni\\colon X\\to {\\mathbf P}\n={\\mathbf P}(E^\\vee)=\n{\\rm Proj}\\ S^\\bullet E.\n\\label{eqiXP}\n\\end{equation}\nWe use a contra-Grothendieck notation\nfor a projective space\n${\\mathbf P}(E)(k)=(E\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} \\{0\\})\/k^\\times$.\n\nThe dual ${\\mathbf P}^\\vee=\n{\\mathbf P}(E)$\nof ${\\mathbf P}$ parametrizes\nhyperplanes $H\\subset {\\mathbf P}$.\nThe universal hyperplane\n$Q=\\{(x,H)\\mid x\\in H\\}\n\\subset \n{\\mathbf P}\\times\n{\\mathbf P}^\\vee$\nis defined by\nthe identity ${\\rm id}\\in\n{\\rm End}(E)$\nregarded as a section \n$F\\in \\Gamma(\n{\\mathbf P}\\times\n{\\mathbf P}^\\vee,\n{\\cal O}(1,1))\n=E\\otimes E^\\vee$.\nBy the canonical injection\n$\\Omega^1_{{\\mathbf P}\/k}(1)\n\\to E\\otimes {\\cal O}_{\\mathbf P}$,\nthe universal hyperplane\n$Q$ is identified \nwith the covariant projective\nspace bundle\n${\\mathbf P}(T^*{\\mathbf P})$\nassociated to the cotangent bundle\n$T^*{\\mathbf P}$.\nThe image of the conormal bundle\n$T^*_Q(\n{\\mathbf P}\\times {\\mathbf P}^\\vee)\n\\to\nQ\\times_\n{{\\mathbf P}\\times {\\mathbf P}^\\vee}\nT^*({\\mathbf P}\\times {\\mathbf P}^\\vee)\n\\to\nQ\\times_\n{\\mathbf P}\nT^*{\\mathbf P}$\nby the projection\nis identified\nwith the universal sub line bundle\nof the pull-back \n$Q\\times_\n{\\mathbf P}T^*{\\mathbf P}$\non \n$Q={\\mathbf P}(T^*{\\mathbf P})$.\n\n\nThe fibered product\n$X\\times_{\\mathbf P}Q\n={\\mathbf P}(X\\times_{\\mathbf P}T^*{\\mathbf P})$\nis the intersection\nof  $X\\times {\\mathbf P}^\\vee$\nwith $Q$\nin ${\\mathbf P}\\times {\\mathbf P}^\\vee$\nand is the universal family\nof hyperplane sections.\nWe consider\nthe universal family of hyperplane sections\n\\begin{equation}\n\\begin{CD}\nX@<p<<X\\times_{\\mathbf P}Q\n@>{p^\\vee}>> {\\mathbf P}^\\vee={\\mathbf P}(E).\n\\end{CD}\n\\label{eqhsfb}\n\\end{equation}\n\nLet $C\\subset T^*X$ be a\nclosed conical subset.\nDefine a closed conical subset\n$\\widetilde C\\subset\nX\\times_{\\mathbf P}T^*{\\mathbf P}$\nto be the pull-back of $C$ by\nthe surjection\n$X\\times_{\\mathbf P}T^*{\\mathbf P}\n\\to T^*X$\nand let\n\\begin{equation}\n{\\mathbf P}(\\widetilde C)\\subset\n{\\mathbf P}(X\\times_{\\mathbf P}T^*{\\mathbf P})\n=\nX\\times_{\\mathbf P}Q\n\\label{eqPS}\n\\end{equation}\nbe the projectivization.\nThe subset\n${\\mathbf P}({\\widetilde C})\n\\subset\nX\\times_{\\mathbf P}Q\n\\subset X\\times {\\mathbf P}^\\vee$\nconsists of the points $(x,H)$\nsuch that the fiber\n$T^*_{X\\times_{\\mathbf P}Q}\n(X\\times{\\mathbf P}^\\vee)\n\\times_{X\\times_{\\mathbf P}Q}(x,H)\n\\subset\n(X\\times_{\\mathbf P}T^*{\\mathbf P})\\times_Xx$ \nis a subset of $\\widetilde C$\nsince the conormal\nbundle $T^*_{X\\times_{\\mathbf P}Q}\n(X\\times{\\mathbf P}^\\vee)\n\\subset \nX\\times_{\\mathbf P} T^*{\\mathbf P}$\nis the universal sub line bundle\non the projective bundle\n$X\\times_{\\mathbf P}Q=\n{\\mathbf P}(X\\times_{\\mathbf P} T^*{\\mathbf P})$.\nIf $C=T^*_XX$ is the $0$-section,\nthe lifting $\\widetilde C$\nis the conormal bundle\n$T^*_X{\\mathbf P}$.\nFurther if $i\\colon X\\to {\\mathbf P}$\nis a closed immersion,\nthe image $p^\\vee(\n{\\mathbf P}(\\widetilde C))\n\\subset {\\mathbf P}^\\vee$\nis the dual variety of $X$.\n\nIf $V\\subset {\\mathbf P}$ is\nan open subscheme such that\n$i\\colon X\\to {\\mathbf P}$\ninduces a closed immersion\n$i^{\\circ}\\colon X\\to V$,\nthen\n$\\widetilde C\\subset\nX\\times_{\\mathbf P}T^*{\\mathbf P}\n=X\\times_VT^*V\n\\subset T^*V$\nis identified with $i^{\\circ}_{\\circ}C$\n(\\ref{eqf!C}).\n\n\n\\begin{lm}\\label{lmPC}\nLet $C\\subset T^*X$ be a\nclosed conical subset.\nThe complement \n$X\\times_{\\mathbf P}Q\n\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} {\\mathbf P}(\\widetilde C)$\nis the largest open subset\nwhere the pair\n$X\\gets X\\times_{\\mathbf P}Q\n\\to {\\mathbf P}^\\vee$\nis $C$-transversal.\n\\end{lm}\n\n\n\\proof{\nBy Lemma \\ref{lmCf}.9,\nthe largest open subset\n$U\\subset X\\times _{\\mathbf P}Q$\nwhere the pair\n$(p,p^\\vee)$\nis $C$-transversal equals\nthe largest open subset\nwhere $(p,p^\\vee)\\colon \nX\\times_{\\mathbf P}Q\\to X\\times\n{\\mathbf P}^\\vee$\nis $C\\times T^*{\\mathbf P}^\\vee$-transversal.\nThe kernel \n$L={\\rm Ker}\\bigl(\n(X\\times_{\\mathbf P}Q)\\times_XT^*X\n\\oplus\n(X\\times_{\\mathbf P}Q)\\times_\n{{\\mathbf P}^\\vee}T^*{\\mathbf P}^\\vee\n\\to \nT^*(X\\times_{\\mathbf P}Q)\\bigr)$\nis canonically identified with\nthe restriction of\nthe universal sub line bundle of\n$T^*{\\mathbf P}$ on\n$Q={\\mathbf P}(T^*{\\mathbf P})$.\nHence,  $U$ is the complement\nof ${\\mathbf P}(\\widetilde C)$.\n\\qed}\n\\medskip\n\nWe consider a similar \nbut slightly different situation.\nLet $X$ be a smooth\nscheme over a field $k$\nand $h\\colon X\\to {\\mathbf P}\n={\\mathbf P}^n$\nbe an immersion over $k$\nto a projective space.\nLet $Q={\\mathbf P}(T^*{\\mathbf P})\n\\subset {\\mathbf P}\n\\times {\\mathbf P}^\\vee$\nbe the universal family\nof hyperplanes\nand consider the\ncommutative diagram \n\\begin{equation}\n\\xymatrix{\nX\\times_{\\mathbf P}Q\n\\ar[r]\\ar[d]_p\n\\ar[rrrd]^{\\!\\! p^{\\vee}}\n& \nQ\\ar[rrd]^{{\\bms p}^{\\vee}}\n\\ar[d]_{\\!\\!\\!\\!\\!\\! {\\bms p}}&&\n\\\\\nX\\ar[r]^h&\n{\\mathbf P}\n&&{\\mathbf P}^\\vee}\n\\label{eqXPh}\n\\end{equation}\nwith cartesian square.\nWe have $Q\n={\\mathbf P}(X\\times_{\\mathbf P}T^*{\\mathbf P})$.\n\n\n\\begin{lm}\\label{lmh!}\nLet ${\\mathbf P}={\\mathbf P}^n$\nbe a projective space \nand let ${\\mathbf P}^\\vee$\nbe the dual projective space.\nLet $C^\\vee\n\\subset T^*{\\mathbf P}^\\vee$\nbe a closed conical subset \nsuch that every irreducible\ncomponent is of dimension $n$.\nDefine a closed conical subset $C\n\\subset T^*{\\mathbf P}$ by\n$C={\\bm p}_{\\circ}\n{\\bm p}^{\\vee \\circ}C^\\vee$.\nThen, every irreducible\ncomponent of $C$ is of dimension $n$.\n\\end{lm}\n\n\\proof{\nIt suffices to consider\nthe case where $C^\\vee$ is irreducible.\nWe have ${\\mathbf P}(C)=\n{\\mathbf P}(C^\\vee)$\nin $Q={\\mathbf P}(T^*{\\mathbf P})\n={\\mathbf P}(T^*{\\mathbf P}^\\vee)$.\nUnless the base of $C^\\vee$\nis finite,\n$C$ contains the $0$-section\n$T^*_{\\mathbf P}{\\mathbf P}$.\nIf the base of $C^\\vee$\nconsists of a closed point\ncorresponding to a hyperplane\n$H\\subset {\\mathbf P}$,\nwe have $C=T^*_H{\\mathbf P}$.\nThus the assertion follows.\n\\qed}\n\n\\begin{pr}\\label{prhCt}\nLet the notation be as in\n{\\rm (\\ref{eqXPh})} and\nlet $C^\\vee\n\\subset T^*{\\mathbf P}^\\vee$\nbe a closed conical subset.\nDefine a closed conical subset \n$C\\subset T^*{\\mathbf P}$\nby $C={\\bm p}_{\\circ}{\\bm p}^{\\vee \\circ}C^\\vee$\nand \nits projectivization\n${\\mathbf P}(C)\n\\subset\n{\\mathbf P}(T^*{\\mathbf P})$.\nThen, the complement\n$X\\times_{\\mathbf P}Q\n\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} (X\\times_{\\mathbf P}Q)\n\\cap {\\mathbf P}(C)$\nis the largest open subset $U\n\\subset X\\times_{\\mathbf P}Q$\nwhere the pair\n$(p^\\vee, p)$\nis $C^\\vee$-transversal.\n\\end{pr}\n\n\\proof{\nThe proof is similar to that of\nLemma \\ref{lmPC}.1.\nBy Lemma \\ref{lmCf}.9,\nthe largest open subset\n$U\\subset X\\times _{\\mathbf P}Q$\nwhere the pair\n$(p^\\vee, p)$\nis $C^\\vee$-transversal equals\nthe largest open subset\nwhere $(p,p^\\vee)\\colon \nX\\times_{\\mathbf P}Q\\to X\\times\n{\\mathbf P}^\\vee$\nis $T^*X\\times C^\\vee$-transversal.\nThe kernel \n$L={\\rm Ker}\\bigl(\n(X\\times_{\\mathbf P}Q)\\times_XT^*X\n\\oplus\n(X\\times_{\\mathbf P}Q)\\times_\n{{\\mathbf P}^\\vee}T^*{\\mathbf P}^\\vee\n\\to \nT^*(X\\times_{\\mathbf P}Q)\\bigr)$\nis canonically identified with\nthe restriction of\nthe universal sub line bundle of\n$T^*{\\mathbf P}^\\vee$ on\n$Q={\\mathbf P}(T^*{\\mathbf P}^\\vee)$.\nHence,  $U$ is the complement\nof the intersection\n$(X\\times_{\\mathbf P}Q)\n\\cap {\\mathbf P}(C^\\vee)$.\nSince $L$ is also canonically identified with\nthe restriction of\nthe universal sub line bundle\nof $T^*{\\mathbf P}$ on\n$Q={\\mathbf P}(T^*{\\mathbf P})$,\nwe have\n${\\mathbf P}(C^\\vee)\n={\\mathbf P}(C)$\nand the assertion follows.\n\\qed}\n\\medskip\n\nLet $\\Delta_X\\subset X\\times_{\\mathbf P}Q$\nbe the sub projective space\nbundle\n\\begin{equation}\n\\Delta_X={\\mathbf P}(T^*_X{\\mathbf P})\n\\subset\n{\\mathbf P}(X\\times_{\\mathbf P}T^*{\\mathbf P})\n=X\\times_{\\mathbf P}Q.\n\\label{eqRX}\n\\end{equation}\n\n\n\\begin{cor}\\label{corhCt}\nLet the notation be as in\nProposition {\\rm \\ref{prhCt}}.\n\n{\\rm 1.}\nThe following conditions are\nequivalent:\n\n{\\rm (1)}\nThe immersion $h\\colon X\\to\n{\\mathbf P}$ is $C$-transversal.\n\n\n{\\rm (2)}\nThe pair $(p^\\vee, p)$\nis $C^\\vee$-transversal\non a neighborhood\nof $\\Delta_X\\subset X\\times_{\\mathbf P}Q$.\n\n{\\rm 2.}\nAssume that\nthe immersion $h\\colon X\\to\n{\\mathbf P}$ is $C$-transversal.\nThen, $p^\\vee\\colon X\\times_{\\mathbf P}Q\n\\to {\\mathbf P}^\\vee$ is\n$C^\\vee$-transversal and we have\n$h^{\\circ}C=p_{\\circ}p^{\\vee \\circ}C^\\vee$.\n\\end{cor}\n\n\\proof{\n1.\nBy Proposition \\ref{prhCt},\nthe condition (2) is equivalent to\n${\\mathbf P}(C)\\cap\n\\Delta_X=\\varnothing$.\nThis is equivalent to\nthat $T^*_X{\\mathbf P}\n\\cap C$\nis a subset of the $0$-section\nand hence to (1).\n\n2.\nBy Lemma \\ref{lmChf}.3\napplied to $C={\\bm p}_{\\circ}\n{\\bm p}^{\\vee \\circ}C^\\vee$,\nthe assumption that\n$h\\colon X\\to {\\mathbf P}$\nis $C$-transversal\nimplies that the immersion\n$i\\colon X\\times_{\\mathbf P}Q\n\\to Q$ is ${\\bm p}^{\\vee \\circ}C^\\vee$-transversal\nand $h^{\\circ}C=\nh^{\\circ}{\\bm p}_{\\circ}\n{\\bm p}^{\\vee \\circ}C^\\vee=\np_{\\circ}i^{\\circ}{\\bm p}^{\\vee \\circ}C^\\vee$.\nHence by Lemma \\ref{lmCh}.3,\n$p^\\vee\\colon X\\times_{\\mathbf P}Q\n\\to {\\mathbf P}^\\vee$ is\n$C^\\vee$-transversal and\nwe have\n$i^{\\circ}{\\bm p}^{\\vee \\circ}C^\\vee=\np^{\\vee \\circ}C^\\vee$.\nThus the assertion is proved.\n\\qed}\n\n\n\n\n\\begin{pr}\\label{prfp}\nLet the notation be as in\nProposition {\\rm \\ref{prhCt}} and\nlet $f\\colon X\\to {\\mathbf A}^1$\nbe a smooth morphism over $k$.\nDefine sub vector bundles\nof $X\\times_{\\mathbf P}T^*{\\mathbf P}$\nby the cartesian diagram\n$$\\begin{CD}\nT^*_X{\\mathbf P}\n@>{\\subset}>> V@>{\\subset}>> \nX\\times_{\\mathbf P}T^*{\\mathbf P}\\\\\n@VVV@VVV@VVV\\\\\nT^*_XX\n@>{\\subset}>> \nX\\times_{{\\mathbf A}^1}T^*{\\mathbf A}^1\n@>{\\subset}>> T^*X\n\\end{CD}$$\nand closed subsets\n$\\Delta_X={\\mathbf P}(T^*_X{\\mathbf P})\n\\subset \\Delta_f={\\mathbf P}(V)\n\\subset \nX\\times_{\\mathbf P}Q\n={\\mathbf P}(X\\times_{\\mathbf P}T^*{\\mathbf P})$\nto be the associated projective subspace bundles.\nThen, the complement \n$X\\times_{\\mathbf P}Q\n\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} (\\Delta_f\\cap {\\mathbf P}(C))$\nis the largest open subset\n$U\\subset X\\times_{\\mathbf P}Q$\nwhere the pair\n$(p^\\vee,f\\circ p)$\nis $C^\\vee$-transversal.\n\\end{pr}\n\n\\proof{\nBy Lemma \\ref{lmCf}.9,\nthe largest open subset\n$U\\subset X\\times_{\\mathbf P}Q$\nwhere the pair\n$(p^\\vee,f\\circ p)$\nis $C^\\vee$-transversal\nequals\nthe largest open subset\nwhere $(f\\circ p,p^\\vee)\\colon\nX\\times_{\\mathbf P}Q\n\\to {\\mathbf A}^1\\times{\\mathbf P}^\\vee$\nis $T^*{\\mathbf A}^1\n\\times C^\\vee$-transversal.\nHence, similarly as in the proof\nof Proposition \\ref{prhCt},\n$U\\subset X\\times_{\\mathbf P}Q$ \nis the complement of\n$\\Delta_f\\cap \n((X\\times_{\\mathbf P}Q)\n\\cap {\\mathbf P}(C^\\vee))=\n\\Delta_f\\cap \n{\\mathbf P}(C)$.\n\\qed}\n\n\n\\begin{cor}\\label{corfp}\nLet the notation be as in\nPropositions {\\rm \\ref{prhCt}}\nand {\\rm \\ref{prfp}} and assume that\n$h\\colon X\\to {\\mathbf P}$\nis $C$-transversal.\nThen,\nthe composition $f\\circ p\\colon \nX\\times_{\\mathbf P}Q\\to X\\to {\\mathbf A}^1$\nis $p^{\\vee \\circ}C^\\vee$-transversal\non the complement\nof the intersection \nwith $\\Delta_f\\cap {\\mathbf P}(C)$.\nFurther, the intersection\n$\\Delta_f\\cap {\\mathbf P}(C)\n\\subset \nX\\times_{\\mathbf P}Q$\nis finite over the complement\nof the largest open subset\nof $X$ where\n$f\\colon X\\to {\\mathbf A}^1$\nis $h^{\\circ}C$-transversal.\n\\end{cor}\n\n\\proof{\nThe first assertion is clear from\nProposition \\ref{prfp}.\n\nWe show the second assertion.\nBy Corollary \\ref{corhCt}.2,\n$p^\\vee\\colon X\\times_{\\mathbf P}Q\n\\to {\\mathbf P}^\\vee$ is\n$C^\\vee$-transversal and\nwe have $h^{\\circ}C=\np_{\\circ}p^{\\vee \\circ}C^\\vee$.\nHence by Lemma \\ref{lmncf},\nif $f\\colon X\\to {\\mathbf A}^1$\nis $h^{\\circ}C$-transversal,\nthen $fp\\colon X\\times_{\\mathbf P}Q\n\\to {\\mathbf A}^1$ is\n$p^{\\vee \\circ} C^\\vee$-transversal.\nThus, by Proposition \\ref{prfp},\nthe intersection\n$\\Delta_f\\cap {\\mathbf P}(C)$\nis a subset of the inverse\nimage of\nthe complement\nof the largest open subset\nof $X$ where\n$f\\colon X\\to {\\mathbf A}^1$\nis $h^{\\circ}C$-transversal.\nTherefore, it suffices to show that\n$\\Delta_f\\cap {\\mathbf P}(C)$\nis finite over $X$.\n\n\nBy the assumption that\n$h\\colon X\\to {\\mathbf P}$\nis $C$-transversal,\nthe morphism\n$p\\colon X\\times_{\\mathbf P}Q\n\\to X$ is $p^{\\vee \\circ}C^\\vee$-transversal\non a neighborhood of $\\Delta_X$\nby Corollary \\ref{corhCt}.1.\nSince $f\\colon X\\to {\\mathbf A}^1$\nis smooth, the composition\n$fp\\colon X\\to {\\mathbf A}^1$\nis $p^{\\vee \\circ}C^\\vee$-transversal\non a neighborhood of $\\Delta_X$\nby Lemma \\ref{lmCf}.4.\nHence, by Proposition \\ref{prfp},\nthe intersection\n$\\Delta_f\\cap {\\mathbf P}(C)$\ndoes not meet $\\Delta_X$.\nSince $\\Delta_X$ is a hyperplane bundle\nof a projective space bundle\n$\\Delta_f$ over $X$,\nthe intersection\n$\\Delta_f\\cap {\\mathbf P}(C)$\nis finite over $X$.\n\\qed}\n\n\n\n\n\\subsection{Image of the projectivization}\n\nWe further recall from \\cite{Be}\na definition and properties.\n\n\\begin{df}[{\\rm cf.\\ \\cite[4.2]{Be}}]\\label{dfwi}\nLet $f\\colon X\\to S$ be a morphism\nof separated schemes of finite type over a field $k$\nand $Y,Z\\subset X$ be closed subsets.\nWe say that $Y$ and $Z$ {\\em well intersect}\nwith respect to $f$ if we have\n\\begin{equation}\n\\dim (Y\\times_SZ\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Y\\times_XZ)\n\\leqq \\dim Y+\\dim Z-\\dim S.\n\\label{eqdfwi}\n\\end{equation}\n\\end{df}\n\nWe slightly modified the\noriginal definition by replacing\nthe equality by an inequality.\n\n\\begin{lm}\\label{lmJ}\nLet $f\\colon X\\to S$ be a morphism of\nseparated schemes of finite type\nover a field $k$\nand $Y,Z\\subset X$ be closed subsets.\n\n{\\rm 1.}\nLet $g\\colon P\\to S$ be\na morphism of\nseparated schemes of finite type\nover a field $k$\nand $X\\to P$ be an immersion\nover $k$.\nThen, $Y$ and $Z$ well intersect\nwith respect to $f$\nif and only if they\nwell intersect\nwith respect to $g$.\n\n{\\rm 2.}\nLet $S'\\to S$ be a faithfully flat morphism of\nrelative constant dimension of\nseparated schemes of finite type\nover a field $k$\nand let $f'$ and $Y',Z'\\subset X'$\ndenote the base changes of $f$ and $Y,Z\\subset X$\nby $S'\\to S$.\nThen, $Y$ and $Z$ well-intersects\nwith respect to $f$ if and only if\n$Y'$ and $Z'$ well-intersects\nwith respect to $f'$.\n\n{\\rm 3.}\nLet $X\\to U\\to T$ be a morphism of\nseparated schemes of finite type\nover a field $k$\nand $V,W\\subset U$ be closed subsets.\nAssume that $X\\to S\\times T$\nis an immersion and that\n$Y,Z\\subset X$ are inverse images of\n$V,W\\subset U$.\nAssume further that\nthe morphisms\n$X\\to S$ and $X\\to U$\nand the base change\n\\begin{equation}\n(X\\times_SX)\n\\times_{(T\\times T)}\n(T\\times T)^{\\circ}\\to\n(U\\times U)\n\\times_{(T\\times T)}\n(T\\times T)^{\\circ}\n=U\\times U\n\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} U\\times_T U\n\\label{eqXXUU}\n\\end{equation}\nof the morphism\n$X\\times_SX\\to U\\times U$\nby the open immersion\n$(T\\times T)^{\\circ} =\nT\\times T\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} T\n\\to T\\times T$ of the complement\nof the diagonal are faithfully flat of relative\nconstant dimensions.\nThen, $Y$ and $Z$ well-intersects\nwith respect to $f$.\n\\end{lm}\n\n\\proof{1.\nSince $X\\to P$ is an immersion,\nwe have $Y\\times_XZ=\nY\\times_PZ$\nand the assertion follows.\n\n2.\nBy the assumption, every term in (\\ref{eqdfwi})\nwith $'$ equals the corresponding term\nwithout $'$ added the relative dimension\nand the assertion follows.\n\n3.\nBy the assumption that\n$X\\to U$ is faithfully \nflat of relative constant dimension,\nwe have\n\\begin{equation}\n\\dim Y=\\dim V+(\\dim X-\\dim U),\\quad\n\\dim Z=\\dim W+(\\dim X-\\dim U).\n\\label{eqYV}\n\\end{equation}\nBy the assumption that $X\\to S\\times T$\nis an immersion,\nthe complement of the diagonal\n$X\\times_SX\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} X$ \nequals the base change \n$(X\\times_SX)\n\\times_{(T\\times T)}\n(T\\times T)^{\\circ}$\nand \n$Y\\times_SZ\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Y\\times_XZ$\nis the inverse image of\n$V\\times W\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~}\nV\\times_T W$\nby the morphism\n(\\ref{eqXXUU}).\nSince the morphism\n(\\ref{eqXXUU}) is faithfully flat\nof relative constant dimension,\nwe have\n\\begin{align}\n&\\dim (Y\\times_SZ\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Y\\times_XZ)\n\\label{eqVW}\n\\\\\n&\\leqq\n\\dim V+\\dim W\n+(\\dim (X\\times_SX\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} X)\n-\\dim (U\\times U\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} U\\times_TU)).\n\\nonumber\n\\end{align}\nBy the assumption that\n$X\\to S$ is faithfully flat\nof relative constant dimension, \nwe have \n$\\dim X\\times_SX=2\\dim X-\\dim S$\nand the right hand side of\n(\\ref{eqVW}) equals\n$\\dim V+\\dim W\n+2(\\dim X-\\dim U)-\\dim S$\nand the assertion follows\nby (\\ref{eqYV}).\n\\qed}\n\n\n\\begin{lm}[{\\rm cf. \\cite[Lemma 4.2]{Be}}]\\label{lmwi}\nLet $Y,Z\\subset X$ be irreducible\nclosed subsets.\nAssume that \n$Y$ and $Z$ well intersects\nwith respect to $f$\nand we have\n$\\dim Y,\\dim Z<\\dim S$.\n\n{\\rm 1.}\nIf $Y=Z$,\nthen $Y\\to \\overline{f(Y)}$ \nis generically radicial.\n\n{\\rm 2.}\nIf $Y\\not \\subset Z$,\nthen we have\n$f(Y)\\not \\subset\n\\overline{f(Y)\\cap f(Z)}$\nand \n$f(Y)\\not \\subset f(Z)$.\n\\end{lm}\n\n\\proof{\n1.\nWe have\n$\\dim (Y\\times_SY\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Y)\n\\leqq 2\\dim Y-\\dim S< \\dim Y.$\n\n2.\nBy 1, we have\n$\\dim Y=\\dim \\overline{f(Y)}$.\nWe have $\\dim \\overline{f(Y)\\cap f(Z)}\n\\leqq\n\\dim Y\\times_SZ\n\\leqq\n\\max(\n\\dim (Y\\times_SZ\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Y\\times_XZ),\n\\dim Y\\times_XZ)$.\nSince $Y$ and $Z$ assumed to\nwell intersect,\nwe have\n$\\dim (Y\\times_SZ\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Y\\times_XZ)\n\\leqq \\dim Y+\\dim Z-\\dim S\n< \\dim Y$.\nIf $Y\\not\\subset Z$,\nwe have\n$\\dim Y\\times_XZ\n< \\dim Y$.\nThus, we have\n$\\dim \\overline{f(Y)\\cap f(Z)}\n<\\dim Y=\\dim \\overline{f(Y)}$.\n\nIf we had $f(Y)\\subset f(Z)$,\nwe would have\n$f(Y)\\subset\n\\overline{f(Y)\\cap f(Z)}$\nand a contradition.\n\\qed}\n\\medskip\n\nFor a $k$-vector space $E$\nof finite dimension\nand a $k$-linear mapping\n$E\\to \\Gamma(X,{\\cal L})$\ndefining an immersion\n$X\\to {\\mathbf P}={\\mathbf P}(E^\\vee)$,\nwe consider the following condition:\n\\begin{itemize}\n\\item[{\\rm (E)}]\nFor every pair of distinct closed\npoints $u\\neq v$ of the base change\n$X_{\\bar k}$ to an algebraic closure\n$\\bar k$ of $k$,\nthe composition\n\\begin{equation}\nE\\to \\Gamma(X,{\\cal L})\n\\otimes_k\\bar k\n\\to \n{\\cal L}_u\/{\\mathfrak m}_u^2{\\cal L}_u\n\\oplus\n{\\cal L}_v\/{\\mathfrak m}_v^2{\\cal L}_v\n\\label{eqE}\n\\end{equation}\nis a surjection.\n\\end{itemize}\nFor an integer $d\\geqq 1$,\nthe linear mapping\n$S^dE\n\\to\\Gamma(X,{\\cal L}^{\\otimes d})$ \nof the symmetric power\ndefines an immersion\n$X\\to {\\mathbf P}(S^dE^{\\vee})$.\n\n\\begin{lm}\\label{lmloc}\nLet $X$ be a quasi-projective\nscheme over a field\n$k$ and\n${\\cal L}$ be an ample\ninvertible ${\\cal O}_X$-module.\nAssume that $E\\to \\Gamma(X,{\\cal L})$ \ndefines an immersion $X\\to \n{\\mathbf P}={\\mathbf P}(E^{\\vee})$.\nFor $d\\geqq 3$, \nthe symmetric power\n$S^dE\\to \\Gamma(X,{\\cal L}^{\\otimes d})$\nsatisfies the condition {\\rm (E)} above.\n\\end{lm}\n\n\\proof{\nWe may assume $k=\\bar k,\\\nX={\\mathbf P}^n,\\\n{\\cal L}={\\mathcal O}(1),\\\nE=\\Gamma(X,{\\cal L})$\nand $u=(0,\\cdots,0,1),\nv=(1,0,\\cdots,0)$.\nThen, the assertion is clear.\n\\qed}\n\\medskip\n\nFor an immersion \n$i\\colon X\\to {\\mathbf P}$\nto a projective space\nand a closed conical\nsubset $C\\subset T^*X$,\nwe consider the following condition:\n\\begin{itemize}\n\\item[(C)]\nFor every irreducible component\n$C_a\\subset T^*X$ of $C=\\bigcup_aC_a$,\nthe inverse image\n$\\widetilde C_a\n\\subset X\\times_{\\mathbf P}T^*{\\mathbf P}$\nis not a subset of the\n$0$-section.\n\\end{itemize}\nIf the condition (C) is satisfied,\nthere is a one-to-one correpspondence\nbetween the irreducible components\nof $C$ and those of ${\\mathbf P}(\\widetilde C)$ sending $C_a$ to ${\\mathbf P}(\\widetilde C_a)$.\n\n\\begin{pr}[{\\rm \\cite[Lemma 4.3]{Be}}]\\label{prwi}\nLet $X$ be a quasi-projective\nsmooth scheme of dimension $n$\nover a field $k$ and\n${\\cal L}$ be an ample\ninvertible ${\\cal O}_X$-module.\nLet $E$ be a $k$-vector space\nof finite dimension and \n$E\\to \\Gamma(X,{\\cal L})$\nbe a $k$-linear mapping\ndefining an immersion\n$X\\to {\\mathbf P}={\\mathbf P}(E^\\vee)$\nand \nsatisfying the condition {\\rm (E)}\nbefore Lemma {\\rm \\ref{lmloc}}.\nLet $C\\subset T^*X$\nbe a closed conical subset\nsatisfying the condition {\\rm (C)} above.\nThen, for irreducible\ncomponents $C_a,C_b$ of $C$,\nthe projectivizations\n${\\mathbf P}(\\widetilde C_a),\n{\\mathbf P}(\\widetilde C_b)\n\\subset {\\mathbf P}(X\\times_{\\mathbf P}\nT^*{\\mathbf P})=X\\times_{\\mathbf P}Q$\nwell intersects with respect to\n$p^\\vee\\colon \nX\\times_{\\mathbf P}Q\\to {\\mathbf P}^\\vee$.\n\\end{pr}\n\\medskip\n\nThe proof is essentially the same as\nthat of \\cite[Lemma 4.3]{Be}.\nFor the sake of completeness,\nwe rephrase the proof.\n\n\n\\proof{\nLet ${\\cal I}\\subset {\\cal O}_{X\\times X}$\ndenote the ideal sheaf defining\nthe diagonal immersion $X\\to X\\times X$\nand let $P\\subset X\\times X$ denote\nthe closed subscheme defined by \n${\\cal I}^2$.\nThe projections $p_1,\\ p_2\\colon P\\to X$\nare finite flat of degree $n+1$.\nDefine a vector bundle $J$\nof rank $n+1$ on $X$ to be that associated\nto the invertible ${\\cal O}_P$-module \n$p_2^*{\\cal L}$ \nregarded as a locally free \n${\\cal O}_X$-module by $p_1$.\nThe exact sequence\n$0\\to \\Omega^1_X\\to {\\cal O}_P\n\\to {\\cal O}_X\\to 0$\ntensored with $p_2^*{\\cal L}$ \ndefines an exact sequence\n\\begin{equation}\n\\begin{CD}\n0@>>>\nT^*X\\otimes L\n@>>>\nJ\n@>>> L@>>>0\n\\end{CD}\n\\label{eqLJ}\n\\end{equation}\nof vector bundles on $X$.\n\nDefine a morphism $X\\times E\\to J$\nof vector bundles on $X$\nto be that induced by the pull-back\nby $p_2$ of the canonical morphism\n${\\cal O}_X\\otimes E\\to {\\cal L}$.\nFor a closed geometric point $x$,\nthe fiber of\n$X\\times E\\to J$\nis identified with the morphism\n$E\\to {\\cal L}_x\/{\\mathfrak m}_x^2{\\cal L}_x$\ninduced by the canonical morphism\n${\\cal O}_X\\otimes E\\to {\\cal L}$.\n\nWe regard\n${\\mathbf P}(\\widetilde C_a),\n{\\mathbf P}(\\widetilde C_b)\n\\subset X\\times_{\\mathbf P}Q$\nas subsets of\n$X\\times {\\mathbf P}^\\vee$\nas in Lemma \\ref{lmJ}.1.\nWe consider the cartesian diagram\n\\begin{equation}\n\\begin{CD}\n&X\\times {\\mathbf P}^\\vee\n@<<<\nX\\times E^{\\circ}\n@>>>\nJ\n@>>> X\n\\\\\n&@VVV@VVV\\\\\n{\\mathbf P}(E)=&\n{\\mathbf P}^\\vee\n@<<<\nE^{\\circ}&=E\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} \\{0\\}.\n\\end{CD}\n\\label{eqEJ}\n\\end{equation}\nThe lower horizontal arrow is\nthe canonical surjection\nand is faithfully flat of\nrelative dimension 1.\nFor an irreducible component $C_a$\nof $C\\subset T^*X$,\ndefine a closed conical\nsubset $C_a\\otimes L\n\\subset T^*X\\otimes L\n\\subset J$\nas a twist of $C_a$ by using\nthe exact sequence (\\ref{eqLJ}).\nThe inverse image\n$\\widetilde C_a^{\\circ}\\subset\nX\\times E^{\\circ}$\nof $C_a\\otimes L\n\\subset J$ by the second upper arrow\nin (\\ref{eqEJ})\nequals the inverse image of\n${\\mathbf P}(\\widetilde C_a)\n\\subset X\\times_{\\mathbf P}Q\n\\subset X\\times {\\mathbf P}^\\vee$\nby the first upper arrow\nby the condition (C).\nBy Lemma \\ref{lmJ}.2\napplied to ${\\mathbf P}(\\widetilde C_a),\n{\\mathbf P}(\\widetilde C_b)\n\\subset\nX\\times_{\\mathbf P}Q\\subset\nX\\times {\\mathbf P}^\\vee$\nand the base change\n$E^{\\circ} \\to{\\mathbf P}^\\vee$,\nit suffices to show\nthat\n$\\widetilde C_a^{\\circ}$\nand\n$\\widetilde C_b^{\\circ}\n\\subset\nX\\times E^{\\circ}$\nwell-intersect with respect to\n$X\\times E^{\\circ} \\to E^{\\circ}$.\n\nThe morphism\n$X\\times E\\to J$\ninduces a surjection\nof vector bundles\n$(X\\times X)^{\\circ} \\times E\\to \n(J\\times J)\\times_{(X\\times X)}(X\\times X)^{\\circ}$\non $(X\\times X)^{\\circ}$\nby the condition (E).\nThus, by Lemma \\ref{lmJ}.3 applied\nto $E^{\\circ}\\gets X\\times E^{\\circ}\\to J\\to X$\ntaken as $S\\gets X\\to U\\to T$,\nthe assertion follows.\n\\qed}\n\n\n\\begin{cor}\\label{corrad}\nLet $X$ be a quasi-projective\nsmooth scheme of dimension $n$\nover a field $k$ and\n${\\cal L}$ be an ample\ninvertible ${\\cal O}_X$-module.\nLet $E$ be a $k$-vector space\nof finite dimension and \n$E\\to \\Gamma(X,{\\cal L})$\nbe a $k$-linear mapping\ndefining an immersion\n$X\\to {\\mathbf P}={\\mathbf P}(E^\\vee)$\nsatisfying the condition {\\rm (E)} \nbefore Lemma {\\rm \\ref{lmloc}}.\nLet $C\\subset T^*X$\nbe a closed conical subset\nsatisfying the condition {\\rm (C)} before\nProposition {\\rm \\ref{prwi}}.\nAssume that every irreducible component\n$C_a$ of\n$C=\\bigcup_aC_a$ is of dimension $n$. \n\n{\\rm 1.}\nFor every irreducible component\n$C_a$ of $C$,\nthe restriction\n${\\mathbf P}(\\widetilde C_a)\\to {\\mathbf P}^\\vee$\nof $p^\\vee\\colon X\\times_{\\mathbf P}Q\n\\to {\\mathbf P}^\\vee$\nis generically radicial\nand the closure \n$D_a=\\overline{\np^\\vee({\\mathbf P}(\\widetilde C_a))}\n\\subset {\\mathbf P}^\\vee$ is a divisor.\n\n{\\rm 2.}\nFor distinct irreducible\ncomponents $C_a\\neq C_b$ of $C$,\nwe have $D_a\\neq D_b$.\n\\end{cor}\n\n\\proof{\n1.\nSince the closed subset\n${\\mathbf P}(\\widetilde C)\\to X\\times_{\\mathbf P}Q$\nis of codimension $n$\nand $p^\\vee\\colon X\\times_{\\mathbf P}Q\n\\to {\\mathbf P}^\\vee$\nis of relative dimension $n-1$,\nwe have\n$\\dim {\\mathbf P}(\\widetilde C)\n=\\dim {\\mathbf P}^\\vee-1$.\nHence it follows from\nProposition \\ref{prwi}\nand Lemma \\ref{lmwi}.1.\n\n2.\nSimilarly as the proof of 1,\nthe assertion follows from\nProposition \\ref{prwi}\nand Lemma \\ref{lmwi}.2.\n\\qed}\n\n\n\n\\section{Singular support}\\label{sss}\n\nWe recall definitions and results from\n\\cite{Be} in Section \\ref{ssss}.\nWe give a description of\nsingular support of the $0$-extension\nof a locally constant sheaf\non the complement of\na divisor with simple normal crossings\nunder a certain assumption\nin Section \\ref{ssram}.\n\n\n\n\\subsection{Singular support}\\label{ssss}\n\n\n\nWe recall definitions and results from\n\\cite{Be}.\nWe assume that $X$\nis a smooth scheme over a field $k$.\n\nLet $\\Lambda$ be a finite local ring\nsuch that the characteristic $\\ell$\nof the residue field is\ninvertible in $k$\nand let ${\\cal F}$ be a constructible complex of\n$\\Lambda$-modules on \nthe \\'etale site of $X$.\nRecall that a complex \n${\\cal F}$ of $\\Lambda$-modules\nis {\\em constructible} if every cohomology sheaf\n${\\cal H}^q{\\cal F}$ is constructible\nand if ${\\cal H}^q{\\cal F}=0$ except\nfor finitely many $q$.\nWe say that a constructible complex\n${\\cal F}$ of $\\Lambda$-modules\nis {\\em locally constant} if every cohomology sheaf\nis locally constant.\n\nWe say that a constructible complex\n${\\cal F}$ of $\\Lambda$-modules\nis of {\\em finite tor-dimension} \nif there exists an integer $a$\nsuch that ${\\cal H}^q({\\cal F}\n\\otimes^L_{\\Lambda}M)=0$\nfor every $q<a$ \nand for every $\\Lambda$-module $M$.\nThe dualizing complex ${\\cal K}_X$\ndefined as $Rf^!\\Lambda$\nfor the canonical morphism\n$f\\colon X\\to {\\rm Spec}\\ k$\nis canonically isomorphic to\n$\\Lambda(n)[2n]$ if\nevery irreducible component of $X$\nis of dimension $n$.\nFor a constructible complex\n${\\cal F}$ of finite tor-dimension,\nthe dual $D_X{\\cal F}\n=R{\\cal H}om({\\cal F},{\\cal K}_X)$\nis also constructible of finite tor-dimension.\n\n\\begin{df}\\label{dfms}\nLet $X$ be a smooth scheme\nover a field $k$ and\nlet $C\\subset T^*X$ be a closed conical\nsubset. Let\n${\\cal F}$ be a constructible\ncomplex of $\\Lambda$-modules on $X$.\n\n{\\rm 1. \\cite[1.3]{Be}}\nWe say that ${\\cal F}$ is\n{\\em micro-supported} on $C$\nif for every $C$-transversal pair\n$(f,h)$ of morphisms\n$h\\colon W\\to X$ and $f\\colon W\\to Y$\nof smooth schemes,\nthe morphism $f\\colon W\\to Y$\nis locally acyclic relatively to \n$h^*{\\cal F}$.\n\n\n{\\rm 2. \\cite[1.5]{Be}}\nWe say that ${\\cal F}$ is\n{\\em weakly micro-supported} on $C$\nif for every $C$-transversal pair\n$(f,j)$ of an \\'etale morphism\n$j\\colon W\\to X$ satisfying the\ncondition {\\rm (W)} below\nand a morphism\n$f\\colon W\\to Y={\\mathbf A}^1_k$,\nthe morphism $f\\colon W\\to Y$\nis locally acyclic relatively to \n$h^*{\\cal F}$:\n\n{\\rm (W)}\nIf $k$ is infinite,\nthen $j\\colon W\\to X$\nis an open immersion.\nIf $k$ is finite,\nthen $j\\colon W\\to X$\nis the composition of\nan open immersion $V\\to X$\nwith $W=V_{k'}\n=V\\times_kk'\\to V$\nfor a finite extension $k'$ of $k$.\n\\end{df}\n\n\\begin{lm}\\label{lmmc}\nLet $C$ and $C'$ be closed\nconical subsets of $T^*X$\nand ${\\cal F}$ be a constructible\ncomplex of $\\Lambda$-modules on $X$.\n\n{\\rm 1.}\nAssume that $C\\subset C'$.\nIf ${\\cal F}$ is micro-supported\n(resp.\\ weakly micro-supported) on $C$,\nthen\n${\\cal F}$ is micro-supported\n(resp.\\ weakly micro-supported) on $C'$.\n\n{\\rm 2.}\nThe complex ${\\cal F}$ is weakly micro-supported\non $C$ if and only if\n$C$ contains the images\nof the fibers $(U\\times_YT^*Y)\n\\times_Ux$ for all \n\\'etale morphisms $j\\colon U\\to X$\nand morphisms\n$f\\colon U\\to Y$ to smooth curves\nsatisfying the condition {\\rm (W)}\nin Definition {\\rm \\ref{dfms}}\nand for closed points $x$ of $U$\nwhere $\\phi_x(j^*{\\cal F},f)\\neq 0$.\n\n\n{\\rm 3.}\nIf ${\\cal F}$ is weakly micro-supported on $C$,\nthen the base $B$ of $C$ contains\nthe support of ${\\cal F}$\nas a subset.\n\n{\\rm 4.}\nWe consider the following conditions\nfor a morphism $h\\colon W\\to X$\nof smooth schemes over $k$:\n\n{\\rm (1)}\n${\\cal F}$ is micro-supported on $C$.\n\n{\\rm (2)}\n$h^*{\\cal F}$ is micro-supported\non $h^{\\circ}C$.\n\nIf $h$ is $C$-transversal,\nwe have\n{\\rm (1)}$\\Rightarrow${\\rm (2)}.\nIf $h$ is \\'etale and surjective,\nconversely we have\n{\\rm (2)}$\\Rightarrow${\\rm (1)}.\n\n{\\rm 5. (\\cite[Lemma 2.1 (ii)]{Be})}\nAssume that \n${\\cal F}$ is micro-supported on $C$.\nThen, for $C$-transversal pair\n$(f,h)$ of morphisms\n$h\\colon W\\to X$ and $f\\colon W\\to Y$\nof smooth schemes,\nthe morphism $f\\colon W\\to Y$\nis {\\em universally} locally acyclic relatively to \n$h^*{\\cal F}$.\n\n\n{\\rm 6.} {\\rm ({\\cite[Lemma 2.2 (ii)]{Be}})}\nAssume ${\\cal F}$ is micro-supported\n(resp.\\ weakly micro-supported) on $C$\nand let $f\\colon X\\to Y$ be a proper morphism\nof smooth schemes over $k$.\nThen $Rf_*{\\cal F}$ is micro-supported\n(resp.\\ weakly micro-supported)\non $f_{\\circ}C$.\n\n{\\rm 7.}\nLet $\\Lambda_0$\nbe the residue field of $\\Lambda$.\nThen,\n${\\cal F}$ is micro-supported on $C$\nif and only if\n${\\cal F}\\otimes_\\Lambda^L\n\\Lambda_0$ is micro-supported on $C$.\n\\end{lm}\n\n\\proof{\n1.\nIf a pair $(f,h)$\nof morphisms $h\\colon W\\to X$\nand $f\\colon X\\to Y$ is $C'$-transversal,\nthen $(f,h)$\nis $C$-transversal.\nHence the assertion follows.\n\n2.\nSince the vanishing cycles\ncomplex $\\phi(j^*{\\cal F},f)$\nis constructible,\nit follows from Lemma \\ref{lmCf}.7.\n\n3.\nDefine an open subset $U=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} B$\nto be the complement of\nthe base $B$ of $C$.\nLet $j\\colon U\\to X$\nbe the open immersion and\n$f\\colon U\\to {\\mathbf A}^1$\nbe the morphism collapsing to $0$.\nThen the pair $(f,j)$\nis $C$-transversal.\nHence the morphism\n$f\\colon U\\to {\\mathbf A}^1$ is locally acyclic\nrelatively to $j^*{\\cal F}$\nand we have ${\\cal F}|_U=0$.\nThus the base $B$\ncontains ${\\rm supp}\\ {\\cal F}$.\n\n4.\n{\\rm (1)}$\\Rightarrow${\\rm (2)}:\nAssume $h\\colon W\\to X$ is\n$C$-transversal.\nIf the pair $(f,g)$ of morphisms $g\\colon V\\to W$ \nand $f\\colon V\\to Y$ \nof smooth schemes over $k$\nis $h^{\\circ}C$-transversal,\nthen the pair $(f,h\\circ g)$ is\n$C$-transversal by Lemma \\ref{lmCh}.3.\nHence the assertion follows.\n\n{\\rm (2)}$\\Rightarrow${\\rm (1)}:\nThe local acyclicity is an \\'etale local condition.\n\n7.\nIt follows from Lemma \\ref{lmctf}.2.\n\\qed}\n\\medskip\n\nBy Lemma \\ref{lmmc}.2,\nthere exists a smallest closed conical\nsubset $C\\subset T^*X$\nsuch that ${\\cal F}$ is weakly micro-supported\non $C$.\nThis smallest $C$ is called the\n{\\em weak singular support} and is denoted by\n$SS^w{\\cal F}$.\nFor a closed conical subset $C$ of $T^*X$,\na constructible complex\n${\\cal F}$ is weakly micro-supported on $C$\nif and only if\n$SS^w{\\cal F}$ is a subset of $C$.\nBy Lemma \\ref{lmmc}.2 and 3,\nthe base of\n$SS^w{\\cal F}$ equals the support of ${\\cal F}$.\n\nIf there exists a smallest closed conical\nsubset $C\\subset T^*X$\nsuch that ${\\cal F}$ is micro-supported\non $C$, then we call such $C$\nthe {\\em singular support}\nof ${\\cal F}$ and let it denoted by\n$SS{\\cal F}$.\nIf the singular support exists,\nwe have\n$SS^w{\\cal F}\n\\subset\nSS{\\cal F}$ by definition.\nIn fact, Theorem \\ref{thmss} below\nincludes the existence of\n$SS{\\cal F}$ and the equality\n$SS^w{\\cal F}\n=SS{\\cal F}$.\n\n\n\n\\begin{lm}\\label{lmlcst}\nLet $X$ be a smooth scheme over $k$\nand ${\\cal F}$ be a constructible\ncomplex of $\\Lambda$-modules on $X$.\n\n{\\rm 1.}\n{\\rm (\\cite[Lemma 2.1 (iii)]{Be})}\nThe following conditions are equivalent:\n\n{\\rm (1)}\n${\\cal F}$ is micro-supported on \nthe $0$-section $T^*_XX$.\n\n{\\rm (2)}\n${\\cal F}$ is locally constant.\n\nIf ${\\cal F}$ is locally constant\nand if the support of ${\\cal F}$ is $X$,\nthen the singular support\n$SS{\\cal F}$ exists and \nboth $SS{\\cal F}$ and $SS^w{\\cal F}$\nare equal to the $0$-section\n$T^*_XX$.\n\n{\\rm 2.}\nAssume $\\dim X=1$.\nLet $U\\subset X$ be the largest open\nsubset where the restriction ${\\cal F}|_U$\nis locally constant.\nThen, ${\\cal F}$ is micro-supported\non the union\n\\begin{equation}\nT^*_XX\\cup \\bigcup_{x\\in {X\\!\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} \\!U}}\n(T^*X\\times_Xx)\n\\label{eqScv}\n\\end{equation}\nof the $0$-section and\nthe fibers of the complement.\nFurther if $X={\\rm supp}\\ {\\cal F}$, \nthen the singular support $SS{\\cal F}$\nand $SS^w{\\cal F}$ \nequal the closed conical subset\n{\\rm (\\ref{eqScv})}.\n\n\n{\\rm 3.}\nLet $D=\\bigcup_{i=1}^mD_i$\nbe a divisor with simple normal crossings\nof $X$ relatively to $k$.\nLet ${\\cal G}$ be a locally constant \nconstructible sheaf\nof $\\Lambda$-modules\non $U=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} D$ tamely ramified along $D$\nand let $j\\colon U\\to X$ be the open immersion.\nThen ${\\cal F}=j_!{\\cal G}$\nis micro-supported on the union\n\\begin{equation}\nC_D=\\bigcup_{I\\subset\\{1,\\ldots,m\\}}\nT^*_{D_I}X\n\\label{eqStame}\n\\end{equation}\nof the conormal bundles\nof the intersections\n$D_I=\\bigcap_{i\\in I}D_i$\nof irreducible components\nfor all subsets $I\\subset\\{1,\\ldots,m\\}$\nof indices.\n\n{\\rm 4.}\nAssume that ${\\cal F}$ is micro-supported\non $C$ and \nlet $f\\colon X\\to Y$ be a $C$-transversal\nproper morphism.\nThen $Rf_*{\\cal F}$ is locally constant.\n\n{\\rm 5.}\n{\\rm ({\\cite[Lemma 2.5 (i)]{Be}})}\nFor a closed immersion\n$i\\colon X\\to Y$\nof smooth schemes over $k$,\nwe have $SS^wi_*{\\cal F}=i_{\\circ}SS^w{\\cal F}$.\n\\end{lm}\n\n\\proof{1.\n(1)$\\Rightarrow$(2)\nSince the identity ${\\rm id}_X\\colon X\\to X$\nis $T^*_XX$-transversal by Lemma \\ref{lmCf}.3,\nthe identity ${\\rm id}_X\\colon X\\to X$\nis locally acyclic relatively to ${\\cal F}$.\nThis means that ${\\cal F}$ is locally constant\nby \\cite[Proposition 2.11]{cst}.\n\n(2)$\\Rightarrow$(1)\nSince a $T^*_XX$-transversal\nmorphism $f\\colon X\\to Y$\nof smooth schemes \nis smooth by Lemma \\ref{lmCf}.2,\nit follows from \nthe local acyclicity of smooth morphism\n\\cite[Th\\'eor\\`eme 2.1]{Artin}.\n\n\nIf ${\\cal F}$ is locally constant\nand if the support of ${\\cal F}$ is $X$,\nLemma \\ref{lmmc}.3 and\n(2)$\\Rightarrow$(1)\nimplies that the singular support\n$SS{\\cal F}$ exists and \nthat the equality $SS{\\cal F}=SS^w{\\cal F}\n=T^*_XX$ holds.\n\n\n\n2.\nIt follows from 1.\\ that\n${\\cal F}$ is micro-supported on the union\nin (\\ref{eqScv}).\n\nAssume that ${\\cal F}$ is \nweakly micro-supported on $C$.\nIf ${\\cal F}$ is not locally constant at $x$,\nthen $\\phi_x({\\cal F},{\\rm id}_X)\\neq 0$.\nHence $C$ contains the fiber $x\\times_XT^*X$\nby Lemma \\ref{lmmc}.2.\nIf ${\\rm supp}\\ {\\cal F}=X$,\nthen $C$ contains the $0$-section\n$T^*_XX$ by Lemma \\ref{lmmc}.3.\n\n3.\nBy Lemma \\ref{lmCh}.5,\nit suffices to show that\nany $C$-transversal morphism\n$f\\colon X\\to Y$\nof smooth schemes is universally\nlocally acyclic relatively to ${\\cal F}$.\nBy Lemma \\ref{lmCf}.8,\nthe $C$-transversality of\n$f\\colon X\\to Y$ is equivalent to\nthat the divisor $D$ has\nsimple normal crossing relatively to $Y$.\nHence, the assertion follows from\na variant of \\cite[1.3.3 (i)]{app}\nwith $Rf_*$ replaced by $f_!$\nproved similarly.\n\n4.\nSince $f$ is proper,\n$Rf_*{\\cal F}$ is\nmicro-supported on $f_{\\circ}C$\nby Lemma \\ref{lmmc}.6.\nSince $f$ is $C$-transversal,\n$f_{\\circ}C\\subset T^*Y$ is a subset of the \n$0$-section.\nHence\n$Rf_*{\\cal F}$ is locally constant\nby 1 and Lemma \\ref{lmmc}.1.\n\\qed}\n\n\n\\begin{thm}\\label{thmss}\nLet $X$ be a smooth scheme of finite type\nover a field $k$ and\nlet ${\\cal F}$ be a constructible complex\nof $\\Lambda$-modules on $X$.\n\n{\\rm 1. (\\cite[Theorem 1.3 (i)]{Be})}\nThe singular support\n$SS{\\cal F}$ exists.\n\n{\\rm 2. (\\cite[Theorem 1.3 (ii)]{Be})} \nIf every irreducible component of $X$\nis of dimension $n$,\nthen\nevery irreducible component of $SS{\\cal F}$\nis of dimension $n=\\dim X$.\n\n{\\rm 3. (\\cite[Theorem 1.5]{Be})}\nWe have $SS{\\cal F}=\nSS^w{\\cal F}$.\n\n\n{\\rm 4. (\\cite[Theorem 1.4 (ii)]{Be})}\nFor an exact sequence\n$0\\to {\\cal F}'\\to {\\cal F}\\to {\\cal F}''\\to 0$\nof perverse sheaves,\nwe have\n\\begin{equation}\nSS{\\cal F}\n=\nSS{\\cal F}'\\cup\nSS{\\cal F}''.\n\\label{eqsump}\n\\end{equation}\nWe have\n\\begin{equation}\nSS{\\cal F}\n=\n\\bigcup_q\nSS\\ ^p\\!{\\cal H}^q{\\cal F}.\n\\label{eqssu}\n\\end{equation}\n\\end{thm}\n\n\\begin{cor}\\label{corss}\n\n{\\rm 1.}\nFor a closed conical subset $C$ of $T^*X$,\nthe following conditions are\nequivalent:\n\n{\\rm (1)}\n${\\cal F}$ is micro-supported on $C$.\n\n{\\rm (2)}\n${\\cal F}$ is weakly micro-supported on $C$.\n\n{\\rm 2. (\\cite[Theorem 1.4 (i)]{Be})}\nFor a smooth morphism $h\\colon W\\to X$,\nwe have\n$SSh^*{\\cal F}=h^{\\circ}SS{\\cal F}$.\n\n{\\rm 3. (\\cite[Lemma 2.1 (i)]{Be})}\nThe base of $SS{\\cal F}=SS^w{\\cal F}$\nequals the support of ${\\cal F}$.\n\n{\\rm 4.}\nAssume that ${\\cal F}$\nis of finite tor-dimension and\nlet $\\Lambda_0$\nbe the residue field of $\\Lambda$.\nThen, for \n${\\cal F}_0=\n{\\cal F}\\otimes_\\Lambda^L\\Lambda_0$\nwe have\n$SS{\\cal F}=SS{\\cal F}_0$.\n\\end{cor}\n\n\\proof{\n1.\nBy Theorem \\ref{thmss}.3,\nboth conditions are equivalent to that\n$SS{\\cal F}$ is a subset of $C$.\n\n2.\nBy Lemma \\ref{lmmc}.4 \n{\\rm (2)}$\\Rightarrow${\\rm (1)}\nand by 1,\nwe have $SSh^*{\\cal F}=h^*SS{\\cal F}$\nfor an \\'etale morphism $h\\colon W\\to X$.\nHence we may assume $W={\\mathbf A}^n\\times X$.\nWe have\n$SSh^*{\\cal F}\\subset h^{\\circ}SS{\\cal F}$\nby Lemma \\ref{lmmc}.4\n{\\rm (1)}$\\Rightarrow${\\rm (2)}.\nHence the $0$-section $i\\colon X\\to \nW={\\mathbf A}^n$ is $SSh^*{\\cal F}$-transversal\nand further \nwe have\n$SS{\\cal F}\\subset i^{\\circ}SSh^*{\\cal F}$\nby Lemma \\ref{lmmc}.4\n{\\rm (1)}$\\Rightarrow${\\rm (2)}.\nThus we have\n$SSh^*{\\cal F}=h^*SS{\\cal F}$.\n\n4.\nIt follows from Lemma \\ref{lmctf}.2.\n\\qed}\n\\medskip\n\n\n\n\nLet $X$ be a smooth scheme over \na field $k$ and let\n$i\\colon X\\to {\\mathbf P}$\nbe an immersion to a projective space\nas in (\\ref{eqiXP}).\nLet ${\\cal F}$ be\na constructible complex\nof $\\Lambda$-modules\nmicro-supported on\na closed conical subset $C\n\\subset T^*X$\nof the cotangent bundle.\nIf $X$ is connected,\nthe condition (C) \nfor $C=SS{\\cal F}$ means\nthat the support of $i_*{\\cal F}$\nis not equal to ${\\mathbf P}$\nby Corollary \\ref{corss}.3.\n\n\nBy Lemma \\ref{lmPC} \nand Lemma \\ref{lmmc}.5,\nthe morphism $p^\\vee \\colon\nX\\times_{\\mathbf P}Q\n={\\mathbf P}(X\\times_{\\mathbf P}T^*{\\mathbf P})\n\\to {\\mathbf P}^\\vee$\nis universally locally acyclic relatively to\n$p^*{\\cal F}$ on the complement\nof the projectivization \n${\\mathbf P}(\\widetilde C)\\subset\n{\\mathbf P}(X\\times_{\\mathbf P}T^*{\\mathbf P})$.\nMore precisely, the following holds.\n\n\n\\begin{thm}[{\\rm cf.\\\n\\cite[Theorem 3.2, Lemma 3.3]{Be}}]\n\\label{thmRn}\nLet $i\\colon X\\to {\\mathbf P}$\nbe an immersion and let\n$\\widetilde C\\subset\nX\\times_{\\mathbf P}T^*{\\mathbf P}$\nbe the inverse image\nof the singular support\n$C=SS{\\cal F}\\subset T^*X$\nby the surjection\n$X\\times_{\\mathbf P}T^*{\\mathbf P}\\to\nT^*X$.\nThen, the complement\n$U=X\\times_{\\mathbf P}Q\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~}\n{\\mathbf P}(\\widetilde C)$\nof the projectivization \n${\\mathbf P}(\\widetilde C)\\subset\n{\\mathbf P}(X\\times_{\\mathbf P}T^*{\\mathbf P})\n=X\\times_{\\mathbf P}Q$\nis the largest open subset\nwhere $p^\\vee \\colon\nX\\times_{\\mathbf P}Q\n={\\mathbf P}(X\\times_{\\mathbf P}T^*{\\mathbf P})\n\\to {\\mathbf P}^\\vee$\nis universally locally acyclic relatively to\n$p^*{\\cal F}$.\n\\end{thm}\n\n\n\\proof{\nFirst, we consider the case where\n$X={\\mathbf P}$ is a projective space.\nApplying \\cite[Theorem 3.2]{Be} to \nthe Radon transform $R{\\cal F}$ \\cite{Rn},\nwe obtain\n$E_{p^{\\vee}}(p^{*}R^\\vee R({\\cal F}))=\n{\\mathbf P}(SSR({\\cal F}))$\nin the notation loc.\\ cit.\nSince, $R^\\vee R({\\cal F})$ is isomorphic to ${\\cal F}$\nexcept locally constant sheaf \\cite{Rn},\nwe have\n$E_{p^{\\vee}}(p^{*}R^\\vee R({\\cal F}))=\nE_{p^{\\vee}}(p^{*}{\\cal F})$.\nBy \\cite[Lemma 3.3]{Be}, we have\n${\\mathbf P}(SS{\\cal F})=\n{\\mathbf P}(SSR({\\cal F}))\n\\subset Q$.\nHence the assertion follows\nfor $X={\\mathbf P}$.\n\nLet $V\\subset {\\mathbf P}$ be\nan open subset including $X$ as a closed subset and\nlet $i^{\\circ}\\colon X\\to V$ be the closed immersion.\nSince the assertion is proved for ${\\mathbf P}$,\nit holds also for an open subscheme $V$.\nSince $SS(i^{\\circ}_*{\\cal F})=\\widetilde C$,\nthe complement\n$V\\times_{\\mathbf P}Q\n\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} {\\mathbf P}(\\widetilde C)$\nis the largest open subset\nof $V\\times_{\\mathbf P}Q$\nwhere $p^{\\vee}\\colon V\\times_{\\mathbf P}Q\n\\to {\\mathbf P}^{\\vee}$ is universally locally \nacyclic relatively to $p^{*}i^{\\circ}_*{\\cal F}$\nby the assertion already proved for $V$.\nSince $p^{*}i^{\\circ}_*{\\cal F}=0$\noutside $X\\times_{\\mathbf P}Q$,\nthe assertion follows.\n\\qed}\n\n\n\n\\begin{cor}\\label{corRn}\nLet $i\\colon X\\to {\\mathbf P}$\nbe an immersion and\n$C\\subset T^*X$\nbe a closed conical subset\nsuch that the base $B\\subset X$\ncontains the support of ${\\cal F}$.\nAssume that\nthere exists a closed subset\n$Z\\subset X\\times_{\\mathbf P}Q\n={\\mathbf P}(X\\times_{\\mathbf P}T^*{\\mathbf P})$\nof codimension $> \\dim X$\nsuch that \n$p^\\vee \\colon\nX\\times_{\\mathbf P}Q\n={\\mathbf P}(X\\times_{\\mathbf P}T^*{\\mathbf P})\n\\to {\\mathbf P}^\\vee$\nis universally locally acyclic on the complement\nof ${\\mathbf P}(\\widetilde C)\\cup Z\\subset \n{\\mathbf P}(X\\times_{\\mathbf P}T^*{\\mathbf P})$.\nThen ${\\cal F}$ is micro-supported on $C$.\n\\end{cor}\n\n\\proof{\nLet $C_0=SS{\\cal F}$\ndenote the singular support.\nBy Corollary \\ref{corss}.3,\nthe base $B_0$ of $C_0$\nequals the support of ${\\cal F}$.\nBy Theorem \\ref{thmRn},\nwe have ${\\mathbf P}(\\widetilde C)\\cup Z\n\\supset {\\mathbf P}(\\widetilde C_0)$.\nSince $Z\\subset \nX\\times_{\\mathbf P}Q$ is of codimension $>\n\\dim X$ by the assumption and\nevery irreducible component\nof ${\\mathbf P}(\\widetilde C_0)\\subset \nX\\times_{\\mathbf P}Q$ is of codimension $\\dim X$\nby Theorem \\ref{thmss}.2,\nwe have ${\\mathbf P}(\\widetilde C)\n\\supset {\\mathbf P}(\\widetilde C_0)$.\nThus, we have\n$C\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} B\\supset C_0\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} B_0$.\nBy the assumption $B\\supset B_0$,\nwe have $C\\supset C_0=SS{\\cal F}$\nand ${\\cal F}$ is micro-supported on $C$\nby Lemma \\ref{lmmc}.1.\n\\qed}\n\\medskip\n\n\n\n\\begin{thm}[{\\rm \\cite[Theorem 1.7]{Be}}]\\label{thmssD}\nLet $X$ be a {\\em projective}\nsmooth scheme of dimension $n$\nover a field $k$ and\n${\\cal L}$ be an ample\ninvertible ${\\cal O}_X$-module.\nLet $E$ be a $k$-vector space\nof finite dimension and \n$E\\to \\Gamma(X,{\\cal L})$\nbe a $k$-linear mapping\ndefining a {\\em closed} immersion\n$X\\to {\\mathbf P}={\\mathbf P}(E^\\vee)$\nand \nsatisfying the condition {\\rm (E)} \nbefore Lemma {\\rm \\ref{lmloc}}.\nThen, $D=p^\\vee({\\mathbf P}(SS{\\cal F}))\n\\subset {\\mathbf P}^\\vee$\nis a divisor and the complement\n${\\mathbf P}^\\vee\n\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} D$ is the largest open\nsubset where\n$Rp^\\vee_*p^*{\\cal F}$\nis locally constant.\n\\end{thm}\n\n\\proof{\nIt suffices to apply\n\\cite[Theorem 1.7]{Be}\nto $i_*{\\cal F}$.\\qed}\n\n\n\n\\begin{cor}\\label{corssD}\nLet ${\\cal F}$ be a constructible\ncomplex of finite tor-dimension.\nThen, we have\n\\begin{equation}\nSS{\\cal F}\n=\nSS D_X{\\cal F}.\n\\label{eqssD}\n\\end{equation}\n\\end{cor}\n\nOne could deduce \ndirectly Corollary \\ref{corssD}\nand its consequence Lemma \\ref{lmRf}.4\nfrom the compatibility of vanishing cycles\nwith duality which seems missing\nin the literature.\n\n\\proof{\nWe may assume that $\\Lambda=\\Lambda_0$\nis a finite field by Corollary \\ref{corss}.4.\nSince the assertion is local on $X$,\nwe may take a closed immersion\n$i\\colon X\\to U$\nand an open immersion\n$j\\colon U\\to {\\mathbf P}$\nto a projective space.\nBy Lemma \\ref{lmlcst}.5 and\nCorollary \\ref{corss}.2,\nwe may assume $X$ is projective.\n\nLet $C=SS{\\cal F}$\nand $C'=SS D_X{\\cal F}$\nbe the singular supports.\nWe take a projective embedding\n$i\\colon X\\to {\\mathbf P}$\ndefined by $E$ satisfying\nthe conditions (E) \nbefore Lemma {\\rm \\ref{lmloc}}\nand (C) before\nProposition {\\rm \\ref{prwi}}.\nLet $D=p^\\vee({\\mathbf P}(\\widetilde C)),\nD'=p^\\vee({\\mathbf P}(\\widetilde C'))\n\\subset {\\mathbf P}^\\vee$\nbe the images.\nThen, by Theorem \\ref{thmssD},\nthe complement\n${\\mathbf P}^\\vee\n\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} D$\nis the largest open subset\nwhere the (shifted and twisted) Radon transform\n$Rp^\\vee_*p^*{\\cal F}$\nis locally constant.\nSimilarly,\nthe complement\n${\\mathbf P}^\\vee\n\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} D'$\nis the largest open subset\nwhere \n$Rp^\\vee_*p^*D_X{\\cal F}$\nis locally constant.\nSince the Radon transform commutes\nwith duality up to shift and twist,\nwe have\n$D=D'$. Hence we have $C=C'$\nby Corollary \\ref{corrad}.2.\n\\qed}\n\\medskip\n\n\n\nIn the proof of Proposition \\ref{prperv},\nwe will use the following fact\nproved in the course of \nthe proof of \\cite[Theorem 1.5]{Be}.\nFor a line $L\\subset {\\mathbf P}^\\vee$,\nthe axis $A_L\\subset {\\mathbf P}$\nis a linear subspace of codimension $2$\ndefined as the intersections\nof hyperplanes parametrized by $L$.\nOn the complement\n$X_L^{\\circ}=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} (X\\cap A_L)$,\na canonical morphism\n\\begin{equation}\n\\begin{CD}\np_L^{\\circ}\\colon\nX_L^{\\circ} @>>> L\n\\end{CD}\n\\label{eqpLo}\n\\end{equation}\nis defined by sending a point\n$x\\in X_L^{\\circ}$\nto the unique hyperplane $H\\in L$\ncontaining $x$.\n\n\n\n\n\\begin{lm}[{\\rm \\cite[4.9 (ii), (iii)]{Be}}]\\label{lmRn}\nLet $i\\colon X\\to {\\mathbf P}$\nbe an immersion and\nlet ${\\cal F}$ be\na perverse sheaf of\n$\\Lambda$-modules on $X$.\nAfter replacing \nthe immersion $i\\colon X\\to {\\mathbf P}$\nby the composition with\n$d$-th Veronese embedding\nfor $d\\geqq 3$,\nfor an irreducible component \n$C_a$ of $C$,\nlet $D_a=\\overline{p^\\vee({\\mathbf P}\n(\\widetilde C_a))}\\subset {\\mathbf P}^\\vee$\ndenote the closure of the image.\nThen there exist\ndense open subsets\n$D_a^{\\circ} \\subset D_a$\nsatisfying the following conditions:\n\nFor $C_a\\neq C_b$,\nwe have $D_a^{\\circ}\\cap D_b=\\varnothing$.\nThe inverse image\n${\\mathbf P}(\\widetilde C_a)^{\\circ}\n=\n{\\mathbf P}(\\widetilde C_a)\n\\times_{D_a}\nD_a^{\\circ}$\nis finite and radicial over\n$D_a^{\\circ}$.\nFor every $(x,H)\n\\in \n{\\mathbf P}(\\widetilde C_a)^{\\circ}$\nand for every line $L\\subset\n{\\mathbf P}^\\vee$\nsuch that $x\\in X_L^{\\circ}$,\nthat \n$L$ meets $D_a^{\\circ}$\nproperly at $H=p_L^{\\circ}(x)$\nand that\nthe tangent line \n$TL\\times_L \\{ H\\}$ of $L$\nat $H=p_L^{\\circ}(x)\n\\in {\\mathbf P}^\\vee$\nis not perpendicular to\nthe fiber \n$T^*_Q({\\mathbf P}\\times{\\mathbf P}^\\vee)\n\\times_Q(x,H)\n\\subset\nT^*{\\mathbf P}^\\vee\n\\times_{{\\mathbf P}^\\vee}\\{H\\}$,\nwe have \n$\\varphi_x^{-1}({\\cal F},p_L^{\\circ})\\neq 0$\nand\n$\\varphi_x^q({\\cal F},p_L^{\\circ} )= 0$\nfor $q\\neq -1$.\n\\end{lm}\n\n\n\n\n\\subsection{Singular support and ramification}\\label{ssram}\n\nWe assume $k$ is perfect.\nIn this subsection, we describe the\nsingular support $SS{\\cal F}$\nof ${\\cal F}=j_!{\\cal G}$\nfor a locally constant sheaf\n${\\cal G}$ on the complement\n$U=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} D$ of a divisor $D$\nwith simple normal crossings\nof a smooth scheme $X$\nover $k$\nand the open immersion $j\\colon U\\to X$.\nFirst, we study the tamely ramified case.\n\n\\begin{pr}\\label{prtame}\nLet ${\\cal G}\\neq 0$ be\na locally constant constructible sheaf\nof $\\Lambda$-modules\non the complement\n$U=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} D$ of a divisor $D=\\bigcup_{i=1}^mD_i$\nwith simple normal crossings\nof a smooth scheme $X$\nover a perfect field $k$.\nAssume that ${\\cal G}$\nis {\\em tamely ramified} along $D$.\nFor the open immersion $j\\colon U\\to X$\nand ${\\cal F}=j_!{\\cal G}$,\nwe have \n\\begin{equation}\nSS{\\cal F}\n=\n\\bigcup_{I\\subset\\{1,\\ldots,m\\}}\nT^*_{D_I}X\n\\label{eqFtame}\n\\end{equation}\n\\end{pr}\n\n\\proof{\nBy Lemma \\ref{lmlcst}.3,\nwe have\n$SS{\\cal F}\n\\subset\n\\bigcup_{I\\subset\\{1,\\ldots,m\\}}\nT^*_{D_I}X$.\nBy Corollary \\ref{corss}.2,\nit is reduced to\nthe case where $X={\\mathbf A}^n$\nand $U={\\mathbf G}_m^n$.\nBy the induction on $n=\\dim X$\nand further by the\ncompatibility with smooth pull-back,\nit suffices to show that the fiber\n$T^*_0X$ at the origin $0\\in X=\n{\\mathbf A}^n={\\rm Spec}\\ k[T_1,\\ldots,T_n]$\nis a subset of $SS^w{\\cal F}$.\nIf $n=0$,\nsince ${\\cal G}\\neq 0$,\nwe have \n$SS{\\cal F}=T^*X$ by Lemma \\ref{lmmc}.3.\n\nAssume $n>0$.\nLet $D_i=(T_i=0)$\n$\\subset X$ and\n$C=\\bigcup_{I\\subsetneqq\\{1,\\ldots,n\\}}\nT^*_{D_I}X$\nbe the union except the fiber\n$T^*_0X$.\nSince the morphism\n$f\\colon X\\to Y=\n{\\rm Spec}\\ k[T]$ defined by\n$T\\mapsto T_1+\\cdots+T_n$\nis $C$-transversal,\nit suffices to show the following.\n\\qed}\n\n\n\\begin{lm\n\\label{lmtame}\nLet $S={\\rm Spec}\\ {\\cal O}_{K}$ be the spectrum\nof a henselian discrete valuation \nring with algebraically closed residue field $k$,\n$X$ be a smooth\nscheme of finite type \nof relative dimension $n-1$ over $S$\nand $D$ be a divisor of $X$\nwith simple normal crossings.\nLet $x$ be a closed point\nof the closed fiber of $X$\ncontained in $D$.\nLet $t_1,\\ldots,t_n\\in {\\mathfrak m}_x$\nbe elements of the maximal ideal such that\n$\\bar t_1,\\ldots,\\bar t_n\\in \n{\\mathfrak m}_x\/{\\mathfrak m}_x^2$\nis a basis.\nAssume that $D$\nis defined by $t_1\\cdots t_n$\nand that the class of \na uniformizer $\\pi$ of $S$\nin ${\\mathfrak m}_x\/\n{\\mathfrak m}_x^2$\nis not contained in any subspaces\ngenerated by $n-1$ elements\nof the basis $\\bar t_1,\\ldots,\\bar t_n$.\n\nLet $\\Lambda$ be a finite local ring\nwith residue characteristic $\\ell$ invertible on $S$\nand let ${\\cal G}$ be a locally constant\nconstructible sheaf of $\\Lambda$-modules\non the complement $U=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} D$\ntamely ramified along $D$.\nLet $j\\colon U\\to X$ denote\nthe open immersion.\n\n{\\rm 1.}\nOn a neighborhood of $x$,\nthe complex $\\phi(j_!{\\cal G})$\nis acyclic except at $x$\nand at degree $n-1$\nand the action of the inertia group\n$I_K={\\rm Gal}(\\bar K\/K)$\non $\\phi^{n-1}_x(j_!{\\cal G})$\nis tamely ramified.\n\n{\\rm 2.}\nIf ${\\cal G}$ is a sheaf of\nfree $\\Lambda$-modules,\nthen the $\\Lambda$-module\n$\\phi^{n-1}_x(j_!{\\cal G})$\nis free of rank ${\\rm rank}\\ {\\cal G}$.\n\\end{lm}\n\n\\proof{\n1.\nWe may write $\\pi=\\sum_{i=1}^n\nu_{i}t_{i}$ in ${\\mathfrak m}_x$\nand $u_{i}$ are invertible.\nBy replacing $t_{i}$ by \n$u_{i}t_{i}$,\nwe may assume that \n$X$ is \\'etale over\n${\\rm Spec} \\ {\\cal O}_{K}\n[t_{1},\\ldots,t_{n}]\/\n(\\pi-(t_{1}+\\cdots+t_n))$.\nBy Abhyankar's lemma,\nwe may assume that ${\\cal G}$\nis trivialized by the abelian covering\n$s_{i}^{m}=t_{i}$ for an integer\n$m$ invertible on $S$.\nSince the assertion is \\'etale local on $X$,\nwe may assume\n$X={\\rm Spec} \\ {\\cal O}_{K}\n[t_{1},\\ldots,t_n]\/\n(\\pi-(t_{1}+\\cdots+t_n))$.\nHence by the variant of\n\\cite[1.3.3 (i)]{app} for $f_!$,\nthe complex $\\phi(j_!{\\cal G})$\nis acyclic outside $x$.\nSince the complex\n$\\phi(j_!{\\cal G})[n-1]$ is a perverse \nsheaf by \\cite[Corollaire 4.6]{au},\nthe complex\n$\\phi(j_!{\\cal G})$ is acyclic except at \ndegree $n-1$.\n\n\nLet $p\\colon X'\\to X$ be the blow-up at $x$\nand $j'\\colon U\\to X'$ be the open immersion.\nLet $D'$ be the proper transform of $D$\nand $E$ be the exceptional divisor.\nThen, the union of $D'$ with the closed\nfiber $X'_s$ has simple normal crossings.\nHence, the action of the inertia group \n$I_K$ on\n$\\phi(j'_!{\\cal G})$ is tamely ramified\nby \\cite[Proposition 6]{epsilon}\nand $\\phi_x(j_!{\\cal G})=R\\Gamma(E,\\phi(j'_!{\\cal G}))$\nis also tamely ramified.\n\n2.\nBy 1,\nwe may assume $\\Lambda$ is a field.\nWe consider the stratification of\n$E$ defined by the intersections\nwith the intersections of\nirreducible components of $D'$.\nThen, on each stratum,\nthe restriction of the cohomology sheaves\n$\\phi^q(j'_!{\\cal G})$ are locally constant\nand are tamely ramified along the boundary\nby \\cite[Proposition 6]{epsilon}.\nFurther, the alternating sum of\nthe rank is $0$ except for\n$E^{\\circ} =E\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} (E\\cap D')$\nand equals ${\\rm rank}\\ {\\cal G}$ on\n$E^{\\circ}$.\nHence, we have\n$$\\dim \\phi_x(j_!{\\cal G})\n=\\chi(E,\\phi(j'_!{\\cal G}))\n=\\chi_c(E^{\\circ},\\phi(j'_!{\\cal G}))\n={\\rm rank}\\ {\\cal G}\\cdot\n\\chi_c(E^{\\circ}).$$\nSince $E^{\\circ}=E_{n-1}^{\\circ}\n\\subset {\\mathbf G}_m^{n-1}\n={\\rm Spec}\\ k[t_1^{\\pm 1},\\ldots,\nt_{n-1}^{\\pm 1}]$\nis the complement\nof the intersection\nwith the hyperplane\n$\\sum_it_i+1=0$\nand the intersection \nis isomorphic to $E_{n-2}^{\\circ}$,\nwe have\n$\\chi_c(E^{\\circ}_{n-1})=\n\\chi_c({\\mathbf G}_m^{n-1})-\n\\chi_c(E^{\\circ}_{n-2})=\n(-1)^{n-1}$\nby induction on $n$.\nThus the assertion follows.\n\\qed}\n\\medskip\n\nTo give a description of the singular\nsupport in some wildly ramified case\nusing ramification theory,\nwe briefly recall ramification theory\n\\cite{AS}, \\cite{nonlog}.\nLet $K$ be a henselian discrete valuation field\nwith residue field of characteristic $p>0$\nand $G_K={\\rm Gal}(K_{\\rm sep}\/K)$\nbe the absolute Galois group.\nThen, the (non-logarithmic)\nfiltration $(G_K^r)_{r\\geqq 1}$\nby ramification groups\nis defined in \\cite[Definition 3.4]{AS}.\nIt is a decreasing filtration by closed\nnormal subgroups\nindexed by rational numbers $\\geqq 1$.\n\nFor a real number $r\\geqq 1$,\nwe define subgroups\n$G_K^{r+}\\subset G_K^{r-}$\nby $G_K^{r+}=\\overline{\\bigcup_{s>r}\nG_K^s}$\nand $G_K^{r-}=\\bigcap_{s<r}\nG_K^s.$\nIt is proved in \\cite[Proposition 3.7 (1)]{AS} that\n$G_K^1$ is the inertia group\n$I={\\rm Ker}(G_K\\to G_F)$ where\n$G_F$ denotes the absolute Galois\ngroup of the residue field $F$\nand $G_K^{1+}$ is the wild inertia group $P$\nthat is the pro-$p$ Sylow subgroup of $I$.\nIt is also proved in \\cite[Theorem 3.8]{AS} that\n$G_K^{r-}=G_K^{r}$\nfor rational numbers $r>1$\nand \n$G_K^{r-}=G_K^{r+}$\nfor irrational numbers $r>1$.\n\nLet $\\Lambda$ be a finite field\nof characteristic $\\neq p$\nand let $V$ be a continuous representation\nof $G_K$ on a $\\Lambda$-vector space\nof finite dimension.\nThen, since $P=G_K^{1+}$ is a pro-$p$ group\nand since \n$G_K^{r-}=G_K^{r}$\nfor rational $r$ and\n$G_K^{r-}=G_K^{r+}$\nfor irrational $r$,\nthere exists a unique decomposition\n\\begin{equation}\nV=\\bigoplus_{r\\geqq 1}V^{(r)}\n\\label{eqslp}\n\\end{equation}\ncalled the slope decomposition\ncharacterized by the condition that\nthe $G_K^{r+}$ fixed part\n$V^{G_K^{r+}}$ is equal to the sum\n$\\bigoplus_{s\\leqq r}V^{(s)}$.\n\n\n\nWe study a geometric case\nwhere $X$ is a smooth scheme\nover a perfect field $k$ of characteristic\n$p>0$.\nLet $D$ be a reduced and irreducible divisor\nand $U=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} D$ be the complement.\nLet ${\\cal G}$ be a \nlocally constant constructible sheaf of\n$\\Lambda$-vector spaces on $U$.\nLet $\\xi$ be the generic point of\nan irreducible component of $D$.\nThen, the local ring\n${\\cal O}_{X,\\xi}$ \nis a discrete valuation\nring and the fraction field $K$\nof its henselization is called\nthe local field at $\\xi$.\nThe stalk of ${\\cal H}^q{\\cal G}$\nat the geometric point of $U$\ndefined by a separable closure\n$K_{\\rm sep}$ defines\na $\\Lambda$-vector space $V^q$\nwith an action\nof the absolute Galois group $G_K$.\n\nFor a rational number $r>1$,\nthe graded quotient\n${\\rm Gr}^rG_K\n=G_K^r\/G_K^{r+}$ is \na profinite abelian group annihilated by $p$\n\\cite[Corollary 2.28]{nonlog}\nand its dual group is related\nto differential forms as follows.\nWe define ideals\n${\\mathfrak m}_{K_{\\rm sep}}^{(r)}$\nand ${\\mathfrak m}_{K_{\\rm sep}}^{(r+)}$\nof the valuation ring ${\\cal O}_{K_{\\rm sep}}$ by\n${\\mathfrak m}_{K_{\\rm sep}}^{(r)}\n=\\{x\\in K_{\\rm sep}\\mid {\\rm ord}_Kx\\geqq r\\}$\nand ${\\mathfrak m}_{K_{\\rm sep}}^{(r+)}\n=\\{x\\in K_{\\rm sep}\\mid {\\rm ord}_Kx>r\\}$\nwhere ${\\rm ord}_K$ denotes the valuation \nnormalized by ${\\rm ord}_K(\\pi)=1$\nfor a uniformizer $\\pi$ of $K$.\nThe residue field $\\bar F$ of ${\\cal O}_{K_{\\rm sep}}$\nis an algebraic closure of $F$ and \nthe quotient\n${\\mathfrak m}_{K_{\\rm sep}}^{(r)}\n\/{\\mathfrak m}_{K_{\\rm sep}}^{(r+)}$\nis an $\\bar F$-vector space of dimension 1.\nA canonical injection\n\\begin{equation}\nch\\colon\nHom_{{\\mathbf F}_p}(\n{\\rm Gr}^rG_K,\n{\\mathbf F}_p)\n\\to \nHom_{\\bar F}\n({\\mathfrak m}_{\\bar K}^{(r)}\/\n{\\mathfrak m}_{\\bar K}^{(r+)},\n\\Omega^1_{X\/k,\\xi}\n\\otimes{\\bar F})\n\\label{eqcf}\n\\end{equation}\nis also defined\n\\cite[Corollary 2.28]{nonlog}.\n\nWe say that the ramification of\n${\\cal G}$ along $D$ is\nisoclinic of slope $r\\geqq 1$\nif $V=V^{(r)}$ in the slope decomposition\n(\\ref{eqslp}).\nThe ramification of\n${\\cal G}$ along $D$ is\nisoclinic of slope $1$\nif and only if\nthe corresponding Galois representation $V$\nis tamely ramified.\nAssume that the ramification of\n${\\cal G}$ along $D$ is\nisoclinic of slope $r>1$.\nAssume also that \n$\\Lambda$ contains a primitive\n$p$-th root of 1 and\nidentify ${\\mathbf F}_p$ with\na subgroup of \n$\\Lambda^\\times$.\nThen, $V=V^{(r)}$ is further decomposed\nby characters\n\\begin{equation}\nV=\\bigoplus_{\\chi\\colon\n{\\rm Gr}^rG_K\\to \n{\\mathbf F}_p}\n\\chi^{\\oplus m(\\chi)}.\n\\label{eqchd}\n\\end{equation}\nFor a character $\\chi$ appearing\nin the decomposition (\\ref{eqchd}),\nthe twisted differential form\n$ch(\\chi)$ defined on a\nfinite covering of a dense open\nscheme of $D$ is\ncalled a characteristic form of ${\\cal G}$.\n\n\nAssume that \n$U=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} D$ is\nthe complement of a divisor\nwith simple normal crossings $D$\nand let $D_1,\\ldots,D_m$\nbe the irreducible components of $D$.\nWe say the ramification of\n${\\cal G}$ along $D$ is\nisoclinic of slope $R=\\sum_ir_iD_i$\nif the ramification of\n${\\cal G}$ along $D_i$ is\nisoclinic of slope $r_i$\nfor every irreducible component\n$D_i$ of $D$.\n\nIn \\cite[Definition 3.1]{nonlog},\nwe define the condition for\nramification of ${\\cal G}$ along $D$\nto be non-degenerate.\nThe condition implies that\nthe characteristic forms\nare extended to differential forms\non the boundary without zero.\nWe say that \nthe ramification of ${\\cal G}$ is non-degenerate along $D$ \nif it admits \\'etale locally\na direct sum decomposition\n${\\cal G}=\\bigoplus_j{\\cal G}_j$\nsuch that each ${\\cal G}_j$\nis isoclinic of slope $R_j\\geqq D$\nfor a ${\\mathbf Q}$-linear combination $R_j$\nof irreducible components of $D$\nand that the ramification of ${\\cal G}_j$\nis non-degenerate along $D$ at multiplicity $R_j$. Note that \nthere exists \na closed subset of codimension at least 2\nsuch that on its complement,\nthe ramification of ${\\cal G}$ along $D$\nis non-degenerate.\n\nWe introduce a slightly stronger\ncondition that implies local acyclicity.\nWe say that \nthe ramification of ${\\cal G}$ is \n{\\em strongly} non-degenerate along $D$\nif it satisfies the condition above\nwith $R_j\\geqq D$\nreplaced by $R_j=D$\nor $R_j>D$.\nThe inequality $R_j>D$\nmeans that the coefficient in $R_j$\nof every irreducible component of $D$\nis $>1$.\nNote that \nif the ramification of ${\\cal G}$ along $D$\nis non-degenerate,\non the complement of the singular locus\nof $D$, \nthe ramification of ${\\cal G}$ along $D$\nis strongly non-degenerate.\n\nLet $j\\colon U=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} D\\to X$\ndenote the open immersion.\nWe define a closed conical subset\n$S(j_!{\\cal G})\\subset T^*X$\nfollowing the definition \nof the characteristic cycle given\n\\cite[Definition 3.5]{nonlog}\nin the strongly non-degenerate case.\nWe will show that $S(j_!{\\cal G})$\nis in fact equal to the singular support\nin Proposition \\ref{prnd} below.\n\nAssume first that \nthe ramification of ${\\cal G}$\nalong $D$ is isoclinic of\nslope $R=D$.\nThen, the locally constant sheaf\n${\\cal G}$ on $U$ is tamely ramified along $D$.\nIn this case,\nlet the singular support $SSj_!{\\cal G}$\n(\\ref{eqFtame}) denoted by\n\\begin{equation}\nS(j_!{\\cal G})\n=\n\\bigcup_IT^*_{D_I}X\n\\label{eqtameS}\n\\end{equation}\nwhere $T^*_{D_I}X$\ndenotes the conormal bundle\nof the intersection $D_I=\\bigcap_ID_i$\nfor a set of indices\n$I\\subset \\{1,\\ldots,m\\}$.\n\nAssume the ramification of ${\\cal G}$\nalong $D$ is isoclinic of\nslope $R=\\sum_ir_iD_i>\nD=\\sum_iD_i$. \nFor each irreducible component,\nwe have a decomposition by \ncharacters \n$V=\\bigoplus_{\\chi\\colon\n{\\rm Gr}^{r_i}G_{K_i}\\to \n{\\mathbf F}_p}\n\\chi^{\\oplus m(\\chi)}$.\nFurther, the characteristic form\nof each character $\\chi$\nappearing in the decomposition\ndefines a sub line bundle $L_\\chi$\nof the pull-back $D_\\chi\\times_XT^*X$\nof the cotangent bundle \nto a finite covering $\\pi_\\chi\n\\colon D_\\chi\\to D_i$\nby the non-degenerate assumption.\nThen, define \na closed conical subset\n$C=S(j_!{\\cal G})\\subset T^*X$ in\nthe case $R>D$ by \n\\begin{equation}\nS(j_!{\\cal G})\n=\nT^*_XX\\cup\n\\bigcup_i\n\\bigcup_{\\chi}\n\\pi_\\chi(L_\\chi).\n\\label{eqwildS}\n\\end{equation}\nIn the general strongly non-degenerate case,\nwe define a closed conical subset\n$C=S(j_!{\\cal G})\\subset T^*X$\nby the additivity\nand \\'etale descent.\n\n\\begin{pr}[{\\rm \\cite[Proposition 3.15]{nonlog}}]\\label{prnd}\nAssume that the ramification of\na locally constant constructible sheaf\n${\\cal G}$ of $\\Lambda$-modules\non the complement $U=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} D$ \nof a divisor with simple normal crossings\nis strongly non-degenerate\nalong $D$.\nThen, \n$S(j_!{\\cal G})$ defined by \n{\\rm (\\ref{eqtameS})},\n{\\rm (\\ref{eqwildS})},\nthe additivity and by \\'etale descent\nsatisfies\n\\begin{equation}\nSS j_!{\\cal G}\n=\nS(j_!{\\cal G}).\n\\label{eqSSj}\n\\end{equation}\n\\end{pr}\n\n\n\n\\proof{\nSince the assertion is \\'etale local,\nwe may assume that ${\\cal G}$\nis isoclinic of slope $R=D$ or $R>D$\nand that the ramification of ${\\cal G}$\nis non-degenerate along $D$ at multiplicity $R$.\nIf $R=D$, it is proved in Proposition \\ref{prtame}.\nTo prove the case $R>D$,\nwe use the following Lemma.\n\n\\begin{lm}\\label{lmDf2}\nLet $E$ be a $k$-vector space\nof finite dimension and \n$E\\to \\Gamma(X,{\\cal L})$\nbe a $k$-linear mapping\ndefining an immersion\n$X\\to {\\mathbf P}={\\mathbf P}(E^\\vee)$.\nLet $T\\subset X$ be\nan integral closed subscheme\nand define a subspace $E'={\\rm Ker}(E\\to \n\\Gamma(T,{\\cal L}\\otimes_{{\\cal O}_X}{\\cal O}_T))$\nand ${\\mathbf P}^{\\prime\\vee}\n={\\mathbf P}(E')\\subset\n{\\mathbf P}^{\\vee}\n={\\mathbf P}(E)$.\n\n{\\rm 1.}\n$T\\times {\\mathbf P}^{\\prime\\vee}\n\\subset\nX\\times {\\mathbf P}^{\\vee}$\nis a subset of\n$T\\times_{\\mathbf P}Q\n\\subset X\\times_{\\mathbf P}Q$\nand the complement\n$(T\\times_{\\mathbf P}Q)\n\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~}\n(T\\times {\\mathbf P}^{\\prime\\vee})$\nis the largest open subscheme where\n$T\\times_{\\mathbf P}Q\n\\to {\\mathbf P}^\\vee$\nis flat.\n\n{\\rm 2.}\nThe codimension of \n$E'\\subset E$\nis strictly larger than $\\dim T$.\n\\end{lm}\n\n\\proof{1.\nFor a hyperplane $H\\subset {\\mathbf P}$,\nwe have $T\\subset H$ \nif $H\\in {\\mathbf P}^{\\prime\\vee}$\nand\n$T\\cap H$ is a divisor of $T$\nif otherwise.\nHence we have\n$T\\times {\\mathbf P}^{\\prime\\vee}\n\\subset\nT\\times_{\\mathbf P}Q$\nand on the complement of\n$T\\times {\\mathbf P}^{\\prime\\vee}$,\nthe divisor \n$T\\times_{\\mathbf P}Q$\nof $T\\times{\\mathbf P}^\\vee$\nis flat over ${\\mathbf P}^\\vee$.\n\n\n\n2.\nThe immersion $X\\to {\\mathbf P}$\ninduces an immersion\n$T\\to {\\mathbf P}(E\/E')$.\nHence we have\n$\\dim T<\\dim E\/E'$.\n\\qed}\n\\medskip\n\nWe assume $R>D$.\nFirst, we show the inclusion\n$SSj_!{\\cal G}\n\\subset S(j_!{\\cal G})$.\nSince the assertion is local on $X$,\nwe may assume that $X$ is quasi-projective.\nLet $i\\colon X\\to {\\mathbf P}$\nbe an immersion \nsatisfying the condition {\\rm (E)}\nbefore Lemma {\\rm \\ref{lmloc}}\nand the condition {\\rm (C)} before\nProposition {\\rm \\ref{prwi}}.\nFor irreducible component $D_i$ of $D$,\nlet ${\\mathbf P}^\\vee_i={\\mathbf P}(E_i)\n\\subset\n{\\mathbf P}^\\vee={\\mathbf P}(E)$\nbe the subspace defined by\n$E_i={\\rm Ker}(E\\to \\Gamma(D_i,{\\cal L}))$.\nLet $Z\\subset X\\times_{\\mathbf P}Q$\nbe the union \n$Z=\\bigcup_{i=1,\\ldots,m}\nD_i\\times {\\mathbf P}^\\vee_i$.\nThen, by Lemma \\ref{lmDf2}.1,\nthe divisor \n$D\\times_{\\mathbf P}Q\n\\subset \nX\\times_{\\mathbf P}Q$ with\nsimple normal crossings\nis flat over ${\\mathbf P}^\\vee$\noutside $Z$.\nHence, by \\cite[Proposition 3.15]{nonlog},\n$p^\\vee\\colon \nX\\times_{\\mathbf P}Q\n\\to {\\mathbf P}^\\vee$\nis universally locally acyclic relatively to\nthe pull-back of $j_!{\\cal G}$\noutside the union\n$Z\\cup {\\mathbf P}(\\widetilde C)$\nfor $C=S(j!{\\cal F})$.\nFurther by Lemma \\ref{lmDf2}.2,\nthe closed subset\n$Z\\subset X\\times_{\\mathbf P}Q$\nis of codimension $>\\dim X$.\nHence we have\n$SSj_!{\\cal G}\n\\subset S(j_!{\\cal G})$\nby\nCorollary \\ref{corRn}.\n\nWe show the equality\n$SSj_!{\\cal G}\n= S(j_!{\\cal G})$\nkeeping the assumption $R>D$.\nBy Theorem \\ref{thmss}.2,\nwe may assume $D$ is irreducible\nsince the assertion is local.\nFurther we may assume \nthere exists a unique $L_\\chi$\nby the additivity\nsince the assertion is \\'etale local.\nThen, the assertion follows from\nthe contraposition of\nLemma \\ref{lmlcst}.1 (1)$\\Rightarrow$(2).\n\\qed}\n\n\n\n\n\\section{Characteristic cycle and \nthe Milnor formula}\\label{sCC}\n\nIn this section,\n$X$ denotes a smooth scheme\nover a perfect field $k$.\nAssume that every irreducible component\nof $X$ is of dimension $n$,\nunless otherwise stated.\n\n\n\\subsection{Morphisms defined by pencils}\n\\label{sspl}\n\n\nLet $E$ be a $k$-vector space of\nfinite dimension and\n${\\mathbf P}={\\mathbf P}(E^\\vee)$\nand its dual\n${\\mathbf P}^\\vee={\\mathbf P}(E)$\nbe as in Section \\ref{ssRn}.\nLet $X$ be a smooth scheme\nover $k$\nand let $i\\colon X\\to {\\mathbf P}$\nbe an immersion.\nLet $L\\subset {\\mathbf P}^\\vee$\nbe a line.\nThe morphism $p_L\\colon X_L\\to L$\ndefined by pencil\nis defined by the cartesian diagram\n\\begin{equation}\n\\begin{CD}\nX_L@>>>\nX\\times_{\\mathbf P}Q\\\\\n@V{p_L}VV @VV{p^\\vee}V\\\\\nL@>>> {\\mathbf P}^\\vee.\n\\end{CD}\n\\label{eqpL}\n\\end{equation}\nThe axis $A_L\\subset {\\mathbf P}$\nis the intersection of hyperplanes\nparametrized by $L$.\nIf the axis $A_L$ meets $X$ properly,\nthe scheme $X_L$ is the blow-up of $X$\nat the intersection\n$X\\cap A_L$.\nThe morphism\n$p_L^{\\circ}\\colon\nX_L^{\\circ} \\to L$\n(\\ref{eqpLo})\nis the restriction of\n$p_L\\colon X_L\\to L$\nto the complement \n$X_L^{\\circ} =X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} (X\\cap A_L)$.\n\n\\begin{lm}\\label{lmAL}\nLet $C\\subset T^*X$ be a\nclosed conical subset\nand \n${\\mathbf P}(\\widetilde C)\\subset\nX\\times_{\\mathbf P}Q=\n{\\mathbf P}(X\\times_{\\mathbf P}T^*{\\mathbf P})$\nbe the projectivization.\nAssume that every irreducible component \nof $X$ and every irreducible component\n$C_a$ of $C$ are of dimension $n$.\nLet $L\\subset {\\mathbf P}^\\vee$\nbe a line and\n$p_L^{\\circ}\\colon X_L^{\\circ}\\to L$\nbe the morphism defined\nby the pencil.\n\n\nThe complement\n$X_L^{\\circ}\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~}( X_L^{\\circ} \\cap\n{\\mathbf P}(\\widetilde C))\n\\subset X_L^{\\circ}$\nis the largest open subset where\n$p_L^{\\circ}\\colon X_L^{\\circ}\\to L$\nis $C$-transversal.\n\\end{lm}\n\n\\proof{\nSince\n$X\\times_{\\mathbf P}Q\\to X$\nis smooth,\nthe immersion $X_L^{\\circ}\\to\nX\\times_{\\mathbf P}Q$ is $C$-transversal\nby Lemma \\ref{lmCh}.3.\nHence, for $x\\in X_L^{\\circ}$, the morphism\n$p_L^{\\circ}\\colon X_L^{\\circ}\\to L$\nis $C$-transversal at $x$ if and only if\n$(x,p_L(x))\\in X\\times_{\\mathbf P}Q$\nis not contained in ${\\mathbf P}(\\widetilde C)$\nby Lemma \\ref{lmPC}.1 and by\nLemma \\ref{lmChf}.1\napplied to the cartesian diagram (\\ref{eqpL}).\nHence the assertion follows.\n\\qed}\n\\medskip\n\nWe construct the universal family of\n(\\ref{eqpL}).\nLet ${\\mathbf G}=\n{\\rm Gr}(1,{\\mathbf P}^\\vee)$ be\nthe Grassmannian variety\nparametrizing lines in\n${\\mathbf P}^\\vee$.\nThe universal line\n${\\mathbf D}\\subset\n{\\mathbf P}^\\vee\\times\n{\\mathbf G}$ \nis canonically identified with the flag variety\nparametrizing\npairs $(H,L)$ of points $H$ of\n${\\mathbf P}^\\vee$ \nand lines $L$ passing through $H$.\nIt is the same as the flag variety\n${\\rm Fl}(1,2,E)$ \nparametrizing pairs\nof a line and a plane including the line in $E$.\n\nThe projective space\n${\\mathbf P}^\\vee$ and the\nGrassmannian variety\n${\\mathbf G}$\nare also equal to\nthe Grassmannian varieties\n${\\rm Gr}(1,E)$ and\n${\\rm Gr}(2,E)$ \nparametrizing lines and planes\nin $E$ respectively.\nThe projections\n${\\mathbf P}^\\vee\n\\gets {\\mathbf D}\n\\to {\\mathbf G}$\nsending a pair $(H,L)$ to the line $L$\nand to the hyperplane $H$\nare the canonical morphisms\n${\\rm Gr}(1,E)\\gets {\\rm Fl}(1,2,E)\\to {\\rm Gr}(2,E)$.\nBy the projection ${\\mathbf D}\\to\n{\\mathbf P}^\\vee$,\nit is also canonically identified\nwith the projective space bundle\nassociated to the tangent bundle\n${\\mathbf D}={\\mathbf P}(T{\\mathbf P}^\\vee)$\nby sending a line passing through\na point to the tangent line\nof the line at the point.\nLet ${\\mathbf A}\\subset {\\mathbf P}\n\\times {\\mathbf G}$\nbe the universal family\nof the intersections\nof hyperplanes parametrized\nby lines.\nThe scheme  ${\\mathbf A}$\nis also canonically identified\nwith the Grassmann bundle\n${\\rm Gr}(2,T^*{\\mathbf P})$\nover ${\\mathbf P}$.\n\nLet $X$ be a smooth scheme\nover $k$\nand let $i\\colon X\\to {\\mathbf P}$\nbe an immersion.\nWe construct a commutative diagram\n\\begin{equation}\n\\begin{CD}\nX\\times_{\\mathbf P}Q\n@<<<\n(X\\times {\\mathbf G})'\n@>>> \nX\\times {\\mathbf G}\n\\\\\n@V{p^\\vee}VV @VVV @VVV\\\\\n {\\mathbf P}^\\vee\n @<<<\n{\\mathbf D}\n@>>>\n{\\mathbf G}.\n\\end{CD}\n\\label{eqGLP}\n\\end{equation}\nwhere the left square is cartesian\nas follows.\nThe left vertical arrow and the lower line\nare the canonical morphisms\n${\\rm Gr}(1,X\\times_{\\mathbf P}T^*{\\mathbf P})\n\\to\n{\\rm Gr}(1,E)\n\\gets\n{\\rm Fl}(1,2,E)\n\\to\n{\\rm Gr}(2,E)$.\nThe fiber product\n${\\mathbf D}\\times_{{\\mathbf P}^{\\vee}} \n(X\\times_{\\mathbf P}Q)$\nis canonically identified with \nthe blow-up \n$(X\\times {\\mathbf G})'\n\\to X\\times {\\mathbf G}$\nat the intersection ${\\mathbf A}\n\\cap (X\\times {\\mathbf G})\n=\nX\\times_{\\mathbf P} {\\mathbf A}\n=\n{\\rm Gr}(2,X\\times_{\\mathbf P} {\\mathbf A})$\nby the elementary lemma below.\n\n\nThe canonical morphism \n$(X\\times {\\mathbf G})'\n\\to X\\times {\\mathbf G}$\nis an isomorphism on \nthe complement\n\\begin{equation}\n(X\\times {\\mathbf G})^{\\circ}\n=\n(X\\times {\\mathbf G})\n\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~}\n(X\\times_{\\mathbf P} {\\mathbf A}).\n\\label{eqXGo}\n\\end{equation}\nThe scheme\n$(X\\times {\\mathbf G})^{\\circ}$\nregarded as an open subscheme of\n$(X\\times {\\mathbf G})'$\nis the complement of\n${\\rm Fl}(1,2,X\\times_{\\mathbf P}T^*{\\mathbf P})$\nregarded as a closed subscheme.\nFor a line\n$L\\subset {\\mathbf P}^\\vee$,\nthe diagram (\\ref{eqpL}) is the fiber of \nthe middle vertical arrow \n$(X\\times {\\mathbf G})'\n\\to {\\mathbf D}$ of (\\ref{eqGLP}) at the\npoint of ${\\mathbf G}$ corresponding to $L$.\nThe complement\n$X_L^{\\circ}$ \nis identified with \n$X_L\\cap (X\\times {\\mathbf G})^{\\circ}$.\n\n\n\\begin{lm}\\label{lmFEL}\nLet $0\\to F\\to E\\to L\\to 0$\nbe an exact sequence of\nvector bundles on a scheme $X$\nsuch that $L$ is a line bundle\nand let $1\\leqq r\\leqq {\\rm rank}\\ F-1$ \nbe an integer.\nThen, the Grassmannian bundles\nparametrizing subbundles\nof rank $r$ and of $r-1$\nand the flag bundles\nparametrizing pairs of subbundles\nof rank $r$ and $r-1$\nwith inclusion form a cartesian diagram\n\\begin{equation}\n\\begin{CD}\n{\\rm Fl}(r-1,r,F)\n@>>>\n {\\rm Gr}(r,F)\\\\\n@VVV@VVV\\\\\nP={\\rm Gr}(r-1,F)\\times_\n{{\\rm Gr}(r-1,E)}\n{\\rm Fl}(r-1,r,E)\n@>>>\n {\\rm Gr}(r,E).\n\\end{CD}\n\\label{eqFlG}\n\\end{equation} \nThe right vertical arrow\nis a regular immersion\nof codimension $r$\nand the lower horizontal arrow\nis the blow-up at the\nclosed subscheme\n${\\rm Gr}(r,F)\\subset\n{\\rm Gr}(r,E)$.\n\\end{lm}\n\n\\proof{\nLet $p\\colon P={\\rm Gr}(r-1,F)\\times_\n{{\\rm Gr}(r-1,E)}\n{\\rm Fl}(r-1,r,E)\\to X$\nand\n$$\\xymatrix{\nD={\\rm Fl}(r-1,r,E)\\ar[rd]_f\n\\ar[rr]^d\n&&\nG={\\rm Gr}(r,E)\\ar[ld]^g\n\\\\&X\n}$$\ndenote the canonical morphisms. \nLet $0\\to {\\cal E}\\to\n{\\cal F}\\to{\\cal L}\\to 0$\nbe the corresponding exact sequence\nof locally free ${\\cal O}_X$-modules.\nLet ${\\cal V}\\subset g^*{\\cal E}$\nand ${\\cal W}\\subset d^*{\\cal V}\n\\subset f^*{\\cal E}$\ndenote the universal sub  \n${\\cal O}_G$-module of rank $r$\nand the universal sub  \n${\\cal O}_D$-module of rank $r-1$\nrespectively.\n\n\nOn $P\\times_{\n{\\rm Gr}(r,E)}\n{\\rm Gr}(r,F)$,\nthe pull-back of\n${\\cal W}\\subset d^*{\\cal V}$\ndefines a flag on the pull-back of\n${\\cal F}$.\nHence, the diagram (\\ref{eqFlG})\nis cartesian.\nOn the complement\n${\\rm Gr}(r,E)\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~}\n{\\rm Gr}(r,F)$,\nthe restriction\nof ${\\cal V}$ and its\nintersection with\nthe pull-back of\n${\\cal F}$ \ndefine a flag on the restriction of\n$g^*{\\cal E}$.\nHence the restriction\n$P\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} {\\rm Fl}(r-1,r,F)\n\\to {\\rm Gr}(r,E)\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~}\n{\\rm Gr}(r,F)$\nis an isomorphism.\n\n\nThe ideal sheaf\n${\\cal I}\\subset {\\cal O}_G$ \ndefining the closed subscheme ${\\rm Gr}(r,F)\n\\subset{\\rm Gr}(r,E)$\nis characterized by the condition that\n${\\cal I}\\cdot g^*{\\cal L}$\nequals the image of the\ncomposition ${\\cal V}\\to g^*{\\cal E}\n\\to g^*{\\cal L}$.\nHence, the immersion\n${\\rm Gr}(r,F)\n\\to{\\rm Gr}(r,E)$ is a regular\nimmersion of codimension $r={\\rm rank}\\ {\\cal V}$.\n\nSince the image of the\ncomposition ${\\rm pr}_2^*d^*{\\cal V}\n\\to p^*{\\cal E}\n\\to p^*{\\cal L}$ is isomorphic to\nthe invertible ${\\cal O}_P$-module\n${\\rm pr}_2^*(d^*{\\cal V}\/{\\cal W})$,\nthe ideal \n${\\cal I}\\cdot {\\cal O}_P$ \ndefining the closed subscheme\n${\\rm Fl}(r-1,r,F)\\subset P$\nis locally generated by one element.\nSince ${\\rm Fl}(r-1,r,F)$ and $P$\nare smooth over $X$,\n${\\rm Fl}(r-1,r,F)$ is a divisor of $P$\nflat over $X$ and \nthe ideal \n${\\cal I}\\cdot {\\cal O}_P$ is\ninvertible.\nThus the morphism $P\\to {\\rm Gr}(r,E)$\nis canonically lifted to the blow-up\n$P\\to {\\rm Gr}(r,E)'$\nby the universality of blow-up.\n\nSince the pull-back $\\pi^*{\\cal I}$\non the blow-up $\\pi\\colon {\\rm Gr}(r,E)'\n\\to {\\rm Gr}(r,E)$ is an invertible\nideal, the image $\\pi^*{\\cal I}\\cdot\n\\pi^*g^*{\\cal L}$ of\nthe composition\n$\\pi^*{\\cal V}\\to\n\\pi^*g^*{\\cal E}\\to\n\\pi^*g^*{\\cal L}$ is an invertible module.\nHence the kernel\n$\\pi^*{\\cal V}\\cap\n\\pi^*g^*{\\cal F}$ is locally \na direct summand of\n$\\pi^*g^*{\\cal F}$ of rank $r-1$\nand \n$\\pi^*{\\cal V}\\cap\n\\pi^*g^*{\\cal F}\n\\subset \\pi^*{\\cal V}$ \ndefines a flag on\n$\\pi^*g^*{\\cal E}$.\nThus \nthe inverse morphism\n${\\rm Gr}(r,E)'\\to P$\nis defined\nand the assertion follows.\n\\qed}\n\n\\subsection{Isolated characteristic points and\nintersection numbers}\n\\label{ssic}\n\n\nThe characteristic cycle will\nbe defined as a cycle\ncharacterized by the Milnor formula\nat isolated characteristic points.\nWe will introduce the notion of\nisolated characteristic point\nand study the intersection number\nappearing in the Milnor formula.\n\n\n\\begin{df}\\label{dfisoc}\nLet $X$ be a smooth scheme over a\nfield $k$.\nLet $C\\subset T^*X$ be a \nclosed conical subset\nof the cotangent bundle $T^*X$.\nLet $h\\colon W\\to X$\nbe an \\'etale morphism\nand $f\\colon W\\to Y$ be a morphism\nover $k$ to a smooth curve over $k$.\n\n{\\rm 1.}\nWe say that a closed point $u$ of $W$\nis at most an {\\em isolated \n$C$-characteristic point} of \n$f\\colon W\\to Y\nif there exists an open neighborhood \n$V\\subset W$ of $u$ such that the pair \n$X\\gets V\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} \\{u\\}\\to Y$ is $C$-transversal.\nWe say that a closed point $u\\in W$\nis an {\\em isolated $C$-characteristic point} of $f$\nif it is at most an isolated $C$-characteristic\npoint but \n$X\\gets W\\to Y$ is not $C$-transversal\nat $u$.\n\n\n{\\rm 2.}\nAssume that every irreducible component \nof $X$ and every irreducible component\n$C_a$ of $C$ are of dimension $n$.\nLet $u$ be at most an isolated $C$-characteristic point of \n$f\\colon W\\to Y$ with respect to $C$\nand $A=\\sum_am_a[C_a]$ be\na linear combination of irreducible components of $C$.\nThen, we define the intersection number\n\\begin{equation}\n(A,df)_{T^*W,u}\n\\label{eqAdf}\n\\end{equation}\nas the intersection number\n$\\sum_am_a(j^*C_a,f^*\\omega)_{T^*W,u}$\nsupported on the fiber of $u$\nfor the section $f^*\\omega$ of $T^*W$\ndefined by the pull-back of\na basis $\\omega$ of $T^*Y$\non a neighborhood of $f(u)\\in Y$.\n\\end{df}\n\nThe cotangent bundle $T^*W$\nis of dimension $2n$ and\nits closed subsets\n$j^*C_a,f^*\\omega$\nare of dimension $n$.\nTheir intersections\n$j^*C_a \\cap f^*\\omega$\nconsist of at most a unique isolated\npoint $f^*\\omega(u)\n\\in T^*_uW$\non the fiber of $u$\nand the intersection numbers\n$(j^*C_a,f^*\\omega)_{T^*W,u}$\nare defined\nif $u$ is at most an isolated \n$C$-characteristic point.\nFurther, since $C$ is conical,\nthe  intersection numbers\n$(j^*C_a,f^*\\omega)_{T^*W,u}$\nare independent of the choice of\nbasis $\\omega$\nand the intersection number\n$(A,df)_{T^*W,u}$\nis well-defined.\n\n\nWe compute the intersection number\n(\\ref{eqAdf}) for\nmorphisms defined by pencils.\n\n\\begin{lm}\\label{lmApL}\nLet $C\\subset T^*X$ be a\nclosed conical subset\nand \n${\\mathbf P}(\\widetilde C)\\subset\nX\\times_{\\mathbf P}Q=\n{\\mathbf P}(X\\times_{\\mathbf P}T^*{\\mathbf P})$\nbe the projectivization.\nAssume that every irreducible component \nof $X$ and every irreducible component\n$C_a$ of $C$ are of dimension $n$.\nLet $L\\subset {\\mathbf P}^\\vee$\nbe a line and\n$p_L^{\\circ}\\colon X_L^{\\circ}\\to L$\nbe the morphism defined\nby the pencil.\n\n\n{\\rm 1.}\nLet $u\\in X_L^{\\circ}$ be a closed point.\nThen, $u$ is (resp.\\ at most) an isolated\ncharacteristic point of \n$p_L^{\\circ}\\colon X_L^{\\circ}\\to L$\nif and only if (resp.\\ either)\n$u$ is an isolated point of\n(resp.\\ or not contained in)\nthe intersection\n$X_L^{\\circ} \\cap\n{\\mathbf P}(\\widetilde C)\\subset\nX\\times_{\\mathbf P}Q$.\n\nFurther, if $u$ is at most an isolated\ncharacteristic point of \n$p_L^{\\circ}\\colon X_L^{\\circ}\\to L$,\nwe have an equality\n\\begin{equation}\n(C,dp_L^{\\circ})_{T^*X,u}\n=\n({\\mathbf P}(\\widetilde C),X_L^{\\circ})_{\nX\\times_{\\mathbf P}Q,u}\n\\label{eqAL}\n\\end{equation}\nof the intersection numbers.\n\n{\\rm 2.}\nAssume that $k$ is algebraically closed\nand \nthat $C$ is irreducible.\nSuppose that $p^\\vee\\colon \nX\\times_{\\mathbf P}Q\n\\to {\\mathbf P}^\\vee$ is generically \nfinite on ${\\mathbf P}(\\widetilde C)$\nand let $\\xi\\in {\\mathbf P}(\\widetilde C)$\nand $\\eta\\in \\Delta\n=\\overline{p^\\vee({\\mathbf P}(\\widetilde C))}\n\\subset {\\mathbf P}^\\vee$\ndenote the generic points.\nThen, there exists a smooth dense\nopen subscheme $\\Delta^{\\circ}\n\\subset \\Delta$ satisfying\nthe following condition:\n\nFor a line $L\\subset {\\mathbf P}^\\vee$\nmeeting $\\Delta$ transversally\nat $H\\in \\Delta^{\\circ}$ and \nfor an isolated $C$-characteristic point $u$\nof $p_L^{\\circ}\\colon \nX_L^{\\circ}\\to L$ such that $(u,H)\\in \n{\\mathbf P}(\\widetilde C)^{\\circ}\n={\\mathbf P}(\\widetilde C)\\times\n_\\Delta\\Delta^{\\circ}$, \nthe both sides of {\\rm (\\ref{eqAL})}\nare equal to the inseparable\ndegree $[\\xi:\\eta]_{\\rm insep}$.\n\\end{lm}\n\nSince ${\\mathbf P}(\\widetilde C)\n\\subset\nX\\times_{\\mathbf P}Q$ \nis of codimension $n$,\nthe intersection product in the right hand\nside of (\\ref{eqAL}) is defined.\n\n\n\\proof{\n1.\nThe first assertion follows\nimmediately from\nLemma \\ref{lmAL}.\nWe show the equality (\\ref{eqAL}).\nLet $\\tilde p_L^{\\circ}\\colon \n{\\mathbf P}_L^{\\circ}=\n{\\mathbf P}\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} {\\mathbf A}_L\n\\to L$ denote the morphism\n$p_L^{\\circ}$ defined with\n$X$ replaced by ${\\mathbf P}$.\nLet $\\omega$ be a basis of\n$T^*L$ on the image $p_L(u)$\nand let $\\tilde p_L^{\\circ*}\\omega$\ndenote the section\n$X_L^{\\circ} \\to \nX_L^{\\circ} \\times_{\\mathbf P}T^*{\\mathbf P}$\ndefined on a neighborhood of $u$ by\nthe pull-back of $\\omega$\nby $\\tilde p_L^{\\circ}$.\nThen, we have\n\\begin{equation}\n(C,dp_L^{\\circ})_{T^*X,x}\n=\n(C,p_L^{\\circ *}\\omega)_{T^*X,x}\n=\n(\\widetilde C,\\tilde\np_L^{\\circ *}\\omega)_\n{X\\times_{\\mathbf P}T^*{\\mathbf P},x}\n=\n({\\mathbf P}(\\widetilde C),\n\\overline{\\widetilde p_L^{\\circ *}\\omega})_{\nX\\times_{\\mathbf P}Q,x}.\n\\label{eqCpL}\n\\end{equation}\n\nThe graph ${\\mathbf P}_L^{\\circ} \\to \n{\\mathbf P}\\times L$\nof $\\tilde p_L^{\\circ}$\nis a regular immersion of codimension 1\nand defines an exact sequence\n\\begin{equation}\n\\begin{CD}\n0\\to\n{\\mathbf P}_L^{\\circ}\\times_Q\nT^*_Q\n({\\mathbf P}\\times {\\mathbf P}^\\vee)\n@>>> \n({\\mathbf P}_L^{\\circ}\\times_{\\mathbf P}T^*{\\mathbf P})\n\\times_{{\\mathbf P}_L^{\\circ}}\n({\\mathbf P}_L^{\\circ}\\times_LT^*L)\n@>>>T^*{\\mathbf P}_L^{\\circ} \\to0\\end{CD}\n\\label{eqTPL}\n\\end{equation}\nof vector bundles on ${\\mathbf P}_L^{\\circ}$.\nHence the left arrow induces\nan isomorphism\n${\\mathbf P}_L^{\\circ}\\times_Q\nT^*_Q\n({\\mathbf P}\\times {\\mathbf P}^\\vee)\n\\to\n{\\mathbf P}_L^{\\circ}\\times_LT^*L$\nto the second factor.\nSince \n$T^*_Q\n({\\mathbf P}\\times {\\mathbf P}^\\vee)\n\\to \nQ\\times_{\\mathbf P}T^*{\\mathbf P}$\nis the universal sub line bundle\non\n$Q={\\mathbf P}(T^*{\\mathbf P})$,\nthe morphism\n${\\mathbf P}_L^{\\circ}\\times_LT^*L\n\\to\nT^*{\\mathbf P}_L^{\\circ}$\nalso defines the restriction\nof the universal sub line bundle\non ${\\mathbf P}_L^{\\circ}\\subset Q$.\nHence\nthe image of the section\n$\\overline{\\widetilde p_L^{\\circ *}\\omega}\n\\colon X_L^{\\circ} \\to \nX \\times_{\\mathbf P}Q=\n{\\mathbf P}(X\\times_{\\mathbf P}T^*{\\mathbf P})$ equals \n$X_L^{\\circ} \\subset\nX\\times_{\\mathbf P}Q$\nand (\\ref{eqCpL}) implies (\\ref{eqAL}).\n\n2.\nLet $\\Delta^{\\circ}\\subset \\Delta$\nbe a smooth dense open subscheme \nsuch that the base change\n${\\mathbf P}(\\widetilde C)^{\\circ}\n=\n{\\mathbf P}(\\widetilde C)\n\\times_\\Delta\\Delta^{\\circ}\n\\to \\Delta^{\\circ}$\nis the composition of\na finite flat radicial morphism\nof degree $[\\xi:\\eta]_{\\rm insep}$\nand a finite \\'etale morphism.\nThen, for a line $L$ and $(u,H)\\in \n{\\mathbf P}(\\widetilde C)^{\\circ}$ as in the assumption,\nthe intersection number\n$({\\mathbf P}(\\widetilde C),X_L^{\\circ})_{\nX\\times_{\\mathbf P}Q,u}$\nequals the degree of\nthe localization at $u$\nof the fiber of the finite flat morphism\n${\\mathbf P}(\\widetilde C)^{\\circ}\\to\n\\Delta^{\\circ}$\nat $H\\in \\Delta^{\\circ}$\nand is equal to\n$[\\xi:\\eta]_{\\rm insep}.$\n\\qed}\n\\medskip\n\nWe give a condition \nfor a function on isolated characteristic\npoints to be given \nas an intersection number.\n\n\\begin{df}\\label{dfflZ}\nLet $k$ be an algebraically closed field.\nLet $f\\colon Z\\to S$ be a {\\em quasi-finite} morphism\nof schemes of finite type \nover $k$.\nWe say that a function $\\varphi\n\\colon Z(k)\\to {\\mathbf Q}$\non the set of closed points\nis {\\em flat over} $S$\nif for every closed point $x\\in Z$,\nthere exists a commutative diagram\n$$\\xymatrix\n{U\\ar[r]\\ar[dr]_g&\nV\\times_SZ\\ar[r]^h\n\\ar[d]&\nZ\\ar[d]^f\\\\\n& V\\ar[r]&S\n}$$\nsatisfying the following conditions\n{\\rm (1)--(4):}\n\n{\\rm (1)} $V\\to S$ is an \\'etale neighborhood\nof  $s=f(x)$.\n\n{\\rm (2)} $U\\subset V\\times_SZ$ is an open neighborhood\nof $x$.\n\n{\\rm (3)} $g\\colon\nU\\to V$ is finite\nand $g^{-1}(s)=\\{x\\}$.\n\n{\\rm (4)} The function $g_*\\varphi$ on $V(k)$\ndefined by $g_*\\varphi(t)=\n\\sum_{z\\in g^{-1}(t)}\\varphi(hz)$\nis {\\em constant}.\n\\end{df}\n\n\n\\begin{lm}\\label{lmfla}\nLet $f\\colon Z\\to S$ be a {\\em quasi-finite} morphism\nof schemes of finite type over\nan algebraically closed field $k$.\n\n{\\rm 1.}\nLet $\\varphi\n\\colon Z\\to {\\mathbf Q}$ be\na function.\nIf $\\varphi$ is flat over $S$\nin the sense of Definition {\\rm \\ref{df11}},\nits restriction on $Z(k)$ is flat over $S$\nin the sense of Definition {\\rm \\ref{dfflZ}}.\n\n{\\rm 2.}\nLet $\\varphi\n\\colon Z(k)\\to {\\mathbf Q}$\nbe a function flat over $S$\nin the sense of Definition {\\rm \\ref{dfflZ}}.\nThen, it is constructible on $Z(k)$.\nIf $\\varphi=0$ on $U(k)\\subset\nZ(k)$ for a dense open subset\n$U\\subset Z$,\nwe have\n$\\varphi=0$ on $Z(k)$.\n\\end{lm}\n\n\\proof{1.\nLet the notation be as in \nDefinition \\ref{dfflZ}.\nThen, by Lemma \\ref{lmscZ}.4,\n$g_*\\varphi$ is locally constant\nand the assertion follows.\n\n2.\nIf $Z\\to S$ is finite, surjective and\nradiciel, then $\\varphi$ is locally\nconstant.\nThe constructibility follows from this\nby devissage.\n\nSince the second assertion is\n\\'etale local on $Z$,\nwe may assume $Z$ is finite over $S$.\nThen $\\varphi$ on $Z(k)$\nis determined by its restriction\nto a dense open subset by\nthe condition (4) in Definition \\ref{dfflZ}.\n\\qed}\n\n\\begin{df}\\label{dffla}\nLet $X$ be a smooth scheme\nover $k$ and\n$C\\subset T^*X$\nbe a closed conical subset.\n\n{\\rm 1.}\nWe say that $\\varphi$ is\na {\\em function on \nisolated $C$-characteristic points}\nif a number $\\varphi(f,u)\\in {\\mathbf Q}$\nis defined\nfor every \nmorphism $f\\colon U\\to Y$\nover $k$\ndefined on an open subscheme $U\\subset X$\nto a smooth curve $Y$\nwith at most \nan isolated $C$-characteristic point $u\\in U$\nand if we have $\\varphi(f,u)=0$\nif $u\\in U$ is not\nan isolated $C$-characteristic point.\n\n\n{\\rm 2.}\nLet $\\varphi$ be\na function on \nisolated $C$-characteristic points.\nWe say that $\\varphi$ is {\\em flat}\nif for every commutative diagram\n\\begin{equation}\n\\xymatrix{\nZ\\ar[r]^{\\subset} &\nU\\ar[rr]^f\\ar[rd]\n\\ar[d]_{{\\rm pr}_1}&\n&Y\\ar[ld]^g\\\\\n&X&S&}\n\\label{eqfla}\n\\end{equation}\nof schemes over $k$\nsatisfying the conditions \n{\\rm (1)--(5)} below,\nthe function \n$\\varphi_f$ on $Z(k)$\ndefined by $\\varphi_f(u)=\\varphi(f_s,u)$ \nfor the base change\n$f_s\\colon U_s\n\\to Y_s$ of $f$ at $s={\\rm pr}_2(u)\\in S$\nis {\\rm flat} in the sense of \nDefinition {\\rm \\ref{dfflZ}:}\n\n{\\rm (1)} $S$ is smooth over $k$.\n\n{\\rm (2)} $g\\colon Y\\to S$\nis a smooth curve.\n\n{\\rm (3)} $U$ is an open subscheme\nof $X\\times S$.\n\n{\\rm (4)} $Z\\subset U$\nis a closed subset {\\em quasi-finite}\nover $S$.\n\n{\\rm (5)} The pair\n$({\\rm pr}_1,f)$\nis $C$-transversal\non the complement $U\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z$.\n\\end{df}\n\nNote that \nthe morphisms\n$f_s\\colon U_s\\to Y_s$\nare $C$-transversal except\npossibly at at most isolated\ncharacteristic points at $Z_s$\nunder the conditions\n(1)--(5) on the diagram (\\ref{eqfla})\nby Lemma \\ref{lmChf}.1.\n\n\n\\begin{pr}\\label{prfla}\nLet $X$ be a smooth scheme\nover\nan algebraically closed field\n$k$ of characteristic $p\\geqq 0$\nand\n$C\\subset T^*X$\nbe a closed conical subset.\nAssume that every irreducible\ncomponent of $X$\nand of $C=\\bigcup_aC_a$ are of dimension $n$.\nLet $\\varphi$ be\na ${\\mathbf Z}[\\frac 1p]$-valued\n(resp.\\ ${\\mathbf Z}$-valued)\nfunction on isolated $C$-characteristic points\nif $p>0$\n(resp.\\ if $p=0$).\n\n{\\rm 1.}\nThe following conditions\nare equivalent:\n\n{\\rm (1)}\n$\\varphi$ is flat\nin the sense of Definition {\\rm \\ref{dffla}.2}.\nIf $g\\colon Y\\to Z$ is an \\'etale morphism\nof smooth curves over $k$,\nwe have\n$\\varphi(f,u)=\\varphi(gf,u)$.\n\n{\\rm (2)}\nThere exists a\n${\\mathbf Z}[\\frac 1p]$-linear\n(resp.\\ ${\\mathbf Z}$-linear)\ncombination\n$A=\\sum_am_aC_a$ satisfying\n\\begin{equation}\n\\varphi(f,u)=(A,df)_{T^*X,u}\n\\label{eqafu}\n\\end{equation}\nfor every  $f\\colon U\\to Y$\nwith at most\nan isolated $C$-characteristic point $u\\in U$.\n\nFurther $A$ in {\\rm (2)} is unique.\nFurther $A$ is independent of $C$\nin the sense that if $C'\\supset C$\nis a closed conical subset\nsuch that\nevery irreducible\ncomponent of $C'=\\bigcup_bC'_b$ is of dimension $n$,\nthen the linear combination\n$A'=\\sum_bm'_bC'_b$ satisfying\n{\\rm (\\ref{eqafu})}\nfor every  $f\\colon U\\to Y$\nwith at most\nan isolated $C'$-characteristic point $u\\in U$\nequals $A$.\n\n{\\rm 2.}\nLet $h\\colon W\\to X$ be an \\'etale\nmorphism of smooth schemes\nover $k$ and\nlet $\\psi$ be\na ${\\mathbf Z}[\\frac 1p]$-valued\n(resp.\\ ${\\mathbf Z}$-valued)\nfunction on isolated $h^* C$-characteristic points\nif $p>0$\n(resp.\\ if $p=0$).\nWe assume that \n$\\varphi$ and $\\psi$ satisfy\nthe equivalent condition in {\\rm 1.}\\\nand let $A$ and $A'$\nbe the linear combinations\nof irreducible components\nof $C$ and of $h^* C$\nsatisfying {\\rm (\\ref{eqafu})}\nfor $\\varphi$ and $\\psi$\nrespectively.\nIf $\\varphi(f,hv)=\\psi(fh,v)$\nfor isolated characteristic points\n$v$ of $fh\\colon W\\times_XU\\to U\\to Y$,\nwe have $A'=h^*A$.\n\\end{pr}\n\n\n\n\\proof{\n1.\n(2)$\\Rightarrow$(1):\nLet the notation be as in\n(\\ref{eqfla}).\nWe show that\nthe function\n$\\varphi_f$ on $Z(k)$\ndefined by\n$\\varphi_f(u)=(A,df_s)_{T^*X,u}$\nis flat over $S$ in the sense of Definition\n\\ref{dffla}.2.\nWe may assume  $A=C_a=C$.\nSince the assertion is local,\nwe may assume that\n$\\Omega^1_{Y\/S}$\nis free of rank $1$\nand the section\n$df\\colon U\\to T^*X$\ndefined by a basis\nis globally defined on $U$.\nSince $T^*X$ is regular,\nthe ${\\cal O}_{T^*X}$-module\n${\\cal O}_C$ is of finite tor-dimension.\nSince $U\\to S$ is flat,\nthe complex of \n${\\cal O}_U$-modules \n${\\cal O}_C\n\\otimes^L_{{\\cal O}_{T^*X}}\n{\\cal O}_U$ defined\nas the pull-back by $df$\nis of finite tor-dimension\nas a complex of \n${\\cal O}_S$-modules.\nHence the function\n$\\varphi_f$ on $Z(k)$\nis flat over $S$ by Lemma \\ref{lmA}.1\nand Lemma \\ref{lmfla}.1.\n\n\nIf $g\\colon Y\\to Z$ is an \\'etale morphism\nof smooth curves over $k$,\nwe have\n$(A,df)_{T^*X,u}\n=(A,dgf)_{T^*X,u}$.\n\n(1)$\\Rightarrow$(2):\nSince the question is local,\nwe may assume $X$ is affine.\nWe take a closed immersion\n$X\\to {\\mathbf A}^n\n={\\rm Spec}\\ k[T_1,\\ldots,T_n]\n\\subset\n{\\mathbf P}^n$.\nLet $E=\\Gamma({\\mathbf P}^n,\n{\\cal O}(1))$ and let\n${\\cal L}$ be the pull-back to\n$X$ of ${\\cal O}(1)$.\nAfter replacing $E\\to \n\\Gamma(X,{\\cal L})$\nby $S^dE\\to \n\\Gamma(X,{\\cal L}^{\\otimes d})$\nfor $d\\geqq 3$,\nwe may assume that\nthe conditions (E) and (C)\nbefore and after Lemma \\ref{lmloc}\nare satisfied.\nWe may identify\n$E$ with the $k$-linear subspace\nof $k[T_1,\\ldots,T_n]$\nconsisting of polynomials of degree $\\leqq 1$.\nSimilarly,\n$S^dE$ is identified\nwith the $k$-linear subspace\nof $k[T_1,\\ldots,T_n]$\nconsisting of polynomials of degree $\\leqq d$.\n\n\nWe consider the universal family \nas in Section \\ref{sspl}.\nDefine an open subset\n\\begin{equation}\n(X\\times {\\mathbf G})^\\triangledown\n\\subset\n(X\\times {\\mathbf G})^{\\circ}\n\\label{eqXGt}\n\\end{equation}\nof $(X\\times {\\mathbf G})^{\\circ}\n\\subset (X\\times {\\mathbf G})'$\n(\\ref{eqXGo}) \nto be the largest open subset such that\n${\\mathbf Z}(\\widetilde C)$\ndefined by the left cartesian square in\n\\begin{equation}\n\\begin{CD}\n{\\mathbf Z}(\\widetilde C)\n@>>>\n(X\\times {\\mathbf G})^\\triangledown\n@>f>> {\\mathbf D}\n@>>> {\\mathbf G}\\\\\n@VVV@VVV@VVV\\\\\n{\\mathbf P}(\\widetilde C)\n@>>> X\\times_{\\mathbf P}Q\n@>>> {\\mathbf P}^{\\vee}\n\\end{CD}\n\\label{eqZPC}\n\\end{equation}\nis quasi-finite over\n${\\mathbf G}$.\nBy Lemma \\ref{lmPC}.1,\nthe complement\n$(X\\times {\\mathbf G})^\\triangledown\n\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~}\n{\\mathbf Z}(\\widetilde C)$\nis the largest open subset\nwhere the pair of\n${\\rm pr}_1\\colon\n(X\\times {\\mathbf G})^\\triangledown\n\\to X$\nand \n$f\\colon\n(X\\times {\\mathbf G})^\\triangledown\n\\to {\\mathbf D}$\nis $C$-transversal.\nFurther by Lemma \\ref{lmAL},\nfor the pair $(u,L)\\in (X\\times {\\mathbf G})^{\\circ}$\nof a line $L\\subset {\\mathbf P}^\\vee$\nand $u\\in X_L^{\\circ}$,\nthe pair $(u,L)\\in (X\\times {\\mathbf G})^{\\circ}$ is a\npoint of ${\\mathbf Z}(\\widetilde C)$\n(resp.\\ of $(X\\times {\\mathbf G})^\\triangledown$)\nif and only if\n$u\\in X_L^{\\circ}$\nis (resp.\\ at most)\nan isolated $C$-characteristic point\nof $p_L^{\\circ} \\colon\nX_L^{\\circ} \\to L$.\n\n\nWe consider the diagram\n\\begin{equation}\n\\xymatrix{{\\mathbf Z}(\\widetilde C)\n\\ar[r]^{\\!\\!\\!\\!\\!\\!\\!\\!\\subset}&\n(X\\times {\\mathbf G})^\\triangledown\n\\ar[rr]^f\\ar[rd]_{p^\\triangledown}\n\\ar[d]&&\n{\\mathbf D}\\ar[ld]^g\\\\\n&X&{\\mathbf G}}\n\\label{eqMfg}\n\\end{equation}\nas (\\ref{eqfla}).\nFor a point $L$ of ${\\mathbf G}$,\nthe fiber of $f\\colon \n(X\\times {\\mathbf G})^\\triangledown\n\\to {\\mathbf D}$\nis a restriction of $p_L^{\\circ}\\colon \nX_L^{\\circ} \\to L$.\nBy the condition (1),\nthe function $\\varphi_f$ on \n${\\mathbf Z}(\\widetilde C)$\nis flat.\nHence, there exists a dense\nopen subscheme ${\\mathbf Z}(\\widetilde C)^{\\circ}\n\\subset {\\mathbf Z}(\\widetilde C)$\nwhere the function\n$\\varphi_f$\nis locally constant by Lemma \\ref{lmfla}.2.\n\nFor each irreducible component\n$C_a$ of $C=\\bigcup_aC_a$,\nthe function $\\varphi_f$\nis constant on a dense open subscheme\n${\\mathbf Z}(\\widetilde C_a)^{\\circ}\n=\n{\\mathbf Z}(\\widetilde C_a)\n\\cap {\\mathbf Z}(\\widetilde C)^{\\circ}\n\\subset\n{\\mathbf Z}(\\widetilde C_a)$.\nDefine a number $\\varphi_a\n\\in {\\mathbf Z}[\\frac1p]$ to be\nthe value of $\\varphi_f$\non ${\\mathbf Z}(\\widetilde C_a)^{\\circ}$.\nThe restriction \n${\\mathbf P}(\\widetilde C_a)\n\\to D_a=\n\\overline {p^\\vee({\\mathbf P}(\\widetilde C_a))}$\nof\n$p^\\vee\\colon X\\times_{\\mathbf P}Q\n\\to {\\mathbf P}^\\vee$\nis generically finite\nby Corollary \\ref{corrad}.1\nsince $E$ is assumed to satisfy the conditions \n(E) and (C) before \nand after Lemma {\\rm \\ref{lmloc}}.\nLet $\\xi_a\\in \n{\\mathbf P}(\\widetilde C_a)$\nand $\\eta_a\\in D_a$\nbe the generic points.\nWe define\n\\begin{equation}\nA=\n\\sum_a\n\\dfrac{\\varphi_a}\n{[\\xi_a:\\eta_a]_{\\rm insep}}[C_a].\n\\label{eqChE}\n\\end{equation}\nSince the inseparable degree\n$[\\xi_a:\\eta_a]_{\\rm insep}$ is a power of $p$\nif $p>0$ (resp. is $1$ if $p=0$),\nthe coefficients\nin $A$ are in ${\\mathbf Z}[\\frac1p]$\n(resp. in ${\\mathbf Z}$).\n\n\nWe show that $A$\nsatisfies (\\ref{eqafu})\nfor morphisms defined by pencils.\nLet $L\\subset {\\mathbf P}^\\vee$ be a line\nand\n$u\\in X_L^{\\circ}$ be at most an isolated\ncharacteristic point of\n$p_L^{\\circ}\\colon \nX_L^{\\circ}\\to L$.\nIf $u\\in X_L^{\\circ}$ is not an isolated\ncharacteristic point of\n$p_L^{\\circ}$,\nthen the both sides of (\\ref{eqafu}) are $0$.\nAssume $u\\in X_L^{\\circ}$ is an isolated\ncharacteristic point of\n$p_L^{\\circ}$.\nThen we have \n$(u,L)\\in {\\mathbf Z}(\\widetilde C)$\nby Lemma \\ref{lmApL}.1.\n\nShrinking ${\\mathbf Z}(\\widetilde C)^{\\circ}$\nif necessary,\nwe may assume that\n${\\mathbf Z}(\\widetilde C)^{\\circ}\n=\n\\coprod_a\n{\\mathbf Z}(\\widetilde C_a)^{\\circ}$.\nIf $(u,L)\\in\n{\\mathbf Z}(\\widetilde C_a)^{\\circ}$,\nthe left hand side of (\\ref{eqafu})\nis $\\varphi_a$\nby the definition of $\\varphi_a$.\nBy Lemma \\ref{lmApL}.2,\nthe left hand side of (\\ref{eqafu})\nis $\\dfrac{\\varphi_a}{[\\xi_a:\\eta_a]_{\\rm insep}}\n\\cdot [\\xi_a:\\eta_a]_{\\rm insep}\n=\\varphi_a$.\nHence the equality (\\ref{eqafu})\nholds on the dense open subset\n${\\mathbf Z}(\\widetilde C)^{\\circ}\n\\subset\n{\\mathbf Z}(\\widetilde C)$.\nThis also proves the uniqueness of $A$\nand implies the independence of $C$.\n\nThe left hand side\nof (\\ref{eqafu}) is \nthe function $\\varphi_f$\non ${\\mathbf Z}(\\widetilde C)$\nand is constructible and\nflat over ${\\mathbf G}$\nby assumption.\nThe right hand side\nof (\\ref{eqafu}) is also a function on \n${\\mathbf Z}(\\widetilde C)$\nflat over ${\\mathbf G}$\nas is proved in (2)$\\Rightarrow$(1).\nHence the equality (\\ref{eqafu})\nholds for every $(u,L)\n\\in {\\mathbf Z}(\\widetilde C)$\nby Lemma \\ref{lmfla}.2.\n\nWe show that the linear combination\n$A$ defined for $X\\to {\\mathbf A}^n\n\\subset {\\mathbf P}^n\n={\\mathbf P}(E^\\vee)$\nequals that defined for\n$X\\to {\\mathbf P}(S^dE^\\vee)$\nfor $d\\geqq 2$.\nSince $E\\subset\nk[T_1,\\ldots,T_n]$ consisting of\npolynomials of degree $\\leqq 1$\nis canonically identified\nwith a subspace of $S^dE\\subset\nk[T_1,\\ldots,T_n]$ consisting of\npolynomials of degree $\\leqq d$,\nthe uniqueness of $A$\nimplies the independence of $d$.\n\nWe show that the equality (\\ref{eqafu})\nholds for every morphism\n$f\\colon U\\to Y$\nwith at most\nan isolated $C$-characteristic point $u\\in U$.\nBy taking an \\'etale morphism\nto ${\\mathbf A}^1$\ndefined on a neighborhood\nof $f(u)\\subset Y$,\nwe may assume $Y={\\mathbf A}^1$\nand $f$ is defined by\na function on $U\\subset X$.\nWe may assume that\n$f$ is defined by a ratio of\npolynomials in $k[T_1,\\ldots,T_n]$ \nof degree $d\\geqq 1$.\nIn other words,\n$f$ equals a morphism\ndefined by a pencil\n$L$ and the assertion is proved.\n\n2.\nWe may assume $X$ is affine\nand take an immersion $X\\to\n{\\mathbf P}^n$ as in\nthe proof of 1.\nLet $C'_b\\subset h^*C$ be an irreducible\ncomponent \nand let $C_a\\subset C$ be the closure of\nits image.\nWe take a closed point\n$(u,L)\\in {\\mathbf Z}(\\widetilde C_a)^\\circ$\nas in the notation in the proof of 1.\\\nsuch that $u=h(v)$ for \na point $v\\in W$.\nThen, since $\\psi(p_Lh,v)\n=\\varphi(p_L,u)$,\nthe coefficient of\n$C'_b$ in $A'$ equals\nthat of\n$C_a$ in $A$.\n\\qed}\n\n\n\n\\subsection{Characteristic cycle}\\label{ssCC}\n\n\nWe state and prove the existence\nof characteristic cycle\nsatisfying the Milnor formula.\n\n\n\n\\begin{thm}[{\\rm cf.{} \\cite[Principe p.\\ 7]{bp}}]\\label{thmM}\nLet $X$ be a smooth scheme over \na perfect field $k$\nof characteristic $p>0$ (resp.\\ $p=0$)\nand ${\\cal F}$ be a constructible\ncomplex of $\\Lambda$-modules \nof finite tor-dimension on $X$.\nLet $C=\\bigcup_aC_a$ be a closed conical subset\nof the cotangent bundle $T^*X$\nsuch that ${\\cal F}$\nis micro-supported on $C$.\nAssume that every irreducible component of\n$X$ and  every irreducible component $C_a$\nof $C$ are of dimension $n$.\nThen, there exists a unique \n${\\mathbf Z}[\\frac1p]$-linear \n(resp.\\ ${\\mathbf Z}$-linear) combination\n$CC_C{\\cal F}=\n\\sum_a m_a[C_a]$ satisfying the following\ncondition:\n\nFor every \\'etale morphism\n$j\\colon W\\to X$,\nevery morphism $f\\colon W\\to Y$\nto a smooth curve and \nevery at most {\\em isolated $C$-characteristic\npoint} $u\\in W$ of $f$,\nwe have\n\\begin{equation}\n-\\dim{\\rm tot}\\ \\phi_u\n(j^*{\\cal F},f)\n=\n(CC_C {\\cal F},df)_{T^*W,u}.\n\\label{eqMil}\n\\end{equation}\nFurther,\nthe linear combination\n$CC_C{\\cal F}$\nis independent of $C$\non which ${\\cal F}$ is micro-supported.\n\\end{thm\n\nWe will give a proof\nby Beilinson\nof the fact that\nthe characteristic cycle \nhas ${\\mathbf Z}$-coefficients\nin Section \\ref{sZ}.\nThis is a generalization of \nthe Hasse-Arf theorem\n\\cite{CL} in the case\n$\\dim X\\leqq 1$. \n\n\\proof{\nWe may assume $k$ is algebraically\nclosed by replacing $k$ by\nan algebraic closure.\nLet $\\Lambda_0$ denote\nthe residue field of the\nfinite local ring $\\Lambda$\nand set ${\\cal F}_0=\n{\\cal F}\\otimes_\\Lambda^L\\Lambda_0$.\nThen, we have\n$\\dim{\\rm tot} \\phi_u\n(j^*{\\cal F},f)\n=\n\\dim{\\rm tot} \\phi_u\n(j^*{\\cal F}_0,f)$\nand ${\\cal F}_0$ is\nmicro-supported on $C$\nif and only if\nand ${\\cal F}$ is\nmicro-supported on $C$\nby Lemma \\ref{lmmc}.7.\nThus, we may assume\n$\\Lambda$ is a field.\n\nWe regard the left hand side\nof (\\ref{eqMil})\nas a function $\\varphi$ on\nisolated $C$-characteristic points\nin the sense of\nDefinition \\ref{dffla}.1.\nIn fact,\nif the pair\nof $j\\colon W\\to X$ and $f\\colon\nW\\to Y$ is $C$-transversal,\nthen $f\\colon W\\to Y$ \nis universally locally acyclic\nrelatively to $j^*{\\cal F}$\nand the left hand side of\n(\\ref{eqMil}) is 0.\nIf $g\\colon Y\\to Z$\nis an \\'etale morphism,\nthe function $\\varphi$ satisfies \nthe condition \n$\\varphi(f,u)=\\varphi(gf,u)$ in (1)\nin Proposition \\ref{prfla}.1.\nIf $h\\colon W\\to X$\nis an \\'etale morphism,\nit also satisfies the condition \n$\\varphi(f,hv)=\\psi(fh,v)$\nin Proposition \\ref{prfla}.2.\nThus by Proposition \\ref{prfla},\nit suffices to show that\n$\\varphi$ is flat in the sense\nof Definition \\ref{dffla}.2.\n\nLet the notation be as in (\\ref{eqfla})\nand we apply \nProposition \\ref{prMsc}.\nThe morphism $f\\colon\nU\\to Y$\nis locally acyclic relatively to\nthe pull-back of ${\\cal F}$\non the complement of\n$Z$ by Lemma \\ref{lmPC}.2.\nThe projection\n${\\rm pr}_2\\colon\nU\\to S$\nis locally acyclic relatively to\nthe pull-back of ${\\cal F}$\nby the generic universal local acyclicity\n\\cite[Th\\'eor\\`eme 2.13]{TF}.\nSince $Z$ is quasi-finite over $S$,\nthe function\n$\\varphi_f$ is constructible and flat\nover $S$ by \nProposition \\ref{prMsc}\nand Lemma \\ref{lmfla}.1.\n\\qed}\n\n\\begin{df}\\label{dfCC}\nWe define the {\\em characteristic cycle}\n$CC {\\cal F}$ \nto be $CC_C{\\cal F}$\nindependent of $C$\non which ${\\cal F}$ is micro-supported.\n\\end{df}\n\nThe Milnor formula \\cite{Milnor}\nand (\\ref{eqMil})\nimply that for the constant sheaf $\\Lambda$,\nwe have\n$$CC \\Lambda\n=(-1)^n\\cdot [T^*_XX].$$\nThus, the formula (\\ref{eqMil})\nis a generalization of\nthe Milnor formula\nproved by Deligne in \\cite{Milnor}\nand shall be also called a Milnor formula.\nWe will give more examples \nand properties of characteristic cycles \nin the rest of this\nsubsection and in Section \\ref{spb}.\nWe keep assuming that $k$ is\nperfect and\n$X$ is smooth of dimension $n$ over $k$.\nLet ${\\cal F}$ be a constructible\ncomplex of $\\Lambda$-modules of finite tor-dimension on $X$.\n\n\\begin{lm}\\label{lmele}\n{\\rm 1.}\nIf ${\\cal F}$\nis locally constant,\nwe have \n\\begin{equation}\nCC {\\cal F}=(-1)^n\n{\\rm rank}\\ {\\cal F}\\cdot [T^*_XX].\n\\end{equation}\n\n{\\rm 2.}\nFor an \\'etale morphism\n$j\\colon U\\to X$,\nwe have\n\\begin{equation}\nCC j^*{\\cal F}\n=j^*CC {\\cal F}.\n\\end{equation}\n\n{\\rm 3.}\nAssume that $\\dim X=1$\nand let $U\\subset X$\nbe a dense open subscheme\nwhere ${\\cal F}$ is locally constant.\nFor $x\\in X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} U$,\nlet $\\bar \\eta_x$\ndenote a geometric generic\npoint of the strict localization\nat a geometric point $\\bar x$\nabove $x$ and let \n\\begin{equation}\na_x({\\cal F})=\n{\\rm rank}\\ {\\cal F}_{\\bar \\eta_x}\n-\n{\\rm rank}\\ {\\cal F}_{\\bar x}\n+\n{\\rm Sw}_x {\\cal F}_{\\bar \\eta_x}\n\\label{eqaxF}\n\\end{equation}\nbe the Artin conductor.\nThen, we have\n\\begin{equation}\nCC {\\cal F}=\n-\\Bigl(\n{\\rm rank}\\ {\\cal F}\\cdot [T^*_XX]\n+\n\\sum_{x\\in X\\!\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} \\!U}a_x({\\cal F})\n\\cdot [T^*_xX]\n\\Bigr)\n\\label{eqdim1}\n\\end{equation}\n\\end{lm}\n\n\\proof{\n1.\nIt follows from the Milnor formula \\cite{Milnor}.\nIt will also follow immediately\nfrom the compatibility of\nthe characteristic cycles\nwith smooth pull-back\nProposition \\ref{prsm*}.\n\n2.\nSince the characterization (\\ref{eqMil})\nis an \\'etale local condition,\nthe assertion follows.\n\n3.\nBy Lemma \\ref{lmlcst}.2,\nit suffices to determine the coefficients.\nFor the $0$-section $T^*_XX$,\nit follows from 1 and 2.\nFor the fibers,\nsince \n$\\dim{\\rm tot}_x\\phi_x\n({\\cal F},{\\rm id})\n=\na_x({\\cal F})$,\nit follows from the Milnor formula (\\ref{eqMil})\nfor the identity $X\\to X$.\n\\qed}\n\\medskip\n\nFor surfaces,\nthe characteristic cycle\nis studied in \\cite{surface}.\n\n\\begin{df}\\label{dfi*A}\nLet $i\\colon X\\to Y$\nbe a closed immersion \nof smooth schemes over $k$\nand let \n\\begin{equation}\n\\begin{CD}\nT^{*}X@<<< X\\times_{Y}T^{*}Y\n@>>> T^{*}Y\n\\end{CD}\n\\label{eqi*A}\n\\end{equation}\nbe the canonical morphisms.\nLet $C\\subset T^*X$\nbe a closed conical subset.\nAssume that every irreducible component of $X$\nand every irreducible component \n$C_a$ of $C=\\bigcup_aC_a$\nare of dimension $n$\nand \nthat every irreducible component of $Y$\nis of dimension $m$.\nThen, for a linear combination\n$A=\\sum_am_a[C_a]$,\nwe define $i_*A$ to be \n$(-1)^{n-m}$-{\\em times} the push-forward\nby the second arrow\n$X\\times_{Y}T^{*}Y\\to T^{*}Y$\nin {\\rm (\\ref{eqi*A})}\nof the pull-back of $A$ by the first arrow\n$X\\times_{Y}T^{*}Y\\to T^{*}X$\nin the sense of intersection theory.\n\\end{df}\n\n\\begin{lm}\\label{lmRf}\nLet ${\\cal F}$ be a constructible complex\nof $\\Lambda$-modules \nof finite tor-dimension on $X$.\n\n{\\rm 1.}\nFor a distinguished triangle\n$\\to {\\cal F}'\\to {\\cal F}\\to {\\cal F}''\\to$\nin $D_{\\rm ctf}(X,\\Lambda)$,\nwe have\n\\begin{equation}\nCC {\\cal F}=\nCC {\\cal F}'+\nCC {\\cal F}''.\n\\label{eqadd}\n\\end{equation}\n\n{\\rm 2.}\nFor a closed immersion \n$i\\colon X\\to Y$\nof smooth schemes over $k$,\nwe have\n\\begin{equation}\nCC i_*{\\cal F}=\ni_*CC {\\cal F}.\n\\label{eqi*F}\n\\end{equation}\n\n\n\n{\\rm 3.}\nFor a morphism $f\\colon X\\to Y$\nof separated \nsmooth schemes of finite type over $k$, \nwe have\n\\begin{equation}\nCC Rf_*{\\cal F}=\nCC Rf_!{\\cal F}.\n\\label{eqRf}\n\\end{equation}\n\n{\\rm 4.}\nWe have\n\\begin{equation}\nCC D_X{\\cal F}=\nCC {\\cal F}.\n\\label{eqChD}\n\\end{equation}\n\\end{lm}\n\n\n\\proof{\n1.\nBy the characterization\nof characteristic cycle\nby the Milnor formula (\\ref{eqMil}),\nit follows from the additivity\nof the total dimension.\n\n2.\nLet $C\\subset T^*X$ be the singular\nsupport of ${\\cal F}$.\nThen,\n$i_*{\\cal F}$ is micro-supported\non $i_{\\circ}C\\subset T^*Y$\nby Lemma \\ref{lmmc}.6.\nLet $Y\\to {\\mathbf P}$\nbe an immersion satisfying the condition\n(E) before Lemma \\ref{lmloc}\nand the condition (C) before\nProposition {\\rm \\ref{prwi}}\nfor $i_{\\circ}C$.\nThen, by the description of\nthe characteristic cycle\n$CC {\\cal F}\n=CC_C^E {\\cal F}$\nin (\\ref{eqChE}),\nit follows from the canonical isomorphism\n$\\phi({\\cal F},p_L^{\\circ} \\circ i)\n\\to \\phi(i_*{\\cal F},p_L^{\\circ} )$\nfor the morphism\n$p_L^{\\circ} \\colon Y_L^{\\circ}\\to L$\ndefined by a pencil $L$.\n\n3.\nBy 1,\nit follows from \\cite{La}.\n\n4. \nWe have $SS{\\cal F}=SSD_X{\\cal F}$\nby Corollary \\ref{corssD}.\nBy 2 and Lemma \\ref{lmele}.2,\nwe may assume $X$ is projective\nas in the proof of Corollary \\ref{corssD}.\nLet $C=SS{\\cal F}=SSD_X{\\cal F}$\nbe the singular support.\nLet $X\\to {\\mathbf P}$\nbe a closed immersion\nsatisfying the condition\n(E) and (C) before \nand after Lemma \\ref{lmloc}.\nThen, for a point\n$(u,L)$\nin the dense open subset\n${\\mathbf P}(\\widetilde C)^{\\circ}\n\\subset\n{\\mathbf P}(\\widetilde C)\n\\subset X\\times_{\\mathbf P}Q$\nand $v=p_L(u)\\in L$,\nwe have\n$\\dim{\\rm tot}\\phi_u({\\cal F},p_L^{\\circ})\n=\na_v(Rp^\\vee_*p^*{\\cal F})|_L$\nand similarly for $D_X{\\cal F}$.\nSince\n$a_v(Rp^\\vee_*p^*{\\cal F})|_L\n=\na_vD_L(Rp^\\vee_*p^*{\\cal F})|_L\n=\na_v(Rp^\\vee_*p^*D_X{\\cal F})|_L$,\nit follows from the description of\nthe characteristic cycle\n$CC {\\cal F}\n=CC_C^E {\\cal F}$\nin (\\ref{eqChE}).\n\\qed}\n\\medskip\n\nFor the residue field $\\Lambda_0$ of\n$\\Lambda$,\na constructible complex\n${\\cal F}$ of $\\Lambda$-modules\nof finite tor-dimension on $X$\nis a perverse sheaf if and only if\n${\\cal F}\\otimes^L_\\Lambda\\Lambda_0$\nis a perverse sheaf.\n\n\\begin{pr}\\label{prperv}\nAssume ${\\cal F}$ is a perverse sheaf on $X$.\n\n{\\rm 1.} {\\rm (\\cite[Question p.\\ 7]{bp})}\nWe have\n\\begin{equation}\nCC {\\cal F}\\geqq0\n\\label{eqperv}\n\\end{equation}.\n\n{\\rm 2.}\nThe support of\n$CC {\\cal F}$\nequals $SS{\\cal F}$.\n\\end{pr}\n\n\\proof{\nBy the description of\nthe characteristic cycle\n$CC {\\cal F}\n=CC_C^E {\\cal F}$\nin (\\ref{eqChE}),\nit follows from Lemma \\ref{lmRn}.\n\\qed\n\n\n\n\n\\begin{cor}\\label{corRj}\nLet $j\\colon U=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} D\\to X$\nbe the open immersion of\nthe complement of a Cartier divisor.\nThen, for a perverse sheaf\n${\\cal F}$ of $\\Lambda$-modules on $U$,\nwe have\n\\begin{equation}\nSS Rj_*{\\cal F}=\nSS j_!{\\cal F}.\n\\label{eqSSRj}\n\\end{equation}\n\\end{cor}\n\n\\proof{\nSince the open immersion\n$j\\colon U=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} D\\to X$ is\nan affine morphism,\n$Rj_*{\\cal F}$ and \n$j_!{\\cal F}$ are perverse sheaves on $X$\nby \\cite[Corollaire 4.1.10]{BBD}.\nHence, it follows from\nLemma \\ref{lmRf}.3 and Proposition \\ref{prperv}.2.\n\\qed}\n\\medskip\n\nWe show the compatibility with\nsmooth pull-back.\n\n\n\\begin{df}\\label{dfCsm}\nLet $h\\colon W\\to X$ be \na smooth morphism of\nsmooth schemes over\na perfect field $k$ and let\n$C\\subset T^*X$ be a closed conical\nsubset.\nAssume that every irreducible\ncomponent of $X$ and\nevery irreducible\ncomponent of $C$\nare of dimension $n$\nand that\nevery irreducible\ncomponent of $W$ is of dimension $m$.\nLet \n\\begin{equation}\n\\begin{CD}\nT^*W@<<< W\\times_XT^*X\n@>>> T^*X\n\\end{CD}\n\\label{eqhsm}\n\\end{equation}\nbe the canonical morphisms.\nThen, for a linear combination\n$A=\\sum_am_a[C_a]$\nof irreducible components of\n$C=\\bigcup_aC_a$,\nwe define $h^!A$ to be \n$(-1)^{n-m}$-{\\em times} the push-forward\nby the first arrow\n$W\\times_XT^*X\\to T^*W$\nin {\\rm (\\ref{eqh!A})}\nof the pull-back of $A$ by the second arrow\n$W\\times_XT^*X\\to T^{*}X$\nin the sense of intersection theory.\n\\end{df}\n\n\n\n\n\\begin{pr}\\label{prsm*}\nLet $h\\colon W\\to X$ be a smooth morphism of \nsmooth schemes over a perfect field $k$ and let\n$C\\subset T^*X$ be a closed conical\nsubset.\nAssume that every irreducible\ncomponent of $X$ and\nevery irreducible\ncomponent of $C$\nare of dimension $n$\nand that\nevery irreducible\ncomponent of $W$ is of dimension $m$.\n\nLet ${\\cal F}$ be\na constructible complex of\n$\\Lambda$-modules on $X$\nof finite tor-dimension\nmicro-supported on\n$C\\subset T^*X$.\nThen, we have \n\\begin{equation}\nCC h^*{\\cal F}=\nh^!CC {\\cal F}.\n\\label{eqsm*}\n\\end{equation}\n\\end{pr}\n\n\\proof{\nSince the assertion is \\'etale local\non $W$, we may assume\n$W=X\\times {\\mathbf A}^n$\nand $h$ is the first projection.\nThen, this is the case\nwhere $Y={\\mathbf A}^n$\nand ${\\cal G}=\\Lambda$\nof \\cite[Theorem 3.6.2]{notes},\nwhich is proved \nusing the Thom-Sebastiani formula\n\\cite{TS}\nwithout\nusing the results in the rest of\nthis article.\n\\qed}\n\n\\subsection{Integrality}\\label{sZ}\n\nIn this section, we give a proof\nby Beilinson\nof the integrality of characteristic cycles.\n\n\\begin{thm}[Deligne]\\label{thmZ}\nThe coefficients\nof $CC{\\cal F}$ are integers.\n\\end{thm}\n\nThis is a generalization of \nthe Hasse-Arf theorem\n\\cite{CL} in the case\n$\\dim X\\leqq 1$. \nWe will deduce Theorem from\nthe Milnor formula\nand the following Proposition.\n\n\\begin{pr}[{\\rm \\cite[Proposition 4.12]{Be}}]\\label{prT}\nLet $X$ be a smooth scheme\nof dimension $n$\nover an algebraically closed\nfield $k$\nand $C\\subset T^*X$\nbe a closed irreducible conical subset\nof dimension $n$.\nLet $u\\in X$ be a closed\npoint and\nlet $(u,\\omega)\\in C$\nbe a closed smooth point of\n$C\\subset T^*X$ regarded as\na reduced closed subscheme.\n\nThen,\nthere exists a function\n$f$ defined on a neighborhood of $u$\nsuch that the section $df$\nof $T^*X$ meets $C$ transversely\nat $(u,\\omega)$\nif one of the following conditions {\\rm (1)}\nand {\\rm (2)} is satisfied:\n\n{\\rm (1)}\nThe characteristic $p$ of $k$ is\ndifferent from $2$.\n\n{\\rm (2)}\nLet $T$ be the tangent space\n$T_{(u,\\omega)}C$ of $C$ at \nthe smooth point $(u,\\omega)$.\nThen, \nthere exists a function\n$g$ defined on a neighborhood of $u$\nsuch that the section $dg$\nof $T^*X$ meets $C$ at \n$(u,\\omega)$\nand that\nthe dimension of\nthe intersection\n$(dg)_*(T_uX)\\cap T\\subset\nT_{(u,\\omega)}(T^*X)$\nis even.\n\\end{pr}\n\n\\proof\nTake a local coordinate\n$x_1,\\ldots,x_n$ of $X$ at $u$\nand \nwrite $\\omega=\\sum_ia_idx_i$\nas a $k$-linear combination\nof the basis $dx_1,\\ldots,dx_n$ of\nthe cotangent space $T^*_uX$.\nThen, the function $g=\\sum_ia_ix_i$\nsatisfies $dg(u)=\\omega$.\nWe will modify \n$g$ as $f=g+\\sum_{i,j}b_{ij}x_ix_j$\nto find a function $f$ satisfying\nthe required condition.\n\nLet $V=T_uX$\nand\n$W=T_{(u,\\omega)}(T^*X)$\ndenote the tangent spaces\nat $u\\in X$ and at\n$(u,\\omega)\\in T^*X$ respectively.\nThe tangent space $W$\nis decomposed as the direct sum of\nthe image $(dg)_*(V)$ of\nthe morphism $(dg)_*\\colon V\\to W$\ndefined by the section $dg\\colon X\\to T^*X$\nwith the tangent space\n$T_\\omega(T^*_uX)$ of the fiber\n$T^*_uX\\subset T^*X$.\nThe latter $T_\\omega(T^*_uX)$\nis naturally identified\nwith the cotangent space\n$T^*_uX\\subset T^*X$ \nthat is the dual\n$V^\\vee$ of $V=T_uX$.\nFurther identifying $V$ with\nits image $(dg)_*(V)$, we identify\n$W=V\\oplus V^\\vee$.\n\nThe required transversality condition means that\nthe intersection \n$(df)_*(V)\\cap T\\subset W$ is $0$.\nSince $df=dg+\\sum_{i,j}(b_{ij}+b_{ji})x_idx_j$,\nwe have $(df)_*=(dg)_*+(B+B^\\vee)$,\nwhere $B\\colon V\\to V^\\vee$\ndenotes the bilinear form on $V$\ndefined by the matrix $(b_{ij})$\nwith respect to the basis $(dx_i)$.\nConsequently, under the identification\n$W=V\\oplus V^\\vee$\nabove, the image of\n$(df)_*\\colon V\\to W$\nis identified with the graph $\\Gamma$ of\n$A=B+B^\\vee\\colon V\\to V^\\vee$.\nThus, the assertion is\na consequence of the\nfollowing lemma on linear algebra.\n\\qed}\n\n\\begin{lm}\\label{lmWV}\nLet $V$ be a $k$-vector space\nof finite dimension\nand $V^\\vee$ be the dual.\n\n{\\rm 1.}\nLet $T\\subset W=V\\oplus V^\\vee$\nbe a linear subspace of\n$\\dim T=\\dim V$\nand set\n$V_1=V\\cap T$.\nThen there exists\na direct sum decomposition\n$V=V_1\\oplus V_2$\nsatisfying the following property:\n\nFor a non-degenerate bilinear form\n$A_1\\colon V_1\\to V_1^\\vee$,\nwe extend it as $A=A_1\\oplus 0 \n\\colon V=V_1\\oplus V_2\\to \nV^\\vee=V_1^\\vee\\oplus V_2^\\vee$\nand let $\\Gamma\\subset\nW=V\\oplus V^\\vee$ denote the graph.\nThen we have\n$\\Gamma\\cap T=0$.\n\n{\\rm 2.}\nEither if $\\dim V$ is even or\nif the characteristic $p$ of $k$ is\ndifferent from $2$,\nthere exists a bilinear form\n$B\\colon V\\to V^\\vee$ such that\n$A=B+B^\\vee$ is non-degenerate.\n\\end{lm}\n\n\\proof{1.\nLet $\\bar T$ denote the\nimage of the morphism $T\\to \nW\\to W\/V=V^\\vee$\nand $\\bar T^\\perp\\subset V$\nbe the orthogonal subspace.\nThen by the assumption $\\dim T=\\dim V$,\nthe subspaces \n$V_1=V\\cap T$\nand $\\bar T^\\perp\\subset V$\nhave the same dimension.\nHence, there exists a\ndirect sum decomposition\n$V=V_1\\oplus V_2$ satisfying\n$V_2\\cap \\bar T^\\perp=0$.\nSince $V_2=(V_1^\\vee)^\\perp$,\nwe have $V_1^\\vee+\\bar T=V^\\vee$.\n\nBy the assumption that\n$A$ is non-degenerate,\nwe have $\\Gamma+T\\supset V_1+V_1^\\vee+V_2$.\nHence, further\nby $V_1^\\vee+\\bar T=V^\\vee$,\nwe obtain\n$\\Gamma+T=W$.\nThus $\\dim \\Gamma=\\dim V=\\dim T$\nimplies \n$\\Gamma\\cap T=0$.\n\n\n2.\nIf $p\\neq 2$, it suffices\nto take a non-degenerate symmetric \nbilinear form $B$.\nIf $p=2$ and if $\\dim V$ is even,\nit suffices to take\na non-degenerate alternating \nbilinear form $A$\nand to write $A=B+B^\\vee$\nby taking a symplectic basis.\n\\qed}\n\n\n\\proof[Proof of Theorem {\\rm \\ref{thmZ}}\n{\\rm (Beilinson)}]{\nWe may assume $k$ is algebraically closed.\nWrite $C=\\bigcup_aC_a$\nas the union of irreducible components.\nWe show that each\ncoefficient $m_a$ in\n$CC{\\cal F}=\\sum_am_a[C_a]$ is\nan integer.\nLet $(u,\\omega)$ be\na smooth point of $C_a$\nnot contained in any other \nirreducible component\n$C_b, (b\\neq a)$\nof $C$.\n\nIf one of the conditions\n(1) and (2) in Proposition \\ref{prT}\nis satisfied, \nfor $f$ as in Proposition \\ref{prT},\nthe coefficient\n$m_a$ equals\n$(CC{\\cal F},df)_{T^*X,u}$\nsince \n$(C_a,df)_{T^*X,u}=1$ and\n$(C_b,df)_{T^*X,u}=0$ for $b\\neq a$.\nBy the Hasse-Arf theorem\n\\cite{CL},\nthe total dimension\n$\\dim{\\rm tot} \\phi_u({\\cal F},f)$\nis an integer and\nthe Milnor formula\n$(CC{\\cal F},df)_{T^*X,u}=\n-\\dim{\\rm tot} \\phi_u({\\cal F},f)$\n(\\ref{eqMil})\nimplies that the coefficient\n$m_a=(CC{\\cal F},df)_{T^*X,u}$\nis an integer in this case.\n\nIn the exceptional case in $p=2$,\nwe take $X\\times {\\mathbf A}^1$\nand the pull-back ${\\rm pr}_1^*{\\cal F}$.\nIf the original $g$ does\nnot satisfies the condition (2)\nin Proposition \\ref{prT}\nat a smooth point $(u,\\omega)$\nof $C_a$,\nthen the composition\n${\\rm pr}_1^*g$ satisfies\nthe condition\nat the smooth point $((u,0),{\\rm pr}_1^*\\omega)$\nof ${\\rm pr}_1^oC_a\n\\subset T^*(X\\times {\\mathbf A}^1)$.\nThus, the assertion follows from\nthe compatibility Proposition \\ref{prsm*}\nof characteristic cycle with smooth pull-back.\n\\qed}\n\n\n\\section{Characteristic class}\\label{scc}\n\n\\subsection{Cycle classes\non projective space bundles}\\label{ssCh}\n\nIn this preliminary subsection,\nwe recall some basic facts\non the Chow groups of\n${\\mathbf P}^n$-bundles.\nIn this section,\nwe assume that $X$ is a\nscheme of finite type over a field $k$.\nWe can replace this assumption\nby some condition\nwhich assures necessary properties\non Chow groups.\n\n\nTo describe the Chow group\nof a projective space bundle, \nwe introduce some notation.\nLet $A=\\bigoplus_iA_i$\nbe a graded module\nand let $c_q, q=0,\\ldots, n+1$\nbe endomorphisms of\n$A$ of degree $-q$\nsending $A_i$ to $A_{i-q}$.\nWe assume that $c_0$ is the identity.\nWe formally set $f=\\sum_{q=0}^{n+1}\nc_qh^{n+1-q}$ and \ndefine a graded module\n\\begin{equation}\nA[h]\/(f)=A^{\\oplus n+1}=\n\\bigoplus_{q=0}^n\nAh^q\n\\label{eqAhf}\n\\end{equation}\nwhere $A_ih^q$\nhas degree $i+n-q$.\nWe define an endomorphism\n$h$ of $A[h]\/(f)$ of degree $-1$\nby sending\non $ah^q$ to $ah^{q+1}$\nfor $a\\in A$ and $q<n$\nand sending $a\\cdot h^n$ to\n$-(\\sum_{q=1}^{n+1}c_qa\\cdot h^{n+1-q})\n=(h^{n+1}-f)\\cdot a$.\n\n\n\\begin{lm}\\label{lmCHP}\nLet $X$ be a scheme of\nfinite type over $k$.\nLet $E$ be a vector bundle\nover $X$ of rank $n+1$\nand let $p\\colon {\\mathbf P}(E)\\to X$\nbe the associated ${\\mathbf P}^n$-bundle.\nLet $c_q(E)$ denote the $q$-th Chern classes of\n$E$ and let $c_h(E)$ denote $\\sum_{q=0}^{n+1}\nc_q(E)h^{n+1-q}$.\n\n{\\rm 1 (\\cite[Theorem 3.3 (b)]{Ful}).}\nThe morphism\n\\begin{equation}\n{\\rm CH}_\\bullet (X)[h]\/(c_h(E))\n\\to\n{\\rm CH}_\\bullet ({\\mathbf P}(E))\n\\label{eqCHP}\n\\end{equation}\nsending $ah^q$\nto $c({\\cal O}(1))^q\\cap p^*a$\nis an isomorphism\nof graded modules.\n\n{\\rm 2.}\nLet $i\\geqq 0$ be an integer.\nThen, the inverse of the degree $i$-part\n\\begin{equation}\n\\bigoplus_{q=0}^n\n{\\rm CH}_{i-n+q}(X)\n=\n\\bigl({\\rm CH}_\\bullet(X)[h]\/(c_h(E))\\bigr)_i\n\\to \n{\\rm CH}_i ({\\mathbf P}(E))\n\\label{eqCHPn}\n\\end{equation}\nof the isomorphism {\\rm (\\ref{eqCHP})}\nsends $b\\in {\\rm CH}_i ({\\mathbf P}(E))$\nto $c(E)\\cap p_*(c_1({\\cal O}(-1))^{-1}b)$.\n\\end{lm}\n\n\\proof{\n2.\nBy the isomorphism\n(\\ref{eqCHP}),\nwe identify ${\\rm CH}_{\\bullet}({\\mathbf P}(E))$\nand its degree $i$-part \nwith ${\\rm CH}_\\bullet (X)[h]\/\n(c_h(E))$ and $\\bigoplus_{q=0}^n\n{\\rm CH}_{i-n+q}(X)h^q$.\nBy the projection formula, the morphism\n$p_*\\colon {\\rm CH}_{\\bullet}({\\mathbf P}(E))\n\\to {\\rm CH}_{\\bullet}(X)$\nis the morphism\n${\\rm CH}_\\bullet (X)[h]\/\n(c_h(E))\\to {\\rm CH}_\\bullet (X)$\ntaking the coefficient of $h^n$.\n\nWe may assume $b=ah^q$ for $q=0,\\ldots,n$\nand $a\\in {\\rm CH}_{i-n+q}(X)$.\nThe difference \n$c(E)(1-h)^{-1}$ $h^q\n-c_h(E)(1-h)^{-1}\n=\\sum_{i=0}^{n+1}c_i(E)(h^q-h^{n+1-i})\/(1-h)$\nis of degree $\\leqq n$ in $h$\nand the top term is $h^n$.\nSince $c_h(E)$ acts as $0$ on \n${\\rm CH}_\\bullet (X)[h]\/(c_h(E))$,\nthe assertion follows.\n\\qed}\n\n\n\n\\begin{lm}\\label{lmincl}\nLet $i\\colon F\\to E$ be an injection\nof vector bundles on $X$\nof ${\\rm rank}\\ E=n+1\\geqq\n{\\rm rank}\\ F=m+1$\nand let $i\\colon \n{\\mathbf P}(F)\n\\to\n{\\mathbf P}(E)$\nalso denote the induced morphism.\nLet $E'$ denote the cokernel.\n\n{\\rm 1.}\nThe diagram \n\\begin{equation}\n\\begin{CD}\n{\\rm CH}_\\bullet({\\mathbf P}(F))\n@<{i^*}<<\n{\\rm CH}_\\bullet({\\mathbf P}(E))\\\\\n@AAA@AAA\\\\\n{\\rm CH}_\\bullet(X)[h]\/(c_h(F))\n@<{\\rm can}<<\n{\\rm CH}_\\bullet(X)[h]\/(c_h(E))\n\\end{CD}\n\\label{eqincl}\n\\end{equation}\nis commutative.\n\n{\\rm 2.}\nThe diagram \n\\begin{equation}\n\\begin{CD}\n{\\rm CH}_\\bullet({\\mathbf P}(F))\n@>{i_*}>>\n{\\rm CH}_\\bullet({\\mathbf P}(E))\\\\\n@AAA@AAA\\\\\n{\\rm CH}_\\bullet(X)[h]\/(c_h(F))\n@>{c_h(E')\\cap }>>\n{\\rm CH}_\\bullet(X)[h]\/(c_h(E))\n\\end{CD}\n\\label{eqincl*}\n\\end{equation}\nis commutative.\n\\end{lm}\n\n\\proof{\n1.\nSince the pull-back of ${\\cal O}(1)$\nby ${\\mathbf P}(F)\\to {\\mathbf P}(E)$\nis also\n${\\cal O}(1)$ on ${\\mathbf P}(F)$,\nthe assertion follows from the definition.\n\n2.\nBy 1, it suffices to\nshow that the endomorphism $i_*i^*$\nof\n${\\rm CH}_\\bullet({\\mathbf P}(E))$\nis identified with the multiplication by $c_h(E')$\non ${\\rm CH}_\\bullet(X)[h]\/(c_h(E))$.\nBy the self-intersection formula,\nthe endomorphism $i_*i^*$\nequals the action of\nthe top Chern class\n$c_{n-m}\n(T_{{\\mathbf P}(F)}{\\mathbf P}(E))$\nof the normal bundle.\n\nBy the exact sequence\n$0\\to \\Omega_{{\\mathbf P}(E)\/X}\n\\to E^\\vee\\otimes {\\cal O}(-1)\\to {\\cal O}_{{\\mathbf P}(E)}\\to 0$\nand the corresponding one for\n${\\mathbf P}(F)$,\nwe obtain an isomorphism\n$N_{{\\mathbf P}(F)\/{\\mathbf P}(E)}\n\\to \nE^{\\prime \\vee}\\otimes {\\cal O}(-1)$\nfor the conormal sheaf.\nHence, the top Chern class\n$c_{n-m} (T_{{\\mathbf P}(F)}{\\mathbf P}(E))$\nequals \n$c_{n-m}(E'\\otimes {\\cal O}(1))$\nand is identified with $c_h(E')$.\n\\qed}\n\n\n\\begin{lm}\\label{lmEF1}\nAssume\n$E=F\\oplus {\\mathbf A}^1_X$.\nLet $i\\colon \n{\\mathbf P}(F)\n\\to\n{\\mathbf P}(E)$ denote the injection\nand let $p\\colon \n{\\mathbf P}(E)\\to X$ be the projection.\n\n\n{\\rm 1.}\nThe morphism\n\\begin{equation}\ni^*\\oplus p_*\\colon\n{\\rm CH}_\\bullet({\\mathbf P}(E))\n\\to\n{\\rm CH}_\\bullet({\\mathbf P}(F))\n\\oplus\n{\\rm CH}_\\bullet(X)\n\\label{eqQZ}\n\\end{equation}\nis an isomorphism.\nIf $X$ is irreducible\nof dimension $d\\leqq n$\nand $i_x\\colon x\\to X$ is\nthe immersion of a smooth\n$k$-rational point, \nthe second projection\n$p_*\\colon\n{\\rm CH}_d({\\mathbf P}(E))\n\\to\n{\\rm CH}_d(X)=\n{\\mathbf Z}$\nequals the pull-back $i_x^*\n\\colon\n{\\rm CH}_d({\\mathbf P}(E))\n\\to\n{\\rm CH}_0({\\mathbf P}^n)=\n{\\mathbf Z}$.\n\n\n{\\rm 2.}\nThe morphism\n\\begin{equation}\n i_*+ p^*\\colon\n{\\rm CH}_\\bullet({\\mathbf P}(F))\n\\oplus\n{\\rm CH}_\\bullet(X)\n\\to\n{\\rm CH}_\\bullet({\\mathbf P}(E))\n\\label{eqQZ2}\n\\end{equation}\nis an isomorphism.\nThe composition \n${\\rm CH}_\\bullet ({\\mathbf P}(E))\n\\to {\\rm CH}_{\\bullet}(X)$\nof the inverse of\nthe isomorphism {\\rm (\\ref{eqQZ2})}\nwith the second projection\nequals the pull-back by\nthe $0$-section\n$s\\colon X\\to F\\subset {\\mathbf P}(E)$.\n\\end{lm}\n\n\\proof{1.\nWe identify\n${\\rm CH}_\\bullet({\\mathbf P}(E))\n=\n{\\rm CH}_\\bullet(X)[h]\/(c_h(E))$\nand \n${\\rm CH}_\\bullet({\\mathbf P}(F))\n=\n{\\rm CH}_\\bullet(X)[h]\/(c_h(F))$.\nThen, $c_h(E)=c_h(F)h$\nand the morphism\n$i^*\\colon\n{\\rm CH}_\\bullet({\\mathbf P}(E))\n\\to\n{\\rm CH}_\\bullet({\\mathbf P}(F))$\nis identified with the surjection\n${\\rm CH}_\\bullet(X)[h]\/(c_h(E))\n\\to\n{\\rm CH}_\\bullet(X)[h]\/(c_h(F))$\nsending $h^n$ to $0$\nby Lemma \\ref{lmincl}.1.\nSince $p_*\\colon\n{\\rm CH}_\\bullet({\\mathbf P}(E))\n=\n{\\rm CH}_\\bullet(X)[h]\/(c_h(E))\n\\to\n{\\rm CH}_\\bullet(X)$\nis the morphism taking\nthe coefficient of\n$h^n$,\nthe morphism (\\ref{eqQZ}) is\nan isomorphism.\n\nThe second assertion follows from\nthe commutative diagram\n$$\\begin{CD}\n{\\rm CH}_d({\\mathbf P}(E))\n@>{i_x^*}>>\n{\\rm CH}_0({\\mathbf P}^n)\\\\\n@V{p_*}VV@VV{\\deg}V\\\\\n{\\rm CH}_d(X)\n@>{i_x^*}>>\n{\\rm CH}_0(x).\n\\end{CD}$$\n\n\n2.\nThe morphism\n$i_*\\colon\n{\\rm CH}_\\bullet({\\mathbf P}(F))\n\\to\n{\\rm CH}_\\bullet({\\mathbf P}(E))$\nis identified with the multiplication\n$h\\cap \\colon\n{\\rm CH}_\\bullet(X)[h]\/(c_h(F))\n\\to\n{\\rm CH}_\\bullet(X)[h]\/(c_h(E))$\nby Lemma \\ref{lmincl}.2.\nHence\nthe morphism (\\ref{eqQZ2}) is\nan isomorphism.\n\nSince $s^*i_*=0$ and\n$s^*p^*$ is the identity of\n${\\rm CH}_\\bullet(X)$,\nthe second assertion follows\nfrom the isomorphism (\\ref{eqQZ2}).\n\\qed}\n\n\\begin{lm}\\label{lmsurj}\nLet $\\theta\\colon E\\to F$ be a surjection\nof vector bundles on $X$\nof ${\\rm rank}\\ E=n+1\\geqq\n{\\rm rank}\\ F=m+1$ and\nlet $K$ denote the kernel.\nLet $\\pi\\colon \n{\\mathbf P}(E)'\n\\to\n{\\mathbf P}(E)$\nbe the blow-up at\n${\\mathbf P}(K)\n\\subset\n{\\mathbf P}(F)$\nand let $\\theta'\\colon \n{\\mathbf P}(E)'\n\\to\n{\\mathbf P}(F)$\ndenote the morphism induced by $\\theta$.\n\n{\\rm 1.}\nLet \n$c_h(K)^{-1}\\cap \n\\colon\n{\\rm CH}_\\bullet(X)[h]\/(c_h(E))\n\\to\n{\\rm CH}_\\bullet(X)[h]\/(c_h(F))$\ndenote the composition \nof the first projection\nwith the inverse of the isomorphism\n$$(c_h(K)\\cap)+ {\\rm can}\\colon\n{\\rm CH}_\\bullet(X)[h]\/(c_h(F))\n\\oplus\n{\\rm CH}_\\bullet(X)[h]\/(c_h(K))\n\\to\n{\\rm CH}_\\bullet(X)[h]\/(c_h(E)).$$\nThen, the diagram \n\\begin{equation}\n\\begin{CD}\n{\\rm CH}_\\bullet({\\mathbf P}(E))\n@>{\\theta'_*\\pi^*}>>\n{\\rm CH}_\\bullet({\\mathbf P}(F))\\\\\n@AAA@AAA\\\\\n{\\rm CH}_\\bullet(X)[h]\/(c_h(E))\n@>{c_h(K)^{-1}\\cap }>>\n{\\rm CH}_\\bullet(X)[h]\/(c_h(F))\n\\end{CD}\n\\label{eqsurj*}\n\\end{equation}\nis commutative.\n\n\n{\\rm 2.}\nThe diagram \n\\begin{equation}\n\\begin{CD}\n{\\rm CH}_\\bullet({\\mathbf P}(E))\n@<{\\pi_*\\theta^{\\prime *}}<<\n{\\rm CH}_\\bullet({\\mathbf P}(F))\\\\\n@AAA@AAA\\\\\n{\\rm CH}_\\bullet(X)[h]\/(c_h(E))\n@<{\\rm can}<<\n{\\rm CH}_\\bullet(X)[h]\/(c_h(F))\n\\end{CD}\n\\label{eqsurj}\n\\end{equation}\nis commutative.\n\\end{lm}\n\n\\proof{\nLet $A={\\rm CH}_\\bullet(X)$, $c_h(F)=f$ and $c_h(E)=f\\cdot g$\nand identify \n${\\rm CH}_\\bullet({\\mathbf P}(E))\n=A[h]\/(f\\cdot g)$ and\n${\\rm CH}_\\bullet({\\mathbf P}(F))\n=A[h']\/(f)$.\nLet $L\\subset {\\mathbf P}(F)\\times_XF$\nbe the universal sub line bundle\nand let\n$V\\subset {\\mathbf P}(F)\\times_XE$\nbe its pull-back by the base change of $\\theta$.\nThen, \n${\\mathbf P}(E)'\n\\to\n{\\mathbf P}(F)$\nis canonically identified with\nthe ${\\mathbf P}^{n-m}$-bundle\n${\\mathbf P}(V)$\nby Lemma \\ref{lmFEL}.\nThus, we identify\n${\\rm CH}_\\bullet({\\mathbf P}(E)')\n=A[h']\/(f)[h]\/((h-h')\\cdot g)$.\nLet $e$ denote the polynomial\nin $h$ and $h'$ defined\nby $f(h)-f(h')=(h-h')e$.\n\nWe have a cartesian diagram\n$$\\begin{CD}\n{\\mathbf P}(K)\n\\times \n{\\mathbf P}(F)\n@>>>\n{\\mathbf P}(E)'\\\\\n@VVV@VV{\\pi}V\\\\\n{\\mathbf P}(K)\n@>>>\n{\\mathbf P}(E).\n\\end{CD}$$\nWe show that this induces a cocartesian\ndiagram\n\\begin{equation}\n\\begin{CD}\nA[h]\/(g(h))[h']\/(f(h'))\n@>{(h-h')\\times}>>\nA[h']\/(f(h'))[h]\/((h-h')\\cdot g(h))\\\\\n@A{e\\times}AA@AAA\\\\\nA[h]\/(g(h))\n@>{f(h)\\times}>>\nA[h]\/(f(h)\\cdot g(h)).\n\\end{CD}\n\\label{eqPE'}\n\\end{equation}\non the Chow groups. \nThe diagram is cocartesian by\n\\cite[Proposition 6.7 (e)]{Ful}.\nBy Lemma \\ref{lmincl}.2,\nthe lower horizontal arrow is\nthe multiplication by $f(h)$.\nThe descriptions of\nthe left vertical arrow\nand the upper horizontal arrow\nfollow similarly from\nthe excess intersection formula\nand the self-intersection formula\nrespectively.\nSince $f(h')=0$ \nand hence $f(h)=(h-h')e$ in\n$A[h']\/(f(h'))[h]\/((h-h')\\cdot g(h))$,\nthe right vertical arrow\nis well-defined as\nthe canonical morphism.\n\n1.\nThe morphism $\\theta'_*\\pi^*$\nis the composition of\nthe right vertical arrow in (\\ref{eqPE'})\nwith the morphism\n$A[h']\/(f(h'))[h]\/((h-h')\\cdot g(h))\\to\nA[h']\/(f(h'))$\ntaking the coefficient of $h^{n-m}$.\nFor $i< n-m$,\nwe have $\\theta'_*\\pi^* h^i=0$.\nFor $i\\geqq 0$,\nwe have $h^ig(h)=\nh^{\\prime i}g(h)$ in\n$A[h']\/(f(h'))[h]\/((h-h')\\cdot g(h))$\nand hence $\\theta'_*\\pi^* \nh^ig(h)=h^{\\prime i}$.\nThus, \nthe commutativity of\n(\\ref{eqsurj*}) is proved.\n\n\n2.\nBy the cocartesian diagram\n(\\ref{eqPE'}),\nwe identify \n$A[h']\/(f(h'))[h]\/((h-h')\\cdot g(h))$\nwith the amalgamated sum.\nThen, the morphism\n$\\pi_*\\colon\nA[h']\/(f(h'))[h]\/((h-h')\\cdot g(h))\n\\to\nA[h]\/(f(h)\\cdot g(h))$\nis induced by the\nidentity of $A[h]\/(f(h)\\cdot g(h))$\nand \nthe morphism\n$A[h]\/(g(h))[h']\/(f(h'))\n\\to \nA[h]\/(g(h))$\ntaking the coefficient of $h^{\\prime m}$.\nThe morphism $\\pi_*\\theta^{\\prime *}$\nis the composition of\nthe canonical morphism\n$A[h']\/(f(h'))\\to\nA[h']\/(f(h'))[h]\/((h-h')$\nwith the above morphism.\nFor $0\\leqq i\\leqq m$,\nwe have $h^{\\prime i}\n=h^i+(h'-h)(\\deg <m)$.\nHence, we have\n$\\pi_*\\theta^{\\prime *}h^{\\prime i}\n=h^i$.\nThus, \nthe commutativity of\n(\\ref{eqsurj}) is proved.\n\\qed}\n\n\n\\begin{lm}\\label{lmtheta}\nLet $\\theta\\colon E\\to F$ be\na surjection of vector bundles\nof rank $n\\geqq m$ over $X$\nand \nlet $K={\\rm Ker}(\\theta\n\\colon E\\to F)$ be the kernel of\n$\\theta$.\n\nLet $A=\\sum_am_aC_a$\nbe a ${\\mathbf Z}$-linear combination\nof irreducible closed\nconical subsets $C_a\\subset E$\nof dimension $i\\leqq m$\nand define\n$\\bar A\n=\n\\sum_am_a\\bar C_a\n\\in {\\rm CH}_i(\n{\\mathbf P}(E\\oplus {\\mathbf A}^1_X))\n=\n{\\rm CH}_{\\bullet\\leqq i}(X)$.\nAssume that the intersection\n$C\\cap K$ of the union $C=\\bigcup_aC_a$\nwith $K$\nis a subset of the $0$-section.\nDefine a linear combination $\\theta(A)$ \nof closed conical subsets of $F$ to be\nthe direct image of $A$ by $\\theta\\colon\nE\\to F$ which is finite on $C$.\nThen,\nwe have\n\\begin{equation}\n[\\overline {\\theta(A)}]=\nc_h(K)^{-1}\\cap\n[\\bar A]\n\\label{eqPhC}\n\\end{equation}\nin ${\\rm CH}_i(\n{\\mathbf P}(F\\oplus {\\mathbf A}^1_X))\n=\n{\\rm CH}_{\\bullet\\leqq i}(X)$.\n\\end{lm}\n\n\\proof{\nLet $\\pi\\colon\n{\\mathbf P}(E\\oplus {\\mathbf A}^1_X)'\n\\to\n{\\mathbf P}(E\\oplus {\\mathbf A}^1_X)$\nand $\\theta'\\colon\n{\\mathbf P}(E\\oplus {\\mathbf A}^1_X)'\n\\to\n{\\mathbf P}(F\\oplus {\\mathbf A}^1_X)$\nbe as in Lemma \\ref{lmsurj}.\nThe assumption that\n$C\\cap K$ \nis a subset of the $0$-section\nmeans that ${\\mathbf P}(K)$ does not\nmeet the closure \n$\\bar C\\to {\\mathbf P}(E\\oplus {\\mathbf A}^1_X)$.\nHence we have $[\\overline {\\theta(A)}]\n= \\theta'_*\\pi^* ([\\bar A])$.\nThus,  the assertion follows\nby Lemma \\ref{lmsurj}.1.\n\\qed}\n\n\n\n\\subsection{Characteristic class\n}\\label{ssccc}\n\nIn this subsection,\n$k$ denotes a perfect field of\ncharacteristic $p\\geqq 0$\nand $\\Lambda$ denotes\na finite field of characteristic\n$\\neq p$.\n\nLet $X$ be a scheme of\nfinite type over $k$\nand $i\\colon X\\to M$\nbe a closed immersion \nto a smooth scheme $M$ \nof dimension $n$ over $k$.\nWe identify\n${\\rm CH}_n({\\mathbf P}(X\\times_MT^*M\n\\oplus {\\mathbf A}^1_X))$\nwith\n${\\rm CH}_\\bullet(X)\n=\\bigoplus_{i=0}^n\n{\\rm CH}_i(X)$\nby the canonical isomorphism\n\\begin{equation}\n{\\rm CH}_\\bullet(X)\n=\\bigoplus_{i=0}^n\n{\\rm CH}_i(X)\n\\to \n{\\rm CH}_n({\\mathbf P}(X\\times_MT^*M\n\\oplus {\\mathbf A}^1_X))\n\\label{eqCCH}\n\\end{equation}\nsending $(a_i)_i$\nto $\\sum_ip^*a_ih^i$\nwhere $p\\colon \n{\\mathbf P}(X\\times_MT^*M\n\\oplus {\\mathbf A}^1_X)\n\\to X$ is the canonical projection\nand \n$h=c_1(L^\\vee)$\nbe the first Chern class of the dual \n$L^\\vee$ of\nthe universal sub line bundle\n$L\\subset\n{\\mathbf P}(X\\times_MT^*M\n\\oplus {\\mathbf A}^1_X)\\times_X\n(X\\times_MT^*M\n\\oplus {\\mathbf A}^1_X)$.\nThe inverse mapping sends $a$\nto $c(T^*X)\\cap p_*(c(L)^{-1}\n\\cap a)$.\n\n\\begin{lm}\\label{lmwell}\nLet $X$ be a scheme of\nfinite type over $k$\nand ${\\cal F}$ be a constructible\ncomplex of $\\Lambda$-modules on $X$.\nLet $i\\colon X\\to M$\nbe a closed immersion \nto a smooth scheme $M$ \nof dimension $n$ over $k$.\nThen, the class of the closure\n$\\overline{CCi_*{\\cal F}}\n={\\mathbf P}(\nCCi_*{\\cal F}\\oplus {\\mathbf A}^1_X)$\nregarded as an element of\n${\\rm CH}_n(X\\times_MT^*X\n\\oplus {\\mathbf A}^1_X)=\n{\\rm CH}_\\bullet (X)$\nis independent of the\nchoice of $M$ or $i$.\n\\end{lm}\n\n\\proof{\nLet $j\\colon X\\to N$ be\nanother closed immersion\nto a smooth scheme $N$ over $k$.\nBy considering the product\nand the projections,\nwe may assume that there exists\na smooth morphism\n$f\\colon M\\to N$ compatible\nwith the immersions $i$ and $j$.\nIt suffices to show that\n$CCi_*{\\cal F}$ meets properly\nthe image of the injection \n$X\\times_NT^*N\\to\nX\\times_MT^*M$\nand that \n$CCj_*{\\cal F}$\nis the pull-back of\n$CCi_*{\\cal F}$\nby Lemma \\ref{lmincl}.1.\nSince the claim is \\'etale local on $X$,\nwe may assume that\nthere exists a section $s\\colon N\\to M$\nof $f$ such that $i=s\\circ j$.\nThen the claim follows from\n$CCi_*{\\cal F}=s_*CCj_*{\\cal F}$\nLemma \\ref{lmRf}.2.\n\\qed}\n\nFollowing the construction\nin \\cite[A.4]{Gi} in an analytic context,\nwe make the definition below.\n\n\n\n\\begin{df}\\label{dfccX}\nLet $X$ be a scheme of\nfinite type over a perfect field $k$\nand let $\\Lambda$ be a\nfinite field of characteristic invertible in $k$.\n\n{\\rm 1.}\nWe say that $X$ is {\\em embeddable}\nif there exists a closed immersion \n$i\\colon X\\to M$\nto a smooth scheme $M$ \nover $k$.\n\n{\\rm 2. (cf.\\ \\cite[A.4]{Gi})}\nAssume that $X$ is embeddable and\nlet ${\\cal F}$ be a constructible\ncomplex of $\\Lambda$-modules\non $X$.\nLet $i\\colon X\\to M$\nbe a closed immersion \nto a smooth scheme $M$ \nof dimension $n$ over $k$.\nWe call the class \n\\begin{equation}\ncc_X{\\cal F}\n=\n\\overline{CCi_*{\\cal F}}\n={\\mathbf P}(\nCCi_*{\\cal F}\\oplus {\\mathbf A}^1_X)\n\\in\n{\\rm CH}_{n}({\\mathbf P}(X\\times_MT^*M\n\\oplus {\\mathbf A}^1_X))\n=\n{\\rm CH}_\\bullet(X)\n\\label{eqccXF}\n\\end{equation}\nof the closure of the characteristic cycle\n$CCi_*{\\cal F}$\nregarded as an element of\n${\\rm CH}_\\bullet(X)$\nthe {\\em characteristic class}\nof ${\\cal F}$.\n\n\nLet $K(X,\\Lambda)\n=K(X)$ denote the Grothendieck\ngroup of the triangulated category\nof constructible complexes\nof $\\Lambda$-modules on $X$.\nWe define a linear morphism\n\\begin{equation}\ncc_X\\colon\nK(X,\\Lambda)\n\\to\n{\\rm CH}_\\bullet(X).\n\\label{eqch}\n\\end{equation}\nto be that sending the class of \n${\\cal F}$ to the characteristic class\n$cc_X{\\cal F}$.\n\n{\\rm 3.}\nAssume that $X$ is embeddable and\nlet $j\\colon U\\to X$\nbe an open immersion.\nLet \n${\\cal F}$ be a locally constant\nconstructible sheaf of $\\Lambda$-modules\non $U$.\nThen, we define the \n(total) {\\em Swan class}\n${\\rm Sw}_X{\\cal F}\n\\in {\\rm CH}_\\bullet (X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} U)$\nto be the class of\nthe closure of the difference\n$CC j_!{\\cal F}\n-{\\rm rank}\\ {\\cal F}\\cdot\nCC j_!\\Lambda_U$.\n\\end{df}\n\nIf $X$ is quasi-projective,\nthen $X$ is embeddable.\n\n\\begin{cn}\\label{cncc}\nLet $X$ be an embeddable\nscheme of finite type over $k$.\nLet $a\\colon X\\to {\\rm Spec}\\ k$\nbe the canonical morphism\nand define ${\\cal K}_X=Ra^!\\Lambda$.\n\n{\\rm 1.}\nThe cohomology class \n$cl(cc_{X,0}{\\cal F})\n\\in H^0(X,{\\cal K}_X)$ \nof the dimension $0$-part of\nthe characteristic class\nequals the characteristic class\n$C({\\cal F})$ defined in \n{\\rm \\cite{AS}}.\n\n{\\rm 2.}\nThe dimension $0$-part\n${\\rm Sw}_{X,0}{\\cal F}\n\\in {\\rm CH}_0(X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} U)$\nof the total Swan class\nequals the Swan class\ndefined in {\\rm \\cite{KS}}.\n\\end{cn}\n\n\\begin{lm}\\label{lmccd}\n{\\rm 1.}\nThe dimension $0$-part\n$cc_{X,0}{\\cal F}$\nis the intersection\nproduct $(CC{\\cal F},T^*_XX)_{T^*X}\n\\in {\\rm CH}_0(X)$\nwith the $0$-section.\n\n{\\rm 2.}\nAssume that $X$ is irreducible\nof dimension $d$.\nFor a constructible\ncomplex ${\\cal F}$\nof $\\Lambda$-modules\non $X$,\nlet ${\\rm rank}^\\circ {\\cal F}$\ndenote the rank of its\nrestriction on\na dense open subset of $X$\nwhere ${\\cal F}$ is locally constant.\nThen, the dimension $d$-part\n$cc_{X,d}{\\cal F}\\in\n{\\rm CH}_d(X)={\\mathbf Z}$\nis $(-1)^d{\\rm rank}^\\circ {\\cal F}$.\n\\end{lm}\n\n\\proof{\n1.\nIt follows from Lemma \\ref{lmEF1}.1.\n\n2.\nAfter shrinking $X$,\nwe may assume that\n${\\cal F}$ is locally constant\nand $X$ is smooth over $k$.\nThen $CC{\\cal F}$\nis\n$(-1)^d{\\rm rank}^\\circ {\\cal F}$-times\nthe $0$-section\nof $T^*X$ and the assertion follows.\n\\qed}\n\\medskip\n\nWe discuss some functorial\nproperties of the morphism\n$cc_X$.\nFor $k={\\mathbf C}$,\nthe commutativity of the diagram\n(\\ref{eqCHf}) below in analytic context\nis proved for\nprojective morphisms in \\cite[Theorem 9.4]{Gi}.\nThis implies \\cite[Theorem A.6]{Gi}\na commutative diagram\n\\begin{equation}\n\\xymatrix{\nK(X,\\Lambda)\\ar[rr]^{cc_X}\n\\ar[d]&&\n{\\rm CH}_\\bullet(X)\\\\\nF(X)\\ar[rru]_{(-1)^\\bullet c_X}&& }\n\\label{eqcX}\n\\end{equation}\nfor the MacPherson-Chern class\n$c_X\\colon F(X)\\to {\\rm CH}_\\bullet(X)$\n\\cite{Mac} with multiplied $(-1)^i$ \non degree $i$ and the canonical morphism\n$K(X,\\Lambda)\\to F(X)$\nto the module of ${\\mathbf Z}$-valued\nconstructible functions\ndefined by the rank of stalks.\n\nHowever in positive characteristic,\none cannot expect\nto have a functoriality\n\\begin{equation}\n\\begin{CD}\nK(X,\\Lambda)\n@>{cc_X}>>\n{\\rm CH}_\\bullet(X)\\\\\n@V{f_*}VV@VV{f_!}V\\\\\nK(Y,\\Lambda)\n@>{cc_Y}>>\n{\\rm CH}_\\bullet(Y)\n\\end{CD}\n\\label{eqCHf}\n\\end{equation}\nfor proper morphism\n$f\\colon X\\to Y$ over $k$\nin full generality (cf.\\ \\cite[Note 87$_1$]{RS})\nexcept for the dimension $0$-part,\nas the following counterexample shows.\n\n\\begin{ex}\\label{exctr}\nWe identify ${\\rm CH}_\\bullet({\\mathbf P}^n)$\nwith $\\bigoplus_{i=0}^n{\\mathbf Z}$.\n\n{\\rm 1.}\nThe characteristic class $cc_{{\\mathbf P}^n}\n\\Lambda=\n(-1)^n[T^*_{{\\mathbf P}^n}\n{\\mathbf P}^n]=\n(-1)^n\nc(\\Omega^1_{{\\mathbf P}^n})=\n((-1)^i\\binom {n+1}{i+1})_{\\dim i}$\nis non-trivial on every degree.\nThe endomorphism\n$f_*$ of ${\\rm CH}_\\bullet({\\mathbf P}^n)$\ninduced by the Frobenius morphism\n$f\\colon {\\mathbf P}^n\\to {\\mathbf P}^n$\nis the multiplication by\n$p^i$ on the dimension $i$-part.\nSince $f_*\\Lambda=\\Lambda$,\nthe diagram \n{\\rm (\\ref{eqCHf})}\nis {\\em not} commutative for\nthe Frobenius morphism\n$f\\colon {\\mathbf P}^n\\to {\\mathbf P}^n$\nexcept possibly for the dimension $0$-part.\n\n{\\rm 2.}\nLet $j\\colon \n{\\mathbf A}^1\n={\\rm Spec}\\ k[x]\n\\to {\\mathbf P}^1$\nbe the open immersion\nand let $i\\colon \n{\\rm Spec}\\ k\n\\to {\\mathbf P}^1$\nbe the closed immersion\nof the complement.\nLet ${\\cal L}$ on ${\\mathbf A}^1$ be\nthe locally constant constructible\nsheaves of rank $1$ defined by\nthe Artin-Schreier equation\n$t^p-t=x$ and define\n${\\cal F}=j_!{\\cal L}\n\\oplus \\Lambda$\nand \n${\\cal G}=j_!\\Lambda^{\\oplus 2}$.\n\nThen, we have\n$CC{\\cal F}\n=\nCC{\\cal G}\n=-2(T^*_{{\\mathbf P}^1}{\\mathbf P}^1\n+T^*_\\infty{\\mathbf P}^1)$\nand hence\n$cc_{{\\mathbf P}^1}\n{\\cal F}\n=\ncc_{{\\mathbf P}^1}\n{\\cal G}\n=(-2,2)$.\nSince\n$cc_{{\\rm Spec} k}\ni^*{\\cal F}=1\\neq\ncc_{{\\rm Spec} k}\ni^*{\\cal G}=0$,\nthere exists {\\em no}\nright vertical arrow that makes the diagram\n\\begin{equation}\n\\begin{CD}\nK(X,\\Lambda)\n@>{cc_X}>>\n{\\rm CH}_\\bullet(X)\\\\\n@V{h^*}VV@VV{?}V\\\\\nK(W,\\Lambda)\n@>{cc_W}>>\n{\\rm CH}_\\bullet(W)\n\\end{CD}\n\\label{eqCHh}\n\\end{equation}\ncommutative\nfor the immersion $\\infty\\to {\\mathbf P}^1$.\n\\end{ex}\n\nWe consider the functoriality\nwith respect to push-forward.\nBy Lemma \\ref{lmccd}.1,\nthe commutative diagram\n(\\ref{eqCHf}) for \nproper smooth morphism $f\\colon\nX\\to {\\rm Spec}\\ k$\nmeans that the index formula\n$(CC{\\cal F},T^*_XX)=\\chi(X_{\\bar k},{\\cal F})$\nholds for\nconstructible complexes ${\\cal F}$ on $X$.\n\n\\begin{lm}\\label{lmclim}\nThe diagram {\\rm (\\ref{eqCHf})}\nis commutative if $f\\colon X\\to Y$\nis a closed immersion\nof embeddable schemes\nof finite type over $k$.\n\\end{lm}\n\n\\proof{\nLet $i\\colon Y\\to M$\nbe a closed immersion to\na smooth scheme over $k$.\nThen, since both $cc_X{\\cal F}$\nand $cc_Yf_*{\\cal F}$\nare defined by\n$CC(if)_*{\\cal F}$,\nthe assertion follows.\n\\qed}\n\\medskip\n\nLet $f\\colon X\\to Y$\nbe a proper morphism\nof embeddable schemes\nof finite type over $k$.\nAs in the proof of Corollary\n\\ref{corcnsm},\nwe obtain a commutative diagram\n\\begin{equation}\n\\begin{CD}\nX@>i>> M\\\\\n@VfVV @VV{g}V\\\\\nY@>j>> N.\n\\end{CD}\n\\label{eqXYMN}\n\\end{equation} where\n$M$ and $N$ are smooth,\nthe right vertical arrow is smooth\nand the horizontal arrows are\nclosed immersions.\nWe consider the morphisms\n\\begin{equation}\n\\begin{CD}\nY\\times_NT^*N\n@<{\\tilde f}<<\nX\\times_NT^*N\n@>h>>\nX\\times_NT^*M\n\\end{CD}\n\\label{eqfh}\n\\end{equation}\nwhere the left arrow is induced\nby $f$ and the right arrow is\nthe canonical injection.\n\nLet $C\n\\subset X\\times_MT^*M$\nbe a closed conical subset.\nAssume that every irreducible \ncomponent of $M$ and of $C=\\bigcup_aC_a$\nis of dimension $m$.\nAssume also that every irreducible \ncomponent of $N$ \nis of dimension $\\leqq n$\nand that every irreducible \ncomponent of\nthe closed conical subset\n$f_\\circ C=\\tilde f(h^{-1}C)\n\\subset Y\\times_NT^*N$\nis of dimension $\\leqq n$.\nThen for a linear combination\n$A=\\sum_am_aC_a$ of\nirreducible components\nof $C$,\nthe push-forward $f_!A=\n\\sum_a\nm_a\\tilde f_*h^!C_a$\nis defined as a linear combination of\nirreducible components of dimension $n$\nof $f_\\circ C$.\n\n\n\\begin{lm}\\label{lmcnpr}\nLet $f\\colon X\\to Y$\nbe a proper morphism\nof embeddable schemes\nof finite type over $k$\nand let {\\rm (\\ref{eqXYMN})}\nbe a commutative diagram\nof schemes over $k$.\nAssume that\n$M$ and $N$ are smooth\nof dimension $m$ and $n$\nrespectively,\nthat the right vertical arrow is smooth\nand that the horizontal arrows are\nclosed immersions.\n\nLet ${\\cal F}$\nbe a constructible complex of\n$\\Lambda$-modules on $X$ \nand $C=SS{\\cal F}\n\\subset X\\times_MT^*M$ \nbe the singular support of\n${\\cal F}$.\nAssume that \n$f_\\circ C=\\tilde f(h^{-1}C)\n\\subset Y\\times_NT^*N$\nis of dimension $\\leqq n$\nand that we have\n$CCR(gi)_*{\\cal F}=f_!CCi_*{\\cal F}$.\nThen, we have\n\\begin{equation}\ncc_YRf_*{\\cal F}\n=f_*cc_X{\\cal F}.\n\\label{eqprcc}\n\\end{equation}\n\\end{lm}\n\n\\proof{\nWe consider the morphisms\n\\begin{equation}\n\\begin{CD}\n{\\mathbf P}(Y\\times_NT^*N\n\\oplus {\\mathbf A}^1_Y)\n@<\\bar f<<\n{\\mathbf P}(X\\times_NT^*N\n\\oplus {\\mathbf A}^1_X)\n@>\\bar h>>\n{\\mathbf P}(X\\times_MT^*M\n\\oplus {\\mathbf A}^1_X)\n\\end{CD}\n\\label{eqfhb}\n\\end{equation}\nextending (\\ref{eqfh})\nand let $\\bar f_!\\colon\n{\\rm CH}_m({\\mathbf P}(X\\times_MT^*M\n\\oplus {\\mathbf A}^1_X))\n\\to\n{\\rm CH}_n({\\mathbf P}(Y\\times_NT^*N\n\\oplus {\\mathbf A}^1_Y))$\n denote the composition \n$\\bar f_*\\bar h^!$.\nBy Lemma \\ref{lmincl}.2,\nthe diagram\\begin{equation}\n\\begin{CD}\n{\\rm CH}_n({\\mathbf P}(X\\times_MT^*M\n\\oplus {\\mathbf A}^1_X))\n@>>>\n{\\rm CH}_\\bullet(X)\n\\\\\n@V{\\bar f_!}VV@VV{f_*}V\\\\\n{\\rm CH}_m({\\mathbf P}(Y\\times_NT^*N\n\\oplus {\\mathbf A}^1_Y))\n@>>>\n{\\rm CH}_\\bullet(Y)\n\\end{CD}\n\\label{eqCCpr}\n\\end{equation}\nis commutative.\n\nLet $A=CC{\\cal F}$.\nBy the assumption\n$\\dim f_\\circ C\\leqq n$,\na cycle $\\overline{f_!A}$ is \ndefined as that\nsupported on the closure of\n$f_\\circ C$ in\n${\\mathbf P}(Y\\times_NT^*N\n\\oplus {\\mathbf A}^1_Y)$\nby taking the closure of\nthe cycle $f_!A$.\nFurther, we have\n$\\overline{f_!A}=\n\\bar f_!\\bar A$.\nThus, by the assumption\n$CCj_*Rf_*{\\cal F}=f_!CCi_*{\\cal F}$\nand (\\ref{eqCCpr}),\nwe obtain\n$\\overline{CCj_*Rf_*{\\cal F}}\n=\\bar f_!\\overline{CCi_*{\\cal F}}\n=f_*\\overline{CCi_*{\\cal F}}$.\nHence, the assertion follows.\n\\qed}\n\\medskip\n\n\n\n\\section{Pull-back of characteristic cycle\nand the index formula}\\label{spb}\n\nWe prove that the construction\nof characteristic cycles\nis compatible with the pull-back\nby properly transversal morphisms\nin Section \\ref{sspb}.\nWe will derive from this\nan index formula for the Euler number\nat the end of Section \\ref{ssREP}.\n\nIn this section,\n$k$ denotes a field\nof characteristic $\\geqq0$.\nWe assume that\nirreducible components of\na smooth scheme over $k$\nhave the same dimension unless otherwise\nstated.\nWe also assume that\nirreducible components of\na closed conical subset \nof the cotangent of\na smooth scheme over $k$\nhave the same dimension\nas the base scheme.\n\n\\subsection{Pull-back of characteristic cycle}\\label{sspb}\n\nIn this subsection,\nwe assume that\n$X$ and $W$ are\nsmooth schemes over a field $k$.\nWe assume that\nevery irreducible component\nof $X$ (resp.\\ of $W$) is of dimension \n$n$ (resp.\\ $m$) and that\nevery irreducible component of\na closed conical subset\n$C\\subset T^*X$ is of dimension $n$.\nWe assume that\na constructible complex\n${\\cal F}$ of $\\Lambda$-modules\non $X$ is of finite tor-dimension.\n\n\n\\begin{df}\\label{dfCt}\nLet $X$ and $W$ be \nsmooth schemes over\na field $k$ and let\n$C\\subset T^*X$ be a closed conical\nsubset.\nAssume that every irreducible\ncomponent of $X$ and\nevery irreducible\ncomponent of $C$\nare of dimension $n$\nand that\nevery irreducible\ncomponent of $W$ is of dimension $m$.\n\n{\\rm 1.}\nWe say that a $C$-transversal\nmorphism $h\\colon W\\to X$ over $k$\nis {\\em properly $C$-transversal}\nif every irreducible component of\n$h^*C=W\\times_XC$\nis of dimension $m$.\n\n{\\rm 2.}\nLet $h\\colon W\\to X$ be \na properly $C$-transversal morphism\nand let \n\\begin{equation}\n\\begin{CD}\nT^*W@<<< W\\times_XT^*X\n@>>> T^*X\n\\end{CD}\n\\label{eqh!A}\n\\end{equation}\nbe the canonical morphisms.\nThen, for a linear combination\n$A=\\sum_am_a[C_a]$\nof irreducible components of\n$C=\\bigcup_aC_a$,\nwe define $h^!A$ to be \n$(-1)^{n-m}$-{\\em times} the push-forward\nby the first arrow\n$W\\times_XT^*X\\to T^*W$\nin {\\rm (\\ref{eqh!A})}\nof the pull-back of $A$ by the second arrow\n$W\\times_XT^*X\\to T^{*}X$\nin the sense of intersection theory.\n\\end{df}\n\n\\begin{lm}\n\\label{lmdimP}\nLet $h\\colon W\\to X$ be a morphism\nof smooth schemes over $k$\nand $C\\subset T^*X$ be a closed\nconical subset.\nAssume that every irreducible component\nof $X$ is of dimension $n$ and\nthat every irreducible component\nof $W$ is of dimension $m=n-c$.\nLet $\\dim_{h(W)}C$ denote\nthe minimum of $\\dim C\\cap T^*U$\nwhere $U$ runs through\nopen neighborhoods of the image $h(W)$.\n\n{\\rm 1.}\nFor $h^*C=W\\times_XC$,\nwe have\n$\\dim h^*C\\geqq \\dim C_{h(W)}-c$.\n\n{\\rm 2.}\nLet $g\\colon V\\to W$ be a morphism\nof smooth schemes over $k$.\nAssume that every irreducible component\nof $C$ is of dimension $n$ and\nthat every irreducible component\nof $V$ is of dimension $l=m-c'$.\nThen, \nthe following conditions are equivalent:\n\n{\\rm (1)}\n$h$ is properly $C$-transversal\non a neighborhood of\n$g(V)\\subset W$ and \n$g\\colon V\\to W$ is properly \n$h^{\\circ}C$-transversal.\n\n{\\rm (2)}\nThe composition $h\\circ g\\colon V\\to X$\nis properly $C$-transversal.\n\\end{lm}\n\n\\proof{\n1.\nIf $h$ is smooth, we have\n$\\dim h^*C= \\dim_{h(W)} C-c$.\nHence, it suffices to consider the\ncase where $h$ is a regular immersion\nof codimension $c$.\nThen it follows from\n\\cite[Chap.\\ 0 Proposition (16.3.1)]{EGA4}\n\n2.\nBy Lemma \\ref{lmCh}.3,\nit suffices to verify the conditions\non the dimension.\nThe implication (1)$\\Rightarrow$(2)\nis clear. We show\n(2)$\\Rightarrow$(1).\nBy 1, we have\n$\\dim g^*(h^*C)\\geqq \\dim_{g(V)} h^*C-c'\n\\geqq n-c-c'$.\nHence (2) implies\nthe equalities\n$\\dim_{g(V)} h^*C= n-c$ and\n$\\dim g^*(h^*C)=\\dim_{g(V)} h^*C-c'$.\nThus we have\n(2)$\\Rightarrow$(1).\n\\qed}\n\\medskip\n\nIf $h\\colon W\\to X$ is \nproperly $C$-transversal,\nthen every irreducible component of\n$h^{\\circ}C$\nis of dimension $\\dim W$\nby Lemma \\ref{lmnc}.\nA smooth morphism\n$h\\colon W\\to X$ is properly $C$-transversal\nby Lemma \\ref{lmCh}.1.\n\n\\begin{ex}\n{\\rm (\\cite[Example 2.18]{nonlog})}\nAssume that $k$ is a perfect\nfield of characteristic $p>2$.\nLet $X={\\mathbf A}^2\n={\\rm Spec}\\ k[x,y]\n\\supset\nU={\\mathbf G}_m\\times\n{\\mathbf A}^1\n={\\rm Spec}\\ k[x^{\\pm1},y]$.\nLet ${\\cal G}$ be a locally free sheaf\nof rank $1$ on $U$\ndefined by the Artin-Schreier equation\n$t^p-t=y\/x^p$.\nThen, the singular support\n$C=SSj_!{\\cal G}$\nfor the open immersion $j\\colon\nU\\to X$ equals the union\nof the $0$-section\n$T^*_XX$ with the line bundle\n$\\langle dy\\rangle_D$\non the $y$-axis $D=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} U$\nspanned by the section $dy$.\nHence, the immersion\n$D\\to X$ is $C$-transversal\nbut is {\\em not} properly\n$C$-transversal.\n\\end{ex}\n\n\n\n\n\nIn the rest of this section,\nwe assume that $k$ is perfect.\n\n\n\\begin{pr}[{Beilinson}]\\label{prh!}\nLet ${\\mathbf P}={\\mathbf P}^n$\nbe a projective space \nand let ${\\mathbf P}^\\vee$\nbe the dual projective space.\nLet ${\\cal G}$ be\na constructible complex of\n$\\Lambda$-modules\non ${\\mathbf P}^\\vee$\nand let ${\\cal F}$ denote\nthe naive inverse Radon transform\n$R{\\bm p}_*{\\bm p}^{\\vee *}{\\cal G}$.\nLet $C^\\vee\n\\subset T^*{\\mathbf P}^\\vee$\nbe a closed conical subset \nsuch that every irreducible\ncomponent is of dimension $n$\nand let\n$C={\\bm p}_{\\circ}\n{\\bm p}^{\\vee \\circ}C^\\vee\n\\subset T^*{\\mathbf P}$.\nAssume that ${\\cal G}$ is\nmicro-supported on $C^\\vee\\subset \nT^*{\\mathbf P}^\\vee$.\n\n{\\rm 1.}\nWe have\n\\begin{equation}\n{\\mathbf P}(CC {\\cal F})=\n{\\mathbf P}({\\bm p}_!CC {\\bm p}^{\\vee*}{\\cal G})\n=\n{\\mathbf P}({\\bm p}_!{\\bm p}^{\\vee!}CC {\\cal G}).\n\\label{eqp!}\n\\end{equation}\n\n{\\rm 2.}\nLet $X$ be a smooth subscheme\nof ${\\mathbf P}$\nand assume that the immersion\n$h\\colon X\\to {\\mathbf P}$\nis {\\em properly $C$-transversal}.\nThen, we have\n\\begin{equation}\nCC h^*{\\cal F}=\nh^!CC{\\cal F}.\n\\label{eqh!}\n\\end{equation}\n\\end{pr}\n\n\\proof{\nFirst, we prove\n\\begin{equation}\n{\\mathbf P}(CC Rp_*p^{\\vee *}{\\cal G})=\n{\\mathbf P}(p_!CC p^{\\vee*}{\\cal G})\n\\label{eqph!}\n\\end{equation}\nfor properly $C$-transversal\nimmersion\n$h\\colon X\\to {\\mathbf P}$.\nWe may assume that $k$ is algebraically closed.\nBoth $CC Rp_*p^{\\vee *}{\\cal G}$\nand $p_!CC p^{\\vee*}{\\cal G}$\nare supported on \n$p_{\\circ}p^{\\vee \\circ}C^\\vee\n=h^{\\circ}C$ by Corollary \\ref{corhCt}.2.\nBy Lemma \\ref{lmh!}\nand the assumption that\n$h$ is properly $C$-transversal,\nevery irreducible component\nof $h^\\circ C$ is of dimension $X$.\nHence it suffices to show the equality\n\\begin{equation}\n(CC Rp_*p^{\\vee *}{\\cal G},df)_u\n=(p_!CC p^{\\vee*}{\\cal G},df)_u\n\\label{eqMilR}\n\\end{equation}\nfor smooth\nmorphisms $f\\colon U\\to {\\mathbf A}^1$\ndefined on open subschemes $U\\subset X$\nwith at most an isolated $C$-characteristic point $u$.\n\nBy the Milnor formula,\nthe left hand side of (\\ref{eqMilR})\nequals\n$-\\dim{\\rm tot}\\\n\\phi_u(Rp_*p^{\\vee *}{\\cal G},f)$.\nBy Corollary \\ref{corfp},\nthere exist at most finitely many\nisolated $p^{\\vee \\circ}C$-characteristic points of\n$f p\\colon U\\times_{\\mathbf P}Q\n\\to {\\mathbf A}^1$.\nHence, \nthe right hand side of (\\ref{eqMilR})\nequals\n$\\sum_v\n(CC p^{\\vee*}{\\cal G},d(f p))_v$\nwhere $v$ runs through\nisolated $p^{\\vee \\circ}C$-characteristic points of\n$fp$.\nFurther by the Milnor formula,\nthis equals\n$-\\sum_v\\dim{\\rm tot}\\\n\\phi_v(p^{\\vee *}{\\cal G},fp)$.\nThus, the equality (\\ref{eqMilR})\nfollows from the isomorphism\n$$\\phi_u(Rp_*p^{\\vee *}{\\cal G},f)\n\\to\nR\\Gamma(Q\\times_Xu,\n\\phi(p^{\\vee *}{\\cal G},fp))\n\\to\n\\bigoplus_v\n\\phi_v(p^{\\vee *}{\\cal G},fp).$$\n\n1.\nFor the first equality,\nit suffices to take\n$X={\\mathbf P}$ in (\\ref{eqph!}).\nThe second follows from\nProposition \\ref{prsm*}.\n\n2.\nBy proper base change theorem,\nwe have an isomorphism\n$h^*{\\cal F}\n\\to Rp_*p^{\\vee *}{\\cal G}$.\nHence by (\\ref{eqp!})\nand (\\ref{eqph!}),\nwe have\n${\\mathbf P}(CCh^*{\\cal F})\n={\\mathbf P}(h^!CC{\\cal F})$.\nBy the assumption\nthat the immersion\n$h$ is properly $C$-transversal,\nthe coefficients\nof the $0$-section\n$T^*_XX$ in $CCh^*{\\cal F}$\nand in $h^!CC{\\cal F}$\nare both equal to $(-1)^{\\dim X}\n{\\rm rank}\\ {\\cal F}$.\nThus the assertion follows.\n\\qed}\n\\medskip\n\n\nFor a linear combination\n$A=\\sum_am_aC_a$\nof irreducible closed  conical subsets\nof dimension $n$,\nwe define the Legendre transform\n$LA=(-1)^{n-1}{\\bm p}_!{\\bm p}^!A$.\nThis is also a linear combination\nof irreducible closed  conical subsets\nof dimension $n$\nby Lemma \\ref{lmh!}.\nSince the definition of ${\\bm p}^!A$\ninvolves the sign $(-1)^{n-1}$,\nthat of the Legendre transform\ndoes not involve sign and\nwe have\n${\\mathbf P}(L(A))=\n\\sum_{a;C_a\\not\\subset \nT^*_{\\mathbf P}{\\mathbf P}}\nm_a{\\mathbf P}(C_a^\\vee)$.\n\n\n\n\\begin{cor}[Beilinson]\\label{corCCR}\nLet ${\\cal F}$\nbe a constructible complex\nof $\\Lambda$-modules on ${\\mathbf P}$.\nThen, for the Radon transform\n$R{\\cal F}$, we have\n\\begin{equation}\n{\\mathbf P}(CCR{\\cal F})=\n{\\mathbf P}(LCC{\\cal F}).\n\\label{eqCCR}\n\\end{equation}\n\\end{cor}\n\n\nWe will remove ${\\mathbf P}$\nin (\\ref{eqCCR})\nin Corollary \\ref{corLR}.\n\n\n\\proof{\nBy Proposition \\ref{prh!}.1,\nwe have \n$${\\mathbf P}(CCR{\\mathcal F})\n=\n{\\mathbf P}((-1)^{n-1}{\\bm p}^\\vee_!\nCC{\\bm p}^*{\\mathcal F})\n=\n{\\mathbf P}((-1)^{n-1}{\\bm p}^\\vee_!\n{\\bm p}^!\nCC{\\mathcal F})\n=\n{\\mathbf P}\n(LCC{\\mathcal F}).$$\n\\qed}\n\n\n\\begin{thm}\\label{thmi*}\nLet $X$ and $W$ be \nsmooth schemes over a perfect field $k$ and let\n$C\\subset T^*X$ be a closed conical\nsubset.\nAssume that every irreducible\ncomponent of $X$ and\nevery irreducible\ncomponent of $C$\nare of dimension $n$\nand that\nevery irreducible\ncomponent of $W$ is of dimension $m$.\n\nLet ${\\cal F}$ be\na constructible complex of\n$\\Lambda$-modules on $X$\nof finite tor-dimension\nmicro-supported on\n$C\\subset T^*X$\nand\nlet $h\\colon W\\to X$ be a \nproperly $C$-transversal morphism.\nThen, we have \n\\begin{equation}\nCC h^*{\\cal F}=\nh^!CC {\\cal F}.\n\\label{eqii}\n\\end{equation}\n\\end{thm}\n\n\\proof[Proof {\\rm (Beilinson)}]{\nLet $h\\colon W\\to X$\nbe a properly $C$-transversal morphism.\nSince $h$ is decomposed\nas the composition\nof the graph $W\\to W\\times X$\nand the projection $W\\times X\\to X$\nand since it is proved for\nsmooth morphisms in Proposition \\ref{prsm*},\nit is sufficient to show\nthe case where $h$ is an immersion.\n\n\nFirst, we show the case\nwhere $X$ is the projective space \n${\\mathbf P}={\\mathbf P}^n$.\nThe case where ${\\cal F}=R{\\bm p}_*\n{\\bm p}^{\\vee*}{\\cal G}$\nis the naive inverse Radon transform\nhas been proved in Proposition\n\\ref{prh!}.2.\nLet ${\\cal F}$ be a constructible\ncomplex on ${\\mathbf P}^n$.\nSince ${\\cal F}$ is isomorphic\nto $R^\\vee R{\\cal F}$ upto\nconstant sheaf\nand the assertion is clear for\nthe constant sheaf,\nit follows in the case\n$h$ is an immersion  \nto ${\\mathbf P}$.\n\nWe show the general case.\nSince the assertion is local,\nwe may assume that $X$ is\naffine and take an immersion\n$i\\colon X\\to {\\mathbf P}$.\nFurther, we may assume that\nthere is a smooth subscheme\n$V\\subset {\\mathbf P}$\nsuch that $X\\cap V=W$\nand that the intersection\nis transversal.\nThen, the immersion $\\tilde h\\colon\nV\\to {\\mathbf P}$\nis $i_{\\circ}C$-transversal\non a neighborhood of $W$.\nHence, it follows from the case\nwhere $h$ is an immersion  \nto ${\\mathbf P}$.\n\\qed}\n\\smallskip\n\n\nWe study the compatibility\nof characteristic classes with pull-back.\nLet $F\\to E$ be\nan injection of vector bundles\nover a scheme $W$ of finite type over $k$\nand let $p\\colon E\\to E\/F$\nbe the canonical surjection\nLet $C\\subset E$\nbe a closed conical subset\nsuch that\nthe intersection\n$C\\cap F$\nis a subset of the $0$-section.\nThen, for a linear combination\n$A=\\sum_am_aC_a$ of irreducible\ncomponents of $C=\\bigcup_aC_a$,\nthe intersection theory defines\na cycle $p^!p_*A$ supported\non $p^{-1}(p(C))=C+F\n\\subset E$.\n\n\\begin{df}\\label{dfprim}\nLet $X$ be an embeddable scheme of\nfinite type over $k$\nand let $i\\colon X\\to M$\nbe a closed immersion to\na smooth scheme over $k$.\nLet $C\\subset X\\times_MT^*M$\nbe a closed conical subset.\nLet $h\\colon W\\to X$\nbe a regular immersion\nof codimension $c$.\n\n{\\rm 1.}\nWe say that $h$ is \n{\\em properly $C$-transversal}\nif the following conditions are\nsatisfied:\nThe canonical morphism\n$T^*_WX\\to W\\times_MT^*M$\nis an injection of vector bundles on $W$.\nFor every irreducible\ncomponents $C_a$ of $C=\\bigcup_aC_a$,\nthe pull-back $W\\times_XC_a\\subset C_a$\nis of codimension $c$.\nThe intersection \n$h^*C\\cap T^*_WX$\nfor $h^*C=W\\times_XC\n\\subset W\\times_MT^*M$\nis a subset of the $0$-section.\n\n\n{\\rm 2.}\nAssume that $h\\colon W\\to X$\nis properly $C$-transversal.\nWe regard the pull-back\n$h^*C=\nW\\times_XC$ as a subset\nof $W\\times_MT^*M$\nand the conormal bundle\n$T^*_WX$ as a sub vector bundle\nof $W\\times_MT^*M$.\nThen, we define a closed conical\nsubset \n$h^!C\\subset W\\times_MT^*M$\nto be the sum\n$h^*C+T^*_WX$.\n\nFor a linear combination\n$A=\\sum_am_aC_a$ of\nirreducible components of\n$C=\\bigcup_aC_a$,\nlet $h^*C_a$ denote the\npull-back of $C_a$ by\nthe regular immersion\n$W\\times_MT^*M\\to \nX\\times_MT^*M$ in the sense of\nintersection theory\nand let $p\\colon W\\times_MT^*M\n\\to (W\\times_MT^*M)\/T^*_WX$\nbe the canonical surjection.\nThen,\nwe define\n$h^!A=(-1)^c\\sum_am_a\np^!p_*h^*C_a$.\n\\end{df}\n\n\n\n\\begin{pr}\\label{prprim}\nLet $X$ be an embeddable scheme of\nfinite type over $k$\nand let $i\\colon X\\to M$\nbe a closed immersion to\na smooth scheme over $k$.\nLet ${\\cal F}$\nbe a constructible complex of\n$\\Lambda$-modules on $X$ \nand $C=SS{\\cal F}\n\\subset X\\times_MT^*M$ \nbe the singular support of\n${\\cal F}$.\n\nLet $h\\colon W\\to X$\nbe a {\\em properly\n$C$-transversal} closed regular immersion\nof codimension $c$.\nThen, we have\n\\begin{equation}\nCC(i\\circ h)_*h^*{\\cal F}\n=h^!CCi_*{\\cal F},\n\\label{eqprim}\n\\end{equation}\n\\begin{equation}\ncc_Wh^*{\\cal F}\n=\n(-1)^c\nc(T^*_WX)^{-1}\\cap h^!cc_X{\\cal F}.\n\\label{eqprim2}\n\\end{equation}\n\\end{pr}\n\n\\proof{\nWe show (\\ref{eqprim}).\nSince the assertion is local on\n$W$, we may assume that\nthere exists a cartesian diagram\n$$\\begin{CD}\nW@>h>> X\\\\\n@VVV@VViV\\\\\nN@>j>>M\n\\end{CD}$$\nwhere the lower\nhorizontal arrow\n$j\\colon N\\to M$ \nis a regular immersion\nof codimension $c$\nof smooth schemes over $k$.\nFurther, we may assume that\n the immersion $j\\colon N\\to M$ \nis properly $C$-transversal.\nThen, by Theorem \\ref{thmi*},\nwe have $CCj^*i_*{\\cal F}\n=j^!CCi_*{\\cal F}$.\nSince\n$(i\\circ h)_*h^*{\\cal F}\n=\nj_*j^*i_*{\\cal F}$, \nwe obtain\n$CC(i\\circ h)_*h^*{\\cal F}\n=\nCCj_*j^*i_*{\\cal F}\n=\nj_*CCj^*i_*{\\cal F}\n=\nj_*j^!CCi_*{\\cal F}\n=\nh^!CCi_*{\\cal F}$\nby Lemma \\ref{lmRf}.2.\n\n\nThe equality (\\ref{eqprim2})\nfollows from (\\ref{eqprim}) and\nLemma \\ref{lmtheta}\nsince the definition of $h^!CCi^*{\\cal F}$\ninvolves the sign $(-1)^c$.\n\\qed}\n\n\\begin{cor}\\label{corcnsm}\nLet $f\\colon X\\to Y$\nbe a smooth morphism of \nrelative dimension $d$ of\nembeddable schemes \nof finite type over $k$\nand let $T^*X_{\/Y}$ \nand $c(T^*X_{\/Y})$ denote\nthe relative cotangent bundle\nand its total Chern class.\nThen,\nthe diagram\n\\begin{equation}\n\\begin{CD}\nK(Y,\\Lambda)\n@>{cc_Y}>>\n{\\rm CH}_\\bullet(Y)\\\\\n@V{f^*}VV@VV{(-1)^dc(T^*X_{\/Y})\\cap f^!}V\\\\\nK(X,\\Lambda)\n@>{cc_X}>>\n{\\rm CH}_\\bullet(X)\n\\end{CD}\n\\label{eqCHhf}\n\\end{equation}\nis commutative.\nIn particular, for $Y={\\rm Spec}\\ k$\nand for a geometrically constant\nsheaf ${\\cal F}$ on $X$ of dimension $n$,\nwe have\n\\begin{equation}\ncc_X{\\cal F}\n=\n(-1)^n\n{\\rm rank}\\ {\\cal F}\\cdot\nc(T^*X).\n\\label{eqY=k}\n\\end{equation}\n\\end{cor}\n\n\\proof{\nLet $i\\colon X\\to M$\nand $j\\colon Y\\to N$\nbe closed immersions\nto smooth schemes over $k$.\nAfter replacing $i\\colon X\\to M$\nby $(i,fj)\\colon X\\to M\\times N$,\nwe obtain a commutative diagram\n\\begin{equation}\n\\xymatrix{\nX\\ar[rdd]_f\\ar[rd]^{\\!\\!\\! h}\\ar[rrd]^i&&\\\\\n&Y\n\\ar[r]_{ j'}\\ar[d]^{g'}&M\\ar[d]^g\\\\\n&Y\\ar[r]^j&N\n}\n\\label{eqXYMNg}\n\\end{equation}\nwhere $g\\colon M\\to N$ is smooth\nand the square is cartesian.\nSince $g$ is smooth\nand $j'_*g^{\\prime *}{\\cal F}=\ng^*j_*{\\cal F}$,\nwe have\n$CCj'_*g^{\\prime *}{\\cal F}=\ng^!CCj_*{\\cal F}$\nby Theorem \\ref{thmi*}.\nLet $C=SSj_*{\\cal F}\n\\subset Y\\times_NT^*N$\nbe the singular support\nand let $C'=g^oC\n\\subset Y'\\times_MT^*M$.\nThen, since $g\\colon M\\to N$ is smooth,\nthe closed immersion\n$h\\colon X\\to Y'$\nis properly $C'$-transversal.\nHence by Proposition \\ref{prprim},\nwe have\n$CCi_*h^*g^{\\prime *}{\\cal F}=\nh^!CCj'_*g^{\\prime *}{\\cal F}=\nh^!g^!CCj_*{\\cal F}$.\nNamely, we have\n$CCi_*f^*{\\cal F}=\nf^!CCj_*{\\cal F}$.\nSince the definition of\nthe right hand side\ninvolves the sign $(-1)^d$,\nwe obtain\n\\begin{equation}\ncc_Xf^*{\\cal F}=\nc(T^*X_{\/Y})\\cap\n(-1)^df^!cc_Y{\\cal F}\n\\label{eqccTXY}\n\\end{equation}\nby Lemma \\ref{lmincl}.2.\nThus the assertion follows.\n\\qed}\n\n\n\n\\subsection{Radon transform and \nthe index formula}\\label{ssREP}\n\n\n\nAssume $X={\\mathbf P}^n$.\nWe identify \n$Q={\\mathbf P}(T^*{\\mathbf P})$\nand let\n${\\bm p}\\colon Q\\to {\\mathbf P}$ and\n${\\bm p}^\\vee\\colon Q\\to {\\mathbf P}^\\vee$\nbe the projections.\nThe Radon transforms\n$R=R{\\bm p}^\\vee_!\n{\\bm p}^*[n-1]$ and \n$R^\\vee=R{\\bm p}_!\n{\\bm p}^{\\vee*}[n-1](n-1)$\ndefine morphisms\n\\begin{equation}\nR\\colon\nK({\\mathbf P},\\Lambda)\n\\to\nK({\\mathbf P}^\\vee,\\Lambda),\n\\quad\nR^\\vee\\colon\nK({\\mathbf P}^\\vee,\\Lambda)\n\\to\nK({\\mathbf P},\\Lambda).\n\\label{dfKR}\n\\end{equation}\nDefine also morphisms $\\chi\\colon \nK({\\mathbf P},\\Lambda)\n\\to {\\mathbf Z}$\nand $\\chi\\colon \nK({\\mathbf P},\\Lambda)\n\\to {\\mathbf Z}$\nby $\\chi{\\cal F}=\n\\chi({\\mathbf P}_{\\bar k},{\\cal F})$\nand\nby $\\chi{\\cal G}=\n\\chi({\\mathbf P}^\\vee_{\\bar k},{\\cal G})$.\n\n\\begin{lm}\\label{lmPn}\nLet $n\\geqq 1$ be an integer\nand ${\\mathbf P}={\\mathbf P}^n$.\n\n{\\rm 1.}\nFor $a=0,\\ldots,n$,\nlet ${\\mathbf P}^a\n\\subset {\\mathbf P}\n={\\mathbf P}^n$\ndenote a linear subspace\nof dimension $a$.\nThen, the images\n$cc_{{\\mathbf P}^n}(\n\\Lambda_{{\\mathbf P}^a}[a])$\nfor $a=0,\\ldots,n$\ndo not depend on\nthe choice of linear subspaces\nand \nform a basis of\na free ${\\mathbf Z}$-module\n${\\rm CH}_\\bullet ({\\mathbf P})$.\n\n\n\n{\\rm 2.}\nThe diagram\n\\begin{equation}\n\\begin{CD}\nK({\\mathbf P},\\Lambda)\n@>{\\chi}>>\n{\\mathbf Z}\n\\\\\n@VRVV@VV{(-1)^{n-1}n\\times}V\\\\\nK({\\mathbf P}^\\vee,\\Lambda)\n@>{\\chi}>>\n{\\mathbf Z}\n\\end{CD}\n\\label{eqRchi}\n\\end{equation}\nand that with $R$ replaced\nby $R^\\vee$ and with\n${\\mathbf P}$ and\n${\\mathbf P}^\\vee$ switched\nare commutative.\n\n\n{\\rm 3.}\nAssume $k$ is algebraically closed.\nThe composition $R^\\vee R\\colon\nK({\\mathbf P},\\Lambda)\\to\nK({\\mathbf P},\\Lambda)$\nmaps ${\\cal F}$ to\n${\\cal F}+(n-1)\\chi{\\cal F}\\Lambda$.\nIf $n\\neq 1$\nand if $f\\colon\nK({\\mathbf P},\\Lambda)\n\\to {\\mathbf Z}$\nis a morphism satisfying\n$fR^\\vee R=n^2f$,\nthen we have\n$f=f([\\Lambda_{{\\mathbf P}^0}])\\cdot\n\\chi$.\n\\end{lm}\n\n\\proof{1.\nIt follows from\n$CC\n\\Lambda_{{\\mathbf P}^a}[a]=\nT^*_{{\\mathbf P}^a}{\\mathbf P}^n$.\n\n2.\nFor constructible complexes ${\\cal F}$\non ${\\mathbf P}$,\nwe have\n$\\chi R{\\cal F}\n=\n(-1)^{n-1}\\chi(Q_{\\bar k},{\\bm p}^*{\\cal F})\n=\n(-1)^{n-1}\\chi \nR{\\bm p}_*{\\bm p}^*{\\cal F}\n=\n(-1)^{n-1}\\chi\nR{\\bm p}_*{\\bm p}^*{\\Lambda}\\otimes{\\cal F}$\nby the projection formula.\nHence the assertion follows from\n$R^q{\\bm p}_*{\\bm p}^*{\\Lambda}\n=\\Lambda(-q\/2)$\nfor $0\\leqq q\\leqq 2(n-1)$ even\nand $=0$ for otherwise.\n\n3.\nSince $R^\\vee R{\\cal F}$\nis isomorphic to ${\\cal F}$\nup to geometrically constant sheaves,\nwe have\n$R^\\vee R{\\cal F}-{\\cal F}\n={\\rm rank}^\\circ\n(R^\\vee R{\\cal F}-{\\cal F})\\cdot\n\\Lambda$.\nBy taking $\\chi$ and applying 2,\nwe obtain\n$(n^2-1)$\n$\\chi{\\cal F}\n=\n{\\rm rank}^\\circ\n(R^\\vee R{\\cal F}-{\\cal F})\\cdot\n\\chi\\Lambda$.\nSince\n$\\chi\\Lambda=n+1$,\nwe obtain\n${\\rm rank}^\\circ\n(R^\\vee R{\\cal F}-{\\cal F})$\n$=\n(n-1)\\chi{\\cal F}$\nand the first assertion follows.\n\nIf $f$ satisfies the condition,\nsimilarly we obtain\n$(n^2-1)\nf({\\cal F})\n=\n(n-1)\\chi{\\cal F}\\cdot\nf(\\Lambda)$.\nThus, $f$ is a constant multiple of\n$\\chi$.\nSince $\\chi\\Lambda_{{\\mathbf P}^0}=1$,\nthe constant is given by\n$f(\\Lambda_{{\\mathbf P}^0})$.\n\\qed}\n\\smallskip\n\nWe define the Legendre transform\n\\begin{equation}\nL\\colon\n{\\rm CH}_\\bullet ({\\mathbf P})\n\\to\n{\\rm CH}_\\bullet ({\\mathbf P}^\\vee)\n\\label{eqbarL}\n\\end{equation}\nby $L={\\bm p}^\\vee_*{\\bm p}^*$\nand $L^\\vee={\\bm p}_*{\\bm p}^{\\vee*}$\nfor the projections\n${\\bm p}\\colon Q\\to {\\mathbf P}$ and\n${\\bm p}^\\vee\\colon Q\\to {\\mathbf P}^\\vee$.\n\n\\begin{pr}[Beilinson]\\label{prRR}\nLet $n\\geqq 1$ be an integer\nand ${\\mathbf P}={\\mathbf P}^n$.\n\n{\\rm 1.}\nThe diagram\n\\begin{equation}\n\\begin{CD}\nK({\\mathbf P},\\Lambda)\n@>{cc_{\\mathbf P}}>>\n{\\rm CH}_\\bullet ({\\mathbf P})\n\\\\\n@VRVV@VVLV\\\\\nK({\\mathbf P}^\\vee,\\Lambda)\n@>{cc_{{\\mathbf P}^\\vee}}>>\n{\\rm CH}_\\bullet ({\\mathbf P}^\\vee)\n\\end{CD}\n\\label{eqR}\n\\end{equation}\nand that with $R$ and $L$ replaced\nby $R^\\vee$ and $L^\\vee$ and with\n${\\mathbf P}$ and\n${\\mathbf P}^\\vee$ switched\nare commutative.\n\n{\\rm 2.}\nThe diagram\n\\begin{equation}\n\\xymatrix{\nK({\\mathbf P},\\Lambda)\\ar[rrd]_{\\chi}\n\\ar[rr]^{\\!\\!\\!\\!\\!\\!\\!\ncc_{\\mathbf P}}&&\n{\\rm CH}_\\bullet ({\\mathbf P})\n\\ar[d]^{\\deg}\\\\\n&&{\\mathbf Z}}\n\\label{eqchP}\n\\end{equation}\nis commutative.\n\\end{pr}\n\n\\proof{\nWe may assume that $k$ is algebraically closed.\nWe prove the assertions by \ninduction on $n$.\nIf $n=1$, the projections\n${\\bm p}\\colon Q\\to {\\mathbf P}$\nand\n${\\bm p}^\\vee\\colon Q\\to {\\mathbf P}^\\vee$\nare isomorphisms\nand the assertion 1 is obvious.\nSince $\\deg cc_{\\mathbf P}{\\cal F}\n=\n(CC{\\cal F},T^*_{\\mathbf P}{\\mathbf P})_\n{T^*{\\mathbf P}}$,\nthe assertion 2 for $n=1$ is nothing but\nthe Grothendieck-Ogg-Shafarevich\nformula for ${\\mathbf P}={\\mathbf P}^1$.\n\nWe show that the assertion 2 for $n-1\\geqq 1$\nimplies the assertion 1 for $n$.\nWe show the commutativity of\nthe diagram (\\ref{eqR})\nby using the direct sum decomposition\n\\begin{equation}\n\\begin{CD}\n{\\rm CH}_\\bullet({\\mathbf P}^\\vee)\n=\n{\\rm CH}_n({\\mathbf P}(T^*{\\mathbf P}^\\vee\n\\oplus {\\mathbf A}^1_{{\\mathbf P}^\\vee}))\n@>>>\n{\\rm CH}_{n-1}({\\mathbf P}(T^*{\\mathbf P}^\\vee))\n\\oplus\n{\\rm CH}_n({\\mathbf P}^\\vee)\\\\\n@.=\n{\\rm CH}_{n-1}(Q)\n\\oplus {\\mathbf Z}\n\\end{CD}\n\\label{eqCHQZ}\n\\end{equation}\n(\\ref{eqQZ}).\nThe compositions with the\nfirst projection $\n{\\rm CH}_\\bullet({\\mathbf P}^\\vee)\n\\to {\\rm CH}_{n-1}(Q)$\nare equal by Corollary \\ref{corCCR}.\nWe show that the compositions with the\nsecond projection ${\\rm pr}_2\\colon\n{\\rm CH}_\\bullet({\\mathbf P}^\\vee)\n\\to {\\mathbf Z}$\ninduced by the projection\n${\\mathbf P}(T^*{\\mathbf P}^\\vee\n\\oplus {\\mathbf A}^1_{{\\mathbf P}^\\vee})\n\\to \n{\\mathbf P}^\\vee$\nare the same.\n\nLet ${\\cal F}$ be a constructible\ncomplex of $\\Lambda$-modules\non ${\\mathbf P}$\nand $C=SS{\\cal F}$ \nbe the singular support of ${\\cal F}$.\nLet $H\\subset {\\mathbf P}$\nbe a hyperplane\nsuch that the immersion\n$h\\colon H\\to {\\mathbf P}$\nis properly $C$-transversal\nand let $i\\colon {\\rm Spec}\\ k\n\\to {\\mathbf P}^\\vee$ \nbe the immersion of the $k$-rational\npoint of ${\\mathbf P}^\\vee$ \ncorresponding to $H$.\n\nBy the hypothesis of induction,\nwe have\n$\\deg cc_Hh^*{\\cal F}=\n\\chi h^*{\\cal F}$.\nBy Proposition \\ref{prprim},\nwe have\n$cc_Hh^*{\\cal F}=\n-h^!cc_{\\mathbf P}{\\cal F}$.\nHence by the commutative diagram\n\\begin{equation}\n\\begin{CD}\n{\\rm CH}_\\bullet ({\\mathbf P})\n@>{h^!}>>\n{\\rm CH}_\\bullet (H)\n\\\\\n@VLVV@VV{\\deg}V\\\\\n{\\rm CH}_\\bullet ({\\mathbf P}^\\vee)\n@>{i^!}>>{\\mathbf Z}\n\\end{CD}\n\\label{eqH}\n\\end{equation}\nand by the last assertion in\nLemma \\ref{lmCHP}.2,\nwe obtain\n${\\rm pr}_2L\ncc_{\\mathbf P}{\\cal F}\n=i^!L\ncc_{\\mathbf P}{\\cal F}\n=-\\deg cc_Hh^*{\\cal F}\n=-\\chi h^*{\\cal F}$.\nWe also have\n${\\rm pr}_2\ncc_{{\\mathbf P}^\\vee} R{\\cal F}\n=\n(-1)^n\n{\\rm rank}^\\circ R{\\cal F}\n=\n-\\chi h^*{\\cal F}$\nby Lemma \\ref{lmccd}\nin the notation ${\\rm rank}^\\circ$\ndefined there.\nHence the assertion 1 follows.\n\nWe show that the assertion 1 for $n\\geqq 2$\nimplies the assertion 2 for $n$.\nBy the commutative diagrams\n(\\ref{eqR}),\nthe endomorphism $R^\\vee R$\nof $K({\\mathbf P},\\Lambda)$\npreserves the kernel\n$K({\\mathbf P},\\Lambda)^0$  of \n$cc_{\\mathbf P}\n\\colon K({\\mathbf P},\\Lambda)\n\\to \n{\\rm CH}_\\bullet({\\mathbf P})$.\nWe show that $R^\\vee R$ acts\non $K({\\mathbf P},\\Lambda)^0$ \nas the identity.\nSince $R^\\vee R{\\cal F}$\nis isomorphic to ${\\cal F}$\nup to constant sheaf,\nwe have\n$R^\\vee R{\\cal F}-{\\cal F}\n=\n{\\rm rank}^\\circ(R^\\vee R{\\cal F}-{\\cal F})\n\\cdot \\Lambda$.\nSince \n${\\rm rank}^\\circ=(-1)^n\n{\\rm pr}_2cc_{\\mathbf P}$\nby Lemma \\ref{lmccd},\nwe have\n${\\rm rank}^\\circ {\\cal F}=0$\nfor ${\\cal F}\\in K({\\mathbf P},\\Lambda)^0$.\nFurther by the commutative diagrams\n(\\ref{eqR}),\nwe also have\n${\\rm rank}^\\circ R^\\vee R{\\cal F}=0$\nfor ${\\cal F}\\in K({\\mathbf P},\\Lambda)^0$.\nHence\n$R^\\vee R$ acts\nas the identity on\n$K({\\mathbf P},\\Lambda)^0$.\n\nOn the other hand,\nby Lemma \\ref{lmPn}.3,\nwe have\n$\\chi R^\\vee R=n^2\\chi $.\nSince $n^2>1$,\n$\\chi$ annihilates\n$K({\\mathbf P},\\Lambda)^0=\n{\\rm Ker}\\ cc_{\\mathbf P}$\nand \n$\\chi$ induces a unique morphism\n${\\rm CH}_\\bullet({\\mathbf P})\n\\to {\\mathbf Z}$ by Lemma \\ref{lmPn}.1.\n\nFor the classes of\n${\\cal F}=\\Lambda_{{\\mathbf P}^a}[a]$,\nwe have\n$\\chi {\\cal F}=(CC{\\cal F},T^*_{\\mathbf P}\n{\\mathbf P})_{T^*{\\mathbf P}}\n=\\deg cc_{\\mathbf P}{\\cal F}$.\nThus, the diagram (\\ref{eqchP}) is commutative\nby Lemma \\ref{lmPn}.1.\n\\qed}\n\n\n\\begin{cor}[Beilinson]\\label{corLR}\nLet ${\\cal F}$\nbe a constructible complex\nof $\\Lambda$-modules on ${\\mathbf P}$.\nThen, for the Radon transform\n$R{\\cal F}$, we have\n\\begin{equation}\nCCR{\\cal F}=\nLCC{\\cal F}.\n\\label{eqCCL}\n\\end{equation}\n\\end{cor}\n\n\\proof{\nExcept for the coefficient\nof the $0$-section, it is proved in\nCorollary \\ref{corCCR}.\nSince the coefficient\nof the $0$-section\nis given by \n${\\rm pr}_2\\colon\n{\\rm CH}_\\bullet({\\mathbf P}^\\vee)\n\\to {\\mathbf Z}$,\nit follows from\nProposition \\ref{prRR}.1.\n\\qed}\n\n\nCorollary \\ref{corLR} means that\n\\cite[Conjecture 2.2]{notes} holds\nfor $p^\\vee\\colon Q\\to {\\mathbf P}^\\vee$\nand $p^*{\\cal F}$\nfor ${\\cal F}$ on ${\\mathbf P}$ \nby Lemma \\ref{lmh!}.\n\n\nWe state and prove\nthe index formula\nfor the Euler-Poincar\\'e characteristic.\n\n\n\\begin{thm}\\label{thmEP}\nLet $X$ be a projective smooth\nvariety over an algebraically closed\nfield.\nThen, we have\n\\begin{equation}\n\\chi(X,{\\cal F})\n=\n(CC {\\cal F},\nT^{*}_{X}X)_{T^{*}X}.\n\\label{eqEP}\n\\end{equation}\n\\end{thm}\n\n\n\\proof[Proof {\\rm (Beilinson)}]{\nSince $X$ is assumed projective,\nit follows from Lemma \\ref{lmclim}\nand Proposition \\ref{prRR}.2.\n\\qed}\n\n\n\n\n\\subsection{Characteristic cycle\nand ramification}\\label{ssramc}\n\nWe briefly recall the definition \nof the characteristic cycle\n\\cite[Definition 3.5]{nonlog}\nin the strongly non-degenerate case.\nWe use the notation in Section \\ref{ssram}.\n\nLet $X$ be a smooth scheme\nof dimension $n$ over a perfect \nfield $k$ and\n$U=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} D$ be\nthe complement of a divisor $D$\nwith normal crossings.\nLet $j\\colon U\\to X$ be\nthe open immersion and\n${\\cal G}$ be a locally constant\nconstructible sheaf of free $\\Lambda$-modules\non $U$.\nAssume that the ramification of\n${\\cal G}$ along $D$ is strongly non-degenerate.\n\nAssume first that $R=D$.\nThen, the locally constant sheaf\n${\\cal G}$ on $U$ is tamely ramified along $D$\nand we define a linear combination by\n\\begin{equation}\nC( j_!{\\cal G})\n=(-1)^n{\\rm rank}\\ {\\cal G}\\cdot\n\\sum_I[T^*_{D_I}X]\n\\label{eqtame}\n\\end{equation}\nwhere $T^*_{D_I}X$\ndenotes the conormal bundle\nof the intersection $D_I=\\bigcap_ID_i$\nfor sets of indices\n$I\\subset \\{1,\\ldots,m\\}$.\n\nAssume $R=\\sum_ir_iD_i>\nD=\\sum_iD_i$. \nFor each irreducible component,\nwe have a decomposition by \ncharacters \n$V=\\bigoplus_{\\chi\\colon\n{\\rm Gr}^{r_i}G_{K_i}\\to \n{\\mathbf F}_p}\n\\chi^{\\oplus m(\\chi)}$.\nFurther, the characteristic form\nof each character $\\chi$\nappearing in the decomposition\ndefines a sub line bundle $L_\\chi$\nof the pull-back $D_\\chi\\times_XT^*X$\nof the cotangent bundle \nto a finite covering $\\pi_\\chi\n\\colon D_\\chi\\to D_i$\nby the non-degenerate assumption.\nThen we define\n\\begin{equation}\nC(j_!{\\cal G})\n=(-1)^n\\Bigl(\n{\\rm rank}\\ {\\cal G}\\cdot\n[T^*_XX]\n+\n\\sum_i\n\\sum_{\\chi}\n\\dfrac{r_i\\cdot m(\\chi)}{[D_\\chi\\colon D_i]}\n\\pi_*[L_\\chi]\\Bigr).\n\\label{eqwild}\n\\end{equation}\nIn the general strongly non-degenerate case,\nwe define $C(j_!{\\cal G})$ by the additivity\nand \\'etale descent.\n\n\\begin{thm}\\label{thmram}\nLet $X$ be a smooth scheme\nof dimension $n$ over a perfect \nfield $k$ and\nlet $j\\colon U\\to X$ be the\nopen immersion of\nthe complement\n$U=X\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} D$ of\na divisor $D$ with simple normal crossings.\nLet ${\\cal G}$\nbe a locally constant constructible\nsheaf of free $\\Lambda$-modules\non $U$ such that\nthe ramification along $D$\nis {\\em strongly non-degenerate}.\nThen we have\n\\begin{equation}\nCC j_!{\\cal G}\n=\nC(j_!{\\cal G}),\n\\label{eqCCj}\n\\end{equation}\nIn other words, \n$C(j_!{\\cal G})$ defined by\n{\\rm (\\ref{eqtame})},\n{\\rm (\\ref{eqwild})},  the additivity\nand by \\'etale descent\nusing\nramification theory\nsatisfies the Milnor formula\n{\\rm (\\ref{eqMil})}.\n\\end{thm}\n\nTheorem \\ref{thmram} is\nproved for dimension $\\leqq 1$\nin Lemma \\ref{lmele}.1 and 2.\nRecall that Theorem \\ref{thmram} is\nproved in dimension $2$\nin \\cite[Proposition 3.20]{surface}\nusing a global argument, as in \\cite{bp}.\nThe tamely ramified case\nof Theorem \\ref{thmram} has been proved\nin \\cite{Yang} by a different method.\nTheorem \\ref{thmram} gives\nan affirmative answer to\n\\cite[Conjecture 3.16]{nonlog}.\n\n\n\\proof{\nIt suffices to show the\nequality of the coefficients\nfor each irreducible component\nof $C=S(j_!{\\cal F})$ by Proposition \\ref{prnd}.\nBy the additivity of characteristic cycles\nand the compatibility with \\'etale pull-back,\nit suffices to show the\ntamely ramified case\nand the totally wildly ramified\ncase separately.\n\nFirst, we prove the tamely ramified case.\nIt suffices to determine the\ncoefficients of $[T^*_{D_I}X]$\nby induction on the number $d$ of\nelements of $I$.\nBy Proposition \\ref{prsm*},\nwe may assume $X={\\mathbf A}^n$\nand $D$ is the complement of\n${\\mathbf G}_m^n$.\nIf $n=0$ and $X$ consists of a\nsingle point, we have\n$CC {\\cal F}=\n{\\rm rank}\\ {\\cal F}\\cdot [T^*_XX]$.\nIf $n\\geqq 1$,\nit follows from Lemma \\ref{lmtame}.\n\nWe prove the totally wildly ramified case.\nIf $\\dim X=0$, it is proved above.\nIt follows from (\\ref{eqdim1}) if\n$\\dim X\\leqq 1$.\nIt suffices to compare\nthe coefficients in\n$CC j_!{\\cal G}$\nand $C(j_!{\\cal G})$\nassuming $\\dim X\\geqq 2$.\nSince the assertion is\n\\'etale local, we may \nassume that $C(j_!{\\cal G})$\nhas a unique irreducible\ncomponent different from the $0$-section.\nBy Theorem \\ref{thmi*}\nand Proposition \\cite[Proposition 3.8]{nonlog},\nfor every properly $C$-transversal immersion $i\\colon W\\to X$\nof a smooth curve,\nwe have \n$i^!CC j_!{\\cal G}=\nCC i^* j_!{\\cal G}$\nand $i^!C(j_!{\\cal G})\n=C(i^*j_!{\\cal G})$\nrespectively.\nHence it is reduced to the case\n$\\dim X=1$.\n\\qed}\n\n\n\\section{${\\cal F}$-transversality and singular support}\\label{sFt}\n\nWe introduce and study a notion of\n${\\cal F}$-transversality\nfor a morphism $h\\colon W\\to X$\nwith respect to a constructible\ncomplex ${\\cal F}$ on $X$\nin Section \\ref{ssFt}\nusing a canonical morphism (\\ref{eqprpg})\ndefined and studied in Section \\ref{ssFcan}.\nWe study the relation between\nthe ${\\cal F}$-transverality\nand the local acyclicity in Section \\ref{ssFla}\nand deduce a relation\nwith the singular support in Section \\ref{ssFs}.\n\n\\subsection{A canonical morphism}\\label{ssFcan}\n\n\nLet $k$ be a field\nand $\\Lambda$ be a finite\nlocal ring such that\nthe residue characteristic\n$\\ell$ is invertible in $k$.\nFor a separated morphism $h\\colon W\\to X$\nof schemes of finite type\nover $k$,\nthe functor\n$Rh^!\\colon D^b_c(X,\\Lambda)\n\\to D^b_c(W,\\Lambda)$\nis defined as the adjoint\nof\n$Rh_!\\colon D^b_c(W,\\Lambda)\n\\to D^b_c(X,\\Lambda)$.\nOne should be able to define\nthe functors $Rh_!,Rh^!$\nwithout assuming separatedness\nusing cohomological descent\nbut we will not go further in this\ndirection.\n\n\n\nLet ${\\cal F}$ and ${\\cal G}$ \nbe constructible complexes\nof $\\Lambda$-modules\non $X$ and on $W$\nrespectively\nand assume that\n${\\cal G}$ is of finite tor-dimension.\nThen, we have an isomorphism\n\\begin{equation}\n{\\cal F}\\otimes^L_\\Lambda\nRh_!{\\cal G} \n\\to Rh_!(h^*{\\cal F}\\otimes^L_\\Lambda{\\cal G} )\n\\label{eqprj}\n\\end{equation}\nof projection formula\n\\cite[(4.9.1)]{Rapport}.\n\\if{This is defined as the adjoint\n$h^*{\\cal F}\\otimes^L_\\Lambda\nh^*Rh_!{\\cal G} \n\\to h^*{\\cal F}\\otimes^L_\\Lambda{\\cal G}$\nof the morphism induced\nby the adjunction\n$h^*Rh_!{\\cal G} \n\\to {\\cal G}$\nif $h$ is proper.\nIt is defined as the inverse of\nthe isomorphism\n$h^*{\\cal F}\\otimes^L_\\Lambda\nh^*Rh_!{\\cal G} \n\\gets h^*{\\cal F}\\otimes^L_\\Lambda{\\cal G}$\nif $h$ is an open immersion.}\\fi\n\n\\begin{df}\\label{dfAB}\nLet $h\\colon W\\to X$\nbe a morphism of finite\ntype of schemes.\nLet ${\\cal F}$ and ${\\cal E}$ be\nconstructible complexes\nof $\\Lambda$-modules on $X$\nand let ${\\cal G}$ be\na constructible complex\nof $\\Lambda$-modules \nof finite tor-dimension on $W$.\nWe say that morphisms\n\\begin{equation}\n{\\cal F}\n\\otimes^L\nRh_!{\\cal G}\n\\to {\\cal E},\\quad\nh^*{\\cal F}\n\\otimes^L\n{\\cal G}\n\\to Rh^!{\\cal E}\n\\label{eqAB}\n\\end{equation}\ncorrespond to each other\nif the first one\nis the composition \nof the isomorphism {\\rm (\\ref{eqprj})}\nwith the adjoint\n$Rh_!(h^*{\\cal F}\n\\otimes^L\n{\\cal G})\n\\to\n{\\cal E}$\nof the second one.\n\\end{df}\n\nSince {\\rm (\\ref{eqprj})}\nis an isomorphism,\nthe correspondence\n{\\rm (\\ref{eqAB})}\nis a one-to-one correspondence.\nIf $h\\colon W\\to X$\nis a smooth separated morphism\nof relative dimension $d$,\nthe morphism\n${\\cal F}\n\\otimes^L\nRh_!\\Lambda(d)[2d]\n\\to {\\cal F}$\ninduced by\nthe trace mapping\n$Rh_!\\Lambda(d)[2d]\\to \\Lambda$\ncorresponds to the canonical isomorphism\n\\begin{equation}\nh^*{\\cal F}\n\\otimes^L\n\\Lambda(d)[2d]\n\\to Rh^!{\\cal F}\n\\label{eqPD}\n\\end{equation}\nof Poincar\\'e duality\n\\cite[Th\\'eor\\`eme 3.2.5]{DP}.\n\n\nRecall that a morphism\n${\\cal F}\\otimes^L_\\Lambda\nRh_!{\\cal G} \n\\to {\\cal E}$\ncorresponds to a morphism\n${\\cal F}\\to\nR{\\cal H}om(\nRh_!{\\cal G},{\\cal E})$\nbijectively.\nSimilarly\na morphism\n$h^*{\\cal F}\\otimes^L_\\Lambda\n{\\cal G} \n\\to Rh^!{\\cal E}$\ncorresponds to a morphism\n$h^*{\\cal F}\\to\nR{\\cal H}om({\\cal G},Rh^!{\\cal E})$\nbijectively.\n\n\\begin{lm}\\label{lmAB}\nLet $h\\colon W\\to X$\nbe a morphism of finite\ntype of schemes.\nLet ${\\cal F}$ and ${\\cal E}$ be\nconstructible complexes\nof $\\Lambda$-modules on $X$\nand let ${\\cal G}$ be\na constructible complex\nof $\\Lambda$-modules \nof finite tor-dimension on $W$.\nLet\n\\begin{equation}\n{\\cal F}\n\\otimes^L\nRh_!{\\cal G}\n\\to {\\cal E},\\quad\nh^*{\\cal F}\n\\otimes^L\n{\\cal G}\n\\to Rh^!{\\cal E}\n\\label{eqAB0}\n\\end{equation}\nbe morphisms corresponding\nto each other.\nThen,\nthe morphism\n\\begin{equation}\nh^*{\\cal F}\n\\to R{\\cal H}om({\\cal G},Rh^!{\\cal E})\n\\label{eqAB1}\n\\end{equation}\ninduced by the second one\nequals the adjoint of\nthe composition\n\\begin{equation}\n{\\cal F}\n\\to R{\\cal H}om(Rh_!{\\cal G},{\\cal E})\n\\to Rh_*\nR{\\cal H}om({\\cal G},Rh^!{\\cal E})\n\\label{eqAB2}\n\\end{equation}\nof that induced by the first\none and the inverse of the\ncanonical isomorphism of adjunction.\n\\end{lm}\n\n\n\n\\proof{\nWe consider the diagram\n\\begin{equation}\n\\xymatrix{\n{\\cal F}\n\\ar[d]\\ar[rd]&\n\\\\\nRh_*\nR{\\cal H}om\n({\\cal G},\nh^*{\\cal F}\\otimes^L{\\cal G})\n\\ar[d]\\ar[r]\n&\nR{\\cal H}om\n(Rh_!{\\cal G},\nRh_!(h^*{\\cal F}\\otimes^L{\\cal G}))\n\\ar[d]\\\\\nRh_*\nR{\\cal H}om\n({\\cal G},\nRh^!{\\cal E})\n\\ar[rd]\\ar[r]\n&\nR{\\cal H}om\n(Rh_!{\\cal G},\nRh_!Rh^!{\\cal E})\n\\ar[d]\\\\\n&\nR{\\cal H}om\n(Rh_!{\\cal G},\n{\\cal E}).}\n\\label{eqdgAB}\n\\end{equation}\nThe top vertical arrow is\nthe adjoint of $h^*{\\cal F}\n\\to \nR{\\cal H}om\n({\\cal G},\nh^*{\\cal F}\\otimes^L{\\cal G})$\ninduced by the identity of\n$h^*{\\cal F}\\otimes^L{\\cal G}$\nand the horizontal arrows\nare defined by the functoriality\nof $Rh_!$.\nThe upper slant arrow is induced\nby the isomorphism of projection formula \n(\\ref{eqprj}) and\nthe upper triangle is commutative.\nThe middle vertical arrows are\ninduced by the second morphism\nof (\\ref{eqAB0})\nand the middle square is commutative.\nThe lower slant arrow\nis the isomorphism of adjunction\nand the lower vertical arrow\nis induced by the adjunction map.\nThus, the diagram (\\ref{eqdgAB})\nis commutative.\n\nThe composition of \nthe left column in\nthe diagram (\\ref{eqdgAB})\nis the adjoint of \n(\\ref{eqAB1}).\nSince the composition \nof the upper slant arrow\nand the right column \nis induced by\nthe first morphism of (\\ref{eqAB0}),\nthe assertion follows.\n\\qed}\n\\medskip\n\nLet ${\\cal F}$ and ${\\cal G}$ \nbe constructible complexes\nof $\\Lambda$-modules\non $X$\nand assume that\n${\\cal G}$ is of finite tor-dimension.\nWe define a canonical morphism\n\\begin{equation}\nc_{h,{\\cal F},{\\cal G}}\\colon\nh^*{\\cal F}\\otimes^L_\\Lambda\nRh^!{\\cal G} \n\\to Rh^!({\\cal F}\\otimes^L_\\Lambda{\\cal G} )\n\\label{eqFG}\n\\end{equation}\nto be that corresponding\nto the morphism\n${\\cal F}\n\\otimes^L_\\Lambda\nRh_!Rh^!{\\cal G} \\to\n{\\cal F}\n\\otimes^L_\\Lambda\n{\\cal G}$ induced by the adjunction\n$Rh_!Rh^!{\\cal G} \\to\n{\\cal G}$.\n\n\n\\begin{lm}\\label{lmhRh}\nLet $h\\colon W\\to X$\nbe a separated\nmorphism of schemes.\nLet ${\\cal F}$ and ${\\cal G}$ \nbe constructible complexes\nof $\\Lambda$-modules\non $X$\nand assume that\n${\\cal G}$ is of finite tor-dimension.\n\n{\\rm 1.}\nLet $g\\colon V\\to W$\nbe a separated\nmorphism of schemes\nand $f\\colon V\\to X$\nbe the composition.\nThen the morphisms {\\rm (\\ref{eqFG})}\nfor $h,g$ and $f$ form\na commutative diagram\n\\begin{equation}\n\\xymatrix{\nf^*{\\cal F}\\otimes^L Rf^!{\\cal G}\n\\ar[d]\\ar[rrrr]^{c_{f,{\\cal F},{\\cal G}}}&&&&\nRf^!({\\cal F}\\otimes^L{\\cal G})\\ar[d]\\\\\ng^*h^*{\\cal F}\\otimes^L Rg^!Rh^!{\\cal G}\n\\ar[rr]^{c_{g,h^*{\\cal F},Rh^!{\\cal G}}}&&\nRg^!(h^*{\\cal F}\\otimes^L Rh^!{\\cal G})\n\\ar[rr]^{Rg^!(c_{h,{\\cal F},{\\cal G}})}&&\nRg^!Rh^!({\\cal F}\\otimes^L{\\cal G}).}\n\\label{eqhgf}\n\\end{equation}\nThe vertical arrows are the\ncanonical isomorphisms.\n\n{\\rm 2.}\nLet ${\\cal E}$ \nbe another constructible complex\nof $\\Lambda$-modules\non $X$\nand assume that ${\\cal F}$\nis of finite tor-dimension.\nThen the morphisms {\\rm (\\ref{eqFG})}\nform a commutative diagram\n\\begin{equation}\n\\xymatrix{\nh^*{\\cal E}\\otimes^L h^*\n{\\cal F}\\otimes^L Rh^!{\\cal G}\n\\ar[d]\n\\ar[rr]\n^{{\\rm id}\\otimes c_{h,{\\cal F},{\\cal G}}}\n&&\nh^*{\\cal E}\\otimes^L Rh^!\n({\\cal F}\\otimes^L {\\cal G})\n\\ar[d]^{c_{h,{\\cal E},{\\cal F}\\otimes{\\cal G}}}\n\\\\\nh^*({\\cal E}\\otimes^L\n{\\cal F})\\otimes^L Rh^!{\\cal G}\n\\ar[rr]^{c_{h,{\\cal E}\\otimes{\\cal F},{\\cal G}}}&&\nRh^!({\\cal E}\\otimes^L\n{\\cal F}\\otimes^L{\\cal G}).}\n\\label{eqEFG}\n\\end{equation}\n\n\n{\\rm 3.}\nAssume that $h$ is an immersion.\nThen the morphism {\\rm (\\ref{eqFG})}\nis induced by the restriction of\n\\begin{equation}\n{\\cal F}\\otimes^L Rh_*Rh^!{\\cal G}\n=\n{\\cal F}\\otimes^L \nR{\\cal H}om (h_!\\Lambda,\n{\\cal G})\n\\to \nRh_*Rh^!(\n{\\cal F}\\otimes^L {\\cal G})\n= \nR{\\cal H}om (h_!\\Lambda,\n{\\cal F}\\otimes^L{\\cal G}).\n\\label{eqhim}\n\\end{equation}\n\\end{lm}\n\n\\proof{\n1.\nIt follows from the commutative \ndiagram \n\\begin{equation}\n\\xymatrix{\nRf_!(f^*{\\cal F}\\otimes^L Rf^!{\\cal G})\n\\ar[d]\\ar[rr]&&\n{\\cal F}\\otimes^L\nRf_!Rf^!{\\cal G}\\ar[d]\\\\\nRh_!Rg_!(g^*h^*{\\cal F}\\otimes^L Rg^!Rh^!{\\cal G})\n\\ar[r]&\nRh_!(h^*{\\cal F}\\otimes^L Rg_!Rg^!Rh^!{\\cal G})\n\\ar[r]&\n{\\cal F}\\otimes^L\nRh_!Rg_!Rg^!Rh^!{\\cal G}.}\n\\label{eqhgpf}\n\\end{equation}\nfor the isomorphisms\nof projection formula.\n\n2.\nWe consider the diagram\n$$\\begin{CD}\nRh_!(h^*{\\cal F}\n\\otimes^LRh^!{\\cal G})\n@>>>\nRh_!Rh^!({\\cal F}\n\\otimes^L{\\cal G})\\\\\n@AAA@VVV\\\\\n{\\cal F}\n\\otimes^LRh_!Rh^!{\\cal G}\n@>>>\n{\\cal F}\n\\otimes^L{\\cal G}.\n\\end{CD}$$\nThe right vertical arrow\nis the adjunction morphism\nand the bottom horizontal arrow\nis induced by the adjunction morphism.\nThe left vertical arrow is\nthe isomorphism (\\ref{eqprj})\nof the projection formula \nand the top horizontal arrow is\ninduced by (\\ref{eqFG}).\nSince (\\ref{eqFG}) is the adjoint\nof the diagonal composition and is defined\nby the commutativity of\nthe lower left triangle,\nthe diagram is commutative.\nTensoring ${\\cal E}$\nand applying the projection formula,\nwe obtain the adjoint of\n(\\ref{eqEFG}).\n\n\n\n\n\n3.\nWe may assume that $h$ is a closed immersion.\nThen, the composition of the\nmorphism\n(\\ref{eqhim}) with\nthe adjunction\n$Rh_*Rh^!({\\cal F}\\otimes^L{\\cal G})\n\\to {\\cal F}\\otimes^L{\\cal G}$\nis the same as\n${\\cal F}\\otimes^LRh_*Rh^!{\\cal G}\n\\to {\\cal F}\\otimes^L{\\cal G}$\ndefining (\\ref{eqFG}) and the assertion follows.\n\\qed}\n\\medskip\n\nAssume ${\\cal G}=\\Lambda$\nand we consider\nthe canonical morphism\n\\begin{equation}\nc_{h,{\\cal F}}\n=c_{h,{\\cal F},\\Lambda}\n\\colon \nh^*{\\cal F}\\otimes^L_\\Lambda Rh^!\\Lambda\n\\to Rh^!{\\cal F}.\n\\label{eqprpg}\n\\end{equation}\nFor another morphism\n$g\\colon V\\to W$ and\nthe composition $f$,\nThe canonical morphisms\n(\\ref{eqFG})\nform a commutative diagram\n\\begin{equation}\n\\xymatrix{\nf^*{\\cal F}\n\\otimes^L \nRf^!\\Lambda\n\\ar[r]^{c_{f,{\\cal F}}}\\ar[d]&\nRf^!{\\cal F}\n\\ar[r]&\nRg^!Rh^!{\\cal F}\n\\\\\ng^*h^*{\\cal F}\n\\otimes^L \nRg^!Rh^!\\Lambda\n\\ar[rr]\n^{c_{g,h^*{\\cal F},Rh^!\\Lambda}}\n&&\nRg^!(h^*{\\cal F}\n\\otimes^L \nRh^!\\Lambda)\\ar[u]\n_{Rg^!(c_{h,{\\cal F}})}\n\\\\\ng^*h^*{\\cal F}\n\\otimes^L \nRg^!\\Lambda\n\\otimes^L \ng^*Rh^!\\Lambda\n\\ar[rr]^{c_{g,h^*{\\cal F}}\n\\otimes{\\rm id}}\\ar[u]\n^{{\\rm id}\\otimes\nc_{g,Rh^!\\Lambda}}&&\nRg^!h^*{\\cal F}\n\\otimes^L \ng^*Rh^!\\Lambda\\ar[u]\n_{c_{g,h^*{\\cal F},Rh^!\\Lambda}}\n}\n\\label{eqhRh}\n\\end{equation}\nby Lemma \\ref{lmhRh}.1 and 2.\n\n\nWe give another description\nof the morphism\n$c_{h,{\\cal F}}\\colon\nh^*{\\cal F}\\otimes^LRh^!\\Lambda\n\\to Rh^!{\\cal F}$\nassuming further that ${\\cal F}$\nis of finite tor-dimension\nand that $X$ is separated.\nRecall that the dual $D_X{\\cal F}$\nis defined as $R{\\cal H}om_\\Lambda\n({\\cal F},{\\cal K}_X)$ where\n${\\cal K}_X=Ra^!\\Lambda$ \nfor the structure morphism \n$a\\colon X\\to {\\rm Spec}\\ k$.\nFor a separated morphism $h\\colon W\\to X$\nover $k$,\nwe have a canonical isomorphism\n$Rh^!{\\cal K}_X=\nRh^!Ra^!\\Lambda\n\\to R(ah)^!\\Lambda={\\cal K}_W$.\nThe isomorphism of adjunction\n$Rh_*D_W\\to D_XRh_!$\ninduces an isomorphism\n$D_Wh^*D_X\\to Rh^!$.\n\n\\begin{lm}\\label{lmAB2}\nAssume that ${\\cal F}$\nis of finite tor-dimension and\nwe consider the morphisms\n\\begin{equation}\nh^*D_X{\\cal F}\n\\otimes^L\nh^*{\\cal F}\n\\otimes^L\nRh^!\\Lambda\n\\to\nh^*{\\cal K}_X\n\\otimes^L\nRh^!\\Lambda\n\\overset{c_{h,{\\cal K}_X}}\\longrightarrow\nRh^!{\\cal K}_X\\to {\\cal K}_W\n\\label{eqFGDD}\n\\end{equation}\nwhere the first arrow\nis induced by the canonical morphism\n$D_X{\\cal F}\n\\otimes^L{\\cal F}\\to {\\cal K}_X$.\nThen, the composition\n\\begin{equation}\nh^*{\\cal F}\n\\otimes^L\nRh^!\\Lambda\n\\to\nD_Wh^*D_X{\\cal F}\n\\to\nRh^!{\\cal F}\n\\label{eqFGD}\n\\end{equation}\nof the morphism induced\nby {\\rm (\\ref{eqFGDD})}\nwith the canonical isomorphism\nequals\nthe morphism \n$c_{h,{\\cal F}}\\colon\nh^*{\\cal F}\n\\otimes^L\nRh^!\\Lambda\n\\to Rh^!{\\cal F}$.\n\\end{lm}\n\n\\proof{\nLet $h^*D_X{\\cal F}\n\\to D_W(h^*{\\cal F}\n\\otimes^LRh^!\\Lambda)$\nbe the morphism induced by\n(\\ref{eqFGDD})\nand let\n\\begin{equation}\nD_X{\\cal F}\n\\to \nRh_*D_W(h^*{\\cal F}\n\\otimes^LRh^!\\Lambda)\n\\overset \\simeq \\to D_XRh_!(h^*{\\cal F}\n\\otimes^LRh^!\\Lambda)\n\\label{eq717}\n\\end{equation}\nbe its adjoint.\nThen, the morphism (\\ref{eqFGD})\nis the adjoint of\nthe dual $Rh_!(h^*{\\cal F}\n\\otimes^LRh^!\\Lambda)\n\\to {\\cal F}$ of (\\ref{eq717})\nsince the isomorphism\n$D_Wh^*D_X\\to Rh^!$\nis induced by\nthe isomorphism $Rh_*D_W\\to D_XRh_!$.\nThe morphism\n$D_X{\\cal F}\n\\otimes^LRh_!(h^*{\\cal F}\n\\otimes^LRh^!\\Lambda)\n\\to {\\cal K}_X$\ncorresponding to (\\ref{eq717})\ncorresponds\nto (\\ref{eqFGDD})\nby Lemma \\ref{lmAB}.\nThus, it is induced by\nthe isomorphism\nof the projection formula,\nthe canonical morphism\n$D_X{\\cal F}\n\\otimes^L{\\cal F}\\to {\\cal K}_X$\nand \nthe adjunction morphism\n$Rh_!h^!\\Lambda\\to \\Lambda$.\nHence the morphism\n(\\ref{eqFGD}) equals\n$c_{h,{\\cal F}}$.\n\\qed}\n\n\n\n\n\n\\subsection{${\\cal F}$-transversality}\\label{ssFt}\n\n\n\\begin{df}\\label{dfprpg}\nLet $h\\colon W\\to X$ \nbe a morphism of\nschemes of finite type over $k$\nand let ${\\cal F}$ be a constructible complex\nof $\\Lambda$-modules on $X$.\n\nIn the case $h$ is {\\em separated},\nwe say that $h$\nis {\\em ${\\cal F}$-transversal}\nif the canonical morphism \n\\begin{equation*}\nc_{h,{\\cal F}}\\colon\nh^*{\\cal F}\\otimes^L_\\Lambda Rh^!\\Lambda\n\\to Rh^!{\\cal F}\n\\leqno{\\rm(\\ref{eqprpg})}\n\\end{equation*}\nis an isomorphism.\nFor general $h$,\nwe say that $h$\nis {\\em ${\\cal F}$-transversal}\nif there exists an open covering\n$W=\\bigcup_iW_i$ consisting\nof open subschemes\nseparated over $X$\nsuch that\nthe restrictions $h|_{W_i}$\nare ${\\cal F}$-transversal.\n\\end{df}\n\nSince the condition is local, a morphism\n$h\\colon W\\to X$ is ${\\cal F}$-transversal\nif and only if\nthe restriction of $h$ to every open subset\nof $W$ separated over $X$\nis ${\\cal F}$-transversal.\n\n\\begin{lm}\\label{lmprpg}\nLet ${\\cal F}$ be a constructible complex\nof $\\Lambda$-modules on \na scheme $X$ \nof finite type over $k$\nand let $h\\colon W\\to X$\nbe a morphism of schemes\nof finite type over $k$.\n\n{\\rm 1.}\nIf $h$ is smooth,\nthen $h\\colon W\\to X$\nis ${\\cal F}$-transversal.\n\n{\\rm 2.}\nIf ${\\cal F}$ is locally constant,\nthen $h\\colon W\\to X$ is\n${\\cal F}$-transversal.\n\n{\\rm 3.}\nAssume that\n$h\\colon W\\to X$\nis ${\\cal F}$-transversal\nand that \n$Rh^!\\Lambda$ is isomorphic\nto $\\Lambda(c)[2c]$ for a\nlocally constant function $c$ on $W$.\nThen for a morphism\n$g\\colon V\\to W$\nof schemes\nof finite type over $k$,\nthe following conditions are equivalent:\n\n{\\rm (1)}\n$g\\colon V\\to W$ \nis $h^*{\\cal F}$-transversal.\n\n{\\rm (2)}\nThe composition\n$f\\colon V\\to W$ \nis ${\\cal F}$-transversal.\n\n\n{\\rm 4.}\nLet $\\Lambda_0$\nbe the residue field of $\\Lambda$.\nAssume that ${\\cal F}$\nis of finite tor-dimension\nand set ${\\cal F}_0=\n{\\cal F}\\otimes^L_\\Lambda\\Lambda_0$,\nThen,\n$h\\colon W\\to X$ is \n${\\cal F}$-transversal\nif and only if\nit is ${\\cal F}_0$-transversal.\n\n{\\rm 5.}\nAssume that ${\\cal F}$ is a perverse sheaf.\nAssume further that $h\\colon\nW\\to X$ is locally of\ncomplete intersection\nof relative virtual dimension\n$c$\nand that \n$Rh^!\\Lambda$ is isomorphic\nto $\\Lambda(c)[2c]$.\nIf $h\\colon W\\to X$ is\n${\\cal F}$-transversal,\nthen $h^*{\\cal F}[c]$\nand $Rh^!{\\cal F}[-c]$\nare perverse sheaves.\n\n{\\rm 6.}\nAssume that ${\\cal F}$\nis of finite tor-dimension.\nThen the following conditions are\nequivalent\n\n{\\rm (1)}\n$h$ is ${\\cal F}$-transversal.\n\n{\\rm (2)}\nThe morphism\n$h^*{\\cal F}\\otimes^L\nRh^!\\Lambda\n\\to D_Wh^*D_X{\\cal F}$\n{\\rm (\\ref{eqFGD})}\ninduced by the morphism\n$h^*D_X{\\cal F}\\otimes^L\nh^*{\\cal F}\\otimes^L\nRh^!\\Lambda\n\\to {\\cal K}_W$\n{\\rm (\\ref{eqFGDD})}\nis an isomorphism.\n\\end{lm}\n\nWe show a converse of 2.\\\nin Corollary \\ref{corfh}.\n\n\\proof{\nBy the remark after Definition \\ref{dfprpg},\nwe may assume that morphisms\nare separated.\nWe will omit to write\nthis remark in the sequel.\n\n1. \nWe may assume that\n$h$ is smooth of relative dimension $d$.\nWe consider the canonical\nisomorphism\n$\\Lambda(d)[2d]\\to Rh^!\\Lambda$\ndefined as the adjoint of\nthe trace mapping\n$Rh_!\\Lambda(d)[2d]\\to \\Lambda$.\nThen, by the description\nof the isomorphism\n(\\ref{eqPD}) loc.\\ cit.,\nthe diagram\n$$\\begin{CD}\nh^*{\\cal F}\\otimes^L\n\\Lambda(d)[2d]@>{\\rm (\\ref{eqPD})}>>\nRh^!{\\cal F}\\\\\n@VVV @|\\\\\nh^*{\\cal F}\\otimes^L\nRh^!\\Lambda\n@>{c_{h,{\\cal F}}}>>\nRh^!{\\cal F}\n\\end{CD}$$\nis commutative\nand\nthe morphism $c_{h,{\\cal F}}$\nis an isomorphism.\n\n2.\nSince the assertion is \\'etale local on $X$,\nit is reduced to the case where ${\\cal F}$\nis constant by devissage.\n\n3.\nWe consider the \ncommutative diagram\n$$\\xymatrix{\nf^*{\\cal F}\n\\otimes^L \nRf^!\\Lambda\n\\ar[r]^{c_{f,{\\cal F}}}\\ar[d]&\nRf^!{\\cal F}\n\\ar[r]&\nRg^!Rh^!{\\cal F}\n\\\\\ng^*h^*{\\cal F}\n\\otimes^L \nRg^!Rh^!\\Lambda\n&&\nRg^!(h^*{\\cal F}\n\\otimes^L \nRh^!\\Lambda)\\ar[u]\n_{Rg^!(c_{h,{\\cal F}})}\n\\\\\ng^*h^*{\\cal F}\n\\otimes^L \nRg^!\\Lambda\n\\otimes^L \ng^*Rh^!\\Lambda\n\\ar[rr]^{c_{g,h^*{\\cal F}}\n\\otimes{\\rm id}}\\ar[u]\n^{{\\rm id}\\otimes\nc_{g,Rh^!\\Lambda}}&&\nRg^!h^*{\\cal F}\n\\otimes^L \ng^*Rh^!\\Lambda\\ar[u]\n_{c_{g,h^*{\\cal F},Rh^!\\Lambda}}\n.}\n\\leqno{\\rm (\\ref{eqhRh})}$$\nSince \n$Rh^!\\Lambda$ is assumed to be isomorphic\nto $\\Lambda(c)[2c]$ for a\nlocally constant function $c$ on $W$,\nthe lower vertical arrows are isomorphisms.\nBy the assumption that\n$h\\colon W\\to X$\nis ${\\cal F}$-transversal,\nthe upper right vertical arrow\n$Rg^!(c_{h,{\\cal F}})$ is\nan isomorphism.\nThe condition (1) means that\nthe top horizontal arrow\n$c_{f,{\\cal F}}$ is\nan isomorphism\nand the condition (2) is equivalent\nto that the bottom horizontal arrow\n$c_{g,h^*{\\cal F}}\n\\otimes{\\rm id}$ is\nan isomorphism by \nthe assumption\nthat \n$Rh^!\\Lambda$ is isomorphic\nto $\\Lambda(c)[2c]$.\nHence the equivalence follows.\n\n4.\nSimilarly as Lemma \\ref{lmctf}.1,\nthe canonical morphism\n$Rf^!{\\cal F}\\otimes_\\Lambda^L\n\\Lambda_0\n\\to\nRf^!{\\cal F}_0$\nis an isomorphism.\nHence, similarly as the proof\nof Lemma \\ref{lmctf}.2,\nthe morphism $c_{h,{\\cal F}}$\nis an isomorphism\nif and only if \n$c_{h,{\\cal F}_0}$\nis an isomorphism.\n\n5. \nIf $h$ is smooth, $h^*(c)[c]=Rh^![-c]$\nis $t$-exact by \\cite[4.2.4]{BBD}\nand the assertion follows in this case.\nHence, we may assume\nthat $h$ is a regular immersion\nof codimension $-c$.\nThen, by \\cite[Corollary 4.1.10]{BBD}\nand induction on $-c$,\nwe have \n$h^*{\\cal F}(c)[c]\\in{^p}\\!D_c^{[0,-c]}(W,\\Lambda)$\nand\n$Rh^!{\\cal F}[-c]\\in{ ^p}\\!D_c^{[c,0]}(W,\\Lambda)$.\nBy the assumption,\nwe have an isomorphism\n$h^*{\\cal F}(c)[c]\\to Rh^!{\\cal F}[-c]$\nand the assertion follows.\n\n6.\nIt follows from Lemma \\ref{lmAB2}.\n\\qed}\n\\medskip\n\n\n\n\\begin{pr}\\label{prRhom}\nLet ${\\cal F}$ be a constructible complex\nof $\\Lambda$-modules of\nfinite tor-dimension on \na scheme $X$ of finite type over $k$.\n\n{\\rm 1.}\nLet $h\\colon W\\to X$\nbe a morphism of schemes \nof finite type over $k$\nand assume that\n$Rh^!\\Lambda$ is isomorphic\nto $\\Lambda(c)[2c]$ for a\nlocally constant function $c$ on $W$.\nThen, the following conditions are equivalent:\n\n{\\rm (1)}\nThe morphism $h\\colon W\\to X$ \nis ${\\cal F}$-transversal.\n\n\n{\\rm (2)}\nThe morphism $h\\colon W\\to X$ \nis $D_X{\\cal F}$-transversal.\n\n\\noindent\nFurther if $h\\colon W\\to X$ \nis ${\\cal K}_X$-transversal,\nthey are \nequivalent to the following condition:\n\n{\\rm (3)}\nThe canonical morphism\n$h^*R{\\cal H}om_X({\\cal F},{\\cal K}_X)\n\\to \nR{\\cal H}om_W(h^*{\\cal F},h^*{\\cal K}_X)$\nis an isomorphism.\n\n{\\rm 2.}\nLet ${\\cal G}$ be a constructible complex\nof $\\Lambda$-modules on \na scheme $X$ of finite type over $k$.\nThen, the following conditions are equivalent:\n\n{\\rm (1)}\nThe diagonal morphism\n$\\delta\\colon X\\to X\\times X$\nis \n$R{\\cal H}om_{X\\times X}({\\rm pr}_2^*{\\cal F},\n{\\rm pr}_1^!{\\cal G})$-transversal.\n\n{\\rm (2)}\nThe canonical morphism\n${\\cal G}\\otimes^L\nR{\\cal H}om_X({\\cal F},\\Lambda)\n\\to\nR{\\cal H}om_X({\\cal F},{\\cal G})$\nis an isomorphism.\n\\end{pr}\n\nThe assumptions in Proposition \\ref{prRhom}.1\nare satisfied if $X$ and $W$ are\nsmooth over $k$.\nThe canonical morphism\nin Proposition \\ref{prRhom}.1(3)\nis an analogue of \n$h^*Sol\\ \\!{\\cal M}\\to Sol h^*{\\cal M}$\nfor a ${\\cal D}$-module ${\\cal M}$.\n\n\\proof{\n1.\nThe equivalence of (1) and (2)\nfollows from\nLemma \\ref{lmprpg}.6,\nthe assumption\n$Rh^!\\Lambda\n\\simeq \\Lambda(c)[2c]$ and\nthe isomorphism\n${\\rm id}\\to D_WD_W$ of biduality\n\\cite[Th\\'eor\\`eme 4.3]{TF}.\n\n\n\nThe assumption that $h\\colon W\\to X$ \nis ${\\cal K}_X$-transversal\nmeans that the canonical morphism\n$c_{h,{\\cal K}_X}\\colon\nh^*{\\cal K}_X\\otimes^L Rh^!\\Lambda\n\\to Rh^!{\\cal K}_X={\\cal K}_W$\nis an isomorphism.\nHence by the definition of\n(\\ref{eqFGDD}),\nthe conditions (1) and (3)\nare equivalent.\n\n2.\nWe consider the commutative diagram\n$$\n\\begin{CD}\n\\delta^*R{\\cal H}om_X({\\rm pr}_2^*{\\cal F},\n{\\rm pr}_1^!{\\cal G})\n\\otimes^L R\\delta^!\\Lambda\n@>{c_{\\delta,R{\\cal H}om_X({\\rm pr}_2^*{\\cal F},\n{\\rm pr}_1^!{\\cal G})}} >>\nR\\delta^!\nR{\\cal H}om_X({\\rm pr}_2^*{\\cal F},\n{\\rm pr}_1^!{\\cal G})\n\\\\\n@AAA@VVV\\\\\n{\\cal G}\\otimes^L\nR{\\cal H}om_X({\\cal F},\\Lambda)\n@>>>\nR{\\cal H}om_X({\\cal F},{\\cal G})\n\\end{CD}$$\ndefined as follows.\nThe bottom horizontal arrow\nis the canonical morphism\nin the condition (2).\nThe canonical isomorphism\n${\\cal G}\\boxtimes^L\nD_X{\\cal F}\\to\nR{\\cal H}om_{X\\times X}({\\rm pr}_2^*{\\cal F},\n{\\rm pr}_1^!{\\cal G})$\n\\cite[(3.1.1)]{FL}\ninduces an isomorphism\n${\\cal G}\\otimes^L\nR{\\cal H}om_X({\\cal F},{\\cal K}_X)\n=\\delta^*(\n{\\cal G}\\boxtimes^L\nD_X{\\cal F})\n\\to \n\\delta^*\nR{\\cal H}om_{X\\times X}({\\rm pr}_2^*{\\cal F},\n{\\rm pr}_1^!{\\cal G})$.\nThis together with the canonical isomorphism\n$R\\delta^!\\Lambda \\otimes {\\cal K}_X\n\\to \\Lambda$\ndefines the left vertical arrow.\nThe right vertical arrow is defined by\n\\cite[3.1.12.2]{DP}.\nSince the condition (1) is equivalent\nto that the top horizontal arrow\n$c_{\\delta,R{\\cal H}om_X({\\rm pr}_2^*{\\cal F},\n{\\rm pr}_1^!{\\cal G})}$\nis an isomorphism,\nthe assertion follows.\n\\qed}\n\n\\begin{pr}\\label{prpgj}\nLet $$\n\\begin{CD}\nW@>h>> X\\\\\n@A{j'}AA@AAjA\\\\\nV@>{h'}>>U\n\\end{CD}$$\nbe a cartesian diagram\nof schemes \nof finite type over $k$\nsuch that the vertical arrows\nare open immersions.\nLet ${\\cal G}$ be a constructible complex\nof $\\Lambda$-modules on $U$.\nWe consider the conditions:\n\n\n{\\rm (1)}\nThe morphism $h\\colon W\\to X$ \nis $j_!{\\cal G}$-transversal.\n\n\n{\\rm (2)}\nThe morphism $h'\\colon V\\to U$\nis ${\\cal G}$-transversal.\n\n\n\n{\\rm 1.}\nThe condition {\\rm (1)}\nimplies {\\rm (2)}.\nConversely,\nif $Rh^!\\Lambda$ is isomorphic to\n$\\Lambda(c)[2c]$ for a\nlocally constant function $c$\nand if the canonical morphisms\n\\begin{equation}\nj_!{\\cal G}\\to Rj_*{\\cal G},\\quad\nj'_!h^{\\prime *}{\\cal G}\\to \nRj'_*h^{\\prime *}{\\cal G}\n\\label{eqj!j*}\n\\end{equation}\nare isomorphisms,\nthe condition {\\rm (2)}\nimplies {\\rm (1)}.\n\n{\\rm 2.}\nAssume that ${\\cal G}$ is of finite\ntor-dimension on $U$.\nThen, the condition\n{\\rm (1)} is equivalent\nto the combination of\n{\\rm (2)}\nand the following condition:\n\n{\\rm (3)}\nThe base change morphism\n\\begin{equation}\n\\begin{CD}\nh^*Rj_*R{\\cal H}om({\\cal G},{\\cal K}_U)\n@>>>\nRj'_*h^{\\prime *}\nR{\\cal H}om({\\cal G},{\\cal K}_U)\n\\end{CD}\n\\label{eqbcg}\n\\end{equation}\nis an isomorphism.\n\\end{pr}\n\n\\proof{1.\nThe implication {\\rm (1)}$\\Rightarrow\n${\\rm (2)} is clear.\n\nWe consider the commutative diagram\n$$\\xymatrix{\nj'_!h^{\\prime*}{\\cal G}\\otimes^L\nRh^!\\Lambda\n\\ar[r]\\ar[d]\n&\nh^*j_!{\\cal G}\\otimes^L\nRh^!\\Lambda\n\\ar[rr]^{c_{h,j_!{\\cal G}}}&&\nRh^!j_!{\\cal G}\\ar[d]\\\\\nRj'_*h^{\\prime*}{\\cal G}\\otimes^L\nRh^!\\Lambda\\ar[dr]&\n&&\nRh^!Rj_*{\\cal G}\\ar[d]\\\\\n&\nRj'_*(h^{\\prime*}{\\cal G}\\otimes^L\nRh^{\\prime !}\\Lambda)\n\\ar[rr]^{Rj'_*(c_{h',{\\cal G}})}&&\nRj'_*Rh^{\\prime!}{\\cal G}\n}$$\ndefined as follows.\nThe upper vertical arrows\nare induced by the\ncanonical morphisms (\\ref{eqj!j*}).\nThe top left horizontal arrow\nis induced by the isomorphism\n$j'_!h^{\\prime*}\\to \nh^*j_!$ and is an isomorphism.\nThe slant arrow is defined \nas the adjoint of the isomorphism\n$j^{\\prime*}\n(Rj'_*h^{\\prime*}{\\cal G}\\otimes^L\nRh^!\\Lambda)\\to \nRj'_*(h^{\\prime*}{\\cal G}\\otimes^L\nRh^{\\prime !}\\Lambda)$\nand is an isomorphism\nif the assumption on\n$Rh^!\\Lambda$ is satisfied.\nThe lower right vertical arrow\nis the adjoint of the isomorphism\n$j^*Rh_!\\to Rh'_!j^{\\prime *}$\nand is an isomorphism.\nThus under the assumptions,\nthe implication {\\rm (2)}$\\Rightarrow\n${\\rm (1)} holds.\n\n\n\n2.\nSince the condition (1)\nimplies (2),\nit suffices to show that\n(\\ref{eqbcg}) is an isomorphism\nif and only if\nthe condition (3) for ${\\cal F}=j_!{\\cal G}$\nin Proposition \\ref{prRhom}.1 is satisfied,\nassuming (2).\nWe consider the\ncommutative diagram\n$$\\begin{CD}\nh^*R{\\cal H}om_X(j_!{\\cal G},{\\cal K}_X)\n@>{(3)}>>\nR{\\cal H}om_W(h^*j_!{\\cal G},\nh^*{\\cal K}_X)\n\\\\\n@VVV@VVV\\\\\nh^*Rj_*R{\\cal H}om_U({\\cal G},{\\cal K}_U)\n@.\nR{\\cal H}om_W(j'_!h^{\\prime*}{\\cal G},\nh^*{\\cal K}_X)\n\\\\\n@V{(\\ref{eqbcg})}VV@VVV\\\\\nRj_*h^{\\prime*}R{\\cal H}om_U({\\cal G},{\\cal K}_U)\n@>>>\nRj'_*R{\\cal H}om_V(h^{\\prime*}\n{\\cal G},h^{\\prime*}{\\cal K}_U)\n\\end{CD}$$\ndefined as follows.\nThe upper left and the\nlower right vertical arrows are\nthe adjunction morphisms\nand are isomorphisms.\nThe upper right vertical arrow\nis induced by the isomorphism\n$h^*j_!{\\cal G}\\to\nj'_!h^{\\prime*}{\\cal G}$\nand is an isomorphism.\n\nThe top horizontal arrow (3) is\nthe canonical morphism in\nthe condition (3) in Proposition \n\\ref{prRhom}.1 for ${\\cal F}=j_!{\\cal G}$.\nThe lower one is induced by\nthat for ${\\cal G}$\nand is an isomorphism\nif (2) is satisfied,\nby Proposition \\ref{prRhom}.1\n(1)$\\Rightarrow$(3).\nThus the assertion follows.\n\\qed}\n\n\\subsection{${\\cal F}$-transversality\nand local acyclicity}\\label{ssFla}\n\nIn Proposition \\ref{prla},\n$X$ and $Y$ denote arbitrary schemes\nand $\\Lambda$ denotes a finite local ring\nsuch that the characteristic of the\nresidue field is invertible on $Y$.\n\n\\begin{pr}[{\\rm \\cite[Proposition 2.10]{app}}]\n\\label{prla}\nLet \n\\begin{equation}\n\\begin{CD}\nX@<{j'}<< U\\\\\n@VfVV @VV{f_V}V\\\\\nY@<j<<V\n\\end{CD}\n\\label{eqXYi}\n\\end{equation}\nbe a cartesian diagram of schemes\nsuch that $j\\colon V\\to Y$\nis an open immersion.\nLet ${\\cal F}$ \nbe a complex of $\\Lambda$-modules\non $X$\nand let ${\\cal G}$\nbe a complex of $\\Lambda$-modules\non $V$.\nAssume that \n$f$ is strongly locally acyclic relatively to ${\\cal F}$.\nThen, the base change morphism\n\\begin{equation}\n{\\cal F}\\otimes f^*Rj_*{\\cal G}\n\\to \nRj'_*(j^{\\prime*}{\\cal F}\n\\otimes f_V^*{\\cal G})\n\\label{eqla1}\n\\end{equation}\nis an isomorphism.\n\\end{pr}\n\n\n\\begin{cor}\\label{corlat}\nLet $f\\colon X\\to Y$ be a \n{\\em smooth} morphism\nof schemes of finite type\nover a field $k$,\n$i\\colon Z\\to Y$\nbe an immersion\nand \n\\begin{equation}\n\\begin{CD}\nW@>h>>X\\\\\n@VgVV @VfVV\\\\\nZ@>i>>Y\n\\end{CD}\n\\label{eqZiY}\n\\end{equation}\nbe a cartesian diagram of schemes.\nLet ${\\cal F}$\nbe a complex of $\\Lambda$-modules\non $X$\nand assume that\n$f$ is strongly locally acyclic relatively to ${\\cal F}$.\nThen \n$h\\colon W\\to X$\nis ${\\cal F}$-transversal.\n\\end{cor}\n\n\\proof{\nWe may assume that $i\\colon Z\\to Y$\nis a closed immersion.\nLet $V=Y\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} Z$\nand consider the cartesian diagram\n\\begin{equation}\n\\begin{CD}\nW@>h>>X@<{j'}<< U\\\\\n@VgVV @VfVV @VV{f_V}V\\\\\nZ@>i>>Y@<j<<V.\n\\end{CD}\n\\label{eqZYV}\n\\end{equation}\nBy Proposition \\ref{prla} applied\nto the right square,\nwe obtain an isomorphism\n${\\cal F}\\otimes f^*Rj_*\\Lambda\n\\to \nRj'_*j^{\\prime*}{\\cal F}$.\nSince $f$ is smooth,\nthis induces an isomorphism\n${\\cal F}\\otimes Rj'_*\\Lambda\n\\to \nRj'_*j^{\\prime*}{\\cal F}$\nby smooth base change theorem\n\\cite{smbc}.\nBy the distinguished triangle\n$\\to h_*Rh^!\\to{\\rm id}\\to\nRj'_*j^{\\prime*}$,\nwe obtain an isomorphism\n$c_{h,{\\cal F}}\\colon\nh^*{\\cal F}\\otimes Rh^!\\Lambda\n\\to \nRh^!{\\cal F}$.\n\\qed}\n\n\\begin{pr}\\label{prF}\nLet $f\\colon X\\to Y$ be a \nmorphism of schemes of finite\ntype over a field $k$\nand let ${\\cal F}$\nbe a complex of $\\Lambda$-modules\non $X$.\nThe morphism\n$f$ is locally acyclic relatively to ${\\cal F}$\nif the following \ncondition is satisfied:\n\nLet $Y'$ and $Z$ be smooth\nschemes \nover a finite extension of $k$\nand \n\\begin{equation}\n\\begin{CD}\nX@<{p'}<< X'@<h<<W\\\\\n@VfVV @VV{f'}V@VVgV\\\\\nY@<p<<Y'@<i<<Z\n\\end{CD}\n\\label{eqXYpi}\n\\end{equation}\nbe a cartesian diagram of\nschemes where\n$p\\colon Y'\\to Y$ is\nproper and generically finite and\n$i\\colon Z\\to Y'$ is a closed immersion.\nThen, \nfor the pull-back\n${\\cal F}'=p^{\\prime *}{\\cal F}$ on $X'$\nthe composition\n\\begin{equation}\nh^*{\\cal F}'\\otimes g^*Ri^!\\Lambda\n\\to \nh^*{\\cal F}'\\otimes Rh^!\\Lambda\n\\overset{c_{h,{\\cal F}'}}\\longrightarrow \nRh^!{\\cal F}'\n\\label{eqF}\n\\end{equation}\nwhere the first arrow\nis induced by the base change morphism\nis an isomorphism.\n\\end{pr}\n\nThough\nProposition \\ref{prF}\nis stated for schemes of finite\ntype over a field $k$,\nit can be generalized to\nmore general schemes\nusing more general versions  of\nalteration.\n\n\\proof{\nLet\n\\begin{equation}\n\\begin{CD}\nX@<{p'}<< X'@<{j'}<<V\\\\\n@VfVV @VV{f'}V@VV{f'_U}V\\\\\nY@<p<<Y'@<j<<U\n\\end{CD}\n\\label{eqXYUpi}\n\\end{equation}\nbe a cartesian diagram of\nschemes where\n$p\\colon Y'\\to Y$ is\nproper and generically finite\nand $j\\colon U=Y\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} D\n\\to Y'$ is the open immersion\nof the complement of\na divisor with simple normal crossings.\nThen, by the assumption\nand the relative property,\nthe canonical morphism\n\\begin{equation}\n{\\cal F}'\\otimes f^{\\prime*}Rj_*\\Lambda\n\\to \nRj'_*j^{\\prime*}{\\cal F}'\n\\label{eqY'U}\n\\end{equation}\nis an isomorphism.\nBy projection formula\nand the proper base change theorem,\n(\\ref{eqY'U}) shows that\nthe canonical morphism\n\\begin{equation}\n{\\cal F}\\otimes f^*R(pj)_*\\Lambda\n\\to \nR(p'j')_*(p'j')^*{\\cal F}\n\\label{eqYU}\n\\end{equation}\nis an isomorphism.\n\n\n\nLet $s\\gets t$ be a specialization\nof geometric points of $Y$,\nlet $Y_{(s)}$ denote the strict localization\nand let\n\\begin{equation}\n\\begin{CD}\nX_s@>{i'_s}>>X\\times_YY_{(s)}@<{j'_t}<< X_t\\\\\n@V{f_s}VV@V{f_{(s)}}VV @VV{f_t}V\\\\\ns@>{i_s}>>Y_{(s)}@<{j_t}<<t\n\\end{CD}\n\\label{eqXYtpi}\n\\end{equation}\nbe the cartesian diagram.\nWe assume that $t$ is\nthe spectrum of an algebraic\nclosure of the residue field\nof the point of $Y$ below $t$.\nIt suffices to show\nthat the canonical morphism \n\\begin{equation}\ni^{\\prime*}_s{\\cal F}\n\\to \ni^{\\prime*}_sRj'_{t*}j^{\\prime*}_t{\\cal F}\n\\label{eqijst}\n\\end{equation}\nis an isomorphism.\n\nBy \\cite[Theorem 4.1]{dJ},\nwe may write $t$ as a limit\n$\\varprojlim_\\lambda\nU_\\lambda$\nof the complements $U_\\lambda=Y_\\lambda\\raisebox{2.33pt}{~\\rule{6.4pt}{1.3pt}~} \nD_\\lambda$,\nin proper and generically\nfinite schemes $Y_\\lambda$\nover $Y$ \nsmooth over finite extensions\nof $k$ of divisors $D_\\lambda\\subset\nY_\\lambda$ with simple normal crossings.\nThen, as the limit of\n(\\ref{eqYU}), the canonical morphism\n\\begin{equation}\n{\\cal F}\\otimes f_{(s)}^*\nRj_{t*}j^*_t\\Lambda\n\\to \nRj'_{t*}j^{\\prime*}_t{\\cal F}\n\\label{eqijst2}\n\\end{equation}\nis an isomorphism.\nSince\n$\\Lambda\\to\ni_s^*Rj_{t*}j^*_t\\Lambda$\nis an isomorphism,\nthe isomorphism\n(\\ref{eqijst2})\ninduces an isomorphism\n(\\ref{eqijst}).\n\\qed}\n\n\\begin{cor}\\label{corF}\nLet $f\\colon X\\to Y$ be a \n{\\em smooth} morphism of \nschemes of finite type over\na field $k$\nand let ${\\cal F}$\nbe a complex of $\\Lambda$-modules\non $X$.\nAssume that\nfor every cartesian diagram \n{\\rm (\\ref{eqXYpi})} satisfying the condition\nthere,\nthe immersion $h\\colon W\\to X'$\nis ${\\cal F}'$-transversal\nfor the pull-back ${\\cal F}'=p^{\\prime*}{\\cal F}$\nby $p'\\colon X'\\to X$.\nThen,\n$f$ is locally acyclic relatively to ${\\cal F}$.\n\\end{cor}\n\n\\proof{\nAssume that a cartesian diagram \n{\\rm (\\ref{eqF})} satisfies the condition\nthere.\nSince $f'$ is smooth,\nthe base change morphism\n$g^*Ri^!\\Lambda\\to Rh^!\\Lambda$\nis an isomorphism\nby the smooth base change theorem\n\\cite{smbc}.\nHence the isomorphism\n(\\ref{eqF}) means that\n$h$ is ${\\cal F}'$-transversal.\nThus it suffices to apply\nProposition \\ref{prF}.\n\\qed}\n\n\\subsection{${\\cal F}$-transversality and singular support}\\label{ssFs}\n\nLet $k$ be a field and\n$\\Lambda$ be a finite field\nof characteristic invertible in $k$.\n\n\\begin{pr}\\label{prfh}\nLet $X$ be a smooth scheme\nover a field $k$\nand \n$\\Lambda$ be a finite field\nof characteristic invertible in $k$.\nLet ${\\cal F}$ be a constructible complex\nof $\\Lambda$-modules on $X$.\nFor a closed conical subset $C\n\\subset T^*X$,\nwe consider the following conditions:\n\n{\\rm (1)}\n${\\cal F}$ is micro-supported on $C$.\n\n{\\rm (2)}\nThe support of ${\\cal F}$\nis a subset of\nthe base $B=C\\cap T^*_XX\\subset X$ \nof $C$ and\nevery $C$-transversal\nmorphism $h\\colon W\\to X$\nof smooth schemes\nof finite type\nis ${\\cal F}$-transversal.\n\n{\\rm 1.}\nIf ${\\cal F}$ is \nof finite tor-dimension,\nthen the condition {\\rm (1)} implies {\\rm (2)}.\n\n{\\rm 2.}\nIf $k$ is {\\em perfect},\nthe condition {\\rm (2)} implies {\\rm (1)}.\n\\end{pr}\n\n\\proof{\n1. \nBy Lemma \\ref{lmlcst}.3,\nthe condition that\n${\\cal F}$ is micro-supported on $C$\nimplies that the support of ${\\cal F}$\nis a subset of the base $B$ of $C$.\n\nWe show that a $C$-transversal morphism \n$h\\colon W\\to X$ of smooth schemes of finite type\nis ${\\cal F}$-transversal\nassuming that\n${\\cal F}$ is micro-supported on $C$.\nBy applying Lemmas \\ref{lmCh}.3 \nand \\ref{lmprpg}.3\nto the graph \n$W\\to W\\times X\\to X$ of $h$ and\nby replacing $h\\colon W\\to X, C$\nand ${\\cal F}$ by \n$W\\to W\\times X, {\\rm pr}_2^*C$ \nand ${\\rm pr}_2^*{\\cal F}$,\nwe may assume that $h$ is an immersion.\nSince the assertion is local on $W$,\nwe may assume that\nthere exists a cartesian diagram\n$$\\begin{CD}\n&W@>h>> X\\\\\n&@VVV @VVfV\\\\\n0=&y@>>>Y&={\\mathbf A}^d\n\\end{CD}$$\nwhere $f\\colon X\\to Y$ is smooth.\nSince $h\\colon W\\to X$ is\n$C$-transversal,\nby Lemma \\ref{lmCf}.1,\nwe may assume that $f\\colon X\\to Y$\nis $C$-transversal.\n\nSince ${\\cal F}$ is assumed\nto be micro-supported on $C$,\nthe morphism $f\\colon X\\to Y$ is locally\nacyclic relatively to ${\\cal F}$.\nHence the assertion follows from\nCorollary \\ref{corlat}.\n\n2.\nLet $h\\colon W\\to X$ and\n$f\\colon W\\to Y$\nbe a $C$-transversal pair of\nsmooth morphisms.\nIt suffices to show that\n$f\\colon W\\to Y$ is locally acyclic\nrelatively to $h^*{\\cal F}$.\nBy replacing ${\\cal F}$\nand $C$ by $h^*{\\cal F}$\nand $h^\\circ C$,\nwe may assume $W=X$.\nSince the base $B$ of $C$\ncontains the support of ${\\cal F}$,\nafter replacing $X$ by a neighborhood\nof the support of ${\\cal F}$,\nwe may assume that $f\\colon X\\to Y$\nis smooth by Lemma \\ref{lmCf}.6.\n\n\nWe show that the condition\nin Corollary \\ref{corF} is satisfied.\nLet (\\ref{eqF}) be a cartesian diagram\nsatisfying the condition there.\nSince $f\\colon X\\to Y$\nis smooth,\n$Y'$ and $W$ are smooth over \na finite extension of $k$\nand since $k$ is assumed perfect,\nthe schemes $X'$ and $W$ are smooth over $k$.\nBy Lemma \\ref{lmChf}.2,\nthe proper morphisms $p'\\colon X'\\to X$\nand $p'h\\colon W\\to X$ are\n$C$-transversal.\nThus, by the condition (2),\nthe morphisms $p'\\colon X'\\to X$\nand $p'h\\colon W\\to X$ are\n${\\cal F}$-transversal.\nHence by Lemma \\ref{lmprpg}.3,\n$h\\colon W\\to X'$ is \n$p^{\\prime *}{\\cal F}$-transversal.\nThus by Corollary \\ref{corF},\nthe morphism $f\\colon X\\to Y$ is\nlocally acyclic relatively to ${\\cal F}$.\n\\qed}\n\n\n\\begin{cor}\\label{corfh}\nAssume that\n$k$ is perfect\nand let ${\\cal F}$\nbe a constructible complex \nof $\\Lambda$-modules \nof finite tor-dimension on $X$.\nThen, \nthe following conditions are equivalent:\n\n{\\rm (1)}\n${\\cal F}$ is locally constant.\n\n{\\rm (2)}\nEvery morphism $h\\colon W\\to X$\nof finite type \nof smooth schemes\nis ${\\cal F}$-transversal.\n\\end{cor}\n\n\\proof{\nThe implication (1)$\\Rightarrow$(2)\nis Lemma \\ref{lmprpg}.2.\nBy Lemma \\ref{lmlcst},\nthe condition (1) is equivalent to\nthat ${\\cal F}$ is micro-supported\non the $0$-section $T^*_XX$.\nHence, it follows from Proposition \\ref{prfh}.\n\\qed}\n\n\n\\begin{cor}\\label{corpSS}\nLet ${\\cal F}$ be a constructible\ncomplex of $\\Lambda$-modules\nof finite tor-dimension on $X$\nand let $C=SS{\\cal F}$ denote the\nsingular support.\nThen, for a properly $C$-transversal\nmorphism $h\\colon W\\to X$\nof smooth schemes of\nfinite type over a perfect $k$,\nwe have \n\\begin{equation}\nSS h^*{\\cal F}\n=SS Rh^!{\\cal F}=h^{\\circ}SS{\\cal F}.\n\\label{eqpSS}\n\\end{equation}\n\\end{cor}\n\n\\proof{\nWe may assume that $\\Lambda$ is a field\nby Corollary \\ref{corss}.2.\nBy Proposition \\ref{prfh}.1,\n$h$ is ${\\cal F}$-transversal. \nFirst, we assume that ${\\cal F}$ is \na perverse sheaf.\nThen $h^*{\\cal F}[\\dim W-\\dim X]$\nis also a perverse sheaf\nby Lemma \\ref{lmprpg}.5\nand $SS h^*{\\cal F}$\nis the support of\n$CC h^*{\\cal F}$ by Proposition \\ref{prperv}.2.\nSince\n$CC h^*{\\cal F}=h^!CC{\\cal F}$ by Theorem \\ref{thmi*}\nand $CC{\\cal F}\\geqq 0$ by\nProposition \\ref{prperv}.1, the assertion\nfollows in this case.\nThe general case \nis reduced to the case where\n${\\cal F}$ is a perverse sheaf by \nthe equality (\\ref{eqssu}) for \n${\\cal F}$ and $h^*{\\cal F}$.\n\\qed}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThroughout, we work over the complex numbers ${\\mathbb{C}}$. \n\n\\subsection{The P=W conjecture}\nThe purpose of this paper is to present a proof of the $P=W$ conjecture by de Cataldo--Hausel--Migliorini \\cite{dCHM1} for arbitrary rank $n$ and genus $g \\geq 2$.\n\nLet $C$ be a nonsingular irreducible projective curve of genus $g \\geq 2$. For two coprime integers $n \\in {\\mathbb{Z}}_{\\geq 1}$ and $d \\in {\\mathbb{Z}}$, there are two moduli spaces $M_{\\mathrm{Dol}}$ and $M_B$, called the Dolbeault and the Betti moduli spaces, attached to $C,n,$ and $d$. \n\n\nThe Dolbeault moduli space parameterizes stable Higgs bundles $({\\mathcal E}, \\theta)$ on $C$, where ${\\mathcal E}$ is a vector bundle on $C$ of rank $n$ and degree $d$, $\\theta: {\\mathcal E} \\to {\\mathcal E} \\otimes \\Omega_C^1$ is a Higgs field, and stability is defined with respect to the slope $\\mu({\\mathcal E},\\theta) ={\\mathrm{deg}({\\mathcal E})}\/{\\mathrm{rk({\\mathcal E})}}$. The moduli space $M_{\\mathrm{Dol}}$ admits the structure of a completely integrable system\n\\[\nh: M_{\\mathrm{Dol}} \\to A := \\bigoplus_{i=1}^n H^0(C, {\\Omega_C^{1}}^{\\otimes i}), \\quad ({\\mathcal E}, \\theta) \\mapsto \\mathrm{char.polynomial}(\\theta),\n\\]\nwhich is referred to as the Hitchin system \\cite{Hit, Hit1}. The Hitchin map $h$ is surjective and proper; it is also Lagrangian with respect to the canonical holomorphic symplectic form on $M_{\\mathrm{Dol}}$ induced by the hyper-K\\\"ahler metric. The \\emph{perverse filtration} is an increasing filtration \n\\[\nP_0H^*(M_{\\mathrm{Dol}}, {\\mathbb{Q}})\\subset P_1H^*(M_{\\mathrm{Dol}}, {\\mathbb{Q}}) \\subset \\cdots \\subset H^*(M_{\\mathrm{Dol}}, {\\mathbb{Q}})\n \\]\non the (singular) cohomology of $M_{\\mathrm{Dol}}$ governed by the topology of the Hitchin system $h$; see Section \\ref{sec1.1} for a brief review.\n\nThe Betti moduli space $M_B$ is the (twisted) character variety associated with $\\mathrm{GL}_n({\\mathbb{C}})$ and degree $d$.  It parametrizes isomorphism classes of irreducible local systems\n\\[\n\\rho: \\pi_1(C\\backslash \\{p\\}) \\rightarrow \\mathrm{GL}_n({\\mathbb{C}})\n\\]\nwhere $\\rho$ sends a loop around a chosen point $p$ to \n$e^{\\frac{2\\pi \\sqrt{-1} d}{n}}\\mathrm{Id}_n$.  Concretely, we have\n\\begin{equation*}\nM_B := \\Big{\\{}a_k, b_k \\in \\mathrm{GL}_n({\\mathbb{C}}),~k=1,2,\\dots,g: ~~\\prod_{j=1}^g [a_j, b_j] = e^{\\frac{2\\pi \\sqrt{-1} d}{n}}\\mathrm{Id}_n \\Big{\\}}\\git\\mathrm{GL}_n({\\mathbb{C}}).\n\\end{equation*}\nIt is an affine variety whose mixed Hodge structure admits a nontrivial weight filtration \n\\[\nW_0 H^*(M_B, {\\mathbb{Q}}) \\subset W_1H^*(M_B, {\\mathbb{Q}}) \\subset \\cdots \\subset H^*(M_B, {\\mathbb{Q}}).\n\\]\n\nNon-abelian Hodge theory \\cite{Simp, Si1994II} gives a canonical diffeomorphism between the two very different algebraic varieties $M_{\\mathrm{Dol}}$ and $M_B$, which identifies their cohomology:\n\\begin{equation}\\label{NAH}\n H^*(M_{\\mathrm{Dol}}, {\\mathbb{Q}})  = H^*(M_{B}, {\\mathbb{Q}}).\n\\end{equation}\nThe $P=W$ conjecture by de Cataldo--Hausel--Migliorini \\cite{dCHM1} \nrefines the identification (\\ref{NAH}); it predicts that the perverse filtration associated with the Hitchin system is matched with the (double-indexed) weight filtration associated with the Betti moduli space. This establishes a surprising connection between topology of Hitchin systems and Hodge theory of character varieties.\n\n\\begin{conj}\\label{conj}[The P=W conjecture for $\\mathrm{GL}_n$, \\cite{dCHM1}] For any $k,m\\in {\\mathbb{Z}}_{\\geq 0}$, we have\n\\[\nP_kH^m(M_{\\mathrm{Dol}}, {\\mathbb{Q}}) = W_{2k}H^m(M_{B}, {\\mathbb{Q}}) = W_{2k+1}H^m(M_B, {\\mathbb{Q}}).\n\\]\n\\end{conj}\n\nConjecture \\ref{conj} has previously been proven for $n=2$ and arbitrary genus $g\\geq 2$ by de Cataldo--Hausel--Migliorini \\cite{dCHM1}, and recently for arbitrary rank $n$ and genus 2 by de Cataldo--Maulik--Shen \\cite{dCMS}. The compatibility between the $P=W$ conjecture and Galois conjugation on the Betti side was proven in \\cite{dCMSZ}, which implies that $P=W$ does not depend on the degree $d$ as long as it is coprime to $n$. We refer to \\cite[Section 1]{dCHM1} and the paragraphs following \\cite[Theorem 0.2]{dCMS} for discussions concerning connections between Conjecture \\ref{conj} and other directions. In particular, by \\cite{CDP} and \\cite[Section 9.3]{MT}, Conjecture \\ref{conj} implies the correspondence between Gopakumar--Vafa invariants and Pandharipande--Thomas invariants \\cite[Conjecture 3.13]{MT} for the local Calabi--Yau 3-fold $T^*C \\times {\\mathbb{C}}$ in any curve class $n[C]$. \n\n\nThe main result of this paper is a full proof of Conjecture \\ref{conj}.\n\n\\begin{thm}\\label{thm}\n    Conjecture \\ref{conj} holds.\n\\end{thm}\n\n\nConjecture \\ref{conj} has several variants which, to the best of our knowledge, are still open. These include the version formulated for possibly singular moduli spaces, intersection cohomology, and general reductive groups \\cite{dCHM1, dCM, FM}, and the version formulated for moduli stacks \\cite{Davison}. There also exist parabolic versions of Conjecture \\ref{conj} and we expect that our argument applies to these settings using parabolic variants of the ingredients here \\cite{Mellit, OY}.\n\n\nFor the case of type $A_{n-1}$ ($\\mathrm{GL}_n, \\mathrm{PGL}_n, \\mathrm{SL}_n$) and a degree $d$ coprime to $n$, Conjecture \\ref{conj} (\\emph{i.e.} the $P=W$ conjecture for $\\mathrm{GL_n}$) is equivalent to the $P=W$ conjecture for $\\mathrm{PGL}_n$; see the paragraph after \\cite[Theorem 0.2]{dCMS}. The case of $\\mathrm{SL}_n$ is more subtle --- it is closely related to the endoscopic decomposition of the $\\mathrm{SL}_n$ Hitchin moduli spaces. A systematic discussion on this aspect can be found in \\cite[Section 5]{MS}. In the special case when $n=p$ a prime number, \\cite[Theorem 0.2]{SLn} implies that the $P=W$ conjecture for $\\mathrm{GL}_p$, $\\mathrm{SL}_p$, and $\\mathrm{PGL}_p$ are all equivalent. In particular, we have the following immediate consequence of Theorem \\ref{thm}, which generalizes the result of \\cite{dCHM1} for $\\mathrm{SL}_2$.\n\n\\begin{thm}\nThe $P=W$ conjecture holds for a curve $C$ of any genus at least two and $\\mathrm{SL}_p$ with $p$ a prime number.\n\\end{thm}\n\n\n\n\n\n\n\\subsection{Idea of the proof}\nOur proof of Conjecture \\ref{conj} has 4 major steps.\n\n\\begin{enumerate}\n    \\item[Step 1.] \\emph{Strong perversity of Chern classes}: Using work of Markman \\cite{Markman}, Shende \\cite{Shende}, and Mellit \\cite{Mellit}, we may reduce the $P=W$ conjecture to a statement about the interaction between Chern classes of the universal family and the perverse filtration. This allows us to reduce the $P=W$ conjecture using a sheaf-theoretic statement which concerns strong perversity of Chern classes.\n    \\item[Step 2.] \\emph{Vanishing cycle techniques}: We apply the formalism of vanishing cycles to reduce the sheaf-theoretic formulation of Step 1 for the Hitchin system, to the case of twisted Hitchin systems associated with meromorphic Higgs bundles. Our motivation is from the key observation by Ng\\^o \\cite{Ngo} and Chaudouard-Laumon \\cite{CL}, that the decomposition theorem for such twisted Hitchin systems is more manageable. \n    \n    Steps 1 and 2 are carried out in Sections 1 and 2.  \n\n    \\item[Step 3.]\\emph{Global Springer theory}: The global Springer theory of Yun \\cite{Yun1, Yun2,Yun3} produces rich symmetries for certain Hitchin moduli spaces; this proves the sheaf-theoretic formulation of Step 1 over the \\emph{elliptic locus}. For our purpose, we need a version of global Springer theory for the stable locus over the total Hitchin base. This is described in Section 3.\n\n    \\item[Step 4.] \\emph{A support theorem}: Lastly, in order to extend part of Yun's symmetries induced by Chern classes from the elliptic locus to the total Hitchin base, we prove a support theorem parallel to \\cite{CL} for certain parabolic Hitchin moduli spaces. This is completed in Section 4.\n\\end{enumerate}\n\nThe idea of combining vanishing cycle functors and support theorems was also applied in \\cite{MS} to give a proof of the topological mirror symmetry conjecture of Hausel--Thaddeus \\cite{HT, GWZ}.\n\n\\subsection{Acknowledgements}\nWe would like to thank Mark Andrea de Cataldo, Bhargav Bhatt, and Max Lieblich for various discussions.  We are especially grateful to Zhiwei Yun for explaining his thesis to us, both in his office and at the playground. J.S. was supported by the NSF grant DMS-2134315. \n\n\n\n\\section{Perverse filtrations and vanishing cycles}\n\nIn this section, we introduce the notion of \\emph{strong perversity} and show its compatibility with vanishing cycle functors (Proposition \\ref{prop1.4}). This plays a crucial role in Section 2 in lifting the $P=W$ conjecture sheaf-theoretically and reduce it to the twisted case.\n\n\\subsection{Perverse filtrations}\\label{sec1.1}\nLet $f: X \\to Y$ be a proper morphism between irreducible nonsingular quasi-projective varieties (or Deligne--Mumford stacks) with $\\mathrm{dim}X = a$ and $\\mathrm{dim}Y = b$. Let $r$ be the defect of semismallness of $f$:\n\\[\nr: = \\mathrm{dim} X \\times_YX - \\mathrm{dim}X.\n\\]\nIn particular, we have $r = a-b$ when $f$ has equi-dimensional fibers. The perverse filtration \n\\[\nP_0H^m(X, {\\mathbb{Q}}) \\subset P_1H^m(X, {\\mathbb{Q}}) \\subset \\dots \\subset H^m(X, {\\mathbb{Q}})\n\\]\nis an increasing filtration on the cohomology of $X$ governed by the topology of the morphism $f$; it is defined to be\n\\[\nP_iH^m(X, {\\mathbb{Q}}) := \\mathrm{Im}\\left\\{ H^{m-a+r}(X, {^\\mathbf{p}\\tau_{\\leq i}} (Rf_* {\\mathbb{Q}}_X[a-r])) \\to H^m(X, {\\mathbb{Q}})\\right\\}\n\\]\nwhere $^\\mathbf{p}\\tau_{\\leq * }$ is the perverse truncation functor \\cite{BBD}.\n\nA cohomology class $\\gamma \\in H^l(X, {\\mathbb{Q}})$ can be viewed as a morphism $\\gamma: {\\mathbb{Q}}_X \\to {\\mathbb{Q}}_X[l]$, which naturally induces \n\\begin{equation}\\label{induced}\n\\gamma: Rf_* {\\mathbb{Q}}_X \\to Rf_* {\\mathbb{Q}}_X [l]\n\\end{equation}\nafter pushing forward along $f$. For an integer $c\\geq 0$, we say that $\\gamma \\in H^l(X, {\\mathbb{Q}})$ has \\emph{strong perversity} $c$ with respect to $f$ if its induced morphism (\\ref{induced}) satisfies\n\\begin{equation}\\label{assumption}\n\\gamma \\left({^\\mathbf{p}\\tau_{\\leq i} Rf_* {\\mathbb{Q}}_X}\\right) \\subset {^\\mathbf{p}\\tau_{\\leq i+(c-l)}} \\left(Rf_* {\\mathbb{Q}}_X  [l] \\right), \\quad \\forall i;\n\\end{equation}\nmore precisely, the condition (\\ref{assumption}) says that the composition\n\\[\n{^\\mathbf{p}\\tau_{\\leq i} Rf_* {\\mathbb{Q}}_X} \\hookrightarrow Rf_* {\\mathbb{Q}}_X \\xrightarrow{\\gamma} Rf_* {\\mathbb{Q}}_X [l]\n\\]\nfactors through ${^\\mathbf{p}\\tau_{\\leq i+(c-l)}} \\left(Rf_* {\\mathbb{Q}}_X  [l] \\right) \\hookrightarrow Rf_* {\\mathbb{Q}}_X  [l]$ for any $i$.  Notice that $\\gamma$ automatically has strong perversity $l$, so this condition is interesting only when $c < l$.\n\n\\begin{lem}\\label{lem1.1}\nIf $\\gamma\\in H^l(X, {\\mathbb{Q}})$ has strong perversity $c$ with respect to $f$, then taking cup-product with $\\gamma$ satisfies\n\\[\n\\gamma\\cup - : H^m(X, {\\mathbb{Q}}) \\to H^{m+l}(X, {\\mathbb{Q}}), \\quad P_iH^m(X, {\\mathbb{Q}}) \\mapsto P_{i+c}H^{m+l}(X, {\\mathbb{Q}}).\n\\]\n\\end{lem}\n\n\\begin{proof}\nThis follows from taking global cohomology for (\\ref{assumption}) and noticing that\n\\begin{align*}\n    H^{m-a+r}\\left(Y, {^\\mathbf{p}\\tau_{\\leq i+(c-l)}} \\left(Rf_* {\\mathbb{Q}}_X  [l+a-r] \\right)\\right) & \\\\ = H^{(m-a+r)+l}&\\left(Y, {^\\mathbf{p}\\tau_{\\leq i+c}} \\left(Rf_* {\\mathbb{Q}}_X  [a-r] \\right)\\right). \\qedhere\n    \\end{align*}\n\\end{proof}\n\nIn general, the perverse filtration $P_\\bullet H^*(X, {\\mathbb{Q}})$ may not be multiplicative, \\emph{i.e.}, for $\\gamma_j \\in P_{c_j}H^*(X, {\\mathbb{Q}})$ (j=1,2,\\dots,s), it may not be true that\n\\[\n\\gamma_1 \\cup \\gamma_2 \\cup \\cdots \\cup \\gamma_s \\in P_{c_1+\\cdots+c_s}H^*(X, {\\mathbb{Q}});\n\\]\nsee \\cite[Exercise 5.6.8]{Park}. In fact, it was proven in \\cite[Theorem 0.6]{dCMS} that Conjecture \\ref{conj} is equivalent to the multiplicativity of the perverse filtration associated with the Hitchin system. The following easy observation illustrates the advantage of considering strong perversity in view of the multiplicativity issue.\n\n\\begin{lem}\\label{lem1.2}\nIf the class $\\gamma_j \\in H^{l_j}(X, {\\mathbb{Q}})$ ($j=1,2,\\dots, s$) has strong perversity $c_j$ with respect to $f$, then the cup product \n\\[\n\\gamma_1 \\cup \\gamma_2 \\cup \\cdots \\cup \\gamma_s \\in H^{l_1+\\cdots+l_s}(X, {\\mathbb{Q}})\n\\]\nhas strong perversity $\\sum_jc_j$.\n\\end{lem}\n\n\n\\subsection{Vanishing cycles}\\label{Sec1.2}\n\nThroughout section \\ref{Sec1.2}, we let $g: X \\to {\\mathbb{A}}^1$ be a morphism such that $X$ is nonsingular and irreducible with $X_0 = g^{-1}(0)$ the closed fiber over $0 \\in {\\mathbb{A}}^1$. We consider the vanishing cycle functor \n\\[\n\\varphi_g: D^b_c(X) \\rightarrow D^b_c(X_0)\n\\]\nwhich preserves the perverse $t$-structures,\n\\[\n\\varphi_g: \\mathrm{Perv}(X) \\rightarrow \\mathrm{Perv}(X_0).\n\\]\nHere $\\mathrm{Perv}(-)$ stands for the abelian category of perverse sheaves. We denote by \n\\[\n\\varphi_g : = \\varphi_g(\\mathrm{IC}_X) = \\varphi_g({\\mathbb{Q}}_X[\\mathrm{dim}X]) \\in \\mathrm{Perv}(X_0) \\]\nthe perverse sheaf of vanishing cycles. We use $X' \\subset X$ to denote the support of the vanishing cycle complex $\\varphi_g$ so that $\\varphi_g \\in \\mathrm{Perv}(X')$.\n\nRecall that for any bounded constructible object ${\\mathcal K} \\in D^b_c(X)$, a class $\\gamma \\in H^l(X, {\\mathbb{Q}})$ induces a morphism\n\\[\n\\gamma: {\\mathcal K} \\to {\\mathcal K} [l] \n\\]\nvia taking the tensor product with $\\gamma: {\\mathbb{Q}}_X \\to {\\mathbb{Q}}_X[l]$. The following lemma shows the compatibility between the vanishing cycle functor and restriction of cohomology classes.\n\n\\begin{lem}\\label{lem1.3}\nWith the same notation as above, let $i: X'\\hookrightarrow X$ be the closed embedding. The morphism \\[\ni^*\\gamma: \\varphi_g \\rightarrow \\varphi_g[l] \\in D^b_c(X')\n\\]\ninduced by the class $i^*\\gamma$ (applied to the object ${\\mathcal K} = \\varphi_g$) coincides with \n\\[\n\\varphi_g(\\gamma): \\varphi_g \\to \\varphi_g[l] \\in D_c^b(X')\n\\]\nobtained by applying the functor $\\varphi_g$ to $\\gamma: {\\mathbb{Q}}_X \\to {\\mathbb{Q}}_X[l]$.\n\\end{lem}\n\n\\begin{proof}\nLet $\\iota: X_0 \\hookrightarrow X$ be the closed embedding of the closed fiber over $0$. By \\cite[Definition 8.6.2]{KS}, the vanishing cycle functor can be written as\n\\begin{equation}\\label{vanishing}\n\\varphi_g(-) = \\iota^* R{\\mathcal H}{om}({\\mathcal C}, -) \\in D^b_c(X_0)\n\\end{equation}\nwith ${\\mathcal C}$ a fixed complex of sheaves. The morphism obtained\nby applying $R{\\mathcal H}{om}({\\mathcal C}, -)$ to $\\gamma: {\\mathbb{Q}}_X \\to {\\mathbb{Q}}_X[l]$ \nis equivalent to the morphism induced by applying $\\gamma$ to $R{\\mathcal H}{om}({\\mathcal C}, {\\mathbb{Q}}_X)$.\nSimilarly, the functor $\\iota^*$ sends the morphism induced by $\\gamma$\nto the morphism induced by $\\iota^*\\gamma$.\n\nTherefore, if we denote by $\\iota': X' \\hookrightarrow X$ the closed embedding and view $\\varphi_g$ as a perverse sheaf on $X'$, we have an equivalent morphisms\n\\begin{equation}\\label{321}\n\\varphi_g(\\gamma) = \\iota^*\\gamma:  \\iota'_*\\varphi_g \\to \\iota'_*\\varphi_g[l] \\in D^b_c(X_0). \n\\end{equation}\n\n\nFinally, after applying $\\iota'^*: D^b_c(X_0) \\rightarrow D^b_c(X')$ to (\\ref{321}) and noticing $\\iota'^*\\iota'_*= \\mathrm{id}$, we obtain that the class $i^*\\gamma = \\iota'^*\\iota^*\\gamma$ induces\n\\[\n\\varphi_g(\\gamma): \\varphi_g \\to \\varphi_g[l] \\in D_c^b(X').\n\\]\nThis completes the proof.\n\\end{proof}\n\n\n\\begin{prop}\\label{prop1.4}\nLet $g: X \\to {\\mathbb{A}}^1$ and $X'$ be as above. Assume that $X'$ is nonsingular and\n\\begin{equation}\\label{4.1_0}\n\\varphi_g \\simeq \\mathrm{IC}_{X'} = {\\mathbb{Q}}_{X'}[\\mathrm{dim}X'] \\in \\mathrm{Perv}(X').\n\\end{equation}\nAssume further that we have the commutative diagram\n\\begin{equation*}\n\\begin{tikzcd}\nX' \\arrow[r, \"i\"] \\arrow[d, \"f'\"]\n& X \\arrow[d, \"f\"] \\\\\nY' \\arrow[r]\n& Y\n\\end{tikzcd}\n\\end{equation*}\nsuch that $f$ is proper and $g =  \\nu \\circ f$ with $\\nu: Y \\to {\\mathbb{A}}^1$. Then if a class $\\gamma \\in H^l(X, {\\mathbb{Q}})$ has strong perversity $c$ with respect to $f$, its restriction $i^*\\gamma \\in H^l(X', {\\mathbb{Q}})$ has strong perversity $c$ with respect to $f'$. \n\\end{prop}\n\n\n\\begin{proof}\nBy definition, the morphism \n\\begin{equation}\\label{1.4_1}\n\\gamma: Rf_* {\\mathbb{Q}}_X \\to Rf_* {\\mathbb{Q}}_X[l] \\in D^b_c(Y)\n\\end{equation}\ninduced by $\\gamma$ satisfies\n\\begin{equation}\\label{1.4_2}\n\\gamma \\left({^\\mathbf{p}\\tau_{\\leq i} Rf_* {\\mathbb{Q}}_X}\\right) \\subset {^\\mathbf{p}\\tau_{\\leq i+(c-l)}} (Rf_* {\\mathbb{Q}}_X[l]), \\quad \\forall i.\n\\end{equation}\nNow we apply the vanishing cycle functor $\\varphi_\\nu$ to (\\ref{1.4_1}). On one hand, we have the base change $Rf'_* \\circ \\varphi_g \\simeq \\varphi_\\nu\\circ Rf_*$ and the fact that the vanishing cycle functor preserves the perverse $t$-structures. Therefore (\\ref{1.4_2}) implies  that the morphism\n\\[\n\\varphi_g(\\gamma): Rf'_* \\varphi_g \\to Rf'_* \\varphi_g[l] \\in D^b_c(X')\n\\]\nsatisfies\n\\[\n\\varphi_g(\\gamma) \\left({^\\mathbf{p}\\tau_{\\leq i}} Rf'_* \\varphi_g\\right) \\subset {^\\mathbf{p}\\tau_{\\leq i+(c-l)}} (Rf'_* \\varphi_g[l]), \\quad \\forall i.\n\\]\nOn the other hand, using the isomorphism (\\ref{4.1_0}) and Lemma \\ref{lem1.3}, the above equation means precisely that the morphism $i^*\\gamma : {\\mathbb{Q}}_{X'} \\to {\\mathbb{Q}}_{X'}[l]$ satisfies\n\\begin{equation*}\n(i^*\\gamma) \\left({^\\mathbf{p}\\tau_{\\leq i} Rf'_* {\\mathbb{Q}}_{X'}}\\right) \\subset {^\\mathbf{p}\\tau_{\\leq i+(c-l)}} (Rf'_* {\\mathbb{Q}}_{X'}[l]), \\quad \\forall i;\n\\end{equation*}\nthat is, the class $i^*\\gamma$ has strong perversity $c$ with respect to $f'$.\n\\end{proof}\n\n\n\n\n\n\n\\section{Strong perversity for Chern classes}\n\nIn this section, we fix the rank $n$ and the degree $d$ with $(n,d)=1$. In Theorem \\ref{conj2.7}, we rephrase and then enhance Conjecture \\ref{conj} to a statement involving ${\\mathcal L}$-twisted Hitchin systems and strong perversity of Chern classes. It will be proven in Sections 3 and 4.\n\n\\subsection{Tautological classes}\n\nAs discussed in \\cite[Section 0.3]{dCMS}, the $P=W$ conjecture for $\\mathrm{GL}_n$ can be reduced to a statement involving tautological classes on $M_{\\mathrm{Dol}}$ and the perverse filtration associated with $h: M_{\\mathrm{Dol}} \\to A$, without reference to the Betti moduli $M_B$. In this subsection, we recall this reduction step.\n\n\nFor convenience, we work with the $\\mathrm{PGL}_n$ Dolbeault moduli space to avoid normalization of a universal family as in \\cite{dCMS}; we refer to \\cite{Shende} for a detailed discussion concerning the formulation of the $P=W$ conjecture in terms of tautological classes for the $\\mathrm{PGL}_n$ Dolbeault moduli space.\n\nFix ${\\mathcal N} \\in \\mathrm{Pic}^d(C)$. Let $\\widecheck{M}_{\\mathrm{Dol}}$ be \nthe moduli stack of stable Higgs bundles $({\\mathcal E}, \\theta)$ with $\\mathrm{det}({\\mathcal E}) \\simeq {\\mathcal N}$ and $\\mathrm{trace}(\\theta) = 0$, rigidified with respect to the generic $\\mu_n$-stabilizer; this is the same as taking its coarse moduli space.  We refer to this (nonsingular) variety as the $\\mathrm{SL}_n$ Dolbeault moduli space of degree $d$. The finite group $\\Gamma: = \\mathrm{Pic}^0(C)[n]$ acts naturally on $\\widecheck{M}_{\\mathrm{Dol}}$ via tensor product. The $\\mathrm{PGL}_n$ Dolbeault moduli space of degree $d$ is recovered by taking the quotient stack\n\\begin{equation}\\label{233}\n\\widehat{M}_{\\mathrm{Dol}} : = \\widecheck{M}_{\\mathrm{Dol}}\/\\Gamma\n\\end{equation}\nwhich is a nonsingular Deligne--Mumford stack. The $\\mathrm{PGL}_n$ Hitchin system \n\\[\n\\widehat{h}: \\widehat{M}_{\\mathrm{Dol}} \\to \\widehat{A}: = \\bigoplus_{i=2}^n H^0(C, {\\Omega_C^1}^{\\otimes i})\n\\]\nis induced by the Hitchin map associated with $\\widecheck{M}_{\\mathrm{Dol}}$ as the $\\Gamma$-action is fiberwise with respect to $h$. Analogous to the $\\mathrm{GL}_n$  case, we have the perverse filtration $P_*H^*(\\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}})$ associated with $\\widehat{h}$. The universal $\\mathrm{PGL}_n$-bundle ${\\mathcal U}$ on $C \\times \\widehat{M}_{\\mathrm{Dol}}$ induces Chern characters\n\\[\n\\mathrm{ch}_k({\\mathcal U}) \\in H^{2k}(C \\times \\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}}),\\quad k \\geq 2. \n\\]\nThe tautological classes $c_k(\\gamma)$ are defined to be\n\\[\nc_k(\\gamma): = \\int_\\gamma \\mathrm{ch}_k({\\mathcal U}) = q_{M*}(q_C^*\\gamma \\cup \\mathrm{ch}_k({\\mathcal U})) \\in H^*(\\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}}), \\quad \\gamma \\in H^*(C,{\\mathbb{Q}}),\n\\]\nwhere $q_{(-)}$ are the projections from $C \\times \\widehat{M}_{\\mathrm{Dol}}$.\n\nNow we consider the $\\mathrm{PGL}_n$ Betti moduli space of degree $d$ with the isomorphism on cohomology provided by non-abelian Hodge theory (\\emph{c.f.}\\cite[Theorem 1.2.4]{dCHM1}):\n\\begin{equation}\\label{NAH2}\nH^*(\\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}}) = H^*(\\widehat{M}_{B}, {\\mathbb{Q}}).\n\\end{equation}\nDefine the Hodge sub-vector space\n\\[\n^k\\mathrm{Hdg}^m(\\widehat{M}_B): = W_{2k}H^m(M_B, {\\mathbb{Q}}) \\cap F^kH^m(M_B, {\\mathbb{C}}) \\subset H^m(\\widehat{M}_{B}, {\\mathbb{Q}}).\n\\]\n\nThe following theorem, collecting results of Markman and Shende, provides a complete description of $H^*(\\widehat{M}_B, {\\mathbb{Q}})$ in terms of the Chern classes $\\mathrm{ch}_k({\\mathcal U})$ and the weight filtration.\n\n\\begin{thm}[\\cite{Markman, Shende}]\\label{thm2.1}\nWe use the same notation as above.\n\\begin{enumerate}\n    \\item[(i)] The tautological classes $c_k(\\gamma)\\in H^*(\\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}})$ generate $H^*(\\widehat{M}_\\mathrm{Dol}, {\\mathbb{Q}})$ as a ${\\mathbb{Q}}$-algebra.\n    \\item[(ii)] The class $c_k(\\gamma)$, passing through the non-abelian Hodge correspondence (\\ref{NAH2}), lies in $^k\\mathrm{Hdg}^*(\\widehat{M}_B)$. In particular, we have a canonical decomposition\n    \\[\nH^*(\\widehat{M}_B, {\\mathbb{Q}}) = \\bigoplus_{m,k} {^k\\mathrm{Hdg}^m(\\widehat{M}_B)}.    \\]\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\nThe first part was proven in \\cite{Markman}, and the second part was proven in \\cite{Shende}.\n\\end{proof}\n\n\nTheorem \\ref{thm2.1} (ii) yields immediately that \n\\[\nW_{2k}H^m(\\widehat{M}_B, {\\mathbb{Q}}) = W_{2k+1}H^m(\\widehat{M}_B, {\\mathbb{Q}}).\n\\]\nMoreover, by Theorem \\ref{thm2.1}, the $P=W$ conjecture implies that each class $\\prod_{i=1}^s c_{k_i}(\\gamma_i)$ lies in the perverse piece $P_{\\Sigma_i{k_i}}H^*(\\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}})$; the latter is in fact equivalent to the $P=W$ conjecture. Indeed, suppose we have \n\\begin{equation}\\label{taut}\n\\prod_{i=1}^s c_{k_i}(\\gamma_i) \\in P_{\\Sigma_i{k_i}}H^*(\\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}})\n\\end{equation}\nfor any product of tautological classes.\nThen we know that $W_{2k}H^*(\\widehat{M}_B, {\\mathbb{Q}}) \\subset P_kH^*(\\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}})$. The curious hard Lefschetz theorem proven by Mellit \\cite{Mellit} forces the two filtrations $P_\\bullet$ and $W_{2\\bullet}$ to coincide as long as one contains the other. This was mentioned in the last paragraph of \\cite[Section 1]{Mellit}; we include its proof here for the reader's convenience.\n\n\\begin{lem}\nWe denote by $V$ the ${\\mathbb{Q}}$-vector space (\\ref{NAH2}) with the perverse and the weight filtrations $P_\\bullet$ and $W_{2\\bullet}$. If $W_{2k} \\subset P_k$ for all $k$, then $W_{2k} = P_k$ for all $k$.\n\\end{lem}\n\n\\begin{proof}\nAssume $r = \\mathrm{dim}\\widehat{M}$; both filtrations terminate at the $r$-th pieces, \\emph{i.e.}, $W_{2r} = P_{r} = V$. We first show that \n\\[\nW_0 = P_0, \\quad W_{2(r-1)} = P_{r-1}. \n\\]\nIn fact, since $W_\\bullet  \\subset P_\\bullet$, we have\n\\[\n\\mathrm{dim} W_0 \\leq \\mathrm{dim} P_0 = \\mathrm{dim}V\/P_{r-1} \\leq \\mathrm{dim} V\/W_{2(r-1)};\n\\]\nmoreover, by the curious hard Lefschetz theorem, each inequality has to be an equality. So our claim follows. \n\nThen we proceed by applying the same argument to $W_1, P_1$ and $W_{2(r-2)}, P_{r-1}$. The lemma follows by a simple induction.\n\\end{proof}\n\n\nIn conclusion, we have reduced the $P=W$ conjecture to the following:\n\n\n\\begin{conj}[Equivalent version of P=W]\\label{conj2.2}\nCondition (\\ref{taut}) holds for all products of tautological classes.\n\\end{conj}\n\n\\subsection{Strong perversity for Chern classes for ${\\mathcal L}$-twisted Hitchin systems}\n\nFor our purposes, it is important to consider Dolbeault moduli spaces of Higgs bundles, twisted by an effective line bundle ${\\mathcal L}$ (\\emph{i.e.}, $H^0(C, {\\mathcal L}) \\neq 0$). These moduli spaces have already appeared in \\cite{CL, HT2, HT, Yun1, Yun2}; we review the construction here briefly. \n\nSet $\\Omega_{\\mathcal L}$ to be the line bundle $\\Omega^1_C \\otimes {\\mathcal L}$ on the curve $C$. We denote by $M^{\\mathcal L}_{\\mathrm{Dol}}$ the moduli space of stable twisted Higgs bundles\n\\[\n({\\mathcal E}, \\theta), \\quad \\theta: {\\mathcal E} \\to {\\mathcal E} \\otimes\\Omega_{\\mathcal L}, \\quad \\mathrm{rk}({\\mathcal E}) = n, ~~\\mathrm{deg}({\\mathcal E})=d,\n\\]\nwith respect to the slope stability condition. The corresponding Hitchin map\n\\[\nh^{\\mathcal L}: M^{\\mathcal L}_{\\mathrm{Dol}} \\to A^{\\mathcal L}: = \\bigoplus_{i=1}^n H^0\\left(C, {\\Omega_{\\mathcal L}}^{\\otimes i} \\right) ,\\quad ({\\mathcal E}, \\theta) \\mapsto \\mathrm{char.polynomial}(\\theta),\n\\]\nis still proper as in the untwisted case; but it fails to be a Lagrangian fibration when $\\mathrm{deg}({\\mathcal L})>0$. The ${\\mathcal L}$-twisted $\\mathrm{SL}_n$ and $\\mathrm{PGL}_n$ Dolbeault moduli spaces $\\widecheck{M}^{\\mathcal L}_{\\mathrm{Dol}}$ and $\\widehat{M}^{\\mathcal L}_{\\mathrm{Dol}}$ can be constructed similarly. The moduli space\n\\[\n\\widecheck{M}^{\\mathcal L}_{\\mathrm{Dol}}:= \\{({\\mathcal E}, \\theta) \\in M^{\\mathcal L}_{\\mathrm{Dol}}|~~ \\mathrm{det}({\\mathcal E}) \\simeq {\\mathcal N} \\in \\mathrm{Pic}^d(C), \\mathrm{trace}(\\theta) = 0\\}\n\\]\nadmits a Hitchin map\n\\[\n\\widecheck{h}^{\\mathcal L}: \\widecheck{M}^{\\mathcal L}_{\\mathrm{Dol}} \\to \\widehat{A}^{\\mathcal L}:= \\bigoplus_{i=2}^nH^0\\left(C, {\\Omega_{\\mathcal L}}^{\\otimes i} \\right)\\]\nand a fiberwise $\\Gamma = \\mathrm{Pic}^0(C)[n]$ action by tensor product. Taking the $\\Gamma$-quotient recovers the ${\\mathcal L}$-twisted $\\mathrm{PGL}_n$ Hitchin map\n\\[\n\\widehat{h}^{\\mathcal L}: \\widehat{M}^{\\mathcal L}_{\\mathrm{Dol}} =\\widecheck{M}^{\\mathcal L}_{\\mathrm{Dol}}\/\\Gamma \\to \\widehat{A}^{\\mathcal L}. \n\\]\n\n\nAn observation in \\cite[Section 4]{MS} is that, for a fixed closed point $p\\in C$, the ${\\mathcal L}$-twisted and ${\\mathcal L}(p)$-twisted $\\mathrm{SL}_n$ Dolbeault moduli spaces can be related via critical loci and vanishing cycles, which we recall in the following.\n\nBy viewing a ${\\mathcal L}$-twisted Higgs bundle naturally as a ${\\mathcal L}(p)$-twisted Higgs bundle, we have the natural embedding $i: \\widecheck{M}^{\\mathcal L}_{\\mathrm{Dol}} \\hookrightarrow \\widecheck{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}}$ which induces the commutative diagram\n\\begin{equation}\\label{2.3_1}\n\\begin{tikzcd}\n\\widecheck{M}^{\\mathcal L}_{\\mathrm{Dol}}\\arrow[r, hook,\"i\"] \\arrow[d, \"\\widecheck{h}^{\\mathcal L}\"]\n&\\widecheck{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}} \\arrow[d, \"\\widecheck{h}^{{\\mathcal L}(p)}\"] \\\\\n\\widehat{A}^{\\mathcal L} \\arrow[r,hook]\n& \\widehat{A}^{{\\mathcal L}(p)}.\n\\end{tikzcd}\n\\end{equation}\n\nWe recall the following theorem from \\cite{MS}.\n\n\\begin{thm}[\\cite{MS} Theorem 4.5]\\label{thm2.5}\nThere exists a regular function $g: \\widecheck{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}} \\to {\\mathbb{A}}^1$ factorized as $g = \\nu \\circ \\widecheck{h}^{{\\mathcal L}(p)}$ with $\\nu: \\widehat{A}^{{\\mathcal L}(p)} \\to {\\mathbb{A}}^1$ such that\n\\[\n\\varphi_g \\simeq \\mathrm{IC}_{\\widecheck{M}^{{\\mathcal L}}_{\\mathrm{Dol}}} = {\\mathbb{Q}}_{\\widecheck{M}^{{\\mathcal L}}_{\\mathrm{Dol}}}[\\mathrm{dim} \\widecheck{M}^{{\\mathcal L}}_{\\mathrm{Dol}}].\n\\]\n\\end{thm}\n\nSince the embedding $i: \\widecheck{M}^{\\mathcal L}_{\\mathrm{Dol}} \\hookrightarrow \\widecheck{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}}$ is $\\Gamma$-equivariant, taking $\\Gamma$-quotients in the diagram (\\ref{2.3_1}) yields the following commutative diagram:\n\\begin{equation}\\label{diagram3}\n\\begin{tikzcd}\n\\widehat{M}^{\\mathcal L}_{\\mathrm{Dol}}\\arrow[r, hook, \"\\widehat{i}\"] \\arrow[d, \"\\widehat{h}^{\\mathcal L}\"]\n&\\widehat{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}} \\arrow[d, \"\\widehat{h}^{{\\mathcal L}(p)}\"] \\\\\n\\widehat{A}^{\\mathcal L} \\arrow[r, hook]\n& \\widehat{A}^{{\\mathcal L}(p)}.\n\\end{tikzcd}\n\\end{equation}\n\n\n\\begin{prop}\\label{prop2.6}\nThe vanishing cycle complex $\\varphi_{\\widehat{g}}$ associated with the regular function\n\\[\n\\widehat{g}: = \\nu \\circ \\widehat{h}^{{\\mathcal L}(p)}: \\widehat{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}} \\to {\\mathbb{A}}^1\n\\]\nsatisfies\n\\[\n\\varphi_{\\widehat{g}} \\simeq \\mathrm{IC}_{\\widehat{M}^{{\\mathcal L}}_{\\mathrm{Dol}}} = {\\mathbb{Q}}_{\\widehat{M}^{{\\mathcal L}}_{\\mathrm{Dol}}}\\left[\\mathrm{dim} \\widehat{M}^{{\\mathcal L}}_{\\mathrm{Dol}}\\right].\\]\n\\end{prop}\n\n\\begin{proof}\nWe reduce Proposition \\ref{prop2.6} to Theorem \\ref{thm2.5}. Consider the $\\Gamma$-quotient map $r: \\widecheck{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}} \\to \\widehat{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}}$. The direct image $r_* {\\mathbb{Q}}_{\\widecheck{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}}}$ admits a natural $\\Gamma$-equivariant structure, whose invariant part recovers \n\\begin{equation}\\label{2.6_1}\n\\left(r_* {\\mathbb{Q}}_{\\widecheck{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}}}\\right)^{\\Gamma} \\simeq {\\mathbb{Q}}_{\\widehat{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}}}.\n\\end{equation}\nSince $g = r \\circ \\widehat{g}$, we have\n\\begin{align*}\n\\varphi_{\\widehat{g}} & = \\varphi_{\\widehat{g}}\\left({\\mathbb{Q}}_{\\widehat{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}}}\\left[\\mathrm{dim} \\widehat{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}}\\right]\\right) \\\\ & \\simeq \\varphi_{\\widehat{g}}\\left(r_* {\\mathbb{Q}}_{\\widecheck{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}}}\\left[\\mathrm{dim} \\widecheck{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}}\\right]\\right)^{\\Gamma}\\\\\n& \\simeq (r_* \\varphi_g)^\\Gamma \\\\ & \\simeq  {\\mathbb{Q}}_{\\widehat{M}^{{\\mathcal L}}_{\\mathrm{Dol}}}\\left[\\mathrm{dim} \\widehat{M}^{{\\mathcal L}}_{\\mathrm{Dol}}\\right].\n\\end{align*}\nHere the first equation follows by definition, the second uses (\\ref{2.6_1}), the third follows from the base change, and the last is given by Theorem \\ref{thm2.5}.\n\\end{proof}\n\n\nNow we formulate a sheaf-theoretic enhancement of Conjecture \\ref{conj2.2}.\n\n\\begin{thm}\\label{conj2.7}\nThere exists an effective line bundle ${\\mathcal L}$ such that the class\n\\[\n\\mathrm{ch}_k({\\mathcal U}^{\\mathcal L}) \\in H^{2k}(C\\times \\widehat{M}^{\\mathcal L}_{\\mathrm{Dol}}, {\\mathbb{Q}})\n\\]\nhas strong perversity $k$ with respect to \n\\[\n\\mathbf{h}^{\\mathcal L}: = \\mathrm{id} \\times \\widehat{h}^{\\mathcal L}:C\\times \\widehat{M}^{\\mathcal L}_{\\mathrm{Dol}} \\to C\\times \\widehat{A}^{\\mathcal L}. \n\\]\nHere ${\\mathcal U}^{\\mathcal L}$ be the universal $\\mathrm{PGL}_n$-bundle over $C\\times \\widehat{M}_{\\mathrm{Dol}}$.\n\\end{thm}\n\nNotice that the perversity bound here is stronger than the trivial one of $2k$.  We prove in the rest of this section that Theorem \\ref{conj2.7} indeed implies Conjecture \\ref{conj2.2}; therefore it implies Conjecture \\ref{conj}. We prove Theorem \\ref{conj2.7} in Sections 3 and 4.\n\n\\begin{rmk}\nNote that although for our purposes, we only need \\emph{one} effective line bundle ${\\mathcal L}$ satisfying Theorem \\ref{conj2.7}, our proof actually shows that it is true for \\emph{any} effective ${\\mathcal L}$.\n\\end{rmk}\n\n\\subsection{Theorem \\ref{conj2.7} implies Conjecture \\ref{conj2.2}} We start with the following claim.\n\n\n\\medskip\n\\noindent {\\bf Claim.} For a point $p\\in C$, if the class\n\\[\n\\mathrm{ch}_k({\\mathcal U}^{{\\mathcal L}(p)}) \\in H^{2k}(C\\times \\widehat{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}}, {\\mathbb{Q}}) \n\\]\nhas strong perversity $k$ with respect to $\\mathbf{h}^{{\\mathcal L}(p)}: C\\times \\widehat{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}} \\to C\\times \\widehat{A}^{{\\mathcal L}(p)}$, then the class\n\\[\n\\mathrm{ch}_k({\\mathcal U}^{{\\mathcal L}}) \\in H^{2k}(C\\times \\widehat{M}^{{\\mathcal L}}_{\\mathrm{Dol}}, {\\mathbb{Q}})\n\\]\nhas strong perversity $k$ with respect to $\\mathbf{h}^{{\\mathcal L}}: C\\times \\widehat{M}^{\\mathcal L}_{\\mathrm{Dol}} \\to C\\times \\widehat{A}^{\\mathcal L}$.\n\n\\begin{proof}[Proof of Claim]\n\nWe consider the commutative diagram obtained from (\\ref{diagram3}):\n\n\\begin{equation}\\label{diagram4}\n\\begin{tikzcd}\nC \\times \\widehat{M}^{\\mathcal L}_{\\mathrm{Dol}}\\arrow[r,hook, \"\\mathbf{i}\"] \\arrow[d, \"\\mathbf{h}^{\\mathcal L}\"]\n&C\\times \\widehat{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}} \\arrow[d, \"\\mathbf{h}^{{\\mathcal L}(p)}\"] \\\\\nC\\times \\widehat{A}^{\\mathcal L} \\arrow[r,hook]\n& C\\times \\widehat{A}^{{\\mathcal L}(p)}.\n\\end{tikzcd}\n\\end{equation}\nHere $\\mathbf{i} = \\mathrm{id} \\times \\widehat{i}$. By definition,  we have $\\mathbf{i}^* {\\mathcal U}^{{\\mathcal L}(p)} = {\\mathcal U}^{{\\mathcal L}}$. Therefore \n\\[\n\\mathbf{i}^* \\mathrm{ch}_k({\\mathcal U}^{{\\mathcal L}(p)}) = \\mathrm{ch}_k({\\mathcal U}^{\\mathcal L}).\n\\]\nWe know from Proposition \\ref{prop2.6} that the vanishing cycle complex associated with the regular function\n\\[\n\\mathbf{g}: =  \\widehat{g} \\circ \\mathrm{pr}_M: C\\times \\widehat{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}} \\to\\widehat{M}^{{\\mathcal L}(p)}_{\\mathrm{Dol}}\\to {\\mathbb{A}}^1\n\\]\nsatisfies\n\\[\n\\varphi_{\\mathbf{g}} \\simeq \\mathrm{IC}_{C\\times \\widehat{M}^{{\\mathcal L}}_{\\mathrm{Dol}}} = {{\\mathbb{Q}}}_{C\\times \\widehat{M}^{{\\mathcal L}}_{\\mathrm{Dol}}}\\left[\\mathrm{dim}~ (C\\times \\widehat{M}^{{\\mathcal L}}_{\\mathrm{Dol}})\\right].\n\\]\nOur claim then follows from Proposition \\ref{prop1.4} (applied to the diagram (\\ref{diagram4}), the class $\\mathrm{ch}_k({\\mathcal U}^{{\\mathcal L}(p)})$, and the map $\\mathbf{g}$).\n\\end{proof}\n\n\n\n\n\nAs a consequence of the claim, the statement of Theorem \\ref{conj2.7} holds for ${\\mathcal L} \\simeq {\\mathcal O}_C$, \\emph{i.e.}, the class \n\\[\n\\mathrm{ch}_k({\\mathcal U}) \\in H^{2k}(C\\times \\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}})\n\\] \nhas strong perversity $k$ with respect to \n\\[\n\\mathbf{h}:= \\mathrm{id} \\times \\widehat{h}: C\\times \\widehat{M}_{\\mathrm{Dol}} \\to C\\times \\widehat{A}.\n\\]\n\n\n\nFinally, to deduce the cohomological statement, we note that since $\\mathbf{h}$ is trivial at the $C$-factor, its induced perverse filtration can be described as:\n\\begin{equation}\\label{2.2_1}\nP_kH^*(C \\times \\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}}) = H^*(C, {\\mathbb{Q}}) \\otimes P_kH^*(\\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}}).\n\\end{equation}\nChoose a homogeneous basis of $H^*(C, {\\mathbb{Q}})$:\n\\[\n\\Sigma=\\{\\sigma_0, \\sigma_1, \\dots, \\sigma_{2g+2}\\}\n\\]\nwith Poincar\\'e-dual basis $\\{\\sigma^{\\vee}_{0}, \\dots, \\sigma^\\vee_{2g+2}\\}$.\n\nWe may express $\\mathrm{ch}_k({\\mathcal U})$ as\n\\begin{equation}\\label{2.4_1}\n\\mathrm{ch}_k({\\mathcal U}) = \\sum_{\\sigma\\in \\Sigma} \\sigma^\\vee \\otimes c_k(\\sigma) \\in H^*(C, {\\mathbb{Q}}) \\otimes H^*(\\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}}).\n\\end{equation}\nSince $\\mathrm{ch}_k({\\mathcal U})$ has strong perversity $k$, Lemma \\ref{lem1.1} implies that\n\\[\n\\mathrm{ch}_k({\\mathcal U}) \\cup - : P_sH^*(C\\times \\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}}) \\to P_{s+k}H^{*}(C\\times \\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}}).\n\\]\nApplying this operator to $\\sigma \\otimes P_sH^*(\\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}}) \\subset H^*(C\\times \\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}})$ with $\\sigma \\in \\Sigma$, we obtain from (\\ref{2.2_1}) and (\\ref{2.4_1}) that\n\\[\nc_k(\\sigma) \\cup -: P_sH^*(\\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}}) \\to P_{s+k}H^*(\\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}}). \n\\]\nIn particular, for any class $\\gamma \\in H^*(C, {\\mathbb{Q}})$ we have\n\\[\nc_k(\\gamma): P_sH^*(\\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}}) \\to P_{s+k}H^*(\\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}})\\]\nwhich further yields\n\\[\n\\prod_ic_{k_1}(\\gamma_i) = \\left( \\prod_ic_{k_1}(\\gamma_i) \\right) \\cup 1 \\in P_{\\Sigma_ik_i}H^*(\\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}}).\n \\]\n Here we used $1\\in P_0H^0(\\widehat{M}_{\\mathrm{Dol}}, {\\mathbb{Q}})$ in the last equation.\n\nThis completes the proof that Theorem \\ref{conj2.7} implies Conjecture \\ref{conj2.2}. \\qed\n\n\n\n\n\\section{Global Springer theory}\n\nIn this section, we review Yun's global Springer theory \\cite{Yun1, Yun2, Yun3} and use it to deduce Theorem \\ref{conj2.7}, assuming a support theorem (Theorem \\ref{thm3.2}). Global Springer theory was previously used to study perverse filtrations for affine Springer fibers in terms of Chern classes \\cite{OY, OY2}.  In our setting, we require a partial extension of Yun's results from the elliptic locus to the entire Hitchin base.\n\n\\subsection{Notations}\nWe fix ${\\mathcal L}$ to be an effective line bundle of sufficiently large degree\n($\\mathrm{deg}\\left({\\mathcal L}\\right)> 2g$ is enough for us).\n Since from now on we only concern the ${\\mathcal L}$-twisted moduli spaces, we will use $M, \\widecheck{M}$, and $\\widehat{M}$ to denote the ${\\mathcal L}$-twisted $\\mathrm{GL}_n$, $\\mathrm{SL}_n$, and $\\mathrm{PGL}_n$ Dolbeault moduli spaces $M^{\\mathcal L}_{\\mathrm{Dol}}, \\widecheck{M}^{\\mathcal L}_{\\mathrm{Dol}}$, and $\\widehat{M}^{\\mathcal L}_{\\mathrm{Dol}}$ respectively. For the same reason, we will uniformly use the term \\emph{Higgs bundles} to call ${\\mathcal L}$-twisted Higgs bundles.\n\nFrom now on we let $G = \\mathrm{PGL}_n$, $\\mathfrak{g}$ its Lie algebra, $B \\subset G$ the Borel subgroup induced by upper triangular matrices with $\\mathfrak{b}$ the Lie algebra, $T\\subset B$ the maximal torus given by diagonal matrices, and $W \\simeq \\mathfrak{S}_n$ the Weyl group. We denote by ${\\mathbb{X}}^*(T)$ the character group of $T$; it is isomorphic to ${\\mathbb{Z}}^{n-1}$ as an abelian group.\n\n\n\n\n\\subsection{Parabolic moduli stacks}\n\nLet $\\widehat{{\\mathfrak{M}}}$ be the moduli stack of $G$-Higgs bundles on $C$; we do not impose any stability condition on $\\widehat{{\\mathfrak{M}}}$ so that it is only a (singular) algebraic stack. The stable locus of $\\widehat{{\\mathfrak{M}}}$ is a nonsingular Deligne--Mumford substack\n\\[\n\\widehat{M} \\hookrightarrow \\widehat{{\\mathfrak{M}}}.\n\\]\n\nYun's global Springer theory \\cite{Yun1} constructs an algebraic stack $\\widehat{{\\mathfrak{M}}}^\\mathrm{par}$ over $C \\times \\widehat{{\\mathfrak{M}}}$:\n\\begin{equation}\\label{PI}\n\\pi: \\widehat{{\\mathfrak{M}}}^\\mathrm{par} \\to C \\times \\widehat{{\\mathfrak{M}}}\n\\end{equation}\nwhich is a global analog of the Grothendieck simultaneous resolution. There are two equivalent constructions of $\\widehat{{\\mathfrak{M}}}^{\\mathrm{par}}$ given in \\cite[Section 2.1]{Yun1}. \n\n\nThe first is to construct $\\widehat{{\\mathfrak{M}}}^{\\mathrm{par}}$ as the moduli stack of parabolic Higgs bundles, which are quadruples $(x, {\\mathcal E}, \\theta, {\\mathcal E}_x^B)$, with $({\\mathcal E}, \\theta)$ is a $G$-Higgs bundle, $x\\in C$ a closed point, and ${\\mathcal E}_x^B$ a $B$-reduction of ${\\mathcal E}$ at $x$, satisfying the constraint that $\\theta$ is compatible with ${\\mathcal E}_x^B$; see \\cite[Definition 2.1.1]{Yun1}. Then the morphism $\\pi$ is given by forgetting the $B$-reduction:\n\\[\n\\pi (x, {\\mathcal E}, \\theta, {\\mathcal E}_x^B)  = (x, ({\\mathcal E},\\theta)) \\in C \\times \\widehat{{\\mathfrak{M}}}.\n\\]\n\nThe second construction is via the Grothendieck simultaneous resolution $\\pi_G: [\\mathfrak{b}\/B] \\to [\\mathfrak{g}\/G]$. More precisely, let $\\rho_{\\mathcal L}$ be the ${\\mathbb{G}}_m$-torsor over $C$ associated with the line bundle $\\Omega_{\\mathcal L}$. Denote by $[\\mathfrak{g}\/G]_{\\mathcal L}$ (resp. $[\\mathfrak{b}\/B]_{\\mathcal L}$) the family of $[\\mathfrak{g}\/G]$ (resp. $[\\mathfrak{b}\/B]$) over the curve $C$ twisted by the torsor $\\rho_{\\mathcal L}$. We have a tautological evaluation map\n\\begin{equation}\\label{evaluation}\n    \\begin{tikzcd}[column sep=small]\n    C\\times \\widehat{{\\mathfrak{M}}} \\arrow[dr, \"\"] \\arrow[rr, \"\\mathrm{ev}\"] & & {[\\mathfrak{g}\/G]_{\\mathcal L}} \\arrow[dl, \"\"] \\\\\n       & C  & \n\\end{tikzcd}\n\\end{equation}\nwhich is a natural $C$-morphism; after base change to a closed point $x\\in C$, the map $\\mathrm{ev}$ sends a $G$-Higgs bundle to the evaluation of its Higgs field at $x$. The morphism (\\ref{PI}) is then induced by the base change of the Grothendieck simultaneous resolution along the evaluation map (see \\cite[Lemma 2.1.2]{Yun1}):\n\\begin{equation*}\n\\begin{tikzcd}\n\\widehat{{\\mathfrak{M}}}^{\\mathrm{par}} \\arrow[r, \"\\mathrm{ev}^p\"] \\arrow[d, \"\\pi\"]\n&{[\\mathfrak{b}\/B]_{\\mathcal L}} \\arrow[d, \"\\pi_G\"] \\\\\nC\\times \\widehat{{\\mathfrak{M}}} \\arrow[r, \"\\mathrm{ev}\"]\n& {[\\mathfrak{g}\/G]_{\\mathcal L}}.\n\\end{tikzcd}\n\\end{equation*}\n\nThe parabolic Hitchin system\n\\[\n\\widehat{h}^p: \\widehat{{\\mathfrak{M}}}^{\\mathrm{par}} \\to C\\times \\widehat{A}\n\\]\nis the composition of $\\pi:\\widehat{{\\mathfrak{M}}}^{\\mathrm{par}} \\to C\\times \\widehat{{\\mathfrak{M}}}$ and the morphism $\\mathbf{h}: C\\times \\widehat{{\\mathfrak{M}}} \\to C\\times \\widehat{A}$\ninduced by the standard (stacky) Hitchin map $\\widehat{h}: \\widehat{{\\mathfrak{M}}} \\to \\widehat{A}$.\\footnote{Recall that we always use the ${\\mathcal L}$-twisted version.}\n\nIn general the moduli stack $\\widehat{{\\mathfrak{M}}}^{\\mathrm{par}}$ is singular, and the parabolic Hitchin map is not proper. In the next section we will impose a stability condition and then restrict to the stable locus. \n\n\\subsection{Stable loci}\n\nWe define the \\emph{stable locus} of the moduli of parabolic $G$-Higgs bundles to be\n\\[\n\\widehat{M}^{\\mathrm{par}}: = (C \\times \\widehat{M}) \\times_{C \\times \\widehat{{\\mathfrak{M}}}} \\widehat{{\\mathfrak{M}}}^{{\\mathrm{par}}};\n\\]\nequivalently, it fits into the Cartesian diagrams\n\\begin{equation}\\label{diag0}\n\\begin{tikzcd}\n\\widehat{M}^{\\mathrm{par}} \\arrow[r,hook] \\arrow[d, \"\\pi\"]&  \\widehat{{\\mathfrak{M}}}^{\\mathrm{par}} \\arrow[r, \"\\mathrm{ev}^p\"] \\arrow[d, \"\\pi\"]\n&{[\\mathfrak{b}\/B]_{\\mathcal L}} \\arrow[d, \"\\pi_G\"] \\\\ C \\times \\widehat{M} \\arrow[r,hook]\n& C\\times \\widehat{{\\mathfrak{M}}} \\arrow[r, \"\\mathrm{ev}\"]\n& {[\\mathfrak{g}\/G]_{\\mathcal L}}.\n\\end{tikzcd}\n\\end{equation}\n\nWe use the same notation as for the stacky case to denote the parabolic Hitchin map restricted to the stable locus:\n\\begin{equation}\\label{parah}\n\\widehat{h}^p: \\widehat{M}^{\\mathrm{par}} \\to C\\times \\widehat{A}.\n\\end{equation}\n\n\\begin{prop}\\label{prop3.1}\n\\begin{enumerate}\n    \\item[(i)] The moduli stack $\\widehat{M}^{\\mathrm{par}}$ is nonsingular and Deligne--Mumford; and\n    \\item[(ii)] the parabolic Hitchin map (\\ref{parah}) is proper.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nThe Deligne--Mumford part of (i) follows from the proof of \\cite[Proposition 2.5.1 (3)]{Yun1}. Indeed, the left vertical arrow in the diagram (\\ref{diag0}) is the pullback of $\\pi_G$, therefore it is schematic and of finite type. This implies that $\\widehat{M}^{\\mathrm{par}}$ is Deligne--Mumford since $C \\times \\widehat{M}$ is Deligne--Mumford.\n\nTo prove the smoothness part of (i), we use the evaluation map (\\ref{evaluation}). Recall that \\cite[Proposition 4.1]{MS} (which was proven via deformation theory) shows that the $C$-morphism $\\mathrm{ev}$ is smooth after restricting over the stable locus $C \\times \\widehat{M}$. Hence by the Cartesian diagrams of (\\ref{diag0}), the evaluation map\n\\[\n\\mathrm{ev}: \\widehat{M}^{\\mathrm{par}} \\to {[\\mathfrak{b}\/B]_{\\mathcal L}}\n\\]\nis also smooth. Since the target is a nonsingular algebraic stack, so is the source.\n\n(ii) follows directly from (\\ref{diag0}) and that $\\widehat{h}: \\widehat{M} \\to \\widehat{A}$ is proper.\n\\end{proof}\n\nAs a consequence of Proposition \\ref{prop3.1}, the direct image complex\n\\[\nR{\\widehat{h}^p}_* ~{\\mathbb{Q}}_{\\widehat{M}^{\\mathrm{par}}} \\in D^b_c(C\\times \\widehat{A})\n\\]\nsatisfies the decomposition theorem \\cite{BBD}.\n\n\\begin{thm}[Support theorem for parabolic Hitchin map]\\label{thm3.2}\nThe decomposition theorem for the parabolic Hitchin map has full support, \\emph{i.e.}, any non-trivial simple perverse summand of \\[\n^\\mathfrak{p}{\\mathcal H}^i\\left(R{\\widehat{h}^p}_* ~{\\mathbb{Q}}_{\\widehat{M}^{\\mathrm{par}}} \\right), \\quad  \\forall i\\in {\\mathbb{Z}}\\]\nhas support $C\\times \\widehat{A}$.\n\\end{thm}\n\nWe will postpone the proof of Theorem \\ref{thm3.2} to Section 4.\n\n\n\\subsection{Proof of Theorem \\ref{conj2.7}}\n\nWe first prove Theorem \\ref{conj2.7} assuming Theorem \\ref{thm3.2}.\n\nThere are three ingredients of global Springer theory\n\\[\n\\pi: \\widehat{M}^\\mathrm{par} \\to C \\times \\widehat{M}\n\\]\nwhich are important in the proof of Theorem \\ref{conj2.7}. We summarize them as follows.\n\\begin{enumerate}\n    \\item[(A)] \\emph{(Splitting of the universal $G$-bundle)} As explained in the second bullet point of \\cite[Construction 6.1.4]{Yun1}, the Chern roots of $\\pi^*{\\mathcal U}$ are given by $c_1(L(\\xi))$ where $L(\\xi)$ is the tautological line bundle on $\\widehat{M}^{\\mathrm{par}}$ associated with $\\xi \\in {\\mathbb{X}}^*(T)$. \n    \n    More precisely, $\\pi^*{\\mathcal U}$ is the universal $G$-bundle on $\\widehat{M}^{\\mathrm{par}}$ whose fiber over $(x,{\\mathcal E},\\theta, {\\mathcal E}^B_x)$ is ${\\mathcal E}_x$; the $B$-reduction ${\\mathcal E}_x^B$ yields a $T$-torsor over $\\widehat{M}^{\\mathrm{par}}$ which induces for each $\\xi \\in {\\mathbb{X}}^*(T)$ a line bundle $L(\\xi)$.\n\n    \n    \\item[(B)] \\emph{(Strong perversity for $c_1(L(\\xi))$)} By \\cite[Lemma 3.2.3]{Yun2}, there exists a Zariski dense open subset of $C\\times \\widehat{A}$ over which the operator\n    \\[\nc_1(L(\\xi)): R\\widehat{h}^p_*{\\mathbb{Q}}_{\\widehat{M}^{\\mathrm{par}}} \\to R\\widehat{h}^p_*{\\mathbb{Q}}_{\\widehat{M}^{\\mathrm{par}}}[2]    \\]\nhas strong perversity $1$ for any $\\xi \\in {\\mathbb{X}}^*(T)$. \nSince $c_1(L(\\xi))$ automatically has strong perversity $2$, showing it has strong perversity $1$ is equivalent to showing\nthe induced morphism of perverse cohomology sheaves\n\\[\n{}^{p}{\\mathcal H}^i\\left( R\\widehat{h}^p_*{\\mathbb{Q}}_{\\widehat{M}^{\\mathrm{par}}}\\right)\\rightarrow\n{}^{p}{\\mathcal H}^i\\left(R\\widehat{h}^p_*{\\mathbb{Q}}_{\\widehat{M}^{\\mathrm{par}}}[2]\\right)\n\\]\nvanishes for each $i$.  Once we have Theorem \\ref{thm3.2}, these sheaves have full support over the entire base, so this vanishing (and thus strong perversity $1$) extends over the total base $C \\times \\widehat{A}$ as well. \nIn fact, \\cite[Lemma 3.2.3]{Yun2} was proven in exactly this way using a support theorem \\cite[Section 4.6.2]{Yun2} for the elliptic locus.\n\n    \n    \\item[(C)] \\emph{(Springer's Weyl group action)} By the Cartesian diagrams (\\ref{diag0}), we may pullback Springer's sheaf-theoretic Weyl group action from the Grothendieck simultaneous resolution; in particular, the object $R\\pi_* {\\mathbb{Q}}_{\\widehat{M}^{\\mathrm{par}}}$ admits a canonical $W$-action whose invariant part recovers the trivial local system\n    \\[\n\\left(R\\pi_* {\\mathbb{Q}}_{\\widehat{M}^{\\mathrm{par}}} \\right)^W = {\\mathbb{Q}}_{C\\times \\widehat{M}}.  \\]\nBy taking global cohomology we have\n\\[\nH^*(\\widehat{M}^{\\mathrm{par}}, {\\mathbb{Q}}) = H^*(C\\times \\widehat{M}, {\\mathbb{Q}}) \\oplus \\left(\\mathrm{variant~~part~~of~~} W \\right).\n\\]\n\\end{enumerate}\n\nNow we prove Theorem \\ref{conj2.7}. We first prove its parabolic version.\n\n\\medskip\n\\noindent {\\bf Claim.} The class\n\\[\n\\mathrm{ch}_k\\left(\\pi^* {\\mathcal U}\\right) \\in H^{2k}(\\widehat{M}^{\\mathrm{par}}, {\\mathbb{Q}})\n\\]\nhas strong perversity $k$ with respect to the parabolic Hitchin system (\\ref{parah}).\n\n\\begin{proof}[Proof of Claim]\nBy (A), the Chern character $\\mathrm{ch}_k(\\pi^*{\\mathcal U})$ can be expressed in terms of $c_1(L(\\xi))$. Hence the claim follows from Lemma \\ref{lem1.2} and the strong perversity of $c_1(L(\\xi))$ given by (B).\n\\end{proof}\n\n\nNext, we reduce Theorem \\ref{conj2.7} to the claim above via the following lemma.\n\n\\begin{lem}\nFor a class $\\gamma \\in H^l(C \\times \\widehat{M}, {\\mathbb{Q}})$, if $\\pi^*\\gamma \\in H^l(\\widehat{M}^{\\mathrm{par}}, {\\mathbb{Q}})$ has strong perversity $k$ with respect to $\\widehat{h}^p: \\widehat{M}^{\\mathrm{par}} \\to C\\times \\widehat{A}$, then $\\gamma$ has strong perversity $k$ with respect to ${\\mathbf{h}} : C\\times \\widehat{M} \\to C\\times \\widehat{A}$.\n\\end{lem}\n\n\\begin{proof}\n    This is a consequence of (C). We write the class $\\gamma$ as a map \n    \\begin{equation}\\label{3.3_1}\n\\gamma: {\\mathbb{Q}}_{C\\times \\widehat{M}} \\to {\\mathbb{Q}}_{C\\times \\widehat{M}}[l],   \n\\end{equation}\nwhose pullback \n\\begin{equation}\\label{3.3_2}\n\\pi^*\\gamma: {\\mathbb{Q}}_{\\widehat{M}^{\\mathrm{par}}} \\to  {\\mathbb{Q}}_{\\widehat{M}^{\\mathrm{par}}}[l]\n\\end{equation}\nrecovers the class $\\pi^*\\gamma \\in H^l(\\widehat{M}^{\\mathrm{par}}, {\\mathbb{Q}})$. By the projection formula $\\pi_*\\pi^*\\gamma = |W|\\cdot \\gamma$, we may recover the morphism (\\ref{3.3_1}) from (\\ref{3.3_2}) by derived pushing forward to $C \\times \\widehat{M}$ and taking the $W$-invariant part.\n\nNow by the assumption we know that the action of $\\pi^*\\gamma$ on the object $R\\widehat{h}^p_*{\\mathbb{Q}}_{\\widehat{M}^{\\mathrm{par}}}$ satisfies\n\\begin{equation}\\label{3.3_4}\n\\pi^*\\gamma: {^\\mathfrak{p}\\tau_{\\leq i}}R\\widehat{h}^p_*{\\mathbb{Q}}_{\\widehat{M}^{\\mathrm{par}}} \\to {^\\mathfrak{p}\\tau_{\\leq i+(k-l)}}\\left(R\\widehat{h}^p_*{\\mathbb{Q}}_{\\widehat{M}^{\\mathrm{par}}}[l]\\right).\n\\end{equation}\nSince we have\n\\[\nR\\widehat{h}^p_*{\\mathbb{Q}}_{\\widehat{M}^{\\mathrm{par}}} = R\\mathbf{h}_*\\left(R\\pi_* {\\mathbb{Q}}_{\\widehat{M}^{\\mathrm{par}}}\\right),\n\\] \nthe ingredient (C) produces a natural $W$-action on this object whose invariant part recovers $R\\mathbf{h}_*{\\mathbb{Q}}_{C\\times \\widehat{M}}$; the operator\n\\begin{equation}\\label{3.3_3}\n\\gamma:  R\\mathbf{h}_*{\\mathbb{Q}}_{C\\times \\widehat{M}} \\to R\\mathbf{h}_*{\\mathbb{Q}}_{C\\times \\widehat{M}}[l]  \n\\end{equation}\nis then recovered from the $W$-invariant part of the action of $\\pi^*\\gamma$ on $R\\widehat{h}^p_*{\\mathbb{Q}}_{\\widehat{M}^{\\mathrm{par}}}$. In particular, the desired property concerning (\\ref{3.3_3}) follows from the $W$-invariant part of (\\ref{3.3_4}).\n\\end{proof}\n\n\n\n\nThus we have completed the proof of Theorem \\ref{conj2.7}. \\qed\n\n\n\n\n\n\n\\section{Parabolic support theorem}\n\nIn the last section, we prove Theorem \\ref{thm3.2}. This is the parabolic version of the Chaudouard--Laumon support theorem \\cite{CL}. For the proof, we ultimately reduce it to a relative dimension bound which we establish in Section \\ref{sec4.4}.\n\n\\subsection{Review of support theorems}\\label{Sec4.1}\n\nWe start with a review of support theorems for Hitchin systems.\n\nFor a proper morphism $f: X\\to Y$ with $X,Y$ nonsingular Deligne--Mumford stacks, the decomposition theorem \\cite{BBD} yields\n\\[\nRf_* {\\mathbb{Q}}_X \\simeq \\bigoplus_i {^\\mathfrak{p}{\\mathcal H}^i(Rf_* {\\mathbb{Q}}_X)[-i]}\n\\]\nwith ${^\\mathfrak{p}{\\mathcal H}^i}(Rf_* {\\mathbb{Q}}_X)$ semisimple perverse sheaves on $Y$; we say that a closed $Z \\subset Y$ is a support of $f$ if it is a support of a simple summand of some ${^\\mathfrak{p}{\\mathcal H}^i}(Rf_* {\\mathbb{Q}}_X)$. A particularly interesting case is that $f$ has full support, that is, $Y$ is the only support of $f$; in this case the cohomology of any closed fiber of $f$ is governed by the nonsingular fibers.\n\nThe study of supports for Hitchin systems was initiated by B.C. Ng\\^o which is crucial in his proof of the fundamental lemma of the Langlands program \\cite{Ngo}. He determines all the supports for the Hitchin system (including the ${\\mathcal L}$-twisted cases) after restricting to the elliptic locus of the Hitchin base; that is the subset formed by integral spectral curves.\n\nAfter Ng\\^o's work, Chaudouard--Laumon \\cite{CL} observed that, if we consider the moduli space of ${\\mathcal L}$-twisted $\\mathrm{GL}_n$ stable Higgs bundles with $\\mathrm{deg}({\\mathcal L}) >0$, then Ng\\^o's support theorem can be extended to the total Hitchin base; in particular they showed that the ${\\mathcal L}$-twisted $\\mathrm{GL}_n$ Hitchin system has full support. Chaudouard--Laumon's idea was extended to the $\\mathrm{SL}_n$ case \\cite{dC_SL}, the endoscopic moduli spaces \\cite{MS}, and singular cases involving strictly semistable Higgs bundles \\cite{MS2, MS3}. See also \\cite{dCHM2, MM} concerning the supports over the open subset of reduced spectral curves for the untwisted (\\emph{i.e.} ${\\mathcal L} = 0$) Hitchin system.\n\nThe idea of Chaudouard--Laumon \\cite{CL} is to show that each support of the ${\\mathcal L}$-twisted Hitchin system has generic point lying in the elliptic locus; this is achieved by combining two constraints: (I) the support inequality for a weak abelian fibration, and (II) $\\delta$-regularity for spectral curves. We describe them in more detail.\n\\begin{enumerate}\n    \\item[(I)] \\emph{(Support inequaltiy)} The Hitchin system admits the structure of a weak abelian fibration, that is, there is a commutative group scheme $P$ over the Hitchin base acting on the moduli space which satisfies certain properties. Then an argument generalizing the Goresky--MacPherson inequality leads to a codimension estimate for the supports. More concretely, it says that any support $Z$ has codimension bounded above by the $\\delta$-function of the group $P$. This part was already carried out in Ng\\^o \\cite[Section 7]{Ngo}; see \\cite[Section 1]{MS2} for a summary.\n    \\item[(II)] \\emph{($\\delta$-regularity)} In the case of type $A$ and for the stable locus, the group scheme is obtained from the multi-degree 0 relative Picard (or, for $\\mathrm{SL}_n$, Prym) variety associated with the family of spectral curves. Then we have the Severi inequality, referred to as \\emph{$\\delta$-regularity}, for the spectral curves. We refer to \\cite[Theorem 5.4.4]{dC_SL} for the precise statement.\n\\end{enumerate}\n\nUsing (I) and (II), one can deduce that no support is allowed to have generic point lying outside the open subset of integral curves; this was explained in \\cite[Section 11]{CL} for $\\mathrm{GL}_n$, in \\cite[Section 6.2]{dC_SL} for $\\mathrm{SL}_n$, and in \\cite[Section 4.5]{MS2} for a general complete linear system in a del Pezzo surface. The argument is to combine the two inequalities above to deduce a numerical contradiction if there is a support that appears outside the elliptic locus.\n\n\n\\subsection{Parabolic Hitchin systems}\nNow we focus on the ${\\mathcal L}$-twisted parabolic Hitchin system (\\ref{parah}). Let $\\widehat{A}^{\\mathrm{ell}} \\subset \\widehat{A}$ be the open subset parameterizing integral spectral curves (the elliptic locus). Following Ng\\^o's method, Yun \\cite{Yun2, Yun3} proved a parabolic support theorem over $C \\times \\widehat{A}^{\\mathrm{ell}}$ and determined all the supports of (\\ref{parah}) in $C \\times \\widehat{A}^\\mathrm{ell}$. More precisely, by \\cite[Section 2]{Yun3} any strict subset of $C \\times \\widehat{A}^{\\mathrm{ell}}$, which is a support of $\\widehat{h}^{\\mathrm{par}}|_{C \\times \\widehat{A}^{\\mathrm{ell}}}$, is a component of the \\emph{endoscopic loci} governed by the endoscopic theory of $G$. As we only consider the special case $G = \\mathrm{PGL}_n$, there are no nontrivial endoscopic loci and the restricted Hitchin map $\\widehat{h}^{\\mathrm{par}}|_{C \\times \\widehat{A}^{\\mathrm{ell}}}$ has full support.\n\nTo prove Theorem \\ref{thm3.2}, it suffices to show that there is no support of (\\ref{parah}) whose generic point lying outside $C \\times \\widehat{A}^{\\mathrm{ell}}$.\n\nSince stability for a $\\mathrm{PGL}_n$ Higgs bundle is described by its corresponding vector bundle (see (\\ref{233})), it is more convenient to work with the $\\mathrm{SL}_n$ moduli spaces. We consider the following Cartesian diagram\n\\begin{equation}\\label{diag321}\n\\begin{tikzcd}\n\\widecheck{M}^{\\mathrm{par}} \\arrow[r, \"(-)\/\\Gamma\"] \\arrow[d, \"\\pi'\"]\n& \\widehat{M}^{\\mathrm{par}} \\arrow[d, \"\\pi\"] \\\\\nC\\times \\widecheck{M} \\arrow[r, \"(-)\/\\Gamma\"]\n& C \\times \\widehat{M}.\n\\end{tikzcd}\n\\end{equation}\nHere the horizontal maps are given by the natural quotient maps by $\\Gamma = \\mathrm{Pic}^0(C)[n]$. To describe the map $\\pi'$ at the left-side of the diagram (see \\cite[Example 2.2.5]{Yun1}),  we recall that $\\widecheck{M}$ parameterizes traceless Higgs bundles with fixed determinant  \n\\[\n({\\mathcal E}, \\theta) , \\quad \\theta: {\\mathcal E} \\to {\\mathcal E}\\otimes \\Omega_{\\mathcal L}, \\quad \\mathrm{rk}({\\mathcal E}) = n, ~~\\mathrm{det}({\\mathcal E}) \\simeq {\\mathcal N},~~ \\mathrm{trace}(\\theta) = 0\n\\]\nwith respect to slope stability. Similarly $\\widecheck{M}^{\\mathrm{par}}$ parameterizes \n\\[\n(x, {\\mathcal E} = {\\mathcal E}_0 \\supset {\\mathcal E}_1 \\supset \\cdots \\supset {\\mathcal E}_n =  {\\mathcal E}_0(-x), \\theta)\n\\]\nwhere $x\\in C$, $({\\mathcal E}, \\theta) \\in \\widecheck{M}$, each ${\\mathcal E}_i$ in the flag is of rank $n$ with ${\\mathcal E}_i\/{\\mathcal E}_{i+1}$ a length 1 skyscraper sheaf supported at $x$, the Higgs field preserves the flag $\\theta({\\mathcal E}_i) \\subset {\\mathcal E}_i\\otimes \\Omega_{\\mathcal L}$, and the map $\\pi'$ in the diagram is the forgetful map. We note that the stability condition for a parabolic Higgs bundle is determined by the stability of the underlying Higgs bundle. \n\nFollowing \\cite[Example 2.2.5]{Yun1} we may also describe $\\widecheck{M}^{\\mathrm{par}}$ via spectral curves. If we present a point in $\\widecheck{M}$ as a 1-dimensional sheaf ${\\mathcal F}_\\alpha$ supported on a spectral curve $C_\\alpha \\subset \\mathrm{Tot}(\\Omega_{\\mathcal L})$ with $p_\\alpha: C_\\alpha \\to C$ the projection, then for any $x\\in C$, a closed point in the fiber $\\pi'^{-1}(x, {\\mathcal F}_\\alpha)$ is represented by\n\\[\n{\\mathcal F}_\\alpha = {\\mathcal F}_0 \\supset {\\mathcal F}_1 \\supset \\cdots \\supset {\\mathcal F}_n = {\\mathcal F}_0\\otimes p_\\alpha^* {\\mathcal O}_C(-x), \\quad \\mathrm{lengh}({\\mathcal F}_i\/{\\mathcal F}_{i+1}) = 1;\n\\]\nsee \\cite[(2.4)]{Yun1}.\n\nNow we consider the $\\mathrm{SL}_n$ parabolic Hitchin system\n\\begin{equation}\\label{SLh}\n\\widecheck{h}^p: \\widecheck{M}^{\\mathrm{par}} \\to C\\times \\widecheck{M} \\to C\\times \\widehat{A}.\n\\end{equation}\nFrom the diagram (\\ref{diag321}), it recovers the $\\mathrm{PGL}_n$ Hitchin system $\\widehat{h}^p: \\widehat{M}^{\\mathrm{par}} \\to C \\times \\widehat{A}$ by taking the quotient of the source by $\\Gamma$. In particular, we have\n\\[\nR{\\widehat{h}^p}_* ~{\\mathbb{Q}}_{\\widehat{M}^{\\mathrm{par}}} = \\left(R{\\widecheck{h}^p}_* ~{\\mathbb{Q}}_{\\widecheck{M}^{\\mathrm{par}}}\\right)^\\Gamma \\in D_c^b(C\\times \\widehat{A}).\n\\]\nHence in order to prove Theorem \\ref{thm3.2}, it suffices to show that there is no support of (\\ref{SLh}) with generic point lying outside $C \\times \\widehat{A}^{\\mathrm{ell}}$. In the next two sections, we adapt the strategy of Chaudouard--Laumon \\cite{CL} and de Cataldo \\cite{dC_SL} (for the $\\mathrm{SL}_n$-version) to this parabolic setting, and verify the parabolic analogs of (I) and (II) of Section \\ref{Sec4.1}. \n\n\n\n\n\\subsection{Weak abelian fibrations and $\\delta$-regularity}\n\nA general setup for support theorems was given in \\cite[Theorem 1.8]{MS2}. We now check that all the assumptions there are satisfied by (\\ref{SLh}).\n\n\nWe first construct the $(C\\times \\widehat{A})$-group scheme $P$ which acts on $\\widecheck{M}^{\\mathrm{par}}$. Recall that in \\cite{dC_SL} de Cataldo showed that $\\widecheck{M}$ admits a weak abelian fibration structure $(\\widecheck{P}, \\widecheck{M}, \\widehat{A})$.\\footnote{We refer to \\cite[Section 1]{MS2} for an introduction to weak abelian fibrations.} Here the $\\widehat{A}$-group scheme $\\widecheck{P}$ which acts on $\\widecheck{M}$ is given by the identity component of the relative Prym variety associated with the spectral curves \\cite[(43)]{dC_SL}. For a spectral curve $p_\\alpha: C_\\alpha \\to C$ with a Higgs bundle given by ${\\mathcal F}_\\alpha$ supported on $C_\\alpha$, the Prym action is induced by tensor product\n\\[\n{\\mathcal Q} \\cdot {\\mathcal F}_{\\alpha} = {\\mathcal Q} \\otimes {\\mathcal F}_\\alpha, \\quad {\\mathcal Q} \\in \\mathrm{Prym}^0(C_\\alpha\/C) = \\widecheck{P}_{[C_\\alpha]},~ \\quad [C_\\alpha] \\in \\widehat{A}. \n\\]\nWe denote by $P$ the $(C\\times \\widehat{A})$-group scheme obtained by the pullback of $\\widecheck{P}$. The $P$-action on $C\\times \\widecheck{M}$ can be lifted to $\\widecheck{M}^{\\mathrm{par}}$:\n\\[\n{\\mathcal Q} \\cdot (x, {\\mathcal F}_\\alpha = {\\mathcal F}_0 \\supset {\\mathcal F}_1 \\supset \\cdots \\supset {\\mathcal F}_n) =  (x, {\\mathcal Q}\\otimes{\\mathcal F}_\\alpha = {\\mathcal Q} \\otimes {\\mathcal F}_0 \\supset {\\mathcal Q}\\otimes{\\mathcal F}_1 \\supset \\cdots \\supset {\\mathcal Q} \\otimes {\\mathcal F}_n),\n\\] \nsince the stability condition does not rely on the flag.\n\n\n\\begin{prop}\nThe triple $(P, \\widecheck{M}, C\\times \\widehat{A})$ forms an weak abelian fibration, \\emph{i.e.},  it satisfies (i,ii,ii) of \\cite[Section 1.1]{MS2}.\n\\end{prop}\n\n\\begin{proof}\n    (i) is clear from (\\ref{diag0}), and (iii) only depends on the group scheme $P$ which follows from the corresponding property for $\\widecheck{P}$ proved in \\cite[Theorem 4.7.2]{dC_SL}. To prove (ii): since the $P$-action on $\\widecheck{M}^{\\mathrm{par}}$ is a lifting of the $P$-action on $C\\times \\widecheck{M}$, and the latter has affine stabilizers by \\cite{dC_SL}, we obtain that the $P$-stabilizer for any point $z \\in \\widecheck{M}^{\\mathrm{par}}$ is contained in the (affine) $P$-stabilizer for $\\pi(z) \\in C \\times \\widecheck{M}$. This proves (ii).\n\\end{proof}\n\n\n\n\nFrom this proposition, in order to show the support inequality ((I) of Section \\ref{Sec4.1}) for $(P, \\widecheck{M}, C\\times \\widehat{A})$, by \\cite[Theorem 1.8]{MS2}, it suffices to estabish the relative dimension bound\n\\begin{equation}\\label{relative}\n   \\tau_{> 2\\mathbf{d}}\\left( R\\widecheck{h}^p_* {\\mathbb{Q}}_{\\widecheck{M}^{\\mathrm{par}}}\\right) = 0, \\quad \\mathbf{d}: = \\mathrm{dim} \\widecheck{M}^{\\mathrm{par}} - \\mathrm{dim} (C\\times \\widehat{A})\n\\end{equation}\nwith $\\tau_{>*}$ the standard truncation functor. \n\nOnce we have (\\ref{relative}),\nwe can combine the support inequality with $\\delta$-regularity of \n$\\widecheck{P}$ \\cite[corollary 5.4.4]{dC_SL} to prove \nTheorem \\ref{thm3.2} as follows.  If $Z \\subset C \\times \\widehat{A}$ is an irreducible support of $\\widecheck{h}^p$, the projection $W = \\mathrm{pr}_{\\widehat{A}}(Z) \\subset \\hat{A}$ also satisfies the support inequality\n\\[\n\\mathrm{codim}_{\\widehat{A}} W \\leq \\mathrm{codim}_{C\\times \\widehat{A}}Z \\leq \\delta_Z = \\delta_W.\n\\]\nHere we used in the second inequality the support inequality for $\\widecheck{h}^p$, and used in the last equality that the $\\delta$-function on $C\\times \\widehat{A}$ is pulled back from $\\widehat{A}$. The argument of \\cite[Section 6.2]{dC_SL} then shows that the generic point of $W$ lies in $\\widehat{A}^{\\mathrm{ell}}$. Hence the generic point of $Z$ lies in $C\\times \\widehat{A}^{\\mathrm{ell}}$ as we desired.\n\n\n\n\nIn conclusion, this reduces Theorem \\ref{thm3.2} to (\\ref{relative}).\n\n\n\n\\subsection{Proof of the relative dimension bound (\\ref{relative})}\\label{sec4.4}\n\nWe need to show that each closed fiber of the morphism\n$$\\widecheck{h}^p: \\widecheck{M}^{\\mathrm{par}} \\rightarrow C\\times \\widehat{A}$$\nhas dimension\n\\[\n\\mathbf{d} = \\mathrm{dim}\\widecheck{M}^{\\mathrm{par}} - \\mathrm{dim} (C\\times \\widehat{A}) =    \\frac{n(n-1)}{2}\\deg({\\mathcal L}) + (n^2-1)(g-1).\\]\nHere the last equation is obtained by the dimension formula of \\cite[Proposition 2.4.6]{dC_SL} directly:\n\\[\n\\mathbf{d} =  \\mathrm{dim}\\widecheck{M} - \\mathrm{dim}\\widehat{A} = \\frac{n(n-1)}{2} \\deg({\\mathcal L}) + (n^2-1)(g-1).\n\\]\n\n\n\nFirst,  we note that the morphism $\\widecheck{h}^p$ is surjective since the usual Hitchin map $\\widecheck{h}: \\widecheck{M} \\to \\widehat{A}$ is surjective. Furthermore, as $\\widecheck{h}^p$ is equivariant with respect to the scaling ${\\mathbb{G}}_m$-action on the Higgs fields, it suffices to bound from above the dimension of the fiber over $(x, 0) \\in C\\times \\widehat{A}$ for each point $x \\in C$ by $\\mathbf{d}$, since upper semicontinuity will force all other fibers to have the same dimension.  We fix the point $p$ from now on.\n\nUsing the assumption $\\deg({\\mathcal L})>2g$, we may express ${\\mathcal L}$ as ${\\mathcal O}_C(D)$ with $D = x_0 + x_1 + \\dots + x_t$ an effective reduced divisor containing $x = x_0$. In particular, a Higgs bundle $({\\mathcal E}, \\theta: {\\mathcal E} \\to {\\mathcal E}\\otimes \\Omega_{\\mathcal L})$ can be viewed as a meromorphic (un-twisted) Higgs bundles with at most simple poles along $D$:\n\\[\n({\\mathcal E}, \\theta), \\quad \\theta: {\\mathcal E} \\to {\\mathcal E} \\otimes \\Omega_C^1(D).\n\\]\nWe will control the fiber dimension using an analogous bound for the nilpotent cone for \\emph{strongly parabolic} Higgs bundles for the pair $(C,D)$.\n\nRecall that, given $D$ as above, a strongly parabolic Higgs bundle consists of an $\\mathrm{SL}_n$ Higgs bundle of degree $d$:\n\\[\n({\\mathcal E}, \\theta), \\quad \\theta: {\\mathcal E}\\to {\\mathcal E} \\otimes \\Omega^1_C(D),~~~\\mathrm{rk}({\\mathcal E}) =n,~~\\mathrm{det}({\\mathcal E})\\simeq {\\mathcal N}, ~~\\mathrm{trace}(\\theta) = 0,\n\\]\nalong with a flag at each point $x_i \\in D$:\n\\[\n{\\mathcal E}|_{x_i}=F_{x_i,0}  \\supset F_{x_i,1} \\supset F_{x_i,2}, \\supset \\cdots \\supset F_{x_i,n} = 0, \\quad \\mathrm{dim} F_{x_i,j}\/F_{x-i, j+1} = 1,\n\\]\nsuch that the Higgs field satisfies $\\theta(F_{x_i, j}) \\subset F_{x_i, j+1}$.\\footnote{One may compare this with the (non-strong) parabolic condition $\\theta(F_{x,j}) \\subset F_{x,j}$ we used earlier in this paper for the global Springer theory.}\n\n\n\n\nLet $\\widecheck{M}^{\\mathrm{spar}}(D)$ denote the moduli space of strongly parabolic Higgs bundles associated with the pair $(C,D)$, such that the underlying twisted Higgs bundle is stable. Let\n$$A(D) = \\bigoplus_{i=2}^{n} H^0(C, \\Omega_{\\mathcal L}^{\\otimes i}(-D)) \\subset \\widehat{A}$$\ndenote the linear subspace consisting of sections which vanish along the points of $D$. There is a Hitchin map for the strongly parabolic Higgs moduli space\n$$\\widecheck{h}^{\\mathrm{spar}}:\\widecheck{M}^{\\mathrm{spar}}(D) \\rightarrow A(D)$$\nwhich is proper, surjective, and Lagrangian with respect to a natural holomorphic symplectic form \\cite{Faltings}. In particular, the dimension of the zero fiber is given by\n\\begin{align*}\n\\dim \\widecheck{M}^{\\mathrm{spar}}(D)_0 = \\dim A(D) &= \\sum_{j=2}^{n} \\left((j-1)(\\deg(\\Omega_L)) +g-1\\right)\\\\\n&= \\frac{n(n-1)}{2}\\deg ({\\mathcal L}) + (n^2-1)(g-1).\n\\end{align*}\nSee also \\cite[Theorem 6.9]{SWW} for a direct proof of the above dimension formula.\\footnote{The formula above has $g$ less than the dimension formula obtained in \\cite[Theorem 6.9]{SWW}, since we consider the $\\mathrm{SL}_n$ case where we fixed the determinant on $C$.}\n\nWe now consider the following diagram relating the two types of parabolic Higgs moduli spaces $\\widecheck{M}^{\\mathrm{spar}}(D)$ and \n$\\widecheck{M}^{\\mathrm{par}}$:\n\n\\begin{equation}\\label{diagram}\n\\begin{tikzcd}\n\\widecheck{M}^{\\mathrm{spar}}(D)_0 \\arrow{d}{q_0}\\arrow[r,hook]\n&\\widecheck{M}^{\\mathrm{spar}}(D)\\arrow{d}{q}\n& \n\\\\\n\\widecheck{M}^{\\mathrm{par}}_{(x,0)}\\arrow{d}\\arrow[r, hook]&\n{\\widecheck{M}^{\\mathrm{par}}}|_{x \\times A(D)} \\arrow{d}\\arrow[r,hook]&\n\\widecheck{M}^{\\mathrm{par}}\\arrow[d,\"\\widecheck{h}^p\"]\\\\\n\\{(x,0)\\} \\arrow[r, hook] & x \\times A(D)  \\arrow[r, hook] & C\\times \\widehat{A}\n\\end{tikzcd}\n\\end{equation}\nwhere all squares are Cartesian.  \n\nWe first claim that the natural map $q$ (sending a strong parabolic Higgs bundle to a parabolic Higgs bundle) is surjective. Indeed, given a point $z \\in \\widecheck{M}^{\\mathrm{par}}$ lying over $C\\times {A}(D)$,\nchoosing a point in its preimage $q^{-1}(z)$ only consists of fixing a flag at each point of $x_i$ with $i>0$, preserved by the Higgs field --- since the characteristic polynomial already has zero roots at all points of $D$, the Higgs field is automatically strongly parabolic.\n\n\nBy base change, this implies that $q_0$ is surjective as well, so we get the desired dimension upper bound\n\\[\n\\dim {\\widecheck{M}^{\\mathrm{par}}}_{(x,0)} \\leq \\dim \\widecheck{M}^{\\mathrm{spar}}(D)_0 = \\frac{n(n-1)}{2}\\deg ({\\mathcal L}) + (n^2-1)(g-1),\n\\]\nwhich completes the proof. \\qed\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{figure}[!ht]\n  \\centering\n  \\begin{subfigure}{1.0\\columnwidth}\n     \\includegraphics[draft=false,trim={0 200px 0 200px},clip,width=\\columnwidth]{figures\/smoke_stable0027.jpeg}\n  \\end{subfigure} \\\\\n  \\begin{subfigure}{1.0\\columnwidth}\n     \\includegraphics[draft=false,trim={0 200px 0 200px},clip,width=\\columnwidth]{figures\/smoke_stable0126.jpeg}\n  \\end{subfigure} \\\\\n  \\begin{subfigure}{1.0\\columnwidth}\n     \\includegraphics[draft=false,trim={0 200px 0 200px},clip,width=\\columnwidth]{figures\/smoke_stable0260.jpeg}\n  \\end{subfigure}\n \n\\caption{{\\textbf{Colorful smoke jets}}. Multicolored jets of smoke are simulated with BSLQB. Intricate mixing is induced as the flows collide at the spherical boundary.}\\label{fig:rainbowsmoke}\n\\end{figure}\n\nWhether it be billowing smoke, energetic explosions, or breaking waves, simulation of incompressible flow is an indispensable tool for modern visual effects. Ever since the pioneering works of Foster and Metaxas \\shortcite{foster:1996:liquids}, Stam \\shortcite{stam:1999:stable} and Fedkiw et al. \\shortcite{fedkiw:2001:visual,foster:2001:practical}, the Chorin \\shortcite{chorin:1967:numerical} splitting of advective and pressure projection terms  has been the standard in computer graphics applications \\cite{bridson:2008:fluid-simulation}. Most techniques use regular grids of Marker-And-Cell (MAC) \\cite{harlow:1965:viscous-flow} type with pressure and velocity components staggered at cell centers and faces respectively. Furthermore, advection is most often discretized using semi-Lagrangian techniques originally developed in the atmospheric sciences \\cite{stam:1999:stable,robert:1981:stable}. Although well-established, these techniques are not without their drawbacks. For example, the staggering utilized in the MAC grids is cumbersome since variables effectively live on four different grids. This can complicate many algorithms related to incompressible flow. E.g. Particle-In-Cell (PIC) \\cite{harlow:1964:pic} techniques like FLIP \\cite{brackbill:1986:flip-pic,zhu:2005:sand-fluid}, Affine\/Polynomial Particle-In-Cell (APIC\/PolyPIC) \\cite{jiang:2015:apic,fu:2017:poly} and the Material Point Method (MPM) \\cite{sulsky:1994:history-materials,stomakhin:2014:augmented-mpm} must transfer information separately to and from each individual grid. Similarly, semi-Lagrangian techniques must separately solve for upwind locations at points on each of the velocity component grids. Moreover, while semi-Lagrangian techniques are renowned for the large time steps they admit (notably larger than the Courant-Friedrichs-Levy (CFL) condition), their inherent stability is plagued by dissipation that must be removed for most visual effects phenomena. Another limitation of the MAC grid arises with free-surface water simulation. In this case, the staggering prevents many velocity components near the fluid free surface from receiving a correction during projection (see e.g. \\cite{bridson:2008:fluid-simulation}). Each of these velocity components must then be separately extrapolated to from the interior to receive a pressure correction. \\\\\n\\\\\nMAC grids are useful because the staggering prevents pressure null modes while allowing for accurate second order central differencing in discrete grad\/div operators. However, there are alternatives in the computational physics literature. Many mixed Finite Element Method (FEM) techniques use collocated velocities \\cite{hughes:2000:book} without suffering from pressure mode instabilities. For example, Taylor-Hood elements \\cite{taylor:1973:TH} use collocated multi-quadratic velocity interpolation and multilinear pressure interpolation to enforce incompressiblity. Recently, B-spline interpolation \\cite{deboor:1978:splines} has been used with Taylor-Hood \\cite{bressan:2010:isogeometric}. We build on this work and develop an approach based on collocated multi-quadratic B-spline interpolation for velocities. This choice is motivated by the simplicity of collocated grids compared to staggering, but also because of the ease of attaining continuous derivatives with B-spline interpolation. For example, this interpolation is often chosen with MPM applications since $C^1$ interpolation is essential for stability \\cite{steffen:2008:analysis}. In the context of fluids, we show that this allows for extremely stable and accurate advection.\n\\\\\n\\\\\nWe develop a new approach for Chorin splitting \\shortcite{chorin:1967:numerical} based on the collocated multiquadratic B-spline velocity, multilinear pressure Taylor-Hood element \\cite{bressan:2010:isogeometric}. However, unlike the fully collocated technique of Bressan \\shortcite{bressan:2010:isogeometric}, we stagger pressures on the nodes of the grid and velocities at cell centers as in \\cite{ando:2013:surfacing}, since it reduces coupling in the pressure projection system and naturally accommodates particle-based definition of the flow domain for free-surface simulation of water. Notably, our formulation does not require velocity extrapolation after pressure projection for free-surface flow calculations as is typically needed with MAC grids. We use regular grids, but as in \\cite{batty:2007:solid-fluid,batty:2008:buckling,larionov:2017:stokes}, we allow for irregular domains in a variational way using cut cells. However, rather than a weighted finite different approach, we use an FEM approach as in XFEM \\cite{belytschko:2009:review,koschier:2017:xfem} and virtual node (VNA) \\cite{schroeder:2014:vna} techniques. In VNA and XFEM approaches, integrals arising in the variational formulation are carried out over the intersection of the grid with the domain geometry. \\\\\n\\\\\n\\begin{figure}[!ht]\n  \\centering\n  \\begin{subfigure}{\\columnwidth}\n     \\includegraphics[draft=false,trim={0 100px 0 50px},clip,width=\\columnwidth]{figures\/dambreak_1.jpeg}\n  \\end{subfigure} \\\\\n  \\begin{subfigure}{\\columnwidth}\n     \\includegraphics[draft=false,trim={0 100px 0 50px},clip,width=\\columnwidth]{figures\/dambreak_14.jpeg}\n  \\end{subfigure} \\\\\n  \\begin{subfigure}{\\columnwidth}\n     \\includegraphics[draft=false,trim={0 100px 0 50px},clip,width=\\columnwidth]{figures\/dambreak_0090.jpeg}\n  \\end{subfigure}\n\\caption{{\\textbf{Dam break}}. A block of water falls in a rectangular domain with obstacles.  Dynamic splashing behavior is followed by settling of the water in the tank. White water rendering effects are added based on \\cite{ihmsen:2012:unified}.}\\label{fig:dambreak_final}\n\\end{figure}\nWe leverage $C^1$ continuity guaranteed by our quadratic B-spline velocity interpolation to develop BSLQB, a novel Backward Semi-Lagrangian (BSL) \\cite{robert:1981:stable} technique that achieves second order accuracy in space and time. BSL techniques utilize the implicit form of semi-Lagrangian advection. We show that our novel BSL method for quadratic B-splines dramatically reduces numerical dissipation with only a small modification to the widely-adopted explicit semi-Lagrangian formulations typically used in graphics applications. Semi-Lagrangian techniques for velocity advection utilize the implicit relation associated with solution of Burgers' equation\n\\begin{align}\\label{eq:impBurg}\n\\mb{u}(\\mb{x},t)=\\mb{u}(\\mb{x}-(t-s)\\mb{u}(\\mb{x},t),s)\\iff \\ \\frac{D\\mb{u}}{Dt}=\\frac{\\partial \\mb{u}}{\\partial t}+\\frac{\\partial \\mb{u}}{\\partial \\mb{x}}\\mb{u}=\\mathbf{0}\n\\end{align}\nfor $s\\leq t$ \\cite{evans:2010:pde}. Traditionally, graphics applications have preferred the explicit variant of semi-Lagragian advection whereby grid velocities are updated through the expression\n\\begin{align}\\label{eq:SL}\n\\mb{u}_\\mb{i}^{n+1}=\\mb{u}(\\mb{x}_\\mb{i}-\\Delta t \\mb{u}_\\mb{i}^n,t^n)\n\\end{align}\nwhere $\\mb{x}_\\mb{i}$ is the location of grid node $\\mb{i}$, $\\mb{u}_\\mb{i}^n,\\mb{u}_\\mb{i}^{n+1}$ are velocities at the node at times $t^n$ and $t^{n+1}$ respectively and interpolation over the velocity grid is used to estimate $\\mb{u}(\\mb{x}_\\mb{i}-\\Delta t \\mb{u}_\\mb{i}^n,t^n)$ at non-grid node locations \\cite{sawyer:1963:semi,stam:1999:stable}. In contrast, BSL techniques leverage Equation ~\\eqref{eq:impBurg} directly \n\\begin{align}\\label{eq:BSL}\n\\mb{u}_\\mb{i}^{n+1}=\\mb{u}(\\mb{x}_\\mb{i}-\\Delta t \\mb{u}_\\mb{i}^{n+1},t^n)\n\\end{align}\nwhich requires the solution of an implicit equation for $\\mb{u}_\\mb{i}^{n+1}$ \\cite{robert:1981:stable}. Since our grid interpolation is naturally $C^1$, we show that this can be done very efficiently using a few steps of Newton's method. While this is more expensive than the explicit semi-Lagrangian formulations, we note that each node can still be updated in parallel since the implicit equations for $\\mb{u}_\\mb{i}^{n+1}$ are decoupled in $\\mb{i}$. We show that solution of the implicit Equation~\\eqref{eq:BSL}, rather than the traditionally used explicit Equation~\\eqref{eq:SL} improves the order of convergence from first to second (in space and time). Notably, this does not require use of multiple time steps for backward\/forward estimations of error, as is commonly done \\cite{kim:2006:advections,kim:2005:bfecc,selle:2008:unconditionally,xiu:2001:semi,schroeder:2014:vna}. Furthermore, our method allows for larger-than-CFL time steps and is as stable or more so than explicit semi-Lagrangian formulations.\\\\\n\\\\\n\\begin{figure}[t]\n  \\centering\n  \\begin{subfigure}{.49\\columnwidth}\n     \\includegraphics[draft=false,trim={0 60px 0 60px},clip,width=\\columnwidth]{figures\/Inner_circle_8.jpeg}\n  \\end{subfigure}\n  \\begin{subfigure}{.49\\columnwidth}\n     \\includegraphics[draft=false,trim={0 60px 0 60px},clip,width=\\columnwidth]{figures\/Inner_circle52.jpeg}\n  \\end{subfigure}\\\\\n  \\begin{subfigure}{.49\\columnwidth}\n     \\includegraphics[draft=false,trim={0 60px 0 60px},clip,width=\\columnwidth]{figures\/Inner_circle_401.jpeg}\n  \\end{subfigure}\n  \\begin{subfigure}{.49\\columnwidth}\n     \\includegraphics[draft=false,trim={0 60px 0 60px},clip,width=\\columnwidth]{figures\/Inner_circle_600.jpeg}\n  \\end{subfigure}\n\\caption{{\\textbf{SL vs. BSLQB}}. We compare semi-Lagrangian (left) and BSLQB  (right) in a vorticity-intensive example. BSLQB breaks symmetry and exhibits a more turbulent flow pattern. Note we only use particles for flow visualization and not for PolyPIC advection in this example.}\\label{fig:inner_circle}\n\\end{figure}\nLastly, we develop a hybrid particle\/BSLQB advection technique that utilizes PolyPIC \\cite{fu:2017:poly} in portions of the domain covered by particles and BSLQB in portions without particles. Our formulation naturally leverages the strengths of both approaches. Dense concentrations of particles can be added to regions of the domain where more detail is desired. Also, if particle coverage becomes too sparse because of turbulent flows, BSLQB can be used in the gaps. We demonstrate the efficacy of this technique with smoke simulation and narrow banding of particles near the fluid surface with water simulations as in \\cite{chentanez:2015:coupling,ferstl:2016:narrow,sato:2018:nb}. In this case, level set advection naturally enabled with our BSLQB formulation is preferred in deeper water regions. We summarize our contributions as:\n\\begin{itemize}\n\\item A novel cut-cell collocated velocity B-spline mixed FEM method for Chorin \\shortcite{chorin:1967:numerical} splitting discretization of the incompressible Euler equations.\n\\item BSLQB: a novel BSL technique designed for collocated multiquadratic B-spline velocity interpolation that achieves second order accuracy in space and time.\n\\item A hybrid BSLQB\/PolyPIC method for narrow band free-surface flow simulations and concentrated-detail smoke simulations.\n\\end{itemize}\n\n\\begin{figure*}[!ht]\n  \\centering\n  \\begin{subfigure}{.33\\textwidth}\n     \\includegraphics[draft=false,trim={0 0 0 0},clip,width=\\columnwidth]{figures\/bunnydrown_0003.jpeg}\n  \\end{subfigure} \n  \\begin{subfigure}{.33\\textwidth}\n     \\includegraphics[draft=false,trim={0 0 0 0},clip,width=\\columnwidth]{figures\/bunnydrown_0024.jpeg}\n  \\end{subfigure} \n  \\begin{subfigure}{.33\\textwidth}\n     \\includegraphics[draft=false,trim={0 0 0 0},clip,width=\\columnwidth]{figures\/bunnydrown_0114.jpeg}\n  \\end{subfigure}\n\\caption{{\\textbf{Dam break with bunny}}: Opposing blocks of water collapse in a tank and flow around the irregular domain boundary placed in the middle of the tank.  Particles are colored from slow (blue) to fast (white) speed. }\\label{fig:bunny_dambreak_final}\n\\end{figure*}\n\n\\section{Previous Work}\n\n\\subsection{Advection}\nStam \\shortcite{stam:1999:stable} first demonstrated the efficacy of semi-Lagrangian techniques for graphics applications and they have since become the standard, largely due to the large time steps they engender and their simple interpolatory nature. Many modifications to the original approach of Stam \\shortcite{stam:1999:stable} have been developed, often inspired by approaches in the engineering literature. Fedkiw et al. \\shortcite{fedkiw:2001:visual} use vorticity confinement \\cite{steinhoff:1994:modification} to counterbalance vorticity lost to dissipation and cubic grid interpolation. Kim et al. \\shortcite{kim:2006:advections,kim:2005:bfecc} and Selle et al. \\cite{selle:2008:unconditionally} combine forward and backward semi-Lagrangian steps to estimate and remove dissipative errors. Constrained Interpolation Profile \\cite{kim:2008:semi,yabe:2001:multiphase-analysis,song:2009:derivative-particles} techniques additionally advect function derivatives to reduce dissipation. Molemaker et al. \\shortcite{molemaker:2008:low} use the QUICK technique of Leonhard \\shortcite{leonard:1979:stable} which is essentially upwinding with quadratic interpolation and Adams-Bashforth temporal discretization, although this does not have the favorable stability properties of semi-Lagrangian. Backward Difference Formula techniques are useful because they use an implicit multistep formulation for higher-order semi-Lagrangian advection yet still only require one projection per time step \\cite{xiu:2001:semi,schroeder:2014:vna}.\\\\\n\\\\\nThe main idea in semi-Lagrangian techniques is to interpolate data from a characteristic point. This idea goes back to the Courant-Issaacson-Rees \\shortcite{courant:1952:solution} method. However, as noted in \\cite{fedkiw:2001:visual} semi-Lagrangian advection is very popular in atmospheric science simulation and the variants used in graphics that account for characteristics traveling beyond the local cell in one time step go back to Sawyer \\shortcite{sawyer:1963:semi}. The first BSL approach utilizing Equation~\\eqref{eq:BSL} was done by Robert \\shortcite{robert:1981:stable} in which they use fixed point iteration to solve the nonlinear equation. They fit a bicubic function to their data over $4\\times4$ grid patches, then use that function in the fixed point iteration. If the upwind point leaves the grid, they clamp it to the boundary of the $4\\times4$ patch. This clamping will degrade accuracy for larger time steps. In this case, more general interpolation is typically used (see \\cite{staniforth:1991:semi,falcone:1998:convergence} for useful reviews). Pudykiewicz and Staniforth \\shortcite{pudykiewicz:1984:some} investigate the effects of BSL versus explicit semi-Lagrangian. Specifically, they compare Bates and McDonald \\shortcite{bates:1982:multiply} (explicit) versus Robert \\shortcite{robert:1981:stable} (BSL). They show that keeping all things equal, the choice of Equation~\\eqref{eq:SL} (explicit) instead of Equation~\\eqref{eq:BSL} (BSL) leads to more dissipation and mass loss. This is consistent with our observations with BSLQB.\\\\\n\\\\\nInterestingly, multiquadratic B-splines have not been adopted by the semi-Lagrangian community, despite their natural regularity. Hermite splines, multicubic splines and even Lagrange polynomials are commonly used \\cite{staniforth:1991:semi}. Preference for Hermite splines and Lagrange polynomials is likely due to their local nature (they do not require solution of a global system for coefficients) and preference for multicubic splines (over multi-quadratic) is possibly due to the requirement of odd degree for natural splines (odd degree splines behave like low pass filters and tend to be smoother than even degree splines \\cite{cheng:2001:quadratic,cheney:2012:numerical}). Cubic splines are considered to be more accurate than Hernite splines and Lagrange interpolation \\cite{staniforth:1991:semi,makar:1996:basis}. Interestingly, Riish{\\o}jgaard et al. \\shortcite{riishojgaard:1998:use} found that cubic spline interpolation gave rise to a noisier solution than cubic Lagrange interpolation with a technique analogous to that of Makar and Karpik \\shortcite{makar:1996:basis}. However, they also note that addition of a selective scale diffusion term helps reduce noise associated with cubic splines. Wang and Layton \\shortcite{wang:2010:new} use linear B-splines with BSL but only consider one space dimension which makes Equation~\\eqref{eq:BSL} linear and easily solvable.\\\\\n\\\\\nDissipation with explicit semi-Lagrangian advection is so severe that many graphics researchers have resorted to alternative methods to avoid it. Mullen et al. \\shortcite{mullen:2009:energy} develop energy preserving integration to prevent the need for correcting dissipative behavior. Some authors \\cite{qu:2019:mcm,tessendorf:2011:MCM,sato:2017:long,sato:2018:spatially} resolve the flow map characteristics for periods longer than a single time step (as opposed to one step with semi-Lagrangian) to reduce dissipation. Hybrid Lagrange\/Eulerian techniques like PIC (and related approaches) \\cite{bridson:2008:fluid-simulation,jiang:2015:apic,fu:2017:poly,zhu:2005:sand-fluid} explicitly track motion of particles in the fluid, which is nearly dissipation-free, but can suffer from distortion in particle sampling quality. Vorticity formulations are also typically less dissipative, but can have issues with boundary conditions enforcement \\cite{selle:2005:vortex,angelidis:2005:simulation,chern:2016:schrodinger,elcott:2007:stable,park:2005:vortex,weissmann:2010:filament}. Zehnder et al., Zhang et al. and Mullen et al. \\shortcite{mullen:2009:energy,zehnder:2018:advection,narain:2019:ref,zhang:2015:restoring} have noted that the Chorin projection itself causes dissipation. Zhang et al. \\shortcite{zhang:2015:restoring} reduced artificial dissipation caused by the projection step by estimating lost vorticity and adding it back into the fluid. Zehnder et al. \\shortcite{zehnder:2018:advection,narain:2019:ref} propose a simple, but very effective modification to the splitting scheme that is similar to midpoint rule integration to reduce the projection error.\n\n\\subsection{Pressure projection}\n\nGraphics techniques utilizing pressure projection typically use voxelized MAC grids with boundary conditions enforced at cell centers and faces, however many methods improve this by taking into account sub-cell geometric detail. Enright et al. \\shortcite{enright:2003:using} showed that enforcing the pressure free surface boundary condition at MAC grid edge crossings (rather than at cell centers) dramatically improved the look of water surface waves and ripples. Batty, Bridson and colleagues developed variational weighted finite difference approaches to enforce velocity boundary conditions with MAC grids on edge crossings and improved pressure boundary conditions at the free surface in the case of viscous stress \\cite{batty:2007:solid-fluid,batty:2008:buckling,larionov:2017:stokes}. XFEM \\cite{belytschko:2009:review,koschier:2017:xfem} and virtual node (VNA) \\cite{schroeder:2014:vna} techniques also use cut cell geometry with variational techniques. Schroeder et al. \\shortcite{schroeder:2014:vna} use cut cells with MAC grids, but their technique is limited to moderate Reynolds numbers. \\\\\n\\\\\n\\begin{figure}[!t]\n  \\centering\n  \\begin{subfigure}{0.24\\columnwidth}\n     \\includegraphics[draft=false,trim={300px 0 300px 0},clip,width=\\columnwidth]{figures\/globe0000.jpeg}\n  \\end{subfigure\n  \\begin{subfigure}{0.24\\columnwidth}\n     \\includegraphics[draft=false,trim={300px 0 300px 0},clip,width=\\columnwidth]{figures\/globe0007.jpeg}\n  \\end{subfigure}\n  \\begin{subfigure}{0.24\\columnwidth}\n     \\includegraphics[draft=false,trim={300px 0 300px 0},clip,width=\\columnwidth]{figures\/globe0013.jpeg}\n  \\end{subfigure\n  \\begin{subfigure}{0.24\\columnwidth}\n     \\includegraphics[draft=false,trim={300px 0 300px 0},clip,width=\\columnwidth]{figures\/globe0062.jpeg}\n  \\end{subfigure}\n\\caption{{\\textbf{Water in a globe}}. A block of water splashes and naturally slides along cut cell boundaries in an irregular domain interior to one large sphere and exterior to one small sphere.}\\label{fig:snowglobe}\n\\end{figure}\nThere is a vast literature on enforcing incompressibility in the FEM community \\cite{hughes:2000:book}. Our approach is most similar to the B-spline Taylor-Hood element of Bressan \\cite{bressan:2010:isogeometric}. Adoption of B-spline interpolation in FEM is part of the isogeometric movement \\cite{hughes:2005:isogeometric,ruberg:2012:subdivision}. Originally motivated by the desire to streamline the transition from computer-aided design (CAD) to FEM simulation, isogeometric analysis explores the use of CAD-based interpolation (e.g. B-splines and nonuniform rational B-splines (NURBS)) with FEM methodologies. Hughes et al. \\shortcite{hughes:2005:isogeometric} show that in addition to simplifying the transition from CAD to simulation, the higher regularity and spectral-like properties exhibited by these splines makes them more accurate than traditionally used interpolation. We enforce Dirichlet boundary conditions weakly as in XFEM and VNA approaches \\cite{belytschko:2009:review,koschier:2017:xfem,schroeder:2014:vna}. Bazilevs et al. \\shortcite{bazilevs:2007:weak} show that weak Dirichlet enforcement with isogeometric analysis can be more accurate than strong enforcement.\\\\\n\\\\\nGraphics applications are typically concerned with turbulent, high-Reynolds numbers flows. Interestingly, B-splines have proven effective for these flows by researchers in the Large Eddy Simulation (LES) community \\cite{kim:1998:mixed,kravchenko:1999:bspline}. Kravchenko et al. \\shortcite{kravchenko:1999:bspline} use a variational weighted residuals approach with B-splines for turbulent LES and show that the increased regularity significantly reduces computational costs. Boatela et al. \\shortcite{botella:2002:collocation} use a similar approach, but apply a collocation technique where the strong form of the div-grad formulation of incompressibility is enforced point wise. They show that their B-spline approach attains optimal order of accuracy with accurate resolution of quadratic flow invariants. Boatela et al. \\shortcite{botella:2002:collocation} also introduce a notion of sparse approximation to the inverse mass matrix to avoid dense systems of equations in the pressure solve.\n\n\\begin{figure}[t]\n  \\centering\n  \\begin{subfigure}{0.5\\columnwidth}\n     \\includegraphics[draft=false,trim={300px 0 300px 0},clip,width=\\columnwidth]{figures\/bunny_smoke_thin40_0001.jpeg}\n  \\end{subfigure\n  \\begin{subfigure}{0.5\\columnwidth}\n     \\includegraphics[draft=false,trim={300px 0 300px 0},clip,width=\\columnwidth]{figures\/bunny_smoke_thin40_0026.jpeg}\n  \\end{subfigure} \\\\[-1ex]\n  \\begin{subfigure}{0.5\\columnwidth}\n     \\includegraphics[draft=false,trim={300px 0 300px 0},clip,width=\\columnwidth]{figures\/bunny_smoke_thin40_0065.jpeg}\n  \\end{subfigure\n  \\begin{subfigure}{0.5\\columnwidth}\n     \\includegraphics[draft=false,trim={300px 0 300px 0},clip,width=\\columnwidth]{figures\/bunny_smoke_thin40_0090.jpeg}\n  \\end{subfigure}\n\\caption{{\\textbf{Smoke in an irregular domain}}. Multicolored spheres of smoke with non-zero initial velocity conditions flow and collide inside the Stanford bunny. Zero normal velocity is enforced with our cut cell formulation.}\\label{fig:bunnysmoke}\n\\end{figure}\n\n\\section{Governing Equations and Operator Splitting}\nWe solve the incompressible Euler equations that describe the evolution of a fluid in terms of its mass density $\\rho$, velocity $\\mb{u}$, pressure $p$ and gravitational constant $\\gg$ as\n\\begin{align}\n\\rho\\frac{D\\mb{u}}{Dt} &=\\rho\\left(\\frac{\\partial \\mb{u}}{\\partial t} + \\frac{\\partial \\mb{u}}{\\partial \\mb{x}}\\mb{u}\\right)=-\\nabla p + \\rho\\gg, \\ \\mb{x}\\in\\Omega \\label{eq:mom_cont}\\\\\n\\nabla\\cdot\\mb{u} &= 0, \\ \\mb{x}\\in\\Omega \\label{eq:div_cont} \\\\\n\\mb{u}\\cdot\\mb{n}&=a, \\ \\mb{x}\\in\\partial \\Omega_D \\label{eq:bcv_cont}\\\\\np&=0, \\ \\mb{x}\\in\\partial \\Omega_N \\label{eq:bcp_cont}\n\\end{align}\nwhere Equation~\\eqref{eq:mom_cont} is balance of linear momentum, Equation~\\eqref{eq:div_cont} is the incompressibility constraint, Equation~\\eqref{eq:bcv_cont} is the boundary condition for the normal component of the velocity and Equation~\\eqref{eq:bcp_cont} is the free surface boundary condition. We use $\\Omega$ to denote the region occupied by the fluid, $\\partial \\Omega_D$ to denote the portion of the boundary of the fluid domain on which velocity is prescribed to be $a$ (which may vary over the boundary) and $\\partial \\Omega_N$ is the surface of the water where the pressure is zero (see Figure~\\ref{fig:dAndg}).\\\\\n\\\\\n\\begin{figure}[h]\n\\centering\n\\begin{tabular}{cc}\n\\subf{\\includegraphics[draft=false,width=.45\\columnwidth]{figures\/domain}}\n&\n\\subf{\\includegraphics[draft=false,width=.45\\columnwidth]{figures\/cutcell.eps}}\n\\\\\n\\end{tabular}\n\\caption{{\\textbf{Flow domain and grid}}. {\\textbf{Left}}: we use $\\Omega$ to denote the fluid domain, with $\\partial\\Omega_D$ used to indicate the portion of the fluid domain subject to velocity boundary conditions and $\\partial\\Omega_N$ to indicate the free-surface portion of the boundary with pressure condition $p=0$. {\\textbf{Right}}: We use multiquadratic interpolation for velocity ($\\bar{\\mb{u}}_\\mb{i}$ at cell centers, blue) and multilinear for pressure ($p_\\mb{c}$ at nodes, red). The fluid domain is defined with sub-grid-cell accuracy.}\\label{fig:dAndg}\n\\end{figure}\nIn a Chorin \\shortcite{chorin:1967:numerical} operator splitting of the advective and pressure terms, velocity is first updated to an intermediate field $\\mb{w}$ under the convective $\\rho\\frac{D\\mb{u}}{Dt}=\\mathbf{0}$, followed by an update from the pressure and gravitational body forcing under $\\rho\\frac{\\partial \\mb{u}}{\\partial t}=-\\nabla p + \\rho\\gg$ where the pressure is determined to enforce $\\nabla\\cdot\\mb{u} = 0$. Dividing by the mass density, the convective step is seen to be an update under Burgers' equation~\\eqref{eq:impBurg}. Burgers' equation governs temporally constant Lagrangian velocity (zero Lagrangian acceleration). The characteristic curves for flows of this type are straight lines (since the Lagrangian acceleration is zero), on which the velocity is constant (see Figure~\\ref{fig:burgers}). This gives rise to the implicit relation $\\mb{u}(\\mb{x},t)=\\mb{u}(\\mb{x}-(t-s)\\mb{u}(\\mb{x},t),s)$ for $s\\leq t$. Intuitively, if we want to know the velocity $\\mb{u}(\\mb{x},t)$ at point $\\mb{x}$ at time $t$, we look back along the characteristic passing through $\\mb{x}$ at time $t$ to any previous time $s$; however, the characteristic is the straight line defined by the velocity $\\mb{u}(\\mb{x},t)$ that we want to know. Hence we take an implicit approach to the solution of this equation, which when combined with the operator splitting amounts to\n\\begin{align}\n\\frac{\\mb{w}-\\tilde{\\mb{u}}^n}{\\Delta t} &=\\mathbf{0} \\label{eq:split_a} \\\\\n\\rho\\frac{\\mb{u}^{n+1}-\\mb{w}}{\\Delta t} &=-\\nabla p^{n+1} + \\rho\\gg \\label{eq:split_p}\\\\\n\\nabla\\cdot\\mb{u}^{n+1} &= \\mathbf{0} \\label{eq:split_div}\n\\end{align}\nwhere we use the notation $\\mb{u}^{n+\\alpha}(\\mb{x})=\\mb{u}(\\mb{x},t^{n+\\alpha})$, $\\alpha=0,1$ to denote the time $t^{n+\\alpha}$ velocities. Furthermore, the intermediate velocity $\\mb{w}$ is related to $\\tilde{\\mb{u}}^n$ through $\\tilde{\\mb{u}}^n(\\mb{x})=\\mb{u}(\\mb{x}-\\Delta t \\mb{w}(\\mb{x}),t^n)$.\n\n\\begin{figure*}[ht!]\n\\includegraphics[draft=false,width=\\textwidth]{figures\/burgers}\n\\caption{{\\textbf{BSL versus SL}}. We illustrate the difference between explicit semi-Lagrangian and BSL in 1D. {\\textbf{Left}}: The exact solution of Burgers' equation has straight line characteristics shown in blue, green and red on which velocity (plotted above the plane in gray) is constant. {\\textbf{Center}}: BSL (green) uses Newton's method to solve for the exact characteristic going through $x_i$ at time $t^{n+1}$ to determine $u_i^{n+1}$. {\\textbf{Right}}: explicit semi-Lagrangian (red) uses a stale, time $t^n$ approximation of the characteristic which over shoots, resulting in an underestimate of the velocity and energy loss.}\\label{fig:burgers}\n\\end{figure*}\n\n\\section{Spatial Discretization}\nWe discretize in space by first representing velocity and pressure in terms of mulitquadratic and multilinear B-splines for velocity and pressure respectively. We use a regular grid with spacing $\\Delta x$ and define pressure degrees of freedom at grid vertices and velocity degrees of freedom at grid cell centers as in \\cite{ando:2013:surfacing} (see Figure~\\ref{fig:dAndg}). This efficiently aligns the support of the multiquadratic and multilinear interpolating functions which naturally allows for a grid-cell-wise definition of the flow domain (see Figure~\\ref{fig:fsdnb}). We use $N_\\mb{i}(\\mb{x})$ to represent the multiquadratic B-spline basis function associated with velocity degree of freedom $\\bar{\\mb{u}}_\\mb{i}$ at grid cell center $\\mb{x}_\\mb{i}$ and $\\chi_\\mb{c}(\\mb{x})$ for the multilinear basis function associated with pressure $p_\\mb{c}$ at grid node $\\mb{x}_\\mb{c}$. These are defined as\n\\begin{align}\nN_\\mb{i}(\\mb{x})&=\\prod_{\\alpha}\\hat{N}(\\frac{x_\\alpha-x_{\\alpha \\mb{i}}}{\\Delta x}), \\ \\chi_\\mb{c}(\\mb{x})=\\prod_{\\alpha}\\hat{\\chi}(\\frac{x_\\alpha-x_{\\alpha \\mb{c}}}{\\Delta x})\\\\\n\\hat{N}(\\eta)&=\\left\\{\\begin{array}{lcc}\n\\frac{\\left(\\eta +\\frac{3}{2}\\right)^2}{2},&\\eta\\in(-\\frac{3}{2},-\\frac{1}{2})\\\\\n-\\eta^2+\\frac{3}{4},&\\eta\\in[-\\frac{1}{2},\\frac{1}{2}]\\\\\n\\frac{\\left(\\eta -\\frac{3}{2}\\right)^2}{2},&\\eta\\in(\\frac{1}{2},\\frac{3}{2})\\\\\n0,&\\textrm{otherwise}\n\\end{array}\\right. \\ \n\\hat{\\chi}(\\nu)=\\left\\{\\begin{array}{lcc}\n1+\\nu,&\\nu\\in(-1,0)\\\\\n1-\\nu,&\\nu\\in[0,1)\\\\\n0,&\\textrm{otherwise}\n\\end{array}\\right.\n\\end{align}\nwhere we use Greek indices $\\alpha$ to indicate components of the vectors $\\mb{x}$, $\\mb{x}_\\mb{i}$ and $\\mb{x}_\\mb{c}$. With this convention we interpolate to define velocity and pressure fields \n\\begin{align}\\label{eq:interp}\n\\mb{u}(\\mb{x})=\\sum_\\mb{i} \\bar{\\mb{u}}_\\mb{i} N_\\mb{i}(\\mb{x}), \\ p(\\mb{x})=\\sum_\\mb{c} p_\\mb{c} \\chi_\\mb{c}(\\mb{x}).\n\\end{align}\nWe use the notation $\\bar{\\mb{u}}_\\mb{i}$ to distinguish it from the velocity at the grid node $\\mb{u}(\\mb{x}_\\mb{i})=\\sum_\\mb{j} \\bar{\\mb{u}}_\\mb{j} N_\\mb{j}(\\mb{x}_\\mb{i})$ since the multiquadratic B-splines are not interpolatory and these will in general be different. Note that multilinear interpolation is interpolatory and $p_\\mb{c}=\\sum_\\mb{d} p_\\mb{d} \\chi_\\mb{d}(\\mb{x}_\\mb{c})$.\n\\subsection{BSLQB Advection}\\label{sec:bslqb}\nWith this interpolation choice, we first solve for intermediate grid node velocity values $\\mb{w}(\\mb{x}_\\mb{i})$ from Equation~\\eqref{eq:split_a} as\n\\begin{align}\\label{eq:ad_disc}\n\\mb{w}(\\mb{x}_\\mb{i})=\\sum_\\mb{j} \\bar{\\mb{u}}^n_\\mb{j} N_\\mb{j}\\left(\\mb{x}_\\mb{i}-\\Delta t \\mb{w}(\\mb{x}_\\mb{i})\\right).\n\\end{align}\nWe can solve this equation using Newton's method since the multiquadratic B-splines are $C^1$. We use $\\mb{w}^k_\\mb{i}$ to denote the $k^\\textrm{th}$ Newton approximation to $\\mb{w}(\\mb{x}_\\mb{i})$. Explicit semi-Lagrangian is used as an initial guess with $\\mb{w}^0_\\mb{i}=\\sum_\\mb{j} \\bar{\\mb{u}}^n_\\mb{j} N_\\mb{j}\\left(\\mb{x}_\\mb{i}-\\Delta t \\sum_\\ll \\bar{\\mb{u}}^n_\\ll N_\\ll(\\mb{x}_\\mb{i})\\right)$ and then we update iteratively via $\\mb{w}^k_\\mb{i}\\mathrel{+}=\\boldsymbol\\delta\\mb{u}^k$ with Newton increment $\\boldsymbol\\delta\\mb{u}^k$ satisfying\n\\begin{align*}\n\\boldsymbol\\delta\\mb{u}^k&=\\left(\\mb{I}+\\Delta t \\frac{\\partial \\mb{u}^n}{\\partial \\mb{x}}\\left(\\mb{x}_\\mb{i}-\\Delta t \\mb{w}^k_\\mb{i}\\right)\\right)^{-1}\\left(\\sum_\\mb{j} \\bar{\\mb{u}}^n_\\mb{j} N_\\mb{j}\\left(\\mb{x}_\\mb{i}-\\Delta t \\mb{w}^k_\\mb{i}\\right)-\\mb{w}^k_\\mb{i}\\right)\n\\end{align*}\nwhere $\\frac{\\partial \\mb{u}^n}{\\partial \\mb{x}}\\left(\\mb{x}_\\mb{i}-\\Delta t \\mb{w}^k_\\mb{i}\\right)=\\sum_\\mb{j} \\bar{\\mb{u}}^n_\\mb{j}\\frac{\\partial N_\\mb{j}}{\\partial \\mb{x}}\\left(\\mb{x}_\\mb{i}-\\Delta t \\mb{w}^k_\\mb{i}\\right)$. It is generally observed \\cite{kuo:1990:semi,pudykiewicz:1984:some} that with BSL approaches of this type, this iteration will converge as long as $\\mb{I}+\\Delta t \\sum_\\mb{j} \\bar{\\mb{u}}^n_\\mb{j} \\frac{\\partial N_\\mb{j}}{\\partial \\mb{x}}\\left(\\mb{x}_\\mb{i}-\\Delta t \\mb{w}^k_\\mb{i}\\right)$ is non-singular. We note that this condition holds as long as no shocks form under Burgers' equation \\cite{evans:2010:pde} (forward from time $t^n$). This is a safe assumption since we are modeling incompressible flow with which shock formation does not occur, but it may be a problem for compressible flows. In practice, this iteration converges in 3 or 4 iterations, even with CFL numbers larger than 4 (see Section~\\ref{sec:ex_hybrid}). When it does fail (which occurs less than one percent of the time in the examples we run), it is usually for points near the boundary with characteristics that leave the domain (since we cannot estimate $\\frac{\\partial \\mb{u}^n}{\\partial \\mb{x}}$ using grid interpolation if the upwind estimate leaves the grid). In this case we use explicit semi-Lagrangian and interpolate from the boundary conditions if the characteristic point is off the domain.\\\\\n\\\\\nOnce we have obtained the grid node values of the intermediate velocity $\\mb{w}(\\mb{x}_\\mb{i})$, we must determine interpolation coefficients $\\bar{\\mb{w}}_\\mb{j}$ such that $\\mb{w}(\\mb{x}_\\mb{i})=\\sum_\\mb{j} \\bar{\\mb{w}}_\\mb{j} N_\\mb{j}(\\mb{x}_\\mb{i})$. On the boundary of the grid, we set $\\bar{\\mb{w}}_\\mb{j} = \\mb{w}(\\mb{x}_\\mb{j})$ since we can only interpolate to $\\mb{x}_\\mb{i}$ if all of its neighbors have data. This yields a square, symmetric positive definite system of equations for the remaining $\\bar{\\mb{w}}_\\mb{j}$. The system is very well conditioned with sparse, symmetric matrix $N_\\mb{j}(\\mb{x}_\\mb{i})$ consisting of non-negative entries and rows that sum to one.  The sparsity and symmetry of the system arises from the compact support and geometric symmetry, respectively, of the B-spline basis functions $N_\\mb{j}$. The system can be solved to a residual of machine precision in one iteration of PCG (or tens of iterations of unpreconditioned CG). In practice, we have noticed that for some flows, determining the coefficients $\\bar{\\mb{w}}_\\mb{j}$ can lead to increasingly oscillatory velocity fields. This is perhaps due to the unfavorable filtering properties of even order B-splines  \\cite{cheng:2001:quadratic,cheney:2012:numerical}. However, we found that a simple stabilization strategy can be obtained as\n\\begin{align}\\label{eq:BSLQBsys}\n\\sum_\\mb{j} \\left(\\lambda N_\\mb{j}(\\mb{x}_\\mb{i}) + (1-\\lambda)\\delta_{\\mb{i}\\mb{j}}\\right)\\bar{\\mb{w}}_\\mb{j}=\\mb{w}(\\mb{x}_\\mb{i})\n\\end{align}\nwhere $\\lambda\\in[0,1]$ and $\\delta_{\\mb{i}\\mb{j}}$ is the Kronecker delta. A value of $\\lambda=0$ is very stable, but extremely dissipative. Stable yet energetic behavior is achieved by decreasing the value of $\\lambda$ under grid refinement. In practice we found that $\\lambda\\in (.95,1 ]$ with $\\lambda =c\\Delta x$ for constant $c$ provided a good balance without compromising second order accuracy of the method (see Section~\\ref{sec:ex_hybrid}). We note that Riish{\\o}jgaard et al. \\shortcite{riishojgaard:1998:use} also added diffusion to cubic spline interpolation based semi-Lagrangian to reduce noise.\n\\subsection{Hybrid BSLQB-PolyPIC Advection}\\label{sec:poly}\nIn some portions of the domain, we store particles with positions $\\mb{x}_p^n$ and PolyPIC \\cite{fu:2017:poly} velocity coefficients $\\mb{c}^n_p$. In the vicinity of the particles, we use PolyPIC \\cite{fu:2017:poly} to update the intermediate velocity field $\\bar{\\mb{w}}_\\mb{j}$. First we update particle positions as $\\mb{x}^{n+1}_p=\\mb{x}^{n}_p + \\Delta t \\mb{v}_p^{n}$ (where the velocity $\\mb{v}_p^{n}$ is determined from $\\mb{c}^n_p$ following \\cite{fu:2017:poly}). Then the components $\\bar{w}_{\\mb{j}\\alpha}$ of the coefficients $\\bar{\\mb{w}}_\\mb{j}$ are determined as\n\\begin{align}\\label{eq:polyp2g}\n\\bar{w}_{\\mb{j}\\alpha}=\\frac{\\sum_p m_pN_\\mb{j}(\\mb{x}_p^{n+1})\\left(\\sum_{r=1}^{N_r} s_r(\\mb{x}_\\mb{j}-\\mb{x}^{n+1}_p)c^n_{pr\\alpha}\\right)}{\\sum_p m_pN_\\mb{j}(\\mb{x}_p^{n+1})}\n\\end{align}\nwhere $N_r$ is the number of polynomial modes $s_r(\\mb{x})$, as in Fu et al. \\shortcite{fu:2017:poly}. To create our hybrid approach, we update  $\\bar{w}_{\\mb{j}\\alpha}$ from Equation~\\eqref{eq:polyp2g} whenever the denominator is greater than a threshold $\\sum_p m_pN_\\mb{j}(\\mb{x}_p^{n+1})>\\tau^m$, otherwise we use the BSLQB update from Equation~\\eqref{eq:BSLQBsys}. We use this threshold because the grid node update in Equation\\eqref{eq:polyp2g} loses accuracy when the denominator is near zero and in this case the BSLQB approximation is likely more accurate. Note that the polynomial mode coefficients for the next time step $\\mb{c}^{n+1}_p$ are determined from the grid velocities at the end of the time step (using particle positions $\\mb{x}_p^{n+1}$ and after pressure projection).\n\n\\section{Pressure Projection}\nWe solve Equations~\\eqref{eq:split_p}-\\eqref{eq:split_div} and boundary condition Equations~\\eqref{eq:bcv_cont}-\\eqref{eq:bcp_cont} in a variational way. To do this, we require that the dot products of Equations~\\eqref{eq:split_p}, \\eqref{eq:split_div} and Equations~\\eqref{eq:bcv_cont} with arbitrary test functions $\\mb{r}$, q and $\\mu$ respectively integrated over the domain are always equal to zero. The free surface boundary condition in Equation~\\eqref{eq:bcp_cont} is naturally satisfied by our treatment of Equation~\\eqref{eq:split_p}. We summarize this as \n\\begin{align}\n\\int_\\Omega \\mb{r}\\cdot\\rho\\left(\\frac{\\mb{u}^{n+1}-\\mb{w}}{\\Delta t}\\right)d\\mb{x}&=\\int_\\Omega p^{n+1}\\nabla\\cdot\\mb{r} + \\rho\\mb{r}\\cdot \\gg d\\mb{x} \\label{eq:var_mom}\\\\\n&-\\int_{\\partial \\Omega}p^{n+1} \\mb{r}\\cdot \\mb{n} ds(\\mb{x})\\nonumber\\\\\n\\int_{\\Omega} q \\nabla \\cdot \\mb{u}^{n+1} d\\mb{x}&=0 \\label{eq:var_div}\\\\\n\\int_{\\partial \\Omega_D} \\mu \\left(\\mb{u}^{n+1}\\cdot\\mb{n}-a\\right) ds(\\mb{x})&=0.\\label{eq:var_bcv}\n\\end{align}\nHere we integrate by parts in the integral associated with Equation~\\eqref{eq:split_p}. Furthermore, we modify the expression $\\int_{\\partial \\Omega}p^{n+1} \\mb{r}\\cdot \\mb{n} ds(\\mb{x})$ in Equation~\\eqref{eq:var_mom} in accordance with the boundary conditions. We know that the pressure is zero on $\\partial \\Omega_N$, however we do not know its value on $\\partial \\Omega_D$. We introduce the pressure on this portion of the domain as a Lagrange multiplier $\\lambda^{n+1}$ associated with satisfaction of the velocity boundary condition in Equation~\\eqref{eq:var_bcv}. Physically, this is the external pressure we would need to apply on $\\partial \\Omega_D$ to ensure that $\\mb{u}^{n+1}\\cdot\\mb{n}=a$. With this convention, we have $\\int_{\\partial \\Omega}p^{n+1} \\mb{r}\\cdot \\mb{n} ds(\\mb{x})=\\int_{\\partial \\Omega_D}\\lambda^{n+1} \\mb{r}\\cdot \\mb{n} ds(\\mb{x})$. We note that unlike Equation~\\eqref{eq:var_bcv} (and its strong form counterpart $\\eqref{eq:bcv_cont}$) that requires introduction of a Lagrange multiplier, Equation~\\eqref{eq:bcp_cont} is naturally enforced through the weak form simply by setting $p^{n+1}=0$ in the integral over $\\partial \\Omega_N$ in Equation~\\eqref{eq:var_mom}.\\\\\n\\\\\nTo discretize in space, we introduce interpolation for the test functions $\\mb{r}$, $q$ and $\\mu$. We use the same spaces as in Equation~\\eqref{eq:interp} for velocity and pressure for $\\mb{r}=\\sum_\\mb{i} \\bar{\\mb{r}}_\\mb{i} N_\\mb{i}$ and $q=\\sum_\\mb{d} q_\\mb{d} \\chi_\\mb{d}$. For the test functions $\\mu$, we choose the same space as $q,p$, but with functions restricted to $\\partial \\Omega_D$, $\\mu=\\sum_\\mb{b} \\mu_\\mb{b} \\chi_\\mb{b}$ for $\\mb{b}$ with grid cell $\\Omega_\\mb{b} \\cap \\partial \\Omega_D \\neq \\emptyset$ (see Figure~\\ref{fig:fsdnb}). We choose the same space for $\\lambda^{n+1}=\\sum_\\mb{b}\\lambda^{n+1}_\\mb{b} \\chi_\\mb{b}$ to close the system. With these choices for the test functions, the variational problem is projected to a finite dimensional problem defined by the interpolation degrees of freedom. This is expressed as a linear system for velocities $\\bar{\\mb{u}}_\\mb{j}^{n+1}$, internal pressures $p^{n+1}_\\mb{c}$, and external pressures $\\lambda_\\mb{b}^{n+1}$ that is equivalent to \n\\begin{align}\n\\left(\\begin{array}{ccc}\n\\mb{M}&-\\mb{D}^T&\\mb{B}^T\\\\\n-\\mb{D}&&\\\\\n\\mb{B}&&\n\\end{array}\\right)\n\\left(\n\\begin{array}{c}\n\\mb{U}^{n+1}\\\\\n\\mb{P}^{n+1}\\\\\n\\boldsymbol\\Lambda^{n+1}\n\\end{array}\n\\right)=\n\\left(\n\\begin{array}{c}\n\\mb{M}\\mb{W}  + \\hat{\\gg}\\\\\n\\mathbf{0}\\\\\n\\AA\n\\end{array}\n\\right).\n\\end{align}\nHere $\\mb{U}^{n+1}$, $\\mb{P}^{n+1}$ and $\\boldsymbol\\Lambda^{n+1}$ are the vectors of all unknown $\\bar{\\mb{u}}_\\mb{j}^{n+1}$, $p_\\mb{c}^{n+1}$ and $\\lambda_\\mb{b}^{n+1}$ respectively. Furthermore $\\mb{M}$ is the mass matrix, $\\mb{B}$ defines the velocity boundary conditions and $\\mb{D}$ defines the discrete divergence condition. Lastly, $\\mb{W} $ is the vector of all $\\bar{\\mb{w}}_\\mb{i}$ that define the intermediate velocity, $\\hat{\\gg}$ is from gravity and $\\AA$ is the variational boundary condition. Using the convention that Greek indices $\\alpha,\\beta$ range from $1-3$, these matrices and vectors have entries \n\\begin{align}\nM_{\\alpha\\mb{i}\\beta\\mb{j}}=\\delta_{\\alpha\\beta}\\int_\\Omega \\frac{\\rho}{\\Delta t} N_\\mb{i} N_\\mb{j} d\\mb{x}, \\ D_{\\mb{d} \\beta \\mb{j}}&=\\int_\\Omega \\chi_\\mb{d}\\frac{\\partial N_\\mb{j}}{\\partial x_\\beta}d\\mb{x}, \\ \\hat{g}_{\\alpha \\mb{i}}=\\int_\\Omega \\rho g_\\alpha N_\\mb{i} d\\mb{x} \\label{eq:vol_int} \\\\\nB_{\\mb{b} \\beta\\mb{j}}=\\int_{\\Omega_D} \\chi_\\mb{b} N_\\mb{j} n_\\beta ds(\\mb{x}), \\ A_\\mb{b} &= \\int_\\Omega a\\chi_\\mb{b} ds(\\mb{x}).\\label{eq:b_int}\n\\end{align}\nIf we define $\\mb{G}=[-\\mb{D}^T,\\mb{B}^T]$, we can convert this system into a symmetric positive definite one for $\\mb{P}^{n+1}$ and $\\boldsymbol\\Lambda^{n+1}$ followed by a velocity correction for $\\mb{U}^{n+1}$\n\\begin{align}\n\\left(\n\\begin{array}{c}\n\\label{eq:spd_system}\n\\mb{P}^{n+1}\\\\\n\\boldsymbol\\Lambda^{n+1}\n\\end{array}\n\\right)&=\\left(\\mb{G}^{T}\\mb{M}^{-1}\\mb{G}\\right)^{-1}\n\\left(\\mb{G}^T\\left(\\mb{W}+\\mb{M}^{-1}\\hat{\\gg}\\right)-\\left(\\begin{array}{c}\\mathbf{0}\\\\\\AA\\end{array}\\right)\\right)\\\\\n\\mb{U}^{n+1}&=-\\mb{M}^{-1}\\mb{G}\\left(\n\\begin{array}{c}\n\\mb{P}^{n+1}\\\\\n\\boldsymbol\\Lambda^{n+1}\n\\end{array}\n\\right)+\\mb{W}+\\mb{M}^{-1}\\hat{\\gg}.\n\\end{align}\nUnfortunately, this system will be dense in the current formulation since the full mass matrix $M_{\\alpha\\mb{i}\\beta\\mb{j}}$ is non-diagonal with dense inverse \\cite{botella:2002:collocation}. However, a simple lumped mass approximation\n\\begin{align}\\label{eq:mass_lump}\nM^l_{\\alpha\\mb{i}\\beta\\mb{j}}=\\left\\{\n\\begin{array}{lcc}\n\\delta_{\\alpha\\beta}\\int_\\Omega \\frac{\\rho}{\\Delta t} N_\\mb{i} d\\mb{x},&\\mb{i}=\\mb{j}\\\\\n0,&\\textrm{otherwise}\n\\end{array}\n\\right.\n\\end{align}\ngives rise to a sparse matrix in Equation~\\eqref{eq:spd_system}.\n\n\\begin{figure}[!ht]\n\\centering\n\\begin{tabular}{cc}\n\\subf{\\includegraphics[draft=false,width=.45\\columnwidth]{figures\/narrow_band}}\n&\n\\subf{\\includegraphics[draft=false,width=.4\\columnwidth]{figures\/narrow_band2}}\n\\\\\n\\end{tabular}\n\\caption{{\\textbf{Discrete free surface fluid domain}}. {\\textbf{Left}}: We define the fluid domain to consist of cells that either have (1) a particle (dark blue) in it or (2) a node with non-positive level set value (light blue). {\\textbf{Right}}: Boundary Lagrange multiplier external pressure $\\lambda_\\mb{b}$ (orange circles) are like the interior pressures $p_\\mb{c}$ except only defined on fluid domain cells that intersect $\\partial \\Omega_D$.}\\label{fig:fsdnb}\n\\end{figure}\n\n\\begin{figure}[!ht]\n  \\centering\n  \\begin{subfigure}{.49\\columnwidth}\n     \\includegraphics[draft=false,clip,width=1.0\\columnwidth]{figures\/narrow_band_2d0000_cropped.jpeg}\n    \n  \\end{subfigure} \n  \\begin{subfigure}{.49\\columnwidth}\n     \\includegraphics[draft=false,clip,width=1.0\\columnwidth]{figures\/narrow_band_2d0048_cropped.jpeg}\n    \n  \\end{subfigure}\\\\\n  \\begin{subfigure}{0.33\\columnwidth}\n     \\includegraphics[draft=false,trim={920px 430px 830px 470px},clip,width=1.0\\columnwidth]{figures\/narrowband3d_0.jpeg}\n  \\end{subfigure\n  \\begin{subfigure}{0.33\\columnwidth}\n     \\includegraphics[draft=false,trim={920px 430px 830px 470px},clip,width=1.0\\columnwidth]{figures\/narrowband3d_12.jpeg}\n  \\end{subfigure}\n  \\begin{subfigure}{0.33\\columnwidth}\n     \\includegraphics[draft=false,trim={920px 430px 830px 470px},clip,width=1.0\\columnwidth]{figures\/narrowband3d_26.jpeg}\n  \\end{subfigure\n\\caption{{\\textbf{Narrow band free surface}}. A circle\/sphere falls in a tank of water under gravity. Using only a narrow band of particles saves computational cost and enables increased resolution of the free surface. {\\textbf{Top}}: In 2D we illustrate the hybrid particle(dark blue)\/level set (light blue) representation. {\\textbf{Bottom}}: Particles are colored based on velocity magnitude.}\\label{fig:narrow_band_2d}\n\\end{figure}\n\n\n\\subsection{Cut cells}\n\\label{sec:cutcells}\nAs in XFEM and VNA approaches \\cite{belytschko:2009:review,koschier:2017:xfem,schroeder:2014:vna}, we resolve sub-grid-cell geometry by simply performing the integrations in Equations~\\eqref{eq:vol_int}-\\eqref{eq:b_int} over the geometry of the fluid domain. We use a level set to define solid boundaries (green in Figure~\\ref{fig:fsdnb}) on which velocity boundary conditions are defined. We triangulate the zero isocontour using marching cubes \\cite{chernyaev:1995:marching} (see Figure~\\ref{fig:cutcell}). The integrals in Equations~\\eqref{eq:vol_int}-\\eqref{eq:b_int} all involve polynomials over volumetric polyhedra (Equations~\\eqref{eq:vol_int}, blue in Figure~\\ref{fig:cutcell}) or surface polygons (Equations~\\eqref{eq:b_int}, green in Figure~\\ref{fig:cutcell}) and we use Gauss quadrature of order adapted to compute the integrals with no error (see \\cite{gagniere:2020:tech_doc}). For free surface flows, we use particles (and additionally a level set function in the case of narrow banding, see Section~\\eqref{sec:nb}) to denote grid cells with fluid in them. Cells near the solid boundary are clipped by the marching cubes geometry. The fluid domain $\\Omega$ is defined as the union of all clipped and full fluid cells (see Figure~\\ref{fig:fsdnb}). \\\\ \n\\\\\nNotably, taking a cut cell approach with our variational formulation allows us to prove that our method can resolve a standing pool of water exactly without producing numerical currents. We know that with gravitational force $\\rho\\gg$ (e.g. with $\\gg$ pointing in the $y$ direction with magnitude $g$), steady state is maintained if the pressure increases with depth as $p=\\rho g\\left(y_0-y\\right)$ where $y_0$ is the height of the water surface at rest, since $-\\nabla p + \\rho\\gg = \\mathbf{0}$. Since we use multilinear interpolating functions for $p$, the exact solution is representable in our discrete space and with a short proof we show (see \\cite{gagniere:2020:tech_doc}) that this means our method will choose it to maintain a standing pool of water, independent of fluid domain boundary geometry.\n\n\\begin{figure}[!ht]\n\\includegraphics[draft=false,width=\\columnwidth]{figures\/cut_cell_combo}\n\\caption{{\\textbf{Cut cells}}. We show the 14 essential cases used in determining the cut cell fluid domain geometry. Blue faces indicate the intersection of the grid cell with the fluid domain. Green faces indicate the velocity boundary condition faces on $\\partial \\Omega_D$.}\\label{fig:cutcell}\n\\end{figure}\n\n\\section{Narrow band free surface}\\label{sec:nb}\nFor free surface flows, we develop a narrow band approach as in \\shortcite{chentanez:2015:coupling,ferstl:2016:narrow,sato:2018:nb}. We represent the fluid domain with a level set and seed particles in a band of width $W$ from the zero isocontour (see Figure~\\ref{fig:fsdnb}). Particles are advected and used to augment BSLQB advection as detailed in Section~\\ref{sec:poly}. We also advect the level set by interpolating its value at the previous step from the upwind location $\\mb{x}_\\mb{i}-\\Delta t \\mb{w}(\\mb{x}_\\mb{i})$ determined in Equation~\\eqref{eq:ad_disc}. We then use the updated particle locations to compute a narrow band level set from the particles based on the method of Boyd and Bridson \\cite{boyd:2012:multiflip}. We update the level set to be the union of that defined by the narrow band and that from advection. This is done by taking the minimum of the two level set values and then redistancing with the method of Zhao \\shortcite{zhao:2005:fast}.\n\\section{Examples}\n\\begin{figure}[h]\n\\includegraphics[draft=false,width=\\columnwidth]{figures\/vonKarman_cropped}\n\\caption{{\\textbf{Von Karman vortex shedding}}. We demonstrate the accuracy of our Hybrid BSLQB\/PolyPIC with vortex shedding past a notch in 2D. Note the smooth transition between regions with particles (PolyPIC) and those without (BSLQB).}\\label{fig:vonK}\n\\end{figure}\n\n\n\\subsection{Hybrid BSLQB\/PolyPIC}\\label{sec:ex_hybrid}\nWe demonstrate our hybrid BSLQB\/PolyPIC advection with water simulation. We prevent excessive run times by utilizing a narrow band of particles near the free surface and a level set (with BSLQB advection) in deeper levels. Figure~\\ref{fig:narrow_band_2d} Top shows a disc of water splashing in a rectangular tank with dimension $1\\times2$ and grid cell size $\\Delta x = 1\/255$. The time step is restricted to be in the range $\\Delta t \\in \\left[0.005, 0.01\\right]$.  20 particles are initialized in every cell that is initially in a narrow band of $7 \\Delta x$ below the zero isocontour of the level set. Figure~\\ref{fig:narrow_band_2d} Bottom shows an analogous 3D example where a sphere of water splashes in a tank.  A cell size of $\\Delta x = \\frac{1}{63}$ is used in a domain with dimensions $1\\times 2 \\times 1$.  We take a fixed time step of $\\Delta t = 0.01$ and demonstrate that narrow banding does not prevent larger-than-CFL time steps.  1,008,187 particles are used to resolve the free surface in a narrow band of width $5\\Delta x$.  As in 2D, the particles capture highly-dynamic behavior of the free surface while the level set is sufficient to represent the bulk fluid in the bottom half of the domain.\\\\\n\\\\\nWe also demonstrate our hybrid advection with a vortex shedding example (see Figure~\\ref{fig:vonK}). The flow domain $\\Omega$ is a $3\\times 1$ rectangle with circle of radius $0.05$. We seed a band of particles of width $.2$ above the midline $y=.5$ for PolyPIC advection. Advection in the rest of the domain is done with BSLQB. The vorticity plot illustrates a seamless transition between the two advection schemes. The simulation was run with a grid resolution of $\\Delta x=\\frac{1}{255}$, CFL number of 4 (i.e. $\\Delta t = \\frac{4\\Delta x}{v_\\textrm{max}}$), and inlet speed of $1.5$.\n\\begin{figure}[!b]\n  \\centering\n  \\begin{subfigure}{.49\\columnwidth}\n     \\includegraphics[draft=false,clip,width=\\columnwidth]{figures\/cutcell_compare1.jpeg}\n  \\end{subfigure}\n  \\begin{subfigure}{.49\\columnwidth}\n     \\includegraphics[draft=false,clip,width=\\columnwidth]{figures\/cutcell_compare2.jpeg}\n  \\end{subfigure}\n \n\\caption{{\\textbf{Cut cell vs.\\ voxelized domain}}. Using a cut cell domain (right) instead of a voxelized domain (left) yields marked improvements in simulation quality.}\\label{fig:voxelized_vs_cut_cell}\n\\end{figure}\n\n\\subsection{BSLQB comparison with explicit semi-Lagrangian}\nWe demonstrate improved resolution of flow detail with BSLQB compared to explicit semi-Lagrangian in a 2D example of smoke flowing past a circle (see Figure~\\ref{fig:interp_correction}) and with a 2D spinning circle example (see Figure~\\ref{fig:inner_circle}). Note that particles are only used for flow visualization and not for PolyPIC advection in these examples. BSLQB exhibits more energetic, turbulent flows than semi-Lagrangian advection. Notably, the BSLQB result breaks symmetry sooner. In Figure~\\ref{fig:interp_correction} we also examine the effect of extremal values of the $\\lambda$ parameter described in Equation~\\eqref{eq:BSLQBsys}.  A zero value of $\\lambda$ is quite dissipative compared to a full value of $\\lambda = 1$ for both semi-Lagrangian and BSLQB.  As mentioned in Section~\\ref{sec:bslqb}, we generally found that keeping $\\lambda$ close to 1 provided the least dissipative behavior, while setting the value slightly less than 1 helped restore stability when necessary (one can also dynamically adjust this value over the course of a simulation). In Figure~\\ref{fig:inner_circle}, we initially set the angular velocity to 4 radians per second in a circle of radius $.2$ (with $\\Omega=[0,1]\\times[0,1]$). The simulation is run with $\\Delta x=\\frac{1}{511}$ and a $\\Delta t = .02$ (CFL number of 3).\\\\\n\\\\\nWe examine the convergence behavior of BSLQB for the 2D Burgers' equation $\\frac{D\\mb{u}}{Dt}=\\mathbf{0}$ with initial data $\\mb{u}(\\mb{x})=\\mb{x}\\cdot\\left(\\AA\\mb{x}\\right)$ for $\\AA=\\mb{R}\\boldsymbol\\Lambda\\mb{R}^T$ for diagonal $\\boldsymbol\\Lambda$ with entries $1$ and $.25$ and rotation (of $.1$ radians) $\\mb{R}$ (see Figure~\\ref{fig:conv}). We examine the convergence behavior under refinement in space and time with $\\Delta t=\\Delta x$. We compute the best fit line to the plot of the logarithm of the $L^\\infty$ norm of the error versus the logarithm of $\\Delta x$ for a number of grid resolutions. We observe slopes of approximately 2 for BSLQB with interpolation parameter $\\lambda=1$ and $\\lambda=1-c\\Delta x$ (with $c = 2.95$), indicating second order accuracy in space and time under refinement. We observe slopes of approximately 1 for explicit semi-Lagrangian, indicating first order.\n\\begin{figure}[ht]\n  \\centering\n  \\begin{subfigure}{1.0\\columnwidth}\n     \\includegraphics[draft=false,trim={0 60px 0 60px},clip,width=\\columnwidth]{figures\/interp_correction_0300.pdf}\n  \\end{subfigure}\n  \\\\\n  \\begin{subfigure}{1.0\\columnwidth}\n     \\includegraphics[draft=false,trim={0 40px 0 300px},clip,width=\\columnwidth]{figures\/interp_correction_0060.jpeg}\n  \\end{subfigure}\n\\caption{{\\textbf{Interpolation correction}}. BSLQB exhibits more fine-scale flow detail and vorticity than semi-Lagrangian for extremal values of interpolation parameter $\\lambda$ (Equation~\\eqref{eq:BSLQBsys}). From left to right: semi-Lagrangian with $\\lambda = 0$, BSLQB with $\\lambda = 0$, semi-Lagrangian with $\\lambda = 1$, BSLQB with $\\lambda = 1$.}\\label{fig:interp_correction}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\includegraphics[draft=false,width=\\columnwidth]{figures\/Convergence}\n\\caption{{\\textbf{Convergence}}. We compare explicit semi-Lagrangian (SL, red), with BSLQB (blue) and interpolation coefficient $\\lambda=1$ (Equation~\\eqref{eq:BSLQBsys}) and BSLQB with interpolation coefficient $\\lambda=1-c\\Delta x$ (orange). We plot $\\log(\\Delta x)$ versus $\\log(e)$ (where $e$ is the infinity norm of the error) for a variety of grid resolutions $\\Delta x$ and compute the best fit lines. The slope of the line provides empirical evidence for the convergence rate of the method.}\\label{fig:conv}\n\\end{figure}\n\\begin{figure}[!ht]\n  \\centering\n  \\begin{subfigure}{\\columnwidth}\n     \\includegraphics[draft=false,trim={0 200px 0 200px},clip,width=\\columnwidth]{figures\/smoke_fast0104.jpeg}\n  \\end{subfigure}\\\\\n  \\begin{subfigure}{\\columnwidth}\n     \\includegraphics[draft=false,trim={0 200px 0 200px},clip,width=\\columnwidth]{figures\/smoke_fast0161.jpeg}\n  \\end{subfigure}\\\\\n  \\begin{subfigure}{\\columnwidth}\n     \\includegraphics[draft=false,trim={0 200px 0 200px},clip,width=\\columnwidth]{figures\/\/smoke_fast0350.jpeg}\n  \\end{subfigure}\n\\caption{{\\textbf{Smoke jet}}. A plume of smoke is simulated with BSLQB. Zero normal velocity boundary conditions are enforced on the irregular boundary of the sphere inducing intricate flow patterns as the smoke approaches it.}\\label{fig:smokejet}\n\\end{figure}\n\n\\subsection{Cut cell examples}\nWe demonstrate the ability of our cut cell method to produce detailed flows in complicated irregular domains for smoke and free surface water examples. Figure~\\ref{fig:rainbowsmoke} demonstrates the subtle and visually interesting behavior that arises as two plumes of multicolored smoke flow to the center of a cubic domain colliding with a spherical boundary. We use $\\Delta x = 1\/63$ and $\\Delta t = .02$. We demonstrate a more complex domain in Figure~\\ref{fig:bunnysmoke}. Puffs of colored smoke with converging initial velocities are placed in a bunny shaped clear domain. We use grid size $\\Delta x = 1\/127$ and a fixed time step of $\\Delta t = 0.01$ (CFL number $>1$). In Figure~\\ref{fig:snowglobe}, we demonstrate water splashing, while accurately conforming to the walls of an irregular domain defined as the interior of a large sphere and exterior of a small inner sphere. The spatial resolution of the domain is $\\Delta x = 1\/127$, and 30 particles per cell are seeded in the initial fluid shape. A minimum time step of $\\Delta t=0.001$ is enforced, which is often larger than the CFL condition. We also consider dam break simulations in rectangular domains with column obstacles (Figure~\\ref{fig:dambreak_final}) and a bunny obstacle (Figure~\\ref{fig:dambreak_final}). Both examples use a grid cell size of $\\Delta x = 1\/127$, 8 particles per cell and a fixed time step of $\\Delta t = 0.003$. Lastly, we demonstrate the benefits of our cut cell formulation over a more simplified, voxelized approach in Figure~\\ref{fig:voxelized_vs_cut_cell}. Notice the water naturally sliding in the cut cell domain compared with the jagged flow in the voxelized domain.\n\n\n\n\\subsection{Performance considerations}\n\n\\begin{table}[]\n\\begin{tabular}{@{}llll@{}}\n\\toprule\nExample         & Seconds & \\# Particles & $\\Delta x^{-1}$   \\\\ \\midrule\nSmoke Jet (Fig.~\\ref{fig:smokejet})      & 1,212             & 12,502,349 & 127   \\\\\nMultiple Jets (Fig.~\\ref{fig:rainbowsmoke})  & 53             & 25,004,699             & 63   \\\\\nBunny Smoke (Fig.~\\ref{fig:bunnysmoke})    & 160             & 24,000,000           & 127    \\\\\nSmoke Spheres$\\text{*}$ (Fig.~\\ref{fig:bigsmoke})  & 428             & 64,000,000             & 255    \\\\\nNarrow Band (Fig.~\\ref{fig:narrow_band_2d})    & 396             & 1,008,187             & 63   \\\\\nWater Globe (Fig.~\\ref{fig:snowglobe})    & 242             & 524,415             & 127  \\\\\nDam Break (Fig.~\\ref{fig:dambreak_final})       & 870            & 3,251,409            & 127  \\\\\nBunny Dam Break (Fig.~\\ref{fig:bunny_dambreak_final}) & 1,171             & 4,797,535           & 127 \\\\ \\bottomrule\n\\end{tabular}\n\\caption{Average time per frame (in seconds) for each of the 3D examples shown in the paper.  Examples were run on workstations with 16-core CPUs running at 2.20 GHz, except for the smoke spheres example, which was run on a cluster equipped with CPUs running at 3.07 GHz and Nvidia Tesla V100 GPUs which were used for the linear solves.}\n\\label{tbl:perf}\n\\end{table}\n\nThe implementation of our method takes advantage of hybrid parallelism (MPI, OpenMP, and CUDA\/OpenCL) on heterogeneous compute architectures in order to achieve practical runtime performance (see Table~\\ref{tbl:perf} for 3D example performance numbers). The spatial domain is uniformly divided into subdomains assigned to distinct MPI ranks, which distributes much of the computational load at the expense of synchronization overhead exchanging ghost information across ranks. On each rank, steps of our time integration loop such as BSLQB advection are multithreaded using OpenMP or CUDA when appropriate. The dominant costs per time step are the solution of the pressure projection system and, in the case of free surface simulation, assembly of the pressure system and its preconditioner. We permute Equation~\\eqref{eq:spd_system} so that each rank's degrees of freedom are contiguous in the solution vector then solve the system using AMGCL \\cite{demidov:2019:amgcl} using the multi-GPU VexCL backend (or the OpenMP CPU backend on more limited machines). Using a strong algebraic multigrid preconditioner with large-degree Chebyshev smoothing allows our system to be solved to desired tolerance in tens of iterations, even at fine spatial resolution.  An important step in minimizng the cost of system assembly is to scalably parallelize sparse matrix-matrix multiplication, for which we use the algorithm of Saad \\shortcite{saad:2003:sparse}.  In the future, we are interested in implementing load balancing strategies such as the simple speculative load balancing approach of \\cite{shah:2018:balancing}, particularly for free surface flows. We note that our implementation enables high-resolution simulations such as that in Figure~\\ref{fig:bigsmoke} at relatively modest computational cost (see Table~\\ref{tbl:perf}).\n\n\\section{Discussion and Limitations}\nOur approach has several key limitations that could be improved. First, our adoption of collocated multiquadratic velocity and multilinear pressure is a significant departure from most fluid solvers utilized in graphics applications. We note that BSLQB and BSLQB\/PolyPIC could be used with a MAC grid; however, each velocity face component would have to be solved for individually. Another drawback for our multiquadratic velocity and multilinear pressure formulation is that it gives rise to a very wide pressure system stencil consisting of 49 non-zero entries per row in 2D and 343 in 3D. Collocated approaches that make use of multilinear velocities and constant pressure give rise to 9 (2D) and 27 (3D) entries per row \\cite{zhang:2017:impm}, however they do not allow for $C^1$ continuity and require spurious pressure mode damping. Our wide stencils likely negatively affect the efficacy of preconditioning techniques as well, however we were very pleased with the efficiency of the AMGCL \\cite{demidov:2019:amgcl} library. Also, while the use of mass lumping in Equation~\\eqref{eq:mass_lump} is necessary to ensure a sparse pressure projection system, Boatella et al. \\shortcite{botella:2002:collocation} note that this has been shown to degrade accuracy. In fact, Boatela et al. \\shortcite{botella:2002:collocation} introduce a sparse approximate inverse to the full mass matrix to avoid dense systems of equations in the pressure solve without degrading accuracy. Split cubic interpolation, which approximates similar systems with tridiagonal ones could also possibly be used for this \\cite{huang:1994:semi}. Adoption of one of these approaches with our formulation would be an interesting area of future work. Also, we note that the more sophisticated transition criteria for narrow banding techniques in Sato et al. \\shortcite{sato:2018:nb} could naturally be used with our method.  Finally, we note that the work of Zehnder et al. \\shortcite{zehnder:2018:advection,narain:2019:ref} could be easily applied to our technique to further reduce dissipation since it is based on the Chorin \\shortcite{chorin:1967:numerical} splitting techniques (Equations~\\eqref{eq:split_a}-\\eqref{eq:split_div}) that we start from.\n\n\\begin{figure}[!hb]\n  \\centering\n  \\begin{subfigure}{.49\\columnwidth}\n  \\includegraphics[draft=false,trim={0 0 0 0},clip,width=\\columnwidth]{figures\/bigsmoke_0001.jpeg}\n  \\end{subfigure}  %\n  \\begin{subfigure}{.49\\columnwidth}\n  \\includegraphics[draft=false,trim={0 0 0 0},clip,width=\\columnwidth]{figures\/bigsmoke_0064.jpeg}\n  \\end{subfigure} \\\\ %\n  \\begin{subfigure}{.49\\columnwidth}\n  \\includegraphics[draft=false,trim={0 0 0 0},clip,width=\\columnwidth]{figures\/bigsmoke_0128.jpeg}\n  \\end{subfigure}  %\n  \\begin{subfigure}{.49\\columnwidth}\n  \\includegraphics[draft=false,trim={0 0 0 0},clip,width=\\columnwidth]{figures\/bigsmoke_0240.jpeg}\n  \\end{subfigure}\n\n\\caption{{\\textbf{High-resolution smoke}}: Two spheres of smoke collide in a high-resolution 3D simulation ($\\Delta x = 1\/255$). BSQLB accurately resolves vorticial flow detail.}\\label{fig:bigsmoke}\n\\end{figure}\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn their paper ``AndrODet: An adaptive Android obfuscation detector'' \\cite{Mirzaei2019}, Mirzaei at al. present the modular machine learning system AndrODet, which is capable of detecting three types of obfuscation in Android apps using statically extracted features: string encryption, identifier renaming, and control-flow obfuscation. AndrODet uses the MOA \\cite{bifet2010moa} framework to perform on-line learning. The authors also compare their on-line approach with batch learning. In this comment paper, we have only considered the string encryption (SE) detection capabilities of AndrODet. Therefore, we restrict the discussion to SE detection for the remainder of the paper.\n\nThe authors propose several features for classifying apps as string encrypted, such as the average string entropy, average string length, etc., and evaluate on-line and batch models based on these features. For on-line learning, a combination of the AMD \\cite{Wei2017} and PraGuard \\cite{Maiorca2015} datasets are used for training and evaluation, whereas for batch-learning the model was trained on the AMD dataset and evaluated on the PraGuard dataset. The AMD set consists of 24,553 malware samples labeled with the obfuscation methods used by each sample (if any). PraGuard was constructed by collecting samples from two malware databases and running them through the DexGuard \\cite{DexGuard} obfuscation tool.\n\nThe evaluation by Mirzaei et al.\\ shows that, using the features proposed in the paper, both batch learning and on-line learning approaches can achieve over 80\\% accuracy in SE detection. However, upon closer examination, we discovered a significant methodological problem in the way their evaluation is performed. While the AMD dataset has almost 25,000 malware samples, the samples in the dataset all belong to one of 71 malware families. Therefore, many samples share unique characteristics due to the fact that they belong to the same family. Moreover, for most malware families, all samples are in the same class, i.e., either all samples use SE or none of the samples use SE. (Only 3,883 samples belong to a family that has both SE and non-SE members.) This introduces a risk of \\emph{memorization}, i.e., that the classifier learns a set of unique signatures of \\emph{malware families}, instead of a generalizable model for detecting SE. Also, during evaluation of their approach, Mirzaei et al.\\ make no effort to organize the dataset in a way that prevents malware of the same family from appearing in both the training and testing data. Therefore, memorization would also risk artificially inflating the measured accuracy of the model. Our own evaluation strongly suggests that the high accuracy that they report is indeed due to memorization. We trained and evaluated models on the AMD dataset with the same configuration and feature set as Mirzaei et al., using both batch learning and on-line learning. In the first experiment, 100 training and testing subsets were created by randomly sampling apps from the AMD dataset. In the second experiment, we made sure that the datasets were split so that samples from the same family never appeared in both the training and testing subsets. In the first experiment, we achieved an average accuracy of 92\\% and 89\\% for the batch and on-line cases respectively, which is similar to the accuracy reported by Mirzaei et al.\\ However, in the second experiment, wherein we eliminated the possibility of learning a model based on the malware family, the respective accuracies dropped to 50\\% and 51\\%.\n\nThe dramatic drop in accuracy was surprising to us at first, as the accuracy when evaluating the batch learning model on the PraGuard dataset was very high according to the paper, which in turn seems to imply that the model can generalize well across different datasets. Unfortunately, however, the good performance on the PraGuard dataset appears to be coincidental. Since the DexGuard tool that was used to generate the PraGuard dataset employs a SE method that is fundamentally different from what is used by most other obfuscators, this dataset is not well-suited to evaluate how well an SE detector generalizes. Furthermore, the PraGuard dataset is particularly ill-suited to evaluate the AndrODet system, since the DexGuard tool effectively makes it impossible to statically extract the features used by AndrODet from encrypted strings. We elaborate further on this issue in Section \\ref{sec:PraGuard}.\n\n\\section{Background}\n\nIn this section, we first provide some technical background on Android apps, which is required to understand the following discussion. We then briefly describe the AndrODet system.\n\nAndroid apps are distributed in the form of Android application packages (APKs), which can in turn contain several DEX-files. Each DEX file contains static data and bytecode of one or more classes. Of particular interest to our discussion here is the \\emph{string sectio}n of a DEX file, where all identifiers (method names, etc.) and constant strings used by the contained classes are stored. Each unique constant string has one entry in the string section. Obfuscation tools that implement SE replace the plaintext constant strings stored in the string section of an app with encrypted or scrambled versions and insert special decryption logic into the program code to decrypt strings just prior to their use. (Only non-identifier strings are relevant in the context of SE, as identifiers must be statically resolvable. Therefore, identifiers are obfuscated through renaming.)\n\nThe SE detection component of AndrODet retrieves all (non-identifier) strings from the string section of all DEX-files in an APK and extracts several features from the string material. The features are then fed to a machine learning model that determines whether or not the app uses SE. The following features are computed by AndrODet from the strings in the string section:\n\n\\begin{itemize}\n\t\\item Average entropy\n\t\\item Average wordsize\n\t\\item Average length\n\t\\item Average number of '\\texttt{=}' characters\n\t\\item Average number of '\\texttt{-}' characters\n\t\\item Average number of '\\texttt{\/}' characters\n\t\\item Average number of '\\texttt{+}' characters\n\t\\item Average number of repeated characters\n\\end{itemize}\n\nOn a side note, we believe that the authors' use of the term \"average wordsize\" is misleading since, based on the published source code of AndrODet, it turns out that the \"wordsize\" does not refer to the average size of words in strings. Instead, it is the size of the underlying Python string object used to represent strings in their implementation. It is unclear to us why the authors chose to use this as a feature, since the size of a string object is not solely determined by the number of bytes required to represent a string, but also depends on, for example, the underlying Python implementation and the memory allocator used by the OS. Also, the size of a string object is not guaranteed to be the same for two strings with the same content, or to be consistent between runs.\n\nAndrODet uses the MOA platform for on-line learning, in which samples are fed to the on-line learning system in a streaming fashion. The \\emph{prequential} evaluation mode of MOA was used for evaluation: When a new sample arrives from the stream, it is classified by the current model and the result is recorded. The model is then immediately updated with the new sample, and the process repeats for the next sample. The final accuracy is computed as the average percentage of correct classifications.  The authors also train a model using batch learning for comparison. For batch learning, they use the ATM framework \\cite{Swearingen2017} for automatic tuning of model hyperparameters.\n\n\\section{Empirical Verification of the Problem}\n\nWe have evaluated both the batch and on-line learning approaches described by Mirzaei et al., using the published AndrODet source code. We used all of the samples from the AMD dataset, except for 135 samples in which the strings in the string section could not be decoded. We created the datasets for training and testing in the following way: First, 100 training and testing sets were constructed by splitting the AMD dataset into two sets of equal size using completely random sampling. Next, we created another 100 pairs of training and testing sets, by repeating the above procedure, with the added constraint that samples from the same malware family should never appear in both training and testing data. These two collections of training and testing sets will be referred to as the \\emph{random} and \\emph{non-overlapping} sets, respectively, for the remainder of this section. The exact procedure for creating the non-overlapping sets is outlined in Algorithm \\ref{alg:nonoverlapping}.\n\n\\begin{algorithm}\n\t$F$: set of all families\\\\\n\t$S$: set of all samples\\\\\n\t$T_r$: set of samples in the training set\\\\\n\t$T_e$: set of samples in the testing set\\\\\n\ts($f$): a function returning all the samples belonging to family $f$\\\\\n\trand($s$): a function returning a random sample from set $s$\n\t\\BlankLine\n\t\\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output}\n\t\\Input{$S,F$}\n\t\n\t$T_r = \\emptyset$\\\\\n\t\\While{$|T_r| \\leq \\frac{|S|}{2}$}{\n\t\t$f$ = rand($F$)\\\\\n\t\t$F = F - f$\\\\\n\t\t$Tr = Tr \\cup s(f)$\n\t}\n\t$T_e = S - T_r$\n\t\n\t\\Output{$Tr,Te$}\n\t\n\t\\caption{Procedure for constructing training and testing sets without overlapping malware families.}\n\t\\label{alg:nonoverlapping}\n\\end{algorithm}\n\nWe repeated the experiments for each of the 100 train\/test sets of the respective splitting strategies. Like Mirzaei et al., we used an SVM classifier for batch learning, and leveraging bagging for on-line learning. For the batch learning case we ran the ATM framework to find the best hyperparameters for each train\/test set, using a budget of 200 configuration trials. For the on-line case, we first trained a model on the training set, and then used the model to classify the samples in the testing data (without updating the model). The results for batch and on-line learning are shown in Figure \\ref{fig:boxplot-full}. For the on-line case, we get a mean accuracy of 89\\% with the random sets, which is similar to the accuracy reported by Mirzaei et al.\\ For the non-overlapping case, however, the accuracy drops to only 51\\% on average (with a much greater variance across the 100 models). For the batch learning case we get similar results, with an average accuracy of 92\\% for the random case, and 50\\% for the non-overlapping case.\n\n\\begin{figure}\n\t\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\naxis on top,\ntick pos=both,\nxmin=0.5, xmax=4.5,\nxtick style={color=black},\nxtick={1,2,3,4},\nxticklabel style = {rotate=90.0},\nxticklabels={MOA-random,MOA-nonoverlapping,ATM-random,ATM-nonoverlapping},\nylabel={Accuracy},\nymin=0, ymax=1,\nytick style={color=black}\n]\n\\addplot [blue]\ntable \n\t0.775 0.88768531411254\n\t1.225 0.88768531411254\n\t1.225 0.893173069047424\n\t0.775 0.893173069047424\n\t0.775 0.88768531411254\n};\n\\addplot [blue, dashed]\ntable \n\t1 0.88768531411254\n\t1 0.881726595134737\n};\n\\addplot [blue, dashed]\ntable \n\t1 0.893173069047424\n\t1 0.897534605618806\n};\n\\addplot [black]\ntable \n\t0.8875 0.881726595134737\n\t1.1125 0.881726595134737\n};\n\\addplot [black]\ntable \n\t0.8875 0.897534605618806\n\t1.1125 0.897534605618806\n};\n\\addplot [blue, mark=*, mark size=1, mark options={solid,draw=black}, only marks]\ntable \n\t1 0.879351298222623\n\t1 0.878532230321894\n\t1 0.878614137111967\n};\n\\addplot [blue]\ntable \n\t1.775 0.448953868188499\n\t2.225 0.448953868188499\n\t2.225 0.565457452789083\n\t1.775 0.565457452789083\n\t1.775 0.448953868188499\n};\n\\addplot [blue, dashed]\ntable \n\t2 0.448953868188499\n\t2 0.331892251413369\n};\n\\addplot [blue, dashed]\ntable \n\t2 0.565457452789083\n\t2 0.718708718626156\n};\n\\addplot [black]\ntable \n\t1.8875 0.331892251413369\n\t2.1125 0.331892251413369\n};\n\\addplot [black]\ntable \n\t1.8875 0.718708718626156\n\t2.1125 0.718708718626156\n};\n\\addplot [blue, mark=*, mark size=1, mark options={solid,draw=black}, only marks]\ntable \n\t2 0.193103448275862\n\t2 0.211149571871015\n\t2 0.19775890746739\n\t2 0.245456808563605\n\t2 0.172118618159584\n\t2 0.80905030757563\n\t2 0.813487972508591\n\t2 0.830046082949309\n\t2 0.807444061962134\n};\n\\addplot [blue]\ntable \n\t2.775 0.915369809157179\n\t3.225 0.915369809157179\n\t3.225 0.919690392333524\n\t2.775 0.919690392333524\n\t2.775 0.915369809157179\n};\n\\addplot [blue, dashed]\ntable \n\t3 0.915369809157179\n\t3 0.910803505610615\n};\n\\addplot [blue, dashed]\ntable \n\t3 0.919690392333524\n\t3 0.923007617331477\n};\n\\addplot [black]\ntable \n\t2.8875 0.910803505610615\n\t3.1125 0.910803505610615\n};\n\\addplot [black]\ntable \n\t2.8875 0.923007617331477\n\t3.1125 0.923007617331477\n};\n\\addplot [blue, mark=*, mark size=1, mark options={solid,draw=black}, only marks]\ntable \n\t3 0.896387910557785\n\t3 0.926939143254976\n};\n\\addplot [blue]\ntable \n\t3.775 0.441470797594011\n\t4.225 0.441470797594011\n\t4.225 0.556582811685542\n\t3.775 0.556582811685542\n\t3.775 0.441470797594011\n};\n\\addplot [blue, dashed]\ntable \n\t4 0.441470797594011\n\t4 0.309025650004377\n};\n\\addplot [blue, dashed]\ntable \n\t4 0.556582811685542\n\t4 0.726636475916015\n};\n\\addplot [black]\ntable \n\t3.8875 0.309025650004377\n\t4.1125 0.309025650004377\n};\n\\addplot [black]\ntable \n\t3.8875 0.726636475916015\n\t4.1125 0.726636475916015\n};\n\\addplot [blue, mark=*, mark size=1, mark options={solid,draw=black}, only marks]\ntable \n\t4 0.202896551724138\n\t4 0.0830752413918747\n\t4 0.177315805564048\n};\n\\addplot [red]\ntable \n\t0.775 0.89102301580801\n\t1.225 0.89102301580801\n};\n\\addplot [red]\ntable \n\t1.775 0.510450114148141\n\t2.225 0.510450114148141\n};\n\\addplot [red]\ntable \n\t2.775 0.917642722581702\n\t3.225 0.917642722581702\n};\n\\addplot [red]\ntable \n\t3.775 0.493219442035339\n\t4.225 0.493219442035339\n};\n\\end{axis}\n\\end{tikzpicture}\n\t\\caption{Box plot of classifier accuracy for the random and non-overlapping train\/test configurations, using the batch (ATM) and on-line (MOA) machine learning frameworks.}\n\t\\label{fig:boxplot-full}\n\\end{figure}\n\nOne potential concern for us was that the distribution of samples from different families in the AMD dataset is highly skewed. For example, about one third of all samples belong to one family. This could, potentially, result in some training or testing sets containing samples from only a handful of families when we use our non-overlapping approach. To rule out the possibility that this was the cause of the reduced accuracy in the non-overlapping cases, we conducted another set of experiments where a classifier was trained on samples from all families except one, and the model was evaluated on the left-out family. We repeated this process for all families and computed the average accuracy, weighted by the number of samples in each family. The experiment was performed both for batch (ATM) and on-line learning (MOA), using the same parameters and setup as described above. Similarly to our first set of experiments, the average accuracy was 51\\% for batch learning and 59\\% for on-line learning. Moreover, the F-scores for the respective cases were 0.14 and 0.23. Figure \\ref{fig:acc-families} shows the individual classification accuracy for each family. As can be seen from the figure, the accuracy varies widely between different families.\n\n\\begin{figure*}\n\t\\includegraphics[width=\\linewidth]{acc-families}\n\t\\caption{Individual classification accuracy of each family, when a classifier was trained on all other families in the AMD dataset.}\n\t\\label{fig:acc-families}\n\\end{figure*}\n\n\\section{Shortcomings of the PraGuard Dataset for Evaluating AndrODet}\n\\label{sec:PraGuard}\n\nThe PraGuard dataset was created by obfuscating all samples from the combined MalGenome and Contagio malware datasets using the obfuscation tool DexGuard. Mirzaei et al.\\ used 1495 string-encrypted samples from PraGuard, along with the same number of non-obfuscated samples, to evaluate their batch-learned model. The reason that we believe PraGuard to be a poor choice for evaluating AndrODet is the particular way in which DexGuard implements SE. Most obfuscators that support SE work by substituting a plaintext string in the string section with an encrypted version. Therefore, the encrypted string can still be readily extracted from the string section and analyzed. However, DexGuard implements a more sophisticated form of SE that stores encrypted strings as byte arrays, rather than as actual strings in the string section. Since such byte arrays cannot be easily told apart from regular arrays used by the obfuscated app, DexGuard effectively prevents extraction and analysis of encrypted strings. Although DexGuard is closed source, independent analysis of DexGuard-obfuscated apps has confirmed this method of applying SE \\cite{Moses2018}. Because DexGuard prevents AndrODet from extracting strings from the string section to compute features used for classification, we believe PraGuard is a poor choice for evaluating the generalizability of AndrODet. In fact, detecting DexGuard-obfuscated apps in which all (or almost all) constant strings have been removed from the string section is a trivial task that can be achieved with a simple heuristic, and does not require an advanced machine learning approach. We have analyzed the PraGuard dataset and confirmed that for 90\\% of the SE samples \\emph{all} non-identifier strings were removed from the string section\\footnote{Additionally, for about 3\\% of the samples, the AndrODet code for extracting strings failed to process the APK.}. The vast majority of the remaining samples had only a small number of non-identifier strings. We speculate that the relatively good performance on the PraGuard dataset reported by Mirzaei et al.\\ is due to some DexGuard-encrypted samples appearing in the AMD dataset, allowing the classifier to learn to identify this particular type of SE. (For example, it would be trivial for a machine learning system to learn that an average string length of 0 indicates obfuscation.) However, as our experiments in the preceding section show, AndrODet fails to accurately detect SE in the general case.\n\n\\section{Discussion and Conclusion}\n\nOur experimental evaluation of the SE detection capabilities of AndrODet strongly indicates that the system fails to learn a generalizable model for detecting SE, and instead bases its decision on the malware family to which a sample belongs. Since we obtained similar results with both the on-line and batch learning approaches proposed by Mirzaei et al., we believe the main problem to be the features used by AndrODet, rather than limitations of a specific machine learning approach. A potential explanation for the weak discriminative power of the features used to detect SE in AndrODet is that they are all based on compound statistics over all strings in an app. Since the percentage of strings that are actually subjected to encryption can vary dramatically between SE-obfuscated apps (i.e., it cannot be assumed that all strings in an SE-obfuscated app are encrypted) features such as the \\emph{average} string entropy or length are not very informative. A more viable approach might be to detect SE on a per-string basis, and determine, for example, a threshold on the number of obfuscated strings for classifying an app as using SE.\n\nFinally, it should be noted that we have only considered SE detection in this comment paper. Since we have not evaluated the performance of AndrODet's detection of identifier renaming and control flow obfuscation, we can neither confirm nor rule out the possibility of similar problems when detecting these kinds of obfuscations. Since Mirzaei et al.\\ use compound statistics for the entire app also when detecting these other two types of obfuscation, one potential concern is that AndrODet may fail to generalize to cases where only a subset of the identifiers, or only a fraction of the code of an app, has been obfuscated.\n\n\\smallskip\n\\noindent\n\\emph{In the interest of open science, we have made the code for running our experiments available at}\\\\ \n\\begin{minipage}{\\linewidth}\n{\\small \\url{https:\/\/github.com\/alirezamohammadinodooshan\/androdet-se-eval}}\n\\end{minipage}\n\n\\section*{Acknowledgments}\nThe calculations were performed using computer resources provided by the Swedish National Infrastructure for Computing (SNIC) at the National Supercomputer Centre (NSC).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \nDetection of cosmic rays arriving at the Earth with energies above \n$10^{20} $ eV questions the presence of the GZK cutoff~\\cite{gzk}.\nThis cutoff determines the energy where the cosmic ray spectrum\nis expected to abruptly steepen. Cosmic rays with ultra high energies (above \n$\\sim 5 \\times 10^{19} $ eV) lose energy through photoproduction of pions\nwhen transversing the Cosmic Microwave Background Radiation (CMB). As\nthe CMB attenuates ultra high energy cosmic rays (UHECR) on a 50 Mpc\nscale (or characteristic distance) at $10^{20}$ eV,\none can determine it's production maximum distance. An event\nof $10^{20} {\\rm eV}$ has to be produced within $\\sim 100$ Mpc, unless it is\na non standard particle~\\cite{cfk,afk}.\nThe absence of any powerful source located within this \nrange~\\cite{sommers} --- that could accelerate a cosmic ray to such an \nenergy --- turns the existence \nof these events into a mystery, the so called GZK puzzle. \n\nThe results of two important cosmic ray experiments AGASA~\\cite{agasa} and \nHiRes~\\cite{hires} are not consistent. Not only is the energy \nspectrum measured by HiRes systematically below the one measured by AGASA, \nbut also the \nHiRes spectrum steepens around $10^{20} $ eV while AGASA's spectrum flattens \naround this energy region. The steepening in the HiRes spectrum may be in \nagreement with a GZK cutoff, while AGASA's is thought not to be. \n\nThere are many possible ways to understand this discrepancy\n~\\cite{demarco,stanev}. \nThe Pierre \nAuger Observatory~\\cite{auger} will soon have a statistically significant \ndata sample and\nwill certainly shed light into understanding these events.\n\nIn this article we focus on the role\nof the shape of the error distribution in the energy determination. \nWe show that the intrinsic \nfeatures of an air shower results in a lognormal error distribution on the\nenergy determination. \nThe minimum standard deviation of this distribution ($\\sigma$) is set by \nphysical properties of the shower. \nIf additional errors due to detection --\nwhich increases the $\\sigma$ -- are not kept to minimum, the end of the\nenergy spectrum will be smeared in a way that the GZK feature might not be seen.\n\nUnderstanding the energy error is crucial in order to determine whether \nor not the GZK cutoff is present. \nA lognormal error distribution on the reconstructed primary cosmic ray\nenergy is to be expected due to fluctuations\nboth in the shower starting point as well as from the cascade development \n\\cite{gaisser}. According to simulations by the AUGER collaboration \n\\cite{desrep} the depth of first interaction affects the rate of development\nof the particle cascade of the shower which results in a fluctuation\nof about 15\\% on the number of muons and about 5\\% on the eletromagnetic\ncomponent. Auger also predicts that the number of muons in a proton induced \nshower increases with primary energy as E$^{0.85}$ \\cite{desrep}. \nThe 15\\% fluctuation will then contribute as a fixed fractional error and\nthe fluctuation on the number of muons on the ground will be \n$N = (1 \\pm 0.15) N_0 (E\\,\/\\,E_0)^{0.85}$. Therefore one has to add a 15\\% \ncontribution to the error in estimating the primary energy in addition \nto the $\\sqrt{N}$ error factor.\nAs this shower starting fluctuation error is a percentage of the \nenergy it results in a lognormal error distribution.\n\nThere are mainly two ways of determining the energy: ground detectors reconstruct\nthe energy based on the particle density at a certain distance from the\nshower core and fluorescence detectors which \ndetermine the energy through the shower longitudinal profile~\\cite{hires}.\nThe longitudinal profile determines the number of particles in the shower\nper depth and it is well known to have large fluctuations. \nAs mentioned above, the fluctuations  arise both from the shower starting point \nas well as from the cascade  development. \nThe same is expected for the  energy determination in ground detectors, \nsince the particle density depends on the number of particles.\n\nThe inherent fluctuations and  resulting lognormal error distribution will affect\ncrucially the analysis of data collected in ground arrays since their data\nsample is collected at one particular depth. It does also affect fluorescence\ndata but as the energy reconstruction uses the full longitudinal profile of\nthe shower, there is more potential information to estimate the original \nenergy.\n\nFigure~\\ref{fig:grdp} shows the distribution\nof particles at ground level for $2 \\times 10^4$ simulated \nshowers~(using \\cite{aires}) \nfrom $10^{20}$ eV protons. A lognormal fit with $\\sigma = 0.08$ is\nsuperimposed and it is clear that the distribution has a lognormal shape.\nThe same distribution for showers from $10^{18}$~eV\nprotons is shown in Figure~\\ref{fig:grdp18}. The poor fit is due to an excess\nof simulated events relative to the lognormal at the high end.\nThe standard deviation of the fit is 0.14. Effects due to the errors with\nasymmetrical and non-gaussian tails are shown in \\cite{vaz}.\n\n\\begin{figure} \n\\centering\\leavevmode \\epsfxsize=220pt \\epsfbox{pgrd_e20.eps}\n\\caption{Distribution of total number of particles at ground level.\nRatio of number of particles over average is shown.\n$2\\times 10^4$ showers were simulated with the Aires~\\cite{aires} package.\nPrimary particles are $10^{20}$ eV protons and $\\langle {\\rm N_{part}}\n\\rangle = 2.7 \\times 10^{10}$. Superimposed is a lognormal curve with $\\sigma = 0.08$.}\n\\label{fig:grdp}\n\\end{figure}\n\\begin{figure} \n\\centering\\leavevmode \\epsfxsize=220pt \\epsfbox{pgrd_e18.eps}\n\\caption{Same as Figure~\\ref{fig:grdp} but for $10^{18}$ eV protons\nas the primaries. Here $\\langle {\\rm N_{part}} \\rangle = 14.7 \\times 10^7$.\nSuperimposed is a lognormal curve with $\\sigma = 0.14$. The poor fit is due to \nan excess of simulated events relative to the lognormal at the high end.}\n\\label{fig:grdp18}\n\\end{figure}\n\nThe simulated showers used Sybill interaction model and assumed that the ground \nwas at sea level (defined in Aires \\cite{aires} as 0\\,m or 1036\\,g\/cm$^2$). A more\nthorough analysis is under way to understand why the error distribution for lower\nenergies (as in Figure~\\ref{fig:grdp18}) deviates from the lognormal shape.\nHowever it is clear that most of the events in excess come from the tail of the\nmaximum depth (XMAX) of the shower distribution. In Figure~\\ref{fig:xmax18} we\nshow the XMAX distribution for the same events used in Figure~\\ref{fig:grdp18}.\nIf we cut events with XMAX $>$ 890 g\/cm$^2$ from this data set, \nthe ground particles distribution will lose part of the excess events. This \ndistribution is shown in Figure~\\ref{fig:cut18}. \nThese excess events, if included, would only make more exaggerated the effect we discuss here.\n\n\n\\begin{figure} \n\\centering\\leavevmode \\epsfxsize=220pt \\epsfbox{xmax_e18.eps}\n\\caption{Maximum shower depth (XMAX) distribution for $2 \\times 10^{4}$ showers \nwith $10^{18}$ eV protons as the primaries. The arrow in the XMAX axis indicates\nwhere an analysis cut will be applied.}\n\\label{fig:xmax18}\n\\end{figure}\n\\begin{figure} \n\\centering\\leavevmode \\epsfxsize=220pt \\epsfbox{pgrde18cut.eps}\n\\caption{Same as Figure~\\ref{fig:grdp18} but events with XMAX $>$ 890 g\/cm$^2$\nwere removed. Superimposed is a lognormal curve with $\\sigma = 0.13$. \nThe fit improves in relation to Figure~\\ref{fig:grdp18}.}\n\\label{fig:cut18}\n\\end{figure}\n\n\nThe results shown in \nFigures~\\ref{fig:grdp} and~\\ref{fig:grdp18} are also dependent on the location of\nthe ground level. We have also simulated events with ground above\nsea level at 950\\,g\/cm$^2$. The lognormal continues to fit well the\n$10^{20}$ eV distribution and its $\\sigma$ improves to 0.05. The $10^{18}$ eV \ndistribution still has an excess but the chisquare improves to 4 and the $\\sigma$\nto 0.10.\n\nBelow we will describe how we determine the UHECR spectrum\nassuming a injection power spectrum from cosmologically distributed sources. \nWe account for energy loss due to propagation through the CMB. \nWe then describe how the energy error is\nevaluated and how it affects the energy reconstruction and the determination\nof the GZK cutoff.\n\n\\section{Analytical determination of UHECR propagation and energy \nspectrum}\nOur analytical approach assumes a cosmological cosmic ray flux. We \nassume extragalactic sources isotropically distributed at different \nredshifts~\\cite{blan}. These\nsources produce a power law energy spectrum (injection spectrum) which is\nassumed to be:\n\\begin{equation}\nF(E) = k E^{-\\alpha} \\exp\\left(-\\frac{E}{E_{max}}\\right)\n\\label{eq:flux}\n\\end{equation}\nwhere $E$ is the cosmic ray energy, $k$ is a normalization factor, $\\alpha$ \nis the spectral index and $E_{max}$ is the maximum energy at the source.\n\nThe energy degradation of protons through the CMB includes losses \ndue to pair production~\\cite{blu,geddes}, expansion of the universe~\\cite{bere} \nand photopion production~\\cite{bere}. These losses at present\nepoch are shown in Figure~\\ref{fig:enloss}.\n\\begin{figure} \n\\centering\\leavevmode \\epsfxsize=250pt \\epsfbox{enloss_t.eps}\n\\caption{Energy losses (as labeled) of a proton transversing the CMB at \npresent epoch.}\n\\label{fig:enloss}\n\\end{figure}\nWe include current values\nfor the matter and dark energy density ($\\Omega_M$ and $\\Omega_\\Lambda$)\nwhen considering the energy loss due to expansion of the universe \n($\\beta_z$):\n\n\\begin{equation}\n\\beta_z(E,z) = H_0 \\sqrt{\\Omega_M (1 + z)^3 + \\Omega_\\Lambda}\n\\end{equation}\nwhere $\\beta$ is defined as $\\beta = 1\/E \\times dE\/dt$ and \n$\\Omega_M = 0.3$, $\\Omega_\\Lambda = 0.7$ and $H_0 = 75 $\n~km~s$^{-1}$~Mpc$^{-1}$.\n\nAlso the energy losses due to pair or to photopion production ($\\beta(E,z)$)\nat a certain epoch with redshift z, is corrected. Since the number density\nof the cosmic background photons varies as $n = n_0 \\, (1+z)^3$ the energy loss\nat z differs from the energy loss today ($\\beta_0(E)$) in the following way:\n\n\\begin{equation}\n\\beta(E,z) = (1 + z)^3 \\beta_0((1+z)E)\n\\end{equation}\n\nOnce all energy loss mechanisms are known, the energy with which a\nproton has to be generated in order to account for the energy observed\ntoday can be determined. The generated energy depends on the distance\nor epoch (redshift) from today. This can be well determined by a\nmodification factor $\\eta(E,z)$~\\cite{bere} which relates the generated\nenergy spectrum to the modified (and measured) one.\n\nThe cosmological flux assumes the observer in the center of a sphere\nof large radius and an isotropic density of sources \\cite{blan}.\nThe flux at the Earth is given by:\n\\begin{eqnarray*}\nj(E) & = & \\frac{c}{4 \\pi H_0} \\int_0^z F(E_g) \n\\left(\\frac{E_g}{E}\\right)^{-\\alpha} (1 + z)^m \n\\frac{dE_g}{dE} \\\\\n& & \\times \\frac{dz}\n{(1 + z)\\left[\\Omega_M (1 + z)^3 + \\Omega_\\Lambda\\right]^{1\/2}}\n\\end{eqnarray*}\nwhere $E_g$ is the generated cosmic ray energy (at a source located with\nredshift $z$); $F(E)$ is given by Equation~\\ref{eq:flux}; $E$ is the cosmic \nray energy determined at the Earth; $\\alpha$ is the same spectral index as\nin Equation~\\ref{eq:flux}; $m$ accounts for the luminosity evolution of the\nsources and $c$ is the speed of light. We assume $m = 0$ and therefore do not\ntake luminosity evolution into account.  \nThe modification factor $\\eta$ is given by:\n\\begin{equation}\n\\eta = \\int_0^z \\left(\\frac{E_g}{E}\\right)^{-\\alpha} \\frac{dE_g}{dE} \\frac{dz}\n{(1 + z)\\left[\\Omega_M (1 + z)^3 + \\Omega_\\Lambda\\right]^{1\/2}}\n\\end{equation}\n\nFor comparison, we determine the modification factor for arbitrary\nredshifts and assuming no cosmological constant. Our results match those\nof~\\cite{bere,demarco} and are shown in Figure~\\ref{fig:mod}.\n\n\\begin{figure} \n\\centering\\leavevmode \\epsfxsize=250pt \\epsfbox{mf.eps}\n\\caption{Modification factor~\\cite{bere} from our analytical calculation\nversus cosmic ray energy. Curves are for different redshifts (top: \n$z = 0.002$; middle: $z = 0.02$; bottom $z = 0.2$)\nand assuming no cosmological constant in order to compare results to \n\\cite{bere,demarco}.}\n\\label{fig:mod}\n\\end{figure}\n\nIn Figure~\\ref{fig:specg} is shown (black solid curve) the cosmic ray expected\nflux versus energy ($E$) at Earth multiplied by $E^3$ \nfrom a cosmological injection spectrum with $\\alpha = 2.6$. The expected GZK\nfeature is present.\n\n\\begin{figure} \n\\centering\\leavevmode \\epsfxsize=250pt \\epsfbox{fluxg.8.eps}\n\\caption{Cosmic ray energy spectrum ($\\times E^3$) from a cosmological \nflux (solid black line) with spectral index $\\alpha = 2.6$. The other curves \nare the energy spectrum convoluted with a lognormal\nerror with standard deviations $\\sigma$ as shown.}\n\\label{fig:specg}\n\\end{figure}\n\n\\section{Errors effects on the energy reconstruction}\nWe will now assume that the cosmic ray energy spectrum from a cosmological\nisotropic distribution of sources \nis the true spectrum. To understand how an error in the reconstructed energy\naffects the spectrum, we convolute the cosmological flux assuming a lognormal\nerror on the energy.\n\nThe lognormal distribution is given by\n\\begin{equation}\n\\frac{dP(E',E)}{d\\ln E} = k \n\\exp\\left[-\\frac{1}{2\\sigma^2} \\log^2\\frac{E'}{E}\\right]\n\\end{equation}\nwhere $k = 1\\, \/\\, \\sqrt{2 \\pi}\\sigma$ is a normalization to unit area and \n$\\sigma$  is the standard deviation of $\\log_{10}E$. \nWhen a lognormal error in the energy reconstruction is assumed, the flux will be \nconvoluted in the following way:\n\\begin{equation}\ndF'(E) = F(E') \\frac{dP(E',E)}{dE} dE' \n\\end{equation}\nwhere F is given by Equation~\\ref{eq:flux}.\n\nThe expected flux ($\\times E^3$) for energies reconstructed with a lognormal\nerror distribution is shown in Figure~\\ref{fig:specg}. The curves are for \na spectral index\n$\\alpha = 2.6$ and $\\sigma = 0.08$, 0.14, 0.25 and 0.5 as labeled.\n\nIt is very clear that not only the flux increases by a constant factor but\nalso the GZK feature is smeared. As shown in Figures~\\ref{fig:grdp}\nand \\ref{fig:grdp18}, the standard deviation in the lognormal \ndistribution will be obtained in an ideal case where thousands of events \nare detected depends on the energy of the primary particle. It is\n0.08 for a $10^{20}$ eV proton and 0.14 for a $10^{18}$ eV proton.\nFigure~\\ref{fig:specg} shows that if the standard deviation is above\n0.14 the GZK cutoff will show up at higher energies than in the true \nspectrum.\n\n\\section{Results and Conclusions}\nFigure~\\ref{fig:specg} shows how the energy spectrum from a cosmological\nflux is smeared due to a lognormal error in the energy reconstruction.\nIntrinsic shower fluctuations leads to a lognormal distribution of\nobserved energy deposition and number of particles in the shower.\nA standard deviation of $log_{10}E$ equal to 0.25 is enough to modify not only the \nshape as\nwell as the normalization of the spectrum measured at the Earth.\nAs a consequence the GZK feature will be smeared and might not be detected \nat all. Such a $\\sigma$ (standard deviation of $\\log_{10}E$) can\neasily result from a detector that only samples a small portion of\nthe total number of particles. This will be more crucial\nto ground detectors since their particle sample is detected all at one \nheight. The standard deviation of the intrinsic \nenergy error distribution for ground detectors is expected to be larger \nthan for fluorescence detectors.\n\nThe air fluorescence\ndetectors will have lower intrinsic lognormal standard deviation as they\nobserve the full development of the shower through the range of view.\nThey miss observing only what goes into the ground or is out of the field\nof view.\n\nAs the Pierre Auger Observatory will not only increase the data sample\nto a significant level, but also combine both ground and fluorescence\ntechniques, it will have constraints to understand and better control\nthe errors in the energy reconstruction. In this way it is\npossible to keep the standard deviation of the intrinsic lognormal\nenergy error to its minimum value.\n\nThe lognormal curves shown in \nFigure~\\ref{fig:grdp} and \\ref{fig:grdp18} have $\\sigma = 0.08$ and 0.14\nrespectively. However one can expect a larger value from an observed distribution\nsince the detectors sample only a portion of the total number of particles.\nOn the other hand\nthe standard deviation of the distribution depends on the ground level altitude\nand\ntherefore an analysis equivalent to ours has to be done for a specific\ndepth.\n\nFigure~\\ref{fig:2spec} shows that the lognormal error in the energy\nis also affected by the spectral index of the injection spectra. However\nthe error in the energy reconstruction will smear the flux in a\nsignificant way independently of the spectral index.\n\n\\begin{figure} \n\\centering\\leavevmode \\epsfxsize=300pt \\epsfbox{flux2.eps}\n\\vspace*{-2.5cm}\n\\caption{Same as Figure~\\ref{fig:specg} but with spectral index $\\alpha = 3.0$\n(top) and $\\alpha = 2.3$ (bottom). Curve with crosses (x) has \n$\\sigma = 0.5$; circles (o), $\\sigma = 0.25$ and squares, $\\sigma = 0.1$.}\n\\label{fig:2spec}\n\\end{figure}\n\nWe have shown that a lognormal error in the energy reconstruction of\nthe UHECR spectra will affect not only the shape but also the normalization\nof the measured energy spectra. A standard deviation equal to or greater \nthan 0.25 will smear the GZK feature. As a consequence this feature will \nnot be seen. \nThis result is independent of the spectral index of the injection spectra.\n\nIn conclusion, the establishment of the presence or not of the GZK cutoff in \nthe \nUHECR spectrum depends not only in a larger data sample but also in the \ndetermination of the shape of the energy error distribution. The standard \ndeviation of this distribution has to be kept to its intrinsec value. If\nit is equal or greater to 0.25 the GZK feature will be smeared and not be\ndetected.\n\n{\\em Acknowledgements --} \nWe thank Don Groom for useful comments.\nThis work was partially supported by NSF Grant Physics\/Polar Programs\nNo. 0071886 and in part\nby the Director, Office of Energy Research, Office of High Energy and\nNuclear Physics, Division of High Energy Physics of the U.S. Department\nof Energy under Contract Num. DE-AC03-76SF00098 through the Lawrence\nBerkeley National Laboratory.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $\\sigma\\in(0,1)$ with $\\sigma\\neq\\frac{1}{2}$. We consider the Cauchy problem for the fractional nonlinear Schr\\\"odinger equation\n\\begin{equation}\\tag{$\\textup{NLS}_\\sigma$}\ni\\partial_tu+(-\\Delta)^\\sigma u+\\mu|u|^{p-1}u=0,\\ u(0)=u_0\\in H^s,\n\\end{equation}\nwhere $\\mu= \\pm 1$ depending on the focusing or defocusing case. The operator $(-\\Delta)^{\\sigma}$ is the so-called fractional laplacian, a Fourier multiplier of $|\\xi|^{2\\sigma}$. The fractional laplacian is the infinitesimal generator of some Levy processes \\cite{B}. A rather extensive study of the potential theoretic aspects of this operator can be found in \\cite{landkof}. \n\n\nThe previous equation is a fundamental equation of fractional quantum mechnics, a generalization of the standard quantum mechanics extending the Feynman path integral to Levy processes \\cite{laskin}. \n\nThe purpose of the present paper is to develop a general well-posedness and ill-posedness theory in Sobolev spaces. The one-dimensional case has been treated in \\cite{CHKL} for cubic nonlinearities, i.e. $p=3$, and $\\sigma\\in(\\frac{1}{2},1)$. Here, we consider a higher-dimensional version and other types of nonlinear terms. We also include all $\\sigma\\in (0,1)$ except $\\sigma=\\frac{1}{2}$; furthermore, contrary to \\cite{CHKL} where the use of Bourgain spaces was crucial (since the main goal of their paper was to derive well-posedness theory on the flat torus), we rely only on standard Strichartz estimates and functional inequalities in $\\mathbb{R}^d.$ In the case of Hartree-type nonlinearities, the local well-posedness and blow-up have been investigated in \\cite{ozawa}. \n\nIn the present paper, we will not consider global aspects with large data. For that, we refer the reader to \\cite{sire2} for a study of the energy-critical equation in the radial case, following the seminal work of Kenig and Merle \\cite{km1,km2}. As a consequence, we do not consider blow-up phenomena, an aspect we will treat in a forthcoming work.\n\nWe introduce two important exponents for our purposes: \n$$\ns_c=\\frac{d}{2}-\\frac{2\\sigma}{p-1}\n$$\nand \n$$s_g=\\frac{1-\\sigma}{2}.$$\n\nHere, $s_c$ is the scaling-critical regularity exponent in the following sense: for $\\lambda>0$, the transformation\n\\[u(t,x)\\mapsto \\frac{1}{\\lambda^{2\\sigma\/(p-1)}} u\\Big(\\frac{t}{\\lambda^{2\\sigma}},\\frac{x}{\\lambda}\\Big), \\quad u_0(x)\\mapsto \\frac{1}{\\lambda^{2\\sigma\/(p-1)}} u_0\\Big(\\frac{x}{\\lambda}\\Big)\\]\nkeeps the equation invariant and one can expect local-wellposedness for $s \\geq s_c$, since the scaling leaves the $\\dot H^{s_c}$ norm invariant. On the other hand, $s_g$ is the critical regularity in the ``pseudo\"-Galilean invariance (see the proof of ill-posedness below). \nUnder the flow of the equation ($\\textup{NLS}_\\sigma$), the following\nquantities are conserved:\n\\begin{align*}\nM[u]=&\\int_{\\mathbb R^d} |u(t,x)|^2dx&&\\textup{(mass)},\\\\\nE[u]=&\\int_{\\mathbb R^d}\\frac{1}{2}|\\,|\\nabla|^{\\sigma}u(t,x)|^2+\\frac{\\mu\n}{p+1}|u(t,x)|^{p+1}dx.&&\\textup{(energy)}.\n\\end{align*}\n\nAn important feature of the equation under study is a loss of derivatives for the Strichartz estimates as proved in \\cite{COX}. Unless additional assumptions are met such as radiality as in \\cite{zihua}, one has a loss of $d(1-\\sigma)$ derivatives in the dispersion (see \\eqref{dispersive estimate with loss}). This happens to be an issue in several arguments. \n\n\n\\subsection*{Main results} The goal of this paper is to show that $(\\textup{NLS}_\\sigma)$ is locally well-posed in $H^s$ for $s\\geq \\max(s_c,s_g, 0)$, and it is ill-posed in $H^s$ for $s\\in (s_c, 0)$. We start with well-posedness results. \n\\begin{theorem}[Local well-posedness in subcritical cases]\\label{subcritical LWP}\nLet\n\\begin{align*}\n\\left\\{\\begin{aligned}\n&s\\geq s_g&&\\text{ when }d=1\\textup{ and }2\\leq p<5,\\\\\n&s>s_c&&\\text{ when }d=1\\textup{ and }p\\geq 5,\\\\\n&s>s_c&&\\text{ when }d\\geq 2\\textup{ and }p\\geq3.\n\\end{aligned}\n\\right.\n\\end{align*}\nThen, $(\\textup{NLS}_\\sigma)$ is locally well-posed in $H^s$. \n\\end{theorem}\n\n\n\n\\begin{theorem}[Local well-posedness in critical cases]\\label{critical LWP}\nSuppose that\n\\begin{align*}\n\\left\\{\\begin{aligned}\n&p>5&&\\text{ when }d=1,\\\\\n&p>3&&\\text{ when }d\\geq 2.\n\\end{aligned}\n\\right.\n\\end{align*}\nThen, $(\\textup{NLS}_\\sigma)$ is locally well-posed in $H^{s_c}$. \n\\end{theorem}\n\nThe proof of Theorem \\ref{critical LWP} is based on a new method, improving on estimates in \\cite{CKSTT}. This improvement, based on controlling the nonlinearity in a suitable space, is necessary due to the loss of derivatives in the Strichartz estimates. \n\nAs a by-product, we also prove small data scattering. \n\\begin{theorem}[Small data scattering]\\label{scattering}\nSuppose that\n\\begin{align*}\n\\left\\{\\begin{aligned}\n&p>5&&\\text{ when }d=1,\\\\\n&p>3&&\\text{ when }d\\geq 2.\n\\end{aligned}\n\\right.\n\\end{align*}\nThen, there exists $\\delta>0$ such that if $\\|u_0\\|_{H^{s_c}}<\\delta$, then $u(t)$ scatters in $H^{s_c}$. Precisely, there exist $u_\\pm \\in H^{s_c}$ such that\n$$\\lim_{t\\to\\pm\\infty}\\|u(t)-e^{it(-\\Delta)^\\sigma}u_\\pm\\|_{H^{s_c}}=0.$$\n\\end{theorem}\n\n\\begin{remark}\nContrary to the case $\\sigma\\neq\\frac{1}{2}$, when $\\sigma=\\frac{1}{2}$, the fractional NLS does not have small data scattering. See \\cite{krieger}. \n\\end{remark}\n\nFinally, our last theorem is the ill-posedness result. Note that our result is not optimal, since one should expect ill-posedness in $H^s$ up to $s_g=\\frac{1-\\sigma}{2}$, which is nonnegative. We hope to come back to this issue in a forthcoming work. \n\n\\begin{theorem}[Ill-posedness]\\label{ill-posedness}\nLet $d=1,2$ or $3$ and $\\sigma\\in(\\frac{d}{4},1)$. If $p$ is not an odd integer, we further assume that $p\\geq k+1$, where $k$ is an integer larger than $\\frac{d}{2}$. Then, $(\\textup{NLS}_\\sigma)$ is ill-posed in $H^s$ for $s\\in (s_c,0)$.\n\\end{theorem}\n\nAn interesting feature of the previous ill-posedness result is the fact that, contrary to the standard NLS equation ($\\sigma=1$) there is no exact Galilean invariance. However, one can introduce a new ``pseudo-Galilean invariance\" which is enough to our purposes. More precisely, for $v\\in\\mathbb{R}^d$, we define the  transformation\n$$\\mathcal{G}_vu(t,x)=e^{-iv\\cdot x}e^{it|v|^{2\\sigma}}u(t,x-2t\\sigma|v|^{2(\\sigma-1)}v).$$\nNote that when $\\sigma=1$, $\\mathcal{G}_v$ is simply a Galilean transformation, and that NLS is invariant under this transformation, that is, if $u(t)$ solves NLS, so does $\\mathcal{G}_vu(t)$. However, when $\\sigma\\neq1$, $(\\textup{NLS}_\\sigma)$ is not exactly symmetric with respect to pseudo-Galilean transformations. This opens the construction of solitons for $(\\textup{NLS}_\\sigma)$ which happen to be different from the ones constructed in the standard case $\\sigma=1$. Indeed, if we search for exact solutions of the type\n\\begin{equation}\nu(t,x)=e^{it(|v|^{2\\sigma}-\\omega^{2\\sigma})} e^{-iv\\cdot x}Q_\\omega(x-2t\\sigma|v|^{2(\\sigma-1)}v),\n\\end{equation}\nthen the profile $Q_\\omega$ solves the pseudo-differential equation\n\\begin{equation}\\label{Q equation}\n\\mathcal{P}_vQ_\\omega+\\omega^{2\\sigma}Q_\\omega-|Q_\\omega|^{p-1}Q_\\omega=0,\n\\end{equation}\nwhere\n\\begin{equation}\n\\mathcal{P}_v=e^{iv\\cdot x}(-\\Delta)^\\sigma e^{-iv\\cdot x}-|v|^{2\\sigma}-2i\\sigma|v|^{2\\sigma-2}v\\cdot\\nabla,\n\\end{equation}\ni.e., $\\mathcal{P}_v$ is a Fourier multiplier $\\widehat{\\mathcal{P}_v f}(\\xi)=p_v(\\xi)\\hat{f}(\\xi)$, wiht symbol\n\\begin{equation}\np_v(\\xi)=|\\xi-v|^{2\\sigma}-|v|^{2\\sigma}+2\\sigma|v|^{2\\sigma-2}v\\cdot\\xi.\n\\end{equation}\nWe plan to come back to this issue in future works. \n\n\n\n\n\n\n\\section{Strichartz Estimates}\n\nIn this section, we review Strichartz estimates for the linear fractional Schr\\\"odinger operators. \nWe say that $(q,r)$ is \\textit{admissible} if\n$$\\frac{2}{q}+\\frac{d}{r}=\\frac{d}{2},\\quad 2\\leq q,r\\leq\\infty,\\quad (q,r,d)\\neq(2,\\infty,2).$$\nWe define the Strichartz norm by\n$$\\|u\\|_{S_{q,r}^s(I)}:=\\||\\nabla|^{-d(1-\\sigma)(\\frac{1}{2}-\\frac{1}{r})}u\\|_{L_{t\\in I}^qW_x^{s,r}},$$\nwhere $I=[0,T)$. Let $\\psi: \\mathbb{R}^d\\to [0,1]$ be a compactly supported smooth function such that $\\sum_{N\\in 2^{\\mathbb{Z}}}\\psi_N=1$, where $\\psi_N(\\xi)=\\psi(\\frac{\\xi}{N})$. For dyadic $N\\in 2^{\\mathbb{Z}}$, let $P_N$ be a Littlewood-Paley projection, that is, $\\widehat{P_N f}(\\xi)=\\psi(\\frac{\\xi}{N})\\hat{f}(\\xi)$.  Then, we define a slightly stronger Strichartz norm by \n$$\\|u\\|_{\\tilde{S}_{q,r}^s(I)}:=\\Big(\\sum_{N\\in2^\\mathbb{Z}}\\|P_N(|\\nabla|^{-d(1-\\sigma)(\\frac{1}{2}-\\frac{1}{r})}u)\\|_{L_{t\\in I}^qW_x^{s,r}}^2\\Big)^{1\/2}.$$\n\n\\begin{proposition}[Strichartz estimates \\cite{COX}]\\label{Strichartz}\nFor an admissible pair $(q,r)$, we have\n\\begin{align*}\n\\|e^{it(-\\Delta)^\\sigma}u_0\\|_{S_{q,r}^s(I)}, \\|e^{it(-\\Delta)^\\sigma}u_0\\|_{\\tilde{S}_{q,r}^s(I)}&\\lesssim\\|u_0\\|_{H^s},\\\\\n\\Big\\|\\int_0^t e^{i(t-s)(-\\Delta)^\\sigma}F(s)ds\\Big\\|_{S_{q,r}^s(I)}&\\lesssim \\|F\\|_{L_{t\\in I}^1H_x^{s}},\\\\\n\\Big\\|\\int_0^t e^{i(t-s)(-\\Delta)^\\sigma}F(s)ds\\Big\\|_{\\tilde{S}_{q,r}^s(I)}&\\lesssim \\|F\\|_{L_{t\\in I}^1H_x^{s}}.\n\\end{align*}\n\\end{proposition}\n\n\\begin{proof}[Sketch of Proof]\nBy the standard stationary phase estimate, one can show that\n$$\\|e^{it(-\\Delta)^\\sigma}P_1\\|_{L^1\\to L^\\infty}\\lesssim|t|^{-\\frac{d}{2}},$$\nand by scaling,\n\\begin{equation}\\label{dispersive estimate with loss}\n\\|e^{it(-\\Delta)^\\sigma}P_N\\|_{L^1\\to L^\\infty}\\lesssim N^{d(1-\\sigma)}|t|^{-\\frac{d}{2}}.\n\\end{equation}\nThen, it follows from the argument of Keel-Tao \\cite{KT} that for any $I\\subset\\mathbb{R}$,\n\\begin{align*}\n\\|e^{it(-\\Delta)^\\sigma}P_N(|\\nabla|^{-d(1-\\sigma)(\\frac{1}{2}-\\frac{1}{r})}u_0)\\|_{L_{t\\in I}^q W_x^{s,r}}&\\lesssim\\|P_Nu_0\\|_{H^s},\\\\\n\\Big\\|\\int_0^t e^{i(t-s)(-\\Delta)^\\sigma}P_N(|\\nabla|^{-d(1-\\sigma)(\\frac{1}{2}-\\frac{1}{r})}F)(s)ds\\Big\\|_{L_{t\\in I}^qW_x^{s,r}}&\\lesssim \\|P_NF\\|_{L_{t\\in I}^1H_x^{s}}.\n\\end{align*}\nSquaring the above inequalities and summing them over all dyadic numbers in $2^{\\mathbb{Z}}$, we prove Strichartz estimates.\n\\end{proof}\n\n\nThe loss of derivatives is due to the Knapp phenomenon (see \\cite{zihua}). However, in the radial case, one can overcome this loss as proved in \\cite{zihua}, restricting then the admissible powers of the fractional laplacian. Indeed, in \\cite{zihua}, this is proved that one has optimal Strichartz estimates if $\\sigma \\in (d\/(2d-1),1)$. In particular, the number $d\/(2d-1)$ is larger than $1\/2$ and there is a gap between the Strichartz estimates for the wave operator $\\sigma=1\/2$ and the one occuring for higher powers. This issue suggests that a new phenomenon might occur for this range of powers. \n\n\n\n\n\\section{Local Well-posedness}\n\nWe establish local well-posedness of the fractional NLS by the standard contraction mapping argument based on Strichartz estimates. Due to loss of regularity in Strichartz estimates, our proof relies on the $L_x^\\infty$ bounds (see Lemma 3.2 and 3.3).\n\n\\subsection{Subcritical cases}\n\nFirst, we consider the case that $d=1$ and $2\\leq p<5$. In this case, the equation is scaling-subcritical in $H^s$ for $s>s_g$, since $s_g>s_c$. We remark that in the proof, we control the $L_{t\\in I}^4L_x^\\infty$ norm simply by Strichartz estimates (see \\eqref{1d Strichartz 1} and \\eqref{1d Strichartz 2}).\n\n\\begin{proof}[Proof of Theorem \\ref{subcritical LWP} when $d=1$ and $2\\leq p<5$]\nWe define\n$$\\Phi_{u_0}(u):=e^{it(-\\Delta)^\\sigma}u_0+ i\\mu\\int_0^t e^{i(t-s)(-\\Delta)^\\sigma}(|u|^{p-1}u)(s)ds.$$\nLet\n$$\\|u\\|_{X^s}:=\\|u\\|_{L_{t\\in I}^\\infty H_x^s\\cap L_{t\\in I}^4 L_x^\\infty},$$\nwhere $I=[0,T)$. Then, applying the 1d Strichartz estimates\n\\begin{align}\n\\|e^{it(-\\Delta)^\\sigma}u_0\\|_{L_{t\\in I}^4 L_x^\\infty}&\\lesssim\\|u_0\\|_{\\dot{H}^{s_g}}\\label{1d Strichartz 1},\\\\\n\\|e^{it(-\\Delta)^\\sigma}u_0\\|_{L_{t\\in I}^\\infty H_x^s}&\\lesssim\\|u_0\\|_{H^s},\\nonumber\\\\\n\\Big\\|\\int_0^t e^{i(t-s)(-\\Delta)^\\sigma}F(s)ds\\Big\\|_{L_{t\\in I}^4L_x^\\infty}&\\lesssim \\|F\\|_{L_{t\\in I}^1\\dot{H}_x^{s_g}}\\label{1d Strichartz 2},\\\\\n\\Big\\|\\int_0^t e^{i(t-s)(-\\Delta)^\\sigma}F(s)ds\\Big\\|_{L_{t\\in I}^\\infty H_x^s}&\\lesssim \\|F\\|_{L_{t\\in I}^1H_x^s}\\nonumber,\n\\end{align}\nwe get\n$$\\|\\Phi_{u_0}(u)\\|_{X^s}\\lesssim \\|u_0\\|_{H^s}+\\||u|^{p-1}u\\|_{L_{t\\in I}^1H_x^s}.$$\nBy the fractional chain rule\n\\begin{equation}\\label{chain rule}\n\\||\\nabla|^s F(u)\\|_{L^q}\\lesssim \\|F'(u)\\|_{L^{p_1}}\\||\\nabla|^s u\\|_{L^{p_2}},\n\\end{equation}\nwhere $s>0$ and $\\frac{1}{q}=\\frac{1}{p_1}+\\frac{1}{p_2}$, and H\\\"older inequality, we obtain\n$$\\||u|^{p-1}u\\|_{L_{t\\in I}^1H_x^s}\\lesssim \\Big\\| \\||u|^{p-1}\\|_{L_x^\\infty}\\|u\\|_{H_x^s}\\Big\\|_{L_{t\\in I}^1} \\leq T^{\\frac{5-p}{4}}\\|u\\|_{L_{t\\in I}^4L_x^\\infty}^{p-1}\\|u\\|_{L_{t\\in I}^\\infty H_x^s}.$$\nFor the fractional chain rule \\eqref{chain rule}, we refer \\cite{CW}, for example. We remark that one can choose $p_1=\\infty$ in \\eqref{chain rule}. Indeed, this can be proved by a little modification of the last step in the proof of Proposition 3.1 in \\cite{CW}. Thus, we have\n$$\\|\\Phi_{u_0}(u)\\|_{X^s}\\lesssim \\|u_0\\|_{H^s}+T^{\\frac{5-p}{4}} \\|u\\|_{X^s}^p.$$\nSimilarly, by Strichartz estimates,\n$$\\|\\Phi_{u_0}(u)-\\Phi_{u_0}(v)\\|_{X^s}\\lesssim \\||u|^{p-1}u-|v|^{p-1}v\\|_{L_{t\\in I}^1H_x^s}.$$\nThen, applying the fractional Leibniz rule and the fractional chain rule in \\cite{CW}, we get \n\\begin{align*}\n\\||u|^{p-1}u-|v|^{p-1}v\\|_{H_x^s}&=\\Big\\|\\int_0^1 p|v+t(u-v)|^{p-1}(u-v) dt\\Big\\|_{H_x^s}\\\\\n&\\leq p\\int_0^1\\||v+t(u-v)|^{p-1}(u-v) \\|_{H_x^s}dt\\\\\n&\\lesssim\\int_0^1\\|v+t(u-v)\\|_{L_x^\\infty}^{p-1}\\|u-v\\|_{H_x^s}\\\\\n&\\ \\ \\ +\\||v+t(u-v)|^{p-1}\\|_{H_x^s}\\|u-v\\|_{L_x^\\infty}dt\\\\\n&\\lesssim\\int_0^1\\|v+t(u-v)\\|_{L_x^\\infty}^{p-1}\\|u-v\\|_{H_x^s}\\\\\n&\\ \\ \\ +\\|v+t(u-v)\\|_{L_x^\\infty}^{p-2}\\|v+t(u-v)\\|_{H_x^s}\\|u-v\\|_{L_x^\\infty}dt\\\\\n&\\leq (\\|u\\|_{L_x^\\infty}^{p-1}+\\|v\\|_{L_x^\\infty}^{p-1})\\|u-v\\|_{H_x^s}\\\\\n&\\ \\ \\ +(\\|u\\|_{L_x^\\infty}^{p-2}+\\|v\\|_{L_x^\\infty}^{p-2})(\\|u\\|_{H_x^s}+\\|v\\|_{H_x^s})\\|u-v\\|_{L_x^\\infty}.\n\\end{align*}\nThus, it follows that\n\\begin{align*}\n&\\|\\Phi_{u_0}(u)-\\Phi_{u_0}(v)\\|_{X^s}\\\\\n&\\lesssim T^{\\frac{5-p}{4}}\\Big\\{(\\|u\\|_{L_{t\\in I}^4L_x^\\infty}^{p-1}+\\|v\\|_{L_{t\\in I}^4L_x^\\infty}^{p-1})\\|u-v\\|_{L_{t\\in I}^\\infty H_x^s}\\\\\n&\\quad\\quad\\quad+(\\|u\\|_{L_{t\\in I}^4L_x^\\infty}^{p-2}+\\|v\\|_{L_{t\\in I}^4L_x^\\infty}^{p-2})(\\|u\\|_{L_{t\\in I}^\\infty H_x^s}+\\|v\\|_{L_{t\\in I}^\\infty H_x^s})\\|u-v\\|_{L_{t\\in I}^4L_x^\\infty}\\Big\\}\\\\\n&\\lesssim T^{\\frac{5-p}{4}} (\\|u\\|_{X^s}^{p-1}+\\|v\\|_{X^s}^{p-1})\\|u-v\\|_{X^s}.\n\\end{align*}\nChoosing sufficiently small $T>0$, we conclude that $\\Phi_{u_0}$ is a contraction on a ball\n$$B:=\\{u: \\|u\\|_{X^s}\\leq 2\\|u_0\\|_{H^s}\\}$$\nequipped with the norm $\\|\\cdot\\|_{X^s}$.\n\\end{proof}\n\nNext, we will prove Theorem \\ref{subcritical LWP} when $d=1$ and $p\\geq 5$, or $d\\geq 2$ and $p\\geq3$. In this case, we do not have a good control on the $L_x^\\infty$ norm from Strichartz estimates. Instead, we make use of Sobolev embedding.\n\n\\begin{lemma}[$L_{t\\in I}^{p-1}L_x^\\infty$ bound]\nSuppose that $d=1$ and $p\\geq5$, or $d\\geq 2$ and $p\\geq3$. Let $s>s_c$. Then, we have\n\\begin{equation}\\label{subcritical Lx-infty bound}\n\\|u\\|_{L_{t\\in I}^{p-1}L_x^\\infty}\\lesssim T^{0+}\\|u\\|_{S_{q_0,r_0}^{s}(I)},\n\\end{equation}\nwhere $(q_0,r_0) =((p-1)^+,\\Big(\\tfrac{2d(p-1)}{d(p-1)-4}\\Big)^-)$ is an admissible pair. Here, we denote by $c^+$ a number larger than $c$ but arbitrarily close to $c$, and similarly for $c^-$.\n\\end{lemma}\n\n\\begin{proof}\nWe observe that\n$$\\frac{1}{r_0}-\\frac{s-d(1-\\sigma)(\\frac{1}{2}-\\frac{1}{r_0})}{d}<0.$$\nThus, by Sobolev inequality,\n$$\\|u\\|_{L_{t\\in I}^{p-1}L_x^\\infty}\\leq T^{0+}\\|u\\|_{L_{t\\in I}^{q_0}L_x^\\infty}\\lesssim\\||\\nabla|^{-d(1-\\sigma)(\\frac{1}{2}-\\frac{1}{r_0})}u\\|_{L_{t\\in I}^{q_0}W_x^{s,r_0}}=\\|u\\|_{S_{q_0,r_0}^s(I)}.$$\n\\end{proof}\n\nWe also employ a standard persistence of regularity argument.\n\\begin{lemma}[Persistence of regularity]\\label{persistence of regularity}\nLet $1<q\\leq\\infty$, $1<r<\\infty$ and $s_1\\geq s_2$. Then, $B=\\{u: \\|u\\|_{L_{t\\in I}^q W_x^{s_1,r}}\\leq R\\}$, equipped with the norm $\\|\\cdot\\|_{L_{t\\in I}^q W_x^{s_2,r}}$, is a complete metric space.\n\\end{lemma}\n\n\\begin{proof}\nWe recall:\n\\begin{theorem}[Theorem 1.2.5 in \\cite{Caz}]Consider two Banach spaces $X\\hookrightarrow Y$ and $1<p,q\\leq \\infty$. Let $(f_n)_{n\\geq 0}$ be a bouned sequence in $L^q(I, Y)$ and let $f: I\\to Y$ be such that $f_n(t)\\rightharpoonup f(t)$ in $Y$ as $n\\to\\infty$, for a.e. $t\\in I$. If $(f_n)_{n\\geq 0}$ is bounded in $L^p(I;X)$ and if $X$ is reflexive, then $f\\in L^p(I;X)$ and $\\|f\\|_{L^p(I;X)}\\leq \\liminf_{n\\to\\infty}\\|f_n\\|_{L^p(I;X)}$.\n\\end{theorem}\n\nSuppose that $(f_n)_{n=1}^\\infty$ be a Cauchy sequence in $B$. Then, $f_n$ converges to $f$ in $L_{t\\in I}^q W_x^{s_2,r}$. Moreover, it follows from Theorem 1.2.5 in \\cite{Caz} that\n$$\\|f\\|_{L_{t\\in I}^q W_x^{s_1,r}}\\leq \\liminf_{n\\to\\infty}\\|f_n\\|_{L_{t\\in I}^q W_x^{s_1,r}}\\leq R,$$\nand thus $f\\in B$. Therefore, we conclude that $B$ is complete.\n \\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{subcritical LWP} when $d=1$ and $p\\geq 5$, or $d\\geq 2$ and $p\\geq3$]\nDefine the map $\\Phi_{u_0}$ as above, and let \n$$X^\\alpha:=L_{t\\in I}^\\infty H_x^\\alpha\\cap S_{q_0,r_0}^\\alpha(I),$$\nwhere $(q_0, r_0)$ is an admissible pair in Lemma 3.2. Then, by Strichartz estimates, the fractional chain rule and \\eqref{subcritical Lx-infty bound}, we get\n\\begin{align*}\n\\|\\Phi_{u_0}(u)\\|_{X^s}&\\lesssim \\|u_0\\|_{H^s}+\\||u|^{p-1}u\\|_{L_{t\\in I}^1H_x^s}\\\\\n&\\lesssim \\|u_0\\|_{H^s}+\\|u\\|_{L_{t\\in I}^{p-1}L_x^\\infty}^{p-1}\\|u\\|_{L_{t\\in I}^\\infty H_x^s}\\\\\n&\\lesssim \\|u_0\\|_{H^s}+T^{0+}\\|u\\|_{S_{q_0,r_0}^{s}(I)}^{p-1}\\|u\\|_{L_{t\\in I}^\\infty H_x^s}\\\\\n&\\leq \\|u_0\\|_{H^s}+T^{0+}\\|u\\|_{X^s}^p,\n\\end{align*}\nand similarly\n\\begin{equation}\\label{difference in LWP proof}\n\\begin{aligned}\n\\|\\Phi_{u_0}(u)-\\Phi_{u_0}(v)\\|_{X^0}&\\lesssim\\||u|^{p-1}u-|v|^{p-1}v\\|_{L_{t\\in I}^1L_x^2}\\\\\n&\\lesssim\\|(|u|^{p-1}+|v|^{p-1})|u-v|\\|_{L_{t\\in I}^1L_x^2}\\\\\n&\\lesssim(\\|u\\|_{L_{t\\in I}^{p-1}L_x^\\infty}^{p-1}+\\|v\\|_{L_{t\\in I}^{p-1}L_x^\\infty}^{p-1})\\|u-v\\|_{L_{t\\in I}^\\infty L_x^2}\\\\\n&\\lesssim T^{0+}(\\|u\\|_{S_{q_0,r_0}^{s}(I)}^{p-1}+\\|v\\|_{S_{q_0,r_0}^{s}(I)}^{p-1})\\|u\\|_{L_{t\\in I}^\\infty L_x^2}\\\\\n&\\lesssim T^{0+}(\\|u\\|_{X^s}^{p-1}+\\|v\\|_{X^s}^{p-1})\\|u-v\\|_{X^0}.\n\\end{aligned}\n\\end{equation}\nThus, for sufficiently small $T>0$, $\\Phi_{u_0}$ is contractive on a ball\n$$B:=\\{u: \\|u\\|_{X^s}\\leq 2\\|u_0\\|_{H^s}\\}$$\nequipped with the norm $\\|\\cdot\\|_{X^0}$, which is complete by Lemma \\ref{persistence of regularity}.\n\\end{proof}\n\n\\begin{remark}\nThe standard persistence of regularity argument allows us to avoid derivatives in \\eqref{difference in LWP proof}. Indeed, for $u\\in B$, $\\|\\langle\\nabla\\rangle^su\\|_{L_{t\\in I}^{p-1}L_x^\\infty}$ is not necessarily bounded.\n\\end{remark}\n\n\\subsection{Scaling-critical cases}\nIn the scaling-critical case, we use the following lemma, which plays the same role as \\eqref{subcritical Lx-infty bound}. We note that the norms in the lemma are defined via the Littlewood-Paley projection in order to overcome the failure of the Sobolev embedding $W^{s,p}\\hookrightarrow L^q$, $\\frac{1}{q}=\\frac{1}{p}-\\frac{s}{d}$, when $q=\\infty$. Lemma 3.3 generalizes \\cite[Lemma 3.1]{CKSTT}.\n\n\\begin{lemma}[Scaling-critical $L_{t\\in I}^{p-1}L_x^\\infty$ bound]\n\\begin{equation}\\label{critical Lx-infty bound}\n\\|u\\|_{L_{t\\in I}^{p-1}L_x^\\infty}^{p-1}\\lesssim \\left\\{\\begin{aligned}\n&\\|u\\|_{\\tilde{S}_{4,\\infty}^{s_c}(I)}^4\\|u\\|_{\\tilde{S}_{\\infty,2}^{s_c}(I)}^{p-5}&&\\text{ when }d=1\\textup{ and }p>5,\\\\\n&\\|u\\|_{\\tilde{S}_{2+,\\infty-}^{s_c}(I)}^{2}\\|u\\|_{\\tilde{S}_{\\infty,2}^{s_c}(I)}^{p-3}&&\\text{ when }d=2\\textup{ and }p>3,\\\\\n&\\|u\\|_{\\tilde{S}_{2,\\frac{2d}{d-2}}^{s_c}(I)}^{2}\\|u\\|_{\\tilde{S}_{\\infty,2}^{s_c}(I)}^{p-3}&&\\text{ when }d\\geq 3\\textup{ and }p>3.\n\\end{aligned}\n\\right.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nWe will prove the lemma only when $d\\geq3$. By interpolation $\\|f\\|_{L^{p_\\theta}}\\leq \\|f\\|_{L^{p_0}}^\\theta \\|f\\|_{L^{p_1}}^{1-\\theta}$, $\\frac{1}{p_\\theta}=\\frac{\\theta}{p_0}+\\frac{1-\\theta}{p_1}$, $0<\\theta<1$, it suffices to show the lemma for rational $(p-1)=\\frac{m}{n}>2$ with $\\gcd(m,n)=1$. First, we estimate\n$$A(t)=\\Big[\\sum_N\\|P_{N}u(t)\\|_{L_x^\\infty}\\Big]^m\\sim\\sum_{N_1\\geq\\cdots\\geq N_m}\\prod_{i=1}^m\\|P_{N_i}u(t)\\|_{L_x^\\infty}.$$\nObserve from Bernstein's inequality that \n\\begin{align}\n\\|P_Nu(t)\\|_{L_x^\\infty}&\\lesssim N^{-\\frac{\\sigma(p-3)}{p-1}}d_N\\label{Bernstein1},\\\\\n\\|P_Nu(t)\\|_{L_x^\\infty}&\\lesssim N^{\\frac{2\\sigma}{p-1}}d_N'\\label{Bernstein2},\n\\end{align}\nwhere\n$$d_N=\\|P_Nu(t)\\|_{\\dot{W}_x^{s_c-(1-\\sigma),\\frac{2d}{d-2}}},\\ d_N'=\\|P_Nu(t)\\|_{\\dot{H}_x^{s_c}}.$$\nAs a consequence, we have\n\\begin{equation}\\label{Bernstein3}\n\\|P_Nu(t)\\|_{L_x^\\infty}\\lesssim\\Big(N^{-\\frac{\\sigma(p-3)}{p-1}}d_N\\Big)^\\theta\\Big(N^{\\frac{2\\sigma}{p-1}}d_N'\\Big)^{1-\\theta}=N^{\\frac{\\sigma(p-3)}{(p-1)(p-2)}}(d_N)^\\theta (d_N')^{1-\\theta},\n\\end{equation}\nwhere $\\theta=\\frac{1}{p-2}$. Hence, applying \\eqref{Bernstein1} for $i=1,\\cdots,n$ and \\eqref{Bernstein3} for $i=n+1,\\cdots,m$, we bound $A(t)$ by\n$$\\lesssim\\sum_{N_1\\geq \\cdots\\geq N_m} \\Big(\\prod_{i=1}^nN_i^{-\\frac{\\sigma(p-3)}{p-1}}d_{N_i}\\Big) \\Big(\\prod_{i=n+1}^m N_i^{\\frac{\\sigma(p-3)}{(p-1)(p-2)}}(d_{N_i})^\\theta (d_{N_i}')^{1-\\theta}\\Big).$$\nFor an arbitrarily small $\\epsilon>0$, we let\n$$\\tilde{d}_N=\\sum_{N'\\in 2^{\\mathbb{Z}}} \\min\\Big(\\frac{N}{N'},\\frac{N'}{N}\\Big)^\\epsilon d_{N'},\\quad \\tilde{d}_N'=\\sum_{N'\\in 2^{\\mathbb{Z}}} \\min\\Big(\\frac{N}{N'},\\frac{N'}{N}\\Big)^\\epsilon d_{N'}'.$$\nThen, since $d_N\\leq \\tilde{d}_N$ and $\\tilde{d}_{N_i}\\leq (\\frac{N_1}{N_{i}})^\\epsilon \\tilde{d}_{N_1}$ and similarly for primes, $A(t)$ is bounded by\n$$\\lesssim\\sum_{N_1\\geq \\cdots\\geq N_m} \\Big(\\prod_{i=1}^n N_i^{-\\frac{\\sigma(p-3)}{p-1}}\\Big(\\frac{N_1}{N_{i}}\\Big)^\\epsilon \\tilde{d}_{N_1}\\Big) \\Big(\\prod_{i=n+1}^m N_i^{\\frac{\\sigma(p-3)}{(p-1)(p-2)}}\\Big(\\frac{N_1}{N_{i}}\\Big)^\\epsilon (\\tilde{d}_{N_1})^\\theta (\\tilde{d}_{N_1}')^{1-\\theta}\\Big).\n$$\nSumming in $N_m, N_{m-1}, ..., N_{n+1}$ and using $m-n=(p-2)n$,\n\\begin{align*}\nA(t)&\\lesssim\\sum_{N_1\\geq \\cdots\\geq N_n} \\Big(\\prod_{i=1}^n N_i^{-\\frac{\\sigma(p-3)}{p-1}}\\Big(\\frac{N_1}{N_{i}}\\Big)^\\epsilon \\tilde{d}_{N_1}\\Big)\\\\\n&\\quad\\quad\\quad\\quad\\times N_n^{\\frac{\\sigma(p-3)(m-n)}{(p-1)(p-2)}}\\Big(\\frac{N_1}{N_{n}}\\Big)^{(m-n)\\epsilon} (\\tilde{d}_{N_1})^{(m-n)\\theta} (\\tilde{d}_{N_1}')^{(m-n)(1-\\theta)}\\\\\n&=\\sum_{N_1\\geq \\cdots\\geq N_n} \\Big(\\prod_{i=1}^n N_i^{-\\frac{\\sigma(p-3)}{p-1}}\\Big(\\frac{N_1}{N_{i}}\\Big)^\\epsilon \\tilde{d}_{N_1}\\Big)\\\\\n&\\quad\\quad\\quad\\quad\\times N_n^{\\frac{\\sigma(p-3)n}{p-1}}\\Big(\\frac{N_1}{N_{n}}\\Big)^{(p-2)n\\epsilon} (\\tilde{d}_{N_1})^{(m-n)\\theta} (\\tilde{d}_{N_1}')^{(m-n)(1-\\theta)},\n\\end{align*}\nand then summing in $N_n, N_{n-1}, ..., N_1$, we obtain that\n$$A(t)\\lesssim\\sum_{N_1} (\\tilde{d}_{N_1})^{n+(m-n)\\theta} (\\tilde{d}_{N_1}')^{(m-n)(1-\\theta)}=\\sum_{N_1} (\\tilde{d}_{N_1})^{2n} (\\tilde{d}_{N_1}')^{m-2n},$$\nwhich is, by H\\\"older inequality and Young's inequality, bounded by\n\\begin{align*}\n&\\lesssim\\|(\\tilde{d}_{N})^{2n} \\|_{\\ell_N^2}\\|(\\tilde{d}_{N}')^{m-2n}\\|_{\\ell_N^{2}}=\\|\\tilde{d}_{N}\\|_{\\ell_N^{4n}}^{2n}\\|\\tilde{d}_{N}'\\|_{\\ell_N^{2(m-2n)}}^{m-2n}\\\\\n&\\leq \\|\\tilde{d}_{N}\\|_{\\ell_N^2}^{2n}\\|\\tilde{d}_{N}'\\|_{\\ell_N^2}^{m-2n}\\lesssim \\|d_{N}\\|_{\\ell_N^2}^{2n}\\|d_{N}'\\|_{\\ell_N^2}^{m-2n}=\\|d_{N}\\|_{\\ell_N^2}^{2n}\\|d_{N}'\\|_{\\ell_N^2}^{(p-3)n}.\n\\end{align*}\nFinally, by the estimate for $A(t)$, we prove that \n\\begin{align*}\n\\|u\\|_{L_{t\\in I}^{p-1}L_x^\\infty}^{p-1}&\\leq \\int_I A(t)^{p-1}dt=\\int_I A(t)^\\frac{m}{n}dt\\\\\n&\\lesssim\\int_I \\|d_N\\|_{\\ell_N^2}^2\\|d_N'\\|_{\\ell_N^2}^{p-3} dt\\leq \\|d_N\\|_{L_{t\\in I}^2\\ell_N^2}^2\\|d_N'\\|_{L_{t\\in I}^\\infty\\ell_N^2}^{p-3}\\\\\n&\\leq \\|u\\|_{\\tilde{S}_{2,\\frac{2d}{d-2}}^{s_c}(I)}^{2}\\|u\\|_{\\tilde{S}_{\\infty,2}^{s_c}(I)}^{p-3}.\n\\end{align*}\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of theorem \\ref{critical LWP}]\nFor simplicity, we assume that $d\\geq3$. Indeed, with little modifications, we can prove the theorem when $d=1,2$. We define $\\Phi_{u_0}(u)$ as in the proof of Theorem \\ref{subcritical LWP}. Then, by Strichartz estimates, the fractional chain rule and \\eqref{critical Lx-infty bound}, we have\n\\begin{align*}\n\\|\\Phi_{u_0}(u)\\|_{\\tilde{S}_{2,\\frac{2d}{d-2}}^{s_c}(I)}&\\leq \\|e^{it(-\\Delta)^\\sigma}u_0\\|_{\\tilde{S}_{2,\\frac{2d}{d-2}}^{s_c}(I)}+c_0\\||u|^{p-1}u\\|_{L_{t\\in I}^1H^{s_c}}\\\\\n&\\leq \\|e^{it(-\\Delta)^\\sigma}u_0\\|_{\\tilde{S}_{2,\\frac{2d}{d-2}}^{s_c}(I)}+c_1\\|u\\|_{L_{t\\in I}^{p-1}L_x^\\infty}^{p-1}\\|u\\|_{L_{t\\in I}^\\infty H^{s_c}}\\\\\n&\\leq \\|e^{it(-\\Delta)^\\sigma}u_0\\|_{\\tilde{S}_{2,\\frac{2d}{d-2}}^{s_c}(I)}+c\\|u\\|_{\\tilde{S}_{2,\\frac{2d}{d-2}}^{s_c}(I)}^2\\|u\\|_{\\tilde{S}_{\\infty,2}^{s_c}(I)}^{p-2}.\n\\end{align*}\nSimilarly, one can show that\n\\begin{align*}\n\\|\\Phi_{u_0}(u)\\|_{\\tilde{S}_{\\infty,2}^{s_c}(I)}&\\leq c\\|u_0\\|_{H^{s_c}}+c\\|u\\|_{\\tilde{S}_{2,\\frac{2d}{d-2}}^{s_c}(I)}^2\\|u\\|_{\\tilde{S}_{\\infty,2}^{s_c}(I)}^{p-2}\n\\end{align*}\nand\n\\begin{align*}\n&\\|\\Phi_{u_0}(u)-\\Phi_{u_0}(v)\\|_{\\tilde{S}_{2,\\frac{2d}{d-2}}^0(I)}+\\|\\Phi_{u_0}(u)-\\Phi_{u_0}(v)\\|_{\\tilde{S}_{\\infty, 2}^0(I)}\\\\\n&\\leq c_0(\\|u\\|_{L_{t\\in I}^{p-1}L_x^\\infty}^{p-1}+\\|v\\|_{L_{t\\in I}^{p-1}L_x^\\infty}^{p-1})\\|u-v\\|_{L_{t\\in I}^\\infty L_x^2}\\\\\n&\\leq c(\\|u\\|_{\\tilde{S}_{2,\\frac{2d}{d-2}}^{s_c}(I)}^2\\|u\\|_{\\tilde{S}_{\\infty,2}^{s_c}(I)}^{p-2}+\\|v\\|_{\\tilde{S}_{2,\\frac{2d}{d-2}}^{s_c}(I)}^2\\|v\\|_{\\tilde{S}_{\\infty,2}^{s_c}(I)}^{p-2})\\|u-v\\|_{L_{t\\in I}^\\infty L_x^2}.\n\\end{align*}\nNow we let $\\delta=\\delta(c,\\|u_0\\|_{H^{s_c}})>0$ be a sufficiently small number to be chosen later, and then we pick $T=T(u_0,\\delta)>0$ such that\n$$\\|e^{it(-\\Delta)^\\sigma}u_0\\|_{\\tilde{S}_{2,\\frac{2d}{d-2}}^{s_c}(I)}\\leq\\delta,$$\nDefine \n$$B=\\Big\\{u: \\|u\\|_{\\tilde{S}_{2,\\frac{2d}{d-2}}^{s_c}(I)}\\leq 2\\delta\\textup{ and }\\|u\\|_{\\tilde{S}_{\\infty,2}^{s_c}(I)}\\leq 2c\\|u_0\\|_{H^{s_c}}\\Big\\}$$\nequipped with the norm\n$$\\|u\\|_X:=\\|u\\|_{\\tilde{S}_{2,\\frac{2d}{d-2}}^{0}(I)}+\\|u\\|_{\\tilde{S}_{\\infty,2}^{0}(I)}.$$\nThen, for $u\\in B$, we have\n\\begin{align*}\n\\|\\Phi_{u_0}(u)\\|_{\\tilde{S}_{2,\\frac{2d}{d-2}}^{s_c}(I)}&\\leq \\delta+c (2\\delta)^2(2c\\|u_0\\|_{{H}^{s_c}})^{p-2}\\leq 2\\delta,\\\\\n\\|\\Phi_{u_0}(u)\\|_{\\tilde{S}_{\\infty,2}^{s_c}(I)}&\\leq c\\|u_0\\|_{{H}^{s_c}}+c (2\\delta)^2(2c\\|u_0\\|_{{H}^{s_c}})^{p-2}\\leq 2c\\|u_0\\|_{{H}^{s_c}}.\n\\end{align*}\nChoosing sufficiently small $\\delta>0$, we prove that $\\Phi_{u_0}$ maps  $B$ to itself. Similarly, one can show \n$$\\|\\Phi_{u_0}(u)-\\Phi_{u_0}(v)\\|_X\\leq\\frac{1}{2}\\|u-v\\|_X.$$\nTherefore, it follows that $\\Phi_{u_0}$ is a contraction mapping in $B$.\n\\end{proof}\n\n\\begin{remark}\n$(i)$ In the proofs, the $L_x^\\infty$ norm bounds are crucial for the following reason. In Proposition \\ref{Strichartz}, there is a loss of regularity except the trivial ones,\n$$\\|e^{it(-\\Delta)^\\sigma}u_0\\|_{L_{t\\in I}^\\infty L_x^2}=\\|u_0\\|_{L^2}$$\nand\n$$\\Big\\|\\int_0^t e^{it(-\\Delta)^\\sigma}F(s)ds\\Big\\|_{L_{t\\in I}^\\infty L_x^2}\\leq \\|F\\|_{L_{t\\in I}^1L_x^2}.$$\nHence, when we estimate the $L_{t\\in I}^\\infty H_x^s$ norm of the integral term in $\\Phi_{u_0}(u)$, we are forced to use the trivial one\n\\begin{equation}\n\\Big\\|\\int_0^t e^{it(-\\Delta)^\\sigma}|u|^{p-1}u(s)ds\\Big\\|_{L_{t\\in I}^\\infty H_x^s}\\leq \\||u|^{p-1}u\\|_{L_{t\\in I}^1H_x^s}.\n\\end{equation}\nIndeed, otherwise, we have a higher regularity norm on the right hand side. Then, we cannot close the contraction mapping argument. Moreover, if $u_0\\in H^s$, there is no good bound for $\\|e^{it(-\\Delta)^\\sigma}u_0\\|_{L_{t\\in I}^qW_x^{s,r}}$ except the trivial one $(q,r)=(\\infty,2)$. Thus, we are forced to bound the right hand side of (3.10) by\n$$\\|u\\|_{L_{t\\in I}^{p-1}L_x^\\infty}^{p-1}\\|u\\|_{L_{t\\in I}^\\infty H_x^s}.$$\nTherefore, we should have a good control on $\\|u\\|_{L_{t\\in I}^{p-1}L_x^\\infty}$.\\\\\n$(ii)$ When $p<3$, the $L_{t\\in I}^{p-1}L_x^\\infty$ norm is scaling-supercritical. Thus, based on our method, the assumptions on $p$ in Theorem \\ref{subcritical LWP} and \\ref{critical LWP} are optimal except $p=3$ in the critical case.\n\\end{remark}\n\n\n\n\n\n\n\n\n\\section{Small Data Scattering}\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{scattering}]\nFor simplicity, we consider the case $d\\geq 3$ only. It follows from the estimates in the proof of Theorem \\ref{critical LWP} that if $\\|u_0\\|_{H^s}$ is small enough, then\n$$\\|u(t)\\|_{L_{t\\in\\mathbb{R}}^{p-1}L_x^\\infty}+\\|u(t)\\|_{L_{t\\in\\mathbb{R}}^\\infty H_x^{s_c}}\\leq\\|u(t)\\|_{\\tilde{S}_{2,\\frac{2d}{d-2}}^{s_c}(\\mathbb{R})}+\\|u(t)\\|_{\\tilde{S}_{\\infty,2}^{s_c}(\\mathbb{R})}\\lesssim\\|u_0\\|_{H^{s_c}}<\\infty.$$\nBy Strichartz estimates, the fractional chain rule and \\eqref{critical Lx-infty bound}, we prove that\n\\begin{align*}\n&\\|e^{-iT_1(-\\Delta)^\\sigma}u(T_1)-e^{-iT_2(-\\Delta)^\\sigma}u(T_2)\\|_{H^{s_c}}\\\\\n&=\\Big\\|\\int_{T_1}^{T_2} e^{-is(-\\Delta)^\\sigma}(|u|^{p-1}u)(s)ds\\Big\\|_{H^{s_c}}\\\\\n&\\leq \\|u(t)\\|_{L_{t\\in[T_1,T_2)}^{p-1}L_x^\\infty}^{p-1}\\|u(t)\\|_{L_{t\\in[T_1,T_2)}^\\infty H_x^{s_c}}\\to 0\n\\end{align*}\nas $T_1,T_2\\to\\pm\\infty$. Thus, the limits\n$$u_\\pm=\\lim_{t\\to\\pm\\infty} e^{-it(-\\Delta)^\\sigma}u(t)$$\nexist in $H^{s_c}$. Repeating the above estimates, we show that\n$$\\|u(t)-e^{it(-\\Delta)^\\sigma}u_\\pm\\|_{H^{s_c}}=\\|e^{-it(-\\Delta)^\\sigma}u(t)-u_\\pm\\|_{H^{s_c}}\\to 0$$ \nas $t\\to\\pm\\infty$.\n\\end{proof}\n\n\\section{Ill-posedness}\n\nWe will prove Theorem \\ref{ill-posedness} following the strategy in \\cite{CCT}. Throughout this section, we assume that $d=1,2$ or $3$ and $\\frac{d}{4}<\\sigma<1$. If $p$ is not an odd integer, we further assume that $p\\geq k+1$, where $k$ is the smallest integer greater than $\\frac{d}{2}$. \n\nFirst, we construct an almost non-dispersive solution by small dispersion analysis.\n\n\\begin{lemma}[Small dispersion analysis]\\label{small dispersion analysis}\nGiven a Schwartz function $\\phi_0$, let $\\phi^{(\\nu)}(t,x)$ be the solution to the fractional NLS\n\\begin{equation}\\label{small dispersion}\ni\\partial_t u+\\nu^{2\\sigma}(-\\Delta)^\\sigma u+\\mu|u|^{p-1}u=0,\\ u(0)=\\phi_0,\n\\end{equation}\nand $\\phi^{(0)}(t,x)$ be the solution to the ODE with no dispersion\n$$i\\partial_t u+\\mu|u|^{p-1}u=0,\\ u(0)=\\phi_0,$$\nthat is,\n\\begin{equation}\\label{no dispersion}\n\\phi^{(0)}(t,x)=\\phi_0(x)e^{it\\omega |\\phi_0(x)|^{p-1}}.\n\\end{equation}\nThen there exist $C, c>0$ such that if $0<\\nu\\leq c$ is sufficiently small, then\n\\begin{equation}\\label{small dispersion estimate}\n\\|\\phi^{(\\nu)}(t)-\\phi^{(0)}(t)\\|_{H^k}\\leq C\\nu^{2\\sigma}\n\\end{equation}\nfor all $|t|\\leq c |\\log \\nu|^c$.\n\\end{lemma}\n\n\\begin{proof}\nThe proof closely follows the proof of Lemma 2.1 in \\cite{CCT}.\n\\end{proof}\n\nObviously, $\\phi^{(\\nu)}(t,\\nu x)$ is a solution to $(\\textup{NLS}_\\sigma)$. Moreover, $\\phi^{(\\nu)}(t,\\nu x)$ is bounded and almost flat in the following sense.\n\n\\begin{corollary}\\label{small dispersion corollary}\nLet $\\phi^{(\\nu)}$, $\\nu$ and $c$ be in Lemma \\ref{small dispersion analysis}. Let $s\\geq 0$. Then,\n\\begin{equation}\\label{L^infty control}\n\\|\\phi^{(\\nu)}(t,\\nu x)\\|_{L_x^\\infty}\\sim 1\n\\end{equation}\nand\n\\begin{equation}\\label{H^s control}\n\\|\\phi^{(\\nu)}(t,\\nu x)\\|_{\\dot{H}_x^s}\\sim \\nu^{s-\\frac{d}{2}}(c |\\log \\nu|^c)^s\n\\end{equation}\nfor all $|t|\\leq c |\\log \\nu|^c$.\n\\end{corollary}\n\n\\begin{proof}\nSince $k>\\frac{d}{2}$, by the Sobolev inequality, we have\n\\begin{align*}\n\\|\\phi^{(\\nu)}(t,\\nu x)-\\phi^{(0)}(t,\\nu x)\\|_{L_x^\\infty}&=\\|\\phi^{(\\nu)}(t, x)-\\phi^{(0)}(t,x)\\|_{L_x^\\infty}\\\\\n&\\lesssim\\|\\phi^{(\\nu)}(t)-\\phi^{(0)}(t)\\|_{H^k}\\lesssim\\nu^{2\\sigma}.\n\\end{align*}\nThen, \\eqref{L^infty control} follows from the explicit formula \\eqref{no dispersion} for $\\phi^{(0)}(t,x)$. \nIt follows from \\eqref{small dispersion estimate} and \\eqref{no dispersion} that\n$$\\|\\phi^{(\\nu)}(t, \\nu x)\\|_{\\dot{H}_x^s}\\leq \\nu^{s-\\frac{d}{2}}(\\|\\phi^{(0)}(t)\\|_{\\dot{H}^s}+\\|\\phi^{(\\nu)}(t)-\\phi^{(0)}(t)\\|_{\\dot{H}^s})\\sim \\nu^{s-\\frac{d}{2}}(c |\\log \\nu|^c)^s.$$\n\\end{proof}\n\nFor $v\\in\\mathbb{R}^d$, we define the pseudo-Galilean transformation by\n$$\\mathcal{G}_vu(t,x)=e^{-iv\\cdot x}e^{it|v|^{2\\sigma}}u(t,x-2t\\sigma|v|^{2(\\sigma-1)}v).$$\nNote that when $\\sigma=1$, $\\mathcal{G}_v$ is simply a Galilean transformation, and that NLS is invariant under this transformation, that is, if $u(t)$ solves NLS, so does $\\mathcal{G}_vu(t)$. However, when $\\sigma\\neq1$, $(\\textup{NLS}_\\sigma)$ is not exactly symmetric with respect to pseudo-Galilean transformations. Indeed, if $u(t)$ solves $(\\textup{NLS}_\\sigma)$, then $\\tilde{u}(t)=\\mathcal{G}_vu(t)$ obeys $(\\textup{NLS}_\\sigma)$ with an error term\n\\begin{equation}\\label{fNLS with error}\ni\\partial_t\\tilde{u}+(-\\Delta)^\\sigma\\tilde{u}+\\omega|\\tilde{u}|^{p-1}\\tilde{u}=e^{it|v|^{2\\sigma}}e^{-iv\\cdot x}(\\mathcal{E}u)(t,x-2\\sigma t|v|^{2(\\sigma-1)}v),\n\\end{equation}\nwhere\n$$\\widehat{\\mathcal{E}u}(\\xi)=E(\\xi)\\hat{u}(\\xi)$$\nwith \n$$E(\\xi)=|\\xi-v|^{2\\sigma}-|\\xi|^{2\\sigma}-|v|^{2\\sigma}+2\\sigma|v|^{2(\\sigma-1)}v\\cdot\\xi.$$\nHowever, we note that\n\\begin{equation}\\label{E bound}\n|E(\\xi)|\\lesssim |\\xi|^{2\\sigma}.\n\\end{equation}\nIndeed, if $|\\xi|\\leq\\frac{|v|}{100}$, then\n$$E(\\xi)=\\Big||v|^{2\\sigma}\\Big(|\\tfrac{v}{|v|}-\\tfrac{\\xi}{|v|}|^{2\\sigma}-1+2\\sigma \\tfrac{v}{|v|}\\cdot\\tfrac{\\xi}{|v|}\\Big)-|\\xi|^{2\\sigma}\\Big|\\lesssim|v|^{2\\sigma}\\tfrac{|\\xi|^2}{|v|^2}+|\\xi|^{2\\sigma}\\lesssim|\\xi|^{2\\sigma}.$$\nOtherwise, \n$$E(\\xi)\\lesssim|\\xi|^{2\\sigma}+|v|^{2\\sigma}+|\\xi|^{2\\sigma}+|v|^{2\\sigma}+2\\sigma|v|^{2\\sigma-1}|\\xi|\\lesssim|\\xi|^{2\\sigma}.$$\nTherefore, one would expect an \\textit{almost} symmetry for an almost flat solution $u(t)$, such as $\\phi^{(\\nu)}(t,\\nu x)$ in Lemma \\ref{small dispersion analysis}. Precisely, we have the following lemma.\n\n\\begin{lemma}[Pseudo-Galilean transformation]\\label{Pseudo-Galilean transformation}\nLet $\\phi^{(\\nu)}$, $\\nu$ and $c$ be in Lemma \\ref{small dispersion analysis}. For $v\\in\\mathbb{R}^d$, we define\n$$\\tilde{u}(t,x)=(\\mathcal{G}_v\\phi^{(\\nu)}(\\cdot, \\nu\\cdot))(t,x)=e^{-iv\\cdot x}e^{it|v|^{2\\sigma}}\\phi^{(\\nu)}\\big(t,\\nu(x-2t\\sigma|v|^{2(\\sigma-1)}v)\\big),$$\nand let $u(t,x)$ be the solution to $(\\textup{NLS}_\\sigma)$ with the same initial data\n\\begin{equation}\\label{initial data}\ne^{-iv\\cdot x}\\phi^{(\\nu)}(0,\\nu x)=e^{-iv\\cdot x}\\phi_0(0,\\nu x).\n\\end{equation}\nThen, there exists $\\delta>0$ such that\n\\begin{equation}\\label{estim}\n\\|e^{iv\\cdot x}(u(t)-\\tilde{u}(t))\\|_{H_x^k}\\lesssim \\nu^{\\delta}\n\\end{equation}\nfor all $|t|\\leq c |\\log \\nu|^c$.\n\\end{lemma}\n\n\\begin{remark}\nWhen $p=3$, in \\cite{CHKL} the authors could use the counterexample in  \n\\cite{CCTamer}. This counterexample is constructed by  \npseudo-conformal symmetry and Galilean transformation. A good thing is  \nthat this solution is very small in high Sobolev norms, too. Somehow,  \nthis smallness allows \\cite{CHKL} to show that the error in pseudo-Galilean  \ntransformation is also small. However, when $p>3$, the counterexample in  \n\\cite{CCTamer} does not work. Later, Christ, Colliander and Tao \\cite{CCT} constructed  \na different counterexample which works for more general $p$.  \nUnfortunately, this counterexample is not small in high Sobolev norms.  \nIt is very large instead. In particular, for our purposes, it is hard to control the error from  \npseudo-Galilean transformation. But, our new  \ncounterexample still has small high Sobolev norm after translating it  \nto its frequency center; this is the term $e^{iv\\cdot x}$ in equation \\eqref{estim}. Using this smallness, we can prove that pseudo-Galilean  \ntransformation is almost invariant. We also remark that the condition $\\sigma>\\frac{d}{4}$ is to guarantee smallness of the error (see \\eqref{II(s) estimate}).\n\\end{remark}\n\n\\begin{proof}[Proof of Lemma \\ref{Pseudo-Galilean transformation}]\nLet $R(t)=(u-\\tilde{u})(t)$. Then, $R(t)$ satisfies\n$$i\\partial_tR+(-\\Delta)^\\sigma R=\\mu\\big(|\\tilde{u}|^{p-1}\\tilde{u}-|u|^{p-1}u\\big)-e^{it|v|^{2\\sigma}}(\\mathcal{E}\\phi^{(\\nu)}\\big(t,\\nu(x-2\\sigma t|v|^{2(\\sigma-1)}v)\\big),$$\nor equivalently\n\\begin{align*}\nR(t)&=i\\int_0^t e^{i(t-s)(-\\Delta)^{\\sigma}}\\Big\\{\\mu\\big(|u|^{p-1}u-|\\tilde{u}|^{p-1}\\tilde{u}\\big)(s)\\\\\n&\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad+e^{is|v|^{2\\sigma}}(\\mathcal{E}\\phi^{(\\nu)}\\big(s,\\nu(x-2\\sigma s|v|^{2(\\sigma-1)}v)\\big)\\Big\\}ds.\n\\end{align*}\nHence, by a trivial estimate, we get\n\\begin{align*}\n\\|e^{iv\\cdot x}R(t)\\|_{H^k}&\\leq\\int_0^t \\big\\|e^{iv\\cdot x}\\big(|u|^{p-1}u-|\\tilde{u}|^{p-1}\\tilde{u}\\big)(s)\\big\\|_{H^k}+\\|\\mathcal{E}\\phi^{(\\nu)}(s,\\nu\\cdot)\\|_{H^k}ds\\\\\n&=\\int_0^t I(s)+II(s)ds.\n\\end{align*}\nFirst, by \\eqref{E bound} and \\eqref{H^s control}, we show that\n\\begin{equation}\\label{II(s) estimate}\n\\int_0^tII(s)ds\\lesssim \\int_0^t\\sum_{j=0}^k\\|\\phi^{(\\nu)}(s,\\nu\\cdot)\\|_{\\dot{H}^{j+2\\sigma}}ds\\sim (c |\\log \\nu|^c)^{1+2\\sigma-\\frac{d}{2}}\\nu^{2\\sigma-\\frac{d}{2}}.\n\\end{equation}\nFor $I(s)$, expanding $u=\\tilde{u}+R$ and then applying H\\\"older inequality and Sobolev inequalities, we bound $I(s)$ by\n\\begin{equation}\\label{I(s) bound}\n\\lesssim\\sum_{j=1}^p\\|e^{iv\\cdot x}R\\|_{H^k}^j.\n\\end{equation}\nFor example, when $p=3$,\n\\begin{align*}\nI(s)&\\leq2\\| |e^{iv\\cdot x}\\tilde{u}|^2e^{iv\\cdot x}R\\|_{H^k}+\\|(e^{iv\\cdot x}\\tilde{u})^2\\overline{e^{iv\\cdot x}R}\\|_{H^k}+2\\|e^{iv\\cdot x}\\tilde{u}|e^{iv\\cdot x}R|^2\\|_{H^k}\\\\\n&\\quad+\\|\\overline{e^{iv\\cdot x}\\tilde{u}}(e^{iv\\cdot x}R)^2\\|_{H^k}+\\||e^{iv\\cdot x}R|^2e^{iv\\cdot x}R\\|_{H^k}\\\\\n&=:I_1(s)+I_2(s)+I_3(s)+I_4(s)+I_5(s).\n\\end{align*}\nConsider\n$$I_1(s)=\\sum_{|\\alpha|\\leq k}\\|\\nabla_{x_1}^{\\alpha_1}\\cdots\\nabla_{x_d}^{\\alpha_d}(|e^{iv\\cdot x}\\tilde{u}|^2e^{iv\\cdot x}R)(s)\\|_{L^2}=:\\sum_{|\\alpha|\\leq k}I_{1,\\alpha}(s),$$\nwhere $\\alpha=(\\alpha_1,\\alpha_2,\\cdots,\\alpha_d)$ is a multi-index with $|\\alpha|=\\sum_{i=1}^d\\alpha_i$. Observe that whenever a derivative hits\n$$e^{iv\\cdot x}\\tilde{u}(s)=e^{is|v|^{2\\sigma}}\\phi^{(\\nu)}\\big(s,\\nu(x-2s\\sigma|v|^{2(\\sigma-1)}v)\\big),$$\nwe get a small factor $\\nu$. Hence, after distributing derivatives by the Leibniz rule,  the worst term we have in $I_{1,\\alpha}(s)$ is\n$$\\| |e^{iv\\cdot x}\\tilde{u}(s)|^2\\nabla^\\alpha e^{iv\\cdot x}R(s)\\|_{L^2},$$\nwhich is, by \\eqref{L^infty control}, bounded by \n$$\\| e^{iv\\cdot x}\\tilde{u}(s)\\|_{L^\\infty}^2\\|\\nabla^\\alpha e^{iv\\cdot x}R(s)\\|_{L^2}\\sim \\|\\nabla^\\alpha e^{iv\\cdot x}R(s)\\|_{L^2}.$$\nLikewise, we estimate other terms.\n\nCollecting all,\n$$\\|e^{iv\\cdot x}R(t)\\|_{H^k}\\lesssim (c |\\log \\nu|^c)^{1+2\\sigma-\\frac{d}{2}}\\nu^{2\\sigma-\\frac{d}{2}}+\\int_0^t \\sum_{j=1}^p\\|e^{iv\\cdot x}R(s)\\|_{H^k}^jds$$\nfor $|t|\\leq c|\\log \\nu|^c$. Then, by the standard nonlinear iteration argument, we prove the lemma.\n\\end{proof}\n\nSince we have solutions almost symmetric with respect to the pseudo-Galilean transformations, we can make use of the following decoherence lemma to construct counterexamples for local well-posedness.\n\n\\begin{lemma}[Decoherence]\\label{Decoherence}\nLet $s<0$. Fix a nonzero Schwartz function $w$. For $a,a'\\in[\\frac{1}{2},1]$, $0<\\nu\\leq\\lambda\\ll 1$ and $v\\in\\mathbb{R}^d$ with $|v|\\geq 1$, we define \n$$\\tilde{u}^{(a,\\nu,\\lambda,v)}(t,x):=\\mathcal{G}_v\\Big(\\lambda^{-\\frac{2\\sigma}{p-1}}\\phi^{(a,\\nu)}(\\lambda^{-2\\sigma}\\cdot,\\lambda^{-1}\\nu\\cdot)\\Big)(t,x),$$\nwhere $\\phi^{(a,\\nu)}$ is the solution to \\eqref{small dispersion} with initial data $aw$. Then, we have\n\\begin{align*}\n\\|\\tilde{u}^{(a,\\nu,\\lambda, v)}(0)\\|_{H^s}, \\|\\tilde{u}^{(a',\\nu,\\lambda, v)}(0)\\|_{H^s}&\\leq C|v|^s\\lambda^{-\\frac{2\\sigma}{p-1}}(\\tfrac{\\lambda}{\\nu})^{d\/2},\\\\\n\\|\\tilde{u}^{(a,\\nu,\\lambda, v)}(0)-\\tilde{u}^{(a',\\nu,\\lambda, v)}(0)\\|_{H^s}&\\leq C|v|^s\\lambda^{-\\frac{2\\sigma}{p-1}}(\\tfrac{\\lambda}{\\nu})^{d\/2}|a-a'|\n\\end{align*}\nand \n\\begin{align*}\n&\\|\\tilde{u}^{(a,\\nu,\\lambda, v)}(t)-\\tilde{u}^{(a',\\nu,\\lambda, v)}(t)\\|_{H^s}\\\\\n&\\geq c|v|^s\\lambda^{-\\frac{2\\sigma}{p-1}}(\\tfrac{\\lambda}{\\nu})^{d\/2}\\Big\\{\\|(\\phi^{(a,\\nu)}(\\tfrac{t}{\\lambda^{2\\sigma}})-\\phi^{(a',\\nu)}(\\tfrac{t}{\\lambda^{2\\sigma}})\\|_{L^2}-C|\\log \\nu|^C(\\tfrac{\\lambda}{\\nu})^{-k}|v|^{-s-k}\\Big\\}\n\\end{align*}\nfor all $|t|\\leq c|\\log \\nu|^c\\lambda^{2\\sigma}$.\n\\end{lemma}\n\n\\begin{proof}\nThe proof closely follows the proof of Lemma 3.1 in \\cite{CCT}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{ill-posedness}]\nThe proof is very similar to that of Theorem 1 in \\cite{CCT} except that in the last step, we need to use Lemma \\ref{Pseudo-Galilean transformation} due to lack of exact symmetry. We give a proof for the readers' convenience.\n\nLet $\\epsilon>0$ be a given but arbitrarily small number. Let $\\nu=\\lambda^{\\alpha}$, where $\\alpha>0$ is a small number to be chosen later. Then, we pick $v\\in \\mathbb{R}^d$ such that\n$$\\lambda^{-\\frac{2\\sigma}{p-1}}|v|^s(\\lambda\/\\nu)^{d\/2}=\\epsilon\\Leftrightarrow |v|=\\nu^{\\frac{1}{s}(\\frac{d(1-\\alpha)}{2}+\\frac{2\\alpha\\sigma}{p-1})}\\epsilon^{1\/s}.$$\nNote that since $s<0$, $\\frac{1}{s}(\\frac{d(1-\\alpha)}{2}+\\frac{2\\alpha\\sigma}{p-1})=\\frac{1}{s}(\\frac{d}{2}-\\alpha s_c)<0$ for sufficiently small $\\alpha$, and thus $|v|\\geq 1$. Hence, it follows from Lemma \\ref{Decoherence} that\n \\begin{align}\n\\|\\tilde{u}^{(a,\\nu,\\lambda, v)}(0)\\|_{H^s}, \\|\\tilde{u}^{(a',\\nu,\\lambda, v)}(0)\\|_{H^s}&\\leq C\\epsilon,\\\\\n\\|\\tilde{u}^{(a,\\nu,\\lambda, v)}(0)-\\tilde{u}^{(a',\\nu,\\lambda, v)}(0)\\|_{H^s}&\\leq C\\epsilon|a-a'|,\n\\end{align}\nand \n\\begin{align*}\n&\\|\\tilde{u}^{(a,\\nu,\\lambda, v)}(t)-\\tilde{u}^{(a',\\nu,\\lambda, v)}(t)\\|_{H^s}\\\\\n&\\geq c\\epsilon\\Big\\{\\|(\\phi^{(a,\\nu)}(\\tfrac{t}{\\lambda^{2\\sigma}})-\\phi^{(a',\\nu)}(\\tfrac{t}{\\lambda^{2\\sigma}})\\|_{L^2}-C|\\log \\nu|^C(\\tfrac{\\lambda}{\\nu})^{-k}|v|^{-s-k}\\Big\\}\n\\end{align*}\nfor all $|t|\\leq c|\\log \\nu|^c\\lambda^{2\\sigma}$. Now we observe from the explicit formula \\eqref{no dispersion} for $\\phi^{(a,0)}$ and \\eqref{small dispersion} that there exists $T>0$ such that $\\|\\phi^{(a,\\nu)}(T)-\\phi^{(a',\\nu)}(T)\\|_{L^2}\\geq c$. Moverover, if $\\alpha>0$ is sufficiently small, $C|\\log \\nu|^C(\\tfrac{\\lambda}{\\nu})^{-k}|v|^{-s-k}\\to 0$ as $\\nu \\to0$. Therefore, for $\\nu$ small enough, we have\n\\begin{equation}\n\\|\\tilde{u}^{(a,\\nu,\\lambda, v)}(\\lambda^{2\\sigma}T)-\\tilde{u}^{(a',\\nu,\\lambda, v)}(\\lambda^{2\\sigma}T)\\|_{H^s}\\geq c\\epsilon.\n\\end{equation}\nNext, we replace $\\tilde{u}^{(a,\\nu,\\lambda, v)}$ and $\\tilde{u}^{(a',\\nu,\\lambda, v)}$ in $(6.11)$, $(6.12)$ and $(6.13)$ by $u^{(a,\\nu,\\lambda, v)}$ and $u^{(a',\\nu,\\lambda, v)}$ by Lemma \\ref{Pseudo-Galilean transformation} with $O(\\nu^{\\delta})$ erorr. Then, making $|a-a'|$ arbitrarily small and then sending $\\nu\\to 0$ (so, $\\lambda^{2\\sigma}T\\to 0$), we complete the proof.\n\\end{proof}\n\n\\section*{Acknowledgements}\nY.H. would like to thank IH\\'ES for their hospitality and support while he visited in the summer of 2014. Y.S. would like to thank the hospitality of the Department of Mathematics at University of Texas at Austin where part of the work was initiated. Y.S. acknowledges the support of ANR grants \"HAB\" and \"NONLOCAL\".  \n\n\\bibliographystyle{alpha} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe study of learning systems with concepts borrowed from statistical mechanics and thermodynamics has a long history reaching back to Maxwell's demon and the ensuing debate on the relation between physics and information \\cite{parrondo2015thermodynamics}. Over the last 20 years, the informational view of thermodynamics has experienced great developments, which has allowed to broaden its scope form equilibrium to non-equilibrium phenomena \\cite{jarzynski2011equalities,de2013non}. Of particular importance are the so-called fluctuation theorems \\cite{seifert2012stochastic,jarzynski2000hamiltonian,crooks1999entropy}, which relate equilibrium quantities to non-equilibrium trajectories allowing, thus, to approximate equilibrium quantities via experimental realizations of non-equilibrium processes \\cite{ytreberg2004efficient,park2003free}. Among the fluctuation theorems, two results stand out, Jarzynski's equality \\cite{jarzynski1997equilibrium,cohen2004note,jarzynski2004nonequilibrium} and Crooks' fluctuation theorem \\cite{crooks1998nonequilibrium,crooks2000path}, as they aim to bridge the apparent chasm between reversible microscopic laws \nand irreversible macroscopic phenomena \\cite{loschmidt1876ueber}.\n\nThe advances in non-equilibrium thermodynamics have recently also led to new theoretical insights into simple learning systems \\cite{goldt2017stochastic,perunov2016statistical,england2015dissipative,still2012thermodynamics,ortega2013thermodynamics,grau2018non}.\nAbstractly, thermodynamic quantities like energy, entropy or free energy can be thought to define order relations between states \\cite{lieb1991,gottwald2019}, which makes them applicable to a wide range of problems. In the economic sciences, for example, such order relations are typically used to define a decision-maker's preferences over states \\cite{mascollel1995}. Accordingly, a decision-maker or a learning system can be thought to maximize a utility function, analogous to a physical system that aims to minimize an energy function.\nMoreover, in the presence of uncertainty in stochastic choice, such decision-makers can be thought to operate under entropy constraints reflecting the decision-maker's precision \\cite{ortega2013thermodynamics,parrondo2015thermodynamics}, resulting in soft-maximizing the corresponding utility function instead of perfectly maximizing it. This is formally equivalent to following a Boltzmann distribution with energy given by the utility. Therefore, in this picture, the physical concept of work corresponds to utility changes caused by the environment, whereas the physical concept of heat corresponds to utility gains due to internal adaptation \\cite{still2012thermodynamics}. Like a thermodynamic system is driven by work, such learning systems are driven by changes in the utility landscape (e.g. changes in an error signal). By exposing learning systems to varying environmental conditions, it has been hypothesized that adaptive behavior can be studied in terms of fluctuation theorems \\cite{grau2018non,england2015dissipative}, which are not necessarily tied to physical processes but are broadly applicable to stochastic processes satisfying certain constraints \\cite{hack2022jarzyskis}.\n\nAlthough fluctuation theorems have been empirically observed in numerous experiments in the physical sciences \\cite{douarche2005experimental,collin2005verification,saira2012test,liphardt2002equilibrium,an2015experimental,smith2018verification},  there have been no reported experimental results relating fluctuation theorems to adaptive behavior in humans or other living beings. Here, we test Jarzynski's equality and Crooks'  fluctuation theorem experimentally in a human sensorimotor adaptation task.\nIn this context, the fluctuation theorem establishes a linear relationship between the externally imposed utility changes driving the learning process (which are directly related to non-predicted information and energy dissipation \\cite{still2012thermodynamics}) and the log-probability ratio between forward and backward adaptation trajectories, when exposing participants to the sequence of environments either in the forward or reverse order. Accordingly, such learners can be quantitatively characterized by a hysteresis effect that can also be observed in simple physical systems.\n\n\\section{Results}\n\\label{sect:results}\n\nIn a visuomotor adaptation task, human participants controlled a cursor on a screen towards a single stationary target by moving a mechanical manipulandum that was obscured from their vision under an overlaid screen---see Figure~\\ref{methods exp}\\textbf{A}. Crucially, in each trial $n$, the  position  of  the  cursor  could be rotated with angle $\\theta_n$ relative to the actual hand position so that participants had to adapt when moving the cursor from the start position to the target. To measure participants' adaptive state, we recorded their movement position at the time of crossing a certain distance from the start position, so that  their response could be characterized by an angle $x_n$. \nThe deviation between participants' response and the required movement incurs a sensorimotor loss \\cite{kording2004loss}, that can be quantified as an exponential quadratic error \n\\begin{equation}\n\\label{utility}\n E_n(x)= 1 - e^{- (x-(\\theta_n+b))^2},\n\\end{equation}\nwhere $\\theta_n$ is the true rotation angle set by the trial $n$ and $b$ is a participant-specific parameter for the bias, respectively---see Figure \\ref{methods exp}\\textbf{D}. This loss is taken to be the energy (or negative utility) of a participant's stochastic response $X_n = x_n$. Therefore the pointing behavior after a suitably long adaption time can be described by a Boltzmann equilibrium distribution $p_n^{eq}$ of the form\n\\begin{equation}\n\\label{Boltzmann}\n    p_{n}^{eq}(x_n) =  \\exp\\big(-\\beta( E_n(x_n) - F_n )\\big),\n\\end{equation}\nfor all $x_n\\in A_n$, where the sensorimotor error $E_n(x_n)$ plays the role of an energy, the free energy term $F_n = \\frac{1}{\\beta} \\log \\int_{A_n} \\exp\\left( -\\beta E_n(x_n) \\right) dx_n$ is caused by the normalization, and $A_n$ is the support of the equilibrium distribution $p_{n}^{eq}$, which will vary for each participant, as we explain in Section \\ref{exp design}. Moreover, the softness-parameter $\\beta$, also known as \\textit{inverse temperature}, controls the trade-off between entropy maximization and energy minimization, essentially interpolating between a purely stochastic choice ($\\beta = 0$) and a purely rational choice ($\\beta \\to \\infty$) minimizing the energy perfectly. \n\n\\begin{figure}[!tb]\n\\centering\n    \\includegraphics[scale=.32]{lastI.jpg}\n    \\caption{\\textbf{A} Schematic representation of an experimental trial with deviation angle $\\theta$. The dotted line represents the participant's hand movement and the continuous line represents the rotated movement observed on the screen. \\textbf{B} Experimental protocol. The continuous line represents the deviation angles $\\theta$ imposed during the first experimental cycle, where trials 1 to 25 constitute the forward process and trials 34 to 58 constitute the backward process. The dotted line represents the second cycle. \\textbf{C} Illustration of the equilibrium distributions \\eqref{Boltzmann} with $b,\\theta_n=0$ resulting from the exponential quadratic error \\eqref{utility} and, respectively, $\\beta=1,1.5,2$.\n    The shaded area represents the target, which tolerates, at most, an error of $2 \\degree$. \\textbf{D} As participants have to equilibrate between forward and backward protocol, we compare their performance in the $0 \\degree$ plateaus between protocols with the equilibrium distribution recorded before the start of the protocol,  here shown exemplarily for participant 7 (red: normalized error histogram for the in-between plateaus,  green: equilibrium histogram). The same comparison for each participant can be found in Figure \\ref{plateaus all}.}\n    \\label{methods exp}\n\\end{figure}\n\nThe task consisted in a sequence of target reaching trials, where the rotation angle $\\theta_n$ changed from one trial $n$ to the next trial $n+1$ according to a given up-down protocol---see Figure~\\ref{methods exp}\\textbf{B}---, so that participants' responses could be represented by a trajectory  $\\vect{x}=(x_0,x_1,..,x_N)$.\nWhen the environment is changing over many time steps, we can distinguish error changes $\\Delta E_{ext}(\\vect{x}) \\coloneqq \\sum_{n=0}^{N-1} (E_{n+1}(x_n)-E_{n}(x_n))$ that are induced externally by changes in the environment, from error changes $\\Delta E_{int}(\\vect{x}) \\coloneqq \\sum_{n=1}^{N} (E_{n}(x_n)-E_{n}(x_{n-1}))$ due to internal adaptation when changing from $x_{n-1}$ to $x_{n}$. Crucially, it is exactly the externally induced changes in error, $\\Delta E_{ext}(\\vect{x})$, analogous to the physical concept of work, that drive the adaptation process: if $\\Delta E_{ext}(\\vect{x})$ is large, the system is more surprised and has to adapt more. In the following, we thus refer to $\\Delta E_{ext}(\\vect{x})$ as \\emph{driving error} or \\emph{driving signal}. When applying Crooks' fluctuation theorem for general adaptive systems \\cite{hack2022jarzyskis} to the above setting, we obtain the linear relation\n\\begin{equation}\n\\label{prediction}\n    \\Delta E_{ext}(\\vect{x}) - \\Delta F = \\frac{1}{\\beta}\\log \\left( \\frac{\\rho^F(\\Delta E_{ext}(\\vect{x}))}{\\rho^B(-\\Delta E_{ext}(\\vect{x}))} \\right),\n\\end{equation}\nwhere $\\Delta F$ denotes the free energy difference $F_N-F_0$, and $\\rho^F$ and $\\rho^B$ are the learner's probability densities over possible driving errors after sequentially exposing the learning system to the $N+1$ environments in forward and reverse order, respectively. In equation \\eqref{prediction}, these densities are evaluted at the actual driving errors $\\Delta E_{ext}(\\vect{x})$ and $-\\Delta E_{ext}(\\vect{x})$, respectively, for a particular adaptive trajectory $\\vect{x}$.\n \n\n\nA direct consequence of \\eqref{prediction} is  Jarzynski's equality \\cite{crooks1998nonequilibrium}, which states that\n\\begin{equation}\n\\label{prediction II}\n    \\big\\langle e^{-\\beta \\Delta E_{ext}(\\vect{X})}\\big\\rangle = e^{-\\beta \\Delta F},\n\\end{equation}\nwhere $\\langle ~ \\cdot ~ \\rangle \\coloneqq \\mathbb E[~\\cdot~]$ denotes the expectation operator, considering $\\vect{X} = (X_n)_{n=0}^N$ a Markov chain with transition densitites $\\Pi_n$ that have $p^{eq}_n$ as stationary distributions. In our experiment, $\\vect{X}$ represents participants' responses that are repeated over multiple repetitions of the forward-backward protocol.    \nIn the following, we will test the relationships \\eqref{prediction} and \\eqref{prediction II} experimentally with $\\Delta F = 0$ as our human learners start and end in the same environmental state.\n\n\nIn our experiment the task is divided into 20 cycles of 66 trials each, following the protocol  \\eqref{protocol}  illustrated in  Figure \\ref{methods exp}\\textbf{B}. We refer to trials 1 to 25 of each cycle as a realization of the \\emph{forward process} and trials 34 to 58 as a realization of the \\emph{backward process}. Notice the backward process consists of the same angles as the forward process, that is, the same utility functions, but in reversed order. Thus, we record for each participant 20 values for $\\Delta E_{ext}(\\vect{x})$ in both the forward and backward processes\nthat we use to estimate participants' probability densities of the forward and backward processes, $\\rho^F$ and $\\rho^B$, respectively, using kernel density estimation. As the amount of data is limited to test the linear relation in \\eqref{prediction}, we will use simulation results in the following to compare against participants' behavior.\n\n\n\\begin{figure}[!tb]\n\\centering\n    \\includegraphics[scale=.2]{fig-2-last-IV.jpg}\n    \\caption{Simulation of Crooks' fluctuation theorem. \\textbf{A} Simulation with 1000 cycles. In black, the theoretical prediction; in red, the linear regression for the simulated data and, in green, the simulated points. Since the simulated data set adjusts pretty well to Crooks' fluctuation theorem \\eqref{prediction}, Jarzynski's equality \\eqref{prediction II} is fulfilled. \\textbf{B} Simulation with 20 cycles and bootstrapping. The black line is the theoretical prediction \\eqref{prediction} while the red line and shaded area are, respectively, the mean and the 99 \\% confidence interval of \\eqref{prediction} after 1000 bootstraps of the driving error values obtained in a single run (which consists of 20 cycles).}\n    \\label{2 simus}\n\\end{figure}\n\nWhen simulating an artificial decision-maker based on a stochastic optimization scheme with Markovian dynamics, for example a Metropolis-Hasting algorithm with target distribution $p_n^{eq}\\propto \\exp(-\\beta E_n)$, it is clear that we can recover the linear relationship \\eqref{prediction}, provided that sufficient samples are collected \\cite{hack2022jarzyskis}---see, for example, a simulation with 1000 cycles in Figure \\ref{2 simus}\\textbf{A} where we can see a good adjustment between the theoretical prediction (in black) and the linear regression of the observed data (in red). As a result, \\eqref{prediction II} also holds in this scenario. The more critical question is what happens when only few samples are available. To this end, we use the stochastic optimization algorithm to simulate the protocol of our experiment, that is, 20 cycles, and indicate confidence intervals using 1000 bootstraps. It can be seen in Figure \\ref{2 simus}\\textbf{B} that the theoretical prediction is consistent with the $99\\%$ confidence interval in the region where $|\\Delta E_{\\textrm{ext}}| \\leq 4$ (which is the region where our experimental data lies).\nUsing the same bootstrapped data, we obtain several estimates of $\\langle e^{- \\Delta E_{ext}(\\vect{X})}\\rangle$ (the mean of $e^{- \\Delta E_{ext}(\\vect{X})}$ for the observed values of $\\Delta E_{ext}(\\vect{X})$ at each bootstrap) which we use to calculate a confidence interval for it. This results in the $99\\%$ confidence interval for $\\langle e^{-\\Delta E_{ext}(\\vect{X})}\\rangle$ being $(0.48,\\text{ }1.64)$, which is consistent with the theoretical prediction \\eqref{prediction II} for $\\Delta F = 0$.  Accordingly, we will expect a similar behavior for our experimental data. Note we take, for simplicity, $b=0$, $\\beta=1$ and, for all $n$, $A_n=[-90,90]$ in these simulations.\n\n\n\n\\begin{comment}\n\\begin{figure}[!tb]\n\\centering\n    \\includegraphics[scale=.35]{long-simu2.eps}\n    \\caption{Linear relation long simulation with 1000 cycles. In black, the theoretical prediction; in red, the linear regression for the simulated data and, in blue, the simulated points. Since the simulated dataset adjusts pretty well to Crooks' fluctuation theorem, Jarzynski's equality is fulfilled.}\n    \\label{long simu}\n\\end{figure}\n\n\\begin{figure}[!tb]\n\\centering\n    \\includegraphics[scale=.8]{short.simu-boot2.eps}\n    \\caption{Linear relation short simulation. In black, the theoretical line; in blue, two thin lines stand for the boundaries of the 99.7 \\% confidence interval after 1000 bootstraps of the driving error values obtained in one run of the short simulation and a thick line stands for the mean of these bootstraps. (we should color between the lines)}\n    \\label{short simu}\n\\end{figure}\n\\end{comment}\n\n\\begin{figure}[!tb]\n\\centering\n   \n    \\includegraphics[scale=0.21]{lastIII-II.jpg}\n    \\caption{Hysteresis effect. The filled triangles are the mean of the observed angles for every deviation in both the forward process, in green, and the backward process, in red. The black line is the forward protocol. Participants that achieve at least $50\\%$ adaptation are shaded by a green background color. Hysteresis can be observed between trials 1 and 5, 9 and 17 and 21 and 25. Notice,  as expected, the forward means are below the backward in the first region, above in the second and below again in the third.}\n    \\label{hyste plot}\n\\end{figure}\n\nParticipants' average adaptive responses can be seen in Figure \\ref{hyste plot} compared to the experimentally imposed true parameter values (the trial-by-trial responses can be seen in Figure \\ref{forward all}). The green and red lines distinguish the forward and backward trajectories, respectively, so that, from the contrast between the two curves, hysteresis becomes apparent, as common in simple physical systems \\cite{jarzynski2011equalities} and as reported previously in similar experiments for sensorimotor adaptation \\cite{turnham2012facilitation}. Participants that achieve at least $50\\%$ adaptation are shaded by a green background color and are our participants of interest. The three participants that fail to achieve this minimum adaptation level are marked by a red shade. Instead of excluding these participants entirely from the analysis, we keep them in to show the contrast to the well-adapted participants and to highlight that the results reported for the well-adapted participants do not hold trivially for any participant producing inconsistent behavior.\n\nFigure \\ref{together} shows participants' data compared to the theoretical prediction from \\eqref{prediction} and the 99 \\% confidence interval after 1000 bootstraps as in the case of the simulations in Figure \\ref{2 simus}\\textbf{B}.\nThere, we see that our data follow the trend of the theoretical prediction and lie within or close to the confidence interval bounds of the prediction for ample regions for several participants.\nThis is not a trivial result, as can be easily seen, when randomizing the temporal order of the trajectory points or when replacing the utility function with another one that does not fit the setup. Figure \\ref{togetherB}\\textbf{B} and \\ref{togetherB}\\textbf{C} show this, for example, for an inverted Mexican hat (\\eqref{mex hat} with $\\sigma=4$) that assigns low utility to the target region, and for resamples of the trajectory points in a random order, respectively. Both results are clearly incompatible with the theoretical prediction. \n\n\nWhen conducting an additional robustness analysis in Figure~\\ref{graph distances}, we found that, under the proposed utility function, participants' behavior is compatible with Crooks' fluctuation theorem for a broad neighbourhood of parameter settings, but breaks down when choosing implausible parameters. Regarding Jarzynski's equality \\eqref{prediction II}, the confidence intervals for the majority of participants are consistent with the theoretical prediction when using the bootstrapped values to calculate $\\langle e^{-\\beta \\Delta E_{ext}(\\vect{X})}\\rangle$ (cf. Table \\ref{jarz participants}). In contrast, when following the same procedure for both the inverted Mexican hat and the randomized procedure, we obtain consistency for a considerably smaller number of participants. In particular, for the inverted Mexican hat, we obtain consistency for only two participants. Moreover, these participants are $S_8$ and $S_9$, which belong to the group that did not reach at least $50\\%$ adaptation (indicated by the red background area in the figures). For the randomized procedure, the expected number of participants that show consistency is also close to two, although the specific subjects which are consistent vary with the realization of the randomized procedure. More specifically, after 1000 runs of the randomized procedure, the mean number of consistent subjects we observed was 2.33.   \n\n\n\\begin{comment}\n\\begin{table}[!htb]\n\\centering\n \\begin{tabular}{||c| c| c ||} \n \\hline\n participant & Mean & Standard deviation \\\\ [0.5ex] \n \\hline\\hline\n 1 & 1.02 & 0.35  \\\\ \n 2 & 0.72 & 0.30  \\\\\n 3 & 0.43 & 0.09 \\\\\n 4 & 1.36 & 0.51  \\\\\n 5 & 0.57 & 0.27 \\\\\n 6 & 0.48 & 0.15 \\\\\n 7 & 0.31 & 0.10 \\\\\n 8 & 4.80 & 2.60 \\\\\n 9 & 1.55 & 0.51 \\\\\n 10 & 2.28 & 1.30 \\\\\n \\hline\n \\end{tabular}\n \\caption{Jarzynski's equality experimental results. Notice the results should be one according to the theoretical prediction. To get the results, we bootstrap the observed value of $\\Delta E_{ext}(\\vect{x})$ for the forward process 1000 times and calculate $\\int p(W) e^{-W}dW$, where $p(W)$ is obtained using kernel density estimation on each bootstrap. Aside from participants 3 and 7, all participants are almost consistent with 3 sigmas.}\n \\end{table}\n\\end{comment}\n \n \\begin{table}[!tb]\n\\centering\n \\begin{tabular}{||c| c|| c| c ||} \n \\hline\n participant & Confidence interval & participant & Confidence interval \\\\ [0.5ex] \n \\hline\\hline\n 1 & \\cellcolor[RGB]{175,234,180}(0.03,\\text{ }48.59) & 6 & \\cellcolor[RGB]{175,234,180} (0.04,\\text{ }3.75) \\\\ \n 2 & \\cellcolor[RGB]{175,234,180} (0.03,\\text{ }137.58) & 7  &\\cellcolor[RGB]{175,234,180} (0.01,\\text{ }0.50)\\\\\n 3 & \\cellcolor[RGB]{175,234,180} (0.01,\\text{ }3.63) & 8 &\\cellcolor[RGB]{255, 182, 193}(1.98,\\text{ }518130.21)\\\\\n 4 & \\cellcolor[RGB]{175,234,180}(0.49,\\text{ }63.48) & 9  & \\cellcolor[RGB]{255, 182, 193}(0.76,\\text{ }77.24)\\\\\n 5 &\\cellcolor[RGB]{175,234,180} (0.46,\\text{ }1.37) & 10  &\\cellcolor[RGB]{255, 182, 193}(0.26,\\text{ }48758.33)\\\\\n \\hline\n \\end{tabular}\n \\caption{Experimental results for Jarzynski's equality. We include the confidence intervals for the left hand side of \\eqref{prediction II}, which we obtain after bootstrapping the observed values of $\\Delta E_{ext}(\\vect{x})$ for the forward process 1000 times and estimating $\\langle e^{-\\beta \\Delta E_{ext}(\\vect{X})}\\rangle$ by its mean for each set of bootstrapped data.\n\n Note, in our setup, the theoretical prediction fulfills $\\Delta F=0$ in the right hand side of \\eqref{prediction II}, resulting in values around $1.0$ for this estimate. Participants that achieve at least $50\\%$ adaptation (c.f. Figure \\ref{hyste plot}) are shaded by a green background color.}\n \\label{jarz participants}\n \\end{table}\n\n\n\n\n\n\\section{Discussion}\n\n\n\\begin{figure}[!tb]\n\\centering\n    \\includegraphics[scale=0.21]{lastIV.jpg}\n    \\caption{Experimental results for Crooks' fluctuation theorem when the sensorimotor loss behaves as an exponential quadratic error \\eqref{utility}. The black line is the theoretical prediction of Crooks' fluctuation theorem \\eqref{prediction} while the curves stand for the mean path after 1000 bootstraps of the observed driving error values. Participants that achieve at least $50\\%$ adaptation (c.f. Figure \\ref{hyste plot}) are shaded by a green background color. The shaded areas inside the graphs are the 99\\% confidence intervals which result from bootstrapping. Note we fit the parameters for each participant according to Section \\ref{exp design}.}\n    \\label{together}\n\\end{figure}\n\n\n\\begin{figure}[!tb]\n\\centering\n    \\includegraphics[scale=0.21]{lastV-II.jpg}\n    \\caption{Control results for Crooks' fluctuation theorem in two scenarios: \\textbf{A} the sensorimotor loss behaves like a Mexican hat function and \\textbf{B} the sensorimotor loss behaves as an exponential quadratic error but we sample the observed angles randomly with repetition. The black line is the theoretical prediction of Crooks' fluctuation theorem \\eqref{prediction} while the curves stand for the mean path after 1000 bootstraps of the observed driving error values. The shaded areas inside the graphs are the 99\\% confidence intervals which result from bootstrapping. Note, for simplicity, we assume $\\beta=1$ for all participants when using the Mexican hat to demonstrate that the result in (A) does not trivially hold for any cost function. For \\textbf{B}, we fit the parameters for each participant according to Section \\ref{exp design}.}\n    \\label{togetherB}\n\\end{figure}\n\nIn our experiment we have investigated the hypothesis that human sensorimotor adaptation may be participant to the thermodynamic fluctuation theorems first reported by Crooks \\cite{crooks1999entropy} and Jarzynski \\cite{jarzynski2000hamiltonian}. In particular, we tested whether changes in sensorimotor error induced externally by an experimental protocol are linearly related to the log-ratio of the  probabilities of behavioral trajectories under a given forward and time-reversed backward protocol of a sequence of visuomotor rotations. We found that participants' data, in all cases where participants showed an appropriate adaptive response, was consistent with this prediction\nor close to its confidence interval bounds, as expected from our simulations with finite sample size.\nMoreover, we found that the exponentiated error averaged over the path probabilities was statistically compatible with unity for these participants, in line with Jarzynski's theorem. \n\nTogether these results not only extend the experimental evidence of \\linebreak Boltzmann-like relationships between the probabilities of behavior and the corresponding order-inducing functions---such as energy, utility, or sensorimotor error---from the equilibrium to the non-equilibrium domain, but also from simple physical systems to more complex learning systems when studying adaptation in changing environments, deepening, thus, the parallelism between thermodynamics in physics and decision-making systems \\cite{ortega2013thermodynamics}.\n\n\\begin{comment}\n\\begin{figure}[!tb]\n\\centering\n    \\includegraphics[scale=0.65]{fig_5.png}\n    \\caption{(NEED TO CHANGE) Close-up of Figure \\ref{together} \\textbf{A} in the region where the simulations of the experiment show a better fit regarding Crooks' fluctuation theorem \\eqref{prediction}, namely, the $\\Delta E_{\\textrm{ext}}$-interval $[-1,1]$ (cf. Figure \\ref{2 simus}\\textbf{B}).}\n    \\label{together zoom}\n\\end{figure}\n\\end{comment}\n\nWhen testing for the validity of thermodynamic relations, one of the most critical issues is the choice of the energy function, that is, in our case, the error cost function. In physical systems, the energy function is usually hypothesized following from simple models involving point masses, springs, rigid bodies, etc., and generally requires knowledge of the degrees of freedom of the system under consideration. Here we have used an exponential quadratic error as a utility function, as it has been suggested previously that human pointing behavior can be best captured by loss functions that approximately follow a negative parabola for small errors and then level off for large errors \\cite{kording2004loss}. In the absence of very large errors, many studies in the literature on sensorimotor learning have only used the quadratic loss term  \\cite{wolpert1995internal,todorov2002optimal}. Thus, our assumptions are in line with the literature. Crucially, the reported results fail when assuming non-sensical cost functions, like the Mexican hat.\n\n\nExperimental tests of both Jarzynski's equality \\eqref{prediction II} and Crooks fluctuation theorem \\eqref{prediction} have been previously reported  in classical physics \\cite{douarche2005experimental,collin2005verification,toyabe2010experimental,saira2012test,liphardt2002equilibrium} and also, in the case of Jarzynski's equality, in quantum physics \\cite{an2015experimental,smith2018verification}. Importantly, these results have been successfully tested in several contexts: unfolding and refolding processes involving RNA \\cite{collin2005verification,liphardt2002equilibrium}, electronic transitions between electrodes manipulating a charge parameter \\cite{saira2012test}, rotation of a macroscopic object inside a fluid surrounded by magnets where the current of a wire attached to the macroscopic object is manipulated \\cite{douarche2005experimental}, and a trapped ion \\cite{an2015experimental,smith2018verification}. Despite differences in physical realization, protocols, and energy functions (and thus work functions), all the above experiments follow the same basic design behind the approach presented here. This supports the claim that \nfluctuation theorems do not necessarily rely on involved physical assumptions but are simple mathematical properties of certain stochastic processes \\cite{hack2022jarzyskis}, although originally they were derived in the context of non-equilibrium thermodynamics \\cite{jarzynski1997equilibrium,crooks1998nonequilibrium}. \n\nMathematically, Crooks theorem  \\eqref{prediction} holds for any Markov process (i), whose initial distribution is in equilibrium (ii), and whose transition probabilities satisfy detailed balance with respect to the corresponding equilibrium distributions (iii) \\cite{hack2022jarzyskis}. \nOur experimental test of Equation~\\eqref{prediction} can be seen, thus, as a test for the hypothesis that human sensorimotor adaptation processes satisfy conditions (i), (ii), and (iii). Condition (i) requires adaptation to be  Markovian, which is in line with most error-driven models of sensorimotor adaptation  \\cite{shadmehr2012} that assume some internal state update of the form $x_{t+1}=f(x_t, e)$ with adaptive state $x$ and error $e$. While such models have proven fruitful for simple adaptation tasks like ours, they also have clear limitations, for example when it comes to meta-learning processes that have been reported in more complex learning scenarios \\cite{braun2010,lieder2019}. Condition (ii) is supported by our data in the second and last rows of Figure \\ref{plateaus all},\nwhere it can be seen that participants' behavior at the beginning of each cycle is roughly consistent with the equilibrium behavior recorded prior to the start of the experiment. Condition (iii) requires  that the adaptive process converges to the equilibrium distribution \\eqref{Boltzmann} dictated by the environment and that the behaviour remains statistically unchanged when staying in that environment. Moreover, it requires that the equilibrium behavior at each energy level is time-reversible, that means, once adaption has ceased the trial-by-trial behavior would have the same statistics when played forward or backward in a video recording. Note, however, that does not imply time-reversibility over the entire adaptation trajectory, but is only required locally for each transition step. \nThe usual noise-driven models of adaptation fulfil this requirement, like the Metropolis-Hastings scheme that has been proposed to simulate human sensorimotor adaptation \\cite{SANBORN2016883,grau2018non}.\n\nWhile Jarzynski's equality \\eqref{prediction II} directly follows from Crooks theorem, weaker assumptions are sufficient to derive it \\cite{hack2022jarzyskis,jarzynski1997equilibrium}. In particular, condition (iii) regarding detailed balance is not necessary, as it is only required that the behavioral distribution does not change anymore once the equilibrium distribution is reached. Thus, Equation~\\eqref{prediction II} can be used as a test for the weaker hypothesis that human sensorimotor adaptation satisfies conditions (i), (ii) and stationarity after convergence. While Jarzynski's equality only requires samples from the forward process, Crooks theorem also tests the relation between the forward and the backward processes. In particular, Crooks theorem\ndecouples the information processing with respect to any particular environment from the biases introduced by the adaptation history, that is, it assumes the transition probabilities for any given environment are independent of the history. Hence, the observed difference in behaviour after having adapted to the same environment, the hysteresis, is solely explained in terms of the information processing history before encountering the environment. Such hysteresis effects are not only common in simple physical systems like magnets or elastic bands, but have also been reported for sensorimotor tasks \\cite{kelso1994,schack2011,turnham2012facilitation}. The hysteresis effects we report in Figure~\\ref{hyste plot} are in line with a system obeying Crooks theorem and can be replicated using Markov Chain Monte Carlo simulations of adaptation \\cite{grau2018non}.\n\n\n\n\n\n\n\n\n\n\nOur study is part of a number of recent studies that have tried to harness equilibrium and non-equilibrium thermodynamics to gain new theoretical insights into simple learning systems \\cite{goldt2017stochastic,perunov2016statistical,england2015dissipative,still2012thermodynamics,ortega2013thermodynamics}.\nFor example, the information that can be acquired by learning in simple forward neural networks has been shown to be bounded by thermodynamic costs given by the entropy change in the weights and the heat dissipated into the environment \\cite{seifert2012stochastic}. More generally, when interpreting a system's response to a stochastic driving signal in terms of computation, the amount of non-predictive information contained in the state about past environmental fluctuations is directly related to the amount of thermodynamic dissipation \\cite{still2012thermodynamics}. This suggests that thermodynamic fundamentals, like the second law, can be carried over to learning systems.\nConsider, for example, a Bayesian learner where the utility is given by the log-likelihood model and where the data are presented either in one chunk for a single update, or consecutively in little batches with many little updates. In the latter case, the cumulative surprise is much smaller and lower bounded by the log-likelihood of the data, which corresponds to the free energy difference before and after learning \\cite{grau2018non}. Finally, it has even been suggested that the dissipation of absorbed work as it is studied in a generalized Crooks theorem may underlie a general thermodynamic mechanism for self-organization and adaptation in living matter \\cite{england2015dissipative}, raising the question of whether such a general principle of adaptive dissipation could also govern biological learning processes \\cite{perunov2016statistical}.\n\n\n\n\n\\begin{comment}\n\\begin{figure}[!htb]\n\\centering\n    \\includegraphics[scale=.8]{jarz-simu.eps}\n    \\caption{In blue, the histogram for Jarzynski's equality. The values are obtained after 1000 bootstraps of one run of the short simulation. The mean is 0.91 and the standard deviation is 0.15. In red, the best fit by a normal distribution. The Lilliefors normality test is not rejected. The best fit mean is 0.93.}\n    \\label{jarzynski simu}\n\\end{figure}\n\\end{comment}\n\n\\begin{comment}\n\\section*{Other narrative}\n\n\\paragraph{Narrative} We aim to study quantitatively the difficulty of adaptation in human decision-making. Given a set of $N+1$ environments where, for each of them, the decision-maker has to choose, after a finite amount of time, some element of a set $S$, we are interested in the distribution of the choices $(x_0,x_1,..,x_N) \\in S^{N+1}$. We could, for example, measure somehow the discrepancy between the observed distribution and the distribution we would get if the decision-maker adapted perfectly. Alternatively, we could measure it by comparing the observed distribution $p$ with the one we get when presenting the decision-maker with the same set of environments in reversed order $p^\\dagger$. In case the decision-maker learned perfectly every environment, both distributions would coincide. The discrepancy between them is, thus, a way of measuring how well the decision-maker adapts to the presented environments. In physics, this discrepancy follows a specific equation, namely\n\\begin{equation}\n\\label{ave eq}\n    \\langle W_d(\\vect{x})\\rangle_{\\rho(\\vect{x})} = \\frac{1}{\\beta} D_{KL}(p||p^\\dagger)\n\\end{equation}\nwhere $p(\\vect{x})$ is the probability of observing some vector $\\vect{x}$ when facing the sequence of environments and $p^\\dagger(\\vect{x})$ is the probability of observing the same vector, in reversed order, when facing the sequence of environments reversed and $W_d$ is the dissipated work (explain). Actually, \\eqref{ave eq} has a point wise counterpart (from which it can be derived) which states $\\forall \\vec{x} \\in S^{N+1}$\n\\begin{equation}\n\\label{point eq}\n W_d(\\vect{x})=\\frac{1}{\\beta} \\log \\frac{\\rho(X_N = x_N,..,X_0=x_0)}{\\rho(Y_N=x_0,..,Y_0=x_N)} \n\\end{equation}\nwhere $X_i$ are the random variables corresponding to the choices when the energies are given in the original order and $Y_i$ the random variables when the environment order in inverted. Given the fact we have a small sample, instead of testing directly \\eqref{point eq}, we test a consequence of it, namely,\n\\begin{equation}\n    W_0= \\log \\Big( \\frac{\\rho^F(W=W_0)}{\\rho^B(W=-W_0)} \\Big)\n\\end{equation}\n$\\forall W_0 \\in W(S^{N+1}$ where $W$ is the work (explain).\n\\end{comment}\n\n\\begin{comment}\n\\clearpage\n\\begin{figure}[ht!]\n\\centering\n    \\includegraphics[scale=0.3]{initial.s1.eps} \\hfill\n     \\includegraphics[scale=0.3]{initial.s2.eps} \n     \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{initial.s3.eps} \\hfill\n     \\includegraphics[scale=0.3]{initial.s4.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{initial.s5.eps} \\hfill\n     \\includegraphics[scale=0.3]{initial.s6.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{initial.s7.eps} \\hfill\n     \\includegraphics[scale=0.3]{initial.s8.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{initial.s9.eps} \\hfill\n     \\includegraphics[scale=0.3]{initial.s10.eps}\n    \\caption{In blue, histogram initial 100 trials (no deviation). In red, equilibrium distribution where the width and center are chosen to represent the histogram.}\n    \\label{plateaus all}\n\\end{figure}\n\n\\clearpage \n\\begin{figure}[ht!]\n\\centering\n    \\includegraphics[scale=0.3]{plat.s1.eps} \\hfill\n     \\includegraphics[scale=0.3]{plat.s2.eps} \n     \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{plat.s3.eps} \\hfill\n     \\includegraphics[scale=0.3]{plat.s4.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{plat.s5.eps} \\hfill\n     \\includegraphics[scale=0.3]{plat.s6.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{plat.s7.eps} \\hfill\n     \\includegraphics[scale=0.3]{plat.s8.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{plat.s9.eps} \\hfill\n     \\includegraphics[scale=0.3]{plat.s10.eps}\n    \\caption{Histogram 0 deviation plateaus.}\n    \\label{plateaus all}\n\\end{figure}\n\n\\clearpage\n\\begin{figure}[ht!]\n\\centering\n    \\includegraphics[scale=0.3]{forw.s1.eps} \\hfill\n     \\includegraphics[scale=0.3]{forw.s2.eps} \n     \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{forw.s3.eps} \\hfill\n     \\includegraphics[scale=0.3]{forw.s4.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{forw.s5.eps} \\hfill\n     \\includegraphics[scale=0.3]{forw.s6.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{forw.s7.eps} \\hfill\n     \\includegraphics[scale=0.3]{forw.s8.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{forw.s9.eps} \\hfill\n     \\includegraphics[scale=0.3]{fow.s10.eps}\n    \\caption{Observed angles in the forward trajectories.}\n    \\label{forward all}\n\\end{figure}\n\\end{comment}\n\n\\begin{comment}\n\\clearpage \n\\begin{figure}[!tb]\n\\centering\n    \\includegraphics[scale=0.3]{hyste-s1.eps} \\hfill\n     \\includegraphics[scale=0.3]{hyste-s2.eps} \n     \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{hyste-s3.eps} \\hfill\n     \\includegraphics[scale=0.3]{hyste-s4.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{hyste-s5.eps} \\hfill\n     \\includegraphics[scale=0.3]{hyste-s6.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{hyste-s7.eps} \\hfill\n     \\includegraphics[scale=0.3]{hyste-s8.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{hyste-s9.eps} \\hfill\n     \\includegraphics[scale=0.3]{hyste-s10.eps}\n    \\caption{Hysteresis plot. In green, the mean of the observed angles for the forward process. In red, the mean of the observed angles for the backward process. In black, the forward protocol. Hysteresis can be observed between trials 1 and 5, 9 and 17 and 21 and 25. Notice the forward means are below the backward in the first region, above in the second and below again in the third, as expected.}\n    \\label{hyste plot}\n\\end{figure}\n\\end{comment}\n\n\n\\begin{comment}\n\\clearpage \n\\begin{figure}[!tb]\n\\centering\n    \\includegraphics[scale=0.3]{jarz-s1.eps} \\hfill\n     \\includegraphics[scale=0.3]{jarz-s2.eps} \n     \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{jarz-s3.eps} \\hfill\n     \\includegraphics[scale=0.2]{jarz-s4.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{jarz-s5.eps} \\hfill\n     \\includegraphics[scale=0.2]{jarz-s6.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{jarz-s7.eps} \\hfill\n     \\includegraphics[scale=0.3]{jarz-s8.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{jarz-s9.eps} \\hfill\n     \\includegraphics[scale=0.3]{jarz-s10.eps}\n    \\caption{Histogram Jarzynski's equation data. The (mean,standard deviation) pair for each participant are: (0.8,0.28), (0.57,0.24), (0.33,0.07), (1.05,0.42), (0.45,0.2), (0.38,0.12), (0.25,0.08), (3.83,1.97), (1.22,0.42) and (1.81,1.03). 7 of the 10 participants show results consistent with the Jarzynski equality with 3 sigmas or less.}\n    \\label{jarz data}\n\\end{figure}\n\\end{comment}\n\n\\begin{comment}\n\\clearpage\n\\begin{figure}[ht!]\n\\centering\n    \\includegraphics[scale=0.3]{back.s1.eps} \\hfill\n     \\includegraphics[scale=0.3]{back.s2.eps} \n     \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{back.s3.eps} \\hfill\n     \\includegraphics[scale=0.3]{back.s4.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{back.s5.eps} \\hfill\n     \\includegraphics[scale=0.3]{back.s6.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{back.s7.eps} \\hfill\n     \\includegraphics[scale=0.3]{back.s8.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{back.s9.eps} \\hfill\n     \\includegraphics[scale=0.3]{back.s10.eps}\n    \\caption{Observed angles in the backward trajectories.}\n    \\label{forward all}\n\\end{figure}\n\n\\clearpage\n\\begin{figure}[ht!]\n\\centering\n    \\includegraphics[scale=0.3]{boot.s1.eps} \\hfill\n     \\includegraphics[scale=0.3]{boot.s2.eps} \n     \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{boot.s3.eps} \\hfill\n     \\includegraphics[scale=0.3]{boot.s4.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{boot.s5.eps} \\hfill\n     \\includegraphics[scale=0.3]{boot.s6.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{boot.s7.eps} \\hfill\n     \\includegraphics[scale=0.3]{boot.s8.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{boot.s9.eps} \\hfill\n     \\includegraphics[scale=0.3]{boot.s10.eps}\n    \\caption{Experimental linear relation. In black, the theoretical line, and, in blue, the boundaries of the 99.7 \\% confidence interval after 1000 bootstraps of the measured driving error values.}\n    \\label{result}\n\\end{figure}\n\n\\clearpage\n\\begin{figure}[ht!]\n\\centering\n    \\includegraphics[scale=0.3]{mex.s1.eps} \\hfill\n     \\includegraphics[scale=0.3]{mex.s2.eps} \n     \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{mex.s3.eps} \\hfill\n     \\includegraphics[scale=0.3]{mex.s4.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{mex.s5.eps} \\hfill\n     \\includegraphics[scale=0.3]{mex.s6.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{mex.s7.eps} \\hfill\n     \\includegraphics[scale=0.3]{mex.s8.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{mex.s9.eps} \\hfill\n     \\includegraphics[scale=0.3]{mex.s10.eps}\n    \\caption{Experimental linear relation using a mexican hat as energy function. In black, the theoretical line, and, in blue, the boundaries of the 99.7 \\% confidence interval after 1000 bootstraps of the measured driving error values.}\n    \\label{mex}\n\\end{figure}\n\n\\clearpage\n\\begin{figure}[ht!]\n\\centering\n    \\includegraphics[scale=0.3]{rando.s1.eps} \\hfill\n     \\includegraphics[scale=0.3]{rando.s2.eps} \n     \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{rando.s3.eps} \\hfill\n     \\includegraphics[scale=0.3]{rando.s4.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{rando.s5.eps} \\hfill\n     \\includegraphics[scale=0.3]{rando.s6.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{rando.s7.eps} \\hfill\n     \\includegraphics[scale=0.3]{rando.s8.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.3]{rando.s9.eps} \\hfill\n     \\includegraphics[scale=0.3]{rando.s10.eps}\n    \\caption{Experimental linear relation sampling with repetition the observed angles. In black, the theoretical line, and, in blue, the boundaries of the 99.7 \\% confidence interval after 1000 bootstraps of the measured driving error values.}\n    \\label{random}\n\\end{figure}\n\\end{comment}\n\n\\begin{comment}\n\\begin{figure}[!tb]\n\\centering\n    \\includegraphics[scale=0.6]{allImages.png}\n    \\caption{Experimental linear relation for three cases: original utility function (rows 1 and 2), inverted Mexican hat utility function (rows 3 and 4) and original utility function after sampling with repetition the observed angles (rows 5 and 6). In black, the theoretical line, and, in blue, the mean path after 1000 bootstraps of the observed driving error values. The colored areas represent the 99\\% confidence interval after 1000 bootstraps of the measured driving error values in each case.}\n    \\label{together}\n\\end{figure}\n\\end{comment}\n\n\\begin{comment}\n\\begin{figure}[!tb]\n\\centering\n    \\includegraphics[scale=0.6]{fig_5.png}\n    \\caption{Close-up of Figure \\ref{together} in the region where the simulations of the experiment show a better fit of the theoretical prediction: driving error values between 1 and -1.}\n    \\label{together zoom}\n\\end{figure}\n\\end{comment}\n\n\\begin{comment}\n\\clearpage\n\\begin{figure}[ht!]\n\\centering\n    \\includegraphics[scale=0.4]{mex.s1_zoom.eps} \\hfill\n     \\includegraphics[scale=0.4]{mex.s2_zoom.eps} \n     \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.4]{mex.s3_zoom.eps} \\hfill\n     \\includegraphics[scale=0.4]{mex.s4_zoom.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.4]{mex.s5_zoom.eps} \\hfill\n     \\includegraphics[scale=0.4]{mex.s6_zoom.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.4]{mex.s7_zoom.eps} \\hfill\n     \\includegraphics[scale=0.4]{mex.s8_zoom.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.4]{mex.s9_zoom.eps} \\hfill\n     \\includegraphics[scale=0.4]{mex.s10_zoom.eps}\n    \\caption{Zoom mexican hat}\n    \\label{mexa zoom}\n\\end{figure}\n\n\\begin{figure}[ht!]\n\\centering\n    \\includegraphics[scale=0.4]{rando.s1_zoom.eps} \\hfill\n     \\includegraphics[scale=0.4]{rando.s2_zoom.eps} \n     \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.4]{rando.s3_zoom.eps} \\hfill\n     \\includegraphics[scale=0.4]{rando.s4_zoom.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.4]{rando.s5_zoom.eps} \\hfill\n     \\includegraphics[scale=0.4]{rando.s6_zoom.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.4]{rando.s7_zoom.eps} \\hfill\n     \\includegraphics[scale=0.4]{rando.s8_zoom.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.4]{rando.s9_zoom.eps} \\hfill\n     \\includegraphics[scale=0.4]{rando.s10_zoom.eps}\n    \\caption{Zoom resampling}\n    \\label{resamp zoom}\n\\end{figure}\n\n\\begin{figure}[ht!]\n\\centering\n    \\includegraphics[scale=0.4]{boot.s1_zoom.eps} \\hfill\n     \\includegraphics[scale=0.4]{boot.s2_zoom.eps} \n     \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.4]{boot.s3_zoom.eps} \\hfill\n     \\includegraphics[scale=0.4]{boot.s4_zoom.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.4]{boot.s5_xoom.eps} \\hfill\n     \\includegraphics[scale=0.4]{boot.s6_zoom.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.4]{boot.s7_zoom.eps} \\hfill\n     \\includegraphics[scale=0.4]{boot.s8_zoom.eps}\n      \\\\[\\smallskipamount]\n     \\includegraphics[scale=0.4]{boot.s9_zoom.eps} \\hfill\n     \\includegraphics[scale=0.4]{boot.s10_zoom.eps}\n    \\caption{Zoom original}\n    \\label{orig zoom}\n\\end{figure}\n\\end{comment}\n\n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFederated learning~\\citep{kairouz2021advances} enables collaborative learning from distributed data located at multiple clients without the need to share the data among the different clients or with a central server. Much progress has been made in recent work on various aspects of this problem setting, such as improved optimization at each client~\\citep{li2020federatedheterogenous}, improved aggregation of client models at the server~\\citep{chen2020fedbe}, handling the heterogeneity in clients' data distributions~\\citep{zhu2021data}, and also efforts towards personalization of the client models~\\citep{mansour2020three}.\n\nMost existing formulations of federated learning view it as an optimization problem where the global loss function is optimized over multiple rounds, with each round consisting of point estimation of a loss function defined over the client's local data, followed by an aggregation of the client models on a central server. Point estimation, however, is prone to overfitting especially if the amount of training data on clients is very small. Moreover, crucially, such an approach ignores the uncertainty in the client models. Indeed, taking into account the model uncertainty has been shown to be useful not just for improved accuracy and robustness of predictions when the amount of training data is limited, as well as in other tasks, such as out-of-distribution (OOD) detection~\\citep{salehi2021unified} and active learning~\\citep{ahn2022federated}. In this work, we present a Bayesian approach for federated learning which takes into account the model uncertainty, and also demonstrate its effectiveness for other tasks in federated settings where accurate estimates of model uncertainty are crucial, such as OOD detection and active learning in federated setting.\n\nDespite its importance, federated learning in a Bayesian setting is inherently a challenging problem. Unlike standard federated learning, in the Bayesian setting, each client needs to estimate the posterior distribution over its weights (and also the posterior predictive distribution which needed at the prediction stage) which is an intractable problem.\nTypical ways to address this intractability of Bayesian inference for deep learning models include (1) Approximate Bayesian inference where the posterior distribution of model parameters is usually estimated via approximate inference methods, such as MCMC~\\citep{zhang2019cyclical,izmailov2021bayesian}, variational inference~\\citep{zhang2018advances}, or other faster approximations such as modeling the posterior via a Gaussian distribution constructed using the SGD iterates~\\citep{maddox2019simple}, or (2) ensemble methods, such as deep ensembles~\\citep{lakshminarayanan2017simple} where the model is trained using different initialization to yield an ensemble whose diversity represents the model uncertainty.\n\nThe other key challenge for Bayesian federated learning is efficiently communicating the client model parameters, which are represented by a probability distribution, to the server, and their aggregation at the server. Note that, unlike standard federated learning, in the Bayesian settings, each client would maintain either a probability distribution over its model weights or an ensemble over the model weights. Both of these approaches make it difficult to efficiently communicate the client models and aggregate them at the server. Some recent attempts towards Bayesian federated learning have relied on simplifications such as assuming that the posterior distribution of each client's weights is a Gaussian~\\citep{al2020federated,linsner2021approaches}, which makes model communication and aggregation at the server somewhat easier. However, this severely restricts the expressiveness of the client models. In our work, we do not make any assumption on the form of the posterior distribution of the client weights. Another appealing aspects of our Bayesian federated learning model is that, at test time, it does not require Monte-Carlo averaging~\\citep{bishop2006pattern,korattikara2015bayesian} which is usually required by Bayesian methods (especially for non-conjugate models, such as deep learning models) at test time, making them slow (essentially, using $S$ Monte-Carlo samples from the posterior makes prediction $S$ times slower). In contrast, our approach leverages ideas from distillation of posterior predictive distribution~\\citep{korattikara2015bayesian}, using which we are able to represent the entire posterior predictive distribution using a single deep neural network, resulting in fast predictions at test time.\n\nOur contributions are summarized below\n\\begin{itemize}\n    \\item We present a novel and efficient approach to Bayesian federated learning in which each client performs a distillation of its posterior predictive distribution into a single deep neural network. This allows solving the Bayesian federated learning problem using ideas developed for standard federated learning methods, while still capturing and leveraging model uncertainty. \n    \\item Our approach does not make any strict assumptions on the form of the clients' posterior distributions (e.g., Gaussian~\\citep{al2020federated}) or predictive distributions. Moreover, despite being Bayesian, our approach is still fast at test time since it does not require Monte-Carlo averaging (which is akin to averaging over an ensemble) but uses the idea of distribution distillation to represent the PPD via a single deep neural network.\n    \\item We present various ways to aggregate the clients' predictive distributions at the server, both with as well as without requiring publicly available (unlabeled) data at the server. \n    \\item In addition to tasks such as classification and out-of-distribution (OOD) detection, we also show a use case of our approach for the problem of active learning in federated setting~\\citep{ahn2022federated} where our approach outperforms existing methods.\n\\end{itemize}\n\n\n\\begin{comment}\nThe popularity of machine and deep learning approaches is a result of availability of better computation and storage mediums in the ecosystem. In many practical scenarios, the pool of data is readily available in the form of a repository or some other storage medium. However, there are also ample cases, where the sharing of data is of concern if it is of private in nature e.g. medical health records. This, coupled with the increasing trend of computational power in mobile and personal devices has brought attention to the concept of leveraging resources on local devices for computation of learning models. It has led an increase in interest in the field of \\textit{federated learning}.\n\nFederated learning explores the possibility of training the learning models using local resources present on remote devices. It uses local computation resources, thereby increasing computation efficiency due to a distributed training fashion. Also, it uses data in a secure way by retaining it with the clients. This eliminates the need of a central data repository and also saves the storage cost and the expensive communication cost between any central server and the distributed systems.  Federated learning process usually iterates through many rounds of communication alternating between the server and client level computations. Usually, in each round, the clients initializes their model with the global server model and trains it using local data, which is then sent to the server. The server then aggregates these models to form the global server model, which can be used for the next round.\n\nBesides the formulation stated above in simpler words, the problem becomes more challenging due to a few more aspects.\n\n\\begin{enumerate}\n    \\item \\textbf{Imbalanced data} Different devices may be able to generate or store different volumes of data. This can be either due to storage characteristics of device or the variability in usage of service\/application generating the data. This means that based on the available data, local learning models may have a small or large base of data to learn from and so their performance can vary with huge gaps.\n    \\item \\textbf{Non-IID data} The data generated by various devices may have different distribution depending on the usage. For ex: Depending on the genre of conversation, the data for next word prediction task may differ.\n    \\item \\textbf{Inactive participation} Many devices do not participate actively because of their availability and data generation activity. This may lead to different number of devices and different active devices present in each round.\n\\end{enumerate}\n\\end{comment}\n\n\\section{Bayesian Federated Learning via Predictive Distribution Distillation}\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=0.35]{figures\/FedDiagram.png}\n    \\caption{\\small{The above figure summarizes our framework. Each client infers the (approximate) posterior distribution by generating the posterior samples (teacher models) which are distilled to give the PPD (student model parameterized by a deep neural network). Each client communicates its MAP teacher sample and the PPD to the server which aggregates them to yield a global teacher sample and the global PPD, both of which are sent back to the clients, which use these quantities for the next round of learning.}}\n    \\label{fig:fedppd_diagram}\n\\end{figure}\n\nUnlike standard federated learning where the client model is represented by a single neural network whose weights are estimated by minimizing a loss function using client's data, we consider the Bayesian setting of federated learning where each client learns a posterior distribution over its weights. The goal is to efficiently communicate the clients' local posteriors to the server and aggregate these local posteriors to learn a global posterior.\n\nHowever, since we usually care about predictive tasks, the actual quantity of interest in the Bayesian setting is not the posterior distribution per se, but the posterior predictive distribution (PPD). Given a set of $S$ samples $\\theta^{(1)},\\ldots,\\theta^{(S)}$ from the posterior, estimated using some training data $\\mathcal{D}$, the (approximate) PPD of a model is defined as $p(y|x,\\mathcal{D}) = \\frac{1}{S}\\sum_{i=1}^S p(y|x,\\theta^{(i)})$. Note that the PPD can be thought of as an ensemble of $S$ models.\n\nSince the PPD is the actual quantity of interest, in our Bayesian federated learning setting, we aim to directly estimate the PPD at each client. However, even estimating and representing the PPD has challenges. In particular, since the PPD is essentially an ensemble of models, storing and communicating such an ensemble from each client to the server can be challenging. To address this issue, we leverage the idea of distribution\/ensemble distillation~\\citep{korattikara2015bayesian}, where the PPD of a deep learning model can be efficiently distilled and stored as a single deep neural network. We leverage this distillation idea on each client to represent the client's PPD using a single neural network which can then be communicated and aggregated at the server in pretty much the same way as it is done in standard federated learning.\n\n\nOur approach can be summarized as follows (and is illustrated in Fig.~\\ref{fig:fedppd_diagram})\n\n\\begin{enumerate}\n    \\item For each client, we perform approximate Bayesian inference for the posterior distribution of the client model weights using Markov Chain Monte Carlo (MCMC) sampling. This gives us a set of samples from the client's posterior and these samples will be used as as teacher models which we will distill into a student model. We use stochastic gradient Langevin dynamics (SGLD) sampling~\\citep{welling2011bayesian} since it gives us an online method to efficiently distill these posterior samples into a student model (step 2 below).\n    \\item For each client, we distill the MCMC samples (teacher models) directly into the posterior predictive distribution (PPD), which is the student model. Notably, in this distillation based approach~\\citep{korattikara2015bayesian}, the PPD for each client is represented succinctly by a \\emph{single} deep neural network, instead of via an ensemble of deep neural network. This makes prediction stage much faster as compared to typical Bayesian approaches. \n    \\item For each client, the teacher model with largest posterior probability (i.e., the MAP sample) from its posterior distribution and the student model representing the client's PPD (both of which are deep neural networks), are sent to the server. \n    \\item The server aggregates the teacher and student models it receives from all the clients. For the aggregation, we consider several approaches which we describe in Sec.~\\ref{sec:aggr}. \n    \\item The aggregated teacher and student models are sent back to each client, and the process continues for the next round.\n    \\item We continue steps 1-5 till convergence.\n\\end{enumerate}\n\n\\subsection{Posterior Inference and Distillation of Client's PPD}\n\\label{sec:fedbdk-1}\nWe assume there are $K$ clients with labeled data $\\mathcal{D}_1,\\ldots,\\mathcal{D}_K$, respectively. On each client, we take the Monte Carlo approximation of its posterior predictive distribution (PPD) and distill it into a single deep neural network using an online Bayesian inference algorithm, as done by the Bayesian Dark Knowledge (BDK) approach in~\\citep{korattikara2015bayesian}. Each iteration of this distillation procedure first generates a sample from the client's posterior distribution using the stochastic Langevin gradient descent (SGLD) algorithm~\\citep{welling2011bayesian} and incrementally ``injects'' into a deep neural network $\\mathcal{S}$ (referred to as ``student'') with parameters $w$, representing a succinct form of the client's (approximate) PPD. This is illustrated by each of the client blocks shown in Fig.~\\ref{fig:fedppd_diagram}. For client $k$, assuming the set of samples generated by SGLD to be $\\theta_k^{(1)},\\ldots,\\theta_k^{(S)}$, this distillation procedure can be seen as learning the parameters $w_k$ of the client $k$'s student model $\\mathcal{S}_k$ by minimizing the following loss function~\\citep{korattikara2015bayesian}\n\\begin{equation}\n    \\hat{L}(w_k) = - \\frac{1}{S} \\sum_{i=1}^S \\sum_{x^\\prime \\in \\mathcal{D}^\\prime_k} \\mathbb{E}_{p(y | x', \\theta_k^{(i)})} \\log \\mathcal{S}_k\n\n\\end{equation}\nNote that, in the above equation, to compute the loss, we use an unlabeled distillation dataset $\\mathcal{D}_k^\\prime$ at client $k$. This unlabeled dataset can be generated from the original labeled dataset $\\mathcal{D}_k$ by adding perturbations to the inputs, as suggested in~\\citep{korattikara2015bayesian}. \n\nWe sketch the full algorithm for optimizing for $w_k$ in the Supplementary Material. We use this algorithm at each client to learn the student model $\\mathcal{S}_k$ which represents a compact approximation of the client $k$'s PPD in form of a single deep neural network (as shown in the client block in Fig.~\\ref{fig:fedppd_diagram}), which can be now communicated to the server just like client models are communicated in standard federated learning algorithms. Note that, as shown in Fig.~\\ref{fig:fedppd_diagram}, in our federated setting, in addition to weights $w_k$ of its PPD approximation (the student model), each client $k$ also sends the posterior sample $\\theta_k^{MAP}$ (the sample with the largest posterior probability), to the server.\n\\subsection{Aggregation of Client Models}\n\\label{sec:aggr}\nAs described in Sec.~\\ref{sec:fedbdk-1}, the server receives two models from client $k$ - the (approximate) MAP sample $\\theta_k^{MAP}$ (the teacher) as well as the (approximate) PPD $w_k$ (the student). We denote the teacher models (approximate MAP samples) from the $K$ clients as $\\{\\theta_1^{MAP},\\ldots,\\theta_K^{MAP}\\}$ and the respective student models (approximate PPD) as $\\{w_1,\\ldots,w_K\\}$. These models need to be aggregated and then sent back to each client for the next round. We denote the server aggregated quantities for the teacher and student models as $\\theta_g$ and $w_g$ (we use $g$ to refer to ``global'').\n\nIn this work, we consider and experiment with three aggregation schemes on the server.\n\n\\textbf{Simple Aggregation of Client Models:} Our first aggregation scheme (shown in Algorithm~\\ref{algo-agg}) computes dataset-size-weighted averages of all the teacher models and all the student models received at the server. Denoting the number of training examples at client $k$ as $n_k$ and $N = \\sum_{k=1}^K n_k$, we compute $\\theta_g = \\frac{1}{N}\\sum_{k=1}^K n_k\\theta_k^{MAP}$ and $w_g = \\frac{1}{N}\\sum_{k=1}^K n_k w_k$, similar to how FedAvg algorithm~\\citep{mcmahan2017communication} aggregates client models on the server.\n\n\\textbf{Client-entropy-based Aggregation of Client Models:} Our second aggregation scheme (shown in Algorithm~\\ref{algo-ent}) uses an estimate of each client model's uncertainty to perform an importance-weighted averaging of the student models from all the clients. For each client $k$, we apply its student model $w_k$ on an unlabeled dataset available at the server and compute the average entropy of predictions on this entire dataset. Denoting the average predictive entropy of client $k$ to be $e_k$, we calculate an importance weight for client $k$ as $I_k = n_k\/e_k$. Essentially, a client with larger predictive entropy will receive a smaller importance weight. Using these importance weights, the student models (PPD weights) are aggregated as $w_g = \\frac{1}{\\sum_{k=1}^K I_k}\\sum_{k=1}^K I_k w_k$. For teacher models, however, we still use the simple dataset-size-weighted average as $\\theta_g = \\frac{1}{N}\\sum_{k=1}^K n_k\\theta_k^{MAP}$.\n\n\\textbf{Distillation-based Aggregation of Client Models:} Our third aggregation scheme goes beyond computing (weighted) averages of models received only from the clients. The motivation behind this approach is that the client models (both teachers as well as students) received at the server may not be diverse enough to capture the diversity and heterogeneity of the clients~\\citep{chen2020fedbe}. To address this issue, this approach (shown in Algorithm~\\ref{algo-distill})  first fits two probability distributions, one over the $K$ teacher models and the other over the $K$ student models received from the clients. It then uses these distributions to generate $M$ \\emph{additional} client-\\emph{like} teacher models and student models. Using the actual teacher models (resp. student models) and the additionally \\emph{generated} teacher models (resp. student models), we perform knowledge distillation on the server to compute the global teacher model $\\theta_g$ and the global student model $w_g$. This server-side distillation procedure requires an \\emph{unlabeled} dataset $\\mathcal{U}$ on the server. Applying the actual and generated teacher models (resp. student models) on the unlabeled dataset $\\mathcal{U}$ gives us pseudo-labeled data $\\mathcal{T}$ where each pseudo-label is defined as the averaged prediction (softmax probability vector) obtained by applying the actual and generated teacher models (resp. student models) to an unlabeled input. For the distillation step, we finally run the Stochastic Weighted Averaging (SWA) algorithm~\\citep{izmailov2018averaging} using the pseudo-labeled data $\\mathcal{T}$ and the simple aggregation of the client models as initialization. Both $\\theta_g$ and $w_g$ can be obtained by following this procedure in an identical manner. Recently, this idea was also used in Federated Bayesian Ensemble (FedBE)~\\citep{chen2020fedbe}. However, FedBE is \\emph{not} Bayesian in the sense that the clients still perform point estimation and it is only the distillation step where a distribution is fit over the client models to generate a more diverse ensemble of models which are distilled using the SWA algorithm to get the global model. \n\n\\begin{comment}\nOur method proposes to use and experiments with three forms of aggregation based on availability of unlabelled data at the server. The aggregation process for version 1 is based on Federated Averaging and is highlighted in Algorithm \\ref{alg:alg2}.\n\nThe aggregation process for version 2 is based on Federated Bayesian Ensemble and is highlighted in Algorithm \\ref{alg:alg3}. It creates a probability distribution over the given samples by treating the individual samples as points from the distribution. Then, using a few more points from the distribution, it approximates an ensemble from the given model.\n\nThe third approach utilizes the perspective that the student model can be a good estimator for entropy since it outputs Posterior Predictive Distribution. Hence, we can use entropy estimates of some sample data along with the original data size (used in Federated Averaging) for aggregation at server. The procedure for the same is highlighted in \\ref{alg:alg4}.\n\n\\end{comment}\n\n\\begin{comment}\nWe aim to learn a global model in federated setting with focus on the underlying uncertainty and propose to learn Predictive Posterior Distribution (PPD) on clients. We sketch our approach in Algorithm~\\ref{algo} and illustrate it in Figure~\\ref{fig:fedppd_diagram}.\n\nIn federated learning, each client holds a local dataset which cannot be shared with other clients\/server. This gets further challenging if data is distributed in non-iid fashion across clients, leading to a possibility of clients models differing substantially. Thus, instead of a point-estimate on clients local dataset, we propose to consider clients' predictions averaged over its posterior distribution. Unfortunately, computing posterior distribution is intractable and computationally inefficient given the large space of model parameters. So, we follow~\\citep{korattikara2015bayesian} to approximate the true posterior by drawing samples from it and distilling them into a single model representing PPD. Basically, each client updates its local model (a.k.a teacher model) for $k$ steps using SGLD optimizer and distills the knowledge of the updated model into a second model (a.k.a student model); $k$ being the hyperparameter. This local training continues for fixed number of update steps or until the local model has converged. Note that any sampling method can be used to draw samples from the posterior and is not limited to SGLD used in~\\citep{korattikara2015bayesian}.\n\n\nThe method proposed in the paper aims to perform Bayesian inference in a federated learning setting. This is done by constructing a model at local clients and global server that outputs Posterior Predictive Distribution instead of predictions derived from a single MAP\/MLE estimate.\n\nHowever, the posterior and predictive posterior distribution for models like neural networks are intractable. So, through many works, there has been an attempt to form approximation over the posterior to derive Posterior Predictive Distribution. An interesting approach to this is to model the Posterior Predictive Distribution using a distillation based training. The method, Bayesian Dark Knowledge, does not aim to approximate the posterior but to output Posterior Predictive Distribution for a given input. The objective function of this approach is to train a student model that approximates the Posterior Predictive Distribution of the teacher network. It can be expressed as below:\n\\begin{gather*}\n    \\hat{L}(w | x') = - \\frac{1}{| \\Theta |} \\sum_{\\theta^S \\in \\Theta} \\mathbb{E}_p(y | x', \\theta^S) \\log S(y | x, w')\n\\end{gather*}\nThis loss function is on a single data point x', but to integrate it over the given domain, a dataset $D'$ is taken. The teacher model continuously generates samples from posterior $p(\\Theta | D)$ by training on the original dataset $D$ which are distilled into the student model $S$ using $D'$ making it behave as probabilistic weighted aggregate of output from samples frandom posterior. The updates for teacher and student model are highlighted in the Bayesian Dark Knowledge (BDK) algorithm demonstrated in \\{cite paper\\}. The Bayesian Dark Knowledge approach in their work used SGLD for generating posterior samples.\n\nIn the proposed approach, we assume two models present globally that learn the posterior and posterior predictive distribution using updates from local clients. At the beginning of each round, the local clients initialize their teacher and student network from weights of the central server teacher and student models. They update their individual teacher and student models following the methodology of Bayesian Dark Knowledge by using the data generated\/stored within them locally. At the end of the round, they communicate the teacher and student models to the central server, where the central server aggregates the weights from these models, which are then used by the clients in the next round. The framework for the same is outlined in Algorithm \\ref{alg:alg1}.\n\nOur method proposes to use and experiments with three forms of aggregation based on availability of unlabelled data at the server. The aggregation process for version 1 is based on Federated Averaging and is highlighted in Algorithm \\ref{alg:alg2}.\n\nThe aggregation process for version 2 is based on Federated Bayesian Ensemble and is highlighted in Algorithm \\ref{alg:alg3}. It creates a probability distribution over the given samples by treating the individual samples as points from the distribution. Then, using a few more points from the distribution, it approximates an ensemble from the given model.\n\nThe third approach utilizes the perspective that the student model can be a good estimator for entropy since it outputs Posterior Predictive Distribution. Hence, we can use entropy estimates of some sample data along with the original data size (used in Federated Averaging) for aggregation at server. The procedure for the same is highlighted in \\ref{alg:alg4}.\n\n\\end{comment}\n\n\\begin{comment}\nWe have performed experiments using the above proposed approaches and a few baselines on classification task for a few datasets. A few additional experiments have also been performed to measure the performance of the above approaches on federated learning related tasks. The description of experiments, experimental settings and results for all the experiments are shown in the Experimental Results section.\n\\end{comment}\n\n\\begin{minipage}[t]{0.5\\textwidth}\n\\begin{algorithm}[H]\n\\caption{FedPPD}\n\\label{algo}\n\\begin{algorithmic}[1]\n\\Require {Number of communication rounds $T$, \\newline\nTotal clients $K$, \\newline Unlabeled dataset $\\mathcal{U}  = \\{x_i\\}_{i=1}^P$ \\newline Server teacher model weights $\\theta_g$, \\newline Server student model weights $w_g$, \\newline Client teacher model weights $\\{\\theta_i\\}_{i=1}^{K}$, \\newline Client student model weights $\\{w_i\\}_{i=1}^K$ \\newline Number of training samples at client $\\{n_i\\}_{i=1}^K$ \\newline}\n\n\\For{ each round $t = 0, \\ldots, T-1$}\n    \\State Server broadcasts $\\theta_g^{(t)}$ and $w_g^{(t)}$ \\newline \n    \\For{ each client $i \\in \\{1, \\dots, K\\}$}\n        \\State $\\theta_i   = \\theta_g^{(t)}$\n        \\State $w_i = w_g^{(t)}$\n        \\State Update $\\theta_i$ and $w_i$ locally as per \\citep{korattikara2015bayesian}\n    \\EndFor\n    \\State Communicate $\\{\\theta_i^{MAP}\\}_{i=1}^K$ and $\\{w_i\\}_{i=1}^K$ to server \\newline\n    \\State $\\theta_g^{(t+1)}$, $w_g^{(t+1)}$ = \\newline Server\\_Update($\\{\\theta_i^{MAP}\\}_{i=1}^K,  \\{w_i\\}_{i=1}^K, \\{n_i\\}_{i=1}^K$) \\newline\n\\EndFor \\newline\n\\State \\Return $w_g$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}[H]\n\\caption{Server\\_Update(Average)}\n\\label{algo-agg}\n\\begin{algorithmic}[1]\n\\Require $\\{\\theta_i^{MAP}\\}_{i=1}^{K}$, $\\{w_i\\}_{i=1}^K$, $\\{n_i\\}_{i=1}^K$ \\newline\n\\State $N = \\sum_{i=1}^K n_i$\n\\State \\Return $\\frac{1}{N}\\sum_{i=1}^K n_i \\theta_i^{MAP}, \\frac{1}{N}\\sum_{i=1}^K n_i w_i$ \\newline\n\\end{algorithmic}\n\\end{algorithm}\n\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{0.45\\textwidth}\n\n\n\\begin{algorithm}[H]\n\\caption{Server\\_Update(Entropy)}\n\\label{algo-ent}\n\\begin{algorithmic}[1]\n\\Require $\\mathcal{U}, \\{\\theta_i^{MAP}\\}_{i=1}^{K}, \\{w_i\\}_{i=1}^K, \\{n_i\\}_{i=1}^K$ \\newline\n\\State $\\theta_g = \\frac{1}{\\sum_{i=1}^K n_i}\\sum_{i=1}^K n_i \\theta_i^{MAP}$ \\newline\n\\State $I = [ ]$ \\Comment{Clients' importance weights}\n\\For {client $i = 1, \\dots, K$}\n    \\State $I[i] = n_i\/Entropy(w_i, \\mathcal{U})$\n\\EndFor \\newline\n\\State $w_g = \\frac{1}{\\sum_{i=1}^K I[i]}\\sum_{i=1}^K I[i] w_i$ \\newline\n\\State \\Return $\\theta_g, w_g$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}[H]\n\\caption{Server\\_Update(Distill)}\n\\label{algo-distill}\n\\begin{algorithmic}[1]\n\\Require $\\mathcal{U}, \\{\\theta_i^{MAP}\\}_{i=1}^{K}, \\{w_i\\}_{i=1}^K, \\{n_i\\}_{i=1}^K$ \\newline\n\\State $\\overline{\\theta}, \\overline{w} = $ Server\\_Update(Average)\n\\newline\n\n\\State Construct global teacher model distribution $p(\\theta | D)$ from $\\{\\theta_i^{MAP}\\}_{i=1}^K$\n\\State Sample $M$ additional teachers and form teacher ensemble\n\\newline $E_T=\\{\\theta_m \\sim p(\\theta | \\mathcal{D})\\}_{m=1}^{M} \\cup \\{\\overline{\\theta}\\} \\cup \\{\\theta_i\\}_{i=1}^{K}$\n\\State Annotate $\\mathcal{U}$ using $E_T$ to generate pseudo-labeled dataset $\\mathcal{T}$ \n\\State Distill $E_T$ knowledge to $\\overline{\\theta}$ using SWA \n\\newline\n$\\theta_g = SWA(\\overline{\\theta}, E_T, \\mathcal{T})$ \n\\newline\n\n\\State Similarly follow steps 2-5 with $\\{w_i\\}_{i=1}^K$ to get $w_g$\n\\newline\n\\State \\Return $\\theta_g, w_g$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\end{minipage}\n\n\n\nThe overall sketch of our Bayesian federated learning procedure, which we call FedPPD (Federated Learning via Posterior Predictive Distributions), is shown in Algorithm~\\ref{algo}. The three aggregation schemes for server-side updates are shown in Algorithm~\\ref{algo-agg}, \\ref{algo-ent}, and \\ref{algo-distill}. Note that, among the 3 aggregation schemes, Algorithm~\\ref{algo-agg} does not require any unlabeled data at the server whereas \\ref{algo-ent}, and \\ref{algo-distill} assume that the server has access to unlabeled data. Also, owing to high computation capacity, server can compute $\\theta_g$ and $w_g$ in parallel for all the aggregation schemes; incurring no additional delays in communication rounds.\n\n\\section{Related Work}\n\nFederated learning has received considerable research interest recently. The area is vast and we refer the reader to excellent surveys~\\citep{li2020federated,kairouz2021advances} on the topic for a more detailed overview. In this section, we discuss the works that are the most relevant to our work.\n\nWhile standard federated learning approaches assume that each client does point estimation of its model weights by optimizing a loss function over its own data, recent work has considered posing federated learning as a posterior inference problem where a global posterior distribution is inferred by aggregating local posteriors computed at each client. FedPA~\\citep{al2020federated} is one such recent approach which performs approximate inference for the posterior distribution of each client's weights. However, it assumes a restrictive form for the posterior (Gaussian). Moreover, the method needs to estimate the covariance matrix of the Gaussian posterior, which is difficult in general and approximations are needed. Moreover, although FedPA estimates the (approximate) posterior on each client, due to efficiency\/communication concerns, at the server, it only computes a point estimate (the mean) of the global posterior. Thus, even though the approach is motivated from a Bayesian setting, in the end, it does not provide a posterior distribution or a PPD for the global model.\n\nRecently, \\citep{linsner2021approaches} presented methods for uncertainty quantification in federated learning using a variety of posterior approximation methods for deep neural networks, such as Monte Carlo dropout~\\citep{gal2016dropout}, stochastic weight averaging Gaussian (SWAG)~\\citep{maddox2019simple}, and deep ensembles~\\citep{lakshminarayanan2017simple}. These approaches, however, also suffer from poor quality of approximation of the posterior at each client. \\citep{lee2020bayesian} also propose a Bayesian approach for federated learning. However, their approach also makes restrictive assumptions, such as the distribution of the gradients at each of the clients being jointly Gaussian.\n\nInstead of a simple aggregation of client models at the server, FedBE~\\citep{chen2020fedbe} uses the client models to construct a distribution at the server and further distills this distribution into a single model using distillation. This model is then sent by the server to each client for the next round. Though maintaining a distribution over client models and distilling it in a single model is more robust than a simple aggregation like federated averaging, FedBE only performs point estimation ignoring any uncertainty in the client models. Another probabilistic approach to federated learning \\cite{thorgeirsson2020probabilistic} fits a Gaussian distribution using the client models, and sends the mean of this Gaussian to each client for the next round of client model training. This approach too does not estimate a posterior at each client, and thus ignores the uncertainty in client models. \n\nIn the context of Bayesian learning, recent work has also explored federated versions of Markov Chain Monte Carlo sampling algorithms, such as stochastic gradient Langevin dynamics sampling~\\citep{lee2020bayesian,el2021federated}. While interesting in their own right in terms of performing MCMC sampling in federated settings, these methods are not designed with the goal of real-world applications of federated learning, where fast prediction and compact model sizes are essential.\n\nAmong other probabilistic approaches to federated learning, recent work has explored the use of latent variables in federated learning. In \\cite{louizos2021expectation}, a hierarchical prior is used on client model's weights where the prior's mean is set to the server's global model, and additional latent variables can also be used to impose other structures, such as sparsity of client model weights. However, these approaches do not model the uncertainty in the client model.\n\nSome of the recent work on federated learning using knowledge distillation is also relevant. Note that our work leverages the ideas of teacher-student distillation, both at the clients (when learning a representation of the PPD using a single deep neural network), as well as in our third aggregation strategy where server-side distillation is used for learning the global model. In federated learning, the idea of distillation has been used in other works as well, such as federated learning when the client models are of different sizes and the (weighted) averaging does not make sense due to the different size\/architecture of different client models~\\citep{zhu2021data}.\n\n\\section{Experiments}\nIn this section, we compare our Bayesian federated learning approach with various relevant baselines on several benchmark datasets. We report results on the following tasks: (1) Classification in federated setting, (2) Active Learning in federated setting, and (3) OOD detection on each client. In this section, we refer to our approach with simple averaging on the server side as FedPPD, the variant with entropy-weighted averaging on the server side as FedPPD+Entropy, and the variant with distillation based aggregation on the server side as FedPPD+Distill.  \n\n\\subsection{Experimental Setup}\n\\subsubsection{Baselines} We compare our methods with the following baselines\n\n(1) \\textbf{FedAvg}~\\citep{mcmahan2017communication} is the standard federated algorithm in which the local models of the participating clients are aggregated at server to compute a global model which is then sent back to all the clients for initialization in the next round.\n\n(2) \\textbf{FedBE}~\\citep{chen2020fedbe} is another state-of-the-art baseline which provides a more robust aggregation scheme in which instead of only averaging the client models at the server, a probability distribution is fit using the client models, several other models are generated from this probability distribution, and then the client models as well as the generated models are distilled into a single model to yield the global model at the server, which is sent to all the clients for initialization in the next round. Note however that the clients in FedBE only perform point estimation of their weights unlike our approach which estimates the posterior distribution and the PPD of each client. \n\n(3) \\textbf{Federated SWAG}~\\citep{linsner2021approaches} is a Bayesian federated learning algorithm which is essentially based on a federated extension of the SWAG~\\cite{maddox2019simple} which is an efficient Bayesian inference algorithm for deep neural networks. However, Federated SWAG relies on a simplification that it executes standard federated averaging for all except the last round and in the last round, the SWAG algorithm is invoked at each client to yield a posterior. Also note that Federated SWAG requires Monte-Carlo sampling at test time (thus relying on ensemble based slow prediction) unlike our method which only requires a single neural network to make the prediction.\n\nWe also considered a comparison with \\textbf{FedPA}~\\citep{al2020federated} which estimates a posterior (assumed to be Gaussian) over the client weights. However, in our experiments (using the author-provided code and suggested experimental settings) on the benchmark dataset, FedPA performed comparably or worse than FedAvg. We therefore omit those results from the main text and report them in the Supplementary Material. \n    \n\\begin{comment}\n\n\\end{comment}\n\n\\subsubsection{Datasets}\nWe evaluate and compare our approach with baseline methods on four datasets: MNIST~\\citep{lecun-mnisthandwrittendigit-2010}, FEMNIST~\\citep{cohen2017emnist}, and CIFAR-10\/100~\\citep{krizhevsky2009learning}. MNIST comprises of images of handwritten digits categorized into 10 classes. It has a total of 60,000 images for training and 10,000 images for testing. FEMNIST consists of images of handwritten characters (digits, lowercase, and uppercase alphabets resulting in total of 62 classes) written by multiple users. It has a total of 80,523 images written by 3,550 users. CIFAR-10 consists of $32\\times32$ dimensional RGB images categorised into 10 different classes. It has a total of 50,000 images for training and 10,000 images for testing. CIFAR-100 is similar to CIFAR-10 but has 100 distinct classes.\n\n\\subsubsection{Model Architecture and Configurations}\n\\label{sec:config}\nIn all our experiments, the student model has a larger capacity compared to teacher model as it is modeling the PPD by distilling multiple models drawn from the posterior distribution. We have used a customized CNN architecture for both teacher and student model on MNIST, FEMNIST and CIFAR-10 dataset, with student model being deeper and\/or wider than its corresponding teacher model. For CIFAR-100, ResNet-18 and ResNet-34 are used as the teacher and student model, respectively.\n\nIn all our experiments, we consider $K=10$ clients with data heterogeneity. Each client holds a small non-i.i.d. subset of training data - approximately 2000 samples for FEMNIST, CIFAR-10 and CIFAR-100 and around 500 samples for MNIST. We use the Leaf~\\citep{caldas2018leaf} benchmark to distribute FEMNIST data across clients based on the writer. We have also excluded digits and considerd only alphabets to increase the class imbalance. However, we let clients have data from multiple writers ensuring that no two clients have same writer assigned. This results in almost similar class distribution but with different styled handwritten images distributed across clients. Also, this setting is different from data distribution on other datasets where each client strictly maintains a small subset of all the classes. In case of MNIST and CIFAR-10, we ensure that there are atmost 2 major classes per client and upto 20 distinct major classes in case of CIFAR-100. For a fair comparison, we run our method and all the baselines for 200 rounds on all the datasets (except MNIST where we run it for 100 rounds) and train local client model for 10 epochs in each round. Also, we assume complete client participation i.e. all the clients are considered in each round. However, we tune the learning rate, momentum and weight decay for each method independently. For FedBE and FedPPD, we run additional 20 and 50 epochs at the server for distillation on CIFAR\/MNIST and FEMNIST datasets, respectively. \n\n\n\n\\begin{wrapfigure}{R}{0.4\\textwidth}\n\\vspace{-5em}\n    \\centering\n    \\includegraphics[scale=0.4]{figures\/CIFAR10_Acc_Plot.pdf}\n    \\caption{Convergence of all the methods on CIFAR-10 dataset}\n    \\label{fig:convergence_plot}\n    \\vspace{-1em}\n   \n   \n   \n   \n   \n   \n   \n   \n\\end{wrapfigure}\n\n\\subsection{Tasks}\n\\textbf{Classification} \nWe evaluate FedPPD (its three variants) and the baselines on several classification datasets and report the accuracy on the respective test datasets. The results are shown in Table~\\ref{tab:classification_acc}. We also show the convergence of all the methods in on CIFAR-10 in Figure~\\ref{fig:convergence_plot} (we show similar plots for other datasets in Supplementary Material). All the three variants of FedPPD outperform the other baselines on all the datasets. As compared to the best performing baseline, our approach yields an improvement of $4.44\\%$ and $7.08\\%$ in accuracy on CIFAR-10 and CIFAR-100, respectively. On MNIST and FEMNIST datasets too, we observe noticeable improvements. The improvements across the board indicate that FedPPD and its variants are able to leverage model uncertainty to yield improved predictions especially when the amount of training data per client is small, which is the case with the experimental settings that we follow (as we mention in Sec.~\\ref{sec:config}). We also observe that in cases where there is a significant heterogeneity in the data distribution across the different clients (on CIFAR-10 and CIFAR-100), the performance gains offered by FedPPD and its variants are much higher as compared to the baselines. On other datasets (MNIST and FEMNIST), the data distributions are roughly similar across different clients and even though the accuracies are higher, the performance gains are not as significant, but reasonable nevertheless.\n\n\n\n\n\\begin{table}[!htbp]\n\\vspace{-2em}\n    \\setlength\\tabcolsep{1pt}\n   \n    \\begin{minipage}{0.5\\linewidth}\n    \\begin{tabular}{ccccc}\n        \\toprule\n        Model & MNIST & FEMNIST & CIFAR-10 & CIFAR-100 \\\\\n        \\midrule\n        FedAvg         & 97.74 & 87.40 & 57.20 & 47.02 \\\\\n        FedAvg+SWAG    & 97.75 & 87.45 & 57.34 & 47.07 \\\\\n        FedBE          & 97.82 & 88.12 & 60.18 & 47.52 \\\\\n        FedPPD         & 97.85 & \\textbf{88.81} & 61.86 & 53.00 \\\\\n        FedPPD+Entropy & 97.93 & 88.65 & 62.19 & 52.72 \\\\ \n        FedPPD+Distill & \\textbf{98.08} & 88.80 & \\textbf{64.62} & \\textbf{54.60} \\\\\n       \n        \\bottomrule\n    \\end{tabular}\n    \\caption{Federated classification test accuracies on benchmark datasets}\n    \\label{tab:classification_acc}\n    \\end{minipage}\n    \\hfill\n    \\begin{minipage}{0.38\\textwidth}\n        \\begin{figure}[H]\n            \\small\n            \\includegraphics[height=4cm,width=5cm]{figures\/AL_Acc_Plot.pdf}\n            \\caption{Federated Active Learning on CIFAR-10 dataset. Note: FedAvg+SWAG performed almost similarly to FedAvg on this task as well so we skip it from the plot.}\n            \\label{fig:al_results}\n        \\end{figure}\n    \\end{minipage}\n    \\vspace{-0.5cm}\n\\end{table}\n\n\\textbf{Federated Active Learning} We further show the usefulness of our Bayesian approach by using it for the problem of federated active learning. In active learning, the goal of the learner is to iteratively request the labels of the most informative inputs instances, add these labeled instances to the training pool, retrain the model, and repeat the process until the labeling budget remains. Following~\\citep{ahn2022federated}, we extend our method and the baselines to active learning in federated setting using entropy of the predictive distribution of an input $x$ as the acquisition function, defined as $I(x) = -\\sum_{i=1}^{k} p(y=y_i|x) \\log p(y=y_i|x)$  as a score function. In federated active learning setting (we provide a detailed sketch of the federated active learning algorithm in the Supplementary Material), each client privately maintains a small amount of labeled data and a large pool of unlabeled examples. In each round of active learning, clients participate in federated learning with their currently labeled pool of data until the global model has converged. Now, each client uses the global model to identify a fixed number (budget) of the most informative inputs among its pool of unlabeled input based on the highest predictive entropies $I(x)$, which are then annotated (locally maintaining data privacy) and added to the pool of labeled examples. Now, with this, next round of active learning begins, where clients will participate in federated learning and use the global model to expand their labeled pool. This process continues until either the unlabeled dataset has been exhausted completely or desired accuracy has been achieved. For a fair comparison, we have run federated active learning on CIFAR-10 dataset with same parameters for all the approaches. We start active learning with 400 labeled and 3200 unlabeled samples at each client and use a budget of 400 samples in every round of active learning. For federated learning, we use the same hyperparameters as for the classification experiments. We stop federated active learning once all the clients have exhausted their unlabeled dataset and show the results in Figure~\\ref{fig:al_results}. FedPPD and its variants attain the best accuracies among all the methods compared, which shows that our Bayesian approach provides more robust estimates of the model and predictive uncertainty as compared to the other baselines, and thus outperforms them on the problem of federated active learning.\n\n\\textbf{Out-of-distribution (OOD) detection} We also evaluate FedPPD and its variants, and the other baselines, in terms of their ability to distinguish between Out-of-Distribution (OOD) data and data used during training phase (in-distribution data). For this, given any sample $x$ to be classified among $k$ distinct classes and model weights $\\theta$ (or PPD for our approach), we compute Shannon entropy of the model's predictive distribution for the input $x$ and compute the AUROC (Area Under ROC curve) metric. We use KMNIST as OOD data for models trained on FEMNIST, and SVHN for CIFAR-10\/CIFAR-100 models. Note that, to avoid class imbalance, we sample an equal amount of data for both the distributions (out and in) and repeat it 5 times. We report the results in Table~\\ref{tab:auroc_score}. FedPPD and its variants consistently result in better AUROC scores on all the datasets validating its robustness and accurate estimates of model uncertainty. In addition to OOD detection, we also apply all the methods for the task of identifying the correct predictions and incorrect predictions based on the predictive entropies. For this task too, FedPPD and its variants outperform the other baselines. \n\n\\begin{table}[!htbp]\n\\setlength\\tabcolsep{4pt}\n    \\centering\n    \\scriptsize\n    \\makebox[\\textwidth][c]{\n    \\begin{tabular}{|c|c|c|c|c|c|c|}\n    \\toprule      & \\multicolumn{3}{c|}{Out of Domain Detection} & \\multicolumn{3}{c|}{Correct\/Incorrect Prediction}\\\\\n\\midrule\n        Model & FEMNIST & CIFAR-10 & CIFAR-100  & FEMNIST & CIFAR-10 & CIFAR-100 \\\\\n        \\hline\n        FedAvg         & $0.957 \\pm 0.003$ & $0.728 \\pm 0.013$  & $0.703 \\pm 0.011$ & $0.846 \\pm 0.011$ & $0.742 \\pm 0.011$  & $0.792 \\pm 0.003$ \\\\\n        FedAvg+SWAG    & $0.956 \\pm 0.003$ & $0.728 \\pm 0.013$  & $0.704 \\pm 0.011$ & $0.845 \\pm 0.009$ & $0.743 \\pm 0.010$  & $0.800 \\pm 0.004$\\\\\n        FedBE          & $0.959 \\pm 0.002$ & $0.728 \\pm 0.006$  & $0.669 \\pm 0.009$ & $\\mathbf{0.863 \\pm 0.005}$ & $0.753 \\pm 0.007$  & $0.789 \\pm 0.005$\\\\\n        FedPPD         & $\\mathbf{0.983 \\pm 0.003}$ & $0.701 \\pm 0.007$  & $0.698 \\pm 0.009$ & $0.862 \\pm 0.008$ & $0.755 \\pm 0.007$  & $0.814 \\pm 0.003$\\\\\n        FedPPD+Entropy & $0.982 \\pm 0.002$ & $\\mathbf{0.768 \\pm 0.009}$  & $0.721 \\pm 0.014$ & $0.856 \\pm 0.006$ & $0.749 \\pm 0.007$  & $0.817 \\pm 0.004$\\\\ \n        FedPPD+Distill & $0.975 \\pm 0.002$ & $0.765 \\pm 0.006$  &  $\\mathbf{0.784 \\pm 0.008}$ & $0.853 \\pm 0.013$ & $\\mathbf{0.769 \\pm 0.006}$  &  $\\mathbf{0.823 \\pm 0.002}$\\\\\n       \n        \\bottomrule\n    \\end{tabular}\n    }\n    \\caption{AUROC scores for OOD detection and correct\/incorrect predictions}\n    \\label{tab:auroc_score}\n\\end{table}\n\\vspace{-1cm}\n\n\\begin{comment}\n\\begin{table}[!htbp]\n    \\tiny\n    \\begin{tabular}{cccc}\n        \\toprule\n        Model & FEMNIST & CIFAR-10 & CIFAR-100 \\\\\n        \\midrule\n        FedAvg         & $0.957 \\pm 0.003$ & $0.728 \\pm 0.013$  & $0.703 \\pm 0.011$ \\\\\n        FedAvg+SWAG    & $0.956 \\pm 0.003$ & $0.728 \\pm 0.013$  & $0.704 \\pm 0.011$\\\\\n        FedBE          & $0.966 \\pm 0.003$ & $0.728 \\pm 0.006$  & $0.669 \\pm 0.009$ \\\\\n        FedPPD         & $\\mathbf{0.983 \\pm 0.003}$ & $0.701 \\pm 0.007$  & $0.698 \\pm 0.009$ \\\\\n        FedPPD+Entropy & $0.982 \\pm 0.002$ & $\\mathbf{0.768 \\pm 0.009}$  & $0.721 \\pm 0.014$ \\\\ \n        FedPPD+Distill & $0.949 \\pm 0.003$ & $0.765 \\pm 0.006$  &  $\\mathbf{0.784 \\pm 0.008}$ \\\\\n       \n        \\bottomrule\n    \\end{tabular}\n    \\caption{AUROC score for OOD data detection}\n    \\label{tab:auroc_correct}\n\\end{table}\n\n\\begin{table}[t]\n    \\centering\n    \\begin{tabular}{ >{\\centering\\arraybackslash}p{4cm} >{\\centering\\arraybackslash}p{4cm} }\n        \\toprule\n        Model & CIFAR-10\\\\\n        \\midrule\n        FedAvg         &  51.54\\\\\n        FedBE          &  54.29\\\\\n        FedPPD         & 58.87\\\\\n        FedPPD+Distill & 57.44\\\\\n       \n        \\bottomrule\n    \\end{tabular}\n    \\caption[caption]{\\centering Classification accuracy on test \\hspace{\\linewidth} dataset\n     for federated active learning experiment}\n    \\label{tab:active_learning}\n\\end{table}\n\\end{comment}\n\n\\vspace{0.5em}\n\\section{Conclusion and Discussion}\n\\vspace{-0.5em}\nCapturing and leveraging model uncertainty in federated learning has several benefits as we demonstrate in this work. To achieve this, we developed a Bayesian approach to federated learning by leveraging the idea of distilling the posterior predictive into a single deep neural network. The Bayesian approach not only yields more accurate and robust predictions in federated learning in situations with limited training data at each client and heterogeneity across clients, but is also helpful for tasks, such as OOD detection and active learning in federated setting. Our work provides a general framework to solve Bayesian federated learning. In this work, we consider a specific scheme to distill the PPD at each client. However, other methods that can distill the posterior distribution into a single neural network~\\citep{wang2018adversarial,vadera2020generalized} are also worth leveraging for Bayesian federated learning. Another interesting future work will be to extend our approach to settings where different clients could possibly be having different model architectures. Finally, our approach first generates MCMC samples (using SGLD) and then uses these samples to obtain the PPD in form of a single deep neural network. Recent work has shown that it is possible to distill an ensemble into a single model without explicitly generating samples from the distribution~\\citep{ratzlaff2019hypergan}. Using these ideas for Bayesian federated learning would also be an interesting future work.  \n\n\\begin{comment}\n\\begin{ack}\nUse unnumbered first level headings for the acknowledgments. All acknowledgments\ngo at the end of the paper before the list of references. Moreover, you are required to declare\nfunding (financial activities supporting the submitted work) and competing interests (related financial activities outside the submitted work).\nMore information about this disclosure can be found at: \\url{https:\/\/neurips.cc\/Conferences\/2022\/PaperInformation\/FundingDisclosure}.\n\n\nDo {\\bf not} include this section in the anonymized submission, only in the final paper. You can use the \\texttt{ack} environment provided in the style file to automatically hide this section in the anonymized submission.\n\\end{ack}\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAt the frontier of computational statistics there is growing interest\nin parallel implementation of Monte Carlo algorithms using multi-processor\nand distributed architectures. However, the resampling step of sequential\nMonte Carlo (SMC) methods \\citep{gordon1993novel} (see \\citep{kunsch2013particle}\nfor a recent overview) which involves a degree of interaction between\nsimulated ``particles'', hinders their parallelization. So, whilst\nmulti-processor implementation offers some speed up for SMC, the potential\nbenefits of distributed computing are not fully realized \\citep{lee2010utility}. \n\nPerforming resampling only occasionally, a technique originally suggested\nfor the somewhat different reason of variance reduction \\citep{liu1995blind},\nalleviates this problem to some extent, but the collective nature\nof the resampling operation remains the computational bottleneck.\nOn the other hand, crude attempts to entirely do away with the resampling\nstep may result in unstable or even non-convergent algorithms. With\nthese issues in mind we seek a better understanding of the relationship\nbetween the interaction structure of SMC algorithms and theoretical\nproperties of the approximations they deliver. Our overall aim is\nto address the following question:\n\n\\smallskip{}\n\n\n\\emph{To what extent can the degree of interaction between particles\nbe reduced, whilst ensuring provable stability of the algorithm?}\n\n\\smallskip{}\nOur strategy is to introduce and study an unusually general type of\nSMC algorithm featuring a parameterized resampling mechanism. This\nprovides a flexible framework in which we are ultimately able to attach\nmeaning to \\emph{degree of interaction} in terms of graph-theoretic\nquantities. To address the matter of \\emph{provable stability}, we\nseek conditions under which the algorithm yields time-uniformly convergent\napproximations of prediction filters, and approximations of marginal\nlikelihoods whose relative variance can be controlled at a linear-in-time\ncost. \n\nThe general algorithm we study is defined in terms of a family of\nMarkov transition matrices, $\\alpha$, and we refer to the algorithm\nitself as $\\alpha$SMC. We shall see that through particular choices\nof $\\alpha$ one obtains, as instances of $\\alpha$SMC, well known\nalgorithms including sequential importance sampling (SIS), the bootstrap\nparticle filter (BPF) and the adaptive resampling particle filter\n(ARPF) in which resampling is triggered by monitoring some functional\ncriterion, such as the Effective Sample Size (ESS) \\citep{liu1995blind}. \n\nAlthough the ESS does not necessarily appear in the definition of\nthe general $\\alpha$SMC algorithm, we find that it does appear quite\nnaturally from the inverse quadratic variation of certain martingale\nsequences in its analysis. This allows us to make precise a sense\nin which algorithmic control of the ESS can guarantee stability of\nthe algorithm. Our results apply immediately to the ARPF, but our\nstudy has wider-reaching methodological consequences: in our framework\nit becomes clear that the standard adaptive resampling strategy is\njust one of many possible ways of algorithmically controlling the\nESS, and we can immediately suggest new, alternative algorithms which\nare provably stable, but designed to avoid the type of complete interaction\nwhich is inherent to the ARPF and which hinders its parallelization.\nThe structure of this paper and our main contributions are as follows. \n\nSection~\\ref{sec:aSMC} introduces the general algorithm, $\\alpha$SMC.\nWe explain how it accommodates several standard algorithms as particular\ncases and comment on some other existing SMC methods.\n\nSection~\\ref{sec:Martingale-approximations-and} presents Theorem\n\\ref{thm:convergence}, a general convergence result for $\\alpha$SMC.\nWe give conditions which ensure unbiased approximation of marginal\nlikelihoods and we elucidate connections between certain invariance\nproperties of the matrices  $\\alpha$ and the negligibility of increments\nin a martingale error decomposition, thus formulating simple sufficient\nconditions for weak and strong laws of large numbers. We also discuss\nsome related existing results.\n\nSection~\\ref{sec:stability} presents our second main result, Theorem\n\\ref{thm:L_R_mix}. We show, subject to regularity conditions on the\nhidden Markov model (HMM) under consideration, that enforcement of\na strictly positive lower bound on a certain coefficient associated\nwith ESS of $\\alpha$SMC is sufficient to guarantee non-asymptotic,\ntime-uniform bounds on: 1) the exponentially normalized relative second\nmoment of error in approximation of marginal likelihoods, and 2) the\n$L_{p}$ norm of error in approximation of prediction filters. The\nformer implies a linear-in-time variance bound and the latter implies\ntime-uniform convergence. These results apply immediately to the ARPF.\n\nSection~\\ref{sec:Discussion} houses discussion and application of\nour results. We point out the pitfalls of some naive approaches to\nparallelization of SMC and discuss what can go wrong if the conditions\nof Theorem~\\ref{thm:convergence} are not met. Three new algorithms,\nwhich adapt the degree of interaction in order to control the ESS\nand which are therefore provably stable, are then introduced. We discuss\ncomputational complexity and through numerical experiments examine\nthe degree of interaction involved in these algorithms and the quality\nof the approximations they deliver compared to the ARPF.\n\n\n\\section{$\\alpha$SMC\\label{sec:aSMC}}\n\nA hidden Markov model (HMM) with measurable state space $\\left(\\mathsf{X},\\mathcal{X}\\right)$\nand observation space $\\left(\\mathsf{Y},\\mathcal{Y}\\right)$ is a\nprocess $\\left\\{ \\left(X_{n},Y_{n}\\right);n\\geq0\\right\\} $ where\n$\\left\\{ X_{n};n\\geq0\\right\\} $ is a Markov chain on $\\mathsf{X}$,\nand each observation $Y_{n}$, valued in $\\mathsf{Y}$, is conditionally\nindependent of the rest of the process given $X_{n}$. Let $\\mu_{0}$\nand $f$ be respectively a probability distribution and a Markov kernel\non $\\left(\\mathsf{X},\\mathcal{X}\\right)$, and let $g$ be a Markov\nkernel acting from $\\left(\\mathsf{X},\\mathcal{X}\\right)$ to $\\left(\\mathsf{Y},\\mathcal{Y}\\right)$,\nwith $g(x,\\cdot)$ admitting a density, denoted similarly by $g(x,y)$,\nwith respect to some dominating $\\sigma$-finite measure. The HMM\nspecified by $\\mu_{0}$, $f$ and $g$, is\n\\begin{eqnarray}\n &  & X_{0}\\sim\\mu_{0}(\\cdot),\\quad\\left.X_{n}\\right|\\{X_{n-1}=x_{n-1}\\}\\sim f(x_{n-1},\\cdot),\\quad n\\geq1,\\label{eq:HMM}\\\\\n &  & \\;\\quad\\quad\\quad\\quad\\hspace{1.1em}\\quad\\left.Y_{n}\\right|\\left\\{ X_{n}=x_{n}\\right\\} \\sim g(x_{n},\\cdot),\\quad\\quad\\quad\\quad n\\geq0.\\nonumber \n\\end{eqnarray}\n\n\nWe shall assume throughout that we are presented with a fixed observation\nsequence $\\left\\{ y_{n};n\\geq0\\right\\} $ and write \n\\[\ng_{n}(x):=g(x,y_{n}),\\quad n\\geq0.\n\\]\nThe following assumption imposes some mild regularity which ensures\nthat various objects appearing below are well defined. It shall be\nassumed to hold throughout without further comment.\n\\begin{assumption*}\n$\\mathbf{\\mathbf{(A1)}}$ For each $n\\geq0$, $\\sup_{x}g_{n}(x)<+\\infty$\nand $g_{n}(x)>0$ for all $x\\in\\mathsf{X}$.\n\\end{assumption*}\nWe take as a recursive definition of the \\emph{prediction filters},\nthe sequence of distributions $\\left\\{ \\pi_{n};n\\geq0\\right\\} $ given\nby \n\\begin{eqnarray}\n &  & \\pi_{0}:=\\mu_{0},\\nonumber \\\\\n &  & \\pi_{n}\\left(A\\right):=\\frac{\\int_{\\mathsf{X}}\\pi_{n-1}\\left(dx\\right)g_{n-1}(x)f(x,A)}{\\int_{\\mathsf{X}}\\pi_{n-1}\\left(dx\\right)g_{n-1}(x)},\\quad A\\in\\mathcal{X},\\quad n\\geq1,\\label{eq:filtering_recursion}\n\\end{eqnarray}\nand let $\\left\\{ Z_{n};n\\geq0\\right\\} $ be defined by\n\\begin{equation}\nZ_{0}:=1,\\quad\\quad Z_{n}:=Z_{n-1}\\int_{\\mathsf{X}}\\pi_{n-1}\\left(dx\\right)g_{n-1}\\left(x\\right),\\quad n\\geq1.\\label{eq:Z_recusion}\n\\end{equation}\nDue to the conditional independence structure of the HMM, $\\pi_{n}$\nis the conditional distribution of $X_{n}$ given $Y_{0:n-1}=y_{0:n-1}$;\nand $Z_{n}$ is the marginal likelihood of the first $n$ observations,\nevaluated at the point $y_{0:n-1}$. Our main computational objectives\nare to approximate $\\left\\{ \\pi_{n};n\\geq0\\right\\} $ and $\\left\\{ Z_{n};n\\geq0\\right\\} $. \n\n\n\\subsection{The general algorithm}\n\nWith population size $N\\geq1$, we write $[N]:=\\{1,\\ldots,N\\}$. \\emph{To\nsimplify presentation, whenever a summation sign appears without the\nsummation set made explicit, the summation set is taken to be $[N]$,\nfor example we write $\\Sigma_{i}$ to mean $\\Sigma_{i=1}^{N}$. }\n\nThe $\\alpha$SMC algorithm involves simulating a sequence $\\left\\{ \\zeta_{n};n\\geq0\\right\\} $\nwith each $\\zeta_{n}=\\left\\{ \\zeta_{n}^{1},\\ldots,\\zeta_{n}^{N}\\right\\} $\nvalued in $\\mathsf{X}^{N}$. Denoting $\\mathbb{X}:=\\left(\\mathsf{X}^{N}\\right)^{\\mathbb{N}}$,\n$\\mathcal{F}^{\\mathbb{X}}:=\\left(\\mathcal{X}^{\\otimes N}\\right)^{\\otimes\\mathbb{N}}$,\nwe shall view $\\left\\{ \\zeta_{n};n\\geq0\\right\\} $ as the canonical\ncoordinate process on the measurable space $\\left(\\mathbb{X},\\mathcal{F}^{\\mathbb{X}}\\right)$,\nand write $\\mathcal{F}_{n}$ for the $\\sigma$-algebra generated by\n$\\left\\{ \\zeta_{0},\\ldots,\\zeta_{n}\\right\\} $. By convention, we\nlet $\\mathcal{F}_{-1}:=\\{\\mathbb{X},\\emptyset\\}$ be the trivial $\\sigma$-algebra.\nThe sampling steps of the $\\alpha$SMC algorithm, described below,\namount to specifying a probability measure, say $\\mathbb{P}$, on\n$\\left(\\mathbb{X},\\mathcal{F}^{\\mathbb{X}}\\right)$. Expectation w.r.t.~$\\mathbb{P}$\nshall be denoted by $\\mathbb{E}$. \n\nLet $\\mathbb{A}_{N}$ be a non-empty set of Markov transition matrices,\neach of size $N\\times N$. For $n\\geq0$ let $\\alpha_{n}:\\mathbb{X}\\rightarrow\\mathbb{A}_{N}$\nbe a matrix-valued map, and write $\\alpha_{n}^{ij}$ for the $i$th\nrow, $j$th column entry so that for each $i$ we have $\\sum_{j}\\alpha_{n}^{ij}=1$\n(with dependence on the $\\mathbb{X}$-valued argument suppressed).\nThe following assumption places a restriction on the relationship\nbetween $\\alpha$ and the particle system $\\left\\{ \\zeta_{n};n\\geq0\\right\\} $.\n\\begin{assumption*}\n\\textbf{\\emph{(A2)}} For each $n\\geq0$, the entries of $\\alpha_{n}$\nare all measurable with respect to $\\mathcal{F}_{n}$\n\\end{assumption*}\nIntuitively, the members of $\\mathbb{A}_{N}$ will specify different\npossible interaction structures for the particle algorithm and under\n\\textbf{(A2)}, each $\\alpha_{n}$ is a random matrix chosen from $\\mathbb{A}_{N}$\naccording to some deterministic function of $\\left\\{ \\zeta_{0},\\ldots,\\zeta_{n}\\right\\} $.\nExamples are given below. We shall write $\\mathbf{1}_{1\/N}$ for the\n$N\\times N$ matrix which has $1\/N$ as every entry and write $Id$\nfor the identity matrix of size apparent from the context in which\nthis notation appears. We shall occasionally use $Id$ also to denote\nidentity operators in certain function space settings. Let $\\mathcal{M}$,\n$\\mathcal{P}$ and $\\mathcal{L}$ be respectively the collections\nof measures, probability measures and real-valued, bounded, $\\mathcal{X}$-measurable\nfunctions on $\\mathsf{X}$. We write\n\\[\n\\left\\Vert \\varphi\\right\\Vert :=\\sup_{x}\\left|\\varphi(x)\\right|,\\quad\\quad\\text{osc}(\\varphi):=\\sup_{x,y}\\left|\\varphi(x)-\\varphi(y)\\right|,\n\\]\nand\n\\begin{equation}\n\\mu(\\varphi):=\\int_{\\mathsf{X}}\\varphi(x)\\mu(dx),\\quad\\text{for any}\\quad\\varphi\\in\\mathcal{L},\\;\\mu\\in\\mathcal{M}.\\label{eq:mu(phi)_notation}\n\\end{equation}\n\n\\begin{rem*}\nNote that $\\mathbb{X}$, $\\mathcal{F}^{\\mathbb{X}}$, $\\mathcal{F}_{n}$,\n$\\mathbb{P}$, $\\alpha$ and various other objects depend on $N$,\nbut this dependence is suppressed from the notation. Unless specified\notherwise, any conditions which we impose on such objects should be\nunderstood as holding for all $N\\geq1$.\n\\end{rem*}\nLet $\\left\\{ W_{n}^{i};i\\in[N],n\\geq0\\right\\} $ be defined by the\nfollowing recursion:\n\n\\begin{equation}\nW_{0}^{i}:=1,\\quad\\quad W_{n}^{i}:=\\sum_{j}\\alpha_{n-1}^{ij}W_{n-1}^{j}g_{n-1}(\\zeta_{n-1}^{j}),\\quad i\\in[N],n\\geq1.\\label{eq:W_n_defn}\n\\end{equation}\n\n\nThe following algorithm implicitly specifies the law $\\mathbb{P}$\nof the $\\alpha$SMC particle system. For each $n\\geq1$, the ``Sample''\nstep should be understood as meaning that the variables $\\zeta_{n}=\\left\\{ \\zeta_{n}^{i}\\right\\} _{i\\in[N]}$\nare conditionally independent given $\\left\\{ \\zeta_{0},\\ldots,\\zeta_{n-1}\\right\\} $.\nThe line of Algorithm~\\ref{alg:aSMC} marked $(\\star)$ is intentionally\ngeneric, it amounts to a practical, if imprecise restatement of \\textbf{(A2).\n}In the sequel we shall examine instances of $\\alpha$SMC which arise\nwhen we consider specific $\\mathbb{A}_{N}$ and impose more structure\nat line $(\\star)$.\n\n\\begin{algorithm}[H]\n\\begin{raggedright}\n\\qquad{}For $n=0$,\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}For $i=1,\\ldots,N$,\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}\\qquad{}Set\\quad{} $W_{0}^{i}=1$\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}\\qquad{}Sample\\quad{} $\\left\\{ \\zeta_{0}^{i}\\right\\} _{i\\in[N]}\\iid\\mu_{0}$ \n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}For $n\\geq1$,\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n$(\\star)$\\qquad{}\\enskip{}\\hspace{0.25em}Select $\\alpha_{n-1}$\nfrom $\\mathbb{A}_{N}$ according to some functional of $\\left\\{ \\zeta_{0},\\ldots,\\zeta_{n-1}\\right\\} $\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}For $i=1,\\ldots,N$,\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}\\qquad{}Set\\quad{} $W_{n}^{i}=\\sum_{j}\\alpha_{n-1}^{ij}W_{n-1}^{j}g_{n-1}(\\zeta_{n-1}^{j})$\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}\\qquad{}Sample\\quad{} $\\zeta_{n}^{i}|\\mathcal{F}_{n-1}\\;\\sim\\;\\dfrac{\\sum_{j}\\alpha_{n-1}^{ij}W_{n-1}^{j}g_{n-1}(\\zeta_{n-1}^{j})f(\\zeta_{n-1}^{j},\\cdot)}{W_{n}^{i}}$\n\\par\\end{raggedright}\n\n\\protect\\caption{$\\alpha$SMC\\label{alg:aSMC}}\n\\end{algorithm}\n\n\nWe shall study the objects \n\n\\begin{equation}\n\\pi_{n}^{N}:=\\frac{\\sum_{i}W_{n}^{i}\\;\\delta_{\\zeta_{n}^{i}}}{\\sum_{i}W_{n}^{i}},\\quad\\quad\\quad Z_{n}^{N}:=\\frac{1}{N}\\sum_{i}W_{n}^{i},\\quad n\\geq0,\\label{eq:pi^N_andZ^N}\n\\end{equation}\nwhich as the notation suggests, are to be regarded as approximations\nof $\\pi_{n}$ and $Z_{n}$, respectively. We shall also be centrally\nconcerned with the following coefficient, which is closely related\nto the ESS,\n\n\\begin{equation}\n\\mathcal{E}_{n}^{N}:=\\frac{\\left(N^{-1}\\sum_{i}W_{n}^{i}\\right)^{2}}{N^{-1}\\sum_{i}\\left(W_{n}^{i}\\right)^{2}}=\\frac{\\left(N^{-1}\\sum_{i}\\sum_{j}\\alpha_{n-1}^{ij}W_{n-1}^{j}g_{n-1}(\\zeta_{n-1}^{j})\\right)^{2}}{N^{-1}\\sum_{i}\\left(\\sum_{j}\\alpha_{n-1}^{ij}W_{n-1}^{j}g_{n-1}(\\zeta_{n-1}^{j})\\right)^{2}},\\quad n\\geq1,\\label{eq:ESS_defn_front}\n\\end{equation}\nand by convention $\\mathcal{E}_{0}^{N}:=1$. The second equality in\n(\\ref{eq:ESS_defn_front}) is immediate from the definition of $W_{n}^{i}$,\nsee (\\ref{eq:W_n_defn}). Note that $\\mathcal{E}_{n}^{N}$ is always\nvalued in $[0,1]$, and if we write\n\\begin{equation}\nN_{n}^{\\text{eff}}:=N\\mathcal{E}_{n}^{N},\\label{eq:N_eff}\n\\end{equation}\nwe obtain the ESS of \\citet{liu1995blind}, although of course in\na generalized form, since $\\mathcal{E}_{n}^{N}$ is defined in terms\nof the generic ingredients of $\\alpha$SMC. A few comments on generality\nare in order. Firstly, for ease of presentation, we have chosen to\nwork with a particularly simple version of $\\alpha$SMC, in which\nnew samples are proposed using the HMM Markov kernel $f$. The algorithm\nis easily generalized to accommodate other proposal kernels. Secondly,\nwhilst we focus on the application of SMC methods to HMM's, our results\nand methodological ideas are immediately transferable to other contexts,\nfor example via the framework of \\citep{smc:meth:DDJ06}.\n\n\n\\subsection{Instances of $\\alpha$SMC\\label{sub:Instances-of-SMC}}\n\nWe now show how $\\alpha$SMC admits SIS, the BPF and the ARPF, as\nspecial cases, through particular choices of $\\mathbb{A}_{N}$. Our\npresentation is intended to illustrate the structural generality of\n$\\alpha$SMC, thus setting the scene for the developments which follow.\nThe following lemma facilitates exposition by ``unwinding'' the\nquantities $\\left\\{ W_{n}^{i}\\right\\} _{i\\in[N]}$ defined recursively\nin (\\ref{eq:W_n_defn}). It is used throughout the remainder of the\npaper.\n\\begin{lem}\n\\label{lem:W_n_representation}For $n\\geq1$, $0\\leq p<n$ and $i_{n}\\in[N]$,\n\\begin{equation}\nW_{n}^{i_{n}}=\\sum_{\\left(i_{p},\\ldots,i_{n-1}\\right)\\in[N]^{n-p}}W_{p}^{i_{p}}\\prod_{q=p}^{n-1}g_{q}(\\zeta_{q}^{i_{q}})\\alpha_{q}^{i_{q+1}i_{q}},\\label{eq:unwind}\n\\end{equation}\nand in particular\n\\begin{equation}\nW_{n}^{i_{n}}=\\sum_{\\left(i_{0},\\ldots,i_{n-1}\\right)\\in[N]^{n}}\\prod_{p=0}^{n-1}g_{p}(\\zeta_{p}^{i_{p}})\\alpha_{p}^{i_{p+1}i_{p}}.\\label{eq:unwind2}\n\\end{equation}\n\n\\end{lem}\nThe proof of (\\ref{eq:unwind})--(\\ref{eq:unwind2}) is a simple induction\nand is therefore omitted. From (\\ref{eq:unwind2}) and definitions\nabove we immediately observe:\n\\begin{cor}\n\\label{cor:measurability}If $\\mathbf{(A2)}$ holds, then $W_{n}^{i}$\nmust be measurable w.r.t.~$\\mathcal{F}_{n-1}$ for every $n\\geq0$\nand $i\\in[N]$.\n\\end{cor}\n\n\\subsubsection*{Sequential importance sampling: $\\mathbb{A}_{N}=\\{Id\\}$}\n\nSince in this case $\\mathbb{A}_{N}$ consists of only a single element,\n$\\alpha$ is actually a deterministic sequence, \\textbf{(A2)} is trivially\nsatisfied and at line $(\\star)$ of Algorithm~\\ref{alg:aSMC} we\nhave $\\alpha_{n}=Id$ fixed for all $n\\geq0$. In this situation Lemma\n\\ref{lem:W_n_representation} gives $W_{n}^{i}=\\prod_{p=0}^{n-1}g_{p}(\\zeta_{p}^{i})$\nfor $n\\geq1$, so in turn\n\\[\n\\pi_{n}^{N}=\\frac{\\sum_{i}\\delta_{\\zeta_{n}^{i}}\\prod_{p=0}^{n-1}g_{p}(\\zeta_{p}^{i})}{\\sum_{i}\\prod_{p=0}^{n-1}g_{p}(\\zeta_{p}^{i})},\\quad\\quad\\quad Z_{n}^{N}=\\frac{1}{N}\\sum_{i}\\prod_{p=0}^{n-1}g_{p}(\\zeta_{p}^{i}),\\quad n\\geq1,\n\\]\n and $\\alpha$SMC reduces to: \n\\begin{algorithm}[H]\n\\begin{raggedright}\n\\qquad{}For $n=0$,\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}For $i=1,\\ldots,N$,\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}\\qquad{}Set\\quad{} $W_{0}^{i}=1$.\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}\\qquad{}Sample\\quad{} $\\zeta_{0}^{i}\\sim\\mu_{0}$, \n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}For $n\\geq1$,\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}For $i=1,\\ldots,N$,\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}\\qquad{}Set\\quad{} $W_{n}^{i}=W_{n-1}^{i}g_{n-1}(\\zeta_{n-1}^{i})$.\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}\\qquad{}Sample\\quad{} $\\zeta_{n}^{i}|\\mathcal{F}_{n-1}\\;\\sim\\; f(\\zeta_{n-1}^{i},\\cdot)$.\n\\par\\end{raggedright}\n\n\\protect\\caption{Sequential importance sampling\\label{alg:SIS}}\n\\end{algorithm}\n\n\n\n\\subsubsection*{Bootstrap particle filter: $\\mathbb{A}_{N}=\\{\\mathbf{1}_{1\/N}\\}$}\n\nIn this case $\\alpha$ is again a deterministic sequence and \\textbf{(A2)}\nis trivially satisfied. At line $(\\star)$ we have $\\alpha_{n}=\\mathbf{1}_{1\/N}$\nfixed for all $n\\geq0$. Lemma~\\ref{lem:W_n_representation} gives,\nfor all $i_{n}\\in[N]$,\n\\begin{equation}\nW_{n}^{i_{n}}=\\sum_{\\left(i_{0},\\ldots,i_{n-1}\\right)\\in[N]^{n}}\\prod_{p=0}^{n-1}\\frac{g_{p}(\\zeta_{p}^{i_{p}})}{N}=\\prod_{p=0}^{n-1}\\left(\\frac{1}{N}\\sum_{i_{p}}g_{p}(\\zeta_{p}^{i_{p}})\\right),\\quad n\\geq1.\\label{eq:bootstrap_W_n^i}\n\\end{equation}\nNote that then $W_{n}^{i}=W_{n}^{j}$ for all $i,j$, so $NW_{n}^{i}=\\sum_{j}W_{n}^{j}$\nand we obtain, according to (\\ref{eq:pi^N_andZ^N}), \n\\begin{equation}\n\\pi_{n}^{N}=\\frac{1}{N}\\sum_{i}\\delta_{\\zeta_{n}^{i}},\\quad\\quad\\quad Z_{n}^{N}=\\prod_{p=0}^{n-1}\\left(\\frac{1}{N}\\sum_{i_{p}}g_{p}(\\zeta_{p}^{i_{p}})\\right),\\quad n\\geq1,\\label{eq:bootstrap_Z_n^N}\n\\end{equation}\nand $\\alpha$SMC algorithm reduces to the BPF. Since $W_{n}^{i}=W_{n}^{j}$\nfor all $i,j$, we write by convention the weight update steps only\nfor $W_{n}^{1}$. \n\n\\begin{algorithm}[H]\n\\begin{raggedright}\n\\qquad{}For $n=0$,\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}Set\\quad{} $W_{0}^{1}=1$.\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}For $i=1,\\ldots,N$,\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}\\qquad{}Sample\\quad{} $\\zeta_{0}^{i}\\sim\\mu_{0}$, \n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}For $n\\geq1$,\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}Set\\quad{} $W_{n}^{1}=W_{n-1}^{1}\\cdot\\dfrac{1}{N}\\sum_{i}g_{n-1}(\\zeta_{n-1}^{i})$.\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}For $i=1,\\ldots,N$,\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}\\qquad{}Sample\\quad{} $\\zeta_{n}^{i}|\\mathcal{F}_{n-1}\\;\\sim\\;\\dfrac{\\sum_{j}g_{n-1}(\\zeta_{n-1}^{j})f(\\zeta_{n-1}^{j},\\cdot)}{\\sum_{j}g_{n-1}(\\zeta_{n-1}^{j})}$.\n\\par\\end{raggedright}\n\n\\protect\\caption{Bootstrap particle filter\\label{alg:boot_pf}}\n\\end{algorithm}\n\n\n\n\\subsubsection*{Adaptive resampling particle filter: $\\mathbb{A}_{N}=\\{Id,\\mathbf{1}_{1\/N}\\}$}\n\nIn this case each $\\alpha_{n}$ is allowed to take only the value\n$Id$ or $\\mathbf{1}_{1\/N}$, with the latter corresponding to resampling,\nand vice-versa. The choice between $Id$ and $\\mathbf{1}_{1\/N}$ is\nmade by comparing some functional of the particle system to a threshold\nvalue. We consider the case of the popular ESS-based resampling rule\n\\citep{liu1995blind}, partly for simplicity, but also because monitoring\nof the ESS is especially pertinent to the discussions which follow.\nThis ARPF arises as an instance of $\\alpha$SMC if we take as line\n$(\\star)$ of Algorithm~\\ref{alg:aSMC} the rule:\n\\begin{equation}\n\\alpha_{n-1}:=\\begin{cases}\n\\mathbf{1}_{1\/N}, & \\quad\\text{ if }\\quad\\frac{\\left(N^{-1}\\sum_{i}W_{n-1}^{i}g_{n-1}(\\zeta_{n-1}^{i})\\right)^{2}}{N^{-1}\\sum_{i}\\left(W_{n-1}^{i}g_{n-1}(\\zeta_{n-1}^{i})\\right)^{2}}<\\tau.\\\\\nId, & \\quad\\text{ otherwise},\n\\end{cases}\\label{eq:alpha_ARPF}\n\\end{equation}\nwhere $\\tau\\in(0,1]$ is a threshold value. Lemma~\\ref{lem:ARPF_A2}\nin the appendix shows by an inductive argument that the adaptation\nrule (\\ref{eq:alpha_ARPF}) satisfies $\\mathbf{(A2)}$. The ARPF is\ntraditionally expressed in terms of the random times at which resampling\noccurs. For completeness, the appendix contains derivations of expressions\nfor $\\pi_{n}^{N}$ and $Z_{n}^{N}$ in terms of such times and similar\nmanipulations can be used to write out the form of $\\alpha$SMC in\nthis case. \n\nLooking back to the expression for $\\mathcal{E}_{n}^{N}$ in (\\ref{eq:ESS_defn_front}),\nwe find:\n\\begin{eqnarray}\n\\alpha_{n-1}=\\mathbf{1}_{1\/N}\\quad & \\Rightarrow & \\quad\\mathcal{E}_{n}^{N}=1,\\label{eq:alpha_ARPF_1}\\\\\n\\alpha_{n-1}=Id\\;\\;\\;\\quad & \\Rightarrow & \\quad\\mathcal{E}_{n}^{N}=\\frac{\\left(N^{-1}\\sum_{i}W_{n-1}^{i}g_{n-1}(\\zeta_{n-1}^{i})\\right)^{2}}{N^{-1}\\sum_{i}\\left(W_{n-1}^{i}g_{n-1}(\\zeta_{n-1}^{i})\\right)^{2}}.\\label{eq:alpha_ARPF_2}\n\\end{eqnarray}\nWe then adopt the point of view that according to (\\ref{eq:alpha_ARPF})--(\\ref{eq:alpha_ARPF_2}),\nthe ARPF \\emph{enforces} the condition: $\\inf_{n\\geq0}\\mathcal{E}_{n}^{N}\\geq\\tau>0$,\nor equivalently\n\\[\n\\inf_{n\\geq0}N_{n}^{\\text{eff}}\\geq N\\tau>0,\n\\]\n by construction. This seemingly trivial observation turns out to\nbe crucial when we address time-uniform convergence of the ARPF in\nSection~\\ref{sec:stability}, and the condition $\\inf_{n\\geq0}\\mathcal{E}_{n}^{N}>0$\nwill appear repeatedly in discussions which lead to the formulation\nof new, provably stable algorithms in Section~\\ref{sec:Discussion}. \n\n\n\\subsubsection*{Comments on other algorithms}\n\nIn the engineering literature, a variety of algorithmic procedures\ninvolving distributed computing have been suggested \\citep{bolic2005resampling}.\n``Local'' particle approximations of Rao--Blackwellized filters\nhave been devised in \\citep{chen2011decentralized} and \\citep{johansen2012exact}\n. \\citet{Verge_island_particle} have recently suggested an ``island''\nparticle algorithm, designed for parallel implementation, in which\nthere are two levels of resampling and the total population size $N=N_{1}N_{2}$\nis defined in terms of the number of particles per island, $N_{1}$,\nand the number of islands, $N_{2}$. Interaction at both levels occurs\nby resampling, at the island level this means entire blocks of particles\nare replicated and\/or discarded. They investigate the trade-off between\n$N_{1}$ and $N_{2}$ and provide asymptotic results which validate\ntheir algorithms. In the present work, we provide some asymptotic\nresults in Section~\\ref{sec:Martingale-approximations-and} but it\nis really the non-asymptotic results in Section~\\ref{sec:stability}\nwhich lead us to suggest specific novel instances of $\\alpha$SMC\nin Section~\\ref{sec:Discussion}. Moreover, in general $\\alpha$SMC\nis distinct from all these algorithms and, other than in some uninteresting\nspecial cases, none of them coincide with the adaptive procedures\nwe suggest in Section~\\ref{sub:Algorithms-with-adaptive}. \n\n\n\\section{Convergence\\label{sec:Martingale-approximations-and}}\n\nIn this section our main objective is to investigate, for general\n$\\alpha$SMC (Algorithm~\\ref{alg:aSMC}), conditions for convergence\n\\begin{equation}\nZ_{n}^{N}-Z_{n}\\rightarrow0\\quad\\text{and}\\quad\\pi_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi)\\rightarrow0,\\label{eq:as_convergence_intro}\n\\end{equation}\nat least in probability, as $N\\rightarrow\\infty$. \n\nIn the case of SIS, i.e.~$\\mathbb{A}_{N}=\\{Id\\}$, it is easy to\nestablish (\\ref{eq:as_convergence_intro}), since the processes $\\left\\{ \\zeta_{n}^{i};n\\geq0\\right\\} _{i\\in[N]}$\nare independent Markov chains, of identical law. On the other hand,\nfor the bootstrap filter, i.e.~$\\mathbb{A}_{N}=\\{\\mathbf{1}_{1\/N}\\}$,\nthe convergence $\\pi_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi)\\rightarrow0$,\ncan be proved under very mild conditions, by decomposing $\\pi_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi)$\nin terms of ``local'' sampling errors, see amongst others \\citep{smc:theory:Dm04,smc:the:DM08}\nfor this type of approach. For instance, for $A\\in\\mathcal{X}$ we\nmay write \n\\begin{eqnarray}\n\\pi_{1}^{N}(A)-\\pi_{1}(A) & = & \\frac{1}{N}\\sum_{i}\\delta_{\\zeta_{1}^{i}}(A)-\\frac{\\sum_{i}g_{0}(\\zeta_{0}^{i})f(\\zeta_{0}^{i},A)}{\\sum_{i}g_{0}(\\zeta_{0}^{i})}\\label{eq:intro_boot_decomp1}\\\\\n & + & \\frac{\\sum_{i}g_{0}(\\zeta_{0}^{i})f(\\zeta_{0}^{i},A)}{\\sum_{i}g_{0}(\\zeta_{0}^{i})}-\\pi_{1}(A).\\label{eq:intro_boot_decomp2}\n\\end{eqnarray}\nHeuristically, the term on the r.h.s.~of (\\ref{eq:intro_boot_decomp1})\nconverges to zero because given $\\mathcal{F}_{0}$, the samples $\\left\\{ \\zeta_{1}^{i}\\right\\} _{i\\in[N]}$\nare conditionally i.i.d.~according $\\frac{\\sum_{i}g_{0}(\\zeta_{0}^{i})f(\\zeta_{0}^{i},\\cdot)}{\\sum_{i}g_{0}(\\zeta_{0}^{i})}$,\nand the term in (\\ref{eq:intro_boot_decomp2}) converges to zero because\nthe samples $\\left\\{ \\zeta_{0}^{i}\\right\\} _{i\\in[N]}$ are i.i.d.~according\nto $\\mu_{0}$. A similar argument ensures that $\\pi_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi)\\rightarrow0$,\nfor any $n\\geq0$ and therefore by the continuous mapping theorem\n$Z_{n}^{N}-Z_{n}\\rightarrow0$, since \n\\[\nZ_{n}=\\prod_{p=0}^{n-1}\\pi_{p}(g_{p}),\\quad\\text{and}\\quad Z_{n}^{N}=\\prod_{p=0}^{n-1}\\pi_{p}^{N}(g_{p}).\n\\]\nIn the case of $\\alpha$SMC, $\\left\\{ \\zeta_{n}^{i}\\right\\} _{i\\in[N]}$\nare conditionally independent given $\\mathcal{F}_{n-1}$, but we do\nnot necessarily have either the unconditional independence structure\nof SIS, or the conditionally i.i.d.~structure of the BPF to work\nwith. \n\n\\citet{smc:the:DM08} have established a CLT for the ARPF using an\ninductive approach w.r.t.~deterministic time periods. \\citet{arnaud2009smc}\nhave obtained a CLT for the ARPF based on an alternative multiplicative\nfunctional representation of the algorithm. Convergence of the ARPF\nwas studied in \\citep{del2012adaptive} by coupling the adaptive algorithm\nto a reference particle system, for which resampling occurs at deterministic\ntimes. One of the benefits of their approach is that existing asymptotic\nresults for non-adaptive algorithms, such as central limit theorems\n(CLT), can then be transferred to the adaptive algorithm with little\nfurther work. Their analysis involves a technical assumption \\citep[Section 5.2]{del2012adaptive}\nto deal with the situation where the threshold parameters coincide\nwith the adaptive criteria. Our analysis of $\\alpha$SMC does not\nrest on any such technical assumption, and in some ways is more direct,\nbut we do not obtain concentration estimates or a CLT. Some more detailed\nremarks on this matter are given after the statement of Theorem~\\ref{thm:convergence}. \n\n\\citet{crisan2012particle} studied convergence and obtained a CLT\nfor an adaptive resampling particle filter in continuous time under\nconditions which they verify for the case of ESS-triggered resampling,\nwithout needing the type of technical assumption of \\citep{del2012adaptive}.\nTheir study focuses, in part, on the random times at which resampling\noccurs and dealing with the subtleties of the convergence in continuous\ntime. Our asymptotic $N\\rightarrow\\infty$ analysis is in some ways\nless refined, but in comparison to this and the other existing works,\nwe analyze a more general algorithm, and it is this generality which\nallows us to suggest new adaptive algorithms in Section~\\ref{sec:Discussion},\ninformed by the time-uniform non-asymptotic error bounds in our Theorem~\\ref{thm:L_R_mix}. \n\nTo proceed, we need some further notation involving $\\alpha$. Let\nus define the matrices: $\\alpha_{n,n}:=Id$ for $n\\geq0$, and recursively\n\\begin{equation}\n\\alpha_{p,n}^{ij}:=\\sum_{k}\\alpha_{p+1,n}^{ik}\\alpha_{p}^{kj},\\quad\\quad(i,j)\\in[N]^{2},\\;0\\leq p<n,\\label{eq:a_pn_defn}\n\\end{equation}\nand the vectors: \n\\begin{equation}\n\\beta_{n,n}^{i}:=N^{-1},\\quad\\quad n\\geq0,\\; i\\in[N],\\label{eq:beta_n_n_defn}\n\\end{equation}\nand recursively \n\\begin{equation}\n\\beta_{p,n}^{i}:=\\sum_{j}\\beta_{p+1,n}^{j}\\alpha_{p}^{ji}\\quad\\quad i\\in[N],\\;0\\leq p<n.\\label{eq:beta_defn}\n\\end{equation}\nNote that since each $\\alpha_{n}$ is a random Markov transition matrix,\nso is each $\\alpha_{p,n}$, and each $\\left\\{ \\beta_{p,n}^{i}\\right\\} _{i\\in[N]}$\ndefines a random probability distribution on $[N]$. Moreover, from\nthese definitions we immediately have the identity\n\\begin{equation}\n\\beta_{p,n}^{i}=N^{-1}\\sum_{j}\\alpha_{p,n}^{ji},\\quad\\quad i\\in[N],\\;0\\leq p\\leq n.\\label{eq:beta_in_terms_of_alpha}\n\\end{equation}\n\n\\begin{assumption*}\n$\\;$\n\n$\\mathbf{(B)}$ - for all $0\\leq p\\leq n$ and $i\\in[N]$, \\textbf{$\\beta_{p,n}^{i}$}\nis measurable w.r.t.~the trivial $\\sigma$-algebra $\\mathcal{F}_{-1}$.\n\n$\\;$\n\n$\\mathbf{(B^{+})}$ -\\textbf{ }assumption $\\mathbf{(B)}$ holds and,\nfor all $0\\leq p\\leq n$, $\\lim_{N\\rightarrow\\infty}\\max_{i\\in[N]}\\beta_{p,n}^{i}=0$.\n\n$\\;$\n\n$\\mathbf{(B^{++})}$ -\\textbf{ }every member of $\\mathbb{A}_{N}$\nadmits the uniform distribution on $[N]$ as an invariant distribution\n\\end{assumption*}\nWe note the following:\n\\begin{itemize}\n\\item Intuitively, $\\mathbf{(B)}$ ensures that even though $\\alpha$ is\na sequence of random Markov transition matrices, the elements of the\nprobability vector $\\left\\{ \\beta_{p,n}^{i}\\right\\} _{i\\in[N]}$ are\nall constants. $\\mathbf{(B)}$ holds, trivially, when every element\nof every $\\alpha_{n}$ is measurable w.r.t.~$\\mathcal{F}_{-1}$,\ni.e.~the sequence $\\alpha$ is completely pre-determined. This is\ntrue, for example, when the set $\\mathbb{A}_{N}$ consists of only\na single element, as is the case for SIS and the BPF. \n\\item The $\\lim_{N\\rightarrow\\infty}\\max_{i\\in[N]}\\beta_{p,n}^{i}=0$ part\nof $\\mathbf{(B^{+})}$ is an asymptotic negligibility condition. In\nSection~\\ref{sub:Ensuring-convergence} we describe what can go wrong\nwhen this assumption does not hold. \n\\item $\\mathbf{(B^{++})}$ does not require the members of $\\mathbb{A}_{N}$\nto be irreducible, for example it is satisfied with $\\mathbb{A}_{N}=\\{Id\\}$\n\\item $\\mathbf{(B^{++})}$\\textbf{$\\Rightarrow$}$\\mathbf{(B^{+})}$. To\nsee this, observe that when $\\mathbf{(B^{++})}$ holds, every random\nmatrix $\\alpha_{p,n}$, defined in (\\ref{eq:a_pn_defn}), also admits\nthe uniform distribution on $[N]$ as invariant, then using (\\ref{eq:beta_in_terms_of_alpha})\nwe have $\\beta_{p,n}^{i}=N^{-1}\\sum_{j}\\alpha_{p,n}^{ji}=N^{-1}$\nfor all $i\\in[N]$. The reverse implication is clearly not true in\ngeneral.\n\\item $\\mathbf{(B^{++})}$ holds when every member of $\\mathbb{A}_{N}$\nis doubly-stochastic, because such matrices always leave the uniform\ndistribution invariant. $\\mathbf{(B^{++})}$\\textbf{ }therefore holds\nfor the ARPF, which has $\\mathbb{A}_{N}=\\{Id,\\mathbf{1}_{1\/N}\\}$.\n\\end{itemize}\nThe main result of this section is \n\\begin{thm}\n\\label{thm:convergence} $\\;$Assume $\\mathbf{(A2)}$. For any $n\\geq0$,\n$\\varphi\\in\\mathcal{L}$ and $r\\geq1$,\n\n$\\;$\n\n1) if $\\mathbf{(B)}$ holds, then $\\mathbb{E}[Z_{n}^{N}]=Z_{n}$ for\nany $N\\geq1$,\n\n$\\;$\n\n2) if $\\mathbf{(B^{+})}$ holds, then \n\\begin{eqnarray}\n &  & \\lim_{N\\rightarrow\\infty}\\mathbb{E}\\left[\\left|Z_{n}^{N}-Z_{n}\\right|^{r}\\right]=0,\\label{eq:convergence_L_r_statement_gam_weak}\\\\\n &  & \\lim_{N\\rightarrow\\infty}\\mathbb{E}\\left[\\left|\\pi_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi)\\right|^{r}\\right]=0,\\label{eq:convergence_L_r_statement_pi_weak}\n\\end{eqnarray}\n \n\n\\quad{}and therefore $Z_{n}^{N}\\rightarrow Z_{n}$ and $\\pi_{n}^{N}(\\varphi)\\rightarrow\\pi_{n}(\\varphi)$\nin probability as $N\\rightarrow\\infty$,\n\n$\\;$\n\n3) if $\\mathbf{(B^{++})}$\\textbf{ }holds\\textbf{, }then\n\\begin{eqnarray}\n &  & \\sup_{N\\geq1}\\sqrt{N}\\mathbb{E}\\left[\\left|Z_{n}^{N}-Z_{n}\\right|^{r}\\right]^{1\/r}<+\\infty,\\label{eq:convergence_L_r_statement_gamm}\\\\\n &  & \\sup_{N\\geq1}\\sqrt{N}\\mathbb{E}\\left[\\left|\\pi_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi)\\right|^{r}\\right]^{1\/r}<+\\infty,\\label{eq:convergence_L_r_statement_pi}\n\\end{eqnarray}\n\n\n\\quad{}and therefore $Z_{n}^{N}\\rightarrow Z_{n}$ and $\\pi_{n}^{N}(\\varphi)\\rightarrow\\pi_{n}(\\varphi)$\nalmost surely, as $N\\rightarrow\\infty$.\\end{thm}\n\\begin{rem}\nThe lack-of-bias property $\\mathbb{E}[Z_{n}^{N}]=Z_{n}$ is desirable\nsince it could be used to validate the use of $\\alpha$SMC within\ncomposite SMC\/MCMC algorithms such as those of \\citep{andrieu2010particle}. \n\\end{rem}\n\n\\begin{rem}\n\\label{rem_alpha_conv}Theorem~\\ref{thm:convergence} holds without\nany sort of requirement that the entries of each $\\alpha_{n}$ converge\nas $N\\rightarrow\\infty$. For example, $\\mathbf{(B^{++})}$ holds\nif for $N$ odd we choose $\\mathbb{A}_{N}=\\{Id\\}$ and for $N$ even\nwe choose $\\mathbb{A}_{N}=\\{\\mathbf{1}_{1\/N}\\}$. As a reflection\nof this, and as is apparent upon inspection of the proof, without\nfurther assumption we cannot in general replace $\\sup_{N\\geq1}$ in\n(\\ref{eq:convergence_L_r_statement_gamm})--(\\ref{eq:convergence_L_r_statement_pi})\nwith $\\lim_{N\\rightarrow\\infty}$, because such limits may not exist.\n\\end{rem}\n\n\\begin{rem}\nThe proof of Theorem~\\ref{thm:convergence} hinges on a martingale\ndecomposition of errors associated with $Z_{n}^{N}$ and $\\pi_{n}^{N}(\\varphi)$,\nsee Proposition~\\ref{prop:martingale}. Our overall approach is inspired\nby some of the ideas of \\citep[Chapters 7 and 9]{smc:theory:Dm04},\nbut the path we take and the details are necessarily different since\nthe analysis of \\citep{smc:theory:Dm04} does not apply to $\\alpha$SMC\nin general. \n\\end{rem}\nThe following notation is used throughout the remainder of the paper.\nIntroduce the non-negative kernels \n\\begin{equation}\nQ_{n}:\\mathsf{X}\\times\\mathcal{X}\\rightarrow\\mathbb{R}_{+},\\quad\\quad Q_{n}(x,dx^{\\prime}):=g_{n-1}(x)f(x,dx^{\\prime}),\\quad n\\geq1,\\label{eq:Q_GM_defn}\n\\end{equation}\nthe corresponding operators on functions and measures: \n\\begin{eqnarray}\nQ_{n}(\\varphi)(x) & := & \\int_{\\mathsf{X}}Q_{n}(x,dx^{\\prime})\\varphi(x^{\\prime}),\\quad\\varphi\\in\\mathcal{L},\\label{eq:Q_op_defn}\\\\\n\\mu Q_{n}(\\cdot) & := & \\int_{\\mathsf{X}}\\mu(dx)Q_{n}(x,\\cdot),\\quad\\mu\\in\\mathcal{M},\\label{eq:Q_op_defn-1}\n\\end{eqnarray}\nand for $n\\geq1$ and $0\\leq p<n$, \n\\begin{equation}\nQ_{p,p}:=Id,\\quad\\quad Q_{p,n}:=Q_{p+1}\\cdots Q_{n}.\\label{eq:Q_semigroup}\n\\end{equation}\nWe shall also consider the following scaled versions of these operators:\n\\begin{equation}\n\\overline{Q}_{n}:=\\frac{Q_{n}}{\\pi_{n-1}(g_{n-1})},\\quad\\quad\\overline{Q}_{p,p}:=Id,\\quad\\quad\\overline{Q}_{p,n}:=\\overline{Q}_{p+1}\\cdots\\overline{Q}_{n}.\\label{eq:Q_bar_defn}\n\\end{equation}\n\n\nThen define the non-negative measures\n\\[\n\\gamma_{n}:=\\mu_{0}Q_{0,n}(\\cdot),\\quad n\\geq0,\n\\]\nunder $\\mathbf{\\mathbf{(A1)}}$ we are assured that $\\gamma_{n}(1)>0$.\nDue to the conditional independence structure of the HMM, it can easily\nbe checked that \n\\[\n\\pi_{n}=\\frac{\\gamma_{n}}{\\gamma_{n}(1)},\\quad\\quad Z_{n}=\\gamma_{n}(1),\\quad n\\geq0,\n\\]\nand\n\\[\n\\overline{Q}_{p,n}=\\frac{Q_{p,n}}{\\pi_{p}Q_{p,n}(1)}.\n\\]\n\n\nFor $i\\in[N]$ and $0\\leq p\\leq n$, introduce the random measures\n\\begin{equation}\n\\Gamma_{p,n}^{N}:=\\sum_{i}\\beta_{p,n}^{i}W_{p}^{i}\\delta_{\\zeta_{p}^{i}},\\quad\\quad\\overline{\\Gamma}_{p,n}^{N}:=\\frac{\\Gamma_{p,n}^{N}}{\\gamma_{p}(1)}.\\label{eq:Gamma_defn}\n\\end{equation}\nwhere $W_{p}^{i}$ is as in (\\ref{eq:W_n_defn}). For simplicity of\nnotation, we shall write $\\Gamma_{n}^{N}:=\\Gamma_{n,n}^{N},\\;\\overline{\\Gamma}_{n}^{N}:=\\overline{\\Gamma}_{n,n}^{N}$.\nIf we define \n\\begin{equation}\n\\overline{W}_{n}^{i}:=\\frac{W_{n}^{i}}{\\gamma_{n}(1)},\\quad n\\geq0,\\label{eq:W_bar_defn}\n\\end{equation}\nthen we have from (\\ref{eq:Gamma_defn}), \n\\[\n\\overline{\\Gamma}_{p,n}^{N}=\\sum_{i}\\beta_{p,n}^{i}\\overline{W}_{p}^{i}\\delta_{\\zeta_{p}^{i}}.\n\\]\n\n\nFinally, we observe from (\\ref{eq:beta_n_n_defn}) that\n\\[\n\\Gamma_{n}^{N}=\\sum_{i}\\beta_{n,n}^{i}W_{n}^{i}\\delta_{\\zeta_{n}^{i}}=N^{-1}\\sum_{i}W_{n}^{i}\\delta_{\\zeta_{n}^{i}}.\n\\]\n\n\n\n\\subsection{Error decomposition}\n\nThroughout this section let $\\varphi\\in\\mathcal{L}$, $n\\geq0$ and\n$N\\geq1$ be arbitrarily chosen, but then fixed. Define, for $1\\leq p\\leq n$\nand $i\\in[N]$,\n\\[\n\\Delta_{p,n}^{i}:=\\overline{Q}_{p,n}(\\varphi)(\\zeta_{p}^{i})-\\frac{\\sum_{j}\\alpha_{p-1}^{ij}W_{p-1}^{j}\\overline{Q}_{p-1,n}(\\varphi)(\\zeta_{p-1}^{j})}{\\sum_{j}\\alpha_{p-1}^{ij}W_{p-1}^{j}\\overline{Q}_{p}(1)(\\zeta_{p-1}^{j})},\n\\]\nand $\\Delta_{0,n}^{i}:=\\overline{Q}_{0,n}(\\varphi)(\\zeta_{0}^{i})-\\mu_{0}\\overline{Q}_{0,n}(\\varphi)$,\nso that $\\mathbb{E}\\left[\\left.\\Delta_{p,n}^{i}\\right|\\mathcal{F}_{p-1}\\right]=0$\nfor any $i\\in[N]$ and $0\\leq p\\leq n$. Then for $0\\leq p\\leq n$\nand $i\\in[N]$ set $k:=pN+i$, and\n\n\\[\n\\xi_{k}^{N}:=\\sqrt{N}\\beta_{p,n}^{i}\\overline{W}_{p}^{i}\\Delta_{p,n}^{i},\n\\]\nso as to define a sequence $\\left\\{ \\xi_{k}^{N};k=1,\\ldots,(n+1)N\\right\\} $.\nFor $k=1,\\ldots,(n+1)N$, let $\\mathcal{F}^{(k)}$ be the $\\sigma$-algebra\ngenerated by $\\left\\{ \\zeta_{p}^{i};\\; pN+i\\leq k,\\; i\\in[N],0\\leq p\\leq n\\right\\} $.\nSet $\\mathcal{F}^{(-1)}:=\\{\\mathbb{X},\\emptyset\\}$.\n\nThe following proposition is the main result underlying Theorem~\\ref{thm:convergence}.\nThe proof is given in the appendix.\n\\begin{prop}\n\\label{prop:martingale} Assume $\\mathbf{(A2)}$ and $\\mathbf{(B)}$.\nWe have the decomposition\n\n\\begin{equation}\n\\sqrt{N}\\left[\\overline{\\Gamma}_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi)\\right]=\\sum_{k=1}^{(n+1)N}\\xi_{k}^{N},\\label{eq:Gamma_telescope}\n\\end{equation}\nwhere for $k=1,\\ldots,(n+1)N$, the increment $\\xi_{k}^{N}$ is measurable\nw.r.t.~$\\mathcal{F}^{(k)}$ and satisfies \n\\begin{equation}\n\\mathbb{E}\\left[\\left.\\xi_{k}^{N}\\right|\\mathcal{F}^{(k-1)}\\right]=\\mathbb{E}\\left[\\left.\\xi_{k}^{N}\\right|\\mathcal{F}_{p-1}\\right]=0,\\quad\\text{with}\\quad p:=\\left\\lfloor (k-1)\/N\\right\\rfloor .\\label{eq:xi_cond_exp}\n\\end{equation}\nFor each $r\\geq1$ there exists a universal constant $B(r)$ such\nthat\n\\begin{eqnarray}\n &  & \\mathbb{E}\\left[\\left|\\overline{\\Gamma}_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi)\\right|^{r}\\right]^{1\/r}\\nonumber \\\\\n &  & \\leq B(r)^{1\/r}\\sum_{p=0}^{n}\\mathrm{osc}\\left(\\overline{Q}_{p,n}(\\varphi)\\right)\\mathbb{E}\\left[\\left|\\sum_{i}\\left(\\beta_{p,n}^{i}\\overline{W}_{p}^{i}\\right)^{2}\\right|^{r\/2}\\right]^{1\/r}.\\label{eq:martingale_burkholder_bound}\n\\end{eqnarray}\n \n\\end{prop}\nThe proof of Theorem~\\ref{thm:convergence}, which is mostly technical,\nis given in the appendix. Here we briefly discuss our assumptions\nand sketch some of the main arguments. Part 1) of Theorem~\\ref{thm:convergence}\nfollows immediately from (\\ref{eq:Gamma_telescope}) and (\\ref{eq:xi_cond_exp})\napplied with $\\varphi=1$. In turn, the martingale structure of\\textbf{\n}(\\ref{eq:Gamma_telescope}) and (\\ref{eq:xi_cond_exp}) is underpinned\nby the measurability conditions \\textbf{(A2)} and $\\mathbf{(B)}$.\nThe proofs of parts 2) and 3) of Theorem~\\ref{thm:convergence},\ninvolve applying Proposition~\\ref{prop:martingale} in conjunction\nwith the identities\n\\begin{eqnarray}\nZ_{n}^{N}-Z_{n} & = & \\Gamma_{n}^{N}(1)-\\gamma_{n}(1),\\nonumber \\\\\n\\pi_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi) & = & \\frac{\\Gamma_{n}^{N}(\\varphi)}{\\Gamma_{n}^{N}(1)}-\\frac{\\gamma_{n}(\\varphi)}{\\gamma_{n}(1)}.\\label{eq:convergence_sketch_id}\n\\end{eqnarray}\nIn order to prove that these errors convergence to zero in probability\nwe show that the quadratic variation term in (\\ref{eq:martingale_burkholder_bound})\nconverges to zero. In general, we cannot hope for the latter convergence\nwithout some sort of negligibility hypothesis on the product terms\n$\\left\\{ \\mathrm{osc}\\left(\\overline{Q}_{p,n}(\\varphi)\\right)\\beta_{p,n}^{i}\\overline{W}_{p}^{i};i\\in[N]\\right\\} $.\nAssumption $\\mathbf{\\mathbf{(A1)}}$ allows us to crudely upper-bound\n$\\mathrm{osc}\\left(\\overline{Q}_{p,n}(\\varphi)\\right)$ and $\\overline{W}_{p}^{i}$;\nthe measurability condition $\\mathbf{(B)}$ allows us to dispose of\nthe expectation in (\\ref{eq:martingale_burkholder_bound}); then via\nMarkov's inequality and the classical equivalence: \n\\[\n\\lim_{N\\rightarrow\\infty}\\max_{i\\in[N]}\\beta_{p,n}^{i}=0\\quad\\Leftrightarrow\\quad\\lim_{N\\rightarrow\\infty}\\sum_{i}\\left(\\beta_{p,n}^{i}\\right)^{2}=0,\n\\]\nwhich holds since $\\left(\\max_{i\\in[N]}\\beta_{p,n}^{i}\\right)^{2}\\leq\\sum_{i}\\left(\\beta_{p,n}^{i}\\right)^{2}\\leq\\max_{i\\in[N]}\\beta_{p,n}^{i}$,\nthe negligibility part of $\\mathbf{(B^{+})}$ guarantees that $\\left|\\Gamma_{n}^{N}(\\varphi)-\\gamma_{n}(\\varphi)\\right|$\nconverges to zero in probability. The stronger condition $\\mathbf{(B^{++})}$\nbuys us the $\\sqrt{N}$ scaling displayed in part 3). In Section~\\ref{sub:Ensuring-convergence}\nwe discuss what can go wrong when $\\mathbf{(B^{+})}$ does not hold.\n\n\n\\section{Stability\\label{sec:stability}}\n\nIn this section we study the stability of approximation errors under\nthe following regularity condition.\n\\begin{assumption*}\n$\\mathbf{(C)}$ There exists $\\left(\\delta,\\epsilon\\right)\\in[1,\\infty)^{2}$\nsuch that\n\\[\n\\sup_{n\\geq0}\\sup_{x,y}\\frac{g_{n}(x)}{g_{n}(y)}\\leq\\delta,\\quad\\quad f(x,\\cdot)\\leq\\epsilon f(y,\\cdot),\\quad(x,y)\\in\\mathsf{X}^{2}.\n\\]\n\\end{assumption*}\n\\begin{rem}\n\\label{rem:assumption_C}Assumption $\\mathbf{(C)}$ is a standard\nhypothesis in studies of non-asymptotic stability properties of SMC\nalgorithms. Similar conditions have been adopted in \\citep[Chapter 7]{smc:theory:Dm04}\nand \\citep{smc:the:LGO04}, amongst others. $\\mathbf{(C)}$ guarantees\nthat $Q_{p,n}$, and related objects, obey a variety of regularity\nconditions. In particular, we immediately obtain\n\n\\begin{equation}\n\\sup_{p,n}\\sup_{x}\\overline{Q}_{p,n}(1)(x)\\leq\\sup_{p,n}\\sup_{x,y}\\frac{Q_{p,n}(1)(x)}{Q_{p,n}(1)(y)}\\leq\\delta\\epsilon<+\\infty.\\label{eq:Q_p,n_bounded}\n\\end{equation}\nFurthermore if we introduce the following operators on probability\nmeasures:\n\n\\begin{equation}\n\\Phi_{n}:\\mu\\in\\mathcal{P}\\mapsto\\frac{\\mu Q_{n}}{\\mu(g_{n-1})}\\in\\mathcal{P}\\quad\\quad n\\geq1,\\label{eq:Phi_defn}\n\\end{equation}\n\n\n\\begin{equation}\n\\Phi_{p,n}:=\\Phi_{n}\\circ\\cdots\\circ\\Phi_{p+1},\\quad0\\leq p<n,\\label{eq:Phi_defn_semigroup}\n\\end{equation}\nthen by \\citep[Theorem 4.3.1]{smc:theory:Dm04}, there exists a finite\nconstant $C$ and $\\rho\\in[0,1)$ such that \n\\begin{equation}\n\\sup_{\\mu,\\mu^{\\prime}\\in\\mathcal{P}}\\left\\Vert \\Phi_{p,n}(\\mu)-\\Phi_{p,n}(\\mu^{\\prime})\\right\\Vert \\leq C\\rho^{n-p}.\\label{eq:Phi_stability}\n\\end{equation}\nIt follows from (\\ref{eq:filtering_recursion}), (\\ref{eq:Phi_defn})\nand (\\ref{eq:Phi_defn_semigroup}) that \n\\[\n\\pi_{n+1}=\\Phi_{n+1}(\\pi_{n})=\\Phi_{p,n+1}(\\pi_{p})=\\Phi_{0,n+1}(\\mu_{0}),\\quad0\\le p\\leq n,\n\\]\nso (\\ref{eq:Phi_stability}) can be used to describe the forgetting\nof the initial distribution of the non-linear filter. Properties similar\nto (\\ref{eq:Q_p,n_bounded}) and (\\ref{eq:Phi_stability}) can be\nobtained under conditions weaker and more realistic than $\\mathbf{(C)}$,\nsee e.g. \\citep{whiteley2013} but the developments involved are substantially\nmore technical, lengthy and complicated to present. Our aim is to\nexpedite the presentation of stability properties of $\\alpha$SMC,\nand $\\mathbf{(C)}$ allows this to be achieved whilst retaining some\nof the essence of more realistic hypotheses on $g_{n}$ and $f$. \n\\end{rem}\nThe main result of this section is the following theorem, whose proof\nwe briefly postpone. \n\\begin{thm}\n\\label{thm:L_R_mix}Assume $\\mathbf{(A2)}$, $\\mathbf{(B^{++})}$\\textbf{\\textup{\\emph{\n}}}\\textup{\\emph{and }}$\\mathbf{(C)}$\\textup{\\emph{.}}\\textbf{\\textup{\\emph{\n}}}\\textup{\\emph{Then there exist finite constants, $c_{1}$ and for\neach $r\\geq1$, $c_{2}(r)$, such that for any $\\tau\\in(0,1]$, $N\\geq1$,\nand $\\varphi\\in\\mathcal{L}$,}}\n\\begin{equation}\n\\inf_{n\\geq0}\\mathcal{E}_{n}^{N}\\geq\\tau\\quad\\Rightarrow\\quad\\begin{cases}\n\\;\\;\\sup_{n\\geq1}\\;\\mathbb{E}\\left[\\left(\\dfrac{Z_{n}^{N}}{Z_{n}}\\right)^{2}\\right]^{1\/n}\\;\\leq\\;1+\\dfrac{c_{1}}{N\\tau},\\\\\n\\\\\n\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\mathrm{and}\\\\\n\\\\\n\\;\\;\\sup_{n\\geq0}\\;\\mathbb{E}\\left[\\left|\\pi_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi)\\right|^{r}\\right]^{1\/r}\\;\\leq\\;\\dfrac{c_{2}(r)}{\\sqrt{N\\tau}}.\n\\end{cases}\\label{eq:them_Stability_Statement}\n\\end{equation}\n\n\\end{thm}\n\n\\begin{rem}\n\\label{rem:linear_variance}Using very similar arguments to those\nof \\citet[Proof of Corollary 5.2]{smc:the:CdMG11}, it follows from\nthe first inequality of (\\ref{eq:them_Stability_Statement}) that\n\n\\begin{eqnarray*}\n\\left.\\begin{array}{c}\n\\inf_{n\\geq0}\\mathcal{E}_{n}^{N}\\geq\\tau\\\\\n\\\\\n\\text{and}\\\\\n\\\\\nN\\tau\\geq nc_{1}\n\\end{array}\\right\\}  & \\quad\\Rightarrow\\quad & \\mathbb{E}\\left[\\left(\\dfrac{Z_{n}^{N}}{Z_{n}}-1\\right)^{2}\\right]\\leq\\frac{2nc_{1}}{N\\tau}.\n\\end{eqnarray*}\n\n\\end{rem}\n\n\\begin{rem}\nIt follows immediately from the second inequality in (\\ref{eq:them_Stability_Statement})\nthat when $\\inf_{n\\geq0}\\mathcal{E}_{n}^{N}\\geq\\tau$ for all $N\\geq1$,\nthe prediction filter errors are time-uniformly convergent in the\nsense \n\\[\n\\lim_{N\\rightarrow\\infty}\\sup_{n\\geq0}\\mathbb{E}\\left[\\left|\\pi_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi)\\right|^{r}\\right]^{1\/r}=0.\n\\]\n\n\\end{rem}\n\n\\begin{rem}\nFurther to the discussion of Section~\\ref{sub:Instances-of-SMC},\nin the case of the BPF we have $\\mathcal{E}_{n}^{N}=1$ and hence\n$\\inf_{n\\geq0}\\mathcal{E}_{n}^{N}\\geq\\tau$ always, and for the ARPF\nwe also have $\\inf_{n\\geq0}\\mathcal{E}_{n}^{N}\\geq\\tau$ always, by\nvirtue of the ESS rule for selection of $\\alpha_{n}$. In Section~\\ref{sec:Discussion}\nwe shall introduce new algorithms designed to guarantee $\\inf_{n\\geq0}\\mathcal{E}_{n}^{N}\\geq\\tau$.\n\\end{rem}\n\n\\begin{rem}\nIt is possible to deduce estimates for the constants $c_{1}$ and\n$c_{2}(r)$ using the statements and proofs of Propositions~\\ref{prop:norm_const_bound}\nand~\\ref{prop:L_p_bound_mixing}, which are the main ingredients\nin the proof of Theorem~\\ref{thm:L_R_mix}. We omit such expressions\nonly for simplicity of presentation. \n\\end{rem}\nThe proofs of Propositions~\\ref{prop:norm_const_bound} and~\\ref{prop:L_p_bound_mixing}\nare given in the appendix.\n\\begin{prop}\n\\label{prop:norm_const_bound}Assume $\\mathbf{(A2)}$, $\\mathbf{(B^{++})}$\\textbf{\\textup{\\emph{\n}}}\\textup{\\emph{and}}\\textup{ }$\\mathbf{(C)}$\\textup{\\emph{. If\nfor some sequence of constants $\\left\\{ \\tau_{n};n\\geq0\\right\\} \\in(0,1]^{\\mathbb{N}}$\nand $N\\geq1$, \n\\[\n\\mathcal{E}_{n}^{N}\\geq\\tau_{n},\n\\]\nthen for any $n\\geq1$, \n\\[\n\\mathbb{E}\\left[\\left(\\frac{Z_{n}^{N}}{Z_{n}}-1\\right)^{2}\\right]\\leq\\sum_{p=0}^{n-1}\\frac{\\mathrm{osc}\\left(\\overline{Q}_{p,n}(1)\\right)^{2}}{N\\tau_{p}}\\left(\\mathbb{E}\\left[\\left(\\frac{Z_{p}^{N}}{Z_{p}}-1\\right)^{2}\\right]+1\\right).\n\\]\n}}\n\\end{prop}\n\\medskip{}\n\n\n\n\\begin{prop}\n\\label{prop:L_p_bound_mixing}Consider the constants and Markov kernels:\n\\[\n\\delta_{p,n}:=\\sup_{x,y}\\frac{Q_{p,n}(1)(x)}{Q_{p,n}(1)(y)},\\quad\\quad P_{p,n}(x,A):=\\frac{Q_{p,n}(\\mathbb{I}_{A})(x)}{Q_{p,n}(1)(x)},\\quad x\\in\\mathsf{X},A\\in\\mathcal{X},\\;0\\leq p\\leq n.\n\\]\nAssume $\\mathbf{(A2)}$,\\emph{ }$\\mathbf{(B)}$\\textbf{\\textup{\\emph{\n}}}\\textup{\\emph{and }}$\\mathbf{(C)}$\\textup{\\emph{.}}\\textbf{\\textup{\\emph{\n}}}\\textup{\\emph{Then for any $r\\geq1$ there exists a finite constant\n$B(r)$ such that for any $N\\geq1$, $n\\geq0$, and $\\varphi\\in\\mathcal{L}$,}}\\textup{\n\\begin{equation}\n\\mathbb{E}\\left[\\left|\\pi_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi)\\right|^{r}\\right]^{1\/r}\\leq4B(r)^{1\/r}\\sum_{p=0}^{n}\\delta_{p,n}\\left\\Vert P_{p,n}(\\bar{\\varphi})\\right\\Vert \\mathbb{E}\\left[\\left|\\mathcal{C}_{p,n}^{N}\\right|^{r}\\right]^{1\/r}.\\label{eq:L_p_bound_mixing_statement}\n\\end{equation}\n }\\textup{\\emph{where $\\bar{\\varphi}:=\\varphi-\\pi_{n}(\\varphi)$ and}}\n\\[\n\\mathcal{C}_{p,n}^{N}:=\\frac{\\sqrt{\\sum_{i}\\left(\\beta_{p,n}^{i}W_{p}^{i}\\right)^{2}}}{\\sum_{i}\\beta_{p,n}^{i}W_{p}^{i}}.\n\\]\n\n\\end{prop}\n\\medskip{}\n\n\\begin{proof}\n\\emph{(of Theorem~\\ref{thm:L_R_mix})} For the first bound on the\nright of (\\ref{eq:them_Stability_Statement}) under the conditions\nof the Theorem we apply Proposition~\\ref{prop:norm_const_bound}\nto give the following recursive bound:\n\\begin{equation}\nv_{n}\\leq\\sum_{p=0}^{n-1}\\frac{C}{N\\tau}\\left(v_{p}+1\\right),\\label{eq:v_n_recursion}\n\\end{equation}\nwhere $v_{n}:=\\mathbb{E}\\left[\\left(Z_{n}^{N}\/Z_{n}-1\\right)^{2}\\right]$\nand\n\\[\nC:=\\sup_{p,n}\\text{osc}\\left(\\overline{Q}_{p,n}(1)\\right)^{2}\\leq4\\sup_{p,n}\\left\\Vert \\overline{Q}_{p,n}(1)\\right\\Vert ^{2}<+\\infty,\n\\]\nunder $\\mathbf{(C)}$; see Remark~\\ref{rem:assumption_C}, (\\ref{eq:Q_p,n_bounded}).\nWe shall now prove\n\\begin{equation}\nv_{n}\\leq\\left(1+\\frac{C}{N\\tau}\\right)^{n}-1,\\quad\\forall n\\geq0,\\label{eq:v_n_induction_hyp}\n\\end{equation}\nwhich holds trivially if $C=0$, since in that case $v_{n}=0$ by\n(\\ref{eq:v_n_recursion}). Therefore suppose $C>0$. The argument\nis inductive. To initialize, note that since by definition $Z_{0}^{N}=Z_{0}=1$,\nwe have $v_{0}=0$. Now assume (\\ref{eq:v_n_induction_hyp}) holds\nat all ranks strictly less than some fixed $n\\geq1$. Using (\\ref{eq:v_n_recursion}),\nwe then have at rank $n$,\n\\begin{eqnarray*}\nv_{n} & \\leq & \\frac{C}{N\\tau}\\sum_{p=0}^{n-1}\\left(v_{p}+1\\right)\\\\\n & \\leq & \\frac{C}{N\\tau}\\sum_{p=0}^{n-1}\\left(1+\\frac{C}{N\\tau}\\right)^{p}\\\\\n & = & \\frac{C}{N\\tau}\\frac{\\left(1+\\frac{C}{N\\tau}\\right)^{n}-1}{\\left(1+\\frac{C}{N\\tau}\\right)-1}\\\\\n & = & \\left(1+\\frac{C}{N\\tau}\\right)^{n}-1.\n\\end{eqnarray*}\nThis completes the proof of (\\ref{eq:v_n_induction_hyp}), from which\nthe second equality on the right of (\\ref{eq:them_Stability_Statement})\nfollows immediately upon noting that by Theorem~\\ref{thm:convergence},\n$\\mathbb{E}[Z_{n}^{N}]=Z_{n}$.\n\nFor the second bound on the right of (\\ref{eq:them_Stability_Statement}),\nfirst note that as per Remark~\\ref{rem:assumption_C}, under $\\mathbf{(C)}$\\textbf{\n}we have \n\\begin{eqnarray*}\n\\left\\Vert P_{p,n}(\\bar{\\varphi})\\right\\Vert  & = & \\sup_{x}\\left|P_{p,n}(\\varphi)(x)-\\pi_{n}(\\varphi)\\right|\\\\\n & = & \\sup_{x}\\left|\\Phi_{p,n}(\\delta_{x})(\\varphi)-\\Phi_{p,n}(\\pi_{p})(\\varphi)\\right|\\\\\n & \\leq & \\sup_{\\mu,\\nu\\in\\mathcal{P}}\\left\\Vert \\Phi_{p,n}(\\mu)-\\Phi_{p,n}(\\nu)\\right\\Vert \\left\\Vert \\varphi\\right\\Vert \\leq\\left\\Vert \\varphi\\right\\Vert C\\rho^{n-p},\n\\end{eqnarray*}\nand \n\\[\n\\sup_{n\\geq0}\\sup_{p\\leq n}\\;\\delta_{p,n}<+\\infty,\n\\]\n Using these upper bounds, the fact that under $\\mathbf{(B^{++})}$\nwe have $\\beta_{p,n}^{i}=1\/N$, and Proposition~\\ref{prop:L_p_bound_mixing},\nwe find that there exists a finite constant $\\widetilde{B}(r)$ such\nthat for any $N\\geq1$, $n\\geq0$, $\\varphi\\in\\mathcal{L}$, \n\n\\emph{\n\\[\n\\mathbb{E}\\left[\\left|\\pi_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi)\\right|^{r}\\right]^{1\/r}\\leq\\left\\Vert \\varphi\\right\\Vert \\frac{\\tilde{B}(r)}{\\sqrt{N}}\\sum_{p=0}^{n}\\rho^{n-p}\\mathbb{E}\\left[\\left|\\mathcal{E}_{p}^{N}\\right|^{-r\/2}\\right]^{1\/r},\n\\]\n}where\n\\[\n\\mathcal{E}_{n}^{N}=\\frac{\\left(N^{-1}\\sum_{i}W_{n}^{i}\\right)^{2}}{N^{-1}\\sum_{i}\\left(W_{n}^{i}\\right)^{2}}.\n\\]\n\n\\end{proof}\n\n\\section{Discussion\\label{sec:Discussion}}\n\n\n\\subsection{Why not just run independent particle filters and average?\\label{sub:Why-not-just}}\n\nOne obvious approach to parallelization of SMC is to run a number\nof independent copies of a standard algorithm, such as the BPF, and\nthen in some sense simply average their outputs. Let us explain possible\nshortcomings of this approach. \n\nSuppose we want to run $s\\geq1$ independent copies of Algorithm~\\ref{alg:boot_pf},\neach with $q\\geq1$ particles. For purposes of exposition, it is helpful\nto express this collection of independent algorithms as a particular\ninstance of $\\alpha$SMC: for the remainder of Section~\\ref{sub:Why-not-just},\nwe set $N=sq$ and consider Algorithm~\\ref{alg:aSMC} with $\\mathbb{A}_{N}$\nchosen to consist only of the block diagonal matrix:\n\\begin{equation}\n\\left[\\begin{array}{cccc}\n\\mathbf{q^{-1}} & \\mathbf{0} & \\cdots & \\mathbf{0}\\\\\n\\mathbf{0} & \\mathbf{q^{-1}} & \\cdots & \\mathbf{0}\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\n\\mathbf{0} & \\mathbf{0} & \\cdots & \\mathbf{q^{-1}}\n\\end{array}\\right]\\label{eq:alpha_block}\n\\end{equation}\nwhere $\\mathbf{q}^{-1}$ is a $q\\times q$ submatrix with every entry\nequal to $q^{-1}$ and $\\mathbf{0}$ is a submatrix of zeros, of the\nsame size. In this situation, a simple application of Lemma~\\ref{lem:W_n_representation}\nshows that for any $n\\geq1$ and $\\ell\\in[s]$, if we define $B(\\ell):=\\{(\\ell-1)q+1,(\\ell-1)q+2,\\ldots,\\ell q\\}$,\nthen\n\\begin{equation}\n\\text{for all}\\quad i_{n}\\in B(\\ell),\\quad\\quad W_{n}^{i_{n}}=\\prod_{p=0}^{n-1}\\left(N^{-1}\\sum_{i_{p}\\in B(\\ell)}g_{p}\\left(\\zeta_{p}^{i_{p}}\\right)\\right)=:\\mathbb{W}_{n}^{\\ell},\\label{eq:W_n^i_blocks}\n\\end{equation}\nc.f. (\\ref{eq:bootstrap_W_n^i})--(\\ref{eq:bootstrap_Z_n^N}), and\nfurthermore upon inspection of Algorithm~\\ref{alg:aSMC}, we find\n\\begin{equation}\n\\text{for all }\\ell\\in[s]\\text{ and }i\\in B(\\ell),\\quad\\quad\\mathbb{P}\\left(\\left.\\zeta_{n}^{i}\\in A\\right|\\mathcal{F}_{n-1}\\right)=\\frac{\\sum_{j\\in B(\\ell)}g_{n-1}\\left(\\zeta_{n-1}^{j}\\right)f\\left(\\zeta_{n-1}^{j},A\\right)}{\\sum_{j\\in B(\\ell)}g_{n-1}\\left(\\zeta_{n-1}^{j}\\right)},\\label{eq:blocks_law}\n\\end{equation}\nfor any $A\\in\\mathcal{X}$. It follows that the blocks of particles\n\\[\n\\hat{\\zeta}_{n}^{k}:=\\left\\{ \\zeta_{n}^{i}\\right\\} _{i\\in B(\\ell)},\\quad\\ell\\in[s],\n\\]\nare independent, and for each $\\ell\\in[s]$, the sequence $\\left\\{ \\hat{\\zeta}_{n}^{\\ell};n\\geq0\\right\\} $\nevolves under the same law as a BPF, with $q$ particles. Furthermore\nwe notice \n\\[\n\\pi_{n}^{N}=\\pi_{n}^{sq}=\\frac{\\sum_{i}W_{n}^{i}\\;\\delta_{\\zeta_{n}^{i}}}{\\sum_{i}W_{n}^{i}}=\\frac{\\sum_{\\ell\\in[s]}\\sum_{i\\in B(\\ell)}W_{n}^{i}\\;\\delta_{\\zeta_{n}^{i}}}{\\sum_{\\ell\\in[s]}\\sum_{i\\in B(\\ell)}W_{n}^{i}}=\\frac{\\sum_{\\ell\\in[s]}\\mathbb{W}_{n}^{\\ell}\\left(q^{-1}\\sum_{i\\in B(\\ell)}\\delta_{\\zeta_{n}^{i}}\\right)}{\\sum_{\\ell\\in[s]}\\mathbb{W}_{n}^{\\ell}},\n\\]\nwhere $q^{-1}\\sum_{i\\in B(\\ell)}\\delta_{\\zeta_{n}^{i}}$ may be regarded\nas the approximation of $\\pi_{n}$ obtained from the $\\ell$th block\nof particles. Since we have assumed that $\\mathbb{A}_{N}$ consists\nonly of the matrix (\\ref{eq:alpha_block}), $\\mathbf{(A2)}$ and$\\mathbf{(B^{++})}$\nhold, and by Theorem~\\ref{thm:convergence} we are assured of the\na.s.~convergence $\\pi_{n}^{sq}(\\varphi)\\rightarrow\\pi_{n}(\\varphi)$\nwhen $q$ is fixed and $s\\rightarrow\\infty$. In words, we have convergence\nas the total number of bootstrap algorithms tends to infinity, even\nthough the number of particles within each algorithm is fixed. On\nthe other hand, simple averaging of the output from the $m$ independent\nalgorithms would entail reporting:\n\\begin{equation}\n\\frac{1}{sq}\\sum_{i\\in[sq]}\\delta_{\\zeta_{n}^{i}}\\label{eq:naive}\n\\end{equation}\nas an approximation of $\\pi_{n}$; the problem is that (\\ref{eq:naive})\nis biased, in the sense that in general it is not true that, with\n$q$ fixed, $(sq)^{-1}\\sum_{i\\in[sq]}\\varphi(\\zeta_{n}^{i})\\rightarrow\\pi_{n}(\\varphi)$\nas $s\\rightarrow\\infty$ (although obviously we do have convergence\nif $q\\rightarrow\\infty$). In summary, simple averages across independent\nparticle filters do not, in general, converge as the number of algorithms\ngrows. \n\nWe can also discuss the quality of an approximation of $Z_{n}$ obtained\nby simple averaging across the $s$ independent algorithms; let us\nconsider the quantities\n\\[\n\\mathbb{Z}_{n}^{(q,\\ell)}:=\\frac{1}{\\ell}\\sum_{j\\in[\\ell]}\\mathbb{W}_{n}^{j},\\quad\\ell\\in[s].\n\\]\nComparing (\\ref{eq:W_n^i_blocks}) with (\\ref{eq:bootstrap_Z_n^N}),\nand noting (\\ref{eq:blocks_law}) and the independence properties\ndescribed above, we have\n\\begin{equation}\n\\mathbb{E}\\left[\\mathbb{Z}_{n}^{(q,s)}\\right]=Z_{n},\\quad\\quad\\mathbb{E}\\left[\\left(\\frac{\\mathbb{Z}_{n}^{(q,s)}}{Z_{n}}-1\\right)^{2}\\right]=\\frac{1}{s}\\mathbb{E}\\left[\\left(\\frac{\\mathbb{Z}_{n}^{(q,1)}}{Z_{n}}-1\\right)^{2}\\right],\\label{eq:Z_naive_average}\n\\end{equation}\nwhere the first equality holds due to the first part of Theorem~\\ref{thm:convergence}:\nin this context the well known lack-of-bias property of the BPF. Under\ncertain ergodicity and regularity conditions, \\citep[Proposition 4]{WhiteleyTPF}\nestablishes that $\\mathbb{E}\\left[\\left(\\mathbb{Z}_{n}^{(q,1)}\/Z_{n}\\right)^{2}\\right]$\ngrows exponentially fast along observation sample paths when $q$\nis fixed and $n\\rightarrow\\infty$. When that occurs, it is clear\nfrom (\\ref{eq:naive}) that $s$ must be scaled up exponentially fast\nwith $n$ in order to control the relative variance of $\\mathbb{Z}_{n}^{(q,s)}$.\nOn the other hand, by Theorem~\\ref{thm:L_R_mix} and Remark~\\ref{rem:linear_variance},\nit is apparent that if we design an instance of $\\alpha$SMC so as\nto enforce $\\inf_{n\\geq0}\\mathcal{E}_{n}^{N}>0$, then we can control\n$\\mathbb{E}\\left[\\left(Z_{n}^{N}\/Z_{n}\\right)^{2}\\right]$ at a more\nmodest computational cost. When $\\mathbb{A}_{N}$ consists only of\nthe matrix (\\ref{eq:alpha_block}) we do not have a guarantee that\n$\\inf_{n\\geq0}\\mathcal{E}_{n}^{N}>0$, but in Section~\\ref{sub:Algorithms-with-adaptive}\nwe shall suggest some novel algorithms which do guarantee this lower\nbound and therefore enjoy the time-uniform convergence and linear-in-time\nvariance properties of Theorem~\\ref{thm:L_R_mix}. Before addressing\nthese stability issues we discuss the conditions under which the $\\alpha$SMC\nalgorithm converges.\n\n\n\\subsection{Ensuring convergence\\label{sub:Ensuring-convergence}}\n\nThroughout Section~\\ref{sub:Ensuring-convergence}, we consider the\ngeneric Algorithm~\\ref{alg:aSMC}. We describe what can go wrong\nif the conditions of Theorem~\\ref{thm:convergence} do not hold:\nas an example of a situation in which $\\mathbf{(B^{+})}$ does not\nhold; suppose that $\\mathbb{A}_{N}$ consists only of the transition\nmatrix of a simple random walk on the star graph with $N$ vertices,\ncall it $\\mathcal{S}_{N}$. That is, for $N>2$, $\\mathcal{S}_{N}$\nis an undirected tree with one internal vertex and $N-1$ leaves,\nand for $N\\leq2$, all vertices are leaves. Examples of $\\mathcal{S}_{N}$\nare illustrated in Figure~\\ref{fig:Star-graphs-of}. It is elementary\nthat a simple random walk on $\\mathcal{S}_{N}$ has unique invariant\ndistribution given by \n\\[\n\\frac{d_{N}^{i}}{\\sum_{j}d_{N}^{j}},\\quad i\\in[N],\\quad\\quad\\text{ where}\\quad d_{N}^{i}:=\\text{ degree of vertex }i\\text{ in }\\mathcal{S}_{N}.\n\\]\nAssuming that for every $N>2$ the internal vertex of $\\mathcal{S}_{N}$\nis labelled vertex $1$, then for all $0\\leq p<n$, we have $\\beta_{p,n}^{1}=1\/2$\nfor all $N>2$, so $\\mathbf{(B^{+})}$ does not hold, and thus part\n2) of Theorem~\\ref{thm:convergence} does not hold.\n\n\\begin{figure}\n\\hfill{}\\includegraphics[width=1\\textwidth]{stars2}\\hfill{}\n\n\\protect\\caption{\\label{fig:Star-graphs-of}Star graphs. }\n\n\n\\end{figure}\n\n\nAs a more explicit example of convergence failure, suppose that $\\mathbb{A}_{N}$\nconsists only of the matrix which has $1$ for every entry in its\nfirst column, and zeros for all other entries. This is the transition\nmatrix of a random walk on a directed graph of which all edges lead\nto vertex $1$. It follows that for all $0\\leq p<n$, we have $\\beta_{p,n}^{1}=1$\nand $\\beta_{p,n}^{i}=0$ for all $i\\in[N]\\setminus\\{1\\}$, so $\\mathbf{(B^{+})}$\nclearly does not hold. If additionally $f(x,\\cdot)=\\delta_{x}(\\cdot)$\n, then by inspection of Algorithm~\\ref{alg:aSMC} we have $\\mathbb{P}\\left(\\left\\{ \\zeta_{n}^{i}=\\zeta_{0}^{1}\\right\\} \\right)=1$\nfor all $i\\in[N]$ and all $n\\geq1$. We then also have $\\mathbb{P}\\left(\\left\\{ \\pi_{n}^{N}=\\delta_{\\zeta_{0}^{1}}\\right\\} \\right)=1$,\nso that we obtain a generally poor and non-convergent approximation\nof $\\pi_{n}$. \n\nIn both these situations vertex $1$ is, in graph theoretic terms,\na \\emph{hub} and an intuitive explanation of the convergence failure\nis that the contribution of particle $1$ to $\\pi_{n}^{N}$ does not\nbecome negligible as $N\\rightarrow\\infty$, so that no ``averaging''\ntakes place. Assumption $\\mathbf{(B^{+})}$ ensures enough negligibility\nto prove the weak laws of large numbers in Theorem~\\ref{thm:convergence}.\nAssumption $\\mathbf{(B^{++})}$ may be viewed as ensuring negligibility,\nand in such a way as to ensure the $\\sqrt{N}$ rate of convergence\nand strong law in the final part of Theorem~\\ref{thm:convergence}.\nAs a practical summary, we recommend verifying \\emph{$\\mathbf{(B^{++})}$},\nor at least\\textbf{ }avoid using graphs with hubs, since otherwise\n$\\alpha$SMC may not converge.\n\n\n\\subsection{Provably stable algorithms with adaptive interaction\\label{sub:Algorithms-with-adaptive}}\n\nThere are of course many choices of $\\mathbb{A}_{N}$ which do satisfy\n$\\mathbf{(B^{++})}$. In this section we provide some guidance and\nsuggestions on this matter. In order to focus our attention we consider\nin addition to $\\mathbf{(B^{++})}$, the following criteria against\nwhich to assess candidates for $\\mathbb{A}_{N}$ and whatever functional\nis used at line $(\\star)$ of Algorithm~\\ref{alg:aSMC}: \n\n\\begin{enumerate}[\\hspace{5 mm}(a)]\n\n\\item\\label{enu:criterion1}the condition $\\inf_{n\\geq0}\\mathcal{E}_{n}^{N}>0$\nshould be enforced, so as to ensure stability\n\n\\item\\label{enu:criterion2}the computational complexity of associated\nsampling, weight and ESS calculations should not be prohibitively\nhigh\n\n\\end{enumerate} The motivation for (\\ref{enu:criterion1}) is the\ntheoretical assurance given by Theorem~\\ref{thm:L_R_mix}. The motivation\nfor (\\ref{enu:criterion2}) is simply that we do not want an algorithm\nwhich is much more expensive than any of the standard SMC methods,\nAlgorithms~\\ref{alg:SIS}--\\ref{alg:boot_pf} and the ARPF. It is\neasily checked that the complexity of SIS is $O(N)$ per unit time\nstep, which is the same as the complexity of the BPF \\citep{carpenter1999improved}\nand the ARPF.\n\nThroughout the remainder of Section~\\ref{sub:Algorithms-with-adaptive}\nwe shall assume that $\\mathbb{A}_{N}$ consists only of transition\nmatrices of simple random walks on regular undirected graphs. We impose\na little structure in addition to this as per the following definition,\nwhich identifies an object related to the standard notion of a block-diagonal\nmatrix.\n\\begin{defn*}\nA \\textbf{B-matrix} is a Markov transition matrix which specifies\na simple random walk on a regular undirected graph which has a self-loop\nat every vertex and whose connected components are all complete subgraphs.\n\\end{defn*}\nNote that due to the graph regularity appearing in this definition,\nif $\\mathbb{A}_{N}$ consists only of B-matrices, then $\\mathbf{(B^{++})}$\nis immediately satisfied. This regularity is also convenient for purposes\nof interpretation: it seems natural to use graph degree to give a\nprecise meaning to ``degree of interaction''. Indeed $Id$ and $\\mathbf{1}_{1\/N}$\nare both B-matrices, respectively specifying simple random walks on\n$1$-regular and $N$-regular graphs, and recall for the ARPF, $\\mathbb{A}_{N}=\\left\\{ Id,\\mathbf{1}_{1\/N}\\right\\} $;\nthe main idea behind the new algorithms below is to consider an instance\nof $\\alpha$SMC in which $\\mathbb{A}_{N}$ is defined to consist of\nB-matrices of various degrees $d\\in[N]$, and define adaptive algorithms\nwhich select the value of $\\alpha_{n-1}$ by searching through $\\mathbb{A}_{N}$\nto find the graph with the smallest $d$ which achieves $\\mathcal{E}_{n}^{N}\\geq\\tau>0$\nand hence satisfies criterion 1. In this way, we ensure provable stability\nwhilst trying to avoid the complete interaction which occurs when\n$\\alpha_{n-1}=\\mathbf{1}_{1\/N}$. \n\nAnother appealing property of B-matrices is formalized in the following\nlemma; see criterion (\\ref{enu:criterion2}) above. The proof is given\nin the appendix.\n\\begin{lem}\n\\label{lem:complexity}Suppose that $A=\\left(A^{ij}\\right)$ is a\nB-matrix of size $N$. Then given the quantities $\\left\\{ W_{n-1}^{i}\\right\\} _{i\\in[N]}$\nand $\\left\\{ g_{n-1}(\\zeta_{n-1}^{i})\\right\\} _{i\\in[N]}$, the computational\ncomplexity of calculating $\\left\\{ W_{n}^{i}\\right\\} _{i\\in[N]}$\nand simulating $\\left\\{ \\zeta_{n}^{i}\\right\\} _{i\\in[N]}$ as per\nAlgorithm~\\ref{alg:aSMC}, using $\\alpha_{n-1}=A$, is $O(N)$.\n\\end{lem}\nWhen calculating the overall complexity of Algorithm~\\ref{alg:aSMC}\nwe must also consider the complexity of line $(\\star)$, which in\ngeneral depends on $\\mathbb{A}_{N}$ and the particular functional\nused to choose $\\alpha_{n}$. We resume this complexity discussion\nafter describing the specifics of some adaptive algorithms. \n\n\n\\subsubsection*{Adaptive interaction}\n\nThroughout this section we set $m\\in\\mathbb{N}$ and then $N=2^{m}$.\nConsider Algorithm~\\ref{alg:aSMC} with $\\mathbb{A}_{N}$ chosen\nto be the set of B-matrices of size $N$. We suggest three adaptation\nrules at line $(\\star)$ of Algorithm~\\ref{alg:aSMC}: Simple, Random,\nand Greedy, all implemented via Algorithm~\\ref{alg:generic adaptation}\n(note that dependence of some quantities on $n$ is suppressed from\nthe notation there), but differing in the way they select the index\nlist $\\mathcal{I}_{k}$ which appears in the ``while'' loop of that\nprocedure. The methods for selecting $\\mathcal{I}_{k}$ are summarised\nin Table~\\ref{tab:Choosing_I_k}: the Simple rule needs little explanation,\nthe Random rule implements an independent random shuffling of indices\nand the Greedy rule is intended, heuristically, to pair large weights,\n$\\mathbb{W}_{k}^{i}$, will small weights in order to terminate the\n``while'' loop with as small a value of $k$ as possible. Note that,\nformally, in order for our results for $\\alpha$SMC to apply when\nthe Random rule is used, the underlying probability space must be\nappropriately extended, but the details are trivial so we omit them. \n\n\\begin{algorithm}\n\\begin{raggedright}\n\\qquad{}at iteration $n$ and line $(\\star)$ of Algorithm~\\ref{alg:aSMC},\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}for $i=1,\\ldots,N$,\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}\\qquad{}set $B(0,i)=\\{i\\}$, $\\mathbb{W}_{0}^{i}=W_{n-1}^{i}g_{n-1}(\\zeta_{n-1}^{i})$,\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}set $k=0$\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}set $\\overline{\\mathbb{W}}_{0}=N^{-1}\\sum_{i}\\mathbb{W}_{0}^{i}$\n, $\\mathcal{E}=\\frac{\\left(\\overline{\\mathbb{W}}_{0}\\right)^{2}}{N^{-1}\\sum_{i}\\left(\\mathbb{W}_{0}^{i}\\right)^{2}}$, \n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}while $\\mathcal{E}<\\tau$\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}\\qquad{}set $\\mathcal{I}_{k}$ according to the\nSimple, Random or Greedy scheme of Table~\\ref{tab:Choosing_I_k}.\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}\\qquad{}for $i=1,\\ldots,N\/2^{k+1}$ \n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}\\qquad{}\\qquad{}set $B(k+1,i)=B(k,\\mathcal{I}_{k}(2i-1))\\cup B(k,\\mathcal{I}_{k}(2i))$\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}\\qquad{}\\qquad{}set $\\mathbb{W}_{k+1}^{i}=\\mathbb{W}_{k}^{\\mathcal{I}_{k}(2i-1)}\/2+\\mathbb{W}_{k}^{\\mathcal{I}_{k}(2i)}\/2$\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}\\qquad{}set $k=k+1$\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}\\qquad{}set $\\mathcal{E}=\\frac{\\left(\\overline{\\mathbb{W}}_{0}\\right)^{2}}{N^{-1}2^{k}\\sum_{i\\in[N\/2^{k}]}\\left(\\mathbb{W}_{k}^{i}\\right)^{2}}$\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}set $K_{n-1}=k$\n\\par\\end{raggedright}\n\n\\begin{raggedright}\n\\qquad{}\\qquad{}set $\\alpha_{n-1}^{ij}=\\begin{cases}\n1\/2^{K_{n-1}}, & \\text{if }i\\sim j\\text{ according to \\ensuremath{\\left\\{ B(K_{n-1},i)\\right\\} _{i\\in[N\/2^{K_{n-1}}]}}}\\\\\n0, & \\text{otherwise}.\n\\end{cases}$\n\\par\\end{raggedright}\n\n\\protect\\caption{\\label{alg:generic adaptation}Adaptive selection of $\\alpha_{n-1}$}\n\\end{algorithm}\n\n\nFollowing the termination of the ``while'' loop, Algorithm~\\ref{alg:generic adaptation}\noutputs an integer $K_{n-1}$ and a partition $\\left\\{ B(K_{n-1},i)\\right\\} _{i\\in[N\/2^{K_{n-1}}]}$\nof $[N]$ into $N\/2^{K_{n-1}}$ subsets, each of cardinality $2^{K_{n-1}}$;\nthis partition specifies $\\alpha_{n-1}$ as a B-matrix and $2^{K_{n-1}}$\nis the degree of the corresponding graph (we keep track of $K_{n-1}$\nfor purposes of monitoring algorithm performance in Section~\\ref{sub:Numerical-illustrations}).\nProposition~\\ref{prop:Upon-termination-of} is a formal statement\nof its operation and completes our complexity considerations. The\nproof is given in the appendix. It can be checked by an inductive\nargument similar to the proof of Lemma~\\ref{lem:ARPF_A2}, also in\nthe appendix, that when $\\alpha_{n}$ is chosen according to Algorithm~\\ref{alg:generic adaptation}\ncombined with any of the adaptation rules in Table~\\ref{tab:Choosing_I_k},\n\\textbf{(A2)} is satisfied. \n\\begin{prop}\n\\label{prop:Upon-termination-of}The weights $\\left\\{ \\mathbb{W}_{k}^{i}\\right\\} _{i\\in[N\/2^{k}]}$\ncalculated in Algorithm~\\ref{alg:generic adaptation} obey the expression\n\\begin{equation}\n\\mathbb{W}_{k}^{i}=2^{-k}\\sum_{j\\in B(k,i)}W_{n-1}^{j}g_{n-1}(\\zeta_{n-1}^{j}).\\label{eq:W_k_explicit}\n\\end{equation}\nMoreover, $\\alpha_{n-1}$ delivered by Algorithm~\\ref{alg:generic adaptation}\nis a B-matrix and when this procedure is used at line $(\\star)$ of\nAlgorithm~\\ref{alg:aSMC}, the weights calculated in Algorithm~\\ref{alg:aSMC}\nare given, for any $i\\in[N\/2^{K_{n-1}}]$, by\n\\begin{equation}\nW_{n}^{j}=\\mathbb{W}_{K_{n-1}}^{i},\\quad\\quad\\text{for all \\quad}j\\in B(K_{n-1},i)\\label{eq:W_equals_bb_W}\n\\end{equation}\nand $\\mathcal{E}_{n}^{N}\\geq\\tau$ always. The overall worst-case\ncomplexity of Algorithm~\\ref{alg:aSMC} is, for the three adaptation\nrules in Table~\\ref{tab:Choosing_I_k}, Simple: $O(N)$, Random:\n$O(N)$, and Greedy: $O(N\\log_{2}N)$. \n\\end{prop}\n\\begin{table}[h]\n\\begin{tabular}[c]{>{\\raggedright}p{1.2cm}l}\n\\toprule  \n\\addlinespace \n\\textbf{\\footnotesize{Simple}} & {\\footnotesize{set $\\mathcal{I}_{k}=(1,\\ldots,N\/2^{k})$}}\\tabularnewline\\addlinespace\n\\midrule  \n\\addlinespace \n\\textbf{\\footnotesize{Random}} & \n{\\footnotesize{if $k=0$, set $\\mathcal{I}_{k}$ to a random permutation of $[N\/2^{k}]$, otherwise $\\mathcal{I}_{k}=(1,\\ldots,N\/2^{k})$}}\\tabularnewline\\addlinespace\n\\midrule  \n\\addlinespace \n\\multirow{2}{1.2cm}{\\textbf{\\footnotesize{Greedy}}} &\n{\\footnotesize{set $\\mathcal{I}_{k}$ such that}}\\tabularnewline  &\n{\\hspace{\\bigskipamount}\\footnotesize{$\\mathbb{W}_{k}^{\\mathcal{I}_{k}(1)}\\geq\\mathbb{W}_{k}^{\\mathcal{I}_{k}(3)}\\geq\\cdots\\geq\\mathbb{W}_{k}^{\\mathcal{I}_{k}(N\/2^{k}-1)}\\geq\\mathcal{\\mathbb{W}}_{k}^{\\mathcal{I}_{k}(N\/2^{k})}\\cdots\\geq\\mathbb{W}_{k}^{\\mathcal{I}_{k}(4)}\\geq\\mathbb{W}_{k}^{\\mathcal{I}_{k}(2)}$}}\\tabularnewline\\addlinespace \\bottomrule\\addlinespace\n\\end{tabular}\\protect\\caption{\\label{tab:Choosing_I_k}Adaptation rules for choosing $\\mathcal{I}_{k}$}\n\\end{table}\n\n\n\n\\subsection{Numerical illustrations\\label{sub:Numerical-illustrations}}\n\nWe consider a stochastic volatility HMM: \n\\begin{eqnarray*}\n &  & X_{0}\\sim\\mathcal{N}(0,1),\\quad X_{n}=aX_{n-1}+\\sigma V_{n},\\\\\n &  & Y_{n}=\\varepsilon W_{n}\\exp(X_{n}\/2).\n\\end{eqnarray*}\n where $\\left\\{ V_{n}\\right\\} _{n\\in\\mathbb{N}}$ and $\\left\\{ W_{n}\\right\\} _{n\\in\\mathbb{N}}$\nare sequences of mutually i.i.d.~$\\mathcal{N}(0,1)$ random variables,\n$\\left|a\\right|<1$, and $\\sigma,\\varepsilon>0$. To study the behaviour\nof the different adaptation rules in terms of effective sample size,\na sequence of $3\\cdot10^{4}$ observations were generated from the\nmodel with $a=0.9$, $\\sigma=0.25$, and $\\varepsilon=0.1$. This\nmodel obviously does not satisfy $\\mathbf{(C)}$, but $\\mathbf{(A1)}$\nis satisfied as long as the observation record to does include the\nvalue zero.\n\nThe ARPF and $\\alpha$SMC with the Simple, Random and Greedy adaptation\nprocedures specified in Section~\\ref{sub:Algorithms-with-adaptive}\nwere run on this data with $N=2^{10}$ and threshold $\\tau=0.6$.\nTo give some impression of ESS and interaction behaviour, Figure~\\ref{fig:ESS-and-interaction}\nshows snapshots of $N_{n}^{\\text{eff}}$ and $K_{n}$ versus $n$,\nfor $575\\leq n\\leq825$. The sample path of $N_{n}^{\\text{eff}}$\nfor ARPF displays a familiar saw-tooth pattern, jumping back up to\n$N=2^{10}$ when resampling, i.e.~when $K_{n}=10$. The Simple adaptation\nscheme keeps $N_{n}^{\\text{eff}}$ just above the threshold $\\tau N=0.6\\times2^{10}$,\nwhereas the Greedy strategy is often able to keep $N_{n}^{\\text{eff}}$\nwell above this threshold, with smaller values of $K_{n}$, i.e.~with\na lower degree of interaction. The results for the Random adaptation\nrule, not shown in this plot, where qualitatively similar to those\nof the Greedy algorithm but slightly closer to the Simple adaptation. \n\nIn order to examine the stationarity of the particle processes as\nwell as the statistical behavior of the degree of interaction over\ntime, Figure~\\ref{fig:histograms_and_E_vs_k} shows two histograms\nof $K_{n}$ for each of the adaptation rules. One histogram is based\non the sample of $K_{n}$ where $100<n\\leq15050$, and the other is\nbased on $K_{n}$ where $15050<n\\leq30000$. For each algorithm, the\nsimilarity between the histograms for the two time intervals suggests\nthat the process $\\left\\{ K_{n}\\right\\} _{n\\geq0}$ is stationary.\nAs expected, the distribution of $K_{n}$ for ARPF is dichotomous\ntaking only values equal to $K_{n}=0$ when there is no interaction,\ni.e.~the resampling is skipped or $K_{n}=10$ for the complete interaction,\ni.e.~resampling. It is apparent that the Simple, Random and Greedy\nalgorithms move the distribution of $K_{n}$ towards smaller values\nand almost always manage to avoid the complete interaction. For the\nRandom and Greedy algorithms, $K_{n}$ rarely exceeds $1$, i.e.~in\norder to guarantee $\\mathcal{E}_{n}^{N}$ it is rarely necessary to\nconsider anything more than pair-wise interaction. \n\\begin{figure}\n\\hfill{}\\includegraphics[width=1\\textwidth]{figure1}\\hfill{}\\protect\\caption{\\label{fig:ESS-and-interaction}Snapshots of ESS and degree of interaction.\nTop: $N_{n}^{\\text{eff}}$ vs.~$n$ (solid) and threshold $\\tau N$\n(dashed). Bottom: $K_{n}$ vs.~$n$ (stems) and the base two logarithm\nof the time-average of $2^{K_{n}}$ (dashed). Recall from Section\n\\ref{sub:Algorithms-with-adaptive} that $2^{K_{n}}$ is the degree\nof the graph corresponding to the matrix $\\alpha_{n}$ selected by\nAlgorithm~\\ref{alg:generic adaptation}, and returned to line $(\\star)$\nof Algorithm~\\ref{alg:aSMC}. }\n\\end{figure}\n\n\n\n\nThe plot on the right of Figure~\\ref{fig:histograms_and_E_vs_k}\nshows, for each of the Simple, Random and Greedy adaptation rules,\nthe relationship between the intermediate variables $\\mathcal{E}$\nand $k$ appearing in the ``while'' loop of Algorithm~\\ref{alg:generic adaptation}.\nIn order to obtain equal sample sizes for plotting purposes, Algorithm~\\ref{alg:generic adaptation}\nwas modified slightly so as to evaluate $\\mathcal{E}$ for every value\n$k\\in\\{0,\\ldots,m\\}$, whilst still outputting $K_{n-1}$ as the smallest\nvalue of $k$ achieving $\\mathcal{E}\\geq\\tau$. The plotted data were\nthen obtained, for each $k$, by averaging the corresponding values\nof $\\mathcal{E}$ over the time steps of the algorithm. It is apparent\nthat, for small values of $k$, the Random and Greedy strategies achieve\na faster increase in $\\mathcal{E}$ than the Simple strategy, and\nthis explains the shape of the histograms on the left of Figure~\\ref{fig:histograms_and_E_vs_k}.\n\\begin{figure}\n\\hfill{}\\includegraphics[width=1\\textwidth]{figure32}\\hfill{}\n\n\\protect\\caption{\\label{fig:histograms_and_E_vs_k}Left: Histograms of $K_{n}$ for\nthe ARPF and the three adaptation rules of Table~\\ref{tab:Choosing_I_k}.\nThe light bars were obtained from $\\left\\{ K_{n};n=101,\\ldots,15050\\right\\} $\nand the dark bars from $\\left\\{ K_{n};n=15051,\\ldots,30000\\right\\} $\nRight: Growth of $\\mathcal{E}$ vs.~$k$ for the Simple (solid),\nRandom (dash-dot) and Greedy (dashed). }\n\\end{figure}\n\n\nFigure~\\ref{fig:Mean-square-error} shows a comparison of the mean\nsquared errors (MSE) of approximating the conditional expectation\nof $\\phi(X_{p})$ with respect to the underlying stochastic volatility\nHMM given the observations $\\{y_{n};0\\leq n\\leq p+\\ell\\}$, where\n$\\ell\\in\\{-5,0,1\\}$ and $\\phi$ is some test function. The cases,\n$\\ell=-5$, $\\ell=0$, and $\\ell=1$ correspond to the lag 5 smoother,\nfilter and one step predictor, respectively. The lag 5 smoother results\nwere obtained by tracing back ancestral lineages. In order to estimate\nthe approximation error, a reference value for the conditional expectation\nwas evaluated by running a BPF with a large sample size $N=2^{17}$.\nApproximation errors were evaluated for $N_{\\mathrm{MC}}=1000$ Monte\nCarlo runs of 1000 time steps each with $N=2^{9}$, and MSE was obtained\nby averaging over the time steps and the Monte Carlo runs. First 30\ntime steps were excluded in the calculations to avoid any non-stationary\neffects due to initialization. The results show that the Random and\nGreedy algorithms produce consistently smaller errors than the Simple\nalgorithm and for large values of $\\tau$ the Greedy algorithm appears\nto consistently outperform ARPF.\n\n\\begin{figure}\n\\hfill{}\\includegraphics[width=1\\textwidth]{figure4}\\hfill{}\\protect\\caption{\\label{fig:Mean-square-error}MSE vs.~$\\tau$ for the lag 5 smoother,\nfilter, and one step predictor using the four algorithms ARPF (solid),\nSimple ($\\triangle$), Random ($\\circ$), and Greedy ($\\times)$ and\nthree test functions $\\phi$.}\n\\end{figure}\n\n\n\n\\subsection{Concluding remarks}\n\\begin{itemize}\n\\item The martingale decomposition presented in Proposition~\\ref{prop:martingale}\nmay also be exploited to pursue central limit theorems. A study of\nthis will be conducted elsewhere, but we believe, further to Remark~\\ref{rem_alpha_conv},\nthat it will in general involve some further hypotheses in order to\nensure convergence of the covariance of this martingale and thus prove\nthe existence of a well-defined asymptotic variance.\n\\item It is worth pointing out that there are also SMC algorithms other\nthan those listed in Section~\\ref{sub:Instances-of-SMC} that can\nbe formulated as instances of $\\alpha$SMC, e.g.~the stratified resampling\nalgorithm of \\citet{Kitagawa1996} and the auxiliary particle filter\nof \\citet{pitt1999filtering}. It should be kept in mind, however,\nthat the successful formulation of any algorithm as an instance of\n$\\alpha$SMC does not necessarily imply that the assumptions $\\mathbf{(B)}$,\n$\\mathbf{(B^{+})}$ or $\\mathbf{(B^{++})}$ hold, and the validity\nof Theorems~\\ref{thm:convergence} and~\\ref{thm:L_R_mix} is in\nthat sense not automatic.\n\\end{itemize}\n\n\\section{Appendix\\label{sec:Appendix}}\n\\begin{lem}\n\\label{lem:ARPF_A2}If for every $n\\geq0$, $\\alpha_{n}$ is chosen\naccording to the ESS thresholding rule (\\ref{eq:alpha_ARPF}), then\n$\\mathbf{(A2)}$ is satisfied. \\end{lem}\n\\begin{proof}\nThe proof is by induction. To initialize, we have at rank $n=0$,\n\\begin{equation}\n\\alpha_{0}:=\\begin{cases}\n\\mathbf{1}_{1\/N}, & \\quad\\text{ if }\\quad\\frac{\\left(N^{-1}\\sum_{i}W_{0}^{i}g_{0}(\\zeta_{0}^{i})\\right)^{2}}{N^{-1}\\sum_{i}\\left(W_{0}^{i}g_{0}(\\zeta_{0}^{i})\\right)^{2}}<\\tau,\\\\\nId, & \\quad\\text{ otherwise},\n\\end{cases}\\label{eq:alpha_ARPF-1}\n\\end{equation}\nnoting that by definition $W_{0}^{i}=1$, we find that the entries\nof $\\alpha_{0}$ are measurable w.r.t.~$\\mathcal{F}_{0}$. For the\ninduction hypothesis, suppose that for some $n\\geq0$ and all $p\\leq n$,\nthe entries of $\\alpha_{p}$ are measurable w.r.t.~$\\mathcal{F}_{n}$.\nIt follows immediately from Lemma~\\ref{lem:W_n_representation},\nequation (\\ref{eq:unwind2}), that $\\left\\{ W_{n+1}^{i}\\right\\} _{i\\in[N]}$\nare all measurable w.r.t.~$\\mathcal{F}_{n+1}$, and it follows from\n(\\ref{eq:alpha_ARPF}) applied at rank $n+1$ that the entries of\n$\\alpha_{n+1}$ are measurable w.r.t.~$\\mathcal{F}_{n+1}$, and hence\nthe induction hypothesis holds at rank $n+1$. This completes the\nproof. \n\\end{proof}\n\n\\subsubsection*{Resampling times description of the ARPF}\n\nIn order to derive expressions for $\\pi_{n}^{N}$ and $Z_{n}^{N}$\nin the case of the ARPF, define a family of random sets $\\left\\{ \\sigma_{n};n\\geq1\\right\\} $,\nand random times $\\left\\{ T_{n};n\\geq1\\right\\} $ as follows \n\\begin{eqnarray}\n\\sigma_{n} & := & \\left\\{ m;\\;1\\leq m\\leq n\\text{\\;\\ and }\\;\\alpha_{m-1}=\\mathbf{1}_{1\/N}\\right\\} ,\\nonumber \\\\\nT_{n} & := & \\max\\left(\\sigma_{n}\\right),\\label{eq:T_n_defn}\n\\end{eqnarray}\nwith $T_{n}:=0$ one the event $\\left\\{ \\sigma_{n}=\\emptyset\\right\\} $.\nIntuitively, $T_{n}$ can be thought of as the last resampling time\nbefore $n$. Then by construction, using the recursive definition\nof $W_{n}^{i}$ in (\\ref{eq:W_n_defn}), and (\\ref{eq:T_n_defn}),\nwe have on the event $\\left\\{ \\sigma_{n}\\neq\\emptyset\\right\\} $,\n\\begin{eqnarray}\n & W_{T_{n}}^{i} & =\\sum_{j}\\alpha_{T_{n}-1}^{ij}W_{T_{n}-1}^{j}g_{T_{n}-1}(\\zeta_{T_{n}-1}^{j})\\nonumber \\\\\n &  & =\\frac{1}{N}\\sum_{j}W_{T_{n}-1}^{j}g_{T_{n}-1}(\\zeta_{T_{n}-1}^{j})\\;=:\\;\\widetilde{W}_{n},\\quad n\\geq1,\\label{eq:W_adapt}\n\\end{eqnarray}\nwhich is independent of $i$. On the event $\\{\\sigma_{n}=\\emptyset\\}$,\ndefine $\\widetilde{W}_{n}:=1$. \n\nOn the event $\\{\\sigma_{n}\\neq\\emptyset\\}\\cap\\{T_{n}=n\\}$, we trivially\nhave $W_{n}^{i}=W_{T_{n}}^{i}=\\widetilde{W}_{n}$, by (\\ref{eq:W_adapt}).\nOn the event $\\{\\sigma_{n}\\neq\\emptyset\\}\\cap\\{T_{n}<n\\}$, applying\nequation (\\ref{eq:unwind}) of Lemma~\\ref{lem:W_n_representation}\nwith $p=T_{n}$, and (\\ref{eq:W_adapt}), yields\n\\begin{eqnarray*}\nW_{n}^{i_{n}} & = & \\sum_{\\left(i_{T_{n}},\\ldots,i_{n-1}\\right)\\in[N]^{n-T_{n}}}W_{T_{n}}^{i_{T_{n}}}\\prod_{q=T_{n}}^{n-1}g_{q}(\\zeta_{q}^{i_{q}})\\alpha_{q}^{i_{q+1}i_{q}}\\\\\n & = & \\widetilde{W}_{n}\\sum_{\\left(i_{T_{n}},\\ldots,i_{n-1}\\right)\\in[N]^{n-T_{n}}}\\prod_{q=T_{n}}^{n-1}g_{q}(\\zeta_{q}^{i_{q}})\\mathbb{I}[i_{q+1}=i_{q}]\\\\\n & = & \\widetilde{W}_{n}\\prod_{p=T_{n}}^{n-1}g_{p}(\\zeta_{p}^{i_{n}}).\n\\end{eqnarray*}\nCollecting the above definitions and substituting into (\\ref{eq:pi^N_andZ^N})\ngives\n\\[\n\\pi_{n}^{N}=\\frac{\\sum_{i}\\delta_{\\zeta_{n}^{i}}\\prod_{p=T_{n}}^{n-1}g_{p}(\\zeta_{p}^{i})}{\\sum_{i}\\prod_{p=T_{n}}^{n-1}g_{p}(\\zeta_{p}^{i})},\\quad\\quad Z_{n}^{N}=\\widetilde{W}_{n}\\cdot\\frac{1}{N}\\sum_{i}\\prod_{p=T_{n}}^{n-1}g_{p}(\\zeta_{p}^{i}),\n\\]\nwith the convention $\\prod_{p=n}^{n-1}g_{p}(\\zeta_{p}^{i})=1$. Similar\nelementary calculations can be used to derive expressions for the\nsampling steps of the ARPF, in the interests of brevity we leave it\nto the reader to write out the details.\n\n\n\\subsection*{Proofs and auxiliary results for Section~\\ref{sec:Martingale-approximations-and}}\n\\begin{proof}\n\\emph{(of Theorem~\\ref{thm:convergence})}. \n\nFor part 1), note\n\\[\n\\overline{\\Gamma}_{n}^{N}(1)-\\pi_{n}(1)=\\frac{Z_{n}^{N}}{Z_{n}}-1,\n\\]\nthen applying Proposition~\\ref{prop:martingale} with $\\varphi=1$\nand using (\\ref{eq:Gamma_telescope})--(\\ref{eq:xi_cond_exp}) gives\n\\[\n\\mathbb{E}[Z_{n}^{N}]=Z_{n}.\n\\]\n\n\nMoving to the proof of part 2), let us assume for now, only $\\mathbf{(A1)},\\mathbf{(A2)}$\\textbf{\n}and $\\mathbf{\\mathbf{(B)}}$, but not necessarily $\\mathbf{(B^{+})}$.\nDefine $c_{n}:=\\sup_{x}g_{n}(x)\/\\pi_{n}(g_{n})$. Under $\\mathbf{\\mathbf{(A1)}}$,\nwe have\n\\[\n\\mathrm{osc}\\left(\\overline{Q}_{p,n}(\\varphi)\\right)\\leq2\\left\\Vert \\varphi\\right\\Vert \\sup_{x}\\overline{Q}_{p,n}(1)(x)\\leq2\\left\\Vert \\varphi\\right\\Vert \\prod_{q=p}^{n-1}c_{q}<+\\infty\n\\]\nand also using Lemma~\\ref{lem:W_n_representation}, (\\ref{eq:W_bar_defn})\nand the fact that each $\\alpha_{p}$ is a Markov transition matrix,\nwe obtain \n\\[\n0<\\overline{W}_{p}^{i_{p}}\\leq\\sum_{\\left(i_{0},\\ldots,i_{p-1}\\right)\\in[N]^{p}}\\prod_{q=0}^{p-1}c_{q}\\alpha_{q}^{i_{q+1}i_{q}}=\\prod_{q=0}^{p-1}c_{q}<+\\infty.\n\\]\nFrom (\\ref{eq:martingale_burkholder_bound}) we then obtain\n\\begin{eqnarray}\n &  & \\mathbb{E}\\left[\\left|\\overline{\\Gamma}_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi)\\right|^{r}\\right]^{1\/r}\\nonumber \\\\\n &  & \\leq2\\left\\Vert \\varphi\\right\\Vert B(r)^{1\/r}\\left(\\prod_{p=0}^{n-1}c_{p}\\right)\\sum_{p=0}^{n}\\left|\\sum_{i}\\left(\\beta_{p,n}^{i}\\right)^{2}\\right|^{1\/2}\\nonumber \\\\\n &  & \\leq2\\left\\Vert \\varphi\\right\\Vert B(r)^{1\/r}\\left(\\prod_{p=0}^{n-1}c_{p}\\right)\\sum_{p=0}^{n}\\left|\\max_{i\\in[N]}\\beta_{p,n}^{i}\\right|^{1\/2},\\label{eq:L_r_bound_proof_max}\n\\end{eqnarray}\nwhere the final inequality holds because $\\left\\{ \\beta_{p,n}^{i}\\right\\} _{i\\in[N]}$\nis a probability vector. Then invoking $\\mathbf{(B^{+})}$, the convergence\nin (\\ref{eq:convergence_L_r_statement_gam_weak}) follows from (\\ref{eq:L_r_bound_proof_max})\napplied with $\\varphi=1$. For (\\ref{eq:convergence_L_r_statement_pi_weak}),\nwe apply Minkowski's inequality, the fact $\\left|\\Gamma_{n}^{N}(\\varphi)\/\\Gamma_{n}^{N}(1)\\right|\\leq\\left\\Vert \\varphi\\right\\Vert $\nand (\\ref{eq:L_r_bound_proof_max}) twice to obtain \n\\begin{eqnarray}\n\\mathbb{E}\\left[\\left|\\pi_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi)\\right|^{r}\\right]^{1\/r} & \\leq & \\mathbb{E}\\left[\\left|\\overline{\\Gamma}_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi)\\right|^{r}\\right]^{1\/r}\\nonumber \\\\\n & + & \\mathbb{E}\\left[\\left|\\frac{\\Gamma_{n}^{N}(\\varphi)}{\\Gamma_{n}^{N}(1)}\\right|^{r}\\left|\\overline{\\Gamma}_{n}^{N}(1)-1\\right|^{r}\\right]^{1\/r}\\nonumber \\\\\n & \\leq & 4\\left\\Vert \\varphi\\right\\Vert \\left[B(r)\\right]^{1\/r}\\left(\\prod_{p=0}^{n-1}c_{p}\\right)\\sum_{p=0}^{n}\\left|\\max_{i\\in[N]}\\beta_{p,n}^{i}\\right|^{1\/2},\\label{eq:L_r_bound_proof_pi}\n\\end{eqnarray}\nThe convergence in probability then follows from Markov's inequality,\ncompleting the proof of part 2).\n\nFor part 3), under $\\mathbf{(B^{++})}$ we have $\\beta_{p,n}^{i}=1\/N$,\nand therefore $\\left|\\max_{i\\in[N]}\\beta_{p,n}^{i}\\right|^{1\/2}=N^{-1\/2}$.\nSubstituting this into (\\ref{eq:L_r_bound_proof_max}) with $\\varphi=1$,\nand into (\\ref{eq:L_r_bound_proof_pi}), gives (\\ref{eq:convergence_L_r_statement_gamm})-(\\ref{eq:convergence_L_r_statement_pi}).\nThe almost sure convergence follows from the Borel--Cantelli lemma.\n\\end{proof}\n\n\\begin{proof}\n\\emph{(of Proposition~\\ref{prop:martingale})}. Applying the identities\n$\\beta_{p-1,n}^{i_{p-1}}=\\sum_{i_{p}}\\beta_{p,n}^{i_{p}}\\alpha_{p-1}^{i_{p}i_{p-1}}$,\nsee (\\ref{eq:beta_defn}), and $\\overline{W}_{p}^{i_{p}}=\\sum_{i_{p-1}}\\alpha_{p-1}^{i_{p}i_{p-1}}\\overline{W}_{p-1}^{i_{p-1}}\\overline{Q}_{p}(1)(\\zeta_{p-1}^{i_{p-1}})$,\nsee (\\ref{eq:W_n_defn}), (\\ref{eq:Q_bar_defn}), (\\ref{eq:W_bar_defn});\nwith the conventions $\\alpha_{-1}:=Id$, and $\\overline{\\Gamma}_{-1}^{N}\\overline{Q}_{-1,n}(\\varphi)=\\overline{W}_{-1}^{i}\\overline{Q}_{-1,n}(\\varphi)(\\zeta_{-1}^{i}):=\\mu_{0}\\overline{Q}_{0,n}(\\varphi)=\\pi_{n}(\\varphi)$,\nwe have\n\\begin{eqnarray}\n &  & \\overline{\\Gamma}_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi)\\nonumber \\\\\n &  & =\\sum_{p=0}^{n}\\left[\\overline{\\Gamma}_{p,n}^{N}\\overline{Q}_{p,n}(\\varphi)-\\overline{\\Gamma}_{p-1,n}^{N}\\overline{Q}_{p-1,n}(\\varphi)\\right]\\nonumber \\\\\n &  & =\\sum_{p=0}^{n}\\left[\\sum_{i_{p}}\\beta_{p,n}^{i_{p}}\\overline{W}_{p}^{i_{p}}\\overline{Q}_{p,n}(\\varphi)(\\zeta_{p}^{i_{p}})-\\sum_{i_{p-1}}\\sum_{i_{p}}\\beta_{p,n}^{i_{p}}\\alpha_{p-1}^{i_{p}i_{p-1}}\\overline{W}_{p-1}^{i_{p-1}}\\overline{Q}_{p-1,n}(\\varphi)(\\zeta_{p-1}^{i_{p-1}})\\right]\\nonumber \\\\\n &  & =\\sum_{p=0}^{n}\\sum_{i_{p}}\\beta_{p,n}^{i_{p}}\\overline{W}_{p}^{i_{p}}\\left[\\overline{Q}_{p,n}(\\varphi)(\\zeta_{p}^{i_{p}})-\\frac{\\sum_{i_{p-1}}\\alpha_{p-1}^{i_{p}i_{p-1}}\\overline{W}_{p-1}^{i_{p-1}}\\overline{Q}_{p-1,n}(\\varphi)(\\zeta_{p-1}^{i_{p-1}})}{\\overline{W}_{p}^{i_{p}}}\\right]\\nonumber \\\\\n &  & =\\sum_{p=0}^{n}\\sum_{i_{p}}\\beta_{p,n}^{i_{p}}\\overline{W}_{p}^{i_{p}}\\Delta_{p,n}^{i_{p}}=N^{-1\/2}\\sum_{k=1}^{(n+1)N}\\xi_{k}^{N}.\\label{eq:Gamma-gamma_decomp}\n\\end{eqnarray}\nEach $\\xi_{k}^{N}$ is measurable w.r.t.~$\\mathcal{F}^{(k)}$ because,\nusing Corollary~\\ref{cor:measurability}, \\textbf{(A2)} and $\\mathbf{\\mathbf{(B)}}$\nwe have that for any $k=1,\\ldots,(n+1)N$, if we set $p:=\\left\\lfloor (k-1)\/N\\right\\rfloor $\nand $i:=k-pN$, the quantity $\\Delta_{p,n}^{i}$ is measurable w.r.t.~$\\mathcal{F}^{(k)}$\nand $\\beta_{p,n}^{i_{p}}\\overline{W}_{p}^{i_{p}}$ is measurable w.r.t.~$\\mathcal{F}_{p-1}$.\n\nTo verify (\\ref{eq:xi_cond_exp}), again use the fact that for any\n$i\\in[N]$ and $0\\leq p\\leq n$, $\\beta_{p,n}^{i}\\overline{W}_{p}^{i}$\nis measurable w.r.t.~$\\mathcal{F}_{p-1}$, and note that given $\\mathcal{F}_{p-1}$,\nthe particles $\\left\\{ \\zeta_{p}^{i}\\right\\} _{i=1}^{N}$ are conditionally\nindependent, and distributed as specified in Algorithm~\\ref{alg:aSMC}.\nHence for any $k=1,\\ldots,(n+1)N$ and $p:=\\left\\lfloor (k-1)\/N\\right\\rfloor $\nand $i:=k-pN$, we have $\\mathbb{E}\\left[\\left.\\xi_{k}\\right|\\mathcal{F}^{(k-1)}\\right]=\\sqrt{N}\\beta_{p,n}^{i}\\overline{W}_{p}^{i}\\mathbb{E}\\left[\\left.\\Delta_{p,n}^{i}\\right|\\mathcal{F}_{p-1}\\right]=0$. \n\nFor the inequality (\\ref{eq:martingale_burkholder_bound}), by Minkowski's\ninequality and (\\ref{eq:Gamma-gamma_decomp}),\n\n\\begin{equation}\n\\mathbb{E}\\left[\\left|\\overline{\\Gamma}_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi)\\right|^{r}\\right]^{1\/r}\\leq\\sum_{p=0}^{n}\\mathbb{E}\\left[\\left|\\overline{\\Gamma}_{p,n}^{N}\\overline{Q}_{p,n}(\\varphi)-\\overline{\\Gamma}_{p-1}^{N}\\overline{Q}_{p-1,n}(\\varphi)\\right|^{r}\\right]^{1\/r}.\\label{eq:Gamma-gamma_mink}\n\\end{equation}\nFor each term in (\\ref{eq:Gamma-gamma_mink}), using the above stated\nconditional independence and measurability properties, we may apply\n\\citep[Lemma 7.3.3]{smc:theory:Dm04} to establish the existence of\nan independent constant $B(r)$, depending only on $r$ and such that\n\\begin{eqnarray}\n &  & \\mathbb{E}\\left[\\left.\\left|\\overline{\\Gamma}_{p,n}^{N}\\overline{Q}_{p,n}(\\varphi)-\\overline{\\Gamma}_{p-1}^{N}\\overline{Q}_{p-1,n}(\\varphi)\\right|^{r}\\right|\\mathcal{F}_{p-1}\\right]\\nonumber \\\\\n &  & =\\mathbb{E}\\left[\\left.\\left|\\sum_{i}\\beta_{p,n}^{i}\\overline{W}_{p}^{i}\\Delta_{p}^{i}\\right|^{r}\\right|\\mathcal{F}_{p-1}\\right]\\nonumber \\\\\n &  & \\leq B(r)\\text{osc}\\left(\\overline{Q}_{p,n}(\\varphi)\\right)^{r}\\left(\\sum_{i}\\left(\\beta_{p,n}^{i}\\overline{W}_{p}^{i}\\right)^{2}\\right)^{r\/2},\\label{eq:GammaN-GammaN}\n\\end{eqnarray}\nalmost surely. The proof is completed upon combining this estimate\nwith (\\ref{eq:Gamma-gamma_mink}).\n\\end{proof}\n\n\\subsection*{Proofs for Section~\\ref{sec:stability}}\n\n\n\\begin{proof}\n\\emph{(of Proposition~\\ref{prop:norm_const_bound})} The proof follows\na similar line of argument to \\citep[Proof of Theorem 16.4.1.]{DelMoral2013book},\nbut applies to a more general algorithm than considered there. To\nstart, we apply Proposition~\\ref{prop:martingale}, equation (\\ref{eq:Gamma_telescope})\nwith $\\varphi=1$ and (\\ref{eq:xi_cond_exp}), we obtain\n\\[\n\\mathbb{E}\\left[\\left(\\frac{Z_{n}^{N}}{Z_{n}}-1\\right)^{2}\\right]=\\sum_{p=0}^{n}\\sum_{i_{p}}\\mathbb{E}\\left[\\left(\\beta_{p,n}^{i_{p}}\\overline{W}_{p}^{i_{p}}\\Delta_{p,n}^{i_{p}}\\right)^{2}\\right].\n\\]\n Under $\\mathbf{(B^{++})}$ we have $\\beta_{p,n}^{i_{p}}=1\/N$, then\nusing the other hypotheses of the Proposition and noting $\\text{osc}\\left(Q_{n,n}(1)\\right)=\\text{osc}\\left(1\\right)=0$,\nwe have for $n\\geq1$,\n\\begin{eqnarray*}\n\\mathbb{E}\\left[\\left(\\frac{Z_{n}^{N}}{Z_{n}}-1\\right)^{2}\\right] & = & \\sum_{p=0}^{n}\\sum_{i}\\mathbb{E}\\left[\\frac{1}{N^{2}}\\left(\\overline{W}_{p}^{i}\\right)^{2}\\left(\\Delta_{p,n}^{i}\\right)^{2}\\right]\\\\\n & \\leq & \\sum_{p=0}^{n-1}\\text{osc}\\left(\\overline{Q}_{p,n}(1)\\right)^{2}\\mathbb{E}\\left[\\frac{1}{N^{2}}\\sum_{i}\\left(\\overline{W}_{p}^{i}\\right)^{2}\\right]\\\\\n & = & \\sum_{p=0}^{n-1}\\text{osc}\\left(\\overline{Q}_{p,n}(1)\\right)^{2}\\frac{1}{N}\\mathbb{E}\\left[\\frac{1}{\\mathcal{E}_{p}^{N}}\\left(\\frac{1}{N}\\sum_{i}\\overline{W}_{p}^{i}\\right)^{2}\\right]\\\\\n & \\leq & \\sum_{p=0}^{n-1}\\frac{\\text{osc}\\left(\\overline{Q}_{p,n}(1)\\right)^{2}}{N\\tau_{p}}\\mathbb{E}\\left[\\left(\\frac{1}{N}\\sum_{i}\\overline{W}_{p}^{i}-1\\right)^{2}+1\\right]\\\\\n & = & \\sum_{p=0}^{n-1}\\frac{\\text{osc}\\left(\\overline{Q}_{p,n}(1)\\right)^{2}}{N\\tau_{p}}\\left(\\mathbb{E}\\left[\\left(\\frac{Z_{p}^{N}}{Z_{p}}-1\\right)^{2}\\right]+1\\right),\n\\end{eqnarray*}\nwhere last two lines use $N^{-1}\\sum_{i}\\overline{W}_{p}=Z_{p}^{N}\/Z_{p}$\nand by Theorem~\\ref{thm:convergence}, $\\mathbb{E}\\left[Z_{p}^{N}\\right]=Z_{p}$.\n\\end{proof}\n\n\\begin{proof}\n\\emph{(of Proposition~\\ref{prop:L_p_bound_mixing})} First note that\nby exactly the same arguments as in the proof of Proposition~\\ref{prop:martingale},\nequation (\\ref{eq:GammaN-GammaN}), we have for any $\\phi\\in\\mathcal{L}$,\n$0\\leq p\\leq n$,\n\n\\begin{eqnarray}\n &  & \\mathbb{E}\\left[\\left.\\left|\\Gamma_{p,n}^{N}Q_{p,n}(\\phi)-\\Gamma_{p-1}^{N}Q_{p-1,n}(\\phi)\\right|^{r}\\right|\\mathcal{F}_{p-1}\\right]\\nonumber \\\\\n &  & \\leq B(r)\\text{osc}\\left(Q_{p,n}(\\phi)\\right)^{r}\\left(\\sum_{i}\\left(\\beta_{p,n}^{i}W_{p}^{i}\\right)^{2}\\right)^{r\/2},\\label{eq:GammaN-GammaN_mixing_proof}\n\\end{eqnarray}\nwith the convention $\\Gamma_{-1}^{N}Q_{-1,n}(\\phi)=\\gamma_{n}(\\phi)$. \n\nFor the remainder of the proof, fix $\\varphi\\in\\mathcal{L}$ arbitrarily,\nand set $\\bar{\\varphi}:=\\varphi-\\pi_{n}(\\varphi)$. Defining \n\\[\nD_{p,n}^{N}:=\\frac{\\Gamma_{p,n}^{N}Q_{p,n}\\left(\\bar{\\varphi}\\right)}{\\Gamma_{p,n}^{N}Q_{p,n}(1)},\\quad0\\leq p\\leq n,\n\\]\nand then noting\n\\[\nD_{n,n}^{N}=\\frac{\\Gamma_{n}^{N}\\left(\\bar{\\varphi}\\right)}{\\Gamma_{n}^{N}(1)}=\\pi_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi),\n\\]\nwe shall focus on the decomposition:\n\\begin{eqnarray}\n\\pi_{n}^{N}(\\varphi)-\\pi_{n}(\\varphi) & = & D_{0,n}^{N}+\\sum_{p=1}^{n}D_{p,n}^{N}-D_{p-1,n}^{N},\\label{eq:pi_n^N_decomp}\n\\end{eqnarray}\nwith the convention that the summation is zero when $n=0$.\n\nFor $1\\leq p\\leq n$, write\n\\[\nD_{p,n}^{N}-D_{p-1,n}^{N}=T_{p,n}^{\\left(N,1\\right)}+T_{p,n}^{\\left(N,2\\right)},\n\\]\n where\n\\begin{eqnarray*}\nT_{p,n}^{\\left(N,1\\right)} & := & \\frac{1}{\\Gamma_{p,n}^{N}Q_{p,n}(1)}\\left[\\Gamma_{p,n}^{N}Q_{p,n}\\left(\\bar{\\varphi}\\right)-\\Gamma_{p-1,n}^{N}Q_{p-1,n}\\left(\\bar{\\varphi}\\right)\\right]\\\\\nT_{p,n}^{\\left(N,2\\right)} & := & \\frac{\\Gamma_{p-1,n}^{N}Q_{p-1,n}(\\bar{\\varphi})}{\\Gamma_{p-1,n}^{N}Q_{p-1,n}(1)}\\frac{\\left[\\Gamma_{p-1,n}^{N}Q_{p-1,n}\\left(1\\right)-\\Gamma_{p,n}^{N}Q_{p,n}\\left(1\\right)\\right]}{\\Gamma_{p,n}^{N}Q_{p,n}\\left(1\\right)}.\n\\end{eqnarray*}\n\n\nWe have the estimates\n\\begin{equation}\n\\frac{\\text{osc}\\left(Q_{p,n}(\\phi)\\right)}{\\inf_{x}Q_{p,n}\\left(1\\right)(x)}\\leq2\\delta_{p,n}\\left\\Vert P_{p,n}(\\phi)\\right\\Vert ,\\label{eq:osc\/inf_bound}\n\\end{equation}\n(which is finite under assumption $(\\mathbf{C})$ - see also (\\ref{eq:Q_p,n_bounded})),\nand\n\\begin{eqnarray}\n\\left|\\frac{\\Gamma_{p-1,n}^{N}Q_{p-1,n}(\\phi)}{\\Gamma_{p-1,n}^{N}Q_{p-1,n}(1)}\\right| & \\leq & \\left\\Vert P_{p-1,n}(\\phi)\\right\\Vert .\\label{eq:Gamma\/Gamma_bound}\n\\end{eqnarray}\nApplying (\\ref{eq:GammaN-GammaN_mixing_proof}) with $\\phi=\\bar{\\varphi}$,\nusing (\\ref{eq:osc\/inf_bound}) and noting that $\\Gamma_{p,n}^{N}(1)$\nis measurable w.r.t.~$\\mathcal{F}_{p-1}$, we obtain\n\\begin{eqnarray*}\n\\mathbb{E}\\left[\\left.\\left|T_{p,n}^{\\left(N,1\\right)}\\right|^{r}\\right|\\mathcal{F}_{p-1}\\right]^{1\/r} & \\leq & B(r)^{1\/r}2\\delta_{p,n}\\frac{\\left\\Vert P_{p,n}(\\bar{\\varphi})\\right\\Vert }{\\Gamma_{p,n}^{N}(1)}\\left(\\sum_{i}\\left(\\beta_{p,n}^{i}W_{p}^{i}\\right)^{2}\\right)^{1\/2}.\n\\end{eqnarray*}\nApplying (\\ref{eq:GammaN-GammaN_mixing_proof}) with $\\phi=1$, using\n(\\ref{eq:Gamma\/Gamma_bound}) and the same measurability condition,\nwe obtain \n\\begin{eqnarray*}\n\\mathbb{E}\\left[\\left.\\left|T_{p,n}^{\\left(N,2\\right)}\\right|^{r}\\right|\\mathcal{F}_{p-1}\\right]^{1\/r} & \\leq & B(r)^{1\/r}2\\delta_{p,n}\\frac{\\left\\Vert P_{p-1,n}(\\bar{\\varphi})\\right\\Vert }{\\Gamma_{p,n}^{N}(1)}\\left(\\sum_{i}\\left(\\beta_{p,n}^{i}W_{p}^{i}\\right)^{2}\\right)^{1\/2}.\n\\end{eqnarray*}\nTherefore, via Minkowski's inequality and using\n\\[\n\\left\\Vert P_{p-1,n}(\\bar{\\varphi})\\right\\Vert =\\left\\Vert Q_{p-1,n}(\\bar{\\varphi})\/Q_{p-1,n}(1)\\right\\Vert =\\left\\Vert Q_{p}Q_{p,n}(\\bar{\\varphi})\/Q_{p-1,n}(1)\\right\\Vert \\leq\\left\\Vert P_{p,n}(\\bar{\\varphi})\\right\\Vert ,\n\\]\nwe have\n\\begin{equation}\n\\mathbb{E}\\left[\\left|D_{p,n}^{N}-D_{p-1,n}^{N}\\right|^{r}\\right]^{1\/r}\\leq B(r)^{1\/r}4\\delta_{p,n}\\left\\Vert P_{p,n}(\\bar{\\varphi})\\right\\Vert \\mathbb{E}\\left[\\left|\\mathcal{C}_{p,n}^{N}\\right|^{r}\\right]^{1\/r}.\\label{eq:D_p-D_p_proof}\n\\end{equation}\n\n\nFor the remaining term, $D_{0,n}^{N}$, we have \n\\begin{eqnarray*}\nD_{0,n}^{N} & = & \\frac{1}{\\Gamma_{0,n}^{N}Q_{0,n}(1)}\\left(\\Gamma_{0,n}^{N}Q_{0,n}\\left(\\bar{\\varphi}\\right)-\\gamma_{n}(\\bar{\\varphi})\\right),\n\\end{eqnarray*}\nwhere the final equality holds since $\\gamma_{n}(\\bar{\\varphi})=\\gamma_{n}(\\varphi)-\\gamma_{n}(1)\\pi_{n}(\\varphi)=0$.\nUsing (\\ref{eq:GammaN-GammaN_mixing_proof}) and (\\ref{eq:osc\/inf_bound})\nin a similar fashion to above we obtain \n\\begin{equation}\n\\mathbb{E}\\left[\\left|D_{0,n}^{N}\\right|^{r}\\right]^{1\/r}\\leq B(r)^{1\/r}2\\delta_{0,n}\\left\\Vert P_{0,n}(\\bar{\\varphi})\\right\\Vert \\mathbb{E}\\left[\\left|\\mathcal{C}_{0,n}^{N}\\right|^{r}\\right]^{1\/r}.\\label{eq:D_0_proof}\n\\end{equation}\nThe proof is complete upon using Minkowski's inequality to bound the\nmoments of (\\ref{eq:pi_n^N_decomp}) using (\\ref{eq:D_p-D_p_proof})\nand (\\ref{eq:D_0_proof}). \n\\end{proof}\n\n\\subsection*{Proofs for Section~\\ref{sec:Discussion}}\n\\begin{proof}\n\\emph{(of Lemma~\\ref{lem:complexity})} Label the vertices of the\ngraph corresponding to $A$ arbitrarily with the integers $[N]$.\nLet $s\\geq1$ be the number of connected components of this graph.\nThen for each $\\ell\\in[s]$ let $B(\\ell)$ be the set of labels of\nthe $\\ell$'th connected component. Since $A$ is a B-matrix, each\nconnected component is complete, so we have for any $\\ell\\in[s]$\nand $i\\in B(\\ell)$,\n\\begin{equation}\nW_{n}^{i}=\\sum_{j}\\alpha_{n-1}^{ij}W_{n-1}^{j}g_{n-1}(\\zeta_{n-1}^{j})=\\left|B(\\ell)\\right|^{-1}\\sum_{j\\in B(\\ell)}W_{n-1}^{j}g_{n-1}(\\zeta_{n-1}^{j}).\\label{eq:W_n_complexity}\n\\end{equation}\nThe complexity of calculating $W_{n}^{i}$ is thus $O(\\left|B(\\ell)\\right|)$,\nand since $W_{n}^{i}=W_{n}^{j}$ for all $i,j\\in B(\\ell)$, the complexity\nof calculating $\\left\\{ W_{n}^{i}\\right\\} _{i\\in[N]}$ is $O\\left(\\sum_{\\ell\\in[s]}\\left|B(\\ell)\\right|\\right)=O(N)$.\nArguing similarly to (\\ref{eq:W_n_complexity}), with $\\alpha_{n-1}=A$\nwe find that under Algorithm~\\ref{alg:aSMC}, for each $\\ell\\in[s]$,\nthe $\\left\\{ \\zeta_{n}^{i}\\right\\} _{i\\in B(\\ell)}$ are conditionally\ni.i.d.~according to \n\\begin{equation}\n\\frac{\\sum_{j\\in B(\\ell)}W_{n-1}^{j}g_{n-1}(\\zeta_{n-1}^{j})f(\\zeta_{n-1}^{j},\\cdot)}{\\sum_{j\\in B(\\ell)}W_{n-1}^{j}g_{n-1}(\\zeta_{n-1}^{j})}.\\label{eq:complexity_block_distbn}\n\\end{equation}\nBy the same arguments used in \\citep{carpenter1999improved} to address\nthe BPF, drawing $\\left|B(\\ell)\\right|$ samples from (\\ref{eq:complexity_block_distbn})\ncan be achieved at $O(\\left|B(\\ell)\\right|)$ complexity, and thus\nthe overall complexity of the sampling part of Algorithm~\\ref{alg:aSMC}\nis $O(\\sum_{\\ell\\in[s]}\\left|B(\\ell)\\right|)=O(N)$.\n\\end{proof}\n\n\\begin{proof}\n\\emph{(of Proposition~\\ref{prop:Upon-termination-of})} We prove\n(\\ref{eq:W_k_explicit}) by induction. We have \n\\[\n\\mathbb{W}_{0}^{i}=W_{n-1}^{i}g_{n-1}(\\zeta_{n-1}^{i})=\\sum_{j\\in B(0,i)}W_{n-1}^{j}g_{n-1}(\\zeta_{n-1}^{j})\n\\]\n and, when (\\ref{eq:W_k_explicit}) holds at rank $k$, we have at\nrank $k+1$, \n\\begin{eqnarray*}\n\\mathbb{W}_{k+1}^{i} & = & \\mathbb{W}_{k}^{\\mathcal{I}_{k}(2i-1)}\/2+\\mathbb{W}_{k}^{\\mathcal{I}_{k}(2i)}\/2\\\\\n & = & 2^{-(k+1)}\\sum_{j\\in B(k,\\mathcal{I}_{k}(2i-1))}W_{n-1}^{j}g_{n-1}(\\zeta_{n-1}^{j})\\\\\n & + & 2^{-(k+1)}\\sum_{j\\in B(k,\\mathcal{I}_{k}(2i))}W_{n-1}^{j}g_{n-1}(\\zeta_{n-1}^{j})\\\\\n & = & 2^{-(k+1)}\\sum_{j\\in B(k,\\mathcal{I}_{k}(2i-1))\\cup B(k,\\mathcal{I}_{k}(2i))}W_{n-1}^{j}g_{n-1}(\\zeta_{n-1}^{j})\\\\\n & = & 2^{-(k+1)}\\sum_{j\\in B(k+1,i)}W_{n-1}^{j}g_{n-1}(\\zeta_{n-1}^{j}).\n\\end{eqnarray*}\n Finally, for any $i\\in[N\/2^{k}]$ and $j\\in B(k,i)$ \n\\[\nW_{n}^{j}=\\sum_{\\ell}\\alpha_{n-1}^{i\\ell}W_{n-1}^{\\ell}g_{n-1}(\\zeta_{n-1}^{\\ell})=2^{-k}\\sum_{\\ell\\in B(k,i)}W_{n-1}^{\\ell}g_{n-1}(\\zeta_{n-1}^{\\ell})=\\mathbb{W}_{k}^{i},\n\\]\nwhich establishes (\\ref{eq:W_k_explicit})--(\\ref{eq:W_equals_bb_W}). \n\nNo matter what adaptation rule of Table~\\ref{tab:Choosing_I_k} is\nused, each quantity $\\left\\{ B(k,i)\\right\\} _{i\\in[N\/2^{k}]}$ obtained\nby Algorithm~\\ref{alg:generic adaptation} is, by construction, a\npartition of $[N]$ and thus the $\\alpha_{n-1}$ output by Algorithm\n\\ref{alg:generic adaptation} is a B-matrix. Noting that a B-matrix\nalways admits the uniform distribution on $[N]$ as an invariant distribution,\nwe have for any B-matrix, say $A$, the identity $\\overline{\\mathbb{W}}_{0}=N^{-1}\\sum_{i}\\sum_{j}A^{ij}W_{n-1}^{i}g_{n-1}(\\zeta_{n-1}^{i})$\nand so upon termination of the ``while'' loop in Algorithm~\\ref{alg:generic adaptation},\n$\\mathcal{E}=\\mathcal{E}_{n}^{N}$ and hence $\\mathcal{E}_{n}^{N}\\geq\\tau$\nalways. \n\nFor the Simple and Random adaptation rules, the worst case complexity\nof Algorithm~\\ref{alg:generic adaptation} is as follows. The part\nof the algorithm preceding the ``while'' loop is $O(N)$. The complexity\nof iteration $k$ of the ``while'' loop is $O(N\/2^{k})$, the worst\ncase is when the loop terminates with $k=m$, in which case the complexity\nof the ``while'' loop is $O(\\sum_{k=0}^{m}N\/2^{k})$, thus the overall\ncomplexity is no more than $O(N)$. \n\nFor the Greedy procedure, the sort operation required to obtain $\\mathcal{I}_{k}$\nis of complexity $O\\left(N\/2^{k}\\log_{2}\\left(N\/2^{k}\\right)\\right)$,\nand so in the worst case, the complexity of the ``while'' loop is\nof the order\n\\[\nt(N):=\\sum_{k=0}^{m}\\frac{N}{2^{k}}\\log_{2}\\left(\\frac{N}{2^{k}}\\right),\n\\]\nor expressed recursively, $t(N)=t(N\/2)+N\\log_{2}N$, and $t(2)=2$.\nA simple induction shows that this recursion has solution $t(N)=2[1+N(\\log_{2}N-1)]$,\nhence the overall worst case complexity of the ``while'' loop is\n$O(N\\log_{2}N)$. The proof is complete since by Lemma~\\ref{lem:complexity},\nthe complexity of operations in Algorithm~\\ref{alg:aSMC} other than\nline $(\\star)$ is $O(N)$. \n\\end{proof}\n\\bibliographystyle{imsart-nameyear}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Statement of results}\n\n\\noindent\nLet $X$ be a smooth, projective variety over a characteristic $0$\nfield $k$, and denote by $\\Kbm{0,r}{X,\\beta}$ the Kontsevich moduli\nspace of genus $0$, $r$-pointed stable maps to $X$ of class $\\beta$.\nBehrend and Fantechi defined a \\emph{perfect obstruction theory} for\n$\\Kbm{0,r}{X,\\beta}$, i.e., a complex $E^\\bullet$ perfect of amplitude\n$[-1,0]$ together with a map to the cotangent complex $\\phi:E^\\bullet\n\\rightarrow L^\\bullet_{\\Kbm{0,r}{X,\\beta}}$ such that $h^0(\\phi)$ is\nan isomorphism and $h^{-1}(\\phi)$ is surjective.  In many cases $\\phi$\nis a quasi-isomorphism, and then the dualizing sheaf on\n$\\Kbm{0,r}{X,\\beta}$ is the determinant $\\text{det}(E^\\bullet)$.  For\nthis reason $\\text{det}(E^\\bullet)$ is called the \\emph{virtual\ncanonical bundle}.  This paper gives a formula,\nProposition~\\ref{prop-vc}, for the virtual canonical bundle in terms\nof tautological divisor classes on $X$, modulo torsion.\n\n\\medskip\\noindent\nGrothendieck-Riemann-Roch applies in a straightforward manner, but the\nresulting formula is not optimal: it is difficult to determine whether\nthe virtual canonical bundle is NEF, ample, etc.  The main work in\nSection~\\ref{sec-div} proves divisor class relations yielding a\nsuccinct formula for the virtual canonical bundle.  The proof reduces\nto local computations for the universal family over the Artin stack of\nall prestable curves of genus $0$, cf. Section~\\ref{sec-local}.\nBecause of this, most results are stated for Artin stacks.  This leads\nto one \\emph{ad hoc} consruction: since there is as yet no theory of\ncycle class groups for Artin stacks admitting Chern classes for all\nperfect complexes of bounded amplitude, a Riemann-Roch theorem for all\nperfect morphisms relatively representable by proper algebraic spaces,\nand arbitrary pullbacks for all cycles coming from Chern classes, a\nstand-in $Q_\\pi$ is used, cf. Section~\\ref{sec-Qpi} (also by avoiding\nRiemann-Roch, this allows some relations to be proved ``integrally''\nrather than ``modulo torsion'').  Also, although the relative Picard\nfunctor of the universal family of genus $0$ curves is well-known, a\ndescription is included in Sections ~\\ref{sec-dec} and ~\\ref{sec-cpct}\nfor completeness.\n\n\\medskip\\noindent  \nIn the special case $X={\\mathbb P}^n_k$, Pandharipande proved most of\nthese divisor class relations, ~\\cite{QDiv}, and the formula for the\nvirtual canonical bundle, ~\\cite{Pand97}, modulo numerical\nequivalence.  This was certainly our inspiration, but our proofs are\ncompletely different, yield a more general virtual canonical bundle\nformula, and hold modulo torsion (and sometimes ``integrally'') rather\nthan modulo numerical equivalence.\n\n\\section{Decorated prestable curves} \\label{sec-dec}\n\\marpar{sec-dec}\n \n\\noindent\nThere exists an Artin stack parametrizing prestable curves whose dual\ngraph is a given modular graph.  This section describes a variant\nArtin stack obtained by ``decorating'' the modular graph.  Although\nthe variation is simple, it arises often enough to warrant a few\nwords.  This variant is used in the next section to describe the\nclosure of the identity section in the relative Picard functor for the\nuniversal family of prestable curves of compact type.\nThe reference for this section is ~\\cite{BM}.\n\n\\medskip\\noindent\nA \\emph{modular graph} is a (not necessarily connected) graph $\\sigma$\n-- edges are undirected and \\emph{tails} or half-edges are allowed --\ntogether with a \\emph{genus function} $g:\\text{Vertex}(\\sigma)\n\\rightarrow \\mathbb{Z}_{\\geq 0}$.  There are 2 collections of morphisms\nbetween modular graphs: \\emph{contractions} are surjective on\nvertices, roughly contracting subgraphs of the domain to vertices of\nthe target, and \\emph{graph inclusions} are inclusions of\nsubgraphs. (In ~\\cite{BM}, the \\emph{combinatorial morphisms} are\nobtained from graph inclusions by adjoining formal inverses of certain\n``stabilizing'' contractions; stability is not an issue here, so graph\ninclusions are more appropriate).  Also, for every diagram,\n$$\n\\begin{CD}\n& & \\sigma_3 \\\\ \n& & @VV a V \\\\\n\\sigma_2 @> \\phi >> \\sigma_1\n\\end{CD}\n$$\nof a contraction $\\phi$ and a graph inclusion $a$, there is a\n\\emph{pullback diagram},\n$$\n\\begin{CD}\n\\sigma_4 @> a^*\\phi >> \\sigma_3 \\\\\n@V \\phi^* a VV @VV a V \\\\\n\\sigma_2 @> \\phi >> \\sigma_1\n\\end{CD}\n$$\nof a contraction $a^*\\phi$ and a graph inclusion $\\phi^*a$ such that\nthe maps on vertices, $a\\circ a^*\\phi, \\phi \\circ\n\\phi^*a:\\text{Vertex}(\\sigma_4) \\rightarrow \\text{Vertex}(\\sigma_1)$\nare equal.  The diagram is unique up to a unique isomorphism (both as\na contraction and a graph inclusion) of $\\sigma_4$.  The category of\nmodular graphs is denoted $\\mathfrak{G}$.\n\n\\medskip\\noindent\nTo each prestable curve there is an associated modular graph, and to\neach modular graph $\\sigma$ there is an Artin stack $\\mathfrak{M}(\\sigma)$\nparametrizing prestable curves along with a contraction of the\nassociated modular graph to $\\sigma$.  This defines a lax 2-functor\nfrom $\\mathfrak{G}$ to the 2-category of Artin stacks, covariant for\ncontractions, contravariant for graph inclusions, and such that for\nevery pullback diagram there is a 2-equivalence $\\mathfrak{M}(a)\\circ\n\\mathfrak{M}(\\phi) \\Rightarrow \\mathfrak{M}(a^*\\phi) \\circ \\mathfrak{M}(\\phi^*a)$.\n\n\\begin{defn} \\label{defn-decg}\n\\marpar{defn-decg}\nA \\emph{category of decorated modular graphs} is a category $\\mathfrak{H}$\nwith 2 sets of morphisms -- $\\mathfrak{H}$-contractions and $\\mathfrak{H}$-graph\ninclusions -- together with a functor $p:\\mathfrak{H} \\rightarrow \\mathfrak{G}$\ncompatible with both contractions and graph inclusions satisfying the\nfollowing axioms,\n\\begin{enumerate}\n\\item[(i)]\nfor every $\\mathfrak{H}$-contraction $\\phi:\\tau_2 \\rightarrow \\tau_1$ and\n$\\mathfrak{H}$-graph inclusion $a:\\tau_3 \\rightarrow \\tau_1$, there exists\nan object $\\tau_4$, an $\\mathfrak{H}$-contraction $a^*\\phi:\\tau_4\n\\rightarrow \\tau_3$ and an $\\mathfrak{H}$-graph inclusion $\\phi^*a:\\tau_4\n\\rightarrow \\tau_2$ mapping under $p$ to a pullback diagram, moreover\nthis is unique up to unique isomorphism of $\\tau_4$, and\n\\item[(ii)]\nfor every object $\\tau$ in $\\mathfrak{H}$ and every contraction $\\phi:p(\\tau)\n\\rightarrow \\sigma$ in $\\mathfrak{G}$, there is a $\\mathfrak{H}$-contraction $\\psi$\nsuch that $p(\\psi)=\\phi$, and $\\psi$ is unique up to unique isomormphism.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{constr} \\label{constr-semigp}\n\\marpar{constr-semigp}\nLet $A$ be an Abelian semigroup.  Define $\\mathfrak{G}_A$ to be the category\nwhose objects are pairs $(\\sigma,\\alpha)$ of a modular graph $\\sigma$\ntogether with a function $\\alpha:\\text{Vertex}(\\sigma) \\rightarrow A$,\nwhere $\\mathfrak{G}_A$-contractions, $\\phi:(\\sigma_1,\\alpha_1) \\rightarrow\n(\\sigma_2,\\alpha_2)$, are contractions $\\phi:\\sigma_1 \\rightarrow\n\\sigma_2$ such that $\\alpha_2(v) = \\sum_{w\\in \\phi^{-1}(v)}\n\\alpha_1(w)$ for every $v\\in \\text{Vertex}(\\sigma_2)$, and where\n$\\mathfrak{G}_A$-graph inclusions, $a:(\\sigma_1,\\alpha_1) \\rightarrow\n(\\sigma_2,\\alpha_2)$, are graph inclusions $a:\\sigma_1 \\rightarrow\n\\sigma_2$ such that $\\alpha_2(a(v)) = \\alpha_1(v)$ for every vertex\n$v\\in \\text{Vertex}(\\sigma_1)$.  Define $p:\\mathfrak{G}_A \\rightarrow\n\\mathfrak{G}$ to be the obvious forgetful functor.  This is a category of\ndecorated modular graphs; the only one used in the rest of this paper.\n\\end{constr}\n\n\\medskip\\noindent\nThe aim of this section is to construct for every object $\\tau$ of\n$\\mathfrak{H}$ an Artin stack $\\mathfrak{M}_{\\mathfrak{H}}(\\tau)$ parametrizing\nprestable curves along with a lifting of the associated modular graph\nto an object of $\\mathfrak{H}$ contracting to $\\tau$.  The association $\\tau\n\\mapsto \\mathfrak{M}_{\\mathfrak{H}}(\\tau)$ should define a lax 2-functor from\n$\\mathfrak{H}$ to the 2-category of Artin stacks, covariant for\n$\\mathfrak{H}$-contractions, contravariant for $\\mathfrak{H}$-graph inclusions,\nand such that for every pullback diagram there is an associated\n2-equivalence.\n\n\\begin{defn} \\label{defn-sat}\n\\marpar{defn-sat}\nLet $\\mathfrak{H}$ be a category of decorated modular graphs, considered as\na usual category whose morphisms are $\\mathfrak{H}$-contractions.  A\nsubcategory $\\mathfrak{H}'$ is \\emph{saturated} if $\\mathfrak{H}'$ contains every\n$\\mathfrak{H}$-contraction whose domain is in $\\mathfrak{H}'$.  A subcategory\n$\\mathfrak{H}'$ is \\emph{$p$-embedding} if the functor of categories with\ncontractions as morphisms, $p:\\mathfrak{H}' \\rightarrow \\mathfrak{G}$, is an\nequivalence to a (necessarily full) subcategory of $\\mathfrak{G}$.\n\n\\noindent\nLet $\\tau$ be an object of $\\mathfrak{H}$ and denote by $\\mathfrak{H}_\\tau$ the\ncategory whose objects are contractions $\\phi:\\tau_\\phi \\rightarrow\n\\tau$ and whose morphisms are commutative diagrams of contractions.  A\nsubcategory $\\mathfrak{H}'$ of $\\mathfrak{H}_\\tau$ is \\emph{saturated} if\n$\\mathfrak{H}'$ contains every morphism in $\\mathfrak{H}_\\tau$ whose domain is in\n$\\mathfrak{H}'$.  A subcategory $\\mathfrak{H}'$ of $\\mathfrak{H}_\\tau$ is\n\\emph{$p$-embedding} if the functor $p:\\mathfrak{H}' \\rightarrow\n\\mathfrak{G}_{p(\\tau)}$ is an equivalence to a (necessarily full)\nsubcategory of $\\mathfrak{G}_{p(\\tau)}$.  Denote by\n$\\text{Sat}(\\mathfrak{H}_\\tau)$ the set of saturated, $p$-embedding\nsubcategories of $\\mathfrak{H}_\\tau$ directed by reverse inclusion of\nsubcategories.\n\\end{defn}\n\n\\medskip\\noindent\nLet $\\tau$ be an object of $\\mathfrak{H}$ and let $\\mathfrak{H}'$ be a saturated,\n$p$-embedding subcategory of $\\mathfrak{H}_\\tau$.  Define\n$U_{\\mathfrak{H}'}(\\tau)$ to be the open substack of $\\mathfrak{M}(p(\\tau))$ whose\ncomplement is the union of the images of all 1-morphisms\n$\\mathfrak{M}(\\phi):\\mathfrak{M}(\\sigma) \\rightarrow \\mathfrak{M}(p(\\tau))$ such that\n$\\phi$ is not in the image of $p:\\mathfrak{H}' \\rightarrow\n\\mathfrak{G}_{p(\\tau)}$.  It is straightforward that $u_{\\mathfrak{H}'}(\\tau)$ is\nopen: the intersection with every quasi-compact open substack of\n$\\mathfrak{M}(p(\\tau))$ is open, and $U_{\\mathfrak{H}'}(\\tau)$ is the union of\nthese open sets.\n\n\\medskip\\noindent\nLet $\\mathfrak{H}' \\subset \\mathfrak{H}''$ be saturated, $p$-embedding\nsubcategories of $\\mathfrak{H}_\\tau$.  Then $U_{\\mathfrak{H}'}(\\tau) \\subset\nU_{\\mathfrak{H}''}(\\tau)$ as subsets of $\\mathfrak{M}(p(\\tau))$.  Therefore\n$\\mathfrak{H}' \\mapsto U_{\\mathfrak{H}'}(\\tau)$ is a directed system of open\nimmersions of Artin stacks indexed by $\\text{Sat}(\\mathfrak{H}_\\tau)$.\nBecause this is a directed system of open immersions, the direct limit\nis an Artin stack.\n\n\\begin{notat} \\label{notat-MHtau}\n\\marpar{notat-MHtau}\nDenote by $\\mathfrak{M}_{\\mathfrak{H}}(\\tau)$ the direct limit of the directed\nsystem $\\mathfrak{H}' \\mapsto U_{\\mathfrak{H}'}(\\tau)$.  Denote by\n$\\mathfrak{M}_p(\\tau): \\mathfrak{M}_{\\mathfrak{H}}(\\tau) \\rightarrow \\mathfrak{M}(p(\\tau))$\nthe natural 1-morphism.  If $\\mathfrak{H}=\\mathfrak{G}_A$, also denote\n$\\mathfrak{M}_{\\mathfrak{H}}(\\tau)$ by $\\mathfrak{M}_A(\\tau)$.\n\\end{notat}\n\n\\medskip\\noindent\nThe ``points'' of $\\mathfrak{M}_{\\mathfrak{H}}(\\tau)$ have a simple description.\n\n\\begin{defn} \\label{defn-strict}\n\\marpar{defn-strict}\nFor every modular graph $\\sigma$, define\n$\\mathfrak{M}^{\\text{strict}}(\\sigma)$ to be the open substack of\n$\\mathfrak{M}(\\sigma)$ that is the complement of the images of all\n$\\mathfrak{M}(\\phi)$ where $\\phi:\\sigma' \\rightarrow \\sigma$ is a\nnon-invertible contraction.\n\\end{defn}\n\n\\begin{lem} \\label{lem-pts}\n\\marpar{lem-pts}\nLet $\\tau$ be an object of $\\mathfrak{H}$ and let $\\phi:\\sigma \\rightarrow\np(\\tau)$ be a contraction.  The 2-fibered product\n$\\mathfrak{M}^{\\text{strict}}(\\sigma) \\times_{\\mathfrak{M}(p(\\tau))}\n\\mathfrak{M}_{\\mathfrak{H}}(\\tau)$ is equivalent to a disjoint union of copies of\n$\\mathfrak{M}^{\\text{strict}}(\\sigma)$ indexed by equivalence classes of\ncontractions $\\psi$ in $\\mathfrak{H}_\\tau$ such that $p(\\psi)=\\phi$.\n\\end{lem}\n\n\\begin{proof}\nLet $(\\eta,\\zeta,\\theta)$ be an object of the 2-fibered product, i.e.,\na triple of an object of $\\mathfrak{M}^{\\text{strict}}(\\sigma)$, an object\nof $\\mathfrak{M}_{\\mathfrak{H}}(\\tau)$ and an equivalence\n$\\theta:\\mathfrak{M}(\\phi)(\\eta) \\rightarrow \\mathfrak{M}_{p}(\\tau)(\\zeta)$.\nThere is a saturated, $p$-embedding subcategory $\\mathfrak{H}'$ such that\n$\\mathfrak{M}_p(\\tau)(\\zeta)$ is in $U_{\\mathfrak{H}'}(\\tau)$.  Because this is in\nthe image of $\\mathfrak{M}(\\phi)$, there is a contraction $\\psi:\\tau_\\psi\n\\rightarrow \\tau$ in $\\mathfrak{H}'$ such that $\\phi=p(\\psi)$.  Because\n$\\mathfrak{H}'$ is $p$-embedding, $\\psi$ is unique up to unique isomorphism.\nBy the nature of the direct limit, $\\psi$ is independent of the choice\nof $\\mathfrak{H}'$.\n\n\\medskip\\noindent\nConversely, given an object $\\eta$ of $\\mathfrak{M}^{\\text{strict}}(\\sigma)$\nand a contraction $\\psi:\\tau_\\psi \\rightarrow \\tau$ in $\\mathfrak{H}_\\tau$\nsuch that $p(\\psi) = \\phi$, define $\\mathfrak{H}'$ to be the subcategory of\n$\\mathfrak{H}_\\tau$ consisting of all contractions through which $\\psi$\nfactors.  By Definition~\\ref{defn-decg}(ii), this is a saturated,\n$p$-embedding subcategory.  And $\\mathfrak{M}(\\phi)(\\eta)$ is in\n$U_{\\mathfrak{H}}(\\tau)$.  The image in the direct limit is an object\n$\\zeta$, and there is a canonical isomorphism $\\theta:\n\\mathfrak{M}(\\phi)(\\eta) \\rightarrow \\mathfrak{M}_p(\\tau)(\\zeta)$.  Thus\n$(\\eta,\\zeta,\\theta)$ is an object of the 2-fibered product.\n\n\\medskip\\noindent\nIt is left to the reader to verify these operations give an\nequivalence of stacks.\n\\end{proof}\n\n\\medskip\\noindent\n\\textbf{Note:}  \nThe functorialities are only sketched.  Given an $\\mathfrak{H}$-contraction\n$\\phi:\\tau_1 \\rightarrow \\tau_2$, Definition~\\ref{defn-decg}(ii) gives\na map of directed sets $\\text{Sat}(\\phi):\\text{Sat}(\\mathfrak{H}_{\\tau_1})\n\\rightarrow \\text{Sat}(\\mathfrak{H}_{\\tau_1})$, and composition with\n$\\mathfrak{M}(p(\\phi))$ gives a compatible family of 1-morphisms of directed\nsystems.  This defines the 1-morphism $\\mathfrak{M}_{\\mathfrak{H}}(\\phi)$.  Given\nan $\\mathfrak{H}$-graph inclusion $a:\\tau_1 \\rightarrow \\tau_2$, existence\nof pullback diagrams, Definition~\\ref{defn-decg}(i), gives a map of\ndirected sets $\\text{Sat}(a):\\text{Sat}(\\mathfrak{H}_{\\tau_2}) \\rightarrow\n\\text{Sat}(\\mathfrak{H}_{\\tau_1})$, and composition with $\\mathfrak{M}(p(a))$\ngives a compatible family of 1-morphisms of directed systems.  This\ndefines the 1-morphism $\\mathfrak{M}_{\\mathfrak{H}}(a)$.  The rest of the\ncompatibilities are straightforward.\n\n\\section{The universal relative Picard for curves of compact type}\n\\label{sec-cpct}\n\\marpar{sec-cpct}\n\n\\noindent\nThe results in this section are well-known, and easily follow from\n~\\cite{Raynaud} and ~\\cite{Ner}.  It is useful in the rest of the\npaper to gather the results here.\n\n\\begin{notat} \\label{notat-H}\n\\marpar{notat-H}\nDenote by $\\mathfrak{H} \\subset \\mathfrak{G}_\\mathbb{Z}$ the full subcategory of objects\n$(\\sigma,\\alpha)$ such that $\\sigma$ is a forest of trees, i.e., the\ngraph has no cycles.  For each triple of integers $g,n \\geq 0$ and\n$e$, denote by $\\tau_{g,n}(e)$ the object of $\\mathfrak{H}$ consisting of a\ntree $\\sigma_{g,n}$ with a single vertex of genus $g$ and $n$ flags,\nsuch that $\\alpha(v)=n$.\n\\end{notat}\n\n\\medskip\\noindent\nDenote by $\\pi:\\mathcal{C} \\rightarrow \\mathfrak{M}_{\\mathfrak{H}}(\\tau_{g,0}(0))$ the\npullback from $\\mathfrak{M}(\\sigma_{g,0})$ of the universal curve.  For each\n4-tuple $A=((g',g''),(e',e''))$ of integers $g',g''\\geq 0$,\n$g'+g''=g$, and integers $e',e''$, $e'+e''=0$, denote by $\\tau_A$ the\ntree with vertices $v',v''$ such that $g(v')=g', \\alpha(v')=e'$ and\n$g(v'')=g'', \\alpha(v'')=e''$.  Denote by $\\phi:\\tau_A \\rightarrow\n\\tau_{g,0}(0)$ the canonical contraction.  The 2-fibered product\n$\\mathfrak{M}_{\\mathfrak{H}}(\\tau_A) \\times_{\\mathfrak{M}_{\\mathfrak{H}}(\\tau_{g,0}(0))}\n\\mathcal{C}$ has 2 irreducible components $\\mathcal{C}', \\mathcal{C}''$ corresponding\nto the vertices $v', v''$.  There is a unique effective Cartier\ndivisor $\\mathcal{D} \\subset \\mathcal{C}$ such that for every\n$A=((g',g''),(e',e''))$,\n$$\n\\mathfrak{M}^{\\text{strict}}_{\\mathfrak{H}}(\\tau_A)\n\\times_{\\mathfrak{M}_{\\mathfrak{H}}(\\tau_{g,0}(0))} \\mathcal{D}\n$$ \nis empty if $e'=e''=0$ and is $e'\\mathcal{C}''$ if\n$e'>0$.  \n\n\\medskip\\noindent\nLet $U\\subset \\mathfrak{M}(\\sigma_{g,0})$ denote the open substack that is\nthe image of $\\mathfrak{M}_p(\\tau_{g,0}(0))$, i.e., $U$ is the Artin stack\nof $n$-pointed, genus $g$ curves of \\emph{compact type}.  The\n1-morphism $\\pi:\\mathcal{C}_U \\rightarrow U$ is cohomologically flat, so by\n~\\cite[Thm. 7.3]{Artin} the relative Picard functor of the universal\ncurve over $U$ is a 1-morphism $\\text{pr}:\\text{Pic}_{\\mathcal{C}_U\/U}\n\\rightarrow U$ relatively representable by \\emph{non-separated}\nalgebraic spaces.  The closure $E_{\\mathcal{C}_U\/U}$ of the identity\nsection gives a closed substack of $\\text{Pic}_{\\mathcal{C}_U\/U}$ which is\nrelatively representable over $U$ by non-separated group algebraic\nspaces, ~\\cite[Prop. 5.2]{Raynaud}.  The quotient $Q_{\\mathcal{C}_U\/U}$ of\n$\\text{Pic}_{\\mathcal{C}_U\/U}$ by $E_{\\mathcal{C}_U\/U}$ is a stack that is\nrelatively representable over $U$ by a countable disjoint union of\nsmooth, proper group algebraic spaces, ~\\cite[Thm. 4.1.1]{Raynaud}\n(properness requires a bit more, see ~\\cite[Ex. 8, p. 246]{Ner}).  The\nnext lemma describes $E_{\\mathcal{C}_U\/U}$.\n\n\\medskip\\noindent\nThe invertible sheaf $\\mathcal O_{\\mathcal{C}}(\\mathcal{D})$ defines a\n1-morphism $f:\\mathfrak{M}_{\\mathfrak{H}}(\\tau_{g,0}(0)) \\rightarrow\n\\text{Pic}_{\\mathcal{C}_U\/U}$, and there is a natural 2-equivalence of\n$\\text{pr}\\circ f$ with $\\mathfrak{M}_{\\mathfrak{H}}(\\tau_{g,0}(0))$.\n\n\\begin{lem} \\label{lem-subgp} \n\\marpar{lem-subgp}\nThe 1-morphism $f$ defines an equivalence to $E_{\\mathcal{C}_U\/U}$, the\nclosure of the \nidentity section of $\\text{Pic}_{\\mathcal{C}_U\/U}$.  \nDenoting by\n$Q^0_{\\mathcal{C}_U\/U}$ the identity component of the quotient and by\n$\\text{Pic}^0_{\\mathcal{C}_U\/U}$ the preimage, there are 1-morphisms\n$$\n\\text{Pic}^0_{\\mathcal{C}_U\/U} \\rightleftarrows\n\\mathfrak{M}_{\\mathfrak{H}}(\\tau_{g,0}) \\times\nQ^0_{\\mathcal{C}_U\/U}\n$$ \ngiving an equivalence of stacks\nover $U$, and splitting the extension of group algebraic spaces over\n$U$.\n\\end{lem}  \n\n\\begin{proof}\nIt is easy to see $f$ is an equivalence to its image which is a\nsubgroup of $E_{\\mathcal{C}_U\/U}$.  To prove the image of $f$ is all of\n$E_{\\mathcal{C}_U\/U}$, by the valuative criterion of closedness it suffices\nto check equality of pullbacks for every map of a DVR to\n$\\mathfrak{M}_{g,0}$ sending the generic point to\n$\\mathfrak{M}_{g,0}^{\\text{strict}}$. By ~\\cite[Prop. 6.1.3]{Raynaud}, the\nsections of $E_{\\mathcal{C}_U\/U}$ over are a DVR are just the quotient of\nthe free Abelian group on the irreducible components of the closed\nfiber by the subgroup generated by the entire fiber.  By\nLemma~\\ref{lem-pts} the same is true for the pullback of\n$\\mathfrak{M}_{\\mathfrak{H}}(\\tau_{g,0}(0))$, and it is clear the map between them\nis an isomorphism.\n\n\\medskip\\noindent\nThe splitting of $\\text{Pic}^0_{\\mathcal{C}_U\/U} \\rightarrow\nQ^0_{\\mathcal{C}_U\/U}$ is given by the subfunctor of\n$\\text{Pic}^0_{\\mathcal{C}_U\/U}$ of invertible sheaves whose degree on\nevery irreducible component of every fiber is $0$, denoted by $P^0$ in\n\\cite{Raynaud}.\n\\end{proof}\n\n\\medskip\\noindent\nIn the special case that $g=0$, more is true.  First of all,\n$\\mathfrak{M}_{\\mathbb{Z}}(\\tau) = \\mathfrak{M}_{\\mathfrak{H}}(\\tau)$ for every $\\tau$ of genus\n$0$.  Secondly,  $U=\\mathfrak{M}_{0,n}$.  \n\n\\begin{cor}[Raynaud, Prop. 9.3.1, ~\\cite{Raynaud}] \\label{cor-subgp}\n\\marpar{cor-subgp}\nFor $g=0$, $\\text{Pic}^0_{\\mathcal{C}\/\\mathfrak{M}_{0,0}}$ is equivalent\nto $\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(0))$.\n\\end{cor}\n\n\\medskip\\noindent\nMoreover, the union $\\cup_{e\\in \\mathbb{Z}} \\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(e))$ is a\ngroup algebraic space over $\\mathfrak{M}_{0,0}$ containing\n$\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(0))$ as a subgroup algebraic space over\n$\\mathfrak{M}_{0,0}$.  Essentially, given a contraction\n$\\phi:\\sigma\\rightarrow \\sigma_{0,0}$ and given liftings\n$\\psi_i:(\\sigma,\\alpha_i) \\rightarrow \\tau_{0,0}(e_i)$ for $i=1,2$,\naddition is determined by $\\psi_1+\\psi_2=\\psi:(\\sigma,\n\\alpha_1+\\alpha_2) \\rightarrow \\tau_{0,0}(e_1+e_2)$.  The \\emph{total\ndegree map} gives an isomorphism of $\\text{Pic}\/\\text{Pic}^0$ with\n$\\mathbb{Z}$.  The following result is easy.\n\n\\begin{lem} \\label{lem-g0}\n\\marpar{lem-g0}\nFor $g=0$, for every $e$ there is an equivalence of stacks over\n$\\mathfrak{M}_{0,0}$, $\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(e)) \\rightleftarrows\n\\text{Pic}^e_{\\mathcal{C}\/\\mathfrak{M}_{0,0}}$, such that the equivalence\n$\\cup_{e\\in \\mathbb{Z}} \\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(e)) \\rightleftarrows\n\\text{Pic}_{\\mathcal{C}\/\\mathfrak{M}_{0,0}}$ is an equivalence of group algebraic\nspaces over $\\mathfrak{M}_{0,0}$ and is compatible with the equivalence in\nCorollary~\\ref{cor-subgp}.\n\\end{lem}\n\n\\subsection{Notation for boundary divisor classes}\n\\label{subsec-bound}\n\\marpar{subsec-bound}\n\n\\noindent\nLet $r\\geq 0$ be an integer, and let $e_1,\\dots,e_r$ be integers.\nDenote by $\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(e_1,\\dots,e_r))$ the 2-fibered\nproduct,\n$$\n\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(e_1)) \\times_{\\mathfrak{M}_{0,0}}\n\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(e_2)) \\times_{\\mathfrak{M}_{0,0}} \\dots\n\\times_{\\mathfrak{M}_{0,0}} \\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(e_r)).\n$$\nFor each $i=1,\\dots,r$, let $(e'_i,e''_i)$ be a pair of integers such\nthat $e'_i + e''_i = e_i$.  Let $\\sigma$ be the modular graph with two\nvertices $v',v''$ with $g(v')=g(v'')=0$, one edge connecting $v',\nv''$, and no tails.  Let $\\phi:\\sigma \\rightarrow \\sigma_{0,0}$ be the\ncanonical contraction.  Denote by\n$\\zeta:\\mathfrak{M}^{\\text{strict}}(\\sigma) \\rightarrow\n\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{g,0}(e_1,\\dots,e_r))$ the 1-morphism whose\nprojection to the $i^{\\text{th}}$ factor is determined via\nLemma~\\ref{lem-pts} by the lifting $\\psi_i:(\\sigma,\\alpha_i)\n\\rightarrow \\tau_{g,0}(e_i)$ of $\\phi$ such that $\\alpha_i(v') = e'_i,\n\\alpha_i(v'')=e_i''$.  Define $\\Delta_{(e'_1,e''_1,\\dots,e'_r,e''_r)}$\nto be the effective Cartier divisor on\n$\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(e_1,\\dots,e_r))$ that is the closure of the\nimage of $\\zeta$.\n\n\\medskip\\noindent\nLet $\\pi:C\\rightarrow M$ be a flat 1-morphism relatively represented\nby proper algebraic spaces whose geometric fibers are connected,\nat-worst-nodal curves of arithmetic genus $0$.  Let $D_1,\\dots,D_r$ be\nCartier divisor classes on $C$ of relative degrees $e_1,\\dots,e_r$.\nLet $f(e'_1,e''_1,\\dots,e'_r,e''_r)$ be a function on $\\mathbb{Z}^{2r}$ with\nvalues in $\\mathbb{Z}$, resp. $\\mathbb{Q}$, etc.  Denote by $\\xi: M \\rightarrow\n\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(e_1,\\dots,e_r))$ the 1-morphism whose\nprojection to the $i^\\text{th}$ factor,\n$\\text{Pic}^{e_i}_{\\mathfrak{C}\/\\mathfrak{M}_{0,0}}$ is determined by\n$\\mathcal O_C(D_i)$.\n\n\\begin{notat} \\label{notat-boundary}\n\\marpar{notat-boundary}\nDenote by,\n$$\n\\sum_{(\\beta',\\beta'')} f(\\langle D_1,\\beta' \\rangle, \\langle\nD_1, \\beta'' \\rangle, \\dots, \\langle D_r, \\beta' \\rangle, \\langle D_r,\n\\beta'' \\rangle) \\Delta_{\\beta',\\beta''}\n$$\nthe Cartier divisor class, resp. $\\mathbb{Q}$-Cartier divisor class, etc., \nthat is the pullback by $\\xi$ of the Cartier divisor class, etc.,\n$$\n\\sum_{(e'_1,e''_1,\\dots,e'_r,e''_r)} f(e'_1,e''_1,\\dots,e'_r,e''_r)\n\\Delta_{(e'_1,e''_1,\\dots,e'_r,e''_r)},\n$$\nthe summation over all sequences $(e'_1,e''_1,\\dots,e'_r,e''_r)$ with\n$e'_i + e''_i = e_i$.  \nIf $f(e'_1,e''_1,\\dots,e'_r,e''_r) = f(e''_1,e'_1,\\dots,e''_r,e'_r)$,\ndenote by,\n$$\n{\\sum_{(\\beta',\\beta'')}}^\\prime f(\\langle D_1,\\beta' \\rangle, \\langle\nD_1, \\beta'' \\rangle, \\dots, \\langle D_r, \\beta' \\rangle, \\langle D_r,\n\\beta'' \\rangle) \\Delta_{\\beta',\\beta''}\n$$\nthe pullback by $\\xi$ of,\n$$\n{\\sum_{(e'_1,e''_1,\\dots,e'_r,e''_r)}}^\\prime\nf(e'_1,e''_1,\\dots,e'_r,e''_r)\n\\Delta_{(e'_1,e''_1,\\dots,e'_r,e''_r)},\n$$\nwhere the summation is over equivalence classes of sequences\n$(e'_1,e''_1,\\dots,e'_r,e''_r)$ such that $e'_i+e''_i=e_i$ \nunder the equivalence relation\n$(e'_1,e''_1,\\dots,e'_r,e''_r) \\sim (e''_1,e'_1,\\dots,e''_r,e'_r)$.  \n\\end{notat}\n\n\\begin{ex} \\label{ex-boundary}\n\\marpar{ex-boundary}\nLet $n\\geq 0$ be an integer and let $(A,B)$ be a partition of\n$\\{1,\\dots,n\\}$.  For the universal family over $\\mathfrak{M}_{0,n}$, denote\nby $s_1,\\dots,s_n$ the universal sections.  Then,\n$$\n\\sum_{\\beta',\\beta''} \\prod_{i\\in A} \\langle s_i,\\beta' \\rangle\n\\cdot \\prod_{j\\in B} \\langle s_j, \\beta'' \\rangle\n\\Delta_{\\beta',\\beta''} \n$$\nis the Cartier divisor class of the boundary divisor $\\Delta_{(A,B)}$.  \n\\end{ex}\n\n\n\\section{The functor $Q_\\pi$} \\label{sec-Qpi}\n\\marpar{sec-Qpi}\n\n\\noindent\nLet $M$ be an Artin stack, and let $\\pi:C \\rightarrow M$ be a flat\n1-morphism, relatively representable by proper algebraic spaces whose\ngeometric fibers are connected, at-worst-nodal curves of arithmetic\ngenus $0$.  There exists an invertible dualizing sheaf $\\omega_\\pi$,\nand the relative trace map, $\\text{Tr}_\\pi: R\\pi_* \\omega_\\pi[1]\n\\rightarrow \\mathcal O_M$ is a quasi-isomorphism.  In particular,\n$\\text{Ext}^1_{\\mathcal O_C}(\\omega_\\pi,\\mathcal O_C)$ is canonically isomorphic to\n$H^0(M,\\mathcal O_M)$.  Therefore $1\\in H^0(M,\\mathcal O_M)$ determines an extension\nclass, i.e., a short exact sequence,\n$$\n\\begin{CD}\n0 @>>> \\omega_\\pi @>>> E_\\pi @>>> \\mathcal O_C @>>> 0.\n\\end{CD}\n$$\nThe morphism $\\pi$ is perfect, so for every complex $F^\\bullet$ perfect\nof bounded amplitude on $C$, $R\\pi_* F^\\bullet$ is a perfect complex of\nbounded amplitude on $M$.  By ~\\cite{detdiv}, the\ndeterminant of a perfect complex of bounded amplitude is defined.\n\n\\begin{defn} \\label{defn-E}\n\\marpar{defn-E}\nFor every complex $F^\\bullet$ perfect of bounded amplitude on $C$, \ndefine $Q_\\pi(F^\\bullet) = \\text{det}(R\\pi_* E_\\pi\\otimes F^\\bullet)$.\n\\end{defn}\n\n\\medskip\\noindent\nThere is another interpretation of $Q_\\pi(F^\\bullet)$.\n\n\\begin{lem} \\label{lem-interp}\n\\marpar{lem-interp}\nFor every complex $F^\\bullet$ perfect of bounded amplitude on $C$, \n$$\nQ_\\pi(F^\\bullet) \\cong\n\\text{det}(R\\pi_*(F^\\bullet)) \\otimes \n\\text{det}(R\\pi_*((F^\\bullet)^\\vee))^\\vee.\n$$\n\\end{lem}\n\n\\begin{proof}\nBy the short exact sequence for $E_\\pi$, $Q_\\pi(F^\\bullet) \\cong\n\\text{det}(R\\pi_*(F^\\bullet)) \\otimes \\text{det}(R\\pi_*(\\omega_\\pi\n\\otimes F^\\bullet))$.  The lemma follows by duality.\n\\end{proof}\n\n\\medskip\\noindent\nIt is straightforward to compute $F^\\bullet$ whenever there exist\ncycle class groups for $C$ and $M$ such that Chern classes are defined\nfor all perfect complexes of bounded amplitude and such that\nGrothendieck-Riemann-Roch holds for $\\pi$.\n\n\\begin{lem} \\label{lem-GRR}\n\\marpar{lem-GRR}\nIf there exist cycle class groups for $C$ and $M$ such that Chern\nclasses exist for all perfect complexes of bounded amplitude and such\nthat Grothendieck-Riemann-Roch holds for $\\pi$, then modulo $2$-power\ntorsion, the first Chern class of $Q_\\pi(F^\\bullet)$ is\n$\\pi_*(C_1(F^\\bullet)^2 - 2C_2(F^\\bullet))$.\n\\end{lem}\n\n\\begin{proof}\nDenote the Todd class of $\\pi$ by $\\tau = 1 + \\tau_1 + \\tau_2 +\n\\dots$.  Of course $\\tau_1 = -C_1(\\omega_\\pi)$.  By GRR,\n$\\text{ch}(R\\pi_* \\mathcal O_C) = \\pi_*(\\tau)$.  The canonical map $\\mathcal O_M\n\\rightarrow R\\pi_*\\mathcal O_C$ is a quasi-isomorphism.  Therefore\n$\\pi_*(\\tau_2)=0$, modulo $2$-power torsion.  By additivity of the\nChern character, $\\text{ch}(E_\\pi) = 2 + C_1(\\omega_\\pi) +\n\\frac{1}{2}C_1(\\omega_\\pi)^2 + \\dots$.  Therefore,\n$$\n\\text{ch}(E_\\pi)\\cdot \\tau = 2 + 2\\tau_2 + \\dots\n$$\nSo for any complex $F^\\bullet$ perfect of bounded amplitude,\n$$\n\\begin{array}{c}\n\\text{ch}(E_\\pi\\otimes F^\\bullet)\\cdot \\tau = \\text{ch}(F^\\bullet) \\cdot\n\\text{ch}(E_\\pi) \\cdot \\tau = \\\\\n(\\text{rk}(F^\\bullet) + C_1(F^\\bullet) +\n\\frac{1}{2}(C_1(F^\\bullet)^2 - 2C_2(F^\\bullet)) +\\dots)(2 + 2\\tau_2+\n\\dots). \n\\end{array}\n$$\nApplying $\\pi_*$ gives,\n$$\n2\\pi_*(C_1(F^\\bullet)) + \\pi_*(C_1(F^\\bullet)^2 - 2C_2(F^\\bullet)) + \\dots\n$$\nTherefore the first Chern class of $\\text{det}(R\\pi_*(E_\\pi\\otimes\nF^\\bullet))$ is $\\pi_*(C_1(F^\\bullet)^2 - 2C_2(F^\\bullet))$, modulo\n$2$-power torsion.\n\\end{proof}\n\n\\begin{rmk} \\label{rmk-Q}\nThe point is this.  In every reasonable case, $Q_\\pi$ is just\n$\\pi_*(C_1^2-2C_2)$.  Moreover $Q_\\pi$ is compatible with base-change\nby arbitrary 1-morphisms.  This allows to reduce certain computations\nto the Artin stack of all genus $0$ curves.  As far as we are aware,\nno one has written a definition of cycle class groups for all locally\nfinitely presented Artin stacks that has Chern classes for all perfect\ncomplexes of bounded amplitude, has pushforward maps and\nGrothendieck-Riemann-Roch for perfect 1-morphisms representable by\nproper algebraic spaces, and has pullback maps by arbitrary\n1-morphisms for cycles coming from Chern classes.  Doubtless such a\ntheory exists; whatever it is, $Q_\\pi = \\pi_*(C_1^2-2C_2)$.\n\\end{rmk}\n\n\\medskip\\noindent\nLet the following diagram be 2-Cartesian,\n$$\n\\begin{CD}\nC' @> \\zeta_C >> C \\\\\n@V \\pi' VV @VV \\pi V \\\\\nM' @> \\zeta_M >> M\n\\end{CD}\n$$\ntogether with a 2-equivalence $\\theta:\\pi\\circ \\zeta_C \\Rightarrow\n\\zeta_M \\circ \\pi'$.  \n\n\\begin{lem} \\label{lem-pullback}\n\\marpar{lem-pullback}\nFor every complex $F^\\bullet$ perfect of bounded amplitude on $C$, \n$\\zeta_M^* Q_\\pi(F^\\bullet)$ is isomorphic to $Q_{\\pi'}(\\zeta_C^*\nF^\\bullet)$.\n\\end{lem}\n\n\\begin{proof}\nOf course $\\zeta_C^* E_\\pi = E_{\\pi'}$.  And $\\zeta_M^* R\\pi_*$ is\ncanonically equivalent to $R(\\pi')_* \\zeta_C^*$ for perfect\ncomplexes of bounded amplitude.  Therefore $\\zeta_M^*\nQ_\\pi(F^\\bullet)$ equals $\\text{det}(\\zeta_M^* R\\pi_*(E_\\pi\\otimes\nF^\\bullet))$ equals $\\text{det}(R(\\pi')_* \\zeta_C^*(E_\\pi \\otimes\nF^\\bullet))$ equals $\\text{det}(R(\\pi')_* E_{\\pi'}\\otimes \\zeta_C^*\nF^\\bullet)$ equals $Q_{\\pi'}(\\zeta_C^* F^\\bullet)$.\n\\end{proof}\n\n\\begin{lem} \\label{lem-inv}\n\\marpar{lem-inv}\nLet $L$ be an invertible sheaf on $C$ of relative degree $e$ over\n$M$.  For every invertible sheaf $L'$ on $M$, $Q_\\pi(L\\otimes \\pi^*L')\n\\cong Q_\\pi(L)\\otimes (L')^{2e}$.  In particular, if $e=0$,\n$Q_\\pi(L\\otimes \\pi^* L') \\cong Q_\\pi(L)$.\n\\end{lem}\n\n\\begin{proof}\nTo compute the rank of $R\\pi_*(E_\\pi\\otimes F^\\bullet)$ over any\nconnected component of $M$, it suffices to base-change to the spectrum\nof a field mapping to that component.  Then, by\nGrothendieck-Riemann-Roch, the rank is $2\\text{deg}(C_1(F^\\bullet))$.\nIn particular, $R\\pi_*(E_\\pi \\otimes L)$ has rank $2e$.  \n\n\\medskip\\noindent\nBy the projection formula, $R\\pi_*(E_\\pi\\otimes L\\otimes \\pi^* L')\n\\cong R\\pi_*(E_\\pi \\otimes L)\\otimes L'$.  Of course\n$\\text{det}(R\\pi_*(E_\\pi \\otimes L)\\otimes L') = Q_\\pi(L)\\otimes\n(L')^\\text{rank}$.  This follows from the uniqueness of $\\text{det}$:\nfor any invertible sheaf $L'$ the association $F^\\bullet \\mapsto\n\\text{det}(F^\\bullet \\otimes L')\\otimes\n(L')^{-\\text{rank}(F^\\bullet)}$ also satisfies the axioms for a\ndeterminant function and is hence canonically isomorphic to\n$\\text{det}(F^\\bullet)$.  Therefore $Q_\\pi(L\\otimes \\pi^*L') =\nQ_\\pi(L)\\otimes (L')^{2e}$.\n\\end{proof}\n\n\\section{Local computations} \\label{sec-local}\n\\marpar{sec-local}\n\n\\noindent\nThis section contains 2 computations: $Q_\\pi(\\omega_\\pi)$ and\n$Q_\\pi(L)$ for every invertible sheaf on $C$ of relative degree $0$.\nBecause of Lemma~\\ref{lem-pullback} the first computation reduces to\nthe universal case over $\\mathfrak{M}_{0,0}$.  Because of\nLemma~\\ref{lem-pullback} and Lemma~\\ref{lem-inv}, the second\ncompuation reduces to $\\mathcal O_{\\mathcal{C}}(\\mathcal{D})$ over\n$\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(0))$.  In each case the computation is\nperformed locally.\n\n\\subsection{Computation of $Q_\\pi(\\omega_\\pi)$} \\label{subsec-Qomega}\n\\marpar{subsec-Qomega}\nAssociated to $\\pi_C:C\\rightarrow M$, there is a 1-morphism $\\zeta_M:M\n\\rightarrow \\mathfrak{M}_{0,0}$, a 1-morphism $\\zeta_C: C \\rightarrow\n\\mathcal{C}$, and a 2-equivalence $\\theta:\\pi_{\\mathcal{C}} \\circ \\zeta_C\n\\Rightarrow \\zeta_M\\circ \\pi_C$ such that the following diagram is\n2-Cartesian,\n$$\n\\begin{CD}\nC @> \\zeta_C >> \\mathcal{C} \\\\\n@V \\pi_C VV @VV \\pi_{\\mathcal{C}} V \\\\\nM @> \\zeta_M >> \\mathfrak{M}_{0,0}\n\\end{CD}\n$$\nOf course $\\omega_{\\pi_C}$ is isomorphic to $\\zeta_C^*\n\\omega_{\\pi_\\mathcal{C}}$.  By Lemma~\\ref{lem-pullback},\n$Q_{\\pi_C}(\\omega_{\\pi_C}) \\cong \\zeta_M^*\nQ_{\\pi_{\\mathcal{C}}}(\\omega_{\\pi_{\\mathcal{C}}})$.  So the computation of\n$Q_{\\pi_C}(\\omega_{\\pi_C})$ is reduced to the universal family.\n\n\\medskip\\noindent\nLet the open substack $U_1\\subset \\mathfrak{M}_{0,0}$ be the complement of\nthe union of the images of $\\mathfrak{M}(\\phi):\\mathfrak{M}(\\sigma) \\rightarrow\n\\mathfrak{M}_{0,0}$ as $\\phi:\\sigma \\rightarrow \\sigma_{0,0}$ ranges over\nall contractions such that $\\#\\text{Vertex}(\\sigma) \\geq 3$.  Let $U_2\n\\subset U_1$ be the open substack\n$\\mathfrak{M}^{\\text{strict}}(\\sigma_{0,0})$.\n\n\\begin{prop} \\label{prop-Qomega}\n\\marpar{prop-Qomega}\n\\begin{enumerate}\n\\item[(i)]\nOver the open substack $U_1$, $\\omega_\\pi^\\vee$ is $\\pi$-relatively\nample.  \n\\item[(ii)]\nOver $U_1$,\n$R^1\\pi_*\\omega_\\pi^\\vee|_{U_1} = (0)$ and $\\pi_*\n\\omega_\\pi^\\vee|_{U_1}$ is locally free of rank 3.  \n\\item[(iii)]\nOver $U_2$, there is\na canonical isomorphism $i:\\text{det}(\\pi_* \\omega_\\pi^\\vee|_{U_2})\n\\rightarrow \\mathcal O_{U_2}$. \n\\item[(iv)]\nThe image of $\\text{det}(\\pi_*\\omega_\\pi^\\vee|_{U_1}) \\rightarrow\n\\text{det}(\\pi_*\\omega_\\pi^\\vee|_{U_2}) \\xrightarrow{i} \\mathcal O_{U_2}$ is\n$\\mathcal O_{U_1}(-\\Delta) \\subset \\mathcal O_{U_2}$.\n\\item[(v)]\nOver $U_1$, $Q_\\pi(\\omega_\\pi)|_{U_1} \\cong \\mathcal O_{U_1}(-\\Delta)$.\nTherefore on all of $\\mathfrak{M}_{0,0}$, $Q_\\pi(\\omega_\\pi) \\cong\n\\mathcal O_{\\mathfrak{M}_{0,0}}(-\\Delta)$.  \n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nOver $\\mathbb{Z}$, let $V = \\mathbb{Z}\\{\\mathbf{e}_0,\\mathbf{e}_1\\}$ be a free module of\nrank $2$.  Choose dual coordinates $y_0,y_1$ for $V^\\vee$.  Let\n${\\mathbb P}^1_\\mathbb{Z} = {\\mathbb P}(V)$ be the projective space with homogeneous\ncoordinates $y_0, y_1$.  Let $\\mathbb{A}^1_\\mathbb{Z}$ be the affine space with\ncoordinate $x$.  Denote by $Z\\subset \\mathbb{A}^1_\\mathbb{Z} \\times {\\mathbb P}^1_\\mathbb{Z}$ the\nclosed subscheme $\\mathbb{V}(x,y_1)$, i.e., the image of the section\n$(0,(1,0))$.  Let $\\nu:C\\rightarrow \\mathbb{A}^1_\\mathbb{Z} \\times {\\mathbb P}^1_\\mathbb{Z}$ be\nthe blowing up of $Z$.  Denote by $E\\subset C$ the exceptional\ndivisor.\n\n\\medskip\\noindent\nDefine $\\pi:C\\rightarrow \\mathbb{A}^1_\\mathbb{Z}$ to be $\\text{pr}_{\\AA^1}\\circ\n\\nu$.  This is a flat, proper morphism whose geometric fibers are\nconnected, at-worst-nodal curves of arithmetic genus $0$.  Moreover,\nno geometric fiber has more than 1 node.  Thus there is a 1-morphism\n$\\zeta:\\mathbb{A}^1_\\mathbb{Z} \\rightarrow U_1$ such that the pullback of $\\mathcal{C}$\nis equivalent to $C$.  It is straightforward that $\\zeta$ is smooth\nand is surjective on geometric points.  Thus (i) and (ii) can be\nchecked after base-change by $\\zeta$.  Also (iv) will reduce to a\ncomputation after base-change by $\\zeta$.\n\n\\medskip\\noindent\n\\textbf{(i) and (ii):} Denote by ${\\mathbb P}^2_\\mathbb{Z}$ the projective space with\ncoordinates $u_0,u_1,u_2$.  There is a rational transformation\n$f:\\mathbb{A}^1_\\mathbb{Z} \\times {\\mathbb P}^1_\\mathbb{Z} \\dashrightarrow \\mathbb{A}^1_\\mathbb{Z} \\times\n{\\mathbb P}^2_\\mathbb{Z}$ by\n$$\n\\begin{array}{ccc}\nf^*x & = & x, \\\\\nf^*u_0 & = & xy_0^2, \\\\\nf^*u_1 & = & y_0y_1, \\\\\nf^*u_2 & = & y_1^2\n\\end{array}\n$$\nBy local computation, this extends to a morphism $f:C\\rightarrow\n\\mathbb{A}^1_\\mathbb{Z} \\times {\\mathbb P}^2_\\mathbb{Z}$ that is a closed immersion and whose\nimage is $\\mathbb{V}(u_0u_2-xu_1^2)$.  By the adjunction formula,\n$\\omega_\\pi$ is the pullback of $\\mathcal O_{{\\mathbb P}^2}(-1)$.  In particular,\n$\\omega_\\pi^\\vee$ is very ample. Moreover, because\n$H^1({\\mathbb P}^2_\\mathbb{Z},\\mathcal O_{{\\mathbb P}^2}(-1)) = (0)$, also $H^1(C,\\omega_\\pi^\\vee) =\n(0)$.  By cohomology and base-change results,\n$R^1\\pi_*(\\omega_\\pi^\\vee) = (0)$ and $\\pi_*(\\omega_\\pi^\\vee)$ is\nlocally free of rank 3.  \n\n\\medskip\\noindent\n\\textbf{(iii):} The curve ${\\mathbb P}^1_\\mathbb{Z} = {\\mathbb P}(V)$ determines a morphism\n$\\eta: \\text{Spec }(\\mathbb{Z}) \\rightarrow U_2$.  This is smooth and surjective on\ngeometric points.  Moreover it gives a realization of $U_2$ as the\nclassifying stack of the group scheme $\\text{Aut}({\\mathbb P}(V)) =\n\\textbf{PGL}(V)$.  Taking the exterior power of the Euler exact\nsequence, $\\omega_{{\\mathbb P}(V)\/\\mathbb{Z}} = \\bigwedge^2(V^\\vee)\\otimes\n\\mathcal O_{{\\mathbb P}(V)}(-2)$.  Therefore $H^0({\\mathbb P}(V),\\omega_{{\\mathbb P}(V)\/\\mathbb{Z}}^\\vee)$\nequals $\\bigwedge^2(V)\\otimes \\text{Sym}^2(V^\\vee)$ as a\nrepresentation of $\\textbf{GL}(V)$.  The determinant of this\nrepresentation is the trivial character of $\\textbf{GL}(V)$.\nTherefore it is the trivial character of $\\textbf{PGL}(V)$.  This\ngives an isomorphism of $\\text{det}(\\pi_*\\omega_\\pi|_{U_2})$ with\n$\\mathcal O_{U_2}$.\n\n\\medskip\\noindent\n\\textbf{(iv):} This can be checked after pulling back by $\\zeta$.  The\npullback of $U_2$ is $\\mathbb{G}_{m,\\mathbb{Z}} \\subset \\mathbb{A}^1_\\mathbb{Z}$.  The\npullback of $i$ comes from the determinant of\n$H^0(\\mathbb{G}_{m,\\mathbb{Z}}\\times {\\mathbb P}^1_\\mathbb{Z},\\omega_\\pi^\\vee) =\n\\bigwedge^2(V) \\otimes \\text{Sym}^2(V^\\vee) \\otimes\n\\mathcal O_{\\mathbb{G}_m}$.  By the adjunction formula, $\\omega_{C\/\\mathbb{A}^1} =\n\\nu^*\\omega_{\\mathbb{A}^1\\times {\\mathbb P}^1\/\\mathbb{A}^1}(E)$.  Hence\n$\\nu_*\\omega_{C\/\\mathbb{A}^1}^\\vee = I_Z \\omega_{\\mathbb{A}^1\\times\n{\\mathbb P}^1\/\\mathbb{A}^1}$.  Therefore the canonical map,\n$$\nH^0(C,\\omega_{C\/\\mathbb{A}^1}^\\vee) \\rightarrow H^0(\\mathbb{A}^1_\\mathbb{Z} \\times\n{\\mathbb P}^1_\\mathbb{Z}, \\omega^\\vee_{\\mathbb{A}^1\\times {\\mathbb P}^1\/\\mathbb{A}^1}),\n$$ \nis given by,\n$$\n\\begin{array}{l}\n\\mathcal O_{\\mathbb{A}^1}\\{\\mathbf{f}_0,\\mathbf{f}_1,\\mathbf{f}_2\\} \\rightarrow\n\\bigwedge^2(V)\\otimes \\text{Sym}^2(V^\\vee) \\otimes \\mathcal O_{\\mathbb{A}^1}, \\\\\n\\mathbf{f}_0 \\mapsto x\\cdot (\\mathbf{e}_0\\wedge\\mathbf{e}_1)\\otimes y_0^2, \\\\\n\\mathbf{f}_1 \\mapsto  \\ \\ \\ \\     (\\mathbf{e}_0\\wedge \\mathbf{e}_1) \\otimes y_0y_1, \\\\\n\\mathbf{f}_2 \\mapsto  \\ \\ \\ \\      (\\mathbf{e}_0\\wedge \\mathbf{e}_1) \\otimes y_1^2\n\\end{array}\n$$\nIt follows that $\\text{det}(\\pi_*\\omega_\\pi^\\vee) \\rightarrow\n\\mathcal O_{\\mathbb{G}_m}$ has image $\\langle x \\rangle \\mathcal O_{\\mathbb{A}^1}$, i.e.,\n$\\eta^* \\mathcal O_{U_1}(-\\Delta)$.\n\n\\medskip\\noindent\n\\textbf{(v):} By the short exact sequence for $E_\\pi$,\n$Q_\\pi(\\omega_\\pi) = \\text{det}(R\\pi_*\\omega_\\pi)\\otimes\n\\text{deg}(R\\pi_* \\omega_\\pi^2)$.  Because the trace map is a\nquasi-isomorphism, $\\text{det}(R\\pi_* \\omega_\\pi) = \\mathcal O_{U_1}$.  By\n(ii) and duality, $$ \\text{det}(R\\pi_* \\omega_\\pi^2) \\cong\n\\text{det}(R^1\\pi_*\\omega_\\pi^2)^vee \\cong \\text{det}(\\pi_*\n\\omega_\\pi^\\vee).\n$$  \nBy (iv), this is $\\mathcal O_{U_1}(-\\Delta)$.  Therefore $Q_\\pi(\\omega_\\pi)\n\\cong \\mathcal O_{U_1}(-\\Delta)$ on $U_1$.  Because $\\mathfrak{M}_{0,0}$ is\nregular, and because the complement of $U_1$ has codimension $2$, this\nisomorphism of invertible sheaves extends to all of $\\mathfrak{M}_{0,0}$.\n\\end{proof}\n\n\\medskip\\noindent\nThe sheaf of relative differentials $\\Omega_\\pi$ is a pure coherent\nsheaf on $\\mathcal{C}$ of rank $1$, flat over $\\mathfrak{M}_{0,0}$ and is\nquasi-isomorphic to a perfect complex of amplitued $[-1,0]$.  \n\n\\begin{lem} \\label{lem-Omega}\n\\marpar{lem-Omega}\nThe perfect complex $R\\pi_*\\Omega_\\pi$ has rank $-1$ and determinant\n$\\cong \\mathcal O_{\\mathfrak{M}_{0,0}}(-\\Delta)$.  The perfect complex $R\\pi_*\nR\\textit{Hom}_{\\mathcal O_{\\mathcal{C}}}(\\Omega_\\pi,\\mathcal O_{\\mathcal{C}})$ has rank $3$\nand determinant $\\cong \\mathcal O_{\\mathfrak{M}_{0,0}}(-2\\Delta)$.\n\\end{lem}\n\n\\begin{proof}\nThere is a canonical injective sheaf homomorphism $\\Omega_\\pi\n\\rightarrow \\omega_\\pi$ and the support of the cokernel, $Z\\subset\n\\mathcal{C}$, is a closed substack that is smooth and such that\n$\\pi:Z\\rightarrow \\mathfrak{M}_{0,0}$ is unramified and is the normalization\nof $\\Delta$.  Over $U_1$, the lemma immediately follows from this and\nthe arguments in the proof of Proposition~\\ref{prop-Qomega}.  As in\nthat case, it suffices to establish the lemma over $U_1$.\n\\end{proof}\n\n\\subsection{Computation of $Q_\\pi(L)$ for invertible sheaves of degree\n  $0$} \\label{subsec-QL}\n\\marpar{subsec-QL}\n\n\\noindent\nLet $M$ be an Artin stack, let $\\pi:C\\rightarrow M$ be a flat\n1-morphism, relatively representable by proper algebraic spaces whose\ngeometric fibers are connected, at-worst-nodal curves of arithmetic\ngenus $0$.  Let $L$ be an invertible sheaf on $C$ of relative degree\n$0$ over $M$.  This determines a morphism to the relative Picard of\nthe universal curve over $\\mathfrak{M}_{0,0}$, i.e., $\\zeta_M:M \\rightarrow\n\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(0))$ such that the pullback of $\\mathcal{C}$ is\nequivalent to $C$, and such that the pullback of\n$\\mathcal O_{\\mathcal{C}}(\\mathcal{D})$ differs from $L$ by $\\pi^*L'$ for an invertible\nsheaf $L'$ on $M$.  By Lemma~\\ref{lem-pullback} and\nLemma~\\ref{lem-inv}, $Q_\\pi(L) \\cong \\zeta_M^*\nQ_\\pi(\\mathcal O_{\\mathcal{C}}(\\mathcal{D}))$.\n\n\\medskip\\noindent\nLet $\\pi:\\mathcal{C} \\rightarrow \\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(0))$ be the\nuniversal curve.\n\n\\begin{prop} \\label{prop-lcomp}\n\\marpar{prop-lcomp}\nOver $\\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(0))$, $\\pi_*E_\\pi(\\mathcal{D}) = (0)$ and \n$R^1\\pi_*\nE_\\pi(\\mathcal{D})$ is a sheaf supported on $\\Delta$.  The stalk of\n$R^1\\pi_* E_\\pi(\\mathcal{D})$ at the generic point of $\\Delta_a$ is a\ntorsion sheaf of length $a^2$.  The filtration by order of vanishing\nat the generic point has associated graded pieces of length $2a-1,\n2a-3, \\dots, 3,1$.  \n\\end{prop}\n\n\\begin{proof}\nOver the open complement of $\\Delta$, the divisor $\\mathcal{D}$ is $0$.  So\nthe first part of the proposition reduces to the statement that\n$R\\pi_* E_\\pi$ \nis quasi-isomorphic to $0$.  By definition of $E_\\pi$, there is an\nexact triangle,\n$$\n\\begin{CD}\nR\\pi_* E_\\pi @>>> R\\pi_* \\mathcal O_{\\mathcal{C}} @> \\delta >> R\\pi_* \\omega_\\pi\n[1] @>>> \nR\\pi_* E_\\pi[1].\n\\end{CD}\n$$\nOf course the canonical isomorphism $R\\pi_* \\mathcal O_{\\mathcal{C}} \\cong\n\\mathcal O_{\\mathfrak{M}}$, and \n$E_\\pi$ were defined so that the composition of $\\delta$ with the trace\nmap, which is a quasi-isomorphism in this case, would be the\nidentity.  Therefore $\\delta$ is a quasi-isomorphism, so $R\\pi_*\nE_\\pi$ is quasi-isomorphic to $0$.  \n\n\\medskip\\noindent\nThe second part can be proved, and to an extent only makes sense,\nafter smooth base-change to a scheme.  Let ${\\mathbb P}^1_s$ be a copy of\n${\\mathbb P}^1$ with homogeneous coordinates $S_0,S_1$.  Let ${\\mathbb P}^1_x$ be a\ncopy of ${\\mathbb P}^1$ with homogeneous coordinates $X_0,X_1$.  Let ${\\mathbb P}^1_y$\nbe a copy of ${\\mathbb P}^1$ with homogeneous coordinates $Y_0,Y_1$.  Denote\nby $C \\subset {\\mathbb P}^1_s \\times {\\mathbb P}^1_x \\times {\\mathbb P}^1_y$ the divisor with\ndefining equation $F=S_0X_0Y_0 - S_1X_1Y_1$.  The projection\n$\\text{pr}_s:C \\rightarrow {\\mathbb P}^1_s$ is a proper, flat morphism whose\ngeometric fibers are connected, at-worst-nodal curves of arithmetic\ngenus $0$.  Denote by $L$ the invertible sheaf on $C$ that is the\nrestriction of $\\text{pr}_x^*\\mathcal O_{{\\mathbb P}^1_x}(a)\\otimes\n\\text{pr}_y^*\\mathcal O_{{\\mathbb P}^1_y}(-a)$.  This is an invertible sheaf of\nrelative degree $0$.  Therefore there is an induced $1$-morphism\n$\\zeta:{\\mathbb P}^1_s \\rightarrow \\mathfrak{M}_{\\mathbb{Z}}(\\tau_{0,0}(0))$.  \n\n\\medskip\\noindent\nIt is straightforward that $\\zeta$ is smooth, and the image intersects\n$\\Delta_b$ iff $b=a$.  Moreover, $\\zeta^* \\Delta_a$ is the reduced\nCartier divisor $\\mathbb{V}(S_0S_1) \\subset {\\mathbb P}^1_s$.  There is an\nobvious involution $i:{\\mathbb P}^1_s\\rightarrow {\\mathbb P}^1_s$ by\n$i(S_0,S_1)=(S_1,S_0)$, and $\\zeta\\circ i$ is $2$-equivalent to\n$\\zeta$.  Therefore the length of the $R^1\\text{pr}_{s,*}\nE_{\\text{pr}_s}\\otimes L$ is $2$ times the length of the stalk of\n$R^1\\pi_* E_{\\pi}(\\mathcal{D})$ at the generic point of $\\Delta_a$; more\nprecisely, the length of the stalk at each of $(1,0),(0,1)\\in {\\mathbb P}^1_s$\nis the length of the stalk at $\\Delta_a$.  Similarly for the lengths\nof the associated graded pieces of the filtration.\n\n\\medskip\\noindent\nBecause $E_{\\text{pr}_s}$ is the extension class of the Trace mapping,\n$R^1\\text{pr}_{s,*}E_{\\text{pr}_s} \\otimes L$ is the cokernel of the\n$\\mathcal O_{{\\mathbb P}^1_s}$-homomorphisms,\n$$\n\\gamma:\\text{pr}_{s,*}(L) \\rightarrow\n\\text{Hom}_{\\mathcal O_{{\\mathbb P}^1_s}}(\\text{pr}_{s,*}(L^\\vee), \\mathcal O_{{\\mathbb P}^1_s}),\n$$\ninduced via adjointness from the multiplication map,\n$$\n\\text{pr}_{s,*}(L) \\otimes \\text{pr}_{s,*}(L^\\vee) \\rightarrow\n\\text{pr}_{s,*}(\\mathcal O_C) = \\mathcal O_{{\\mathbb P}^1_s}.\n$$\n\n\\medskip\\noindent\nOn ${\\mathbb P}^1_s\\times {\\mathbb P}^1_x \\times {\\mathbb P}^1_y$ there is a locally free\nresolution of the push-forward of $L$, resp. $L^\\vee$,\n$$\n\\begin{array}{c}\n0 \\rightarrow \\mathcal O_{{\\mathbb P}^1_s}(-1)\\boxtimes \\mathcal O_{{\\mathbb P}^1_x}(a-1)\\boxtimes\n\\mathcal O_{{\\mathbb P}^1_y}(-a-1) \\xrightarrow{F} \\mathcal O_{{\\mathbb P}^1_s}(0)\\boxtimes\n\\mathcal O_{{\\mathbb P}^1_x}(a) \\boxtimes \\mathcal O_{{\\mathbb P}^1_y}(-a) \\rightarrow L \\rightarrow\n0, \\\\\n0 \\rightarrow \\mathcal O_{{\\mathbb P}^1_s}(-1)\\boxtimes \\mathcal O_{{\\mathbb P}^1_x}(-a-1)\\boxtimes\n\\mathcal O_{{\\mathbb P}^1_y}(a-1) \\xrightarrow{F} \\mathcal O_{{\\mathbb P}^1_s}(0)\\boxtimes\n\\mathcal O_{{\\mathbb P}^1_x}(-a) \\boxtimes \\mathcal O_{{\\mathbb P}^1_y}(a) \\rightarrow L^\\vee\n\\rightarrow 0\n\\end{array}\n$$\nHence $R\\text{pr}_{s,*}L$ is the complex,\n$$\n\\mathcal O_{{\\mathbb P}^1_s}(-1)\\otimes_k H^0({\\mathbb P}^1_x,\\mathcal O_{{\\mathbb P}^1_x}(a-1)) \\otimes_k\nH^1({\\mathbb P}^1_y,\\mathcal O_{{\\mathbb P}^1_y}(-a-1)) \\xrightarrow{F} \\mathcal O_{{\\mathbb P}^1_s}\n\\otimes_k H^0({\\mathbb P}^1_x,\\mathcal O_{{\\mathbb P}^1_x}(a))\\otimes_k\nH^1({\\mathbb P}^1_y,\\mathcal O_{{\\mathbb P}^1_y}(-a)).\n$$\nSimilarly for $R\\text{pr}_{s,*}L^\\vee$.  It is possible to write out\nthis map explicitly in terms of bases for $H^0$ and $H^1$, but for the\nmain statement just observe the complex has rank $1$ and degree $-a^2$.\nSimilarly for $R\\text{pr}_{s,*}L^\\vee$.  Therefore $R^1\\pi_* E_\\pi(L)$\nis a torsion sheaf of length $2a^2$.  Because it is equivariant for\n$i$, the localization at each of $(0,1)$ and $(1,0)$ has length $a^2$.\n\n\\medskip\\noindent\nThe lengths of the associated graded pieces of the filtration by order\nof vanishing at $\\mathbb{V}(S_0S_1)$ can be computed from the\ncomplexes for $R\\text{pr}_{s,*}L$ and $R\\text{pr}_{s,*}L^\\vee$.  This\nis left to the reader.\n\\end{proof}\n\n\\begin{cor} \\label{cor-lcomp}\n\\marpar{cor-lcomp}\nIn the universal case, $Q_\\pi(\\mathcal{D}) = -\\sum_{a\\geq 0} a^2 \\Delta_a$.\nTherefore in the general case of $\\pi:C\\rightarrow M$ and an\ninvertible sheaf $L$ of relative degree $0$,\n$$\nQ_\\pi(L) = {\\sum_{\\beta',\\beta''}}^\\prime \\langle C_1(L),\\beta'\n\\rangle \\langle \nC_1(L), \\beta'' \\rangle \\Delta_{\\beta',\\beta''}.\n$$\n\\end{cor}\n\n\\section{Some divisor class relations} \\label{sec-div}\n\\marpar{sec-div}\n\n\\noindent\nIn this section, Proposition~\\ref{prop-Qomega} and\nProposition~\\ref{prop-lcomp} are used to deduce several other divisor\nclass relations.  As usual, let $M$ be an Artin stack and let\n$\\pi:C\\rightarrow M$ be a flat 1-morphism, relatively representable by\nproper algebraic spaces whose geometric fibers are connected,\nat-worst-nodal curves of genus $0$.\n\n\\begin{hyp} \\label{hyp-GRR}\n\\marpar{hyp-GRR}\nThere are cycle class groups for $C$ and $M$ admitting Chern classes\nfor locally free sheaves, and such that Grothendieck-Riemann-Roch\nholds for $\\pi$.\n\\end{hyp}\n\n\\begin{lem} \\label{lem-rel1}\nFor every Cartier divisor class $D$ on $C$ of relative degree $\\langle\nD, \\beta \\rangle$ over $M$, modulo $2$-power torsion,\n$$\n\\pi_*(D\\cdot D) + \\langle D,\\beta \\rangle \\pi_*(D\\cdot\nC_1(\\omega_\\pi)) = {\\sum_{\\beta',\\beta''}}^\\prime \\langle D,\\beta'\n\\rangle \\langle D,\\beta'' \\rangle \\Delta_{\\beta',\\beta''}.\n$$\n\\end{lem}\n\n\\begin{proof}\nDefine $D' = 2D + \\langle D,\\beta \\rangle C_1(\\omega_\\pi)$.  This is a\nCartier divisor class of relative degree $0$.  By\nCorollary~\\ref{cor-lcomp}, \n$$\nQ_\\pi(D') = {\\sum_{\\beta',\\beta''}}^\\prime (\\langle 2D,\\beta' \\rangle\n-\\langle D,\\beta \\rangle)(\\langle 2D,\\beta'' \\rangle - \\langle D,\\beta\n\\rangle) \\Delta_{\\beta',\\beta''}.\n$$\nBy Lemma~\\ref{lem-GRR} this is,\n$$\n\\begin{array}{c}\n4\\pi_*(D\\cdot D) +4\\langle D,\\beta \\rangle \\pi_*(D\\cdot\nC_1(\\omega_\\pi) + (\\langle D,\\beta \\rangle)^2 Q_\\pi(C_1(\\omega_\\pi)) =\n\\\\\n{\\sum_{\\beta',\\beta''}}^\\prime (4\\langle D,\\beta' \\rangle \\langle\nD,\\beta'' \\rangle - (\\langle D,\\beta \\rangle)^2)\n\\Delta_{\\beta',\\beta''}.\n\\end{array}\n$$\nBy Proposition~\\ref{prop-Qomega}, $Q_\\pi(\\omega_\\pi) =\n-{\\sum_{\\beta',\\beta''}}' \\Delta_{\\beta',\\beta''}$.  Substituting this\ninto the equation, simplifying, and dividing by 4 gives the relation.\n\\end{proof}\n\n\\begin{lem} \\label{lem-rel2}\n\\marpar{lem-rel2}\nFor every pair of Cartier divisor classes on $C$, $D_1,D_2$, of\nrelative degrees $\\langle D_1,\\beta \\rangle$, resp. $\\langle D_2,\\beta\n\\rangle$, modulo $2$-power torsion,\n$$\n\\begin{array}{c}\n2\\pi_*(D_1\\cdot D_2) + \\langle D_1,\\beta \\rangle \\pi_*(D_2\\cdot\nC_1(\\omega_\\pi)) + \\langle D_2,\\beta \\rangle \\pi_*(D_1\\cdot\nC_1(\\omega_\\pi)) = \n\\\\\n\\\\\n{\\sum_{\\beta',\\beta''}}^\\prime (\\langle D_1,\\beta' \\rangle \\langle\nD_2,\\beta'' \n\\rangle + \\langle D_2,\\beta' \\rangle \\langle D_1,\\beta'' \\rangle)\n\\Delta_{\\beta',\\beta''}.\n\\end{array}\n$$\n\\end{lem}\n\n\\begin{proof}\nThis follows from Lemma~\\ref{lem-rel1} and the polarization identity\nfor quadratic forms.\n\\end{proof}\n\n\\begin{lem} \\label{lem-rel3}\n\\marpar{lem-rel3}\nFor every section of $\\pi$, $s:M\\rightarrow C$, whose image is\ncontained in the smooth locus of $\\pi$,\n$$\ns(M)\\cdot s(M) + s(M)\\cdot C_1(\\omega_\\pi).\n$$\n\\end{lem}\n\n\\begin{proof}\nThis follows by adjunction since the relative dualizing sheaf of\n$s(M)\\rightarrow M$ is trivial.\n\\end{proof}\n\n\\begin{lem} \\label{lem-rel4}\n\\marpar{lem-rel4}\nFor every section of $\\pi$, $s:M\\rightarrow C$, whose image is\ncontained in the smooth locus of $\\pi$ and for every Cartier divisor\nclass $D$ on $C$ of relative degree $\\langle D,\\beta \\rangle$ over\n$M$, modulo $2$-power torsion,\n$$\n\\begin{array}{c}\n2\\langle D,\\beta \\rangle s^*D  -\\pi_*(D\\cdot D) - \\langle D,\\beta\n\\rangle^2 \\pi_*(s(M)\\cdot s(M)) = \\\\\n\\\\\n{\\sum_{\\beta',\\beta''}}^\\prime\n(\\langle D,\\beta' \\rangle^2 \\langle s(M),\\beta'' \\rangle + \\langle\nD,\\beta'' \\rangle^2 \\langle s(M),\\beta' \\rangle )\n\\Delta_{\\beta',\\beta''}.\n\\end{array}\n$$\n\\end{lem}\n\n\\begin{proof}\nBy Lemma~\\ref{lem-rel2},\n$$\n\\begin{array}{c}\n2s^* D + \\pi_*(D\\cdot C_1(\\omega_\\pi)) + \\langle D,\\beta \\rangle\n\\pi_*(s(M)\\cdot C_1(\\omega_\\pi)) = \\\\\n\\\\\n{\\sum}^\\prime (\\langle D,\\beta' \\rangle \\langle s(M),\\beta'' \\rangle +\n\\langle D,\\beta'' \\rangle \\langle s(M),\\beta' \\rangle\n)\\Delta_{\\beta',\\beta''}.\n\\end{array}\n$$\nMultiplying both sides by $\\langle D,\\beta \\rangle$,\n$$\n\\begin{array}{c}\n2\\langle D,\\beta \\rangle s^* D + \\langle D,\\beta \\rangle \\pi_*(D \\cdot\nC_1(\\omega_\\pi)) + \\langle D, \\beta \\rangle^2 \\pi_*(s(M)\\cdot\nC_1(\\omega_\\pi)) = \\\\\n\\\\\n{\\sum}^\\prime(\\langle D,\\beta \\rangle \\langle D,\\beta' \\rangle \\langle\ns(M),\\beta'' \\rangle + \\langle D,\\beta \\rangle \\langle D,\\beta''\n\\rangle \\langle s(M),\\beta' \\rangle ) \\Delta_{\\beta',\\beta''}.\n\\end{array}\n$$\nFirst of all, by Lemma~\\ref{lem-rel4}, $\\langle D,\\beta \\rangle^2\n\\pi_*(s(M) \\cdot C_1(\\omega_\\pi)) = - \\langle D,\\beta \\rangle^2\n\\pi_*(s(M)\\cdot s(M))$.  Next, by Lemma~\\ref{lem-rel1},\n$$\n\\langle D,\\beta \\rangle \\pi_*(D\\cdot C_1(\\omega_\\pi)) = -\\pi_*(D\\cdot\nD) + {\\sum}^\\prime \\langle D,\\beta' \\rangle \\langle D,\\beta''\n\\Delta_{\\beta',\\beta''}.\n$$\nFinally,\n$$\n\\begin{array}{c}\n\\langle D,\\beta \\rangle \\langle D,\\beta' \\rangle \\langle s(M),\\beta''\n\\rangle + \\langle D,\\beta \\rangle \\langle D,\\beta'' \\rangle \\langle\ns(M),\\beta' \\rangle = \\\\ \\\\\n(\\langle D,\\beta' \\rangle + \\langle D,\\beta'' \\rangle) \\langle\nD,\\beta' \\rangle \\langle s(M),\\beta'' \\rangle + (\\langle D,\\beta'\n\\rangle + \\langle D,\\beta'' \\rangle) \\langle D,\\beta'' \\rangle \\langle\ns(M), \\beta' \\rangle = \\\\ \\\\\n\\langle D,\\beta' \\rangle^2 \\langle s(M),\\beta'' \\rangle + \\langle\nD,\\beta'' \\rangle^2 \\langle s(M),\\beta' \\rangle + \\langle D,\\beta'\n\\rangle \\langle D,\\beta'' \\rangle (\\langle s(M),\\beta' \\rangle +\n\\langle s(M),\\beta'' \\rangle) = \\\\ \\\\\n\\langle D,\\beta' \\rangle^2 \\langle s(M),\\beta'' \\rangle + \\langle\nD,\\beta'' \\rangle^2 \\langle s(M),\\beta' \\rangle + \\langle D,\\beta'\n\\rangle \\langle D,\\beta'' \\rangle.\n\\end{array}\n$$\nPlugging in these 3 identities and simplifying gives the relation.\n\\end{proof}\n\n\\medskip\\noindent\nLet $\\mathcal{C}$ be the universal curve over $\\mathfrak{M}_{0,0}$.  Let\n$\\mathcal{C}_\\text{smooth}$ denote the smooth locus of $\\pi$.  The\n2-fibered product $\\text{pr}_1:\\mathcal{C}_\\text{smooth}\n\\times_{\\mathfrak{M}_{0,0}} \\mathcal{C} \n\\rightarrow \\mathcal{C}_\\text{smooth}$ together with the diagonal\n$\\Delta:\\mathcal{C}_\\text{smooth} \\rightarrow \\mathcal{C}_\\text{smooth}\n\\times_{\\mathfrak{M}_{0,0}} \\mathcal{C}$ determine a 1-morphism\n$\\mathcal{C}_\\text{smooth} \\rightarrow \\mathfrak{M}_{1,0}$.  This extends to a\n1-morphism $\\mathcal{C} \\rightarrow \\mathfrak{M}_{1,0}$.  The pullback of the\nuniversal curve is a 1-morphism $\\pi':\\mathcal{C}' \\rightarrow \\mathcal{C}$ that\nfactors through $\\text{pr}_1:\\mathcal{C}\\times_{\\mathfrak{M}_{0,0}} \\mathcal{C}\n\\rightarrow \\mathcal{C}$.  Denote the pullback of the universal section by\n$s:\\mathcal{C} \\rightarrow \\mathcal{C}'$.  Now $\\mathcal{C}$ is regular, and the\ncomplement of $\\mathcal{C}_\\text{smooth}$ has codimension $2$.  In\nparticular, $s^*\\mathcal O_{\\mathcal{C}'}(s(\\mathcal{C}))$ can be computed on\n$\\mathcal{C}_\\text{smooth}$.  But the restriction to $\\mathcal{C}_\\text{smooth}$\nis clearly $\\omega^\\vee_\\pi$.  Therefore $s^*\\mathcal O_{\\mathcal{C}'}(s(\\mathcal{C}))\n\\cong \n\\omega_\\pi^\\vee$ on all of $\\mathcal{C}$.\n\n\\medskip\\noindent\nPulling this back by $\\zeta_C:C\\rightarrow \\mathfrak{C}$ gives a 1-morphism\n$\\pi':C'\\rightarrow C$ that factors through $\\text{pr}_1:C\\times_M\nC\\rightarrow C$.  Let $D$ be a Cartier divisor class on $C$ and\nconsider the pullback to $C'$ of $\\text{pr}_2^* D$ on $C\\times_M C$.\nThis is a Cartier divisor class $D'$ on $C'$.  Of course $s^* D' =\nD$.  Moreover, by the projection formula the pushforward to $C\\times_M\nC$ of $D'\\cdot D'$ is $\\text{pr}_2^*(D\\cdot D)$.  Therefore\n$(\\pi')_*(D'\\cdot D')$ is $(\\text{pr}_1)_*\\text{pr}_2^*(D\\cdot D)$,\ni.e., $\\pi^*\\pi_*(D\\cdot D)$.  Finally, denote by,\n$$\n\\sum_{\\beta',\\beta''} \\langle D,\\beta'' \\rangle^2\n\\widetilde{\\Delta}_{\\beta',\\beta''},\n$$\nthe divisor class on $C$,\n$$\n{\\sum_{\\beta',\\beta''}}^\\prime (\\langle D,\\beta'' \\rangle^2 \\langle\ns,\\beta' \\rangle + \\langle D,\\beta' \\rangle^2 \\langle s,\\beta''\n\\rangle)\\Delta_{\\beta',\\beta''}.\n$$\nThe point is this: if $\\pi$ is smooth over every generic point of $M$,\nthen the divisor class $\\widetilde{\\Delta}_{\\beta',\\beta''}$ is the\nirreducible component of $\\pi^{-1}(\\Delta_{\\beta',\\beta''})$\ncorresponding to the vertex $v'$, i.e., the irreducible component with\n``curve class'' $\\beta'$.\nPutting this all together and applying Lemma~\\ref{lem-rel4} gives the\nfollowing.\n\n\\begin{lem} \\label{lem-rel5}\n\\marpar{lem-rel5}\nFor every Cartier divisor class $D$ on $C$ of relative degree $\\langle\nD,\\beta \\rangle$ over $M$,\n$$\n\\begin{array}{c}\n2\\langle D,\\beta \\rangle D - \\pi^*\\pi_*(D\\cdot D) + \\langle D,\\beta\n\\rangle^2 C_1(\\omega_\\pi) = \\\\ \\\\\n\\sum_{\\beta',\\beta''} \\langle D,\\beta'' \\rangle^2\n\\widetilde{\\Delta}_{\\beta',\\beta''}.\n\\end{array}\n$$\nIn particular, the relative Picard group of $\\pi$ is generated by\n$C_1(\\omega_\\pi)$ and the boundary divisor classes\n$\\widetilde{\\Delta}_{\\beta',\\beta''}$.  \n\\end{lem}\n\n\\begin{rmk} \\label{rmk-rel5}\n\\marpar{rmk-rel5}\nIf $\\langle D,\\beta \\rangle \\neq 0$ then, at least up to torsion,\nLemma~\\ref{lem-rel1} follows from Lemma~\\ref{lem-rel5} by intersecting\nboth sides of the relation by $D$ and then applying $\\pi_*$.  This was\npointed out by Pandharipande, who also proved Lemma~\\ref{lem-rel4} up\nto numerical equivalence in ~\\cite[Lem. 2.2.2]{QDiv} (by a very\ndifferent method).   \n\\end{rmk}\n\n\\begin{lem} \\label{lem-rel6}\n\\marpar{lem-rel6}\nLet $s,s':M\\rightarrow C$ be sections with image in the smooth locus\nof $\\pi$ such that $s(M)$ and $s'(M)$ are disjoint.  Then,\n$$\n\\pi_*(s(M)\\cdot s(M)) + \\pi_*(s'(M)\\cdot s'(M)) = -\n\\sum_{\\beta',\\beta''} \\langle s(M),\\beta' \\rangle \\langle\ns'(M),\\beta'' \\rangle \\Delta_{\\beta',\\beta''}.\n$$\n\\end{lem}\n\n\\begin{proof}\nApply Lemma~\\ref{lem-rel2} and use $s(M)\\cdot s'(M)=0$ and\nLemma~\\ref{lem-rel3}.  \n\\end{proof}\n\n\\begin{lem} \\label{lem-rel7}\nLet $r\\geq 2$ and $s_1,\\dots,s_r:M\\rightarrow C$ be sections with image in the\nsmooth locus of $\\pi$ and which are pairwise disjoint.  Then,\n$$\n-\\sum_{i=1}^r \\pi_*(s_i(M)\\cdot s_i(M)) = (r-2)\\pi_*(s_1(M)\\cdot\ns_1(M)) + \\sum_{\\beta',\\beta''} \\langle s_1(M),\\beta' \\rangle \\langle\ns_2(M) + \\dots + s_r(M),\\beta'' \\rangle \\Delta_{\\beta',\\beta''}.\n$$\n\\end{lem}\n\n\\begin{proof}\nThis follows from Lemma~\\ref{lem-rel6} by induction.\n\\end{proof}\n\n\\begin{lem} \\label{lem-rel8}\n\\marpar{lem-rel8}\nLet $r\\geq 2$ and let $s_1,\\dots,s_r:M \\rightarrow C$ be sections with\nimage in the smooth locus of $\\pi$ and which are pairwise disjoint.\nThen,\n$$\n-\\sum_{i=1}^r \\pi_*(s_i(M)\\cdot s_i(M)) = r(r-2) \\pi_*(s_1(M)\\cdot\ns_1(M)) + \\sum_{\\beta',\\beta''} \\langle s_1(M),\\beta' \\rangle \\langle\ns_2(M) + \\dots + s_r(M),\\beta'' \\rangle^2 \\Delta_{\\beta',\\beta''}.\n$$\nCombined with Lemma~\\ref{lem-rel7} this gives,\n$$\n\\begin{array}{c}\n(r-1)(r-2)\\pi_*(s_1(M)\\cdot s_1(M)) = \\\\ \\\\\n-\\sum_{\\beta',\\beta''} \\langle\ns_1(M),\\beta' \\rangle \\langle s_2(M) + \\dots + s_r(M),\\beta'' \\rangle\n(\\langle s_2(M)+\\dots +s_r(M), \\beta'' \\rangle - 1)\n\\Delta_{\\beta',\\beta''},\n\\end{array}\n$$\nwhich in turn gives,\n$$\n\\begin{array}{c}\n-(r-1)\\sum_{i=1}^r\\pi_*(s_i(M)\\cdot s_i(M)) = \\\\ \\\\\n\\sum_{\\beta',\\beta''} \\langle s_1(M),\\beta' \\rangle \\langle s_2(M) +\n\\dots + s_r(M), \\beta'' \\rangle (r- \\langle s_2(M) +\\dots + s_r(M),\n\\beta'' \\rangle) \\Delta_{\\beta',\\beta''}.\n\\end{array}\n$$\nIn the notation of Example~\\ref{ex-boundary}, this is,\n$$\n-(r-1)(r-2)\\pi_*(s_1(M)\\cdot s_1(M)) = \\sum_{(A,B), \\ 1\\in A}\n\\#B(\\#B-1) \\Delta_{(A,B)},\n$$\nand\n$$\n-(r-1)\\sum_{i=1}^r \\pi_*(s_i(M)\\cdot s_i(M)) = \\sum_{(A,B), \\ 1\\in A}\n\\#B(r-\\#B) \\Delta_{(A,B)}.\n$$\n\\end{lem}\n\n\\begin{proof}\nDenote $D=\\sum_{i=2}^r s_i(M)$.  Apply Lemma~\\ref{lem-rel4} to get,\n$$\n\\begin{array}{c}\n2(r-1)\\cdot 0 - \\sum_{i=2}^r \\pi_*(s_i(M)\\cdot s_i(M)) -\n(r-1)^2\\pi_*(s_1(M)\\cdot s_1(M)) = \\\\ \\\\\n\\sum_{\\beta',\\beta''} \\langle s_1(M),\\beta' \\rangle \\langle\ns_2(M)+\\dots + s_r(M),\\beta'' \\rangle^2 \\Delta_{\\beta',\\beta''}.\n\\end{array}\n$$\nSimplifying,\n$$\n-\\sum_{j=1}^r \\pi_*(s_i(M)\\cdot s_i(M)) = r(r-2) \\pi_*(s_1(M)\\cdot\ns_1(M)) + \\sum \\langle s_1(M),\\beta' \\rangle \\langle s_2(M)+ \\dots\n+s_r(M),\\beta'' \\rangle^2 \\Delta_{\\beta',\\beta''}.\n$$\nSubtracting from the relation in Lemma~\\ref{lem-rel7} gives the\nrelation for $(r-1)(r-2)\\pi_*(s_1(M)\\cdot s_1(M))$.  \nMultiplying the first relation by $(r-1)$, plugging in the second\nrelation and simplifying gives the third relation.\n\\end{proof}\n\n\\begin{lem} \\label{lem-rel9}\n\\marpar{lem-rel9}\nLet $r\\geq 2$ and let $s_1,\\dots,s_r:M\\rightarrow C$ be everywhere\ndisjoint sections with image in the smooth locus.  For every $1\\leq i\n< j \\leq r$, using the notation from Example~\\ref{ex-boundary},\n$$\n\\sum_{(A,B), \\ i\\in A} \\#B(r-\\#B) \\Delta_{(A,B)} = \\sum_{(A',B'), j \\in\n  A} \\#B'(r-\\#B') \\Delta_{(A',B')}.\n$$\n\\end{lem}\n\n\\begin{proof}\nThis follows from Lemma~\\ref{lem-rel8} by permuting the roles of $1$\nwith $i$ and $j$.\n\\end{proof}\n\n\\begin{lem} \\label{lem-rel10}\n\\marpar{lem-rel10}\nLet $r \\geq 2$ and let $s_1,\\dots,s_r:M \\rightarrow C$ be everywhere\ndisjoint sections with image in the smooth locus of $\\pi$.  For every\nCartier \ndivisor class $D$ on $C$ of relative degree $\\langle D,\\beta \\rangle$,\n$$\n2(r-1)(r-2)\\langle D,\\beta \\rangle s_1^*D = (r-1)(r-2)\\pi_*(D\\cdot D)\n+ \\sum_{\\beta',\\beta''} \\langle s_1(M),\\beta' \\rangle a(D,\\beta'')\n\\Delta_{\\beta',\\beta''},\n$$\nwhere, \n$$\na(D,\\beta'') = (r-1)(r-2)\\langle D,\\beta'' \\rangle^2 -\n\\langle D,\\beta \\rangle^2 \\langle s_2(M)+\\dots +s_r(M), \\beta'' \\rangle (\n\\langle s_2(M)+\\dots + s_r(M),\\beta'' \\rangle -1).\n$$\nIn particular, if $r\\geq 3$, then modulo torsion $s_i^*D$ is in the\nspan of $\\pi_*(D\\cdot D)$ and boundary divisors for every\n$i=1,\\dots,r$.\n\\end{lem}\n\n\\begin{proof}\nThis follows from Lemma~\\ref{lem-rel4} and Lemma~\\ref{lem-rel8}.\n\\end{proof}\n\n\\begin{lem} \\label{lem-rel11}\n\\marpar{lem-rel11}\nLet $r\\geq 2$ and let $s_1,\\dots,s_r:M\\rightarrow C$ be everywhere\ndisjoint sections with image in the smooth locus of $\\pi$.  Consider\nthe sheaf $\\mathcal{E} = \\Omega_\\pi(s_1(M)+\\dots+s_r(M))$.  The perfect\ncomplex $R\\pi_*R\\textit{Hom}_{\\mathcal O_C}(\\mathcal{E},\\mathcal O_C)$ has rank $3-r$ and\nthe first Chern class of the determinant is $-2\\Delta\n-\\sum_{i=1}^r(s_i(M)\\cdot s_i(M))$.  In particular, if $r\\geq 2$, up\nto torsion,\n$$\n\\begin{array}{c}\nC_1(\\text{det}R\\pi_*\nR\\textit{Hom}_{\\mathcal O_C}(\\Omega_\\pi(s_1(M)+\\dots+s_r(M)),\\mathcal O_C)) = \\\\ \\\\\n-2\\Delta + \\frac{1}{r-1}\\sum_{(A,B),\\ 1\\in A} \\#B(r-\\#B)\n\\Delta_{(A,B)}.\n\\end{array}\n$$\n\\end{lem}\n\n\\begin{proof}\nThere is a short exact sequence,\n$$\n\\begin{CD}\n0 @>>> \\Omega_\\pi @>>> \\Omega_\\pi(s_1(M)+\\dots+s_r(M))\n@>>> \\oplus_{i=1}^r (s_i)_*\\mathcal O_M @>>> 0.\n\\end{CD}\n$$\nCombining this with Lemma~\\ref{lem-Omega}, Lemma~\\ref{lem-rel8}, and\nchasing through exact sequences gives the lemma.\n\\end{proof}\n\n\\section{The virtual canonical bundle} \\label{sec-vircan}\n\\marpar{sec-vircan}\n\n\\noindent\nLet $k$ be a field, let $X$ be a connected, smooth algebraic space\nover $k$ of dimension $n$, let\n$M$ be an Artin stack over $k$, let $\\pi:C\\rightarrow M$ be a flat\n1-morphism, \nrelatively representable by proper algebraic spaces whose geometric\nfibers are connected, at-worst-nodal curves of arithmetic genus $0$,\nlet $s_1,\\dots,s_r:M\\rightarrow C$ be pairwise disjoint sections with\nimage contained in the smooth locus of $\\pi$ (possibly $r=0$, i.e.,\nthere are no sections), and let $f:C\\rightarrow\nX$ be a 1-morphism of $k$-stacks.  In this setting, Behrend and\nFantechi introduced a perfect complex $E^\\bullet$ on $M$ of amplitude\n$[-1,1]$ and a morphism to the cotangent complex,\n$\\phi:E^\\bullet \\rightarrow L_M^\\bullet$, ~\\cite{BM}.  \nIf $\\text{char}(k)=0$ and $M$ is the Deligne-Mumford stack of stable\nmaps to $X$, Behrend and Fantechi prove $E^\\bullet$ has amplitude\n$[-1,0]$,\n$h^0(\\phi)$ is an isomorphism\nand $h^{-1}(\\phi)$ is surjective.  In\nmany interesting cases, $\\phi$ is a quasi-isomorphism.  Then\n$\\text{det}(E^\\bullet)$ is an invertible \ndualizing sheaf for $M$. \nBecause of this, \n$\\text{det}(E^\\bullet)$ is called the \\emph{virtual canonical\n  bundle}.  In this section the relations from Section~\\ref{sec-div}\nare used to give a formula for the divisor class of the virtual\ncanonical bundle.  Hypothesis ~\\ref{hyp-GRR} holds for $\\pi$.\n\n\\medskip\\noindent\nDenote by $L_{(\\pi,f)}$ the cotangent complex of the morphism\n$(\\pi,f):C\\rightarrow M\\times X$.  This is a perfect complex of\namplitude $[-1,0]$.  There is a distinguished triangle,\n$$\n\\begin{CD}\nL_\\pi @>>> L_{(\\pi,f)} @>>> f^* \\Omega_X[1] @>>> L_\\pi[1].\n\\end{CD}\n$$\nThere is a slight variation $L_{(\\pi,f,s)}$ taking into account the\nsections which fits into a distinguished triangle,\n$$\n\\begin{CD}\nL_\\pi(s_1(M)+\\dots+s_r(M)) @>>> L_{(\\pi,f,s)} @>>> f^*\\Omega_X[1] @>>>\nL_\\pi(s_1(M)+\\dots+s_r(M))[1]. \n\\end{CD}\n$$\nThe complex $E^\\bullet$ is defined\nto be $(R\\pi_*(L_{(\\pi,f,s)}^\\vee)[1])^\\vee$, where $(F^\\bullet)^\\vee$\nis $R\\textit{Hom}(F^\\bullet,\\mathcal O)$.  In particular,\n$\\text{det}(E^\\bullet)$ is the determinant of\n$R\\pi_*(L_{(\\pi,f,s)}^\\vee)$.  From the distinguished triangle,\n$\\text{det}(E^\\bullet)$ is\n$$\n\\text{det}(R\\pi_*\nR\\textit{Hom}_{\\mathcal O_C}(\\Omega_\\pi(s_1(M)+\\dots+s_r(M)),\\mathcal O_C)) \n\\otimes \\text{det}(R\\pi_* f^*T_X)^\\vee.\n$$\nBy Lemma~\\ref{lem-rel11}, the first term is known.  \nThe second term\nfollows easily from Grothendieck-Riemann-Roch.\n\n\\begin{lem} \\label{lem-TX}\n\\marpar{lem-TX}\nAssume that the relative degree of $f^*C_1(\\Omega_X)$ is nonzero.\nThen $R\\pi_*f^*T_X[-1]$ has rank $\\langle -f^*C_1(\\Omega_X),\\beta\n\\rangle + n$, and up to torsion the first Chern class of the\ndeterminant is,\n$$\n\\begin{array}{c}\n\\frac{1}{2\\langle -f^*C_1(\\Omega_X),\\beta \\rangle} \\left[ 2\\langle\n-f^*C_1(\\Omega_X), \\beta \\rangle \\pi_* f^* C_2(\\Omega_X) \\right. \\\\\n\\\\\n - (\\langle\n-f^* C_1(\\Omega_X),\\beta \\rangle + 1) \\pi_* f^* C_1(\\Omega_X)^2 +\n\\\\ \\\\\n\\left.\n{\\sum}' \\langle -f^*C_1(\\Omega_X),\\beta' \\rangle \\langle\n-f^*C_1(\\Omega_X),\\beta'' \\rangle \\Delta_{\\beta',\\beta''}\\right].\n\\end{array}\n$$\n\\end{lem}\n\n\\begin{proof}\nThe Todd class $\\tau_\\pi$ of $\\pi$ is $1 - \\frac{1}{2}C_1(\\omega_\\pi)\n+ \\tau_2 + \\dots$, \nwhere $\\pi_*\\tau_2 = 0$.  The Chern character of $f^*T_X$ is,\n$$\nn - f^*C_1(\\Omega_X) + \\frac{1}{2}(f^*C_1(\\Omega_X)^2 -\n2f^*C_2(\\Omega_X)) + \\dots\n$$\nTherefore $\\text{ch}(f^*T_X)\\cdot \\tau_\\pi$ equals,\n$$\nn -\n\\left[f^*C_1(\\Omega_X)+\\frac{n}{2}C_1(\\Omega_\\pi) \\right] + \\frac{1}{2}\\left[\nf^*C_1(\\Omega_X)^2 - 2f^*C_2(\\Omega_X) + f^*C_1(\\Omega_X)\\cdot\nC_1(\\omega_\\pi) \\right] + n\\tau_2 + \\dots\n$$\nApplying $\\pi_*$ and using that $\\pi_*\\tau_2 = 0$, the rank is\n$n+\\langle -f^*C_1(\\Omega_X),\\beta \\rangle$, and the determinant has\nfirst Chern class,\n$$\n\\frac{1}{2}\\pi_*\\left[ f^*C_1(\\Omega_X)^2 - 2f^*C_2(\\Omega_X) \\right] +\n\\frac{1}{2} \\pi_*(f^*C_1(\\Omega_X)\\cdot C_1(\\omega_\\pi)).\n$$\nApplying Lemma~\\ref{lem-rel1} and simplifying gives the relation.\n\\end{proof}  \n\n\\begin{prop} \\label{prop-vc}\n\\marpar{prop-vc}\nThe rank of $E^\\bullet$ is $\\langle -f^*C_1(\\Omega_X),\\beta \\rangle +\nn + r - 3.$  The following divisor class reations hold modulo torsion.\nIf $\\langle -f^*C_1(\\Omega_X),\\beta \\rangle\\neq 0$ and $r=0$,\nthe first Chern class of the virtual canonical bundle\nis,\n\\begin{equation}\n\\begin{array}{c}\n\\frac{1}{2\\langle -f^*C_1(\\Omega_X),\\beta \\rangle} \\left[ 2\\langle\n-f^*C_1(\\Omega_X), \\beta \\rangle \\pi_* f^* C_2(\\Omega_X) \\right. \\\\\n\\\\\n- (\\langle\n-f^* C_1(\\Omega_X),\\beta \\rangle + 1) \\pi_* f^* C_1(\\Omega_X)^2 +\n\\\\ \\\\\n\\left.\n{\\sum}' (\\langle -f^*C_1(\\Omega_X),\\beta' \\rangle \\langle\n-f^*C_1(\\Omega_X),\\beta'' \\rangle -4\\langle -f^*C_1(\\Omega_X),\\beta\n\\rangle) \\Delta_{\\beta',\\beta''}\\right].\n\\end{array}\n\\end{equation}\nIf $\\langle -f^*C_1(\\Omega_X),\\beta \\rangle \\neq 0$ and $r=1$, \nthe first Chern class of the virtual canonical bundle is,\n\\begin{equation}\n\\begin{array}{c}\n\\frac{1}{2\\langle -f^*C_1(\\Omega_X),\\beta \\rangle} \\left[ 2\\langle\n-f^*C_1(\\Omega_X), \\beta \\rangle \\pi_* f^* C_2(\\Omega_X) \\right. \\\\\n\\\\\n- (\\langle\n-f^* C_1(\\Omega_X),\\beta \\rangle + 1) \\pi_* f^* C_1(\\Omega_X)^2 +\n\\\\ \\\\\n\\left.\n{\\sum}' (\\langle -f^*C_1(\\Omega_X),\\beta' \\rangle \\langle\n-f^*C_1(\\Omega_X),\\beta'' \\rangle -4\\langle -f^*C_1(\\Omega_X),\\beta\n\\rangle) \\Delta_{\\beta',\\beta''}\\right] \\\\\n\\\\\n-\\pi_*(s_1(M)\\cdot s_1(M)).\n\\end{array}\n\\end{equation}\nIf $\\langle -f^*C_1(\\Omega_X),\\beta \\rangle \\neq 0$ and $r\\geq 2$, \nthe first Chern class of the virtual canonical bundle is,\n\\begin{equation}\n\\begin{array}{c}\n\\frac{1}{2\\langle -f^*C_1(\\Omega_X),\\beta \\rangle} \\left[ 2\\langle\n-f^*C_1(\\Omega_X), \\beta \\rangle \\pi_* f^* C_2(\\Omega_X) \\right. \\\\\n\\\\\n- (\\langle\n-f^* C_1(\\Omega_X),\\beta \\rangle + 1) \\pi_* f^* C_1(\\Omega_X)^2 +\n\\\\ \\\\\n\\left.\n{\\sum}' (\\langle -f^*C_1(\\Omega_X),\\beta' \\rangle \\langle\n-f^*C_1(\\Omega_X),\\beta'' \\rangle -4\\langle -f^*C_1(\\Omega_X),\\beta\n\\rangle) \\Delta_{\\beta',\\beta''}\\right] \\\\\n\\\\\n+ \\frac{1}{r-1} \\sum_{(A,B), 1\\in\n  A} \\#B(r-\\#B)\\Delta_{(A,B)}.\n\\end{array}\n\\end{equation}\n\\end{prop}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Background}\n\nThe Parkes-MIT-NRAO (PMN) 4.85~GHz survey (Griffith~\\&~Wright~1993) covered the entire southern sky to a limiting flux density of 30~mJy (varying slightly with declination); it increased the number of known southern radio sources by a factor of approximately 6. Wright et al. (1996) made simultaneous 4.80\/8.64~GHz Australia Telescope Compact Array (ATCA) measurements of a complete sample of 8068 PMN sources defined by the flux density limits $S_{4.85~\\rm{GHz}} \\geq 70$~mJy $(-73^\\circ < \\delta < -38^\\circ.5)$, $S_{4.85~\\rm{GHz}} \\geq 50$~mJy $(-87^\\circ < \\delta < -73^\\circ)$ and the requirement that galactic latitude $|b| \\geq 2^\\circ$. The Wright et al. ATCA observations resulted in: positions accurate to (standard error) $0.6''$; spectral data and structural information of resolution approximately $1''$. This dataset is the largest deep, complete, unbiased, dual-frequency, interferometric catalogue of southern radio sources. It covers $19~\\%$ of the sky, yet this area contains only 1 of the 39 previously known cases of gravitational lensing classed as \\emph{probable\\ lenses} by the \\emph{CASTLES}\\footnote{$\\rm{http:\/\/cfa-www.harvard.edu\/castles\/castles.edu}$} collaboration (Figure 1). The resolution ($\\sim 1''$) samples the peak of the image separation histogram for the 14 JVAS\/CLASS lenses published prior to these proceedings (Browne~et~al.,~1998, Koopmans~et~al.,~1999, Myers~et~al.,~1999). The 2178 flat--spectrum sources in these data constitute our finding survey.\n\n\nCandidate gravitational lenses were selected using an automated routine, based on objective criteria, designed to reduce sample contamination by radio galaxies and objects within, or behind, the Galactic plane. Radio galaxies are often observed to exhibit steepening radio spectral indices with increasing separation from the core (e.g. Wiita \\& Gopal-Krishna, 1990), objects showing such spectral steepening were not included in the sample.  Simulated ATCA observations of known northern hemisphere lenses (transposed to declinations within the surveyed region) were used to test and refine candidate selection.\n\nThe 110 flat-spectrum objects satisfying our selection criteria of component fluxes $S_{\\nu}> 5 \\sigma~(\\sim~30$~mJy) (at both 4.80 and 8.64~GHz) and $|b| \\geq 5^\\circ$ were chosen for further study at radio, infrared and optical wavelengths. New simultaneous 4.80\/8.64 GHz, dual linear polarisation ATCA observations of each lens candidate have been made. These new data (of resolution $\\sim 1''$ at 8.64~GHz) exhibit a factor 8 improvement in S:N, and vastly improved coverage of the interferometric u-v plane, over data in hand. Total intensity and polarisation maps of all objects have been made and examined. Re-running the automated lens candidate finder rejected half of the observed objects. We have obtained deep near infrared $K_N$-Band images of 35 sources remaining in the lens candidate sample, deep optical B-Band images (and colours) for 19 of these objects and 1 spectrum. \n\n\n\\begin{figure}\n\\caption{\\small{Aitoff projection showing the positions of known gravitational lenses, their discovery method and the sky coverage of the current major surveys for flat radio spectrum gravitational lenses. The hatched region is covered by this survey (the region shaded black represents the un-surveyed area of radius $3^\\circ$ centered on the southern celestial pole). The region bounded by solid black lines is the area ($0^\\circ > \\delta > -40^\\circ$, $|b| > 10^\\circ$) surveyed by the MIT group  (Winn et al., these proceedings), the regions of the northern sky for which $|b| > 10^\\circ$ (dashed line) are covered by CLASS (Browne et al., these proceedings).}}\n\\plotfiddle{oprouton1.eps}{5.15cm}{90}{40}{40}{125}{-50}\n\\end{figure}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\n\nBlandford and Payne \\cite{blandpayne} showed angular momentum transfer via large magnetic stresses, \nin the absence of alpha-viscosity, in the framework of self-similar\nKeplerian disk flows. Here, we show in a simpler 1.5-dimensional, vertically averaged\ndisk model that the Maxwell stresses due to strong magnetic fields are adequate enough \nfor angular momentum transfer even in advective accretion flows, without self-similar assumption, \ndescribing the hard spectral states of black hole sources. \n\nThe idea of exploring magnetic stress in order to explain astrophysical systems is not\nreally new. This was implemented, for example, in the solar wind which was understood to \nhave decreased Sun's angular momentum through the effect\nof magnetic stresses (see, e.g., Ref.~\\refcite{waber}), in the proto-stellar gas clouds \nwhich might have been contracted by magnetic effects \\cite{Mouschovias}. \nOzernoy and Usov \\cite{usov} and Blandford \\cite{bland} showed that the energy is possible to \nextract continuously by electromagnetic torques and\ntwisted field lines in accretion disks. By linear stability analysis of the accretion disks,\nit was shown by Cao and Spruit \\cite{cao} that angular momentum is possible to remove by the \nmagnetic torque exerted by a centrifugally driven wind. \nHowever, by solving the local vertical structure\nof a geometrically thin accretion disk threaded by a poloidal magnetic field, \nOgilvie and Livio \\cite{ogil} showed the shortcoming of launching an outflow and suggested\nfor an existence of additional source of energy for its successful launching.\n\nHere we demonstrate, semi-analytically, the effects of strong magnetic field, stronger than\nthat needed for magnetorotational instability (MRI) \\cite{sujit}, with \nplasma$-\\beta > 1$ yet, on to the vertically averaged advective accretion flows in \nvertical equilibrium in order to transport matter. \nTherefore, we consider the flow variables to depend on the radial coordinate only. \nAlthough, in reality, a non-zero vertical magnetic field should induce a vertical motion,\nin the platform of the present assumption, any vertical motion will be featured as an outward motion. \nIndeed, our aim here is to furnish removal of angular momentum from the flow via\nmagnetic stresses, independent of its vertical or outward transport.\n\n\n\n\n\\section{Basic model equations}\\label{model}\n\nWe describe optically thin, magnetized, viscous, axisymmetric, advective, vertically averaged, steady-state accretion\nflow, in the pseudo-Newtonian framework with the Mukhopadhyay \\cite{m02} potential. Hence, the equation \nof continuity, vertically averaged hydromagnetic equations for energy-momentum balance in different directions are \ngiven by (assuming that the dimensionless variables do not vary significantly in the vertical direction such that \n$\\partial\/ \\partial z \\sim s_i\/h$ and, as a consequence, the vertical component of velocity is zero),\n\\begin{eqnarray}\n\\nonumber\n&&\\dot{M}=4\\pi  x \\rho h \\vartheta,~~~ \n\\vartheta\\frac{d\\vartheta}{dx}+\\frac{1}{\\rho}\\frac{dP}{dx}-\\frac{\\lambda^{2}}{x^3}+F=\\frac{1}{4\\pi \\rho}\\left(B_x\\frac{dB_x}{dx}+s_1\\frac{B_xB_z}{h}-\\frac{B_{\\phi}^2}{x}\\right),\\\\\n\\nonumber\n&&\\vartheta\\frac{d\\lambda}{dx}=\\frac{1}{x\\rho}\\frac{d}{dx}\\left(x^2W_{x\\phi}\\right)+\\frac{x}{4\\pi \\rho}\\left(B_x\\frac{dB_{\\phi}}{dx}+s_2\\frac{B_zB_{\\phi}}{h}+\\frac{B_xB_{\\phi}}{x}\\right),\\\\\n&&\\frac{P}{\\rho h}=\\frac{Fh}{x}-\\frac{1}{4\\pi \\rho}\\left(B_x\\frac{dB_z}{dx}+s_3\\frac{B_z^2}{h}\\right),\\\\\n\\nonumber\n&&\\vartheta T\\frac{ds}{dx}=\\frac{\\vartheta}{\\Gamma_3-1}\\left(\\frac{dP}{dx}-\\frac{\\Gamma_1P}{\\rho}\\frac{d\\rho}{dx}\\right)\n=Q^+-Q^-=Q^+_{vis}+Q^+_{mag}-Q^-_{vis}-Q^-_{mag},\n\\end{eqnarray}\nwhere $W_{x\\phi}=\\alpha(P+\\rho \\vartheta^2)$ and $\\alpha$ the Shakura-Sunyaev viscosity parameter \\cite{ss}.\nNote that $s_1$, $s_2$ and $s_3$ are the degrees of vertical scaling for the radial, azimuthal and vertical components of the \nmagnetic field respectively. \nHere $\\dot{M}$ is the \nconserved mass accretion rate, $\\rho$ the mass density of the flow, $\\vartheta$ the radial velocity, $P$ the total \npressure including the magnetic contribution, $F$ the force corresponding to the pseudo-Newtonian potential for \nrotating black holes \\cite{m02}, $\\lambda$ the angular momentum per unit mass, $W_{x\\phi}$ the viscous shearing stress written \nfollowing the Shakura-Sunyaev prescription \\cite{ss} with appropriate modification \\cite{mg03},\n$h \\sim z$, the half-thickness and $x$ the radial coordinate of the disk, when both of \nthem are expressed in units of $GM\/c^2$, where $G$ the Newton's gravitation constant, $M$ the mass of black hole, \n$c$ the speed of light, $s$ the entropy per unit volume, \n$T$ the (ion) temperature of the flow, $Q^+$ and $Q^-$ are the net rates of energy released and radiated out per unit \nvolume in\/from the flow respectively (when $Q^+_{vis}$, $Q^+_{mag}$, $Q^-_{vis}$, $Q^-_{mag}$ are the respective \ncontributions from viscous and magnetic parts). All the variables are made dimensionless in the spirit of \ndimensionless $x$ and $z$. We further assume, for the present purpose, the heat radiated \nout proportional to the released rate with the proportionality constants $(1 - f_{vis})$ and $(1 - f_m)$, respectively, \nfor viscous and magnetic parts of the radiations. $\\Gamma_1$, $\\Gamma_3$, which are functions of polytropic constant $\\gamma$, \nindicate the polytropic indices depending \non the gas and radiation content in the flow (see, e.g., Ref.~\\refcite{rm1}, for exact expressions) \nand $B_x$, $B_{\\phi}$ and $B_z$ are the components of magnetic field. The model for $Q^+_{vis}$ is taken from the \nprevious work \\cite{rm1}, and the relation for $Q^+_{mag}$ is taken from that by Bisnovatyi-Kogan and \nRuzmaikin \\cite{bis}.\n\nHydromagnetic flow equations must be supplemented by (for the present purpose, steady-state) equations of induction\nand no magnetic monopole at the limit of very large Reynolds number, given by\n\\begin{eqnarray}\n\\nabla \\times \\vec{v} \\times \\vec{B}=0,~~~\n\\frac{d}{dx}(xB_x)+s_3\\frac{B_z}{h}=0,\n\\end{eqnarray}\nwhen $\\vec{v}$ and $\\vec{B}$ are respectively the velocity and magnetic field vectors and $\\nu_m$ is the magnetic \ndiffusivity. \n\n\\section{Solutions and Results}\\label{sol}\n\nWe take into account two situations. (1) Flows with a relatively higher $\\dot{M}$ and, hence, lower $\\gamma$, \nmodelled around stellar mass black holes: such flows may or may not form Keplerian accretion disks. (2) Flows with \na lower $\\dot{M}$ and, hence, higher $\\gamma$, modelled around supermassive black holes: such flows are necessarily \nhot gas dominated advective (or advection dominated) accretion flows.\n\nWe find that the flows with plasma-$\\beta > 1$, but $\\alpha=0$, exhibit adequate matter transport, as efficient as the \n$\\alpha$-viscosity with $\\alpha= 0.08$, but without magnetic stresses, would do. This is interesting as the origin of $\\alpha$ (and the corresponding \ninstability and turbulence) is\nitself not well understood. The maximum required large scale magnetic field is $\\sim 10^5$G in a disk\naround $10M_{\\odot}$ black holes and $\\sim10$G in a disk around $10^7M_{\\odot}$ supermassive black holes,\nwhere $M_\\odot$ is solar mass. \nThe presence of such a field, in particular for a stellar mass black hole disk when the binary companion supplying \nmass is a Sun-like star with\nthe magnetic field on average 1G, may be understood, if the field is approximately frozen with the disk fluids (or the\nsupplied fluids from the companion star remain approximately frozen with the magnetic field) or disk fluids exhibit very\nlarge Reynolds number. Indeed, all the present computations are done at the limit of large Reynolds number, as really\nis the case in accretion flows, such that the term associated with the magnetic diffusivity in the induction equation is\nneglected. The size of a disk around supermassive black holes is proportionately larger compared to that around\na stellar mass black hole. Hence, from the equipartition theory, indeed the magnetic field is expected to be decreased\nhere compared to that around stellar mass black holes. Figure 1 shows a typical set of accretion solutions and confirms that\nflows around a stellar mass black hole with $\\alpha$-viscosity but without large scale magnetic fields (viscous flow)\nshow similar behavior to that with large scale magnetic fields \nwithout $\\alpha$-viscosity (magnetic flow). The flows around a supermassive black hole show very similar features, except\nwith reduced field strength. For other details, see Ref.~\\refcite{kou}.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[angle=0,width=2.5in]{f1}\n\\end{center}\n\\caption{\n(a) Mach number, (b) angular momentum per unit mass in $GM\/c$, (c) inverse of plasma-$\\beta$,\n(d) azimuthal component of magnetic field in G,\nwhen solid and dotted lines are for magnetic flows around\nSchwarzschild ($a=0, \\lambda_c=3.2$) and Kerr ($a=0.998, \\lambda_c=1.8$) black holes respectively,\nand dashed and long-dashed lines are for viscous flows\naround Schwarzschild ($a=0, \\alpha=0.017, \\lambda_c=3.15$) and Kerr ($a=0.998, \\alpha=0.012, \\lambda_c=1.8$)\nblack holes respectively, where $a$ is the dimensionless spin parameter of black hole and $\\lambda_c$ the quantity at critical radius.\nOther parameters are $M=10M_\\odot$, $\\dot{M}=0.1$ Eddington rate,\npolytropic constant $\\gamma=1.335, f_{vis}=f_m=0.5, s_2=-0.5$.\n}\n\\label{stmasm}\n\\end{figure}\n\n\nLet us now explore in more details, how exactly various components of magnetic stress lead to\nangular momentum transfer in the flows. \nFigure \\ref{stmasstr}a shows that the stress component $B_x B_z$ around a Schwarzschild\nblack hole increases almost throughout\nas matter advances towards the black hole. This implies that the flow is prone\nto outflow through the field lines, which effectively helps in\ninfalling matter towards the black hole. However, in the near vicinity of black hole, $B_x B_z$ decreases,\nas indeed outflow is not possible therein. \nIn this flow zone, the angular momentum becomes very small which practically does\nnot affect the infall. The magnitude of $B_\\phi B_z$ decreases till the\ninner region of accretion flow, implying the part of matter to be spiralling out\nand, hence, removing angular momentum leading to infall of matter.\nFinally, the magnitude of $B_x B_\\phi$\nincreases at a large and a small distances from the black hole (except around\nthe transition radius), which helps removing angular momentum and further infall. \nThis is the same as the Shakura-Sunyaev viscous stress would do with the increase of matter pressure.\nHowever, at the intermediate zone,\nthe angular momentum transfer through $B_x B_\\phi$ reverses and a part of\nthe matter outflows. At the Keplerian to sub-Keplerian transition zone,\ndue to the increase of flow thickness, matter is vertically kicked effectively,\nshowing a decrease of $B_x B_\\phi$. Majority of the features remain similar\nfor the flow around a rotating black hole, as shown in Fig. \\ref{stmasstr}b.\nHowever, a rotating black hole reveals a stronger\/efficient outflow\/jet in general. Hence, except at the inner zone,\n$B_x B_\\phi$ decreases throughout, which helps kicking the matter outwards by \ntransferring the angular momentum inwards. \n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[angle=0,width=2.5in]{f2}\n\\end{center}\n\\vskip-4cm\n\\caption{\nComponents of magnetic stress: $B_x B_\\phi$ (solid line),\n$B_x B_z$ (dotted line), $B_\\phi B_z$ (dashed line), for (a) Schwarzschild\nmagnetic flow of Fig. \\ref{stmasm}, (b) Kerr magnetic flow of Fig. \\ref{stmasm}.\n}\n\\label{stmasstr}\n\\end{figure}\n\nDifferent components of the magnetic stress tensor have different roles: $B_xB_{\\phi}$ controls the infall in the disk plane,\nwhereas $B_{\\phi}B_z$ renders the flow to spiral outwards and, hence, outflow. Moreover, $B_xB_z$ helps to kick the \nmatter out vertically. Larger the field strength, larger is the power of magnetic stresses. Interestingly, \nthe magnitude of magnetic\nfield decreases, as the steady-state matter advances towards the black hole. This is primarily because $B_{\\phi}B_z$ \n(and also $B_xB_{\\phi}$ for a rotating black hole) decreases inwards almost entirely in order to induce inflow\nvia angular momentum transfer through outflow. \nThis further reveals\na decreasing $|B_{\\phi}|$ as the output of self-consistent solutions of the coupled set of equations.\n\n\n\\section{Discussion and Conclusions}\\label{last}\n\nIs there any observational support for the existence of such a magnetic field, as required for the magnetic accretion\nflows discussed here? Interestingly, the polarization measurements in the hard state of Cyg X-1 imply that it should\nhave at least 10mG field at the source of emission \\cite{lau}. In order to explain such high polarization, a\njet model was suggested by Zdziarski et al. \\cite{zd}, which requires a magnetic field $\\sim (5 - 10) \\times 10^5$G \nat the base of jet and hence in the underlying accretion disk. \n\nIn the present computations, we have assumed the flow to be vertically averaged without allowing any vertical\ncomponent of the flow velocity. The most self-consistent approach,\nin order to understand vertical transport of matter through the magnetic effects which in turn leads to the radial\ninfall of rest of the matter, is considering the flow to be moving in the vertical direction from the disk plane as well.\nSuch an attempt, in the absence of magnetic and viscous effects, was made earlier by one of the present authors \\cite{deb} in the model\nframework of coupled disk-outflow systems. In such a framework, the authors further showed that the outflow\npower of the correlated disk-outflow systems increases with the increasing spin of black holes. Our future goal is now\nto combine that model with the model of present work, so that the coupled disk-outflow systems can be investigated\nmore self-consistently and rigorously, when the magnetic field plays indispensable role in order to generate vertical\nflux in the three-dimensional flows.\n\n\n\n\n\\section*{Acknowledgments}\nK.C. thanks the Academies' Summer Research Fellowship Programme of India for offering him a Fellowship to\npursue his internship in Indian Institute of Science, Bangalore, \nwhen most of the calculations of this project were done.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecently, high resolution X-ray images using {\\it Chandra} have\nrevealed 49 point sources in the Antennae \\citep{zez02a}.  We will\nassume a distance to the Antennae of 19.3 Mpc (for $H_{0}$=75 km\ns$^{-1}$ Mpc$^{-1}$), which implies 10 sources have X-ray luminosities\ngreater than $10^{39}$ ergs s$^{-1}$.  Considering new observations of\nred giant stars in the Antennae indicate a distance of 13.8 Mpc\n\\citep{sav04}, we point out this ultraluminous X-ray source population\ncould decrease by roughly a half.  Typically, masses of black holes\nproduced from standard stellar evolution are less than $\\sim20$ \n$M_{\\odot}$ \\citep[e.g.,][]{fry01}.  The Eddington luminosity limit\nimplies that X-ray luminosities $>10^{39}$ ergs s$^{-1}$ correspond to\nhigher-mass objects not formed from a typical star.  Several authors\n\\citep[e.g.,][]{fab89,zez99,rob00,mak00} suggest these massive ($10\n\\sbond 1000$ $M_{\\sun}$) compact sources outside galactic nuclei are\nintermediate mass black holes (IMBHs), a new class of BHs.  While\nIMBHs could potentially explain the observed high luminosities, other\ntheories exist as well, including beamed radiation from a stellar mass\nBH \\citep{kin01}; super-Eddington accretion onto lower-mass objects\n\\citep[e.g.,][]{moo03,beg02}; or supernovae exploding in dense\nenvironments \\citep{ple95,fab96}.\n\nCompact objects tend to be associated with massive star formation,\nwhich is strongly suspected to be concentrated in young stellar\nclusters \\citep{lad03}.  Massive stars usually end their lives in\nsupernovae, producing a compact remnant.  This remnant can be kicked\nout of the cluster due to dynamical interactions, stay behind after\nthe cluster evaporates, or remain embedded in its central regions.\nThis last case is of particular interest to us as the compact object\nis still {\\it in situ}, allowing us to investigate its origins via the\nambient cluster population.  The potential for finding such\nassociations is large in the Antennae due to large numbers of both\nX-ray point sources and super star clusters; a further incentive for\nstudying these galaxies.\n\nIn \\citet[henceforth Paper I]{bra05} we presented $J$ and $K_s$\nphotometry of $\\sim220$ clusters in the Antennae.  Analysis of ($J -\nK_s$) colors indicated that many clusters in the overlap region suffer\nfrom 9--10 mag of extinction in the $V$-band.  This result contrasts\nwith previous work by \\citet{whi02} who associated optical sources\nwith radio counterparts in the Antennae \\citep{nef00} and argued that\nextinction is not large in this system.  Here, we continue our\nanalysis of these Antennae IR images by making a frame-tie between the\nIR and {\\it Chandra} X-ray images from \\citet{fab01}.  Utilizing the\nsimilar dust-penetrating properties of these wavelengths, we\ndemonstrate the power of this approach to finding counterparts to\nX-ray sources.  By comparing the photometric properties of clusters\nwith and without X-ray counterparts, we seek to understand the cluster\nenvironments of these X-ray sources.  In \\S2 we discuss the IR\nobservations of the Antennae. \\S3 explains our matching technique and\nthe photometric properties of the IR counterparts.  We conclude with a\nsummary of our results in \\S4.\n\n\\section{Observations and Data Analysis}\n\n\\subsection{Infrared Imaging}\n\nWe obtained near-infrared images of NGC 4038\/9 on 2002 March 22 using\nthe Wide-field InfraRed Camera (WIRC) on the Palomar 5-m telescope.\nAt the time of these observations, WIRC had been commissioned with an\nunder-sized HAWAII-1 array (prior to installation of the full-sized\nHAWAII-2 array in September 2002), providing a $\\sim 4.7 \\times\n4.7$-arcminute field of view with $\\sim0\\farcs25$ pixels (``WIRC-1K''\n-- see \\citet{wil03} for details).  Conditions were non-photometric\ndue to patches of cloud passing through.  Typical seeing-limited\nimages had stellar full-width at half-maximum of $1\\farcs0$ in $K_s$\nand $1\\farcs3$ in $J$.  We obtained images in both the $J$- ($1.25\n\\mu$m) and $K_s$-band ($2.15 \\mu$m) independently.  The details of the\nprocessing used to obtain the final images are given in Paper 1.\n\n\\subsection{Astrometric Frame Ties}\n\nThe relative astrometry between the X-ray sources in NGC 4038\/9 and\nimages at other wavelengths is crucial for successful identification\nof multi-wavelength counterparts.  Previous attempts at this have\nsuffered from the crowded nature of the field and confusion between\npotential counterparts \\citet{zez02b}.  However, the infrared waveband\noffers much better hopes for resolving this issue, due to the similar\ndust-penetrating properties of photons in the {\\it Chandra} and $K_s$\nbands.  (See also \\citet{bra05} for a comparison of IR extinction to\nthe previous optical\/radio extinction work of \\citet{whi02}.)  We thus\nproceeded using the infrared images to establish an astrometric\nframe-tie, i.e. matching {\\it Chandra} coordinates to IR pixel\npositions.\n\nAs demonstrated by \\citet{bau00}, we must take care when searching for\nX-ray source counterparts in crowded regions such as the Antennae.\nTherefore, our astrometric frame-tie used a unique approach based on\nsolving a two-dimensional linear mapping function relating right\nascension and declination coordinates in one image with x and y pixel\npositions in a second image.  The solution is of the form:\n\n\\begin{eqnarray}\nr_1 = ax_1+by_1+c, \\\\\nd_1 = dx_1+dy_1+f\n\\end{eqnarray}\n\nHere $r_1$ and $d_1$ are the right ascension and declination,\nrespectively, for a single source in one frame corresponding to the\n$x_1$ and $y_1$ pixel positions in another frame.  This function\nconsiders both the offset and rotation between each frame.  Since we\nare interested in solving for the coefficients $a$--$f$, elementary\nlinear algebra indicates we need six equations or three separate\nmatches.  Therefore, we need at least three matches to fully describe\nthe rms positional uncertainty of the frame-tie.\n\nWe first used the above method to derive an approximate astrometric\nsolution for the WIRC $K_s$ image utilizing the presence of six\nrelatively bright, compact IR sources which are also present in images\nfrom the 2-Micron All-Sky Survey (2MASS).  We calculated pixel\ncentroids of these objects in both the 2MASS and WIRC images, and used\nthe 2MASS astrometric header information to convert the 2MASS pixel\ncentroids into RA and Dec.  These sources are listed in Table 1.\nApplying these six matches to our fitting function we found a small\nrms positional uncertainty of $0\\farcs2$, which demonstrates an\naccurate frame-tie between the 2MASS and WIRC images.\n\nUsing the 2MASS astrometric solution as a baseline, we identified\nseven clear matches between {\\it Chandra} and WIRC sources, which had\nbright compact IR counterparts with no potentially confusing sources\nnearby (listed in Table 2).  We then applied the procedure described\nabove, using the {\\it Chandra} coordinates listed in Table 1 of\n\\citet{zez02a} (see that reference for details on the {\\it Chandra}\nastrometry) and the WIRC pixel centroids, and derived the astrometric\nsolution for the IR images in the X-ray coordinate frame.  For the 7\nmatches, we find an rms residual positional uncertainty of $\\sim\n0\\farcs5$ which we adopt as our $1 \\sigma$ position uncertainty.  We\nnote that the positional uncertainty is an entirely {\\it empirical}\nquantity.  It shows the achieved uncertainty in mapping a target from\none image reference frame to the reference frame in another band, and\nautomatically incorporates all contributing sources of uncertainty in\nit.  These include, but are not limited to, systematic uncertainties\n(i.e. field distortion, PSF variation, etc. in both {\\it Chandra} and\nWIRC) and random uncertainties (i.e. centroid shifts induced by photon\nnoise, flatfield noise, etc. in both {\\it Chandra} and WIRC).  Thus,\ngiven the empirical nature of this uncertainty, we expect it to\nprovide a robust measure of the actual mapping error -- an expectation\nwhich seems to be borne out by the counterpart identification in the\nfollowing section.\n\nTo further test the accuracy of our astrometric solution we explored the range\nin rms positional uncertainties for several different frame-ties.\nSpecifically we picked ten IR\/X-ray matches separated by\n$<$1$\\arcsec$, which are listed in Table 3 (see \\S3.1).  Of these ten\nwe chose 24 different combinations of seven matches resulting in 24\nunique frame-ties.  Computing the rms positional uncertainty for each\nwe found a mean of 0$\\farcs$4 with a 1$\\sigma$ uncertainty of\n0$\\farcs$1.  Considering the rms positional uncertainty for the\nframe-tie used in our analysis falls within 1$\\sigma$ of this mean\nrms, this indicates we made an accurate astrometric match between the\nIR and X-ray.\n\n\n\\subsection{Infrared Photometry}\n\nWe performed aperture photometry on 222 clusters in the $J$-band and\n221 clusters in the $K_s$-band (see also Paper 1).  We found that the\nfull width at half maximum (FWHM) was 3.5 pixels ($0\\farcs9$) in the\n$K_s$ image and 4.6 pixels ($1\\farcs2$) in the $J$ band.  We used a\nphotometric aperture of 5-pixel radius in $K_s$ band, and 6-pixel\nradius in $J$ band, corresponding to $\\sim 3\\sigma$ of the Gaussian\nPSF.\n\nBackground subtraction is both very important and very difficult in an\nenvironment such as the Antennae due to the brightness and complex\nstructure of the underlying galaxies and the plethora of nearby\nclusters.  In order to address the uncertainties in background\nsubtraction, we measured the background in two separate annuli around\neach source: one from 9 to 12 pixels and another from 12 to 15 pixels.\nDue to the high concentration of clusters, crowding became an issue.\nTo circumvent this problem, we employed the use of sky background arcs\ninstead of annuli for some sources.  These were defined by a position\nangle and opening angle with respect to the source center.  All radii\nwere kept constant to ensure consistency.  In addition, nearby bright\nsources could shift the computed central peak position by as much as a\npixel or two.  If the centroid position determined for a given source\ndiffered significantly ($>1$ pixel) from the apparent brightness peak\ndue to such contamination, we forced the center of all photometric\napertures to be at the apparent brightness peak.  For both annular\nregions, we calculated the mean and median backgrounds per pixel.\n\nMultiplying these by the area of the central aperture, these values\nwere subtracted from the flux measurement of the central aperture to\nyield 4 flux values for the source in terms of DN.  Averaging these\nfour values provided us with a flux value for each cluster.  We\ncomputed errors by considering both variations in sky background,\n$\\sigma_{sky}$, and Poisson noise, $\\sigma_{adu}$.  We computed\n$\\sigma_{sky}$ by taking the standard deviation of the four measured\nflux values.  We then calculated the expected Poisson noise by scaling\nDN to $e^-$ using the known gain of WIRC \\citep[2 $e^{-1}$\nDN$^{-1}$,][]{wil03} and taking the square root of this value.  We\nadded both terms in quadrature to find the total estimated error in\nphotometry.\n\nWe then calibrated our photometry using the bright 2MASS star in Table 1.\n\n\\section{Results and Discussion}\n\n\\subsection{Identification of IR Counterparts to {\\it Chandra} Sources}\n\nWe used the astrometric frame-tie described above to identify IR\ncounterpart candidates to {\\it Chandra} X-ray sources in the WIRC\n$K_s$ image.  We restricted our analysis to sources brighter than\n$K_s\\sim19.4$ mag.  This is our $K_s$ sensitivity limit which we\ndefine in our photometric analysis below (see \\S3.2.1).  Using the\n$0\\farcs5$ rms positional uncertainty of our frame-tie, we defined\ncircles with 2$\\sigma$ and 3$\\sigma$ radii around each {\\it Chandra}\nX-ray source where we searched for IR counterparts (Figure 1).  If an\nIR source lay within a $1\\farcs0$ radius ($2\\sigma$) of an X-ray\nsource, we labeled these counterparts as ``strong''.  Those IR sources\nbetween $1\\farcs0$ and $1\\farcs5$ ($2-3 \\sigma$) from an X-ray source\nwe labeled as ``possible'' counterparts.  We found a total of 13\nstrong and 6 possible counterparts to X-ray sources in the Antennae.\nThese sources are listed in Table 3 and shown in Figure 3.  Of the 19\nX-ray sources with counterparts, two are the nuclei \\citep{zez02a},\none is a background quasar \\citep{cla05}, and two share the same IR\ncounterpart.  Therefore, in our analysis of cluster properties, we\nonly consider the 15 IR counterparts that are clusters.  (While\nX-42 has two IR counterparts, we chose the closer, fainter cluster for\nour analysis.)\n\nWe then attempted to estimate the level of ``contamination'' of these\nsamples due to chance superposition of unrelated X-ray sources and IR\nclusters.  This estimation can be significantly complicated by the\ncomplex structure and non-uniform distribution of both X-ray sources\nand IR clusters in the Antennae, so we developed a simple, practical\napproach.  Given the $<0\\farcs5$ rms residuals in our relative\nastrometry for sources in Table 2, we assume that any IR clusters\nlying in a background annulus with radial size of\n$2\\farcs0$--$3\\farcs0$ ($4-6\\sigma$) centered on all X-ray source\npositions are chance alignments, with no real physical connection (see\nFigure 1).  Dividing the total number of IR sources within the\nbackground annuli of the 49 X-ray source positions by the total area\nof these annuli, we find a background IR source surface density of\n0.02 arcsecond$^{-2}$ near {\\it Chandra} X-ray sources.  Multiplying\nthis surface density by the total area of all ``strong'' regions\n($1\\farcs0$ radius circles) and ``possible'' regions ($1\\farcs0$ --\n$1\\farcs5$ annuli) around the 49 X-ray source positions, we estimated\nthe level of source contamination contributing to our ``strong'' and\n``possible'' IR counterpart candidates.  We expect two with a\n$1\\sigma$ uncertainty of +0.2\/-0.1\\footnote{Found using confidence\nlevels for small number statistics listed in Tables 1 and 2 of\n\\citet{geh86}.} of the 13 ``strong'' counterparts to be due to chance\nsuperpositions, and three with a $1\\sigma$ uncertainty of\n+0.5\/-0.3\\footnotemark[\\value{footnote}] of the six ``possible''\ncounterparts to be chance superpositions.\n\nThis result has several important implications.  First of all, it is\nclear that we have a significant excess of IR counterparts within\n$1\\farcs0$ of the X-ray sources -- 13, where we expect only two in the\nnull hypothesis of no physical counterparts.  Even including the\n``possible'' counterparts out to $1\\farcs5$, we have a total of 19\ncounterparts, where we expect only five from chance superposition.\nSecondly, this implies that for any given ``strong'' IR counterpart,\nwe have a probability of $\\sim$ 85\\% ($11\/13$ with a $1\\sigma$\nuncertainty of 0.3\\footnotemark[\\value{footnote}]) that the\nassociation with an X-ray source is real.  Even for the ``possible''\ncounterparts, the probability of true association is $\\sim$50\\%. These\nlevels of certainty are a tremendous improvement over the\nX-ray\/optical associations provided by \\citet{zez02b}, and are strong\nmotivators for follow-up multi-wavelength studies of the IR\ncounterparts.  Finally, we can also conclude from strong concentration\nof IR counterparts within $\\sim 1 \\arcsec$ of X-ray sources that the\nframe tie uncertainty estimates described above are reasonable.\n\nFigure 2 is a $4\\farcm3\\times4\\farcm3$ $K_s$ image of the Antennae\nwith X-ray source positions overlaid.  We designate those X-ray\nsources with counterparts using red circles.  Notice that those\nsources with counterparts lie in the spiral arms and bridge region of\nthe Antennae.  Since these regions are abundant in star formation,\nthis seems to indicate many of the X-ray sources in the Antennae are\ntied to star formation in these galaxies.\n\n\\subsection{Photometric Properties of the IR Counterparts}\n\n\\subsubsection{Color Magnitude Diagrams}\n\nUsing the 219 clusters that had both $J$ and $K_s$ photometry, we made\n$(J-K_s)$ versus $K_s$ color magnitude diagrams (Figure 4).  We\nestimated our sensitivity limit by first finding all clusters with\nsignal-to-noise $\\sim5\\sigma$.  The mean $J$ and $K_s$ magnitudes for\nthese clusters were computed separately and defined as cutoff values\nfor statistical analyzes.  This yielded 19.0 mag in $J$ and 19.4 mag\nin $K_s$.  We note that the X-ray clusters are generally bright in the\nIR compared to the general population of clusters.  While the IR\ncounterpart for one X-ray source (X-32) falls below our $J$-band\nsensitivity limit, its $K_s$ magnitude is still above our $K_s$\ncutoff.  Therefore, we retained this source in our analysis.\n\nWe then broke down the X-ray sources into three luminosity classes\n(Figure 4).  We took the absorption-corrected X-ray luminosities,\n$L_X$, as listed in Table 1 of \\citet{zez02a} for all sources of\ninterest.  These luminosities assumed a distance to the Antennae of 29\nMpc.  We used 19.3 Mpc (for $H_{0}$=75 km s$^{-1}$ Mpc$^{-1}$) instead\nand so divided these values by 2.25 as suggested in \\citet{zez02a}.\nWe defined the three X-ray luminosities as follows: Low Luminosity\nX-ray sources (LLX's) had $L_{X}$ $<$ 3$\\times$$10^{38}$ ergs\ns$^{-1}$, High Luminosity X-ray sources (HLX's) were between $L_{X}$\nof 3$\\times10^{38}$ergs s$^{-1}$ and 1$\\times10^{39}$ ergs\ns$^{-1}$, while $L_{X}$ $>$ 1$\\times10^{39}$ ergs s$^{-1}$ were\nUltra-Luminous X-ray Sources (ULX's).  In Figure 4 we designate each\nIR counterpart according to the luminosity class of its corresponding\nX-ray source.  There does not appear to be a noticeable trend in the\nIR cluster counterparts between these different groupings.\n\n\\subsubsection{Absolute K Magnitudes}\n\nTo further study the properties of these IR counterparts, we\ncalculated $M_{K_s}$ for all IR clusters.  We calculated reddening\nusing the observed colors, $(J-K_s)_{obs}$, (hence forth the ``color\nmethod'').  Assuming all clusters are dominated by O and B stars,\ntheir intrinsic $(J-K_s)$ colors are $\\sim$0.2 mag.  Approximating\nthis value as 0 mag, this allowed us to estimate $A_{K_s}$ as $\\simeq$\n$(J-K_s)_{obs}$\/1.33 using the extinction law defined in\n\\citet[hereafter CCM]{car89}.  Since these derived reddenings are\nbiased towards young clusters, they will lead to an overestimate of\n$M_{K_s}$ for older clusters.\n\nFor IR counterparts to X-ray sources, we also computed X-ray-estimated\n$A_{K_s}$ using the column densities, $N_{H}$, listed in Table 5 of\n\\citet{zez02a}.  Here, $N_{H}$ is derived by fitting both a Power Law\n(PL) and Raymond-Smith \\citep[RS][]{ray77} model to the X-ray spectra.\nUsing the CCM law, $A_{K_s}$ is defined as 0.12$A_{V}$.  Taking\n$A_{V}$ = $5\\times10^{-22}$ mag cm$^2$ $N_{H}$, we could then derive\n$A_{K_s}$.\n\nThen we compared $A_{K_s}$ calculated using the ``color method'' to\n$A_{K_s}$ found using the above two $N_{H}$ models.  We found the\n``color method'' matched closest to $N_{H}(PL)$ for all except one\n(the cluster associated with {\\it Chandra} source 32 as designated in\n\\citet{zez02b}).\n\nIn Figure 5, we plot histograms of the distribution of $K_s$-band\nluminosity, $M_{K_s}$.  Figure 5 displays all clusters as well as over\nplotting only those with X-ray counterparts.  Notice that the clusters\nwith associated X-ray sources look more luminous.  To study whether\nthis apparent trend in luminosity is real, we compared these two\ndistributions using a two-sided Kolmogorov-Smirnov (KS) test.  In our\nanalysis, we only included clusters below $M_{K_s}$ $<$ -13.2 mag.\nRestricting our study to sources with ``good'' photometry, we first\ndefined a limit in $K_s$, 18.2 mag, using the limiting $J$ magnitude,\n19.0 mag as stated above, and, since the limit in $K_s$ is a function\nof cluster color, the median $(J-K_s)$ of 0.8 mag.  Subtracting the\ndistance modulus to the Antennae, 31.4 mag, from this $K_s$ limit, we\ncomputed our cutoff in $M_{K_s}$.  Since all clusters with X-ray\nsources fall below this cutoff, our subsample is sufficient to perform\na statistical comparison.\n\nThe KS test yielded a D-statistic of 0.37 with a probability of\n$3.2\\times10^{-2}$ that the two distributions of clusters with and\nwithout associated X-ray sources are related.  Considering the\nseparate cluster populations as two probability distributions, each\ncan be expressed as a cumulative distribution.  The D-statistic is\nthen the absolute value of the maximum difference between each\ncumulative distribution.  This test indicates that those clusters with\nX-ray counterparts are more luminous than most clusters in the\nAntennae.\n\n\\subsubsection{Cluster Mass Estimates}\n\n\\citet{whi99} found 70\\% of the bright clusters observed with the {\\it\nHubble Space Telescope} have ages $<$20 Myr.  Therefore, in this study\nwe will assume all clusters are typically the same age, $\\sim$20 Myr.\nThis allows us to make the simplifying assumption that cluster mass is\nproportional to luminosity and ask: Does the cluster mass affect the\npropensity for a given progenitor star to produce an X-ray binary?  We\nestimated cluster mass using $K_s$ luminosity ($M_{K_s}$).  Since\ncluster mass increases linearly with flux (for an assumed constant age\nof all clusters), we converted $M_{K_s}$ to flux.  Using the data as\nbinned in the $M_{K_s}$ histogram (Figure 5), we calculated an average\nflux per bin.  By computing the fraction of number of clusters per\naverage flux, we are in essence asking what is the probability of\nfinding a cluster with a specific mass.  Since those clusters with\nX-ray sources are more luminous, we expect a higher probability of\nfinding an X-ray source in a more massive cluster.  As seen in Figure\n6, this trend does seem to be true.  Applying a KS-test between the\ndistributions for all clusters and those associated with X-ray sources\nfor clusters below the $M_{K_s}$ completeness limit defined in the\nprevious section, we find a D-statistic of 0.66 and a probability of\n$7.2\\times10^{-3}$ that they are the same.  Hence, the two\ndistributions are distinct, indicating it is statistically significant\nthat more massive clusters tend to contain X-ray sources.\n\nWhile we assume all clusters are $\\sim$20 Myr above, we note that the\nactual range in ages is $\\sim$1--100 Myr \\citep{whi99}.\nBruzual-Charlot spectral photometric models \\citep{bru03} indicate\nthat clusters in this age range could vary by a factor of roughly 100\nin mass for a given $K_s$ luminosity.  Thus, we emphasize that the\nanalysis above should be taken as suggestive rather than conclusive\nevidence, and note that in a future paper (Clark, et al. 2006, in\npreparation) we explore this line of investigation and the impacts of\nage variations on the result in depth.\n\n\\subsubsection{Non-detections of IR Counterparts to X-ray Sources}\n\nTo assess whether our counterpart detections were dependent of\nreddening or their intrinsic brightness, we found limiting values for\n$M_{K_s}$ for those X-ray sources without detected IR counterparts.\nWe achieved this by setting all clusters $K_s$ magnitudes equal to our\ncompleteness limit defined for the CMDs (19.4 mag; see \\S3.2.1) and\nthen finding $M_{K_s(lim)}$ for each using $A_{K_s}$ calculated for\nthat cluster.  Since $M_{K_s(lim)}$ is theoretical and only depends on\nreddening, we could now find this limit for all X-ray sources using an\n$A_{K_s}$ estimated from the observed $N_{H}$ values.  Thus we\nconsidered all IR counterparts (detections) and those X-ray sources\nwithout a counterpart (nondetections).  If nondetections are due to\nreddening there should not exist a difference in $M_{K_s(lim)}$\nbetween detections and nondetections.  In contrast, if nondetections\nare intrinsically fainter, we expect a higher $M_{K_s(lim)}$ for these\nsources.  In the case of detections, we considered reddening from both\nthe ``color method'' and the $N_{H}(PL)$ separately.  We could only\nderive nondetections using $N_{H}(PL)$ reddening.  Figure 7 shows\n$M_{K_s(lim)}$ appears higher for all nondetections. To test if this\nobservation is significant, we applied a KS-test to investigate\nwhether detections and nondetections are separate distributions.  We\nfind a D-statistic of 0.82 and probability of $8.8\\times10^{-6}$ that\nthese two distributions are the same using the ``color method'' for\ndetections. Considering the $N_{H}(PL)$ reddening method for\ndetections instead, the D-statistic drops to 0.48 and the probability\nincreases to $3.9\\times10^{-2}$.  Since both tests indicate these\ndistributions are distinct, the observed high $M_{K_s(lim)}$ for\nnondetections seems to be real.  This leads to the conclusion that\nthese sources were undetected because they are intrinsically IR-faint,\nand that reddening does not play the dominant role in nondetections.\n\nWe summarize these statistics in Table 4.  Here we calculated the mean\n$K_s$, $(J-K_s)$, and $M_{K_s}$ for three different categories: 1) all\nclusters, 2) clusters only connected with X-ray sources, and 3) these\nclusters broken down by luminosity class.  We also include\nuncertainties in each quantity.  Notice that the IR counterparts\nappear brighter in $K_s$ and intrinsically more luminous than most\nclusters in the Antennae, although there is no significant trend in\ncolor.  We also summarize the above KS-test results in Table 5.\n\n\\section{Conclusions}\n\nWe have demonstrated a successful method for finding counterparts to\nX-ray sources in the Antennae using IR wavelengths.  We mapped {\\it\nChandra} X-ray coordinates to WIRC pixel positions with a positional\nuncertainty of $\\sim 0\\farcs5$.  Using this precise frame-tie we\nfound 13 ``strong'' matches ($< 1\\farcs0$ separation) and 5\n``possible'' matches ($1 - 1\\farcs5$ separation) between X-ray\nsources and IR counterparts.  After performing a spatial and\nphotometric analysis of these counterparts, we reached the following\nconclusions:\n\n1.  We expect only 2 of the 13 ``strong'' IR counterparts to be chance\nsuperpositions.  Including all 19 IR counterparts, we estimated 5 are\nunrelated associations.  Clearly, a large majority of the X-ray\/IR\nassociations are real.\n\n2.  The IR counterparts tend to reside in the spiral arms and bridge\nregion between these interacting galaxies.  Since these regions\ncontain the heaviest amounts of star formation, it seems evident that\nmany of the X-ray sources are closely tied to star formation in this\npair of galaxies.\n\n3.  A $K_s$ vs. $(J - K_s)$ CMD reveal those clusters associated with\nX-ray sources are brighter in $K_s$ but there does not seem to be a\ntrend in color.  Separating clusters by the X-ray luminosity classes\nof their X-ray counterpart does not reveal any significant trends.\n\n4.  Using reddening derived $(J - K_s)$ colors as well as from\nX-ray-derived $N_H$, we found $K_s$-band luminosity for all clusters.\nA comparison reveals those clusters associated with X-ray sources are\nmore luminous than most clusters in the Antennae.  A KS-test indicates\na significant difference between X-ray counterpart clusters and the\ngeneral population of clusters.\n\n5.  By relating flux to cluster luminosity, simplistically assuming a\nconstant age for all clusters, we estimated cluster mass.  Computing\nthe fraction of number of clusters per average flux, we estimated the\nprobability of finding a cluster with a specific mass.  We find more\nmassive clusters are more likely to contain X-ray sources, even after\nwe normalize by mass.\n\n6.  We computed a theoretical, limiting $M_{K_s}$ for all counterparts\nto X-ray sources in the Antennae using X-ray-derived reddenings.\nComparing detections to non-detections, we found those clusters with\nX-ray source are intrinsically more luminous in the IR.\n\nIn a future paper exploring the effects of cluster mass on XRB\nformation rate (Clark, et al. 2006a, in preparation), we will\ninvestigate the effects of age on cluster luminosity and hence our\ncluster mass estimates.  Another paper will extend our study of the\nAntennae to optical wavelengths (Clark, et al. 2006b, in preparation).\nThrough an in depth, multi-wavelength investigation we hope to achieve\na more complete picture of counterparts to several X-ray sources in\nthese colliding galaxies.\n\n\\acknowledgments\n\nThe authors thank the staff of Palomar Observatory for their excellent\nassistance in commissioning WIRC and obtaining these data.  WIRC was\nmade possible by support from the NSF (NSF-AST0328522), the Norris\nFoundation, and Cornell University.  S.S.E. and D.M.C. are supported\nin part by an NSF CAREER award (NSF-9983830).  We also thank\nJ.R. Houck for his support of the WIRC instrument project.\n\n\\vfill \\eject\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWe consider realizations of linear dynamical systems that are variously characterized as positive real, passive, or port-Hamiltonian.\nWe restrict ourselves to \\emph{linear time-invariant systems} represented as\n\\begin{equation} \\label{statespace}\n \\begin{array}{rcl} \\dot x & = & Ax + B u \\quad \\mbox{with}\\ x(0)=0,\\\\\ny&=& Cx+Du,\n\\end{array}\n\\end{equation}\nwhere $u:\\mathbb R\\to\\mathbb{C}^m$,   $x:\\mathbb R\\to \\mathbb{C}^n$,  and  $y:\\mathbb R\\to\\mathbb{C}^m$  are vector-valued functions denoting, respectively, the \\emph{input}, \\emph{state},\nand \\emph{output} of the system. The coefficient matrices $A\\in \\mathbb{C}^{n \\times n}$,   $B\\in \\mathbb{C}^{n \\times m}$, $C\\in \\mathbb{C}^{m \\times n}$, and  $D\\in \\mathbb{C}^{m \\times m}$ are constant.\nReal and complex $n$-vectors ($n\\times m$ matrices) are denoted by $\\mathbb R^n$, $\\mathbb C^{n}$\n($\\mathbb R^{n \\times m}$, $\\mathbb{C}^{n \\times m}$), respectively.\nWe refer to (\\ref{statespace}) concisely in terms of a four-tuple of matrices describing the realization ${\\mathcal M}:=\\left\\{A,B,C,D\\right\\}$.\n\nOur principal focus is on the structure of \\emph{passive} systems and relationships with positive-real transfer functions and port-Hamiltonian system representations.\nA nice introduction to passive systems can be found in the seminal papers of Willems (\\cite{Wil71},  \\cite{Wil72a}, \\cite{Wil72b}), where a general notion of system passivity is introduced and linked to related system-theoretic notions such as positive realness and stability. Willems refers to the earlier works of Kalman \\cite{Kal63}, Popov \\cite{Pop73}, and Yakubovich \\cite{Yak62}, where versions of what are now called Kalman-Yakubovich-Popov (KYP) conditions were derived.\nRenewed interest in these ideas came from the study of port-Hamiltonian (pH) systems, which may be viewed as particular parameterizations of passive systems that arise from certain energy-based modeling frameworks (see e.g. \\cite{Sch04}, \\cite{Sch13}, \\cite{SchJ14}). The KYP conditions lead to characterizations of system passivity through the solution set of an associated \\emph{linear matrix inequality (LMI)}.  The convexity of this solution set has led to extensive use of\nconvex optimization techniques in systems and control theory (see e.g. \\cite{BoyB90}, \\cite{NesN94}).\n\nThe solution set of the KYP-LMI leads to a natural parametrization of families of pH realizations for a given passive system.\nWith this observation, it is not surprising that some pH realizations of a given system reflect well the underlying robustness of passivity to system perturbations and that some pH realizations will do this better than others.  Our main result shows that the analytic center of certain barrier functions associated with the KYP-LMI leads to favorable pH realizations in this sense; we derive computable bounds for the passivity\nradii for these realizations.\n\nThe paper is organized as follows. In Section~\\ref{sec:prelim}, we recall the KYP conditions and link the solution set of the KYP-LMI to different system realizations that reflect passivity. In Section~\\ref{sec:analytic} we review some basic concepts in convex analysis\n and introduce the concept of an \\emph{analytic center} associated with a barrier function for the KYP-LMI.\nIn Section~\\ref{sec:passrad} we define the passivity radius for a model realization ${\\mathcal M}$, measuring its robustness to system perturbations that may lead to a loss of passivity.  We show that the analytic center of the KYP-LMI yields a model representation with good robustness properties. In Section~\\ref{sec:criteria} we consider other measures that could also serve as criteria for robustness of model passivity.\nIn Section~\\ref{sec:num} we illustrate our analytic results with a few numerical examples. We conclude with Section~\\ref{sec:conclusion},\noffering also some points for further research.\n\n\\section{Positive-realness, passivity, and\n\\hfill \\break \\mbox{port-Hamiltonian systems} }\\label{sec:prelim}\n\n\nWe restrict ourselves to linear time-invariant systems as in \\eqref{statespace} which are \\emph{minimal}, that is, the pair $(A,B)$ is \\emph{controllable}  (for all $s\\in \\mathbb C$, $\\rank \\mbox{\\small $[\\,s I-A \\quad B\\,]$} =n$ ), and the pair $(A,C)$ is \\emph{observable} ($(A^\\mathsf{H},C^\\mathsf{H})$ is controllable). Here, the Hermitian (or conjugate) transpose (transpose) of a vector or matrix $V$ is denoted by\n$V^{\\mathsf{H}}$ ($V^{mathsf{T}}$).  We also assume that $\\rank B=\\rank C=m$.\nWe require input\/output port dimensions to be equal  ($m$) and for convenience we assume that the system is initially in a quiescent state, $x(0)=0$. By applying the Laplace transform to \\eqref{statespace} and eliminating the state, we obtain the  \\emph{transfer function}\n\\begin{equation} \\label{ABCD}\n\\mathcal T(s):=D+C(sI_n-A)^{-1}B,\n\\end{equation}\nmapping the Laplace transform of $u$ to the Laplace transform of $y$.\n$I_n$ is the identity matrix in $\\mathbb{C}^{n \\times n}$\n(subsequently, the subscript may be omitted if the dimension is clear from the context).\nOn the imaginary axis $\\imath \\mathbb R$, $\\mathcal T(\\imath\\omega)$\ndescribes the \\emph{frequency response} of the system.\n\n\nWe denote the set of Hermitian matrices in $\\mathbb{C}^{n \\times n}$ by $\\Hn$.\nPositive definiteness (semidefiniteness) of  $A\\in \\Hn$ is denoted by $A>0$ ($A\\geq 0$).\nThe set of all positive definite (positive semidefinite) matrices in $\\Hn$ is denoted $\\Hnpd$ ($\\Hnpsd$).  The real and imaginary parts of a complex matrix, $Z$, are written as ${\\mathfrak{Re}} (Z)$ and ${\\mathfrak{Im}} (Z)$, respectively.\n\nWe proceed to review briefly some representations of linear systems associated with the notion of passivity.\n\\subsection{Positive-real systems}\\label{sec:posreal}\nConsider a system ${\\mathcal M} $ as in (\\ref{statespace}) and its transfer function $\\mathcal T$ as in \\eqref{ABCD}.\n\\begin{definition}\\label{def:posreal}\nA transfer function $\\mathcal T(s)$ is {\\em positive real}\nif the matrix-valued rational function\n\\begin{equation}\\label{defphi}\n\\Phi(s):= \\mathcal T^{\\mathsf{H}}(-s) + \\mathcal T(s)\n\\end{equation}\nis positive semidefinite for $s$ on the imaginary axis:\n$$\n\\Phi(\\imath\\omega)\\in \\Hmpsd \\quad \\mbox{ for all }\\omega\\in \\mathbb{R}.\n$$\n$\\mathcal T(s)$ is \\emph{strictly positive real} if $\\Phi(\\imath \\omega)\\in \\Hmpd $ for all $\\ \\omega\\in \\mathbb{R}$.\n\\end{definition}\nFor any $X \\in \\Hn$, define the matrix function\n\\begin{eqnarray*} \\label{prls}\nW(X) &:=& \\left[\n\\begin{array}{cc}\n-X\\,A - A^{\\mathsf{H}}X & C^{\\mathsf{H}} - X\\,B \\\\\nC- B^{\\mathsf{H}}X & D+D^{\\mathsf{H}}\n\\end{array}\n\\right]\\\\\n& =& W(0)- \\left[\n\\begin{array}{cc}\nX\\,A + A^{\\mathsf{H}}X &  X\\,B \\\\\nB^{\\mathsf{H}}X & 0\n\\end{array}\n\\right].\n\\end{eqnarray*}\n From \\eqref{ABCD} and (\\ref{defphi}), simple manipulations produce\n\\begin{eqnarray*}\n\\Phi\\,(s) &= &\n\\left[ \\begin{array}{cc} B^{\\mathsf{H}}(-s\\,I_n - A^{\\mathsf{H}})^{-1} & I_m  \\end{array} \\right]\n\\, W(0) \\left[ \\begin{array}{c} (s\\,I_n -A)^{-1}B \\\\ I_m \\end{array} \\right] \\nonumber \\\\\n & =&\n\\left[ \\begin{array}{cc} B^{\\mathsf{H}}(-s\\,I_n - A^{\\mathsf{H}})^{-1} & I_m  \\end{array} \\right]\n\\, W(X) \\left[ \\begin{array}{c} (s\\,I_n -A)^{-1}B \\\\ I_m \\end{array} \\right]. \\label{popovs}\n\\end{eqnarray*}\nDefine the \\emph{matrix pencils}:\n\\[\n\\mathcal L_{0}(s) =\ns \\left[ \\begin{array}{cc|c} 0 & I_n & 0\\\\[1mm]\n\t-I_n & 0 & 0\\\\[1mm]\n\t\\hline \\rule{0mm}{1.1em}\n        0 & 0 & 0 \\end{array} \\right]\n\t-\\left[ \\begin{array}{cc|c} 0 & A & B \\\\[1mm]\n\tA^{\\mathsf{H}} & 0 & C^{\\mathsf{H}} \\\\[1mm]\n        \\hline \\rule{0mm}{1.1em}\n         B^{\\mathsf{H}} & C & D+D^{\\mathsf{H}}  \\end{array} \\right],\n\\]\nand, for any $X\\in \\Hn$,\n\\[\n\\mathcal L_{X}(s) =\n\\left[ \\begin{array}{cc|c} I_n & 0 & 0 \\\\\n-X & I_n & 0 \\\\ \\hline 0 & 0 & I_m  \\end{array} \\right] \\mathcal L_{0}(s)\n\\left[ \\begin{array}{cc|c} I_n & -X & 0 \\\\\n0 & I_n & 0 \\\\ \\hline 0 & 0 & I_m  \\end{array} \\right],\n\\]\nObserve for any $X\\in \\Hn$, $\\mathcal L_{X}(s)$ and $\\mathcal L_{0}(s)$ are equivalent pencils, since they are related\nvia a congruence transformation. Note also that $\\Phi(s)$ is\nthe \\emph{Schur complement} associated with the $(3,3)$ block of $\\mathcal L_{0}(s)$\n(and hence, also of $\\mathcal L_{X}(s)$ for any $X\\in \\Hn$).\n\nIf $\\mathcal T(s)$ is  positive real,\nthen it is known \\cite{Wil71} that there exists $X\\in \\Hn$ such that the KYP-LMI holds, namely\n\\begin{equation} \\label{KYP-LMI}\nW(X) \\geq 0,\n\\end{equation}\nand so a factorization must exist:\n\\begin{equation} \\label{lw}\nW(X) =\n\\left[ \\begin{array}{c} L^{\\mathsf{H}} \\\\ M^{\\mathsf{H}} \\end{array} \\right]\\,\n\\left[ \\begin{array}{cc} L &  M \\end{array} \\right]\n\\end{equation}\nfor $L\\in \\mathbb{C}^{r \\times n}$ and $M \\in \\mathbb{C}^{r \\times m} $,\n where $r=\\rank W(X)$.\n  Introducing $\\mathcal G(s)= L(s\\,I_n - A)^{-1}B + M,$~one may then define the \\emph{spectral factorization} of $\\Phi(s)$ as\n\\begin{equation} \\label{spectral}\n\\Phi(s)= \\mathcal G^{\\mathsf{H}}(- s) \\mathcal G(s).\n\\end{equation}\nDefine the solution set and subsets to the KYP-LMI (\\ref{KYP-LMI}):\n\\begin{subequations}\\label{LMIsolnsets}\n\\begin{align}\n&{\\mathbb X}:=\\left\\{ X\\in \\Hn \\left|\\  W(X) \\geq 0 \\right.\\right\\}, \\label{XsolnWpsd} \\\\[1mm]\n&\\XWpd :=\\left\\{ X\\in \\Hn \\left|   W(X) \\geq 0,\\ X >0 \\right.\\right\\} = \\Hnpd \\cap {\\mathbb X}, \\label{XpdsolnWpsd} \\\\[1mm]\n&\\XWpdpd :=\\left\\{ X\\in \\Hn \\left|   W(X) > 0,\\ X >0 \\right.\\right\\}, \\label{XpdsolWpd}\n\\end{align}\n\\end{subequations}\nFor each $X\\in{\\mathbb X}$, there is a factorization of $W(X)$ of the form (\\ref{lw}) leading to a spectral factorization (\\ref{spectral}) of $\\Phi(s)$. We are mainly interested in $\\XWpd$ and $\\XWpdpd$, which are\nrespectively,  the set of positive-definite solutions to the KYP-LMI (\\ref{KYP-LMI}), and the subset of those solutions for which the KYP-LMI (\\ref{KYP-LMI}) holds \\emph{strictly}.\n\nAn important subset of ${\\mathbb X}$ are those solutions to (\\ref{KYP-LMI}) for which the\nrank $r$ of $W(X)$ is minimal ({i.e.}, for which $r=\\rank\\Phi(s)$). Let $S:= D+D^{\\mathsf{H}}=\\lim_{s\\rightarrow\\infty}\\Phi(s)$.\nIf $S$ is nonsingular, then\nthe minimum rank solutions in $\\XWpd$\nare those for which $\\rank W(X) = \\rank S  = m$, which in turn is the case\nif and only if the Schur complement of $S$ in $W(X)$ is zero.  This Schur\ncomplement is associated with the \\emph{algebraic Riccati equation (ARE)}:\n\\begin{multline}\n\\mathsf{Ricc}(X) := -XA-A^{\\mathsf{H}}X  \\\\ -(C^{\\mathsf{H}}-XB)S^{-1}(C-B^{\\mathsf{H}}X)=0.\\label{riccati}\n\\end{multline}\nSolutions to (\\ref{riccati}) produce directly a spectral factorization of $\\Phi(s)$,\nIndeed,  each solution $X$ of~\\eqref{riccati} corresponds to a\n\\emph{Lagrangian invariant subspace} spanned by the columns of $U:=\\mat{cc} I_n & -X^{\\mathsf{T}} \\rix^{\\mathsf{T}} $\nthat remains invariant under the action of the Hamiltonian matrix\n\\begin{equation}\\label{HamMatrix}\nH:=\\mat{cc} A-B S^{-1} C & - B S^{-1} B^{\\mathsf{H}} \\\\\nC^{\\mathsf{H}} S^{-1} C & -(A-B S^{-1} C)^{\\mathsf{H}} \\rix.\n\\end{equation}\n$U$ satisfies $HU=U A_F$ for a \\emph{closed loop matrix} $A_F=A-BF$ with $F := S^{-1}(C-B^{\\mathsf{H}}X)$ (see e.g., \\cite{FreMX02}).\nEach solution $X$ of~\\eqref{riccati}  could also be associated with an \\emph{extended Lagrangian invariant subspace}\nfor the pencil $\\mathcal{L}_{0}(s)$ (see \\cite{BenLMV15}), spanned by the columns of\n$ \\widehat{U}:=\\mat{ccc} -X^{\\mathsf{T}}\n& I_n & -F^{\\mathsf{T}}  \\rix^{\\mathsf{T}}$.\n In particular, $\\widehat{U}$ satisfies\n\\[\n\\left[ \\begin{array}{ccc} 0 & A & B \\\\\n\tA^{\\mathsf{H}} & 0 & C^{\\mathsf{H}} \\\\  B^{\\mathsf{H}} & C & S  \\end{array} \\right] \\widehat{U}\n  =\\left[ \\begin{array}{ccc} 0 & I_n & 0\\\\\n\t-I_n & 0 & 0\\\\ 0 & 0 & 0 \\end{array} \\right] \\widehat{U} A_F,\n\\]\nsee also {e.g.} \\cite{IonOW98,Wil71}.   The condition that $S$ is invertible is equivalent\nto the condition that the pencil $\\mathcal{L}_{0}(s)$ has \\emph{differentiation index one},\n{i.e.}, all eigenvalues at $\\infty$ are semi-simple, \\cite{KunM06}.\nIf $\\mathcal{L}_{0}(s)$ has no purely imaginary eigenvalues, then there are ${2n}\\choose{n}$ solutions $X\\in \\Hn$ of (\\ref{riccati}), each associated with an appropriate choice of a Lagrangian invariant subspace for $\\mathcal{L}_{0}(s)$. Every choice leads to different spectra for the closed loop matrix, $A_F$ (see \\cite{FreMX02,Wil71} for a parametrization of all possible Lagrangian subspaces).\nAmong the possible solutions of (\\ref{riccati}) there are two extremal solutions, $X_-$ and $X_+$.  $X_-$ leads to a closed loop matrix, $A_F$, with spectra in the (open) left half-plane; $X_+$ leads to $A_F$ with spectra in the (open) right half-plane.  All solutions $X$ of (\\ref{riccati}) are bracketed by $X_-$ and $X_+$:\n\\begin{equation} \\label{Xbounded}\n0\\leq X_- \\leq X \\leq X_+\n\\end{equation}\nand so, in this special case the set ${\\mathbb X}$ is bounded, but it may be empty or the solution may be unique if $X_-=X_+$, see Section~\\ref{sec:analytic}.\n\n\\subsection{Passive systems}\\label{sec:DisSys}\n\\begin{definition}\\label{def:passive}\nA system ${\\mathcal M} :=\\left\\{A,B,C,D\\right\\}$ is \\emph{passive} if there exists a state-dependent\n\\emph{storage function}, $\\mathcal H(x)\\geq 0$, such that for any $\\mu,t_0\\in \\mathbb R$ with $\\mu>t_0$,\n the \\emph{dissipation inequality} holds:\n\\begin{equation} \\label{supply} \\mathcal H(x(\\mu))-\\mathcal H(x(t_0)) \\le \\int_{t_0}^{\\mu} {\\mathfrak{Re}} (y(t)^{\\mathsf{H}}u(t)) \\, dt\n\\end{equation}\nIf for all $\\mu>t_0$, the inequality in \\eqref{supply}\nis strict then the system is \\emph{strictly passive}.\n\\end{definition}\nIn the terminology of (\\cite{Wil72a}),\n${\\mathfrak{Re}} (y(t)^{\\mathsf{H}}u(t))$ is the \\emph{supply rate} of the system.\nA general theory of dissipative systems (of which passive systems are a special case) was developed in the seminal papers \\cite{Wil71,Wil72a,Wil72b}, where links to earlier work by Kalman, Popov, and Yakubovich and the KYP-LMI (\\ref{KYP-LMI}) are given.  Note that the original definition of passivity given by Willems was for real systems; we reformulate it here for complex systems.\n\\begin{theorem}[\\cite{Wil71}]\\label{LMIWil}\nSuppose the system ${\\mathcal M}$ of (\\ref{statespace}) is minimal. Then the KYP-LMI (\\ref{KYP-LMI}): $W(X)\\geq 0$,\nhas a solution $X\\in \\Hnpd$ if and only if ${\\mathcal M}$ is a passive system. If this is the case, then\n\\begin{itemize}\n\\item $\\mathcal H(x):=\\frac{1}{2}x^{\\mathsf{H}}Xx$ defines a storage function associated with the supply rate ${\\mathfrak{Re}}(y^\\mathsf{H}u)$ satisfying the dissipation inequality \\eqref{supply};\\\\\n\\item there exist maximal and minimal solutions $X_- \\leq X_+$  in $\\Hnpd$ of \\eqref{KYP-LMI},\nsuch that for all solutions, $X$, of \\eqref{KYP-LMI}:\n    %\n\\[\n 0 < X_- \\leq X \\leq X_+.\n\\]\n\\end{itemize}\n\\end{theorem}\nRecall that a matrix $A\\in \\mathbb{C}^{n \\times n}$  is \\emph{asymptotically stable} if all its eigenvalues are in the open left half plane and \\emph{(Lyapunov) stable} if all eigenvalues are in the closed left half plane with any eigenvalues occurring on the imaginary axis being semisimple.\nTheorem~\\ref{LMIWil} asserts that if $X>0$  is a solution of $W(X)\\geq 0$, then the system ${\\mathcal M}$ of \\eqref{statespace} is stable and if it satisfies $W(X)> 0$, then it is asymptotically stable, since $\\mathcal H(x)$ is a Lyapunov function for ${\\mathcal M}$ which is strict if $W(X)> 0$, (see {e.g.} \\cite{LanT85}). Note, however, that  for (asymptotic) stability of $A$ it is sufficient if the $(1,1)$ block of $W(X)$ is (positive definite) positive semidefinite.\n\\begin{corollary}\\label{cor:pr}\nConsider a minimal system ${\\mathcal M} $ as in (\\ref{statespace}). ${\\mathcal M}$ is passive if and only if it is positive real\nand stable. It is strictly passive if and only if it is strictly positive real and asymptotically stable. In the latter case, $X_+ -X_- >0$ (\\cite{Wil71}).\n\\end{corollary}\nNote that minimality is not necessary for passivity.  For example, the system\n$\\dot{x}=-x,y = u$  is both stable and passive but not minimal.  In this case, the KYP-LMI (\\ref{KYP-LMI}) is satisfied with any (scalar) $X>0$, the Hamiltonian may be defined as\n$\\mathcal H(x)=\\frac{X}{2} x(t)^2$, and the dissipation inequality evidently holds since for $t_1\\geq t_0$,\n{\\small\n\\vspace{-5mm}\\begin{align*}\n\\mathcal{H}(x(t_1))-\\mathcal{H}(x(t_0)) & = \\frac{X}{2} (x(t_0) e^{-(t_1-t_0)})^2 -\\frac{X}{2} (x(t_0))^2 \\\\\n = \\frac{X}{2} (x(t_0))^2& (e^{-2(t_1-t_0)}-1) \\leq 0 \\leq \\int_{t_0}^{t_1} y(t) \\, u(t) \\, dt\n\\end{align*} }\n\\subsection{Port-Hamiltonian systems}\\label{sec:PH}\n\\begin{definition}\\label{def:ph}\nA linear time-invariant \\emph{port-Hamiltonian (pH) system} is one for which the following realization is possible:\n\\begin{equation} \\label{pH}\n \\begin{array}{rcl} \\dot x  & = & (J-R)Q x + (G-K) u,\\\\\ny&=& (G+K)^{\\mathsf{H}}Q x+Du,\n\\end{array}\n\\end{equation}\nwhere $Q=Q^{\\mathsf{H}} >0$, $J=-J^{\\mathsf{H}}$, and\n\\[\n\\left[ \\begin{array}{lc} R & K \\\\ K^{\\mathsf{H}} & \\mathsf{sym}(D) \\end{array} \\right] \\geq 0 \\quad \\mbox{with}\n\\quad \\mathsf{sym}(D)=\\frac12(D+D^{\\mathsf{H}})\n\\]\n\\end{definition}\nPort-Hamiltonian systems were introduced  in \\cite{Sch04} as a tool for energy-based modeling.  An energy storage function\n$\\mathcal{H}(x)=\\frac12x^{\\mathsf{H}}Qx$ plays a central role and under the conditions given, the dissipation inequality (\\ref{supply}) holds and so pH systems are always passive.  Conversely, any passive system may be represented as a pH system ({\\ref{pH}), see e.g.,  \\cite{BeaMX15_ppt}.\nWe briefly describe the construction of such a representation: Suppose the model ${\\mathcal M}$ of \\eqref{statespace} is minimal and passive and let $X=Q\\in \\XWpd$ be a solution of the KYP-LMI \\eqref{KYP-LMI}.\nFor this $Q$, define $J:=\\frac12 (AQ^{-1}- Q^{-1}A^{\\mathsf{H}})$, $R:=-\\frac 12 (AQ^{-1}+ Q^{-1}A^{\\mathsf{H}})$,\n$K:= \\frac12\\left(Q^{-1}C^{\\mathsf{H}}-B\\right)$, and $G:= \\frac12\\left(Q^{-1}C^{\\mathsf{H}}+B\\right)$.\nDirect substitution shows that (\\ref{statespace}) may be written in the form of (\\ref{pH}), with $J=-J^{\\mathsf{H}}$, and\n\\[\n\\left[ \\begin{array}{lc} R & K \\\\ K^{\\mathsf{H}} & \\mathsf{sym}(D) \\end{array} \\right] =\n\\frac12 \\left[ \\begin{array}{lc} Q^{-1} & 0 \\\\ 0  & I \\end{array} \\right] \\ W(Q)\\\n\\left[ \\begin{array}{lc} Q^{-1} & 0 \\\\ 0  & I \\end{array} \\right]  \\geq 0.\n\\] \n\nAnother possible representation of a passive system as a standard pH system can be obtained by using a symmetric factorization of a solution $X$ of\n\\eqref{KYP-LMI}:  $X=T^{\\mathsf{H}}T$ with $T\\in \\mathbb{C}^{n \\times n}$ (e.g., the Hermitian square root of $X$ or the Cholesky factorization of $X$ are two possibilities).\nOne defines a state-space transformation, $x_T=Tx$, leading to an equivalent realization in $T$-coordinates:\n\\[\n\\{A_T,B_T,C_T,D_T\\} := \\{TAT^{-1}, TB, CT^{-1}, D \\}\n\\]\nThe associated KYP-LMI \\eqref{KYP-LMI} with respect to the new coordinate system can be written as\n\\begin{eqnarray*} \\label{prls}\nW_T(\\hat{X}) &:=& \\left[\n\\begin{array}{cc}\n-\\hat{X}\\,A_T - A_T^{\\mathsf{H}} \\hat{X} & C_T^{\\mathsf{H}} - \\hat{X}\\,B_T \\\\\nC_T- B_T^{\\mathsf{H}} \\hat{X} & D+D^{\\mathsf{H}}\n\\end{array}\n\\right] \\geq 0,\n\\end{eqnarray*}\nbut since $X=T^{\\mathsf{H}}T$ is a solution to the KYP-LMI \\eqref{KYP-LMI} in the original state-space coordinates, we have $\\hat{X}=I$ and\n\\begin{eqnarray}  \\nonumber\nW_T(I)&=&\\left[ \\begin{array}{cccc} T^{-H} & 0\\\\ 0 & I_m\n\\end{array}\n\\right] W(X)\n\\left[ \\begin{array}{cccc} T^{-1} & 0\\\\ 0 & I_m\n\\end{array}\n\\right] \\geq 0.  \\label{PH}\n\\end{eqnarray}\nWe can then use the Hermitian and skew-Hermitian part of $A_T$ to obtain a pH representation in $T$-coordinates:\n$J_T:=\\frac12 (A_T-A_T^{\\mathsf{H}})$, $R_T:=-\\frac 12 (A_T+ A_T^{\\mathsf{H}})$,\n$K_T:= \\frac12\\left(C_T^{\\mathsf{H}}-B_T\\right)$, $G_T:= \\frac12\\left(C_T^{\\mathsf{H}}+B_T\\right)$, and $Q_T=I$, so that\n\\begin{equation} \\label{pHalt}\n \\begin{array}{rcl} \\dot{x}_T  & = & (J_T-R_T)x_T + (G_T-K_T) u,\\\\\ny&=& (G_T+K_T)^{\\mathsf{H}} x_T+Du,\n\\end{array}\n\\end{equation}\nis a valid pH representation in $T$ state-space coordinates.\n\n\nWe have briefly presented three closely related concepts for a minimal linear time-invariant system of the form \\eqref{statespace}, positive realness, passivity, and that the system has pH structure. All three properties can be characterized algebraically via the solutions of linear matrix inequalities, invariant subspaces of special even pencils, or solutions of Riccati inequalities. However, there typically is a lot of freedom in the representation of such systems. This freedom, which results from particular choices of solutions to the KYP-LMI as well as  subsequent  state space transformations, may be used to make the representation more robust\nto perturbations. In many ways the pH representation seems to be the most robust representation \\cite{MehMS16,MehMS17} and it also has many other advantages: it encodes the geometric and algebraic properties directly in the properties of the coefficients \\cite{SchJ14}; it allows easy ways for structure preserving model reduction \\cite{GugPBS12,PolV10}; it easily extends to descriptor systems \\cite{BeaMXZ17_ppt,Sch13}; and it greatly simplifies optimization methods for computing stability and passivity radii \\cite{GilMS18,GilS16,GilS17,OveV05}.\n\n\nThe remainder of this paper will deal with the question of how to make use of this freedom in the state space transformation to determine a 'good' or even `optimal' representation as a pH system. To do this we study in the next section the set of solutions of the KYP-LMI \\eqref{KYP-LMI} and in particular its \\emph{analytic center}.\n\n\\section{The analytic center of the solution set $\\mathbb{X}^>$} \\label{sec:analytic}\nSolutions of the KYP-LMI  \\eqref{KYP-LMI} and of linear matrix inequalities\nare usually obtained via optimization algorithms, see {e.g.} \\cite{BoyEFB94}.  A common approach involves defining a\na \\emph{barrier function} $b:\\mathbb{C}^{n \\times n} \\to \\mathbb C$ that is defined and finite throughout the interior of the constraint set becoming infinite as the boundary is approached, and\nthen using this function in the optimization scheme. The minimum of the barrier function\nitself is of independent interest and is called the \\emph{analytic center} of the constraint set \\cite{GenNV99}.\n\nWe have seen in Section~\\ref{sec:DisSys} that for a system ${\\mathcal M} $ that is minimal and strictly passive there  exists a (possibly large) class of state space transformations that  transform the system to pH form. This class is characterized by the set $\\XWpd$ of positive definite solutions of (the strict version) of the linear matrix inequality \\eqref{KYP-LMI}.\nIf the set $\\XWpd$ is non-empty and bounded, then the \\emph{barrier function}\n\\begin{equation*}\nb(X) := - \\log \\det W(X),\n\\end{equation*}\nis bounded from below, but becomes infinitely large when $W(X)$ becomes singular.\nThe analytic center of the domain $\\XWpd$ is\nthe minimizer of this barrier function. To characterize the analytic center, we analyze the \\emph{interior}  of the set $\\XWpd$\ngiven by\n\\begin{eqnarray*\n\\mathrm{Int} \\, \\XWpd &:=& \\left\\{ X\\in \\XWpd \\; |\n\\mbox{ there exists } \\delta>0 \\mbox{ such that }\\right .\\\\ && \\left .X+\\Delta_X \\in \\XWpd\n \\mbox{for all}\\;  \\Delta_X \\in \\Hn \\mbox{ with } \\|\\Delta_X\\|\\le \\delta  \\right\\}.\n\\end{eqnarray*\nHere $\\|\\Delta_X\\|$ is the spectral norm of $\\Delta_X$ given by the maximal singular value of $\\Delta_X$.\nWe compare $\\mathrm{Int} \\, \\XWpd$ with the open set\n\\\n\\XWpdpd = \\left\\{ X\\in \\XWpd \\; | \\; W(X)> 0 \\right\\}.\n\\\nSince $b(X)$ is finite for all points in $\\XWpdpd$, there is an open neighborhood where it stays bounded, and this implies that $\\XWpdpd \\subseteq \\mathrm{Int} \\, \\XWpd$.\nThe converse inclusion is not necessarily true. For example, consider a $2\\times 2$ transfer function having the form, $\\mathcal T(s)=\\mathrm{diag} (t(s), 0 )$,  where $t(s)$ is a scalar-valued, strictly passive transfer function.  The LMI is rank deficient for all $X\\in \\Hn$ (hence $\\XWpdpd=\\emptyset$) but there is a relative interior, since $t(s)$ is strictly passive. The characterization when both sets are equal is given by the following theorem.\n\\begin{theorem}\\label{thm:interior}\nSuppose the system ${\\mathcal M}$ of (\\ref{statespace}) is passive and $\\rank(B)=m$.  Then $\\XWpdpd \\equiv \\mathrm{Int}\\,\\XWpd$.\n\\end{theorem}\n{\\bf Proof:} If $\\XWpd=\\emptyset$  then $\\XWpdpd=\\emptyset$  as well.\nOtherwise, pick an $X\\in \\mathrm{Int}\\,\\XWpd$ and suppose\nthat $W(X)$ is  positive semidefinite and singular.  Then there exists a nontrivial $0\\neq [z_1^\\mathsf{T},z_2^\\mathsf{T}]^\\mathsf{T}\\in\\mathsf{Ker}\\, W(X)$ and\nan $\\varepsilon>0$ sufficiently small so that\nif $\\Delta X \\in \\Hn$ with $\\|\\Delta X\\|_F\\leq \\varepsilon$ then  $X+\\Delta X\\in \\XWpd$.\nObserve that for all such $\\Delta X $, we have $W(X+\\Delta X)=W(X)+\\Gamma(\\Delta X)\\geq 0$, where\n $ \\Gamma(\\Delta X)=-\\left[\\begin{array}{cc} \\Delta X A+ A^{\\mathsf{H}} \\Delta X &  \\Delta X B \\\\\n  B^{\\mathsf{H}}\\Delta X  & 0 \\end{array} \\right] $,\n and so\n %\n\\begin{equation}\n0\\leq \\left[\\begin{array}{c} z_1\\\\ z_2 \\end{array} \\right]^{\\mathsf{H}}\nW(X+\\Delta X)\\ \\left[\\begin{array}{c} z_1\\\\ z_2 \\end{array} \\right]\n =\n\\left[\\begin{array}{c} z_1\\\\ z_2 \\end{array} \\right]^{\\mathsf{H}}\n\\Gamma(\\Delta X)\\ \\left[\\begin{array}{c} z_1\\\\z_2 \\end{array} \\right] \\label{nonstrict}\n \\end{equation}\n %\nIf there was a choice for $\\Delta X \\in \\Hn$ with $\\|\\Delta X\\|_F\\leq \\varepsilon$ producing strict inequality in \\eqref{nonstrict},\nthen we would arrive at a contradiction, since the choice $-\\Delta X$ satisfies the same requirements yet violates the inequality.  Thus, we must have equality in \\eqref{nonstrict}\nfor all $\\Delta X \\in \\Hn$,\nwhich in turn implies\n\\[\nW(X+\\Delta X)\\left[\\begin{array}{c} z_1\\\\ z_2 \\end{array} \\right]= \\Gamma(\\Delta X)\\left[\\begin{array}{c} z_1\\\\ z_2 \\end{array} \\right]=0.\n\\]\nThis\nmeans that\n $B^{\\mathsf{H}}\\Delta X\\, u=0$ for all $\\Delta X \\in \\Hn$.\n If $z_1=0$, then we find that $\\Delta X B\\, z_2=0$ for all $\\Delta X \\in \\Hn$ which in turn means $B z_2=0$ and so, in light of the initial hypothesis on $B$, means that $z_2=0$ which is a contradiction, and thus,\n \nwe must conclude that $W(X)$ is nonsingular after all, hence positive definite.\n To eliminate the last remaining case, suppose that $z_1\\neq 0$.  Choosing first $\\Delta X=I$, we find that $z_1\\perp \\mathsf{Ran}(B)$.\n Pick $0\\neq b\\in  \\mathsf{Ran}(B)$ and define  $\\Delta X =I-2ww^{\\mathsf{H}}$ with\n $w=\\frac{1}{\\sqrt{2}}(\\frac{z_1}{\\|z_1\\|}-\\frac{b}{\\|b\\|})$.  Then\n $B^{\\mathsf{H}}\\Delta X\\, z_1 = \\frac{\\|z_1\\|}{\\|b\\|}B^{\\mathsf{H}}b =0$\n which implies that $z_1=0$, and so, $z=0$,\n $W(X)>0$, and again the assertion holds.\n\\hfill $\\Box$\n\nTo characterize when $\\XWpdpd \\equiv \\mathrm{Int}\\,\\XWpd\\neq \\emptyset$ is complicated and there is some confusion in the literature, because several factors may influence the solvability of the KYP-LMI.\nIt is clear that $S=D+D^{H}$ must be positive definite for a solution to be in $\\XWpdpd$, but clearly this is not sufficient as is demonstrated by the simple system $\\dot x=u$, $y=x+du$ (with $d>0$) which is minimal and has $X=1\\in  \\XWpd$ as only solution of the KYP-LMI, so  $\\mathrm{Int} \\, \\XWpd=\\emptyset$.\nIn this case the associated pencil $\\mathcal L_0$ has one eigenvalue $\\infty$ and the  purely imaginary eigenvalues $\\pm i  d$. It is passive, but not strictly passive, and stable (but not asymptotically stable), which is in contradiction to many statements in the literature, see e.~g.~\\cite{Gri04}, where unfortunately no distinction between passivity and strict passivity is made. The system is, furthermore, port-Hamiltonian with $J=0$, $R=0$, $B,C=1$, $Q=1$, $P=0$ and $D=1$, and satisfies the dissipation inequality. An analogous example is obtained with $A= \\left [ \\begin{array}{cc} 0 & 1 \\\\ -1 & 0 \\end{array} \\right ]$, $B^{\\mathsf{H}}=C=\\left [ \\begin{array}{cc} 1 & 0  \\end{array} \\right ]$, and $D=1$.\nThen, $X=I_2$ is the unique positive definite solution, $\\mathrm{Int} \\, \\XWpd=\\emptyset$, and there are double eigenvalues of $\\mathcal L_0$ at $\\pm i$. The system is not asymptotically stable and not strictly passive, but stable, passive and pH. If in this example we choose $D=0$, then still $X=I$ is the unique positive definite solution of the KYP-LMI, but now $\\mathcal L_0$ has the only purely imaginary eigenvalue $0$ and two Kronecker blocks of size $2$ for the eigenvalue $\\infty$. In this case the Riccati equation \\eqref{riccati} cannot be be formed and there does not exist a two-dimensional extended Lagrangian invariant space associated with the stable eigenvalues.\n\n\n\\begin{remark}{\\rm\nNote that the solutions $X_+$ and $X_-$ of the Riccati equation \\eqref{riccati} yield singular $W(X_+)$ and $W(X_-)$, and are thus  on the boundary of $\\XWpd$,\neven though they are positive definite, as was pointed out in Theorem \\ref{LMIWil}.\n}\\end{remark}\n\n\nIn the sequel, we  assume that $\\XWpdpd\\neq \\emptyset$, so that the analytic center of $\\XWpd$ is well-defined, see also \\cite{NesN94}, and we can compute it as a candidate for a `good' solution to the LMI (\\ref{KYP-LMI}) yielding a robust representation. This requires the solution of an optimization problem.\nKeep in mind that we have the following the set inclusions\n\\[\n\\mathbb{X}_{W} \\subset   \\mathrm{Int}\\XWpd=\\XWpdpd  \\subset \\XWpd \\subset   \\Hnpd    \\subset \\Hn.\n\\]\nFor $X,Y \\in \\Hn $ we define the \\emph{Frobenius inner product}\n\\[\n \\langle X,Y\\rangle  := \\mathsf{trace}\\!\\left({\\mathfrak{Re}}(Y)^T{\\mathfrak{Re}}(X)+{\\mathfrak{Im}}(Y)^T{\\mathfrak{Im}}(X)\\right),\n\\]\nwhich has the properties $\\langle X,Y \\rangle= \\langle Y,X \\rangle$, $\\|X\\|_F= \\langle X,X \\rangle^{\\frac{1}{2}}$, and $YZ \\rangle= \\langle YX,Z \\rangle = \\langle XZ,Y \\rangle$.\n\nThe \\emph{gradient} of the barrier function $b(X)$ with respect to $W$ is given by\n\\[\n\\partial b(X) \/ \\partial W =  -W(X)^{-1}.\n\\]\nUsing the chain rule and the Frobenius inner product, it follows from \\cite{NesN94}\nthat $X\\in \\mathbb{C}^{n \\times n}$ is an extremal point of $b(X)$ if and only if\n\\[\n\\langle \\partial b(X) \/ \\partial W,  \\Delta W(X)[\\Delta_X] \\rangle = 0 \\quad \\mbox{for all} \\; \\Delta_X \\in \\Hn,\n\\]\nwhere $\\Delta W(X)[\\Delta_X]$ is the incremental step in the direction $\\Delta_X$ given by\n\\[\n\\Delta W(X)[\\Delta_X] = -\\left[ \\begin{array}{cc}\nA^{\\mathsf{H}}\\Delta_X+\\Delta_X A & \\Delta_X B \\\\ B^{\\mathsf{H}}\\Delta_X & 0 \\end{array} \\right].\n\\]\nFor an extremal point it is then necessary that\n\\begin{equation}\\label{orth}\n\\langle W(X)^{-1} , \\left[ \\begin{array}{cc}\nA^{\\mathsf{H}}\\Delta_X + \\Delta_X A & \\Delta_X B \\\\ B^{\\mathsf{H}}\\Delta_X & 0 \\end{array} \\right] \\rangle \\ =0\n\\end{equation}\nfor all $\\Delta_X\\in \\Hn$.\nDefining $F := S^{-1}(C-B^HX)$,\n$P :=-A^HX-XA-F^{\\mathsf{H}}SF$, and $A_F := A-BF$, it has been shown in \\cite{GenNV99} that \\eqref{orth} holds if and only if $P$ is invertible and\n\\begin{equation} \\label{skew}\nA_F^{\\mathsf{H}}P +PA_F =0.\n\\end{equation}\nNote that $P$ is nothing but the evaluation of the Riccati operator \\eqref{riccati} at $X$, and that $A_F$ is the corresponding closed loop matrix. For the solutions of the Riccati equation we have\n$P=\\mathsf{Ricc}(X)=0$ (so $P$ is not invertible) and the corresponding closed loop matrix\nhas all its eigenvalues equal to an adequate   subset of the eigenvalues of the Hamiltonian matrix $H$ in (\\ref{HamMatrix}). For an interior point of $\\XWpd$\nwe have $P=\\mathsf{Ricc}(X)> 0$, and hence $P$ has an invertible\nsquare root $P^{\\frac{1}{2}}\\in \\Hnpd$. Multiplying\n(\\ref{skew}) on both sides with  $P^{-\\frac{1}{2}}$\nwe obtain that\n\\[\nP^{-\\frac{1}{2}}A_F^{\\mathsf{H}}P^{\\frac{1}{2}} + P^{\\frac{1}{2}}A_FP^{-\\frac{1}{2}}=0.\n\\]\nThus, $\\hat A_F:=P^{\\frac{1}{2}}A_F P^{-\\frac{1}{2}}$ is skew-Hermitian and therefore  $\\hat A_F$ as well as $A_F$ have all its eigenvalues on the imaginary axis. Hence the closed loop matrix $A_F$ of the analytic center has a spectrum that is also 'central' in a certain sense.\n\nIt is important to note that\n\\[\n \\det (W(X)) = \\det (\\mathsf{Ricc}(X)) \\det S,\n\\]\nwhich implies that we are also finding a stationary point of\n$\\det(\\mathsf{Ricc}(X))$, since $S$\nis constant and invertible. Since  $P\\in \\Hnpd$, we can rewrite the equations defining the analytic center of\n$\\XWpd$ as the solutions $X\\in \\Hn$, $P\\in \\Hnpd$, $F\\in \\mathbb C^{m,n}$ of the system of matrix equations\n\\begin{eqnarray} \\nonumber\nS F  &=& C-B^{\\mathsf{H}}X,\\\\\n P   &=& -A^{\\mathsf{H}}X-XA-F^{\\mathsf{H}}SF,  \\label{FXP}\\\\ \\nonumber\n 0&=& P(A-BF)+(A^{\\mathsf{H}}-F^{\\mathsf{H}}B^{\\mathsf{H}})P\\\\\n &=&PA_F+A_F^{\\mathsf{H}}P.\\nonumber\n\\end{eqnarray}\nSystem (\\ref{FXP}) can be used to determine a solution via an iterative method, that uses a starting value $X_0$  to compute $P_0,F_0$ and then consecutively solutions $X_i$, $i=1,2,\\ldots$\nfollowed by computing a new $P_i$ and $F_i$.\n\n\\begin{remark}\\label{rem:evp}{\\rm\nFor given matrices $P,F$ the solution $X$ of \\eqref{riccati} can be obtained via\nthe invariant subspace\n\\[\n\\left[ \\begin{array}{ccc} 0 & I_n & 0\\\\\n\t-I_n & 0 & 0\\\\ 0 & 0 & 0 \\end{array} \\right]\\mat{c} -X \\\\ \\phantom{-}I_n \\\\ -F \\rix Z=\n\\left[ \\begin{array}{ccc} 0 & A & B \\\\\n\tA^{\\mathsf{H}} & -P & C^{\\mathsf{H}} \\\\  B^{\\mathsf{H}} & C & S \\end{array} \\right]\\mat{c} -X \\\\ \\phantom{-}I_n \\\\ -F \\rix,\n\\]\nComputing this subspace for $P=\\mathsf{Ricc}(X)=0$ allows to compute the extremal solutions $X_+$ and $X_-$ of \\eqref{riccati}, which then can be used to compute a starting point for an optimization scheme, see \\cite{BanMVN17_ppt} for details.\n}\n\\end{remark}\n\n\n\n\\section{The passivity radius}\\label{sec:passrad}\nOur goal to achieve `good' or even `optimal' pH representations of a passive system can be realized in different ways. A natural measure for optimality is a large \\emph{passivity radius}\n$\\rho_{{\\mathcal M}}$, which is the smallest perturbation (in an appropriate norm) to the coefficients of a model ${\\mathcal M} $ that makes the system non-passive. Computational methods to determine $\\rho_{{\\mathcal M}}$ were introduced in \\cite{OveV05}, while the converse question, what is the nearest passive system to a non-passive system has recently been discussed in \\cite{GilMS18,GilS17}.\n\nOnce we have determined a solution $X\\in \\XWpd$ to the LMI~(\\ref{KYP-LMI}), we can determine the representation \\eqref{pH}\nas in Section~\\ref{sec:PH} and the system is automatically passive (but not necessarily strictly passive). For each such representation we can determine the passivity radius and then choose the most robust solution $X\\in \\XWpd$ under perturbations by maximizing the passivity radius or by minimizing the condition number of $X^{\\frac 12}$, which is the transformation matrix to pH form, see Section~\\ref{sec:cond}.\n\n\\subsection{The $X$-passivity radius}\n\nAlternatively,  for $X\\in \\mathrm{Int}\\XWpd$ we can determine the smallest perturbation ${\\Delta_{\\mathcal M}}$ of the system matrices $A,B,C,D$ of the model ${\\mathcal M} $ that leads to a loss of positive definiteness of $W(X)$, because then we are on the boundary of the set of passive systems. This is a very suitable approach for perturbation analysis, since for fixed $X$ the matrix\n\\begin{equation*} \\label{hx}\nW(X) = \\left[\n\\begin{array}{cc}\n0 & C^{\\mathsf{H}} \\\\\nC & D+D^{\\mathsf{H}}\n\\end{array}\n\\right]\n- \\left[\\begin{array}{cc}\nA^{\\mathsf{H}}X + X\\,A & X\\,B \\\\\nB^{\\mathsf{H}}X & 0\n\\end{array}\n\\right]\n\\end{equation*}\nis linear in the unknowns $A,B,C,D$ and when we perturb the coefficients, then we preserve strict passivity as long as\n\\begin{eqnarray*}\n&& W_{\\Delta_{\\mathcal M}} (X) := \\left[\n\\begin{array}{cc}\n 0 & (C+\\Delta_C)^{\\mathsf{H}} \\\\\n(C+\\Delta_C) & (D+\\Delta_D)+(D+\\Delta_D)^{\\mathsf{H}}\n\\end{array}\n\\right]\\\\\n&& \\qquad - \\left[\\begin{array}{cc}\n(A+\\Delta_A)^{\\mathsf{H}}X + X\\,(A+\\Delta_A) & X\\,(B+\\Delta_B) \\\\\n(B+\\Delta_B)^{\\mathsf{H}}X & 0\n\\end{array}\n\\right]>0.\n\\end{eqnarray*}\nHence, given $X\\in \\mathrm{Int}\\XWpd$, we can look for the smallest perturbation $\\Delta_{\\mathcal M}$ to the model ${\\mathcal M} $ that makes $\\det (W_{\\Delta_{\\mathcal M}}(X)=0$. To measure the size of the perturbation $\\Delta_{\\mathcal M}$ of a state space model ${\\mathcal M} $ ,we use the Frobenius norm\n\\[\n \\|\\Delta_{\\mathcal M} \\| := \\left \\|\\left[\\begin{array}{ccc}\n\\Delta_A & \\Delta_B \\\\\n\\Delta_C & \\Delta_D\n\\end{array}\\right] \\right \\|_F.\n\\]\nDefining for $X\\in \\mathrm{Int}\\XWpd$ the  \\emph{$X$-passivity radius} as\n\\[\n\t\\rho_{\\mathcal M}(X):= \\inf_{\\Delta_{\\mathcal M}\\in \\mathbb C^{n+m,n+m}}\\left\\{ \\| \\Delta_{\\mathcal M} \\| \\; | \\; \\det W_{\\Delta_{\\mathcal M}}(X) = 0\\right\\}.\n\\]\nNote that in order to compute $\\rho_{\\mathcal M}(X)$ for the model ${\\mathcal M} $, we must first find a point $X\\in \\mathrm{Int}\\XWpd$, since $W(X)$  must be positive definite to start with and also $X$ should be positive definite to obtain a state-space transformation to pH form.\n\nWe have the following relation between the $X$-passivity radius and the usual passivity radius.\n\\begin{lemma}\\label{passbound}\nConsider a given model ${\\mathcal M}$ . Then the passivity radius is given by\n\\begin{eqnarray*}\n\\nonumber\n\t\\rho_{{\\mathcal M}}&=& \\sup_{X\\in \\mathrm{Int}\\XWpd}\\inf_{\\Delta_{\\mathcal M}\\in \\mathbb C^{n+m,n+m}}\\{\\| \\Delta_{\\mathcal M} \\| | \\det W_{\\Delta_{\\mathcal M}}(X)=0\\}\\\\ &=& \\sup_{X\\in \\mathrm{Int}\\XWpd} \\rho_{{\\mathcal M}}(X).\\label{passive}\n\t\\end{eqnarray*}\n\\end{lemma}\n{\\bf Proof:}\n\tIf for any given $X \\in \\mathrm{Int} \\XWpd$ we have that  $\\| \\Delta_{\\mathcal M} \\|< \\rho_{{\\mathcal M}}(X)$, then all systems ${\\mathcal M} +\\Delta_{\\mathcal M}$ with $\\| \\Delta_{\\mathcal M} \\| < \\rho_{\\mathcal M}(X)$ are strictly passive.\n\tTherefore $\\rho_{\\mathcal M} \\ge \\sup_{\\mathrm{Int}\\XWpd} \\rho_{\\mathcal M}(X)$. Equality follows, since there exists a perturbation $\\Delta_{\\mathcal M}$ of norm $\\rho_{\\mathcal M}$ for which there does not exist a point $X\\in \\mathrm{Int}\\XWpd$ for which $W_{\\Delta_{\\mathcal M}}(X)> 0$, hence, this system is not strictly passive anymore.\n\\hfill $\\Box$\n\nWe derive an explicit formula for the $X$-passivity radius based on a one parameter  optimization problem. For this, we rewrite the condition $W_{\\Delta_{\\mathcal M}} (X)>0$ as\n\\begin{eqnarray} \\nonumber\n&& \\left[\\begin{array}{cc} -X & 0 \\\\ 0 & I_m \\end{array}\\right]\n\\left[\\begin{array}{cc} A + \\Delta_A & B+\\Delta_B \\\\ C+\\Delta_C & D+\\Delta_D \\end{array}\\right] \\\\\n&& \\label{wdelta} \\qquad +\n\\left[\\begin{array}{cc} A^{\\mathsf{H}}+\\Delta_A^{\\mathsf{H}} & C^{\\mathsf{H}}+\\Delta_C^{\\mathsf{H}} \\\\ B^{\\mathsf{H}}+\\Delta_B^{\\mathsf{H}} & D^{\\mathsf{H}}+\\Delta_D^{\\mathsf{H}} \\end{array}\\right]\n\\left[\\begin{array}{cc} -X & 0 \\\\ 0 & I_m \\end{array}\\right] >0.\n\\end{eqnarray}\nSetting\n\\begin{equation} \\label{defWhatX}\n\\hat W:=W(X), \\; \\hat X := \\left[\\begin{array}{cc} X & 0 \\\\ 0 & I_m \\end{array}\\right], \\; \\Delta_T := \\left[\\begin{array}{cc} -\\Delta_A & -\\Delta_B \\\\ \\Delta_C & \\Delta_D \\end{array}\\right],\n\\end{equation}\ninequality (\\ref{wdelta}) can be written as the LMI\n\\begin{equation} \\label{WDelta}\nW_{\\Delta_{\\mathcal M}}  = \\hat W+\\hat X \\Delta_T + \\Delta_T^{\\mathsf{H}} \\hat X >0\n\\end{equation}\nas long as the system is still passive. In order to violate this condition, we need to find the smallest $\\Delta_T$ such that $\\det W_{\\Delta_{\\mathcal M}} =0$. Factoring out $\\hat W^{-\\frac{1}{2}}$ on both sides of (\\ref{WDelta})  yields the characterization\n\\begin{eqnarray}\n&&\\det\\left(I_{n+m} +\n\\hat W^{-\\frac{1}{2}}\\hat X \\Delta_T \\hat W^{-\\frac{1}{2}}+\n\\hat W^{-\\frac{1}{2}}\\Delta_T^{\\mathsf{H}} \\hat X \\hat W^{-\\frac{1}{2}}\\right) \\nonumber \\\\\n&&\\det\\left(I_{n+m} +\n\\left[ \\begin{array}{cc} \\hat W^{-\\frac{1}{2}}\\hat X & \\hat W^{-\\frac{1}{2}}\\end{array}\\right]\n\\left[ \\begin{array}{cc} 0 & \\Delta_T \\\\ \\Delta_T^{\\mathsf{H}} & 0 \\end{array}\\right]\n\\left[ \\begin{array}{cc} \\hat X\\hat W^{-\\frac{1}{2}} \\\\ \\hat W^{-\\frac{1}{2}}\\end{array}\\right] \\right)\n\\nonumber  \\\\ \\nonumber\n&&\\det\\left(I_{2(n+m)} +\n\\left[ \\begin{array}{cc} 0 & \\Delta_T \\\\ \\Delta_T^{\\mathsf{H}} & 0 \\end{array}\\right]\n\\left[ \\begin{array}{cc} \\hat X \\hat W^{-\\frac{1}{2}} \\\\ \\hat W^{-\\frac{1}{2}}\\end{array}\\right]\n\\left[ \\begin{array}{cc} \\hat W^{-\\frac{1}{2}}\\hat X & \\hat W^{-\\frac{1}{2}}\\end{array}\\right] \\right)\\\\\n&& =0.\\label{rhoX}\n\\end{eqnarray}\nThe minimal perturbation $\\Delta_T$ for which this is the case was described in \\cite{OveV05}\nusing the following theorem, which we have slightly modified in order to take into account the positive semi-definiteness of the considered matrix.\n\\begin{theorem} \\label{thm:OveV} Consider the matrices $\\hat X, \\hat W$ in (\\ref{defWhatX}) and the pointwise positive semidefinite matrix function\n\\begin{equation}\\label{defmgamma}\nM(\\gamma):= \\left[ \\begin{array}{cc} \\gamma \\hat X \\hat W^{-\\frac{1}{2}} \\\\ \\hat W^{-\\frac{1}{2}} \/ \\gamma\\end{array}\\right]\n\\left[ \\begin{array}{cc} \\gamma \\hat W^{-\\frac{1}{2}} \\hat X & \\hat W^{-\\frac{1}{2}} \/ \\gamma \\end{array}\\right]\n\\end{equation}\nin the real parameter $\\gamma$. Then the largest eigenvalue $\\lambda_{\\max}(M(\\gamma))$ is a \\emph{unimodal function} of $\\gamma$ ({i.e.} it is first monotonically decreasing and then monotonically increasing with growing $\\gamma$). At the minimizing value $\\underline \\gamma$,  $M(\\underline{\\gamma})$ has an eigenvector $z$, {i.e.}\n\\[\n M(\\underline{\\gamma}) z = \\underline\\lambda_{\\max} z, \\quad z:=\\left[ \\begin{array}{cc} u \\\\ v \\end{array}\\right],\n \\]\nwhere\n$  \\|u\\|_2^2=\\|v\\|_2^2=\\frac{1}{2}$.\nThe minimum norm perturbation $\\Delta_T$ is of rank $1$ and is given by $\\Delta_T=2uv^{\\mathsf{H}}\/\\underline{\\lambda}_{\\max}$. It has norm $1\/\\underline{\\lambda}_{\\max}$\nboth in the spectral norm and in the Frobenius norm.\n\\end{theorem}\n{\\bf Proof}\nThe proof for a slightly different formulation was presented in \\cite{OveV05}. Here we therefore just present the basic idea in our formulation. Let $\\Gamma:=\\mathrm{diag}(\\gamma I_{m+n}, \\frac{1}{\\gamma} I_{m+n})$, then $M(\\gamma)=\\Gamma H_1 \\Gamma$, while\n\\[\n\\Gamma^{-1}\\left[ \\begin{array}{cc} 0 & \\Delta_T \\\\ \\Delta_T^{\\mathsf{H}} & 0 \\end{array}\\right]\\Gamma^{-1}=\\left[ \\begin{array}{cc} 0 & \\Delta_T \\\\ \\Delta_T^{\\mathsf{H}} & 0 \\end{array}\\right].\n\\]\nSetting\n\\\nK(\\gamma):=\\left( I_{2(n+m)} + \\left[ \\begin{array}{cc} 0 & \\Delta_T \\\\ \\Delta_T^{\\mathsf{H}} & 0 \\end{array}\\right] M(\\gamma) \\right),\n\\\nthen $\\det(K(\\gamma))$ is independent of $\\gamma$. But the vector $z$ is in the kernel of $K(\\gamma)$, which implies that also $K(1)$ is singular. The value of the norm of the constructed\n$\\Delta_T$ follows from the fact that the subvectors $u$ and $v$ must have equal norm at the minimum.\n\\hfill $\\Box$\n\nA bound for $\\underline{\\lambda}_{\\max}$ in Theorem~\\ref{thm:OveV} is obtained by the following result.\n\\begin{corollary}\\label{cor:lev} Consider the matrices $\\hat X, \\hat W$ in (\\ref{defWhatX}) and the pointwise matrix function $M(\\gamma)$ as in (\\ref{defmgamma}). The largest eigenvalue of $M(\\gamma)$ is also the largest eigenvalue of\n\\[\n\\gamma^2 \\hat W^{-\\frac{1}{2}} \\hat X^2 \\hat W^{-\\frac{1}{2}} + \\hat W^{-1}\/\\gamma^2.\n\\]\nA simple upper bound for $\\underline{\\lambda}_{\\max}$ is given by $\\underline{\\lambda}_{\\max}\\le \\frac{2}{\\alpha\\beta}$ where $\\alpha^2:=\\lambda_{\\min}(\\hat W)$ and $\\beta^2=\\lambda_{\\min}(\\hat X^{-1}\\hat W\\hat X^{-1})$. The corresponding lower bound for $\\| \\Delta_T \\|_F$ then becomes\n\\[\n  \\rho_{\\mathcal M}(X) = \\min_{\\gamma} \\| \\Delta_T \\|_F \\ge \\alpha\\beta\/2.\n\\]\n\\end{corollary}\n{\\bf Proof} Clearly $\\|\\hat W^{-1}\\|_2\\le \\frac{1}{\\alpha^2}$ and $\\|\\hat W^{-\\frac{1}{2}} \\hat X^2 \\hat W^{-\\frac{1}{2}} \\|_2\\le \\frac{1}{\\beta^2}$. So if we choose $\\gamma^2=\\frac{\\beta}{\\alpha}$\n\tthen\n\\begin{eqnarray*}\n&& \\min_{\\gamma} \\|\\gamma^2 \\hat W^{-\\frac{1}{2}} \\hat X^2 \\hat W^{-\\frac{1}{2}} + \\hat W^{-1}\/\\gamma^2\\|\\\\\n && \\qquad\\le  \\| (\\beta\/\\alpha)\\hat W^{-\\frac{1}{2}} \\hat X^2 \\hat W^{-\\frac{1}{2}} + (\\alpha\/\\beta)\\hat W^{-1}\\|\\\\\t &&\\qquad \\le  \\frac{2}{\\alpha\\beta} . \\qquad \\Box\n\\end{eqnarray*}\n\n\tWe can construct a perturbation $\\Delta_T=\\epsilon (\\alpha\\beta)vu^{\\mathsf{H}}$ of norm $|\\epsilon|(\\alpha\\beta)$ which makes the matrix  $W_{\\Delta_{\\mathcal M}}$ singular and therefore gives an upper bound for $\\rho_M(X)$. To compute this perturbation, let $u$, $v$ and $w$ be vectors of norm $1$, satisfying\n$\\hat W^{-\\frac{1}{2}}u=u\/\\alpha$, $\\hat W^{-\\frac{1}{2}} \\hat X v=w\/\\beta$, $\\Delta_T=\\epsilon(\\alpha\\beta)vu^{\\mathsf{H}}$, and $\\epsilon u^{\\mathsf{H}}w=-|\\epsilon u^{\\mathsf{H}}w|$,\n{i.e.},  $u$, $v$ and $w$ are the singular vectors associated with the largest singular values\n$1\/\\alpha$ of\t$\\hat W^{-\\frac{1}{2}}$ and $1\/\\beta$ of $\\hat W^{-\\frac{1}{2}}\\hat X$. Inserting these values in (\\ref{rhoX}), it follows that\n\\begin{eqnarray*}\n&& \\det\\left(I_{n+m} +\n\t\\left[ \\begin{array}{cc} \\hat W^{-\\frac{1}{2}}\\hat X & \\hat W^{-\\frac{1}{2}}\\end{array}\\right]\n\t\\left[ \\begin{array}{cc} 0 & \\Delta_T \\\\ \\Delta_T^{\\mathsf{H}} & 0 \\end{array}\\right]\n\t\\left[ \\begin{array}{cc} \\hat X\\hat W^{-\\frac{1}{2}} \\\\ \\hat W^{-\\frac{1}{2}}\\end{array}\\right] \\right) \\\\\n&& \\qquad =\\det\\left(I_{n+m} +\n\t\\left[ \\begin{array}{cc} w & u \\end{array}\\right]\n\t\\left[ \\begin{array}{cc} 0 & \\epsilon \\\\ \\overline \\epsilon & 0 \\end{array}\\right]\n\t\\left[ \\begin{array}{cc} w^{\\mathsf{H}} \\\\ u^{\\mathsf{H}} \\end{array}\\right] \\right) \\\\\n&& \\qquad =\n\t\\det\\left(I_{2} + \\left[ \\begin{array}{cc} 0 & \\epsilon \\\\ \\overline \\epsilon & 0 \\end{array}\\right]\n\t\\left[ \\begin{array}{cc} w^{\\mathsf{H}} \\\\ u^{\\mathsf{H}} \\end{array}\\right] \\left[ \\begin{array}{cc} w &  u \\end{array}\\right]\\right).\n\\end{eqnarray*}\nIf we now choose the argument of the complex number $\\epsilon$ such that $\\epsilon u^{\\mathsf{H}}w$ is real and negative and the amplitude of $\\epsilon$ such that $1=|\\epsilon u^{\\mathsf{H}}w|+ |\\epsilon|$, \tthen\n\\[\n\\det \\left(I_{2} + \\left[ \\begin{array}{cc} \\epsilon u^{\\mathsf{H}}w & \\epsilon\\\\ \\overline \\epsilon & \\overline \\epsilon w^{\\mathsf{H}}u \\end{array}\\right]\\right) = (1-|\\epsilon u^{\\mathsf{H}}w|)^2-|\\epsilon|^2=0.\n\\]\nSince $0\\le |u^{\\mathsf{H}}w| \\le 1$, we have that $\\frac{1}{2} \\le |\\epsilon| \\le 1$ and thus we have the interval $\\alpha\\beta\/2 \\le \\rho_{\\mathcal M}(X) \\le |\\epsilon| \\alpha\\beta$, $\\frac{1}{2} \\le |\\epsilon| \\le 1$ in which the  $X$-passivity radius is located.\nIf $u$ and $w$ are linearly dependent, then this interval shrinks to a point and the estimate is exact.\nWe summarize these observations in the following theorem.\n\\begin{theorem}\\label{thm:Xpassivity}\nLet ${\\mathcal M}=\\{A,B,C,D\\}$ be a given model and let $X\\in \\mathrm{Int}\\XWpd$. Then the $X$-passivity radius $\\rho_{\\mathcal M}(X)$ at this $X$ is bounded by\n\\[\n \\alpha\\beta\/2 \\le \\rho_{\\mathcal M}(X) \\le  \\alpha\\beta\/(1+|u^{\\mathsf{H}}w|),\n \\]\n %\nwhere\n$\\alpha^2:=\\lambda_{\\min}(\\hat W)$, $\\beta^2=\\lambda_{\\min}(\\hat X^{-1}\\hat W\\hat X^{-1})$, $\\hat W^{-\\frac{1}{2}}u=u\/\\alpha$, $\\hat W^{-\\frac{1}{2}} \\hat X v=w\/\\beta$.\nMoreover, if $u$ and $w$ are linear dependent, then $\\rho_{\\mathcal M}(X)=\\alpha\\beta\/2$.\n\\end{theorem}\nIf we use these bounds for the passivity radius in a pH system, we have the following corollary.\n\\begin{corollary}\\label{cor:xeqI} If for a given system $\\mathcal M$ we have that $X=I_n\\in \\mathrm{Int}\\XWpd$   then\nthe corresponding representation of the system is port-Hamiltonian, {i.e.}, it has the representation ${\\mathcal M}:= \\{J-R,G-K,G^{\\mathsf{H}}+K^{\\mathsf{H}},S+N\\}$  and the X-passivity radius\nis given by\n \\[\n \\rho_{\\mathcal M}(I)=\\lambda_{\\min}W(I)=\\lambda_{\\min}\\left[\\begin{array}{cc} R & K \\\\ K^{\\mathsf{H}} & S\t \\end{array}\\right].\n \\]\n %\nMoreover, if $X=I_n$ is the analytic center of $\\mathrm{Int}\\XWpd$, then\n$\\rho_{\\mathcal M}(I)$ equals the passivity radius $\\rho_{\\mathcal M}$ of ${\\mathcal M}$.\n\\end{corollary}\n{\\bf Proof}\n    This follows directly from Theorem~\\ref{thm:Xpassivity}, since then  $\\alpha=\\beta$ and we can choose $u=w$.\n\\hfill $\\Box$\n\\begin{remark}\\label{rem:alphabeta}{\\rm Considering a pH representation,\nthe conditions $\\hat W\\geq \\alpha^2 I_{n+m}$ and $\\hat X^{-1} \\hat W \\hat X^{-1}\\geq \\beta^2 I_{n+m}$ yield the necessary (but not sufficient) condition for passivity that\n\\[\n\\left[ \\begin{array}{cc} \\hat W & \\alpha \\beta I_{n+m} \\\\\n    \t\\alpha \\beta I_{n+m} &  \\hat X^{-1} \\hat W \\hat X^{-1}\t\n    \t\\end{array} \\right] \\geq 0,\n\\]  \t\nUsing the square root $\\hat T:= \\hat X^{\\frac12}$ of  $\\hat X>0$ and a congruence transformation, one finds that this is equivalent to\n\\[\n \\left[ \\begin{array}{cc}  \\hat T^{-1} \\hat W  \\hat T^{-1} & \\alpha \\beta I_{n+m} \\\\\n    \t\\alpha \\beta I_{n+m} &  \\hat T^{-1} \\hat W \\hat T^{-1}\t\n    \t\\end{array} \\right] \\geq 0,\n\\]\nwhich implies that $\\hat T^{-1} \\hat W  \\hat T^{-1} \\geq \\alpha\\beta I_{n+m}$.\n\nDefining $\\xi:=\\lambda_{\\min}(\\hat T^{-1} \\hat W  \\hat T^{-1})$, we then also obtain the inequality $\\xi \\ge \\alpha\\beta$. Because of \\eqref{PH}, this is also equal to\n\\[\n \\xi =\\lambda_{\\min} \\left[\\begin{array}{cc}\nR & K \\\\ K^{\\mathsf{H}} & S\n\\end{array}\\right]\n\\]\nwhich suggests that pH representations\nare likely to guarantee a good passivity margin. In order to compute the optimal product $\\xi=\\alpha\\beta$, we could maximize $\\xi$ under the constraint\n\\[\n\\hat W-\\xi\\hat X = \\left[ \\begin{array}{cc} - XA- A^{\\mathsf{H}} X - \\xi X & C^{\\mathsf{H}}- XB \\\\ C-B^{\\mathsf{H}} X & S-\\xi I_m \\end{array}\\right]> 0.\n\\]\n}\n\\end{remark}\n\n\\subsection{A port-Hamiltonian barrier function}\n\nFrom our previous discussions it appears that if we want to make sure that a state-space representation has a large passivity radius, we should not maximize the determinant of $W(X)$, but maximize instead\n\\begin{equation} \\label{tilde}\n\\det \\left ( \\left[ \\begin{array}{cc} X^{-\\frac12} & 0 \\\\ 0 & I_m \\end{array}\\right]  W(X)\\left[ \\begin{array}{cc} X^{-\\frac12} & 0 \\\\ 0 & I_m \\end{array}\\right] \\right )\n\\end{equation}\nunder the constraint $X>0$ so that its square root $T=X^\\frac12$ exists.\nEquivalently, if $X>0$, we can maximize  the determinant of\n\\begin{eqnarray*} \\tilde W(X)&:=& W(X)\\left[ \\begin{array}{cc} X^{-1} & 0 \\\\ 0 & I_m \\end{array}\\right]\\\\\n&=&  \\left[ \\begin{array}{cc} -XAX^{-1}-A^{\\mathsf{H}} & C^{\\mathsf{H}}-XB \\\\ CX^{-1}-B^{\\mathsf{H}} & S \\end{array}\\right]\n\\end{eqnarray*}\nwhich has the same eigenvalues and the same determinant, but is expressed in terms of the variable $X$.\n\nWith this modified barrier function $\\tilde b(X):=- \\log \\det \\tilde W(X)$ we obtain the following formulas for the gradient of the barrier with respect to $\\tilde W$ and the incremental step of $\\tilde W(X)$ in the direction $\\Delta_X$.\n\\begin{eqnarray*}\n\\partial \\tilde b(X\/\\partial \\tilde W) &=& -\\tilde W(X)^{-{\\mathsf{H}}} = -W(X)^{-1}\\left[ \\begin{array}{cc} X & 0 \\\\ 0 & I_m \\end{array}\\right], \\\\\n\\Delta \\tilde W(X)[\\Delta_X] &=&  \\left[ \\begin{array}{cc}\nXAX^{-1}\\Delta_X  - \\Delta_X A & -\\Delta_X B \\\\ -CX^{-1}\\Delta_X & 0 \\end{array} \\right]\\left[ \\begin{array}{cc} X^{-1} & 0 \\\\ 0 & I_m \\end{array}\\right].\n\\end{eqnarray*}\nUsing again the chain rule, the necessary condition for an extremal point is then that for all $\\Delta_X \\in \\Hn$, $<\\partial \\tilde b(X) \/ \\partial \\tilde W,  \\Delta \\tilde W(X)[\\Delta_X]> = 0$, or equivalently\n\\begin{equation} \\label{orthtilde}\n< W(X)^{-1} , \\left[ \\begin{array}{cc} XAX^{-1}\\Delta_X -\\Delta_X A  & -\\Delta_X B \\\\ -CX^{-1}\\Delta_X & 0\n\\end{array} \\right] > \\ =0 \\; .\n\\end{equation}\nDefining $P$ and $F$ as before, and using that\n\\[\n W(X)^{-1} = \\left[ \\begin{array}{cc} I_n & 0 \\\\ -F & I_m \\end{array}\\right]\n \\left[ \\begin{array}{cc} P^{-1} & 0 \\\\ 0 & S^{-1} \\end{array}\\right]\n  \\left[ \\begin{array}{cc} I_n & -F^{\\mathsf{H}} \\\\ 0 & I_m \\end{array}\\right],\n\\]\nit then follows that \\eqref{orthtilde} holds if and only if $P$ is invertible and for all $\\Delta_X \\in \\Hn$\nwe have\n\\[\n<  P^{-1} , (XAX^{-1}+F^{\\mathsf{H}}CX^{-1})\\Delta_X  - \\Delta_X (A-BF) > \\ =0,\n\\]\nor equivalently\n\\[\n< \\Delta_X , P^{-1} (XAX^{-1}+F^{\\mathsf{H}}CX^{-1}) - (A-BF)P^{-1}   > \\ =0,\n\\]\nwhich can be expressed as\n\\begin{eqnarray*}\n&& P[(X^{-1}A^{\\mathsf{H}}X+X^{-1}C^{\\mathsf{H}}F)-(A-BF)]\\\\\n&& \\qquad + [(XAX^{-1}+F^HCX^{-1})-(A-BF)^{\\mathsf{H}}]P=0.\n\\end{eqnarray*}\nNote that if one performs the coordinate transformation\n\\[\n\\{A_T,B_T,C_T,D_T\\} := \\{TAT^{-1}, TB, CT^{-1}, D \\}\n\\]\nwhere $T^2=X$, then  $P_T=T^{-1}PT^{-1}$ and $F_T=FT^{-1}$, which yields\nthe equivalent condition\n\\begin{eqnarray*}\n&& P_T[(A_T^{\\mathsf{H}}-A_T)+(C^{\\mathsf{H}}_T+B_T)F_T]\\\\\n && \\qquad +[(A_T^{\\mathsf{H}}-A_T)+(B_T+C_T^{\\mathsf{H}})F_T]^{\\mathsf{H}}P_T=0.\n\\end{eqnarray*}\nMoreover, we have that\n\\[\n F_T = S^{-1}(C_T-B_T^{\\mathsf{H}}), \\quad P_T= -A_T-A_T^{\\mathsf{H}}-F_T^{\\mathsf{H}}SF_T.\n\\]\nThus, if we use a pH representation ${\\mathcal M}=\\{A_T,B_T,C_T,D_T\\}=\\{J-R,G-K,(G+K)^{\\mathsf{H}},D\\}$, then at the analytic center of the modified barrier function, we have\n\\begin{eqnarray*}\nSF_T&=&K^{\\mathsf{H}}, \\ P_T= R -F_T^{\\mathsf{H}}SF_T, \\\\\n0&=&  P_T(J-GF_T)+(J-GF_T)^{\\mathsf{H}}P_T,\n\\end{eqnarray*}\nand of course $X_T=I$, which implies that the passivity radius is given by\n\\[\n \\lambda_{\\min} \\left[ \\begin{array}{cc} R & K \\\\ K^{\\mathsf{H}} & S \\end{array}\\right].\n\\]\nOn the other hand, since\nwe have optimized the determinant of $\\tilde W(X)$ which has the same determinant as \\eqref{tilde}, it follows that\n$$  \\det \\tilde W(X)= \\det \\left[ \\begin{array}{cc} R & K \\\\ K^{\\mathsf{H}} & S \\end{array}\\right]\n$$\nand we can expect to have obtained a nearly optimal passivity margin as well.\n\n\n\n\\section{Other Radii} \\label{sec:criteria}\n\t\nEven though in this paper we are focusing on the passivity, we point out that pH representations also have other properties that are important to consider. In this section, we consider two such properties.\n\n\\subsection{The Stability Radius}\n\n\nIf a positive definite solution of the LMI (\\ref{KYP-LMI}) exists, then it follows from the positive definiteness of the  $(1,1)$ block that  the system is asymptotically stable. Hence we can employ the same technique that we have used for the characterization of the passivity radius to bound the  \\emph{stability radius}, {i.e.}, the smallest perturbation $\\Delta_A$ that makes the system loose its asymptotic stability.  Introducing the positive definite matrices\n\\begin{eqnarray}\\nonumber\nV(X)&:=& -XA-A^{\\mathsf{H}}X , \\\\ V_{\\Delta_A}(X)&:=& V(X) - X\\Delta_A-\\Delta_A^{\\mathsf{H}}X,\n\\label{VDelta}\n\\end{eqnarray}\nwe define the \\emph{$X$-stability radius} as the smallest $\\Delta_A$, for which $V_{\\Delta_A}(X)$ looses its positive definiteness, {i.e.} for\n$X\\in {\\mathcal X}^{>0}$ with $V(X)> 0$, the  $X$-stability radius is defined as\n\t%\n\t\\[\n\t\\rho_A(X):= \\inf_{\\Delta_A\\in \\mathbb{C}^{n \\times n}}\\left\\{ \\| \\Delta_A \\| \\; | \\; \\det V_{\\Delta_A}(X) = 0\\right\\}.\n\t\\]\n\t%\nNote that \\eqref{VDelta} is defined similar to  \\eqref{WDelta}, except for a sign change and therefore as for the passivity radius we obtain the bound\n\\[\n\\alpha\\beta\/2 \\le \\rho_A(X) \\le \\alpha\\beta\/(1+|u^{\\mathsf{H}}w|),\n\\]\nwhere $\\alpha^2 = \\lambda_{\\min}(V)$, $\\beta^2 = \\lambda_{\\min}(X^{-1}VX^{-1})$, and where $u$, $v$ and $w$ are normalized vectors satisfying\n\\[\nV^{-\\frac{1}{2}}u=u\/\\alpha, \\quad V^{-\\frac{1}{2}} X v=w\/\\beta.\n\\]\n\\begin{corollary}\\label{cor:xeqIstab} If for a given system $\\mathcal M$ we have that $X=I_n\\in \\mathrm{Int}\\XWpd$   then at this point the corresponding representation of the system is port-Hamiltonian, {i.e.}, it has the representation ${\\mathcal M}:= \\{J-R,G-K,G^{\\mathsf{H}}+K^{\\mathsf{H}},D\\}$  and the X-stability radius of this model is given by\n\\[\n\t\\rho_A(I)=\\lambda_{\\min}V(I)=\\lambda_{\\min} (R).\n\\]\n\\end{corollary}\n{\\bf Proof}\n\tThis follows directly from Theorem~\\ref{thm:Xpassivity}, since then  $\\alpha=\\beta$ and we can choose $u=w$.\n\\hfill $\\Box$\n\n\\begin{remark}\\label{rem:stabrad}{\\rm\n\t\tIt follows from the conditions $V\\geq \\alpha^2 I_{n+m}$ and $X^{-1} V X^{-1}\\geq \\beta^2 I_{n+m}$ that a necessary (but not sufficient) condition for stability is given by $T^{-1}VT^{-1}\\geq \\alpha\\beta I_n$, where $T=X^{\\frac12}$.\n\t}\n\\end{remark}\nAnother robustness measure for the transformation $T$ is to require that the \\emph{field of values} $\\{ x^{\\mathsf{H}}A_Tx | x\\in \\mathbb C^n\\}$ of the transformed matrix $A_T$ is as far left as possible into the left half plane. In other words, we want to minimize the real part of the right most \\emph{Rayleigh quotient} of $A_T$ given by\n\\[\n\\min_{T s.t. T^2\\in \\XWpd} \\{ \\max_{x\\neq 0, x\\in \\mathbb{C}^n} {\\mathfrak{Re}}(\\frac{x^{\\mathsf{H}}A_Tx}{x^Hx}) \\}.\n\\]\nWriting $A_T=J_T-R_T$ with $J_T=-J_T^{\\mathsf{H}}$ and $R_T=R_T^{\\mathsf{H}}$  we clearly only need to $x^{\\mathsf{H}}R_Tx$, since ${\\mathfrak{Re}} (x^{\\mathsf{H}}J_Tx)=0$. In other words, we want to determine\n\\[\n\\min_{T\\in \\XWpd} \\{ \\max_{x\\neq 0, x\\in \\mathbb{C}^n} (\\frac{x^{\\mathsf{H}}R_Tx}{x^{\\mathsf{H}}x}) \\},\n\\]\nwhich amounts to maximizing the smallest eigenvalue of the $(1,1)$ block of the LMI~(\\ref{KYP-LMI}).\nIt therefore is an alternative to maximize the determinant of $W_T(X)$, since this will tend to maximize all of its eigenvalues, including those of the principal submatrices.\n\nWe are not advocating here to use either of these two approaches to compute the stability radius of our system, since we know that it is given explicitly by the formula\n\n\\[ \\rho_A =  \\min_{\\omega\\in \\mathbb{R} } \\sigma_{\\min}(A-\\imath \\omega I_n).\n\\]\nWe just want to stress here that using a pH realization based on the analytic center will also yield a robust stability margin.\n\n\n\\subsection{Well conditioned state-space transformations}\\label{sec:cond}\n\nSince for any solution $X\\in \\XWpd$, $T=X^{\\frac 12}$  yields a state space transformation to pH form, we can also try to optimize the condition number of $T$ or directly the condition number of $X=T^2\\in \\XWpd$ within the set described by $ X_- \\leq X \\leq X_+ $.\n\nLet us first consider the special case that $X_+$ and $X_-$ commute. In this  case there exists a unitary transformation $U$ that  simultaneously diagonalizes both $X_-$ and $X_+$, {i.e.}, $\nU^{\\mathsf{H}}X_-U=\\mathrm{diag}\\{\\lambda^{(-)}_1, \\ldots, \\lambda^{(-)}_n\\}$ and  $U^{\\mathsf{H}}X_+U=\\mathrm{diag}\\{\\lambda^{(+)}_1, \\ldots, \\lambda^{(+)}_n\\}$.\nSince $ X_- \\leq X \\leq X_+ $, it follows that each eigenvalue $\\lambda_i$ of $X$, $i=1,\\ldots,n$ must lie in the closed interval\n$\\lambda_i \\in [\\lambda^{(-)}_i , \\lambda^{(+)}_i]$, and that these intervals are nonempty. If there exists a point $\\lambda$\nin the intersection of all these intervals, then $X=\\lambda I_n$ is an optimal choice and it has condition number $\\kappa(X)=1$. If not, then there are at least two non-intersecting intervals, which implies that\n$\\lambda^{(+)}_{\\min} < \\lambda^{(-)}_{\\max}$ and hence that the closed interval $[\\lambda^{(+)}_{\\min} , \\lambda^{(-)}_{\\max}]$ must then be non-empty. Moreover, it must also intersect each of the intervals $[\\lambda^{(-)}_i , \\lambda^{(+)}_i]$ in at least one point.\nThus, if we choose for any $i=1,\\ldots,n$\n\\[\n \\lambda_i \\in [\\lambda^{(-)}_i , \\lambda^{(+)}_i] \\cap [\\lambda^{(+)}_{\\min} , \\lambda^{(-)}_{\\max}],\n\\]\nthen the resulting matrix will have optimal condition number $\\kappa(X)=\n\\frac{\\lambda^{(-)}_{\\max}}{\\lambda^{(+)}_{\\min}}$, and hence $\\kappa(T)=\\sqrt{\\frac{\\lambda^{(-)}_{\\max}}{\\lambda^{(+)}_{\\min}}}$.\nThe proof that this is optimal follows from the Loewner ordering of positive semidefinite matrices. The largest eigenvalue of $X$ must be larger or equal to $\\lambda_{\\max}^{(-)}$ and the smallest eigenvalue of $X$ must be smaller or equal to  $\\lambda_{\\min}^{(+)}$.\n\nIf  $X_+$ and $X_-$ do not commute, then there still exists a (non-unitary) congruence transformation $L$ which simultaneously diagonalizes both $D^{(-)}:=L^{\\mathsf{H}}X_-L$ and $D^{(+)}:=L^{\\mathsf{H}}X_+L$ but the resulting diagonal elements $d^{(-)}_i$\nand $d^{(+)}_i$ are not the eigenvalues anymore. Nevertheless, the same construction holds for any matrix $X$, but we cannot prove optimality anymore. On the other hand, we can guarantee the bound\n\\[\n\\kappa(T)\\le \\max \\left (\\kappa(L), \\kappa(L) \\sqrt{\\frac{d^{(-)}_{\\max}}{d^{(+)}_{\\min}}}\\right ).\n\\]\n\n\\section{Numerical examples}\\label{sec:num}\nIn this section we present a few numerical examples for realizations that are construct on the basis of the analytic center.\n\nWe first look at a real scalar transfer function of first degree ($m$=$n$=1) because in this case both analytic centers that we proposed earlier, are easy to compute analytically. The transfer functions of interest are given by\n\\begin{eqnarray*} T(s)&=&d + \\frac{cb}{s-a}, \\\\\n \\Phi(s)&=&\\left[ \\begin{array}{cc} b\/(-s-a) & 1  \\end{array} \\right]\n\\left[\n\\begin{array}{cc}\n0 & c \\\\\nc & 2d\n\\end{array}\n\\right].\\left[ \\begin{array}{c} b\/(s-a) \\\\ 1 \\end{array} \\right].\n\\end{eqnarray*}\nIf we assume that it the system is strictly passive, then\n\\[\nW(x) = \\left[\\begin{array}{cc} -2ax & c-bx \\\\ c-bx & 2d \\end{array}\\right]\n\\]\nmust be positive definite for some value of $x$. This implies that $d>0$ and that\n$\\det W(x)=-4adx-(c-bx)^2=-b^2x^2-2(2ad-cb)x-c^2 $ is positive for some value of $x$.\nThis implies that the discriminant $(2ad-cb)^2-b^2c^2 = 4a^2d^2-4abcd = 4ad(ad-bc)$ must be positive. Since the system is also stable, we finally have the following necessary and sufficient conditions for strict passivity of a real first degree scalar function:\n\\[\n  a<0, \\quad d>0, \\quad \\det  \\left[\\begin{array}{cc}\na & b \\\\ c & d \\end{array}\\right] < 0.\n\\]\nIt is interesting to point out that the transfer function $\\Phi(\\jmath\\omega)$ is a non-negative and unimodal function of $\\omega$ with extrema at $0$ and $\\infty$.\nWe thus can check strict passivity by verifying the positivity of $\\Phi(\\jmath\\omega)$ at these two values, and the stability of $a$~:\n\\[ \\Phi(\\infty) = d >0 , \\ \\Phi(0)= \\frac{2(ad-cb)}{a} >0, \\ a<0.\n\\]\nSince the determinant is quadratic in $x$, it is easy to determine the analytic center  $x_*$ of the linear matrix inequality $W(x)>0$ and the corresponding feedback and Riccati operator:\n\\begin{eqnarray*} x_*&=&\\frac{c}{b}-2d\\frac{a}{b^2}, \\quad\nf=\\frac{a}{b}, \\quad  p=2d\\frac{a^2}{b^2}-2c\\frac{a}{b},\\\\\nW(x) &=& \\left[\\begin{array}{cc}\n4d\\frac{a^2}{b^2}-2c\\frac{a}{b} & 2d\\frac{a}{b} \\\\ 2d\\frac{a}{b} & 2d \\end{array}\\right]\n= \\left[\\begin{array}{cc} 1 & \\frac{a}{b} \\\\ 0 & 1 \\end{array}\\right].\n\\left[\\begin{array}{cc}p & 0 \\\\ 0  & 2d \\end{array}\\right]\n\\left[\\begin{array}{cc} 1 & 0 \\\\ \\frac{a}{b} & 1\\end{array}\\right],\n\\end{eqnarray*}\nwhich implies $\\det H(x)=2d\\cdot p$ and the strict passivity condition\n\\[ a < 0, \\quad r > 0 \\quad \\mathrm{and}  \\quad p= \\frac{2a}{b^2}(ad-bc) >0.\n\\]\nThe strict passivity is lost when either one of the following happens\n\\[ d+\\delta_d=0, \\quad a+\\delta_a = 0, \\quad \\det \\left[ \\begin{array}{cc} a+\\delta_a & b+\\delta_b \\\\ c+\\delta_c & d+\\delta_d \\end{array} \\right]=0.\n\\]\nTherefore, it follows that\n\\[\n\\rho = \\min(d, a, \\sigma_2 \\left[ \\begin{array}{ccc} a & b \\\\ c & d  \\end{array} \\right]) =  \\sigma_2 \\left[ \\begin{array}{ccc} a & b \\\\ c & d  \\end{array} \\right].\n\\]\nBut at the analytic center $x_*=(2da-cb)\/b^2$, we have\n\\[\n  \\det\n2d\\left[ \\begin{array}{cc} 2\\frac{a^2}{b^2}-\\frac{ac}{bd} & \\frac{a}{b} \\\\ \\frac{a}{b} & 1 \\end{array} \\right]=2\\frac{da}{b^2}(\\frac{a}{b}-\\frac{c}{d})\n\\]\nwhich shows that the positivity at the analytic center yields the correct condition for strict passivity of the model.\n\nIf we use the port-Hamiltonian barrier function, we have to consider only $x>0$\nsince  $\\det \\tilde W(x)=\\det W(x) \/x$. One easily checks that the derivative of\n$\\det \\tilde W(x)$ has a positive zero at $x_*=|\\frac{c}{b}|$, which eventually yields a balanced realization ${\\mathcal M}_T=\\{a,\\sqrt{|bc|}, bc\/\\sqrt{|bc|},d\\}$, and an improved passivity radius\n\\[\n \\sigma_2 \\left[ \\begin{array}{ccc} a & bc\/\\sqrt{|bc|} \\\\ \\sqrt{|bc|} & d  \\end{array} \\right].\n\\]\n\n\nAs second test case we look at a random numerical  model  $\\{A,B,C,D\\}$ in pH form of state dimension $n=6$ and input\/output dimension $m=3$  via\n\\[\n  \\left[ \\begin{array}{ccc} R & K \\\\ K^{\\mathsf{H}} & S  \\end{array} \\right] := M M^{\\mathsf{H}} ,\n\\]\nwhere $M$ is a random $(n+m)\\times(n+m)$ matrix generated in MATLAB. From this we then identified the model $A:=-R\/2, B:=-C^{\\mathsf{H}}:=-K\/2$ and $D:=S\/2$.\nThis construction guarantees us that $X_0=I_n$ satisfies the LMI positivity constraint for the model ${\\mathcal M}:=\\{A,B,C,D\\}$. We then used the Newton iteration developed in \\cite{BanMVN17_ppt} to compute the analytic center $X_c$ of the LMI\n\\[ W(X):= \\left[\n\\begin{array}{cc}\n-X\\,A - A^{\\mathsf{H}}X & C^{\\mathsf{H}} - X\\,B \\\\\nC- B^{\\mathsf{H}}X & S\n\\end{array}\n\\right] >0,\n\\]\n\nusing the barrier function $b(X):=-\\ln \\det W(X)$. We then determined the quantities $\\alpha^2 := \\lambda_{\\min}()\\hat W)$, $\\beta^2 :=  \\lambda_{\\min}(\\hat X^{-1}\\hat W \\hat X^{-1})$, and $\\xi := \\lambda_{\\min}(\\hat X^{-\\frac12}\\hat W \\hat X^{-\\frac12})$,\nwhere $\\tilde W:=W(X_c)$ and $\\tilde X := \\mathrm{diag} \\{X_c,I_m\\}$. The constructed matrix\n\\[\n\\hat X^{-\\frac12}\\hat W \\hat X^{-\\frac12}=\\left[ \\begin{array}{ccc} R_c & K_c \\\\ K_c^{\\mathsf{H}} & S_c  \\end{array} \\right]\n\\]\nalso contains the parameters of the port-Hamiltonian realization at the analytic center $X_c$. The results are given in the table below\n\\[\n\\begin{array}{c|c|c|c|c|c}\n\t \\alpha^2 & \\beta^2 & \\xi & \\alpha\\beta & \\lambda_{\\min}(R_c)  & \\rho_{stab} \\\\\n\t \\hline\n\t 0.002366 & 0.001065 &  0.002381 & 0.001587 & 0.1254 & 0.1035\n\\end{array}\n\\]\nThey indicate that $\\lambda_{\\min}(R_c)$ at the analytic center is a good approximation of the true stability radius, and that $\\lambda_{\\min}(\\hat X^{-\\frac12}\\hat W \\hat X^{-\\frac12})$ at the analytic center is a good approximation of the $X_c$-passivity radius estimate $\\alpha\\beta$.\n\n\n\\section{Conclusion} \\label{sec:conclusion}\n\nIn this paper we have introduced the concept of the analytic center for a barrier function derived from the KYP LMI $W(X)>0$ for passive systems.\nWe have shown that the analytic center yields very promising results for choosing the  coordinate transformation  to pH form for a given passive system.\n\nSeveral important issues have not been addressed yet. Can we also apply these ideas to the limiting situations where $W(X)$ is singular and\/or $X$ is singular, or when the given system is not minimal.  More importantly, one should also analyze if these ideas can also be adapted to models\nrepresented in descriptor form.\n\nAnother interesting issue is that of finding the nearest passive system to a given system that is not passive. This has an important application in identification,\nwhere may loose the property of passivity, due to computational and round-off errors incurred during the identification.\n\\section*{Acknowledgment}\nThe authors greatly appreciate the help of Daniel Bankmann from TU Berlin for the computation of the numerical example.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nIn many real-world problems, a central objective is to optimally choose a right action to maximize the profit of interest.\nFor example, in marketing, an advertising campaign is designed to promote people to purchase a product~\\citep{renault_chapter_2015}.\nA marketer can choose whether to deliver an advertisement to each individual or not,\nand the outcome is the number of purchases of the product.\nAnother example is personalized medicine, where a treatment is chosen depending on each\npatient to maximize the medical effect and minimize the risk of adverse events\nor harmful side effects~\\citep{katsanis_case_2008, abrahams_case_2009}.\nIn this case, giving or not giving a medical treatment to each individual are the possible actions to choose,\nand the outcome is the rate of recovery or survival from the disease.\nHereafter, we use the word \\emph{treatment} for taking an action, following the personalized medicine example.\n\n\\emph{A\/B testing}~\\citep{kohavi_controlled_2009} is a standard method for such tasks, where two groups of people, A and B, are randomly chosen.\nThe outcomes are measured separately from the two groups after treating all the members of Group A but none of Group B.\nBy comparing the outcomes between the two groups by a statistical test,\none can examine whether the treatment positively or negatively affected the outcome.\nHowever, A\/B testing only compares the two extreme options: treating everyone or no one.\nThese two options can be both far from optimal when the treatment has positive effect on some individuals but negative effect on others.\n\nTo overcome the drawback of A\/B testing, \\emph{uplift modeling} has been investigated recently~\\citep{radcliffe2011real, uplift-icml2012-workshop, rzepakowski2012decision}.\n\\emph{Uplift modeling} is the problem of estimating the \\emph{individual uplift}, the incremental profit brought by the treatment conditioned on features of each individual.\nUplift modeling enables us to design a refined decision rule for optimally determining whether to treat each individual or not, depending on his\/her features.\nSuch a treatment rule allows us to only target those who positively respond to the treatment and avoid treating negative responders.\n\nIn the standard uplift modeling setup, there are two types of labels~\\citep{radcliffe2011real, uplift-icml2012-workshop, rzepakowski2012decision}:\nOne is whether the treatment has been given to the individual and the other is its outcome.\nExisting uplift modeling methods require each individual to be \\emph{jointly}\ngiven these two labels for analyzing the association between outcomes and the\ntreatment~\\citep{radcliffe2011real, uplift-icml2012-workshop, rzepakowski2012decision}.\nHowever, joint labels are expensive or hard (or even impossible) to obtain in many real-world problems.\nFor example, when distributing an advertisement by email,\nwe can easily record to whom the advertisement has been sent.\nHowever, for technical or privacy reasons,\nit is difficult to keep track of those people until we observe the outcomes on whether they buy the product or not.\nAlternatively, we can easily obtain information about purchasers of the product at the moment when the purchases are actually made.\nHowever, we cannot know whether those who are buying the product have been exposed to the advertisement or not.\nThus, every individual always has one missing label.\nWe term such samples \\emph{separately labeled samples}.\n\nIn this paper, we consider a more practical uplift modeling setup\nwhere no jointly labeled samples are available, but only separately labeled samples are given.\nTheoretically, we first show that the individual uplift is identifiable when we have two sets of\nseparately labeled samples collected under \\emph{different} treatment policies.\nWe then propose a novel method that directly estimates the individual uplift only from separately labeled samples.\nFinally, we demonstrate the effectiveness of the proposed method through experiments.\n\n\\section{Problem Setting}\nThis paper focuses on estimation of the \\emph{individual uplift} $u(\\bm x)$, often called \\emph{individual treatment effect (ITE)} in the causal inference literature~\\citep{rubin_causal_2005}, defined as\n$u(\\bm x) := \\mathbf{E}[Y_1\\given \\bm x] - \\mathbf{E}[Y_{-1}\\given \\bm x]$,\nwhere\n$\\mathbf{E}[\\;\\cdot\\given\\cdot\\;]$ denotes the conditional expectation,\nand $\\bm x$ is a $\\mathcal{X}$-valued random variable\n($\\mathcal{X}\\subseteq \\Re^d$) representing features of an individual,\nand\n $Y_1, Y_{-1}$ are $\\mathcal{Y}$-valued \\emph{potential outcome variables}~\\citep{rubin_causal_2005}\n($\\mathcal{Y}\\subseteq\\Re$) representing outcomes that would be observed\nif the individual was treated and not treated, respectively.\nNote that only one of either $Y_1$ or $Y_{-1}$ can be observed for each individual.\nWe denote the $\\{1, -1\\}$-valued random variable of the treatment assignment\nby $t$, where $t = 1$ means that the individual has been treated and $t = -1$ not treated.\nWe refer to the population for which we want to evaluate $u(\\bm x)$ as the \\emph{test population},\nand denote the density of the test population by $p(Y_1, Y_{-1}, \\bm x, t)$.\n\nWe assume that $t$ is \\emph{unconfounded}\nwith either of $Y_1$ and $Y_{-1}$ conditioned on $\\bm x$,\ni.e.\\@ $p(Y_1\\given \\bm x, t) = p(Y_1\\given \\bm x)$\nand $p(Y_{-1}\\given \\bm x, t) = p(Y_{-1}\\given \\bm x)$.\nUnconfoundedness is an assumption commonly made in observational studies~\\citep{ite_gen_bound, causal_uplift_review}.\nFor notational convenience, we denote by $y := Y_t$ the outcome of the treatment assignment $t$.\nFurthermore, we refer to any conditional density of $t$ given $\\bm x$ as a \\emph{treatment policy}.\n\nIn addition to the test population,\nwe suppose that there are two \\emph{training populations} $k = 1, 2$,\nwhose joint probability density $p_k(Y_1, Y_{-1}, \\bm x, t)$\nsatisfy\n\\begin{align}\n    p_k(Y_{t_0}=y_0\\given\\bm x = \\bm x_0)\n    &= p(Y_{t_0}=y_0\\given\\bm x=\\bm x_0)\\quad \\text{(for $k=1, 2$)},\\label{eq:1st_cond}\\\\\n    p_1(t=t_0\\mid \\bm x=\\bm x_0)\n    &\\neq p_2(t=t_0 \\mid \\bm x=\\bm x_0), \\label{eq:conditions_on_p}\n\\end{align}\nfor all possible realizations $\\bm x_0\\in\\mathcal{X}$, $t_0\\in\\{-1, 1\\}$, and $y_0\\in\\mathcal{Y}$.\nIntuitively, Eq.~\\eqref{eq:1st_cond} means that\npotential outcomes depend on $\\bm x$ in the same way as those in the test population,\nand Eq.~\\eqref{eq:conditions_on_p} states that those two policies give a treatment with different probabilities for every $\\bm x = \\bm x_0$.\n\nWe suppose that the following four training data sets, which we call \\emph{separately labeled samples}, are given:\n\\begin{align*}\n\\{(\\bm x^{(k)}_i, y^{(k)}_i)\\}_{i=1}^{n_k}\\stackrel{\\mathrm{i.i.d.}}{\\sim} p_k(\\bm x, y),\\quad\n \\{(\\widetilde{\\bm x}^{(k)}_i, t^{(k)}_i)\\}_{i=1}^{\\widetilde{n}_k}\\stackrel{\\mathrm{i.i.d.}}{\\sim} p_k(\\bm x, t)\\quad \\text{(for $k=1, 2$)},\n\\end{align*}\nwhere $n_k$ and $\\widetilde{n}_k$, $k=1, 2$, are positive integers.\nUnder Assumptions~\\eqref{eq:1st_cond}, \\eqref{eq:conditions_on_p}, and the unconfoundedness, we have\n$p_k(Y_t \\given\\bm x, t=t_0) = p(Y_{t_0}\\given\\bm x, t=t_0) = p(Y_{t_0}\\given\\bm x)$\nfor $t_0\\in\\{-1,1\\}$ and $k\\in\\{1, 2\\}$.\nNote that we can safely denote $p(y \\given \\bm x, t) := p_k(y\\given \\bm x, t)$.\nMoreover, we have $\\mathbf{E}[Y_{t_0}\\given\\bm x] = \\mathbf{E}[y\\given\\bm x, t=t_0]$ for $t_0=1, -1$,\nand thus our goal boils down to the estimation of\n\\begin{align}\n    u(\\bm x) = \\mathbf{E}[y\\given\\bm x,t=1] - \\mathbf{E}[y\\given\\bm x,t=-1]\\label{eq:i_ul}\n\\end{align}\nfrom the separately labeled samples,\nwhere the conditional expectation is taken over $p(y\\given\\bm x, t)$.\n\nEstimation of the individual uplift is important for the following reasons.\n\n\\textbf{It enables the estimation of the average uplift.}\nThe \\emph{average uplift} $U(\\pi)$ of the treatment policy $\\pi(t\\given\\bm x)$ is the\naverage outcome of $\\pi$, subtracted by that of the policy $\\pi_{-}$,\nwhich constantly assigns the treatment as $t=-1$,\ni.e., $\\pi_-(t=\\tau\\given\\bm x) := 1[\\tau=-1]$,\nwhere $1[\\cdot]$ denotes the indicator function:\n\\begin{align}\n  U(\\pi)\n  &:= \\iint \\sum_{t=-1,1}y p(y\\given\\bm x,t)\\pi(t\\given\\bm x) p(\\bm x) \\mathrm{d}y\\dbm x\n  -\\iint \\sum_{t=-1,1} y p(y\\given\\bm x,t)\\pi_-(t\\given\\bm x) p(\\bm x) \\mathrm{d}y\\dbm x\\nonumber\\\\\n  &= \\int u(\\bm x) \\pi(t = 1\\given\\bm x) p(\\bm x) \\dbm x.\\label{eq:AO_AU}\n\\end{align}\nThis quantity can be estimated from samples of $\\bm x$ once we obtain an estimate of $u(\\bm x)$.\n\n\\textbf{It provides the optimal treatment policy.}\nThe treatment policy given by $\\pi(t=1\\given\\bm x) = 1[0 \\le u(\\bm x)]$ is the\noptimal treatment that maximizes the average uplift $U(\\pi)$ and equivalently the average outcome $\\iint \\sum_{t=-1,1}y\np(y\\given\\bm x,t)\\pi(t\\given\\bm x) p(\\bm x) \\mathrm{d}y\\dbm x$ (see Eq.~\\eqref{eq:AO_AU})~\\citep{rzepakowski2012decision}.\n\n\\textbf{It is the optimal ranking scoring function.}\nFrom a practical viewpoint, it may be useful to prioritize\nindividuals to be treated according to some ranking scores\nespecially when the treatment is costly and only a limited number of\nindividuals can be treated due to some budget constraint.\nIn fact, $u(\\bm x)$ serves as the optimal ranking scores for this purpose~\\citep{tuffery_data_2011}.\nMore specifically, we define a family of treatment policies $\\{\\pi_{f,\n  \\alpha}\\}_{\\alpha\\in\\Re}$\n\\emph{associated with scoring function $f$} by $\\pi_{f,\\alpha}(t=1\\given\\bm x) =\n1[\\alpha \\le f(\\bm x)]$. Then, under some technical condition, $f = u$\nmaximizes the \\emph{area under the uplift curve (AUUC)} defined as\n\\begin{align*}\n  \\AUUC(f)\n    &:= \\int_0^1 U(\\pi_{f, \\alpha}) \\mathrm{d}C_\\alpha\\\\\n    &= \\int_0^1 \\int u(\\bm x) 1[\\alpha \\le f(\\bm x)] p(\\bm x) \\dbm x \\mathrm{d}C_\\alpha\\\\\n    &= \\mathbf{E}[1[f(\\bm x) \\le f(\\bm x')] u(\\bm x')],\n\\end{align*}\nwhere $C_\\alpha := \\Pr[f(\\bm x) < \\alpha]$, $\\bm x,\\bm\nx'\\stackrel{\\mathrm{i.i.d.}}{\\sim} p(\\bm x)$, and $\\mathbf{E}$ denotes the expectation\nwith respect to these variables.\nAUUC is a standard performance measure for uplift modeling\nmethods~\\citep{radcliffe_using_2007, radcliffe2011real, uplift-icml2012-workshop, rzepakowski2012decision}.\nFor more details, see Appendix~\\ref{sec:AUUC_ranking} in the supplementary material.\n\n\\textbf{Remark on the problem setting:}\nUplift modeling is often referred to as individual treatment effect estimation or heterogeneous treatment effect estimation\nand has been extensively studied especially in the causal inference literature~\\citep{rubin_causal_2005, pearl2009causality, causal_uplift_review, hill_bayesian_2011, imai_estimating_2013, wager_estimation_2015, johansson_learning_2016, kunzel_meta-learners_2017}.\nIn particular, recent research has investigated the problem under the setting of \\emph{observational studies}, inference using data obtained from \\emph{uncontrolled experiments} because of its practical importance~\\citep{ite_gen_bound}.\nHere, experiments are said to be uncontrolled when some of treatment variables are not controlled to have designed values.\n\nGiven that treatment policies are unknown, our problem setting is also of observational studies\nbut poses an additional challenge that stems from missing labels.\nWhat makes our problem feasible is that we have two kinds of data sets\nfollowing different treatment policies.\n\nIt is also important to note that our setting generalizes the standard setting for observational studies since the former is reduced to the latter when one of the treatment policies always assigns individuals to the treatment group, and the other to the control group.\n\nOur problem is also closely related to individual treatment effect estimation via instrumental variables~\\citep{imbens_instrumental_2014, athey_generalized_2016, hartford_deep_2017, lewis_adversarial_2018}.\\footnote{%\nAmong the related papers mentioned above, the most relevant one is~\\citet{lewis_adversarial_2018},\nwhich is concurrent work with ours.\n}\n\n\\section{Naive Estimators}\\label{sec:naive}\nA naive approach is first estimating the conditional density\n$p_k(y\\given\\bm x)$ and $p_k(t\\given\\bm x)$ from training samples by some\nconditional density estimator \\citep{bishop2006PRML,\n  sugiyama2010conditional}, and then solving the following linear system for\n$p(y\\given\\bm x, t=1)$ and $p(y\\given\\bm x, t=-1)$:\n\\begin{align}\n  \\underbrace{p_k(y\\given \\bm x)}_{\\text{Estimated from $\\{(\\bm x^{(k)}_i, y^{(k)}_i)\\}_{i=1}^{n}$}} &= \\sum_{t=-1,1} p(y\\given\\bm x, t) \\underbrace{p_k(t\\given\\bm x)}_{\\text{Estimated from $\\{(\\widetilde{\\bm x}^{(k)}_i, t^{(k)}_i)\\}_{i=1}^{\\widetilde{n}}$}}\n  \\quad \\text{(for $k = 1, 2$)}.\\label{eq:p_y_given_x}\n\\end{align}\nAfter that, the conditional expectations of $y$ over $p(y\\given\\bm x,\nt=1)$ and $p(y\\given\\bm x, t=-1)$ are calculated by numerical integration, and finally\ntheir difference is calculated to obtain another estimate of $u(\\bm x)$.\n\nHowever, this may not yield a good estimate due to the difficulty of\nconditional density estimation and the instability of numerical integration.\nThis issue may be alleviated by working on the following linear system implied by Eq.~\\eqref{eq:p_y_given_x} instead:\n$\\mathbf{E}_k[y\\given\\bm x] = \\sum_{t=-1,1}\\mathbf{E}[y\\given\\bm x, t]p_k(t\\given\\bm x)$, $k=1, 2$,\nwhere $\\mathbf{E}_k[y\\given\\bm x]$ and $p_k(t\\given\\bm x)$ can be estimated from our samples.\nSolving this new system for $\\mathbf{E}[y\\given\\bm x, t=1]$ and $\\mathbf{E}[y\\given\\bm x, t=-1]$ and taking their difference gives an estimate of $u(\\bm x)$.\nA method called \\emph{two-stage least-squares} for instrumental variable regression takes such an approach~\\citep{imbens_instrumental_2014}.\n\nThe second approach of estimation $E_k[y|x]$ and $p_k(t|x)$ avoids both conditional density estimation and numerical\nintegration, but it still involves post processing of solving the linear system and\nsubtraction, being a potential cause of performance deterioration.\n\n\n\\section{Proposed Method}\nIn this section, we develop a method that can overcome the aforementioned problems\nby directly estimating the individual uplift.\n\n\n\\subsection{Direct Least-Square Estimation of the Individual Uplift}\n\\label{sec:direct_estimation}\nFirst, we will show an important lemma that directly relates the marginal distributions of separately labeled samples to the individual uplift $u(\\bm x)$.\n\n\\begin{lemma}\\label{lem:u_is_diff_ratio}\nFor every $\\bm x$ such that $p_1(\\bm x) \\neq p_2(\\bm x)$, $u(\\bm x)$ can be\nexpressed as\n\\begin{align}\n  u(\\bm x)\n  = 2\\times\\frac{\\mathbf{E}_{y\\sim p_1(y\\given \\bm x)}[y] - \\mathbf{E}_{y\\sim p_2(y\\given \\bm x)}[y]}%\n{\\mathbf{E}_{t\\sim p_1(t\\given\\bm x)}[t] - \\mathbf{E}_{t\\sim p_2(t\\given\\bm x)}[t]}\\label{eq:diff_ratio}.\n\\end{align}\n\\end{lemma}\nFor a proof, refer to Appendix~\\ref{sec:proof_of_lemma1} in the supplementary material.\n\nUsing Eq.~\\eqref{eq:diff_ratio}, we can re-interpret the naive methods described\nin Section~\\ref{sec:naive} as estimating the conditional expectations on the right-hand side by separately performing regression on\n$\\{(\\bm x^{(1)}_i, y^{(1)}_i)\\}_{i=1}^{n_1}$, $\\{(\\bm x^{(2)}_i, y^{(2)}_i)\\}_{i=1}^{n_2}$,\n$\\{(\\widetilde{\\bm x}^{(1)}_i, t^{(1)}_i)\\}_{i=1}^{\\widetilde{n}_1}$, and $\\{(\\widetilde{\\bm x}^{(2)}_i, t^{(2)}_i)\\}_{i=1}^{\\widetilde{n}_2}$.\nThis approach may result in unreliable performance when the denominator\nis close to zero, i.e., $p_1(t\\given\\bm x) \\simeq p_2(t\\given\\bm x)$.\n\nLemma~\\ref{lem:u_is_diff_ratio} can be simplified by introducing auxiliary variables $z$ and $w$, which are $\\mathcal{Z}$-valued and $\\{-1, 1\\}$-valued random variables\nwhose conditional probability density and mass are defined by\n\\begin{align*}\n  \\textstyle p(z=z_0\\given\\bm x) &= \\textstyle\\frac{1}{2} p_1(y=z_0\\mid\\bm x)\n                                     + \\frac{1}{2} p_2(y=-z_0\\given\\bm x),\\\\\n  \\textstyle p(w=w_0\\given\\bm x) &= \\textstyle\\frac{1}{2} p_1(t=w_0\\mid\\bm x) + \\frac{1}{2} p_2(t=-w_0\\given\\bm x),\n\\end{align*}\nfor any $z_0\\in\\mathcal{Z}$ and any $w_0\\in\\{-1, 1\\}$,\nwhere $\\mathcal{Z} := \\{s_0 y_0\\given y_0\\in\\mathcal{Y}, s_0\\in\\{1, -1\\}\\}$.\n\n\\begin{lemma}\\label{lem:u_is_ratio}\nFor every $\\bm x$ such that $p_1(\\bm x) \\neq p_2(\\bm x)$, $u(\\bm x)$ can be\nexpressed as\n\\begin{align*}\nu(\\bm x) = 2\\times\\frac{\\mathbf{E}[z \\given \\bm x]}{\\mathbf{E}[w \\mid \\bm x]},\n\\end{align*}\nwhere $\\mathbf{E}[z\\given\\bm x]$ and $\\mathbf{E}[w\\given\\bm x]$ are the conditional expectations\nof $z$ given $\\bm x$ over $p(z\\given\\bm x)$ and $w$ given $\\bm x$ over\n$p(w\\given\\bm x)$, respectively.\n\\end{lemma}\nA proof can be found in Appendix~\\ref{sec:proof_of_lemma2} in the supplementary material.\n\nLet $w^{(k)}_{i} := (-1)^{k-1} t^{(k)}_i$ and $z^{(k)}_{i} := (-1)^{k-1} y^{(k)}_i$.\nAssuming that $p_1(\\bm x) = p_2(\\bm x) =: p(\\bm x)$, $n_1 = n_2$, and $\\widetilde{n}_1 =\n\\widetilde{n}_2$ for simplicity,\n$\\{(\\widetilde{\\bm x}_i, w_i)\\}_{i=1}^n := \\{(\\widetilde{\\bm x}^{(k)}_i,\nw^{(k)}_{i})\\}_{k=1,2;\\ i=1,\\dots,\\widetilde{n}_k}$ and $\\{(\\bm x_i, z_i)\\}_{i=1}^n := \\{(\\bm x^{(k)}_i,\nz^{(k)}_{i})\\}_{k=1,2;\\ i=1,\\dots,n_k}$ can\nbe seen as samples drawn from $p(\\bm x, z) := p(z\\given\\bm x)p(\\bm x)$\nand $p(\\bm x, w) := p(w\\given\\bm x)p(\\bm x)$, respectively, where $n=n_1+n_2$ and $\\widetilde{n}=\\widetilde{n}_1+\\widetilde{n}_2$.\nThe more general cases where $p_1(\\bm x) \\neq p_2(\\bm x)$, $n_1 \\neq\nn_2$, or $\\widetilde{n}_1 \\neq \\widetilde{n}_2$ are discussed in Appendix~\\ref{sec:different_p_or_n} in the supplementary material.\n\n\\begin{theorem}\\label{thm:u_by_ls}\nAssume that $\\mu_w, \\mu_z \\in L^2(p)$ and $\\mu_w(\\bm x) \\neq 0$ for every $\\bm x$ such that $p(\\bm x) > 0$,\nwhere $L^2(p) := \\{f: \\mathcal{X}\\to\\Re \\given \\mathbf{E}_{\\bm x\\sim p(\\bm x)}[f(\\bm x)^2] < \\infty\\}$.\nThe individual uplift $u(\\bm x)$ equals the\nsolution to the following least-squares problem:\n\\begin{align}\n  u(\\bm x) = \\argmin_{f\\in L^2(p)} \\mathbf{E}[(\\mu_w(\\bm x)f(\\bm x) - 2\\mu_z(\\bm x))^2]\\label{eq:ls},\n\\end{align}\nwhere $\\mathbf{E}$ denotes the expectation over $p(\\bm x)$,\n$\\mu_w(\\bm x) := \\mathbf{E}[w\\given\\bm x]$, and $\\mu_z(\\bm x) := \\mathbf{E}[z\\given\\bm x]$.\n\\end{theorem}\nTheorem~\\ref{thm:u_by_ls} follows from Lemma~\\ref{lem:u_is_ratio}.\nNote that $p_1(\\bm x) \\neq p_2(\\bm x)$ in Eq.~\\eqref{eq:conditions_on_p}\nimplies $\\mu_w(\\bm x)\\neq 0$.\n\nIn what follows, we develop a method that directly estimates $u(\\bm x)$\nby solving Eq.~\\eqref{eq:ls}.\nA challenge here is that it is not straightforward to evaluate the objective\nfunctional since it involves unknown functions, $\\mu_w$ and $\\mu_z$.\n\n\n\\subsection{Disentanglement of $z$ and $w$}\nOur idea is to transform the objective functional in Eq.~\\eqref{eq:ls} into\nanother form in which $\\mu_w(\\bm x)$ and $\\mu_z(\\bm x)$ appear separately and linearly inside the expectation operator so that we can approximate them using our separately labeled samples.\n\nFor any function $g \\in L^2(p)$ and any $\\bm x\\in\\mathcal{X}$, expanding the left-hand side of the inequality $\\mathbf{E}[(\\mu_w(\\bm x)f(\\bm x)\n- 2\\mu_z(\\bm x) - g(\\bm x))^2] \\ge 0$, we have\n\\begin{align}\n  \\mathbf{E}[(\\mu_w(\\bm x)f(\\bm x) - 2\\mu_z(\\bm x))^2]\n  \\ge 2\\mathbf{E}[(\\mu_w(\\bm x) f(\\bm x) - 2\\mu_z(\\bm x)) g(\\bm x)] - \\mathbf{E}[g(\\bm x)^2]\n    =: J(f, g).\\label{eq:lowerbound}\n\\end{align}\nThe equality is attained when $g(\\bm x) = \\mu_w(\\bm x) f(\\bm x) - \\mu_z(\\bm x)$\nfor any fixed $f$.\nThis means that the objective functional of Eq.~\\eqref{eq:ls} can be calculated\nby maximizing $J(f, g)$ with respect to $g$.\nHence,\n\\begin{align}\n  u(\\bm x)\n  &= \\argmin_{f\\in L^2(p)} \\max_{g\\in L^2(p)} J(f, g).\\label{eq:original_prob}\n\\end{align}\n\nFurthermore, $\\mu_w$ and $\\mu_z$ are separately and linearly included in $J(f, g)$,\nwhich makes it possible to write it in terms of $z$ and $w$ as\n\\begin{align}\n  J(f, g) = 2\\mathbf{E}[w f(\\bm x) g(\\bm x)] - 4\\mathbf{E}[z g(\\bm x)] - \\mathbf{E}[g(\\bm x)^2].\\label{eq:Lfg}\n\\end{align}\nUnlike the original objective functional in Eq.~\\eqref{eq:ls},\n$J(f, g)$ can be easily estimated using sample averages by\n\\begin{align}\n  \\widehat{J}(f, g)\n  = \\frac{2}{\\widetilde{n}} \\sum_{i=1}^{\\widetilde{n}} w_i f(\\widetilde{\\bm x}_i) g(\\widetilde{\\bm x}_i)\n  - \\frac{4}{n} \\sum_{i=1}^{n} z_i g(\\bm x_i)\n  - \\frac{1}{2n} \\sum_{i=1}^{n} g(\\bm x_i)^2\n  - \\frac{1}{2\\widetilde{n}} \\sum_{i=1}^{\\widetilde{n}} g(\\widetilde{\\bm x}_i)^2.\\label{eq:approxLfg}\n\\end{align}\n\n\nIn practice, we solve the following regularized empirical optimization problem:\n\\begin{align}\n  \\min_{f\\in F}\\max_{g\\in G} \\widehat{J}(f, g) + \\Omega(f, g),\\label{eq:approx_prob}\n\\end{align}\nwhere $F$, $G$ are models for $f$, $g$ respectively, and\n$\\Omega(f, g)$ is some regularizer.\n\nAn advantage of the proposed framework is that it is model-independent,\nand any models can be trained by optimizing the above objective. \n\nThe function $g$ can be interpreted as a \\emph{critic} of $f$ as follows.\nMinimizing Eq.~\\eqref{eq:Lfg} with respect to $f$ is equivalent to\nminimizing $J(f, g) = \\mathbf{E}[g(\\bm x) \\{\\mu_w(\\bm x) f(\\bm x) - 2\\mu_z(\\bm x)\\}]$.\n$g(\\bm x)$ serves as a good critic of $f(\\bm x)$\nwhen it makes the cost $g(\\bm x) \\{\\mu_w(\\bm x) f(\\bm x) - 2\\mu_z(\\bm x)\\}$ larger for $\\bm x$ at which $f$ makes a larger error $\\vert\\mu_w(\\bm x) f(\\bm x) - 2\\mu_z(\\bm x)\\vert$.\nIn particular, $g$ maximizes the objective above when $g(\\bm x) = \\mu_w(\\bm x) f(\\bm x) - 2\\mu_z(\\bm x)$ for any $f$, and the maximum coincides with the least-squares objective in Eq.~\\eqref{eq:ls}.\n\nSuppose that $F$ and $G$ are linear-in-parameter models:\n$F = \\{f_{\\bm\\alpha}: \\bm x \\mapsto \\bm\\alpha^\\top\\bm\\phi(\\bm x)\\given \\bm\\alpha\\in\\Re^{b_f}\\}$\nand $G = \\{g_{\\bm\\beta}: \\bm x \\mapsto \\bm\\beta^\\top\\bm\\psi(\\bm x)\\given \\bm\\beta\\in\\Re^{b_g}\\}$,\nwhere $\\bm\\phi$ and $\\bm\\psi$ are $b_{\\mathrm{f}}$-dimensional\nand $b_{\\mathrm{g}}$-dimensional vectors of basis functions in $L^2(p)$.\nThen, $\\widehat{J}(f_{\\bm\\alpha}, g_{\\bm\\beta}) = 2\\bm\\alpha^\\top\\bm A\\bm\\beta - 4\\bm b^\\top\\bm\\beta -\\bm\\beta^\\top\\bm C\\bm\\beta$,\nwhere\n\\begin{align*}\n  \\bm A &:= \\frac{1}{\\widetilde{n}} \\sum_{i=1}^{\\widetilde{n}} w_i \\bm\\phi(\\widetilde{\\bm x}_i)\\bm\\psi(\\widetilde{\\bm x}_i)^\\top,\n  \\quad\\bm b := \\frac{1}{n} \\sum_{i=1}^{n} z_i \\bm\\psi(\\bm x_i),\\\\\n  \\bm C &:= \\frac{1}{2n} \\sum_{i=1}^{n} \\bm\\psi(\\bm x_i)\\bm\\psi(\\bm x_i)^\\top\n          + \\frac{1}{2\\widetilde{n}} \\sum_{i=1}^{\\widetilde{n}}\\bm\\psi(\\widetilde{\\bm x}_i)\\bm\\psi(\\widetilde{\\bm x}_i)^\\top.\n\\end{align*}\nUsing $\\ell_2$-regularizers, $\\Omega(f, g) = \\lambda_{\\mathrm{f}} \\bm\\alpha^\\top\\bm\\alpha -\n\\lambda_{\\mathrm{g}}\\bm\\beta^\\top\\bm\\beta$ with some positive constants $\\lambda_{\\mathrm{f}}$ and $\\lambda_{\\mathrm{g}}$,\nthe solution to the inner maximization problem can be obtained in the following\nanalytical form:\n\\begin{align*}\n    \\widehat{\\bm\\beta}_{\\bm\\alpha}\n    := \\argmax_{\\bm\\beta} \\widehat{J}(f_{\\bm\\alpha}, g_{\\bm\\beta})\n    = \\widetilde{\\bm C}^{-1}(\\bm A^\\top\\bm\\alpha - 2\\bm b),\n\\end{align*}\nwhere $\\widetilde{\\bm C} = \\bm C + \\lambda_{\\mathrm{g}}\\bm I_{b_{\\mathrm{g}}}$ and\n$\\bm I_{b_{\\mathrm{g}}}$ is the $b_{\\mathrm{g}}$-by-$b_{\\mathrm{g}}$ identity matrix. Then, we can obtain the solution\nto Eq.~\\eqref{eq:approx_prob} analytically as\n\\begin{align*}\n  \\widehat{\\bm\\alpha}\n  &:= \\argmin_{\\bm\\alpha} \\widehat{J}(f_{\\bm\\alpha}, g_{\\widehat{\\bm\\beta}_{\\bm\\alpha}})\n  = 2(\\bm A\\widetilde{\\bm C}^{-1}\\bm A^\\top + \\lambda_{\\mathrm{f}}\\bm I_{b_{\\mathrm{g}}})^{-1}\\bm A \\widetilde{\\bm C}^{-1}\\bm b.\n\\end{align*}\nFinally, from Eq.~\\eqref{eq:ls},\nour estimate of $u(\\bm x)$ is given as $\\widehat{\\bm\\alpha}^\\top\\bm\\phi(\\bm x)$.\n\n\n\\textbf{Remark on model selection:}\nModel selection for $F$ and $G$ is not straightforward since the test performance measure cannot be directly evaluated with (held out) training data of our problem.\nInstead, we may evaluate the value of $J(\\widehat{f}, \\widehat{g})$, where $(\\widehat{f}, \\widehat{g})\\in F\\times G$ is the optimal solution pair to $\\min_{f\\in F}\\max_{g\\in G} \\widehat{J}(f, g)$.\nHowever, it is still nontrivial to tell if the objective value is small because the solution is good in terms of the outer minimization, or because it is poor in terms of the inner maximization.\nWe leave this issue for future work.\n\n\n\\section{Theoretical Analysis}\nA theoretically appealing property of the proposed method is that its objective\nconsists of simple sample averages.\nThis enables us to establish a generalization error bound in terms of the Rademacher complexity~\\citep{koltchinskii_rademacher_2001, mohri2012foundations}.\n\nDenote\n$\\varepsilon_G(f) := \\sup_{g\\in L^2(p)} J(f, g) - \\sup_{g\\in G} J(f, g)$.\nAlso, let $\\mathfrak{R}_{q}^N(H)$ denote the \\emph{Rademacher complexity} of a set of\nfunctions $H$ over $N$ random variables following probability density $q$ (refer\nto Appendix~\\ref{sec:gen_bound} for the definition).\nProofs of the following theorems and corollary can be found in Appendix~\\ref{sec:gen_bound}, Appendix~\\ref{sec:proof_of_cor1}, and Appendix~\\ref{sec:proof_of_MSE_bound} in the supplementary material.\n\n\\begin{theorem}\\label{thm:err_bound}\n  Assume that $n_1 = n_2$, $\\widetilde{n}_1 = \\widetilde{n}_2$,\n  $p_1(\\bm x) = p_2(\\bm x)$,\n  $W := \\inf_{x\\in\\mathcal{X}}\\abs{\\mu_w(\\bm x)} > 0$,\n  $M_{\\mathcal{Z}} := \\sup_{z\\in\\mathcal{Z}}\\abs{z} < \\infty$,\n  $M_{F} := \\sup_{f\\in F, x\\in\\mathcal{X}}\\abs{f(x)} < \\infty$,\n  and $M_{G} := \\sup_{g\\in G, x\\in\\mathcal{X}}\\abs{g(x)} < \\infty$.\n  Then, the following holds with probability at least $1-\\delta$\n  for every $f\\in F$:\n  \\begin{align*}\n    \\mathbf{E}_{\\bm x\\sim p(\\bm x)}[(f(\\bm x) - u(\\bm x))^2]\n    \\le \\frac{1}{W^2}\\left[\n        \\sup_{g\\in G}\\widehat J(f, g)\n        + \\mathcal{R}_{F, G}^{n, \\widetilde{n}}\n        + \\left(\\frac{M_z}{\\sqrt{2n}}\n        + \\frac{M_w}{\\sqrt{2\\widetilde{n}}}\\right)\\sqrt{\\log\\frac{2}{\\delta}}\n        + \\varepsilon_G(f)\n      \\right],\n  \\end{align*}\n  where $M_z := 4M_{\\mathcal{Y}}M_{\\mathcal{G}} + M_{\\mathcal{G}}^2\/2$,\n  $M_w = 2 M_{\\mathcal{F}}M_{\\mathcal{G}} + M_{\\mathcal{G}}^2\/2$,\n  and\n  $\\mathcal{R}_{F, G}^{n, \\widetilde{n}} := 2(M_F + 4M_Z)\\mathfrak{R}_{p(\\bm x, z)}^{n}(G) + 2(2M_F +\n  M_G)\\mathfrak{R}_{p(\\bm x, w)}^{\\widetilde{n}}(F) + 2(M_F + M_G)\\mathfrak{R}_{p(\\bm x, w)}^{\\widetilde{n}}(G)$.\n\\end{theorem}\n\nIn particular, the following bound holds for the linear-in-parameter models.\n\\begin{corollary}\\label{cor:ls_bound}\n    Let $F = \\{x\\mapsto \\bm\\alpha^\\top\\bm\\phi(\\bm x) \\given \\norm{\\bm\\alpha}_2 \\le \\Lambda_F\\}$,\n    $G = \\{x\\mapsto \\bm\\beta^\\top\\bm\\psi(\\bm x) \\given \\norm{\\bm\\beta}_2 \\le \\Lambda_G\\}$.\n    Assume that $r_F := \\sup_{\\bm x\\in\\mathcal{X}}\\norm{\\bm\\phi(\\bm x)} < \\infty$\n    and $r_G := \\sup_{\\bm x\\in\\mathcal{X}}\\norm{\\bm\\psi(\\bm x)} < \\infty$,\n    where $\\norm{\\cdot}_2$ is the $L_2$-norm.\n    Under the assumptions of Theorem~\\ref{thm:err_bound},\n    it holds with probability at least $1-\\delta$ that for every $f\\in F$,\n    \\begin{align*}\n      \\mathbf{E}_{\\bm x\\sim p(\\bm x)}[(f(\\bm x) - u(\\bm x))^2]\n      \\le \\frac{1}{W^2}\\left[\n          \\sup_{g\\in G}\\widehat J(f, g)\n          + \\frac{C_z\\sqrt{\\log\\frac{2}{\\delta}}+D_z}{\\sqrt{2n}}\n          + \\frac{C_w\\sqrt{\\log\\frac{2}{\\delta}}+D_w}{\\sqrt{2\\widetilde{n}}}\n          + \\varepsilon_G(f)\n      \\right],\n    \\end{align*}\n    where\n    $C_z := r_G^2 \\Lambda_G^2 + 4 r_G \\Lambda_G M_\\mathcal{Y}$,\n    $C_w := 2 r_F^2 \\Lambda_F^2 + 2 r_F r_G \\Lambda_F\\Lambda_G+ r_G^2 \\Lambda_G^2 $,\n    $D_z := r_G^2 \\Lambda_G^2\/2 + 4r_G \\Lambda_G M_\\mathcal{Y}$,\n    and $D_w := r_G^2\\Lambda_G^2\/2 + 4r_F r_G\\Lambda_F \\Lambda_G$.\n\\end{corollary}\nTheorem~\\ref{thm:err_bound} and Corollary~\\ref{cor:ls_bound} imply that minimizing\n$\\sup_{g\\in G} \\widehat{J}(f, g)$, as the proposed method does,\namounts to minimizing an upper bound of the mean squared error.\nIn fact, for the linear-in-parameter models, it can be shown that\nthe mean squared error of the proposed estimator is upper bounded\nby $O(\\nicefrac{1}{\\sqrt{n}} + \\nicefrac{1}{\\sqrt{\\widetilde{n}}})$\nplus some model mis-specification error with high probability as follows.\n\n\\newcommand{O\\left(\\left(\\frac{1}{\\sqrt{n}}+\\frac{1}{\\sqrt{\\widetilde{n}}}\\right)\\log\\frac{1}{\\delta}\\right)}{O\\left(\\left(\\frac{1}{\\sqrt{n}}+\\frac{1}{\\sqrt{\\widetilde{n}}}\\right)\\log\\frac{1}{\\delta}\\right)}\n\\begin{theorem}[Informal]\\label{thm:MSE_bound}\n    Let $\\widehat{f} \\in F$ be any approximate solution to $\\inf_{f\\in F}\\sup_{g\\in G}\\widehat{J}(f, g)$ with sufficient precision.\n    Under the assumptions of Corollary~\\ref{cor:ls_bound},\n    it holds with probability at least $1-\\delta$ that\n\\begin{align}\n    \\mathbf{E}_{\\bm x\\sim p(\\bm x)}[(\\widehat{f}(\\bm x)-u(\\bm x))^2]\n    \\le O\\left(\\left(\\frac{1}{\\sqrt{n}}+\\frac{1}{\\sqrt{\\widetilde{n}}}\\right)\\log\\frac{1}{\\delta}\\right) + \\frac{2\\varepsilon_G^F + \\varepsilon_F}{W^2},\n\\end{align}\nwhere $\\varepsilon_G^F := \\sup_{f\\in F}\\varepsilon_G(f)$ and $\\varepsilon_F := \\inf_{f\\in F} J(f)$.\n\\end{theorem}\nA more formal version of Theorem~\\ref{thm:MSE_bound} can be found in Appendix~\\ref{sec:proof_of_MSE_bound}.\n\n\n\\section{More General Loss Functions}\nOur framework can be extended to more general loss functions:\n\\begin{align}\n  \\inf_{f\\in L^2(p)} \\mathbf{E}[\\ell(\\mu_w(\\bm x)f(\\bm x), 2\\mu_z(\\bm x))],\\label{eq:gen_obj}\n\\end{align}\nwhere $\\ell: \\Re\\times\\Re\\to\\Re$ is a loss function that is lower semi-continuous and convex\nwith respect to both the first and the second arguments,\nwhere a function $\\varphi:\\Re\\to\\Re$ is \\emph{lower semi-continuous} if $\\liminf_{y\\to y_0} \\varphi(y) = \\varphi(y_0)$ for every $y_0\\in\\Re$~\\citep{rockafellar_convex_1970}.\\footnote{%\n$\\liminf_{y\\to y_0} \\varphi(y) := \\lim_{\\delta\\searrow 0}\\inf_{\\abs{y-y_0} \\le \\delta} \\varphi(y)$.\n}\nAs with the squared loss, a major difficulty in solving this optimization\nproblem is that the operand of the expectation has nonlinear dependency on both\n$\\mu_w(\\bm x)$ and $\\mu_z(\\bm x)$ at the same time.\nBelow, we will show a way to transform the objective functional into a form that\ncan be easily approximated using separately labeled samples.\n\nFrom the assumptions on $\\ell$, we have $\\ell(y, y') = \\sup_{z\\in\\Re}yz -\n\\ell^*(z, y')$, where $\\ell^*(\\cdot, y')$ is the convex conjugate of the\nfunction $y\\mapsto \\ell(y, y')$ defined for any $y'\\in\\Re$ as $z \\mapsto\n\\ell^*(z, y') = \\sup_{y\\in\\Re}[yz-\\ell(y, y')]$ (see \\citet{rockafellar_convex_1970}). Hence,\n\\begin{align*}\n  \\mathbf{E}[\\ell(\\mu_w(\\bm x) f(\\bm x), 2\\mu_z(\\bm x))]\n  = \\sup_{g\\in L^2(p)} \\mathbf{E}[\\mu_w(\\bm x)f(\\bm x)g(\\bm x) - \\ell^*(g(\\bm x), 2\\mu_z(\\bm x))].\n\\end{align*}\nSimilarly, we obtain $\\mathbf{E}[\\ell^*(g(\\bm x), 2\\mu_z(\\bm x))] = \\sup_{h\\in L^2(p)}\n2\\mathbf{E}[\\mu_z(\\bm x)h(\\bm x)] - \\mathbf{E}[\\ell^*_*(g(\\bm x), h(\\bm x))]$,\nwhere $\\ell^*_*(y, \\cdot)$ is the convex conjugate of the function $y'\\mapsto\n\\ell^*(y, y')$ defined for any $y, z'\\in\\Re$ by $\\ell^*_*(y, z') := \\sup_{y'\\in\\Re}[y'z-\\ell^*(y, y')]$.\nThus, Eq.~\\eqref{eq:gen_obj} can be rewritten as\n\\begin{align*}\n  \\inf_{f\\in L^2(p)}\\sup_{g\\in L^2(p)} \\inf_{h\\in L^2(p)} K(f, g, h),\n\\end{align*}\nwhere $K(f, g, h) := \\mathbf{E}[\\mu_w(\\bm x)f(\\bm x)g(\\bm x)] - 2\\mathbf{E}[\\mu_z(\\bm x)h(\\bm x)] + \\mathbf{E}[\\ell_*^*(g(\\bm x),\nh(\\bm x))]$. Since $\\mu_w$ and $\\mu_z$ appear separately and linearly, $K(f, g, h)$ can be approximated by sample averages using separately\nlabeled samples.\n\n\n\\section{Experiments}\\label{sec:experiment}\nIn this section, we test the proposed method and compare it with baselines.\n\n\\subsection{Data Sets}\n\\label{sec:experiment_setting}\nWe use the following data sets for experiments.\n\n\\textbf{Synthetic data:}\n    Features $\\bm x$ are drawn from the two-dimensional Gaussian distribution\n    with mean zero and covariance $10\\bm I_2$.\n    We set $p(y\\given\\bm x, t)$ as the following logistic models:\n    $p(y \\given \\bm x, t) = 1\/(1 - \\exp(-y\\bm a_t^\\top \\bm x))$,\n    where $\\bm a_{-1} = (10, 10)^\\top$ and $\\bm a_{1} = (10, -10)^\\top$.\n    We also use the logistic models for $p_k(t\\given\\bm x)$: $p_1(t\\given\\bm x) = 1\/(1-\\exp(-tx_2))$ and $p_2(t\\given\\bm x) = 1\/(1-\\exp(-t\\{x_2 + b\\})$,\n    where $b$ is varied over 25 equally spaced points in $[0, 10]$.\n    We investigate how the performance changes when the difference between $p_1(t\\given\\bm x)$ and $p_2(t\\given\\bm x)$ varies.\n\n\\textbf{Email data:}\nThis data set consists of data collected in an email advertisement campaign for promoting customers to visit a website of a store~\\citep{hillstrom, email_winner}.\nOutcomes are  whether customers visited the website or not.\nWe use $4 \\times 5000$ and $2000$ randomly sub-sampled data points for training and evaluation, respectively.\n\n\\textbf{Jobs data:}\nThis data set consists of randomized experimental data obtained from a job training program called the National Supported Work Demonstration~\\citep{lalonde1986evaluating}, available at \\url{http:\/\/users.nber.org\/~rdehejia\/data\/nswdata2.html}.\nThere are 9 features, and outcomes are income levels after the training program.\nThe sample sizes are $297$ for the treatment group and $425$ for the control group.\nWe use $4\\times 50$ randomly sub-sampled data points for training\nand $100$ for evaluation.\n\n\\textbf{Criteo data:}\nThis data set consists of banner advertisement log data collected by Criteo~\\citep{couterfactual_test_bed} available at \\url{http:\/\/www.cs.cornell.edu\/~adith\/Criteo\/}.\nThe task is to select a product to be displayed in a given banner so that the click rate will be maximized.\nWe only use records for banners with only one advertisement slot.\nEach display banner has 10 features, and each product has 35 features.\nWe take the 12th feature of a product as a treatment variable\nmerely because it is a well-balanced binary variable.\nThe outcome is whether the displayed advertisement was clicked.\nWe treat the data set as the population although it is biased from the actual population since non-clicked impressions were randomly sub-sampled down to $10\\%$ to reduce the data set size.\nWe made two subsets with different treatment policies by appropriately sub-sampling according to the predefined treatment policies (see Appendix~\\ref{sec:subsample} in the supplementary material).\nWe set $p_k(t\\given\\bm x)$ as $p_1(t\\given\\bm x) = 1\/(1+\\exp(-t\\bm 1^\\top \\bm x))$ and $p_2(t\\given\\bm x) = 1\/(1+\\exp(t\\bm 1^\\top\\bm x))$, where $\\bm 1 := (1, \\dots, 1)^\\top$.\n\n\\subsection{Experimental Settings}\nWe conduct experiments under the following settings.\n\n\\textbf{Methods compared:}\nWe compare the proposed method with baselines that separately estimate the four conditional expectations in Eq.~\\eqref{eq:diff_ratio}.\nIn the case of binary outcomes, we use the logistic-regression-based (denoted by FourLogistic) and a neural-network-based method trained with the soft-max cross-entropy loss (denoted by FourNNC).\nIn the case of real-valued outcomes, the ridge-regression-based (denoted by FourRidge) and a neural-network-based method trained with the squared loss (denoted by FourNNR).\nThe neural networks are fully connected ones with two hidden layers each with 10 hidden units.\nFor the proposed method, we use the linear-in-parameter models with Gaussian basis functions centered at randomly sub-sampled training data points (see Appendix~\\ref{sec:gau_basis} for more details).\n\n\\textbf{Performance evaluation:}\nWe evaluate trained uplift models by the area under the uplift curve (AUUC)\nestimated on test samples with joint labels as well as \\emph{uplift curves}~\\citep{radcliffe_differential_1999}.\nThe uplift curve of an estimated individual uplift is the trajectory of the average uplift\nwhen individuals are gradually moved from the control group to the treated group in the descending order according to the ranking given by the estimated individual uplift.\nThese quantities can be estimated when data are randomized experiment ones.\nThe Criteo data are not randomized experiment data unlike other data sets,\nbut there are accurately logged propensity scores available.\nIn this case, uplift curves and the AUUCs can be estimated using the inverse propensity scoring~\\citep{austin2011introduction, unbiased_offline}.\nWe conduct $50$ trials of each experiment with different random seeds.\n\n\n\\subsection{Results}\nThe results on the synthetic data are summarized in Figure~\\ref{fig:toy_auuc}.\nFrom the plots, we can see that all methods perform relatively well in terms of AUUCs when the policies are distant from each other (i.e., $b$ is larger).\nHowever, the performance of the baseline methods immediately declines as the treatment policies get closer to each other (i.e., $b$ is smaller).\\footnote{%\nThe instability of performance of FourLogistic can be explained as follows. FourLogistic uses linear models, whose expressive power is limited.\nThe resulting estimator has small variance with potentially large bias. Since different $b$ induces different $u(\\bm x)$, the bias depends on $b$.\nFor this reason, the method works well for some $b$ but poorly for other $b$.\n}\nIn contrast, the proposed method maintains its performance relatively longer until $b$ reaches the point around $2$.\nNote that the two policies would be identical when $b = 0$, which makes it impossible to identify the individual uplift from their samples by any method since the system in Eq.~\\eqref{eq:p_y_given_x} degenerates.\nFigure~\\ref{fig:toy_result} highlights their performance in terms of the squared error.\nFor FourNNC, test points with small policy difference $\\abs{p_1(t=1\\given\\bm x) - p_2(t=1\\given\\bm x)}$ (colored darker) tend to have very large estimation errors.\nOn the other hand, the proposed method has relatively small errors even for such points.\nFigure~\\ref{fig:result_real} shows results on real data sets.\nThe proposed method and the baseline method with logistic regressors\nboth performed better than the baseline method with neural nets\non the Email data set (Figure~\\ref{fig:Email_ul_curve}).\nOn the Jobs data set, the proposed method again performed better than the baseline methods with neural networks.\nFor the Criteo data set, the proposed method outperformed the baseline methods (Figure~\\ref{fig:Criteo_ul_curve}).\nOverall, we confirmed the superiority of the proposed both on synthetic and real data sets.\n\n\\begin{figure}[t]\n\\centering\n\\begin{minipage}{0.31\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{result_synthetic.pdf}\n\\end{minipage}\n\\hspace{2mm}\n\\begin{minipage}{0.64\\textwidth}\n\\centering\n\\caption{Results on the synthetic data.\n     The plot shows the average AUUCs obtained by the\n     proposed method and the baseline methods\n     for different $b$.\n     $p_1(t\\given\\bm x)$ and $p_2(t\\given\\bm x)$\n     are closer to each other when $b$ is smaller.}\n\\label{fig:toy_auuc}\n\\end{minipage}\n\\end{figure}\n\n\n\\begin{figure*}[t]\n  \\centering\n  \\begin{subfigure}[c]{0.31\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{sqerr_4Logistic_b1.png}\n    \\subcaption{Baseline (FourLogistic).}\n    \\label{fig:error_4logistic_toydata}\n  \\end{subfigure}\n  \\begin{subfigure}[c]{0.31\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{sqerr_4NN_b1.png}\n    \\subcaption{Baseline (FourNNC).}\n    \\label{fig:error_4NN_toydata}\n  \\end{subfigure}\n  \\begin{subfigure}[c]{0.31\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{sqerr_MinMaxGau_b1.png}\n      \\subcaption{Proposed (MinMaxGau).}\n    \\label{fig:error_proposed_toydata}\n  \\end{subfigure}\n  \\caption{The plots show the squared errors of the estimated individual uplifts\n     on the synthetic data with $b=1$.\n     Each point is darker-colored when $\\abs{p_1(t=1\\given\\bm x)-p_2(t=1\\given\\bm x)}$ is smaller, and lighter-colored otherwise.}\n  \\label{fig:toy_result}\n  \\begin{subfigure}[c]{0.31\\textwidth}\n    \\includegraphics[width=\\textwidth]{result_Email.pdf}\n    \\subcaption{The Email data.}\n    \\label{fig:Email_ul_curve}\n  \\end{subfigure}\n  \\begin{subfigure}[c]{0.31\\textwidth}\n    \\includegraphics[width=\\textwidth]{result_Jobs.pdf}\n    \\subcaption{The Jobs data.}\n    \\label{fig:Jobs_ul_curve}\n  \\end{subfigure}\n  \\begin{subfigure}[c]{0.31\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{result_Criteo.pdf}\n      \\subcaption{The Criteo data.}\n    \\label{fig:Criteo_ul_curve}\n  \\end{subfigure}\n  \\caption{Average uplifts as well as their standard errors on real-world data sets.}\n  \\label{fig:result_real}\n\\end{figure*}\n\n\\section{Conclusion}\nWe proposed a theoretically guaranteed and practically useful method for uplift modeling or individual treatment effect estimation under the presence of systematic missing labels.\nThe proposed method showed promising results in our experiments on synthetic and real data sets.\nThe proposed framework is model-independent: any models can be used to approximate the individual uplift including ones tailored for specific problems and complex models such as neural networks.\nOn the other hand, model selection may be a challenging problem due to the min-max structure.\nAddressing this issue would be important research directions for further expanding the applicability and improving the performance of the proposed method.\n\n\n\\clearpage\n\\subsubsection*{Acknowledgments}\nWe are grateful to Marthinus Christoffel du Plessis and Takeshi Teshima for their inspiring suggestions and for the meaningful discussions.\nWe would like to thank the anonymous reviewers for their helpful comments.\nIY was supported by JSPS KAKENHI 16J07970.\nJA and FY would like to thank Adway for its support.\nMS was supported by the International Research Center for Neurointelligence (WPI-IRCN) at The University of Tokyo Institutes for Advanced Study.\n\n\\FloatBarrier\n\n\\small\n\n\\bibliographystyle{plainnat}\n\n\\subsection{Average Uplift in Terms of the Individual Uplift}\n\\begin{align}\nU(\\pi) &= \\iint \\sum_{t=-1,1} y p(y \\given t, \\bm x) \\pi(t \\given\\bm x) p(\\bm x) \\mathrm{d}y \\dbm x\n- \\iint \\sum_{t=-1,1} y p(y \\given t, \\bm x) 1[t = -1] p(\\bm x) \\mathrm{d}y\\dbm x\\nonumber\\\\\n&= \\iint y [p(y \\given t = 1, \\bm x) \\pi(t = 1\\given\\bm x)\n- p(y \\given t = -1, \\bm x)\\pi(t = 1\\given\\bm x)] p(\\bm x) \\mathrm{d}y\\dbm x\\nonumber\\\\\n&= \\iint y [p(y \\given t = 1, \\bm x) - p(y \\given t = -1, \\bm x)]\n\\pi(t = 1\\given\\bm x) p(\\bm x) \\mathrm{d}y\\dbm x\\nonumber\\\\\n&= \\int u(\\bm x) \\pi(t = 1\\given\\bm x) p(\\bm x) \\dbm x.\n\\end{align}\n\n\\subsection{Area Under the Uplift Curve and Ranking}\n\\label{sec:AUUC_ranking}\nDefine the following symbols:\n\\begin{itemize}\n  \\item $C_\\alpha := \\Pr[f(\\bm x) < \\alpha ]$,\n  \\item $U(\\alpha; f) := \\int u(\\bm x)1[\\alpha \\le f(\\bm x)]p(\\bm x)\\dbm x$,\n  \\item $\\Rank(f) :=\\mathbf{E}[1[f(\\bm x') \\le f(\\bm x)][u(\\bm x') - u(\\bm x)]]$,\n  \\item $\\AUUC(f) := \\int_0^1 U(\\alpha; f) \\mathrm{d}C_\\alpha$.\n\\end{itemize}\nThen, we have\n\\begin{align*}\n  \\AUUC(f)\n            &= \\int_{-\\infty}^\\infty U(\\alpha) \\frac{\\mathrm{d}C_\\alpha}{\\mathrm{d}\\alpha}\\mathrm{d}\\alpha\\\\\n            &= \\int_{-\\infty}^\\infty U(\\alpha) p_{f(\\bm x)}(\\alpha)\\mathrm{d}\\alpha\\\\\n            &= \\int_{\\Re^d} U(f(\\bm x)) p(\\bm x)\\dbm x\\\\\n            &= \\iint 1[f(\\bm x) \\le f(\\bm x')]u(\\bm x')p(\\bm x')\\dbm x' p(\\bm x)\\dbm x\\\\\n            &= \\mathbf{E}[1[f(\\bm x) \\le f(\\bm x')]u(\\bm x')]\\\\\n            &(= \\mathbf{E}[1[f(\\bm x) \\le f(\\bm x')][y^+ - y^-]]),\n\\end{align*}\nwhere $y^+ \\sim p(y\\given\\bm x', t=1)$ and $y^-\\sim p(y\\given\\bm x', t=-1)$.\n\nAssuming $\\Pr[f(\\bm x') = f(\\bm x)] = 0$, we have\n\\begin{align*}\n  \\Rank(f)\n  &:=\\mathbf{E}[1[f(\\bm x) \\ge  f(\\bm x')][u(\\bm x) - u(\\bm x')]]\\\\\n  &= \\mathbf{E}[1[f(\\bm x) \\ge f(\\bm x')]u(\\bm x)]\\\\\n  &\\phantom{=}- \\mathbf{E}[1[f(\\bm x) \\ge f(\\bm x')]u(\\bm x')]\\\\\n  &= \\AUUC(f)\n  - \\mathbf{E}[(1 - 1[f(\\bm x) \\le f(\\bm x')])u(\\bm x')]\\\\\n  &= \\mathbf{E}[u(\\bm x)]-2\\AUUC(f).\n\\end{align*}\nThus, $\\Rank(f) = 2(\\AUUC(f) - \\AUUC(r))$, where $r:\\Re^d\\to\\Re$ is the random\nranking scoring function that outputs $1$ or $-1$ with probability $1\/2$ for any\ninput $\\bm x$.\n$\\Rank(f)$ is maximized when $f(\\bm x) \\le f(\\bm x') \\iff u(\\bm x) \\le u(\\bm\nx')$ for almost every pair of $\\bm x\\in\\Re^d$ and $\\bm x\\in\\Re^d$.\nIn particular, $f = u$ is a maximizer of the objective.\n\n\\subsection{Proof of Lemma~\\ref{lem:u_is_diff_ratio}}\n\\label{sec:proof_of_lemma1}\n\\newtheorem*{lem:u_is_diff_ratio}{Lemma~\\ref{lem:u_is_diff_ratio}}\n\\begin{lem:u_is_diff_ratio}\nFor every $\\bm x$ such that $p_1(\\bm x) \\neq p_2(\\bm x)$, $u(\\bm x)$ can be\nexpressed as\n\\begin{align}\n  u(\\bm x)\n  = 2\\times\\frac{\\mathbf{E}_{y\\sim p_1(y\\given \\bm x)}[y] - \\mathbf{E}_{y\\sim p_2(y\\given \\bm x)}[y]}%\n{\\mathbf{E}_{t\\sim p_1(t\\given\\bm x)}[t] - \\mathbf{E}_{t\\sim p_2(t\\given\\bm x)}[t]}.\n\\end{align}\n\\end{lem:u_is_diff_ratio}\n\n\\begin{proof}\n\\begin{align*}\n  \\mathbf{E}_{y\\sim p_1(y\\given \\bm x)}[y] - \\mathbf{E}_{y\\sim p_2(y\\given \\bm x)}[y]\n  &= \\int\\sum_{\\tau=-1,1}y p(y\\given\\bm x,t=\\tau)p_1(t=\\tau\\given\\bm x)\\mathrm{d}y\\\\\n  &\\phantom{=}-\\int\\sum_{\\tau=-1,1}y p(y\\given\\bm x,t=\\tau)p_2(t=\\tau\\given\\bm x)\\mathrm{d}y\\\\\n  &= \\int\\sum_{\\tau=-1,1}y p(y\\given\\bm x,t=\\tau)(p_1(t=\\tau\\given\\bm x)-p_2(t=\\tau\\given\\bm x))\\mathrm{d}y\\\\\n  &= \\sum_{\\tau=-1,1}\\mathbf{E}_{y\\sim p(y\\given\\bm x,t=\\tau)}[y](p_1(t=\\tau\\given\\bm x)-p_2(t=\\tau\\given\\bm x))\\\\\n  &= \\mathbf{E}_{y\\sim p(y\\given\\bm x,t=1)}[y](p_1(t=1\\given\\bm x)-p_2(t=1\\given\\bm x))\\\\\n  &\\phantom{=}+\\mathbf{E}_{y\\sim p(y\\given\\bm x,t=-1)}[y](1-p_1(t=1\\given\\bm x)-1+p_2(t=1\\given\\bm x))\\\\\n  &= u(\\bm x) (p_1(t=1\\mid\\bm x) - p_2(t=1\\mid\\bm x)).\n\\end{align*}\nWhen $p_1(t=1\\mid\\bm x) \\neq p_2(t=1\\mid\\bm x)$, it holds that\n\\begin{align*}\n  u(\\bm x)\n  &= \\frac{\\mathbf{E}_{y\\sim p_1(y\\given \\bm x)}[y] - \\mathbf{E}_{y\\sim p_2(y\\given \\bm x)}[y]}%\n {p_1(t=1\\mid\\bm x) - p_2(t=1\\mid\\bm x)}\\\\\n  &= 2\\times\\frac{\\mathbf{E}_{y\\sim p_1(y\\given \\bm x)}[y] - \\mathbf{E}_{y\\sim p_2(y\\given \\bm x)}[y]}%\n{\\mathbf{E}_{t\\sim p_1(t\\mid\\bm x)}[t] - \\mathbf{E}_{t\\sim p_2(t\\mid\\bm x)}[t]}.\n\\end{align*}\n\\end{proof}\n\n\\subsection{Proof of Lemma~\\ref{lem:u_is_ratio}}\n\\label{sec:proof_of_lemma2}\n\\newtheorem*{lem:u_is_ratio}{Lemma~\\ref{lem:u_is_ratio}}\n\\begin{lem:u_is_ratio}\nFor every $\\bm x$ such that $p_1(\\bm x) \\neq p_2(\\bm x)$, $u(\\bm x)$ can be\nexpressed as\n\\begin{align*}\nu(\\bm x) = 2\\times\\frac{\\mathbf{E}[z \\given \\bm x]}{\\mathbf{E}[w \\mid \\bm x]},\n\\end{align*}\nwhere $\\mathbf{E}[z\\given\\bm x]$ and $\\mathbf{E}[w\\given\\bm x]$ are the conditional expectations\nof $z$ given $\\bm x$ over $p(z\\given\\bm x)$ and $w$ given $\\bm x$ over\n$p(w\\given\\bm x)$, respectively.\n\\end{lem:u_is_ratio}\n\n\n\\begin{proof}\nWe have\n\\begin{align*}\n  \\mathbf{E}[z\\given\\bm x] &= \\int\\zeta\\left[\\frac{1}{2}p_1(y=\\zeta\\given\\bm x)+\\frac{1}{2}p_2(y=-\\zeta\\given\\bm x)\\right]\\mathrm{d}\\zeta\\\\\n                   &= \\frac{1}{2}\\int\\zeta p_1(y=\\zeta\\given\\bm x)\\mathrm{d}\\zeta+\\frac{1}{2}\\int\\zeta p_2(y=-\\zeta\\given\\bm x)\\mathrm{d}\\zeta\\\\\n                   &= \\frac{1}{2}\\int y p_1(y\\given\\bm x)\\mathrm{d}y - \\frac{1}{2}\\int y p_2(y\\given\\bm x)\\mathrm{d}y\\\\\n                   &= \\frac{1}{2}\\mathbf{E}_{y\\sim p_1(y\\given\\bm x)}[y] - \\frac{1}{2}\\mathbf{E}_{y\\sim p_2(y\\given\\bm x)}[y].\n\\end{align*}\nSimilarly, we obtain\n\\begin{align*}\n  \\mathbf{E}[w\\given\\bm x] &= \\frac{1}{2}\\mathbf{E}_{t\\sim p_1(t\\mid\\bm x)}[t] - \\frac{1}{2}\\mathbf{E}_{t\\sim p_2(t\\mid\\bm x)}[t].\n\\end{align*}\nThus, \n\\begin{align*}\n  2\\times\\frac{\\mathbf{E}[z\\given\\bm x]}{\\mathbf{E}[w\\given\\bm x]}\n  &= 2\\times\\frac{\\mathbf{E}_{y\\sim p_1(y\\given \\bm x)}[y] - \\mathbf{E}_{y\\sim p_2(y\\given \\bm x)}[y]}%\n    {\\mathbf{E}_{t\\sim p_1(t\\mid\\bm x)}[t] - \\mathbf{E}_{t\\sim p_2(t\\mid\\bm x)}[t]}\n    = u(\\bm x).\n\\end{align*}\n\\end{proof}\n\n\\subsection{Proof of Theorem~\\ref{thm:err_bound}}\n\\label{sec:gen_bound}\nWe restate Theorem~\\ref{thm:err_bound} below.\n\\newtheorem*{thm:err_bound}{Theorem~\\ref{thm:err_bound}}\n\\begin{thm:err_bound}\n  Assume that $n_1 = n_2$, $\\widetilde{n}_1 = \\widetilde{n}_2$,\n  $p_1(\\bm x) = p_2(\\bm x)$,\n  $W := \\inf_{x\\in\\mathcal{X}}\\abs{\\mu_w(\\bm x)} > 0$,\n  $M_{\\mathcal{Z}} := \\sup_{z\\in\\mathcal{Z}}\\abs{z} < \\infty$,\n  $M_{F} := \\sup_{f\\in F, x\\in\\mathcal{X}}\\abs{f(x)} < \\infty$,\n  and $M_{G} := \\sup_{g\\in G, x\\in\\mathcal{X}}\\abs{g(x)} < \\infty$.\n  Then, the following holds with probability at least $1-\\delta$ that\n  for every $f\\in F$,\n  \\begin{align*}\n    \\mathbf{E}_{\\bm x\\sim p(\\bm x)}[(f(\\bm x) - u(\\bm x))^2]\n    \\le \\frac{1}{W^2}\\left[\n        \\sup_{g\\in G}\\widehat J(f, g)\n        + \\mathcal{R}_{F, G}^{n, \\widetilde{n}}\n        + \\left(\\frac{M_z}{\\sqrt{2n}}\n        + \\frac{M_w}{\\sqrt{2\\widetilde{n}}}\\right)\\sqrt{\\log\\frac{2}{\\delta}}\n        + \\varepsilon_G(f)\n      \\right],\n  \\end{align*}\n  where $M_z := 4M_{\\mathcal{Y}}M_{\\mathcal{G}} + M_{\\mathcal{G}}^2\/2$,\n  $M_w = 2 M_{\\mathcal{F}}M_{\\mathcal{G}} + M_{\\mathcal{G}}^2\/2$,\n  $\\mathcal{R}_{F, G}^{n, \\widetilde{n}} := 2(M_F + 4M_Z)\\mathfrak{R}_{p(\\bm x, z)}^{n}(G) + 2(2M_F +\n  M_G)\\mathfrak{R}_{p(\\bm x, w)}^{\\widetilde{n}}(F) + 2(M_F + M_G)\\mathfrak{R}_{p(\\bm x, w)}^{\\widetilde{n}}(G)$.\n\\end{thm:err_bound}\n\nDefine $J(f, g)$ and $\\widehat{J}(f, g)$ as in Section~3.2 and denote\n\\begin{align*}\n  \\varepsilon_G(f)\n  &:= \\sup_{g\\in L^2(p)} J(f, g) - \\sup_{g\\in G} J(f, g).\n\\end{align*}\n\n\\begin{definition}[Rademacher Complexity]\n  We define the \\emph{Rademacher complexity} of $H$\n  over $N$ random variables following probability distribution $q$ by\n  \\begin{align*}\n    \\mathfrak{R}_p^N(H) = \\mathbf{E}_{V_1, \\dots, V_N, \\sigma_1, \\dots, \\sigma_N}\\left[\\sup_{h\\in H}\\frac{1}{N}\\sum_{i=1}^N\\sigma_ih(V_i)\\right],\n  \\end{align*}\n  where $\\sigma_1, \\dots, \\sigma_N$ are independent, $\\{-1,\n  1\\}$-valued uniform random variables.\n\\end{definition}\n\n\n\\begin{lemma}\\label{lem:obj_bound}\n  Under the assumptions of Theorem~\\ref{thm:err_bound},\n  with probability at least $1-\\delta$, it holds that for every $f\\in F$,\n  \\begin{align*}\n    J(f, g) \\le \\widehat{J}(f, g) + \\mathfrak{R}_{F, G}\n      + \\left(\\frac{M_z}{\\sqrt{n}} + \\frac{M_w}{\\sqrt{\\widetilde{n}}}\\right)\\sqrt{\\log\\frac{2}{\\delta}}.\n  \\end{align*}\n\\end{lemma}\n\nTo prove Lemma~\\ref{lem:obj_bound}, we use the following lemma, which is a slightly modified\nversion of Theorem~3.1 in~\\citet{mohri2012foundations}.\n\\begin{lemma}\\label{lem:rad_bound}\n  Let $H$ be a set of real-valued functions on a measurable space $\\mathcal{D}$.\n  Assume that $M := \\sup_{h\\in H, v\\in\\mathcal{D}} h(v) < \\infty$.\n  Then, for any $h\\in H$ and any $\\mathcal{D}$-valued i.i.d. random variables\n  $V, V_1, \\dots, V_N$ following density $q$, we have\n  \\begin{align}\n    \\mathbf{E}[h(V)] \\le \\frac{1}{N}\\sum_{i=1}^N h(V_i) + 2 \\mathfrak{R}_{q}^N(H) + \\sqrt{\\frac{M^2}{N}\\log\\frac{1}{\\delta}}.\n  \\end{align}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:rad_bound}]\n  We follow the proof of Theorem~3.1 in~\\citet{mohri2012foundations}\n  except that we set the constant $B_\\phi$ in Eq.~\\eqref{eq:diff_bounded} to $M\/m$\n  when we apply McDiarmid's inequality (Section~\\ref{sec:McD}).\n\\end{proof}\n\nNow, we prove Lemma~\\ref{lem:obj_bound}.\n\\begin{proof}[Proof of Lemma~\\ref{lem:obj_bound}]\n  For any $f\\in\\mathcal{F}$, $g\\in\\mathcal{G}$,\n  $\\bm x', \\widetilde{\\bm x}' \\in\\mathcal{X}$,\n    $z'\\in\\mathcal{Z} := \\{y, -y\\given y\\in\\mathcal{Y}\\}$, and $w'\\in\\{-1, 1\\}$, we define $h_z$ and $h_w$ as follows:\n  \\begin{align*}\n    h_z(\\bm x', z'; g) &:= -4z'g(\\bm x') - \\frac{1}{2}g(\\bm x')^2,\\\\\n    h_w(\\widetilde{\\bm x}', w'; f, g) &:= w'f(\\widetilde{\\bm x}')g(\\widetilde{\\bm x}') - \\frac{1}{2} g(\\widetilde{\\bm x}')^2.\n  \\end{align*}\n\n  Denoting $H_z := \\{(\\bm x', z') \\mapsto h_z(\\bm x', z'; g) \\given g\\in G\\}$, we have\n  \\begin{align*}\n    \\sup_{h\\in H_z, \\bm x'\\in\\mathcal{X}, z'\\in\\mathcal{Z}} \\abs{h(\\bm x', z')}\n    &\\le 4M_ZM_G + \\frac{1}{2}M_G^2 =: M_z < \\infty,\n  \\end{align*}\n  and thus, we can apply Lemma~\\ref{lem:rad_bound} to confirm that\n  with probability at least $1-\\delta\/2$,\n  \\begin{align*}\n    \\mathbf{E}_{(\\bm x, z)\\sim  p(\\bm x, z)}[h_z(\\bm x, z; g)]\n    \\le \\frac{1}{n} \\sum_{(\\bm x_i, z_i)\\in S_z} h_z(\\bm x_i, z_i; g)\n      + 2 \\mathfrak{R}_p^{n}(H_z)\n    + \\sqrt{\\frac{M_z^2}{n}\\log\\frac{2}{\\delta}},\n  \\end{align*}\n  where $\\{(\\bm x_i, z_i)\\}_{i=1}^n =: S_z$ are the samples defined in Section~\\ref{sec:direct_estimation}.\n  Similarly, it holds that with probability at least $1-\\delta\/2$,\n  \\begin{align*}\n    \\mathbf{E}_{(\\widetilde{\\bm x}, w)\\sim  p(\\bm x, w)}[h_w(\\widetilde{\\bm x}, w; f, g)]\n    \\le \\frac{1}{\\widetilde{n}} \\sum_{(\\widetilde{\\bm x}, w_i)\\in S_w} h_w(\\widetilde{\\bm x}_i, w_i; f, g)\n  + 2 \\mathfrak{R}_p^{\\widetilde{n}}(H_w)\n    + \\sqrt{\\frac{M_w^2}{\\widetilde{n}}\\log\\frac{2}{\\delta}},\n  \\end{align*}\n  where $H_w := \\{(\\widetilde{\\bm x}', w') \\mapsto h_w(\\widetilde{\\bm x}', w';\n  f, g) \\given f\\in F, g\\in G\\}$, $M_w := M_FM_G + M_G^2\/2$,\n  and $\\{(\\widetilde{\\bm x}_i, w_i)\\}_{i=1}^n =: S_w$ are the samples defined in Section~\\ref{sec:direct_estimation}.\n  By the union bound, we have the following with probability at least\n  $1-\\delta$:\n  \\begin{align}\n    &\\mathbf{E}_{(\\bm x, z)\\sim  p(\\bm x, z)}[h_z(\\bm x, z; g)]\n    + \\mathbf{E}_{(\\widetilde{\\bm x}, w)\\sim  p(\\bm x, w)}[h_w(\\widetilde{\\bm x}, w; f, g)]\\\\\n    &\\le \\frac{1}{n} \\sum_{(\\bm x_i, z_i)\\in S_z} h_z(\\bm x_i, z_i, g)\n    + \\frac{1}{\\widetilde{n}} \\sum_{(\\widetilde{\\bm x}, w_i)} h_w(\\bm x_i, w_i, f, g)\\\\\n      &\\phantom{\\le}+ 2(\\mathfrak{R}_p^{n}(H_z) + \\mathfrak{R}_p^{\\widetilde{n}}(H_w))\n      + \\left(\\frac{M_z}{\\sqrt{n}} + \\frac{M_w}{\\sqrt{\\widetilde{n}}}\\right)\\sqrt{\\log\\frac{2}{\\delta}},\\label{eq:bound_h}\n  \\end{align}\n  Using some properties of the Rademacher complexity including Talagrand's lemma,\n  we can show that\n  \\begin{align}\n      \\mathfrak{R}_p^{n}(H_z) &\\le (M_F + 4M_Z)\\mathfrak{R}_p^{n}(G),\\label{eq:bound_R_z}\\\\\n      \\mathfrak{R}_p^{\\widetilde{n}}(H_w) &\\le (2M_F + M_G)\\mathfrak{R}_p^{\\widetilde{n}}(F) + (M_F + M_G)\\mathfrak{R}_p^{\\widetilde{n}}(G).\\label{eq:bound_R_w}\n  \\end{align}\n  Clearly,\n  \\begin{align*}\n      \\widehat{J}(f, g) &= \\frac{1}{n}\\sum_{(\\bm x_i, z_i)\\in S_z} h(\\bm x_i, z_i;\n  g)\n  + \\frac{1}{\\widetilde{n}} \\sum_{(\\widetilde{\\bm x}_i, w_i)\\in S_w} h(\\widetilde{\\bm x}_i, w_i; f, g),\\\\\n  J(f, g) &= \\mathbf{E}_{(\\bm x, z)\\sim  p(\\bm x, z)}[h_z(\\bm x, z; g)] +\n  \\mathbf{E}_{(\\widetilde{\\bm x}, w)\\sim  p(\\bm x, z)}[h_w(\\widetilde{\\bm x}, w; f,\n  g)].\n  \\end{align*}\n  From Eq.~\\eqref{eq:bound_h}, Eq.~\\eqref{eq:bound_R_z}, and Eq.~\\eqref{eq:bound_R_w}, we obtain\n  \\begin{align}\n    J(f, g) \\le \\widehat{J}(f, g) + \\mathfrak{R}_{F, G}\n      + \\left(\\frac{M_z}{\\sqrt{n}} + \\frac{M_w}{\\sqrt{\\widetilde{n}}}\\right)\\sqrt{\\log\\frac{2}{\\delta}},\\label{eq:JvsJhat}\n  \\end{align}\n  where\n  \\begin{align*}\n      \\mathfrak{R}_{F, G} := 2(M_F + 4M_Z)\\mathfrak{R}_p^{n}(G)\n      + 2(2M_F + M_G)\\mathfrak{R}_p^{\\widetilde{n}}(F) + 2(M_F + M_G)\\mathfrak{R}_p^{\\widetilde{n}}(G).\n  \\end{align*}\n\n\\end{proof}\n\nFinally, we prove Theorem~\\ref{thm:err_bound}.\n\\begin{proof}[Proof of Theorem~\\ref{thm:err_bound}]\n  From Lemma~\\ref{lem:obj_bound}, with probability at least $1-\\delta$,  it holds that for all $f\\in F$\n  \\begin{align}\n      \\sup_{g\\in G} J(f, g)\n      \\le \\sup_{g\\in G} \\widehat{J}(f, g) + \\mathfrak{R}_{F, G}\n      + \\left(\\frac{M_z}{\\sqrt{n}} + \\frac{M_w}{\\sqrt{\\widetilde{n}}}\\right)\\sqrt{\\log\\frac{2}{\\delta}}.\\label{eq:b1}\n  \\end{align}\n    Moreover, recalling $W := \\inf_{\\bm x\\in\\mathcal{X}}\\abs{\\mu_w(\\bm x)}$\n\tand $\\sup_{g\\in L^2(p)} J(f, g) = \\mathbf{E}[(\\mu_w(\\bm x)f(\\bm x) -\\mu_z(\\bm x))^2]$, we have\n  \\begin{align}\n      \\mathbf{E}\\left[(f(\\bm x) - u(\\bm x))^2\\right]\n      &= \\mathbf{E}\\left[\\left(f(\\bm x) - \\frac{\\mu_z(\\bm x)}{\\mu_w(\\bm x)}\\right)^2\\right]\\\\\n      &\\le \\frac{1}{W^2} \\mathbf{E}[(\\mu_w(\\bm x)f(\\bm x) - \\mu_z(\\bm x))^2]\\\\\n      &= \\frac{1}{W^2} \\left[\\varepsilon_G(f) + \\sup_{g\\in G} J(f, g)\\right].\\label{eq:b2}\n  \\end{align}\n    Combining Eq.~\\eqref{eq:b1} and Eq.~\\eqref{eq:b2} yields the inequality of the theorem.\n\\end{proof}\n\n\n\\subsection{Proof of Corollary~\\ref{cor:ls_bound}}\n\\label{sec:proof_of_cor1}\n\\newtheorem*{cor:ls_bound}{Corollary~\\ref{cor:ls_bound}}\n\\begin{cor:ls_bound}\n    Let $F = \\{x\\mapsto \\bm\\alpha^\\top\\bm\\phi(\\bm x) \\given \\norm{\\bm\\alpha}_2 \\le \\Lambda_F\\}$,\n    $G = \\{x\\mapsto \\bm\\beta^\\top\\bm\\psi(\\bm x) \\given \\norm{\\bm\\beta}_2 \\le \\Lambda_G\\}$,\n    and assume that $r_F := \\sup_{\\bm x\\in\\mathcal{X}}\\norm{\\bm\\phi(\\bm x)}_2 < \\infty$\n    and $r_G := \\sup_{\\bm x\\in\\mathcal{X}}\\norm{\\bm\\psi(\\bm x)}_2 < \\infty$,\n    where $\\norm{\\cdot}_2$ is the $L_2$-norm.\n    Under the assumptions of Theorem~\\ref{thm:err_bound},\n    it holds with probability at least $1-\\delta$ that for every $f\\in F$,\n    \\begin{align*}\n      \\mathbf{E}_{\\bm x\\sim p(\\bm x)}[(f(\\bm x) - u(\\bm x))^2]\n      \\le \\frac{1}{W^2}\\left[\n          \\sup_{g\\in G}\\widehat J(f, g)\n          + \\frac{C_z\\sqrt{\\log\\frac{2}{\\delta}}+D_z}{\\sqrt{2n}}\n          + \\frac{C_w\\sqrt{\\log\\frac{2}{\\delta}}+D_w}{\\sqrt{2\\widetilde{n}}}\n          + \\varepsilon_G(f)\n      \\right],\n    \\end{align*}\n    where\n    $C_z := r_G^2 \\Lambda_G^2 + 4 r_G \\Lambda_G M_\\mathcal{Y}$,\n    $C_w := 2 r_F^2 \\Lambda_F^2 + 2 r_F r_G \\Lambda_F\\Lambda_G+ r_G^2 \\Lambda_G^2 $,\n    $D_z := r_G^2 \\Lambda_G^2\/2 + 4r_G \\Lambda_G M_\\mathcal{Y}$,\n    and $D_w := r_G^2\\Lambda_G^2\/2 + 4r_F r_G\\Lambda_F \\Lambda_G$.\n\\end{cor:ls_bound}\n\n\\begin{proof}\nUnder the assumptions, it is known that the Rademacher complexity of the linear-in-parameter model $F$ can be upper bounded as follows~\\citep{mohri2012foundations}:\n\\begin{align*}\n    \\mathfrak{R}_p^N(F) \\le \\frac{r_F \\Lambda_F}{\\sqrt{N}}.\n\\end{align*}\nWe can bound $\\mathfrak{R}_p^N(G)$ similarly.\nApplying these bounds to Theorem~\\ref{thm:err_bound}, we obtain the statement of Corollary~1. \n\\end{proof}\n\n\n\\subsection{Proof of Theorem~\\ref{thm:MSE_bound}}\n\\label{sec:proof_of_MSE_bound}\n\\newtheorem*{thm:MSE_bound}{Theorem~\\ref{thm:MSE_bound}}\nWe prove the following, formal version of Theorem~\\ref{thm:MSE_bound}.\n\\begin{thm:MSE_bound}\nUnder the assumptions of Corollary~\\ref{cor:ls_bound}, it holds with probability at least $1-\\delta$ that $\\mathbf{E}[(\\widehat{f}(\\bm x)-u(\\bm x))^2]\\le(4e_{n, \\delta} + 2\\varepsilon_G^F + \\varepsilon_F)\/W^2$,\nwhere $\\varepsilon_G^F := \\sup_{f\\in F}\\varepsilon_G(f)$,\nand $\\varepsilon_F := \\inf_{f\\in F} J(f)$,\n$\\widehat{f} \\in F$ is any approximate solution to $\\inf_{f\\in F}\\sup_{g\\in G}\\widehat{J}(f, g)$ satisfying $\\sup_{g\\in G} \\widehat{J}(\\widehat{f}, g) \\le \\inf_{f\\in F}\\sup_{g\\in G} \\widehat{J}(f, g) + e_{n, \\delta}$, and\n    \\begin{align*}\n      e_{n, \\delta} := \\frac{C_z\\sqrt{\\log\\frac{2}{\\delta}}+D_z}{\\sqrt{2n}}\n      + \\frac{C_w\\sqrt{\\log\\frac{2}{\\delta}}+D_w}{\\sqrt{2\\widetilde{n}}}.\n    \\end{align*}\n\n\\end{thm:MSE_bound}\n\n\\begin{proof}\n    Let $J(f) := \\sup_{g\\in L^2} J(f, g) = \\mathbf{E}[(\\mu_w(\\bm x)f(\\bm x) - \\mu_z(\\bm x))^2]$, $J_G(f) := \\sup_{g\\in G} J(f, g)$, $\\widehat{J}_G(f) := \\sup_{g\\in G} \\widehat{J}(f, g)$.\n    Let $\\widetilde{f} \\in F$ be any approximate solution to $\\inf_{f\\in F}J(f)$ satisfying $J(\\widetilde{f}) \\le \\varepsilon_F + e_{n, \\delta}$.\n\n    As a special case of Eq.~\\ref{eq:b1}, we can prove that with probability at least $1 - \\delta$, it holds for every $f\\in F$ that $J_G(f) \\le \\widehat{J}_G(f) + e_{n, \\delta}$.\n    From Corollary~\\ref{cor:ls_bound}, it holds that with probability at least $1 - \\delta$,\n\\begin{align*}\nJ(\\widehat{f})\n&\\le\n\\left[J(\\widehat{f}) - J_G(\\widehat{f})\\right]\n+\\left[J_G(\\widehat{f}) - \\widehat{J}_G(\\widehat{f})\\right]\n+\\left[\\widehat{J}_G(\\widehat{f}) - \\widehat{J}_G(\\widetilde{f})\\right]\\\\\n&\\phantom{\\le}\n+\\left[\\widehat{J}_G(\\widetilde{f}) - J_G(\\widetilde{f})\\right]\n+ \\left[J_G(\\widetilde{f}) - J(\\widetilde{f})\\right]\n+ J(\\widetilde{f})\\\\\n&\\le \\varepsilon_G^F + e_{n, \\delta} + e_{n, \\delta}\\\\\n&\\phantom{\\le} + e_{n, \\delta} + \\varepsilon_G^F + \\left[\\varepsilon_F + e_{n, \\delta} \\right]\\\\\n&\\le 4e_{n, \\delta} + 2\\varepsilon_G^F + \\varepsilon_F.\n\\end{align*}\nSince $\\mathbf{E}[(\\widehat{f}(\\bm x)-u(\\bm x))^2] \\le \\frac{1}{W^2}J(\\widehat{f})$, \nwe obtain the bound in Theorem~\\ref{thm:MSE_bound}.\n\\end{proof}\n\n\n\\subsection{Binary Outcomes}\nWhen outcomes $y$ take on binary values, e.g., success or failure, without loss of generality, we can assume that $y\\in\\{-1, 1\\}$.\nThen, by the definition of the individual uplift, $u(\\bm x)\\in[-2, 2]$ for any\n$\\bm x\\in\\Re^d$.\nIn order to incorporate this fact, we may add the following range constraints on $f$: $-2 \\le f(\\bm x) \\le 2$ for every $\\bm x \\in \\{\\bm x_i\\}_{i=1}^n \\cup \\{\\widetilde{\\bm x}_i\\}_{i=1}^{\\widetilde{n}}$.\n\n\n\\subsection[Handling Different Feature Distributions]{Cases Where $p_1(\\bm x) \\neq p_2(\\bm x)$ or $(n_1, \\widetilde{n}_1)\\neq(n_1, \\widetilde{n}_1)$}\n\\label{sec:different_p_or_n}\n\nSo far, we have assumed that $p_1(\\bm x) = p_2(\\bm x)$, $m_1 = m_2$, and $n_1\n= n_2$.\nThe proposed method can be adapted to the more general case where these assumptions may not hold.\n\nLet $r_k(\\bm x) = \\frac{n}{2n_k}\\cdot\\frac{p(\\bm x)}{p_k(\\bm x)}$ and\n$\\widetilde{r}_k(\\bm x) = \\frac{\\widetilde{n}}{2\\widetilde{n}_k}\\cdot\\frac{p(\\bm x)}{p_k(\\bm x)}$, $k=1,2$, for every $\\bm x$ with $p_k(\\bm x) > 0$.\nLet $k_i := 1$ if the sample $\\bm x_i$ originally comes from $p_1(\\bm x)$,\nand $k_i := 2$ if it comes from $p_2(\\bm x)$. Similarly, define\n$\\widetilde{k}_i\\in\\{1, 2\\}$ according to whether $\\widetilde{\\bm x}_i$ comes\nfrom $p_1(\\bm x)$ or $p_2(\\bm x)$.\nThen, unbiased estimators of the three terms in the proposed objective Eq.~\\eqref{eq:Lfg} \nare given as the following weighted sample averages using $r_k$ and $\\widetilde{r}_k$:\n\\begin{align*}\n  \\mathbf{E}_{\\bm x\\sim p(\\bm x)}[w f(\\bm x)g(\\bm x)]\n  &\\approx \n  \\frac{1}{\\widetilde{n}}\\sum_{i=1}^{\\widetilde{n}}[w_i f(\\widetilde{\\bm x}_i)g(\\widetilde{\\bm x}_i)\\widetilde{r}_{\\widetilde{k}_i}(\\widetilde{\\bm x}_i)],\\\\\n  \\mathbf{E}_{\\bm x\\sim p(\\bm x)}[z g(\\bm x)]\n  &\\approx \n  \\frac{1}{n}\\sum_{i=1}^{n}[z_i g(\\bm x_i)r_{k_i}(\\bm x_i)]\\\\\n  \\mathbf{E}_{\\bm x\\sim p(\\bm x)}[g(\\bm x)^2]\n  &\\approx \n  \\frac{1}{2n}\\sum_{i=1}^{n}[g(\\bm x_i)^2r_{k_i}(\\bm x_i)]\n  + \\frac{1}{2\\widetilde{n}}\\sum_{i=1}^{\\widetilde{n}}[g(\\widetilde{\\bm x}_i)^2\\widetilde{r}_{\\widetilde{k}_i}(\\widetilde{\\bm x}_i)].\n\\end{align*}\n\nThe density ratios $p_k(\\bm x)\/p(\\bm x)$ can be accurately estimated using i.i.d.\\@\nsamples from $p_k(\\bm x)$ and $p(\\bm x)$~\\citep{nguyen2010estimating, sugiyama2012density, yamada2013relative, liu2017ratio}.\n\n\n\\subsection{Unbiasedness of the Weighted Sample Average}\nBelow, we show that the weighted sample averages are unbiased estimates.\nWe only prove for $\\mathbf{E}[wf(\\bm x)g(\\bm x)]$ since the other cases can be proven similarly.\nThe expectation of the weighted sample average transforms as follows:\n\\begin{align*}\n  &\\frac{1}{\\widetilde{n}}\\sum_{i=1}^{\\widetilde{n}} \\mathbf{E}_{\\widetilde{\\bm x}^{(k)}_i\\sim p_k(\\bm x), t^{(k)}_i\\sim p_k(t\\given\\widetilde{\\bm x}^{(k)}_i)}\\left[w_if(\\widetilde{\\bm x}_i)g(\\widetilde{\\bm x}_i)\\widetilde{r}_{\\widetilde{k}_i}(\\widetilde{\\bm x}_i)\\right]\\\\\n  &=\\frac{1}{\\widetilde{n}}\\sum_{k=1,2}\\sum_{i=1}^{\\widetilde{n}_k} \\mathbf{E}_{\\bm x\\sim p_k(\\bm x), t\\sim p_k(t\\given\\bm x)}\\left[(-1)^{k-1}t f(\\bm x)g(\\bm x)\\frac{\\widetilde{n}}{2\\widetilde{n}_k}\\cdot\\frac{p(\\bm x)}{p_k(\\bm x)}\\right]\\\\\n  &=\\frac{1}{2}\\sum_{k=1,2} \\mathbf{E}_{\\bm x\\sim p(\\bm x), t\\sim p_k(t\\given\\bm x)}\\left[(-1)^{k-1}t f(\\bm x)g(\\bm x)\\right]\\\\\n  &=\\iint(-1)^{k-1}t\\sum_{k=1,2}\\frac{1}{2}p_k(t\\given\\bm x)f(\\bm x)g(\\bm x)p(\\bm x)\\mathrm{d}t \\dbm x\\\\\n  &=\\iint w p(w\\given\\bm x) f(\\bm x)g(\\bm x)p(\\bm x)\\mathrm{d}t \\dbm x\\\\\n  &=\\mathbf{E}_{\\bm x\\sim p(\\bm x), w\\sim p(w\\given\\bm x)}[w f(\\bm x)g(\\bm x)].\n\\end{align*}\n\n\n\\subsection{Gaussian Basis Functions Used in Experiments}\n\\label{sec:gau_basis}\nThe $l$-th element of $\\bm\\phi(\\bm x) = (\\phi_1(\\bm x), \\dots, \\phi_{b_f}(\\bm x))^\\top$ is defined by\n\\begin{align*}\n    \\phi_l(\\bm x) := \\exp\\left(\\frac{-\\norm{\\bm x - \\bm x^{(l)}}^2}{\\sigma^2}\\right),\n\\end{align*}\nwhere $\\bm x^{(l)}$, $l = 1, \\dots, b_f$, are randomly chosen training data points.\nWe used $b_f = 100$ and $\\sigma = 25$ for all experiments. $\\bm\\psi$ is defined similarly.\n\n\\subsection{Justification of the Sub-Sampling Procedure}\n\\label{sec:subsample}\nSuppose that we want a sample subset $S_k$ following the treatment policy $p_k(t\\mid\\bm x)$.\nFor each sample $(\\bm x_i, t_i, y_i) \\sim p(\\bm x, t, y)$ in the original dataset,\nwe randomly add it into $S_k$ with probability proportional to $p_k(t_i\\mid\\bm x_i) \/ p(t_i\\mid\\bm x_i)$.\nThen,\n\\begin{align*}\n    p(\\bm x_i, t_i, y_i \\mid (\\bm x_i, t_i, y_i) \\in S_k)\n    &= \\frac{p((\\bm x_i, t_i, y_i) \\in S_k \\mid \\bm x_i, t_i, y_i)p(\\bm x_i, t_i, y_i)}%\n    {\\int \\sum_{y_i, t_i} p((\\bm x_i, t_i, y_i) \\in S_k \\mid \\bm x_i, t_i, y_i)p(\\bm x_i, t_i, y_i)\\mathrm{d}\\bm x_i}\\\\\n    &= \\frac{p_k(t_i\\mid\\bm x_i)p(y_i\\mid \\bm x_i, t_i) p(\\bm x_i)}\n    {\\int \\sum_{y_i, t_i} p_k(t_i\\mid\\bm x_i)p(y_i\\mid \\bm x_i, t_i) p(\\bm x_i)\\mathrm{d}\\bm x_i}\\\\\n    &= p_k(t_i\\mid\\bm x_i)p(y_i\\mid \\bm x_i, t_i) p(\\bm x_i).\n\\end{align*}\nThis means that the subsamples $S_k$ preserve the original $p(y\\mid\\bm x, t)$ and $p(\\bm x)$ but follow the desired treatment policy $p_k(t\\mid\\bm x)$.\n\n\n\n\\subsection{McDiarmid's Inequality}\n\\label{sec:McD}\nAlthough McDiarmid's inequality is a well known theorem,\nwe present the statement to make the paper self-contained.\n\\begin{theorem}[McDiarmid's inequality]\n  \\label{theorem:McD}\n  Let $\\varphi: \\mathcal{D}^N \\to \\Re$ be a measurable function.\n  Assume that there exists a real number $B_{\\varphi} > 0$ such that\n  \\begin{align}\n    \\abs{\\varphi(v_1, \\dots, v_N) - \\varphi(v'_1,\n  \\dots, v'_N)} \\le B_{\\varphi}, \\label{eq:diff_bounded}\n  \\end{align}\n  for any $v_i, \\dots, v_N, v_1, \\dots, v'_N\\in\\mathcal{D}$ where $v_i = v'_i$\n  for all but one $i\\in\\{1, \\dots, N\\}$.\n  Then, for any $\\mathcal{D}$-valued independent random variables $V_1,\n  \\dots, V_N$ and any $\\delta > 0$\n  the following holds with probability at least $1 - \\delta$:\n  \\begin{align*}\n    \\varphi(V_1, \\dots, V_N) \\le \\mathbf{E}[\\varphi(V_1, \\dots, V_N)] + \\sqrt{\\frac{B_{\\varphi}^2N}{2} \\log \\frac{1}{\\delta}}.\n  \\end{align*}\n\\end{theorem}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{#1}}\n\n\\newcommand{\\Delta}{\\Delta}\n\\newcommand{\\frac{1}{2}}{\\frac{1}{2}}\n\\newcommand{\\frac{1}{3}}{\\frac{1}{3}}\n\\newcommand{\\frac{1}{4}}{\\frac{1}{4}}\n\\newcommand{\\p}[2]{{\\pi_{#1}^{(#2)}}}\n\\newcommand{{\\rm Tr}\\,}{{\\rm Tr}\\,}\n\\newcommand{\\Lambda}{\\Lambda}\n\\newcommand{\\lambda}{\\lambda}\n\\newcommand{\\begin{equation}}{\\begin{equation}}\n\\newcommand{\\end{equation}}{\\end{equation}}\n\\newcommand{\\gamma_{\\rm str}}{\\gamma_{\\rm str}}\n\n\\hyphenation{pre-print}\n\n\\begin{document}\n\\vspace*{-3.3cm}\n\\vbox{\n\\begin{flushright}\nNBI-HE-94-28 \\\\\nMay 1994\n\\end{flushright}\n\\title{From Trees to Galaxies:\\\\\nThe Potts Model on a Random Surface}\n\\author{Mark Wexler \\\\ \\\\\n      \\small {\\it Niels Bohr Institute, Blegdamsvej 17}\\\\\n      \\small {\\it 2100 Copenhagen \\O, Denmark}\\\\ \n      \\small {\\it wexler@nbivax.nbi.dk}}\n\\date{}\n\\maketitle\n\\begin{abstract}\nThe matrix model of random surfaces with $c=\\infty$ has\nrecently been solved and found to be identical to a random\nsurface coupled to a $q$-states Potts model with $q=\\infty$.\nThe mean field-like solution exhibits a novel type of tree\nstructure.  The natural question is, down to which---if\nany---finite values of $c$ and $q$ does this behavior persist?\nIn this work we develop, for the Potts model, an expansion\nin the fluctuations about the $q=\\infty$ mean field solution.\nIn the lowest---cubic---non-trivial order in this expansion\nthe corrections to mean field theory can be given a nice\ninterpretation in terms of structures (trees and ``galaxies'')\nof spin clusters.  When $q$ drops below a finite $q_c$, the\ngalaxies overwhelm the trees at all temperatures, thus\nsuppressing mean field behavior.  Thereafter the phase\ndiagram resembles that of the Ising model, $q=2$.\n\\end{abstract}}\n\n\\section{Introduction}\n\nThe infinite-state Potts model on random spherical\nsurface can be solved exactly \\cite{Me2,Me3} by resumming all\norders of its low temperature expansion \\cite{Me1}.  The\nproblem and its solution are of interest for the following\nreasons.\n\\begin{itemize}\n\\item\nIt can be shown to be equivalent to conformally invariant\nmodels with $c = \\infty$ (which can be realized, for example,\nby coupling multiple Ising models to the surface).  Thus the\nsolution opens a window onto the mysterious $c>1$ regime,\nhitherto inaccessible via either Liouville theory or matrix\nmodels.\n\\item\nThis equivalence to conformal matter holds in spite of a\nwell known fact that on a fixed two-dimensional lattice\na Potts model with more that four states suffers a\nfirst order---rather than a continuous---phase transition.\nCoupling an $\\infty$-state Potts model to\na random surface changes the order of the transition\nfrom first to third; in the case of $q<4$\nthe order of the transition correspondingly changes\nfrom second to third.\n\\item\nThe reason why the transition for $q=\\infty$ is of third\norder is that it is no longer a spin-ordering transition;\nrather it is a tree-growing transition, where the nodes\nof the trees are spin clusters (equivalently, ``baby\nuniverses'').  The transition in the random surface\nPotts model from spin-ordering (low $q$) to tree-growing\n(high $q$) behavior is a novel one, and deserves attention\nin its own right.\n\\end{itemize}\n\nSome recent work has given support to the mean field\ntheory at $q=c=\\infty$.  Ambj\\o rn, Durhuus and\nJ\\'onsson \\cite{ADJ,Durhuus} (see also \\cite{ABC})\nhave shown that if a random\nsurface model has susceptibility exponent $\\gamma_{\\rm str}$, its\n``polymerized'' version has $\\bar{\\gamma}_{\\rm str}\n=\\gamma_{\\rm str}\/(\\gamma_{\\rm str}-1)$,\nin agreement with the mean field result $\\gamma_{\\rm str} = \\frac{1}{3}$.\nTheir polymerized surfaces of baby universes\nare suggestive of the trees of clusters found in the\nmean field model.\\footnote{Although baby universes\n(geometrical structures) are not {\\it a priori} identical\nto clusters (matter structures), the two seem to fuse\nat large $q$ or $c$.  Note that the trees in \\cite{ADJ}\nare a result of the $N\\to\\infty$ limit, while the\ntrees in \\cite{Me2,Me3} are a result of $q,c\\to\\infty$.}\nHarris and Wheater \\cite{HarrisWheater}\nhave studied multiple Ising models,\nand in the limit $c\\to\\infty$ have identified trees as\ndominant, and found\n$\\gamma_{\\rm str}=\\frac{1}{2}$.\\footnote{Their trees are made of\n{\\it vertices}, not {\\it clusters}, and thus form\na subset of trees-of-clusters.  There is no contradiction:\nto find clusters at the vertices of the trees, one\nhas to take $c\\to\\infty$ and $T\\to0$ simultaneously,\nkeeping $e^{-2\/T}c$ fixed.}\nA recent numerical study \\cite{MonteCarlo} of multiple\n$q=2, 3,$ and $4$ Potts models shows that spikiness of\nthe surfaces increases with increasing central charge,\nand that $\\gamma_{\\rm str}$ may well approach $\\frac{1}{3}$.\nFinally, a current numerical study of the Potts model\nfor various values of $q$ and for surfaces with tadpoles\nand self-energies \\cite{Teaser} is confirming the scaling of the\ncritical temperature $T_c^{-1} = \\frac{1}{2} \\log q + {\\cal O}(1)$\nwith large $q$, and, more significantly, details of\ncluster geometry predicted by the mean field theory.\n\nFor convenience, let us introduce the concept of a\n{\\it skeleton graph}: given a triangulated surface with some spin\nconfiguration, its skeleton is a graph that represents\nthe spin clusters as simple vertices, and whose edges are\nthe links between different spin clusters.\\footnote{So\nbe definition, no vertex of a skeleton may be connected\nto itself.}\nEach loop of skeleton graph introduces a factor $1\/q$\n(or $1\/c$ for conformal models).\nThus at $q=\\infty$ (or at $c=\\infty$),\nthe only skeletons that are present are trees.\nWhen we move down to large but finite $q$, the corrections\nintroduce loops; and while the leading terms are universal,\nthe corrections in $1\/q$ are not---their details depend\non the details of the matter model.\nThis is a familiar picture in $1\/D$ expansions.  For instance\nin the scalar $\\lambda\\phi^4$ theory, the upper critical\ndimension is 4: below this, fluctuations or ``loops''\nbecome essential, and mean field theory breaks down.\n\nIt would be very interesting to know if something similar\nhappens for random surfaces.  One's curiousity is heightened\nby the fact that Liouville theory seems to be silent on the\nmatter.  To investigate the large-but-finite $q$ region,\nwe develop a systematic expansion of the model (in its\nequivalent one-matrix form) about the $q=\\infty$ solution.\nA priori, we would expect that the first non-trivial order\nin this expansion to be quadratic, as in many similar cases.\nIn this order we sum all the skeletons that have one or\nno loops, in other words the $q^0$ and $q^{-1}$ terms in\nthe partition function.  But it is not hard to show that\nthis $q^{-1}$ correction is insignificant, in that it does not\nmodify the critical behavior of the model.\n\nThe next, cubic, order of our approximation is much richer.\nHere we include all skeletons whose clusters have three\nor fewer external legs that lie on loops.  We therefore\nhave terms with arbitrarily many loops, meaning arbitrarily\nhigh powers of $q^{-1}$.  A new structure makes its appearance:\nthe clusters are organized into $\\phi^3$ graphs of spherical\ntopology.  These ``surfaces'' of clusters will be called\n{\\it galaxies}.  The trees are still present: any number\nof them may grow out of any cluster.  On top of that,\na typical skeleton graph includes several galaxies,\nlinked also into a tree.\n\nThis complicated system shows complicated critical behavior.\nFor $q=\\infty$, there is a magnetized low temperature phase where the\nclusters are large, and there are few of them: essentially\npure gravity.  At some point that phase becomes unstable,\nand the tree of clusters blows up.  Above that, the clusters\nare small, but the tree is large.\n\nAt large but finite $q$, the above picture of critical behavior\nis unchanged, except for the appearance of a new phase\nat high temperature,\nthat of large galaxies.  At low temperature we still have\nthe pure gravity phase of large clusters; this still goes\ninto an intermediate phase of large trees.  But now, at an even higher\ntemperature (which is infinite when $q$ is infinite, and\nwhich decreases with $q$), the area of the galaxies---the\nnumber of clusters they contain---diverges.  In this phase,\nthe loops become essential: without loops there could be no\ngalaxies.  The phase of large galaxies is also akin to\npure gravity.\n\nThe tree phase becomes narrower and narrower as $q$ decreases,\nuntil at some $q_c$, it disappears altogether: one goes\nfrom the magnetized phase directly into the phase\nof large galaxies.  Although mean field theory is modified\nat any finite $q$, it is completely destroyed below $q_c$.\nFor $q \\le q_c$ the phase diagram is altogether reasonable,\nin that it resembles the known phase diagrams of the $q<4$\nmodels: there are two pure gravity phases, at low and at\nhigh temperature, separated by a multicritical point.\n\nThis paper is organized as follows.  Section 2 presents the\nmulti-matrix version of the Potts model, its low temperature\nexpansion, its effective one-matrix versions, and its\n$q=\\infty$ solution.\nIn section 3 we introduce an expansion about the $q=\\infty$\nsolutions and its truncated versions, and discuss its\ngeometrical implications.\nIn section 5 we\nanalyze the critical behavior of the simplest nontrivial\ntruncation.\n\n\\section{Definition of the model}\n\nThe $q$-state Potts model on a spherical random surface\nwill be defined using a matrix model (see \\cite{GinspargMoore}\nfor a review):\n\\begin{equation}\n\\label{basicpotts}\nF_q(g,\\zeta) = \\frac{1}{q} \\log \\frac{1}{f_0}\n\\int {\\cal D} \\phi_1 \\cdots {\\cal D} \\phi_q \\:\n\\exp -N \\, {\\rm Tr}\\, \\left( \\frac{1}{2} \\sum_{ij}\n\\phi_i G_{ij} \\phi_j + g \\sum_i \\phi_i^3 \\right)\n\\end{equation}\nwhere the $\\phi_i$ are $N \\times N$ hermitian matrices\nand all matrix integrals are to be understood as being\ndivided by $N^2$ and taken in the limit $N \\to \\infty$.\nThe matrix $G$ defines\nthe matter that is to be coupled to the fluctuating\nsurface; in the case of the Potts model it is\n\\begin{equation}\nG = \\left(\n\\begin{array}{cccc}\n1 & \\zeta & \\cdots & \\zeta \\\\\n\\zeta & 1 & \\ddots & \\vdots \\\\\n\\vdots & \\ddots & \\ddots & \\zeta \\\\\n\\zeta & \\cdots & \\zeta & 1\n\\end{array}\n\\right)^{-1}\n= \\: \\left(\n\\begin{array}{cccc}\na_q & b_q & \\cdots & b_q \\\\\nb_q & a_q & \\ddots & \\vdots \\\\\n\\vdots & \\ddots & \\ddots & b_q \\\\\nb_q & \\cdots & b_q & a_q\n\\end{array} \\right)\n\\end{equation}\n\\begin{equation}\n\\label{aqbq}\na_q = \\frac{1 + (q-2) \\zeta}{(1-\\zeta)(1 + (q-1)\\zeta)},\\qquad\nb_q = \\frac{-\\zeta}         {(1-\\zeta)(1 + (q-1)\\zeta)}\n\\end{equation}\nThe coupling constant $\\zeta$ is related to the matter\ntemperature by $\\zeta = e^{-2\/T}$.\nFinally, the normalization $f_0$ is defined by requiring\n$F_q(0,\\zeta) = 1$.\n\nThe partition function $F_q$ can be expanded in powers of\n$\\zeta$.  This is the so-called low temperature or cluster\nexpansion, as in order $\\zeta^n$ we have a sum over all connected\nconfigurations of clusters with $n$ inter-cluster links;\nas well as over the {\\it internal} geometry of each cluster.\nDefining $\\Delta = q - 1$,\nthe first four orders of the expansion are\n\\begin{eqnarray}\n\\label{lte}\n\\lefteqn{F_q(g,c) = \\pi_0 +\n\\frac{1}{2} \\Delta \\pi_1^2 \\zeta + \n\\left[\\frac{1}{2} \\Delta^2 \\pi_1^2 (\\pi_2 + \\pi_{11}) +\n   \\frac{1}{4} \\Delta \\pi_2^2 \\right] \\zeta^2 } \\\\\n& & \\hspace{1.2cm} \\epsfxsize=.1cm\\epsfbox{z0.eps}\n    \\hspace{.8cm} \\epsfxsize=.7cm\\epsfbox{z11.eps}\n    \\hspace{1.7cm} \\epsfxsize=1cm\\epsfbox{z112.eps}\n    \\hspace{1.7cm} \\epsfxsize=.7cm\\epsfbox{z22.eps} \\nonumber \\\\\n& & \\mbox{\\vspace*{-1cm}} \\nonumber \\\\\n& & \\mbox{} + \\left[\\frac{1}{2} \\Delta^3 \\pi_1^2 (\\pi_2 + \\pi_{11})^2 +\n\\frac{1}{6} \\Delta^3 \\pi_1^3 (\\pi_3 + \\pi_{12} + \\pi_{111})\n+ \\frac{1}{6} \\Delta(\\Delta-1) \\pi_2^3 \\right. \\nonumber \\\\\n& & \\vspace*{-1cm}\n    \\hspace{1.5cm} \\epsfxsize=1.2cm\\epsfbox[0 -160 500 -124]{z1122.eps}\n    \\hspace{2.8cm} \\epsfxsize=.6cm\\epsfbox[0 -50 252 253]{z1113.eps}\n    \\hspace{2.9cm} \\epsfxsize=.7cm\\epsfbox[0 -60 288 192]{z222.eps} \\nonumber \\\\\n& & \\qquad\\left. \\mbox{} + \\frac{1}{2} \\Delta^2 \\pi_1 \\pi_2 \\left(\\pi_3\n+ \\frac{\\pi_{12}}{3}\\right) +\n\\frac{1}{24} \\Delta \\pi_3^2\\right] \\zeta^3 + \\cdots \\nonumber \\\\\n& & \\hspace{2.4cm} \\epsfxsize=1cm\\epsfbox{z123.eps}\n    \\hspace{2.1cm} \\epsfxsize=.7cm\\epsfbox{z33.eps}\n\\end{eqnarray}\nThe picture underneath each term is the corresponding\nskeleton graph---see the Introduction.\nThe various coefficients $\\pi_p$\n($p$ is a partition of an integer $n$, where $n$\nis the cluster's external legs)\nare functions of the cosmological constant $g$ that sum\nover the internal geometries of the clusters.\n\nTo define and to calculate the $\\pi$'s, one introduces\nan external matrix integral\n\\begin{equation}\n\\label{kgndef}\n\\Pi(g,\\Lambda) = -\\frac{1}{2N} {\\rm Tr}\\,\\Lambda^2 -\n\\log \\frac{1}{p_0} \\int {\\cal D}\\phi \\:\n\\exp -N \\, {\\rm Tr}\\, \\left( \\Lambda\\phi + \\frac{1}{2} \\phi^2 +\ng \\phi^3 \\right)\n\\end{equation}\nThis integral has been calculated by Kazakov and by\nGross and Newman.\nIf we expand it in powers of $\\Lambda$, we get\n\\begin{eqnarray}\n\\label{kgnexp}\n\\Pi(g,\\Lambda) & = & \\pi_0 + \\frac{\\pi_1}{N} {\\rm Tr}\\, \\Lambda +\n\\frac{1}{2} \\left[\\frac{\\pi_2}{N} {\\rm Tr}\\, \\Lambda^2 +\n\\frac{\\pi_{11}}{N^2} ({\\rm Tr}\\, \\Lambda)^2 \\right] \\nonumber \\\\\n& & \\qquad + \\left[ \\frac{\\pi_3}{N} {\\rm Tr}\\, \\Lambda^3 +\n\\frac{\\pi_{12}}{N^2} {\\rm Tr}\\, \\Lambda {\\rm Tr}\\, \\Lambda^2 +\n\\frac{\\pi_{111}}{N^3} ({\\rm Tr}\\, \\Lambda)^3 \\right] + \\cdots\n\\end{eqnarray}\nThe coefficients $\\pi_p(g)$ are the ones we encounter\nin the low temperature expansion (\\ref{lte}).\nIn $n$-th order, we find $\\pi_p$ for all partitions\n$p$ of $n$.  The coefficient $\\pi_{n_1 n_2 \\cdots}$\nis a connected Green's function of the one-matrix\nmodel $\\Pi(g,0)$ with $n_1 + n_2 + \\cdots\\:$ external\nlegs, on each of which we place the matrix $\\Lambda$;\nthe graphs can be twisted in various ways, which\ngives rise to the different combinations of traces.\nFor example, $\\pi_2$ sums graphs such as\n$$\n\\ba\\epsfxsize=3cm\\epsfbox{twist0.eps}\\ea \\quad \\leadsto \\quad {\\rm Tr}\\, \\Lambda^2\n$$\nwhile $\\pi_{11}$ sums graphs such as\n$$\n\\ba\\epsfxsize=3cm\\epsfbox{twist1.eps}\\ea \\quad \\leadsto \\quad ({\\rm Tr}\\,\\Lambda)^2\n$$\nThe untwisted coefficients $\\pi_n$ are just\nthe $n$-point connected BIPZ Green's function \\cite{BIPZ},\nwhile the twisted coefficients have not been discussed\nbefore in the literature.  They can all be calculated\nexactly, and the first few are given in\nAppendix A.\n\nThe complicated linear combinations\nof the coefficients $\\pi_p$ appearing\nin the low temperature expansion (\\ref{lte}) are due\nto the large $N$ limit of our model.  It will be seen\nbelow that there is a way to organize these linear\ncombinations that will clarify their nature and simplify\ntheir calculation.\n\nHere we briefly summarize the solution of the Potts\nmodel in the $q\\to\\infty$ limit.\nAs it can easily be seen that the coefficient of $\\zeta^n$\nin the partition function is an $n$-th degree polynomial\nin $\\Delta$, the proper scaling for the temperature is\n$\\zeta \\sim 1\/\\Delta$.  Defining $\\zeta = a\/\\Delta$ and keeping $a$\nfixed\\footnote{This is the most interesting way of taking\nthe $q\\to\\infty$ limit.  If we keep $\\zeta$ fixed instead,\nfor example, we will find only one of the several phases\nof the model.}\nwhile letting $q \\to \\infty$, it was found that\nonly skeleton graphs that are trees contribute to\n$F_\\infty(g,a)$.  These can be resummed to give\n\\begin{eqnarray}\n\\label{qinfint}\nF_\\infty(g,a) & = & \\lim_{\\theta\\to\\infty} \\frac{1}{\\theta}\n\\log \\frac{1}{f_0} \\int_{-\\infty}^{\\infty} dx \\:\n\\exp -\\theta \\left[ \\frac{x^2}{2a} - \\Pi(g, x) \\right]\\\\\n\\label{qinfsol}\n& = & \\Pi(g, y) - \\frac{y^2}{2a},\\qquad\n\\Pi(g,y) \\equiv \\Pi(g,y1)\n\\end{eqnarray}\nwhere the function $y(g,a)$ is a saddle point of\n$F_\\infty$:\n\\begin{equation}\n\\label{qinfsaddle}\ny = a \\frac{\\partial}{\\partial y} \\Pi(g,y) = a \\Pi_1(g,y)\n\\end{equation}\nThe general features of the $q=\\infty$ model's critical\nbehavior have already been described in the Introduction;\nsee \\cite{Me3} for details.\\footnote{This limit can\nbe called a mean field solution because it can be\narrived at by (an {\\it a priori} inexact) procedure\nof substituting $\\phi_i \\phi_j \\to \\Phi \\phi_i$ in\nthe multi-matrix model (\\ref{basicpotts}) and solving\nself-consistently for $\\Phi$; additionally, one\nhas to take the (not {\\it a priori} obvious) ansatz\nthat $\\Phi$ is a multiple of unity, as in Wilson's\nmean field solution of QCD.}\n\nKazakov \\cite{Kazakov} has pointed out\nthat the random surface Potts model\ncan also be formulated as a one-matrix model with the action\n\\begin{equation}\n\\label{natural}\nS_n = \\Delta \\left[\\frac{1}{2\\hat{a}} {\\rm Tr}\\,\\phi^2 -\nN \\left(1 + \\frac{1}{\\Delta}\\right)\n\\Pi\\left((1 - a\/\\Delta)^{3\/2} g, \\phi\\right) \\right],\n\\quad \\hat{a} = \\frac{a}{1 - a\/\\Delta}\n\\end{equation}\nThe simple derivation is given in Appendix B.\nThis effective action will be called the ``natural'' one.\nFor the purposes of the present calculation, it is useful to\nintroduce a slightly different, ``artificial'' effective action\n\\begin{equation}\n\\label{artificial}\nS_a = \\Delta \\left[-\\rho\\, {\\rm Tr}\\,\\phi + \\frac{1}{2a} {\\rm Tr}\\,\\phi^2 -\nN \\Pi(g,\\phi) \\right]\n\\end{equation}\nThe difference between these two effective models, their relative\nmerits and drawbacks, are fully explained in\nAppendix B.\n\nIn going from the multi-matrix model (\\ref{basicpotts})\nto the effective one-matrix models (\\ref{natural}) and\n(\\ref{artificial}), we trade the complexity of having\nseveral matrices for the complexity of the new one-matrix\npotential---see eq.\\ (\\ref{kgnexp}).  But what is really\nhappening is that the vertices of the effective models\nare playing a completely different role than the vertices\nof the original model.  In the original multi-matrix model,\nthe vertices represented points on the discretized\nworldsheet.  In the effective models, the vertices---which\nare complicated functions of the original cosmological\nconstant $g$---actually represent entire spin clusters.\nThis is possible because a twisted cluster is equivalent,\nfrom the large-$N$ fatgraph point of view, to a nonlinear\nvertex\n$$\n\\ba\\epsfxsize=3cm\\epsfbox{twist1.eps}\\ea \\quad \\Longleftrightarrow \\quad \\ba\\epsfxsize=1.5cm\\epsfbox{nonlin.eps}\\ea\n$$\nsuch as $({\\rm Tr}\\,\\Lambda)^2$.  This is why all possible nonlinear\nterms appear in the expansion (\\ref{kgnexp}) of the\npotential $\\Pi(g,\\Lambda)$; or, seen from the other direction,\nwhy clusters of every possible twist appear in the low\ntemperature expansion (\\ref{lte}).\\footnote{The\nprocedure that led from multi-matrix model to the effective\none-matrix model can be applied to any---not only Potts---matter.\nOnly the Potts model, however, is symmetric with respect\nto {\\it any} relabeling of matter states; that is why its\neffective model is a {\\it one}-matrix model.  Effective\ncluster models for other types of matter are multi-matrix\nmodels.}\n\nOne could rederive the $q,\\Delta \\to \\infty$ limit\nin a quick and dirty way by noticing that\nwhen the model (\\ref{natural}) is reduced to an eigenvalue\nproblem, the charge of the eigenvalues will scale as\n$\\Delta^{-1}$, and therefore the width of the eigenvalue\nspectrum will scale as $\\Delta^{-1\/2}$.  At $\\Delta=\\infty$, the\neigenvalue density will be a delta function---precisely\nthe content of equation (\\ref{qinfint}).\n\n\\section{Fluctuations about the mean field}\n\nSo at $q=\\infty$ the eigenvalue density of the effective\nmodels (\\ref{natural}) and (\\ref{artificial}) collapses\nto a delta function.\nWhen $q$ is large but finite, the eigenvalues\nspread into a narrowly peaked distribution.  The main idea\nof this work is to investigate the shape of that distribution\nby expanding the effective models about the scalar solution\n\\begin{equation}\n\\label{phipsi}\n\\phi = y 1 + \\psi\n\\end{equation}\nin powers of $\\psi$.  One type of critical point will result\nfrom singularities of the equations for the eigenvalue\ndistribution of $\\psi$.\nOnce an approximate distribution is found in this way, we will\nplug it back into the full effective model to determine critical\nbehavior;\nanother type of critical point will result from singularities\nof the potential $\\Pi$ for a given $\\psi$.\nFor simplicity, the following calculations will be shown only for\nthe artificial model (\\ref{artificial}); the calculations for the\nnatural model (\\ref{natural}) differ in trivial ways, and their\nresult will be given at the end.\n\nThe action (\\ref{artificial}) now reads\n\\begin{equation}\n\\label{intermact}\nS_a = \\Delta \\left[ N\\left(-\\rho y + \\frac{y^2}{2a}\\right) +\n\\left(-\\rho + \\frac{y}{a}\\right) {\\rm Tr}\\, \\psi\n+ \\frac{1}{2a} {\\rm Tr}\\,\\psi^2\n-N \\Pi(g,y+\\psi) \\right]\n\\end{equation}\nThe reader is reminded \\cite{Me3}\nthat the $q=\\infty$ partition function\n$F_\\infty$ is a sum over free trees, and its saddle point $y(g,a)$\nis a sum over {\\it planted} trees: trees with a marked vertex\nof order one.  Explicitly,\n\\begin{equation}\n\\label{yexpl}\ny(g,a) = \\pi_1 a + \\pi_1 (\\pi_2 + \\pi_{11}) a^2 +\n\\left[\\pi_1 (\\pi_2 + \\pi_{11})^2 +\n\\frac{1}{2} \\pi_1^2 (\\pi_3 + \\pi_{12} + \\pi_{111})\\right] + \\cdots\n\\end{equation}\nThis low temperature expansion of $y$ can be represented\ngraphically as\n$$\n\\epsfxsize=8cm\\epsfbox{y.eps}\n$$\nWe are led to consider the expansion of the external matrix integral\nabout a scalar:\n\\begin{eqnarray}\n\\label{expwithy}\n\\lefteqn{\n\\Pi(g,y + \\psi) = \\Pi_0(g,y)\n+ \\frac{\\Pi_1(g,y)}{N} {\\rm Tr}\\, \\psi}\\\\\n& & \\quad \\mbox{} + \\frac{1}{2} \\left[\n\\frac{\\Pi_2(g,y)}{N} {\\rm Tr}\\, \\psi^2\n+ \\frac{\\Pi_{11}(g,y)}{N^2} ({\\rm Tr}\\, \\psi)^2 \\right] \\nonumber \\\\\n& & \\quad \\mbox{} + \\frac{1}{6} \\left[\n\\frac{\\Pi_3(g,y)}{N} {\\rm Tr}\\, \\psi^3\n+ \\frac{\\Pi_{12}(g,y)}{N^2} {\\rm Tr}\\, \\psi {\\rm Tr}\\, \\psi^2\n+ \\frac{\\Pi_{111}(g,y)}{N^3} ({\\rm Tr}\\,\\psi)^3 \\right]\n+ \\cdots \\nonumber\n\\end{eqnarray}\nThe vertex functions $\\Pi_p$ can be calculated explicitly, and are\ngiven in Appendix A.\nTo understand what they are counting,\nwe can expand several of them in powers of $y$:\n\\begin{eqnarray}\n\\Pi_0(g,y) & = & \\pi_0 + \\pi_1 y + \\frac{1}{2} (\\pi_2 + \\pi_{11}) y^2\n+ \\frac{1}{6} (\\pi_3 + \\pi_{12} + \\pi_{111}) y^3 + \\cdots\\nonumber \\\\\n\\Pi_2(g,y) & = & \\pi_2 + \\left(\\pi_3 + \\frac{\\pi_{12}}{3}\\right) y\n+ \\frac{1}{2} \\left(\\pi_4 + \\frac{\\pi_{13}}{2} + \\frac{\\pi_{22}}{3} +\n\\frac{\\pi_{112}}{6} \\right) y^2 + \\cdots\\nonumber \\\\\n\\label{cubexpy}\n\\Pi_3(g,y) & = & \\pi_3 + \\left(\\pi_4 + \\frac{\\pi_{13}}{4}\n\\right) y + \\cdots,\\: {\\rm etc.}\n\\end{eqnarray}\nThe linear combinations of $\\pi_p$'s in $\\Pi_0, \\Pi_2, \\Pi_3,\n\\ldots$ are precisely those that occur in the low temperature\nexpansion (\\ref{lte}).  The $n$-th coefficient of $\\Pi_0$\ncounts clusters that have $n$ external legs, each of which is\nconnected to a tree, {\\it i.e.}, none of its legs reconnect.\nBy contrast,\nthe $n$-th coefficient of $\\Pi_2$ counts clusters that have\ntwo legs that lie on a loop (and will eventually reconnect),\nand $n$ other legs that are connected to trees; and so on.\nThus a cluster with three external legs, for example, may\nappear in the low temperature expansion as $\\pi_3$, $\\pi_3\n+ \\pi_{12}\/3$, or $\\pi_3 + \\pi_{12} + \\pi_{111}$, depending\non whether it has 3, 2, or 0 legs that lie on loops.\n\nThe expansion of action (\\ref{intermact}) in powers of\n$\\psi$ is therefore\\footnote{Two terms cancel due to\n(\\ref{qinfsol}).}\n\\begin{eqnarray}\n\\label{artfull}\n\\frac{S_a}{\\Delta} & = & -N \\left(\\Pi_0 - \\frac{y^2}{2a} + \\rho y\\right)\n-\\rho\\,{\\rm Tr}\\,\\psi + \\frac{1}{2} (a^{-1} - \\Pi_2) {\\rm Tr}\\, \\psi^2 -\n\\frac{\\Pi_{11}}{2N} ({\\rm Tr}\\,\\psi)^2 \\nonumber \\\\\n& & \\qquad\\mbox{} - \\frac{\\Pi_3}{6} {\\rm Tr}\\, \\psi^3\n- \\frac{\\Pi_{12}}{6 N} {\\rm Tr}\\, \\psi \\, {\\rm Tr}\\, \\psi^2\n- \\frac{\\Pi_{111}}{6 N^2} ({\\rm Tr}\\,\\psi)^3 + \\cdots\n\\end{eqnarray}\nAccording to the above discussion, the vertex $\\Pi_3\\,{\\rm Tr}\\,\\psi^3$\nis really an infinite sum of vertices\n(\\ref{cubexpy}), and can be graphically\nrepresented as\n\\begin{equation}\n\\label{vertfigone}\n\\Pi_3(g,y)\\,{\\rm Tr}\\,\\psi^3 \\quad \\leadsto \\quad\n\\begin{array}{c}\\epsfxsize=7cm\\epsfbox{vert0.eps}\\end{array}\n\\end{equation}\nwhile a non-linear vertex such as $\\Pi_{12}\\,{\\rm Tr}\\,\\psi\\,{\\rm Tr}\\,\\psi^2$\nmay be represented as\n\\begin{equation}\n\\label{vertfigtwo}\n\\Pi_{12}(g,y)\\,{\\rm Tr}\\,\\psi\\,{\\rm Tr}\\,\\psi^2 \\quad \\leadsto \\quad\n\\begin{array}{c}\\epsfxsize=7cm\\epsfbox{vert1.eps}\\end{array}\n\\end{equation}\nThe model (\\ref{artfull}) is an expansion about the tree-like,\nmean field solution $\\phi = y$.  The geometrical meaning of\nthis statement is reflected in the above pictures: trees are\nautomatically included in the vertices of the new matrix model.\nThe graphs of this matrix model can be pictured as a dense\nnetwork (in fact, a spherical surface) of clusters, from the\nvertices of which sprout trees.\n\nTo proceed farther with the matrix\nmodel (\\ref{artfull}) we are obliged to truncate\nthe potential $\\Pi(g,y+\\psi)$.\nThe simplest possibility is to truncate\nafter the quadratic term.\nWe take $\\rho=0$ in this case,\nand immediately evaluate the partition\nfunction\\footnote{The last term in (\\ref{trunctwo})\nis there to cancel cluster self-con\\-nec\\-tions; it should have\nbeen put in by hand as a constant in the action.}\n\\begin{equation}\n\\label{trunctwo}\nF = \\Pi_0(y) - \\frac{y^2}{2a}\n- \\frac{1}{2\\Delta} \\log[1 - a \\Pi_2(y)] - \\frac{a \\Pi_2(y)}{2\\Delta}\n\\end{equation}\nThe new $1\/\\Delta$ terms in the partition function just count the\none-loop contributions to the low temperature expansion\n(\\ref{lte}).\nIt is clear that they cannot effect the critical\nbehavior of the model: a large tree with one loop is\nindistinguishable from just a large tree.  This can\nbe demonstrated by noticing that the only way in which\nthe $1\/\\Delta$ terms can effect the critical behavior of the model\nis through a divergence of the logarithm;\nbut, since $\\Pi_2 < \\partial_y^2\n\\Pi_0 = \\Pi_2 + \\Pi_{11}$ (see Appendix A),\nand $1 - a \\partial_y^2\n\\Pi_0 \\ge 0$ in the physical region of the $q=\\infty$ model\n(see \\cite{Me2}), this is impossible.\n\nWe are therefore obliged to keep higher order terms in the\npotential $\\Pi(g,y+\\psi)$.\nThe meaning of a truncation after $\\psi^n$ terms,\nfor $n \\ge 3$, is as follows.\nConsider a cluster with $\\lambda$ links leading to other clusters.\nOf those links, $\\lambda_t$ are to trees and $\\lambda_\\ell = \\lambda - \\lambda_t$\nare to loops; meaning that if one of the $\\lambda_t$ tree links is\ncut, the skeleton will be disconnected.  For instance, the cluster\nindicated by the arrow\n$$\n\\epsfxsize=2cm\\epsfbox{exvert.eps}\n$$\nhas $\\lambda_t = 2$ and $\\lambda_\\ell = 3$.  So: if we truncate the\naction (\\ref{artfull}) after the $\\psi^n$ terms, we discard\nany skeleton that has a cluster with $\\lambda_\\ell > n$.  Note\nthat $\\lambda_t$ does not matter: as discussed above, each vertex\nof (\\ref{artfull}) automatically includes arbitrary numbers\nof trees.\n\nWe now consider the model (\\ref{artfull}) in the {\\it cubic\napproximation}: we discard all terms of order $\\psi^4$ or\nhigher.  In this approximation the\nclusters---which are themselves spherical $\\varphi^3$\ngraphs---are arranged into spherical $\\varphi^3$ graphs\nthat also have trees of clusters\ngrowing from their vertices (given by\npowers of $y$: see (\\ref{vertfigone}), (\\ref{vertfigtwo})).\nThis structure will be\ncalled a ``galaxy.''  A typical graph in the cubic\nmodel has several galaxies, with the nonlinear terms\n$({\\rm Tr}\\,\\phi)^2$, ${\\rm Tr}\\,\\phi\\,{\\rm Tr}\\,\\phi^2$, and $({\\rm Tr}\\,\\phi)^3$\nconnecting the galaxies into a tree structure (not to\nbe confused with the trees that grow out of the vertices\ninside the galaxies).  The cubic approximation differs\nradically from the quadratic one in that it does not simply\nsum over graphs with one more loop and with one higher\norder in $\\Delta^{-1}$, but rather it sums over a large\nclass of graphs with an arbitrary number of loops.\nIn fact, it sums over all graphs, provided that no\ncluster has more than three legs lying on loops;\nfor example, the following are simple skeleton graphs\nthat are {\\it not} included in the cubic approximation:\n$$\n\\epsfxsize=5cm\\epsfbox{notcubic.eps}\n$$\n\nThe solution of the cubic model will proceed along the\nsame lines as the simpler model solved in \\cite{Indians}.\nFirst off, we set\n\\begin{equation}\n\\rho = -\\frac{\\Pi_3}{2a} - \\frac{\\Pi_{12}}{6a}\n\\end{equation}\nto cancel the cluster self-connections.  Then we follow\nthe standard procedure, reducing the problem to eigenvalues\nand introducing a continuous approximation to the spectrum\n$\\lambda(x)$.  Using \nprocedure that is exact in the $N\\to\\infty$ limit\nwe introduce the two moments\n\\begin{equation}\nc = \\left\\langle\\frac{1}{N}{\\rm Tr}\\,\\psi\\right\\rangle = \\int dx\\,\\lambda(x),\n\\qquad\nd = \\left\\langle\\frac{1}{N}{\\rm Tr}\\,\\psi^2\\right\\rangle=\\int dx\\,\\lambda^2(x)\n\\end{equation}\nas parameters, for which will we solve self-consistently\nat the end.  The saddle point equation for the action reads\n\\begin{eqnarray}\n\\label{saddlepoint}\n\\frac{\\delta S}{\\delta\\lambda} & = & q + m \\lambda(x) + 3 p \\lambda^2(x)\n- \\frac{1}{2} \\int' \\frac{dy}{\\lambda(x)-\\lambda(y)} = 0 \\\\\n-\\frac{p}{\\Delta} & = & \\frac{\\Pi_3}{6} \\\\\n-\\frac{m}{\\Delta} & = & -\\frac{1}{a} + \\Pi_2 + \\frac{c \\Pi_{12}}{3} \\\\\n-\\frac{q}{\\Delta} & = & \\rho +  c \\Pi_{11} +\n\\frac{d\\Pi_{12}}{6} + \\frac{c^2 \\Pi_{111}}{2}\n\\end{eqnarray}\nkeeping in mind that the $\\Pi_p$'s are (complicated) functions\nof $g$ and $y(g,a)$.\nThe eigenvalue density for this problem is\n\\begin{equation}\n\\label{evaldens}\nu(\\lambda) = \\frac{1}{\\pi} (m + \\sigma + 3 p \\lambda)\n\\sqrt{\\frac{4}{m + 2\\sigma} - \\left(\\lambda - \\frac{\\sigma}{3p}\\right)^2}\n\\end{equation}\nwhere the variable $\\sigma$ satisfies the equation\n\\begin{equation}\n\\label{sigma-eq}\n18 p^2 + \\sigma(m+\\sigma)(m+2\\sigma) + 3 p q(m+2\\sigma) = 0\n\\end{equation}\nUsing the eigenvalue density (\\ref{evaldens}), the self-consistency\nconditions on $c$ and $d$ become\n\\begin{eqnarray}\n\\label{c-eq}\nc & = & \\frac{3p}{(m+2\\sigma)^2} + \\frac{\\sigma}{3p} \\\\\n\\label{d-eq}\nd & = & \\frac{m+4\\sigma}{(m+2\\sigma)^2} +\n\\left(\\frac{\\sigma}{3p}\\right)^2\n\\end{eqnarray}\n\nThe reader is reminded that the eigenvalue density (\\ref{evaldens})\nis for the $\\psi$ matrix.  Since $\\sigma$ scales as $\\Delta^0$, for\nlarge $\\Delta$ the eigenvalue density for $\\phi$ in the cubic\napproximation is centered on the $q=\\infty$ value $y$ with\na displacement $\\sigma\/3p \\sim {\\cal O}(\\Delta^{-1})$.  The width\nof the distribution is $\\sqrt{4\/(m+2\\sigma)} \\sim {\\cal O}\n(\\Delta^{-1\/2})$, as previously advertised.\n\n\\section{Critical behavior}\n\nThe cubic approximation to the Potts model is expected\nto have three critical phases: the phase of {\\it large\nclusters}, the phase of {\\it large trees} of clusters,\nand the phase of {\\it large galaxies} of clusters.\n\nFirst, the regime of {\\it large clusters}.\nThis is the magnetized, low temperature phase.\nAs we have an approximate form for the eigenvalue density\n(\\ref{evaldens}), we can plug it back into the full action\n(\\ref{artificial}) and look for the singularities of the\npotential $\\Pi(g,\\phi)$.  The reader is reminded\n\\cite{Me3} that for $q=\\infty$, the eigenvalue density\nis a delta function at $y(g,a)$; the potential in that\ncase can easily be reduced to the ``standard'' form\n$\\phi^2\/2 + G \\phi^3$, with $G = (1 - 12 g y)^{-3\/4} g$.\nAt the critical point\n\\begin{equation}\n\\label{qinfcrit}\nG=G_c=1\/\\sqrt{108\\sqrt{3}}\n\\end{equation}\nthe area of the clusters (but not their number) diverges.\nThe same thing happens in the case of the spread out density\n(\\ref{evaldens}), though the computation is somewhat more\ndifficult: we are looking for the critical point of the\nexternal matrix model (\\ref{kgndef}).\nThe result can be found in \\cite{GrossNewman}.\nThe only dependence on the external matrix is through\nthe moments\n\\begin{equation}\n\\label{defmoments}\n\\sigma_k = \\int d\\lambda \\, u(\\lambda) (x - \\lambda)^{-k\/2}\n\\end{equation}\nwhere the variable $x$ satisfies the implicit equation\n\\begin{equation}\n\\label{eqgn}\nx = \\frac{1}{12g} - \\sqrt{3g} \\sigma_1(x)\n\\end{equation}\nFor the eigenvalue density, these moments can be\ncalculated explicitly (as functions of $x$):\n\\begin{eqnarray}\n\\lefteqn{\\sigma_k = \\left(x - y - \\frac{\\sigma}{3p}\\right)^{-k\/2}\n\\left[\\mbox{}_2 F_1\\!\\left(\\frac{1}{2}+\\frac{k}{4},\\frac{k}{4};2;\n\\frac{4}{(1+2\\sigma) (x - \\sigma\/3p)^2}\n\\right)\\right.} \\\\\n& & \\left.\\qquad\\mbox{} + \\frac{3kp}{2(1+2\\sigma)^2\n(x - \\sigma\/3p)}\n\\,\\mbox{}_2 F_1\\!\\left(\\frac{1}{2}+\\frac{k}{4},1+\\frac{k}{4};3;\n\\frac{4}{(1+2\\sigma) (x - \\sigma\/3p)^2}\n\\right)\\right]\n\\nonumber\n\\end{eqnarray}\nThe criticality condition is the point where the\n$x$ equation (\\ref{eqgn}) ceases to have solutions.\nAlgebraically this can be expressed as\n\\begin{equation}\n\\label{stupidcrit}\n3 g \\sigma_3^2 = 4\n\\end{equation}\nIt can easily be checked that when $q,\\Delta\\to\\infty$,\nand consequently $\\sigma\/3p,4\/(m+2\\sigma) \\to 0$,\nthese new criticality conditions (\\ref{eqgn}) and\n(\\ref{stupidcrit}) reduce to (\\ref{qinfcrit}).\nIt will be seen in the phase diagrams that as\n$q$ approaches infinity, the curve of large clusters\nwill approach the corresponding $q=\\infty$ curve,\ngiven by solving eqs.\\ (\\ref{qinfcrit}) and \n(\\ref{qinfsaddle}) simultaneously.\n\nIn the large cluster phase matter is manifestly\nmagnetized, and is therefore decoupled from the\ngeometry; so this is a phase of pure gravity,\nwith $\\gamma_{\\rm str} = -\\frac{1}{2}$.  Another way of saying the\nsame thing: in this phase the number of clusters\nis small, and few clusters is the same as one\ncluster, as far as critical behavior is concerned.\n\nSecond, the phase of {\\it large trees} of clusters.\nThere are, in principle, two ways of getting there.\nAt $q=\\infty$, the trees become large when the saddle\npoint equation for $y(g,a)$ (\\ref{qinfsaddle})\nceases to have a solution \\cite{Me3}; in other words, when\n\\begin{equation}\n\\label{qinfsaddlecrit}\na \\frac{\\partial^2}{\\partial y^2} \\Pi(g,y)\n= a \\left[\\Pi_2(g,y) + \\Pi_{11}(g,y)\\right] = 1\n\\end{equation}\nThe other way to get large trees is to make the\ntree of {\\it galaxies} (of clusters) blow up.\nThis is not exactly the same as a large tree of\nclusters at $q=\\infty$, but very nearly the same,\nas in general the galaxies at the vertices of this\ntree will be small; and as far as critical behavior\nis concerned, a small galaxy is no different from\na galaxy of size one, in other words, a single\ncluster.  This intuition is supported by the fact\nthat both large tree phases have $\\gamma_{\\rm str} = +\\frac{1}{2}$;\nand as $q\\to\\infty$, the curves of the two phases\napproach each other.\n\nTo calculate the location of the second large tree\nphase, we can look for the point in the cubic model\n(\\ref{saddlepoint}) where the string susceptibility\n$\\partial^2 F\/\\partial p^2$ diverges \\cite{Indians}.\nEquivalently, and computationally simpler, we can\nlook for the divergence of the\n``pseudo-susceptibility''\\footnote{$d'$ or $\\sigma'$\nwould do just as well.}\n\\begin{equation}\n\\label{defpseudo}\nc' \\equiv \\frac{\\partial c}{\\partial p} \\to \\infty\n\\end{equation}\nThis is most simply done by differentiating the\neqs.\\ (\\ref{sigma-eq}--\\ref{d-eq}) with respect to\n$p$, and solving the resulting linear system for $c'$,\n$d'$, and $\\sigma'$.\nAs the result is unwieldy, and in any case can easily\nbe derived, it will not be given here.  The same procedure,\nif applied to the simple quartic model in \\cite{Indians},\nreproduces the result for the large tree phase given in that\nwork.\n\nIn the large tree phase one has an ensemble of clusters\nof different size.  As one moves toward the magnetized\nphase, clusters of higher and higher area begin to be\nincluded.  At the multicritical point, some\nmoments of the cluster area diverge, a sign \nof a continuous spin ordering phase\ntransition.\\footnote{For the Ising model\non a fixed two-dimensional lattice, for example, the\nsecond moment of the cluster size distribution is the\nlowest one to diverge.\\cite{Fisher}}  The susceptibility\nexponent in the bulk of the tree phase is\n$\\gamma_{\\rm str} = \\frac{1}{2}$, the generic exponent for tree growth.\nIn the tree phase, and only in the tree phase, the\nclusters are probably identical to the geometric structures\nknown as ``baby universes'' \\cite{MonteCarlo}.\nTherefore the transition to large trees from high\ntemperature can be interpreted as a magnetization\nof baby universes (but not of the whole surface);\nand the transition to trees from low temperature can\nbe interpreted as a break-up of the magnetized surface\ninto magnetized (but in different directions) baby\nuniverses \\cite{Teaser}.\n\nFinally, the cubic model should also have a phase\nof {\\it large galaxies}.\nThis occurs simply at the normal, BIPZ, critical point \nof the model (\\ref{saddlepoint}).  This is simply given\nby\n\\begin{equation}\n\\label{largegalaxies}\n\\Gamma = \\frac{p}{(m - 12 p q)^{3\/4}} = \\Gamma_c =\n\\frac{1}{\\sqrt{108 \\sqrt{3}}}\n\\end{equation}\nThis phase is radically different from anything that\nexists at $q=\\infty$; it is in this phase that loops\nbecome important.  Interestingly, it bears some resemblance\nto what {\\it must} be the case for small $q$: the spin\nclusters at high temperature do not form trees,\nbut complicated planar networks with many loops.\n\nThe large planar networks of clusters in the galaxy\nphase resemble the configurations of a low $q$ model\nat high temperature.  As we go to higher temperature,\nmoreover, the clusters get smaller, as matter fluctuations\nare once again decoupled from the geometry.  The large\ngalaxy phase is therefore pure gravity, once again.\nThe susceptibility exponent is $\\gamma_{\\rm str} = -\\frac{1}{2}$, the\nstandard pure gravity value for nonlinear models in\nthis phase.\n\nWe are now ready to study the complicated\ntranscendental equations of the cubic approximation.\nThe procedure is as follows.  First, fix the three\nadjustable parameters: $\\Delta = q-1$, $g$, and $a$.\nThen solve the saddlepoint equation (\\ref{qinfsaddle})\nfor $y(g,a)$; this fixes all coefficients in the\ncubic action (\\ref{artfull}).  Next, solve the\ncubic equation (\\ref{c-eq}) for $c$ as a function\nof $\\sigma$, plug into (\\ref{sigma-eq}) and solve\nnumerically for $\\sigma$, taking the root closest\nto zero.  Finally, check the criticality conditions:\n(\\ref{eqgn}) and (\\ref{stupidcrit}) for the phase\nof large clusters; (\\ref{defpseudo}) for large\ntrees; or (\\ref{largegalaxies}) for large galaxies.\n\nThe following is a schematic, not-to-scale sketch\nof the resulting critical curves.\n$$\n\\epsfxsize=12cm\\epsfbox{approx.eps}\n$$\nIf drawn to scale, the curves are rather close to\neach other and hard to distinguish by eye.\nAs $q\\to\\infty$, the curves reassuringly approach\nthe exact $q=\\infty$ critical curve, where the\ntransition from magnetized to tree phases occurs\nat $g = 1\/\\sqrt{288}, a = 1$.  As we move\naway from $q=\\infty$, the new critical regime of\nlarge galaxies appears.  As $q$ gets lower, the\nphases of large galaxies and large clusters\napproach each other, eating away at the mean field,\nlarge tree phase.\n\nThe susceptibility exponent $\\gamma_{\\rm str}$ is $-\\frac{1}{2}$ in the\nmagnetized and galaxy phases, as discussed above,\nand $+\\frac{1}{2}$ in the tree phase.  At the two multicritical\npoints $\\gamma_{\\rm str} = \\frac{1}{3}$.  Further, we find $\\alpha=-1$\nat both multicritical points, as shown in Appendix C.\nThe equality of the exponents at the two multicritical\npoints is not a coincidence: the model at one of the\npoints can be mapped onto the model at the other---see\nthe appendix for details.\n\nAt $q=q_c$ ($q_c \\approx 133$ for the natural model,\nand $q_c \\approx 108$ for the artificial model), the\nphase of large trees ceases to exist.  Thereafter,\nthe magnetized phase of large clusters goes\ndirectly into a phase of large galaxies.\nThe mean field behavior is entirely eliminated.\nLet $g_{c1}(q)$ be the multicritical point between\nphases of large galaxies and large trees, and\n$g_{c2}(q)$ the multicritical point between \nthe phases of large trees and large clusters.\n$\\Delta g_c = g_{c2} - g_{c1}$ is a measure of\n``how much of mean field theory is left'' at a\ngiven value of $q$.  Here it is plotted for both\nthe natural and the artificial models.\n\\epsfxsize=12cm\\epsfbox{qplot2.eps}\n\nFor $q \\le q_c$, the two pure gravity phases\n(magnetized and large galaxies) are joined directly\nonto each other, with a multicritical point in\nthe middle.  This phase diagram bears a striking\nresemblance to the known phase diagram for, say,\n$q=2$ \\cite{KazakovIsing}.\n\n\\section{Discussion}\n\nThe results obtained for $q_c$ cannot be considered\nas anything more than order-of-magnitude estimates.\nThe real result of this calculation is the discovery of\nthe mechanism for the break-down\nof mean field theory: the divergence\nof galaxies.  The resulting critical behavior interpolates\nsmoothly between the somewhat odd phase diagram at $q=c=\\infty$\nand the well-known phase diagram for $q \\le 4, c \\le 1$.\nAnother result of this calculation is the likelihood\nthat $q_c$ is {\\it finite}, that mean field behavior\nholds for some finite Potts\nmodels (with modifications, as has been shown).\n\nThe two natural questions at this stage are: is the\npresent result merely an artifact of the truncation of the\naction (\\ref{artfull}), or are the higher-order terms\nessential?  And does a similar picture hold for conformally\nsymmetric models, which, after all, reduce to the same\nmean field theory at $c=\\infty$?\n\nThe first question should not be too hard to answer.\nIt seems that we would not find any qualitatively new\nbehavior by including the higher-order terms, which\nwould simply add other types of vertices to the galaxies\nbut would not modify the essential structure.\nMean field theory should always be destroyed by the\ndivergence of the galaxies.\nIt is true that quartic or higher models will add\nnew multicritical points; but as we only have the three\nparameters $q, g,$ and $a$ to tune, we should not be\nable to reach them.\nTo repeat the present calculation in the quartic approximation\nshould be somewhat tedious, but not too difficult.\n\nThe second question is more difficult to answer, and the\nauthor does not have any suggestions on how to deal with\nit.\nThe immediate difficulty is that, although effective\ncluster models such as (\\ref{natural}) and (\\ref{artificial})\ncan be introduced for any matter model on the surface,\nthey are multi-matrix rather than one-matrix models.\nStill, they closely resemble the $c=\\infty$ model, so\nperhaps it is the right way to start.\n\n\\vspace{1cm}\nThe author thanks the Niels Bohr Institute for\nits hospitality, and Jan Ambj\\o rn and Gudmar Thorleifsson\nfor stimulating discussions.  Marc Potters provided many\nof the ideas in the early stage of this work.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{section:intro}\nOur current working hypothesis is that the dominant force acting on large scales in the universe is gravity, and that gravity is accurately described by General Relativity (GR). In practice, we often focus on cosmological systems that are smaller than the Hubble scale; this permits us to make a set of simplifications referred to as the quasi-static (QS) limit. \n\n The QS regime corresponds to a window of lengthscales that are considerably smaller than the cosmological horizon, such that ${\\cal H}\/k\\ll1$, but sufficiently large such that linear perturbation theory is still valid.\nIn concrete figures, a reasonable estimate would be distances less than $600h^{-1}$ Mpc but greater than $10-20h^{-1}$ Mpc. The usual argument is that within this window of lengthscales the time derivatives of metric potentials are significantly smaller than their spatial derivatives. In GR this statement is a natural consequence of the subhorizon condition (${\\cal H}\/k\\ll1$), since the linear gravitational potentials evolve on the Hubble timescale. In practical terms, implementing conditions such as \\mbox{$|\\ddot\\Phi|\\ll |\\nabla^2\\Phi|$} makes the linearised field equations easy to work with.\n\nWhen we go over to a modified theory of gravity, the situation is less clear-cut. On one hand we can reason that any gravity theory consistent with current observations must behave in a manner very similar to $\\Lambda$CDM, so we expect our quasistatic limit to be preserved. On the other hand, when we modify GR we naturally introduce new dynamical degrees of freedom (hereafter d.o.f.) which might have evolutionary timescales different from ${\\cal H}$. In this paper we will assume that the new d.o.f. are sufficiently subdominant that a QS limit still exists for most of the history of the universe.\n\nThe largest distances that can be probed by current and next-generation galaxy surveys fall predominantly within the QS regime. To use these experiments to test the laws of gravity, we need to understand the typical behaviour of non-GR  theories in the QS limit. There is a long-standing intuition that modified gravity theories generically lead to a novel dependence of observables on the length scale at which they are measured, e.g. the density-weighted growth rate, $f\\sigma_8(z)$, becomes a function of wavenumber $k$. Work has already begun to search for such signatures \\cite{Johnson2014}.\n\nThe goal of this paper is to make concrete these intuitions. What are the implications of a (non-)detection of scale-dependence for the host of  gravity theories in the current literature \\cite{Clifton2012}? What are the most theoretically-motivated parameters for observers to measure? We present three main results:\n\n\\vspace{2mm}\n\\noindent 1) Considering a frequently-used parameterisation of the linearised gravitational field equations, we show that only a few key physical principles are needed to derive the fixed scale-dependence of many gravity theories in the QS regime. Our results apply to any theory with second-order equations of motion and one new spin-0 degree of freedom, which does \\textit{not} have to be a scalar field. This generalises the results of \\cite{defelice2011, Unobservables2013, Gleyzes2013, Bloomfield2013} beyond single-scalar field theories (see \\cite{Silvestri2013} for related ideas). The derivation is compact and does not require knowledge of a gravitational action -- hence its generality.\n\\vspace{2mm}\n\n\\noindent 2) We show that the detectability of this non-GR scale-dependence hinges crucially on the existence of new physical quantities (characteristic masses, lengths, etc.) that are generically introduced when modifying GR. If all such parameters are tuned to be comparable to the Hubble scale, it is highly unlikely that any novel scale-dependence of observables will be detectable in the QS regime.\n\\vspace{2mm}\n\n\\noindent 3) Turning these ideas around, we isolate the leading-order scale-dependent terms and estimate the constraints placed upon them by next-generation cosmological experiments. The headline results are displayed in Fig.~\\ref{fig:ellipses}.\n\nThe structure of this paper is as follows: in \\S\\ref{section:set-up} we discuss how characteristic physical quantities enter most popular theories of gravity. In \\S\\ref{section:derivation} we derive the result described in 1) above. In \\S\\ref{section:mass_scales} we discuss the implications of a (non-)detection of scale dependence, i.e. point 2) above. We also isolate the leading-order contribution to scale-dependence; in \\S\\ref{section:forecasts} we carry out example forecasts for future constraints on this leading-order term (point 3) above). We conclude in \\S\\ref{section:conclusions}. Some technical details are relegated to the appendices.\n\n\n\\section{New Scales \\& Set-up}\n\\label{section:set-up}\nNew physical scales are a near-universal feature of alternative gravity theories. This is no surprise: the success of GR in describing the Solar System generally forces us to introduce a `transition scale' into gravitational physics, positing that gravity reduces to GR on one side of this transition scale, but receives modifications on the other side. \n\nLet us elaborate with some examples. Probably the most familiar example of a new scale arises in scalar-tensor theories, where a mass scale emerges from second derivatives of the potential, $V(\\phi)_{,\\phi\\phi}$. In $f(R)$ gravity the new scale is more often thought of as a Compton wavelength for the scalaron, but is similarly derived from derivatives of an effective potential (a function of $f_{,RR}$) \\cite{Pogosian2008}. New scales can arise in a different way when there is non-trivial coupling between the matter energy-momentum tensor and the scalar degree of freedom, e.g. in theories which display chameleon screening, the transition scale is marked by a potential well depth, $|\\Phi|\\sim 10^{-6}$ \\cite{Davis2012,Jain2013}.\n\nVector-tensor theories are often endowed with an energy (mass) scale at which violations of Lorentz invariance become manifest. This new scale can be an explicit parameter in the Lagrangian of the theory (as, for example, in Horava-Lifschitz gravity \\cite{Horava2009, Sotiriou_HL_review}), or it can arise implicitly via spontaneous Lorentz violation at the level of the field equations (e.g. in the effective field theory of a vector coupled to gravity \\cite{Gripaios2007}).\n\nIt is well known that current bimetric theories not only have an explicit mass scale (the mass of the graviton), but also a system-dependent length scale, the Vainshtein radius, which signals the onset of screening \\cite{Babichev2013}. \nSimilarly, higher-dimensional theories such as DGP \\cite{Deffayet2002} and other braneworld models can have both an explicit scale such as a warp factor or crossover scale, as well as Vainshtein radii.\n\nFinally, there has been recent interest in non-local theories \\cite{Maggiore22014, Dirian2014, Maggiore2014, Foffa2014} containing Lagrangian terms such as $R\\,\\square^{-2}R$. The solutions for $\\square^{-1}R$ involve integrals over Green's functions for the $\\square^{-1}$ operator, $G(x,x^\\prime)$. This naturally suggests a characteristic scale between the spacetime points $x$ and $x^\\prime$ over which non-local interactions occur.\n\nIn essence, new transition scales can be dependent on a variety of physical quantities such as energy, ambient density, acceleration, potential, etc. Throughout this paper we will be agnostic about the origin of any new transition scale. In many gravity theories the new scale(s) are tuned to be of order the Hubble scale today, in the hope that they might replace the cosmological constant as the driver for accelerated expansion. In other theories, however, an effective cosmological constant is included in the theory (sometimes explicitly, sometimes via a `back door') \\cite{Faulkner2007,HuSawicki}. Then, since cosmic acceleration is already taken care of, new mass or length scales inherent in the theory may take a much wider range of values. \n\n\nLet us define a wavenumber $k_{\\mathrm{gal}}$ that represents the largest perturbation mode that can be reasonably well-measured by next-generation cosmological surveys. Let us also introduce a mass scale $M$ that represents the transition scale accompanying some generic modifications to gravity; in what follows we will also frequently interpret $M$ as an inverse lengthscale. (Note that we can extract a mass or length scale from any of the physical quantities discussed above by using appropriate factors of $c,\\,\\hbar$ etc., and taking appropriate powers).\n\nWe can then envisage three scenarios:\n\\begin{enumerate}[a)]\n\\item $M\\sim {\\cal H}\\Rightarrow M\\ll k_{\\mathrm{gal}}$ in the QS regime. \n\\item ${\\cal H} \\ll M\\lesssim k_{\\mathrm{gal}}$. \n\\item $k_{\\mathrm{gal}}\\ll M$.  \n\\end{enumerate} \nIn this paper we will treat situations a) and b). We will work with one new dynamical degree of freedom, denoting its perturbations by $\\chi$. $\\chi$ does \\textit{not} have to be a scalar field, but could instead be a spin-0 perturbation of a new vector or tensor field, a new d.o.f. excited in the metric (i.e. a metric d.o.f. which is non-dynamical in GR), a Stuckelberg field, or several other possibilities \\cite{PPF2013}. \n\nFor the purposes of this paper we are only interested in the relative orders of magnitude of terms, not in precise factors of order unity. We can therefore write time derivatives of $\\chi$ as $\\dot\\chi={\\Gamma_\\chi}\\chi$, where ${\\Gamma_\\chi}$ is the evolutionary timescale of the new d.o.f. perturbation. There are two possibilities for this timescale: it could either be approximately the Hubble timescale (like for the metric potentials), or it could be a new, shorter timescale determined by $M$ (note that with appropriate factors of $c$ a mass is dimensionally equivalent to an inverse timescale). For now we will maintain generality, but later on we will we see that setting ${\\Gamma_\\chi}\\sim {\\cal H}$ or ${\\Gamma_\\chi}\\sim M$ can have different consequences.\n\nWe will not consider situation c), because in this scenario the existence of a quasistatic limit becomes questionable. If $M$ is very large and ${\\Gamma_\\chi}\\sim M$, then we have that $\\dot\\chi$ is very rapidly evolving, violating quasistaticity. Furthermore, the novel effects that occur at wavelengths close to the lengthscale $M^{-1}$ are likely to occur inside the nonlinear regime, which is beyond the scope of this paper. \n\nConsideration of the Friedmann equation suggests that the background (zeroth-order) values of any new fields present are constrained to evolve on Hubble timescales. Using the example of a scalar field to illustrate, we mean that $\\dot{\\phi}^2\\sim {\\cal H}^2\\phi^2$. Now one might well argue that if a field evolves like ${\\cal H}$ on the background, its perturbations must evolve like ${\\cal H}$ too, that is, only the case ${\\Gamma_\\chi}\\sim{\\cal H}$ is of interest. In \\S\\ref{section:mass_scales} we will argue that if this is true, it seems unlikely that the scale-dependent properties of modified gravity will be measurable any time soon. \n\n\\section{Derivation}\n\\label{section:derivation}\nTo obtain result 1) of \\S\\ref{section:intro}, we first need to take a step back to the gravitational field equations. A linear combination of two components of the tensor field equations gives the gravitational Poisson equation, whilst the transverse spatial component gives the `slip' relation (shown here at late times; our conventions for the metric potentials are given in Appendix~\\ref{app:derivs}):\n\\begin{align}\n-k^2\\Phi&=4\\pi G\\mu(a,k) a^2{\\bar \\rho}_m \\Delta_m \\label{eqNP}\\\\\n\\Phi&=\\gamma(a,k) \\Psi \\label{gamma}\n\\end{align}\nwhere $\\bar{\\rho}_m$ is the mean matter density, $\\Delta_m$ is the gauge-invariant density contrast, and a sum over all matter species is implied.\n\nIn the expressions above we have introduced two functions, $\\mu(a,k)$ and $\\gamma(a,k)$, that have been used extensively as a parameterisation of modified field equations in the QS regime \\cite{BZ2008, Bean2010, Hojjati2013, Simpson2013, Motta2013}. This parameterisation is convenient for theoretical work because it parameterises the `raw' gravitational field equations obtained directly from the action. However, a slightly different parameterisation (`$\\left\\{\\tilde{\\mu},\\,\\Sigma\\right\\}$' -- see eq.(\\ref{convert_params})) is preferred for data analyses, because it leads to minimal parameter degeneracy when combining redshift-space distortions and weak lensing surveys. For this reason we will present our theoretical results in terms of $\\left\\{\\mu,\\,\\gamma\\right\\}$, then rotate to the basis  $\\left\\{\\tilde\\mu,\\,\\Sigma\\right\\}$ for the forecasts in \\S\\ref{section:forecasts}.\n\nTo manipulate a particular gravity theory into the form of eqs.(\\ref{eqNP}) and (\\ref{gamma}), one begins with the full (ie. unparameterised) Poisson equation and slip relation of the theory, and the linearly perturbed equation of motion for the new d.o.f. The steps are as follows:\n\\begin{enumerate}\n\\item Apply the quasistatic approximation to the metric terms in the three equations listed above, that is, drop terms containing $\\ddot\\Phi,\\,\\dot\\Phi,\\,\\ddot\\Psi$ ad $\\dot\\Psi$. Of the remaining terms, discard those with prefactors that evolve on Hubble timescales, i.e. drop ${\\cal H}^2\\Phi$, but keep $\\nabla^2\\Phi$. Time derivatives of $\\chi$ should be replaced by $\\dot\\chi\\approx{\\Gamma_\\chi}\\chi$ and $\\ddot\\chi\\approx{\\Gamma_\\chi}^2\\chi$ but not discarded.\n\\item Take two linear combinations of the unparameterised slip equation and the equation of motion for the d.o.f.: one combination that eliminates $\\chi$, and one that eliminates $\\Psi$. The form of $\\gamma(a,k)$ can be read off from the first linear combination. \n\\item Substitute the ratios $\\Psi\/\\Phi$ and $\\chi\/\\Phi$ obtained in the previous step into the right-hand side of the Poisson equation, so that it is written purely in terms of $\\Phi$ (plus the usual GR term in $\\Delta$). Rearrange this equation into the form of eq.(\\ref{eqNP}) and read off $\\mu(a,k)$.\n\\end{enumerate}\n\nWe wish to avoid laboriously carrying out these steps for many individual gravity theories. So instead we will apply this procedure to a set of `template' field equations that reflect the structure of real theories. A similar derivation was presented first in \\cite{Silvestri2013}; the addition we make is the explicit consideration of a new mass scale, $M$, as discussed in \\S\\ref{section:set-up}.\n\nWe will write down these templates in the conformal Newtonian gauge. To maintain transparent correspondence with the usual linearised Einstein equations we will not use an explicitly gauge-invariant combination of variables to represent the new d.o.f.. However, we will use the fact that the equations must ultimately have a gauge-invariant formulation to guide the construction of our templates.\n\n For example, the usual gauge-invariant Bardeen potentials $\\hat\\Phi$ and $\\hat\\Psi$ contain first- and second-order time derivatives respectively. This means that $\\hat\\Psi$ can only appear in the Poisson equation as part of the combination $\\dot{\\hat\\Phi}+{\\cal H}\\hat\\Psi$ (in which the second time-derivatives cancel out -- see Appendix~\\ref{app:derivs} or \\cite{PPF2013}) to avoid converting the Poisson equation from a constraint into a dynamical equation.\n\nAn example will help to clarify this point and allow us to introduce some notation. For the case where the dimensionless new d.o.f. $\\chi$ is a scalar field or a fluid energy density, the Poisson equation has the form:\n\\begin{align}\n-2k^2\\Phi&=8\\pi G a^2\\bar{\\rho}_m\\Delta_m+\\Phi\\left(h_1k^2+h_2\\left[{\\cal H}^2,M^2,{\\cal H} M\\right]\\right)\\nonumber\\\\\n&+h_3\\left[{\\cal H},M\\right]\\dot\\Phi+m_2\\left[{\\cal H}^2,M^2,{\\cal H} M\\right]\\Psi\\label{Poissonscalar}\\\\\n&+\\chi\\left(g_1k^2+g_2\\left[{\\cal H}^2,M^2,{\\cal H} M\\right]\\right)+g_3\\left[{\\cal H},M\\right]\\dot\\chi\\nonumber\n\\end{align}\nwhere $M$ is the potential new mass scale. Throughout this paper we will use notation like $\\left[{\\cal H}^2, M^2, {\\cal H} M \\right]$ to indicate a function of time which has dimensions of mass-squared. Terms appearing in this function can have three possible order-of-magitudes: ${\\cal H}^2$, $M^2$ or ${\\cal H} M$. The numerical coefficients accompanying these order-of-magnitude terms are unimportant for our purposes. We will use the dimensionless order-unity coefficients $h_i$, $g_i$ etc. simply as a convenient way to refer to individual terms. In complete analogy, $\\left[{\\cal H},M\\right]$ denotes a time-dependent function with the dimension of mass, which can have two possible orders of magnitude: $\\sim{\\cal H}$ or $\\sim M$.\n\nNote that $\\Psi$ appears up to one derivative order lower than $\\Phi$ in eq.(\\ref{Poissonscalar}). This is due to the aforementioned requirement that it must be possible to `repackage' these terms into the combination \\mbox{$\\alpha\\dot\\Phi+\\beta(\\dot\\Phi+{\\cal H}\\Psi)$}, where $\\alpha$ and $\\beta$ are numerical coefficients.\n \nCarrying out step 1 of our procedure, eq.(\\ref{Poissonscalar}) becomes:\n \\begin{align}\n-2k^2\\Phi&=8\\pi G a^2\\rho\\Delta+\\Phi\\left[h_1k^2+h_2M^2\\right]+\\Psi\\left[m_2M^2\\right]\\nonumber\\\\\n&+\\chi\\left[g_1k^2+g_2M^2+g_3M{\\Gamma_\\chi}\\right]\\label{QSPoissonscalar}\n\\end{align}\n\nFor brevity we will not write here the non-quasistatic templates for the slip relation and equation of motion, for this scalar field\/fluid example; they are given in eqs.(\\ref{eomscalar}) and (\\ref{slipscalar}). We move straight to their QS limits, which are:\n\\begin{align}\n\\chi\\Big[d_1 {\\Gamma_\\chi}^2&+d_2 M{\\Gamma_\\chi}+d_3 M^2+d_4 k^2\\Big]\\label{QSeomscalar}\\\\\n&+\\Phi\\left[b_2 M^2+b_3k^2\\right] +\\Psi\\left[c_2 M^2+c_3k^2\\right]=0\\nonumber\\\\\n&\\Phi-\\Psi=e_0\\Phi+j_0\\Psi+f_0\\chi& \\label{QSslipscalar}\n\\end{align}\n\n\\noindent Carrying out steps 2 and 3 described above, we obtain the following forms for $\\mu$ and $\\gamma$:\n\\begin{align}\n\\gamma&=\\frac{p_1+p_2\\frac{M^2}{k^2}+p_3\\frac{{\\Gamma_\\chi} M}{k^2}+p_4\\frac{{\\Gamma_\\chi}^2}{k^2}}{q_1+q_2\\frac{M^2}{k^2}+q_3\\frac{{\\Gamma_\\chi} M}{k^2}+q_4\\frac{{\\Gamma_\\chi}^2}{k^2}}\\label{gamscalar}\\\\\n\\mu&=\\left[p_1+p_2\\frac{M^2}{k^2}+p_3\\frac{{\\Gamma_\\chi} M}{k^2}+p_4\\frac{{\\Gamma_\\chi}^2}{k^2}\\right]\\times\\nonumber\\\\\n\\Big[t_1&+t_2\\frac{M^2}{k^2}+t_3\\frac{{\\Gamma_\\chi} M}{k^2}+t_4\\frac{{\\Gamma_\\chi}^2}{k^2}+t_5\\frac{M^4}{k^4}+t_6\\frac{{\\Gamma_\\chi} M^3}{k^4}\\nonumber\\\\\n&+t_7\\frac{{\\Gamma_\\chi}^2M^2}{k^4}\\Big]^{-1}\\label{muscalar}\n\\end{align}\nwhere the $p_i,\\,q_i$ and $t_i$ are simple algebraic combinations of the order-unity coefficients in eqs.(\\ref{QSPoissonscalar})-(\\ref{QSslipscalar}). Their precise forms are given in Table~\\ref{tab:scalar} in Appendix~\\ref{app:derivs}.\n\nWe immediately recognise that eqs.(\\ref{gamscalar}) and (\\ref{muscalar}) subsume some results already known for scalar field theories \\cite{defelice2011, Unobservables2013, Silvestri2013, Gleyzes2013, Bloomfield2013}, but note that we have not needed to use the complex form of the Horndeski Lagrangian \\cite{Horndeski, Deffayet2011,Gleyzes2014} to obtain them here. For example, we see that $\\mu$ and $\\gamma$ share the same numerator; an equivalent result was proved in \\cite{Silvestri2013, Gleyzes2013} (note that other authors use a slightly different parameterisation variables to ours, equivalent to the set $\\left\\{\\tilde{\\mu},\\,\\gamma\\right\\}$, where $\\tilde\\mu$ is defined in eq.(\\ref{convert_params})).\n  \nOur expression for $\\mu$ contains three terms -- $t_5$, $t_6$ and $t_7$ -- which have not been included in works focused on Horndeski theories. In all Horndeski-type theories we have seen investigated this $k^{-4}$ dependence of $\\mu$ and $\\gamma$ is not present, because the terms represented by $h_2$, $m_2$, $g_2$ and $g_3$ in eq.(\\ref{QSPoissonscalar}) do not feature in the QS limit of their field equations \\footnote{Without the presence of a new mass scale, these terms are of magnitude $\\sim {\\cal H}^2$ and hence are discarded when the QS limit is taken.}. However, we will leave these terms in our general expressions because they could exist in some as-yet-undiscovered gravity theory. It is in the spirit of this paper to remain as agnostic as possible. \n\nWe stress that these forms for $\\mu$ and $\\gamma$ have been derived using purely a few basic principles, such as gauge-invariance and restriction to second-order equations of motion. We have used neither a model-specific action nor a general EFT-inspired one. In fact, if we repeat steps 1-3 for the case where the new d.o.f. is the spatial spin-0 perturbation of a timelike vector field (i.e. $|\\chi|\\sim |k|V$), such as occurs in Einstein-Aether theories \\cite{Jacobson2008,Zlosnik2008}, we reach exactly the same form as eqs.(\\ref{gamscalar}) and (\\ref{muscalar}). The derivation is given in Appendix~\\ref{app:derivs}, and only differs from the one shown here in small details. \n\nNote also that the authors of \\cite{Solomon2014,Koennig} have recently derived expressions equivalent to $\\mu$ and $\\gamma$ in a particular bigravity model, and found them to have a form contained by eqs.(\\ref{gamscalar}) and (\\ref{muscalar}) -- see eqs.(85) and (86) of \\cite{Koennig}. This is not unexpected, since bigravity -- despite its complex field equations (not fully encapsulated by eqs.~\\ref{QSPoissonscalar}-\\ref{QSslipscalar}) -- ultimately has the same physical features as the cases treated explicitly here, i.e. it introduces only one new spin-0 degree of freedom and respects gauge-invariance and locality. If the current stability issues surrounding generic bigravity models \\cite{Comelli2014} can be resolved, then eqs.(\\ref{gamscalar}) and (\\ref{muscalar}) can be used as a universal parameterisation for virtually \\textit{all} theories with second-order equations of motion and a single d.o.f of any type. \n\n\n \\section{REGIMES OF INTEREST} \n \\label{section:mass_scales}\nAs discussed in \\S\\ref{section:set-up}, there are essentially two choices for the scale $M$ that we have introduced. Either we can set $M$ to be of order the Hubble scale, or we can posit a new scale, $M\\gg{\\cal H}$, which marks the transition from the GR limit to some larger theory of gravity. This choice governs whether it makes sense to invest effort searching for scale-dependent signatures of modified gravity \\cite{Johnson2014}.\n \\vspace{5mm}\n \n\\noindent \\textit{Case a).} Many gravity theories explicitly tune their new scale(s) to be of order \\mbox{$H_0\\sim10^{-3}$} eV in order to produce a viable expansion history. For example, in recent bigravity models the graviton mass is taken to be of order the present Hubble scale. In this case there is no choice other than setting $M={\\Gamma_\\chi}={\\cal H}$. If we carry out the full QS approximation \\textit{all} scale-dependent terms in eqs.(\\ref{gamscalar}) and (\\ref{muscalar}) must be discarded and we are left with only the simple time-dependent expressions. Scale-dependence in $\\mu$ and $\\gamma$ only arises if we consider the first-order corrections in $({\\cal H}\/k)^2$ to the QS approximation, which leads to \\footnote{In case a) the precise form of the $p_i$, $q_i$ and $t_i$ differ in very small details from those given in Table~\\ref{tab:scalar}, e.g. factors of two. Given the approximate nature of this work it is not necessary to display another complete set of tables with the modified coefficients.}:\n \\begin{align}\n \\gamma(a)&=\\frac{p_1+\\left(p_2+p_3+p_4\\right)\\frac{{\\cal H}^2}{k^2}}{q_1+\\left(q_2+q_3+q_4\\right)\\frac{{\\cal H}^2}{k^2}} \\\\\n  \\mu(a)&=\\frac{p_1+\\left(p_2+p_3+p_4\\right)\\frac{{\\cal H}^2}{k^2}}{t_1+\\left(t_2+t_3+t_4\\right)\\frac{{\\cal H}^2}{k^2}}\n \\end{align}\n Expressions of precisely this form have been worked out explicitly for numerous theories \\cite{defelice2011, Solomon2014, Koennig}.\n \nIn this scenario any scale-dependence of observables will be very weak. Until we are able to survey a substantial fraction of our Hubble volume, it is arguably better to focus our efforts on tightly constraining the time dependence of $\\{\\mu,\\gamma\\}$ or $\\{\\tilde\\mu,\\Sigma\\}$ by combining information from all scale bins. \n \\vspace{5mm}\n \n  \n \\noindent \\textit{Case b).} In this scenario one posits a new physical transition scale $M^{-1}$ below the cosmological horizon distance, such that $M\\gg{\\cal H}$. For example, in $f(R)$ gravity the new mass scale is approximately given by \\mbox{$M^2\\propto 1\/f_{,RR}$}, and $f_{,RR} R\\ll1\\,\\Rightarrow\\,1\/f_{,RR} \\gg {\\cal H}^2$ is needed to ensure stability in the matter-dominated era \\cite{Hojjati2012}. \n   \nRecall that in \\S\\ref{section:set-up} we introduced a wavenumber $k_{\\mathrm{gal}}$ that typified the maximum distance scale that could be well-constrained by near-future galaxy surveys.  Given the lack of deviations from $\\Lambda$CDM+GR to date, one might naively assume the maximum value for $M$ is of order $k_{\\mathrm{gal}}$. In principle one should then attempt to constrain the full form of eqs.(\\ref{gamscalar}) and (\\ref{muscalar}). However, it has been shown that in practice it will be difficult to constrain all the individual $p_i,\\,q_i$ and $t_i$ \\cite{Hojjati2014}.\n\nThis motivates us to consider a slightly less accurate but simpler approach. We perform a Taylor expansion of eqs.(\\ref{gamscalar}) and (\\ref{muscalar}) in the vicinity of the naive assumption $M\\lesssim k_{\\mathrm{gal}}$, and keep only the leading order terms. We show in Appendix~\\ref{appendix:As_and_Ms} that this gives the following expressions:\n\\begin{align}\n{\\mu}(a,k)&\\simeq1+A_{\\mu}(a)\\left[1+\\left(\\frac{M_{\\mu}(a)}{k}\\right)^2\\right] \\label{iii}\\\\ \n\\gamma(a,k)&\\simeq1+A_\\gamma(a)\\left[1+\\left(\\frac{M_\\gamma(a)}{k}\\right)^2\\right]\\label{kkk}\n\\end{align}\nThe precise content of $M_\\mu(a)$ and $M_{\\gamma}(a)$ depends on whether ${\\Gamma_\\chi}\\sim M$ or ${\\Gamma_\\chi} \\sim {\\cal H}$, but this kind of detail is not important here. The thrust of our argument is that equations (\\ref{iii}) and (\\ref{kkk}) provide a simple, general and theoretically well-motivated description of scale-dependence that should be easily applicable to observations. They include case a) as a limit, if $M_\\mu(a)$ and $M_{\\gamma}(a)$ are set to be of order ${\\cal H}$.\n \n \\begin{figure}[t!]\n\\begin{center}\n\\hspace{-2mm}\n\\includegraphics[scale=0.27]{scales_line_plot.pdf}\n\\caption{Schematic diagram illustrating the arguments of \\S\\ref{section:mass_scales} (case b).}\n\\label{figure:scales_line_plot}\n\\end{center}\n\\end{figure}\n\\vspace{5mm}\n \n From the discussion of this paper, we now understand that a non-zero detection of $M_\\gamma$, $M_\\mu$, $A_\\mu$ or $A_\\gamma$ would signify one of two possible things:\n \\begin{enumerate}\n \\item A breakdown of the QS approximation ${\\cal H}\/k \\ll 1$. The scale-dependence would then be due to first-order corrections in ${\\cal H}^2\/k^2$.\n \\item The existence of a new scale in gravitational physics \\footnote{At the risk of over-emphasis, remember from our discussion above that the converse is not true. A lack of observed scale-dependence does not rule out modified gravity.}.\n \\end{enumerate}\n Either of these scenarios would have profound implications for our understanding of gravity on large scales.\n\n \n \n \n  \n \n\\section{Detecting scale dependence.}\n\\label{section:forecasts}\nWe now speculate on the constraints that can be placed on the kind of parameterisation introduced above with future cosmological surveys. As we mentioned in \\S\\ref{section:derivation}, the function set $\\left\\{\\mu,\\,\\gamma\\right\\}$ is the most convenient for theoretical work. However, a change of basis will enable us to minimise parameter degeneracies when using redshift-space distortion (hereafter RSD) and weak lensing data \\cite{Simpson2013,Linder2013}. We introduce the new function set $\\left\\{\\tilde\\mu,\\,\\Sigma\\right\\}$, related to the old set by:\n\\begin{align}\n\\tilde\\mu(a,k)&=\\frac{\\mu(a,k)}{\\gamma(a,k)} & \\Sigma(a,k)&=\\frac{\\mu(a,k)}{2}\\left(1+\\frac{1}{\\gamma(a,k)}\\right)\n\\label{convert_params}\n\\end{align}\nEffectively, $\\tilde\\mu$ parameterises the geodesic equation for non-relativistic particles that governs the linear collapse of cold dark matter; $\\Sigma$ parameterises the geodesic equation for photons that governs weak gravitational lensing. However, it is important to note that gravitational lensing is also sensitive to $\\tilde\\mu$, because the lensing convergence and shear spectra involve integrals over the matter power spectrum, which is affected by modified structure growth \\cite{Simpson2013,Leonardinprep}.\n\nWe will write the new function set in a form analogous to eqs.(\\ref{iii}) and (\\ref{kkk}), that is:\n\\begin{align}\n{\\tilde{\\mu}}(a,k)&\\simeq1+A_{\\tilde{\\mu}}(a)\\left[1+\\left(\\frac{M_{\\tilde{\\mu}}(a)}{k}\\right)^2\\right] \\label{mutilde} \\\\ \n\\Sigma(a,k)&\\simeq1+A_\\Sigma(a)\\left[1+\\left(\\frac{M_\\Sigma(a)}{k}\\right)^2\\right] \\label{Sigma}\n\\end{align}\nWe stress that we are more interested in the general form of the scale-dependence rather than the precise (and lengthy) expressions relating $A_{\\tilde\\mu},\\,A_\\Sigma, M^2_{\\tilde\\mu}$, and $M^2_\\Sigma$ to the coefficients of the field equations  (though for completeness the relationships between the $\\left\\{\\mu,\\,\\gamma\\right\\}$ and $\\left\\{\\tilde{\\mu},\\,\\Sigma\\right\\}$ parameterisations are given in Appendix \\ref{appendix:conv}).\n\nAn unavoidable feature of model-independent tests of gravity is that ansatzes must be chosen for the time-dependent functions. There must be enough parameters in the ansatz to capture important signatures in the data without weakening the constraints too severely. As a simplicity-motivated test case, we will choose our ansatz to be (partially following \\cite{Simpson2013}):\n\\begin{align}\nA_{\\tilde\\mu}(a)&=\\tilde{\\mu}_0\\frac{\\Omega^{GR}_\\Lambda(a)}{\\Omega^{GR}_{\\Lambda 0}} \\label{Amutilde} \\\\\nA_\\Sigma(a)&=\\Sigma_0\\frac{\\Omega^{GR}_\\Lambda(a)}{\\Omega^{GR}_{\\Lambda 0}} \\label{ASigma}\\\\\nM_{\\tilde\\mu}&=m_{\\tilde\\mu}\\,(20 H_0) \\label{yrt}\\\\\nM_{\\Sigma}&=m_{\\Sigma}\\,(20 H_0)\\label{yrt2}\n\\end{align}\nwhere $m_{\\tilde\\mu}$ and $m_{\\Sigma}$ are constants. Remembering that we will want to be able to interpret the $M_i^{-1}$ as lengthscales, it is convenient to introduce a subhorizon distance unit of $(20H_0)^{-1}$ and express $M_i^{-1}$ in units of this distance. In the simple forecasts here we will focus on perturbative observables, fixing the background expansion history to match that of the $\\Lambda$CDM+GR model and using Planck best-fit cosmological parameters \\cite{Planck_params}. For an analysis that accounts for a modified expansion history see \\cite{Leonardinprep}.\n\nIn principle we should really allow $M_{\\tilde\\mu}$ and $M_\\Sigma$ to be functions of time. Treating them as constants simply corresponds to imposing the same overall time-dependent amplitudes $A_i(a)$ on both the scale-free and scale-dependent modifications to the QS field equations.\n\nThe set of four parameters that we will forecast for is:\n\\begin{align}\n\\tilde{\\mu}_0,\\;\\;\\Sigma_0,\\;\\;\\tilde{\\mu}_0m^2_{\\tilde\\mu},\\;\\;\\Sigma_0m^2_\\Sigma\n\\end{align}\nNote that the scale-dependent parts of the parameterisation are sensitive to a degenerate combination of the time-dependent amplitude and the possible new effective mass\/length scale; we cannot constrain $M_{\\tilde\\mu}$ and $M_\\Sigma$ individually. \n\\begin{center}\n\\begin{figure*}[t]\n\\hspace{-1.2cm}\n\\subfigure{\\label{fig:mu0Sig0}\\includegraphics[scale=0.475]{ellipse_muSig.png}}\n\\hspace{-0.5cm}\n\\subfigure{\\label{fig:MmuMSig}\\includegraphics[scale=0.476]{ellipse_MmuMSig.png}}%\n\\vspace{-0.8cm}%\n\\caption{Forecast constraints on the scale-independent (left panel) and scale-dependent (right panel) parts of the $\\{\\tilde{\\mu},\\Sigma\\}$ parameterisation, using a DETF stage 4-like experiment. Redshift-space distortions (green contours) constrain only the two parameters in the $\\tilde\\mu(z,k)$ function, $\\tilde{\\mu}_0$ and $m_{\\tilde\\mu}$. Gravitational weak lensing (red contours) predominantly constrain the parameters in $\\Sigma(z,k)$, but also have some dependence on $\\tilde{\\mu}_0$. Blue contours show the combined constraints. The parameters $m_{\\tilde{\\mu}}$ and $m_{\\Sigma}$ are $M_{\\tilde{\\mu}}$ and $M_\\Sigma$ expressed in distance units of $\\left(20 H_0\\right)^{-1}$\\,Mpc, see eqs.(\\ref{yrt}) and (\\ref{yrt2}).}\n\\label{fig:ellipses}\n\\end{figure*}\n\\end{center}\n\nWe consider a Dark Energy Task Force stage 4 (DETF4)  experiment that combines a galaxy clustering survey and a dedicated tomographic weak lensing survey. Weak lensing utilises scale-dependent information naturally, as the standard quantities to calculate are angular power spectra. RSD measurements, however, generally do not. Usually we talk about the density-weighted growth rate, $f\\sigma_8(z)$, implicitly assuming data from all scales ($k$-bins) has been combined.\n\nWe modify this situation by dividing each redshift bin of our hypothetical survey into five bins in $k$-space, with edges \\mbox{$\\left[0.005,0.02,0.05,0.08,0.12,0.15\\right]$h\\,Mpc$^{-1}$}; the choice of $k$-binning is analogous to \\cite{Johnson2014}, and the upper limit is chosen to cut off before nonlinearities start to dominate. It seems likely that as our survey sizes increase large-scale measurements will improve, whilst small-scale measurements will remain dominated by a lack of understanding of baryonic physics and the effects of nonlinearities. For this reason we will take our $k$-bins to have the following fractional errors at all redshifts, from large-scale to small-scale:~$\\left[0.01,0.03,0.03,0.09,0.09\\right]$. \n\nFor the tomographic gravitational lensing, we consider five source bins. These are constructed by taking the total distribution of source galaxies as:\n\\begin{equation}\nn(z) \\propto z^{\\alpha} e^{-\\left(\\frac{z}{z_0}\\right)^\\beta}\n\\label{nofz}\n\\end{equation}\nwith $\\alpha=2$, $\\beta=1.5$, and $z_0=z_m\/1.412$ where $z_m$ is the median redshift of the survey \\cite{Thomas2009, Amendola2008}.  $n(z)$ is then divided into five bins between $z=0.5$ and $z=2$, each with equal numbers of galaxies.  The lensing errors are encoded in the covariance matrices:\n\\begin{align}\n{\\bf C}_{ij}(\\ell)=\\sqrt{\\frac{2}{(2\\ell+1)f_{sky}}}\\left( P_{ij}^{\\kappa,\\,GR}(\\ell)+\\delta_{ij}\\,\\frac{\\langle\\gamma_{\\mathrm{int}}^2\\rangle}{\\bar{n}_i}\\right)\n\\end{align}\n where $P_{ij}^{\\kappa,\\,GR}(\\ell)$ is the cross-correlated convergence power spectrum sourced by galaxies in  bins $i$ and $j$, and $f_{sky}$ is the fraction of the sky covered by the survey. $\\langle\\gamma_{\\mathrm{int}}^2\\rangle^{\\frac{1}{2}}$ is the r.m.s. intrinsic shear and $\\bar{n}_i$ is the number of galaxies per steradian in source bin $i$. For further details see \\cite{Hu1999, Leonardinprep}.\n\nOur model-agnostic approach to the field equations (eqs.\\ref{QSPoissonscalar}-\\ref{QSslipscalar}) means that factors of order unity are of no relevance here. For this reason we do not attempt a detailed, experiment-specific forecast (for which the $k$-bin errors and maximum $k$ value would evolve with redshift). More precise forecasts can be found in, for example, \\cite{Amendola_Fogli2013, Taddei2014}. Other model-independent tests of $\\Lambda$CDM using the growth rate have recently appeared in \\cite{Nesseris_Sapone}.\n\nFig.~\\ref{fig:ellipses} shows marginalised 2D constraints on the scale-independent and scale-dependent parts of the parameterisation (eqs.~\\ref{mutilde} and \\ref{Sigma}). RSDs (green contours) constrain only $\\tilde{\\mu}_0$ and $\\tilde{\\mu}_0M^2_{\\tilde\\mu}$, whilst weak lensing (red contours) is sensitive to all four parameters. The authors of \\cite{Simpson2013} have applied a scale-independent parameterisation to CFTHLens+WiggleZ data, finding that lensing is only weakly sensitive to ${\\tilde\\mu}_0$. We agree with these scale\\textit{-independent} results, but find that the scale\\textit{-dependent} parts of the `lensing function' $\\Sigma(a,k)$ and the `RSD function' $\\tilde{\\mu}(a,k)$ are more strongly correlated \\cite{Leonardinprep}, see the right panel of Fig.~\\ref{fig:ellipses}.\n\n A rough estimate of the precision with which we will be able to measure these new effective mass\/length scales in cosmology gives us\n$\\sigma \\left(m^2_{\\tilde\\mu}\\right)\\sim{\\sigma\\left(\\tilde{\\mu}_0 m^2_{\\tilde\\mu}\\right)}\/{\\sigma\\left(\\tilde{\\mu}_0 \\right)}\\sim 6.7$ and \n$\\sigma \\left(m^2_{\\Sigma}\\right)\\sim{\\sigma\\left(\\Sigma_0 m^2_{\\Sigma}\\right)}\/{\\sigma\\left(\\Sigma_0 \\right)}\\sim 13.2$. \nInterpreting these limits as distance scales, we find lower bounds of order $ M^{-1}_{\\tilde\\mu}\\geq 364\\,\\mathrm{Mpc}$ and $M^{-1}_{\\Sigma}\\geq  260\\, \\mathrm{Mpc}$. \nWe see that RSDs are  more sensitive than weak lensing to new fundamental scales. That is, we should be able to pin down a new characteristic distance scale all the way up to $364$~Mpc with growth rate measurements.\n\nGiven that the bounds we have found on the $M_i$ are comparable to $k_{\\mathrm{gal}}$, the Taylor expansion of eqs.(\\ref{gamscalar}) and (\\ref{muscalar}) (see Appendix~\\ref{appendix:As_and_Ms}) may not be accurate enough. Yet, the well-behaved form of eqs.(\\ref{gamscalar}) and (\\ref{muscalar}) (ratios of quadratic polynomials) suggest that subsequent corrections in higher powers of $M^2\/k^2$ might change the bounds placed on $M_{\\tilde\\mu}$ and $M_\\Sigma$ by, at most, a factor of a few. \n\n\n\n\n\\section{Conclusions}\n\\label{section:conclusions}\n\nThe spirit of this work has been to take a step back from detailed model-specific investigations in alternative theories of gravity. Ultimately, the expressions collected in Appendix~\\ref{app:derivs} link the parameterisation we presented in eqs.(\\ref{mutilde}) and (\\ref{Sigma}) to the field equations of a specific gravity theory; but, as we hope has been clear, this is not the strategy we are advocating.  Instead, the goal of this paper has been to highlight the fact that even the exotic plethora of gravity theories on the market today share basic physical features which endow them with the same structure in the quasistatic regime. \n\nWe have found that the scale-dependence of gravity theories is closely linked to an effective mass scale or length scale which the linearised field equations inherit from their parent quadratic action. Our derivation has allowed for the evolutionary timescale of the new degree of freedom to be affected by this new scale in the system, rather than relying too heavily on GR-based intuitions that might suggest $\\dot\\chi$ is negligible in the QS regime.\n\nIn many theories a new mass scale is tuned to be $\\sim{\\cal H}$ in order to produce accelerated expansion. Indeed, the motivation behind much of the current bestiary of modified gravity theories is to render the cosmological constant obsolete. {\\it Generally} this renders the scale-dependendence undetectable in the quasistatic regime. We conjecture that most of the theories giving rise to detectable scale-dependence are those which introduce a new scale much larger than the Hubble scale ($M\\gg{\\cal H}$), and meanwhile rely on a cosmological constant to achieve a viable expansion history.\n\nWe caution, though, that theories with screening mechanisms complicate the issue somewhat. One can envisage a smaller, plausibly detectable length scale emerging if it is a compound of fundamental scales and couplings to the energy-momentum tensor.\n\nNote that at no point in this paper have we needed a concrete action from which to start our calculations: knowledge of the basic physical properties of a theory (eg. second-order equations of motion and a single dynamical spin-0 perturbation) is sufficient. We have trivially recovered results of \\cite{defelice2011,Gleyzes2013, Bloomfield2013, Unobservables2013}, and have found them to be more general than previously realised (see \\cite{Silvestri2013} for a similar analysis along these lines).\n\nWe advocate that measurement of scale-dependent observables is an important and feasible target for next-generation cosmology experiments. They have the potential to unveil a scale at which new physics beyond $\\Lambda$CDM+GR kicks in. More conservatively, scale-dependent measurements would also act as an essential test of the quasistatic approximation that has rapidly grown in popularity over the past few years.\n\n\n\\textit{Acknowledgments.} \nIt is a pleasure to thank L. Amendola, J. Gleyzes, M. Kunz, M. Lagos, L. Miller, J. Noller, F. Piazza, A. Silvestri, F. Vernizzi and H. Winther. TB is supported by All Souls College, Oxford. PGF acknowledges support from Leverhulme, STFC, BIPAC and the Oxford Martin School. DL is supported by the Rhodes Trust. MM acknowledges support from the Swiss National Science Foundation and the Balzan foundation via the University of Oxford.\n\n\\begin{widetext}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nData streams are environments such as network event logs, video streams, call records, and transaction records where vast amounts of data is generated at high speed \\cite{aggarwal2007datastreams}. Classification models in data streams have to work under strict time and memory constraints, and be able to adapt changes in the distribution of data over time \\cite{bifet2009new}. Such changes in the relation between data points and their corresponding classes over time are called concept drifts \\cite{gama2010knowledge}. Robustness against concept drifts is a topic that is extensively studied in the literature \\cite{bonab2018goowe, brzezinski2014reacting}. As data continuously arrive, learned information from the past instances become irrelevant under concept drifts. Classification models need to adapt to the new concepts by not only learning the new concepts but also forgetting the now-obsolete ones \\cite{brzezinski2014reacting}.\n\nEnsembles are common choices for models in stream environments, as they often improve predictive performance while providing robustness against concept-drifting and time-evolving data \\cite{bifet2009new}. Ensembles employ different approaches for dealing with concept drifts, such as replacing the weakest \/ lowest weight classifier with a new one that is trained with the more recent data \\cite{bonab2018goowe}, or updating ensemble weights regularly \\cite{wang2003awe, bonab2018goowe}. Despite the popularity and prevalence of ensemble learning in data streams, ensemble pruning is still a vastly undiscovered area of research in stream mining community.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\linewidth]{method.pdf}\n    \\vspace{-0.4cm}\n    \\caption{Pipeline of Proposed Method. Prediction history of each component in the ensemble are recorded. At each row of records, colored cells indicate relevance scores for corresponding classes. Then, these records are used in calculating class-wise losses for each component, and class-wise rankings of those components. The acquired rankings are combined using a rank fusion algorithm (Modified Borda Count). The resulting top $\\varphi$ components ($\\varphi < K$) are selected, and the rest is pruned.}\n    \\label{fig:method-figure}\n\\end{figure*}\n\n\nIn this study, we tackle the task of ensemble pruning in data streams for multi-class classification. Our contributions are as follows. We propose, \n\\begin{itemize}\n    \\item an explicit pruning technique that can be integrated into any type of streaming ensemble, and called whenever pruning is requested. Our method results in pruned ensembles that are better performing in terms of accuracy and more efficient in terms of memory consumption (as demonstrated in Section \\ref{sec:results}).\n    \\item to the best of our knowledge, the first large-scale \\footnote{The previous studies \\cite{krawczyk2015one, cruz2019fire} experimented on datasets with number of instances in the scales of hundreds or thousands} on-the-fly ensemble pruning method for streaming data.\n    \\item a class imbalance-aware pruning method, where a pruned ensemble does not lose its ability to classify rare or less-frequent classes, as a result of our class-wise component prediction analysis and ranking (as described in Section \\ref{sec:class-wise-analysis-ranking}).\n\\end{itemize}\n\n\n\\section{Problem Definition and Notation}\n\nWe consider the problem of ensemble pruning within the context of supervised classification in data streams. Data stream $\\mathcal{D}$ is a (possibly infinite) sequence of time-ordered data that consists of pairs $(X^{(t)}, y^{(t)})$ where $t$ denotes the arrival time. In multi-class classification, the target $y^{(t)}$ has $L$-many classes where $L>2$. An ensemble model $\\xi$ is a group of $K$ component classifiers, i.e. $\\xi = \\{ C_1, C_2, \\dots, C_K \\}$ where the prediction of the model for an instance is a combination of individual hypotheses ($h_k (X)$) of its component classifiers where $k$ denotes the classifier index. \n\nLet $\\hat{\\rho}_{k,l}$ be the record of the predictions of $k$th component for the $l$th class on a sliding window containing the latest $N$ instances. Similarly, let $\\rho_{l}$ be the record of the ground truth information for the $l$th class on the sliding window.\n\nThe aim of ensemble pruning is selecting a subset of classifiers $\\xi' \\subset \\xi$ such that the ensemble's both efficiency and predictive performance are improved. Here, let us denote the size of the pruned ensemble as $|\\xi'| = \\varphi$ where $\\varphi < K$. \n\n\\section{Related Work}\n\\textbf{\\textit{Online Ensembles}}. The literature on the online ensembles is abundant \\cite{krawczyk2017ensemble}. Here, we only mention the ensembles that are used in our experiments. One of the most commonly used and well-known ensembles in data streams is AWE (Accuracy Weighted Ensemble) \\cite{wang2003awe}. It is a chunk-based ensemble where each classifier are weighted according to their expected accuracy in the latest chunk. Similarly, GOOWE \\cite{bonab2018goowe} is another chunk-based stacking ensemble that assigns optimal weights to its components using a geometric interpretation of target space and solving an optimization problem as the chunks are processed. Both of these ensembles are dynamic and they replace one of their components at the end of each chunk.\n\n\n\\noindent\n\\textbf{\\textit{Pruning in Data Streams}}. A related topic to on-the-fly ensemble pruning is Dynamic Ensemble Selection (DES) \\cite{ko2008dynamic}. In DES, a subset of components from a pool is selected for each test instance. DES assigns estimated levels of competence for each classifier in the pool, so that a selection method selects the classifiers to be assigned to incoming instances. Note that, unlike ensemble pruning, selection of a subset of classifiers for every instance does not reduce the size of the pool of classifiers. Firefly \\cite{krawczyk2015one} introduces a swarm intelligence approach to pruning with \\textit{one-class} classifiers where input classifiers are members of a firefly population. Interactions between fireflies describe the effectiveness of the classifiers, and best representatives of diverse groups in the population are selected as the pruning step. Lastly, FIRE-DES++ \\cite{cruz2019fire} incorporates a pre-selection stage where only the classifiers that can correctly classify at least a pair of instances with different classes can proceed to the next stage of classification. Again, the pre-selection stage in FIRE-DES++ does not change the size of the pool of classifiers.\n\n\\vspace{-0.1cm}\n\\section{Proposed Method}\n\\begin{algorithm}\n    \\caption{\\texttt{CCRP}: Class-wise Component Ranking-based Pruner}\n    \\label{alg:ccrp}\n    \\begin{algorithmic}[1]\n        \\Require $\\mathcal{D}$: data stream, $\\xi$: ensemble, $\\varphi$: pruned ensemble size\n        \\Ensure $\\xi'$: pruned ensemble\n        \\State Initialize $\\rho$ as an empty FIFO buffer of size $N$.          \n        \\State Initialize $\\hat{\\rho}$ as an empty FIFO buffer of size $K \\times N$.\n        \\For{$(X,y) \\in \\mathcal{D}$}\n           \n            \\State $\\hat{\\rho}_k.$ \\texttt{append}$(h_k (X)) \\; $ for $\\forall k$  \\Comment{[ Preliminary Phase ]}\\label{code:preliminary}\n            \\State $\\rho.$ \\texttt{append}$(y)$ \n            \n            \\If{prune}  \\Comment{[ Phase I ]} \\label{code:phase-1} \n                \\For{$l \\in L$}     \\Comment{Start CCRP}\n                    \\For{$k \\in K$}\n                        \\State $ scores.$\\texttt{append}$($\\texttt{MSE}$(\\rho_{l}, \\hat{\\rho}_{k,l}))$\\label{code:mse-line}  \\Comment{Class-wise MSEs}\n                    \\EndFor\n                    \\State $ ranks[l] =$ \\texttt{argsort}$(scores)$ \\Comment{Class-wise order of components}\n                \\EndFor\n                \\State $ccrp\\_ranks =$ \\texttt{ModifiedBorda}$(ranks)$ \\label{code:ModifiedBorda} \\Comment{[ Phase II ]}\\label{code:phase-2} \n                \\State $\\xi' \\leftarrow$ top ranked $\\varphi$ components based on $ccrp\\_ranks$  \n                \\State $\\xi \\leftarrow \\xi'$ \\Comment{Pruned ensemble in effect}\n            \\EndIf\n        \\EndFor\n    \\end{algorithmic}\n\\end{algorithm}\n\nWe introduce, CCRP, \\textbf{C}lass-wise \\textbf{C}omponent \\textbf{R}ankings based \\textbf{P}runer for multi-class online ensembles. CCRP can be integrated to work with both dynamic and static multi-class ensembles to improve the performance and memory consumption. CCRP can be performed at any point in time on the stream. The proposed method consists of three phases: \\textit{\\textbf{Preliminary Phase}}: Recording Component Predictions on the Sliding Window, \\textit{\\textbf{Phase I}}: On-the-Fly Performance Analysis and Class-wise Ranking of Components, and \\textit{\\textbf{Phase II}}: Fusion of Rankings and Component Selection (see Figure \\ref{fig:method-figure}).\n\n\n\\subsubsection*{Preliminary Phase: Recording Component Predictions on the Sliding Window.}  This phase only applies to the ensembles that are not already recording the component predictions on the sliding window (e.g. OzaBagging \\cite{oza2005onlinebb}). In this phase, for the latest $N$ data instances, predictions of classifiers and the ground truth are recorded as  $\\rho$ and $\\hat{\\rho}$ respectively (Alg.\\ref{alg:ccrp} line \\ref{code:preliminary}). Since only the latest $N$ instances are kept in the records, the required memory for this process is fixed, despite being proportional to the number of classes.\n\n\\subsubsection*{Phase I: On-the-Fly Performance Analysis and Class-wise Ranking of Components.} This is the initial phase of CCRP in which per class performances of classifiers are measured. In this phase, for each class $l$, predictions of classifiers and the ground truth are extracted as $\\hat{\\rho}_{k,l}$ and $\\rho_{l}$ respectively (Alg.\\ref{alg:ccrp} line \\ref{code:phase-1}). Then, scores of classifiers are calculated using Mean Square Error (Eqn. \\ref{eq:loss-mse}) for each class (Alg.\\ref{alg:ccrp} line \\ref{code:mse-line}).\n\\label{sec:class-wise-analysis-ranking}\n\n\\vspace{-0.3cm}\n\\begin{equation}\n\\label{eq:loss-mse}    \n    \\mathcal{L}_{k,l}(X) = \\sum^{N}_{i=1} \\Big( \\hat{\\rho}_{k,l}^{(i)} - \\rho_{l}^{(i)} \\Big)^2, \\qquad 1 \\leq k \\leq K, \\; 1 \\leq l \\leq L\n\\end{equation}\n\n\\subsubsection*{Phase II: Fusion of Rankings and Component Selection.} In the final phase, CCRP generates the overall ranking of the classifiers, using a modified version Borda Count \\cite{nuray2006automatic} rank fusion method (Alg.\\ref{alg:ccrp} line \\ref{code:ModifiedBorda}). In Modified Borda Count (MBC, hereafter), CCRP assigns $K \\times L$ points (instead of $K$ points in the regular Borda Count \\cite{nuray2006automatic}) to the highest ranking components in the class-wise rankings. This ensures the winning component for each class to appear at the top $L$ places in the overall ranking. Afterwards, the second classifier gets $K - 1$ points and each proceeding classifier gets 1 point less than its successor. Then, the top $\\varphi$ classifiers from the overall ranking are selected as the members of the pruned ensemble. It is recommended \\cite{bonab2019less} that the pruned ensemble size should be at least equal to the number of classes. Taking this into account, CCRP guarantees that the best performing classifier for each class is included in the pruned ensemble when $\\varphi \\geq L$.\n\n\\section{Experimental Setup and Results} \\label{sec:results}\n\n\\subsection{Setup}\nThe proposed method and modifications to existing ensembles are integrated into scikit-multiflow \\cite{montiel2018scikit} library. The experiments are evaluated prequentially \\cite{gama2009issues}. We report prequential and overall accuracy as performance metrics. In addition, for memory efficiency analysis in the experiments, we define \\textit{Mean Memory Consumption Ratio} ($\\mu$) which indicates the percentage of average memory the pruned ensemble uses with respect to the original ensemble (Eqn. \\ref{eq:mmcr}).\n\n\\vspace{-0.3cm}\n\\begin{equation}\n\\label{eq:mmcr}\n    \\mu = \\frac{\\sum_{\\forall \\text{chunk}} \\; Size(\\xi')}{\\sum_{\\forall \\text{chunk}} \\; Size(\\xi)}\n\\end{equation}\n\nThroughout the experiments, we use two typical dynamic ensembles (AWE \\cite{wang2003awe} and GOOWE \\cite{bonab2018goowe}) with Hoeffding Trees \\cite{domingos2000mining} as their base classifiers. CCRP is performed whenever the ensembles are full, i.e. when the ensemble size reaches $K$. After pruning, the ensembles continue to grow until they reach the maximum size $K$, where pruning takes place again. This process continues indefinitely.\n\n\n\\begin{table}[!h]\n\\caption{Datasets}\n\\vspace{-0.3cm}\n\\begin{tabular}{l|lrrr}\n                                     & \\textbf{Name}                                                    & \\multicolumn{1}{l}{\\textbf{\\# Features}} & \\multicolumn{1}{l}{\\textbf{\\# Classes}} & \\multicolumn{1}{l}{\\textbf{\\# Instances}} \\\\ \\hline\n\\multirow{3}{*}{\\rotatebox{90}{\\textbf{Real}}}       &  COVTYPE \\cite{bifet2013efficient}                                                         & 54                                            & 7                                            & 581.012                                      \\\\\n                                     & Poker Hand \\cite{bifet2013efficient}        & 10                                            & 10                                           & 829.201                                      \\\\\n                                     & Rialto    \\cite{losing2016knn}                                                       & 27                                            & 10                                           & 82.250                                       \\\\ \\hline\n\\multirow{3}{*}{\\rotatebox{90}{\\textbf{Synth}}} & Mov. Squares  \\cite{losing2016knn}     & 2                                             & 4                                            & 200.000                                      \\\\\n                                     & Mov. RBF \\cite{losing2016knn}        & 10                                            & 5                                            & 200.000                                      \\\\\n                                     & Trn. Chssbrd  \\cite{losing2016knn}   & 2                                             & 8                                          & 200.000                                     \n\\end{tabular}\n\\label{tab:datasets}\n\\end{table}\n\nThe models are ran on three real-world and three synthetic well-known datasets with concept drifts \\cite{bifet2013efficient, losing2016knn}. A summary of the dataset information is provided in Table \\ref{tab:datasets}.\n\n\\begin{figure}\n    \\vspace{-0.35cm}\n    \\centering\n    \\includegraphics[width=\\linewidth]{all-in-one-vertical.pdf}\n   \n    \\caption{The impact of CCRP ($\\varphi = L$) on the prequential accuracy of two ensemble models on three datasets over the first few thousand instances. The first occurrence of pruning is denoted with a vertical red line.}\n    \\label{fig:ccrp-with-different-ensembles}\n\\end{figure}\n\n\\subsection{Effect of CCRP on Predictive Performance and Memory Efficiency}\n\\textit{Does CCRP yield more effective and efficient ensemble models?} We respond to this question by taking both accuracy and memory consumption ($\\mu$) into account. We perform experiments with AWE (where $K = 20$) and GOOWE ($K = 30$) to investigate the effect of CCRP on the ensembles. \n\nTo examine the change in the behavior of ensembles, \\textit{ceteris paribus}, we consider the case when CCRP is performed for the first time on the stream. In Figure \\ref{fig:ccrp-with-different-ensembles}, it can be observed that the overall accuracy of the pruned ensemble is always higher than the original ensemble \\textit{after} pruning with CCRP. In addition to the improvement in accuracy, it should be noted that the ensembles after pruning are relatively more robust to the changes in the data stream. This can be observed from the steepness of the declining accuracies of the original ensembles compared to the pruned ensembles.\n\n\n\\begin{table}[]\n\\caption{Effect of CCRP on Accuracy and Mean Memory Consumption Ratio ($\\mu$) for Different Ensembles}\n\\vspace{-0.3cm}\n\\begin{tabular}{l|cc|c|cc|c}\n\\multicolumn{1}{c|}{} & \\multicolumn{3}{c|}{\\textbf{AWE} \\cite{wang2003awe}}                           & \\multicolumn{3}{c}{\\textbf{GOOWE} \\cite{bonab2018goowe}}                         \\\\ \\cline{2-7} \n                      & \\multicolumn{1}{c}{Default} & \\multicolumn{1}{c|}{CCRP} & $\\mu$ & \\multicolumn{1}{c}{Default} & \\multicolumn{1}{c|}{CCRP} & $\\mu$ \\\\ \\hline\nCOVTYPE               &    {\\ul 0.661}     &         0.659                  &  \\textbf{75\\%} &        {\\ul 0.852}          &          0.829     &  \\textbf{10\\%} \\\\\nPokerHand             &    0.525     &        {\\ul 0.585}          & \\textbf{83\\%}  &          {\\ul 0.712}       &            0.651       &  \\textbf{14\\%} \\\\\nRialto                &       0.377        &      {\\ul 0.379}          &  \\textbf{80\\%}  &       {\\ul  0.428}           &      0.420                &  \\textbf{55\\%} \\\\\nMov. Squares          &       0.478        &       {\\ul 0.532}         &  \\textbf{71\\% } &        0.354            &       {\\ul 0.359}     &  \\textbf{52\\%} \\\\\nMov. RBF              &       {\\ul 0.499}     &      {\\ul 0.499}         &  \\textbf{72\\%}  &        {\\ul 0.433}     &         0.396         &  \\textbf{19\\%} \\\\\nTrn. Chssbrd       &      {\\ul 0.146}   &        0.141               &  \\textbf{71\\%}  &        0.685 &             {\\ul 0.791}         &  \\textbf{70\\%}\n\\vspace{-0.4cm}\n\\end{tabular}\n\\label{table:acc-memory}\n\\end{table}\n\nFigure \\ref{fig:ccrp-with-different-ensembles} indicates that the initial calls of CCRP \\textit{locally} improves the ensemble performance. Yet, \\textit{is that the case throughout the stream?} To inspect the effect of CCRP on both accuracy and memory consumption throughout the stream, we experiment with AWE ($K=20$) and GOOWE ($K=30$) on all of the datasets, and report overall accuracies, as well as $\\mu$ (Table \\ref{table:acc-memory}). \n\nIn Table \\ref{table:acc-memory}, notice that CCRP yields $10\\% \\leq \\mu \\leq 83\\%$, which means the pruned ensembles consume at most $90\\%$ and at least $17\\%$ less memory over the course of their run. In 5 out of 12 cases, CCRP improves the performance of ensembles while reducing the memory consumption by $20$ to $25\\%$. In a small number of cases (3 out of 12: GOOWE on COVTYPE, PokerHand and MovingRBF), ensembles without pruning outperforms the ones with pruning. These are also the cases where CCRP yields the most drastic reductions in memory consumption, around $80$ to $90\\%$. This behaviour may occur due to removing old and strong components of the ensemble in the presence of drifts. For the remaining 4 cases, the overall accuracy for the pruned models are on par with the ones that are not pruned. A decrease in memory consumption around $30\\%$ can be observed in these cases, as well. These memory savings imply similar savings in CPU time, since memory consumption of an ensemble is proportional to the complexity of its components.\n\n\\subsection{Superiority of CCRP over Different Baseline Selection Techniques}\n\\textit{Is using CCRP's pruning scheme indeed better than selecting the highest weighted $\\varphi$ components from a weighted ensemble? What difference does using MBC make with respect to the regular Borda Count?} To investigate these questions, we design the following experiment: We choose a weighted ensemble (GOOWE \\cite{bonab2018goowe}) which normally employs a replacement policy where the lowest weight component is replaced at the end of each chunk. We, then, modify GOOWE's component replacement policy so that it removes its component using CCRP, and Borda Count with $\\varphi = K - 1$, and places the newly trained component to the freed-up spot. Overall accuracy values for several datasets are reported in Table \\ref{table:differentPrune}. The winning method(s) for each dataset is underlined.\n \n\\begin{table}[!h] \n\\caption{Accuracy of GOOWE (with $K = 10$) using Different Pruning Schemes (with $\\varphi=9$) for the Selected Datasets}\n\\vspace{-0.3cm}\n\\begin{tabular}{l|ccc}\n             & \\textbf{CCRP} & \\textbf{Weight-based} & \\textbf{Regular Borda} \\\\ \\hline\nCOVTYPE      & {\\ul 0.839}              & 0.796                      & 0.703                     \\\\\nPoker        & {\\ul 0.726}               & 0.667                      & 0.701                     \\\\\nMov. RBF     & {\\ul 0.543}              & 0.537                      & {\\ul 0.543}                    \\\\\nTrn. Chessboard     & {\\ul 0.452}              & 0.450                      & 0.449                    \n\\end{tabular}\n\\vspace{-0.3cm}\n\\label{table:differentPrune}\n\\end{table}\n\nThis experiment shows that the internal component weights of an ensemble do not necessarily indicate their contribution to the overall performance of the ensemble. In addition, CCRP's superiority over using Regular Borda \\cite{nuray2006automatic} shows MBC's capability of handling class imbalance. By assigning $K \\times L$ points to the best performing classifier for a less frequent class, MBC ensures that that classifier is not ranked poorly and selected for the resulting subset of classifiers.\n\n\\vspace{-0.3cm}\n\\section{Conclusion and Future Work}\nIn this work, we present an on-the-fly ensemble pruner for evolving data streams for the first time in the literature. Our approach utilizes predictions of ensembles to generate class-wise rankings and combine these rankings with a rank fusion algorithm. We demonstrate that the proposed pruning scheme yields smaller and more efficient ensembles with on par or better predictive performance consistently. Additionally, we show that the proposed rank fusion algorithm works better than its alternatives. Our experiments on different ensembles provide evidence that our method can be integrated into any streaming ensemble.\n\nFor future work, we plan to do a rigorous analysis of diversity of the ensemble components before and after pruning, considering various diversity measures \\cite{krawczyk2017ensemble}. The proposed approach can be extended by incorporating diversity of components into the rank fusion process. Additionally, \"how to select the optimal prune size ($\\varphi$)\" is still an open question. The effect of ensemble parameters and dataset properties on $\\varphi$ can be examined. Lastly, we observed that the time of the pruning is crucial to obtain high predictive performance. Therefore, integrating a concept drift detector into the proposed method and pruning the ensemble whenever a drift is detected may further improve the performance of models.\n\\vspace{-0.3cm}\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} The impressive progress of nonlinear science in the \nlast several decades revealed a large number of structures and mechanisms \nthat were novel and fascinating to almost all of us. This has, in a sense, \nsuggested that several aspects of our existing statistical theory, whose \nfirst kinetic equation was proposed more than a century ago, need certain \nkinds of `upgrade' or `renewal'. \n\nIn some of our relatively recent papers{\\cite{chen1,chen2}, we were tempted \nto analyze the standard kinetic theory from different perspectives. It was \nargued that (i) in various artificial and realistic situations, distribution \nfunctions of gases may have very unconventional structures: they can be, for \ninstance, discontinuous at every and each spatial point and thus the usual \ndifferential apparatus becomes inapplicable (after all, there are indeed many \nstructures and mechanisms in nature to which the use of the usual \ndifferential apparatus is quite limited); and that (ii) while the left side \nof the Boltzmann equation is symmetric in terms of partial derivatives with \nrespect to the position and velocity, the right side of the equation is \nconstructed entirely in the velocity space (the position coordinates merely \nserve as inactive parameters), which is incomprehensible especially in the \nmathematical sense. \n\nThe major objective of this work is, however, rather practical: to define a \nkinetic gas and, at the same time, to find out an algorithm which can be used \nto calculate the behavior of the gas rather completely. It is hoped that if \nsuch study gets established somehow, further studies, either concerning the \nfoundation of the existing statistical theory or concerning the development \nof more general treatment, will be inspired and promoted. \n\nThe structure of this paper is the following. In Sect.~2, we propose our \nworking model, a gas leaking out of a container, resembling the cavity model \nfor the black-body radiation. In Sect.~3, the zeroth-order distribution \nfunction of the leaking gas is formulated and, in passing, it is shown that \ndistribution functions of real gases may have radically discontinuous \nstructures untreatable for the standard theory. In Sect.~4, the collisional \ncorrection is investigated with help of a methodology that is slightly \ndifferent in form but much different in concept from the standard treatment. \nIn Sect.~5, we further comment on why the distribution function averaged over \nfinite velocity solid-angle ranges needs to be introduced. Sect.~6 summarizes \nthe paper. \n\n\\section{A leaking gas} \n\nAt the kinetic level, one of the customary conceptions about calculating a \npractical gas is to solve the Boltzmann differential-integral equation with \nhelp of a difference scheme. Known difficulties associated with this \nconception may be summarized as follows. \\begin{enumerate} \\item There are \nseven variables: time, three spatial coordinates and three velocity \ncomponents. If we divide each variable into $N$ intervals, `the degrees of \nfreedom' of the system are $N^7$. To reveal the true properties of a \nnonequilibrium phenomenon, such as those related to a turbulence, $N$ has to \nbe rather large and $N^7$ has to be terribly huge, so that no today's \ncomputational means really helps. \\item The complex nature of the collisional \noperator in Boltzmann's equation makes the situation worse. \\item Boundary \nconditions and initial conditions impose extra problems. As to a differential \nequation, well specified boundary and initial conditions usually mean that a \nuniquely and clearly defined solution can be constructed, say, from a \ndifference scheme. (The uniqueness may not be truly desired in view of the \nfact that bifurcations take place in nature.) As to an integral equation, \nboundary and initial conditions may be specified rather loosely and it leaves \nus certain room to `manipulate'. Boltzmann's equation is a \ndifferential-integral equation and it is not entirely clear in which \ndirection we should go. \\end{enumerate} \nDue to these difficulties (and \npossibly many more), almost no realistic problems have been fully treated, \nlet alone conclusive comparison between the kinetic theory and realistic \nexperiments. \n\nInterestingly enough, the aforementioned difficulties, at least some of them, \nare not intrinsic to the dynamics that we wish to study. In some sense, if \nthe standard approach had not dominated our mind too strongly, we might have \nalready had workable alternative approaches, at least for some special cases. \nIn what follows, we shall try to substantiate this viewpoint. \n\nWe consider a gas consisting of hard spheres and confined to a closed \ncontainer. The interactions between the walls and particles, and between the \nparticles themselves, make the gas solidly and quickly in equilibrium. \nNamely, the probability of finding particles in a phase volume element $d{\\bf \nr}d{\\bf v}=dxdy dz dv_x dv_y dv_z$ takes its Maxwell form \n\\begin{equation} \\label{fm0} f_M\\equiv n_0\\left(\\frac m{2\\pi\\kappa \nT}\\right)^{\\frac 32} \\exp\\left( - \\frac{mv^2}{2\\kappa T}\\right), \n\\end{equation}\nwhere $m$ is the mass of particle, $\\kappa$ the Boltzmann constant, $T$ the \ntemperature of the gas and $n_0$ the particle density in the container. \n\\vskip15pt \n\n\\hspace{40pt} \\setlength{\\unitlength}{0.015in} \\begin{picture}(200,100) \n\\put(30,100){\\line(1,0){60}}\n\\put(30,10){\\line(1,0){60}}\n\\put(40,20){\\line(1,0){40}}\n\\put(40,90){\\line(1,0){40}}\n\n\n\\put(90,53){\\line(0,-1){43}}\n\\put(90,57){\\line(0,1){43}}\n\\put(90,57){\\line(-2,3){10}}\n\\put(90,53){\\line(-2,-3){10}}\n\n\\multiput(30,10)(2,0){6}{\\line(0,1){90}}\n\\multiput(40,10)(2,0){21}{\\line(0,1){10}}\n\\multiput(40,100)(2,0){21}{\\line(0,-1){10}}\n\n\\put(80,10){\\line(0,1){28}}\n\\put(82,10){\\line(0,1){31}}\n\\put(84,10){\\line(0,1){34}}\n\\put(86,10){\\line(0,1){37}}\n\\put(88,10){\\line(0,1){40}}\n\n\\put(80,100){\\line(0,-1){28}}\n\\put(82,100){\\line(0,-1){31}}\n\\put(84,100){\\line(0,-1){34}}\n\\put(86,100){\\line(0,-1){37}}\n\\put(88,100){\\line(0,-1){40}}\n\n\\multiput(42,23)(0,4){17}{\\circle*{1}}\n\\multiput(46,23)(0,4){17}{\\circle*{1}}\n\\multiput(50,23)(0,4){17}{\\circle*{1}}\n\\multiput(54,23)(0,4){17}{\\circle*{1}}\n\\multiput(58,23)(0,4){17}{\\circle*{1}}\n\\multiput(62,23)(0,4){17}{\\circle*{1}}\n\\multiput(66,23)(0,4){17}{\\circle*{1}}\n\\multiput(70,23)(0,4){17}{\\circle*{1}}\n\\multiput(74,23)(0,4){17}{\\circle*{1}}\n\\multiput(78,23)(0,4){17}{\\circle*{1}}\n\\multiput(82,47)(0,4){5}{\\circle*{1}}\n\\multiput(86,51)(0,4){3}{\\circle*{1}}\n\n\\multiput(95,55)(1.8,0){20}{\\circle*{0.7}}\n\\multiput(94,57)(1.6,0.9){20}{\\circle*{0.7}}\n\\multiput(92.5,58.5)(0.9,1.6){20}{\\circle*{0.7}}\n\\multiput(94,53)(1.6,-0.9){20}{\\circle*{0.7}}\n\\multiput(92.5,51.5)(0.9,-1.6){20}{\\circle*{0.7}}\\end{picture} \n\n\\vspace{-8pt} \\begin{center} \\begin{minipage}{10cm} { \\vskip-0.3cm Figure~1: \nSchematic of a leaking gas.} \\end{minipage} \\end{center} \n\nWe then suppose that there is a small hole on the wall of the container, as \nshown in Fig.~1. The aim of our formalism is to determine the distribution \nfunction of the leaking gas. For convenience of discussion, it is further \nassumed that (i) the hole is indeed small so that the leaking is relatively \nslow and the relevant distribution function can be considered to be \ntime-independent; (ii) since the regions far from the hole are kept in the \nvacuum state (by a pump for instance), no incoming particles need to be taken \ninto account; and (iii) the leaking gas outside the container is so dilute \nthat the zeroth-order solution can be determined by the collisionless \ntrajectories of particles while the first-order correction can be formulated \nby assuming the particles of the leaking gas to collide with each other once \nand only once. \n\n\\section{The zeroth-order distribution function} \n\nIf we ignore the particle-to-particle interaction and ignore the \ngravitational force acting on each particle, the collisionless motion of \nevery particle of the leaking gas is conceptually simple---moving along a \nstraight line. However, it is still necessary and interesting to express the \ncorresponding distribution function in a proper mathematical form. \n\nWe start our discussion with the conventional theory. The theory states that \nthe distribution function satisfies, with collisions ignored completely, \n\\begin{equation} \\label{bltz} \\frac{\\partial f}{\\partial t}+{\\bf v}\\cdot \n\\frac {\\partial f} {\\partial {\\bf r}}+ {{\\bf F}\\over m} \\cdot \\frac {\\partial \nf} {\\partial {\\bf v}}=0,\n\\end{equation}\nwhere $\\bf F$ represents the external force. Equation (\\ref{bltz}) is \nsometimes termed the collisionless Boltzmann equation\\cite{reif}. A natural \nnotion related to such terminology is that any procedure, a difference scheme \nfor instance, if applicable in solving the regular Boltzmann equation, must \nbe applicable in solving this reduced equation. Strangely, complicated and \nsubtle issues arise from this `natural' notion; and we shall discuss these \nissues at the end of this section. \n\nTo formulate the zeroth-order distribution function, the following `slightly \ndifferent' approach is helpful. Rewriting Equation (\\ref{bltz}) along a \nparticle path in the phase space, we arrive at the path \ninvariance\\cite{reif,harris} \n\\begin{equation} \\label{fm}\\left. \\frac{df}{dt}\\right|_{\\rm path}=0. \n\\end{equation}\nAs far as our leaking gas is concerned, this means $f|_{\\rm path}=f_M$, where \n$f_M$ has been defined by expression (\\ref{fm0}). \\vskip 20pt \n\n\\hspace{90pt} \\setlength{\\unitlength}{0.014in} \\begin{picture}(200,100) \n\n\\put(109,97){\\line(1,-2){5}}\n\\put(107,79){\\makebox(20,8)[l]{$\\Delta \\Omega$}}\n\\put(92,89){\\makebox(20,8)[l]{$\\bf r$}}\n\n\\put(42,100){\\line(0,-1){25.5}}\n\\put(39,100){\\line(0,-1){21}}\n\\put(36,100){\\line(0,-1){16.5}}\n\\put(33,100){\\line(0,-1){12}}\n\\put(30,100){\\line(0,-1){7.5}}\n\\put(42,10){\\line(0,1){25.5}}\n\\put(39,10){\\line(0,1){21}}\n\\put(36,10){\\line(0,1){16.5}}\n\\put(33,10){\\line(0,1){12}}\n\\put(30,10){\\line(0,1){7.5}}\n\n\\put(45,100){\\line(0,-1){30}}\n\\put(45,10){\\line(0,1){30}}\n\\multiput(10,10)(0,5.0){19}{\\circle*{1.5}}\n\\multiput(15,10)(0,5.0){19}{\\circle*{1.5}}\n\\multiput(20,10)(0,5.0){19}{\\circle*{1.5}}\n\\multiput(25,15)(0,5.0){17}{\\circle*{1.5}}\n\\multiput(30,25)(0,5.0){13}{\\circle*{1.5}}\n\\multiput(35,30)(0,5.0){11}{\\circle*{1.5}}\n\\multiput(40,40)(0,5.0){7}{\\circle*{1.5}}\n\n\\multiput(45,43)(0,6){5}{\\line(0,1){3}}\n\\put(45,50){\\line(2,-3){7}}\n\\put(50,32){\\makebox(20,8)[l]{$\\Delta S_0$}}\n\n\\put(45,70){\\line(-2,3){20}}\n\\put(45,40){\\line(-2,-3){20}}\n\\put(45,55){\\vector(3,2){54}}\n\\multiput(45,55)(3,0){15}{\\circle*{1}}\n\\put(56,55.5){\\makebox(20,8)[l]{$\\theta$}}\n\n\\multiput(47,69)(2,0.85){35}{\\circle*{1}}\n\\multiput(47,42)(1.5,1.4){45}{\\circle*{1}}\n\\end{picture} \n\n\\vspace{-5pt}\\begin{center} \\begin{minipage}{10cm} { \\vskip-0.3cm Figure~2: \nThe velocity distribution at the position $\\bf r$.} \\end{minipage} \n\\end{center} \n\nThen, it seems that the zeroth-order distribution function on the right side \nof the hole is uniformly identical since any spatial point there is reachable \nalong a path radiating from the inside of the container (the external force \nis zero and the paths in the phase space and in the spatial space are the \nsame). This notion is, however, rather misleading. By moving together with a \nparticle of the leaking gas, an observer easily realizes that the true \nparticle density around him becomes lower and lower. With this realization in \nmind, it can easily be found that the distribution function of the leaking \ngas is strongly limited in the velocity space: the farther from the hole the \nstronger the limitation is. Namely, as shown in Fig.~2, the true implication \nof (\\ref{fm}) is \n\\begin{equation} \\label{cone} f({\\bf r},{\\bf v}) =\\left\\{ \\begin{array}{ll} \nf_M &{\\rm the{\\hskip 4pt} direction{\\hskip 4pt} of {\\hskip 4pt} {\\bf \nv}{\\hskip 4pt} within{\\hskip 4pt} \\Delta\\Omega} \\\\ 0 & {\\rm otherwise} \n\\end{array} \\right. , \n\\end{equation}\nwhere $\\Delta\\Omega$ is a solid-angle range whose size is defined by the \nsolid-angle range \n\\begin{equation} \\label{cone1} \\frac{\\Delta S_0 \\cos \\theta}{r^2} \\quad{\\rm \nwith}\\quad r=|{\\bf r}| \n\\end{equation}\nand whose direction is defined by the position vector ${\\bf r}$ (with the \norigin at the hole). Heuristically and intuitively, it can be said that the \nvelocity distribution at every point there is a sting-like function. In \nalmost all the regions on the right side of the hole, where $r$ is relatively \nlarge or $\\cos\\theta$ is relatively small, the stings tend to be infinitely \nacute. \n\nThough (\\ref{cone}) and (\\ref{cone1}) correctly describe the distribution \nfunction, there are some kinds of inconvenience in applying them in \nanalytical calculations. For practical and theoretical reasons, we wish to \nexpress the distribution function $f({\\bf r},{\\bf v})$ directly and \nexplicitly in terms of ${\\bf r}$ and ${\\bf v}$. \n\nTo get such expression, we adopt the picture that the sting-like velocity \ndistributions of the leaking gas are indeed infinitely acute, either by \nassuming the hole to be truly small or by assuming the interested region to \nbe truly distant. Under this understanding, we can use a $\\delta$-function to \nreflect the limitation of the distribution function in the velocity space and \nreplace (\\ref{cone}) and (\\ref{cone1}) by \n\\begin{equation} \\label{delta} f({\\bf r},{\\bf v})\\equiv \ng(r,v)\\delta\\left(\\Omega_{\\bf v}- \\Omega_{\\bf r} \\right) \n\\end{equation}\nwith $$ g(r,v) = n_0 \\left(\\frac m{2\\pi\\kappa T}\\right)^{3\/2} e^{- \n\\frac{mv^2}{2\\kappa T}} \\frac{\\Delta S_0 \\cos\\theta}{ r^2}, $$ where \n$\\Omega_{\\bf v}$ is the solid angle in the direction of $\\bf v$ and \n$\\Omega_{\\bf r}$ is the solid angle in the direction of $\\bf r$. Expression \n(\\ref{delta}) enables us to do analytical calculation easily. For instance, \nwith help of it we may compute the zeroth-order particle density outside the \ncontainer by \n\\begin{equation} n({\\bf r})=\\int f({\\bf r},{\\bf v}) d{\\bf v} =\\int f v^2dv \nd\\Omega= n_0\\frac{\\Delta S_0 \\cos\\theta}{4\\pi r^2}, \n\\end{equation}\nwhich is quantitatively consistent with our intuitive notion about the \nparticle density outside the container. \n\nBefore leaving this subject, we go back to discuss several usual concepts \nconcerning the collisionless and regular Boltzmann equations. It is commonly \nbelieved that the collisionless Boltzmann equation (\\ref{bltz}) is completely \nequivalent to the path-invariance expressed by (\\ref{fm}). In our view, this \nequivalence is just a formal one. For one thing, expression (\\ref{fm}) makes \nsense strictly along a particle path, which is governed by the collisionless \nmotion equations of single particle; while Boltzmann's equations are supposed \nto be solved by a difference scheme with help of boundary and initial \nconditions in which trajectories of individual particle do not play any role. \nFor another, unconventional topology and dynamics are related to (\\ref{fm}). \nDistribution functions described by it, such as (\\ref{cone}) and \n(\\ref{delta}), can be shaped like stings; more than that, these stings \n`spread' out along paths of particles, become sharper and sharper when \nspreading, and continue to spread out after being infinitely sharp. \n(Distributions produced by boundaries can be even `sharper', see Sect.~5.) \nBoltzmann's equations, being ones containing differential terms, are not \ncompatible with these `radical' things. \n\n\\section{The collisional correction} In this section, we try to deal with \ncollisions. To make our discussion involve less details, we shall only \nformulate the first-order correction. Namely, it is assumed that the \nparticles expressed by (\\ref{delta}) will collide once and only once \n(although further extension can be made with no principal difficulty). \n\n\\hspace{-25pt} \\setlength{\\unitlength}{0.020in} \\begin{picture}(200,100) \n \n\\put(90,53){\\line(0,-1){23}}\n\\put(90,57){\\line(0,1){23}}\n\\put(90,57){\\line(-2,3){10}}\n\\put(90,53){\\line(-2,-3){10}}\n\n\\put(80,30){\\line(0,1){8}}\n\\put(82,30){\\line(0,1){11}}\n\\put(84,30){\\line(0,1){14}}\n\\put(86,30){\\line(0,1){17}}\n\\put(88,30){\\line(0,1){20}}\n\n\\put(80,80){\\line(0,-1){8}}\n\\put(82,80){\\line(0,-1){11}}\n\\put(84,80){\\line(0,-1){14}}\n\\put(86,80){\\line(0,-1){17}}\n\\put(88,80){\\line(0,-1){20}}\n\n\\multiput(82,47)(0,4){5}{\\circle*{1.5}}\n\\multiput(86,51)(0,4){3}{\\circle*{1.5}}\n\n\\multiput(95,55)(1.8,0){14}{\\circle*{0.7}}\n\\multiput(94,57)(1.6,0.9){14}{\\circle*{0.7}}\n\\multiput(92.5,58.5)(0.9,1.6){14}{\\circle*{0.7}}\n\\multiput(94,53)(1.6,-0.9){14}{\\circle*{0.7}}\n\\multiput(92.5,51.5)(0.9,-1.6){14}{\\circle*{0.7}}\n\n\\multiput(150.2,39.4)(16.6,8.2){2}{\\line(1,-2){8}}\n\\multiput(150.2,39.4)(9.2,4.6){2}{\\line(2,1){7}}\n\\put(158,44.25){\\line(-1,5){10}}\n\\put(158,44.25){\\line(-4,5){33}}\n\\put(157,49){\\line(-1,0){3}}\n\\put(156,54){\\line(-1,0){5.5}}\n\\put(155,59){\\line(-1,0){8.5}}\n\\put(154,64){\\line(-1,0){11}}\n\\put(153,69){\\line(-1,0){14.3}}\n\\put(152,74){\\line(-1,0){17.5}}\n\\put(151,79){\\line(-1,0){20.5}}\n\\put(150,84){\\line(-1,0){24}}\n\\put(149,89){\\line(-1,0){12}}\n\n\\put(126,24){\\makebox(20,8)[l]{Detector}}\n\\put(153,79){\\makebox(20,8)[l]{Effective}}\n\\put(153,72){\\makebox(20,8)[l]{cone $-\\Delta\\Omega$}}\n\\end{picture} \n\n\\vspace{-29pt}\\begin{center} \\begin{minipage}{10cm} { \\vskip-0.3cm Figure~3: \nA particle detector is placed in the leaking gas.} \\end{minipage} \n\\end{center} \n\nWhat can be measured in an experiment should be of our first concern. For \nthis reason, we consider a particle detector placed somewhere in the leaking \ngas and try to determine how many particles will be recorded by the detector, \nas illustrated in Fig.~3. There are several essential things worth mentioning \nabout the detector. \\begin{enumerate} \\item The opening allowing particles to \nenter the detector is considered to be sufficiently small. Or, in the \nmathematical language, we regard the area of the opening, denoted by $\\Delta \nS$, as an infinitesimal quantity throughout this section. \\item Without \nspecifying concrete structure and mechanism of the detector, it is assumed \nthat every particle that enters the detector and is in the velocity range \n\\begin{equation} \\label{vvolume} \\Delta {\\bf v}= \\Delta v v^2 \\Delta \\Omega, \n\\end{equation}\nwhere $\\Delta v$ and $\\Delta \\Omega$ are predetermined before the \nmeasurement, will be recorded, and every other particle will not. \\item While \n$\\Delta S$ and $\\Delta v$ are allowed to shrink to zero, we define $\\Delta \n\\Omega$ as a finite solid-angle range. As will be seen, the discrimination \nagainst $\\Delta\\Omega$ is taken almost entirely by necessity. \\end{enumerate} \n\nIf the detector works as described above, do we know the distribution \nfunction at the entry of the detector? The answer to it is almost a positive \none. If $\\Delta N$ is the number that the detector counts during $\\Delta t$, \nthe particle density in the phase volume element $(\\Delta S v \\Delta \nt)(\\Delta v v^2 \\Delta \\Omega)$ is \n\\begin{equation} \\label{deltaN} f(t,{\\bf r}, {\\bf v},\\Delta \\Omega )\\approx \n\\frac {\\Delta N}{(\\Delta S v \\Delta t)(\\Delta v v^2 \\Delta \\Omega)}.\n\\end{equation}\nThe form of (\\ref{deltaN}) illustrates one of the most distinctive features \nof this approach: it tries to calculate the distribution function directly \nrather than to formulate a dynamic equation concerning how many particles \nenter and leave a phase volume element. Apart from other advantages, this \nbrings to us a lot of convenience in computational terms. \n\nNoticing that only the collisions taking place within the shaded spatial cone \n$-\\Delta\\Omega$, which is equal and opposite to the velocity cone $\\Delta \n\\Omega$, can possibly contribute to $\\Delta N$, as shown in Fig.~3, we call \nthe region inside $-\\Delta\\Omega$ the effective cone. Since this effective \ncone is a finite one ($\\Delta \\Omega$ is finite), our primary task is to \ndivide it into many subregions, denoted by $d{\\bf r}'$ (the origin of the \ncoordinates is still at the center of the hole as in Sect.~3), and to \ncalculate how collisions in each of the subregions give contribution to \n$\\Delta N$. \n\nConsider that two beams of identical particles (but still distinguishable in \nterms of classical mechanics) \n\\begin{equation} f(t',{\\bf v}',{\\bf r}')d{\\bf v}' \\quad {\\rm and} \\quad \nf(t',{\\bf v}'_1,{\\bf r}')d{\\bf v}'_1 \n\\end{equation}\ncollide within $d{\\bf r}'$ and at time $t'$, producing particles with \nvelocities ${\\bf v}$ and ${\\bf v}_1$ respectively. It is noted that there is \na time delay \n\\begin{equation} t-t'=\\frac{|{\\bf r}-{\\bf r}'|} {v}\\quad {\\rm with}\\quad \nv=|{\\bf v}| \n\\end{equation}\nbetween the time of collision and the time of detection. Since the leaking is \nassumed to be relatively slow (or the gas inside the container is supplied by \nan external gas source), we think of our problem as a time-independent one \nand pay no more attention to the time variable hereafter. To better observe \nthe collisions, we define \n\\begin{equation} \\label{cu} \\left\\{\\begin{array}{l} {\\bf v}'+{\\bf \nv}'_1=2{\\bf c}'\\cr {\\bf v}'-{\\bf v}'_1=2{\\bf u}' \\end{array}\\right. \\quad \n{\\rm and}\\quad \\left\\{\\begin{array}{l} {\\bf v}+{\\bf v}_1=2{\\bf c}\\cr {\\bf \nv}-{\\bf v}_1=2{\\bf u} \\end{array}\\right. . \n\\end{equation}\nThat is to say, by virtue of the conservation laws of energy and momentum, \n${\\bf c}={\\bf c}'$ represents the velocity of the center-of-mass and $u=|{\\bf \nu}|=|{\\bf u}'|$ stands for the particle speed relative to the center-of-mass. \nExamining the beam-to-beam collision in the center-of-mass frame, we find \nthat the differential number of the colliding particles is \n\\begin{equation} \\label{twobeams} [d{\\bf r}'f({\\bf r}',{\\bf v}')d{\\bf v}'] \n[f({\\bf r}',{\\bf v}'_1)d{\\bf v}'_1] [2u\\sigma_c({\\bf u}',{\\bf u}) d \\Omega_c \n\\Delta t ],\n\\end{equation}\nwhere $\\Omega_c$ is the solid angle between ${\\bf u}'$ and ${\\bf u}$ and \n$\\sigma_c$ is the cross section associated with particles emerging within the \nsolid-angle range $d\\Omega_c$. By integrating (\\ref{twobeams}) and getting \nhelp from (\\ref{delta}), the right side of (\\ref{deltaN}) is equal to \n\\begin{equation} \\label{entry} \\int_{-\\Delta \\Omega}d{\\bf r}'\\int_{\\Delta \nv\\Delta \\Omega_0} d\\Omega_c \\int_0^\\infty dv'\\int_0^\\infty dv'_1 \n\\frac{2u\\sigma_c g(r',v') g(r',v'_1)} {(|{\\bf r}-{\\bf r}'|^2 \\Delta \\Omega_0 \nv)\\dot (v^2\\Delta v \\Delta \\Omega )}, \n\\end{equation}\nwhere $\\Delta \\Omega_0$ is the solid-angle range formed by the point $d{\\bf \nr}'$ (as the apex) and the detector opening $\\Delta S$ (as the base), and the \nsubindex $\\Delta v\\Delta \\Omega_0$ there states that only particles that can \nbe recorded by the detector will be taken into account. Since the size of \n$\\Delta \\Omega_0$ is `much smaller' than that of $\\Delta\\Omega$, every \nparticle starting its journey from the effective cone and entering the \ndetector will be treated as one within $\\Delta \\Omega$. With help of the \nvariable transformation from $(v',v'_1)$ to $(c',u')$ and finally to $(c,u)$, \nwe rewrite expression (\\ref{entry}) as \n\\begin{equation} \\label{entry1} \\int_{-\\Delta \\Omega}d{\\bf r}'\\int_{\\Delta \nv\\Delta \\Omega_0} d\\Omega_c \\int_0^\\infty dc \\int_{-c}^{c} du \n\\frac{2u\\sigma_c\\cdot \\|J\\| \\cdot g(c+u) g(c-u)} {(|{\\bf r}-{\\bf r}'|^2 \n\\Delta \\Omega_0 v)\\dot (v^2\\Delta v \\Delta \\Omega )}, \n\\end{equation}\nin which the Jacobian between the variable transformation is \n\\begin{equation} \\| J\\|=\\frac{\\partial (v',v'_1)}{\\partial (c,u)}=2.\n\\end{equation}\nIn view of the symmetry of the cross section there, we have \n\\begin{equation} \\int_{-c}^c du\\cdots =2\\int_0^c du \\cdots. \n\\end{equation}\n\\vspace{-0.5cm}\n\n\\hspace{30pt} \\setlength{\\unitlength}{0.013in} \\begin{picture}(200,135) \n\n\\put(135,93){\\makebox(35,8)[l]{${\\bf c}$}}\n\\put(190,50){\\makebox(35,8)[l]{$\\bf u$}}\n\\put(220,90){\\vector(-1,-1){60}}\n\\put(70,90){\\line(1,0){150}}\n\\put(217.5,90.5){\\vector(1,0){1}}\n\\multiput(70,90)(3.0,-2.3){32}{\\circle*{1.2}}\n\\multiput(70,90)(3.0,-1.7){35}{\\circle*{1.2}}\n\\multiput(163,18.7)(1.6,2.4){6}{\\circle*{1.2}}\n\\multiput(154,42.4)(-1.6,-2.4){5}{\\circle*{1.2}}\n\\put(70,60){\\makebox(35,8)[l]{$\\Delta\\Omega_0$}}\n\\put(140,14){\\makebox(35,8)[l]{$\\Delta v$}}\n\n\\put(143.22,53.87){\\circle*{1}}\n\\put(144.80,50.69){\\circle*{1}}\n\\put(146.52,47.57){\\circle*{1}}\n\\put(148.36,44.53){\\circle*{1}}\n\\put(150.32,41.57){\\circle*{1}}\n\\put(152.41,38.70){\\circle*{1}}\n\\put(154.62,35.91){\\circle*{1}}\n\\put(156.94,33.22){\\circle*{1}}\n\\put(159.38,30.63){\\circle*{1}}\n\\put(161.91,28.15){\\circle*{1}}\n\\put(164.56,25.77){\\circle*{1}}\n\\put(167.29,23.50){\\circle*{1}}\n\\put(170.13,21.35){\\circle*{1}}\n\\put(173.04,19.32){\\circle*{1}}\n\\put(176.04,17.42){\\circle*{1}}\n\\put(179.12,15.64){\\circle*{1}}\n\\put(182.27,14.00){\\circle*{1}}\n\\put(185.49,12.48){\\circle*{1}}\n\\end{picture} \n\n\\begin{center} \\begin{minipage}{10cm} { \\vskip-0.3cm Figure~4: The relation \nbetween the velocity element $v^2\\Delta v \\Delta\\Omega_0 $ and the velocity \nelement $u^2 du d\\Omega_c $.} \\end{minipage} \\end{center} \\vspace{5pt} \n\n\\noindent And, by investigating the situation in the velocity space shown in \nFig.~4, the following relation can be found out: \n\\begin{equation} \\int_{\\Delta v \\Delta \\Omega_0}u^2 du d\\Omega_c \\cdots \n\\approx v^2\\Delta v \\Delta\\Omega_0\\cdots .\n\\end{equation}\nTherefore, the first-order distribution function at the entry of the detector \nis \n\\begin{equation} \\label{final} f({\\bf r}, {\\bf v}, \\Delta \\Omega)=\\frac{1}{v \n\\Delta \\Omega} \\int_{-\\Delta \\Omega}d{\\bf r}' \\int_0^\\infty dc \n\\frac{8\\sigma_c({\\bf u}',{\\bf u})g(r',c+u) g(r',c-u)}{u|{\\bf r}-{\\bf r}'|^2}, \n\\end{equation}\nin which, although $u =|{\\bf c}-{\\bf v}|$, $u<c$ needs to be ensured for the \nintegral to make sense and the direction of $\\bf u'$ is the same as that of \n${\\bf c}'$ or ${\\bf c}$. \n\nSince we have, at the beginning of this section, assumed $\\Delta\\Omega$ to be \na finite solid-angle range, expression (\\ref{final}) is nothing but the \ndistribution function averaged over $\\Delta\\Omega$, which is, in a sense, \nstill different from `the true and exact distribution function' there. In the \nnext section, we shall come back to this issue and find out that the true and \nexact distribution function is actually beyond our reach. \n\nIf interested, readers may test the resultant formalism with aid of realistic \nand computational experiments or develop it to cover more practical and more \ncomplicated cases. This is one of the major purposes of the present paper. \n\n\\section{Discussion} \n\nIn this section, we wish to further justify one of our introduced concepts, \nthe distribution function averaged over finite solid-angle ranges of \nvelocity. \\vskip 10pt \n\n\\hspace{30pt} \\setlength{\\unitlength}{0.018in} \\vspace{0.5cm} \n\\begin{picture}(100,100) \\hspace{0.0cm} \n\n\\put(30.797, 87.459){\\circle*{1.0}}\n\\put(31.083, 86.931){\\circle*{1.0}}\n\\put(31.369, 86.404){\\circle*{1.0}}\n\\put(31.656, 85.877){\\circle*{1.0}}\n\\put(31.942, 85.349){\\circle*{1.0}}\n\\put(32.228, 84.822){\\circle*{1.0}}\n\\put(32.514, 84.294){\\circle*{1.0}}\n\\put(32.800, 83.767){\\circle*{1.0}}\n\\put(33.044, 83.219){\\circle*{1.0}}\n\\put(33.288, 82.671){\\circle*{1.0}}\n\\put(33.532, 82.122){\\circle*{1.0}}\n\\put(33.776, 81.574){\\circle*{1.0}}\n\\put(34.020, 81.026){\\circle*{1.0}}\n\\put(34.263, 80.478){\\circle*{1.0}}\n\\put(34.507, 79.930){\\circle*{1.0}}\n\\put(34.751, 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35.080){\\circle*{1.0}}\n\\put(38.605, 34.492){\\circle*{1.0}}\n\\put(38.484, 33.905){\\circle*{1.0}}\n\\put(38.363, 33.317){\\circle*{1.0}}\n\\put(38.243, 32.729){\\circle*{1.0}}\n\\put(38.122, 32.141){\\circle*{1.0}}\n\\put(38.002, 31.554){\\circle*{1.0}}\n\\put(37.881, 30.966){\\circle*{1.0}}\n\\put(37.761, 30.378){\\circle*{1.0}}\n\\put(37.640, 29.790){\\circle*{1.0}}\n\\put(37.479, 29.212){\\circle*{1.0}}\n\\put(37.319, 28.634){\\circle*{1.0}}\n\\put(37.158, 28.056){\\circle*{1.0}}\n\\put(36.997, 27.478){\\circle*{1.0}}\n\\put(36.836, 26.900){\\circle*{1.0}}\n\\put(36.676, 26.322){\\circle*{1.0}}\n\\put(36.515, 25.744){\\circle*{1.0}}\n\\put(36.354, 25.166){\\circle*{1.0}}\n\\put(36.155, 24.600){\\circle*{1.0}}\n\\put(35.956, 24.034){\\circle*{1.0}}\n\\put(35.756, 23.468){\\circle*{1.0}}\n\\put(35.557, 22.902){\\circle*{1.0}}\n\\put(35.358, 22.336){\\circle*{1.0}}\n\\put(35.158, 21.770){\\circle*{1.0}}\n\\put(34.959, 21.204){\\circle*{1.0}}\n\\put(34.723, 20.653){\\circle*{1.0}}\n\\put(34.487, 20.101){\\circle*{1.0}}\n\\put(34.251, 19.549){\\circle*{1.0}}\n\\put(34.015, 18.998){\\circle*{1.0}}\n\\put(33.778, 18.446){\\circle*{1.0}}\n\\put(33.542, 17.895){\\circle*{1.0}}\n\\put(33.306, 17.343){\\circle*{1.0}}\n\\put(33.070, 16.792){\\circle*{1.0}}\n\\put(32.789, 16.261){\\circle*{1.0}}\n\\put(32.508, 15.731){\\circle*{1.0}}\n\\put(32.227, 15.201){\\circle*{1.0}}\n\\put(31.946, 14.671){\\circle*{1.0}}\n\\put(31.666, 14.141){\\circle*{1.0}}\n\\put(65,50){\\vector(1,0){50}}\n\\put(65,56.5){\\vector(4,1){50}}\n\\put(65,43.5){\\vector(4,-1){50}}\n\\multiput(80,90)(-2,2){3}{\\vector(-1,-1){15}}\n\\put(101.5,54){\\makebox(20,8)[l]{${\\bf r}$}}\n\\put(16,42){\\makebox(20,8)[r]{$\\Delta_i S$}}\n\\put(100,56){\\circle*{2}}\n\\put(38.5,50){\\oval(3,5)}\n\n\\end{picture} \n\n\\vspace{-28pt}\\begin{center} \\begin{minipage}{10cm} { \\vskip-0.3cm Figure~5: \nParticles refleted by a surface.} \\end{minipage} \\end{center} \n\nLet's look at the path-invariance theorem expressed by (\\ref{fm}) again. \nAlthough the path-invariance is not equivalent to the collisionless Boltzmann \nequation (see the remark at the end of Sect.~3), the theorem can still be \nthought of as a part of the conventional theory since it is formally the same \nas the collisionless Boltzmann equation and it is consistent with the usual \nnotion that distribution functions of real gases are continuous or, at least, \npiecewise continuous. Then, a crucial question arises. Are all distribution \nfunctions of realistic gases continuous or, at least, piecewise continuous? \nIf this is the case, we will be able, at least in principle, to determine the \nexact distribution function. To see what the real situation is, we examine \nthe case illustrated in Fig.~5, where particles are moving toward a surface \nwith a definite velocity (or, equivalently, let the surface move toward the \nparticles). Since the surface is not uniform geometrically and physically, we \nneed to, in order to formulate the reflected particles, divide it into many \nsmall surface elements, denoted by $\\Delta_i S$ herein. It is easy to see \nthat, corresponding to one of $\\Delta_i S$ the reflected particles are like \nones emitted from a point particle source and the distribution function, at a \nposition ${\\bf r}$ around the surface, is \n\\begin{equation} \\label{delta1} \\Delta_i f = \\frac{\\eta_i({\\bf v}) \\Delta_i \nS }{4\\pi |{\\bf r}-{\\bf r}_i|^2} \\delta\\left(\\Omega_{\\bf v}- \\Omega_{{\\bf \nr}{-\\bf r}_i}\\right), \n\\end{equation}\nwhere ${\\bf r}_i$ represents the position vector of $\\Delta_i S$ and \n$\\eta_i({\\bf v})$ is the reflection (emission) function that can be \ndetermined by experiments\\cite{kogan}. As has been illustrated in Sect.~3, \nthe distribution function expressed by (\\ref{delta1}) is an infinitely thin \n`sting' at every spatial point, and the particle density along a particle \npath is no longer invariant. The total distribution function at ${\\bf r}$ is \n\\begin{equation} \\label{delta2} f=\\sum_i \\Delta_i f = \\sum_i \n\\frac{\\eta_i({\\bf v}) \\Delta_i S }{4\\pi |{\\bf r}-{\\bf r}_i|^2} \n\\delta\\left(\\Omega_{\\bf v}- \\Omega_{{\\bf r}{-\\bf r}_i}\\right). \n\\end{equation}\nIt is then found that, if the true and exact distribution function is indeed \nof our concern, the $\\delta$-functions in (\\ref{delta2}) will stay there and \nno regular functions can be used to replace them because each of \n$\\Omega_{{\\bf r}{-\\bf r}_i}$ in it points in a distinctive direction, which \nmeans, in a more vivid language, the velocity distribution at every spatial \npoint is shaped like an infinite number of infinitely thin stings (similar to \nfunctions dealt with in the studies of fractals). Without introducing the \ndistribution function averaged over velocity solid-angle ranges, we will have \ngreat difficulty in setting up a calculable theory. \\vspace{-33pt} \n\n\\hspace{15pt} \\setlength{\\unitlength}{0.023in} \\begin{picture}(200,105) \n\n\\multiput(40,20)(40,0){2}{\\line(0,1){15}}\n\\multiput(40,35)(22,0){2}{\\line(1,0){18}}\n\\put(60,35){\\line(1,5){6}}\n\\put(60,35){\\line(-1,5){6}}\n\\multiput(60.5,33)(0.4,-2){8}{\\circle*{0.7}}\n\\multiput(59.5,33)(-0.4,-2){8}{\\circle*{0.7}}\n\\put(66,23){\\makebox(35,8)[l]{$\\Delta\\Omega$}}\n\\put(60.5,20){\\line(2,1){7}}\n\n\\multiput(60,73)(0.2,-2){6}{\\circle*{0.7}}\n\\multiput(60,73)(-0.2,-2){6}{\\circle*{0.7}}\n\\put(65,68){\\makebox(35,8)[l]{$\\Delta\\Omega_0$}}\n\\put(60,64){\\line(2,1){7}}\n\\put(61,35.5){\\line(2,1){7}}\n\\put(69,38.5){\\makebox(35,8)[l]{$\\Delta S$}}\n\\put(133,35){\\line(2,1){7}}\n\\put(141,38){\\makebox(35,8)[l]{$\\Delta S$}}\n\n\\multiput(110,20)(40,0){2}{\\line(0,1){15}}\n\\multiput(110,35)(26,0){2}{\\line(1,0){14}}\n\n\\multiput(124,35)(12,0){2}{\\line(0,1){28}}\n\\multiput(126,35)(2,0){5}{\\circle*{0.7}}\n\n\\multiput(130,35)(0.2,-2){9}{\\circle*{0.7}}\n\\multiput(130,35)(-0.2,-2){9}{\\circle*{0.7}}\n\\put(135,24){\\makebox(35,8)[l]{$\\Delta\\Omega$}}\n\\put(130,21){\\line(2,1){7}}\n\n\n\\put(130,70){\\vector(0,-1){14}}\n\\put(126,49){\\makebox(8,8)[c]{$\\bf v$}}\n\\put(55,8){\\makebox(10,8)[c]{$\\bf (a)$}}\n\\put(125,8){\\makebox(10,8)[c]{$\\bf (b)$}}\n\\end{picture} \n\n\\vspace{-23pt}\\begin{center} \\begin{minipage}{10cm} {Figure~6: The effective \nregion above the detector: (a) If $\\Delta\\Omega$ is fixed and $\\Delta S$ \nshrinks to zero, it is cone-shaped. (b) If $\\Delta S$ is fixed and \n$\\Delta\\Omega$ shrinks to zero, it is cylinder-shaped. } \\end{minipage} \n\\end{center} \n\nTreatment of particle-to-particle interactions is always an indispensable \nchapter of statistical mechanics. While referring readers to \nelsewhere\\cite{chen1,chen2} for analyses of perplexing problems in the \ntextbook methodology, we shall here comment on our own treatment of collision \npresented in the last section. At first glance, if $\\Delta \\Omega$ there were \ninfinitely small, both the denominator and numerator of (\\ref{final}) would \ntend to zero simultaneously and the total expression would remain finite and \nvalid. A careful inspection, however, tells us that there are two `competing' \nquantities $\\Delta\\Omega$ and $\\Delta S$. If we fix $\\Delta \\Omega$ and let \n$\\Delta S$ approach zero, the effective region is a cone-shaped one, shown in \nFig.~6a, and the related distribution function is, as has been formulated, \nthe one averaged over the finite solid-angle range $\\Delta\\Omega$. Whereas, \nif we fix $\\Delta S$ and let $\\Delta \\Omega$ approach zero, the situation is \nrather different: the effective region becomes a cylinder-shaped one, shown \nin Fig.~6b, and the related distribution function will be, if formulated, the \none averaged over the finite spatial area $\\Delta S$. In both the above \nsituations, each of the results is related to an integration, each of the \nintegrations is taken over a region infinitely-extended in space. (Even if \nmore collisions are taken into account the above assertion still holds its \nsignificance since particles from the remote regions can still come freely in \nterms of probabilities.) There is no way to ensure the continuity of each \nformalism let alone the consistency between the two formalisms. \n\n\n\nThe discussions in this section clearly show that, in dealing with both \ncollisionless dynamics and collisional dynamics, a finite range of certain \nvariable (whether in the position space or in the velocity space) and the \naverage over this range have to be employed. If the approach is inherently \nand completely that of differentiation, which means that the exact \ndistribution function is of concern and every position and velocity range \nmust shrink to zero, insurmountable conflicts surface automatically in one \nway or another. The repetitive appearances of conflict remind us of the \nsituation in quantum mechanics where the accurate position and momentum \ncannot be determined simultaneously. Possible implication of this issue has \nyet to be explored. \n\n\\section{Summary} We have proposed a gas as our working model and formulated \na feasible method to calculate it. The formalism shows that real gases, at \nleast certain types of them, can be calculated at the kinetic level. It is \nexpected that such calculations will soon be compared to realistic or \ncomputational experiments. \n\nIn this paper, collisional effects were investigated under the assumption \nthat each particle involves, at most, one collision. If more collisions are \ntaken into account, the trajectory of a particle will be very much like that \nof the Brownian motion or, in another sense, similar to that appearing in the \nlogistic map of nonlinear studies. Although the extensions in this direction \nhave been tried\\cite{chen3,chen4}, it is not appropriate to comment on them \nbefore this work is admitted by more in this community. \n\nHelpful discussions with professors Hanying Guo, Ke Wu and Qiang Chen are \ngreatly appreciated. This paper is supported by School of Science, BUAA, PRC. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nInvestigating thermodynamic structures of nonequilibrium steady states (NESSs) has been a topic of active researches in nonequilibrium statistical mechanics~\\cite{Landauer,Oono,Raulle,Komatsu1,Komatsu2,Saito,Sasa,Seifert,Hatano,Speck,Esposito1,Esposito2,Lebowitz,Nemoto,Komatsu0,Maes1,Maes2}. For example, the extension of the relations in equilibrium thermodynamics, such as the Clausius equality, to NESSs is a great challenge~\\cite{Landauer,Oono,Raulle,Komatsu1,Komatsu2,Saito,Sasa,Seifert}.  The extended thermodynamics, which is called steady state thermodynamics (SST)~\\cite{Oono}, is expected to be useful to analyze and to predict the dynamical properties of NESSs.  However, the complete picture of SST has not been  understood.\n\n\nIn equilibrium thermodynamics, the Clausius equality tells us how one can determine thermodynamic potential (entropy) by measuring the heat: \n\\begin{equation}\n\\Delta S - \\sum_\\nu \\beta^\\nu Q^\\nu = 0,\n\\label{Clausius}\n\\end{equation}\nwhich is universally valid for quasi-static transitions between equilibrium states.  Here, $\\nu$ is an index of the heat baths, $\\beta^\\nu$ is the inverse temperature of bath $\\nu$, $Q^\\nu$ is the heat that the system absorbed from bath $\\nu$, and $S$ is the Shannon entropy of the system.   The second term of the left-hand side (LHS) of (\\ref{Clausius}) is called the entropy production in the baths.\nTo generalize the Clausius equality to nonequilibrium situations,  it has been proposed~\\cite{Landauer,Oono} that heat $Q^\\nu$ needs to be replaced by excess heat $Q_{\\rm ex}^\\nu$,  which describes an additional heat induced by a transition between NESSs with time-dependent external control parameters such as the electric field.  Correspondingly,  the total heat can be decomposed as $Q^\\nu = Q_{\\rm ex}^\\nu + Q_{\\rm hk}^\\nu$, where housekeeping heat $Q_{\\rm hk}^\\nu$ describes the steady heat current in a NESS without any parameter change.  \nQuantitative definitions of these quantities will be given later.\nOne may then expect that there exists some thermodynamic potential $S_{\\rm SST}$ which characterizes NESSs such that\n\\begin{equation}\n \\Delta S_{\\rm SST} - \\sum_\\nu \\beta^\\nu Q^{\\nu}_{\\rm ex} = 0,\n\\label{Clausius1}\n\\end{equation}\nholds for quasi-static transitions between NESSs, where the second term of the LHS corresponds to the excess part of the entropy production in the baths.  Komatsu, Nakagawa, Sasa, and Tasaki (KNST) found that $S_{\\rm SST}$ in Eq.~(\\ref{Clausius1}) is a symmetrized version of the Shannon entropy in the lowest order of nonequilibriumness~\\cite{Komatsu1,Komatsu2}.  However, the full order expression of the  extended Clausius equality~(\\ref{Clausius1}) has been elusive.   Then, some fundamental questions  arise:  What is the nonequilibrium thermodynamic potential $S_{\\rm SST}$ in Eq.~(\\ref{Clausius1})  in the full order expression? Does there exist such a potential at all?\n\nIn this paper, we answer these questions, and derive a full order expression of the excess entropy production for Markovian jump processes.   We note that driven lattice gases are special cases of our formulation.  We have found that an extended Clausius equality in the form of (\\ref{Clausius1}) does not hold in general; scalar thermodynamic potential $S_{\\rm SST}$ should be replaced by a  vector potential.   In other words, the first term of the LHS of (\\ref{Clausius1}) should be replaced by  a geometrical quantity that depends only on trajectories in the parameter space.  Our result includes  equilibrium Clausius equality~(\\ref{Clausius}) and the KNST's extended Clausius equality as special cases.   We will also derive the general condition that there exists a thermodynamic potential $S_{\\rm SST}$ such that Eq.~(\\ref{Clausius1}) holds.\n\nWe have used the technique of the full counting statistics~\\cite{Esposito1,Bagrets} to prove our main results.  In the context of the full counting statistics (and also stochastic ratchets), it has been reported~\\cite{Parrondo,Sinitsyn1,Sinitsyn2,Sinitsyn3,Jarzynski1,Ohkubo1,Ren,Ohkubo2} that several phenomena in classical stochastic processes are analogous to the Berry's geometrical phase in quantum mechanics~\\cite{Berry,Samuel}.  In this analogy, the above-mentioned vector potential corresponds to the gauge field that induces the Berry phase.   Our result can also be regarded as a generalization of these previous studies on the classical Berry phase.\n\nThis paper is organized as follows.  In Sec.~II, we formulate the model of our system and define the decomposition of the entropy production into the housekeeping and excess parts based on the full counting statistics.  In Sec.~III, we derive our main results, which consist of the geometrical expressions of the excess parts of the cumulant generating function and the average of the entropy production.  In Sec.~IV, we apply our main results to two special cases; one is equilibrium thermodynamics with the detailed balance, and the other is the KNST's extended Clausius equality.  In Sec.~V, we discuss a quantum dot as a simple example, where  Eq.~(\\ref{Clausius1}) does not hold in general.   In Sec.~VI, we conclude this paper with some discussions.\n\n\n\\section{Setup}\n\nWe first formulate our setup and define the decomposition of the cumulant generating function of the entropy production into the excess and housekeeping parts.\n\n\\subsection{Dynamics}\n\nWe consider Markovian jump processes with $N < \\infty$ microscopic states.  Let $p_x$ be the probability that the system is in state $x$.  The probability distribution of the system is then characterized by vector $| p \\rangle :=  [p_1, p_2, \\cdots, p_N]^T$, where $1, 2, \\cdots, N$ describe the  states, and ``$T$'' describes the transpose of the vector.\nThe time evolution of the probability distribution is given by a master equation $| \\dot p (t) \\rangle = R (\\bm{\\alpha} (t)) | p (t) \\rangle$,\nwhere $| \\dot  p (t) \\rangle$ describes the time derivative of $| p (t) \\rangle$, $R(\\bm{\\alpha})$ is a $N \\times N$ matrix characterizing the transition rate of the dynamics with external parameters $\\bm{\\alpha}$.  Here, the external parameters correspond to, for example, a potential or a nonconservative force applied to a lattice gas, or the temperatures of the heat baths.  We drive the system by changing $\\bm{\\alpha}$.  For simplicity of notations,  we will often omit   ``$(\\bm{\\alpha}(t))$'' or  ``$(t)$'' in the following discussions.\nWe note that $\\sum_{x} R_{xy} = 0$ holds for every $y$, where $R_{xy}$ is the $xy$-component of $R$ that characterizes the transition rate from state $y$ to $x$.\nWe assume that $R$ is irreducible such that $R$ has eigenvalue $0$ without degeneracy  due to the Perron-Frobenius theorem.  We write as $\\langle 1 |$ and $| p^S \\rangle$ the left and right eigenvectors of $R$ corresponding to eigenvalue $0$ such that $\\langle 1 | R = 0$ and $R | p^S \\rangle = 0$ hold.  We note that $\\langle 1 | = [1, 1, \\cdots, 1]$ holds and that $| p^S \\rangle = [p^S_1, p^S_2, \\cdots, p^S_N]^T$ is the unique steady distribution of the dynamics with a given $\\bm \\alpha$.  For simplicity, we assume that $R$ is diagonalizable.\nWe also assume that the transition matrix can be decomposed into the contributions from multiple heat baths, labeled by $\\nu$, as $R_{xy} = \\sum_\\nu R^\\nu_{xy}$.\n\nWe next introduce the entropy production that depends on trajectories of the system.  Such a trajectory-dependent entropy production has been studied in terms of nonequilibrium thermodynamics of stochastic systems~\\cite{Lebowitz,Crooks,Jarzynski2,Seifert}.\nThe entropy production in bath $\\nu$ with transition from $y$ to $x$ is given by\n\\begin{eqnarray}\n\\sigma^\\nu_{xy} =  \\left\\{\n\\begin{array}{l}\n\\ln \\frac{R^\\nu_{xy}}{R^\\nu_{y x}} = - \\beta^\\nu Q^\\nu_{xy} \\  ({\\rm if} \\ R^\\nu_{xy}\\neq 0 \\ {\\rm and} \\ R^\\nu_{y x}\\neq 0),  \\\\\n0 \\  ({\\rm if} \\ R^\\nu_{xy} = 0 \\ {\\rm and}  \\ R^\\nu_{y x} = 0), \n\\end{array}\n\\right.\n\\label{entropy}\n\\end{eqnarray}\nwhere $Q_{xy}^\\nu$ is the heat that is absorbed in the system from bath $\\nu$ during the transition from $y$ to $x$.  Equality (\\ref{entropy}) is consistent with the detailed fluctuation theorem~\\cite{Lebowitz,Crooks,Jarzynski2,Seifert}. \nThe integrated entropy production from time $0$ to $\\tau$ is determined by the trajectory of system's states during the time interval as \n\\begin{equation}\n\\sigma = \\sum_{t : \\ {\\rm jump}} \\sigma^\\nu_{x(t+0)y(t-0)},\n\\end{equation}\nwhere the sum is taken over all times at which the system jumps, and $y(t-0)$ and $x(t+0)$ are  the states immediately before and after the jump at $t$, respectively.  \nWe note that the ensemble average of $\\sigma$ is equivalent to the entropy production in the conventional thermodynamics of macroscopic systems. A reason why we consider the trajectory-dependent entropy production lies in the fact that the entropy production is connected to the heat through Eq.~(\\ref{entropy}) at the level of each trajectories.\n\n\n\n\n\n\\subsection{Full Counting Statistics}\n\nWe then discuss the full counting statistics of $\\sigma$.  Let $P(\\sigma)$ be the probability of $\\sigma$.  Its cumulant generating function is given by \n\\begin{equation}\nS(i\\chi) := \\ln \\int d\\sigma e^{i\\chi \\sigma}P(\\sigma),\n\\end{equation}\nwhere $\\chi \\in \\mathbb R$ is the counting field.  $S(i \\chi)$ leads to the cumulants of $\\sigma$ like $\\langle \\sigma \\rangle = \\partial S (i \\chi) \/ \\partial (i \\chi) |_{\\chi = 0}$,\nwhere $\\langle \\cdots \\rangle$ describes the statistical average.\nTo calculate $S(i\\chi)$, we define matrix $R_\\chi$ as $( R_\\chi )_{xy} := \\sum_\\nu R_{xy}^\\nu \\exp (i\\chi \\sigma_{xy}^\\nu)$,\nand consider the time evolution of vector $| p_\\chi (t) \\rangle$ corresponding to\n\\begin{equation}\n | \\dot p_\\chi (t) \\rangle = R_\\chi (\\bm \\alpha (t)) | p_\\chi (t) \\rangle\n\\label{master_chi}\n\\end{equation}\nwith initial condition $| p_\\chi (0) \\rangle := | p(0) \\rangle$.  The formal solution of Eq.~(\\ref{master_chi}) is given by $| p_\\chi (t) \\rangle = {\\rm T} \\! \\exp_{\\leftarrow} \\left( \\int_0^\\tau R_\\chi (\\bm{\\alpha} (t)) dt \\right) | p(0) \\rangle$, where ${\\rm T} \\! \\exp_{\\leftarrow}$ describes the left-time-ordered exponential.\nThen we can show that\n\\begin{equation}\ne^{S(i\\chi)} = \\langle 1 | p_\\chi (\\tau) \\rangle\n\\end{equation}\nholds, where $\\langle \\cdot | \\cdot \\rangle$ means the inner product of left and right vectors.\n\nWe write the eigenvalues of $R_\\chi$  as $\\lambda_\\chi^n$'s, where  $n=0$ corresponds to the eigenvalue with the maximum real part. \nIf $| \\chi |$  is sufficiently small, $\\lambda_\\chi^0$  is not degenerated and $R_\\chi$ is diagonalizable.  \nWe write as $\\langle \\lambda_\\chi^n |$ and $| \\lambda_\\chi^n \\rangle$ the left and right eigenvectors corresponding to $\\lambda_\\chi^n$,  which we can normalize as $\\langle \\lambda_\\chi^n | \\lambda_\\chi^m \\rangle = \\delta_{nm}$  with $\\delta_{nm}$ being  the Kronecker's delta.  \nIn particular, we write $\\langle \\lambda_\\chi^0 | =: \\langle 1_\\chi |$ and $| \\lambda_\\chi^0 \\rangle =: | p^S_\\chi \\rangle$.\nWe note that, if $\\chi = 0$,  $\\langle 1_\\chi |$ and $| p_\\chi^S \\rangle$ reduce to $\\langle 1 |$ and $| p^S \\rangle$, respectively.   \n\n\\subsection{Decomposition of the Entropy Production}\n\nIt is known that $\\lambda_\\chi^0 (\\bm \\alpha)$ is the cumulant generating function of $\\sigma$ in the steady distribution with parameter $\\bm \\alpha$.\nMore precisely, $\\lambda_\\chi^0 (\\bm \\alpha)$ satisfies\n\\begin{equation}\n\\lambda_\\chi^0 (\\bm \\alpha) = \\lim_{\\tau \\to +\\infty} \\frac{S(i \\chi; \\bm{\\alpha}; \\tau)}{\\tau},\n\\label{hk_average}\n\\end{equation}\nwhere $S(i \\chi; \\bm{\\alpha}, \\tau)$ is the cumulant generating function of $\\sigma$ from $0$ to $\\tau$ with $\\bm \\alpha$ being fixed.\n\n\nWe then decompose the cumulant generating function into two parts:\n\\begin{equation}\nS(i\\chi) = S_{\\rm hk} (i \\chi) + S_{\\rm ex} (i \\chi),\n\\end{equation}\nwhere $S_{\\rm hk} (i\\chi)$ is the house-keeping part defined as\n\\begin{equation}\nS_{\\rm hk} (i\\chi) : = \\int_0^\\tau \\lambda_\\chi^0 (\\bm{\\alpha} (t)) dt, \n\\end{equation}\nand $S_{\\rm ex} (i\\chi)$ is the excess part   defined as $S_{\\rm ex}(i\\chi) := S (i \\chi) - S_{\\rm hk} (i \\chi)$.  \nThe average of the excess entropy production is given by \n\\begin{equation}\n\\langle \\sigma \\rangle_{\\rm ex} = \\frac{\\partial S_{\\rm ex} (i \\chi)}{\\partial (i \\chi)} \\biggr|_{\\chi = 0}.\n\\label{excess_average}\n\\end{equation}\n\n\nWe note that the above decomposition is consistent with that in Refs.~\\cite{Komatsu1,Komatsu2}.  In fact, from Eqs.~(\\ref{hk_average})  and (\\ref{excess_average}), we can show \n\\begin{equation}\n\\langle \\sigma \\rangle_{\\rm ex} = \\langle \\sigma \\rangle - \\int_0^\\tau \\langle \\dot \\sigma  \\rangle_{{\\rm hk}; \\bm{\\alpha} (t)} dt,\n\\end{equation}\nwhere $\\langle \\dot \\sigma \\rangle_{{\\rm hk}; \\bm{\\alpha}} := \\partial \\lambda_\\chi^0 (\\bm \\alpha) \/ \\partial (i \\chi) |_{\\chi = 0}$ is the long-time average of the entropy production per unit time with $\\bm{\\alpha}$ being fixed.\n\n\n\n\\section{Main Results}\n\nWe now discuss the main results of this paper, which we will refer to as  Eqs.~(\\ref{berry2}) and (\\ref{average_e}).  \nFirst of all, we expand $| p_\\chi (t) \\rangle$ as \n\\begin{equation}\\\n| p_\\chi (t) \\rangle = \\sum_n c_n(t) e^{\\Lambda_\\chi^n (t)} | \\lambda_\\chi^n (\\bm{\\alpha} (t)) \\rangle,\n\\label{c_0}\n\\end{equation}\nwhere $\\Lambda_\\chi^n (t) := \\int_0^t \\lambda_\\chi^n (\\bm{\\alpha}(t')) dt'$.  We can show that $\\dot{c}_0 = - \\sum_n c_n \\langle 1_\\chi | \\dot{\\lambda}_\\chi^n \\rangle e^{\\Lambda_\\chi^n - \\Lambda_\\chi^0}$ and  $\\langle 1_\\chi | \\dot \\lambda_\\chi^n \\rangle = \\langle 1_\\chi | \\dot R_\\chi | \\lambda_\\chi^n \\rangle \/ (\\lambda^n_\\chi - \\lambda_\\chi^0)$ hold.  Therefore, if the speed of the change of the external parameters is much smaller than the relaxation speed of the system, we obtain\n\\begin{equation}\n\\dot c_0 (t) \\simeq - c_0 (t) \\langle 1_\\chi (\\bm{\\alpha} (t))  | \\dot p_\\chi^S (\\bm{\\alpha} (t)) \\rangle.\n\\label{equation}\n\\end{equation}\nHere, we have  used that the real part of $\\Lambda^n_\\chi - \\Lambda_\\chi^0$ is negative for all $n \\neq 0$.  \nWe note that this result is similar (but not equivalent) to the adiabatic theorem in quantum mechanics.  \n\n\n\n\nAssume that we quasi-statically change parameter $\\bm \\alpha$  between time $0$ and $\\tau$ along a curve $C$ in the parameter space.   The solution of Eq.~(\\ref{equation}) is  given by\n\\begin{equation}\n\\begin{split}\nc_0 (\\tau) &=  c_0 (0) e^{- \\int_0^\\tau dt  \\langle 1_\\chi (\\bm{\\alpha} (t)) | \\dot p_\\chi^S (\\bm{\\alpha} (t)) \\rangle } \\\\\n&= c_0 (0) e^{- \\int_C \\langle 1_\\chi | d | p_\\chi^S \\rangle },\n\\end{split}\n\\label{berry1}\n\\end{equation}\nwhere  ``$d$'' on the right-hand side (RHS) means the total differential in terms of $\\bm{\\alpha}$ such that $d|p^S_\\chi \\rangle := d\\bm{\\alpha} \\cdot \\frac{\\partial}{\\partial \\bm{\\alpha}}|p^S_\\chi \\rangle$.\nLet the initial distribution be the steady distribution $| p (0) \\rangle = | p^S (\\bm \\alpha (0)) \\rangle$, which leads to $c_0 (0) = \\langle 1_\\chi (\\bm{\\alpha} (0)) | p^S(\\bm{\\alpha}(0)) \\rangle$. We then obtain the excess part of the cumulant generating function as\n\\begin{equation}\n\\begin{split}\nS_{\\rm ex} (i\\chi) =   &\\int_C \\langle 1_\\chi | d | p_\\chi^S \\rangle \\\\\n&+ \\ln \\langle 1_\\chi (\\bm{\\alpha}(0) )| p^S(\\bm{\\alpha}(0)) \\rangle + \\ln \\langle 1 | p_\\chi^S (\\bm{\\alpha}(\\tau)) \\rangle,\n\\end{split}\n\\label{berry2}\n\\end{equation}\nwhere the RHS  is geometrical and analogous to the Berry phase in quantum mechanics~\\cite{Berry}; it only depends on trajectory $C$ in the parameter space.   More precisely, the  RHS of (\\ref{berry2}) is analogous to the non-cyclic Berry phase~\\cite{Ohkubo2,Samuel}.\nWe note that $\\Lambda_\\chi^n (\\tau)$ is analogous to the dynamical phase.   In this analogy, $| p_\\chi^S \\rangle$ and $R_\\chi$ respectively correspond to a state vector and  a Hamiltonian.\nEquality~(\\ref{berry2}) is our first main result.\n\n\nIn terminologies of the Berry phase,  $\\langle 1_\\chi | d |  p_\\chi^S \\rangle$ corresponds to a vector potential or a gauge field whose base space  is the parameter space.  \nThe second and the third terms of the RHS of (\\ref{berry2}) confirms the gauge invariance of $S_{\\rm ex}(i \\chi)$ as is the case for quantum mechanics~\\cite{Samuel}, where the gauge transformation corresponds to the transformation of the left and right eigenvectors of $R_\\chi (\\bm{\\alpha})$ as $\\langle 1_\\chi (\\bm{\\alpha}) |  \\mapsto \\langle 1_\\chi (\\bm{\\alpha}) |  e^{ - \\theta (\\bm{\\alpha})}$ and $| p_\\chi^S (\\bm{\\alpha}) \\rangle \\mapsto e^{\\theta (\\bm{\\alpha})}| p^S (\\bm{\\alpha}) \\rangle$ with $\\theta (\\bm {\\alpha})$ being a scalar.  \nWe note that several formulae that are similar to Eq.~(\\ref{berry2}) have been obtained for different setups~\\cite{Sinitsyn1,Sinitsyn2,Sinitsyn3,Ohkubo1,Ren,Ohkubo2}.\n\n\nBy differentiating Eq.~(\\ref{berry2}) in terms of $i \\chi$, we obtain a simple expression of the average of the excess entropy production:\n\\begin{equation}\n \\int_C \\langle 1' | d | p^S \\rangle + \\langle \\sigma \\rangle_{\\rm ex} = 0,\n\\label{average_e}\n\\end{equation}\nwhere $\\langle 1' | := \\partial  \\langle 1_\\chi | \/  \\partial (i\\chi) |_{\\chi = 0}$.  \nEquality~(\\ref{average_e}) is the second main result, which is the full order expression of the average of the excess entropy production.  On the contrary to Eq.~(\\ref{Clausius1}), the first term of the LHS of (\\ref{average_e}) is not given by the difference of a scalar potential $S_{\\rm SST}$, but by a geometrical  quantity. We also refer to $\\langle 1' | d | p^S \\rangle$ as a vector potential.\n\nWe can explicitly calculate $\\langle 1' |$.  By differentiate the both-hand sides of $\\langle 1_\\chi | R_\\chi = \\lambda_\\chi^0 R_\\chi$ in terms of $i\\chi$, we have $\\langle 1' | = - \\langle 1 | \\partial R_\\chi \/ \\partial (i\\chi) |_{\\chi = 0} R^\\dagger + k \\langle 1 |$,\nwhere $R^\\dagger$ is the Moore-Penrose pseudo-inverse of $R$ and $k$ is an unimportant constant. Therefore, we obtain\n\\begin{equation}\n\\langle \\sigma \\rangle_{\\rm ex} = \\int_C \\sum_{\\nu x y z} \\sigma_{xy}^\\nu R_{xy}^\\nu R_{yz}^\\dagger dp_z^S.\n\\end{equation}\nSome similar formulae for particle currents have been obtained in Refs.~\\cite{Parrondo,Jarzynski1}.\n\n\nWe next consider the condition for the existence of thermodynamic potential $S_{\\rm SST}$ that satisfies Eq.~(\\ref{Clausius1}). For simplicity, we assume that the parameter space is simply-connected, i.e., there is no ``hole'' or singularity.   The necessary and sufficient condition for the existence of $S_{\\rm SST}$ is that the integral in the first term of the LHS of (\\ref{average_e}) is always determined only by the initial and final points of $C$; or equivalently, $\\oint_C \\langle 1' | d | p^S \\rangle = 0$ holds for every closed curve $C$.  On the other hand, the Stokes theorem states that $\\oint_C \\langle 1' | d | p^S \\rangle  = \\int_S d\\left( \\langle 1' | d | p^S \\rangle \\right)$ holds,\nwhere $S$ is a surface whose boundary is $C$, and ``$d$'' means the exterior derivative.  By using the wedge product ``$\\wedge$,'' we have $d\\left( \\langle 1' | d | p^S \\rangle \\right) = d\\langle 1' | \\wedge d | p^S \\rangle := \\sum_x  d 1'_x \\wedge d  p^S_x = \\sum_{xkl} \\frac{\\partial  1'_x}{\\partial \\alpha_k} \\frac{\\partial p^S_x}{\\partial \\alpha_l} d\\alpha_k \\wedge d\\alpha_l$,  where $1'_x$ means the $x$-component of vector $\\langle 1' |$, and $\\alpha_k$ is the $k$-component of $\\bm{\\alpha}$.   Therefore the necessary and sufficient condition is that \n\\begin{equation}\nd\\langle 1' | \\wedge d | p^S \\rangle = 0\n\\label{no_curvature}\n\\end{equation}\nholds in every point of the parameter space.  Equation~(\\ref{no_curvature}) is equivalent to \n\\begin{equation}\n\\sum_x \\left( \\frac{\\partial 1' _x}{\\partial \\alpha_k} \\frac{\\partial  p^S_x}{\\partial \\alpha_l} - \\frac{\\partial  1'_x}{\\partial \\alpha_l} \\frac{\\partial p^S_x}{\\partial \\alpha_k}  \\right) = 0\n\\end{equation}\nfor all $(k, l)$. In terminology of the gauge theory,   $d\\langle 1' | \\wedge d | p^S \\rangle$ corresponds to the strength of the gauge field or the curvature.  For the case of the $U(1)$-gauge theory, the curvature is the magnetic field.\n\n\nIn  equilibrium thermodynamics, Eq.~(\\ref{no_curvature}) holds due to the Maxwell relation, and $\\langle 1' | d |  p^S \\rangle$ becomes the total differential of the Shannon entropy as we will see in the next section.  On the other hand, Eq.~(\\ref{no_curvature}) does not hold for transitions between NESSs in general.  In this sense, vector potential  $\\langle 1' | d |  p^S \\rangle$ plays a fundamental role  instead of  the scalar thermodynamic potential (i.e., the Shannon entropy) in SST.  \n\n\n\n\n\n\n\\section{Special Cases}\n\nIn this section, we discuss two special cases, in which the first term of  the LHS of~(\\ref{average_e}) reduces to the total differential of a scalar thermodynamic potential.\n\n\\subsection{Equilibrium Thermodynamics}\n\nIn general, we can explicitly show that Eq.~(\\ref{average_e}) reduces to the equilibrium Clausius equality if the detailed balance is satisfied.   Let $E_x$ be the energy of state $x$. The transition rate is given by $R_{xy} = e^{\\beta (E_y - W_{xy})}$ with $W_{xy} = W_{y x}$,  the steady distribution by $p_x^S = e^{-\\beta E_x} \/ Z$ with $Z$ being the partition function, and the entropy production in a bath by  $\\sigma_{xy}  = -\\beta (E_x - E_y) = -\\beta Q_{xy}$.  In the quasi-static limit, the system is in contact with a single heat bath with inverse temperature $\\beta$ at each time, while $\\beta$ can be time-dependent.  We then obtain \n\\begin{equation}\n \\langle 1' | d | p^S \\rangle = \\sum_x \\beta E_x dp^S_x = d \\left(- \\sum_x p_x^S \\ln p_x^S \\right),\n\\end{equation}\nwhich means that $\\langle 1' | d | p^S \\rangle $ is the total differential of the Shannon entropy.   \n\n\\subsection{KNST's Extended Clausius Equality}\n\nWe now show that Eq.~(\\ref{average_e}) reduces to the KNST's  extended Clausius equality~\\cite{Komatsu1,Komatsu2} in the lowest order  of nonequilibriumness.  Here we assume that, for every $(x,y)$, there exists at most single $\\nu$ that satisfies $R_{xy}^\\nu \\neq 0$, so that we can remove index $\\nu$. This is the same assumption as in Refs.~\\cite{Komatsu1,Komatsu2}.\nMoreover, we formally introduce the time-reversal of states;  the time-reversal of state $x$, denoted as $x^\\ast$, is assigned in the phase space.  Since we are considering stochastic jump processes that do not have any momentum term usually, we just interpret the correspondence $x \\mapsto x^\\ast$ as a formal mathematical map.   \nCorrespondingly, we should replace $\\ln (R^\\nu_{xy} \/ R^\\nu_{y x})$ in Eq.~(\\ref{entropy}) by $\\ln (R^\\nu_{xy} \/ R^\\nu_{y^\\ast x^\\ast})$.  Only with this replacement, all of the foregoing arguments remain unchanged in the presence of the time-reversal. We also assume that, in thermal equilibrium, $p_{x}^S = p_{x^\\ast}^S$ holds.\nWe define\n\\begin{equation}\n\\eta := \\sum_{xyz} \\ln (p_{x^\\ast}^S \/ p_{y^\\ast}^S )R_{xy}R_{yz}^\\dagger dp_z^S = \\sum_x \\left( \\ln p_{x^\\ast}^S \\right) dp_x^S\n\\end{equation}\nand  $\\tilde{R}_{xy} := R_{y^\\ast x^\\ast} p_{x^\\ast}^S \/ p_{y^\\ast}^S$.\nHere, $\\tilde R$ is the  adjoint of  $R$ for the cases of $x = x^\\ast$~\\cite{Hatano,Esposito2}.  We note that $\\sum_x \\tilde{R}_{xy} = 0$ holds for every $y$.  Since $R = \\tilde R$ holds if the detailed balance is satisfied, we characterize the nonequilibriumness of the dynamics by $\\varepsilon := \\max_{xy} | (\\tilde{R}_{xy} - R_{xy})\/R_{xy} |$.\nWe then obtain \n\\begin{equation}\n\\begin{split}\n\\langle 1' | d | p^S \\rangle +\\eta &= \\sum_{xyz} \\ln (\\tilde{R}_{xy} \/ R_{xy}) R_{xy} R_{yz}^\\dagger dp_z^S \\\\\n &= \\sum_{xyz}  (\\tilde{R}_{xy} - R_{xy})  R_{yz}^\\dagger dp_z^S + O(\\varepsilon^2 \\Delta) \\\\\n&= O(\\varepsilon^2 \\Delta),\n \\end{split}\n \\label{eta1}\n \\end{equation}\nwhere $\\Delta := \\max_x | dp_x^S |$ characterizes the amount of the infinitesimal change of the steady distribution.  \nOn the other hand, \n\\begin{equation}\n\\eta = d \\left( \\sum_x p_x^S \\ln \\sqrt{p_x^S p_{x^\\ast}^S} \\right) +  O(\\varepsilon^2 \\Delta)\n\\label{eta2}\n\\end{equation}\nholds~\\cite{Komatsu2}.  From Eqs.~(\\ref{eta1}) and (\\ref{eta2}),  we obtain\n\\begin{equation}\n\\langle 1' | d | p^S \\rangle =  d \\left( - \\sum_x p_x^S \\ln \\sqrt{p_x^S p_{x^\\ast}^S} \\right) +  O(\\varepsilon^2 \\Delta),\n\\end{equation}\nwhich implies the KNST's extended Clausius equality, where the first term of the RHS is the total differential of the symmetrized Shannon entropy. \nWe note that, if we gradually change parameter $\\bm \\alpha$ from an equilibrium distribution, then the KNST's extended Clausius equality is valid up to the order of $O(\\varepsilon^2)$ because $\\Delta = O(\\varepsilon)$ holds.\n\n\n\n\n\\section{Example} \n\nAs a simple example that illustrates the absence of a scalar thermodynamic potential, we consider a stochastic model of a quantum dot that is in contact with two baths that are labeled by  $\\nu = L$ and $R$ (see also Fig.~1 (a))~\\cite{Bagrets}.  \nThis model describes the stochastic dynamics of the number of  electrons in the dot by a classical master equation.  \n\n\nAn electron is transfered from the baths to the dot  one by one or vise-versa.   We assume that the states of the dot are $x=0$ and $1$, which respectively describe that the electron is absent and occupies the dot.  The probability distribution is described by $| p \\rangle = [p_0, p_1]^T$, and the transition rate is given by $R = \\sum_{\\nu = L, R} R^\\nu$ with\n\\begin{eqnarray}\nR^\\nu = \\left[ \n\\begin{array}{cc}\n- \\gamma_\\nu f_\\nu & \\gamma_\\nu (1 - f_\\nu)   \\\\\n\\gamma_\\nu f_\\nu & -\\gamma_\\nu (1 - f_\\nu) \\\\\n\\end{array} \n\\right],\n\\end{eqnarray}\nwhere $\\gamma_\\nu$ is the tunneling rate between the dot and bath $\\nu$,  and $f_\\nu = (e^{\\beta (E - \\mu_\\nu)}+1)^{-1}$ is the Fermi distribution function with $\\beta$ being the inverse temperature of the baths,  $\\mu_\\nu$ being the chemical potential of bath $\\nu$, and $E$ being the excitation energy of the dot.  The entropy production is given by $\\sigma_{00}^\\nu = \\sigma_{11}^\\nu = 0$ and $\\sigma_{10}^\\nu = - \\sigma_{01}^\\nu = \\sigma_\\nu$ with $\\sigma_\\nu := \\beta (\\mu_\\nu - E)$.  For simplicity, we set $\\gamma_L = \\gamma_R  =: \\gamma$.  Without loss of generality, we assume that the control parameters are  $\\sigma_L$ and $\\sigma_R$.  \n\\begin{figure}[htbp]\n \\begin{center}\n \\includegraphics[width=80mm]{Figure_dot_revised.eps}\n \\end{center}\n \\caption{(a) A schematic of the model of a quantum dot.  A single electron is transfered to\/from the two heat baths with chemical potentials $\\mu_L$ and $\\mu_R$.  (b) $\\langle \\sigma \\rangle_{\\rm ex}$ (the solid line) and $-\\Delta S$ (the dashed line) for quasi-static processes.  They are coincident with each other up to the second order of the nonequilibriumness of the final state that is denoted by $u$.} \n\\end{figure}\n\n\n\nWe can explicitly calculate the vector potential  as\n\\begin{equation}\n\\langle 1' | d | p^S \\rangle = -\\frac{1}{4} (\\sigma_L + \\sigma_R) (f_L(1-f_L) d \\sigma_L + f_R (1-f_R)d\\sigma_R),\n\\end{equation}\nand the curvature as\n\\begin{equation}\nd\\langle 1' | \\wedge d | p^S \\rangle = \\frac{1}{4} (f_L (1 - f_L) - f_R (1-f_R)) d\\sigma_L \\wedge d \\sigma_R.\n\\end{equation}\nTherefore, the curvature vanishes only if  $\\mu_L = \\mu_R$ or $2E = \\mu_L + \\mu_R$ holds.  The former case corresponds to equilibrium thermodynamics.  Since the curvature vanishes only on the two lines in the two-dimensional parameter space, any scalar potential cannot be defined on the entire parameter space.  We note that the quantities that we have calculated  here are different from those in the previous researches~\\cite{Sinitsyn1,Sinitsyn2,Sinitsyn3,Ohkubo1,Ren}.  \n\nAs a simple illustration, we consider the following situation.  The dot is initially in thermal equilibrium with $\\sigma_L = \\sigma_R = 0$. We then quasi-statically change $\\sigma_L$ from $0$ to $u$, while $\\sigma_R$ is not changed.  We calculate $\\langle \\sigma \\rangle_{\\rm ex} =  \\int_0^u \\sigma_L f_L(1-f_L) d\\sigma_L \/ 4$ for this process.  For comparison, we also calculate the difference of the Shannon entropy between the initial and final distributions of the dot, denoted as $\\Delta S$.  Figure 1 (b) shows $\\langle \\sigma \\rangle_{\\rm ex}$ (the solid line) and $-\\Delta S$ (the dashed line) versus $u$.  They are coincident with each other up to the order of $O(u^2)$, which is consistent with the extended Clausius equality discussed in Sec.~IV.~ B with $u = O(\\varepsilon) = O(\\Delta)$.\n\\\n\\section{Conclusions and Discussions}\nWe have derived the geometrical expressions of the excess entropy production for quasi-static transitions between NESSs: Eq.~(\\ref{berry2}) for $S_{\\rm ex}(i \\chi)$ and Eq.~(\\ref{average_e}) for $\\langle \\sigma \\rangle_{\\rm ex}$.  Our results imply that  the vector potentials  $\\langle 1_\\chi | d |  p_\\chi^S \\rangle$  and $\\langle 1' | d | p^S \\rangle$ play  important roles in SST.   We have also derived condition~(\\ref{no_curvature}) that a scalar thermodynamic potential exists. \n\nWe note that the arguments in Secs.~II and III are  not restricted to the case of entropy production $\\sigma_{xy}^\\nu$, but can  be formally applied to an arbitrary quantity $f_{xy}^\\nu$ that satisfies $f_{xx}^\\nu = 0$.  In fact, even if we replace $\\sigma_{xy}^\\nu$ by any $f_{xy}^\\nu$, the formal expressions of the main results in Sec.~III remain unchanged.  However, we have explicitly used the properties of $\\sigma_{xy}^\\nu$ such as Eq.~(\\ref{entropy}) in Sec.~IV.\n\nWe also note that, as is the case for the gauge theory, we can rephrase our results (\\ref{berry1}) and (\\ref{berry2}) in terms of differential geometry~\\cite{Nakahara}.  We consider a trivial vector bundle whose base manifold is parameter space $\\{ \\bm \\alpha \\}$.  The fiber is $\\mathbb C$, and $c_0 (t)$ in Eq.~(\\ref{c_0}) is an element of the fiber.   Then $\\langle 1_\\chi | d | p_\\chi^S \\rangle$ is a connection form,  and Eq.~(\\ref{equation}) describes the parallel displacement of $c_0$  with the connection along curve $C$. \n\n\nIn this paper, we have assumed that nonequilibrium dynamics is modeled by a Markovian jump process with transition rate $R$ being diagonalizable.  To generalize our results to other models of nonequilibrium dynamics is a future issue.  For example, it is worth investigating whether our result can be generalized to Langevin systems.\nMoreover, to investigate the usefulness of our results in nonequilibrium thermodynamics is also a future challenge.\n\n\n\\begin{acknowledgments}\nWe are grateful to T. S. Komatsu, N. Nakagawa, T. Nemoto, J. Ohkubo, S. Okazawa, K. Saito, S. Sasa, and H. Tasaki for valuable discussions.  This work was  supported by the Global COE Program ``The Next Generation of Physics, Spun from Universality and Emergence'' from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan,  the Grantin-Aid of MEXT (Grants No.~21540384),  and the Grant-in-Aid for Research Activity Start-up (Grants No.~11025807).\n\\end{acknowledgments}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\subsection{Pose R-CNN}\nOur base network is derived from the Mask R-CNN~\\cite{Mask-RCNN} with the VGG backbone~\\cite{SimonyanZ14}. \n\\paragraph{Mask R-CNN:} Let us first brief the Mask-RCNN. Mask R-CNN consists of two main component. The first component, called a Region Proposal Network (RPN), proposes candidate object bounding boxes (RoIs). The second component extracts features using RoIAlign layer~\\cite{Mask-RCNN} from each candidate box and performs classification, bounding-box regression, and segmentation.\nWe refer readers to~\\cite{Mask-RCNN} details of Mask-RCNN. \n\n\\junk{\nHere we present our adaptation from Mask-RCNN with VGG backbone. The input to our network is a RGB image with the size $480 \\times 640$. The Region Proposal Network takes the last feature map of VGG (i.e., $conv5\\_3$) as input and output RoIs at different sizes and shapes. We use 15 anchors in the RPN (5 scales which are $16 \\times 16$, $32\\times32$, $64 \\times 64$, $128\\times128$ and $256 \\times 256$ and 3 aspect ratios which are $2:1$, $1:1$, $1:2$). This design allows the network to detect small objects. For each output RoI of RPN, a fixed-size $7\\times7$ feature map is pooled from the $conv5\\_3$ feature map using the RoIAlign layer. This pooled feature map is used to regress box coordinate, classify, segment, and estimate 6D pose for the object inside the box.\n}\n\n\\subsection{Pose R-CNN}\nPose-CNN also consist of two main components, with an identical first component (which is RPN) as Mask R-CNN. In the second component, in parallel to predicting the class, box\ncoordinate, and segment mask, Pose R-CNN also outputs a 6D pose for each RoI.  \n\n\\paragraph{Pose representation}\n\n\\paragraph{Multi-task loss function}\n\n\\paragraph{Network architecture}\n\n\\section{Conclusion}\n\\label{sec:conl}\nIn this paper, we propose \\method{}, a deep learning approach for jointly detecting,  segmenting, and most importantly recovering 6D poses of object instances from an single RGB image. \\method{} is end-to-end trainable and can directly output estimated poses without any post-refinements. The novelty design is at a new pose head branch, which uses the Lie algebra to represent the rotation space. \\method{} compares favorably with the state-of-the-art RGB-based 6D object pose estimation methods. Furthermore, \\method{} also allows a fast inference which is around 10 fps. An interesting future work is to improve the network for handling  with rotationally symmetric objects. \n\n\n\\subsection{Training data}\n\\section{Experiments}\n\\label{sec:exp}\nWe evaluate \\method{} on two widely used datasets, i.e., the single object pose dataset LINEMOD provided by Hinterstoisser et al.~\\cite{ACCV12} and the multiple object instance pose dataset provided by Tejani et al.~\\cite{DBLP:conf\/eccv\/TejaniTKK14}.  \nWe also compare \\method{} to the state-of-the-art methods for 6D object pose estimation from RGB images~\\cite{CVPR16,BB8,SSD-6D}.\n\n\\junk{When evaluating on the dataset of Hinterstoisser et al.~\\cite{ACCV12}, we mainly target to compare our work with the recent state-of-the-art 6D pose estimation method from Brachmann et al.~\\cite{CVPR16} because that work also estimates 6D pose from a single RGB input~\\cite{CVPR16}. For reference purposes, we also mention the results of LINE2D~\\cite{LINE2D}, which is a template-based approach\\footnote{LINE2D~\\cite{LINE2D} is originally proposed for object detection. It is then extended for 6D pose estimation by~\\cite{CVPR16}.}. It is because the method~\\cite{CVPR16} has already improves over LINE2D~\\cite{LINE2D}.}\n\n\\textbf{Metric:}\nDifferent 6D pose measures have been proposed in the past. In order to evaluate the recovered poses, we use the \\textit{standard} metrics used in~\\cite{CVPR16,BB8}. To measure pose error in 2D, we project the 3D object model into the image using the groundtruth pose and the estimated pose. The estimated pose is accepted if the IoU between two project boxes is higher than 0.5. This metric is called as \\textit{2D-pose} metric. To measure the pose error in 3D, the $5cm5^\\circ$ and \\textit{$\\textrm{ADD}$} metrics is used. \nIn $5cm5^\\circ$ metric,  an estimated pose is accepted if it is within $5cm$ translational error and $5^\\circ$ angular error of the ground truth pose.\nIn $\\textrm{ADD}$ metric, an estimated pose is accepted if the average distance between transformed model point clouds by the groundtruth pose and the estimated pose is smaller than $10\\%$ of the object's diameter. We also provide the $F1$ score of the detection and segmentation results. A detection \/ segmentation is accepted if its IoU with the groundtruth box \/ segmentation mask is higher than a threshold. We report results with the widely used thresholds, i.e., $0.5$ and $0.9$~\\cite{Faster-RCNN,Mask-RCNN}.\n\\input{exp_hoistin}\n\n\\input{exp_tenaji}\n\n\\subsection{Timing}\nWhen testing on the LINEMODE dataset~\\cite{ACCV12}, Brachman {{et al.}}~\\cite{CVPR16} reports a running time around $0.45s$ per image. \\method{} is several times faster than their method, i.e., its end-to-end architecture allows the inference at around $0.1s$ per image on a Titan X GPU. SSD-6D~\\cite{SSD-6D} and BB8~\\cite{BB8} report the running time around $0.1s$ and $0.3s$ per image, respectively. \\method{} is comparable to SSD-6D in the inference speed, while it is around 3 times faster than BB8. It is worth noting that due to using post-refinement, the inference time of SSD-6D and BB8 may be increased when the input image contains multiple object instances. We note that although \\method{} is fast, its parameters are still not optimized for the speed. \n As shown in the recent study~\\cite{DBLP:journals\/corr\/HuangRSZKFFWSG016}, better trade-off between speed and accuracy may be achieved by carefully selecting parameters, e.g., varying the number of proposals after RPN, image sizes, which is beyond scope of the current work. \n\n\n\\subsection{Single object pose estimation}\n\\label{exp:hoi}\n\n\\begin{table*}[!t]\n\\vspace{0.5cm}\n   \\centering\n   \\footnotesize\n    \\begin{center}\n    \\begin{tabular}{c| c c c c c c c c c c c c c c} \n&Ape &Bvise &Cam &Can &Cat &Driller &Duck &Box &Glue &Holep &Iron &Lamp &Phone &Average\\\\  \n\\Xhline{0.75pt}\n\t&\\multicolumn{14}{c}{\\textbf{IoU 0.5}} \\\\ \nDetection &99.8 &100 &99.7 &100 &99.5 &100 &99.8 &99.5 &99.2 &99.0 &100 &99.8 &100 &99.7\\\\ \nSegmentation &99.5 &99.8 &99.7 &100 &99.1 &100 &99.4 &99.5 &99.0 &98.6 &99.2 &99.4 &99.7 &99.4\n\\\\  \\hline\n\n\t&\\multicolumn{14}{c}{\\textbf{IoU 0.9}} \\\\\nDetection \t &85.4 &91.7 &93.3 &93.6 &89.3 &87.5 &86.3 &94.2 &81.1 &93.2 &92.5 &91.3 &90.8 &90.0\\\\ \nSegmentation &80.6 &57.0 &91.4 &62.5 &52.1 &74.6 &81.2 &91.9 &73.3 &84.6 &90.3 &85.0 &84.6 &77.6\\\\ \\hline\n    \\end{tabular}\n    \\end{center}\n    \\vspace{-0.2cm}\n    \\caption{F1 score for 2D detection and segmentation of \\method{} on LINEMOD dataset~\\cite{ACCV12} for single object.} \n    \\label{tab:2D_det_seg}\n\\end{table*}\n\n\n\\begin{table*}[!t]\n   \\centering\n   \\footnotesize\n   \\begin{center}\n    \\begin{tabular}{c| c c c c c c c c c c c c c c} \n&Ape &Bvise &Cam &Can &Cat &Driller &Duck &Box &Glue &Holep &Iron &Lamp &Phone &Average\\\\  \n\\Xhline{1pt}\n\t&\\multicolumn{14}{c}{\\textbf{2D-pose metric}}\\\\ \n\\method{} &99.8 &100 &99.7 &100 &99.2 &100 &99.8 &99.0 &97.1 &98.0 &99.7 &99.8 &99.1 &99.3\\\\ \nBrachmann\\cite{CVPR16} &98.2 &97.9 &96.9 &97.9 &98.0 &98.6 &97.4 &98.4 &96.6 &95.2 &99.2 &97.1 &96.0 &97.5\\\\\nSSD-6D\\cite{SSD-6D} &- &- &- &- &- &- &- &- &- &- &- &- &- &99.4\\\\\n\n   \t\\hline\n   \t   \t   \t&\\multicolumn{14}{c}{$\\mathbf{5cm5^\\circ}$ \\textbf{metric}} \\\\\n\\method{} &57.8 &72.9 &75.6 &70.1 &70.3 &72.9 &67.1 &68.4 &64.6 &70.4 &60.7 &70.9 &69.7 &68.5\\\\ \nBrachmann\\cite{CVPR16} &34.4 &40.6 &30.5 &48.4 &34.6 &54.5 &22.0 &57.1 &23.6 &47.3 &58.7 &49.3 &26.8 &40.6\\\\\nBB8\\cite{BB8} &80.2 &81.5 &60.0 &76.8 &79.9 &69.6 &53.2 &81.3 &54.0 &73.1 &61.1 &67.5 &58.6 &69.0\\\\\n   \t\\hline   \t\n   \t   \t&\\multicolumn{14}{c}{$\\mathbf{ADD}$ \\textbf{metric}} \\\\\n\\method{} &38.8 &71.2 &52.5 &86.1 &66.2 &82.3 &32.5 &79.4 &63.7 &56.4 &65.1 &89.4 &65.0 &65.2\\\\ \nBrachmann\\cite{CVPR16}  &33.2 &64.8 &38.4 &62.9 &42.7 &61.9 &30.2 &49.9 &31.2 &52.8 &80.0 &67.0 &38.1 &50.2\\\\\nBB8\\cite{BB8} &40.4 &91.8 &55.7 &64.1 &62.6 &74.4 &44.3 &57.8 &41.2 &67.2 &84.7 &76.5 &54.0 &62.7\\\\\nSSD-6D\\cite{SSD-6D} &- &- &- &- &- &- &- &- &- &- &- &- &- &76.3\\\\\n   \t\\hline   \t\n    \\end{tabular}\n    \\end{center}\n   \\vspace{-0.2cm}\n    \\caption{Pose estimation accuracy on the LINEMOD dataset~\\cite{ACCV12} for single object.}\n    \\label{tab:pose} \n\\end{table*}\n\n\\begin{figure*}[!t]\n\\vspace{0.1cm}\n\\centering\n\\subfigure{\n       \\includegraphics[scale=0.17]{Hinter\/05color00013.jpg}\n}\n\\subfigure{\n      \n       \\includegraphics[scale=0.17]{Hinter\/05color00013_m.jpg} \n}\n\\subfigure{\n       \\includegraphics[scale=0.17]{Hinter\/05color00013_p2.jpg} \n}\n\\\\\n\\vspace{0.1cm}\n\\subfigure{\n       \\includegraphics[scale=0.17]{Hinter\/06color00050.jpg}\n}\n\\subfigure{\n      \n       \\includegraphics[scale=0.17]{Hinter\/06color00050_m.jpg} \n}\n\\subfigure{\n       \\includegraphics[scale=0.17]{Hinter\/06color00050_p2.jpg} \n}\n\\\\\n\\vspace{0.1cm}\n\\subfigure{\n       \\includegraphics[scale=0.17]{Hinter\/08color00002.jpg} \n}\n\\subfigure{\n       \\includegraphics[scale=0.17]{Hinter\/08color00002_m.jpg} \n}\n\\subfigure{\n       \\includegraphics[scale=0.17]{Hinter\/08color00002_p2.jpg} \n}\n\\caption[]{Qualitative results for single object pose estimation on the LINEMOD dataset~\\cite{ACCV12}. From left to right: (i) original images, (ii) the predicted 2D bounding boxes, classes, and segmentations, (iii) 6D poses in which the green boxes are the groundtruth poses and the red boxes are the predicted poses. Best view in color.}\n\\label{fig:singleobj}\n\\vspace{-0.3cm}\n\\end{figure*}\nIn~\\cite{ACCV12}, the authors publish LINEMOD, a RGBD dataset,  which has become a \\textit{de facto} standard benchmark for 6D pose estimation. The dataset contains poorly textured objects in a cluttered scene. We only use the RGB images to evaluate our method. The dataset contains 15 object sequences. For fair comparison to~\\cite{CVPR16,BB8,SSD-6D}, we evaluate our method on the $13$ object sequences for which the 3D models are available. \nThe images in each object sequence contain multiple objects, however, only one object is annotated with the groundtruth class label, bounding box, and 6D pose. The camera intrinsic matrix is also provided with the dataset. Using the given groundtruth 6D poses, the object models, and the camera matrix, we are able to compute the groundtruth segmentation mask for the annotated objects. \nWe follow the evaluation protocol in~\\cite{CVPR16,BB8} which uses RGB images from the object sequences for training and testing.\nFor each object sequence,\n\\red{we randomly select $30\\%$ of the images for training and validation. The remaining images serve as the test set.}   \nA complication when using this dataset for training arises because  \nnot all objects in each image are annotated, i.e., only one object is annotated per sequence, even though multiple objects are present. This is problematic for training a detection\/segmentation network such as~\\cite{Faster-RCNN,Mask-RCNN} because the training may be confused, e.g. slow or fail to converge, if an object is annotated as foreground in some images and as background other images. \nHence, we preprocess the training images as follows. For each object sequence, we use the RefineNet~\\cite{Lin:2017:RefineNet}, a state-of-the-art semantic segmentation algorithm, to train a semantic segmentation model. The trained model is applied on all training images in other sequences. The predicted masks in other sequences are then filtered out, so that the appearance of the objects without annotated information does not hinder the training. \n\n\\junk{\n\\paragraph{Evaluation protocol} \nIn the evaluation protocol of~\\cite{CVPR16}, the authors assume that for each testing image, the object sequence which the testing image belongs to is known. Hence they can select the learned features corresponding to the known object class for computing the pose. It helps to improve the accuracy. \nWe do not require the object sequence to be known in our evaluation. From the detection results which have already filtered by a global threshold, i.e. 0.9, for each object class, we keep only one detection with the highest classification score. This allows us to access the 2D detection and segmentation performance. We count a 2D detection \/ segmentation to be correct if its IoU with the groundtruth box \/ segmentation mask is higher than a threshold. We report the detection and segmentation results with IoU thresholds 0.5 and 0.9. In order to evaluate the recovered poses, we follow the metrics used in~\\cite{CVPR16}. To measure pose error in 2D, we project the 3D object model into the image using the groundtruth pose and the estimated pose. The estimated pose is accepted if the IoU between two project boxes is higher than 0.5. This metric is called as \\textit{2D pose}. To measure the pose error in 3D, the $5cm5^\\circ$ metric is used. An estimated pose is accepted if it is within $5cm$ translational error and $5^\\circ$ angular error of the ground truth pose. \\junk{The angular distance between two rotation matrices $P$ and $Q$ is computed as follows \n\\begin{equation}\n\\theta = \\arccos \\frac{tr(R)-1}{2}\n\\end{equation}\nwhere $R=PQ^T$.}\n}\n\n\\textbf{Results:}\nTable~\\ref{tab:2D_det_seg} presents the 2D detection and segmentation results of \\method{}. At an IoU 0.5, the results show that both detection and segmentation achieve nearly perfect scores for all object categories. This reconfirms the effective design of Faster R-CNN~\\cite{Faster-RCNN} and Mask R-CNN~\\cite{Mask-RCNN} for object detection and instance segmentation. \nWhen increasing the IoU to 0.9, both detection and segmentation accuracy  significantly decrease. The more decreasing is observed for the segmentation, i.e., the dropping around \\red{10\\%} and \\red{22\\%} for detection and segmentation, respectively.\n\nWe put our interest on the pose estimation results, which is the main focus of this work. \nTable~\\ref{tab:pose} presents the comparative pose estimation accuracy between \\method{} and the state-of-the-art works of Brachmann et al.~\\cite{CVPR16}, BB8~\\cite{BB8}, SSD-6D~\\cite{SSD-6D} which also use RGB images as inputs to predict the poses. Under \\textit{2D-pose} metric, \\method{} is comparable to SSD-6D, while outperforms over \\cite{CVPR16} around \\red{2\\%}.\nUnder $5cm5^\\circ$ metric, \\method{} is slightly lower than BB8, while it significantly outperforms~\\cite{CVPR16}, i.e., around \\red{28\\%}. Under $\\textrm{ADD}$ metric, \\method{} outperforms BB8 \\red{2.5\\%}.\nThe results of \\method{} are also more stable than BB8~\\cite{BB8}, e.g., under $5cm5^\\circ$ metric, the standard deviations in the accuracy of \\method{} and BB8~\\cite{BB8} are \\red{5.0} and \\red{10.6}, respectively. \nThe Table~\\ref{tab:pose} also shows that both \\method{} and BB8~\\cite{BB8} are worse than SSD-6D. However, it is worth noting that SSD-6D~\\cite{SSD-6D} does not use images from the object sequences for training. \nThe authors~\\cite{SSD-6D} perform a discrete sampling over \\textit{whole} rotation space and use the known 3D object models to generate synthetic images used for training. By this way, the training data of SSD-6D is able to cover more rotation space than~\\cite{CVPR16}, BB8~\\cite{BB8}, and~\\method{}. \nFurthermore, SSD-6D also further uses an ICP-based refinement to improve the accuracy. Different from SSD-6D, \\method{} directly outputs the pose without any post-processing. \nFigure~\\ref{fig:singleobj} shows some qualitative results of \\method{} for single object pose estimation on the LINEMOD dataset.\n\n\n\n\\subsection{Multiple object instance pose estimation}\n\\label{exp:tejani}\nIn~\\cite{DBLP:conf\/eccv\/TejaniTKK14}, Tejani et al. publish a dataset consisting of six object sequences in which the images in each sequence contain multiple instances of the same object with different levels of occlusion. Each object instance is provided with the groundtruth class label, bounding box, and 6D pose. Using the given groundtruth 6D poses, the object models, and the known camera matrix, we are able to compute the groundtruth segmentation masks for object instances. \nWe use the RGB images provided by the dataset for training and testing. \\red{We randomly split $30\\%$ images in each sequence for training and validation. The remaining images serve as the test set.}\n\n\\junk{\n\\paragraph{Evaluation protocol} \nBecause there are multiple instances of an object in an image, we do not constraint the maximum number of the detection results. We accept all detections whose the classification scores are higher than a threshold. The threshold is fixed to $0.9$ for all sequences. A 2D detection \/ segmentation is accepted if its IoU with the groundtruth box \/ segmentation mask is higher than a threshold. We report the results with IoU thresholds $0.5$ and $0.9$. \nIn order to evaluate the recovered poses, we use {2D pose} and $5cm5^\\circ$ metrics as mentioned in the first dataset. \nBecause there are multiple object instances in each image, follow the original work~\\cite{DBLP:conf\/eccv\/TejaniTKK14}, we report results with $F1$ score, which is the harmonic mean of precision and recall.\n}\n\n\\begin{table*}[!t]\n  \n   \\footnotesize\n   \\begin{center}\n    \\begin{tabular}{c| c c c c c c c} \n&Camera &Coffee &Joystick &Juice &Milk &Shampoo &Average\\\\  \n\\Xhline{0.75pt}\n\t&\\multicolumn{7}{c}{\\textbf{IoU 0.5}} \\\\ \nDetection &99.8 &100 &99.8 &99.2 &99.7 &99.5 &99.6 \\\\ \nSegmentation &99.8 &99.7 &99.6 &99.0 &99.3 &99.5 &99.4 \\\\ \\hline\n\t&\\multicolumn{7}{c}{\\textbf{IoU 0.9}} \\\\ \nDetection     &88.3 &97.2 &97.7 &89.4 &83.5 &82.4 &89.7 \\\\ \nSegmentation  &81.7 &95.0 &95.5 &86.9 &77.8 &74.7 &85.2 \\\\ \\hline\n    \\end{tabular}\n    \\end{center}\n  \\vspace{-0.2cm}\n    \\caption{F1 score for 2D detection and segmentation of \\method{} on the dataset of Tejani et al.~\\cite{DBLP:conf\/eccv\/TejaniTKK14} for multiple object instances.} \n\n   \\label{tab:2D_det_seg_tejani}\n\\end{table*}\n\\begin{table*}[!t]\n\\vspace{-0.2cm}\n   \\centering\n   \\footnotesize\n   \\begin{center}\n    \\begin{tabular}{c| c c c c c c c} \n&Camera &Coffee &Joystick &Juice &Milk &Shampoo &Average\\\\  \\Xhline{1pt}\n\t&\\multicolumn{7}{c}{\\textbf{2D pose metric}}\\\\ \n\\method{}\t    \t&99.2 &100 &99.6 &98.4 &99.5 &99.1 &99.3 \\\\ \nSSD-6D\\cite{SSD-6D} &-    &-   &-    &-    &-    &-    &98.8 \\\\\n   \t\\hline\n   \t&\\multicolumn{7}{c}{$\\mathbf{5cm5^\\circ}$ \\textbf{metric}}\\\\ \n\\method{}   &76.5 &18.7 &60.2 &85.6 &73.5 &72.4 &64.5 \\\\ \\hline\n   \t&\\multicolumn{7}{c}{$\\mathbf{ADD}$ \\textbf{metric}}\\\\ \n\\method{}   &80.4 &35.4 &27.5 &81.2 &71.6 &75.8 &62.0 \\\\ \n\n   \t\\hline \t\n    \\end{tabular}\n    \\end{center}\n    \\vspace{-0.2cm}\n    \\caption{Pose estimation accuracy on the dataset of Tejani et al.~\\cite{DBLP:conf\/eccv\/TejaniTKK14} for multiple object instances.} \n    \\label{tab:pose_tejani}\n\\end{table*}\n\\begin{figure*}[!t]\n\\centering\n\\vspace{-0.3cm}\n\\subfigure{\n       \\includegraphics[scale=0.17]{tejani\/01color00011.jpg}\n}\n\\subfigure{\n       \\includegraphics[scale=0.17]{tejani\/01color00011_m.jpg} \n}\n\\subfigure{\n      \n       \\includegraphics[scale=0.17]{tejani\/01color00011_p2.jpg} \n\n}\\\\\n\\vspace{-0.2cm}\n\\subfigure{\n       \\includegraphics[scale=0.17]{tejani\/04color00291.jpg}\n}\n\\subfigure{\n      \n       \\includegraphics[scale=0.17]{tejani\/04color00291_m.jpg} \n}\n\\subfigure{\n       \\includegraphics[scale=0.17]{tejani\/04color00291_p2.jpg} \n}\n\\\\\n\\vspace{-0.2cm}\n\\subfigure{\n       \\includegraphics[scale=0.17]{tejani\/06color00002.jpg}\n}\n\\subfigure{\n\n        \\includegraphics[scale=0.17]{tejani\/06color00002_m.jpg} \n}\n\\subfigure{\n       \\includegraphics[scale=0.17]{tejani\/06color00002_p2.jpg} \n}\n\\caption[]{Qualitative results for pose estimation on the multiple object instance dataset of Tejani et al.~\\cite{DBLP:conf\/eccv\/TejaniTKK14}. From left to right: (i) the original images, (ii) the predicted 2D bounding boxes, classes, and segmentations (different instances are shown with different colors), (iii) 6D poses in which the green boxes are the groundtruth poses and the red boxes are the predicted poses. Best view in color.}\n\\label{fig:singleobj_tenaji}\n\\vspace{-0.3cm}\n\\end{figure*}\n\n\n\\begin{figure}[!t]\n\\centering\n\\subfigure{\n      \n       \\includegraphics[scale=0.15]{tejani\/02color00083_m.jpg}\n}\n\\subfigure{\n       \\includegraphics[scale=0.15]{tejani\/02color00083_p2.jpg} \n}\n\\caption[]{One fail case on the Coffee sequence in the dataset of Tejani et al.~\\cite{DBLP:conf\/eccv\/TejaniTKK14}. Although the network can correctly predict the bounding box, the estimated 6D is incorrect due to the rotational symmetry of the object. We can see that the incorrect rotations are mainly in the Yaw axis.}\n\\label{fig:fail_tejani}\n\\end{figure}\n\\textbf{Results:} \nThe 2D detection and segmentation results are presented in Table~\\ref{tab:2D_det_seg_tejani}. \nAt an IoU $0.5$, \\method{} achieves nearly perfect scores. When increasing the IoU to $0.9$, the average accuracy decreases around \\red{10\\%} and \\red{14\\%} for detection and segmentation, respectively. We found that the Shampoo category has the most drop. It is caused by its flat shape, e.g., at some certain poses, the projected 2D images only contain a small side edge of the object, resulting the drop of scores at high IoU. \n\n\nThe accuracy of the recovered poses is reported in Table~\\ref{tab:pose_tejani}. Under \\textit{2D-pose} metric, both \\method{} and SSD-6D~\\cite{SSD-6D} achieve mostly perfect scores.  \n Under $5cm5^\\circ$ and $\\textrm{ADD}$ metrics, on the average, \\method{} achieves \\red{64.5\\%} and \\red{62.0\\%} accuracy, respectively. We note that no previous works reports the standard $5cm5^\\circ$ and $\\textrm{ADD}$ metrics on this dataset using only RGB images.\n\nFigure~\\ref{fig:singleobj_tenaji} shows some qualitative results for the predicted bounding boxes, classes, segmentations, and 6D poses for multiple object instances. \n\n\nWe also found that \\method{} is not very robust to nearly rotationally symmetric objects. That is because if an object is rotational symmetry in $z$ axis, any rotation of the 3D object in the Yaw angle will produce the same object appearance in the 2D image. This makes confusion for the network to predict the rotation using only appearance information. \nAs shown in Table~\\ref{tab:pose_tejani}, the Coffee sequence, which is nearly rotational symmetry in both shape and texture, has very low $5cm5^\\circ$ score. Figure~\\ref{fig:fail_tejani} shows a failure case for this object sequence. \n\n\\subsection{Inference}\n\\section{Introduction}\nDetecting objects and their 6D poses (3D location and orientation) is an important task for many robotic applications including object manipulations (e.g., pick and place), parts assembly, to name a few. In clutter environments, objects must be first detected before their poses can be estimated. Accurate object segmentation is also important, especially when objects are occluded. Our goal is to develop a deep learning framework that jointly detects, segments,  and most importantly recovers 6D poses of object instances from a single RGB image. While the first two tasks are getting more and more mature thanks to the power of deep learning, the 6D pose estimation problem remains a challenging problem.\n\n\n\n\n\n\nTraditional object pose estimation methods are mainly based on the matching of hand-crafted local features (e.g. SIFT~\\cite{SIFT_Lowe}). However, these local feature matching based approaches are only suitable for richly textured objects. For poorly textured objects, template-based matching~\\cite{ACCV12,DBLP:conf\/eccv\/TejaniTKK14,DBLP:conf\/iccv\/Rios-CabreraT13} or dense feature learning approaches are usually used~\\cite{ECCV14,ICCV15,CVPR17,CVPR17_2}. However, the template-based methods are usually sensitive to illuminations and occlusions.  \nThe feature learning approaches~\\cite{ECCV14,ICCV15,CVPR17,CVPR17_2} have shown better performances against template-based methods, but they suffer from several disadvantages, i.e., \nthey require a time-consuming multi-stage processing for learning dense features, generating coarse pose hypotheses, and refining the coarse poses. \n\n\n\nWith the rising of deep learning, especially Convolutional Neural Networks (CNN), the object classification~\\cite{krizhevsky2012imagenet}, object detection~\\cite{Fast-RCNN,Faster-RCNN}, and recently object instance segmentation~\\cite{Mask-RCNN,affnet} tasks have achieved remarkable  improvements. However, the application of CNN to 6D object pose estimation problem is still limited. \nRecently, there are few works~\\cite{SSD-6D,BB8,posecnn} which apply deep learning for 6D object pose estimation. These methods, however, are not end-to-end or only estimate a coarse object poses. They require further post-refinements to improve the accuracy, which linearly increases the running time, w.r.t. the number of detected objects. \n\n\\junk{\nThe key idea for recent large improvements in object detection and object instance segmentation problems~\\cite{Faster-RCNN,Mask-RCNN} is the development of a CNN architecture, i.e., Region Proposal Network (RPN)~\\cite{Faster-RCNN}. RPN is actually a CNN which is trained to produce multiple object (bounding boxes) proposals in an image at different shapes and sizes.  \nFaster R-CNN~\\cite{Faster-RCNN} further refines and classify bounding boxes produced by RPN using additional fully connected layers.\nThe recent work Mask R-CNN~\\cite{Mask-RCNN} goes beyond Faster-RCNN, i.e., it performs  binary segmentation in each bounding box produced by RPN.\n}\n\nRecently, Mask R-CNN~\\cite{Mask-RCNN} achieves state-of-the-art results in the instance segmentation problem. The key component of Mask R-CNN is a Region Proposal Network (RPN)~\\cite{Faster-RCNN}, which predicts multiple object (bounding boxes) proposals in an image at different shapes and sizes. \nMask R-CNN further segments instances inside bounding boxes produced from RPN by using additional convolutional layers. \n\nInspired by the impressive results of Mask R-CNN for object instance segmentation, \nwe are motivated to find the answer for the question that, \\textit{can we exploit the merits of RPN to not only segment but also recover the poses of object instances in a single RGB image, in an end-to-end fashion?}\nTo this end, we design a network which simultaneously detects\\footnote{The detection means the prediction of both bounding boxes and class labels.}, segments, and also recovers 6D poses of object instances from a single RGB image. In particular, we propose a \\method{} network, which goes beyond Mask R-CNN by adding a novel branch for regressing the poses for the object instances inside bounding boxes produced by RPN. The proposed pose branch is parallel with the detection and segmentation branches. \n\nOur main contribution is a novel object pose regressor, where the network regresses translation and rotation parameters seperately. Cares must be taken when regressing 3D rotation matrices as not all $3\\times3$ matrices are valid rotation matrices. To work around, we resort to the Lie algebra associated with the $SO(3)$ Lie group for our 3D rotation representation. Compared to other representations such as quaternion or orthonormal matrix, Lie algebra is an optimal choice as it is less parammeters and unconstrained, thus making the training process easier. Although the Lie algebra representation has been widely used in geometry-based robot vision problems~\\cite{DBLP:conf\/iros\/Agrawal06,DBLP:conf\/icra\/RosGSPL13}, to our best knowledge, this is the first work which successfully uses the Lie algebra in a CNN for regressing 6D object poses.\n\n\nDifferent from recent deep learning-based 6D pose estimation methods which are not end-to-end trainable~\\cite{BB8} or only predict a rough pose followed by a pose refinement step~\\cite{SSD-6D,BB8,posecnn}, the proposed \\method{} is a single deep learning architecture. It takes a RGB image as input and directly outputs 6D object poses without any pose post-refinements. Additionally, our system also returns segmentation masks of object instances. The experimental results show that \\method{} is competitive or outperforms the state-of-the-art methods on standard datasets. Furthermore, \\method{} is simple and elegant, allows the inference at the speed of $\\textrm{10 fps}$, which is several times faster than many existing methods.\n\nThe remainder of this paper is organized as follows. Section~\\ref{sec:related} presents related works. Section~\\ref{sec:method} details the proposed \\method{}. Section~\\ref{sec:exp} evaluates and compares \\method{} to the state-of-the-art 6D object pose estimation methods. Section~\\ref{sec:conl} concludes the paper. \n\\section{Method}\n\\label{sec:method}\nOur goal is to simultaneously detect, segment, and estimate the 6D poses of object instances in the input image. \nMask R-CNN performs well for the first two tasks, except the 6D pose estimation. In order to achieve a complete system, we propose a novel branch which takes RoIs from RPN as inputs and outputs the 6D poses of the instances inside the RoIs.\nAlthough the concept is simple, the additional 6D pose is distinct from the other branches. It requires an effective way to represent the 6D pose and a careful design of the loss function. \nIn this paper, we represent a pose by a 4-dimensional vector, in which the first three elements represent the Lie algebra associated with the rotation matrix of the pose; the last element represents the $z$ component of the translation vector of the pose. Given the predicted $z$ component and the predicted bounding box from the box regression branch, we use projective property to recover the full translation vector. The architecture of \\method{} is shown in Figure~\\ref{fig:overview}.\n\n\\subsection{\\method{}}\n\n\\begin{figure*}[!t] \n\\centering   \t\t\t\t\n\\includegraphics[scale=0.4]{figure-1_edit.pdf}    \n    \\caption{An overview of \\method{} framework. {From left to right:} The input to \\method{} is a RGB image. A deep CNN backbone (i.e., VGG) is used to extract features over the whole image. The RPN is attached on the last convolutional layer of VGG (i.e., $conv5\\_3$) and outputs RoIs. For each RoI, the corresponding features from the feature map $conv5\\_3$ are extracted and pooled into a fixed size $7\\times 7$. The pooled features are used as inputs for $4$ head branches. For the box regression and classification heads, we follow Mask-RCNN~\\cite{Mask-RCNN}. The segmentation head is \\textit{adapted} from~\\cite{Mask-RCNN}, i.e., four $3\\times3$ consecutive convolutional layers (denoted as `$\\times4$') are used. The ReLu is used after each convolutional layer. A deconvolutional layer is used to upsample the feature map to $28\\times28$ which is the segmentation mask. The proposed pose head consists of four fully connected layers. The ReLu is used after each of the first three fully connected layers. The last fully connected layer outputs four numbers which represent for the pose. As shown on the right image, the network outputs the detected instances (with classes, i.e., Shampoo), the predicted segmentation masks (different object instances are shown with different colors) and the predicted 6D poses for detected instances (shown with 3D boxes).\n}\n    \\label{fig:overview} \n\\end{figure*}\n\nLet us first briefly recap Mask R-CNN~\\cite{Mask-RCNN}. Mask R-CNN consists of two main components. The first component is a RPN~\\cite{Faster-RCNN} which produces candidate RoIs. The second component extracts features from each candidate RoI using RoIAlign layer~\\cite{Mask-RCNN} and performs the classification, the bounding box regression, and the segmentation. We refer readers to~\\cite{Mask-RCNN} for details of Mask-RCNN. \n\n\\method{} also consists of two main components. The first component is also a RPN. In the second component, in \\textit{parallel} to the existing branches of Mask R-CNN, \\method{} also outputs a 6D pose for the objects inside RoIs. \n\n\\paragraph{Pose representation}\nAn important task when designing the pose branch is the representation space of the output poses. We learn the translation in the Euclidean space. In stead of predicting full translation vector, our network is trained to regress the $z$ component only. The reason is that when projecting a 3D object model into a 2D image, two translation vectors with the same $z$ and the different $x$ and $y$ components may produce two objects which have very similar appearance and scale in 2D image (at different positions in the image -- in the extreme case of parallel projection, there is no difference at all). This causes difficulty for the network to predict the $x$ and $y$ components by using only appearance information as input. However, the object size and the scale of its textures in a 2D image provide strong cues about the $z$-coordinate. \nThis projective property allows the network to learn the $z$ component of the translation using the 2D object appearance only. Given the $z$ component, it is used together with predicted bounding box, which is outputted by the bounding box regression branch, to fully recover the translation. \nThe detail of this recovering process is presented in the following sections.  \n\nRepresenting the rotation part of the pose is more complicated than the translation part. Euler angles are intuitive due to the explicit meaning of parameters.\nHowever, the Euler angles wrap around at 2$\\pi$ radians, i.e., having multiple values representing the same angle. This causes difficulty in learning a uni-modal scalar regression task.  \nFurthermore, the Euler angles-based representation suffers from the well-studied problem of gimbal lock~\\cite{pose}. Another alternative, the use of $3\\times3$ orthonormal matrix is over-parametrised, and creates the problem of enforcing the orthogonality constraint when training the network through back-propagation. A final common representation is the unit length 4-dimensional quaternion. \nOne of the downsides of quaternion representation is its norm should be unit. This constraint may harm the optimization~\\cite{DBLP:conf\/iccv\/KendallGC15}. \n \nIn this work, we use the Lie algebra $so(3)$ associated with the Lie group $SO(3)$ (which is space of 3D rotation matrices) as our rotation representation. \nThe Lie algebra $so(3)$  is known as the tangent space at the identity element of the Lie group $SO(3)$. \nWe choose the Lie algebra $so(3)$ to represent the rotation because an arbitrary element of $so(3)$ admits a skew-symmetric matrix representation parameterized by a vector in ${\\mathbb R}^3$ which is continuous and smooth.\nThis means that the network needs to regress only three scalar numbers for a rotation, without any constraints. \nTo our best knowledge, this paper is the first one which uses Lie algebra for representing rotations in training a deep network for 6D object pose estimation task. \n\nDuring training, we map the groundtruths of rotation matrices to their associated elements in $so(3)$ by the closed form Rodrigues logarithm mapping~\\cite{log-exp-map}.\nThe mapped values are used as regression targets when learning to predict the rotation. \n\nIn summary, the pose branch is trained to regress a 4-dimensional vector, in which the first three elements represent rotation part and the last element represents the $z$ component of the  translation part of the pose. \n\n\\paragraph{Multi-task loss function}\nIn order to train the network, we define a multi-task loss to jointly train the bounding\nbox class, the bounding box position, the segmentation, and the pose of the object inside the box. Formally, the loss function is defined as follows\n\\begin{equation}\n L=\\alpha_1L_{cls} + \\alpha_2L_{box} + \\alpha_3L_{mask} + \\alpha_4L_{pose} \n \\label{eq:allloss}\n\\end{equation}\n  The classification loss $L_{cls}$, the bounding box regression loss $L_{box}$, and the segmentation loss $L_{mask}$ are defined similar as~\\cite{Mask-RCNN}, which are \\textit{softmax} loss, \\textit{smooth L1} loss, and \\textit{binary cross entropy} loss, respectively. The $\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4$  coefficients are scale factors to control the important of each loss during training. \n\nThe pose branch outputs 4 numbers for each RoI, which represents the Lie algebra for the rotation and $z$ component of the translation. It is worth noting that in our design, the output of pose branch is class-agnostic,\nbut the class-specific counterpart (i.e., with a $4C$-dimensional output vector in which $C$ is the number of classes) is also applicable. \nThe pose regression loss $L_{pose}$ is defined as follows\n\\begin{equation}\nL_{pose} = \\norm{r-\\hat{r}}_p + \\beta\\norm{t_z - \\hat{t}_z}_p\n\\label{eq:poseloss}\n\\end{equation}\nwhere $r$ and $\\hat{r}$ are two 3-dimensional vectors representing the regressed rotation and the groundtruth rotation, respectively; $t_z$ and $\\hat{t}_z$ are two scalars representing the regressed $z$ and the groundtruth $z$ of the translation; $p$ is a distance norm; $\\beta$ is a scale factor to control the rotation and translation regression errors.\n\n\\paragraph{Network architecture}\nFigure~\\ref{fig:overview} shows the schematic overview of \\method{}. We differentiate two parts of the network, i.e., the backbone and the head branches. The backbone is used to extract features over the whole image and is shared between head branches. There are four head branches corresponding to the four different tasks, i.e., the bounding box regression, the bounding box classification, the segmentation, and the 6D pose estimation for the object inside the box. \nFor the backbone, we follow Faster R-CNN~\\cite{Faster-RCNN} which uses VGG~\\cite{SimonyanZ14} together with a RPN attached on the last convolutional layer of VGG (i.e., $conv5\\_3$). For each output RoI of RPN, a fixed-size $7\\times7$ feature map is pooled from the $conv5\\_3$ feature map using the RoIAlign layer~\\cite{Mask-RCNN}. This pooled feature map is used as input for head branches. For the network heads of the bounding box regression and classification, we closely follow the Mask R-CNN~\\cite{Mask-RCNN}. \nFor the segmentation head, we adapt from Mask R-CNN. In our design, four $3\\times3$ consecutive convolutional layers (denoted as `$\\times4$' in Figure~\\ref{fig:overview}) are used after the pooled feature map. The ReLu is used after each convolutional layer. A deconvolutional layer is used to upsample the feature map to $28\\times28$ which is the segmentation mask. It is worth noting that for segmentation head, we use the class-agnostic design, i.e., this branch outputs a single mask, regardless of class. We empirically found that this design reduces the model complexity and the inference time, while it is nearly effective as the class-specific design. This observation is consistent with the observation in Mask R-CNN~\\cite{Mask-RCNN}. \n\nIn order to adapt the shared features to the specific pose estimation task, the pose head branch consists of a sequence of 4 fully connected layers in which the number of outputs are $4096 \\to 4096 \\to 384 \\to 4$.  The ReLU is used after each fully layer, except for the last layer. This is because the regressing targets (i.e., the groundtruths) contain both negative and positive values. We note that our pose head has a simple structure. More complex design may have potential improvements. \n\n\\subsection{Training and inference}\n\\paragraph{Training}\nWe implement \\method{} using Caffe deep learning library~\\cite{DBLP:conf\/mm\/JiaSDKLGGD14}. The input to our network is a RGB image with the size $480 \\times 640$. \nThe RPN outputs RoIs at different sizes and shapes. We use 5 scales and 3 aspect ratios, resulting 15 anchors in the RPN. The 5 scales are $16 \\times 16$, $32\\times32$, $64 \\times 64$, $128\\times128$ and $256 \\times 256$; the 3 aspect ratios are $2:1$, $1:1$, $1:2$. This design allows the network to detect small objects. \n\nThe $\\alpha_1, \\alpha_2, \\alpha_3$, and $\\alpha_4$ in (\\ref{eq:allloss}) are empirically set to 1, 1, 2, 2, respectively. The values of $\\beta$ in (\\ref{eq:poseloss}) is empirically set to 1.5. \nAn important choice for the pose loss (\\ref{eq:poseloss}) is the regression norm $p$. Typically, deep learning models use $p=1$ or $p=2$. With the datasets used in this work, we found that $p=1$ give better results and hence is used in our experiments. \n\nWe train the network in an end-to-end manner using stochastic gradient descent with $0.9$ momentum and $0.0005$ weight decay. The network is trained on a Titan X GPU for $350k$ iterations. Each mini batch has $1$ image.\nThe learning rate is set to $0.001$ for the first $150k$ iterations and then decreased by 10 for the remaining iterations. The top $2000$ RoIs from RPN (with a ratio of 1:3 of positive to negative) are subsequently used for computing the multi-task loss. A RoI is considered positive if it has an intersection over union (IoU) with a groundtruth box of at least 0.5 and negative otherwise. The losses $L_{mask}$ and $L_{pose}$ are defined for only positive RoIs.\n\n\\paragraph{Inference}\nAt the test phase, we run a forward pass on the input image.  The top $1,000$ RoIs produced by the RPN are selected and fed into the box regression and classification branches, followed by non-maximum suppression~\\cite{Fast-RCNN}. Based on the outputs of the classification branch, we select the outputted boxes from the regression branch that have classification scores higher than a certain threshold (i.e., $0.9$) as the detection results. The segmentation branch and the pose branch are then applied on the detected boxes, which output segmentation masks and the 6D poses for the objects inside the boxes. \n\n\\paragraph{From the 4-dimensional regressed pose to the full 6D pose}\nGiven the predicted Lie algebra, i.e., the first three elements of the predicted 4-dimensional vector from pose branch, we use the exponential Rodrigues mapping~\\cite{log-exp-map} to map it to the corresponding rotation matrix.  \nIn order to recover the full translation, we rely on the predicted $z$ component ($t_z$ -- the last element of the 4-dimensional predicted vector) and the predicted bounding box coordinates to compute two missing components $t_x$ and $t_y$. We assume that the bounding box center (in 2D image) is the projected point of the 3D object center (the origin of the object coordinate system). Under this assumption, using the 3D-2D projection formulation, we compute $t_x$ and $t_y$ as follows\n\\begin{equation}\nt_x = \\frac{(u_0 - c_x)t_z}{f_x}\n\\end{equation}\n\\begin{equation}\nt_y = \\frac{(v_0 - c_x)t_z}{f_y}\n\\end{equation}\nwhere $u_0$, $v_0$ are the bounding box center in 2D image, and the matrix $[f_x, 0, c_x; 0, f_y, c_y; 0, 0, 1]$ is the known intrinsic camera calibration matrix.\n\n\\section{Related Work}\n\\label{sec:related}\nIn this section, we first review the 6D object pose estimation methods.  We then brief the main design of the recent methods which are based on RPN for object detection and segmentation.\n\n\\textbf{Classical approaches.} The topic of pose estimation has great attention in the past few years. For objects with rich of texture, sparse feature matching approaches have been shown good accuracy~\\cite{DBLP:conf\/clor\/GordonL06,SIFT_Lowe,DBLP:conf\/icra\/MartinezCS10}. Recently, researchers have put more focus on poor texture or texture-less objects. The most traditional approaches for poor texture objects are to use object templates~\\cite{ACCV12,DBLP:conf\/eccv\/TejaniTKK14,DBLP:conf\/iccv\/Rios-CabreraT13}. The most notable work belonging to this category is LINEMOD~\\cite{ACCV12} which is based on stable gradient and normal features. However, LINEMOD is designed to work with RGBD images. Furthermore, template-based approaches are sensitive to the lighting and occlusion. \n\n\\textbf{Feature learning approach.} Recent 6D pose estimation researches have relied on feature learning for dealing with insufficient texture objects~\\cite{ECCV14,ICCV15,CVPR17,CVPR17_2}. In~\\cite{ECCV14,ICCV15}, the authors show that the dense feature learning approach outperforms matching approach. The basic  design of \\cite{ECCV14,ICCV15,CVPR17,CVPR17_2} is a time-consuming multi-stage pipeline, i.e., a random forest is used for jointly learning the object categories for pixels\nand the coordinates of pixels w.r.t. object coordinate systems (known as object coordinates). A set of pose hypotheses is generated by using the outputs of the forest and the depth channel of the input image.\nAn energy function is defined on the generated pose hypotheses to select hypotheses. The selected pose hypotheses are further refined to obtain the final pose. Note that the pipelines in those works heavily depend on the depth channel. The depth information is required in both pose hypothesis generation and refinement. The work~\\cite{CVPR16} also follows a multi-stage approach as~\\cite{ECCV14,ICCV15} but is designed to work with RGB inputs.\nIn order to deal with the missing depth information, the distribution of object coordinates is approximated as a mixture model when generating pose hypotheses. \n\n\\junk{\nThe disadvantage of feature learning approaches~\\cite{ECCV14,ICCV15,CVPR16,CVPR17_2} is that the generation of pose hypotheses uses only local information, i.e., only three or four pixels are used to generate a hypothesis. As result, this may generate bad hypotheses because it does not consider a global context over the whole object. Furthermore, by requiring multiple processing steps, those approaches are are time-consuming, making them unsuitable for real-time applications. \n}\n\n\\textbf{CNN-based approach.} In recent years, CNN has been applied for 6D pose problem, firstly for camera pose~\\cite{DBLP:conf\/iccv\/KendallGC15,kendall2017posenet}, and recently for object pose~\\cite{ICCV15,CVPR17_2,SSD-6D,BB8,posecnn,yolo-6D}. \n\nIn~\\cite{DBLP:conf\/iccv\/KendallGC15,kendall2017posenet}, the authors train CNNs to directly regress 6D camera pose from a single RGB image. The camera pose estimation task is arguably easier than the object pose estimation task, because to estimate object pose, it also requires accurate detection and classification of the object, while these steps are not typically needed for camera pose. \n\nIn~\\cite{ICCV15}, the authors use a CNN in their object pose estimation system. However, \nthe CNN is only used as a probabilistic model to learn to compare the learned information (produced by a random forest) and the rendered image. The CNN outputs an energy value for selecting pose hypotheses which are further refined.\nThe work in~\\cite{CVPR17_2} improves over~\\cite{ICCV15} by using a CNN and reinforcement learning for joint selecting and refining pose hypotheses. \n\nIn SSD-6D~\\cite{SSD-6D}, the authors extend SSD detection framework~\\cite{SSD} to 3D detection and 3D rotation estimation. The authors decompose 3D rotation space into discrete viewpoints and in-plane rotations. They then treat the rotation estimation as a classification problem.\nHowever, to get the good results, it is required to manually find an appropriate sampling for the rotation space. Furthermore, the approach SSD-6D does not directly output the translation, i.e., to estimate  the translation, for each object, an offline stage is required to precomputes bounding boxes w.r.t. all possible sampled rotations. This precomputed information is used together with the estimated bounding box and rotation to estimate the 3D translation. \nIn the recent technical report~\\cite{posecnn}, the authors propose a network, dubbed PoseCNN, which jointly segments objects and estimates the rotation and the distance of segmented objects to camera. However, by relying on a semantic segmentation approach (which is a FCN~\\cite{FCN}) to localize objects, it may be difficult for PoseCNN to deal with input image which contains multiple instances of an object. \nBoth SSD-6D and PoseCNN also require further pose refinement steps to improve the accuracy. \n\nIn BB8~\\cite{BB8}, the authors propose a cascade of multiple CNNs for object pose estimation task. A segmentation network is firstly applied to the input image to localize objects. Another CNN is then used to predict 2D projections of the corners of the 3D bounding boxes around objects. The 6D pose is estimated for the correspondences between the projected 2D coordinates and the 3D ground control points of \nbounding box corners using a PnP algorithm. Finally, a CNN per object is trained to refine the pose. By using multiple separated CNNs, BB8 is not end-to-end and is time-consuming for the inference.\nSimilar to~\\cite{BB8}, in the recent technical report~\\cite{yolo-6D}, the authors extend YOLO object detection network~\\cite{yolo9000} to predict 2D projections of the corners of the 3D bounding boxes around objects. Given the projected 2D coordinates and the 3D ground control points of bounding box corners, a PnP algorithm is further used to estimate the 6D object pose.  \n\n\\textbf{RPN-based detection and segmentation.} One of key components in the recent successful object detection method  Faster R-CNN~\\cite{Faster-RCNN} and instance segmentation method Mask R-CNN~\\cite{Mask-RCNN} is the Region Proposal Network --- RPN. The core idea of RPN is to dense sampling the whole input image by many overlap bounding boxes at different shapes and sizes. The network is trained to produce multiple object proposals (also known as Region of Interest --- RoI). This design of RPN allows to smoothly search over different scales of feature maps. \n\\junk{\nFor each RoI, a fixed-size small feature map (e.g., $7\\times7$) is pooled from the image feature map using the RoIPool layer~\\cite{RCNN} or RoIAlign layer~\\cite{Mask-RCNN}. These layers work by dividing the RoI into a regular grid and then max-pooling the feature map values in each grid cell. In Faster R-CNN, the outputs of the RoIPool layer are used to refine the RoI coordinates and to classify the RoI label. In Mask-RCNN, the outputs of the RoIAlign layer are used not only for refining and recognizing the RoI but also for segmenting the object inside the RoI.\n}\nFaster R-CNN~\\cite{Faster-RCNN} further refines and classifies RoIs with additional fully connected layers,\nwhile Mask R-CNN~\\cite{Mask-RCNN} further improves over Fast R-CNN by segmenting instances inside RoIs with additional convolutional layers. \n\nIn this paper, we go beyond Mask RCNN. In particular, depart from the backbone of Mask R-CNN, we propose a novel head branch which takes RoIs from RPN as inputs to regress the 6D object poses and is parallel with the existing branches. This results a novel end-to-end architecture which is not only detecting, segmenting but also directly recovering the 6D poses of object instances from a single RGB image. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMassive stars play an important role in the evolution of the Universe particularly in the late stages of their evolution. Through their strong stellar winds and supernova explosions, they inject mechanical energy into the interstellar medium (ISM) \\citep{bib:Abbott1982}. Also, they are principal sources of heavy elements in the ISM. In spite of their importance, our understanding of late stages of massive stars' evolution is still poor. One of the reasons for this is that massive stars are extremely rare partly because of their short lifetime. Due to their small numbers, the properties of evolved massive stars are still uncertain. For instance, as discussed in \\citet{bib:Massey2003}, \\citet{bib:MasseyOlsen2003} and \\citet{bib:Levesque2005}, there was a discrepancy between observed and theoretically predicted locations of red supergiants, high-mass evolved stars, on the Hertzsprung-Russel (HR) diagram. Compared with stellar evolutionary models, observed red supergiants appear to be too cool and too luminous.\n\nVY Canis Majoris (VY CMa) is one of the most well-studied red supergiants. Similar to other red supergiants, the location of VY CMa on the HR diagram is also uncertain. For instance, \\citet{bib:Monnier1999}, \\citet{bib:Smith2001} and \\citet{bib:Humphreys2007} obtained the luminosity of (2--5) $\\times$ 10$^5$ L$_{\\odot}$ from the spectral energy distribution (SED) assuming the distance of 1.5 kpc \\citep{bib:Lada1978} and effective temperature of 2800--3000 K based on the stellar spectral type \\citep{bib:LeSidaner1996}. However, \\citet{bib:Massey2006} suggested that the above parameters would make VY CMa cooler and more luminous than what current evolutionary models allow and would place it in the ``forbidden zone\" of the HR diagram, which is on the right-hand side of the Hayashi track in the HR diagram (see figure \\ref{fig:HRDthis}). \\citet{bib:Massey2006} reexamined the effective temperature and obtained a new value of 3650 $\\pm$ 25 K based on optical spectrophotometry combined with a stellar atmosphere model and suggested that the luminosity of VY CMa is only 6.0 $\\times$ 10$^4$ L$_{\\odot}$, which is probably a lower limit \\citep{bib:Massey2008}.\n\n\\citet{bib:Levesque2005} determined the effective temperatures of 74 Galactic red supergiants based on optical spectrophotometry and stellar atmosphere models. They obtained the effective temperatures of the red supergiants with a precision of 50 K. Their new effective temperatures are warmer than those in the literature. \nThese new effective temperature values seem to be consistent with the theoretical stellar evolutionary tracks. However, the luminosities of the red supergiants still had large uncertainty due to possible errors in estimated distances. Since most of red supergiants are very far, it has been difficult to apply the most reliable trigonometric parallax method to their distance measurements.\n\nIn the case of VY CMa, the currently accepted distance of 1.5 kpc \\citep{bib:Lada1978} is obtained by assuming that VY CMa is a member of the NGC 2362 star cluster and that the distance of NGC 2362 is same as that of VY CMa. The distance to the NGC 2362 was determined from the color-magnitude diagram \\citep{bib:Johnson1961} with an accuracy of no better than 30 \\%. Since the luminosity depends on the square of the distance, the accuracy of luminosity estimation is worse than 50 \\%. The proper motions measured with H$_2$O masers in previous studies (\\cite{bib:Richards1998}; \\cite{bib:Marvel1998}) have not been contradictory to the above distance value (1.5 kpc). When the H$_2$O maser region around a star is modeled with a symmetrically expanding spherical shell, the distance to the star from the Sun can be inferred by assuming that the observed mean proper motion multiplied by the distance should be equal to the observed mean \nradial velocity of the H$_2$O masers. The proper motion velocities in \\citet{bib:Richards1998} are well consistent with the H$_2$O maser spectral line velocities at the distance of 1.5 kpc. Also, \\citet{bib:Marvel1998} derived the distance of VY CMa to be 1.4 $\\pm$ 0.2 kpc. However, such a ``statistical parallax'' method depends on kinematical models fitted to the observed relative proper motions. In fact, if we add rotation or anisotropic expansion to the simple spherical expansion model, the inferred distance value could vary beyond quoted error ranges. An accurate distance determination without an assumption is crucial for obtaining the right location of VY CMa on the HR diagram and for determining fundamental parameters of the star.\n\nThanks to the recent progress in the VLBI technique, the distance measurements with trigonometric parallaxes have become possible even beyond 5 kpc (e.g., \\cite{bib:Honma2007}). Since VY CMa has strong maser emission in its circumstellar envelope, we have conducted an astrometric observations of H$_2$O masers around VY CMa to measure an accurate parallax with VLBI Exploration of Radio Astrometry (VERA) and here we present the results.\n\n\\section{Observations and Data Reduction}\n\nWe have observed H$_2$O masers (H$_2$O 6$_{16}$--5$_{23}$ transition at the rest frequency of 22.235080 GHz) in the red supergiant VY CMa with VERA at 10 epochs over 13 months. The epochs are April 24, May 24, September 2, October 30, November 27 in 2006, January 10, February 14, March 26, April 21 and May 27 in 2007 (day of year 114, 144, 245, 303, 331 in 2006, 010, 045, 085, 111 and 147 in 2007, respectively). In each epoch, VY CMa and a position reference source J0725--2640 [$\\alpha$(J2000) = 07h25m24.413135s, $\\delta$(J2000) = --26d40'32.67907\" in VCS 5 catalog, \\citep{bib:Kovalev2007}] were observed simultaneously in dual-beam mode for about 7 hours. The separation angle between VY CMa and J0725--2640 is 1.059 degrees. The instrumental phase difference between the two beams was measured at each station during the observations based on the correlations of artificial noise sources (\\cite{bib:Kawaguchi2000}; \\cite{bib:Honma2008b}). A bright continuum source (DA193 at 1st--3rd epochs and J0530+1330 at other epochs) was observed every 80 minutes for bandpass and delay calibration at each beam.\n\nLeft-handed circularly polarized signals were sampled with 2-bit quantization and filtered with the VERA digital filter unit \\citep{bib;Iguchi2005}. The data were recorded onto magnetic tapes at a rate of 1024 Mbps, providing a total bandwidth of 256 MHz which consists of 16 $\\times$ 16 MHz IF channels. One IF channel was assigned to the H$_2$O masers in VY CMa and the other 15 IF channels were assigned to J0725--2640,  respectively. The correlation processing was carried out on the Mitaka FX correlator \\citep{bib:Chikada1991}. The spectral resolution for H$_2$O maser lines is 15.625 kHz, corresponding to the velocity resolution of 0.21 km s$^{-1}$.\n\nAll data reduction was conducted using the NRAO Astronomical Image Processing System (AIPS) package. The amplitude and the bandpass calibration for target source (VY CMa) and reference source (J0725--2640) were performed independently. The amplitude calibration of each antenna was performed using system temperatures measured during observation. The bandpass calibration was applied using a bright continuum source (DA193 at 1st--3rd epochs and J0530+1330 at other epochs). Doppler corrections were carried out to obtain the radial motions of the H$_2$O masers relative to the Local Standard of Rest (LSR).\n\nThen, we calibrated the clock parameters using the residual delay of a bright continuum calibrator, which is also used for the bandpass calibration. The fringe fitting was made on the position reference source J0725--2640 with integration time of 2 minutes and time interval of 12 seconds to obtain residual delays, rates, and phases. These phase solutions were applied to the target source VY CMa and we also applied the dual-beam phase calibration \\citep{bib:Honma2008b} to correct the instrumental delay difference between the two beams. We calibrated the tropospheric zenith delay offset by using the GPS measurements of the tropospheric zenith delay made at each station \\citep{bib:Honma2008a}. After these calibrations, synthesized clean images were obtained. A typical synthesized beam size (FWHM) was 2 mas $\\times$ 1 mas with a position angle of --23$^{\\circ}$. We measured positions of the H$_2$O maser features relative to the extragalactic source J0725--2640 with two-dimensional Gaussian fitting.\n\n\\section{Results}\n\\begin{figure}\n  \\begin{center}\n    \\FigureFile(75mm,75mm){figure1original.eps}\n   \n  \\end{center}\n  \\caption{The total power spectrum of the H$_2$O masers in VY CMa obtained at Iriki station on April 24, 2006.}\\label{fig:Kspectrum}\n\\end{figure}\nThe autocorrelation spectrum of the H$_2$O masers around VY CMa is shown in figure \\ref{fig:Kspectrum}. The spectrum shows rich maser emission over LSR velocities ranging from --5 to 35 km s$^{-1}$. Though there are variations in flux density, overall structures of maser spectrum are common to all the epochs, indicating that maser spots are surviving over the observing period of 13 months.\n\n\\begin{figure*}\n  \\begin{center}\n    \\FigureFile(100mm,100mm){figure2.eps}\n   \n  \\end{center}\n  \\caption{The distribution of the H$_2$O masers in VY CMa obtained on April 24, 2006. Color, as shown on the scale on the right-hand side, represents the LSR velocities of the maser features.}\\label{fig:Kmap}\n\\end{figure*}\nTo reveal the distribution of the H$_2$O masers, we mapped the H$_2$O maser features in VY CMa at the first epoch (April 24, 2006). We detected 55 maser features that have signal-to-noise ratio larger than 7 in two adjacent channels. The distribution of the H$_2$O maser features is shown in figure \\ref{fig:Kmap}. The most red-shifted component and the most blue-shifted component lie near the apparent center of the distribution and the moderately red-shifted components on a NE-SW line of 150 mas. This distribution well agrees with the previous observational results by \\citet{bib:MarvelPhD}.\n\nAmong the H$_2$O maser features in VY CMa, one maser feature at a LSR velocity of about 0.55 km s$^{-1}$ is analyzed to detect an annual parallax. This feature, the most blue-shifted discrete component, is stably detected at all epochs. It is the third brightest maser component in the total-power spectrum, but in fact it is the second brightest component in the map. We note that while the brightest channel has multiple components, there is only one component in the channel at the LSR velocity of about 0.55 km s$^{-1}$. Because of the simple structure as well as its strength, this velocity channel is most suitable for astrometric measurements. Analyses for the other H$_2$O maser features will be reported in a forthcoming paper.\n\n\\begin{figure*}\n  \\begin{center}\n    \\begin{tabular}{cc}\n      \\FigureFile(85mm,85mm){figure3a.eps} &\n      \\FigureFile(85mm,85mm){figure3b.eps} \\\\\n    \\end{tabular}\n  \\caption{Results of the measured positions of the H$_2$O maser spot at the LSR velocity of 0.55 km s$^{-1}$ in VY CMa using J0725--2640 as a position reference source. The position offsets are with respect to $\\alpha$(J2000.0) = ${\\rm 07^h 22^m 58^s.32906}$, $\\delta$(J2000.0) = --25$^{\\circ}$46'03\".1410. \\textit{The left panel} shows the movements of the maser spot in right ascension as a function of time (day of year). \\textit{The right panel} is the same as \\textit{the left panel} in declination. Solid lines represent the best fit model with an annual parallax and a linear proper motion for the maser spot. Dotted lines represent the linear proper motion (--2.09 $\\pm$ 0.16 mas yr$^{-1}$ in right ascension and 1.02 $\\pm$ 0.61 mas yr$^{-1}$ in declination) and points represent observed positions of maser spot with error bars indicating the positional uncertainties in systematic errors (0.17 mas in right ascension and 0.68 mas in declination).}\\label{fig:para}\t\n  \\end{center}\n\\end{figure*}\n\nFigure \\ref{fig:para} shows the position measurements of the 0.55 km s$^{-1}$ H$_2$O maser component for 13 months. The position offsets are with respect to $\\alpha$(J2000.0) = ${\\rm 07^h 22^m 58^s.32906}$, $\\delta$(J2000.0) = --25$^{\\circ}$46'03\".1410. Assuming that the movements of maser features are composed of a linear motion and the annual parallax, we obtained a proper motion and an annual parallax by the least-square analysis. The declination data have too large errors to be suitable for an annual parallax and a proper motion measurement. Therefore, we obtained the parallax of VY CMa to be 0.88 $\\pm$ 0.08 mas, corresponding to a distance of 1.14 $^{+0.11} _{-0.09}$ kpc using only the data in right ascension. This is the first distance measurement of VY CMa based on an annual parallax measurement with the highest precision. We estimated the positional uncertainties from the least-square analysis, and the values of the errors are 0.17 mas in right ascension and 0.68 mas in declination making reduced $\\chi^{2}$ to be 1.\n\nWe also determined the absolute proper motion in right ascension and declination. The absolute proper motion is --2.09 $\\pm$ 0.16 mas yr$^{-1}$ in right ascension and 1.02 $\\pm$ 0.61 mas yr$^{-1}$ in declination. Compared with the proper motion of VY CMa obtained with Hipparcos, 9.84 $\\pm$ 3.26 mas yr$^{-1}$ in right ascension and 0.75 $\\pm$ 1.47 mas yr$^{-1}$ in declination \\citep{bib:Perryman1997}, there is a discrepancy in the proper motion in right ascension. Nearly 12 mas yr$^{-1}$ difference (more than 65 km s$^{-1}$ at 1.14 kpc) seems too large even when we take into account possible relative motion of the 0.55 km s$^{-1}$ maser feature with respect to the star itself. The difference may be due to the fact that circumstellar envelope of VY CMa has complex small-scale structure at optical wavelengths. This might seriously affect proper motion measurements with Hipparcos.\n\nThe quantitative estimation of the individual error sources in the VLBI astrometry is difficult as previously mentioned in \\citet{bib:Hachisuka}, \\citet{bib:Honma2007}, and \\citet{bib:Hirota2007}. Therefore, we estimated errors from the standard deviations of the least-square analysis to be 0.17 mas in right ascension and 0.68 mas in declination. When we consider that the statistical errors of the position, 0.02--0.09 mas, are estimated from residuals of the Gaussian fitting, the large values of the standard deviations suggest that some systematic errors affect the result of astrometry. In the following, we discuss causes of these astrometric errors, although we cannot separate each factor of error.\n\nFirst, we consider the errors originating from the reference source. The positional errors of the reference source J0725--2640 affect those of the target source. The position of J0725--2640 was determined with an accuracy of 0.34 mas in right ascension  and 0.94 mas in declination, respectively \\citep{bib:Kovalev2007}. Because these offsets are constant at all epochs, it dose not affect the parallax measurements. Also, when the reference source is not a point source, the positional errors of the target source could occur due to the structure and its variation of the reference source. However, since the reference source for our measurements is point-like and shows no structural variations between epochs, this is not likely to be the main source of the positional errors.\n\nSecondly, we consider the baseline errors originated from the positional errors of each VLBI station. The positions of VERA stations are determined with an accuracy of 3 mm by the geodetic observations at 2 and 8 GHz every 2 weeks. The positional errors derived from the baseline errors are 11 $\\mu$as at a baseline of 1000 km with the baseline error of 3 mm. This error is much smaller than our astrometric errors.\n\nThirdly, a variation of the structure of the maser feature could be one of the error sources. \\citet{bib:Hirota2007}, \\citet{bib:Imai2007}, and \\citet{bib:Hirota2008} proposed the maser structure effect as main sources in their trigonometric parallax measurements for nearby star forming regions Orion KL, IRAS 16293--2422 in $\\rho$ Oph East, and SVS 13 in NGC 1333, correspondingly. However, the effect does not seem dominating in our case because, (1) the 0.55 km s$^{-1}$ maser feature showed a stable structure in the closure phase, spectrum and map at all epochs of our observation, (2) this effect is inversely proportional to the distance of the target source and hence should be more than twice less significant for VY CMa as those in the above cases for a given size of the structure variation, (3) it is difficult to explain the large difference between astrometric errors in right ascension and declination by this effect.\n\nFinally, we have to consider the errors by the zenith delay residual due to tropospheric water vapor. These errors originate from the difference of path length through the atmosphere between the target and the reference sources, and generally larger in declination than in right ascension. According to the result of the simulation in \\citet{bib:Honma2008a}, the positional error by the tropospheric delay is 678 $\\mu$as in declination when the atmospheric zenith residual is 3 cm, the declination is --30 degrees, separation angle (SA) is 1 degree, and the position angle (PA) is 0 degrees. This is well consistent with our measurements. Therefore, the atmospheric zenith delay residual is likely to be the major source of the astrometric errors.\n\n\\section{The Location on the HR Diagram}\nWe successfully detected a trigonometric parallax of 0.88 $\\pm$ 0.08 mas, corresponding to a distance of 1.14 $^{+0.11} _{-0.09}$ kpc to VY CMa. Compared with the previously accepted distance 1.5 kpc \\citep{bib:Lada1978}, the distance to VY CMa became 76 \\%. Since the luminosity depends on the square of the distance, the luminosity should become 58 \\% of previous estimates. Hence, here we re-estimate the luminosity of VY CMa with the most accurate distance.\nThe luminosity can be estimated as follows:\n\\begin{equation}\nL = 4 \\pi  d^2  F_{bol},\n\\end{equation}\nwhere $L$ is luminosity, $d$ is distance and $F_{bol}$ is the bolometric flux. To obtain F$_{bol}$, we used the SED of VY CMa. The data are based on HST optical images and near-IR ground based images in \\citet{bib:Smith2001} and IRAS fluxes from 25 to 100 $\\mu$m. The F$_{bol}$ is obtained by integrating the observed fluxes. The estimated luminosity of VY CMa with our distance is (3 $\\pm$ 0.5) $\\times$ 10$^5$ L$_{\\odot}$.\n\nWe re-estimate the luminosities of VY CMa in the previous studies with our distance. \\citet{bib:LeSidaner1996} obtained a luminosity of VY CMa to be 9 $\\times$ 10$^5$ L$_{\\odot}$ from the SED at a kinematic distance of 2.1 kpc. With the distance of 1.14 kpc, the luminosity of \\citet{bib:LeSidaner1996} became 2.6 $\\times$ 10$^5$ L$_{\\odot}$. \\citet{bib:Smith2001} also estimated a luminosity of VY CMa from the SED and their luminosity is 5 $\\times$ 10$^5$ L$_{\\odot}$ at a distance of 1.5 kpc, which is is revised to be 3.0 $\\times$ 10$^5$ L$_{\\odot}$ using our distance. These luminosities are well consistent with each other. \\citet{bib:Massey2006} estimated an effective temperature of VY CMa to be 3650 K, which fitted the MARCS atmosphere model to the observed spectrophotometric data. They estimated an absolute V magnitude $M_V$ using the observed V magnitude, currently known distance of 1.5 kpc and a visible extinction $A_V$. Since the MARCS model presents bolometric corrections as a function of effective temperature, they obtained the luminosity of VY CMa to be 6.0 $\\times$ 10$^4$ L$_{\\odot}$, much lower than our value derived above. When they adopt our distance, their luminosity would be increased. The accurate distance is essential to estimate the accurate luminosity.  \n\nTo place VY CMa on the HR diagram, we adopted an effective temperature from the literatures. As we already mentioned, \\citet{bib:Massey2006} estimated an effective temperature of VY CMa to be 3650 K based on the observed spectrophotometric data. With the previously accepted spectral type of M4-M5, \\citet{bib:LeSidaner1996} obtained an effective temperature of 2800 K and \\citet{bib:Smith2001} also adopted an effective temperature of 3000 K. Since the effective temperature does not depend on the distance, our measurements cannot judge which effective temperature is correct. \n\nAlthough we cannot estimate the effective temperature of VY CMa with our measurements, when we adopt the effective temperature of 3650 K from the MARCS atmosphere model \\citep{bib:Massey2006}, the location of VY CMa on the HR diagram is determined as the filled square in figure \\ref{fig:HRDthis}. For comparison, we also show VY CMa's locations on the HR diagram obtained in the previous studies as filled circles. Our results suggest that the location of VY CMa on the HR diagram is now consistent with the evolutionary track of an evolved star with an initial mass of 25 M$_{\\odot}$. Also, re-scaled luminosity values of \\citet{bib:LeSidaner1996} and \\citet{bib:Smith2001} imply much closer locations to the 25 M$_{\\odot}$ track than their original ones which were deeply inside the ``forbidden zone''. On the other hand, the lower limit of luminosity of 6.0 $\\times$ 10$^4$ L$_{\\odot}$ in \\citet{bib:Massey2006} would be consistent with 15 M$_{\\odot}$ initial mass of VY CMa. According to \\citet{bib:Hirschi2004}, there is an order of magnitude difference in lifetimes between 15 M$_{\\odot}$ and 25 M$_{\\odot}$ in the initial mass. The improvement in mass and age values should affect statistical studies on evolution of massive stars such as the initial mass function \\citep{bib:Salpeter1955}.  \n\nOn the other hand, there is an argument against the effective temperature in \\citet{bib:Massey2006}. \\citet{bib:Humphreys2006} suggested that the spectrum in \\citet{bib:Massey2006} are more like their M4-type reference spectrum than M2-type reference spectrum. \\citet{bib:Humphreys2006} also pointed out that the modeling applicable for ``standard\" stars without mass-loss (MARCS) is simply not valid for an object like VY CMa. When we adopt the previously accepted spectral type, instead of M2.5I \\citep{bib:Massey2006}, the effective temperature is 3000 K \\citep{bib:Smith2001}. The location of VY CMa on the HR diagram is shown as the open square in figure \\ref{fig:HRDthis}. In this case, the position on the HR diagram is still not consistent with the theoretical evolutionary track. Table \\ref{tab:first} summarized our adopted parameters for VY CMa.\n\n\\begin{table*}\n  \\caption{Adopted parameters for VY CMa}\\label{tab:first}\n  \\begin{center}\n    \\begin{tabular}{lll}\n      \\hline\n       Parameter & Value &  Note \\\\ \\hline\n      Parallax & 0.88 $\\pm$ 0.08 mas \\\\\n      Distance & 1.14 $^{+0.11} _{-0.09}$ kpc \\\\\n      Luminosity & (3 $\\pm$ 0.5) $\\times$ 10$^5$ L$_{\\odot}$  \\\\\n      Mass & 25 M$_{\\odot}$ & \\citet{bib:Meynet2003} \\\\\n      Temperature  & 3650 $\\pm$ 25 K & \\citet{bib:Massey2006} \\\\\n                         & 3000 K &\\citet{bib:Smith2001} \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n\\end{table*}\n\n\\begin{figure*}\n  \\begin{center}\n    \\FigureFile(100mm,100mm){figure4.eps}\n   \n  \\end{center}\n  \\caption{The various locations of VY CMa on the HR diagram. The filled and the open squares represent our results. The luminosity of our result is calculated using the distance based on the trigonometric parallax measurements and the bolometric flux from the SED. The filled square adopted the effective temperature of 3650 K \\citep{bib:Massey2006} and the open square adopted 3000 K\\citep{bib:Smith2001}. The circles ``1,''  ``2''  and ``3'' represent the results of \\citet{bib:LeSidaner1996}, \\citet{bib:Smith2001} and \\citet{bib:Massey2006}, respectively. The evolutionary tracks are from \\citet{bib:Meynet2003}.} \\label{fig:HRDthis}\n\\end{figure*}\n\nThe accurate distance measurements of red supergiants, which provide true luminosities, will greatly contribute to the understanding of massive star evolution, though there is still uncertainty in the effective temperature on the stellar surface.  \n\n\\section{Conclusion}\n\nWe have observed the H$_2$O masers around the red supergiant VY CMa with VERA during 10 epochs spread over 13 months. Simultaneous observations for both H$_2$O masers around VY CMa and the position reference source J0725--2640 were carried out. We measured a trigonometric parallax of 0.88 $\\pm$ 0.08 mas, corresponding to a distance of 1.14 $^{+0.11} _{-0.09}$ kpc from the Sun. It is the first result that the distance of VY CMa is determined with an annual parallax measurement. There had been overestimation of the luminosities in previous studies due to the previously accepted distance. Using the most accurate distance based on the trigonometric parallax measurements and the bolometric flux from the observed SED, we estimated the luminosity of VY CMa to be (3 $\\pm$ 0.5) $\\times$ 10$^5$ L$_{\\odot}$. The accurate distance measurements provided the improved luminosity. The location of VY CMa on the HR diagram became much close to the theoretically allowable region, though there is still uncertainty in the effective temperature.  \n \n~\\\\\nWe are grateful to the referee, Dr. Philip Massey, for helpful comments on the manuscript. We would like to thank Prof. Dr. Karl M. Menten for his invaluable comments and for his help improving the manuscript. The authors also would like to thank all the supporting staffs at Mizusawa VERA observatory for their assistance in observations.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe similarity between the $D=5$ simple supergravity (SUGRA) and $D=11$\nSUGRA has been recognized for a long time \\cite{Cr81,ChNi80}.\nThe $D=11$ SUGRA is supposed to play a fundamental role as the\nlow energy limit of the M-theory \\cite{Wi95} --- an expected unified\nspeculation for the well-known five consistent superstring theories.\nThe field contents of the $D=11$ SUGRA theory consist of the metric,\na single Majorana spin-$\\frac32$ fermion along with a (singlet)\nthree-form gauge potential, with neither ``$N>1$'' extensions\nnor matter coupling permitted \\cite{CrJuSc78}.\nThe simple $D=5$ SUGRA, besides the metric $\\hat g_{AB}$, contains a\nspin-$\\frac32$ field $\\hat \\Psi^a_A$ ($a=1,2$ is an internal index)\nand $U(1)$ gauge field (one-form $\\hat B_A$) which replaces the\nthree-form gauge field in the $D=11$ SUGRA.\nThe ``primeval'' likeness comes directly from the fact that\nthe Lagrangians of both SUGRAs are exactly of the same form,\nexcept for the numbering of the gauge field indices.\nIn addition, their dimensional reduction to $D=4$ can be carried out in\na similar way \\cite{CrJu79}. Furthermore,\nthe $D=5$ simple SUGRA can be realized as a Calabi-Yau compactification\nof the $D=11$ SUGRA together with the truncation of the scalar multiplets,\nwhich is always necessary since there arises at least one scalar\nmultiplet for any Calabi-Yau compactification\n\\cite{CaCeAuFe95,FeKhMi96}.\nFurther resemblances between the two SUGRAs are related to the duality\ngroups upon dimensional reduction and the world sheet structure of the\nsolitonic string of the $D=5$ SUGRA \\cite{MiOh98}.\n\nThus the four-dimensional reduced effective action of the $N=2, D=5$\nSUGRA contains an additional Maxwell-like $U(1)$ field and a scalar\nfield regarded as external fields in five dimensions which are contributed\nby $\\hat B$, besides the ones coming from the metric\n$\\hat g_{AB}$ as in the traditional scheme for the Kaluza-Klein theory\n\\cite{AuFrMaRe82,AuMaReFr81,BaFaKe90a,BaFaKe90b}.\nCosmological solutions to this model have been previously considered\nby Balbinot, Fabris and Kerner \\cite{BaFaKe90a,BaFaKe90b}.\nFor the case of spatial homogeneity and isotropy the general\nsolution is non-singular in the scale factor, but unstable due to the\ncollapse to zero of the size of the fifth dimension\n\\cite{BaFaKe90a,BaFaKe90b}.\nBiaxial (with two equal scale factors) anisotropic solutions with a\ncylindrical homogeneous five-dimensional metric lead to singular\nsolutions with positive gravitational coupling \\cite{BaFaKe90b}.\nRecently, an explicit example of a manifestly $U$-duality covariant\nM-theory cosmology in five dimensions resulting from compactification\non a Calabi-Yau three-fold has been obtained in \\cite{LuOvWa98}.\nExact static solutions in $N=2,D=5$ SUGRA have been found by Pimentel\n\\cite{Pi95}, in a metric with cylindrical symmetry, with a particular\ncase corresponding to the exterior of a cosmic string.\n\nThe purpose of the present paper is to construct the general solution\nto the gravitational field equations of the $N=2, D=5$ SUGRA as\nformulated in \\cite{BaFaKe90a,BaFaKe90b} for an anisotropic triaxial\n(all directions have unequal scale factors) Bianchi type I space-time.\nIn this case the general solution of the field equations can be\nexpressed in an exact parametric form.\nFor all cosmological solutions, the singularity at the starting\/ending\ntime of the evolution can not be avoided except in the isotropic limit\nconsidered in \\cite{BaFaKe90a,BaFaKe90b}.\nNevertheless, in the models analyzed in this paper, the anisotropic\nUniverse has non-inflationary evolution for all times and for\nall values of parameters.\n\nThe present paper is organized as follows.\nThe field equations of our model are written down in Section II.\nIn Section III the general solution of the field equations is obtained.\nWe discuss our results and conclusions in Section IV.\n\n\\section{Field Equations, Geometry and Consequences}\nThe bosonic sector of $N=2, D=5$ SUGRA contains the five-dimensional\nmetric $\\hat g_{AB}$ and $U(1)$ gauge field $\\hat B_A$ described by\na Lagrangian which possesses a non-vanishing Chern-Simons term \\cite{Cr81}\n\\begin{eqnarray} \\label{L5}\n\\hat {\\cal L} &=& \\sqrt{-\\hat g} \\left\\{ \\hat R\n   - \\frac14 \\hat F_{AB} \\hat F^{AB} \\right\\} \\nonumber \\\\\n  &-& \\frac1{12\\sqrt{3}} \\epsilon^{ABCDE}\\hat F_{AB}\\hat F_{CD}\\hat B_E,\n\\end{eqnarray}\nwhere $\\hat F_{AB} = 2\\partial_{[A} \\hat B_{B]}$.\nIn this paper we use the following conventions and notations.\nThe variables with hats are five-dimensional objects all other variables\nare four-dimensional. Upper case indices $A,B,...$ are\nused for five-dimensional space-time, greek indices\n$\\mu,\\nu,...$ and low case indices $i,j,...$ are for\nfour-dimensional space-time and three-dimensional space\nrespectively. The signature is $(-,+,+,+,+)$.\n\nAssuming that the five dimensional space-time has locally the structure\nof $M^4 \\times S^1$ with a four-dimensional space-time $M^4$\nwhose spatial sections are homogeneous and asymptotic flat, then\nthe five-dimensional metric can be decomposed along the standard\nKaluza-Klein pattern\n\\begin{equation}\nd \\hat s^2 = \\phi^2 (dx_4 + A_\\mu dx^\\mu)^2 + g_{\\mu\\nu} dx^\\mu dx^\\nu,\n\\end{equation}\nwhere the scale factor $\\phi$ and Kaluza-Klein vector $A_\\mu$ are\nfunctions depending on $x^\\mu$ only.\n\nLooking for a ``ground state'' configuration we set, following\n\\cite{BaFaKe90a,BaFaKe90b}, the Kaluza-Klein vector, $A_\\mu$, equal\nto zero and take the one-form potential $\\hat B_A$ to be $\\hat B_\\mu=0$\nand $\\hat B_4= \\sqrt{3} \\psi(x^\\mu)$.\nUnder this ans\\\"atz, the five-dimensional gravitational field equations\nfor (\\ref{L5}) reduce to a set of four-dimensional equations\n\\begin{eqnarray}\nR_{\\mu\\nu} &-& \\phi^{-1} D_\\mu D_\\nu \\phi \\nonumber\\\\\n  &-& \\frac12 \\phi^{-2} \\left[ 3 \\partial_\\mu \\psi \\partial_\\nu \\psi\n   - g_{\\mu\\nu} (\\partial\\psi)^2 \\right] = 0, \\label{FEA1} \\\\\nD^2 \\phi &+& \\phi^{-1} (\\partial\\psi)^2 = 0, \\label{FEA2} \\\\\nD^2 \\psi &-& \\phi^{-1} \\partial_\\mu \\phi \\partial^\\mu \\psi = 0,\n    \\label{FEA3}\n\\end{eqnarray}\nwhere $D$ denotes the four-dimensional covariant derivative with\nrespect to the metric $g_{\\mu\\nu}$.\nEquivalently, the field equations can be re-derived,\nin the string frame, from the four-dimensional Lagrangian\n\\cite{BaFaKe90a,BaFaKe90b}\n\\begin{equation} \\label{L4}\n{\\cal L} = \\sqrt{-g} \\phi \\left\\{ R\n   - \\frac32 \\phi^{-2} (\\partial\\psi)^2 \\right\\},\n\\end{equation}\nvia variation with respect to the fields $g_{\\mu\\nu}, \\phi$ and $\\psi$.\nIn the Lagrangian (\\ref{L4}), the scale factor $\\phi$ is an analogue\nof the Brans-Dicke field whereas the origin of $\\psi$ is purely\nsupersymmetric.\n\nThe line element of an anisotropic homogeneous flat Bianchi type I\nspace-time is given by\n\\begin{equation}\nds^2 = - dt^2 + a_1^2(t) dx^2 + a_2^2(t) dy^2 + a_3^2(t) dz^2.\n\\end{equation}\nDefining the ``volume scale factor'', $V := \\prod_i a_i$,\n``directional Hubble factors'', $H_i := \\dot a_i\/a_i$,\nand ``average Hubble factor'', $H := \\frac13 \\sum_i H_i$, one can\npromptly find the relation $3H = \\dot V\/V$,\nwhere dot means the derivative with respect to time $t$.\nIn terms of those variables, the field equations\n(\\ref{FEA1}) and the equations of motion for $\\phi$ and $\\psi$\n(\\ref{FEA2},\\ref{FEA3}) coupling with the anisotropic Bianchi type I\ngeometry take the concise forms\n\\begin{eqnarray}\n3 \\dot H + \\sum_i H_i^2 + \\phi^{-1} \\ddot \\phi\n   + \\phi^{-2} \\dot \\psi^2 &=& 0, \\label{FEB1} \\\\\nV^{-1} \\frac{d}{dt} (VH_i) + H_i \\phi^{-1} \\dot\\phi\n   - \\frac12 \\phi^{-2} \\dot \\psi^2 &=& 0, \\; i=1,2,3,\n     \\label{FEB2} \\\\\nV^{-1} \\frac{d}{dt} (V\\dot\\phi) + \\phi^{-1} \\dot \\psi^2\n  &=& 0, \\label{FEB3} \\\\\nV^{-1} \\frac{d}{dt} (V\\dot\\psi) - \\phi^{-1} \\dot\\phi \\dot\\psi\\\n  &=& 0. \\label{FEB4}\n\\end{eqnarray}\n\nThe physical quantities of interest in cosmology are the {\\em expansion\nscalar} $\\theta$, the {\\em mean anisotropy parameter} $A$,\nthe {\\em shear scalar} $\\sigma^2$ and the {\\em deceleration parameter}\n$q$ defined as \\cite{Gr85}\n\\begin{eqnarray}\n\\theta &:=& 3H, \\qquad\nA := \\frac13 \\sum_i \\left( \\frac{H-H_i}{H} \\right)^2, \\nonumber \\\\\n\\sigma^2 &:=&\n     \\frac12 \\left( \\sum_i H_i^2 - 3 H^2 \\right), \\quad\nq := \\frac{d}{dt} H^{-1} - 1. \\label{Def}\n\\end{eqnarray}\n\nThe sign of the deceleration parameter indicates whether the cosmological\nmodel inflates. A positive sign corresponds to standard decelerating\nmodels whereas a negative sign indicates inflationary behavior.\n\n\\section{General Solution of the Field Equations}\nEquation (\\ref{FEB4}) can immediately be integrated to give\n\\begin{equation}\nV \\dot \\psi = \\omega \\phi, \\label{VPP}\n\\end{equation}\nwith $\\omega$ --- a constant of integration.\nFrom equations (\\ref{FEB3}) and (\\ref{FEB4}) one can find that the\nexpressions of the fields $\\phi(t)$ and $\\psi(t)$ have the following form\n\\begin{eqnarray}\n\\phi(t) &=& \\phi_0 \\cos \\left( \\omega \\int \\frac{dt}{V}\n   + \\omega_0 \\right), \\label{phi} \\\\\n\\psi(t) &=& \\psi_0 + \\phi_0 \\sin \\left( \\omega \\int \\frac{dt}{V}\n   + \\omega_0 \\right), \\label{psi}\n\\end{eqnarray}\nwhere $\\phi_0, \\psi_0$ and $\\omega_0$ are constants of integration.\n\nBy summing equations (\\ref{FEB2}) one gets\n\\begin{equation}\nV^{-1} \\frac{d}{dt} (VH) + H \\phi^{-1} \\dot\\phi\n   - \\frac12 \\phi^{-2} \\dot \\psi^2 = 0, \\label{FEB2a}\n\\end{equation}\nwhich can be transformed, by using equations (\\ref{VPP}) and (\\ref{phi}),\ninto the following differential-integral equation\ndescribing the dynamics and evolution\nof a triaxial Bianchi type I space-time in $N=2,D=5$ SUGRA:\n\\begin{equation} \\label{EqV}\n\\ddot V = \\omega \\frac{\\dot V}{V} \\tan \\left( \\omega \\int \\frac{dt}{V}\n   + \\omega_0 \\right) + \\frac32 \\frac{\\omega^2}{V}.\n\\end{equation}\n\nFurthermore, by subtracting equation (\\ref{FEB2a}) from equations\n(\\ref{FEB2}), one can solve for the $H_i$ as\n\\begin{equation}\nH_i = H + \\frac{K_i}{\\phi V}, \\qquad i=1,2,3, \\label{Hi}\n\\end{equation}\nwhere $K_i$ are constants of integration satisfying the following\nconsistency condition\n\\begin{equation}\n\\sum_i K_i = 0. \\label{Ki}\n\\end{equation}\nTherefore the physical quantities of interest (\\ref{Def}) reduce to\n\\begin{equation}\nA = \\frac{K^2}{3\\phi^2 V^2 H^2}, \\qquad\n\\sigma^2 = \\frac32 A H^2,\n\\end{equation}\nwhere $K^2 = \\sum_i K_i^2$.\n\nBy introducing a new variable $\\eta$ related to the physical time $t$\nby means of the transformation $d\\eta := dt\/V$ and by denoting\n$u:=\\dot V = dV\/(V d\\eta)$, equation (\\ref{EqV}) reduces to a first\norder linear differential equation for the unknown function $u$\n\\begin{equation}\n\\frac{du}{d\\eta} = \\omega \\tan( \\omega\\eta + \\omega_0) u\n   + \\frac32 \\omega^2,\n\\end{equation}\nwhose general solution is given by\n\\begin{equation}\nu = C \\cos^{-1}(\\omega\\eta+\\omega_0)\n  + \\frac32 \\omega \\tan(\\omega\\eta+\\omega_0), \\label{dotV}\n\\end{equation}\nwhere $C$ is an arbitrary constant of integration.\n\nDefining a new parameter $\\zeta(\\eta):=\\omega\\eta+\\omega_0$,\nwe can represent the general solution of\nthe field equations for a Bianchi type I space-time in\nthe $N=2,D=5$ SUGRA in the following exact parametric form:\n\\begin{eqnarray}\nt &=& t_0 + \\frac{V_0}{\\omega} \\int\n   \\frac{(1+\\sin\\zeta)^\\beta}{(1-\\sin\\zeta)^\\gamma} d\\zeta,\n   \\label{T} \\\\\nV &=& V_0 \\frac{(1+\\sin\\zeta)^\\beta}{(1-\\sin\\zeta)^\\gamma},\n   \\label{V} \\\\\nH &=& \\frac{\\omega}{2V_0} (\\alpha + \\sin\\zeta)\n   \\frac{(1-\\sin\\zeta)^{\\gamma-\\frac12}}{(1+\\sin\\zeta)^{\\beta+\\frac12}}, \\\\\na_{i} &=& a_{i0}\n   \\frac{(1+\\sin\\zeta)^{\\frac{\\beta}3+\\frac{K_i}{2\\omega\\phi_0}}}\n        {(1-\\sin\\zeta)^{\\frac{\\gamma}3+\\frac{K_i}{2\\omega\\phi_0}}},\n   \\quad i=1,2,3, \\label{ai}\n\\end{eqnarray}\nwhere we have denoted $\\alpha = \\frac{2C}{3\\omega}$,\n$\\beta=\\frac34(\\alpha-1)$, $\\gamma=\\frac34(\\alpha+1)$ and the $a_{i0}$\nare arbitrary constants of integration while $V_0=\\Pi_i a_{i0}$.\nThe observationally important physical quantities are given by\n\\begin{eqnarray}\nA &=& \\frac{4K^2}{3\\phi_0^2 \\omega^2}\n      \\left( \\alpha + \\sin\\zeta \\right)^{-2}, \\label{AA} \\\\\n\\sigma^2 &=& \\frac{K^2}{2\\phi_0^2 V_0^2}\n   \\frac{(1-\\sin\\zeta)^{2\\gamma-1}}{(1+\\sin\\zeta)^{2\\beta+1}}, \\\\\nq &=& 2 \\left\\{ 1 - \\frac{ 1 + \\alpha \\sin\\zeta}\n                         {(\\alpha + \\sin\\zeta)^2} \\right\\}. \\label{qq}\n\\end{eqnarray}\nFinally, the field equation (\\ref{FEB1}) gives a\nconsistency condition relating the constants $K^2, \\omega, \\alpha$ and\n$\\phi_0$:\n\\begin{equation}\nK^2 = \\frac32 \\phi_0^2 \\omega^2 \\left( \\alpha^2 - 1 \\right), \\label{K2}\n\\end{equation}\nleading to\n\\begin{equation}\n\\alpha \\ge 1 \\quad \\hbox{or} \\quad \\alpha \\le -1.\n\\end{equation}\n\nIt is worth noting that these two classes of solutions corresponding to\npositive or negative values of $\\alpha$ and $\\omega$ are not independent.\nIndeed, they can be related via a ``duality'' transformation by\nchanging the signs of $\\omega, \\alpha$ and $\\zeta$ so that\n$t(\\omega,\\alpha,\\zeta)=t(-\\omega,-\\alpha,-\\zeta)$,\n$a_i(\\omega,\\alpha,\\zeta)=a_i(-\\omega,-\\alpha,-\\zeta), i=1,2,3$,\nand $V(\\alpha,\\zeta)=V(-\\alpha,-\\zeta)$ etc.\nThis duality relation can be obtained by a simple inspection of equations\n(\\ref{T})-(\\ref{ai}) and, therefore, all physical quantities are\ninvariant with respect to this transformation.\nMoreover, the physical properties of the cosmological\nmodels presented here are strongly dependent on the signs of the\nparameters $\\alpha$ and $\\omega$.\nNevertheless, due to the duality transformation, hereafter we will consider,\nwithout loss of generality, the cases with positive $\\alpha$ only.\n\nFor some particular values of $\\alpha$, the general solutions\ncan be expressed in an exact non-parametric form, for instance,\nan exact class solutions can be obtained for $\\alpha = \\pm \\frac53$.\nBy introducing a new time variable $\\tau:=\\frac{3\\sqrt{2}\\omega}{V_0}t$,\nand choosing $t_0=\\mp\\frac{V_0}{3\\sqrt{2}\\omega}$,\nthe exact solution in $N=2,D=5$ SUGRA\nfor the Bianchi type I space-time is given by\n\\begin{eqnarray}\n\\tau &=& \\pm \\left[ \\cos^{-3} \\left( \\frac{\\zeta}2 \\pm\\frac{\\pi}4 \\right)\n    -1 \\right], \\\\\nV &=& \\frac{V_0}{2\\sqrt{2}} (1\\pm\\tau) \\left[ (1\\pm\\tau)^\\frac23 - 1\n    \\right]^\\frac12, \\\\\nH &=& \\frac{\\sqrt{2}\\omega}{V_0} \\frac{\\frac43 (1\\pm\\tau)^\\frac23 -1}\n    {(1\\pm\\tau)\\left[ (1\\pm\\tau)^\\frac23 - 1\\right]}, \\\\\na_i &=& \\frac{a_{i0}}{\\sqrt{2}} (1\\pm\\tau)^\\frac13 \\left[\n    (1\\pm\\tau)^\\frac23 - 1 \\right]^{\\frac16\\pm\\frac{K_i}{2\\omega\\phi_0}}, \\\\\nA &=& \\frac89 (1\\pm\\tau)^\\frac43 \\left[ \\frac43 (1\\pm\\tau)^\\frac23 - 1\n    \\right]^{-2}, \\\\\n\\sigma^2 &=& \\frac{8\\omega^2}{3V_0^2} (1\\pm\\tau)^{-\\frac23}\n    \\left[ (1\\pm\\tau)^\\frac23 - 1 \\right]^{-2}, \\\\\nq &=& 2 \\left\\{ 1-\\frac{(1\\pm\\tau)^\\frac23\\left[4(1\\pm\\tau)^\\frac23-5\\right]}\n    {6\\left[\\frac43(1\\pm\\tau)^\\frac23-1\\right]^2} \\right\\}.\n\\end{eqnarray}\n\nThe isotropic limit can be achieved by taking $\\alpha=\\pm 1$\nand, consequently, $K_i=0,i=1,2,3$.\nIt is worth noting that our solutions reduce to two different\ntypes of homogeneous space-times when $\\alpha=\\pm 1$.\n\nFor $\\alpha=1$, we obtain (by denoting $a_1=a_2=a_3=a$)\n\\begin{eqnarray}\nt &=& t_0+\\frac{V_0}{2\\sqrt{2}\\omega}\\left[ \\frac{\\sin\\theta}{\\cos^2\\theta}\n    + \\ln(\\tan\\theta+\\sec\\theta) \\right], \\label{IT1} \\\\\na &=& \\frac{a_0}{2\\sqrt{2}} \\cos^{-3}\\theta, \\label{IA1}\n\\end{eqnarray}\nwhere $\\theta:=\\frac{\\zeta}2 + \\frac{\\pi}4$.\nEqs.(\\ref{IT1}) and (\\ref{IA1}), describing a homogeneous flat isotropic\nspace-time interacting with two scalar fields (Kaluza-Klein and\nsupersymmetric), have been previously obtained by Balbinot, Fabris and\nKerner \\cite{BaFaKe90a} (for an extra choice of the parameter\n$\\omega_0=\\pi\/2$), who extensively studied their physical properties.\nThis isotropic solution also provides a positive gravitational coupling\nat the present time.\n\nIn the isotropic limit corresponding to $\\alpha=-1$, one can obtain\nanother class of isotropic homogeneous flat space-times represented in\nthe following parametric form by\n\\begin{eqnarray}\nt &=& t_0-\\frac{V_0}{2\\sqrt{2}\\omega}\\left[ \\frac{\\cos\\theta}{\\sin^2\\theta}\n    - \\ln(\\csc\\theta-\\cot\\theta) \\right], \\label{IT2} \\\\\na &=& \\frac{a_0}{2\\sqrt{2}} \\sin^{-3}\\theta.  \\label{IA2}\n\\end{eqnarray}\nThis type of flat space-time has not been previously considered.\n\n\\section{Discussions and Final Remarks}\nIn order to study the physical properties of the Bianchi type I Universe\ndescribed by the Eqs. (\\ref{V})-(\\ref{ai}) we need to fix first the range\nof variation of the parameter $\\zeta$. There are no a priori limitations\nin choosing the admissible range of values, thus both positive and negative\nvalues are permitted since the variable $\\eta=\\int dt\/V$ can also\nbe negative.\nBut from a physical point of view it is natural to impose the condition\nsuch that the gravitational coupling $\\phi$ is always positive during\nthe evolution of the Bianchi type I space-time in $N=2,D=5$ SUGRA.\nConsequently, we shall consider $\\zeta\\in (-\\pi\/2, \\pi\/2)$.\nWith this choice, the Universe for $\\alpha<-1$ starts its evolution\nin the infinite past ($t\\to -\\infty$) and ends at a finite moment $t=t_0$.\nFor $\\alpha>1$ the Universe starts at $t=t_0$ and ends in an infinite\nfuture with $t \\to \\infty$. (All discussion here and hereafter are with\nrespect to positive $\\omega$.)\n\nAs can be easily seen from equation (\\ref{dotV}), if $\\alpha<-1$\nwe have $\\dot V < 0$ for all $t$.\nFor these values of the parameters the\nBianchi type I anisotropic Universe collapses from an initial state\ncharacterized by infinite values of the volume scale factor and of the\nscale factors $a_i,i=1,2,3,$ to a singular state\nwith all factors vanishing.\nBut if $\\alpha>1$, the Universe expands and $\\dot V > 0$ for all $t$.\nThe expanding Bianchi type I Universe starts its evolution at the\ninitial moment, $t_0$, from a singular state with zero values of the\nscale factors, $a_i(t_0)=0,i=1,2,3$.\n\nAnother possible way to investigate the singularity behavior at the\ninitial moment is to consider the sign of the quantity\n$R_{\\mu\\nu} u^\\mu u^\\nu$, where $u^\\mu$ is the vector tangent to the\ngeodesics; $u^\\mu=(-1,0,0,0)$ for the present model.\nFrom the gravitational field equations we easily obtain\n\\begin{equation}\nR_{\\mu\\nu} u^\\mu u^\\nu = 3 H^2 (q-A). \\label{Ruu}\n\\end{equation}\nBy using equations (\\ref{AA}), (\\ref{qq}) and (\\ref{K2}) we can\nexpress (\\ref{Ruu}) as\n\\begin{equation}\nR_{\\mu\\nu} u^\\mu u^\\nu\n    = 6H^2 \\frac{\\sin\\zeta}{\\alpha+\\sin\\zeta}.\n\\end{equation}\nFor $\\zeta \\to -\\pi\/2$ the sign of $R_{\\mu\\nu}u^\\mu u^\\nu$ is determined\nby the sign of $1-\\alpha$.\nTherefore we obtain\n\\begin{equation}\nR_{\\mu\\nu} u^\\mu u^\\nu < 0 \\quad \\hbox{for} \\quad \\alpha > 1.\n\\end{equation}\nHence, the energy condition of Hawking-Penrose singularity theorems\n\\cite{HaEl73} is not satisfied for the solutions corresponding to Bianchi\ntype I Universes in the four-dimensional reduced two scalar fields theory of\n$N=2,D=5$ SUGRA with $\\alpha>1$.\nNevertheless, for those solutions an initial singular state is\n{\\em unavoidable} at the initial moment $t_0$.\n\nSince for $\\alpha>1$ the Bianchi type I Universe starts its evolution at\nthe initial moment $t=t_0$ ($\\zeta \\to -\\pi\/2$) from a singular state,\ntherefore, the presence of a variable gravitational coupling, $\\phi$,\nand of a supersymmetric field, $\\psi$, in an anisotropic geometry {\\em can\nnot} remove the initial singularity that mars the big-bang cosmology.\nAt the initial moment the degree of the anisotropy of the space-time\nis maximal, with the initial value of the anisotropy\nparameter $A(t_0)=2(\\alpha+1)\/(\\alpha-1)$.\nFor $t>t_0$ the Universe expands and the anisotropy parameter decreases.\n\nThe behavior of the volume scale factor, of the anisotropy parameter\nand of the deceleration parameter is presented for different values of\n$\\alpha$ in Figs. \\ref{FIG1}-\\ref{FIG3}.\nThe evolution of the Universe is non-inflationary, with $q>0$\nfor all $t>t_0$. Non-inflationary behavior is a generic feature of most\nof the supersymmetric models.\nThis is due to the general fact that the effective potential for the\ninflation field $\\sigma$ in SUGRA typically is too curved,\ngrowing as $\\exp (c \\sigma^2\/m)$, with $c$ a parameter\n(typically $O(1)$) and $m$ the stringy Planck mass \\cite{LiRi97}.\nThe typical values of $c$ make inflation impossible because the\ninflation mass becomes of the order of the Hubble constant.\nSee also \\cite{LiRi97} for a simple realization of hybrid inflation\nin SUGRA.\nIn the present model the presence of the supersymmetric field $\\psi$\nprevents the Bianchi type I Universe from inflating.\n\n\\begin{figure}\n\\epsfxsize=10cm\n\\epsffile{fig1.eps}\n\\caption{Behavior of the volume scale factor of the Bianchi type I\n         space-time for different values of $\\alpha>1$\n         ($V_0=1$ and $\\omega=1$): $\\alpha=3\/2$ (solid curve),\n         $\\alpha=2$ (dotted curve) and $\\alpha=5\/2$ (dashed curve).}\n\\label{FIG1}\n\\end{figure}\n\n\\begin{figure}\n\\epsfxsize=10cm\n\\epsffile{fig2.eps}\n\\caption{Time dependence of the parameter\n         $a=\\frac{3\\phi_0^2\\omega^2}{4K^2} A$ for different values of\n         $\\alpha$: $\\alpha=3\/2$ (solid curve),\n         $\\alpha=2$ (dotted curve) and $\\alpha=5\/2$ (dashed curve).}\n\\label{FIG2}\n\\end{figure}\n\n\\begin{figure}\n\\epsfxsize=10cm\n\\epsffile{fig3.eps}\n\\caption{Evolution of the deceleration parameter $q$ of the Bianchi\n         type I space-time for different values of $\\alpha$:\n         $\\alpha=3\/2$ (solid curve),\n         $\\alpha=2$ (dotted curve) and $\\alpha=5\/2$ (dashed curve).}\n\\label{FIG3}\n\\end{figure}\n\nIn the far future, for $\\zeta \\to \\pi\/2$ and $t\\to \\infty$ ($\\alpha>1$),\nwe have $V \\to \\infty, a_i \\to\\infty, i=1,2,3$.\nIn this limit the anisotropy parameter becomes a non-zero constant and\nthe Universe ends in a still anisotropic phase, but with\na decrease in the value of the anisotropy parameter $A$,\nas compared with the initial one.\nTherefore during its evolution the Bianchi type I Universe cannot\nexperience a transition from the anisotropic phase to the isotropic\nflat geometry.\nThe time evolution of the gravitational coupling $\\phi$ and of the\nsupersymmetric field $\\psi$ is represented in Fig. \\ref{FIG4}.\nThe $\\phi$-field is positive for all values of time.\n\n\\begin{figure}\n\\epsfxsize=10cm\n\\epsffile{fig4.eps}\n\\caption{Time evolution of the gravitational coupling $\\phi$ (solid curve)\n         and of the supersymmetric field $\\psi$ (dashed curve)\n         for $\\alpha=5\/2$ ($\\phi_0=1$, $\\psi_0=0$).}\n\\label{FIG4}\n\\end{figure}\n\nIn the present paper we have investigated the evolution and dynamics\nof a Bianchi type I space-time in a SUGRA toy-model, obtained\nby dimensional reduction of the $N=2,D=5$ SUGRA.\nThe inclusion of the supersymmetric term gives some particular\nfeatures to this cosmological model, by preventing the Universe from\ninflating and attaining completely isotropy.\nBut globally there is a decrease in the degree of anisotropy of the\ngeometry. Hence this model can be used to describe only a specific,\nwell-determined period of the evolution of our Universe.\n\n\\section*{Acknowledgments}\nOne of the authors (CMC) would like to thank prof. J.M. Nester for\nprofitable discussions.\nThe work of CMC was supported in part by the National Science Council\n(Taiwan) under grant NSC 89-2112-M-008-016.\n\nWe are also grateful to prof. Pimentel for calling our attention to the\nresults \\cite{Pi92,PiSo93,PiSo95} about several types of Bianchi\ncosmologies in the framework of $N=2,D=5$ supergravity.\n\n\\begin{appendix}\n\\section{Some Exact Forms for Physical Time}\nFor a large class of values of the parameter $\\alpha$ the general\nsolution of the gravitational field equations can be expressed in a\nclosed explicit form.\nThe variation of the physical time $t$ is determined by the integral\nequation (\\ref{T}). After a trick manipulation, one can rewrite this\nequation in the following form\n\\begin{equation}\nt = t_0 + \\frac{V_0}{\\sqrt{2}\\omega} \\int \\sin^{\\gamma'-3}\\theta\n    \\cos^{-\\gamma'}\\theta d \\theta,\n\\end{equation}\nwhere $\\theta:=\\frac{\\zeta}2+\\frac{\\pi}4$ and\n$\\gamma':=2\\gamma=\\frac32(\\alpha+1)$.\nIn general, for arbitrary $\\gamma'$, this integral can not be closed.\nFortunately, for integer values of $\\gamma'$, the physical time $t$ can\nbe expressed in an explicit form as a function of $\\zeta$. Some of these\nexact forms of the time function are listed in the following.\n(The outcomes for $\\alpha\\pm1$ are given in (\\ref{IT1}) and (\\ref{IT2}) ).\n\n\\noindent{\\bf For $\\alpha < -1$ :}\n\n\\noindent{(i) $\\alpha=-\\frac53, \\; (\\gamma'=-1),$}\n$$ t=t_0-\\frac{V_0}{3\\sqrt{2}\\omega} \\frac1{\\sin^3\\theta}, $$\n\n\\noindent{(ii) $\\alpha=-\\frac73, \\; (\\gamma'=-2),$}\n$$\nt=t_0-\\frac{V_0}{8\\sqrt{2}\\omega}\\Biggl[\n     \\frac{\\cos\\theta(\\cos^2\\theta+1)}{\\sin^4\\theta}\n   + \\ln(\\csc\\theta-\\cot\\theta) \\Biggr],\n$$\n\n\\noindent{(iii) $\\alpha=-3, \\; (\\gamma'=-3),$}\n$$\nt=t_0-\\frac{V_0}{15\\sqrt{2}\\omega}\\Biggl(\n     \\frac{5\\cos^2-2}{\\sin^5\\theta} \\Biggr).\n$$\n\n\\noindent{\\bf For $\\alpha > 1$ :}\n\n\\noindent{(i) $\\alpha=\\frac53, \\; (\\gamma'=4),$}\n$$ t=t_0+\\frac{V_0}{3\\sqrt{2}\\omega} \\frac1{\\cos^3\\theta}, $$\n\n\\noindent{(ii) $\\alpha=\\frac73, \\; (\\gamma'=5),$}\n$$\nt=t_0+\\frac{V_0}{8\\sqrt{2}\\omega}\\Biggl[\n     \\frac{\\sin\\theta(\\sin^2\\theta+1)}{\\cos^4\\theta}\n   - \\ln(\\tan\\theta+\\sec\\theta) \\Biggr],\n$$\n\n\\noindent{(iii) $\\alpha=3, \\; (\\gamma'=6),$}\n$$\nt=t_0+\\frac{V_0}{15\\sqrt{2}\\omega}\\Biggl(\n     \\frac{5\\sin^2\\theta-2}{\\cos^5\\theta} \\Biggr).\n$$\n\n\\end{appendix}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe \\emph{colouring number $\\mathrm{col}(G)$} of a graph $G$ is the minimum\n$k$ for which there is a linear order~$<_L$ on the vertices of $G$\nsuch that each vertex $v$ has \\emph{back-degree} at most $k-1$, that\nis, $v$ has at most $k-1$ neighbours $u$ with $u<_Lv$. The colouring\nnumber is a measure for uniform sparseness in graphs: we have\n$\\mathrm{col}(G)=k$ if and only if every subgraph $H$ of $G$ has a vertex of\ndegree at most $k-1$. Hence, provided $\\mathrm{col}(G)=k$, not only $G$ is\nsparse, but also every subgraph of $G$ is sparse. The colouring number\nminus one is also known as the \\emph{degeneracy}.\n\nRecently, Ne\\v{s}et\\v{r}il and Ossona de Mendez introduced the notions\nof \\emph{bounded expansion}~\\cite{nevsetvril2008grad} and\n\\emph{nowhere density} \\cite{nevsetvril2011nowhere} as very general\nformalisations of uniform sparseness in graphs. Since then, several\nindependent and seemingly unrelated characterisations of these notions\nhave been found, showing that these concepts behave robustly.  For\nexample, nowhere dense classes of graphs can be defined in terms of\nexcluded shallow minors~\\cite{nevsetvril2011nowhere}, in terms of\nuniform quasi-wideness~\\cite{dawar2010homomorphism}, a notion studied\nin model theory, or in terms of a game~\\cite{grohe2014deciding} with\ndirect algorithmic applications. The \\emph{generalised colouring\n  numbers} $\\mathrm{adm}_r$, $\\mathrm{col}_r$, and $\\mathrm{wcol}_r$ were introduced by\nKierstead and Yang~\\cite{kierstead2003orders} in the context of\ncolouring and marking games on graphs. As proved by Zhu\n\\cite{zhu2009colouring}, they can be used to characterise both bounded\nexpansion and nowhere dense classes of graphs.\n\nThe invariants $\\mathrm{adm}_r$, $\\mathrm{col}_r$, and $\\mathrm{wcol}_r$ are defined similarly\nto the classic colouring number: for example, the \\emph{weak\n  $r$-colouring} number $\\mathrm{wcol}_r(G)$ of a graph $G$ is the minimum\ninteger~$k$ for which there is a linear order of the vertices such\nthat each vertex $v$ can reach at most $k-1$ vertices $w$ by a path of\nlength at most~$r$ in which~$w$ is the smallest vertex on the\npath. \n\nThe generalised colouring numbers found important applications in the\ncontext of algorithmic theory of sparse graphs. For example, they play\na key role in Dvo\\v{r}\\'ak's approximation algorithm for minimum\ndominating sets \\cite{dvovrak13}, or in the construction of sparse\nneighbourhood covers on nowhere dense classes, a fundamental step in\nthe almost linear time model-checking algorithm for first-order\nformulas of Grohe et al.~\\cite{grohe2014deciding}.\n  \nIn this paper we study the relation between the colouring numbers and\nthe above mentioned characterisations of nowhere dense classes of\ngraphs, namely with uniform quasi-wideness and the splitter game. We\nuse the generalised colouring numbers to give a new proof that every\nbounded expansion class is uniformly quasi-wide.  This was first\nproved by Ne\\v{s}et\\v{r}il and Ossona de Mendez in\n\\cite{nevsetvril2010first}; however, the constants appearing in the\nproof of~\\cite{nevsetvril2010first} are huge. We present a very simple\nproof which also improves the appearing constants. Furthermore, for\nthe splitter game introduced in~\\cite{grohe2014deciding}, we show that\nsplitter has a very simple strategy to win on any class of bounded\nexpansion, which leads to victory much faster than in general nowhere\ndense classes of graphs.\n\nEvery graph $G$ from a fixed class $\\mathcal{C}$ of bounded expansion\nsatisfies $\\mathrm{wcol}_r(G)\\leq f(r)$ for some function $f$ and all positive\nintegers~$r$. However, the order that witnesses this inequality for\n$G$ may depend on the value $r$. We say that a class $\\mathcal{C}$ admits\n\\emph{uniform orders} if there is a function $f:\\mathbb{N}\\rightarrow\\mathbb{N}$ such\nthat for each $G\\in\\mathcal{C}$ there is one linear order that witnesses\n$\\mathrm{wcol}_r(G)\\leq f(r)$ for every value of $r$.  We show that every\nclass that excludes a fixed topological minor admits uniform orders\nthat can be computed efficiently.\n\nFinally, based on our construction of uniform orders for graphs that\nexclude a fixed topological minor, we provide an alternative proof of\na very recent result of Eickmeyer and\nKawarabayashi~\\cite{eickmeyer2016model}, that the model-checking\nproblem for successor-invariant first-order ($\\mathrm{FO}$) formulas is\nfixed-para\\-meter tractable on such classes (we obtained this result\nindependently of but later than~\\cite{eickmeyer2016model}).\nSuccessor-invariant logics have been studied in database theory and\nfinite model theory, and successor-invariant $\\mathrm{FO}$ is known to be more\nexpressive than plain $\\mathrm{FO}$~\\cite{rossman2007successor}.  The\nmodel-checking problem for successor-invariant $\\mathrm{FO}$ is known to be\nfixed-parameter tractable parameterized by the size of the formula on\nany graph class that excludes a fixed minor~\\cite{eickmeyer2013model}.\nVery recently, this result was lifted to classes that exclude a fixed\ntopological minor by Eickmeyer and\nKawarabayashi~\\cite{eickmeyer2016model}. \nThe key point of their proof\nis to use the decomposition theorem for graphs excluding a fixed\ntopological minor, due to Grohe and Marx~\\cite{grohe2015structure}.\nOur approach is similar to that of~\\cite{eickmeyer2016model}.\nHowever, we employ new constructions based on the generalised\ncolouring numbers and use the decomposition theorem\nof~\\cite{grohe2015structure} only implicitly. In particular, we do not\nconstruct a graph decomposition in order to solve the model-checking\nproblem. Therefore, we believe that our approach may be easier to\nextend further to classes of bounded expansion, or even to nowhere\ndense classes of graphs.\n\n\\section{Preliminaries}\n\n\\subparagraph*{Notation.} We use standard graph-theoretical notation;\nsee e.g.~\\cite{diestel2012graph} for reference.  All graphs considered\nin this paper are finite, simple, and undirected.  For a graph $G$, by\n$V(G)$ and $E(G)$ we denote the vertex and edge sets of $G$,\nrespectively.  A graph~$H$ is a \\emph{subgraph} of~$G$, denoted\n$H\\subseteq G$, if $V(H)\\subseteq V(G)$ and $E(H)\\subseteq E(G)$.  For\nany $M\\subseteq V(G)$, by $G[M]$ we denote the subgraph induced by\n$M$.  We write $G-M$ for the graph $G[V(G)\\setminus M]$ and if\n$M=\\{v\\}$, we write $G-v$ for $G-M$. For a non-negative integer\n$\\ell$, a \\emph{path of length $\\ell$} in~$G$ is a sequence\n$P=(v_1,\\ldots, v_{\\ell+1})$ of pairwise different vertices such that\n$v_iv_{i+1}\\in E(G)$ for all $1\\leq i\\leq \\ell$. We write $V(P)$ for\nthe vertex set $\\{v_1,\\ldots, v_{\\ell+1}\\}$ of~$P$ and $E(P)$ for the\nedge set $\\{v_iv_{i+1} : 1\\leq i\\leq \\ell\\}$ of~$P$ and identify~$P$\nwith the subgraph of $G$ with vertex set $V(P)$ and edge set\n$E(P)$. We say that the path $P$ \\emph{connects} its \\emph{endpoints}\n$v_1,v_{\\ell+1}$, whereas $v_2,\\ldots, v_\\ell$ are the \\emph{internal\n  vertices} of $P$. The {\\em{length}} of a path is the number of its\nedges.  Two vertices $u,v\\in V(G)$ are \\emph{connected} if there is a\npath in $G$ with endpoints $u,v$.  The {\\em{distance}} $\\mathrm{dist}(u,v)$\nbetween two connected vertices $u,v$ is the minimum length of a path\nconnecting $u$ and $v$; if $u,v$ are not connected, we put\n$\\mathrm{dist}(u,v)=\\infty$. The \\emph{radius} of~$G$ is\n$\\min_{u\\in V(G)}\\max_{v\\in V(G)}\\mathrm{dist}(u,v)$. The set of all\nneighbours of a vertex $v$ in $G$ is denoted by $N^G(v)$, and the set\nof all vertices at distance at most $r$ from $v$ is denoted by\n$N^G_r(v)$.  A graph $G$ is $c$-\\emph{degenerate} if every subgraph\n$H\\subseteq G$ has a vertex of degree at most $c$. A $c$-degenerate\ngraph of order $n$ contains an independent set of order at least\n$n\/(c+1)$.\n\nA graph $H$ with $V(H)=\\{v_1,\\ldots, v_n\\}$ is a \\emph{minor} of $G$,\nwritten $H\\preccurlyeq G$, if there are pairwise disjoint connected\nsubgraphs $H_1,\\ldots, H_n$ of $G$, called {\\em{branch sets}}, such\nthat whenever $v_iv_j\\in E(H)$, then there are $u_i\\in H_i$ and\n$u_j\\in H_j$ with $u_iu_j\\in E(G)$. We call $(H_1,\\ldots, H_n)$ a\n{\\em{minor model}} of $H$ in~$G$. The graph $H$ is a \\emph{topological\n  minor} of $G$, written $H\\preccurlyeq^t G$, if there are pairwise\ndifferent vertices $u_1, \\ldots, u_n\\in V(G)$ and a family of paths\n$\\{P_{ij}\\ \\colon\\ v_iv_j\\in E(H)\\}$, such that each $P_{ij}$ connects\n$u_i$ and $u_j$, and paths $P_{ij}$ are pairwise internally\nvertex-disjoint.\n\n\\vspace{-0.5cm}\n\\subparagraph*{Generalised colouring numbers.} Let us fix a graph\n$G$. By $\\Pi(G)$ we denote the set of all linear orders of $V(G)$. For\n$L\\in\\Pi(G)$, we write $u<_L v$ if $u$ is smaller than $v$ in $L$, and\n$u\\le_L v$ if $u<_L v$ or $u=v$.  Let $u,v\\in V(G)$. For a\nnon-negative integer $r$, we say that $u$ is \\emph{weakly\n  $r$-reachable} from~$v$ with respect to~$L$, if there is a path $P$\nof length $\\ell$, $0\\le\\ell\\le r$, connecting $u$ and~$v$ such that\n$u$ is minimum among the vertices of $P$ (with respect to $L$). By\n$\\mathrm{WReach}_r[G,L,v]$ we denote the set of vertices that are weakly\n$r$-reachable from~$v$ w.r.t.\\ $L$.\n\nVertex $u$ is \\emph{strongly $r$-reachable} from $v$ with respect\nto~$L$, if there is a path $P$ of length~$\\ell$, $0\\le\\ell\\le r$,\nconnecting $u$ and $v$ such that $u\\le_Lv$ and such that all internal\nvertices $w$ of~$P$ satisfy $v<_Lw$. Let $\\mathrm{SReach}_r[G,L,v]$ be the set\nof vertices that are strongly $r$-reachable from~$v$ w.r.t.\\ $L$. Note\nthat we have $v\\in \\mathrm{SReach}_r[G,L,v]\\subseteq \\mathrm{WReach}_r[G,L,v]$.\n\nFor a non-negative integer $r$, we define the \\emph{weak $r$-colouring\n  number $\\mathrm{wcol}_r(G)$} of $G$ and the \\emph{$r$-colouring number\n  $\\mathrm{col}_r(G)$} of $G$ respectively as follows:\n\\begin{eqnarray*}\n\\mathrm{wcol}_r(G)& := & \\min_{L\\in\\Pi(G)}\\:\\max_{v\\in V(G)}\\:\n\\bigl|\\mathrm{WReach}_r[G,L,v]\\bigr|,\\\\\n\\mathrm{col}_r(G) & := & \\min_{L\\in\\Pi(G)}\\:\\max_{v\\in V(G)}\\:\n\\bigl|\\mathrm{SReach}_r[G,L,v]\\bigr|.\n\\end{eqnarray*}\n\nFor a non-negative integer $r$, the \\emph{$r$-admissibility}\n$\\mathrm{adm}_r[G,L, v]$ of $v$ w.r.t.\\ $L$ is the maximum size~$k$ of a\nfamily $\\{P_1,\\ldots,P_k\\}$ of paths of length at most $r$ that start\nin~$v$, end at a vertex $w$ with $w\\leq_Lv$, and satisfy\n$V(P_i)\\cap V(P_j)=\\{v\\}$ for all $1\\leq i< j\\leq k$. As for $r>0$ we\ncan always let the paths end in the first vertex smaller than $v$, we\ncan assume that the internal vertices of the paths are larger\nthan~$v$.  Note that $\\mathrm{adm}_r[G,L,v]$ is an integer, whereas\n$\\mathrm{WReach}_r[G,L, v]$ and $\\mathrm{SReach}_r[G,L,v]$ are vertex sets.  The\n\\emph{$r$-admissibility} $\\mathrm{adm}_r(G)$ of~$G$~is\n\\begin{eqnarray*} \n\\mathrm{adm}_r(G) & = & \\min_{L\\in\\Pi(G)}\\max_{v\\in V(G)}\\mathrm{adm}_r[G,L,v].\n\\end{eqnarray*} \nThe generalised colouring numbers were introduced by Kierstead and\nYang \\cite{kierstead2003orders} in the context of colouring and marking\ngames on graphs. The authors also proved that the generalised\ncolouring numbers are related by the following inequalities:\n\\begin{equation}\\label{eq:gen-col-ineq}\n\\mathrm{adm}_r(G)\\leq \\mathrm{col}_r(G)\\le \\mathrm{wcol}_r(G)\\le (\\mathrm{adm}_r(G))^r.\n\\end{equation}\n\n\\subparagraph*{Shallow minors, bounded expansion, and nowhere\n  denseness.} A graph $H$ with $V(H)=\\{v_1,\\ldots, v_n\\}$ is a\n{\\em{depth-$r$ minor}} of $G$, denoted $H\\preccurlyeq_rG$, if there is a\nminor model $(H_1,\\ldots,H_n)$ of $H$ in $G$ such that each $H_i$ has\nradius at most $r$. We write $d(H)$ for the \\emph{average degree}\nof~$H$, that is, for the number $2|E(H)|\/|V(H)|$.  A class $\\mathcal{C}$ of\ngraphs has \\emph{bounded expansion} if there is a function\n$f\\colon\\mathbb{N}\\rightarrow\\mathbb{N}$ such that for all non-negative integers $r$\nwe have $d(H)\\leq f(r)$ for every $H\\preccurlyeq_r G$ with $G\\in\\mathcal{C}$.  A\nclass $\\mathcal{C}$ of graphs is \\emph{nowhere dense} if for every real\n$\\epsilon>0$ and every non-negative integer~$r$, there is an integer\n$n_0$ such that if $H$ is an $n$-vertex graph with $n\\geq n_0$ and\n$H\\preccurlyeq_r G$ for some $G\\in\\mathcal{C}$, then~$d(H)\\leq n^\\epsilon$.\n\nBounded expansion and nowhere dense classes of graphs were introduced\nby Ne\\v{s}et\\v{r}il and Ossona de Mendez as models for uniform\nsparseness of graphs\n\\cite{nevsetvril2008grad,nevsetvril2011nowhere}. As proved by\nZhu~\\cite{zhu2009colouring}, the generalised colouring numbers are\ntightly related to densities of low-depth minors, and hence they can\nbe used to characterise bounded expansion and nowhere dense classes.\n\n\\begin{theorem}[Zhu \\cite{zhu2009colouring}] A class $\\mathcal{C}$ of graphs\n  has bounded expansion if and only if there is a function\n  $f:\\mathbb{N}\\rightarrow\\mathbb{N}$ such that $\\mathrm{wcol}_r(G)\\leq f(r)$ for all $r\\in\\mathbb{N}$\n  and all $G\\in\\mathcal{C}$.\n\\end{theorem}\n\nDue to Inequality~(\\ref{eq:gen-col-ineq}), we may equivalently demand that there\nis a function $f:\\mathbb{N}\\rightarrow\\mathbb{N}$ such that $\\mathrm{adm}_r(G)\\leq f(r)$ or\n$\\mathrm{col}_r(G)\\leq f(r)$ for all non-negative integers $r$ and all\n$G\\in\\mathcal{C}$.\n\nSimilarly, from Zhu's result one can derive a characterisation of\nnowhere dense classes of graphs, as presented in\n\\cite{nevsetvril2011nowhere}.  A class $\\mathcal{C}$ of graphs is called\n\\emph{hereditary} if it is closed under induced subgraphs, that is, if\n$H$ is an induced subgraph of $G\\in\\mathcal{C}$, then $H\\in\\mathcal{C}$.\n\n\\begin{theorem}[Ne\\v{s}et\\v{r}il and Ossona de Mendez\n  \\cite{nevsetvril2011nowhere}] A hereditary class $\\mathcal{C}$ of graphs is\n  no\\-where dense if and only if for every real $\\epsilon>0$ and every\n  non-negative integer $r$, there is a positive integer $n_0$ such\n  that if $G\\in\\mathcal{C}$ is an $n$-vertex graph with $n\\geq n_0$, then\n  $\\mathrm{wcol}_r(G)\\leq n^\\epsilon$.\n\\end{theorem}\n\n\nAs shown in \\cite{dvovrak13}, for every non-negative integer $r$,\ncomputing $\\mathrm{adm}_r(G)$ is fixed-parameter tractable on any class of\nbounded expansion (parameterized by $\\mathrm{adm}_r(G)$). For $\\mathrm{col}_r(G)$ and\n$\\mathrm{wcol}_r(G)$ this is not known; however, by~\\eqref{eq:gen-col-ineq} we\ncan use admissibility to obtain approximations of these numbers.  On\nnowhere dense classes of graphs, for every $\\epsilon>0$ and every\nnon-negative integer $r$, we can compute an order that witnesses\n$\\mathrm{wcol}_r(G)\\leq n^\\epsilon$ in time $\\mathcal{O}(n^{1+\\epsilon})$ if $G$ is\nsufficiently large \\cite{grohe2014deciding}, based on Ne\\v{s}et\\v{r}il\nand Ossona de Mendez's augmentation technique\n\\cite{nevsetvril2008grad}.\n\n\\section{Uniform quasi-wideness and the splitter game}\n\nIn this section we discuss the relation between weak $r$-colouring\nnumbers and two notions that characterise nowhere dense classes:\nuniform quasi-wideness and the splitter game.\n\nFor a graph $G$, a vertex subset $A\\subseteq V(G)$ is called\n\\emph{$r$-independent} in $G$, if $\\mathrm{dist}_G(a,b)>r$ for all different\n$a,b\\in V(G)$. A vertex subset is called \\emph{$r$-scattered}, if it\nis $2r$-independent, that is, if the $r$-neighbourhoods of different\nelements of $A$ do not intersect.\n\nInformally, uniform quasi-wideness means the following: in any large\nenough subset of vertices of a graph from $\\mathcal{C}$, one can find a large\nsubset that is $r$-scattered in $G$, possibly after removing from $G$\na small number of vertices.  Formally, a class $\\mathcal{C}$ of graphs is\n\\emph{uniformly quasi-wide} if there are functions\n$N:\\mathbb{N}\\times\\mathbb{N}\\rightarrow\\mathbb{N}$ and $s:\\mathbb{N}\\rightarrow\\mathbb{N}$ such that for all\n$m, r\\in\\mathbb{N}$, if $W\\subseteq V(G)$ for a graph $G\\in\\mathcal{C}$ with\n$|W|>N(m,r)$, then there is a set $S\\subseteq V(G)$ of size at most\n$s(r)$ such that $W$ contains a subset of size at least $m$ that is\n$r$-scattered in $G-S$.\n\nThe notion of quasi-wideness was introduced by\nDawar~\\cite{dawar2010homomorphism} in the context of homomorphism\npreservation theorems. It was shown in \\cite{nevsetvril2010first} that\nclasses of bounded expansion are uniformly quasi-wide and that uniform\nquasi-wideness characterises nowhere dense classes of graphs.\n\n\\begin{theorem}[Ne\\v{s}et\\v{r}il and Ossona de Mendez\n  \\cite{nevsetvril2010first}] A hereditary class $\\mathcal{C}$ of graphs is\n  no\\-where dense if and only if it is uniformly quasi-wide.\n\\end{theorem}\n\nIt was shown by Atserias et al.\\@ in \\cite{atserias2006preservation} that classes\nthat exclude $K_k$ \nas a minor are uniformly quasi-wide. In fact, in this case we can choose\n$s(r)=k-1$, independent of $r$ (if such a constant function for a class $\\mathcal{C}$\nexists, the class is called \\emph{uniformly almost wide}). However, the function\n$N(m,r)$ that was used in the proof is huge: it comes from an iterated Ramsey\nargument. The same approach was used in \\cite{nevsetvril2010first} to show that\nevery nowhere dense class, and in particular, every class of bounded expansion,\nis uniformly quasi-wide.  We present a new proof that every bounded expansion\nclass is uniformly quasi-wide, which gives us a much better bound on $N(m,r)$\nand which is much simpler than the previously known proof.\n\n\\begin{theorem} Let $G$ be a graph and let $r,m\\in \\mathbb{N}$. Let $c\\in\\mathbb{N}$ be such\n  that $\\mathrm{wcol}_r(G)\\leq c$ and let $A\\subseteq V(G)$ be a set of size at least\n  $(c+1)\\cdot 2^m$. Then there exists a set $S$ of size at most $c(c-1)$ and a set\n  $B\\subseteq A$ of size at least $m$ which is $r$-independent\n  in $G-S$.\n\\end{theorem}\n\\begin{proof} Let $L\\in \\Pi(G)$ be such that\n  $|\\mathrm{WReach}_r[G,L,v]|\\leq c$ for every $v\\in V(G)$. Let $H$ be the\n  graph with vertex set $V(G)$, where we put an edge $uv\\in E(H)$ if\n  and only if $u\\in \\mathrm{WReach}_r[G,L,v]$ or $v\\in \\mathrm{WReach}_r[G,L,u]$. Then\n  $L$ certifies that~$H$ is $c$-degenerate, and hence we can greedily\n  find an independent set $I\\subseteq A$ of size $2^m$ in $H$. By the\n  definition of the graph $H$, we have that\n  $\\mathrm{WReach}_r[G,L,v]\\cap I=\\{v\\}$ for each $v\\in I$.\n\n\\begin{claim}\\label{cl:del}\n  Let $v\\in I$. Then deleting $\\mathrm{WReach}_r[G,L,v]\\setminus \\{v\\}$ from\n  $G$ leaves $v$ at a distance greater than $r$ (in\n  $G-(\\mathrm{WReach}_r[G,L,v]\\setminus\\{v\\})$) from all the other vertices of\n  $I$.\n\\end{claim}\n\\begin{claimproof}\n  Let $u\\in I$ and let $P$ be a path in $G$ that has length at most\n  $r$ and connects $u$ and~$v$. Let $z\\in V(P)$ be minimal with\n  respect to $L$. Then $z<_L v$ or $z=v$. If $z<_L v$, then\n  $z\\in \\mathrm{WReach}_r[G,L,v]$ and hence the path $P$ no longer exists\n  after the deletion of $\\mathrm{WReach}_r[G,L,v]\\setminus\\{v\\}$ from $G$. On\n  the other hand, if $z=v$, then $v\\in \\mathrm{WReach}_r[G,L,u]$,\n  which contradicts the fact that both $u,v\\in I$. \n\\end{claimproof}\n\nWe iteratively find sets\n$B_0\\subseteq \\ldots \\subseteq B_m\\subseteq I$, sets\n$I_0\\supseteq \\ldots \\supseteq I_m$, and sets\n$S_0\\subseteq \\ldots \\subseteq S_m$ such that $B$ is $r$-independent\nin $G-S$, where $B:=B_m$ and $S:=S_m$. We maintain the invariant that\nsets $B_i$, $I_i$, and $S_i$ are pairwise disjoint for each $i$.  Let\n$I_0=I$, $B_0 = \\emptyset$ and $S_0 = \\emptyset$.  In one step\n$i=1,2,\\ldots, m$, we delete some vertices from $I_i$ (thus obtaining\n$I_{i+1}$), shift one vertex from $I_i$ to $B_i$ (obtaining $B_{i+1}$)\nand, possibly, add some vertices from $V(G)\\setminus I_i$ to $S_i$\n(obtaining $S_{i+1}$).  More precisely, let $v$ be the vertex of $I_i$\nthat is the largest in the order $L$.  We set\n$B_{i+1} = B_i\\cup\\{v\\}$, and now we discuss how $I_{i+1}$ and\n$S_{i+1}$ are constructed.\n\nWe distinguish two cases. First, suppose $v$ is connected by a path of\nlength at most~$r$ in $G-S_i$ to at most half of the vertices of $I_i$\n(including $v$). Then we remove these reachable vertices from~$I_i$,\nand set $I_{i+1}$ to be the result. We also set $S_{i+1}=S_i$. Note\nthat $|I_{i+1}|\\geq |I_i|\/2$.\n\nSecond, suppose $v$ is connected by a path of length at most $r$ in\n$G-S_i$ to more than half of the vertices of $I_i$ (including $v$). We\nproceed in two steps. First, we add the at most $c-1$ vertices of\n$\\mathrm{WReach}_r[G,L,v]\\setminus\\{v\\}$ to $S_{i+1}$, that is, we let\n$S_{i+1}=S_i\\cup (\\mathrm{WReach}_r[G,L,v]\\setminus\\{v\\})$. (Recall here that\n$\\mathrm{WReach}_r[G,L,v]\\cap I = \\{v\\}$.) By Claim~\\ref{cl:del}, this leaves\n$v$ at a distance greater than~$r$ from every other vertex of $I_i$ in\n$G-S_{i+1}$. Second, we construct $I_{i+1}$ from $I_i$ by removing the\nvertex $v$ and all the vertices of $I_i$ that are not connected to $v$\nby a path of length at most~$r$ in $G-S_i$, hence we have\n$|I_{i+1}|\\geq\n\\lfloor|I_i|\/2\\rfloor$.\n\nObserve the construction above can be carried out for $m$ steps,\nbecause in each step, we remove at most half of the vertices of $I_i$\n(rounded up) when constructing $I_{i+1}$.  As $|I_0|=|I|=2^m$, it is\neasy to see that the set $I_i$ cannot become empty within $m$\niterations.  Moreover, it is clear from the construction that we end\nup with a set $B=B_m$ that has size~$m$ and is $r$-scattered in $G-S$,\nwhere $S=S_m$.  It remains to argue that $|S_m|\\leq c(c-1)$. For this,\nit suffices to show that the second case cannot apply more than $c$\ntimes in total.\n\nSuppose the second case was applied in the $i$th iteration, when\nconsidering a vertex~$v$.  Every vertex $u\\in I_i$ with $u<_Lv$ that\nwas connected to~$v$ by a path of length at most $r$ in $G-S_i$\nsatisfies $\\mathrm{WReach}_r[G,L,v]\\cap \\mathrm{WReach}_r[G,L,u]\\neq \\emptyset$.\nThus, every remaining vertex~$u\\in I_{i+1}$ has at least one of its\nweakly $r$-reachable vertices deleted (that is, included in\n$S_{i+1}$).  As the number of such vertices is at most $c-1$ at the\nbeginning, and it can only decrease during the construction, this\nimplies that the second case can occur at most~$c$~times.\n\\end{proof}\n\nAs shown in \\cite{siebertz16}, if $K_k\\not\\preccurlyeq G$, then\n$\\mathrm{wcol}_r(G)\\in\\mathcal{O}(r^{k-1})$. Hence, for such graphs we have to delete\nonly a polynomial (in $r$) number of vertices in order to find an\n$r$-independent set of size $m$ in a set of vertices of size single\nexponential in $m$.\n\nWe now implement the same idea to find a very simple strategy for\nsplitter in the splitter game, introduced by Grohe et\nal.~\\cite{grohe2014deciding} to characterise nowhere dense classes of\ngraphs.  Let~${\\ell},r\\in \\mathbb{N}$. The \\emph{simple $\\ell$-round\n  radius-$r$ splitter game} on~$G$ is played by two players,\n\\emph{connector} and \\emph{splitter}, as follows. We let~$G_0:=G$. In\nround~$i+1$ of the game, connector chooses a\nvertex~$v_{i+1}\\in V(G_i)$. Then splitter picks a vertex\n$w_{i+1}\\in N_r^{G_i}(v_{i+1})$. We\nlet~$G_{i+1}:=G_i[N_r^{G_i}(v_{i+1})\\setminus \\{w_{i+1}\\}]$. Splitter\nwins if~$G_{i+1}=\\emptyset$. Otherwise the game continues\nat~$G_{i+1}$. If splitter has not won after~${\\ell}$ rounds, then\nconnector wins.\n\nA \\emph{strategy} for splitter is a function~$\\sigma$ that maps every\npartial play $(v_1, w_1, \\dots,$ $v_s, w_s)$, with associated\nsequence~$G_0, \\dots, G_s$ of graphs, and the next\nmove~$v_{s+1}\\in V(G_s)$ of connector, to a\nvertex~$w_{s+1}\\in N_r^{G_s}(v_{s+1})$ that is the next move of\nsplitter. A strategy~$\\sigma$ is a \\emph{winning strategy} for\nsplitter if splitter wins every play in which she follows the\nstrategy~$f$. We say that splitter \\emph{wins} the simple $\\ell$-round\nradius-$r$ splitter game on~$G$ if she has a winning strategy.\n\n\\begin{theorem}[Grohe et al.\\ \\cite{grohe2014deciding}] A class $\\mathcal{C}$\n  of graphs is nowhere dense if and only if there is a function\n  $\\ell:\\mathbb{N}\\rightarrow\\mathbb{N}$ such that splitter wins the simple\n  $\\ell(r)$-round radius-$r$ splitter game on every graph $G\\in\\mathcal{C}$.\n\\end{theorem}\n\nMore precisely, it was shown in \\cite{grohe2014deciding} that\n$\\ell(r)$ can be chosen as $N(2s(r), r)$, where $N$ and~$s$ are the\nfunctions that characterise $\\mathcal{C}$ as a uniformly quasi-wide class of\ngraphs. We present a proof that on bounded expansion classes, splitter\ncan win much faster.\n\n\\begin{theorem}\\label{thm:splitterwcol} Let $G$ be a graph, let $r\\in\\mathbb{N}$ and let\n  $\\ell=\\mathrm{wcol}_{2r}(G)$. Then splitter wins the $\\ell$-round radius-$r$\n  splitter game.\n\\end{theorem}\n\\begin{proof} Let $L$ be a linear order that witnesses\n  $\\mathrm{wcol}_{2r}(G)=\\ell$. Suppose in round $i+1\\leq \\ell$, connector\n  chooses a vertex $v_{i+1}\\in V(G_i)$. Let $w_{i+1}$ (splitter's\n  choice) be the minimum vertex of $N_r^{G_i}(v_{i+1})$ with respect\n  to $L$. Then for each $u\\in N_r^{G_i}(v_{i+1})$ there is a path\n  between $u$ and $w_{i+1}$ of length at most $2r$ that uses only\n  vertices of $N_r^{G_i}(v_{i+1})$. As $w_i$ is minimum in\n  $N_r^{G_i}(v_{i+1})$, $w_{i+1}$ is weakly $2r$-reachable from each\n  $u\\in N_r^{G_i}(v_{i+1})$. Now let\n  $G_{i+1}:=G_i[N_r^{G_i}(v_{i+1})\\setminus\\{w_{i+1}\\}]$. As $w_{i+1}$\n  is not part of $G_{i+1}$, in the next round splitter will choose\n  another vertex which is weakly $2r$-reachable from every vertex of\n  the remaining $r$-neighbourhood. As $\\mathrm{wcol}_{2r}(G)= \\ell$, the game\n  must stop after at most $\\ell$ rounds.\n\\end{proof}\n\n\n\\section{Uniform orders for graphs excluding a topological minor}\\label{sec:uniform}\n\nIf $\\mathcal{C}$ is a class of bounded expansion such that\n$\\mathrm{wcol}_r(G)\\leq f(r)$ for all $G\\in\\mathcal{C}$ and all $r\\in \\mathbb{N}$, the order\n$L$ that witnesses this inequality for $G$ may depend on the value\n$r$. We say that a class $\\mathcal{C}$ \\emph{admits uniform orders} if there\nis a function $f:\\mathbb{N}\\rightarrow\\mathbb{N}$ such that for each $G\\in\\mathcal{C}$, there\nis a linear order $L\\in \\Pi(G)$ such that\n$|\\mathrm{WReach}_r[G,L,v]|\\leq f(r)$ for all $v\\in V(G)$ and all $r\\in\\mathbb{N}$. In\nother words, there is one order that simultaneously certifies the\ninequality $\\mathrm{wcol}_r(G)\\leq f(r)$ for all $r$.\n\nIt is implicit in \\cite{siebertz16} that every class that excludes a\nfixed minor admits uniform orders, which can be efficiently\ncomputed. We are going to show that the same holds for classes that\nexclude a fixed topological minor. Our construction is similar to the\nconstruction of \\cite{siebertz16}, in particular, our orders can be\ncomputed quickly in a greedy fashion. The proof that we find an order\nof high quality is based on the decomposition theorem for graphs with\nexcluded topological minors, due to Grohe and\nMarx~\\cite{grohe2015structure}.  Note however, that for the\nconstruction of the order we do not have to construct a tree\ndecomposition according to Grohe and Marx~\\cite{grohe2015structure}.\n\n\\subparagraph*{Construction.} Let $G$ be a graph. We present a\nconstruction of an order of $V(G)$ of high quality.  We iteratively\nconstruct a sequence $H_1,\\ldots, H_\\ell$ of pairwise disjoint and\nconnected subgraphs of $G$ such that\n$\\bigcup_{1\\leq i\\leq \\ell}V(H_i)=V(G)$. For $0\\leq i<\\ell$, let\n$G_i \\coloneqq G - \\bigcup_{1\\le j\\le i}V(H_j)$. We say that a\ncomponent $C$ of $G_i$ is \\emph{connected} to a subgraph $H_j$,\n$j\\leq i$, if there is a vertex $u\\in V(H_j)$ and a vertex $v\\in V(C)$\nsuch that $uv\\in E(G)$. For all~$i$, $1\\leq i<\\ell$, we will maintain\nthe following invariant. If $C$ is a component of $G_i$, then the\nsubgraphs $H_{i_1},\\ldots, H_{i_s}\\in \\{H_1,\\ldots, H_i\\}$ that are\nconnected to $C$ form a minor model of the complete graph $K_s$, where\n$s$ is their number.\n\nTo start, we choose an arbitrary vertex $v\\in V(G)$ and let $H_1$ be\nthe connected subgraph $G[\\{v\\}]$. Clearly, $H_1$ satisfies the above\ninvariant.  Now assume that for some $i$, $1\\le i< \\ell$, the sequence\n$H_1,\\ldots,H_i$ has already been constructed. Fix some component $C$\nof $G_i$ and, by the invariant, assume that the subgraphs\n$H_{i_1},\\ldots, H_{i_s} \\in \\{H_1,\\ldots, H_i\\}$ with\n$1\\leq i_1<\\ldots<i_s\\leq i$ that have a connection to $C$ form a\nminor model of~$K_s$. For a vertex $v\\in V(C)$, let $m(v)$ be the\nmaximum cardinality of a family $\\mathcal{P}$ of paths with the\nfollowing properties: each path of~$\\mathcal{P}$ connects $v$ with a\ndifferent subgraph $H_{i_j}$, the internal vertices of each path \nfrom~$\\mathcal{P}$ belong to $G_i$, and the paths of~$\\mathcal{P}$ are\npairwise disjoint apart from sharing $v$. Note that $m(v)$ can be\ncomputed in polynomial time using any maximum flow algorithm.\nPick~$v$ to be a vertex of $C$ with maximum $m(v)$.  Let $T$ be the\ntree of the breadth-first search in $G[C]$ that starts in $v$; thus,\n$T$ is rooted at~$v$.  We choose $H_{i+1}$ to be a minimal connected\nsubtree of $T$ that contains $v$ and, for each~$j$ with\n$1\\leq j\\leq s$, at least one neighbour of $H_{i_j}$ in $C$.\n\nFrom the construction it is easy to see that for every component $C'$\nof $G_{i+1}$, the subgraphs\n$H'_{i_1},\\ldots, H'_{i_{s'}}\\in \\{H_1,\\ldots, H_{i+1}\\}$ that are\nconnected to $C'$ form the minor model of a complete graph, hence the\ninvariant is again established.  Having chosen $H_{i+1}$, we proceed\nto the next iteration. The construction stops when all vertices are\npart of some $H_i$, $1\\leq i\\leq \\ell$.\n\nWe construct an order $L$ of $V(G)$ as follows. Let $v<_Lu$ if\n$v\\in V(H_i)$ and $u\\in V(H_j)$ for some $i<j$. Furthermore, we order\nthe vertices within each $H_i$ arbitrarily.  Obviously, the\nconstruction does not depend on~$r$, hence the produced order is\nuniform for~$G$.\n\n\\subparagraph*{Analysis.} From now on we assume that $G$ excludes $K_k$\nas a topological minor, for some constant~$k$. Furthermore, assume\nthat the graphs $H_1,\\ldots, H_\\ell$ and a corresponding order $L$\nhave been constructed, as described above. We now show that the\nconstructed order has good qualities. Our proof is based on the\nfollowing two key lemmas. The first lemma states that for every\ncomponent $C$ of $G_i$ arising after the construction of\n$H_1,\\ldots, H_i$, every vertex $v$ of $C$ can reach only a bounded\nnumber of subgraphs among $H_1,\\ldots, H_i$ by disjoint paths.\n\n\\begin{lemma}\\label{lem:disjointpaths}\n  There is a constant $\\alpha$ (depending only on $k$) such that for\n  all integers $i$, $1\\leq i<\\ell$, if $C$ is a component of $G_i$,\n  then for every vertex $v\\in V(C)$, we have $m(v)\\leq \\alpha$, where\n  $m(v)$ is defined as in the construction.\n\\end{lemma}\n\nThe second lemma states that from a vertex of $H_{i+1}$, we can reach\nonly a bounded number of vertices of each $H_j$, $1\\leq j\\leq i+1$, by\nshort disjoint paths in $G_i$.\n\n\\begin{lemma}\\label{lem:reachablevertices}\n  There is a constant $\\beta$ (depending only on $k$) such that for\n  all integers $i,j$, where $1\\leq j\\leq i\\leq\\ell$, and all positive\n  integers~$r$, the following holds.  Suppose $v\\in V(H_i)$, and let\n  $\\mathcal{P}$ be any family of paths of length at most $r$ with the\n  following properties: each path from~$\\mathcal{P}$ connects~$v$ with\n  a different vertex of $H_j$, the internal vertices of $\\mathcal{P}$\n  belong to $G_j$, and paths from $\\mathcal{P}$ are internally vertex\n  disjoint.  Then $\\mathcal{P}$ has size not larger than\n  $\\beta\\cdot r$.\n\\end{lemma}\n\nIt is easy to show that the above two lemmas guarantee that $L$ has\nthe required properties.  \n\n\\begin{corollary}\\label{cor:wrapup}\n  If $K_k\\not\\preccurlyeq^tG$, then there exists a constant $\\gamma$\n  (depending only on $k$) and a uniform order $L$ that witnesses\n  $\\mathrm{adm}_r(G)\\leq \\gamma\\cdot r$ for all non-negative integers $r$.\n\\end{corollary}\n\\begin{proof}\n  The $r$-admissibility of a vertex $v$ is determined in the construction when \n  $v$ first appears in some $H_i$, $1\\leq i<\\ell$. More precisely, $\\mathrm{adm}_r(G)$ is \n  upper bounded by the maximum possible number of disjoint paths in $G_i$ of \n  length at most $r$ from $v$ to \n  vertices of $H_1\\cup \\ldots\\cup H_i$. By Lemma~\\ref{lem:disjointpaths},\n  there is a constant $\\alpha$ such that $v$ can reach at most $\\alpha$ distinct\n  $H_j$, $1\\leq j< i$, via internally vertex disjoint paths in $G_i$. Additionally, $v$ can \n  reach other vertices of $H_i$. By Lemma~\\ref{lem:reachablevertices}, there is a constant $\\beta$ such that $v$ \n  reach at most $\\beta\\cdot r$ vertices of each $H_j$ with $j\\leq i$ by internally\n  vertex disjoint paths of length at most~$r$. Let\n  $\\gamma\\coloneqq (\\alpha+1)\\cdot \\beta$, then $v$ can reach at most $\\gamma\\cdot r$ smaller\n  vertices by internally vertex disjoint paths of length at most $r$ whose\n  internal vertices are larger than~$v$ w.r.t.\\ $L$.\n\\end{proof}\n\nThe proof of Lemma \\ref{lem:disjointpaths} is based on the\ndecomposition theorem for graphs with excluded topological minors of\nGrohe and Marx \\cite{grohe2015structure}. Recall that a \\emph{tree\n  decomposition} of a graph~$G$ is a pair $(T,\\beta)$, where $T$ is a\ntree and $\\beta:V(T)\\rightarrow 2^{V(G)}$, such that for every vertex\n$v\\in V(G)$ the set $\\beta^{-1}(v)=\\{t\\in V(T) : v\\in \\beta(t)\\}$ is\nnon-empty and connected in $T$, and for every edge $e\\in E(G)$ there\nis a node $t\\in V(T)$ such that $e\\subseteq \\beta(t)$. The\n\\emph{width} of $(T,\\beta)$ is $\\max\\{|\\beta(t)|-1 : t\\in V(T)\\}$ and\nthe \\emph{adhesion} of $(T,\\beta)$ is\n$\\max\\{|\\beta(s)\\cap \\beta(t)| : st\\in E(T)\\}$.\n\nFor a node $t\\in T$, we call $\\beta(t)$ the \\emph{bag at $t$}. If\n$T'\\subseteq T$, we write $\\beta(T')$ for\n$\\bigcup_{t'\\in V(T')}\\beta(t')$ and if $M\\subseteq V(G)$, we write\n$\\beta^{-1}(M)$ for $\\bigcup_{v\\in M}\\beta^{-1}(v)$. Denote by $K[X]$\nthe complete graph on a vertex set $X$. The \\emph{torso at $t$} is the\ngraph\n$\\tau(t):=G[\\beta(t)]\\cup \\bigcup_{st\\in E(T)}K[\\beta(s)\\cap\n\\beta(t)]$.\n\n\\begin{theorem}[\\cite{grohe2015structure}]\\label{thm:decomposition}\n  For every $k\\in\\mathbb{N}$, there exist constants $a(k),c(k),d(k)$ and\n  $e(k)$ such that the following holds. Let $H$ be a graph on $k$\n  vertices. Then for every graph $G$ with $H\\not\\preccurlyeq^tG$ there is a\n  tree decomposition $(T,\\beta)$ of adhesion at most $a(k)$ such that\n  for all $t\\in V(T)$ one of the following two alternatives hold.\n  \\smallskip\n\\begin{enumerate}\n\\item The torso $\\tau(t)$ has at most $c(k)$ vertices of degree larger\n  than $d(k)$, which we call the \\emph{apex vertices} of\n  $\\tau(t)$. Such a node $t$ will be called a \\emph{bounded degree\n    node}.\n\\item The torso $\\tau(t)$ excludes the complete graph $K_{e(k)}$ as a\n  minor. Such a node $t$ will be called an \\emph{excluded minor node}.\n\\end{enumerate}\n\\end{theorem}\n\nWe will need the following well-known properties of trees and tree\ndecompositions.\n\n\\begin{lemma}[Helly-property for trees]\\label{lem:helly}\n  Let $T$ be a tree and let $(T_i)_{i\\in I}$ be a family of subtrees\n  of $T$. If $V(T_i) \\cap V(T_j)\\not=\\emptyset$, for all $i,j\\in I$,\n  then $\\bigcap_{i\\in I} V(T_i) \\not= \\emptyset$.\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:separator}\n  Let $(T,\\beta)$ be a tree decomposition of a graph $G$. Let $e=st$\n  be an edge of $T$ and let $T_1,T_2$ be the components of $T-e$.\n  Then $\\beta(s)\\cap \\beta(t)$ separates $\\beta(T_1)$ from\n  $\\beta(T_2)$, that is, every path from a vertex of $\\beta(T_1)$ to a\n  vertex of $\\beta(T_2)$ traverses a vertex of\n  $\\beta(s)\\cap \\beta(t)$.\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:connected}\n  If $H\\subseteq G$ is a connected subgraph of $G$, then\n  $\\beta^{-1}(V(H))$ is connected in~$T$.\n\\end{lemma}\n\nFor the proof of Lemma \\ref{lem:disjointpaths}, assume that $G$ is\ndecomposed as described by Theorem \\ref{thm:decomposition}. Assume\nthat $H_1,\\ldots, H_i$ have been constructed and let $C$ be a\ncomponent of $G_i$ that has a connection to the subgraphs\n$H_{i_1},\\ldots, H_{i_s}$.  Recall that throughout the construction we\nguarantee that the subgraphs $H_{i_1},\\ldots, H_{i_s}$ form the minor\nmodel of a complete graph $K_s$.  We first identify one bag of the\ndecomposition as a bag which intersects many distinct branch sets of\nthis minor model.  The following lemma follows easily from the\nseparator properties of tree decompositions, in particular Lemma~\\ref{lem:separator}.\n\n\\begin{lemma}\\label{lem:corebag}\n  There can be at most one node $t$ such that $\\beta(t)$ intersects\n  strictly more than $a(k)$ of the branch sets $H_{i_j}$, for\n  $1\\leq j\\leq s$.\n\\end{lemma}\n\\begin{proof}\n  Assume there are two distinct nodes $t_1$ and $t_2$ with this property, \n  and suppose $\\beta(t_1)$ intersects $R^1_1, \\ldots, R^1_{a(k)+1}\\in \\{H_{i_1},\\ldots, H_{i_s}\\}$, whereas \n  $\\beta(t_2)$ intersects $R^2_1, \\ldots, R^2_{a(k)+1}\\in \\{H_{i_1},\\ldots, H_{i_s}\\}$.\n  Note that some branch sets $R^1_p$ and $R^2_q$ may coincide, but by reorder if necessary, we may assume w.l.o.g. that $R^1_p=R^2_q$ can happen only if $p=q$. \n  Observe that since $H_{i_1},\\ldots, H_{i_s}$ form a minor model of a\n  complete graph, the union of the vertex sets of $R^1_p$ and $R^2_p$ induces a connected subgraph of $G$, for each $p$ with $1\\leq p\\leq a(k)+1$; note that this is true also if $R^1_p=R^2_p$. \n  Consequently, if $e=s_1s_2$ is any edge of $T$ whose removal disconnects $t_1$ from $t_2$, then by Lemma~\\ref{lem:separator} we have that $\\beta(s_1)\\cap \\beta(s_2)$ must contain at least one vertex from \n  $V(R^1_p)\\cup V(R^2_p)$ for each $p$ with $1\\leq p\\leq a(k)+1$. Sets $V(R^1_p)\\cup V(R^2_p)$ are disjoint for distinct $p$, hence we conclude that $|\\beta(s_1)\\cap \\beta(s_2)|\\geq a(k)+1$.\n  This contradicts the fact that $(T,\\beta)$ has adhesion at most $a(k)$.\n\\end{proof}\n\n\nWe now show that there is a bag that intersects every branch set. The\nproof is a simple application of the Helly property of trees\n(Lemma~\\ref{lem:helly}) and Lemma~\\ref{lem:connected}.\n\n\\begin{lemma}\\label{lem:corebag-exists}\n  There is a node $t$ such that $\\beta(t)$ intersects each $H_{i_j}$,\n  for $1\\leq j\\leq s$.\n\\end{lemma}\n\\begin{proof}\nAs each $H_{i_j}$ is connected, by Lemma~\\ref{lem:connected} we have that $\\beta^{-1}(V(H_{i_j}))$ \ninduces a subtree $T_{i_j}$ in $T$. As the $\\{H_{i_1},\\ldots, H_{i_s}\\}$ form a model of a complete graph, they \nare pairwise connected and hence, by the definition of tree decompositions, any two such subtrees \nintersect. By Lemma~\\ref{lem:helly}, they all intersect in one node $t$, which is as desired. \n\\end{proof}\n\nHence, provided $s>a(k)$, there is a node $t$ with $\\beta(t)$\nintersecting at least $a(k)+1$ branch sets~$H_{i_j}$.  By\nLemma~\\ref{lem:corebag}, this node is unique. We call it the\n\\emph{core node} of the minor model.  Next we show that if the model\nis large, then its core node must be a bounded degree node. Shortly\nspeaking, this is because the model $H_{i_1},\\ldots, H_{i_s}$ trimmed\nto the torso of the core node is already a minor model of $K_s$ in\nthis torso.\n\n\\begin{lemma}\\label{lem:degreebag}\n  If $s>\\max\\{a(k), e(k)\\}$, then the core node of the minor model is\n  a bounded degree node.\n\\end{lemma}\n\\begin{proof}\n  As $s>a(k)$, by Lemma~\\ref{lem:corebag} we can identify the \n  unique core node $t$ whose bag intersects all the branch sets $H_{i_j}$. \n  Recall that $\\tau(t)$ is the graph induced by\n  the bag $\\beta(t)$ in which all adjacent separators are turned into cliques. \n  It is easy to see that the subgraphs $H_{i_j}':=\\tau(t)[V(H_{i_j})\\cap \\beta(t)]$ are connected\n  in $\\tau(t)$ and form a minor model of $K_s$. As $s>e(k)$, \n  we infer that $t$ cannot be an excluded minor node, and hence it is a bounded degree node.\n\\end{proof}\n\nFor vertices outside the bag of the core node, the bound promised in\nLemma~\\ref{lem:disjointpaths} can be proved similarly as\nLemma~\\ref{lem:corebag}.\n\n\\begin{lemma}\\label{lem:connectionsfromoutside}\n  Let $C$ be a component of $G_i$ that has a connection to the\n  subgraphs $H_{i_1},\\ldots, H_{i_s}$. If $s>a(k)$, then for every\n  vertex $v\\in V(C)\\setminus\\beta(t)$, where $t$ is the core node of\n  the model, we have that $m(v)\\leq a(k)$.\n\\end{lemma}\n\\begin{proof}\nBy the properties of a tree decomposition, there is an edge $e=tt'$ of $T$ such that\n$\\beta^{-1}(v)$ is contained in the subtree of $T-e$ that contains $t'$.\nSuppose $\\mathcal{P}$ is a family of paths that connect $v$ with distinct branch sets $H_{i_j}$ and are pairwise disjoint apart from $v$.\nRecall that $\\beta(t)$ intersects every branch set $H_{i_j}$.\nTherefore, by extending each path of $\\mathcal{P}$ within the branch set it leads to, we can assume w.l.o.g. that each path of $\\mathcal{P}$ connects $v$ with a vertex of $\\beta(t)$.\nBy Lemma~\\ref{lem:separator}, this implies that each path of $\\mathcal{P}$ intersects $\\beta(t)\\cap \\beta(t')$.\nPaths of $\\mathcal{P}$ share only $v$, which is not contained in $\\beta(t)\\cap \\beta(t')$, and hence we conclude that $|\\mathcal{P}|\\leq |\\beta(t)\\cap \\beta(t')|$.\nAs $\\mathcal{P}$ was chosen arbitrarily, we obtain that $m(v)\\leq |\\beta(t)\\cap \\beta(t')|\\leq a(k)$.\n\\end{proof}\n\n\\vspace{-0.5cm}\nWe now complete the proof of Lemma~\\ref{lem:disjointpaths} by looking\nat the vertices inside the core bag.\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:disjointpaths}]\n  We set $\\alpha:=a(k)+c(k)+d(k)+e(k)$.  Assume towards a\n  contradiction that for some $i$, $1\\leq i<\\ell$, we have that some\n  component $C$ of $G_i$ contains a vertex $v_1$ with $m(v_1)>\\alpha$.\n  Denote the branch sets that have a connection to $C$ by\n  $H_{i_1},\\ldots, H_{i_s}$, where $i_1<i_2<\\ldots<i_s$.  Let\n  $\\mathcal{P}$ be a maximum-size family of paths that pairwise share\n  only $v_1$ and connect $v_1$ with different branch sets $H_{i_j}$.\n  As $m(v_1)>\\alpha$, we have that $|\\mathcal{P}|>\\alpha$, and in\n  particular $s>\\alpha$.  As $\\alpha>a(k)$, by~Lemma~\\ref{lem:corebag} \n  and~Lemma~\\ref{lem:corebag-exists} we can\n  identify the unique core node~$t$ of the minor model. As\n  $s>\\max\\{a(k),e(k)\\}$, by Lemma~\\ref{lem:degreebag} the core node is\n  a bounded degree node.  As $m(v_1)>a(k)$, by\n  Lemma~\\ref{lem:connectionsfromoutside} we have $v_1\\in \\beta(t)$.\n  As $\\mathcal{P}$ contains more than $d(k)$ disjoint paths from $v$\n  to distinct branch sets, the degree of $v_1$ in $G$ must be greater\n  than $d(k)$, hence $v_1$ is an apex vertex of $\\tau(t)$.\n\n  Since $i_1<i_2<\\ldots<i_s$, we have that the component $C$ was\n  created when $H_{i_s}$ was removed from $G_{i_s-1}$. Let $C'$ be the\n  component of $G_{i_s-1}$ that contains $C$ and $H_{i_s}$ (and thus\n  $v_1$).  Observe that $C'$ is still connected to\n  $H_1, \\ldots, H_{i_{s-1}}$, and possibly to some other branch sets.\n  Recall that $H_{i_s}$ was constructed as a subtree of the\n  breadth-first search tree in $G_{i_s}$ that started in a vertex\n  $v_2\\in V(C')$ which, at this point of the construction, had maximum\n  $m(v_2)$ among vertices in $C'$.  However, at this point vertex\n  $v_1$ was also present in $C'$, and $\\mathcal{P}$ certifies that it\n  could send at least $\\alpha-1$ disjoint paths to different branch\n  sets among $H_1, \\ldots, H_{i_{s-1}}$ (in $\\mathcal{P}$, at most one\n  path leads to $H_{i_s}$, and all the other paths are also present in\n  $C'$). We infer that it held that $m(v_2)\\geq \\alpha-1$ at the\n  moment $v_2$ was taken.  Since\n  $\\alpha>a(k)+c(k)+d(k)+e(k)\\geq a(k)+d(k)+e(k)+1$, the same\n  reasoning as above shows that $t$ is also the core vertex of the\n  minor model formed by branch sets connected to $C'$.  Thus, by\n  exactly the same reasoning we obtain that $v_2$ is also an apex\n  vertex of $\\tau(t)$.\n\n  Since $\\alpha>a(k)+c(k)+d(k)+e(k)$, we can repeat this reasoning\n  $c(k)+1$ times, obtaining vertices $v_1,\\ldots, v_{c(k)+1}$, which\n  are all apex vertices of $\\tau(t)$.  This contradicts the fact that\n  $\\tau(t)$ contains at most $c(k)$ apex vertices.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:reachablevertices}]\n  We set $\\beta$ so that $\\beta\\cdot r\\geq (2r+1)\\cdot \\alpha$, where\n  $\\alpha$ is the constant given by Lemma~\\ref{lem:disjointpaths}.\n  For the sake of contradiction, suppose there is a family of paths\n  $\\mathcal{P}$ as in the statement, whose size is larger than\n  $(2r+1)\\cdot \\alpha$.\n\n  Recall that $H_{j}$ was chosen as a subtree of a breadth-first\n  search tree in $G_{j-1}$; throughout the proof, we treat $H_j$ as a\n  rooted tree.  As $H_j$ is a subtree of a BFS tree, every path from a\n  vertex $w$ of the tree to the root $v'$ of the tree is an isometric\n  path in $G_{j-1}$, that is, a shortest path between $w$ and $v'$ in\n  the graph $G_{j-1}$.  If $P$ is an isometric path in a graph $H$,\n  then $|N_r^H(v)\\cap V(P)|\\leq 2r+1$ for all $v\\in V(H)$ and all\n  $r\\in\\mathbb{N}$.  As the paths from $\\mathcal{P}$ are all contained in\n  $G_{j-1}$, and they have lengths at most $r$, this implies that the\n  path family $\\mathcal{P}$ cannot connect $v$ with more than $2r+1$\n  vertices of $H_{j}$ which lie on the same root-to-leaf path in\n  $H_{j}$. Since $|\\mathcal{P}|> (2r+1)\\cdot \\alpha$, we can find a\n  set $X\\subseteq V(H_j)$ such that $|X|>\\alpha$, each vertex of $X$\n  is connected to $v$ by some path from $\\mathcal{P}$, and no two\n  vertices of $X$ lie on the same root-to-leaf path in $H_j$.  Recall\n  that, by the construction, each leaf of $H_j$ is connected to a\n  different branch set $H_{j'}$ for some $j'<j$.  Consequently, we can\n  take the paths of $\\mathcal{P}$ leading to $X$ and extend them\n  within $H_j$ to obtain a family of more than $\\alpha$ disjoint paths\n  in $G_{j-1}$ that connect $v$ with different branch sets $H_{j'}$\n  for $j'<j$.  This contradicts Lemma~\\ref{lem:disjointpaths}.\n\\end{proof}\n\n\\enlargethispage{1cm}\n\\vspace{-0.5cm}\nObserve that the order can be computed in time $\\mathcal{O}(n^5)$:\nfor each vertex, we compute by a standard flow algorithm in time\n$\\mathcal{O}(n^3)$ whether it should be chosen as the next tree root to form\na subgraph $H_{i_j}$. This choice has to be made at most $n$ times.\n\nFinally, we state one property of the construction that follows\nimmediately from Lemma~\\ref{lem:disjointpaths}.\n\n\\begin{lemma}\\label{lem:degbound}\n  Each constructed subgraph $H_i$ has maximum degree at most\n  $\\alpha+1$, where $\\alpha$ is the constant given by\n  Lemma~\\ref{lem:disjointpaths}.\n\\end{lemma}\n\n\\section{Model-checking for successor-invariant first-order\nformulas}\\label{sec:mc}\n\nA \\emph{finite and purely relational signature} $\\tau$ is a finite set\n$\\{R_1,\\ldots, R_k\\}$ of relation symbols, where each relation symbol\n$R_i$ has an associated arity $a_i$. A finite $\\tau$-structure $\\mathfrak{A}$\nconsists of a finite set~$A$, the universe of $\\mathfrak{A}$, and a relation\n$R_i(\\mathfrak{A})\\subseteq A^{a_i}$ for each relation symbol $R_i\\in \\tau$.\nIf $ \\mathfrak{A}$ is a finite $\\tau$-structure, then the \\emph{Gaifman\n  graph} of $\\mathfrak{A}$, denoted $G(\\mathfrak{A})$, is the graph with\n$V(G(\\mathfrak{A}))=A$ and there is an edge $uv\\in E(G(\\mathfrak{A}))$ if and only\nif $u\\neq v$ and $u$ and $v$ appear together in some relation\n$R_i(\\mathfrak{A})$ of $\\mathfrak{A}$. We say that a class $\\mathcal{C}$ of finite\n$\\tau$-structures has \\emph{bounded expansion} if the graph class\n$G(\\mathcal{C}):=\\{G(\\mathfrak{A}) : \\mathfrak{A}\\in \\mathcal{C}\\}$ has bounded\nexpansion. Similarly, for $r\\in\\mathbb{N}$, we write $\\mathrm{adm}_r(\\mathfrak{A})$ for\n$\\mathrm{adm}_r(G(\\mathfrak{A}))$ etc.\n\nLet $V$ be a set. A successor relation on $V$ is a binary relation\n$S\\subseteq V\\times V$ such that $(V,S)$ is a directed path of length\n$|V|-1$. Let $\\tau$ be a finite relational signature. A formula\n$\\phi\\in \\mathrm{FO}[\\sigma\\cup\\{S\\}]$ is \\emph{successor-invariant} if for\nall $\\tau$-structures $\\mathfrak{A}$ and for all successor relations\n$S_1,S_2$ on $V(\\mathfrak{A})$ it holds that\n$(\\mathfrak{A}, S_1)\\models\\phi\\Longleftrightarrow (\\mathfrak{A},S_2)\\models\\phi$.\n\nSuccessor-invariant logics have been studied in database theory and\nfinite model theory in the past. It was shown by Rossman\n\\cite{rossman2007successor} that successor-invariant FO is more\nexpressive than FO without access to a successor relation. It is known\nthat successor-invariant FO (in fact even order-invariant FO) can\nexpress only local queries \\cite{grohe2000locality}, however, the\nproof does not translate formulas into local FO-formulas which could\nbe evaluated algorithmically.  It was shown in\n\\cite{eickmeyer2013model} that the model-checking problem for\nsuccessor-invariant first-order formulas is fixed-parameter tractable\non any proper minor closed class of graphs. Very recently, the same\nresult was shown for classes with excluded topological \nminors~\\cite{eickmeyer2016model}. We give a new proof of the model-checking\nresult of~\\cite{eickmeyer2016model} which is based on the nice\nproperties of the order we have constructed for graphs that exclude a\ntopological minor.\n\nEickmeyer et al.~\\cite{eickmeyer2013model} showed that on well-behaved\nclasses of graphs one can apply the following reduction from the\nmodel-checking problem for successor-invariant formulas to the\nmodel-checking problem for plain first-order formulas.\n\n\\begin{lemma}[Eickmeyer et al.~\\cite{eickmeyer2013model}]\\label{thm:eickmeyer-combine}\n  Let $\\mathcal{C}$ be a class of $\\tau$-structures such that for each\n  $\\mathfrak{A}\\in\\mathcal{C}$ one can compute in polynomial time a graph $H(\\mathfrak{A})$\n  such that\n\\begin{enumerate}\n\\item $V(H(\\mathfrak{A}))=V(G(\\mathfrak{A}))$ and\n  $E(H(\\mathfrak{A}))\\supseteq E(G(\\mathfrak{A}))$.\n\\item $H$ contains a spanning tree $T$ which can be computed in\n  polynomial time and which is of maximum degree $d$ for some fixed\n  integer $d$ depending on $\\mathcal{C}$ only.\n\\item The model-checking problem for first-order formulas on the graph\n  class $\\{H(\\mathfrak{A}) : \\mathfrak{A}\\in\\mathcal{C}\\}$ is fixed-parameter tractable.\n\\end{enumerate}\nThen the model-checking problem for successor-invariant first-order\nformulas is fixed-pa\\-ra\\-me\\-ter tractable on $\\mathcal{C}$.\n\\end{lemma}\n\nWe remark that the original lemma from \\cite{eickmeyer2013model}\nrefers to $k$-walks in $H$, which are easily seen to be equivalent to\nspanning trees of maximum degree $k$. In our view, spanning trees are\nmore intuitive to handle in our graph theoretic context.\n\n\\begin{lemma}\\label{lem:constructingH}\n  Let $k\\in\\mathbb{N}$.  There is a constant $\\delta$, depending only on $k$,\n  and a function $f\\colon \\mathbb{N}\\rightarrow \\mathbb{N}$ such that the following\n  holds.  For every graph $G$ with $K_k\\not\\preccurlyeq^tG$ we can compute\n  in polynomial time a supergraph $H$ with $V(H)=V(G)$ and\n  $E(H)\\supseteq E(G)$ such that $\\mathrm{adm}_r(H)\\leq f(r)$ for all\n  $r\\in \\mathbb{N}$ and such that $H$ contains a spanning tree $T$ with\n  maximum degree at most $\\delta$; furthermore, such a spanning tree\n  $T$ can be also computed in polynomial~time.\n\\end{lemma}\n\\begin{proof}\n  Without loss of generality, we assume that $G$ is\n  connected. Otherwise, we may apply the construction in each\n  connected component separately, and then connect the components\n  arbitrarily using single edges (added to $H$) in a path-like manner.\n  It is easy to see that including the additional edges to the\n  spanning tree increases its maximum degree by at most~$2$, while the\n  admissibility of the graph also increases by at most $2$.\n\n  We perform the construction of the subgraphs $H_1,\\ldots, H_\\ell$\n  almost exactly as in~Lemma~\\ref{sec:uniform}.  However, when\n  constructing the $H_i$'s and the order $L$, we put some additional\n  restrictions that do not change the quality of $L$.  First, recall\n  that when we defined $H_{i+1}$, for some $0\\leq i<\\ell$, we\n  considered a tree of breadth-first search starting at $v_{i+1}$ in a\n  connected component $C$ of $G_i$. Suppose that the subgraphs that\n  $C$ is connected to are $H_{i_1},\\ldots,H_{i_s}$, where\n  $1\\leq i_1<\\ldots<i_s\\leq i$.  Then $H_{i+1}$ was defined as a\n  minimal subtree of the considered BFS tree that contained, for each\n  $1\\leq j\\leq s$, some vertex of $H_{i_j}$ that is adjacent to $C$.\n  Observe that in the construction we were free to choose which\n  neighbour of $H_{i_j}$ will be picked to be included in $H_{i+1}$.\n  For $j<s$ we make an arbitrary choice as before, but the neighbour\n  of $H_{i_s}$ (if exists; note that this is the case for $i>0$) is\n  chosen as follows.  We first select the vertex\n  $w'_{i+1}\\in V(H_{i_s})$ that is the largest in the order $L$ among\n  those vertices of $H_{i_s}$ that are adjacent to $C$ (the vertices\n  of $H_j$ for $j\\leq i$ are already ordered by $L$ at this point).\n  Then, we select any its neighbour $w_{i+1}$ in $C$ as the vertex\n  that is going to be included in $H_{i+1}$ in its construction.\n  Finally, recall that in the construction of $L$, we could order the\n  vertices of $H_{i+1}$ arbitrarily.  Hence, we fix an order of\n  $H_{i+1}$ so that $w_{i+1}$ is the smallest among $V(H_{i+1})$.\n  This concludes the description of the restrictions applied to the\n  construction.\n\n  We now construct $H$ by taking $G$ and adding some edges. During the\n  construction, we will mark some edges of $H$ as {\\em{spanning\n      edges}}.  We start by marking all the edges of all the trees\n  $H_i$, for $1\\leq i\\leq \\ell$, as spanning edges. At the end, we\n  will argue that the spanning edges form a spanning tree of $H$ with\n  maximum degree at most $\\delta$.\n\n  For each $i$ with $1\\leq i<\\ell$, let us examine the vertex\n  $w_{i+1}$, and let us {\\em{charge}} it to $w_{i+1}'$.  Note that in\n  this manner every vertex $w_{i+1}$ is charged to its neighbour that\n  lies before it in the order $L$.  For any $w\\in V(G)$, let $D(w)$ be\n  the set of vertices charged to $w$. Now examine the vertices of $G$\n  one by one, and for each $w\\in V(G)$ do the following. If\n  $D(w)=\\emptyset$, do nothing. Otherwise, if\n  $D(w)=\\{u_1,u_2,\\ldots,u_h\\}$, mark the edge $wu_1$ as a spanning\n  edge, and add edges $u_1u_2,u_2u_3,\\ldots,u_{h-1}u_h$ to $H$,\n  marking them as spanning edges as well.\n\n\\begin{claim}\\label{cl:degbnd}\n  The spanning edges form a spanning tree of $H$ of maximum degree at\n  most $\\alpha+4$, where~$\\alpha$ is the constant given by\n  Lemma~\\ref{lem:disjointpaths}.\n\\end{claim}\n\\begin{claimproof}\nBecause the branch sets partition the graph, the spanning edges form a spanning subgraph of $H$.\nBecause we connect the branch set $H_{i+1}$ only to the largest reachable branch set $H_{i_s}$ \n(and this set is never again the largest reachable branch set for $H_j$, $j>i$), the spanning \nsubgraph is acyclic. It is easy to see that the spanning subgraph is also connected.  \nBy Lemma~\\ref{lem:degbound}, we have that each $H_i$ has maximum degree at most $\\alpha+1$.\nAlso, for every vertex $w\\in V(G)$, at most $3$ additional edges incident to $w$ in $H$ are marked as spanning (two edges are contributed by the path from $u_1$ to $u_h$ (only $u_1$ charges to a different vertex and has degree $1$ on the path) and one edge \nmay be added if a vertex is charged to it). In total, this means that\n$H$ has maximum degree bounded by $\\alpha+4$.\n\\end{claimproof}\n\nIt remains to argue that $H$ has small admissibility. For this, it\nsuffices to prove the following claim.  The proof uses the additional\nrestrictions we introduced in the construction.\n\n\\begin{claim}\\label{cl:adm}\n  Let $r$ be a positive integer. If the order $L$ satisfies \n  $\\max_{v\\in V(G)} |\\mathrm{SReach}_{2r}[G,L,v]|\\leq m$, that is, \n  the order certifies $\\mathrm{col}_{2r}(G)\\leq m$, \n  then $\\mathrm{adm}_r(H)\\leq m+2$.\n\\end{claim}\n\\begin{claimproof}\nWe verify that for each $r$, the order $L$ certifies that $\\mathrm{adm}_r(H)\\leq m+2$. For this, take any vertex $v\\in V(H)=V(G)$, and let $\\mathcal{P}$ be any family of paths of length at most $r$ in $H$\nthat start in $v$, end in distinct vertices smaller than $v$ in $L$, and are pairwise internally disjoint.\nWe can further assume that all the internal vertices of all the paths from $\\mathcal{P}$ are larger than $v$ in $L$.\nLet $i$, $0\\leq i<\\ell$, be such that $v\\in V(H_{i+1})$.\nWe distinguish two cases: either $v=w_{i+1}$ or $v\\neq w_{i+1}$.\n\nWe first consider the case $v\\neq w_{i+1}$; the second one will be very similar. By the construction of the order $L$, it follows that $w_{i+1}<_L v$.\nConsider any path $P\\in \\mathcal{P}$. Then $P$ is a path in $H$; we shall modify it to a walk $P'$ in $G$ as follows.\nSuppose $P$ uses some edge $e$ that is not present in~$H$. By the construction of~$H$, it follows that\n$e=u_1u_2$ is an edge connecting two vertices that are charged to the same vertex $w$; suppose w.l.o.g. $P$ traverses $e$ from $u_1$ to $u_2$. \nDefine $P'$ by replacing the traversal of $e$ on $P$ by a path of length two consisting of $u_1w$ and $wu_2$, and making the same replacement for all other edges on $P$ that do not belong to $G$.\n\nWe claim that all the internal vertices of $P'$ are not smaller, in $L$, than $v$. For this, it suffices to show that whenever some edge $u_1u_2$ is replaced by a path $(u_1w, wu_2)$ as above,\nthen we have that $v\\leq_L w$. Aiming towards a contradiction, suppose that $u_1u_2$ is the first edge on $P$ for which we have $w<_L v$.\nBy the construction, it must be that $(u_1,u_2)=(w_{j_1},w_{j_2})$ for some $j_1,j_2>i+1$, and $w=w_{j_1}'=w_{j_2}'$. \nLet $j$ be such that $w\\in H_j$.\nWhen constructing $H_{j_1}$, we chose $w=w_{j_1}'$ as the largest, w.r.t. $L$, vertex of $H_j$ which was adjacent to the connected component $C'$ of $G_{j_1-1}$ that contains $H_{j_1}$.\nObserve that the prefix of $P'$ up to $w_{j_1}$ is a path in $G$ that, by the choice of $u_1u_2$, contains only vertices not smaller in $L$ than $v$. This prefix has to access the connected component $C'$\nfrom some vertex $q$, for which we of course have $v\\leq_L q$. If $q\\notin V(H_j)$ then, as $H_j$ is the last among subgraphs connected to $C'$, we have that $q\\in V(H_{j'})$ for some $j'<j$ and, consequently,\n$v\\leq_L q<_L w$.\nOtherwise, if $q\\in V(H_j)$, then by the choice of $w=w_{j_1}'$ as the last, in $L$, vertex of the neighbourhood of $C'$ within $H_j$, we also have $v\\leq_L q\\leq_L w$.\nIn both cases we conclude that $v\\leq_L w$, a contradiction.\n\nHence, if we apply the above procedure to all the paths from $\\mathcal{P}$, we obtain a family $\\mathcal{P}'$ of walks in $G$ with the following properties: \neach walk of $\\mathcal{P}'$ has length at most $2r$, it connects $v$ with a different vertex that is smaller in $L$ than $v$, and all its internal vertices are not smaller than $v$.\nNote here that walks from $\\mathcal{P}'$ are not necessarily disjoint, but still their number must be bounded by $\\max_{v\\in V(G)} |\\mathrm{SReach}_{2r}[G,L,v]|\\leq m$. It follows that $|\\mathcal{P}|\\leq m$.\n\nFinally, we consider the second case $v=w_{i+1}$. Observe that in the construction of $H$, we added at most $2$ additional edges incident to $v$ that connect $v$ with other vertices charged to the same vertex.\nAt most two paths from $\\mathcal{P}$ can use these edges, and for the other paths we may apply exactly the same reasoning as in the first case. \nIt follows that $|\\mathcal{P}|\\leq m+2$ in this case; this concludes the proof.\n\\end{claimproof}\n\nThe statement of the lemma now directly follows from Claim~\\ref{cl:degbnd} and Claim~\\ref{cl:adm}.\n\\end{proof}\n\nGiven a graph $G$ that excludes $K_k$ as a topological minor, let us\nwrite $H(G)$ for a graph constructed according to\nLemma~\\ref{lem:constructingH}.\n\n\\begin{corollary}\nThe class $\\{H(G) : K_k\\not\\preccurlyeq^t G\\}$ has bounded expansion. \n\\end{corollary}\n\nWe can now use Theorem~\\ref{thm:eickmeyer-combine} to combine the\nfollowing result of Dvo\\v{r}ak et al.~\\cite{dvovrak2013testing} with\nLemma~\\ref{lem:constructingH}, to prove fixed-parameter tractability\nof successor-invariant $\\mathrm{FO}$ on classes that exclude a fixed\ntopological minor.\n\n\\begin{lemma}[Dvo\\v{r}\\'ak et al.~\\cite{dvovrak2013testing}]\n  The model-checking problem for first-order formulas is\n  fixed-parameter tractable on any class of bounded expansion.\n\\end{lemma}\n\n\\begin{corollary}\n  The model-checking problem for successor-invariant first-order\n  formulas is fixed parameter tractable on any class of graphs that\n  excludes a fixed topological minor.\n\\end{corollary}\n\n\n\\section{Conclusions}\n\nIn this work we gave several new applications of the generalised\ncolouring numbers on classes of bounded expansion.  In particular, we\nhave shown that whenever a graph class $\\mathcal{C}$ excludes some fixed\ntopological minor, then any graph from $\\mathcal{C}$ admits one ordering of\nvertices that certifies the boundedness of the generalised colouring\nnumbers for all radii $r$ at once.  It is tempting to conjecture that\nsuch an ordering exists for any graph class of bounded expansion.\n\nOur construction of the uniform ordering proved to be useful in\nshowing that model-checking successor-invariant $\\mathrm{FO}$ is FPT on any\ngraph class that excludes a fixed topological minor. We believe that\nour construction may be helpful in extending this result to any graph\nclass of bounded expansion, since both the construction of the order,\nand the reasoning of Section~\\ref{sec:mc}, are oblivious to the fact\nthat the graph class excludes some topological minor.  The only place\nwhere we used this assumption is the analysis of the constructed\norder.\n\n\\pagebreak\n\\bibliographystyle{plainurl}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nEver since the seminal observation by Polyakov \\cite{polyakov} that\nLiouville field theory describes two-dimensional gravity as coupled to\na relativistic string, much attention has been devoted to the problem\nof quantizing it, see\ne.g. \\cite{gernev,curtho,fadtak,seiberg,gervais,takhtajan} and\nreferences therein.\n\nDespite the fact that all the arsenals of canonical quantization\n\\cite{gernev,curtho} and path integral quantization\n\\cite{seiberg,takhtajan} have been used, much remains unclear about\nthe quantum theory, especially in the strongly coupled regime with\ncentral charge $1< c < 25$. Some encouraging results have been\nacquired for this region utilizing the underlying $sl_q(2)$\\ symmetry of\nthe quantum theory \\cite{gervais}, and by solving the conformal Ward\nidentities \\cite{takhtajan}.\n\nLiouville theory being a completely integrable model, the powerful\nmethods of the Quantum Inverse Scattering Method\\ \\cite{fadlhou82} should be applicable to\nquantizing it. In the spirit of this method, Liouville theory was put\non lattice in \\cite{fadtak}, but due to peculiarities of the\nintegrable structure of Liouville theory, only partial quantum results\nwere achieved. The concept of lattice Liouville theory, and the\nparallel concept of lattice Virasoro algebra were further developed\nin \\cite{volkov,fadvol,babelon}.\n\nIn this paper we take a somewhat different approach to putting\nLiouville on the lattice, more suitable for applying the methods of\nthe algebraic Bethe Ansatz\\ (see \\cite{fadBA} for recent reviews).  Our\nconventions for classical continuum Liouville theory are explained in\n{\\bf section 2}.\n\nIn order to use the full power of the Quantum Inverse Scattering Method, one needs a $L$-operator\nwhich depends non-trivially of the spectral parameter. The\n$L$-operators used in quantum Liouville theory have so far lacked this\nproperty. In {\\bf Section 3} we shall find a remedy for this\nshortcoming, and thus the methods of the algebraic Bethe Ansatz\\ are in our use.\n\nFollowing the line of thought of \\cite{izekor}, we find in {\\bf Section\n4} a pseudovacuum for the product of $L$-operators on two adjacent\nsites, and derive Bethe Ansatz equations\\ for lattice Liouville theory. The equations\ncan be regarded  as  Bethe Ansatz equations\\ for a XXZ spin chain with spin\n$(-{\\textstyle \\frac{1}{2}}\\mm)$, with an extra phase factor depending on the length of the\nchain $N$. More exactly, the extra phase is related to the $N$:th\npower of the XXZ anisotropy $q=\\e{\\i\\gamma}$, where $\\gamma$ is the\nLiouville coupling constant.\n\nIn {\\bf Section 5} we then map the spin $(-{\\textstyle \\frac{1}{2}}\\mm)$ Bethe equations to\nthe paradigmatic spin $(+{\\textstyle \\frac{1}{2}}\\mm)$ ones with an extra phase factor. This\nis done in the string approach \\cite{taksuz} to exited states in the\nthermodynamic Bethe Ansatz.  The mapping from Liouville to XXZ is successful\nonly for certain root of unity anisotropies $q$; the Liouville\ncoupling has to be of the form $\\gamma=\\pi\\trac{\\nu}{\\nu+1}$, with $\\nu$\nan integer.  There is a reciprocal one to one correspondence between\nBethe states in the lattice Liouville model and the spin ${\\textstyle \\frac{1}{2}}\\mm$ XXZ chain; highest\nstrings are mapped to 1-strings and vice versa.\n\nIn \\cite{blocani,affleck,cardy} it was shown that conformal properties\nof two-dimensional statistical models at criticality can be extracted\n{}from the finite (but large) size corrections to the eigenvalues of the\ntransfer matrix. Based on this, systematic methods for analytical\ncalculation of the finite size corrections\\ have been developed\n\\cite{devewoy,karowski,kluwezi,boizre}.  For a spin chain or lattice\nmodel, the finite size analysis yields $\\operN$ corrections to the\neigenvalues of the transfer matrix, where $N$ is the number of lattice\nsites.\n\nThe calculation by Karowski \\cite{karowski} of the conformal weights\nof string exited states in six-vertex and related Potts models will be\nof particular interest to us. In {\\bf Section 6} we review these\nresults to for the $\\operN$ corrections to a spin ${\\textstyle \\frac{1}{2}}\\mm$ XXZ chain\\ with extra\nphase factor.\n\nFinally, in {\\bf Section 7} we interpret the results in terms of\nlattice Liouville theory. As the extra phase factor,\n$\\exp\\{N\\pi\\trac{\\nu}{\\nu+1}\\}$ is a $\\nu+1$:th root of unity, the\nthermodynamic limit $N\\to\\infty$ of our system of Bethe Ansatz equations\\ should be\ntaken in steps of $\\nu+1$. The phase factor thus becomes a function of\nthe remnant $\\kap = N\/2 \\ \\mbox{mod}\\  (\\nu+1)$.\n\nThe finite size results show that different remnants correspond to\ndifferent exited states, thus generalizing the property of\nspin chains that chains of even and odd length have different spectra.\nThe scaling properties of the antiferromagnetic XXZ vacuum with $\\kap=1$\nyield the central charges of the minimal models \\cite{bpz}\nbelonging to the unitary series of Friedan Qiu and Shenker \\cite{fqs}.\n\nFor $\\kap=0$ we get an ``unrestricted'' sector with central charge $c=1$.\nExcluding this sector from the theory corresponds exactly to the RSOS\nreduction of a critical SOS model \\cite{anbafo} . The spin ${\\textstyle \\frac{1}{2}}\\mm$\nBethe equations with the extra phase  are exactly the Bethe\nequations of a SOS model at criticality. The restriction of these\nmodels is known to produce unitary conformal field theories\\ with $c<1$\n\\cite{huse}.\n\nIt is very plausible that the exclusion of the $\\kap=0$ sector of the\nHilbert space of the theory should be connected to an unitarity\nanalysis in terms of ``good'' representations of the underlying\n$sl_q(2)$\\ symmetry of the theory, c.f. \\cite{jutkar}.\n\nWith the result of \\cite{karowski} that XXZ primary states emerge from\na single string exited over the antiferromagnetic vacuum, we then have\ntwo integers to parameterize exited states: the string length $K$ and\nthe remnant $\\kap$.  Working in the restricted sector, we recover from\nthe finite size corrections\\ the conformal weights corresponding to the whole Ka\\v c\\  table\nof primary states in unitary conformal field theories, parameterized by these two\nintegers.\n\nWe view the results announced here as encouraging when it comes to the\nanalysis of quantum Liouville theory. Generalizing our approach to a\nwider class of Liouville coupling constants, possibly along the lines of\n\\cite{gervais}, might shed more light on the evasive strong coupling regime.\n\nIn addition, doing  the   mapping  of Section 5 in the inverse\ndirection, we see how the Bethe equations of a host of critical\nstatistical models (six-vertex, Potts, III\/IV-critical RSOS) can be\nmapped to the lattice Liouville ones. We regard this as an explicit\nproof of the conformal invariance of these theories at criticality,\nwhich gives a possibility to find a Lagrangian description of the\ncorresponding critical field theories.\n\n\n\\section{Classical Liouville theory}\n\nWe shall be quantizing Liouville theory on a Minkowskian cylinder,\nwith the basic field $\\Phi(x,t)$.  The space coordinate is periodic,\n$x\\in[0, 2\\pi]$, and time is non-compact, as usual: $t \\in [-\\infty,\n\\infty]$.  We take the classical Liouville action in the form\n\\be\n \\S= \\opertgam \\int_0^{2\\pi} dx\\mm\\left\\{{\\textstyle \\frac{1}{2}}\\mm \\dot\\Phi^2 - {\\textstyle \\frac{1}{2}}\\mm \\Phi'^2\n - 2 \\e{-2 \\i \\Phi}   + 2\\i \\Phi''\\right\\} \\.\n \\mabel{claction}\n\\ee\nAs usual,  $\\dot\\Phi \\equiv \\frac{\\partial}{\\partial t} \\Phi(x,t)$ and $\\Phi' \\equiv\n\\frac{\\partial}{\\partial x} \\Phi(x,t)$.  The so called conformal improvement\nterm $\\i \\Phi''$ is required to make the conformal invariance\nmanifest. It can be considered as the flat space residue of the\ncoupling to the scalar curvature $\\sqrt{g} \\mm R(g)\\mm\\Phi$ of the\nLiouville action on a generic Riemann surface, see\ne.g. {\\cite{seiberg}}.\n\nAs it stands, the action of Equation {\\refe{claction}} seems very\nnon-unitary, but as we shall see in the course of this paper, and as\ncan be inferred from {\\cite{gernev,curtho}}, for specific values of\nthe coupling $\\gamma$ the complexities conspire  to yield an\nunitary theory after quantization.\n\nThe equation of motion corresponding to {\\refe{claction}} is\nLiouville's equation (with an imaginary field),\n\\be\n \\ddot\\Phi - \\Phi'' - 4 \\i \\e{-2\\i\\Phi} = 0 \\.\n \\mabel{lioueq}\n\\ee\n In our parameterization the coupling constant does not appear in the\nequation of motion, it only multiplies the action.  By redefining the\nfield it is easy to recover the usual way\n\\cite{gernev,curtho,seiberg} of having the coupling in the\nexponential.\n\nTo unravel the conformal invariance of the theory, it is easiest to move\nover to the Hamiltonian picture.  Taking the conjugate momentum to be\n$\\Pi = {\\textstyle \\frac{1}{2}}\\mm \\dot\\Phi$, with Poisson Brackets\n\\be\n \\{ \\Phi(x),\\mm \\Pi(y) \\} = - \\gamma \\mm\\delta(x-y) \\,\n \\mabel{pbs}\n\\ee\n we get the conformally improved Hamiltonian\n\\[\n H = \\int dx\\ \\H = {\\textstyle \\frac{1}{\\gamma}}\\mm \\int dx\\mm\\left\\{\n \t\\Pi^2 + {\\textstyle \\frac{1}{4}}\\mm \\Phi'^2 + \\e{-2 \\i \\Phi}  - \\i \\Phi'' \\right\\}\n\\]\n and  the improved momentum\n\\[\n P = \\int dx\\ \\P = {\\textstyle \\frac{1}{\\gamma}}\\mm \\int dx\\mm\\left\\{\n \t\\Phi' \\Pi - {\\textstyle \\frac{{\\rm i}}{2}}\\mm\\Pi' \\right\\} \\.\n\\]\n According to the prescriptions of radial quantization \\cite{bpz} we\nknow that on a cylinder the role of the lightcone energy-momentum\ntensor is played by the sum of energy and momentum densities\n\\be\n s_+(x) = \\H(x) + \\P(x) \\. \\mabel{esplus}\n\\ee\n This sum generates the current algebra\n\\[\n \\{s_+(x), \\mm s_+(y)\\} = 2 \\mm (s_+(x) + s_+(y))\n \\mm\\delta'(x-y) \\mm + \\mm {\\textstyle \\frac{1}{\\gamma}}\\mm \\delta'''(x-y)\n\n\\.\n\\]\n Upon Fourier expanding the light-cone energy-momentum, we get a copy\nof the classical version of the Virasoro algebra,\n\\[\n \\i \\{L_n, L_m \\} = (n-m)L_{m+n}\n  - \\frac{\\pi}{2\\gamma}(n^3 - n)\\delta_{n+m,0} \\,\n\\]\n from which we read off the classical central charge\n\\[\n c= - 6 \\pipergam \\.\n\\]\n The difference $s_-(x)\n= \\H(x) -\\P(x)$ generates another, commuting copy of this algebra.\n\n In \\cite{curtho} this system was canonically quantized using a normal\nordering prescription to cope with divergences. The\nquantum corrections shifted the central charge to\n\\[\nc= 1-6(\\pipergam + \\gamperpi -2) \\,\n\\]\n which yields minimal theories for $\\gamperpi$ rational, i.e. when the\ndeformation parameter $q=\\exp{i\\gamma}$ is a root of unity. The subset\n$\\gamperpi = \\frac{\\nu}{\\nu+1}, \\ \\ \\nu=2,3,\\ldots$ corresponds to\nunitary theories. Similar results were obtained in \\cite{gernev}.\n\n\n\\section{An L-matrix for Liouville Theory}\n\nInstead of normal ordering, we shall regularize the ultraviolet\ndivergencies by putting Liouville theory on  a lattice, in a way that\npreserves the integrability of the model.\n\nTo get into a position where the methods of the algebraic Bethe\nAnsatz (see e.g. \\cite{fadBA}) can be used, we have to find a spectral\nparameter dependent quantum Lax operator for lattice Liouville theory.\nTo do this, we shall consider Liouville theory as the massless\nlimit of sine-Gordon theory. Indeed, the sine-Gordon equation of\nmotion\\footnote{the peculiar choice of normalization of sine-Gordon\nmass makes the quantum group structure more transparent in the sequel.}\n\\[\n \\mm\\Box\\mm\\Phi + 8 m^2 \\sin(2\\Phi) = 0\n\\]\n  goes into  the (imaginary) Liouville Equation  \\refe{lioueq}, if we\nrescale the field:\n\\be\n \\Phi \\to \\Phi + \\i\\zeta \\,\n \\mabel{scale}\n\\ee\n and take the limit\n\\be\n m\\to 0 \\; \\zeta\\to\\infty\\, \\  \\ \\mbox{ so that}\\ \\  m\\e{\\zeta} = 1 \\.\n \\mabel{limit}\n\\ee\n\n\n\n\\subsection{The Sine-Gordon L-matrix}\n\nFor lattice sine-Gordon there exists a Lax-operator which is based on\na infinite dimensional representation of the underlying quantum group\n$sl_q(2)$\\ \\cite{izekor,fadBA}. In this approach, lattice sine-Gordon is\ntreated as an inhomogeneous XXZ spin chain.\n\nIn an auxiliary matrix space $a$ of $2\\times2$ matrices, we  define the\n$L$-operator  of the $n$:th site of a XXZ-chain to be the  matrix operator\n\\be\n L_{n,a}^{{\\rm xxz}}(\\lambda) = \\mat{ll}{\n \\sinh(\\lambda + \\i\\gamma\\mm S^{(n)}_3) &  \\i  S^{(n)}_- \\mm \\sin\\gamma\\\\\n \\null&\\null\\\\\n \\i  S^{(n)}_+ \\mm \\sin\\gamma& \\sinh(\\lambda - \\i\\gamma\\mm S^{(n)}_3) } \\,\n \\mabel{Lxxz}\n\\ee\n a function of the spectral parameter $\\lambda$.  If the quantum\noperators defined on the sites generate the quantum group $sl_q(2)$,\n\\be\n \\com{S^{(n)}_+}{S^{(m)}_-} = \\frac{\\sin(2\\gamma S^{(n)}_3)}{\\sin\\gamma}\n\\ \\delta_{n,m}\n\\;  \\com{S^{(n)}_3}{S^{(m)}_\\pm} = \\pm S^{(n)}_\\pm \\ \\delta_{n,m} \\,\n \\mabel{QG}\n\\ee\n the L-operators \\refe{Lxxz} acting on auxiliary spaces $a_1$ and $a_2$\nfulfill the fundamental commutation relations (FCR)\n\\bea\n R_{12}(\\lambda-\\mu) \\stack{1}{L}_n(\\lambda) \\stack{2}{L}_n(\\mu) &=&\n\\stack{2}{L}_n(\\mu) \\stack{1}{L}_n(\\lambda) R_{12}(\\lambda-\\mu)\n \\mabel{FCR} \\\\\n\\stack{1}{L}_n(\\lambda) \\stack{2}{L}_m(\\mu) &=&\n\\stack{2}{L}_m(\\mu) \\stack{1}{L}_n(\\lambda)\\,\\ \\mbox{for}\\ m\\neq n\\.\n \\nonumber\n\\eea\n Here the usual notation for matrices on the tensor product of two\nauxiliary spaces is adopted; $\\stack{1}{L}_n \\equiv L_{n,a_1}\\otimes\n1\\!\\!\\hskip 1pt\\mbox{l}_{a_2}$ etc.\n\nThe FCR encode the integrability of the system, and they are the basis\nof utilizing the algebraic Bethe Ansatz. The XXZ chain belongs to the\nclass where the R-matrix is trigonometric:\n\\be\n R=\\mat{cccc}{\\alpha & \\null& \\null& \\null \\\\\n              \\null & \\beta& \\delta& \\null \\\\\n              \\null & \\delta& \\beta& \\null \\\\\n\t      \\null & \\null& \\null& \\alpha}  \\;\n                               \\begin{array}{l}\\alpha=\\sinh(\\lambda+\\i\\gamma)\\\\\n\t\t\t\t\t       \\beta=\\sinh\\lambda\\\\\n\t\t\t\t\t       \\delta=\\i\\sin\\gamma \\end{array}\\.\n\\mabel{R}\n\\ee\n The ordered product of L-matrices around the periodic chain is the\nmonodromy matrix\n\\be\n T(\\lambda) = L_{\\sN,a}(\\lambda)\\mm L_ {\\sss{ N-1},a}(\\lambda)\n\\mm\\ldots\\mm L_{\\sss{1},a}(\\lambda) \\equiv \\mat{cc}{\\A(\\lambda)&\\B(\\lambda)\\\\\n\t\t\t\t\t    \\C(\\lambda)&\\D(\\lambda)} \\.\n \\mabel{monodromedary}\n\\ee\n The conserved quantities can now be expressed as traces (over the\nauxiliary space $a$) of powers of $T$, which all commute due to the\nfundamental commutation relations. The trace of $T$ over the auxiliary space can be interpreted as\nthe row-to-row transfer matrix of the corresponding two-dimensional\nstatistical model,\n\\be\n \\tau(\\lambda) = \\mbox{Tr}_a\\bigl(T(\\lambda)\\bigr) = \\A \\mm + \\mm \\D \\.\n\\mabel{transfer}\n\\ee\n\n\n\n\n\\vali\n\nTo get a $L$-matrix for sine-Gordon, we use an infinite dimensional\nrepresentation of the quantum group, generated by the canonical\nvariables $\\Phi_n,\\mm \\Pi_n$ with commutation relations\n\\be\n \\com{\\Phi_n}{\\Pi_m} = \\i\\gamma\\mm\\delta_{mn} \\.\n \\mabel{commutator}\n\\ee\n Now we can write  the generators\n\\be\n S_3^{(n)} = - {\\textstyle \\frac{1}{\\gamma}}\\mm\\Phi_n \\;   S_\\pm^{(n)}\n  = \\frac{1}{2 m \\sin\\gamma} \\e{\\pm{\\textstyle \\frac{{\\rm i}}{2}}\\mm\\Pi_n}\n\\Bigl(1 + m^2\\e{2\\i\\Phi_n}\\Bigr) \\e{\\pm{\\textstyle \\frac{{\\rm i}}{2}}\\mm\\Pi_n} \\,\n \\mabel{QGgen}\n\\ee\n which fulfill the commutation relations \\refe{QG}.\n\n\n Using Generators \\refe{QGgen} in Equation \\refe{Lxxz} and multiplying with\nthe matrix $-2 m \\i \\mm \\sigma_1$, we get the $L$-matrix of lattice\nsine-Gordon theory\n\\bea\n L^{SG}_{n,a} &=& -2 m\\i\\mat{rr}{0&1\\\\1&0} L_{n,a}^{{\\rm xxz}} \\cr\n &\\null& \\cr &=&  \\mat{ll}{\n   h_+(\\Phi_n)\\e{\\i\\Pi_n} & -2m\\i\\sinh(\\lambda + \\i\\Phi_n) \\\\\n -2m\\i\\sinh(\\lambda - \\i\\Phi_n)  &   h_-(\\Phi_n)\\e{-\\i\\Pi_n}}  \\.\n \\mabel{sGL}\n\\eea\n Here we have denoted\n\\[\n h_\\pm(\\Phi) \\equiv  1 + m^2 \\e{\\pm 2\\i\\Phi +\\i\\gamma} \\.\n\\]\n Multiplying $L$ with $\\sigma_1$ is a symmetry of the FCR, so\nL-matrix \\refe{sGL} still satisfies Equation \\refe{FCR}.\n\n\n\\subsection{Massless Limit of the  Sine-Gordon L-matrix}\n\nNow we perform the scaling and limiting procedure of Equations\n(\\ref{scale},\\ref{limit}) on the sine-Gordon L-matrix \\refe{sGL}, in\norder to get a $L$-matrix for lattice Liouville theory.\n\nThe functions $h_\\pm$ scale to\n\\[\n h_+ \\to 1 \\; h_- \\to 1 - \\e{-2\\i\\Phi + i\\gamma} \\equiv  h \\.\n\\]\n Performing\n(\\ref{scale},\\ref{limit}) directly on \\refe{sGL}, we thus get\n\\be\n \\tilde L_{n,a}^{\\L} =  \\mat{ll}{\n \\e{\\i\\Pi_n} & -\\e{-\\lambda - \\i\\Phi_n} \\\\\n \\e{\\lambda - \\i\\Phi_n}  &    h(\\Phi_n)\\e{-\\i\\Pi_n}}  \\. \\mabel{trivL}\n\\ee\n This $L$-matrix with spectral parameter was acquired in\n\\cite{fadtak}. The $\\lambda$-dependence of $ \\tilde L_{n,a}^{\\L}$ is\ntrivial, though, it can be removed by a lattice gauge\ntransformation. In other words, the quantum determinant (the central\nelement of the algebra generated by the elements of $\\tilde L_{n,a}$) is\nindependent of $\\lambda$.  If the $\\lambda$ dependence is removed, we are\nleft with the constant $L$-operators inherent in the approaches of\n\\cite{fadtak,gervais,babelon}. As such, this $L$-matrix is not\nviable for the Bethe Ansatz. However, let us comment that work with Lax\noperator {\\refe{trivL}} leads naturally to the lattice deformation of\nVirasoro algebra  \\cite{fadtak,volkov,fadvol,babelon}.\n\nTo get a Liouville $L$-matrix with non-trivial spectral parameter\ndependence, we have to manipulate \\refe{sGL} in a more involved\nway.\\footnote{This prescription was communicated to us by A. Volkov.}\nFirst, we perform a constant lattice gauge transformation on\n\\refe{sGL}:\n\\[\n L^{SG}_{n,a } \\to g \\mm L^{SG}_{n,a} \\mm g^{-1} \\;\n g = \\mat{ll}{m^{{\\textstyle \\frac{1}{2}}\\mm}&\\null\\\\ \\null & m^{-{\\textstyle \\frac{1}{2}}\\mm} }\n\\]\n Then we make the scaling \\refe{scale}, accompanied by a\nrenormalization of the spectral parameter:\n\\be\n \\lambda \\to \\lambda - \\zeta \\.\n \\mabel{renorm}\n\\ee\n After scaling and renormalizing, the limiting procedure \\refe{limit}\nproduces\n\\be\n g \\mm L^{SG}_{n,a} \\mm\\mm g^{-1} \\to \\mat{ll}{\n \\e{\\i\\Pi_n} & -\\i\\e{-\\lambda - \\i\\Phi_n} \\\\\n -2\\i\\sinh(\\lambda - \\i\\Phi_n)  &    h(\\Phi_n)\\e{-\\i\\Pi_n}}\n \\equiv   L^{\\L}_{n,a} \\.\n \\mabel{Lliou}\n\\ee\n\n\nThis is indeed the sought for quantum $L$-matrix for Liouville theory\nwith a non-trivial spectral parameter, corresponding to an integrable\nlattice regularization of the Liouville system described by Action\n\\refe{claction}.\n\nThe easiest way to see that $L$-matrix \\refe{Lliou} corresponds to an\nintegrable lattice version of quantum Liouville theory, is to take the\nclassical continuum limit of \\refe{Lliou} , and find the corresponding\nclassical continuum dynamics.\n\nTo define the classical limit, one has to recover Planck's constant.\nThis is achieved by reinterpreting $\\gamma \\to \\hbar\\gamma$ in all quantum\nexpressions. In the classical limit commutators turn into Poisson\nBrackets according to the usual Heissenberg correspondence,\n$\\frac{\\i}{\\hbar}[,] \\to \\{,\\}$.\n\nSimilarly, to find the continuum limit the lattice spacing $a$ has to\nbe recovered.  The lattice mass in \\refe{sGL} should be $m = a\\mm m'\n$, and the {\\it continuum} mass $m'$ should be taken to zero according\nto Equation  \\refe{limit}.\n\nThe continuum variables are defined by\n\\[\n \\Pi_n \\to a \\mm\\Pi(x)\\; \\Phi_n \\to \\Phi(x) \\;\n \\delta_{mn} \\to a\\mm\\delta(x-y) \\,\n\\]\n which maps the discrete brackets corresponding to \\refe{commutator}\nto the continuous ones of Equation  \\refe{pbs}.\n\nNow we get in the classical continuum limit the matrix\n\\be\n U(x,\\lambda) = \\lim_{a, \\hbar \\to 0} \\mm \\trac{1}{\\i a} \\mm\n(L^\\L - 1\\!\\!\\hskip 1pt\\mbox{l}) = -\\mat{ll}{\n \t-\\Pi(x) & \\e{-\\lambda - \\i\\Phi(x)} \\\\\n\t 2 \\sinh(\\lambda + \\i\\Phi(x))  &   \\Pi(x)}  \\.\n \\mabel{U}\n\\ee\n As the classical limit of  fundamental commutation relations\\ \\refe{FCR}, $U$ satisfies the so called\nfundamental Poisson brackets\n\\[\n \\pois{\\stack{1}{U}(x,\\lambda)}{\\stack{2}{U}(y,\\mu)} =\n \\i\\gamma\\com{r_{12}(\\lambda-\\mu)}{\\stack{1}{U}(x,\\lambda) +\n \t\t\\stack{2}{U}(x,\\mu)} \\mm \\delta(x-y) \\,\n\\]\n  with the trigonometric classical $r$-matrix\n\\[\n r(\\lambda) = \\lim_{\\hbar \\to 0} \\mm\\frac{-1}{\\i\\hbar} \\mm\n\\left(\\frac{R(\\lambda)}{\\sinh\\lambda} - 1\\!\\!\\hskip 1pt\\mbox{l}\\right)\n = \\frac{-1}{\\sinh\\lambda}\\mat{cccc}{\n              \\cosh\\lambda & \\null& \\null& \\null \\\\\n              \\null & 0& 1& \\null \\\\\n              \\null & 1& 0& \\null \\\\\n\t      \\null & \\null& \\null& \\cosh\\lambda} \\.\n\\]\n\nTogether with the matrix\n\\[\n V(x,\\lambda) = - \\mat{ll}{\n \t{\\textstyle \\frac{1}{2}}\\mm\\Phi'(x) & \\e{-\\lambda -\\i\\Phi(x)} \\\\\n\t2 \\sinh(\\lambda - \\i\\Phi(x))  &   -{\\textstyle \\frac{1}{2}}\\mm\\Phi'(x)}   \\,\n\n\\]\n  the classical L-matrix $U$ forms  a Lax-pair for Liouville theory:\n\\[\n \\dot U + V' + i\\com{U}{V}  = 0\n \\iff \\mm\\Box\\mm\\Phi - 4\\i\\e{-2\\i\\Phi} = 0 \\.\n\\]\n This Lax-pair can be acquired in the scaling and limiting procedure\n(\\ref{scale},\\ref{renorm},\\ref{limit}) from the Lax-pair of classical\nsine-Gordon in Reference \\cite{fadkor}.\n\nA. Volkov brought into our attention the fact that Lax-matrix\n{\\refe{U}} is gauge equivalent to\n\\[\n \\tilde U(x,\\mu) = \\mat{cc}{0& \\  \\mu - s_+\\\\\n\t\t          1&0} \\,\n\\]\n where $\\mu = \\exp{2\\lambda}$, and $s_+$ is the energy-momentum density of\nEquation \\refe{esplus}. The Lax Equation turns into the Schr\\\"odinger\nequation\n\\be\n -\\psi'' + s_+\\psi = \\mu \\psi \\,\n \\mabel{scroe}\n\\ee\n which usually is used in connection to the KdV equation. The role of\nEquation \\refe{scroe} for Liouville theory is stressed in {\\cite{dzpopota}}.\n\n\n\\section{Bethe Ansatz for lattice Liouville theory}\n\nIn the algebraic Bethe Ansatz, one tries to triangularize the local $L$-operators in\norder to diagonalize the transfer matrix  over the\nchain, which yields the conserved quantum quantities. This is done by\nfinding a local pseudovacuum, which is annihilated by one of the\noff-diagonal components in $L_{n,a}$. In addition, to get an\neigenstate of the transfer matrix, the pseudovacuum should be an\neigenstate of the diagonal components.\n\nThe $L$-operator \\refe{Lliou} developed in the previous section does\nnot have a local pseudovacuum.  However, as is the case with\nit's ancestor $L^{SG}$ \\cite{izekor}, the product of two $L^\\L$:s from\nadjacent sites indeed has a pseudovacuum.\n\nWe denote $\\Phi_{2n} = \\Phi_2\\mm ;\\ \\Phi_{2n-1} = \\Phi_1$, and\nsimilarly for $\\Pi$.  The product of two Lax-operators is thus\n\\be\n \\L = L^\\L_{2n,a}\\mm L^\\L_{2n-1,a} \\mm =  \\mat{cc}{A&B\\\\ C&D}   \\,\n \\mabel{LL}\n\\ee\n with\n\\beastar\n A &=& \\e{\\i(\\Pi_2+\\Pi_1)} -2 \\e{-\\lambda - \\i\\Phi_2} \\sinh(\\lambda - \\i\\Phi_1) \\\\\n B &=& -\\i\\e{-\\lambda+\\i(\\Pi_2 -\\Phi_1)}\n            -\\i h(\\Phi_1)\\e{-\\lambda - \\i(\\Phi_2 +\\Pi_1)} \\\\\n C &=&  -2\\i\\sinh(\\lambda -\\i\\Phi_2)\\e{\\i\\Pi_1}\n   - 2 \\i h(\\Phi_2)\\e{-\\i\\Pi_2}\\sinh(\\lambda - \\i\\Phi_1) \\\\\n D &=&   h(\\Phi_2)  h(\\Phi_1) \\e{-\\i(\\Pi_2+\\Pi_1)}\n  - 2 \\sinh(\\lambda -\\i\\Phi_2)\\e{-\\lambda-\\i\\Phi_1}\n\\end{eqnarray*}} %%% eeastar \\> {end{eqnarray*}\n\nInspired by \\cite{izekor,fadBA} we  make the Ansatz\n\\[\n \\psi = f(\\Phi_1) \\mm\\delta(\\Phi_1-\\Phi_2 -\\gamma)\n\n\\]\n for the pseudovacuum at site $n$, with $f$ a functional to be\ndefined.  Demanding that the off-diagonal operator $C$ annihilates\nthe vacuum, we get the functional relation\n\\be\n f(\\Phi+\\gamma) = - h(\\Phi) \\mm f(\\Phi) \\.\n \\mabel{functrel}\n\\ee\n This equation sets very stringent conditions on the function $f$. It\nis a  sufficient condition for  the pseudovacuum;\nwhen it is fulfilled, the actions of $A$ and $D$ on $\\psi$ are\ndiagonal, with the eigenvalues:\n\\bea\n A\\mm\\psi &=& (\\!\\e{-2\\lambda +\\i\\gamma} -1)\\mm \\psi \\equiv a(\\lambda)\\mm\\psi \\cr\n D\\mm\\psi &=& (\\!\\e{-2\\lambda -\\i\\gamma} -1)\\mm \\psi \\equiv d(\\lambda)\\mm\\psi  \\.\n \\mabel{aetd}\n\\eea\n\nThe treatment of Functional Relation \\refe{functrel} depends crucially\nof the value of $q~=~\\e{\\i\\gamma}$.  When $|q| < 1$, Equation\n\\refe{functrel} is exactly fulfilled by the quantum dilogarithm, see\nRef. \\cite{fadkas}. Following \\cite{fadcur}, we get an explicit solution\nfor the case $|q| =1$ as well, which  is of interest to this paper:\n\\[\n f(\\Phi) = \\exp{\\mm \\int_{-\\infty}^\\infty\n\\frac{dx}{4x}\\mm\\mm \\frac{\\e{(\\gamma-\\pipert-\\Phi) x}}\n{\\sinh\\pipert x \\sinh{\\textstyle \\frac{\\gamma}{2}}\\mm x}\n} \\mm\\.\n\\]\n  The singularity of the integral at $x=0$ is left under the\nintegration path.  For more discussion on solutions of \\refe{functrel}\nand other similar functional equations we refer to\n\\cite{volkov,fadvol,fadkas,fadcur}.\n\n{}From the local pseudovacuums $\\psi_n$ we can now build up a total\npseudovacuum for the quantum chain:\n\\[\n \\Psi = \\otimes_{n=1}^N \\psi_n \\,\n\\]\n where  the amount of paired sites is denoted by $N$.\n\nThe two-site L-operators \\refe{LL} become triangular when acting on the\nlocal vacuum. Thus the monodromy \\refe{monodromedary} acting on the\ntotal pseudovacuum $\\Psi$ is triangular as well:\n\\[\n T(\\lambda) \\mm\\Psi = \\mat{cc}{a^N(\\lambda)\\mm\\Psi & *\\\\\n\t\t\t    0 & d^N(\\lambda)\\mm\\Psi } \\.\n\\]\n The star in the upper right corner denotes a complicated state\ncreated by various combination of $A$:s, $D$:s and one $B$ acting on\n$\\Psi$.\n\nCorrespondingly, the transfer matrix \\refe{transfer} has the\npseudovacuum eigenvalue\n\\be\n \\tau(\\lambda)\\mm \\Psi = (a^N(\\lambda)\\mm + \\mm  d^N(\\lambda))\\mm\\Psi \\.\n \\mabel{eigentransfer}\n\\ee\n\n\n\\subsection{The Bethe Ansatz Equations}\n\n\nIn the algebraic Bethe Ansatz\\ one makes the assumption that the exited states of the\ntheory can be obtained from the pseudovacuum by acting on it with the\n``pseudoparticle creation operator'' $\\B$.  An arbitrary state of the\nform\n\\be\n \\Psi_m = \\prod_{j=1}^m \\B(\\lambda_j) \\mm\\Psi\n\\mabel{psim}\n\\ee\n is characterized by $m$ values of the spectral parameter $\\lambda_j$.\nThis state is an eigenstate of the trace of the monodromy,\ni.e. $\\A+\\D$, if the spectral parameters $\\set{\\lambda_j}$ satsify a\nset of $m$ coupled trancendental equations known as the Bethe Ansatz equations.\nWritten in terms of  the components\nof $R$-matrix \\refe{R} and the eigenvalues $a^N, d^N$ of $\\A$\nand $\\D$, these equations read\n\\be\n \\left[\\frac{a(\\lambda_k)}{d(\\lambda_k)}\\right]^N =\n\\prod_{\\stackrel{j=1}{ j\\neq k}}^m\n \\frac{\\alpha(\\lambda_k-\\lambda_j)\\beta(\\lambda_j-\\lambda_k)}{\\alpha(\\lambda_j-\\lambda_k)\n                                            \\beta(\\lambda_k-\\lambda_j)}\n \\ \\ \\forall\\ k\\.\n \\mabel{fBAE}\n\\ee\n\n Solutions to \\refe{fBAE} appear in sets of $m$ spectral parameters,\nso called Bethe Ansatz\\ roots, which parameterize different states of the\nsystem. A more refined analysis shows that generically all Bethe Ansatz\\ roots\nare distinct, and that $m\\leq N\/2$, see e.g. \\cite{fadBA}.\n\nFor lattice Liouville theory, the eigenvalues $a$ and $d$\nare given by Equation \\refe{aetd}, and the ratio on the left hand side of\nEquation \\refe{fBAE} is\n\\[\n \\frac{a(\\lambda)}{d(\\lambda)} = \\e{\\i\\gamma} \\BAEspsmi{\\lambda}{\\i{\\textstyle \\frac{\\gamma}{2}}\\mm} \\.\n\\]\n This leads to the Bethe Ansatz equations\n\\be\n \\e{\\i N\\gamma} \\left[\\BAEspsmi{\\lambda_k}{\\i{\\textstyle \\frac{\\gamma}{2}}\\mm}\\right]^N =\n\\prod_{j,\\mm j\\neq k} \\BAEspspl{\\lambda_k-\\lambda_j}{\\i\\gamma} \\ \\ \\forall\\  k\\.\n \\mabel{BAE}\n\\ee\n On the other hand, the Bethe Ansatz equations\\ of a spin-S XXZ chain are\n(see e.g. \\cite{fadBA})\n\\be\n  \\left[\\BAEspspl{\\lambda_k}{\\i S\\gamma}\\right]^N =\n \\prod_{j,\\mm j\\neq k} \\BAEspspl{\\lambda_k-\\lambda_j}{\\i\\gamma} \\ \\ \\forall\\  k\\.\n \\mabel{xxzBAE}\n\\ee\n  Comparing \\refe{BAE} to \\refe{xxzBAE} we see that in terms of the Bethe\nAnsatz, lattice Liouville theory corresponds to a spin $(-{\\textstyle \\frac{1}{2}}\\mm)$ XXZ\nspin chain, with an extra phase factor $\\e{\\i N\\gamma}$.\n\nPhase factors of the form $\\e{\\i \\theta}$ have appeared earlier in the\nliterature on the Bethe Ansatz. In \\cite{karowski} it was used to analyze Bethe Ansatz equations\\\nfor Potts models, related to a seam insertion of a boundary\nfield. Similarly, twisted boundary conditions in Reference\n\\cite{kluwezi} manifest themselves in the form of extra phase factors.\nMost interestingly, in the analysis of the eight-vertex model\n\\cite{baxter,takfad79}, there appears a ``theta-vacuum like'' phase\nfactor in the Bethe equations, related to different SOS-sectors of the\ntheory.\n\nHere, however, the important difference to the situations of\n\\cite{karowski,kluwezi,baxter,takfad79} occurs that the length $N$ of\nthe chain appears in the phase factor. When $q$ is a root of unity,\nthis will give us an extra integer parameter to parametrize\n``primary'' exited states, in addition to the string length used in\n\\cite{karowski}. These two integers then give us enough freedom to\nparameterize the whole Ka\\v c\\  table.\n\nThus different chain lengths occurring in the lattice Liouville model\ncorrespond exactly to the different ``theta-vacuum'' sectors in a\ncritical (R)SOS model.\n\n\n\\section{Mapping  the Lattice Liouville Model\n\t\tto the Spin ${\\textstyle \\frac{1}{2}}\\mm$ XXZ Chain }\\mabel{maptoxxz}\n\n\\noindent\nThe thermodynamics \\cite{taksuz} and finite size effects\n\\cite{devewoy,karowski,kluwezi,boizre} of the Bethe Ansatz\\ of the fundamental spin\n$+{\\textstyle \\frac{1}{2}}\\mm$ representation of XXZ-chains have been widely discussed in\nthe literature. Accordingly it would be a desirable goal to map the\nspin $(-{\\textstyle \\frac{1}{2}}\\mm)$ Liouville Bethe equations \\refe{BAE} to the spin\n$+{\\textstyle \\frac{1}{2}}\\mm$ XXZ ones \\refe{xxzBAE}.  This can be done in the string\npicture \\cite{taksuz} of solutions to the Bethe Ansatz equations.\n\nIn order to do the mapping, we thus make the string hypothesis that in the\nthermodynamic limit $N\\to\\infty$, complex Bethe Ansatz\\ roots $\\set{\\lambda_j}$\ncluster to complexes of the form\n\\be\n \\lambda^{(\\sK)} = \\lambda_\\sK \\mm + \\mm k \\i\\gamma\\, \\ \\ k= -{\\textstyle \\frac{1}{2}}\\mm (K-1), \\mm\\ldots\\mm,\n \\mm{\\textstyle \\frac{1}{2}}\\mm(K-1) \\; \\Im\\lambda_\\sK\\in\\{0,\\pipert\\}\\, \\ \\ K\\in Z_+ \\.\n \\mabel{kstring}\n\\ee\n A collection of $K$ Bethe Ansatz\\ roots with a common center obeying\nEquation \\refe{kstring} is known as  a $K$-string. One-strings are the\nusual real Bethe Ansatz\\ roots. Strings with $\\Im\\lambda_\\sK = 0$ are called\npositive parity strings, if  $\\Im\\lambda_\\sK = \\pipert$ one speaks about\nnegative parity strings.\n\nIt is important to notice that when $q$ is a root of unity there\nis an upper limit to the length of the string $K$\n\\cite{taksuz}. With\n\\[\n \\left[\\e{\\i\\gamma}\\right]^{\\nu+1} = \\pm 1 \\,\n\\]\n the maximal string length is\n\\[\n K_{{\\rm max}} = \\nu.\n\\]\n This limit on the range of $K$ changes the usual concept of\ncombinatorical completeness of Bethe Ansatz\\ states.  In Ref. \\cite{jutkar} it\nwas argued for a $sl_q(2)$\\ invariant spin chain at $q$ root of unity\nthat the Bethe Ansatz\\ states exhibit completeness in the space of ``good''\nrepresentations of the quantum group.\n\n From now on we shall concentrate only on the root of unity case. More\nspecifically, we shall assume that the Liouville coupling constant\ntakes only the values\n\\[\n \\gamma = \\frac{\\pi\\nu}{\\nu +1} \\; \\nu\\in Z_+ +1 \\.\n\n\\]\n Using the string picture, we shall be able to map the states of a\nLiouville chain with ``anisotropy'' $\\gamma$ to a spin ${\\textstyle \\frac{1}{2}}\\mm$ spin\nchain with anisotropy\n\\be\n {\\tilde\\gamma} = \\frac{\\pi}{\\nu +1} \\; \\nu\\in Z_+ +1 \\.\n \\mabel{tilgam}\n\\ee\n\nWe begin with positive parity strings.  The number of $L$-strings is\n$n_\\sL$, and the total number of Bethe Ansatz\\ roots is\n\\[\n m = \\sum_{L=1}^\\nu L \\mm n_\\sL \\.\n\n\\]\n The rapidities of different $L$-strings are denoted $\\lambda_{\\sL,j}\\, \\\nj=1\\mm\\ldots\\mm n_\\sL$.\n\nMultiplying the Bethe Ansatz equations\\ \\refe{BAE} for all the rapidities comprising a\n$K$-string, we get a set of coupled equations of the {\\it\nreal parts} of the strings only.\n\nThe terms in the right hand side of (\\ref{fBAE}--\\ref{xxzBAE}), which\ndescribe the effect on scattering of pseudoparticles on each other,\nnow become scattering matrices of strings on strings.\n\nThe scattering of $K$-strings on 1-strings is described by the\nfunction\n\\bea\n S_{\\sss{ K 1}}(\\lambda) & \\equiv&\n\\prod_{k=-{{\\scriptscriptstyle \\frac{1}{2}(K-1)}}}^{{\\scriptscriptstyle \\frac{1}{2}(K-1)}}  S_{1 1}(\\lambda)\\ =\n\\prod_{k=-{{\\scriptscriptstyle \\frac{1}{2}(K-1)}}}^{{\\scriptscriptstyle \\frac{1}{2}(K-1)}}  \\BAEsps{\\lambda}{\\i k\\gamma} \\cr\n& &\\cr\n& &\\cr\n&=&  \\BAEsps{\\lambda}{{\\textstyle \\frac{1}{2}}\\mm(K+1)\\i\\gamma}\\\n\\BAEsps{\\lambda}{{\\textstyle \\frac{1}{2}}\\mm(K-1)\\i\\gamma} \\. \\mabel{Sk1}\n\\eea\n In terms of this function, we can write the scattering matrix\nof $K$-strings on $L$-strings as\n\\bea\n S_{\\sK\\sL}(\\lambda) &=& \\prod_{k=-{{\\scriptscriptstyle \\frac{1}{2}(K-1)}}}^{{\\scriptscriptstyle \\frac{1}{2}(K-1)}}\\\n\\prod_{l={{\\scriptscriptstyle \\frac{1}{2}(L-1)}}}^{{\\scriptscriptstyle \\frac{1}{2}(L-1)}}\\\n \\BAEsps{\\lambda}{(k-l+1)\\mm\\i\\gamma} \\cr\n& &\\cr\n&=&  \\prod_{k=\\sss{ \\jhalf |K-L|}}^{\\sss{ \\jhalf\n(K+L) - 1}} \\    S_{k1}(\\lambda) \\\n= \\prod_{k=\\sss{ \\jhalf |K-L|}}^{\\sss{ {\\rm\nmin} \\left[\\jhalf (K+L) - 1,\\ \\nu -\\jhalf(K+L)\\right]}} \\\nS_{k1}(\\lambda)\n\\.\n \\mabel{S}\n\\eea\n The first expression in terms of $S_{k1}$ bears close resemblance to\n(an exponential form of) the decomposition of the tensor product of\ntwo irreducible $SU(2)$ representations with spins ${\\textstyle \\frac{1}{2}}\\mm(K-1)$ and\n${\\textstyle \\frac{1}{2}}\\mm(L-1)$. Indeed, the Bethe Ansatz\\ can be viewed as a\ndifferent way of reducing tensor products of spin 1\/2 particles into\nirreducible representations. The factor $S_{\\sK 1}$ attached to a\nspin ${\\textstyle \\frac{1}{2}}\\mm(K-1)$  representation is obtained by fusing $K$\nfactors $S_{11}$ corresponding to spin 1\/2 representations. The\n$S$-matrix then naturally reflects the decomposition of the tensor\nproducts of the representation spaces involved in the scattering.\n\nThe second (``reduced'') expression in terms of $S_{k1}$ is peculiar\nto the root of unity case. It may be related to the decomposition of\nthe tensor product of two ``good'' quantum group representations,\nc.f. \\cite{jutkar}.  The reduced decomposition is related to the\nfusion rules of conformal blocks as treated in {\\cite{verlinde}},\nsuggesting that conformal blocks might be represented by Bethe Ansatz\\\nstrings. In the sequel we will see, how this is to be interpreted.\n\nThe reduced form shows explicitly the symmetry of\n$S_{\\sK\\sL}$ which will allow us to map the Liouville Bethe equations\non the spin ${\\textstyle \\frac{1}{2}}\\mm$ XXZ Bethe equations:\n\\be\n S_\\sss{ \\nu+1-K,\\mm\\nu+1-L}(\\lambda)\n = S_{\\sK,\\sL}(\\lambda) \\.\n \\mabel{Sklsym}\n\\ee\n\n\n Expressing the Liouville Bethe equations \\refe{BAE} in terms of\nstrings we get\n\\be\n \\e{\\i K N\\gamma} \\ \\left[\\BAEspsmi{\\lambda_{\\sK,i}}{\\i\\kpert\\gamma}\\right]^N\n = \\prod_{\\sL; \\ n_\\sL\\neq 0} \\\n   \\prod_{\\stackrel{j=1}{(\\sL,j)\\neq(\\sK,i)}}^{n_\\sL}\n   S_{\\sK\\sL}(\\lambda_{\\sK,i} - \\lambda_{\\sL,j})\\ \\ \\ \\ \\forall\\  (K,i)\\.\n \\mabel{sBAE}\n\\ee\n\nThe  amount of coupled equations is reduced to $\\sum_{\\sL = 1}^\\nu\nn_\\sL$, and all roots of the equations are now taken to be real.\n\nSimilarly, applying the string picture to Equation \\refe{xxzBAE} for\nspin ${\\textstyle \\frac{1}{2}}\\mm$, we get the spin ${\\textstyle \\frac{1}{2}}\\mm$ XXZ \\ Bethe equations in terms\nof strings,\n\\be\n \\left[\\BAEspsmi{\\lambda_{\\sK,i}}{\\i\\kpert\\gamma}\\right]^N\n = \\prod_{\\sL; \\ n_\\sL\\neq 0} \\\n   \\prod_{\\stackrel{j=1}{(\\sL,j)\\neq(\\sK,i)}}^{n_\\sL}\n   \\Bigl(S_{\\sK\\sL}(\\lambda_{\\sK,i} - \\lambda_{\\sL,j})\\Bigr)^{-1} \\ \\ \\ \\ \\forall\\  (K,i)\\.\n \\mabel{xxzsBAE}\n\\ee\n Notice that as compared to {\\refe{xxzBAE}}, we have inverted the equation.\n\nInspired by Equation \\refe{Sklsym}, we parameterize the string lengths\n{}from the maximal string backwards,\n\\[\n K = \\nu +1 - \\tilde K \\; L = \\nu +1 - \\tilde L\\; \\tilde K,\\tilde L =\n1, \\mm\\ldots\\mm , \\nu \\.\n \n\\]\n With this parameterization, it turns out that the Liouville Bethe Ansatz equations\\\n\\refe{sBAE} for the strings $\\set{\\lambda_{\\sL,j}}$ maps to the spin\n${\\textstyle \\frac{1}{2}}\\mm$ XXZ equations for a set of strings $\\set{{\\tilde\\lambda}_{\\tilde\n\\sL,j}}$ with the anisotropy ${\\tilde\\gamma}$.\n\nIndeed, after some algebra we have for the left hand side\n\\be\n \\e{\\i K \\gamma}\\  \\BAEspsmi{\\lambda_{\\sK,i}}{\\i\\kpert\\gamma} =\n \\e{\\i \\tilde K {\\tilde\\gamma}}\\\n\\BAEspsmi{{\\tilde\\lambda}_{\\tilde\\sK,i}}{\\i\\trac{\\tilde K}{2}{\\tilde\\gamma}} \\.\n \\mabel{pmap}\n\\ee\n The spectral parameter is changed in the following way:\n\\be\n {\\tilde\\lambda} = \\lambda - \\i (1+(-1)^\\sK) \\piperf \\, \\mabel{tillam}\n\\ee\n i.e. the parity of even-length strings is changed.  In addition, it\nturns out that\n\\[\n S^\\gamma_\\sss{ K 1}(\\lambda)\n= \\Bigl(S^{\\tilde\\gamma}_\\sss{ \\tilde K 1}(\\tilde\\lambda)\\Bigl)^{-1} \\,\n\\]\n where the upper index denotes whether $\\gamma$ or ${\\tilde\\gamma}$ is\nused when defining $S_\\sss{K1}$ according to Equation \\refe{Sk1}.\n\n Using this, as well as Symmetry Property \\refe{Sklsym}, we get for\nthe string-on-string scattering matrix \\refe{S}\n\\be\n S_{\\sK\\sL}^\\gamma (\\lambda_{\\sK,i} - \\lambda_{\\sL,j})\n =\\Bigl(S_{\\tilde\\sK\\tilde\\sL}^{\\tilde\\gamma}\n   ({\\tilde\\lambda}_{\\tilde\\sK,i}-{\\tilde\\lambda}_{\\tilde\\sL,j})\\Bigr)^{-1} \\.\n \\mabel{smap}\n\\ee\n\nThese results can easily be generalized to  negative parity Liouville\nstrings as well.\n\nNow we are in position to state the equivalence of lattice Liouville\nand spin ${\\textstyle \\frac{1}{2}}\\mm$ XXZ chain\\ Bethe Ans\\\"atze.  Using Equations (\\ref{pmap},\n\\ref{smap}) in \\refe{sBAE} and comparing to Equation \\refe{xxzsBAE},\nwe get {\\it complete equivalence} of the string states in lattice\nLiouville theory and a spin ${\\textstyle \\frac{1}{2}}\\mm$ XXZ chain with an additional phase\nfactor $ \\exp\\{\\i N \\tilde K {\\tilde\\gamma}\\}$. This extra factor will allow\nus to incorporate the remnant of the chain length mod $\\nu+1$ as a\nmeaningful extra parameter in the theory of finite size corrections\\ of an usual XXZ\nchain.\n\nThe spin ${\\textstyle \\frac{1}{2}}\\mm$ XXZ Bethe Ansatz equations\\ we shall be analyzing are thus\n\\be\n \\e{\\i N \\tilde K{\\tilde\\gamma}}\\mm \\left[\n\\BAEspsmi{{\\tilde\\lambda}_{\\tilde\\sK,i}}{\\i\\trac{\\tilde K}{2}\\gamma}\n\\right]^N \\times\n\\prod_{\\sss{\\tilde L}; \\ n_\\sss{\\tilde L}\\neq 0} \\\n   \\prod_{\\stackrel{j=1}{\\sss{(\\tilde L,j)\\neq(\\tilde K,\n  \t\ti)}}}^{n_\\sss{\\tilde L}}\n   S_{\\sss{\\tilde K\\tilde L}}(\\lambda_{\\sss{\\tilde K},i}-\\lambda_{\\sss{\\tilde L},j})\n\\ = \\ 1 \\ \\ \\ \\ \\forall\\  (\\tilde K,i) \\.\n \\mabel{phxxzBAE}\n\\ee\n This equation is valid separately for each $N$, but\nthe thermodynamic limit $N\\to\\infty$ is sensible only if $N$\napproaches $\\infty$ in steps of $\\nu+1$.\n\nAccordingly, we parametrize the (even) chain length  as\n\\be\n \\frac{N}{2} = n (\\nu +1) + \\kap\\, \\ \\ \\kap=0,\\ldots ,\\nu  \\.\n \\mabel{N}\n\\ee\n Taking the thermodynamic limit $N\\to\\infty$ {\\it at fixed} $\\kap$,\nletting $n\\to\\infty$, the limit of Bethe Ansatz equations\\ (\\ref{sBAE}, \\ref{phxxzBAE})\nis well defined.  The extra phase factor in the spin ${\\textstyle \\frac{1}{2}}\\mm$ XXZ chain\\ Bethe\nequations \\refe{phxxzBAE} is thus\n\\be\n \\e{2 \\i\\kap \\tilde K {\\tilde\\gamma}} \\.\n \\mabel{phase}\n\\ee\n\n\\vali\n\nTo summarize, the obtained equivalence of string states in lattice\nLiouville and spin ${\\textstyle \\frac{1}{2}}\\mm$ XXZ is the following:\n\n\\vali\\vali\n\n \\noindent\\begin{tabular}{l|c|c}\n \\null     & Lattice Liouville &\n\t\\begin{minipage}{3cm}\\begin{center}\\vspace*{2mm}\nSpin ${\\textstyle \\frac{1}{2}}\\mm$ XXZ\n\t   \\vspace*{2mm}\\end{center}\\end{minipage} \\\\ \\cline{1-3}\nanisotropy &\n\t\\begin{minipage}{3cm}\\begin{center}\\vspace*{2mm}\n$\\gamma=\\frac{\\pi\\nu}{\\nu+1}$\n\t   \\vspace*{2mm}\\end{center}\\end{minipage} &\n${\\tilde\\gamma} = \\frac{\\pi}{\\nu +1}$\\\\\nstring lengths & $K$                         & $\\tilde K =\\nu +1 -K$ \\\\\nspectral\nparameters &\n\t\\begin{minipage}{3cm}\\begin{center}\\vspace*{2mm}\n$\\lambda$\n\t   \\vspace*{2mm}\\end{center}\\end{minipage}  &\n${\\tilde\\lambda}=\\Re\\lambda+\\i\\left(\\Im\\lambda-(1+(-1)^\\sK)\\piperf\\right)$  \\\\\n \\end{tabular}\n\n \\vali\\vali\n\\noindent\nFor the analysis of the physical vacuum, it is important to note\nthat strings of maximal length are mapped to 1-strings, i.e. ordinary\nBethe Ansatz\\ roots, and vice versa.\n\nWe are intersted in the energy spectrum of the Liouville model.  The\nauxiliary Lax-operator $L_{n,a}$ which was used for the Bethe Ansatz,\nintertwines the $n$:th quantum space and the auxiliary space $sl_2$.\nThe quantum spaces for lattice Liouville model are copies of $L^2({\\rm\nR})$. Thus $L_{n,a}(\\lambda)$ does not degenerate to a permutation\noperator at any value of the spectral parameter $\\lambda$, and it is not\ngood for investigating lattice dynamics.  To get an integrable\nHamiltonian that generates lattice Liouville dynamics, one would have\nto introduce fundamental Lax-operators $L_{n,f}$ that intertwine two\nquantum spaces \\cite{tatafa}.\n\nFortunately, we are here only interested in energy and momentum\neigenvalues, so we do not have to investigate the fundamental\n$L$-operator. Following {\\cite{tatafa}} one can read off the energy\nand momentum eigenvalues from the eigenvalues of the diagonal elements\n$A$ and $D$ of the {\\it auxiliary} Lax-operators $L_{n,a}$.  The\nmomentum eigenvalues are\n\\[\n P^\\L(\\set{\\lambda_j}) =  \\trac{1}{2\\i} \\sum_{j=1}^m  p(\\lambda_j) \\;\n p(\\lambda) =    \\ln \\frac{a^\\gamma(\\lambda)}{d^\\gamma(\\lambda)} \\.\n \\]\n  The upper index for $a$ and $d$ again stresses the particular\nvalue of anisotropy used.\n\nThe energy can be acquired by differentiating:\n\\[\n  E^\\L(\\set{\\lambda_j}) = \\sum_{j=1}^m \\epsilon\\mm(\\lambda_j)\\;\n  \\epsilon\\mm(\\lambda)  = \\frac{\\gamma}{\\pi}\\mm\\frac{d}{d\\lambda}\\mm p(\\lambda)      \\.\n\\]\n\n{}From {\\refe{phxxzBAE}} we see that for the  spin ${\\textstyle \\frac{1}{2}}\\mm$ XXZ chain\\ with extra phase\nfactor, the roles of $a$ and $d$ are interchanged. Accordingly,\nusing  Correspondence {\\refe{pmap}}, we can write the Liouville energy\nand momentum in terms of the  XXZ ones:\n\\bea\n P^{\\L}(\\set{\\lambda_j})\n &=& \\trac{1}{\\i}\\sum_{j=1}^m\n\t\t\\ln\\frac{a^{\\tilde\\gamma}(\\lambda_j)}{d^{\\tilde\\gamma}(\\lambda_j)}\n  = - P^{{\\rm xxz}}(\\set{\\lambda_j})\n \\equiv\n \\i\\ln\\Lambda^{{{\\rm xxz}}}(\\set{\\lambda_j})\\longat{\\lambda=\\i\\trac{{\\tilde\\gamma}}{2}}\n  \\mabel{moment-mal} \\\\\n  E^\\L(\\set{\\lambda_j}) &=& -  E^{{\\rm xxz}}(\\set{\\lambda_j})\n   \\equiv -\\i\\frac{{\\tilde\\gamma}}{\\pi}\\mm \\frac{d}{d\\lambda}\\mm\n  \\ln\\Lambda^{{{\\rm xxz}}}(\\lambda,\\set{\\lambda_j})\n             \\longat{\\lambda=\\i\\trac{{\\tilde\\gamma}}{2}} \\ \\ \\.\n \\mabel{energy-mal}\n\\eea\n Here we have expressed the energy and momentum in terms of\neigenvalues $\\Lambda$ of the transfer matrix \\refe{transfer},\n\\be\n\\left(\\A(\\lambda)+\\D(\\lambda)\\right)\\mm\\Psi_m(\\set{\\lambda_j}) \\equiv\n\\Lambda(\\lambda,\\set{\\lambda_j})\\Psi_m(\\set{\\lambda_j}) \\.\n \\mabel{Lam}\n\\ee\n This is possible for the spin ${\\textstyle \\frac{1}{2}}\\mm$ XXZ chain\\, as the auxiliary\nand fundamental Lax-operators for the spin ${\\textstyle \\frac{1}{2}}\\mm$ XXZ chain\\ coincide.  The XXZ\nLax-operator \\refe{Lxxz} yields local commuting quantities at the\nvalue $\\lambda=\\i\\frac{\\tilde\\gamma}{2}$, for which it becomes a permutation\nmatrix.\n\n\nThe Bethe equations do not have to be completely solved to get the low\nlying spectrum of the Hamiltonian.  We need only the {\\it finite size}\ni.e. $\\operN$ corrections to the eigenvalues of the spin ${\\textstyle \\frac{1}{2}}\\mm$ XXZ\ntransfer matrix corresponding to Bethe Ansatz equations\\ {\\refe{phxxzBAE}}. The reason\nfor this is the following:\n\nIn this paper we started by discretizing Liouville theory in a finite\nvolume $2\\pi = a\\mm N$, with lattice spacing $a$. Thus the\nconformal scaling limit of the spin chain corresponds to the continuum\nlimit of the Liouville field theory.  Moreover, the continuum\nHamiltonian is\n\\be\nH_{{\\rm cont}} = \\trac{1}{a}\\mm H_{\\mm{\\rm lattice}}   \\,\n\\mabel{Hcont}\n\\ee\n so that only $\\operN$ corrections to the eigenvalues of the lattice\nHamiltonian remain finite in the continuum limit.  This is exactly the\nrealm of finite size effects, which in this case are\nsimultaneously finite lattice spacing effects.\n\n\n\n\\section{Finite Size Corrections for the six-vertex model}\n\nAs is well known, the spin ${\\textstyle \\frac{1}{2}}\\mm$ XXZ chain\\ has intimate connections to the\nsix-vertex model of classical statistical mechanics {\\cite{baxter}}.\nFor two dimensional classical statistical models, the $\\operN$\nbehaviour and the conformal properties are closely related.\n\nIn \\cite{blocani,affleck,cardy} it was argued that the central charge\n$c$ of the conformal field theory corresponding to the scaling limit\nof a two-dimensional statistical model is related to the finite size\ncorrections to the free energy, i.e. the logarithm of the maximal\neigenvalue $\\Lambda_o$ of the transfer matrix, in the limit $N\\to\\infty$.\nOn the other hand, the statistical mechanics minimum free energy\nconfiguration corresponds to the ground state energy $E_o$ of the\nscaling conformal theory.  For a model on an infinitely wide strip of\nlength $N$, the behavior of $E_o$ is \\cite{cardy}\n\\be\n E_o = N f_\\infty \\mm\n   - \\mm \\operN \\trac{\\pi}{6}\\mm c \\mm + \\mm \\O(\\trac{1}{N^2}) \\,\n \\mabel{scalec}\n\\ee\n   with $f_\\infty$ the free energy per site in the thermodynamic\nlimit.\n\nSimilarly, the critical indices (conformal weights) $\\Delta,\\Bardelta$\nof operators corresponding to exited states of the system can be read\noff the large $N$ behavior of configurations close to the one\nminimizing the free energy.  In terms of the higher energy and\nmomentum eigenvalues of the corresponding 1+1 dimensional quantum\ntheory, critical indices are:\n \\bea\nE_m - E_o &=& \\trac{2\\pi}{N}\\mm (\\Delta \\mm + \\mm \\Bardelta)\n \t\t\\ +\\ \\O(\\trac{1}{N^2}) \\cr\n & &   \\mabel{scaleweights}\\\\\nP_m - P_o &=& \\trac{2\\pi}{N}\\mm (\\Delta \\mm - \\mm \\Bardelta)\n \t\t\\ +\\ \\O(\\trac{1}{N^2}) \\nonumber\n\\eea\n As opposed to our differential dependence {\\refe{energy-mal}},  the\napproach  of Ref. {\\cite{cardy}} relates  the energy and momentum\ndirectly to lower eigenvalues $\\Lambda_m$ of the transfer matrix\nat criticality; $E_m\\sim -\\Re \\ln\\Lambda_m$, $\\ P_m\\sim -\\Im \\ln\\Lambda_m$.\n\n\\vali\n\nThe finite size corrections\\ of six-vertex models and the corresponding conformal\nproperties have been extensively studied in the literature\n\\cite{devewoy,karowski,kluwezi,boizre}.  For the six-vertex model with an\nextra phase factor of the form \\refe{phase}, and string exitations,\nthe finite size corrections were calculated by Karowski\n\\cite{karowski}.  The results of \\cite{karowski}  relevant for us are\nthe following.\n\nThe ground state of a six-vertex model is described by a filled Dirac\nsea of one-strings, i.e. $n_1= N\/2$, $n_l = 0, \\mm l>1$.  The\nlogarithm of the corresponding maximal eigenvalue of the transfer\nmatrix is \\cite[Eq. 4.3]{karowski}\\footnote{Note that our $\\nu$\ndiffers from the one of \\cite{karowski} by one. The relation between\nour $\\lambda$ and the $\\theta$ of \\cite{karowski} is $\\lambda = -\\i\\theta +\n\\i\\trac{{\\tilde\\gamma}}{2}$. This follows from the differences of the\nrespective Bethe equations \\refe{phxxzBAE} and \\cite[Eq.\n2.4]{karowski}.  }\n\\be\n  \\ln \\Lambda_o \\approx - \\i N f_\\infty(\\lambda) + \\frac{1}{N} \\frac{\\pi}{6}\n\\left(1 - \\frac{6\\kap^2}{\\nu(\\nu+1)} \\right) \\mm\n \\cosh\\left(\\trac{\\pi}{{\\tilde\\gamma}}\\lambda\\right) \\.\n \\mabel{cscale}\n\\ee\n\nExited states consist of higher strings above the vacuum, and holes in\nthe distribution of one-strings.  Low-energy excitations have both the\nnumber of holes and the number of higher strings $\\sum_{\\sL > 1}\nn_\\sL$ of the order $N^0$.\n\nFor a given distribution of strings $\\{n_l\\}$, there is a certain\nnumber of allowed values for the spectral parameters.  This number\ndepends on the behavior of the Bethe Ansatz equations\\ in the limits $\\lambda\\to\\pm\\infty$.\nDue to the dependence of these limits on higher strings, each\n$L$-string gives automatically rise to $2L-2$ holes.\n\nThe ``primary'' exitations correspond to states with one higher\nstring, no extra holes and the holes corresponding to the string\nevenly distibuted between the two surfaces of the Dirac sea of\n1-strings, i.e. between $\\lambda \\sim\\infty$ and $\\lambda \\sim-\\infty$.\n\nFor such states, the finite size behavior of the eigenvalues of the\ntransfer matrix reads\\cite[Eq.  4.5]{karowski}\n\\be\n  \\ln\\Lambda_m - \\ln\\Lambda_o \\approx  -\\trac{2\\pi}{N}\n\\left\\{(\\Delta + \\Bardelta)\n \\cosh\\left(\\trac{\\pi}{{\\tilde\\gamma}}\\lambda\\right)\n\\mm + \\mm (\\Delta - \\Bardelta)\n \\sinh\\left(\\trac{\\pi}{{\\tilde\\gamma}}\\lambda\\right) \\right\\}\\,\n \\mabel{weightscale}\n\\ee\n  where the weights are, if the single higher string is a $\\tilde\nK$-string,\n\\be\n \\Delta = \\Bardelta = \\frac{\\Bigl((\\nu+1)(\\tilde K-1) +\\kap\\Bigr)^2 -\n\\kap^2}{4\\nu(\\nu+1)} \\.\n  \\mabel{delta}\n\\ee\n If more strings and \/or holes are present in the exited state, the\nresulting critical indices differ from the ones above by additional\nintegers.  Accordingly, these states belong to the conformal towers of\ndescendants of the described ``primary states''.\n\n For future use we define the function\n \\[\n  \\delta_{L,\\kap} = \\frac{\\Bigl((\\nu+1)(L-1) +\\kap\\Bigr)^2}\n    {4\\nu(\\nu+1)} \\,\n\n\\]\n in terms of which the critical indices \\refe{delta} read\n \\be\n\\Delta = \\Bardelta = \\delta_{\\tilde K,\\kap} - \\delta_{1,\\kap}\\.\n \\mabel{Deleqdeldel}\n\\ee\n In this form, we explicitely see the subtraction of the part\ncorresponding to the ground state.\n\nAn extra condition on $\\tilde K$ for a primary state was found in\n\\cite{karowski}. There is an upper limit for the length of the strings\nthat contribute to the finite size corrections. For extra phase $\\kap$, only strings\nsatisfying\n\\be\n \\tilde K < \\nu + 1 - \\kap\n\\mabel{Klim}\n\\ee\n contribute. This cutting off of higher strings resembles a result of\nRef. \\cite{jutkar} for a $sl_q(2)$\\ invariant XXZ chain with fixed\nboundary conditions. There it was conjectured that highest strings\ncorrespond to ``bad'' representations of the quantum group (i.e.\nrepresentations with vanishing $q$-dimension), which have to be\nremoved from the spectrum to keep the theory unitary.\n\n\\vali\n\nIn the six-vertex model, one expects conformal invariance at $\\lambda =\n0$, where the $R$-matrix becomes isotropic. At this point, one can use\nEquations (\\ref{scalec}, \\ref{scaleweights}) to recognize the conformal\nproperties of the model.\n\n{}From Equation \\refe{cscale} we read off the central charge:\n\\be\n c =  1 - \\frac{6\\kap^2}{\\nu(\\nu+1)} \\.\n \\mabel{c}\n\\ee\n For the value $\\kap=1$, related to critical Potts models in\n\\cite{karowski}, the central charges of unitary minimal models emerge.\nThe critical indices \\refe{delta} reproduce a row in the Ka\\v c\\ \\ table.\nThe highest string does not contribute to the spectrum, due to\nRestriction \\refe{Klim}.\n\n\n\n\n\n\\section{Conformal Properties of Lattice Liouville Theory}\n\n\nNow we can use the results of Ref. \\cite{karowski} reviewed in the\nprevious section, to calculate the scaling properties of lattice\nLiouville energies and momenta, and to recognize the corresponding\nconformal structures.\n\nWe are interested in Bethe Ansatz\\ states that correspond to the ``primary''\nstates of the six-vertex model, i.e. one $K$-string above the physical\nvacuum, with $K = \\nu + 1 - \\tilde K$.\n\n{}From the scaling forms of the transfer matrix\n(\\ref{weightscale}, \\ref{scalec}) we get Liouville momenta and energies\nusing Prescriptions (\\ref{moment-mal}, \\ref{energy-mal}).  The\nresulting eigenvalues yield critical indices using\nEquation \\refe{scaleweights}.\n\nFor lattice Liouville theory, there is an important difference to the\ncase treated in \\cite{karowski}. The extra phase $\\kap$ is not a\nconstant. Instead we have sectors with different values of $\\kap$,\ncorresponding to different lengths of the chain mod $(\\nu+1)$, as\nindicated by Equation \\refe{N}. In the thermodynamic limit, all values of\n$\\kap$ coexist.\n\n{}From Equation \\refe{cscale} it is easy to see, that the ground state of the\ntheory lies in the $\\kap = 0$ sector, which gives $c=1$, the usual\ncentral charge for a periodic XXZ spin chain \\cite{devewoy,alibaba}.\n\nAs in \\cite{karowski}, minimal models would emerge if the ground state\nwas taken to be in the $\\kap=1$ sector. Here we will adopt this\napproach, discarding the $\\kap=0$ sector alltogether. Later on we will\nreturn to the interpretation of this reduction of the theory.\n\nAccordingly, the central charge is\n \\be\nc = 1 - \\frac{6}{\\nu(\\nu+1)} \\; \\nu = 2,3, \\ldots\n \\mabel{cIII}\n\\ee\n and we recover the unitary minimal conformal field theories\\ of \\cite{bpz,fqs}.\n\nCalculating the exitation energies and the corresponding critical\nindices from Equation \\refe{weightscale}, we have to subtract the ground\nstate energy, not only the minimal energy $\\delta_{1,\\kap}$ within\neach $\\kap$ sector.\n\nWith the ground state lying in the $\\kap=1$ sector,  we get for the\ncritical indices, instead of (\\ref{Deleqdeldel}),\n\\be\n  \\Delta=\\Bardelta = \\delta_{\\tilde K,\\kap} - \\delta_{1,1} =\n\\frac{\\Bigl((\\tilde K - 1) (\\nu+1) + \\kap\\Bigr)^2\n\t\t- 1}{4\\nu(\\nu+1)}\n\\. \\mabel{weightsIII}\n\\ee\n Defining\n\\bea\n p &=& \\tilde K + \\kap -1\\; p = 1, \\ldots, \\nu -1 \\cr\n q &=& \\kap\\; q = 1, \\ldots, \\nu\n \\mabel{pq}\n\\eea\n we can write the critical indices in the form\n\\be\n \\Delta = \\Bardelta \\ =\\ \\frac{\\Bigl( p\\mm (\\nu+1) - q\\mm \\nu\\Bigr)^2 -\n1}{4\\nu(\\nu+1)} \\.\n \\mabel{weightsIV}\n\\ee\n These scaling weights reproduce the whole Ka\\v c\\  table of unitary\nconformal field theories. The ranges of $p$ and $q$ follow accurately from Restriction\n\\refe{Klim} on the maximal string length, and the possible values of\n$\\kap$ with the $\\kap=0$ sector excluded.\n\n\\vali\n\n{}From \\refe{Hcont} we see that in the continuum, the energy and\nmomentum of primary\nBethe Ansatz\\ states are\n\\[\n E = \\Delta \\mm + \\mm \\Bardelta\\; P = \\Delta \\mm - \\mm \\Bardelta \\.\n\\]\n The primary Bethe states are thus products of holomorphic and\nantiholomorphic vectors (right and left movers) with conformal weights\n\\refe{weightsIII}.\n\n\\vali\n\nThe behavior encountered here is exactly the same as the one\nencountered in the restriction of SOS models to RSOS models\n\\cite{anbafo}.  The Bethe Ansatz equations\\ \\refe{phxxzBAE} are the Bethe equations of\nthe so called III\/IV critical limit of the SOS and corresponding eight\nvertex and XYZ models, in the case of root of unity anisotropy.  In\nthe III\/IV critical limit, the extreme off-diagonal term (usually\ndenoted $d$) in the eight-vertex $R$-matrix vanishes, and the elliptic\nfunctions degenerate to trigonometric ones.  Thus (on applying the\nstring picture,) the XYZ Bethe equations of \\cite{baxter,takfad79}\nturn into Equations \\refe{phxxzBAE} at criticality.  For an anisotropy\nof the form \\refe{tilgam}, the SOS Bethe states are parameterized by\nall $0\\leq \\kap \\leq \\nu$. The ground state lies in the $\\kap=0$\nsector, and the corresponding central charge is $c=1$.\n\nIn the root of unity case, it becomes possible to ``restrict'' the SOS\nmodel \\cite{anbafo}. The sectors with $1\\leq\\kap\\leq\\nu$ decouple from\nthe other sectors, and we can restrict our interest to these sectors\nonly.  This decoupled part of the SOS model is known as the RSOS\nmodel.  On the level of Bethe equations, the restriction means leaving\nout the $\\kap=0$ sector. The ground state now lies in the $\\kap=1$\nsector, and in the thermodynamic limit the unitary conformal field theories\\ with $c<1$\nemerge \\cite{huse}.\n\n\\vali\n\nThere is one more subtlety in the interpretation of the results for\nthe lattice Liouville model described above. Remembering the\ncorrespondence between lattice Liouville theory and the spin ${\\textstyle \\frac{1}{2}}\\mm$ XXZ chain\\\ndescribed in Section \\ref{maptoxxz}, it is evident that the physical\nvacuum for Liouville theory consists of maximal strings. This can be\nviewed as the maximal string limit of the fact that the vacuum of\nhigher spin chains consists of higher strings.\n\nDue to the complicated structure of the vacuum, not all\ncombinations of remnant and string length are a priori allowed.\nOn the contrary, we get stringent conditions on $N$ from requiring the\ncoexistence of a specific remnant $N \\ \\mbox{mod}\\  (\\nu+1)$ and a single\n$K$-string (corresponding to a spin ${\\textstyle \\frac{1}{2}}\\mm\\ $ XXZ $\\tilde K$-string)\nabove a sea of maximal $\\nu$-strings.  In fact, these requirements fix\nthe chain length modulo $\\nu(\\nu+1)$.  The parameterization \\refe{N}\nof $N$ has to be extended to\n\\be\n \\frac{N}{2} =  (n(\\nu+1) \\mm-\\mm \\kap \\mm + \\mm K)\\nu + K\n\t     =  (n\\mm\\nu \\mm-\\mm \\kap \\mm + \\mm  K)\n\t\t\t(\\nu +1) + \\kap \\.\n \\mabel{nKr}\n\\ee\n{}From here we see that for this $N$ it is indeed possible to define a\nstate with one $K$-string over a sea of $n(\\nu+1) - \\kap + K$ maximal\nstrings. In addition the remnant is $\\kap$. The thermodynamic limit\nhas to be taken in steps of $\\nu(\\nu+1)$ by taking $n\\to\\infty$ in Equation \n\\refe{nKr}.\n\nAccordingly, the full picture of lattice Liouville primary states is\nthe following.  In a lattice Liouville chain consisting of $N$ sites\n($N$ even), there is a primary state characterized by two integers.\nThese integers are related to the remnants of the chain lenth $\\ \\mbox{mod}\\ \n\\nu(\\nu+1)$ and $\\ \\mbox{mod}\\  (\\nu+1)$,\n\\[\n \\frac{N}2 \\ \\mbox{mod}\\  \\nu(\\nu+1) = (\\nu - p)(\\nu +1) + q \\.\n\\]\n The primary state is the state with a single lower string.  The $q=0$\nsector exhibits the behavior of the SOS ground state, and after the\nrestriction, the RSOS unitary series emerge, with central charges\n\\refe{cIII} and conformal weights \\refe{weightsIV}.  In the\nthermodynamic limit all remnants mod $\\nu(\\nu+1)$ coexist (except\npossibly for the decoupled $q=0$), and the Liouville primary states\ngive the whole Ka\\v c\\  table.\n\n\n\\section{Conclusions}\n\n\nWe have developed a spectral parameter dependent integrable structure\nto quantum Liouville theory on a lattice. Using the ensuing\n$L$-matrix, we have written the Bethe Ansatz equations\\ for Liouville theory.\n\nWe have concentrated on certain Liouville coupling constants $\\gamma$,\nfor which $q=\\e{\\i\\gamma}$ is a root of unity.  Using the string picture\nto describe exited Bethe Ansatz\\ states, we have mapped the Liouville Bethe\nequations to a set of generalized spin ${\\textstyle \\frac{1}{2}}\\mm$ XXZ Bethe Ansatz equations, more exactly\nthe critical SOS Bethe equations.  This mapping takes maximal\nLiouville strings to XXZ one-strings and vice versa. The physical\nLiouville vacuum thus consists of a Dirac sea of maximal strings.\n\nUsing results of Karowski \\cite{karowski} for the finite size corrections\\ to the\neigenvalues of the transfer matrix, we have calculated the central\ncharges and conformal dimensions of the spin ${\\textstyle \\frac{1}{2}}\\mm$ XXZ chain, and accordingly also\nof Liouville theory.\n\nWe found that the continuum limits of lattice Liouville theories with\ncoupling constants $\\gamma = \\pi\\trac{\\nu}{\\nu+1}\\mm, \\ \\nu=2,3,\\ldots$\nreproduce the unitary minimal models of Friedan, Qiu and Shenker\n\\cite{fqs}, on restricting the chain length not to be divisible by\n$\\nu+1$. This restriction is the exact analogue of the RSOS\nrestriction of SOS models at root of unity anisotropies.\n\nPrimary excitations of the Liouville chain are characterized by two\nintegers, the length of a shorter string above the vacuum consisting\nof maximal strings, and the remnant $\\kap$ of the chain length mod\n$(\\nu+1)$. With these two parameters, the conformal weights\ncorresponding to the exited states give all states in the\ncorresponding Ka\\v c\\  table.\n\nTo clarify the structure of the different sectors in the theory, a\nunitarity analysis based on the hidden $sl_q(2)$\\ symmetry is needed. This\nshould illuminate both the RSOS restriction $\\kap\\neq 0$, and the\nresult of \\cite{karowski} that highest XXZ (i.e. lowest Liouville)\nstrings do not contribute to the spectrum. Following \\cite{jutkar} we\nbelieve that these properties are deeply related to properties of root\nof unity representations of $sl_q(2)$. Truncation of the Bethe Ansatz\\ Hilbert\nspace corresponds to excluding ``bad'' representations of the quantum\ngroup, which is  required in order to have a positive metric on\nthe Hilbert space.\n\nIn this work we found equivalence of the Bethe Ansatz equations\\ of the lattice\nLiouville  and  the critical eight-vertex (SOS) models.\nAccordingly, the results presented here can be used to provide a\nLagrangian description of the critical conformal field theories of all\ntwo-dimensional statistical models related to the eight-vertex model.\n\nDue to the intimate connection of Liouville theory to two dimensional\ngravity, it would be very interesting to extend the method of\nquantizing Liouville theory presented here to more general couplings.\nIt is evident that negative couplings $\\gamma$ correspond to the real\nsector of the Liouville model with $c>25$.\n\nWhether the strong coupling results of Gervais \\& al. \\cite{gervais}\nin the regime $1<c<25$ can be reproduced and \\ or extended by Bethe Ansatz\\\nmethods, remains to be seen. For this, one should analyze Liouville\ntheories with imaginary couplings $\\gamma$. This requires formulating\nthe problem in terms of Baxter's equation \\cite{baxter}, and use of\nquantum separation of variables in Sklyanin style \\cite{sklyanin}.\n\n\n\\vali\n\n\\subsubsection*{Acknowledgments}\n\nWe thank A.G. Izergin, R. Kashaev, V. Tarasov and A. Volkov for\nilluminating discussions. In addition, O.T. is indepted to I. Affleck\nfor a discussion.\n\nThe work of LDF was supported by a grant of the Finnish academy and\ngrant R2H000 of ISF. The work of OT  was supported by a NSERC grant.\n\n\n\\setlength{\\baselineskip}{14pt}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nRepresentation learning has gained importance in fields that utilize data generated through physical processes, such as weather forecasting \\cite{mehrkanoon2019deep}, manufacturing \\cite{ringsquandl2017knowledge}, structural health monitoring \\cite{zobeiry2020theory}, acoustics \\cite{mohamed2012understanding}, and medical imaging \\cite{litjens2017survey}. However, in general, the features learned through generic machine learning algorithms typically do not correspond to physically interpretable quantities. Yet, learning physically consistent and interpretable features can improve our understanding of 1) the physically viable information about the data and 2) the composition of the system or process generating it. As a result, in recent years modified learning paradigms, suited to different physical application domains, have begun to draw interest and discussion \\cite{karpatne2017theory}. \nFor example, several researchers have designed physics-informed neural networks to learn approximate solutions to a partial differential equation \\cite{BenjaminErichson2019,Raissi2019,Nabian2018,Tartakovsky2018,Long2018}. These have been recently applied to ultrasonic surface waves to extract velocity parameters and denoise data \\cite{Shukla2020}. Similar physics-guided neural networks \\cite{karpatne2017theory,Karpatne2017a,Lei2018,Mestav2018} use physics-based regularizations to improve data processing, demonstrating that regression tasks (such as estimating the temperature throughout a lake) can be more accurate and robust when compared with purely physics-based solutions or purely data-driven solutions. These approaches demonstrate the strong potential for physics-informed machine learning but remain early in their study.\n\nLikewise, several generative models of waves have recently been studied for applications in electroencephalography \\cite{Luo2020} and seismology \\cite{Mosser2020}, although these learning systems do not assume physical knowledge. Similarly, other prior work has also considered generative models of ultrasonic waves through the use of sparse signal processing \\cite{harley2013sparse}, dictionary learning \\cite{alguri2018baseline,alguri2017model,alguri2021sim}, and neural network sparse autoencoders \\cite{Alguri2019}. These methods utilize the fact that many waves can be expressed as a small, sparse sum of spatial modes and attempt to extract these modes within the data \\cite{alguri2018baseline}. These dictionary learning algorithms have further been combined with wave-informed regularizers to incorporate physical knowledge of wave propagation into the solution \\cite{Tetali2019}.\n\nHowever, despite the wide-ranging applications for representation learning models for physics-constrained problems, one of the key challenges associated with representation learning tasks is that they typically require one to solve a non-convex optimization problem. While this potentially presents considerable challenges due to the general difficulty of non-convex optimization, considerable attention has been devoted recently to the non-convex optimization problems that arise from representation learning problems, and despite non-convex optimization being hard in the general case, certain positive results have begun to emerge in certain settings.  For example, within the context of low-rank matrix recovery, it has been shown that, in certain circumstances, gradient descent is guaranteed to converge to a local minimum and all local minima will be globally optimal \\cite{bhojanapalli2016global, ge2016matrix, park2016non, ge2017no}.\nFurther, other recent work has studied `structured' matrix factorization problems, where one promotes various properties in the matrix factors by imposing some form of regularization on the factorized matrices to promote the desired properties (e.g., an $\\ell_1$ norm on a matrix factor when sparsity is desired in that factor) and shown that such problems can be solved to global optimality in certain circumstances, but whether such guarantees can be made depends critically on how the regularization on the factorized matrices is formulated in the model \\cite{haeffele2014structured,haeffele2019structured,bach2013convex}.\n\n\n\\myparagraph{Paper Contributions and Related Work} In this work, we will specifically focus on data that is obtained from wave-like phenomena and note that many sensing and communications applications for representation learning center around the interpretation of waves and the wave equation to interrogate and characterize systems.  \nFor example, there are numerous applications for representation learning to aid in understanding and designing engineering systems with both electromagnetic waves (e.g., antenna design \\cite{Testolina_2019} and communication systems \\cite{7528397}) and mechanical waves (e.g., structural health monitoring \\cite{alguri2018baseline} and micro-electromechanical device design \\cite{7936037}).  Specifically, we derive a novel matrix factorization model which ensures that the recovered components satisfy wave equation constraints, allowing for direct interpretability of the recovered factors based on known physics and decomposing signals which consist of multiple oscillatory modes into the various components. Although our resulting formulation will also require solving a non-convex optimization problem, we show, using prior work on structured matrix factorization \\cite{haeffele2014structured,haeffele2019structured, bach2013convex}, that our model can be solved to global optimality in polynomial time.  We demonstrate our approach's advantages for two example applications, electrical waves in a multi-segment transmission line and mechanical waves on a fixed string. The first application is inspired from \\textbf{material characterization}, where different materials have different wavenumbers, as arises in the health monitoring of transmission lines and structures \\cite{ernould2020characterization,Shukla2020, shukla2021physics}. The second application provides insights into how enforcing PDE constraints is beneficial in \\textbf{modal analysis}, such as in the study of cantilever beams \\cite{lai2020full}. \n\n\nPrior work along these lines was given in \\cite{Tetali2019}, but in comparison, our algorithm provides global optimality guarantees (along with a certificate of guaranteed optimality), which is difficult to achieve for such non-convex problems. In addition, the formulation in \\cite{Tetali2019} is incomplete, as it does not account for separability in deriving the formulation, which we motivate from separation of variables method. This work also differs in having a low-rank assumption on the matrix factorization which enables us to extend our method to matrix completion strategies on incomplete data.  Further, \\cite{Tetali2019} considers a dictionary learning approach of the data matrix after taking a Fourier transform to obtain a sparse frequency representation, whereas our approach can directly be applied in the time domain. \n\n\n\n\\section{A Model for Wave-Informed Matrix Factorization}\n\\label{others}\n\n\nIn this section, we develop a novel matrix factorization framework which will allow us to decompose a matrix $\\mathbf{Y}$, with each column representing sampling along one dimension (e.g., space), as the product of two matrices $\\mathbf{D} \\mathbf{X}^\\top$, where one of the matrices will be constrained to satisfy wave-equation constraints.  Additionally, our model will learn the number of modes (columns in $\\mathbf{D}$ and $\\mathbf{X}$) directly from the data without needing to specify this \\textit{a priori}.  \n\n\\myparagraph{Problem Formulation}\nTo begin, recall that a key concept in the solution of partial differential equations is the notion of separation.  For example, \ngiven a PDE in space $\\ell$ and time $t$, we assume the solution is of the form $f(\\ell,t) = d(\\ell) x(t)$ and try to solve it. In case of linear PDEs (like the wave equation), for some $N$, if we find that $d_i(\\ell) x_i(t)$ are solutions for all $i \\in [N]$, then we have that $\\sum_{i=1}^{N} d_i(\\ell) x_i(t)$ is also a solution. For a discrete approximation of the solution $\\sum_{i=1}^{N} d_i(\\ell) x_i(t)$, we first note that the product of continuous functions $d_i(\\ell) x_i(t)$ on $\\ell \\in [0,L]$ and $t \\in [0,T]$ can be approximated as an outer product of two vectors, leading directly to a matrix factorization model. We show this as follows: consider matrices $(\\mathbf{D},\\mathbf{X})$ whose $i^\\text{th}$ columns are defined as $\\mathbf{D}_i = \\left[ d_i(0), d_i(\\Delta \\ell), \\cdots, d_i(N_d \\Delta \\ell)  \\right]^{\\top}$ and $\\mathbf{X}_i = \\left[ x_i(0), x_i(\\Delta t), \\cdots, x_i(N_t \\Delta t)  \\right]^{\\top}$ (where $N_d = \\left \\lfloor{L\/\\Delta \\ell}\\right \\rfloor$ and $N_t = \\left \\lfloor{T\/\\Delta t}\\right \\rfloor$), giving $d_i(n \\Delta \\ell) x_i(m \\Delta t) = \\mathbf{D}_{n,i} \\mathbf{X}_{m,i}$. \n\nThus we have that a discrete approximation of $d_i(\\ell) x_i(t)$ is $\\mathbf{D}_i \\mathbf{X}_i^{\\top}$ \nand likewise, the discrete approximation of $\\sum_{i=1}^{N} d_i(\\ell) x_i(t)$ is $ \\sum_{i=1}^{N} \\mathbf{D}_i \\mathbf{X}_i^{\\top} = \\mathbf{D} \\mathbf{X}^{\\top}$. \n\nFrom the above discussion, the fact that we are trying to approximate the wave-data matrix $\\mathbf{Y}$ as a product $\\mathbf{D} \\mathbf{X}^{\\top}$ clearly results in a matrix factorization model. However, beyond simple matrix factorization, we also wish to enforce physical consistency in our model. For this we employ theory-guided regularization (see \\cite{karpatne2017theory}) and the regularizer is derived from the wave equation. Specifically, we need a cost function which 1) captures how well our matrix factorization model matches the observed samples as well as 2) a regularization term which enforces physical consistency in the matrix factors, which we formulated as a problem with the form:\n\\begin{equation}\n    \\underset{\\mathbf{D}, \\mathbf{X}}{\\text{ } \\min \\text{ }} { \\tfrac{1}{2}\\| \\mathbf{Y} - \\mathbf{D}\\mathbf{X}^{\\top} \\|^2_F + \\lambda \\Theta (\\mathbf{D}, \\mathbf{X})}\n    \\label{eqn:obj_final}\n\\end{equation}\nwhere $\\| \\mathbf{Y} - \\mathbf{D}\\mathbf{X}^{\\top} \\|^2_F$ considers the matrix factorization loss, $\\lambda > 0$ is a hyper-parameter to balance the trade-off between fitting the data and satisfying our model assumptions given by $ \\Theta (\\mathbf{D}, \\mathbf{X})$, which is a regularizer that promotes the desired physical consistency. We call this \\emph{wave-informed matrix factorization}. To derive the specific function $\\Theta (\\mathbf{D}, \\mathbf{X})$ which promotes the desired physical consistency, we begin by designing a regularizer to promote wave-consistent signals in the columns of $\\mathbf{D}$.\\footnote{Note that our approach is also generalizable to having wave-consistent regularization in both factors.} \nSpecifically, consider the 1-dimensional wave equation evaluated at $d_i(l)x_i(t)$:\n\\begin{eqnarray}\n    \\frac{\\partial^2 \\left[ d_i(l)x_i(t) \\right] }{\\partial l^2} &=& \\frac{1}{c^2} \\frac{\\partial^2 \\left[ d_i(l)x_i(t) \\right] }{\\partial t^2} \n    \\label{eqn:wave_equation}\n\\end{eqnarray}\nNote that the above equation constrains that each \neach $d_i(l)x_i(t)$ component ($i \\in [N]$) is a wave. This can thus be seen as decomposing the given wave data into a superposition of simpler waves. The Fourier transform in time on both sides of \\eqref{eqn:wave_equation} yields:\n$\\frac{\\partial^2 \\left[ d_i(l) \\right] }{\\partial l^2}X_i(\\omega) = \\frac{-\\omega^2}{c^2} d_i(l) X_i(\\omega)$. Thus, at points where $X_i(\\omega) \\neq 0$, we can enforce the equation,\n\\begin{eqnarray}\n    \\frac{\\partial^2 \\left[ d_i(l) \\right] }{\\partial l^2} &=& \\frac{-\\omega^2}{c^2} d_i(l) \n    \\label{eqn:space_eigen_value}\n\\end{eqnarray}\nto maintain consistency with the wave-equation.\nNote that the above equation is analytically solvable for constant $\\omega\/c$ and the solution takes the form $d_i(l) = A \\sin(\\omega l\/c) + B \\cos(\\omega l\/c)$ and the constants $A$ and $B$ can be determined by initial conditions. Thus, we conclude that the choice of $\\omega\/c$ determines $d_i(l)$ and vice versa.\nWe next discretize equation \\eqref{eqn:space_eigen_value}. From the literature on discretizing differential equations, the above can be discretized to:\n\\begin{eqnarray}\n    \\mathbf{L} \\mathbf{D}_i &=& -k_i^2 \\mathbf{D}_i \n    \\label{eqn:discrete_wave_eqn}\n\\end{eqnarray}\nwhere $k_i = \\omega \/ c$ , also known as the wavenumber and $\\mathbf{L}$ is a discrete second derivative operator, where the specific form of $\\mathbf{L}$ depends on the boundary conditions (e.g., Dirichlet, Dirichlet-Neumann, etc) which can be readily found in the numerical methods literature (see for example \\cite{strang2007computational, golub2013matrix}).\n\nWe also specify the dependence on $i$ owing to the fact that $\\omega\/c$ and $d_i(l)$ determine each other.  \nNow for the factorization to be physically consistent in terms of spatial data, we desire that \\eqref{eqn:discrete_wave_eqn} is satisfied for some (unknown) value of $k_i$, which we promote with the regularizer $\\min_{k_i} \\| \\mathbf{L} \\mathbf{D}_i + k_i^2 \\mathbf{D}_i \\|^2_F$. In addition to requiring that the columns of $\\mathbf{D}$ satisfy the wave equation, we also would like to constrain the number of modes in our factorization (or equivalently find a factorization $\\mathbf{D} \\mathbf{X}^\\top$ with rank equal to the number of modes).  To accomplish this, we add squared Frobenius norms to both $\\mathbf{D}$ and $\\mathbf{X}$, which is known to induce low-rank solutions in the product $\\mathbf{D} \\mathbf{X}^\\top$ due to connections with the variational form of the nuclear norm \\cite{haeffele2019structured,haeffele2014structured,srebro2005rank}.  Taken together, we arrive at a regularization function which decouples along the columns of $(\\mathbf{D},\\mathbf{X})$ given by:\n\\begin{equation}\n\\label{eq:theta}\n\\Theta(\\mathbf{D},\\mathbf{X}) = \\sum_{i=1}^N \\theta(\\mathbf{D}_i,\\mathbf{X}_i), \\text{ where }  \n   \\theta(\\mathbf{D}_i,\\mathbf{X}_i) = \\tfrac{1}{2} \\! \\! \\left( \\|\\mathbf{X}_i\\|_F^2 \\! + \\! \\|\\mathbf{D}_i\\|_F^2 \\right) \\!  +  \\! \\gamma \\min_{k_i} \\|\\mathbf{L} \\mathbf{D}_i \\! + \\! k_i^2 \\mathbf{D}_i \\|_F^2\n\\end{equation}\nIn the supplementary material we show that the above regularization on $\\mathbf{D}$ is equivalent to encouraging the columns of $\\mathbf{D}$ to lie in the passband of a bandpass filter (centered at wavenumber $k$) with the choice of the hyperparameter $\\gamma$ being inversely proportional to the filter bandwidth.  With this regularization function, we then have our final model that we wish to solve:\n\\begin{equation}\n\\label{eq:main_obj}\n\\min_{N \\in \\mathbb{N}_+} \\min_{\\substack{\\mathbf{D} \\in \\mathbb{R}^{L \\times N}, \\mathbf{X} \\in \\mathbb{R}^{T \\times N} \\\\ \\mathbf{k} \\in \\mathbb{R}^N}} \\tfrac{1}{2}\\|\\mathbf{Y}-\\mathbf{D}\\mathbf{X}^\\top\\|_F^2 + \\tfrac{\\lambda}{2} \\sum_{i=1}^N \\left(\\|\\mathbf{X}_i\\|_F^2 + \\|\\mathbf{D}_i\\|_F^2 + \\gamma \\|\\mathbf{L} \\mathbf{D}_i + k_i^2 \\mathbf{D}_i \\|_F^2 \\right).\n\\end{equation}\n\n\\section{Model Optimization with Global Optimality Guarantees}\n\\label{section:model}\nWe note that our model in \\eqref{eq:main_obj} is inherently non-convex in $(\\mathbf{D},\\mathbf{X},\\mathbf{k})$ jointly due to the matrix factorization model as well as the fact that we are additionally searching over the number of columns\/entries in $(\\mathbf{D},\\mathbf{X},\\mathbf{k})$, $N$.  However, despite the potential challenge of non-convex optimization, here we show that we can efficiently solve \\eqref{eq:main_obj} to global optimality in polynomial by leveraging prior results regarding optimization problems for structured matrix factorization \\cite{haeffele2019structured,bach2013convex}.  In particular, the authors of \\cite{haeffele2019structured} consider a general matrix factorization problem of the form:\n\\begin{equation}\n\\label{eq:gen_obj}\n\\min_{N\\in\\mathbb{N}_+}{\\min_{\\mathbf{U} \\in \\mathbb{R}^{m \\times N}, \\mathbf{V} \\in \\mathbb{R}^{n \\times N}}} \\ell(\\mathbf{U} \\mathbf{V}^\\top) + \\lambda \\sum_{i=1}^N \\bar \\theta(\\mathbf{U}_i,\\mathbf{V}_i)\n\\end{equation}\nwhere $\\ell(\\hat \\mathbf{Y})$ is any function which is convex and once differentiable in $\\hat \\mathbf{Y}$ and $\\bar \\theta(\\mathbf{U}_i,\\mathbf{V}_i)$ is any function which satisfies the following 3 conditions:\n\\begin{enumerate}\n\\item $\\bar \\theta(\\alpha \\mathbf{U}_i, \\alpha \\mathbf{V}_i) = \\alpha^2 \\theta(\\mathbf{U}_i, \\mathbf{V}_i), \\ \\forall (\\mathbf{U}_i,\\mathbf{V}_i)$, $\\forall \\alpha \\geq 0$.\n\\item $\\bar \\theta(\\mathbf{U}_i, \\mathbf{V}_i) \\geq 0, \\ \\forall (\\mathbf{U}_i,\\mathbf{V}_i)$.\n\\item For all sequences $(\\mathbf{U}_i^{(n)},\\mathbf{V}_i^{(n)})$ such that $\\|\\mathbf{U}_i^{(n)}(\\mathbf{V}_i^{(n)})^\\top\\| \\rightarrow \\infty$ then $\\bar \\theta(\\mathbf{U}_i^{(n)},\\mathbf{V}_i^{(n)}) \\rightarrow \\infty$.\n\\end{enumerate}\nClearly, our choice of loss function $\\ell$ (squared loss) satisfies the necessary conditions.  However, in \\eqref{eq:main_obj} we wish to optimize over not just the matrix factors $(\\mathbf{D},\\mathbf{X})$ but also the additional $\\mathbf{k}$ parameters, so it is not immediately apparent that the framework from \\cite{haeffele2019structured} can be applied to our problem.  Here we first show that by our design of the regularization function $\\theta$ our formulation will satisfy the needed conditions, allowing us to apply the results from \\cite{haeffele2019structured} to our problem of interest \\eqref{eq:main_obj}.\n\\begin{proposition}\n\\label{prop:equiv_prob}\nThe optimization problem in \\eqref{eq:main_obj} is a special case of the problem considered in \\cite{haeffele2019structured}.\n\\end{proposition}\n\\begin{proof}\nAll proofs can be found in the supplement.\n\\end{proof}\nFrom this, we note that within the framework developed in \\cite{haeffele2019structured} it is shown (Corollary 1) that a given point $(\\tilde \\mathbf{U}, \\tilde \\mathbf{V})$ is a globally optimal solution of \\eqref{eq:gen_obj} iff the following two conditions are satisfied:\n\\begin{equation}\n1) \\ \\  \\langle -\\nabla \\ell( \\tilde \\mathbf{U} \\tilde \\mathbf{V}^\\top), \\tilde \\mathbf{U} \\tilde \\mathbf{V}^\\top \\rangle = \\lambda \\sum_{i=1}^N \\bar \\theta(\\tilde \\mathbf{U}_i, \\tilde \\mathbf{V}_i) \\ \\ \\ \\ \\ \\ 2) \\ \\ \\Omega_{\\bar \\theta}^\\circ (-\\tfrac{1}{\\lambda} \\nabla \\ell(\\tilde \\mathbf{U} \\tilde \\mathbf{V}^\\top)) \\leq 1\n\\end{equation}\nwhere $\\nabla(\\ell(\\hat \\mathbf{Y}))$ denotes the gradient w.r.t. the matrix product $\\hat \\mathbf{Y} = \\mathbf{U} \\mathbf{V}^\\top$ and $\\Omega_{\\bar \\theta}^\\circ(\\cdot)$ is referred to as the `polar problem' which is defined as \n\\begin{equation}\n\\label{eq:polar_def}\n\\Omega_{\\bar \\theta}^\\circ (\\mathbf{Z}) \\equiv \\sup_{\\mathbf{u},\\mathbf{v}} \\mathbf{u}^\\top \\mathbf{Z} \\mathbf{v} \\ \\ \\textnormal{s.t.} \\ \\  \\bar \\theta(\\mathbf{u},\\mathbf{v}) \\leq 1.\n\\end{equation}\nIt is further shown in \\cite{haeffele2019structured} (Proposition 3)  that the first condition above will always be satisfied for any first-order stationary point $(\\tilde \\mathbf{U}, \\tilde \\mathbf{V})$, and that if a given point is not globally optimal then the objective function \\eqref{eq:gen_obj} can always be decreased by augmenting the current factorization by a solution to the polar problem as a new column:\n\\begin{equation}\n(\\mathbf{U}, \\mathbf{V}) \\! \\! \\leftarrow \\! \\! \\left( \\left[ \\tilde \\mathbf{U}, \\ \\tau \\mathbf{u}^* \\right], \\left[ \\tilde \\mathbf{V}, \\ \\tau \\mathbf{v}^* \\right] \\right)  \\! : \\!  \\mathbf{u}^*, \\mathbf{v}^* \\in \\argmax_{\\mathbf{u},\\mathbf{v}} \\mathbf{u}^\\top \\! (-\\tfrac{1}{\\lambda} \\nabla \\ell(\\tilde \\mathbf{U} \\tilde \\mathbf{V}^\\top) \\mathbf{v} \\ \\ \\textnormal{s.t.} \\ \\  \\bar \\theta(\\mathbf{u},\\mathbf{v}) \\leq 1 \n\\end{equation}\nfor an appropriate choice of step size $\\tau > 0$.\nIf one can efficiently solve the polar problem for a given regularization function $\\bar \\theta$, then this provides a means to efficiently solve problems with form \\eqref{eq:main_obj} to global optimality with guarantees of global optimality.  Unfortunately, the main challenge in applying this result from a computational standpoint is that solving the polar problem requires one to solve another challenging non-convex problem in \\eqref{eq:polar_def}, which is often NP-Hard for even relatively simple regularization functions \\cite{bach2013convex}. Here, however, we provide a key positive result, proving that for our designed regularization function \\eqref{eq:theta}, the polar problem can be solved efficiently, in turn enabling efficient and guaranteed optimization of the non-convex model \\eqref{eq:main_obj}.\n\\begin{theorem}\n\\label{thm:polar}\nFor the objective in \\eqref{eq:main_obj}, the associated polar problem is equivalent to:\n\\begin{align}\n\\Omega_\\theta^\\circ (\\mathbf{Z}) &= \\max_{\\mathbf{d} \\in \\mathbb{R}^{L}, \\mathbf{x} \\in \\mathbb{R}^{T}, k\\in \\mathbb{R}} \\mathbf{d}^\\top \\mathbf{Z} \\mathbf{x}\n\\mathrm{s.t.} \\ &\\|\\mathbf{d}\\|^2_F + \\gamma \\|\\mathbf{L}\\mathbf{d} + k^2 \\mathbf{d}\\|_F^2 \\leq 1, \n &\\|\\mathbf{x}\\|^2_F \\leq 1, \\ 0 \\leq k \\leq 2.\n\\end{align}\nFurther, let $\\mathbf{L} = \\Gamma \\Lambda \\Gamma^\\top$ be an eigen-decomposition of $\\mathbf{L}$ and define the matrix $\\mathbf{A}( \\bar k) = \\Gamma(\\mathbf{I} + \\gamma(\\bar k \\mathbf{I} + \\Lambda)^2) \\Gamma^\\top$.  Then, if we define $\\bar k^*$ as%\n\\begin{equation}\n\\bar k^* = \\argmax_{\\bar k \\in [0,4]} \\| \\mathbf{A}(\\bar k)^{-1\/2} \\mathbf{Z} \\|_2\n\\label{eq:k_max}\n\\end{equation}\noptimal values of $\\mathbf{d},\\mathbf{x}, k$ are given as $\\mathbf{d}^* = \\mathbf{A}(\\bar k^*)^{-1\/2} \\bar \\mathbf{d}$, $\\mathbf{x}^* = \\bar \\mathbf{x}$, and $k^* = (\\bar k^*)^{1\/2}$ where $(\\bar \\mathbf{d}, \\bar \\mathbf{x})$ are the left and right singular vectors, respectively, associated with the largest singular value of $\\mathbf{A}(\\bar k^*)^{-1\/2} \\mathbf{Z}$.  Additionally, the above line search over $\\bar k$ is Lipschitz continuous with a Lipschitz constant, $L_{\\bar k}$, which is bounded by:\n\\begin{equation}\nL_{\\bar k} \\leq \\begin{Bmatrix} \\frac{2}{3 \\sqrt{3}} \\sqrt{\\gamma} \\|\\mathbf{Z}\\|_2 & \\gamma \\geq \\frac{1}{32} \\\\ \n4 \\gamma (1 + 16 \\gamma)^{-\\tfrac{3}{2}} \\|\\mathbf{Z}\\|_2 & \\gamma < \\frac{1}{32} \\end{Bmatrix} \\leq \\tfrac{2}{3 \\sqrt{3}} \\sqrt{\\gamma} \\|\\mathbf{Z}\\|_2\n\\end{equation}\n\\end{theorem}\n\nWe also note that the above result implies that we can solve the polar by first performing a (one-dimensional) line search over $k$, and due to the fact that the largest singular value of a matrix is a Lipschitz continuous function, this line search can be solved efficiently by a variety of global optimization algorithms.  For example, we give the following corollary for the simple algorithm given in \\cite{malherbe2017global}, and similar results are easily obtained for other algorithms.\n\n\\begin{corollary}[Adapted from Cor 13 in \\cite{malherbe2017global}]\n\\label{cor:line_search}\nFor the function $f(\\bar k) = \\|\\mathbf{A}(\\bar k)^{-1\/2} \\mathbf{Z} \\|_2$ as defined in Theorem \\ref{thm:polar}, if we let $\\bar k_1, \\ldots \\bar k_r$ denote the iterates of the LIPO algorithm in \\cite{malherbe2017global} then we have $\\forall \\delta \\in (0,1)$ with probability at least $1-\\delta$,\n\\begin{equation}\n\\max_{\\bar k \\in [0,4]} f(\\bar k) - \\max_{i=1\\ldots r} f(\\bar k_i) \\leq \\tfrac{8}{3 \\sqrt{3}} \\sqrt{\\gamma} \\|\\mathbf{Z}\\|_2 \\frac{\\ln(1\/\\delta)}{r}\n\\end{equation}\n\\end{corollary}\nAs a result, we have that the error of the linesearch converges with rate $\\mathcal{O}(1\/r)$, where $r$ is the number of function evaluations of $f(\\bar k)$, to a global optimum, and then, given the optimal value of $k$, the optimal $(\\mathbf{d}, \\mathbf{x})$ vectors can be computed in closed-form via singular value decomposition.  Taken together this allows us to employ the Meta-Algorithm defined in Algorithm \\ref{alg:meta} to solve problem \\eqref{eq:main_obj}.\n\\begin{corollary}\n\\label{cor:poly_time}\nAlgorithm \\ref{alg:meta} produces an optimal solution to \\eqref{eq:main_obj} in polynomial time.\n\\end{corollary}\n\n\\begin{algorithm}\n\\caption{\\bf{Meta-algorithm}}\n\\label{alg:meta}\n\\begin{algorithmic}[1]\n\\State Input $\\mathbf{D}_{init}$, $\\mathbf{X}_{init}$, $\\mathbf{k}_{init}$ \n\\State Initialize $ \\left( \\mathbf{D}, \\mathbf{X}, \\mathbf{k} \\right) \\leftarrow \\left( \\mathbf{D}_{init}, \\mathbf{X}_{init}, \\mathbf{k}_{init} \\right)$\n\\While {global convergence criteria is not met}\n\\State Perform gradient descent on the low-rank wave-informed objective function \\eqref{eq:main_obj} with $N$ fixed to reach a first order stationary point $(\\Tilde{\\mathbf{D}}, \\Tilde{\\mathbf{X}}, \\Tilde{\\mathbf{k}})$\\label{step:grad_desc}\n\\State Calculate the value of $\\Omega^\\circ_\\theta(\\tfrac{1}{\\lambda}(\\mathbf{Y}-\\tilde \\mathbf{D} \\tilde \\mathbf{X}^\\top))$ via Theorem \\ref{thm:polar} above and obtain $\\mathbf{d}^*, \\mathbf{x}^*, k^*$\n\\If {value of polar $\\Omega^\\circ_\\theta(\\tfrac{1}{\\lambda}(\\mathbf{Y}-\\tilde \\mathbf{D} \\tilde \\mathbf{X}^\\top)) = 1$} \\State {Algorithm converged to global minimum}\n\\Else \n\\State {Append  $(\\mathbf{d}^*, \\mathbf{x}^*, k^*)$ to $(\\Tilde{\\mathbf{D}}, \\Tilde{\\mathbf{X}}, \\Tilde{\\mathbf{k}})$ and update $\\left( \\mathbf{D}, \\mathbf{X}, \\mathbf{k} \\right)$\n\\State $ \\left( \\mathbf{D}, \\mathbf{X}, \\mathbf{k} \\right) \\! \\leftarrow \\! \\left( \\left[ \\tilde{\\mathbf{D}}, \\ \\tau \\mathbf{d}^*  \\right], \\left[ \\tilde{\\mathbf{X}}, \\ \\tau \\mathbf{x}^*  \\right] , \\left[ \\tilde{\\mathbf{k}}^\\top, \\ k^*  \\right]^\\top\n\\right)$, $\\tau\\! > \\! 0$ is step size (see supplement).\\label{step:step_size}}\n\\EndIf\n\\State Continue loop.\n\\EndWhile\n\\end{algorithmic}\n\\label{algoblock:meta-algo}\n\\end{algorithm}\n\n\\vspace{-3mm}\n\n\\section{Results \\& Discussions}\n\\label{sec:results}\nIn this section, we evaluate our proposed wave-informed matrix factorization on two synthetic datasets. For each dataset, algorithm iterations are performed until the value of polar is evaluated to be less than a threshold close to 1 to demonstrate convergence to the global minimum (i.e., a polar value of 1).  We note that the distance to the global optimum (in objective value) is directly proportional to the value of the polar value at any point minus 1 (\\cite{haeffele2019structured}, Prop. 4), so choosing a stopping criteria as polar value $\\leq 1 \\! + \\! \\epsilon$ also guarantees optimality within $\\mathcal{O}(\\epsilon)$.\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.25]{figs\/merge_1_diff_strings_middle.png}\n    \\caption{(left) Two cables of different impedances joined end-to-end; (right) cables of two different materials with a slightly alternate configuration, and measurements sampled at \\color{alizarin} \\textbf{X} \\color{black}.}\n    \\label{fig:merge_1}\n    \\vspace{-5mm}\n\\end{figure}\n\n\n\\myparagraph{Characterizing Multi-Segment Transmission Lines} First, we consider the \\textbf{material characterization} problem described in the introduction.\nSpecifically, we consider waves propagating along an electrical transmission line where the impedance of the transmission line changes along the line (as would occur, for example, if the transmission line was damaged or degraded in a region), as depicted in Figure \\ref{fig:merge_1}(left). \nWe simulate wave propagation in this line based on a standard RLGC (resistance, inductance, conductance, capacitance) transmission line model \\cite{Pozar2012}, using an algebraic graph theory engine for one-dimensional waves \\cite{Harley2019}. For this simulation, we combine two transmission line segments of length $0.4$~m (\\color{orange-left}left\\color{black}) and $0.6$~m (\\color{blue-right}right\\color{black}). \n\nThe wavenumber (which is inversely proportional to velocity) of the right segment is $4$~times higher than the wavenumber of the left segment. A modulated Gaussian signal with a center frequency of $10$~MHz and a $3$~dB bandwidth of $10$~MHz is transmitted from the left end of this setup. This impulse travels $0.4$~m till it reaches the interface (i.e., the connection between the two cables). At the interface, part of the propagating signal reflects and another part transmits into the next cable. The signal again reflects at the end of each cable.  \n\nNote the following: 1) there are now two distinct wavenumber regions, 2) the excitation\/forcing function is transient and is therefore a linear combination of nearby frequencies that each have a unique wavenumber, and 3) the end boundaries have loss (i.e., energy exits the system). We show electrical voltage amplitude at a timestamp, see Figure \\ref{fig:Merge_2}(top-left). We observe waves travelling in two regions with different wavenumbers. Note that performing a simple Fourier transform (in space) of the signal will produce unsatisfactory results due to the discrete change in the wavenumber along the line.  However, as we show below, our method automatically recovers the two distinct regions as separated modes in our decomposition. Namely, we solve our wave-informed matrix factorization with $\\gamma=50$ and $\\lambda=0.6$ on the above described wave data. \nInspection of the columns of $\\mathbf{D}$, see Figure \\ref{fig:Merge_2}(top-right), clearly shows that single columns of $\\mathbf{D}$ contain energy largely contained to only one region of the transmission line, automatically providing a natural decomposition of the signal which corresponds to changes in the physical transmission line. \n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.093]{figs\/Merge_2.png}\n    \\caption{(top-left) Electrical amplitude at a timestamp of $0.4115 \\mu$s; (top-right) illustration of two columns of $\\mathbf{D}$ from Wave-informed matrix factorization; (bottom-left) illustration of two columns of $\\mathbf{D}$ from low-rank matrix factorization;(bottom-right) partitioned normalized energy of the first 30 significant columns of Wave-informed matrix factorization and low-rank matrix factorization.}\n    \\label{fig:Merge_2}\n    \\vspace{-5mm}\n\\end{figure}\n\nAs a first baseline comparison, we also perform a low-rank factorization (by setting $\\gamma=0$ and $\\lambda=0.6$), which is equivalent to only using nuclear norm regularization on the matrix product $(\\mathbf{D} \\mathbf{X}^\\top)$.\nWith just low-rank regularization, one observes that while there is a clear demarcation at $0.4$~m, see Figure \\ref{fig:Merge_2}(bottom-left), there is still significant energy in both transmission line segments for each column of $\\mathbf{D}$. \nTo quantitatively evaluate the quality of the decomposition, we normalize the 30 most significant (determined by the corresponding value of $\\|\\mathbf{D}_i \\mathbf{X}_i^{\\top} \\|_F^2$) columns of $\\mathbf{D}$ and plot the energy on each partition, see Figure \\ref{fig:Merge_2}(bottom-right). In case of the low-rank factorization, see Figure \\ref{fig:Merge_2}(bottom-right),\nwe largely observe higher energy levels on regions of higher wavenumber (energy is proportional to the wavenumber for oscillatory quantities) and lower energy levels on regions of lower wavenumber. \nIn the case of our wave-informed matrix factorization model, we see two clear step functions, one indicating the energy on the first segment and the other indicating energy on the second segment, showing a sharp decomposition of the signal into components corresponding to the two distinct regions. We can further encapsulate this into a single quantity representing the decomposition of the signal into components corresponding to the two distinct regions. For this we compute the mean entropy (mean over the columns of $\\mathbf{D}$) of the percentage of signal power in one of the two regions (note the entropy is invariant to the choice of region). In the ideal case, the entropy will be 0, indicating that the learned components have all of the power being exclusively in one of the two line segments, while an entropy of 1 indicates the worst case where the power in the signal of a component column is split equally between the two segments.\n To quantify the performance of our method we also compare against other similar matrix factorization or modal extraction techniques: wave-informed KSVD \\cite{Tetali2019} ({\\color{B}{\\textbf{B}}}), low-rank matrix factorization ({\\color{C}{\\textbf{C}}}), independent component analysis \\cite{hyvarinen2000independent,van2006pp} ({\\color{D}{\\textbf{D}}}), dynamic mode decomposition \\cite{PhysRevFluids.5.054401,SUSUKI2018327} ({\\color{E}{\\textbf{E}}}), ensemble empirical mode decomposition \\cite{wu2009ensemble, fang2011stress, 6707404} ({\\color{F}{\\textbf{F}}}) and principal component analysis \\cite{medina1993three} ({\\color{G}{\\textbf{G}}}). Dynamic mode decomposition ({\\color{E}{\\textbf{E}}}), independent component analysis ({\\color{D}{\\textbf{D}}}), and principal component analysis ({\\color{G}{\\textbf{G}}}) are subspace identification techniques that resemble matrix factorization in many respects. Ensemble empirical mode decomposition is a refined version of empirical mode decomposition, extended to 2D data), which learns a basis from data, similar to our algorithm.\n\n Note that our proposed wave-informed matrix factorization ({\\color{A}{\\textbf{A}}}) achieves the best performance of the seven methods, providing quantitative evidence of our model's ability to isolate distinct materials (and corresponding wavefields). \n\n\\begin{table}[ht]\n\\vspace{-5mm}\n\\caption{\\label{table:entropies} Comparison of entropy based performance for segmentation of a transmission line using algorithms ({\\color{A}{\\textbf{A}}}), ({\\color{B}{\\textbf{B}}}), ({\\color{C}{\\textbf{C}}}), ({\\color{D}{\\textbf{D}}}), ({\\color{E}{\\textbf{E}}}), ({\\color{F}{\\textbf{F}}}) and ({\\color{G}{\\textbf{G}}}), where the mean entropy row represents the mean entropy of partitioned normalized energies for each algorithm.}\n\\centering\n\\begin{tabularx}{\\textwidth}{@{}lXXXXXXX@{}}\n\\toprule\n\\textbf{Algorithm}    & {\\color{A}{\\textbf{A}}}    & {\\color{B}{\\textbf{B}}}     & {\\color{C}{\\textbf{C}}}     & {\\color{D}{\\textbf{D}}}    & {\\color{E}{\\textbf{E}}}    & {\\color{F}{\\textbf{F}}}     & {\\color{G}{\\textbf{G}}}     \\\\ \\midrule\n\\textbf{Mean Entropy} & 0.176 & 0.211 & 0.531 & 0.464 & 0.431 & 0.636 & 0.531 \\\\ \\bottomrule\n\\end{tabularx}\n\\vspace{-5mm}\n\\end{table}\n\n\n\\myparagraph{Characterizing Multi-segment Transmission Lines with Sparsely Sampled Data}\nIn this subsection, we demonstrate the effectiveness of our model when the data is sparsely sampled. Specifically, we reduce the spatial density at which points were sampled (keeping the number of time points the same) by sampling 10 points uniformly on the cable of length $1$m, and we also change the configuration of the transmission lines by assigning the \\color{orange-left} first $0.4$ m \\color{black} and the \\color{orange-left} last $0.5$ m \\color{black} with one material and the \\color{blue-right} middle $0.1$ m \\color{black} with another material (see Fig.~\\ref{fig:merge_1}(right)). We then solve a matrix completion problem with wave-informed matrix factorization to interpolate the wavefield at the remaining 90 points on the $1$m region.  Since the sampling is uniformly done over space, the data matrix contains all zero columns which implies it \\textit{cannot} be completed with standard low-rank matrix completion.  However, we show that wave-informed matrix factorization, on the other hand, fills in those regions appropriately due to the wave-constraint being enforced.\n\nSpecifically, we minimize the following objective\\footnote{The mathematical details of the optimization procedure are almost the same as the algorithm mentioned in Algorithm \\ref{alg:meta} except that the masking operator $\\mathcal{A}(\\cdot)$ also needs to be included (see supplement).}:\n\\begin{equation}\n\\label{eq:main_obj_missing}\n\\!\\!\\min_{N \\in \\mathbb{N}_+} \\!\\!\\min_{\\substack{\\mathbf{D} \\in \\mathbb{R}^{L \\times N}, \\\\ \\mathbf{X} \\in \\mathbb{R}^{T \\times N}, \\mathbf{k} \\in \\mathbb{R}^N}} \\!\\!\\!\\!\\tfrac{1}{2}\\|  \\mathcal{A} \\left( \\mathbf{Y}-\\mathbf{D}\\mathbf{X}^\\top \\right) \\|_F^2  + \\tfrac{\\lambda}{2} \\sum_{i=1}^N \\left(\\|\\mathbf{X}_i\\|_F^2 + \\|\\mathbf{D}_i\\|_F^2 + \\gamma \\|\\mathbf{L} \\mathbf{D}_i + k_i^2 \\mathbf{D}_i \\|_F^2 \\right) \n\\end{equation}\nwhich is a matrix completion problem with a linear masking operator $\\mathcal{A}(\\cdot)$.  We observe, despite the very sparse sampling, the columns of $\\mathbf{D}$ still maintain sufficient structure to identify different material regions. \nIn Figure~\\ref{fig:Merge_3}(left), we plot all of the recovered columns of $\\mathbf{D}$, where we observe that, except for one column of $\\mathbf{D}$, every other column contains very little energy in the region between $0.45$m and $0.55$m. Thus the algorithm again automatically detects a change in this region in the decomposition. Note that the actual change was introduced between $0.4$m and $0.5$m. An error of $0.05$m in estimating the region of material change is analogous to Gibbs phenomenon in signals and system theory -- this especially occurs due to the fact that we impose second derivative constraints on the columns of $\\mathbf{D}$, which imposes a smoothness constraint and shifts the transition. Figure \\ref{fig:Merge_3}(right) is similar to Figure \\ref{fig:Merge_2}(bottom-left) as it quantifies that only one column of $\\mathbf{D}$ is active in the middle region, and the other columns of $\\mathbf{D}$ are active in the other distinct regions, whereas a low-rank factorization model displays significantly more mixing of signal energies in each region. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.09]{figs\/Merge_3.png}\n    \\caption{(left) All the columns of $\\mathbf{D}$ recovered from wave-informed matrix factorization in a single graph, note that the measurements were sampled only at $ \\{ 0,0.1, \\cdots, 0.9 \\}$; (right) normalized energies of various regions, for the first $8$ significant columns of $\\mathbf{D}$ for both algorithms.}\n    \\label{fig:Merge_3}\n    \\vspace{-5mm}\n\\end{figure}\n\n\n\\myparagraph{A Fixed Vibrating String}\nA fixed vibrating string is an example of a wave with fixed boundary conditions. \nAs an example, consider the dynamics of a noisy fixed string, given by: \n\\begin{equation}\ny(\\ell,t) = \\sum_{n=1}^{R} a_n \\sin (n k \\ell) \\sin(n \\omega t) + \\eta(\\ell,t) \\; .\n\\label{eqn:wave_equ}\n\\end{equation}\nThis can be represented in matrix notation as $\\mathbf{Y} = \\mathbf{D} \\mathbf{X}^{\\top} + \\mathbf{N}$, where $\\mathbf{N}$ is an additive noise term.  To demonstrate our framework, we will consider two more challenging variants of \\eqref{eqn:wave_equ} where we add a damping factor (in either time or space) to the amplitude of the wave:\n\\begin{eqnarray}\n    y_t(\\ell,t) = \\sum_{n=1}^{R} a_n e^{-\\alpha_n t} \\sin (n k \\ell) \\sin(n \\omega t) + \\eta(\\ell,t) \n    \\label{eqn:damped_wave_time} \\\\\n     y_s(\\ell,t) = \\sum_{n=1}^{R} a_n e^{-\\beta_n \\ell} \\sin (n k \\ell) \\sin(n \\omega t) + \\eta(\\ell,t)\n    \\label{eqn:damped_wave_space}\n\\end{eqnarray}\n\nThough this does not exactly satisfy the wave equation, waves in practice often have a damped sinusoidal nature.  We first consider the model in \\eqref{eqn:damped_wave_time} (damping in time) where we run our algorithm with $\\mathbf{Y} \\in \\mathbb{R}^{1000 \\times 4000}$, $R=10$, $k = 2 \\pi$, $\\omega = 12 \\pi$, $\\alpha_n = n$, $a_n$ roughly of the order of $10^{1}$, and additive white Gaussian noise $\\eta(\\ell,t)$ of variance $\\approx 10$ and $0$ mean (for the noisy case). We set $\\gamma = 10^9$ and $\\lambda = 200$ (see supplement for the behaviour of the algorithm with respect to changes in $\\gamma$ and $\\lambda$).  From wave-informed matrix factorization, when there is damping in time, we still obtain the columns of $\\mathbf{D}$ (vibrations in space, which is undamped) as clear underlying sinusoids (Figure \\ref{fig:Merge_4}).  We emphasize here that the modes of vibration are exactly recovered from data even in the presence of large amounts of noise (Figure \\ref{fig:Merge_4} (left)), whereas low-rank matrix factorization does not obtain pure sinusoids even in the noiseless case (Figure \\ref{fig:Merge_4} (right) -- the dotted blue curves change in amplitude over space).  To quantify this performance, in the appendix we provide the error between modes recovered by our algorithm and the ground-truth modes. Recall our algorithm also provides guarantees of polynomial-time global optimality (unlike the work of \\cite{Tetali2019}), without using a library matrix as mentioned in \\cite{lai2020full}. \n %\n Next, we run wave-informed matrix factorization on the model in \\eqref{eqn:damped_wave_space} (damping in space) \n with the same parameters as above and $\\beta_n = n\/2$ (also including the additive noise term) and visually compare our algorithms to other modal analysis methods ({\\color{A}{\\textbf{A}}}, {\\color{B}{\\textbf{B}}}, {\\color{C}{\\textbf{C}}}, {\\color{D}{\\textbf{D}}}, {\\color{E}{\\textbf{E}}}, {\\color{F}{\\textbf{F}}}, {\\color{G}{\\textbf{G}}}) (Figure~\\ref{fig:damped_sines}). Here we show recovering damped sinusoids is possible under heavy noise (see two left columns of $\\mathbf{D}$ demonstrated in Figure~\\ref{fig:damped_sines}), where we have chosen the most significant columns of each method. A close look at the recovered modes indicates much cleaner recovery of damped (in space) sinusoids for our method ({\\color{A}{\\textbf{A}}}) compared to others.  For example, ({\\color{B}{\\textbf{B}}}) is the closest in performance to our method, but distortions can be observed in the tails of the components.\n\n\\begin{figure}[ht]\n\\vspace{-4mm}\n    \\centering\n    \\includegraphics[trim=10 10 10 150, clip, scale=0.08]{figs\/Merge_4.png}\n    \\caption{4 columns of $\\mathbf{D}$ from low-rank matrix factorization (blue) and wave-informed matrix factorization (red) with the temporally damped vibrating string model.  Showing with noisy data (left) and the noiseless case (right). \\vspace{-5mm}}\n    \\label{fig:Merge_4}\n\\end{figure}\n\\begin{figure}[ht]\n    \\centering\n    \\vspace{-4mm}\n    \\includegraphics[trim=10 10 10 15, clip, width=\\linewidth]{figs\/dampled_sines.png}\n    \\caption{Two recovered modes (rows) of spatially damped sinusoids ({\\color{A}{\\textbf{A}}}), ({\\color{B}{\\textbf{B}}}), ({\\color{C}{\\textbf{C}}}), ({\\color{D}{\\textbf{D}}}), ({\\color{E}{\\textbf{E}}}), ({\\color{F}{\\textbf{F}}}), ({\\color{G}{\\textbf{G}}}).}    \\label{fig:damped_sines}\n    \\vspace{-5mm}\n\\end{figure}\n\n\n\\myparagraph{Conclusions}\nWe have developed a framework for a wave-informed matrix factorization algorithm with provable, polynomial-time global optimally guarantees. Output from the algorithm was compared with that of low-rank matrix factorization and state-of-the-art algorithms for modal and component analysis. We demonstrated that the wave-informed approach learns representations that are more physically relevant and practical for the purpose of material characterization and modal analysis.  Future work will include 1) generalizing this approach to a variety of linear PDEs beyond the wave equation as well as wave propagation along more than one dimension, 2) applications in baseline-free anomaly detection for structural health monitoring \\cite{alguri2018baseline, alguri2021sim}. \n\n\\textbf{Acknowledgments} This work was partially supported by NIH NIA 1R01AG067396, ARO MURI W911NF-17-1-0304, NSF-Simons MoDL 2031985, and the National Science Foundation under award number ECCS-1839704.\n\n\\bibliographystyle{IEEEtran}\n\n\\section{Supplementary Material}\n\nFollowing is an example Laplacian matrix ($\\mathbf{L}$),\n\\begin{eqnarray}\n    \\mathbf{L} &=& \\frac{1}{(\\Delta l)^2} \\begin{bmatrix}\n    -2 & 1 & 0 & 0 & 0 &\\cdots & 0 \\\\\n    1 & -2 & 1 & 0 & 0 & \\cdots & 0 \\\\\n    0 & 1 & -2 & 1 & 0 & \\cdots & 0 \\\\\n    \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n    0 & 0 & 0 & 0 & 0 & \\cdots & -2 \\\\\n    \\end{bmatrix}.\n    \\label{eqn:Lap_mat}\n\\end{eqnarray}\nTo reduce complexity, all theorems and proofs consider $\\Delta l = 1$ (for $\\Delta l$ defined in equation \\eqref{eqn:Lap_mat} without any loss of generality. The only change that needs to be accommodated is on the values in $\\mathbf{k}$ obtained from equation (\\ref{eq:k_max}) (i.e. while solving the polar). Observing equations (\\ref{eqn:discrete_wave_eqn}) and (\\ref{eqn:Lap_mat}), the only modification that is needed is to note that for $\\Delta l \\neq 1$ we follow Algorithm \\ref{algoblock:meta-algo} and replace the final vector $\\mathbf{k}$ with $\\mathbf{k} \\Delta l$.\n\n\\subsection{Algorithm Details}\n\n\\paragraph{Matrix Completion}\nWe note that our algorithm (and guarantees of polynomial time solutions) is easily generalized to any loss function $\\ell(\\mathbf{D} \\mathbf{X}^\\top)$, provided the loss function is once differentiable and convex w.r.t. $\\mathbf{D} \\mathbf{X}^\\top$.  For example, the following is the modified algorithm for the matrix completion formulation in \\eqref{eq:main_obj_missing}:\n\n\n\\begin{algorithm}\n\\caption{\\bf{Meta-algorithm}}\n\\begin{algorithmic}[1]\n\\State Input $\\mathbf{D}_{init}$, $\\mathbf{X}_{init}$, $\\mathbf{k}_{init}$ \n\\State Initialize $ \\left( \\mathbf{D}, \\mathbf{X}, \\mathbf{k} \\right) \\leftarrow \\left( \\mathbf{D}_{init}, \\mathbf{X}_{init}, \\mathbf{k}_{init} \\right)$\n\\While {global convergence criteria is not met}\n\\State Perform gradient descent on \\eqref{eq:main_obj} with $N$ fixed to reach a first order stationary point $(\\Tilde{\\mathbf{D}}, \\Tilde{\\mathbf{X}}, \\Tilde{\\mathbf{k}})$\\label{step:grad_desc}\n\\State Calculate the value of $\\Omega^\\circ_\\theta \\left(\\tfrac{1}{\\lambda}\\left(\\mathcal{A}^*\\left(\\mathbf{Y}-\\tilde \\mathbf{D} \\tilde \\mathbf{X}^\\top\\right)\\right)\\right)$ via Theorem \\ref{thm:polar} and obtain $\\mathbf{d}^*, \\mathbf{x}^*, k^*$\n\\If {value of polar $\\Omega^\\circ_\\theta \\left(\\tfrac{1}{\\lambda}\\left(\\mathcal{A}^*\\left(\\mathbf{Y}-\\tilde \\mathbf{D} \\tilde \\mathbf{X}^\\top\\right)\\right)\\right) = 1$} \\State {Algorithm converged to global minimum}\n\\Else\n\\State {Append  $(\\mathbf{d}^*, \\mathbf{x}^*, k^*)$ to $(\\Tilde{\\mathbf{D}}, \\Tilde{\\mathbf{X}}, \\Tilde{\\mathbf{k}})$ and update $\\left( \\mathbf{D}, \\mathbf{X}, \\mathbf{k} \\right)$\n\\State $ \\left( \\mathbf{D}, \\mathbf{X}, \\mathbf{k} \\right) \\! \\leftarrow \\! \\left( \\left[ \\tilde{\\mathbf{D}}, \\ \\tau_{\\mathcal{A}} \\mathbf{d}^*  \\right], \\left[ \\tilde{\\mathbf{X}}, \\ \\tau_{\\mathcal{A}} \\mathbf{x}^*  \\right] , \\left[ \\tilde{\\mathbf{k}}^\\top, \\ k^*  \\right]^\\top\n\\right)$, $\\tau_{\\mathcal{A}}\\! > \\! 0$ is step size.}\\label{step:step_size_mising}\n\\EndIf\n\\State Continue loop.\n\\EndWhile\n\\end{algorithmic}\n\\label{algoblock:meta-algo_missing}\n\\end{algorithm}\nwhere $\\mathcal{A}^*$ denotes the adjoint of the linear masking operator.\n\n\\paragraph{Gradient update equations} We set the derivative of the right hand side of equation \\ref{eqn:obj_final} with respect to $\\mathbf{D}$, $\\mathbf{X}$ and $\\mathbf{k}$ and utilize block coordinate descent of Gauss-Seidel type \\cite{xu2013block} to reach a first order stationary point mentioned in step \\ref{step:grad_desc} of Algorithm \\ref{algoblock:meta-algo}. The number of columns of $\\mathbf{D}$ and $\\mathbf{X}$ (denoted by $N$ in equation (\\ref{eq:main_obj})) is fixed and does not change during this step. The following are the gradient update equations (for stepsize $\\alpha_i$):\n\n\n\\begin{gather}\n    \\mathbf{D}_j \\leftarrow \\mathbf{D}_j  - \\alpha_j \\left(  \\left(\\mathbf{D} \\mathbf{X}^{\\top} - \\mathbf{Y} \\right) \\mathbf{X}_j  +  \\lambda \\mathbf{D}_j + 2 \\gamma \\lambda \\left( \\mathbf{L} + k^2_j \\mathbf{I}  \\right)^2 \\mathbf{D}_j \\right) \\label{eq:grad_D} \\\\\n    \\mathbf{X}_j \\leftarrow \\mathbf{X}_j - \\alpha_j \\left( \\left( \\mathbf{D} \\mathbf{X}^{\\top} - \\mathbf{Y} \\right)^{\\top} \\mathbf{D}_j \\right) \\label{eq:grad_X} \\\\\n    k_j \\leftarrow \\sqrt{-\\frac{ \\mathbf{D}_j^\\top \\mathbf{L} \\mathbf{D}_j } {\\|\\mathbf{D}_j\\|_2^2}} \\label{eq:grad_k} \n\\end{gather}\n\n\n\\paragraph{Gradient update equations for the Matrix completion setting}\n\n\\begin{gather}\n    \\mathbf{D}_j \\leftarrow \\mathbf{D}_j  - \\alpha_j \\left(  \\mathcal{A}^* \\left(\\mathbf{D} \\mathbf{X}^{\\top} - \\mathbf{Y} \\right) \\mathbf{X}_j  +  \\lambda \\mathbf{D}_j + 2 \\gamma \\lambda \\left( \\mathbf{L} + k^2_j \\mathbf{I}  \\right)^2 \\mathbf{D}_j \\right) \\label{eq:grad_D} \\\\\n    \\mathbf{X}_j \\leftarrow \\mathbf{X}_j - \\alpha_j \\left( \\mathcal{A}^* \\left( \\mathbf{D} \\mathbf{X}^{\\top} - \\mathbf{Y} \\right)^{\\top} \\mathbf{D}_j \\right) \\label{eq:grad_X} \\\\\n    k_j \\leftarrow \\sqrt{-\\frac{ \\mathbf{D}_j^\\top \\mathbf{L} \\mathbf{D}_j } {\\|\\mathbf{D}_j\\|_2^2}} \\label{eq:grad_k} \n\\end{gather}\n\n\\paragraph{Step size computation} The step size, $\\tau$ mentioned in Step 10 of Algorithm \\ref{algoblock:meta-algo} is computed through the following quadratic minimization problem.\n\\begin{eqnarray}\n    \\min_{\\substack{\\tau \\in \\mathbb{R}}} \\tfrac{1}{2}\\|\\mathbf{Y}- \\mathbf{D}_\\tau \\mathbf{X}_\\tau^\\top\\|_F^2 + \\frac{\\lambda}{2} \\sum_{i=1}^{N+1} \\left(\\|\\left(\\mathbf{X}_{\\tau}\\right)_i\\|_F^2 + \\| \\left( \\mathbf{D}_{\\tau} \\right)_i \\|_F^2 + \\gamma \\|\\mathbf{L} \\left( \\mathbf{D}_{\\tau} \\right)_i + \\left(\\mathbf{k}_{\\tau}^2\\right)_i  \\left( \\mathbf{D}_{\\tau} \\right)_i \\|_F^2 \\right)\n    \\label{eqn:for_tau}\n\\end{eqnarray}\n\nwhere, $\\left(\\mathbf{D}_\\tau, \\mathbf{X}_\\tau, \\mathbf{k}_{\\tau} \\right) = \\left( \\left[ \\tilde{\\mathbf{D}}, \\ \\tau \\mathbf{d}^*  \\right], \\left[ \\tilde{\\mathbf{X}}, \\ \\tau \\mathbf{x}^*  \\right] , \\left[ \\tilde{\\mathbf{k}}^\\top, \\ k^*  \\right]^\\top\n\\right)$ and $N$ is the number of columns in $\\tilde{\\mathbf{D}}$.\n\n\\paragraph{Proposition} The optimal step size $\\tau^*$ that minimizes the expression in (\\ref{eqn:for_tau}) is given by:\n\\begin{eqnarray}\n    \\tau^* = \\frac{\\sqrt{(\\mathbf{d}^*)^\\top \\left( \\mathbf{Y} - \\tilde{\\mathbf{D}} \\tilde{\\mathbf{X}}^{\\top} \\right) \\mathbf{x}^* - \\lambda }}{\\|\\mathbf{d}^*\\|_2 \\|\\mathbf{x}^*\\|_2 }\n    \\label{eq:for_tau_}\n\\end{eqnarray}\n\n\\begin{proof}\nLet $f(\\tau)$ represent the objective function in \\eqref{eqn:for_tau} with everything except for $\\tau$ held fixed:\n\\begin{eqnarray}\n    f(\\tau) = \\tfrac{1}{2}\\|\\mathbf{Y}- \\mathbf{D}_\\tau \\mathbf{X}_\\tau^\\top\\|_F^2 + \\tfrac{\\lambda}{2} \\sum_{i=1}^{N+1} \\left(\\|\\left(\\mathbf{X}_{\\tau}\\right)_i\\|_F^2 + \\| \\left( \\mathbf{D}_{\\tau} \\right)_i \\|_F^2 + \\gamma \\|\\mathbf{L} \\left( \\mathbf{D}_{\\tau} \\right)_i + \\left(\\mathbf{k}_{\\tau}\\right)^2_i  \\left( \\mathbf{D}_{\\tau} \\right)_i \\|_F^2 \\right)\n\\end{eqnarray}\n\nObserve that from the solution to the Polar problem we have by construction that the regularization term $\\theta(\\mathbf{d}^*,\\mathbf{x}^*)=1$, so combined with the positive homogeneity of $\\theta$ we have have that minimizing $f(\\tau)$ w.r.t. $\\tau$ is equivalent to solving:\n\\begin{equation}\n\\min_{\\tau \\geq 0} \\tfrac{1}{2}\\|\\mathbf{Y}- \\tilde \\mathbf{D} \\tilde \\mathbf{X}^\\top - \\tau^2 \\mathbf{d}^* (\\mathbf{x}^*)^\\top \\|_F^2 + \\lambda \\tau^2\n\\end{equation}\n\nTaking the gradient of the above w.r.t. $\\tau^2$ and solving for 0 gives:\n\\begin{equation}\n(\\tau^*)^2 = \\frac{\\langle \\mathbf{Y}-\\tilde \\mathbf{D} \\tilde \\mathbf{X}^\\top, \\mathbf{d}^* (\\mathbf{x}^*)^\\top \\rangle - \\lambda}{ \\|\\mathbf{d}^* (\\mathbf{x}^*)^\\top\\|_F^2}\n\\end{equation}\nThe result is completed by noting that the numerator is guaranteed to be strictly positive due to the fact that the Polar solution has value strictly greater that 1.\n\\end{proof}\n\nThe step size, $\\tau_{\\mathcal{A}}$ for the matrix completion algorithm, mentioned in Step 10 of Algorithm \\ref{algoblock:meta-algo_missing} is computed through the following quadratic minimization problem.\n\\begin{eqnarray}\n    \\min_{\\substack{\\tau_{\\mathcal{A}} \\in \\mathbb{R}}} \\tfrac{1}{2}\\|\\mathbf{Y}- \\mathbf{D}_{\\tau_{\\mathcal{A}}} \\mathbf{X}_{\\tau_{\\mathcal{A}}}^\\top\\|_F^2 + \\frac{\\lambda}{2} \\sum_{i=1}^{N+1} \\left(\\|\\left(\\mathbf{X}_{\\tau_{\\mathcal{A}}}\\right)_i\\|_F^2 + \\| \\left( \\mathbf{D}_{\\tau_{\\mathcal{A}}} \\right)_i \\|_F^2 + \\gamma \\|\\mathbf{L} \\left( \\mathbf{D}_{\\tau_{\\mathcal{A}}} \\right)_i + \\left(\\mathbf{k}_{\\tau_{\\mathcal{A}}}^2\\right)_i  \\left( \\mathbf{D}_{\\tau_{\\mathcal{A}}} \\right)_i \\|_F^2 \\right)\n    \\label{eqn:for_tau_missing}\n\\end{eqnarray}\n\nwhere, $\\left(\\mathbf{D}_{\\tau_{\\mathcal{A}}}, \\mathbf{X}_{\\tau_{\\mathcal{A}}}, \\mathbf{k}_{\\tau_{\\mathcal{A}}} \\right) = \\left( \\left[ \\tilde{\\mathbf{D}}, \\ {\\tau_{\\mathcal{A}}} \\mathbf{d}^*  \\right], \\left[ \\tilde{\\mathbf{X}}, \\ {\\tau_{\\mathcal{A}}} \\mathbf{x}^*  \\right] , \\left[ \\tilde{\\mathbf{k}}^\\top, \\ k^*  \\right]^\\top\n\\right)$ and $N$ is the number of columns in $\\tilde{\\mathbf{D}}$.\n\n\\paragraph{Proposition} The optimal step size $\\tau^*_{\\mathcal{A}}$ that minimizes the expression in (\\ref{eqn:for_tau_missing}) is given by:\n\\begin{eqnarray}\n\\tau^*_{\\mathcal{A}^*} = \\frac{ \\sqrt{ \\langle \\mathcal{A}^* \\left( \\mathbf{Y}-\\tilde \\mathbf{D} \\tilde \\mathbf{X}^\\top \\right), \\mathcal{A}^* \\left( \\mathbf{d}^* {\\mathbf{x}^*}^\\top \\right)\\rangle - \\lambda }}{ \\| \\mathcal{A}^* \\left( \\mathbf{d}^* {\\mathbf{x}^*}^\\top \\right) \\|_F}\n\\end{eqnarray}\n\\begin{proof}\nFollowing the same steps of the previous proof, we obtain:\n\n\\begin{equation}\n\\min_{\\tau_{\\mathcal{A}} \\geq 0} \\tfrac{1}{2}\\| \\mathcal{A}^* \\left( \\mathbf{Y}- \\tilde \\mathbf{D} \\tilde \\mathbf{X}^\\top \\right) - \\tau_{\\mathcal{A}}^2 \\mathcal{A}^* \\left( \\mathbf{d}^* (\\mathbf{x}^*)^\\top \\right) \\|_F^2 + \\lambda \\tau_{\\mathcal{A}}^2\n\\end{equation}\n\nThe proof is completed by minimizing the quadratic for $\\left(\\tau_{\\mathcal{A}}\\right)^2$.\n\n\\end{proof}\n\n\\subsection{Proofs}\n\n{\n\\renewcommand{\\theproposition}{\\ref{prop:equiv_prob}}\n\\begin{proposition}\nThe optimization problem in \\eqref{eq:main_obj} is a special case of the problem considered in \\cite{haeffele2019structured}.\n\\end{proposition}\n\\addtocounter{proposition}{-1}\n}\n\n\n\\begin{proof}\nThe problem considered in \\cite{haeffele2019structured} is stated in \\eqref{eq:gen_obj}. Comparing it with (\\ref{eq:main_obj}) we have:\n\\begin{gather}\n    \\ell \\left( \\mathbf{D}\\mathbf{X}^{\\top} \\right) = \\tfrac{1}{2}\\| \\mathbf{Y} - \\mathbf{D}\\mathbf{X}^{\\top} \\|_F^2\\\\\n    \\bar{\\theta} \\left( \\mathbf{D}_i, \\mathbf{X}_i \\right) = \\tfrac{1}{2} \\left( \\| \\mathbf{D}_i \\|^2_2 + \\| \\mathbf{X}_i \\|^2_2 + \\gamma \\min_{k_i}  \\| \\mathbf{L} \\mathbf{D}_i - k_i \\mathbf{D}_i \\|^2_2 \\right)\n\\end{gather}\nObserve that $\\ell ( \\hat{\\mathbf{Y}} ) = \\tfrac{1}{2}\\| \\mathbf{Y} - \\hat{\\mathbf{Y}} \\|^2_F$ is clearly convex and differentiable w.r.t $\\hat \\mathbf{Y}$. \n Now to realize that the optimization problem in \\eqref{eq:main_obj} is a special case of the problem considered in \\cite{haeffele2019structured}, it suffices to check that $\\bar \\theta(\\mathbf{d},\\mathbf{x})$ satisfies the three conditions of a rank-1 regularizer from \\cite{haeffele2019structured}.\n\n\\begin{enumerate}\n    \\item $\\bar \\theta(\\alpha \\mathbf{d}, \\alpha \\mathbf{x}) = \\alpha^2 \\theta(\\mathbf{d}, \\mathbf{x}), \\ \\forall (\\mathbf{d},\\mathbf{x})$ and $\\forall \\alpha \\geq 0$.\n    \n    For any $\\alpha > 0$, $\\forall (\\mathbf{d}, \\mathbf{x})$ :\n    \\begin{equation}\n         \\begin{split}\n        \\bar{\\theta}\\left( \\alpha \\mathbf{d}, \\alpha \\mathbf{x} \\right)  &= \\| \\alpha \\mathbf{d} \\|^2_2 + \\| \\alpha \\mathbf{x} \\|^2_2 + \\gamma \\min_{k} \\| \\alpha \\mathbf{L} \\mathbf{d} + k^2 \\alpha \\mathbf{d} \\|^2_F \\\\\n        &= \\alpha^2 \\| \\mathbf{d} \\|^2_2 + \\alpha^2 \\| \\mathbf{x} \\|^2_2 + \\alpha^2 \\gamma \\min_{k} \\| \\mathbf{L} \\mathbf{d} + k^2 \\mathbf{d} \\|^2_F \\\\\n       &= \\alpha^2 \\bar{\\theta}(\\mathbf{d},\\mathbf{x})\n    \\end{split}\n    \\end{equation}\n    \n    where we note that scaling $\\mathbf{d}$ by $\\alpha > 0$ does not change the optimal value of $k$ in the third term, allowing $\\alpha^2$ to be moved outside of the norm.\n    \n    \\item $\\bar \\theta(\\mathbf{d}, \\mathbf{x}) \\geq 0, \\ \\forall (\\mathbf{d},\\mathbf{x})$.\n    \n    \n    Clearly, all terms in $\\bar{\\theta}(\\mathbf{d},\\mathbf{x})$ are non-negative, thus, $\\forall (\\mathbf{d},\\mathbf{x})$, we have $\\bar{\\theta}(\\mathbf{d},\\mathbf{x})\\geq 0$.\n    \\item For all sequences $(\\mathbf{d}^{(n)},\\mathbf{x}^{(n)})$ such that $\\|\\mathbf{d}^{(n)}(\\mathbf{x}^{(n)})^\\top\\| \\rightarrow \\infty$ then $\\bar \\theta(\\mathbf{d}^{(n)},\\mathbf{x}^{(n)}) \\rightarrow \\infty$.\n    \n    Here, note that the following is true for all $(\\mathbf{d},\\mathbf{x})$:\n    %\n    \\begin{equation}\n    \\|\\mathbf{d} \\mathbf{x}^\\top \\|_F = \\|\\mathbf{d}\\|_2 \\|\\mathbf{x}\\|_2 \\leq \\tfrac{1}{2}(\\|\\mathbf{d}\\|_2^2 + \\|\\mathbf{x}\\|_2^2) \\leq \\tfrac{1}{2}\\left(\\|\\mathbf{d}\\|_2^2 + \\|\\mathbf{x}\\|_2^2 + \\min_{k} \\| \\mathbf{L} \\mathbf{d} + k^2 \\mathbf{d} \\|^2_2\\right)\n    \\end{equation}\n    As a result we have $\\forall (\\mathbf{d}, \\mathbf{x})$ that $\\|\\mathbf{d}\\mathbf{x}^\\top\\|_F \\leq \\bar \\theta(\\mathbf{d},\\mathbf{x})$, completing the result.\n   \n\\end{enumerate}\n\\end{proof}\n\nBefore proving Theorem \\ref{thm:polar} we first prove an intermediate lemma regarding Lipschitz constants.\n\\begin{lemma}\n\\label{lem:L_bound}\nGiven a set of constants $\\lambda_i, \\ i=1, 2, \\ldots$ such that $\\forall i$, $\\mu_\\lambda \\leq \\lambda_i \\leq 0$, and a constant $\\gamma > 0$, let the function $f$ be defined as:\n\\begin{equation}\nf(x) = \\max_i \\frac{1}{\\sqrt{1+\\gamma(x + \\lambda_i)^2}}.\n\\end{equation}\nThen, over the domain $0 \\leq x \\leq \\mu_x$ $f$ is Lipschitz continuous with Lipschitz constant $L_f$ bounded as follows:\n\\begin{equation}\nL_f \\leq \\left[\\begin{cases} \\frac{2}{3\\sqrt{3}} \\sqrt{\\gamma} & \\gamma\\geq \\frac{1}{2 \\max \\{ \\mu_\\lambda^2, \\mu_x^2 \\}} \\\\\n\\gamma \\max \\{ -\\mu_\\lambda, \\mu_x\\} (1+\\gamma \\max \\{\\mu_\\lambda^2, \\mu_x^2\\} )^{-\\tfrac{3}{2}} & \\mathrm{otherwise}\n\\end{cases}  \\right] \\leq \\frac{2}{3 \\sqrt{3}}\\sqrt{\\gamma}. \n\\end{equation}\n\\end{lemma}\n\n\n\\begin{proof}\nFirst, note that for any two Lipschitz continuous functions $\\psi_a$ and $\\psi_b$, with associated Lipschitz constants $L_a$ and $L_b$, respectively, one has that the point-wise maximum of the two functions, $\\psi(x) = \\max \\{ \\psi_a(x), \\psi_b(x) \\}$, is also Lipschitz continuous with Lipschitz constant bounded by $\\max \\{ L_a, L_b \\}$.  This can be easily seen by the following two inequalities:\n\\begin{align}\n&\\psi_a(x') \\leq \\psi_a(x) + | \\psi_a(x') - \\psi_a(x) | \\leq \\psi(x) + | \\psi_a(x') - \\psi_a(x) | \\leq \\psi(x) + L_a |x' - x| \\\\\n&\\psi_b(x') \\leq \\psi_b(x) + | \\psi_b(x') - \\psi_b(x) | \\leq \\psi(x) + | \\psi_b(x') - \\psi_b(x) | \\leq \\psi(x) + L_b |x' - x|\n\\end{align}\nFrom this we have:\n\\begin{equation}\n\\begin{split}\n&\\psi(x') = \\max \\{ \\psi_a(x'), \\psi_b(x') \\} \\leq \\max \\{ \\psi(x) + L_a |x'-x|, \\psi(x) + L_b |x'-x| \\} = \\psi(x) + \\max \\{L_a, L_b \\} |x'-x| \\\\\n&\\implies \\psi(x') - \\psi(x) \\leq \\max \\{ L_a, L_b \\} |x'-x|\n\\end{split}\n\\end{equation}\nwhich implies the claim from symmetry.\n\nNow, if we define the functions $g$ and $h$ as\n\\begin{equation}\ng(x, \\lambda) = \\frac{1}{\\sqrt{1+\\gamma(x + \\lambda)^2}} \\ \\ \\ \\ \\ \\ \\ h(b) = \\gamma b (1+\\gamma b^2)^{-\\tfrac{3}{2}}\n\\end{equation}\nwe have that the Lipschitz constant of $f$, denoted as $L_f$, is bounded as:\n\\begin{equation}\n\\label{eq:h_max}\n\\begin{split}\nL_f \\leq &\\max_i \\sup_{x \\in [0, u_x]} \\left| \\frac{\\partial}{\\partial x} g(x, \\lambda_i) \\right| \\leq \\sup_{\\lambda \\in [\\mu_\\lambda,0]} \\sup_{x \\in [0, \\mu_x]} \\left| \\frac{\\partial}{\\partial x} g(x, \\lambda) \\right| =  \\\\\n&\\sup_{\\lambda \\in [\\mu_\\lambda,0]} \\sup_{x \\in [0, \\mu_x]} \\left| - \\frac{\\gamma(x+\\lambda)}{ (\\sqrt{1+\\gamma (x+\\lambda)^2})^3 } \\right|\n= \\sup_{b \\in [\\mu_\\lambda, \\mu_x]} \\left| h(b) \\right| = \\sup_{b \\in [0, \\max \\{ -\\mu_\\lambda, \\mu_x \\} ]} h(b)\n\\end{split}\n\\end{equation}\nWhere the first inequality is from the result above about the Lipschitz constant of the point-wise maximum of two functions and the simple fact that the Lipschitz constant of a function is bounded by the maximum magnitude of its gradient, and the final equality is due to the symmetry of $|h(b)|$ about the origin.  Now, finding the critical points of $h(b)$ for non-negative $b$ we have:\n\\begin{equation}\n\\begin{split}\nh'(b) = \\gamma (1+\\gamma b^2)^{-\\tfrac{3}{2}} - 3 \\gamma^2 b^2 (1+\\gamma b^2)^{- \\tfrac{5}{2}} = 0 \n\\implies 3 \\gamma b^2 = (1+ \\gamma b^2) \\implies b^* = \\frac{1}{\\sqrt{2 \\gamma}}\n\\end{split} \n\\end{equation}\nNote that $h(0)=0$, $h(b)>0$ for all $b>0$, and $h'(b)<0$ for all $b > b^*$.  As a result, $b^*$ will be a maximizer of $h(b)$ if it is feasible, otherwise the maximum will occur at the extreme point $b = \\max \\{-\\mu_\\lambda, \\mu_x \\}$.  From this we have the result:\n\\begin{equation}\n\\begin{split}\nL_f &\\leq \\begin{cases} h(b^*) = \\frac{2}{3 \\sqrt{3}} \\sqrt{\\gamma} & \\gamma\\geq \\frac{1}{2 \\max \\{ \\mu_\\lambda^2, \\mu_x^2\\}} \\\\\nh(\\max \\{-\\mu_\\lambda, \\mu_x \\} ) = \\gamma \\max \\{ -\\mu_\\lambda, \\mu_x \\} (1+\\gamma \\max \\{ \\mu_\\lambda^2, \\mu_x^2 \\})^{-\\tfrac{3}{2}} & \\text{otherwise}\n\\end{cases} \\\\\n& \\leq h(b^*) = \\frac{2}{3\\sqrt{3}}\\sqrt{\\gamma}.\n\\end{split}\n\\end{equation}\n\\end{proof}\n\n{\n\\renewcommand{\\thetheorem}{\\ref{thm:polar}}\n\n\\begin{theorem}\nFor the objective in \\eqref{eq:main_obj}, the associated polar problem is equivalent to:\n\\begin{equation}\n\\Omega_\\theta^\\circ (\\mathbf{Z}) = \\max_{\\mathbf{d} \\in \\mathbb{R}^{L}, \\mathbf{x} \\in \\mathbb{R}^{T}, k\\in \\mathbb{R}} \\mathbf{d}^\\top \\mathbf{Z} \\mathbf{x} \\ \\ \\textnormal{s.t.} \\ \\  \\|\\mathbf{d}\\|^2_F + \\gamma \\|\\mathbf{L}\\mathbf{d} + k^2 \\mathbf{d}\\|_F^2 \\leq 1, \\ \\|\\mathbf{x}\\|^2_F \\leq 1, \\ 0 \\leq k \\leq 2.\n\\end{equation}\nFurther, let $\\mathbf{L} = \\Gamma \\Lambda \\Gamma^\\top$ be an eigen-decomposition of $\\mathbf{L}$ and define the matrix $\\mathbf{A}( \\bar k) = \\Gamma(\\mathbf{I} + \\gamma(\\bar k \\mathbf{I} + \\Lambda)^2) \\Gamma^\\top$.  Then, if we define $\\bar k^*$ as%\n\\begin{equation}\n\\bar k^* = \\argmax_{\\bar k \\in [0,4]} \\| \\mathbf{A}(\\bar k)^{-1\/2} \\mathbf{Z} \\|_2\n\\label{eq:k_max}\n\\end{equation}\noptimal values of $\\mathbf{d},\\mathbf{x}, k$ are given as $\\mathbf{d}^* = \\mathbf{A}(\\bar k^*)^{-1\/2} \\bar \\mathbf{d}$, $\\mathbf{x}^* = \\bar \\mathbf{x}$, and $k^* = (\\bar k^*)^{1\/2}$ where $(\\bar \\mathbf{d}, \\bar \\mathbf{x})$ are the left and right singular vectors, respectively, associated with the largest singular value of $\\mathbf{A}(\\bar k^*)^{-1\/2} \\mathbf{Z}$.  Additionally, the above line search over $\\bar k$ is Lipschitz continuous with a Lipschitz constant, $L_{\\bar k}$, which is bounded by:\n\\begin{equation}\nL_{\\bar k} \\leq \\left[ \\begin{cases} \\frac{2}{3 \\sqrt{3}} \\sqrt{\\gamma} \\|\\mathbf{Z}\\|_2 & \\gamma \\geq \\frac{1}{32} \\\\ \n4 \\gamma (1 + 16 \\gamma)^{-\\tfrac{3}{2}} \\|\\mathbf{Z}\\|_2 & \\mathrm{otherwise} \\end{cases} \\right] \\leq \\frac{2}{3 \\sqrt{3}} \\sqrt{\\gamma} \\|\\mathbf{Z}\\|_2\n\\end{equation}\n\\end{theorem}\n\\addtocounter{theorem}{-1}\n}\n\n\\begin{proof}\nThe polar problem associated with the objectives of the form \\eqref{eq:gen_obj} as given in \\cite{haeffele2019structured} is:\n\n\\begin{gather}\n       \\Omega_\\theta^\\circ (\\mathbf{Z}) = \\sup_{\\mathbf{d},\\mathbf{x}} \\mathbf{d}^{\\top} \\mathbf{Z} \\mathbf{x} \\ \\ \\textnormal{s.t.} \\ \\  \\bar \\theta(\\mathbf{d},\\mathbf{x}) \\leq 1 \n\\end{gather}\n\nFor our particular problem, due to the bilinearity between $\\mathbf{d}$ and $\\mathbf{x}$ in the objective the above is equivalent to:\n\\begin{gather}\n         \\Omega_\\theta^\\circ (\\mathbf{Z})   = \\sup_{\\mathbf{d},\\mathbf{x}} \\mathbf{d}^{\\top} \\mathbf{Z} \\mathbf{x} \\ \\ \\textnormal{s.t.} \\ \\  \\| \\mathbf{x} \\|^2_2 \\leq 1, \\|\\mathbf{d}\\|^2_2 + \\gamma \\min_{k} \\| \\mathbf{L} \\mathbf{d} + k^2 \\mathbf{d} \\|^2_2 \\leq 1\n\\end{gather}\nNote that this is equivalent to moving the minimization w.r.t. $k$ in the regularization constraint to a maximization over $k$:\n \\begin{gather}\n         \\Omega_\\theta^\\circ (\\mathbf{Z})   = \\sup_{\\mathbf{d},\\mathbf{x},k} \\mathbf{d}^{\\top} \\mathbf{Z} \\mathbf{x} \\ \\ \\textnormal{s.t.} \\ \\  \\| \\mathbf{x} \\|^2_2 \\leq 1, \\|\\mathbf{d}\\|^2_2 + \\gamma \\| \\mathbf{L} \\mathbf{d} + k^2 \\mathbf{d} \\|^2_2 \\leq 1\n\\end{gather}\n\nNext, note that maximizing w.r.t. $\\mathbf{d}$ while holding $\\mathbf{x}$ and $k$ fixed is equivalent to solving a problem of the form:\n\\begin{align}\n\\max_\\mathbf{d} \\langle \\mathbf{d}, \\mathbf{Z} \\mathbf{x} \\rangle \\ \\ \\textnormal{subject to} \\ \\  \\mathbf{d}^\\top \\mathbf{A} \\mathbf{d} \\leq 1\n\\end{align}\nfor some positive definite matrix $\\mathbf{A}$.  If we make the change of variables $\\bar \\mathbf{d} = \\mathbf{A}^{1\/2} \\mathbf{d}$, this then becomes:\n\\begin{align}\n\\max_{\\bar \\mathbf{d}} \\ \\ &\\langle \\bar \\mathbf{d}, \\mathbf{A}^{-1\/2} \\mathbf{Z} \\mathbf{x} \\rangle \\ \\ \\textnormal{subject to} \\ \\  \\| \\bar \\mathbf{d} \\|_2^2 \\leq 1 = \\| \\mathbf{A}^{-1\/2} \\mathbf{Z} \\mathbf{x} \\|_2 \n\\end{align}\nwhere the optimal $\\bar \\mathbf{d}$ and $\\mathbf{d}$ are obtained at \n\\begin{align}\n\\bar \\mathbf{d}_{opt} &= \\frac{\\mathbf{A}^{-1\/2} \\mathbf{Z} \\mathbf{x}}{\\|\\mathbf{A}^{-1\/2} \\mathbf{Z} \\mathbf{x}\\|_2} \\\\\n\\label{eq:d_opt}\n\\mathbf{d}_{opt} &= \\mathbf{A}^{-1\/2} \\bar \\mathbf{d}_{opt} = \\frac{\\mathbf{A}^{-1} \\mathbf{Z} \\mathbf{x}}{\\|\\mathbf{A}^{-1\/2} \\mathbf{Z} \\mathbf{x}\\|_2}\n\\end{align}\nFor our particular problem, if we make the change of variables $\\bar k = k^2$ we have that $\\mathbf{A}$ is given by:\n\\begin{equation}\n\\mathbf{A}(\\bar k) = (1+\\gamma \\bar k^2) \\mathbf{I} + \\gamma \\mathbf{L}^2 + 2 \\bar k \\gamma \\mathbf{L}\n\\end{equation}\nwhere we have used that $\\mathbf{L}$ is a symmetric matrix.  If we let $\\mathbf{L} = \\Gamma \\Lambda \\Gamma^\\top$ be an eigen-decomposition of $\\mathbf{L}$ then we can also represent $\\mathbf{A}(\\bar k)$ and $\\mathbf{A}(\\bar k)^{-1\/2}$ as:\n\\begin{align}\n\\mathbf{A}(\\bar k) &= \\Gamma \\left( (1+\\gamma \\bar k^2) \\mathbf{I} + \\gamma \\Lambda^2 + 2 \\bar k \\gamma \\Lambda \\right) \\Gamma^\\top \\\\\n &= \\Gamma (\\mathbf{I} + \\gamma ( \\bar k \\mathbf{I} +  \\Lambda)^2) \\Gamma^\\top \\\\ \n \\label{eq:A12}\n\\mathbf{A}(\\bar k)^{-1\/2} &= \\Gamma (\\mathbf{I} + \\gamma ( \\bar k \\mathbf{I} +  \\Lambda)^2)^{-1\/2} \\Gamma^\\top \n\\end{align}\nNow if we substitute back into the original polar problem, we have:\n\\begin{align}\n\\Omega^\\circ(\\mathbf{Z}) &= \\max_{\\mathbf{x}, \\bar k} \\| \\mathbf{A}(\\bar k)^{-1\/2} \\mathbf{Z} \\mathbf{x} \\|_2 \\ \\ \\textnormal{subject to} \\ \\  \\|\\mathbf{x}\\|_2^2 \\leq 1 \\\\\n\\label{eq:k_line}\n&= \\max_{\\bar k} \\|(\\mathbf{A}(\\bar k)^{-1\/2} \\mathbf{Z} \\|_2\n\\end{align}\nwhere $\\|\\cdot\\|_2$ denotes the spectral norm (maximum singular value).  Similarly, for a given $\\bar k$ the optimal $\\mathbf{x}$ is given as the right singular vector of $\\mathbf{A}(\\bar k)^{-1\/2}\\mathbf{Z}$ associated with the largest singular value.\n\nAs a result, we can solve the polar by performing a line search over $\\bar k$, then once an optimal $\\bar k^*$ is found we get $\\mathbf{x}^*$ as the largest right singular vector of $\\mathbf{A}(\\bar k^*)^{-1\/2} \\mathbf{Z}$ and the optimal $\\mathbf{d}^*$ from \\eqref{eq:d_opt} (where $\\mathbf{d}_{opt}$ will be the largest left singular vector of $\\mathbf{A}(\\bar k^*)^{-1\/2} \\mathbf{Z}$ multiplied by $\\mathbf{A}(\\bar k^*)^{-1\/2}$).\n\nNow, an upper bound for $\\bar k$ can be calculated from the fact that the optimal $\\bar k$ is defined using a minimization problem, i.e.\n\\begin{eqnarray}\n    \\bar k^* = \\argmin_{\\bar k} \\| \\mathbf{L} \\mathbf{d} + \\bar k \\mathbf{d} \\|^2\n\\end{eqnarray}\nSo we note for any $\\mathbf{d}$,\n\\begin{eqnarray}\n   \\bar k^* = -\\frac{ \\mathbf{d}^\\top \\mathbf{L} \\mathbf{d}} {\\|\\mathbf{d}\\|_2^2}\n   \\label{eq:opt_k}\n\\end{eqnarray}\nwhich is bounded by the smallest eigenvalue of $\\mathbf{L}$ (note that $\\mathbf{L}$ is negative (semi)definite). We cite the literature on eigenvalues of discrete second derivatives \\cite{chung2000discrete} to note that all eigenvalues of $\\mathbf{L}$ (irrespective of the boundary conditions) lie in the range $[\\frac{-4}{\\Delta l},0]$, since we specifically chose $\\Delta l = 1$ (without loss of generality), we have that all eigenvalues of $\\mathbf{L}$ lie in the range$[-4,0]$.\n\n\n\n\n\n\n\n\nAs a result, we need to only consider $\\bar k$ in the range $[0,4]$:\n\\begin{equation}\n\\bar k^* = \\argmax_{\\bar k \\in [0,4]} \\| \\mathbf{A}(\\bar k)^{-1\/2} \\mathbf{Z} \\|_2\n\\end{equation}\n\nFinally, to show the Lipschitz continuity, we define the function:\n\\begin{equation}\nf_\\mathbf{A}(\\bar k) = \\| \\mathbf{A}(\\bar k)^{-1\/2} \\|_2\n\\end{equation}\nand then note the following:\n\\begin{equation}\n\\begin{split}\n& \\left| \\|\\mathbf{A}(\\bar k)^{-1\/2}\\mathbf{Z} \\|_2 - \\| \\mathbf{A}(\\bar k ')^{-1\/2} \\mathbf{Z} \\|_2 \\right| \\\\\n\\leq & \\left\\| \\left( \\mathbf{A}(\\bar k)^{-1\/2} - \\mathbf{A}(\\bar k')^{-1\/2} \\right) \\mathbf{Z} \\right\\|_2 \\\\\n\\leq & \\left\\| \\mathbf{A}(\\bar k)^{-1\/2} - \\mathbf{A}(\\bar k')^{-1\/2} \\right\\|_2 \\left\\| \\mathbf{Z} \\right\\|_2 \\\\\n\\leq & L_\\mathbf{A} |\\bar k - \\bar k'| \\|\\mathbf{Z}\\|_2\n\\end{split}\n\\end{equation}\nwhere the first inequality is simply the reverse triangle inequality, the second inequality is due to the spectral norm being submultiplicative, and $L_\\mathbf{A}$ denotes the Lipschitz constant of $f_\\mathbf{A}(\\bar k)$.  From the form of $\\mathbf{A}(\\bar k)^{-1\/2}$ in \\eqref{eq:A12} note that we have:\n\\begin{equation}\nf_\\mathbf{A}(\\bar k) \\equiv \\| \\mathbf{A}(\\bar k)^{-1\/2}\\|_2 = \\max_i \\frac{1}{\\sqrt{1 + \\gamma(\\bar k + \\Lambda_{i,i})^2}}\n\\end{equation}\nso the result is completed by recalling from our discussion above that $\\Lambda_{i,i} \\in [-4, 0], \\forall i$ and applying Lemma \\ref{lem:L_bound}.\n\n\\end{proof}\n\n{\n\\renewcommand{\\thecorollary}{\\ref{cor:poly_time}}\n\\begin{corollary}\nAlgorithm \\ref{alg:meta} produces an optimal solution to \\eqref{eq:main_obj} in polynomial time.\n\\end{corollary}\n\\addtocounter{corollary}{-1}\n}\n\\begin{proof}\nThis result is largely a Corollary from Theorem \\ref{thm:polar} and what is known in the literature.  Namely, the authors of \\cite{xu2013block} show that the block coordinate update steps \\eqref{eq:grad_D}, \\eqref{eq:grad_X} and \\eqref{eq:grad_k} in step \\ref{step:grad_desc} of Algorithm \\ref{algoblock:meta-algo} reaches a stationary point in polynomial time because the objective function \\eqref{eq:main_obj} is convex w.r.t. each ($\\mathbf{D}$, $\\mathbf{X}$, $\\mathbf{k}$) if the other terms are held fixed. Next,\nby Theorem \\ref{thm:polar} the optimization problem for solving the polar can be done in polynomial time.  Finally, it has been noted in the literature on structured matrix factorization \\cite{bach2013convex} that the polar update step is equivalent to a generalized conditional gradient step\nand if the conditional gradient steps (i.e., the polar problem) can be solved exactly (as we show in Theorem \\ref{thm:polar}) then the algorithm converges in a polynomial number of such steps.  As a result, due to the fact that the block coordinate update steps reach a stationary in polynomial time we will perform a polar update step (a.k.a., a conditional gradient step) at polynomial time intervals, so the overall algorithm is also guaranteed to converge in polynomial time.\n\n\\end{proof} \n\n\\subsection{Interpreting the Wave-Informed Regularizer as a Bandpass Filter}\n\nNote that when identifying the optimal $k$ value in the polar program, we solve for\n\\begin{equation}\n\\label{eq:k_opt}\n\\argmax_{k \\in [0,4]} \\| \\Gamma (\\mathbf{I} + \\gamma ( \\bar k \\mathbf{I} +  \\Lambda)^2)^{-1\/2} \\Gamma^\\top \\mathbf{Z} \\|_2 \\; .\n\\end{equation}\nThis optimization has an intuitive interpretation from digital signal processing. Given that $\\Gamma$ contains the eigenvectors of a Toeplitz matrix, those eigenvectors have spectral qualities similar to the discrete Fourier transform (the eigenvectors of a circulant matrix would be the discrete Fourier transform). As a result, $\\Gamma^\\top$ transforms the data $\\mathbf{Z}$ into a spectral-like domain and $\\Gamma$ returns the data back to the original domain. Since the other terms are all diagonal matrices, they represent element-wise multiplication across the data in the spectral domain. This is approximately equivalent to a filtering operation, with filter coefficients given by the diagonal entries of $(\\mathbf{I} + \\gamma ( \\bar k \\mathbf{I} +  \\Lambda)^2)^{-1\/2}$.\n\nFurthermore, recall that the transfer function of a 1st-order Butterworth filter is given by:\n\\begin{equation}\nT(\\omega) = \\frac{1}{\\sqrt{1+\\gamma (k_0 + \\omega)^2}}\n\\end{equation}\nwhere $k_0$ is the center frequency of the passband of the filter and $1\/\\sqrt{\\gamma}$ corresponds to the filter's $-3$dB cut-off frequency.  Comparing this to the filter coefficients from \\eqref{eq:k_opt} we note that the filter coefficients are identical to those of the 1st-order Butterworth filter, where $\\Lambda$ corresponds to the angular frequencies.\nAs a result, we can consider this optimization as determining the optimal filter center frequency $(\\bar k)$ with fixed bandwidth $(1\/\\sqrt{\\gamma})$  that retains the maximum amount of signal power from $\\mathbf{Z}$.  Likewise the choice of the $\\gamma$ hyperparameter sets the bandwidth of the filter.  As $\\gamma \\rightarrow \\infty$, the filter bandwidth approaches 0 and thereby restricts us to a single-frequency (i.e., Fourier) solution.\nFurthermore, we can provide a recommended lower bound for $\\gamma$ according to $\\gamma > 1\/k_{bw}^2$, where $k_{bw}$ is the bandwidth of the signal within this spectral-like domain.\n\n\n\\section{Additional Results}\n\n\\subsection{Characterizing Multi-Segment Transmission Lines}\n\n For the simulation considered in \\textbf{Characterizing Multi-Segment Transmission Lines} in \\S~\\ref{sec:results}, Figure \\ref{fig:Merge_2} shows two example columns for $\\gamma = 50$ and $\\lambda = 0.6$. We show in Figure \\ref{fig:polar_objective} the reduction of objective value over iterations, the rate of change of objective value per iteration and the value of polar after each iteration of the overall meta-algorithm. Figures~\\ref{fig:workflow_a}, \\ref{fig:workflow_b}, \\ref{fig:workflow_c} show curves similar to Figure~\\ref{fig:polar_objective} but for different choices of regularization parameters.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.75]{figs\/polar_obj_50_lamb_0_6.png}\n    \\caption{Objective value, rate of change of objective value per iteration and the value of polar after each iteration of the meta-algorithm for the simulation in subsection \\textbf{Characterizing Multi-Segment Transmission Lines} \\S \\ref{sec:results}.}\n    \\label{fig:polar_objective}\n\\end{figure}\n\nWe observe in Figure \\ref{fig:polar_objective} that the change of objective value is almost zero after 70 iterations. At this point, the polar value also goes below 1.1 (which as we note in the main paper also implies we are close to the global minimum) and eventually reaches 1, providing a certificate of global optimality.\n\n \n  \nIn practice, we often stop the algorithm after the polar value reaches below 1.1 as this guarantees a very close to optimal solution.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.95\\textwidth]{figs\/polar_obj_50_lamb_1_6.png}\n    \\caption{Objective value, rate of change of objective value per iteration and the value of polar after each iteration of the meta-algorithm for the simulation in \\textbf{Characterizing Multi-Segment Transmission Lines} \\S \\ref{sec:results} with values of regularization parameters $\\gamma = 50$, $\\lambda = 1.6$}\n    \\label{fig:workflow_a}\n\\end{figure}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.95\\textwidth]{figs\/polar_obj_50_lamb_3_7.png}\n    \\caption{Objective value, rate of change of objective value per iteration and the value of polar after each iteration of the meta-algorithm for the simulation in \\textbf{Characterizing Multi-Segment Transmission Lines} \\S \\ref{sec:results} with values of regularization parameters $\\gamma = 50$, $\\lambda = 3.7$}\n    \\label{fig:workflow_b}\n\\end{figure}\n\\begin{figure}\n    \\centering\n   \\includegraphics[width=0.95\\textwidth]{figs\/polar_obj_50_lamb_5.png}\n    \\caption{Objective value, rate of change of objective value per iteration and the value of polar after each iteration of the meta-algorithm for the simulation in \\textbf{Characterizing Multi-Segment Transmission Lines} \\S \\ref{sec:results} with values of regularization parameters $\\gamma = 50$, $\\lambda = 5$}\n    \\label{fig:workflow_c}\n\\end{figure}\n\n\n\n\n\nIn addition to these optimization results, we also show additional examples of columns of $\\mathbf{D}$ for different values of $\\gamma$ ($500$ and $5000$) with $\\lambda$ fixed at 0.6 to show the variation of the columns of $\\mathbf{D}$ with $\\gamma$.\n\nIn Figures~\\ref{dictatoms_multi_a} and ~\\ref{dictatoms_multi_b}, we observe that with increasing $\\gamma$, the performance of the algorithm in distinguishing the two regions reduces: energy in the dictionary column is no longer confined to the segment containing the same material, it is distributed over the entire column. This can also be observed quantitatively from comparing the partitioned normalized energies (energy on each of the two segments, partitioned by the violet line, of the normalized columns of $\\mathbf{D}$) in Figures~\\ref{energies_a},~\\ref{energies_b} and Figures~\\ref{dictatoms_multi_a},\\ref{dictatoms_multi_b}.\n\n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[width=0.65\\textwidth]{figs\/D_diff_strings_lr_gamma_500_lam_0_6.png}\n    \\caption{Two Columns of $\\mathbf{D}$ obtained from wave-informed matrix factorization when $\\gamma = 500$ $\\lambda = 0.6$.}\n    \\label{dictatoms_multi_a}\n\\end{figure}\n\n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[width=0.65\\textwidth]{figs\/D_diff_strings_lr_gamma_5000_lam_0_6.png}\n    \\caption{Two Columns of $\\mathbf{D}$ obtained from wave-informed matrix factorization when $\\gamma = 5000$ $\\lambda = 0.6$.}\n    \\label{dictatoms_multi_b}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n    \\centering\n   \\includegraphics[width=0.65\\textwidth]{figs\/energies_gamma_500_lamb_0_6.png}\n    \\caption{Two Columns of $\\mathbf{D}$ obtained from wave-informed matrix factorization when $\\gamma = 500$ $\\lambda = 0.6$.}\n    \\label{energies_a}\n\\end{figure}\n\n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[width=0.65\\textwidth]{figs\/energies_gamma_5000_lamb_0_6.png}\n    \\caption{Two Columns of $\\mathbf{D}$ obtained from wave-informed matrix factorization when $\\gamma = 5000$ $\\lambda = 0.6$.}\n    \\label{energies_b}\n\\end{figure}\n\n\n\\subsection{A Vibrating String}\nFor the experiment described in subsection \\textbf{A Fixed Vibrating String} of \\S \\ref{sec:results}, we show the squared difference between the actual matrix $\\mathbf{D}$ and the one estimated from the algorithm in Table \\ref{table:wave_1}, Table \\ref{table:wave_2}, and Table \\ref{table:wave_3} (for different scenarios). We can calculate the percentage error as\n\\begin{gather}\n \\% \\textnormal{error} = \\frac{\\| \\hat{\\mathbf{D}} - \\mathbf{D} \\|_F}{\\|\\mathbf{D}\\|_F} \\times 100\n\\end{gather}\nwhere $\\hat{\\mathbf{D}}$ is the matrix estimated from the algorithm after permuting the columns to optimally align with the ground-truth $\\mathbf{D}$, zero-padding $\\mathbf{D}$ or $\\hat{\\mathbf{D}}$ (whichever has fewer columns) so that the two matrices have the same number of columns, and normalizing the non-zero columns of $\\hat{\\mathbf{D}}$ to have unit $\\ell_2$ norm\n\nSince the matrix is normalized and the actual $\\mathbf{D}$ matrix contains 10 columns (corresponding to 10 modes in the data simulated), we know that $\\| \\mathbf{D} \\|_F = 10$. We define error  $\\| \\hat{\\mathbf{D}} - \\mathbf{D} \\|^2_F$ of the order of $0.01$ ($1 \\%$) to be desirable. We bold error of this order in the tables. Table~\\ref{table:wave_1} describes the performance as a function of regularization as described in \\eqref{eqn:wave_equ} with amplitude of order 10 and noise variance of $0.1$. We see a wide range of values of $\\gamma$ and $\\lambda$ satisfy our error criteria.\n\nTable~\\ref{table:wave_2} describes the performance as a function of regularization as described by the damped vibrations in \\eqref{eqn:damped_wave_time} with amplitude of order 10 and noise variance of $0.1$. We see that the error values satisfy our chosen criteria beyond a particular value of regularization $\\gamma$ ($10^5$) and are also limited in the range of $\\lambda$. The range of $\\lambda$ that satisfies the chosen error condition also reduces with increasing values of $\\gamma$.\n\nTable~\\ref{table:wave_3} describes the performance as a function of regularization as described in \\eqref{eqn:wave_equ} with amplitude of order 10 and noise variance of 10. We observe that a narrow range of regularization values satisfy the error criteria. We thus clearly see the effect of noise in this case. Based on our observations, we believe the large regularization values over fit to the noise with high frequency components, resulting in poor performance. We believe low regularization values do not enforce enough of the wave-constraint, resulting in noisy components.\n\nIn Figures \\ref{fig:accommodations_a}, \\ref{fig:accommodations_b}, we show columns of $\\mathbf{D}$ obtained from low-rank factorization and wave-informed matrix factorization with a signal amplitude of order $10$ and noise variance of $10$. For the low-rank factorization, we choose $\\lambda = 2592$ and for wave-informed matrix factorization we choose $\\gamma = 10^7$ and $\\lambda = 2200$ ($\\gamma$ and $\\lambda$ are chosen corresponding to the minimum value in Table~\\ref{table:wave_3}). The columns of $\\mathbf{D}$ obtained from low-rank matrix factorization in Figure~\\ref{fig:accommodations_a} are noisy and not always decipherable as sinusoids. In contrast, columns of $\\mathbf{D}$ obtained from wave-informed matrix factorization in Figure \\ref{fig:accommodations_b} on the other hand are completely noiseless and have a clearly measurable wavenumber.\n\nFor the case of \\eqref{eqn:damped_wave_space}, when $R=6$, we show an approximate recovery of all the modes in Figure~\\ref{fig:damped_sines_all}.\n\n\\setlength{\\arrayrulewidth}{0.65mm}\n\\setlength{\\tabcolsep}{4pt}\n\\begin{table}[H]\n\n\\captionof{table}{Squared Frobenius norm of the difference between actual $\\mathbf{D}$ and the one obtained from Wave-informed matrix factorization with noise of variance 0.1 and data of the form \\eqref{eqn:wave_equ}}\n\\begin{tabular}{@{}|r|r|r|r|r|r|r|r|r|r|r|r|r|@{}}\n\\hline\n\\multicolumn{1}{|c|}{} & \\multicolumn{12}{|c|}{Regularization $\\left(\\gamma\\right)$} \\\\\n\\hline\n\\multicolumn{1}{|l|}{$\\lambda$} & \\multicolumn{1}{l|}{$10^1$} & \\multicolumn{1}{l|}{$10^{2}$} & \\multicolumn{1}{l|}{$10^3$} & \\multicolumn{1}{l|}{$10^4$} & \\multicolumn{1}{l|}{$10^5$} & \\multicolumn{1}{l|}{$10^6$} & \\multicolumn{1}{l|}{$10^7$} & \\multicolumn{1}{l|}{$10^8$} & \\multicolumn{1}{l|}{$10^9$} & \\multicolumn{1}{l|}{$10^{10}$} & \\multicolumn{1}{l|}{$10^{11}$} & \\multicolumn{1}{l|}{$10^{12}$} \\\\ \\hline\n200                          & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0038}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0035}    & \\cellcolor[HTML]{FFDC7F}1.0023             & \\cellcolor[HTML]{FFDC7F}1.0013             & \\cellcolor[HTML]{FFDC7F}1.0013                  & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         \\\\ \\hline\n300                          & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0038}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0035}    & \\cellcolor[HTML]{FFDC7F}1.0023             & \\cellcolor[HTML]{FFDC7F}1.0013             & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         \\\\ \\hline\n400                          & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0038}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0035}    & \\cellcolor[HTML]{FFEAB4}3.0598             & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         \\\\ \\hline\n500                          & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0038}    & \\cellcolor[HTML]{FFDC7F}1.0035             & \\cellcolor[HTML]{FFEAB3}3.0438             & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         \\\\ \\hline\n600                          & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0038}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0035}    & \\cellcolor[HTML]{FFEAB3}3.0342             & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         \\\\ \\hline\n700                          & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0038}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0035}    & \\cellcolor[HTML]{FFE399}2.0015             & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFDC7F}1.0011                  \\\\ \\hline\n800                          & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0038}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0035}    & \\cellcolor[HTML]{FFDC7F}1.0014             & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFDC7F}1.0011                  \\\\ \\hline\n900                          & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0038}    & \\cellcolor[HTML]{FFDC7F}1.0035             & \\cellcolor[HTML]{FFDC7F}1.0013             & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFDC7F}1.0011                  \\\\ \\hline\n1000                         & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0038}    & \\cellcolor[HTML]{FFDC7F}1.0035             & \\cellcolor[HTML]{FFDC7F}1.0013             & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFE398}2.0009                  \\\\ \\hline\n1100                         & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0038}    & \\cellcolor[HTML]{FFDC7F}1.0035             & \\cellcolor[HTML]{FFDC7F}1.0014             & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFEAB2}3.0009                  \\\\ \\hline\n1400                         & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0038}    & \\cellcolor[HTML]{FFDC7F}1.0035             & \\cellcolor[HTML]{FFEAB2}2.9963             & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFF8E5}5.0005                  \\\\ \\hline\n1800                         & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0038}    & \\cellcolor[HTML]{FFDC7F}1.0035             & \\cellcolor[HTML]{FFDC7F}1.0019             & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFFEFE}6.0004                  \\\\ \\hline\n2000                         & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0038}    & \\cellcolor[HTML]{FFDC7F}1.0035             & \\cellcolor[HTML]{FFDC7F}1.0018             & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFDC7F}1.0011                  & \\cellcolor[HTML]{FFFEFE}6.0004                  \\\\ \\hline\n2100                         & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0038}    & \\cellcolor[HTML]{FFDC7F}\\textbf{1.0035}    & \\cellcolor[HTML]{FFDC7F}1.0017             & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFDC7F}1.0011                  & \\cellcolor[HTML]{FFFEFE}6.0004                     \\\\ \\hline\n2200                         & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0040}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0041}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0038}    & \\cellcolor[HTML]{FFDC7F}\\textbf{1.0035}    & \\cellcolor[HTML]{FFDC7F}1.0017             & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFDC7F}1.0011                  & \\cellcolor[HTML]{FFFEFE}6.0004                  \\\\ \\hline\n\\end{tabular}\n\\label{table:wave_1}\n\\end{table}\n\n\n\\setlength{\\arrayrulewidth}{0.65mm}\n\\setlength{\\tabcolsep}{4pt}\n\\begin{table}[H]\n\\captionof{table}{Squared Frobenius norm of the difference between actual $\\mathbf{D}$ and the one obtained from Wave-informed matrix factorization with noise of variance 0.1 and data of the form \\eqref{eqn:damped_wave_time}}\n\\begin{tabular}{@{}|r|r|r|r|r|r|r|r|r|r|r|r|r|@{}}\n\\hline\n\\multicolumn{1}{|c|}{} & \\multicolumn{12}{|c|}{Regularization $\\left(\\gamma\\right)$} \\\\\n\\hline\n\\multicolumn{1}{|l|}{$\\lambda$} & \\multicolumn{1}{l|}{$10^1$} & \\multicolumn{1}{l|}{$10^{2}$} & \\multicolumn{1}{l|}{$10^3$} & \\multicolumn{1}{l|}{$10^4$} & \\multicolumn{1}{l|}{$10^5$} & \\multicolumn{1}{l|}{$10^6$} & \\multicolumn{1}{l|}{$10^7$} & \\multicolumn{1}{l|}{$10^8$} & \\multicolumn{1}{l|}{$10^9$} & \\multicolumn{1}{l|}{$10^{10}$} & \\multicolumn{1}{l|}{$10^{11}$} & \\multicolumn{1}{l|}{$10^{12}$} \\\\ \\hline\n200                          & \\cellcolor[HTML]{FFD76C}0.2846             & \\cellcolor[HTML]{FFD76C}0.2845             & \\cellcolor[HTML]{FFD76C}0.2829             & \\cellcolor[HTML]{FFD76B}0.2655             & \\cellcolor[HTML]{FFD669}0.1606             & \\cellcolor[HTML]{FFD667}\\textbf{0.0521}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0165}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0018}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFDB7B}1.0011                  & \\cellcolor[HTML]{FFE7A7}3.0006                  \\\\ \\hline\n300                          & \\cellcolor[HTML]{FFD76C}0.2846             & \\cellcolor[HTML]{FFD76C}0.2845             & \\cellcolor[HTML]{FFD76C}0.2829             & \\cellcolor[HTML]{FFD76B}0.2655             & \\cellcolor[HTML]{FFD669}0.1606             & \\cellcolor[HTML]{FFD667}\\textbf{0.0521}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0165}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0018}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFDB7B}1.0011                  & \\cellcolor[HTML]{FFEDBD}4.0005                  \\\\ \\hline\n400                          & \\cellcolor[HTML]{FFD76C}0.2846             & \\cellcolor[HTML]{FFD76C}0.2845             & \\cellcolor[HTML]{FFD76C}0.2829             & \\cellcolor[HTML]{FFD76B}0.2655             & \\cellcolor[HTML]{FFD669}0.1606             & \\cellcolor[HTML]{FFD667}\\textbf{0.0521}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0165}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0018}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFDB7B}1.0011                  & \\cellcolor[HTML]{FFF3D3}5.0005                  \\\\ \\hline\n500                          & \\cellcolor[HTML]{FFD76C}0.2846             & \\cellcolor[HTML]{FFD76C}0.2845             & \\cellcolor[HTML]{FFD76C}0.2829             & \\cellcolor[HTML]{FFD76B}0.2655             & \\cellcolor[HTML]{FFD669}0.1606             & \\cellcolor[HTML]{FFD667}\\textbf{0.0521}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0165}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0018}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFDB7B}1.0011                  & \\cellcolor[HTML]{FFF9E9}6.0004                  \\\\ \\hline\n600                          & \\cellcolor[HTML]{FFD76C}0.2846             & \\cellcolor[HTML]{FFD76C}0.2845             & \\cellcolor[HTML]{FFD76C}0.2829             & \\cellcolor[HTML]{FFD76B}0.2655             & \\cellcolor[HTML]{FFD669}0.1606             & \\cellcolor[HTML]{FFD667}\\textbf{0.0521}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0165}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0018}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFE7A7}3.0006                  & \\cellcolor[HTML]{FFF9E9}6.0004                  \\\\ \\hline\n700                          & \\cellcolor[HTML]{FFD76C}0.2846             & \\cellcolor[HTML]{FFD76C}0.2845             & \\cellcolor[HTML]{FFD76C}0.2829             & \\cellcolor[HTML]{FFD76B}0.2655             & \\cellcolor[HTML]{FFD669}0.1606             & \\cellcolor[HTML]{FFD667}\\textbf{0.0521}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0165}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0018}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}         & \\cellcolor[HTML]{FFE7A7}3.0006                  & \\cellcolor[HTML]{FFF9E9}6.0004                  \\\\ \\hline\n800                          & \\cellcolor[HTML]{FFD76C}0.2846             & \\cellcolor[HTML]{FFD76C}0.2845             & \\cellcolor[HTML]{FFD76C}0.2829             & \\cellcolor[HTML]{FFD76B}0.2655             & \\cellcolor[HTML]{FFD669}0.1606             & \\cellcolor[HTML]{FFD667}\\textbf{0.0521}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0165}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0018}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFDB7B}1.0011                  & \\cellcolor[HTML]{FFEDBD}4.0005                  & \\cellcolor[HTML]{FFF9E9}6.0004                  \\\\ \\hline\n900                          & \\cellcolor[HTML]{FFD76C}0.2846             & \\cellcolor[HTML]{FFD76C}0.2845             & \\cellcolor[HTML]{FFD76C}0.2829             & \\cellcolor[HTML]{FFD76B}0.2655             & \\cellcolor[HTML]{FFD669}0.1606             & \\cellcolor[HTML]{FFD667}\\textbf{0.0521}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0165}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0018}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFDB7B}1.0011                  & \\cellcolor[HTML]{FFEDBD}4.0005                  & \\cellcolor[HTML]{FFF9E9}6.0004                  \\\\ \\hline\n1000                         & \\cellcolor[HTML]{FFD76C}0.2846             & \\cellcolor[HTML]{FFD76C}0.2845             & \\cellcolor[HTML]{FFD76C}0.2829             & \\cellcolor[HTML]{FFD76B}0.2655             & \\cellcolor[HTML]{FFD669}0.1606             & \\cellcolor[HTML]{FFD667}\\textbf{0.0521}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0165}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0018}    & \\cellcolor[HTML]{FFD666}\\textbf{0.0013}    & \\cellcolor[HTML]{FFDB7B}1.0011                  & \\cellcolor[HTML]{FFEDBD}4.0005                  & \\cellcolor[HTML]{FFF9E9}6.0004                  \\\\ \\hline\n1100                         & \\cellcolor[HTML]{FFDD82}1.2836             & \\cellcolor[HTML]{FFDD82}1.2835             & \\cellcolor[HTML]{FFDD81}1.2819             & \\cellcolor[HTML]{FFDD81}1.2644             & \\cellcolor[HTML]{FFDC7F}1.1596             & \\cellcolor[HTML]{FFDC7C}1.0513             & \\cellcolor[HTML]{FFDB7C}1.0162             & \\cellcolor[HTML]{FFDB7B}1.0016             & \\cellcolor[HTML]{FFDB7B}1.0011             & \\cellcolor[HTML]{FFDB7B}1.0011                  & \\cellcolor[HTML]{FFEDBD}4.0005                  & \\cellcolor[HTML]{FFF9E9}6.0004                  \\\\ \\hline\n1400                         & \\cellcolor[HTML]{FFE397}2.2794             & \\cellcolor[HTML]{FFE397}2.2793             & \\cellcolor[HTML]{FFE397}2.2777             & \\cellcolor[HTML]{FFE397}2.2602             & \\cellcolor[HTML]{FFE295}2.1554             & \\cellcolor[HTML]{FFE192}2.0478             & \\cellcolor[HTML]{FFE192}2.0152             & \\cellcolor[HTML]{FFE191}2.0013             & \\cellcolor[HTML]{FFE191}2.0008             & \\cellcolor[HTML]{FFEDBD}4.0003                  & \\cellcolor[HTML]{FFFEFE}7.0002                  & \\cellcolor[HTML]{FFFEFE}7.0002                  \\\\ \\hline\n1800                         & \\cellcolor[HTML]{FFEDBE}4.0460             & \\cellcolor[HTML]{FFEDBE}4.0460             & \\cellcolor[HTML]{FFEDBE}4.0460             & \\cellcolor[HTML]{FFEDBE}4.0459             & \\cellcolor[HTML]{FFEDBE}4.0452             & \\cellcolor[HTML]{FFEDBE}4.0387             & \\cellcolor[HTML]{FFEDBD}4.0127             & \\cellcolor[HTML]{FFEDBD}4.0008             & \\cellcolor[HTML]{FFEDBD}4.0003             & \\cellcolor[HTML]{FFEDBD}4.0003                  & \\cellcolor[HTML]{FFFEFE}7.0002                  & \\cellcolor[HTML]{FFFEFE}7.0002                  \\\\ \\hline\n2000                         & \\cellcolor[HTML]{FFEDBE}4.0460             & \\cellcolor[HTML]{FFEDBE}4.0460             & \\cellcolor[HTML]{FFEDBE}4.0460             & \\cellcolor[HTML]{FFEDBE}4.0459             & \\cellcolor[HTML]{FFEDBE}4.0452             & \\cellcolor[HTML]{FFEDBE}4.0387             & \\cellcolor[HTML]{FFEDBD}4.0127             & \\cellcolor[HTML]{FFEDBD}4.0008             & \\cellcolor[HTML]{FFEDBD}4.0003             & \\cellcolor[HTML]{FFF3D3}5.0002                  & \\cellcolor[HTML]{FFFEFE}7.0002                  & \\cellcolor[HTML]{FFFEFE}7.0002                  \\\\ \\hline\n2100                         & \\cellcolor[HTML]{FFEDBE}4.0460             & \\cellcolor[HTML]{FFEDBE}4.0460             & \\cellcolor[HTML]{FFEDBE}4.0460             & \\cellcolor[HTML]{FFEDBE}4.0459             & \\cellcolor[HTML]{FFEDBE}4.0452             & \\cellcolor[HTML]{FFEDBE}4.0387             & \\cellcolor[HTML]{FFEDBD}4.0127             & \\cellcolor[HTML]{FFEDBD}4.0008             & \\cellcolor[HTML]{FFEDBD}4.0003             & \\cellcolor[HTML]{FFF3D3}5.0002                  & \\cellcolor[HTML]{FFFEFE}7.0002                     & \\cellcolor[HTML]{FFFEFE}7.0002                  \\\\ \\hline\n2200                         & \\cellcolor[HTML]{FFEDBE}4.0460             & \\cellcolor[HTML]{FFEDBE}4.0460             & \\cellcolor[HTML]{FFEDBE}4.0460             & \\cellcolor[HTML]{FFEDBE}4.0459             & \\cellcolor[HTML]{FFEDBE}4.0452             & \\cellcolor[HTML]{FFEDBE}4.0387             & \\cellcolor[HTML]{FFEDBD}4.0127             & \\cellcolor[HTML]{FFEDBD}4.0008             & \\cellcolor[HTML]{FFEDBD}4.0003             & \\cellcolor[HTML]{FFF3D3}5.0002                  & \\cellcolor[HTML]{FFFEFE}7.0002                  & \\cellcolor[HTML]{FFFEFE}7.0002                  \\\\ \\hline\n\\end{tabular}\n\\label{table:wave_2}\n\\end{table}\n\n\n\\setlength{\\arrayrulewidth}{0.65mm}\n\\setlength{\\tabcolsep}{4pt}\n\n\\begin{table}[H]\n\\captionof{table}{Squared Frobenius norm of the difference between actual $\\mathbf{D}$ and the one obtained from Wave-informed matrix factorization with noise of variance 10 and data of the form \\eqref{eqn:wave_equ}}\n\\begin{tabular}{@{}|r|r|r|r|r|r|r|r|r|r|r|r|r|@{}}\n\\hline\n\\multicolumn{1}{|c|}{} & \\multicolumn{12}{|c|}{Regularization $\\left(\\gamma\\right)$} \\\\\n\\hline\n\\multicolumn{1}{|l|}{$\\lambda$} & \\multicolumn{1}{l|}{$10^1$} & \\multicolumn{1}{l|}{$10^{2}$} & \\multicolumn{1}{l|}{$10^3$} & \\multicolumn{1}{l|}{$10^4$} & \\multicolumn{1}{l|}{$10^5$} & \\multicolumn{1}{l|}{$10^6$} & \\multicolumn{1}{l|}{$10^7$} & \\multicolumn{1}{l|}{$10^8$} & \\multicolumn{1}{l|}{$10^9$} & \\multicolumn{1}{l|}{$10^{10}$} & \\multicolumn{1}{l|}{$10^{11}$} & \\multicolumn{1}{l|}{$10^{12}$} \\\\ \\hline\n200                          & \\cellcolor[HTML]{FFFEFE}20.220                & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.216             & \\cellcolor[HTML]{FFFEFE}20.202             & \\cellcolor[HTML]{FFFEFE}20.141             & \\cellcolor[HTML]{FFFEFD}20.063             & \\cellcolor[HTML]{FFFEFD}20.022             & \\cellcolor[HTML]{FFFEFD}20.001             & \\cellcolor[HTML]{FFFEFD}20.001                  & \\cellcolor[HTML]{FFFEFD}20.001                  & \\cellcolor[HTML]{FFE6A2}7.991                   \\\\ \\hline\n300                          & \\cellcolor[HTML]{FFFEFE}20.220             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.216             & \\cellcolor[HTML]{FFFEFE}20.202             & \\cellcolor[HTML]{FFFEFE}20.141             & \\cellcolor[HTML]{FFFEFD}20.063             & \\cellcolor[HTML]{FFFEFD}20.022             & \\cellcolor[HTML]{FFFEFD}20.001             & \\cellcolor[HTML]{FFFEFD}20.001                  & \\cellcolor[HTML]{FFFEFD}20.001                  & \\cellcolor[HTML]{FFE49A}6.991                   \\\\ \\hline\n400                          & \\cellcolor[HTML]{FFFEFE}20.220             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.216             & \\cellcolor[HTML]{FFFEFE}20.202             & \\cellcolor[HTML]{FFFEFE}20.141             & \\cellcolor[HTML]{FFFEFD}20.063             & \\cellcolor[HTML]{FFFEFD}20.022             & \\cellcolor[HTML]{FFFEFD}20.001             & \\cellcolor[HTML]{FFFEFD}20.001                  & \\cellcolor[HTML]{FFFEFD}20.001                  & \\cellcolor[HTML]{FFE49A}6.991                   \\\\ \\hline\n500                          & \\cellcolor[HTML]{FFFEFE}20.220             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.216             & \\cellcolor[HTML]{FFFEFE}20.202             & \\cellcolor[HTML]{FFFEFE}20.141             & \\cellcolor[HTML]{FFFEFD}20.063             & \\cellcolor[HTML]{FFFEFD}20.022             & \\cellcolor[HTML]{FFFEFD}20.001             & \\cellcolor[HTML]{FFFEFD}20.001                  & \\cellcolor[HTML]{FFFEFD}20.001                  & \\cellcolor[HTML]{FFE49A}6.991                   \\\\ \\hline\n600                          & \\cellcolor[HTML]{FFFEFE}20.220             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.216             & \\cellcolor[HTML]{FFFEFE}20.202             & \\cellcolor[HTML]{FFFEFE}20.141             & \\cellcolor[HTML]{FFFEFD}20.063             & \\cellcolor[HTML]{FFFEFD}20.022             & \\cellcolor[HTML]{FFFEFD}20.001             & \\cellcolor[HTML]{FFFEFD}20.001                  & \\cellcolor[HTML]{FFEAB1}10.001                  & \\cellcolor[HTML]{FFE49A}6.991                   \\\\ \\hline\n700                          & \\cellcolor[HTML]{FFFEFE}20.220             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.216             & \\cellcolor[HTML]{FFFEFE}20.202             & \\cellcolor[HTML]{FFFEFE}20.141             & \\cellcolor[HTML]{FFFEFD}20.063             & \\cellcolor[HTML]{FFFEFD}20.022             & \\cellcolor[HTML]{FFFEFD}20.001             & \\cellcolor[HTML]{FFFEFD}20.001                  & \\cellcolor[HTML]{FFE293}6.001                   & \\cellcolor[HTML]{FFE49A}6.991                   \\\\ \\hline\n800                          & \\cellcolor[HTML]{FFFEFE}20.220             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.216             & \\cellcolor[HTML]{FFFEFE}20.202             & \\cellcolor[HTML]{FFFEFE}20.141             & \\cellcolor[HTML]{FFFEFD}20.063             & \\cellcolor[HTML]{FFFEFD}20.022             & \\cellcolor[HTML]{FFFEFD}20.001             & \\cellcolor[HTML]{FFFEFD}20.001                  & \\cellcolor[HTML]{FFE293}6.001                   & \\cellcolor[HTML]{FFE49A}6.991                   \\\\ \\hline\n900                          & \\cellcolor[HTML]{FFFEFE}20.220             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.216             & \\cellcolor[HTML]{FFFEFE}20.202             & \\cellcolor[HTML]{FFFEFE}20.141             & \\cellcolor[HTML]{FFFEFD}20.063             & \\cellcolor[HTML]{FFFEFD}20.022             & \\cellcolor[HTML]{FFFEFD}20.001             & \\cellcolor[HTML]{FFFEFD}20.001                  & \\cellcolor[HTML]{FFE293}6.001                   & \\cellcolor[HTML]{FFE49A}6.991                   \\\\ \\hline\n1000                         & \\cellcolor[HTML]{FFFEFE}20.220             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.216             & \\cellcolor[HTML]{FFFEFE}20.202             & \\cellcolor[HTML]{FFFEFE}20.141             & \\cellcolor[HTML]{FFFEFD}20.063             & \\cellcolor[HTML]{FFFEFD}20.022             & \\cellcolor[HTML]{FFFEFD}20.001             & \\cellcolor[HTML]{FFFEFD}20.001                  & \\cellcolor[HTML]{FFE293}6.001                   & \\cellcolor[HTML]{FFE49A}6.991                   \\\\ \\hline\n1100                         & \\cellcolor[HTML]{FFFEFE}20.220             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.216             & \\cellcolor[HTML]{FFFEFE}20.202             & \\cellcolor[HTML]{FFFEFE}20.141             & \\cellcolor[HTML]{FFFEFD}20.063             & \\cellcolor[HTML]{FFFEFD}20.022             & \\cellcolor[HTML]{FFFEFD}20.001             & \\cellcolor[HTML]{FFFEFD}20.001                  & \\cellcolor[HTML]{FFE293}6.001                   & \\cellcolor[HTML]{FFDD83}3.991                   \\\\ \\hline\n1400                         & \\cellcolor[HTML]{FFFEFE}20.220             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.216             & \\cellcolor[HTML]{FFFEFE}20.202             & \\cellcolor[HTML]{FFFEFE}20.141             & \\cellcolor[HTML]{FFFEFD}20.063             & \\cellcolor[HTML]{FFFEFD}20.022             & \\cellcolor[HTML]{FFFEFD}20.001             & \\cellcolor[HTML]{FFE8A9}9.001                   & \\cellcolor[HTML]{FFE293}6.001                   & \\cellcolor[HTML]{FFE292}5.978                   \\\\ \\hline\n1800                         & \\cellcolor[HTML]{FFFEFE}20.220             & \\cellcolor[HTML]{FFFEFE}20.218             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFFEFE}20.216             & \\cellcolor[HTML]{FFFEFE}20.202             & \\cellcolor[HTML]{FFFEFE}20.141             & \\cellcolor[HTML]{FFFEFD}20.063             & \\cellcolor[HTML]{FFFEFD}20.022             & \\cellcolor[HTML]{FFFEFD}20.001             & \\cellcolor[HTML]{FFE49A}7.001                   & \\cellcolor[HTML]{FFE293}6.001                   & \\cellcolor[HTML]{FFE49A}6.978                   \\\\ \\hline\n2000                         & \\cellcolor[HTML]{FFFEFE}20.220             & \\cellcolor[HTML]{FFFEFE}20.218             & \\cellcolor[HTML]{FFFEFE}20.219             & \\cellcolor[HTML]{FFEAB3}10.216             & \\cellcolor[HTML]{FFDC7D}3.202              & \\cellcolor[HTML]{FFD666}0.141              & \\cellcolor[HTML]{FFD86D}1.063              & \\cellcolor[HTML]{FFDE84}4.021              & \\cellcolor[HTML]{FFDE83}4.001              & \\cellcolor[HTML]{FFDE83}4.001                   & \\cellcolor[HTML]{FFDB7C}3.001                   & \\cellcolor[HTML]{FFE293}6.000                   \\\\ \\hline\n2100                         & \\cellcolor[HTML]{FFFEFE}20.220             & \\cellcolor[HTML]{FFFEFE}20.218             & \\cellcolor[HTML]{FFDE85}4.219              & \\cellcolor[HTML]{FFD667}0.216              & \\cellcolor[HTML]{FFD667}0.202              & \\cellcolor[HTML]{FFD666}0.141              & \\cellcolor[HTML]{FFD666}\\textbf{0.063}     & \\cellcolor[HTML]{FFDC7C}3.021              & \\cellcolor[HTML]{FFDB7C}3.001              & \\cellcolor[HTML]{FFDB7C}3.001                   & \\cellcolor[HTML]{FFD974}2.001                   & \\cellcolor[HTML]{FFE293}6.000                   \\\\ \\hline\n2200                         & \\cellcolor[HTML]{FFFEFE}20.220             & \\cellcolor[HTML]{FFDA76}2.219              & \\cellcolor[HTML]{FFD667}0.219              & \\cellcolor[HTML]{FFD667}0.216              & \\cellcolor[HTML]{FFD667}0.202              & \\cellcolor[HTML]{FFD666}0.141              & \\cellcolor[HTML]{FFD666}\\textbf{0.063}     & \\cellcolor[HTML]{FFDC7C}3.021              & \\cellcolor[HTML]{FFDB7C}3.001              & \\cellcolor[HTML]{FFDB7C}3.001                   & \\cellcolor[HTML]{FFD974}2.001                   & \\cellcolor[HTML]{FFE293}6.000                   \\\\ \\hline\n2300                         & \\cellcolor[HTML]{FFE49C}7.220              & \\cellcolor[HTML]{FFD667}0.219              & \\cellcolor[HTML]{FFD667}0.219              & \\cellcolor[HTML]{FFD667}0.216              & \\cellcolor[HTML]{FFD667}0.202              & \\cellcolor[HTML]{FFD666}0.141              & \\cellcolor[HTML]{FFD666}\\textbf{0.063}     & \\cellcolor[HTML]{FFDC7C}3.021              & \\cellcolor[HTML]{FFDB7C}3.001              & \\cellcolor[HTML]{FFDB7C}3.001                   & \\cellcolor[HTML]{FFD974}2.001                   & \\cellcolor[HTML]{FFE293}6.000                   \\\\ \\hline\n2500                         & \\cellcolor[HTML]{FFD667}0.220              & \\cellcolor[HTML]{FFD667}0.219              & \\cellcolor[HTML]{FFD667}0.219              & \\cellcolor[HTML]{FFD667}0.216              & \\cellcolor[HTML]{FFD667}0.202              & \\cellcolor[HTML]{FFD666}0.141              & \\cellcolor[HTML]{FFD666}\\textbf{0.063}     & \\cellcolor[HTML]{FFDC7C}3.021              & \\cellcolor[HTML]{FFDB7C}3.001              & \\cellcolor[HTML]{FFDB7C}3.001                   & \\cellcolor[HTML]{FFD974}2.001                   & \\cellcolor[HTML]{FFE293}6.000                   \\\\ \\hline\n2700                         & \\cellcolor[HTML]{FFD667}0.220              & \\cellcolor[HTML]{FFD667}0.219              & \\cellcolor[HTML]{FFD667}0.219              & \\cellcolor[HTML]{FFD667}0.216              & \\cellcolor[HTML]{FFD667}0.202              & \\cellcolor[HTML]{FFD666}0.141              & \\cellcolor[HTML]{FFD666}\\textbf{0.063}     & \\cellcolor[HTML]{FFDC7C}3.021              & \\cellcolor[HTML]{FFDB7C}3.001              & \\cellcolor[HTML]{FFDB7C}3.001                   & \\cellcolor[HTML]{FFD974}2.001                   & \\cellcolor[HTML]{FFE293}6.000                   \\\\ \\hline\n3000                         & \\cellcolor[HTML]{FFD667}0.220              & \\cellcolor[HTML]{FFD667}0.219              & \\cellcolor[HTML]{FFD667}0.219              & \\cellcolor[HTML]{FFD667}0.216              & \\cellcolor[HTML]{FFD667}0.202              & \\cellcolor[HTML]{FFD666}0.141              & \\cellcolor[HTML]{FFD666}\\textbf{0.063}     & \\cellcolor[HTML]{FFDC7C}3.021              & \\cellcolor[HTML]{FFDB7C}3.001              & \\cellcolor[HTML]{FFDB7C}3.001                   & \\cellcolor[HTML]{FFD974}2.001                   & \\cellcolor[HTML]{FFE293}6.000                   \\\\ \\hline\n3200                         & \\cellcolor[HTML]{FFD667}0.220              & \\cellcolor[HTML]{FFD667}0.219              & \\cellcolor[HTML]{FFD667}0.219              & \\cellcolor[HTML]{FFD667}0.216              & \\cellcolor[HTML]{FFD667}0.202              & \\cellcolor[HTML]{FFD666}0.141              & \\cellcolor[HTML]{FFD666}\\textbf{0.063}     & \\cellcolor[HTML]{FFDC7C}3.021              & \\cellcolor[HTML]{FFDB7C}3.001              & \\cellcolor[HTML]{FFDB7C}3.001                   & \\cellcolor[HTML]{FFD974}2.001                   & \\cellcolor[HTML]{FFE293}6.000                   \\\\ \\hline\n\\end{tabular}\n\\label{table:wave_3}\n\\end{table}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{figs\/dict_atoms_low_rank.png}\n    \\caption{ \\label{fig:accommodations_a}Example columns of $\\mathbf{D}$ obtained from Low-rank matrix factorization $\\lambda = 2952$. }\n    \\label{fig:my_label}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{figs\/dict_atoms_gamma_50_lambda_0_6.png}\n    \\caption{\\label{fig:accommodations_b}Example columns of $\\mathbf{D}$ obtained from our Wave-informed decomposition with $\\gamma = 10^7$, $\\lambda = 2200$.}\n\\end{figure}\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[trim=15 10 10 15, clip, width=1.1\\linewidth]{figs\/dampled_sines_all.png}\n    \\caption{Recovered modes (rows) of spatially damped sinusoids ({\\color{A}{\\textbf{A}}}), ({\\color{B}{\\textbf{B}}}), ({\\color{C}{\\textbf{C}}}), ({\\color{D}{\\textbf{D}}}), ({\\color{E}{\\textbf{E}}}), ({\\color{F}{\\textbf{F}}}), ({\\color{G}{\\textbf{G}}}) when $R=6$.}    \\label{fig:damped_sines_all}\n    \\vspace{-5mm}\n\\end{figure}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\nThe different disciplines in physics often evolve in isolation from\neach other, developing their own formalisms and terminologies. This\ndifference in concepts, language, and notations hampers the mutual\nunderstanding and deepens the divisions. It has the\neffect of obscuring the circumstance that the different disciplines\nmay have much more in common than meets the eye, and that much of what\nseems to be different is of cultural and not of intrinsic origin. \n\nIn the present paper we highlight the non-trivial circumstance that\nthe mathematical symmetries and structures that govern the Mueller\nmatrices in polarization theory are the same as those of the electromagnetic tensor in Maxwell\ntheory and the Lorentz transformations in relativistic physics, all of\nwhich obey the same Lorentz algebra. Absorption, electric\nfields, and Lorentz boosts are governed by the symmetric part of the\nrespective transformation matrix, while \ndispersion, magnetic fields, and spatial rotations are represented by\nthe antisymmetric part. These two aspects can be\nunified in terms of complex-valued matrices, where\nthe symmetric and antisymmetric aspects represent the real and\nimaginary parts, respectively. \n\nWe can extend this comparison by introducing a Minkowski metric for a\n4D space spanned by the four Stokes parameters $I,Q,U,V$. In this\ndescription Stokes vectors that represent 100\\,\\%\\ polarization\nare null vectors, while partial depolarization causes the Stokes\nvector to lie inside the null cones like the energy-momentum vectors\nof massive particles in ordinary spacetime. This comparison points to an analogy\nbetween depolarization (which can be seen as a symmetry breaking) and\nthe appearance of mass. Another interesting property is that, in\ncontrast to electromagnetism and Lorentz transformations, Stokes\nvectors and Mueller matrices have the rotational symmetry of spin-2\nobjects because they have substructure: they\nare formed from bilinear products of spin-1 objects. Here we try to\nexpose these potentially profound connections and discuss their\nmeanings. \n\nWe start in Sect.~\\ref{sec:emanalog} by showing the\nrelations between the Stokes formalism in polarization physics and the\ncovariant formulation of the Maxwell theory of electromagnetism. After\nclarifying the symmetries it is shown how the introduction of\ncomplex-valued matrices leads to an elegant, unified formulation. In\nSect.~\\ref{sec:lorentz} we show how the Lorentz transformations with\nits boosts and spatial rotations have the same structure. The\nintroduction of a Minkowski metric for polarization space in\nSect.~\\ref{sec:depolmass} reveals a\nnull cone structure for Stokes vectors that represent fully polarized\nlight. It further brings out an analogy between depolarization caused by the\nincoherent superposition of fields and the appearance of what may be\ninterpreted as a mass term. In Sect.~\\ref{sec:spin2} we highlight the\ncircumstance that the Stokes vectors and Mueller matrices in\npolarization physics have the symmetry of spin-2 objects, because they\nhave substructure, being formed from bilinear products of vector\nobjects. Section \\ref{sec:conc} summarizes the conclusions. \n\n\n\\section{The electro-magnetic analogy}\\label{sec:emanalog}\nLet $\\vec{S}_\\nu$ be the 4D Stokes vector for frequency\n$\\nu$. Explicitly, in terms of its transposed form (with superscript\n$T$) and omitting index $\\nu$ for clarity of notation, \n$\\vec{S}^T\\!\\equiv (S_0,\\,S_1,\\,S_2,\\,S_3) \\equiv (I,\\,Q,\\,U,\\,V)$. The equation for\nthe transfer of polarized radiation can then be written as \n\\begin{equation}\\label{eq:transeq}\n{{\\rm d}\\vec{S}_\\nu\\over{\\rm d}\\tau_c}=(\\vec{\\eta}+\\vec{I})\\,\\vec{S}_\\nu\n\\,-\\vec{j}_\\nu\/\\kappa_c\\,,\n\\end{equation}\nwhere $\\tau_c$ is the continuum optical depth, $\\vec{I}$ is the\n$4\\times 4$ identity matrix (representing continuum absorption),\n$\\kappa_c$ is the continuum absorption coefficient, $\\vec{j}_\\nu$ is\nthe emission 4-vector, while the Mueller absorption matrix\n$\\vec{\\eta}$ that represents the polarized processes due to the\natomic line transitions is \n\\begin{equation}\\label{eq:etamat} \n\\vec{\\eta}=\\left(\\matrix{\\,\\,\\eta_I &\\phantom{-}\\eta_Q &\\phantom{-}\\eta_U&\n\\phantom{-}\\eta_V\\,\\,\\cr \\,\\,\\eta_Q&\\phantom{-}\\eta_I&\n\\phantom{-}\\rho_V&-\\rho_U\\,\\,\\cr \\,\\,\\eta_U&-\\rho_V&\\phantom{-}\\eta_I&\\phantom{-}\n\\rho_Q\\,\\,\\cr \\,\\eta_V&\\phantom{-}\\rho_U&-\\rho_Q&\n\\phantom{-}\\eta_I\\,\\, }\\right)\\,.\n\\end{equation}\n For a detailed account of Stokes vector polarization theory with its\nnotations and terminology we refer to the monographs of\n\\citet{stenflo-book94} and \\citet{stenflo-lanlan04}. \n\nWhile $\\eta_{I,Q,U,V}$ represents the absorption terms for the four\nStokes parameters $I,Q,U,V$, the differential phase shifts, generally\nreferred to as anomalous dispersion or\nmagnetooptical effects, are represented by the $\\rho_{Q,U,V}$\nterms. They are formed, respectively, from the imaginary and the real\npart of the complex refractive index that is induced when the atomic\nmedium interacts with the radiation field. \n\nWe notice that $\\vec{\\eta}$ can be expressed as the sum of two\nmatrices: a symmetric matrix that only contains the $\\eta$ terms, and\nan anti-symmetric matrix that only contains the $\\rho$\nterms. Antisymmetric matrices represent spatial rotations, as will be\nparticularly clear in Sect.~\\ref{sec:lorentz} when comparing with\nLorentz transformations. \n\nThese symmetries turn out to be identical to the symmetries that\ngovern both the Maxwell theory for electromagnetism and the Lorentz\ntransformation of the metric in relativity. Here we will compare these\nvarious fields of physics to cast light on the intriguing\nconnections. We begin with electromagnetism. \n\nThe analogy between the Stokes formalism and electromagnetism has\npreviously been pointed out by \\citet{stenflo-lanlan81} using a 3+1\n(space + time) formulation rather than the covariant formulation in\nMinkowski spacetime. It is however only with the covariant formulation that\nthe correspondences become strikingly transparent. \n\nThe Lorentz electromagnetic force law when written in covariant 4D\nform is \n\\begin{equation}\\label{eq:forcelaw} \n{{\\rm d}p^{\\,\\alpha}\\over\\!\\!{\\rm d}\\tau}={e\\over\n  m}\\,F^{\\,\\alpha}_{\\phantom{\\alpha}\\beta}\\,\\,p^{\\,\\beta}\\,,\n\\end{equation}\nwhere $p^{\\,\\alpha}$ are the components of the contravariant energy-momentum\n4-vector, $e$ and $m$ are the electric charge and mass of the\nparticle, and $F^{\\,\\alpha}_{\\phantom{\\alpha}\\beta}$ are the\ncomponents of the electromagnetic tensor, which has the representation\nof a $4\\times 4$ matrix: \n\\begin{equation}\\label{eq:fabmat} \nF^{\\,\\alpha}_{\\phantom{\\alpha}\\beta}\\,=\\left(\\matrix{\\,\\,0&\\phantom{-}E_x&\\phantom{-}E_y&\n\\phantom{-}E_z\\,\\,\\cr \\,\\,E_x&\\phantom{-}0&\n\\phantom{-}B_z&-B_y\\,\\,\\cr \\,\\,E_y&-B_z&\\phantom{-}0&\\phantom{-}\nB_x\\,\\,\\cr \\,\\,E_z&\\phantom{-}B_y&-B_x&\n\\phantom{-}0\\,\\, }\\right)\\,.\n\\end{equation}\n A comparison between Eqs.~(\\ref{eq:etamat}) and (\\ref{eq:fabmat})\nimmediately reveals the correspondence between absorption $\\eta$ and the electric\n$E$ field on the one hand and between anomalous dispersion $\\rho$ and the magnetic $B$\nfield on the other hand: \n\\begin{eqnarray}\\label{eq:emconnect} \n&\\eta_{Q,U,V}\\,\\,\\longleftrightarrow \\,\\, E_{x,y,z}\\nonumber\\\\\n&\\,\\,\\rho_{Q,U,V}\\,\\,\\longleftrightarrow \\,\\, B_{x,y,z}\\,.\n\\end{eqnarray}\n\nLet us follow up this structural comparison in terms of a unified\ndescription, where absorption and dispersion are combined into a\ncomplex-valued absorption. $\\eta$ is proportional to the Voigt\nfunction $H(a,v_q)$, with $\\eta_0$ as the proportionality constant, $a$\nthe dimensionless damping parameter, and $v_q$ the dimensionless\nwavelength or frequency parameter. Index $q$, with $q=0,\\pm 1$,\nindicates the differential shift of the wavelength scale for atomic\ntransitions with magnetic quantum number $m_{\\rm lower}-m_{\\rm\n  upper}=q$. Similarly $\\rho$ is proportional \nto $2F(a,v_q)$, where $F$ is the line dispersion function. \n\\begin{eqnarray}\\label{eq:etarho} \n&\\,\\,\\,\\eta_{I,Q,U,V}=\\,\\,\\eta_0 \\,H_{I,Q,U,V}\\,,\\nonumber\\\\\n&\\rho_{Q,U,V}=2\\eta_0 \\,F_{Q,U,V}\\,.\n\\end{eqnarray}\nIn the unified description the $H$ and $F$ functions are combined into\nthe complex-valued \n\\begin{equation}\\label{eq:hcal}\n{\\cal H}(a,v_q)\\equiv H(a,v_q)-2i\\,F(a,v_q)\\,, \n\\end{equation}\nwhich now represents the building blocks when forming the\ncorresponding quantities with indices $I,Q,U,V$ to refer to the\nrespective Stokes parameters. ${\\cal H}_{I,Q,U,V}$ can be combined\ninto the 4-vector \n\\begin{equation}\\label{eq:hcalvect} \n\\vec{{\\cal H}}\\,\\equiv\\left(\\matrix{\\,\\,{\\cal H}_I\\,\\,\\cr \\,\\,{\\cal H}_Q\\,\\,\\cr \\,\\,{\\cal H}_U\\,\\,\\cr \\,\\,{\\cal H}_V\\,\\, }\\right)\\,\\equiv\\,\\left(\\matrix{\\,\\,{\\cal H}_0\\,\\,\\cr \\,\\,{\\cal H}_1\\,\\,\\cr \\,\\,{\\cal H}_2\\,\\,\\cr \\,\\,{\\cal H}_3\\,\\, }\\right)\\,.\n\\end{equation}\n From the above follows how ${\\cal H}_k$ is related to and unifies the\ncorresponding absorption and dispersion parameters $\\eta_k$ and $\\rho_k$: \n\\begin{equation}\\label{eq:completa}\n\\eta_k -i\\,\\rho_k =\\,\\eta_0\\,{\\cal H}_k\\,. \n\\end{equation}\n\nLet us next define the three symmetric matrices $\\vec{K}^{(k)}$ and three\nantisymmetric matrices $\\vec{J}^{(k)}$ through \n\\begin{eqnarray}\\label{eq:ikj} \n&\\vec{K}^{(k)}_{0j}&\\!\\!\\!\\equiv\\,\\, \\vec{K}^{(k)}_{j\\,0}\\,\\,\\equiv 1\\,,\\nonumber\\\\\n&\\vec{J}^{(k)}_{ij}&\\!\\!\\!\\equiv\\, -\\vec{J}^{(k)}_{j\\,i}\\equiv-\\,\\varepsilon_{ij\\,k}\\,,\n\\end{eqnarray}\nwhere $\\varepsilon_{ij\\,k}$ is the Levi-Civita antisymmetric symbol. \nWe further define the complex-valued matrix \n\\begin{equation}\\label{eq:xmatdef}\n\\vec{T}^{(k)} \\equiv \\vec{K}^{(k)} -i\\,\\vec{J}^{(k)} \\,. \n\\end{equation}\nThen the Mueller matrix from Eq.~(\\ref{eq:etamat}) becomes \n\\begin{equation}\\label{eq:etacompmat} \n\\vec{\\eta} -\\eta_I \\vec{I} =\\,\\eta_0\\,{\\rm Re}\\,(\\,{\\cal\n  H}_k\\,\\vec{T}^{(k)} \\,)\\,. \n\\end{equation}\n\nLet us similarly define the complex electromagnetic vector \n\\begin{equation}\\label{eq:complemvec} \n\\vec{\\cal E} \\equiv \\vec{E}-i\\,\\vec{B}\\,, \n\\end{equation}\nwhich in quantum mechanics represents photons with positive\nhelicity. Then the electromagnetic tensor\n$F^{\\,\\alpha}_{\\phantom{\\alpha}\\beta}$ of Eq.~(\\ref{eq:fabmat}) can be\nwritten as \n\\begin{equation}\\label{eq:compemtensor} \n\\vec{F}\\,=\\,{\\rm Re}\\,(\\,{\\cal\n  E}_k\\,\\vec{T}^{(k)} \\,)\\,. \n\\end{equation}\nComparison with Eq.~(\\ref{eq:etacompmat}) again brings out the\nstructural correspondence between the Mueller matrix and the\nelectromagnetic tensor, this time in a more concise and compact form. It also\nshows how the electric and magnetic fields are inseparably linked, as\nthe real and imaginary parts of the same complex vector. \n\nIt may be argued that the structural similarity between the Mueller\nand electromagnetic formalisms is not unexpected, since the underlying\nphysics that governs the Mueller matrices is the electromagnetic\ninteractions between matter and radiation. The atomic transitions are\ninduced by the oscillating electromagnetic force of the ambient\nradiation field when it interacts with the atomic electrons, and this\ninteraction is governed (in the classical description) by the force\nlaw of Eq.~(\\ref{eq:forcelaw}) with its electromagnetic tensor. This\nis however not the whole story, since there are also profound\ndifferences: As we will see in Sect.~\\ref{sec:spin2} Stokes vectors and Mueller\nmatrices behave like spin-2 objects, while the electromagnetic tensor\nis a spin-1 object. Another interesting aspect in the comparison\nbetween Stokes vectors and the energy-momentum 4-vector is that\ndepolarization of Stokes vectors acts as if the corresponding 4-vector\nhas acquired ``mass'', as will be shown in Sect.~\\ref{sec:depolmass}. Before we turn\nto these topics we will in the next section show the correspondence\nbetween the Mueller matrix and the Lorentz transformation matrix. \n\n\n\\section{Lorentz transformations and the Mueller absorption\n  matrix}\\label{sec:lorentz} \n\nLet $\\vec{X}$ be the spacetime 4-vector: \n\\begin{equation}\\label{eq:xvect} \n\\vec{X}\\,\\equiv\\,\\left(\\matrix{\\,\\,ct\\,\\,\\cr \\,\\,x\\,\\,\\cr \\,\\,y\\,\\,\\cr \\,\\,z\\,\\, }\\right)\\,\\equiv\\,\\left(\\matrix{\\,\\,x_0\\,\\,\\cr \\,\\,x_1\\,\\,\\cr \\,\\,x_2\\,\\,\\cr \\,\\,x_3\\,\\, }\\right)\\,.\n\\end{equation}\n With the Lorentz transformation $\\vec{\\Lambda}$ we transfer to a new\nsystem $\\vec{X}^\\prime$: \n\\begin{equation}\\label{eq:xprimex} \n\\vec{X}^\\prime =\\vec{\\Lambda}\\,\\vec{X}\\,. \n\\end{equation}\n$\\vec{\\Lambda}$ represents rotations in Minkowski space, composed of\nthree spatial rotations $\\phi_k$ and three boosts $\\gamma_k$, which\nmay be regarded as imaginary rotations. Let us combine them into complex\nrotation parameters $\\alpha_k$ through \n\\begin{equation}\\label{eq:compalphagamphi} \n\\alpha_k\\,\\equiv \\,\\gamma_k + i\\,\\phi_k\\,. \n\\end{equation}\nThen the Lorentz transformation $\\vec{\\Lambda}$ can be written as \n\\begin{equation}\\label{eq:explorentz} \n\\vec{\\Lambda}\\equiv e^{\\vec{V}}\\,, \n\\end{equation}\nwhere \n\\begin{equation}\\label{eq:vlormat} \n\\vec{V}\\,=\\,{\\rm Re}\\,(\\,\\alpha_k\\,\\vec{T}^{(k)} \\,)\\,=\\gamma_k\\,\n\\vec{K}^{(k)}+\\,\\phi_k \\,\\vec{J}^{(k)}\\,. \n\\end{equation}\nExplicitly, \n\\begin{equation}\\label{eq:lorentzmi}\n\\vec{V}\\,= \\left(\\matrix{\\,\\,0&\\phantom{-}\\gamma_x&\\phantom{-}\\gamma_y&\n\\phantom{-}\\gamma_z\\,\\,\\cr \\,\\,\\gamma_x&\\phantom{-}0&\n-\\phi_z&\\phantom{-}\\phi_y\\,\\,\\cr \\,\\,\\gamma_y&\\phantom{-}\\phi_z&\\phantom{-}0&-\\phi_x\\,\\,\\cr\n\\,\\,\\gamma_z&-\\phi_y&\\phantom{-}\\phi_x& \n\\phantom{-}0\\,\\, }\\right)\\,. \n\\end{equation}\n Note that in quantum field theory a convention for the definition of\nthe $\\vec{K}$ and $\\vec{J}$ matrices that define the Lorentz algebra\nis used, which differs by the\nfactor of the imaginary unit $i$ from the convention of Eq.~(\\ref{eq:ikj}) used here, in\norder to make $\\vec{K}$ anti-hermitian and $\\vec{J}$ hermitian\n\\citep{stenflo-zee2010}. \n\nComparing with the Mueller matrix and the electromagnetic tensor we see the correspondence \n\\begin{eqnarray}\\label{eq:lorconnect} \n&\\eta_k\\,\\,\\longleftrightarrow \\,\\, \\,\\,\\,\\gamma_k\\nonumber\\,\\,\\,\\longleftrightarrow \\,\\, \\,\\,E_k\\\\\n&\\,\\,\\rho_k\\,\\,\\longleftrightarrow -\\,  \\phi_k\\,\\,\\,\\longleftrightarrow \\,\\, \\,\\,B_k\\,.\n\\end{eqnarray}\nThe minus sign in front of $\\phi_k$ is only due to the convention\nadopted for defining the sense of rotations and is therefore\nirrelevant for the following discussion of the physical meaning of the\nstructural correspondence. \n\nRelations (\\ref{eq:lorconnect}) show how the Lorentz boosts\n$\\gamma_k$, which change the energy and momentum of the boosted object, relate to\nboth absorption $\\eta_k$ and electric field $E_k$, while the spatial\nrotations $\\phi_k$, which do not affect the energy but change the\nphase of the rotated object, relate to both dispersion or phase shift \neffects $\\rho_k$ and to magnetic fields $B_k$. \n\n\n\\section{Analogy between depolarization and the emergence of\n  mass}\\label{sec:depolmass}\nThe structural correspondence between the $4\\times 4$ Mueller\nabsorption matrix, the matrix representation of the covariant\nelectromagnetic tensor, and the Lorentz transformation matrix suggests\nthat there may be a deeper analogy or connection between the 4D Stokes vector\nspace and 4D spacetime. Let us therefore see what happens when we\nintroduce the Minkowski metric to the Stokes vector formalism. The\nusual notation for the Minkowski metric is $\\eta_{\\mu\\nu}$, but to\navoid confusion with the absorption matrix $\\vec{\\eta}$ that we have\nbeen referring to in the present paper, we will use the notation\n$\\vec{g}$ or $g_{\\mu\\nu}$ that is generally reserved for a general\nmetric, but here we implicitly assume that we are only dealing with \ninertial frames, in which $g_{\\mu\\nu}=\\eta_{\\mu\\nu}$. \n\nAssume that $\\vec{I}_\\nu =\\vec{S}_\\nu$ is the 4D Stokes vector, with\nits transpose being $\\vec{I}_\\nu^T\n=(I_\\nu,\\,Q_\\nu,\\,U_\\nu,\\,V_\\nu)$. The scalar product in Minkowski\nspace is then \n\\begin{equation}\\label{eq:itetai}\n\\vec{I}_\\nu^T\n\\vec{g}\\,\\vec{I}_\\nu\\,=\\,I_\\nu^2\\,-\\,(\\,Q_\\nu^2\\,+\\,U_\\nu^2\\,+\\,V_\\nu^2\\,)\\,,  \n\\end{equation}\nwhich also represents the squared length of the Stokes vector in\nMinkowski space. \n\nWe know from polarization physics that the right-hand-side of\nEq.~(\\ref{eq:itetai}) is always $\\geq 0$, and equals zero only when the\nlight beam is 100\\,\\%\\ (elliptically) polarized. Such fully polarized,\npure or coherent states are thus represented by null vectors, in\nexactly the same way as the energy-momentum 4-vector $\\vec{p}$ of\nmassless particles are also null vectors on the surface of null\ncones. The energy-momentum vectors of massive particles live inside\nthe null cones. Similarly the Stokes vectors live inside and not on\nthe surface of null cones only if the light is not fully but partially\npolarized. \n\nThis comparison raises the question whether there is some deeper\nconnection between depolarization and the appearance of mass. In\npolarization physics all individual (coherent) wave packages are\n100\\,\\%\\ polarized, and any coherent superposition of such wave\npackages is also fully polarized. Partial polarization occurs\nexclusively as a result of the {\\it incoherent} superposition of\ndifferent, uncorrelated wave packages. In such cases it is customary to\nrepresent the intensity $I_\\nu$, which represents the energy or the\nnumber of photons carried by the beam, as consisting of two parts, one\nfraction $p_\\nu$ that is fully polarized, and one fraction with\nintensity $I_{\\nu,\\,u}$ that is unpolarized, with transposed Stokes\nvector $I_{\\nu,\\,u}\\,(1,\\,0,\\,0,\\,0)$: \n\\begin{equation}\\label{eq:unpoldecomp}\nI=p_\\nu\\,I \\,+\\,I_u \\,,  \n\\end{equation}\nwhere we have omitted index $\\nu$ for simplicity except for $p_\\nu$ (to\ndistinguish it from the momentum vector $p$ below). This fractional\npolarization $p_\\nu$ is  \n\\begin{equation}\\label{eq:polfrac} \np_\\nu={I-I_u\\over I}=\\,{(Q^2 +U^2 +V^2)^{1\/2}\\over I}\\,. \n\\end{equation}\n\nIn comparison, in particle physics, the scalar product for the 4D\nenergy momentum vector $\\vec{p}$ is \n\\begin{equation}\\label{eq:ptgp} \n\\vec{p}^T\\vec{g}\\,\\vec{p}\\,=\\,m^2\\,c^2\\,, \n\\end{equation}\nfrom which the well-known Dirac equation \n\\begin{equation}\\label{eq:ptetap} \nE^2=p^2\\,c^2\\,+\\,m^2\\,c^4 \n\\end{equation}\nfollows. While the emergence of the mass term corresponds to the\nemergence of the unpolarized component $I_u$,\nEqs.~(\\ref{eq:unpoldecomp}) and (\\ref{eq:ptetap}) look different,\nbecause the decomposition in Eq.~({\\ref{eq:unpoldecomp}) has been done\n  for the unsquared intensity $I$, while in Eq.~({\\ref{eq:ptetap}) it\n    is in terms of the squared components.  Since we have the freedom\n    to choose different ways to mathematically decompose a quantity,\n    this difference is not of particular physical significance. \n\nIn current quantum field theories (QFT) the emergence of mass requires\nthe spontaneous breaking of the gauge symmetry, for which the Higgs\nmechanism has been invented. It is postulated that all of space is\npermeated by a ubiquitous Higgs field, which when interacting with the\nfield of a massless particle breaks the symmetry. When the particle\ngets moved to the non-symmetric state it acquires mass. Because the phases\nof the Higgs field and the field of the initially massless particle\nare uncorrelated, the superposition of the fields is incoherent, which\nmay be seen as one reason for the breaking of the symmetry. \n\nIn polarization physics the emergence of depolarization may also be\ninterpreted as a symmetry breaking, caused by the incoherent\nsuperposition of different wave fields. Incoherence means that the\nphases of the superposed fields are uncorrelated, which has the result that\nthe interference terms, all of which are needed to retain the symmetry,\nvanish. \n\n\n\\section{Stokes vectors as spin-2 objects}\\label{sec:spin2}\nAn object with spin $s$ varies with angle of rotation $\\theta$ as\n$s\\,\\theta$. For $s=\\textstyle{1\\over 2}$ one has to rotate $4\\pi$\nradians to return to the original state, for $s=2$ one only needs to\nrotate $\\pi$ radians,\nand so on. Ordinary vectors, like the electric and magnetic fields\n$\\vec{E}$ and $\\vec{B}$, rotate like spin-1 objects. It may\ntherefore come as a surprise that the Stokes vector rotates with twice\nthe angle, like a spin-2 object, in spite of the identical symmetry\nproperties of the Mueller matrix and the electromagnetic tensor. \n\nThe resolution to this apparent paradox is found by distinguishing\nbetween the kind of spaces in which the rotations are performed. In\nthe Minkowski-type space that is spanned by $I,Q,U,V$ as\ncoordinates, which is the Poincar\\'e\\ space in polarization physics\nfor a fixed and normalized intensity $I$, the transformation properties\nare indeed those of a real vector, a spin-1 object. However, besides\nPoincar\\'e\\ space the Stokes vector also lives in ordinary space, and\na rotation by $\\theta$ of a vector in Poincar\\'e\\ space corresponds to\na rotation in ordinary space by $2\\theta$. While being a spin-1 object\nin Poincar\\'e\\ space, the same object becomes a spin-2 object in\nordinary space. \n\nThe reason why it becomes a spin-2 object is that the Stokes vector\nhas substructure: it is formed from tensor products of Jones\nvectors. Similarly Mueller matrices for coherent (100\\,\\%\\ polarized)\nwave packages are formed from tensor products of Jones matrices. While\nthe Jones vectors and matrices are spin-1 objects in ordinary space,\nthe bilinear products between them become spin-2 objects. \n\nThe fundamental physics that governs the polarization physics does not\nmanifest itself  at the level of these spin-2 objects, because the basic\nprocesses are the electromagnetic interactions between the radiation\nfield and the electrons (which may be bound in atoms), and these\ninteractions are described at the spin-1 level (since the\nelectromagnetic waves represent a spin-1 vector field). The Jones\nmatrices, or, in QM terminology, the Kramers-Heisenberg scattering\namplitudes, contain the fundamental physics. They are the basic\nbuilding blocks for the bilinear products, the spin-2 objects. \n\nThis discussion points to the possibility that the physics of other\ntypes of spin-2 objects, like the metric field in general relativity,\nmay be hidden, because the governing physics may take place within a spin-1\nsubstructure level and would remain invisible if the spin-2 field\nwould be (incorrectly) \nperceived as fundamental, without substructure. \n\n\n\\section{Conclusions}\\label{sec:conc}\nComparison between the Stokes formalism, the covariant formulation of\nelectromagnetism, and the Lorentz transformation shows that they all\nshare the same Lie algebra, namely the algebra of the Lorentz\ngroup. This algebra is 6-dimensional (for instance in the case of\nelectromagnetism we have three electric field components + three magnetic\nfield components). While this is the algebra that is known to govern Lorentz\ntransformations and the related covariant formulation of\nelectromagnetism, it is not obvious why this group algebra should also apply\nto the transformation of Stokes 4-vectors, which have been constructed\nwith the aim of being a powerful tool for the treatment of\npartially polarized light. \n\nIn spite of the common underlying group structure, there is also a\nprofound difference. While the electromagnetic field vectors and \ntensors are objects of a vector space with rotational properties of\nspin-1 objects, Stokes vectors and Mueller matrices have the\nrotational symmetries of spin-2 objects, because they are formed from\ntensor products of spin-1 objects. This \nvector-field substructure contains the governing physics, where\neverything is coherent, and where in the quantum description the probability\namplitudes or wave functions live and get linearly superposed to form mixed states with\ncertain phase relations. When we go to the spin-2 level by\nforming bilinear products between the probability amplitudes, which\ngenerates observable probabilities, or when we form bilinear products\nbetween electric field vectors to generate \nquantities that represent energies or photon numbers, we get\nstatistical quantities (probabilities or energy packets) over which we\ncan form ensemble averages. If the phase relations of the mixed states in\nthe substructure are definite, we get interference effects and 100\\,\\%\\\npolarization for the ensemble averages, while if the phase relations\ncontain randomness (incoherent superposition) we get partial polarization. \n\nWhen comparing the Stokes 4-vector with the energy-momentum 4-vector\nof a particle, 100\\,\\%\\ elliptically polarized light corresponds to\nmassless particles, while the Stokes vector for partially polarized\nlight corresponds to the energy-momentum vector of a massive\nparticle. Depolarization thus has an effect as if the Stokes vector has\nacquired ``mass'' by a symmetry breaking that is caused by the destruction of\ncoherences between mixed states. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\IEEEPARstart{M}{odern} radar systems are able to generate optimal filters matched to increasingly complex target motion, resulting in increased sensitivity to targets exhibiting these motion at the cost of significant processing load. This problem is most difficult for sensors targeting objects in low Earth orbit (LEO), especially sensors with a significant field of regard. This is due to the observation time required to detect smaller targets, combined with significant orbital velocities and large search volumes, increasing the parameter space to impractical levels.\n\nExtending radar processing integration times in order to increase detection sensitivity requires mitigation against range migration, Doppler migration, and angular migration. The correction of these migrations is further complicated by the motion of the Earth, and hence the sensor located on the Earth. The direct implementation of a matched filter in this radar search space may lead to the incorporation of many parameters.\n\nThe nominal trajectory of orbits is well understood and is generally deterministic. The motion of a two-body Keplerian orbit, an idealised case of an object of insignificant mass orbiting around a much larger central body\\footnote{Treated as a single point mass.}, can be expressed entirely by six parameters. Matching the processing to this well-defined orbital motion for the purpose of improved radar detection and space situational awareness is therefore a natural extension.\n\nWhilst the primary aim of this general method is to increase a radar's sensitivity to objects in orbit, detections from a filter matched to a target's orbital trajectory will additionally provide coarse initial orbit determination. Traditionally, performing initial orbit determination requires many radar detections of a pass of an object in space.\n\nAfter briefly covering prior work (\\ref{ssec:prior_work}), Section \\ref{sec:problem_formulation} details the problem formulation, specifically in terms of ambiguity function expressions (\\ref{ssec:ambiguity_surface_generation}) and Keplerian orbital dynamics (\\ref{ssec:orbital_dynamics}). In Section \\ref{sec:odbd}, Orbit Determination Before Detect (ODBD) methods are discussed, including matched processing to orbital parameters, constraining the search volume (\\ref{ssec:odbd_search_constraints}), and constraining the orbit in radar measurement space (\\ref{ssec:odbd_zdc}), particularly for uncued detections. Some specific applications, including single-channel object detection and orbit determination are also discussed (\\ref{ssec:odbd_single_channel}). Section \\ref{sec:results} presents simulated results, with comparison against ephemerides. Section \\ref{sec:conclusion} concludes with a description of future work.\n\\vspace{-1ex}\n\n\\subsection{Prior Work}\n\\label{ssec:prior_work}\nThe motivation for this paper is to further develop techniques for the surveillance of space with the Murchison Widefield Array (MWA) using passive radar. The paper is particularly concerned with developing techniques for uncued detection over a wide field of regard. The MWA is a low frequency (70 - 300 MHz), wide field-of-view, radio telescope located in Western Australia \\cite{2013PASA...30....7T}. The MWA has demonstrated the incoherent detection of the International Space Station (ISS) \\cite{ 2013AJ....146..103T} and other, smaller, objects in orbit \\cite{prabu2020development}. However, for coherent processing, methods compensating for all aspects of motion migration are required in order to detect smaller satellites and space debris \\cite{8835821}. As passive radar systems have no control over the transmitter used for detection, improving processing gain through extended Coherent Processing Intervals (CPIs) is a method used to achieve the required sensitivity \\cite{4653940}. Orbital trajectories are ideal targets for such techniques, as stable and predictable relative motion allows for simpler measurement models. Such techniques have also been used with active radar, for improved sensitivity and processing gain \\cite{markkanen2005real} \\cite{8812975}.\n\nConsisting of 256 tiles spread across many square kilometres, the MWA's sparse layout\\footnote{At FM radio frequencies, even the compact configuration of MWA Phase II is sparse \\cite{pase22018article}.} provides high angular resolution. Objects in orbit will therefore transit many beamwidths per second at the point of closest approach.  Because of this, high angular resolution (normally a desirable attribute) can result in significant angular migration. Highly eccentric orbits will transit significantly faster. This is particularly challenging for the uncued detection of small objects, where longer integration times are needed to achieve sufficient sensitivity.\n\nIndividual radar detections consisting of a single measurement of range, Doppler, azimuth and elevation, only define a broad region of potential orbital parameters \\cite{2007CeMDA..97..289T}. This region may be constrained by incorporating angular rates \\cite{2014demarsradar_regions1}, and even further by including radial acceleration and jerk \\cite{8812975}.  Usually,  many radar detections are required to perform initial orbit determination. The mapping between radar measurement space and orbital parameters is an ongoing area of research \\cite{8448187}.\n\n\n\\section{Problem Formulation}\n\\label{sec:problem_formulation}\n\\subsection{Radar Product Formation} \n\\label{ssec:ambiguity_surface_generation}\nA standard timeseries matched filter is a function to detect reflected copies of a reference signal $d(t)$ in the surveillance signal $s(t)$, specifically copies delayed by $\\tau$ and frequency shifted by $f_D$:\n\\vspace{-1ex}\n\\begin{IEEEeqnarray}{rCL}\n    \\label{eq:woodward1}\n    \\chi(\\tau,f_D) = \\int_T s(t){d}^*(t-\\tau){\\mathrm{e}}^{-j2\\pi f_Dt}\\,dt .\n\\end{IEEEeqnarray}\n\nThis matched filter can be extended to more complicated motions by \\textit{dechirping} (or even applying higher order corrections to) the motion-induced frequency shift. For example, instead of matching to the radial velocity with a Doppler shift of $f_D$, higher order motions could be matched with a time varying frequency (that can be represented as a polynomial phase signal) given by $f_D +\nf_Ct$, where $f_C$ is proportional to the radial acceleration. This can be extended to an arbitrary number of parameters at the cost of adding extra dimensions to the matched filter outputs. \nTo account for any range migration, the delay term $\\tau$ will also need to be a function of time to match the radial motion.\n\nFor a receiver array consisting of $N$ elements, the surveillance signal $s(t)$ can be formed by classical far-field beamforming in a direction of interest such that:\n\\begin{IEEEeqnarray}{rCL}\ns(t) = \\sum_{n=1}^{N}s_{n}(t){\\mathrm{e}}^{-j\\boldsymbol{k}(\\theta,\\phi) \\cdot \\boldsymbol{u}_n } ,\n\\end{IEEEeqnarray}\nwhere $s_{n}(t)$ is the received signal at the $n^{\\text{th}}$ antenna, $\\boldsymbol{u}_n$ is the position of the $n^{\\text{th}}$ antenna, and $\\boldsymbol{k}(\\theta,\\phi)$ is the signal wavevector for azimuth $\\theta$ and elevation $\\phi$. \nTime varying adjustments can be made to every measurement parameter to create a filter, $\\chi$, matched to the exact motion of an object with range $\\rho(t)$ and slant range-rate $\\dot{\\rho}(t)$, in time-varying directions given by azimuth $\\theta(t)$ and elevation $\\phi(t)$:\n\n\\begin{multline}\n  \\label{eq:time_varying_delay_doppler}\n  \\chi\\left(\\theta(t), \\phi(t), \\rho(t), \\dot{\\rho}(t)\\right) = \\int_T \\left[\\sum_{n=1}^{N}{\\mathrm{e}}^{j\\boldsymbol{k}(\\theta(t),\\phi(t)) \\cdot \\boldsymbol{u}_n }s_{n}(t)\\right] \\\\\n  d^*\\left(t-2c^{-1}\\rho(t)\\right){\\mathrm{e}}^{-j\\frac{4\\pi}{\\lambda}\\dot{\\rho}(t)t}dt ,\n\\end{multline}\nwhere the delay to the target is now given by the total path distance scaled by $\\frac{1}{c}$, and the Doppler shift is given by $\\frac{2\\dot{\\rho}}{\\lambda}$.\n\n\\subsection{Orbital Dynamics}\n\\label{ssec:orbital_dynamics}\n\\vspace{-.47ex}\nThe most common elements used to parameterise an orbit are the Keplerian, or \\textit{classical}, orbital elements. These elements directly describe the size, shape, and orientation of an orbital ellipse (with one focus being at the centre of the central body), and the position of an object on this ellipse at some epoch, in the Earth-Centered Inertial (ECI) coordinate frame \\cite{Vallado2001fundamentals}. The ECI coordinate frame has its origin at the centre of the Earth, but it does not rotate with the Earth. It is also worth noting that a Keplerian orbit can, in fact, be any conic section. However, in this paper, it is assumed that orbits describe Earth-captured closed orbits.\n\nThe Keplerian orbital parameters are: the semi-major axis, $a$, and eccentricity, $e$, defining the size and shape of the ellipse; the right-ascension of the ascending node, $\\Omega$, and inclination, $i$, which define the orientation of the elliptical plane to the Earth's equatorial plane; the argument of periapsis, $w$, defining the orientation\/rotation of the ellipse in the orbital plane; and finally, the true anomaly, $\\nu$, defining the position of the object on the ellipse (refer to Figure \\ref{fig:orbitplane1}).\n\n\\vspace{-3ex}\n\\begin{figure}[ht!]\n\\hspace{2ex}\n\\tdplotsetmaincoords{70}{110}\n\\begin{tikzpicture}[tdplot_main_coords,scale=4.3]\n  \\pgfmathsetmacro{\\r}{0.7}\n  \\pgfmathsetmacro{\\O}{45} \n  \\pgfmathsetmacro{\\i}{35} \n  \\pgfmathsetmacro{\\f}{32}\n\n  \\coordinate (O) at (0,0,0);\n\n  \\draw [->] (O) -- (1,0,0) node[anchor=north east] {$\\boldsymbol{I}$};\n  \\draw [->] (O) -- (0,0.9,0) node[anchor=north west] {$\\boldsymbol{J}$};\n  \\draw [->] (O) -- (0,0,0.9) node[anchor=south] {$\\boldsymbol{K}$};\n  \n  \\tdplotdrawarc[dashed]{(O)}{\\r}{0}{360}{}{}\n\n  \\tdplotsetrotatedcoords{\\O}{0}{0}\n\n  \\draw [tdplot_rotated_coords] (0,0,0) -- (\\r,0,0) node [below right] {};\n  \\tdplotdrawarc[->]{(O)}{.28*\\r}{0}{\\O}{anchor=north}{$\\Omega$}\n\n  \\tdplotsetrotatedcoords{-\\O}{\\i}{0}\n  \\tdplotdrawarc[tdplot_rotated_coords]{(O)}{\\r}{0}{360}{}{}  \n  \\begin{scope}[tdplot_rotated_coords]\n    \\draw[->] (O) -- (0,0,0.8) node [above] {$\\boldsymbol{h}$};\n    \\draw (0,0,0) -- (-\\r,0,0);\n    \\tdplotdrawarc[->]{(O)}{.4*\\r}{90}{180}{anchor=west}{$\\omega$}\n    \\coordinate (P) at (180+\\f:\\r);\n    \\draw (O) -- (P);\n    \\tdplotdrawarc[->]{(O)}{.7*\\r}{180}{180+\\f}{anchor=south west}{$\\nu$}\n  \\end{scope}\n\n  \\tdplotsetrotatedcoords{-\\O+\\f}{\\i}{0}\n  \\tdplotsetrotatedcoordsorigin{(P)}\n  \\begin{scope}[tdplot_rotated_coords,scale=.2,thick]\n    \\fill (P) circle (.6ex) node [above] {Celestial Body};\n  \\end{scope}\n\n  \\tdplotsetthetaplanecoords{-\\f}\n  \\tdplotdrawarc[tdplot_rotated_coords,->]{(O)}{0.9*\\r}{0}{\\i}{anchor=south}{$i$}\n\\end{tikzpicture}\n\\vspace{-2ex}\n\\caption{The orbital plane determined by orientation parameters $\\Omega$, $\\omega$, and $i$ relative to the plane of reference in the ECI coordinate frame. These parameters define the direction of the angular momentum vector $\\boldsymbol{h}$. The axes $\\boldsymbol{I}$, $\\boldsymbol{J}$ and $\\boldsymbol{K}$ define the ECI coordinate frame.}\n\\vspace{-1ex}\n\\label{fig:orbitplane1}\n\\end{figure}\n\nIt is also assumed that the only force acting on the object in orbit is due to the gravity of the dominant mass\\footnote{Uniform acceleration does not take into account the ellipsoidal\/oblate nature of the Earth or other forces, such as micro-atmospheric drag, solar weather, and gravity due to other celestial bodies. For the short duration of a single CPI, these factors are generally negligible.}, with the acceleration due to the Earth's gravity $\\boldsymbol{\\ddot{r}}$, given by:\n\\vspace{-0.2ex}\n\\begin{equation}\n\\boldsymbol{\\ddot{r}}= -\\frac{\\mu}{\\lvert\\boldsymbol{r}\\rvert^3}\\boldsymbol{r}\\label{eq:eci_acceleration},\\vspace{-1ex}\n\\end{equation}\nwhere $\\mu$ is the standard gravitational parameter for the Earth.\n\nGiven the orbital parameters, and the acceleration due to the Earth's gravity, the Cartesian position $\\boldsymbol{r}$, and velocity $\\boldsymbol{\\dot{r}}$, for an object in Earth orbit is completely deterministic and is given by:\n\\begin{IEEEeqnarray}{rCL}\n    \\boldsymbol{r} &=& \\frac{a(1-e^2)}{1 + e\\cos{\\nu}}(\\cos{\\nu}\\boldsymbol{P} + \\sin{\\nu}\\boldsymbol{Q})\\label{eq:eci_position}~;\\\\\n    \\boldsymbol{\\dot{r}} &=& \\sqrt{\\frac{\\mu}{a(1-e^2)}}(-\\sin{\\nu}\\boldsymbol{P} + (e + \\cos{\\nu})\\boldsymbol{Q})\\label{eq:eci_velocity}~,\n\\end{IEEEeqnarray}\nwhere $\\boldsymbol{P}$ and $\\boldsymbol{Q}$ represent axes of a coordinate system co-planar with the orbital plane in the Cartesian ECI coordinate frame (given by axes $\\boldsymbol{I}$, $\\boldsymbol{J}$, and  $\\boldsymbol{K}$). The third axis, $\\boldsymbol{W}$, is perpendicular to the orbital plane \\cite{Vallado2001fundamentals}. These vectors are described by:\n\n\\begin{IEEEeqnarray}{rCL}\n\\boldsymbol{P} = & &\n    \\begin{bmatrix} \n    \\cos{\\Omega}\\cos{\\omega} - \\sin{\\Omega}\\cos{i}\\sin{\\omega} \\\\\n    \\sin{\\Omega}\\cos{\\omega} + \\cos{\\Omega}\\cos{i}\\sin{\\omega} \\\\\n    \\sin{i}\\sin{\\omega}\n    \\end{bmatrix}~; \\\\\n    \\boldsymbol{Q} = & & \\begin{bmatrix} \n    -\\cos{\\Omega}\\sin{\\omega} - \\sin{\\Omega}\\cos{i}\\cos{\\omega} \\\\\n    -\\sin{\\Omega}\\sin{\\omega} + \\cos{\\Omega}\\cos{i}\\cos{\\omega} \\\\\n    \\sin{i}\\cos{\\omega}\n    \\end{bmatrix}~;\\\\\n    \\boldsymbol{W} = & & \\begin{bmatrix} \n    \\sin{i}\\sin{\\Omega} \\\\\n    -\\sin{i}\\cos{\\Omega} \\\\\n    \\cos{i}\n    \\end{bmatrix}~.\n\\end{IEEEeqnarray}\nNote that a complicating factor with the ECI reference frame is that a nominally stationary position on the surface of the Earth, such as a fixed radar sensor, will have significant motion.\n\n\\section{{Orbit Determination Before Detect}} \\label{sec:odbd}\n\n\\begin{figure}[ht!]\n\\hspace{7ex}\n\\begin{tikzpicture}[dot\/.style={draw,fill,circle,inner sep=1pt}]\n  \\def5{5}\n  \\def\\a{5}\n  \\def\\angle{10} \n  \\coordinate[] (O) at (0,0) {};\n  \\node[thick, dot,label={\\angle:Celestial Body}] (X) at (20:{4} and {3.2}) {};\n  \\coordinate[label={-30:Sensor Location}] (Q) at (0:{4} and {3}) {};\n  \\draw [->] (O) -- (X) node [midway, above] {$\\boldsymbol{r}$};\n  \\draw [->] (O) -- (Q) node [midway, below] {$\\boldsymbol{q}$};\n  \\draw [dashed,  ->] (X) -- (1.9,2) node [near end, right] [shift=(10:1mm)] {$\\dot{\\boldsymbol{r}}$};\n  \\draw [ ->] (Q) -- (X) node [midway, right] {$\\boldsymbol{\\rho}$};\n  \\fill (X) circle [radius=2pt];\n\n\\end{tikzpicture}\n\\vspace{-2ex}\n\\caption{In the ECI coordinate frame the sensor is at position $\\boldsymbol{q}$, the celestial body at position $\\boldsymbol{r}$ with velocity $\\dot{\\boldsymbol{r}}$ (given by \\eqref{eq:eci_position} and \\eqref{eq:eci_velocity}) and the slant range vector from the sensor to the object given by $\\boldsymbol{\\rho}$.}\n\\label{fig:radar_vectors}\n\\end{figure}\n\nFor a two-body Keplerian orbit, the time-varying terms $\\rho(t)$, $\\dot{\\rho}(t)$, $\\phi(t)$, and $\\theta(t)$ \n\\eqref{eq:time_varying_delay_doppler} can be completely described by an orbit's six independent parameters. Although the position of an object in orbit is given by \\eqref{eq:eci_position}, there are no closed form solutions for the time varying position $\\boldsymbol{r}(t)$. Instead, a Taylor series approximation can be used to calculate an expression for the object's position throughout a CPI such that $\\boldsymbol{r}(t) = \\sum_{n=0}^{\\infty}\\frac{\\boldsymbol{r}^{(n)}(0)t^n}{n!}$ (where $\\boldsymbol{r}^{(n)}(x)$ denotes the $n^{th}$ derivative of $\\boldsymbol{r}$ evaluated at the point $x$), with $t$ being the time through the CPI of length $T$, $t \\in [\\frac{-T}{2}, \\frac{T}{2}]$. With knowledge of the sensor's location, $\\boldsymbol{q}(t)$ (as in Figure \\ref{fig:radar_vectors}), and $\\boldsymbol{\\dot{q}}(t)$ giving the slant range vector from the sensor to the object, as well as the slant-range rate, as $\\boldsymbol{\\rho}(t) = \\boldsymbol{r}(t) - \\boldsymbol{q}(t)$ and $\\boldsymbol{\\dot{\\rho}}(t) = \\boldsymbol{\\dot{r}}(t) - \\boldsymbol{\\dot{q}}(t)$, a polynomial expression for the slant-range and slant-range rate equations of motion over the CPI is possible:\n\\vspace{-1ex}\n\\begin{IEEEeqnarray}{rCL}\n    \\rho(t) &=& \\lvert\\boldsymbol{\\rho}(t)\\rvert = \\lvert \\sum_{n=0}^{\\infty}\\frac{\\boldsymbol{r}^{(n)}(0)t^n}{n!} - \\boldsymbol{q}(t)\\rvert\\label{eq:taylor_position}~;\\\\\n     \\dot{\\rho}(t) &=& \\lvert\\boldsymbol{\\dot{\\rho}}(t)\\rvert = \\lvert \\sum_{n=1}^{\\infty}\\frac{\\boldsymbol{r}^{(n)}(0)t^n}{n!} - \\boldsymbol{\\dot{q}}(t)\\rvert\\label{eq:taylor_rangerate}~.\n\\end{IEEEeqnarray}\nThese expressions can be extended (or truncated) to arbitrary accuracy.\n\nThe directional angles are now calculated as topocentric right ascension and declination, that is right ascension and declination relative to the sensor location, given by $\\alpha$ and $\\delta$, respectively: \n\\vspace{-1.5ex}\n\\begin{IEEEeqnarray}{rCL}\n    \\alpha(t) &=& {\\tan}^{-1}\\left(\\frac{\\rho_{\\boldsymbol{J}}(t)}{\\rho_{\\boldsymbol{I}}(t)}\\right)\\label{eq:alpha_t}~;\\\\\n    \\delta(t) &=& {\\tan}^{-1}\\left(\\frac{\\rho_{\\boldsymbol{K}}(t)}{\\sqrt{{\\rho_{\\boldsymbol{I}}(t)}^2 + {\\rho_{\\boldsymbol{J}}(t)}^2}}\\right)~, \\label{eq:delta_t}\n\\end{IEEEeqnarray}\nnoting that these expressions depend on the individual elements of $\\boldsymbol{\\rho}$ such that $\\boldsymbol{\\rho}(t) = [\\rho_{\\boldsymbol{I}}(t), \\rho_{\\boldsymbol{J}}(t), \\rho_{\\boldsymbol{K}}(t)]^T$.\n\nUsing the expressions in this section, it is possible to form a matched filter to the orbital elements themselves, essentially creating $\\chi(e,a,i,\\Omega,\\omega,\\nu)$ at a given epoch \\eqref{eq:time_varying_delay_doppler}. This enables arbitrarily long CPIs by tracking an orbit throughout the CPI. Additionally, instead of calculating a Taylor Series expression for the orbital position $\\boldsymbol{r}(t)$, and deriving the parameters of interest, it is far more efficient to directly calculate a Taylor Series expression for the parameters of interest. \nFor a sensor at known Cartesian position $\\boldsymbol{q}$, with known instantaneous velocity, acceleration and jerk, given by $\\boldsymbol{\\dot{q}}$, $\\boldsymbol{\\ddot{q}}$, and $\\boldsymbol{\\dddot{q}}$, respectively, and given the slant range vector $\\boldsymbol{\\rho} = \\boldsymbol{r} - \\boldsymbol{q}$, the slant range and its instantaneous derivatives are given by:\n\n\\begin{align}\n  \\rho &= \\lvert\\boldsymbol{\\rho}\\rvert\\label{eq:straight_up_slant_range}~;\\\\\n  \\label{eq:orbit_doppler}\n  \\dot{\\rho} &= \\frac{\\boldsymbol{\\rho}\\cdot\\dot{\\rhovec}}{\\rho}~; \\\\\n  \\label{eq:orbit_chirp}\n  \\ddot{\\rho} &= -\\frac{(\\boldsymbol{\\rho}\\cdot\\dot{\\rhovec})^2}{\\rho^3} +  \\frac{|\\dot{\\rhovec}|^2 + \\boldsymbol{\\rho}\\cdot\\ddot{\\rhovec}}{\\rho}~; \\\\\n  \\label{eq:orbit_jerk}\n  \\dddot{\\rho} &= \\begin{multlined}[t]\n     3\\frac{(\\boldsymbol{\\rho}\\cdot\\dot{\\rhovec})^3}{\\rho^5} \\\\\n     - 3\\frac{(\\boldsymbol{\\rho}\\cdot\\dot{\\rhovec})(|\\dot{\\rhovec}|^2 + \\boldsymbol{\\rho}\\cdot\\ddot{\\rhovec})}{\\rho^3}\\\\\n     + \\frac{3\\dot{\\rhovec}\\cdot\\ddot{\\rhovec} + \\boldsymbol{\\rho}\\cdot\\dddot{\\rhovec}}{\\rho}~,\n  \\end{multlined}\n\\end{align}\n\nwhere $\\dddot{\\boldsymbol{r}}$ is from the derivative of \\eqref{eq:eci_acceleration} and is given by:\n\\begin{IEEEeqnarray}{rCL}\n\\dddot{\\boldsymbol{r}} =  \\frac{3\\mu\\boldsymbol{r}\\cdot\\boldsymbol{\\dot{r}}}{\\lvert\\boldsymbol{r}\\rvert^5}\\boldsymbol{{r}} -\\frac{\\mu}{\\lvert\\boldsymbol{r}\\rvert^3}\\boldsymbol{\\dot{r}}~.\n\\end{IEEEeqnarray}\n\nNow, \\eqref{eq:orbit_doppler}, \\eqref{eq:orbit_chirp}, and \\eqref{eq:orbit_jerk} can be used to directly specify the target's Doppler, chirp rate, and radial jerk. This leads to more efficient expressions (when compared to \n\\eqref{eq:taylor_position} and \\eqref{eq:taylor_rangerate}) for the slant-range, and also slant-range rate, throughout the CPI of length $T$ such that $t \\in [\\frac{-T}{2}, \\frac{T}{2}]$: \n\\begin{IEEEeqnarray}{rCL}\n    \\rho(t) & = & \\rho + \\dot{\\rho}t + \\frac{1}{2}\\ddot{\\rho}t^2 + \\frac{1}{6}\\dddot{\\rho}t^3~; \\\\\n    \\dot{\\rho}(t) & = & \\dot{\\rho} + \\ddot{\\rho}t + \\frac{1}{2}\\dddot{\\rho}t^2~.\n\\end{IEEEeqnarray}\n\nA fourth-order Taylor Series approximation to the slant-range, $\\rho(t)$, was chosen due to previous work, which demonstrated that a third order polynomial phase signal may be required in order to coherently match orbits for CPIs of duration up to 10 seconds \\cite{8835821}. \n\nSimilarly, equivalent approximations can be formed for the angular measurement parameters $\\alpha(t)$ \n\\eqref{eq:alpha_t} and $\\delta(t)$ \\eqref{eq:delta_t}.\n\\subsection{Search-Volume Constraints}\n\\label{ssec:odbd_search_constraints}\nThe methods described above enable coherent processing that matches orbital parameters; however, they are not suitable for searching to perform uncued detections. The parameter space is far too large to be practically searched, and the vast majority of orbits will not correspond to passes within a region of interest above the sensor. Although, as stated earlier in Section \\ref{ssec:orbital_dynamics}, alternatives to the Keplerian parameter set are available. In fact, it is possible to parameterise a Keplerian orbit with the Cartesian position and velocity to constitute the six elements \\cite{Vallado2001fundamentals}. It is also possible to utilise combinations of both sets of elements in other formulations.\n\nInstead of searching through classical orbital parameters, three parameters can be expressed as a hypothesised ECI position within a search volume of interest. This ensures any hypothesised orbit, determined from these initial parameters, will be within the search volume. Given this potential orbital position, $\\boldsymbol{r}$, only three more additional parameters are needed to fully define an elliptical orbit. Although the three elements forming the orbital velocity could be treated as free variables, the majority of possible velocities would not correspond to valid Earth-captured orbits. Instead, given position $\\boldsymbol{r}$ and semi-major axis $a$, the magnitude of the velocity of the corresponding orbit is given by the Vis-Viva equation \\cite{Vallado2001fundamentals}:\n\\begin{IEEEeqnarray}{rCL}\n{\\lvert\\boldsymbol{\\dot{r}}\\rvert}^2\n= \\mu(\\frac{2}{\\lvert\\boldsymbol{r}\\rvert} - \\frac{1}{a})~. \\label{eq:vis_viva}\n\\end{IEEEeqnarray}\n\nFurthermore, given position $\\boldsymbol{r}$ and eccentricity $e$, the semi major axis length will itself be constrained between the potential limits of the orbit's apogee and perigee ranges:\n\\begin{IEEEeqnarray}{rCL}\n\\frac{\\lvert\\boldsymbol{r}\\rvert}{1+e} \\leq a \\leq \\frac{\\lvert\\boldsymbol{r}\\rvert}{1-e}~. \\label{eq:apogiee_and_perigee_ranges}\n\\end{IEEEeqnarray}\n\nThe semi-major axis is also constrained by realistic limits on an orbit's range, as well as a sensor's maximum detection range, represented by minimum and maximum allowable periapsides, ${rp}_{min}$ and ${rp}_{ max}$:\n\\begin{IEEEeqnarray}{rCL}\n\\frac{{rp}_{min}}{1-e} \\leq a \\leq \\frac{{rp}_{max}}{1-e}~. \\label{eq:perigee_ranges_limits}\n\\end{IEEEeqnarray}\n\nAnother constraint is the constant angular momentum of the orbit, $\\boldsymbol{h}$. This vector is perpendicular to the orbital plane, parallel to $\\boldsymbol{W}$\\hspace{-0.5ex}, with a magnitude depending on the size and shape of the ellipse:\n\\begin{IEEEeqnarray}{rCL}\n    \\boldsymbol{h} = \\sqrt{\\mu a(1-e^2)}\\boldsymbol{W} = \\boldsymbol{r}\\times\\boldsymbol{\\dot{r}}~. \\label{eq:angular_momentum}\n\\end{IEEEeqnarray}\n\nThis cross-product may be rewritten to form an expression for the inner product between the position and velocity:\n\\begin{IEEEeqnarray}{rCL}\n    \\boldsymbol{r}\\cdot\\boldsymbol{\\dot{r}} & = & \\pm\\sqrt{\\lvert\\boldsymbol{r}\\rvert^2\\lvert\\boldsymbol{\\dot{r}}\\rvert^2 - \\lvert\\boldsymbol{h}\\rvert^2}~.\n\\end{IEEEeqnarray}\n\nCombined with the magnitude of the velocity, from the Vis-Viva equation \\eqref{eq:vis_viva}, as well as the magnitude of the constant angular momentum \\eqref{eq:angular_momentum}, an expression for this inner product can be formed which depends solely on the position $\\boldsymbol{r}$ and the size and shape of the orbital ellipse:\n\\begin{IEEEeqnarray}{rCL}\n    \\boldsymbol{r}\\cdot\\boldsymbol{\\dot{r}} & = & \\pm\\sqrt{\\lvert\\boldsymbol{r}\\rvert^2\\mu(\\frac{2}{\\lvert\\boldsymbol{r}\\rvert} - \\frac{1}{a}) - \\mu a (1-e^2)}~. \\label{eq:parallel_planes}\n\\end{IEEEeqnarray}\n\nAdditionally, the specific relative angular momentum vector, $\\boldsymbol{h}$, is perpendicular to both the orbital position $\\boldsymbol{r}$ and orbital velocity $\\boldsymbol{\\dot{r}}$. This leads to the expressions $\\boldsymbol{r}\\cdot\\boldsymbol{h}=0$ and $\\boldsymbol{\\dot{r}}\\cdot\\boldsymbol{h}=0$, which result in another constraint on the velocity, dependant on the right ascension of the ascending node, $\\Omega$:\n\\begin{IEEEeqnarray}{rCL}\n    \\begin{bmatrix}\n         r_{\\boldsymbol{K}}\\sin{\\Omega} \\\\ -r_{\\boldsymbol{K}}\\cos{\\Omega} \\\\ r_{\\boldsymbol{J}}\\cos{\\Omega} - r_{\\boldsymbol{I}}\\sin{\\Omega}\n        \\end{bmatrix}\\cdot\\boldsymbol{\\dot{r}} = 0~. \\label{eq:raan_plane}\n\\end{IEEEeqnarray}\n\nThese expressions lead to a simple geometric solution for determining orbits when $\\boldsymbol{r}$ (and other parameters) are known, and $\\boldsymbol{\\dot{r}}$ is unknown. For determining $\\boldsymbol{\\dot{r}}$, \n(\\ref{eq:vis_viva}) defines a  sphere of radius $\\sqrt{\\mu(\\frac{2}{\\lvert\\boldsymbol{r}\\rvert} - \\frac{1}{a})}$, representing valid orbits in the velocity vector's element space. Additionally, \n(\\ref{eq:parallel_planes}) defines two parallel planes of valid orbits, which intersect with (\\ref{eq:vis_viva}) to define two circles. Finally, intersecting these two circles with the plane defined by the position and the right ascension of the ascending node, $\\Omega$, \n\\eqref{eq:raan_plane} will result in a maximum of four intersection points, that is, four velocities, each corresponding to a valid orbit. An example diagram is shown in Figure \\ref{fig:makingshapes}. Although this means that a choice of six orbital parameters will result in up to four potential orbital matched filters, this approach will be far more efficient than methods outlined earlier in this section, as the orbit will be within the search volume, and each parameter choice restricts the range of subsequent parameters. \n\n\\vspace{-3ex}\n\n\\begin{figure}[ht!]\n\\hspace{8ex}\\begin{tikzpicture}[\n  point\/.style = {draw, circle, fill=black, inner sep=0.7pt},\n  scale=0.7\n]\\clip (-4.25,-3.75) rectangle + (8.5,8);\n\\def3cm{3cm}\n\\coordinate (O) at (0,0); \n\n  \\draw[->] (0,0,0) -- (1,0,0) node[anchor=north east]{$\\dot{{r}}_{\\boldsymbol{J}}$};\n  \\draw[->] (0,0,0) -- (0,1,0) node[anchor=north west]{$\\dot{{r}}_{\\boldsymbol{K}}$};\n  \\draw[->] (0,0,0) -- (0,0,1) node[anchor=south east]{$\\dot{{r}}_{\\boldsymbol{I}}$};\n\n\n\\draw[-] (0,0) circle [radius=3cm];\n\n\n\n\\begin{scope}[]\n\\draw[-]\n  (-3.25,2.7) -- (4.25,2.7) -- (3.25,1.2) -- (-4.25,1.2) -- cycle;\n  \n  \\draw[]\n  (-3.25,-1.2) -- (4.25,-1.2) -- (3.25,-2.7) -- (-4.25,-2.7) -- cycle;\n  \n  \\draw[]\n    (-2.3,3.25) -- (-2.3,-3.75) -- (-1.2,-3.25) -- (-1.2,3.75) -- cycle;\n  \n  \\draw[dashed] (0,2) ellipse (2.2 and 0.35);\n  \\draw[dashed] (0,-1.97) ellipse (2.2 and 0.3);\n  \\draw[dashed] (-1.75,0) ellipse (0.3 and 2.4 );\n  \n\\node at (2.5,3.1) {$P_1$};\n\n\\node at (3.6,-0.8) {$P_2$};\n\n\n\\node at (-0.85,3.45) {$P_3$};\n\n\\fill (-1.65,2.23) circle [radius=0.105];\n\n\\fill (-1.93,1.83) circle [radius=0.105];\n\n\\fill (-1.51,-1.73) circle [radius=0.105];\n\n\n\\fill (-1.85,-2.1) circle [radius=0.105];\n\n\n\\end{scope}\n\\end{tikzpicture}\n\\vspace{-1.5ex}\n\\caption{Four valid orbital velocities given by the intersection of the sphere (given by \\eqref{eq:vis_viva}), parallel planes P1 and P2 (given by \\eqref{eq:parallel_planes}), and plane P3 (given by \\eqref{eq:raan_plane} or \\eqref{eq:doppler_plane}).}\n\\label{fig:makingshapes}\n\\end{figure}\n\nTherefore, given an orbital position, $\\boldsymbol{r}$, a choice of eccentricity, $e$, semi-major axis, $a$, and right ascension of the ascending node, $\\Omega$, four potential orbital velocities, $\\boldsymbol{\\dot{r}}$, are calculated, which leads to an expression for the complete matched filter:\n\\vspace{-0.5ex}\n\\begin{IEEEeqnarray}{rCL}\n    \\chi(\\boldsymbol{r}, \\boldsymbol{\\dot{r}}) \n        & = \\hspace{-1ex} \\int\\limits_{-\\frac{T}{2}}^{\\frac{T}{2}} \\hspace{-1ex} [&\\sum_{n=1}^{N}{\\mathrm{e}}^{j\\boldsymbol{k}(\\delta(\\boldsymbol{r}, \\boldsymbol{\\dot{r}},t),\\alpha(\\boldsymbol{r}, \\boldsymbol{\\dot{r}},t)) \\cdot \\boldsymbol{u}_n }s_{n}(t)]\\nonumber\\\\\n     & & ~d^*(t-2c^{-1}\\rho(\\boldsymbol{r}, \\boldsymbol{\\dot{r}},t)){\\mathrm{e}}^{-j\\frac{2\\pi}{\\lambda}\\dot{\\rho}(\\boldsymbol{r}, \\boldsymbol{\\dot{r}},t)t}\\,dt~.~~~\\label{eq:full_OD_right_here}\n\\end{IEEEeqnarray}\n\nThe proposed method tests for only realistic orbits in a given search region. Also, given a set of orbit parameters, this matched filter should maximise a radar's sensitivity to that orbit. Additionally, a detection in this matched filter corresponds to a detection in the orbital element space, providing initial orbit determination from a single detection.\n\nThis style of trajectory-match approach, has several advantages beyond just maximising sensitivity to motion models. Coupling measurement parameters together through a trajectory model can improve achievable resolution compared with using separate independent measurement parameters. As an example, a radar's range resolution is determined solely by the signal bandwidth, but its Doppler and Doppler-rate resolution improve with the CPI length.\n\nThrough coupling the measurement parameters with the trajectory model, as a radar can resolve finer Doppler and Doppler rate measurements it can essentially resolve finer trajectory states. This can potentially improve target localisation as increasingly accurate state measurements could localise a target within a single range bin.\n\\vspace{-0.2ex}\n\n\\subsection{Zero Doppler Crossing}\n\\label{ssec:odbd_zdc}\nThe flexibility of the geometric formulation in Section~\\ref{ssec:odbd_search_constraints} allows radar parameters to be used alongside, and in place of, other orbital parameters to constrain the search space. A Doppler shift $f_D$ will define another plane in $\\dot{\\boldsymbol{r}}$ space, given by:\n\\begin{equation}\n    \\label{eq:doppler_plane}\n    \\frac{\\boldsymbol{\\rho}}{\\rho}\\cdot\\boldsymbol{\\dot{r}} = -\\frac{\\lambda f_D}{2} + \\frac{\\boldsymbol{\\rho}\\cdot\\boldsymbol{\\dot{q}}}{\\rho}~.\n\\end{equation}\nEquation~\\eqref{eq:doppler_plane} can be used to search for a particular Doppler shift instead of one of the orbital parameters. This is useful because it allows a blind search to constrain the search-space solely for objects in orbit at their point of closest approach to the sensor. As an object is passing overhead, its point of closest approach will correspond exactly with it being at zero Doppler, which is when it is most detectable\\footnote{This may not necessarily hold in all instances, depending on particular beampattern and radar cross section factors.}. If a radar is unable to detect an object at its point of closest approach, at its minimum range, there is little value trying to detect it as it moves further away, towards the horizon.\n\nAnother benefit to applying this constraint is that, as Doppler is proportional to the range-rate, this constraint will also restrict the orbit search-space to a point of minimal (or zero) range migration, which greatly simplifies matched-processing\\footnote{Depending on the CPI length, it may be possible to make ${\\rho}(t) \\approx {\\rho}$.}.\n\nThe vast majority of the objects in an Earth-captured orbit are in a circular, or near-circular, orbit. Searching solely for objects in a circular orbit greatly decreases the potential orbital search space. A circular orbit means the eccentricity of the orbital ellipse is zero, $e=0$, and so \\eqref{eq:apogiee_and_perigee_ranges} becomes $a=\\lvert\\boldsymbol{r}\\rvert$. In a circular orbit, the position and velocity vectors will always be perpendicular, so \\eqref{eq:parallel_planes} simplifies to $\\boldsymbol{r}\\cdot\\boldsymbol{\\dot{r}} = 0$, a single plane instead of two parallel planes. The result is that a three-parameter search, within a region of interest, provides sufficient information to match the closest approach of objects in a circular orbit. For a given position\nin a search-region, there will be at most two possible orbits to match against (determined from the intersection of \\eqref{eq:vis_viva}, \\eqref{eq:parallel_planes}, and \\eqref{eq:doppler_plane}). This type of search approach, attempting uncued detection of the most common types of orbit when they are most detectable, is a far more realisable and practical approach than a completely unbounded search through measurement parameters. Additionally, for an eccentric orbit, the orbital velocity and position are perpendicular at perigee \\cite{Vallado2001fundamentals}. For typical radar detection ranges, an object in a highly eccentric orbit is likely to be within a radar's field of regard solely at, or near, perigee. Because of this, the same simplification of $\\boldsymbol{r}\\cdot\\boldsymbol{\\dot{r}} = 0$ could be used to reduce the number of potential orbits.\n\\vspace{-1.1ex}\n\\subsection{Single Channel Orbit Detection}\n\\label{ssec:odbd_single_channel}\nCoupling together measurement parameters is not necessarily new; however, incorporating such techniques into the detection stage offers some significant advantages. By coupling together the measurement parameters using these ODBD methods, it is possible to apply this matched filtering to single beam radar systems. This could be a post-beamformed surveillance signal from an array or even a classic narrowbeam tracking radar. Because the trajectory model determines all measurement parameters, a particular polynomial phase signal which results in a detection is coupled to a particular location and orbit. This is shown in \n\\eqref{eq:full_OD_right_here}.  The beamforming parameters do not determine the location; rather the (hypothesised) location determines the beamforming parameters. Removing the array processing, as in \\eqref{eq:single_channel}, does not remove the ability to localise a target using the algorithm.\n\n\\vspace{-3ex}\n\\begin{IEEEeqnarray}{rCL}\n    \\label{eq:single_channel}\n    \\chi(\\boldsymbol{r},\\boldsymbol{\\dot{r}}) & = \\hspace{-1.5ex} \\int\\limits_{-\\frac{T}{2}}^{\\frac{T}{2}} & \\hspace{-1ex} s(t){d}^*(t-2c^{-1}\\rho(\\boldsymbol{r},\\boldsymbol{\\dot{r}},t)){\\mathrm{e}}^{-j\\frac{2\\pi}{\\lambda}\\dot{\\rho}(\\boldsymbol{r},\\boldsymbol{\\dot{r}},t)t}\\,dt~~~~\n\\end{IEEEeqnarray}\n\nIn the case of a narrow beam radar, the pointing of the beam will be incorporated into the algorithm by determining the search region that is used. Because it handles sensor motion, this type of processing would be ideal for a satellite-based sensor, with the sensor location term $\\boldsymbol{q}(t)$ (or its instantaneous components $\\boldsymbol{q}, \\boldsymbol{\\dot{q}, \\boldsymbol{\\ddot{q}}}$, etc.) themselves determined by a known orbit rather than the motion of the Earth.\n\n\\vspace{-0.2ex}\n\\section{Simulated Results} \\label{sec:results}\nThese methods have been verified by comparing ODBD-derived measurement parameters, described in section \\ref{sec:odbd}, of an object in orbit, against measurement parameters propagated from available ephemerides. These ephemeris tracks consist of the six Keplerian orbital elements, as well as several additional parameters describing drag and orbital decay. These tracks are propagated with the standard SGP-4 propagator used by the USSPACECOM two-line element sets \\cite{USSTRATCOM}.\n\nThe configuration used for these simulations, matching \\cite{8835821}, is a sensor located at the MWA (at a latitude of 27\\degree~south) in a bistatic configuration with a  transmitter in Perth, approximately 600 km further south. This transmitter is taken to be transmitting an FM radio signal at a centre frequency of 100 MHz.\n\nFigure \\ref{fig:standard_circular} shows the path of an object in a near circular orbit at closest approach. The simulated measurement parameters match very well in both angular and delay-Doppler space despite being based on a perfectly circular orbit. Likewise, Figure \\ref{fig:standard-eccentric} also matches with the prediction, noting that the simulation used the matching eccentricity and semi-major axis.\n\nFigure \\ref{fig:eccentricity-mismatch} shows the path of an object in a near circular orbit, but slightly more eccentric than Figure \\ref{fig:standard_circular}  ($e=0.00126$) at point of closest approach. The simulated circular path matches well in the delay-Doppler space but diverges in the angular space. Additionally, several other simulated close eccentricities are shown, resulting in changes to the direction of travel but little difference in the delay-Doppler space. The delay-Doppler results suggest good tolerance to small eccentricity changes, however the sensor's angular resolution may limit potential processing intervals.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{images\/FIGURE1.png}\n\\end{center}\n\\vspace{-2ex}\n\\caption{The measurement parameters of a close pass of an object in a near-circular orbit ($e=0.0007$), as well as the simulation made assuming zero eccentricity at point of closest approach. The left plot is angular space and the right is the delay-Doppler. Twenty seconds of the true pass is shown with ten seconds of the simulated path overlaid.}\n\\label{fig:standard_circular}\n\\end{figure}\n\\vspace{-2ex}\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{images\/FIGURE2.png}\n\\end{center}\n\\vspace{-2ex}\n\\caption{The measurement parameters of a close pass of an object in an eccentric orbit ($e=0.7$), as well as the four simulations made with the correct eccentricity and semi-major axis. The left plot is angular space and the right is the delay-Doppler. Twenty seconds of the true pass is shown with ten seconds of the simulated paths overlaid.}\n\\label{fig:standard-eccentric}\n\\end{figure}\n\\vspace{-2ex}\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{images\/FIGURE3.png}\n\\end{center}\n\\vspace{-2ex}\n\\caption{The measurement parameters of a close pass of an object in a near-circular orbit ($e=0.00126$), as well as several simulations made using different eccentricities. The left plot is angular space and the right is the delay-Doppler. Twenty seconds of the true pass is shown with ten seconds of the simulated paths overlaid.}\n\\label{fig:eccentricity-mismatch}\n\\end{figure}\n\nThe good agreement between the parameters derived from methods described in this paper, when compared with ephemeris derived parameters, suggests that earlier results, \\cite{8835821}, can be practically achieved without requiring apriori information.\n\n\\section{Conclusion} \\label{sec:conclusion}\nModern radars are able form matched-filter products with significant numbers of measurement parameters, especially with digital beamforming and extended processing intervals. Conversely, the motion of an object in a Keplerian orbit is defined by only six parameters. Mapping radar measurement parameters from orbital motion parameters constrains the search space for uncued detection, it additionally allows for other constraints to be applied to further reduce the search-space, most notably when searching for objects in a circular orbit at their point of closest approach to the sensor. For a hypothesised orbit of this type, all range, Doppler, and angular motion parameters can be derived entirely from a three-dimensional position.\nDetections from this matched filter will correspond to the hypothesised orbit. This means that initial orbit determination can be potentially achieved from a single radar detection.\n\nIn future work, these algorithms will be experimentally validated with MWA observations. Noting that although these methods have been developed for the MWA, these methods also apply to conventional active space surveillance radar or even to satellite-based sensors. Additionally, it is planned to investigate the sensitivity of these techniques, characterising their variance by calculating the Cram\\'er-Rao lower bound (CRLB) on the variance of the initial orbital estimates.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro} Introduction}\n\nThere exists an accumulation of studies on quantum dynamics of classically\nchaotic systems\n, e.g.,   kicked rotators,  kicked spin-tops, hydrogen atoms in\ntime-dependent electric field,\nand the standard map\nmodel, to mention a few.~\\cite{NS2001}\n Quantum suppression of energy diffusion,\ndynamical localization\nand other signatures of quantum chaos\nare notable in these dynamics. However, most of the systems treated so far are\nconfined to those with a few degrees-of-freedom, and little attention is\npaid to dynamics of quantum many-body systems\\cite{Jona,Prosen,Flambaum}\n whose adiabatic energy levels\nare characterized by\nGaussian orthogonal ensemble (GOE) spectral statistics, i.e., by a hallmark\nof quantum chaos.\nWhile some important\ncontributions~\\cite{Wilkinson88,WilAus92,WilAus95,Bulgac,Cohen00D,Cohen00M,Machida}\nare devoted to dynamics of a kind of many-body systems,\nthose systems are actually described by the random-matrix models,\nand not by deterministic quantum Hamiltonians.\nIt is highly desirable to explore dynamical behaviors of deterministic\nquantum many-body systems\n     exhibiting GOE or GUE spectral statistics.\n \n     On the other hand, the frustrated quantum spin systems have been receiving\n     a wide attention, and  we can find their realization in $s=\\frac12$\n     antiferromagnetic chains Cu(ampy)Br$_2$~\\cite{Kikuchi} and\n     (N$_2$H$_5$)CuCl$_3$,~\\cite{Hagiwara} and in $s=\\frac12$ triangular\n     antiferromagnets.~\\cite{KPL}\n     The high-lying states of these quantum many-body systems deserve\n     being   studied\n     in the context of \"quantum chaos.\" The advantage of the frustrated quantum\nsystems\n     is that one can expect quantum chaotic behaviors\n     appearing already in the low energy region\n     near the ground state.~\\cite{Nakamura85,Yamasaki04}\n    From the viewpoint of real physics of condensed matters, novel features\nobserved\nin the low-energy region are very important and welcome.\nRecalling that in most of deterministic Hamiltonian systems quantum chaotic\nbehaviors\nappear in high-lying states, the role of\nfrustration is essential in the study of\nquantum dynamics from the ground state of deterministic many-body systems with\nGOE or Gaussian unitary ensemble (GUE) level statistics.\n\n\nIn this paper, we investigate dynamics of\n$XXZ$ quantum spin chains which have antiferromagnetic exchange\ninteractions for the\nnearest-neighbor (NN) and\nthe next-nearest-neighbor (NNN) couplings.  The NNN couplings cause the\nfrustration, i.e., difficulty in achieving the ground state,\nthereby attributing a name of frustrated quantum spin chains to these systems.\nIn fact, the level statistics of the NNN coupled $XXZ$ spin chains\nwithout an applied magnetic field\nhas been studied intensively in Refs.~\\onlinecite{Kudo03,Kudo04}, and\nit has been shown that GOE behavior,\nwhich is typical of quantum chaos, appears already in the low energy\nregion near the ground state.~\\cite{note,note1}\nThe ground-state phase diagram is shown in Ref.~\\onlinecite{Nomura} for\nthe NNN coupled $XXZ$ spin chains without a magnetic field.\n\nA natural extension of the research is to investigate dynamics\nof the frustrated quantum spin chains\nwith an applied periodically oscillating\nmagnetic field.  We calculate a time evolution of the system\nstarting from their ground state and analyze\nthe nature of energy diffusion. We shall numerically exhibit\nthe time dependence of energy variance,\nand show how\nthe diffusion coefficients depend on the coupling constants, the anisotropy\nparameters, the magnetic field and the frequency of the\nfield.  Furthermore, to compare with the energy diffusion in\nthe case of weakened frustrations, we also investigate dynamics of\nthe corresponding energy diffusion in $XXZ$ spin chains with\nsmall NNN couplings.\n \nThe organization of the paper is as follows:\nIn Sec.~\\ref{sec:method}, we briefly describe a numerical approach to\nobtain the time evolution operator. In Sec.~\\ref{sec:variance}\nwe shall show the time dependence of energy variance starting from\nthe ground state of the many-body system and explain a way to evaluate\ndiffusion coefficients. Section \\ref{sec:diffusion} elucidates\nhow diffusion coefficients depend on field strength and driving\nfrequency. \nHere \npower laws are shown to exist in the linear response and non-perturbative\nregions.\nSection \\ref{sec:compare} is devoted to a mechanism of oscillation of energy\ndiffusion. \nConclusions are given in Sec.~\\ref{sec:conc}.\n\n\\section{\\label{sec:method} Numerical Procedure}\n\nWe give Hamiltonian for the NN and NNN exchange-coupled\nspin chain on $L$ sites with a time-periodic oscillating magnetic field as\n\\begin{equation}\n\\mathcal{H}(t)=\\mathcal{H}_0 +\\mathcal{H}_1(t),\n\\label{eq:H}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n     \\mathcal{H}_0 &=& J_1\\sum_{j=1}^{L}(S^x_j S^x_{j+1} +S^y_j S^y_{j+1}\n     +\\Delta S^z_j S^z_{j+1})\n\\nonumber \\\\\n     &+& J_2\\sum_{j=1}^{L}(S^x_j S^x_{j+2} +S^y_j S^y_{j+2}\n     +\\Delta S^z_j S^z_{j+2})   \\nonumber \\\\\n&-& \\sum_{j=1}^{L}B^z_j(0)S^z_j,\n\\end{eqnarray}\n\\begin{equation}\n    \\mathcal{H}_1(t)= \\sum_{j=1}^{L}B^z_j(0)S^z_j -\\sum_{j=1}^{L}B^z_j(t)S^z_j.\n\\end{equation}\nHere, $S_j^{\\alpha}=(1\/2)\\sigma_j^{\\alpha}$ and \n$(\\sigma^x_j, \\sigma^y_j, \\sigma^z_j)$ are the Pauli matrices on the\n$j$th  site;\nthe periodic boundary conditions (P.~B.~C.) are imposed.\nThe magnetic field $B^z_j$ on $j$th site along the $z$ axis is chosen  to\nform a traveling wave:\n\\begin{equation}\n    B^z_j(t)=B_0\\sin\\left( \\omega t-\\frac{2\\pi j}L\\right).\n\\label{eq:jiba}\n\\end{equation}\nThe period of Eq.~(\\ref{eq:H}) as well as Eq.~(\\ref{eq:jiba}) is\n $T=2\\pi\/\\omega$. Because of the coexisting spatial P.~B.~C., however,\n the effective  period of the adiabatic energy spectra is\n given by $T'=T\/L=2\\pi\/(\\omega L)$. In other words, the period of the\nHamiltonian operator is $T$, and the spectral flow of the\n eigenvalues has the effective period $T'$.\nThis periodicity property comes from the\n traveling-wave form of Eq.~(\\ref{eq:jiba}), and is advantageous for our\n getting a sufficient number of relevant data in each period $T$.\n\nWhen $J_1>0$ and $J_2>0$, \nthe unperturbed  Hamiltonian $\\mathcal{H}_0$\nwithout coupling to the magnetic field is translationally invariant and \ncorresponds to a\nfrustrated antiferromagnetic quantum spin model\nexhibiting GOE level statistics.~\\cite{Kudo03,Kudo04} \nIf $J_2=0$ and $B_0=0$, it describes an\nintegrable  and non-frustrated model. Before calculating energy\ndiffusion, we have to consider the symmetries of the model. We divide\nthe Hamiltonian matrix to some sectors which have the same quantum\nnumbers. In the Hamiltonian Eq.(\\ref{eq:H}), total $S^z $ $(S^z_{\\rm tot})$ is\nconserved. The eigenstates with different $S^z_{\\rm tot}$ are\nuncorrelated.\nOn the other hand, the non-uniform magnetic field\nbreaks the translational symmetry, and leads\nto mixing between manifolds of different wave-number values.\n\nBefore proceeding to consider the time evolution of a wave\nfunction, we should note: If we use the original\nHamiltonian\n$\\mathcal{H}(t)=\\mathcal{H}_0 +\\mathcal{H}_1(t)$ as it\nstands, the mean level spacing of eigenvalues would change\ndepending\non $J_2$, $\\Delta$, and $B_0$.\nTo see a universal feature of the energy diffusion, it is\nessential to\nscale the Hamiltonian so that the full range of adiabatic\nenergy eigenvalues becomes almost free from these\nparameters.\nNoting that this energy range for the original Hamiltonian\nis \nof order of $L$  when $J_1=J_2=\\Delta=1$,\nwe define the scaled Hamiltonian  $H(t)=H_0+H_1(t)$\nso that the full energy range equals $L$ at $t=0$, \nwhich will be used throughout in the text.\nThe Sch\\\"odinger equation is then given by\n\\begin{equation}\n    i\\hbar \\frac{\\partial}{\\partial t}|\\psi (t)\\rangle\n    =H(t)|\\psi (t)\\rangle\n    =[H_0+H_1(t)] |\\psi (t)\\rangle.\n\\label{eq:Schrodinger}\n\\end{equation}\nThe solution of Eq. (\\ref{eq:Schrodinger}) consists\nof a sequence of the infinitesimal processes as\n\\begin{eqnarray}\n     |\\psi (t)\\rangle &=& U(t;t-\\Delta t) U(t-\\Delta t;t-2\\Delta t)\n\\nonumber \\\\\n&\\cdots& U(2\\Delta t;\\Delta t)\n    U(\\Delta t;0) |\\psi (0)\\rangle.\n\\end{eqnarray}\nThe initial state $|\\psi(0)\\rangle$ is taken to be the ground state,\nsince our concern lies in the dynamical behaviors starting from the\nmany-body ground state.\nTo calculate a time evolution operator $U(t+\\Delta t;t)$\nfor each short time step $\\Delta t$, we use the\nfourth-order decomposition formula for the exponential\noperator:~\\cite{Suzuki90}\n\\begin{eqnarray}\n    U(t+\\Delta t;t)&=& S(-i p_5\\Delta t\/\\hbar,t_5)\nS(-i p_4\\Delta t\/\\hbar,t_4) \\nonumber\\\\\n&\\cdots& S(-i p_2\\Delta t\/\\hbar,t_2)\n                       S(-i p_1\\Delta t\/\\hbar,t_1),\n\\label{eq:U}\n\\end{eqnarray}\nwhere,\n\\begin{equation}\n    S(x,t)=\\exp\\left( \\frac{x H_1(t)}2 \\right) \\exp(x H_0)\n\\exp\\left( \\frac{x H_1(t)}2 \\right).\n\\end{equation}\nHere, $t_j$'s and $p_j$'s are the following:\n\\begin{eqnarray}\n    t_j &=& t+(p_1+p_2+\\cdots +p_{j-1}+p_j\/2)\\Delta t, \\nonumber\\\\\n    p &=& p_1 =p_2 =p_4=p_5, \\nonumber\\\\\n      &=& 0.4144907717943757\\cdots \\nonumber\\\\\n    p_3&=&1-4p.\n\\end{eqnarray}\nThe numerical procedure based on the above decompositions is quite effective\nwhen $H_1(t)$ and $H_0$\ndo not commute and each time step is very small. \nOur computation below is concerned mainly with\nthe system of $L=10$, whose $S^z_{\\rm tot}=1$ manifold involves 210\nlevels. To check the validity of our assertion, some of the results will be\ncompared to those for the system of $L=14$ and \n$S^z_{\\rm tot}=4$ whose manifold involves 364 levels. \n\n\n\\section{\\label{sec:variance} Time Dependence of Energy Variance}\n\n\\begin{figure}\n\\includegraphics[width=8cm]{fig1.eps}\n\\caption{\\label{fig:1} (Color online) \nTime evolution of energy diffusion for (a) $L=10$ and (b) $L=14$. The\n    parameters are the following: $J_1=J_2=1.0$,\n    $\\Delta=0.3$,  $B_0=1.0$.}\n\\end{figure}\n\nWe calculate time evolution of the state and evaluate\nenergy variances at each integer multiple of\nthe effective period $T'=T\/L=2\\pi \/(\\omega L)$.\nAs mentioned already, we choose the ground state as an initial state,\nfollowing the spirit of real physics of condensed matters.\nThis viewpoint is in contrast to that of\nthe random matrix models where initial states\nare chosen among high-lying\nones.~\\cite{Wilkinson88,WilAus92,WilAus95,Bulgac,Cohen00D,Cohen00M} \nConsequently,\nthe energy variance of our primary concern is\nthe \\textit{variance around the ground state energy} $E_0$\nand is defined by\n\\begin{equation}\n    \\delta E(t)^2= \\langle \\psi (t)|[H(t)-E_0]^2 |\\psi (t) \\rangle .\n\\label{eq:variance}\n\\end{equation}\nTime evolution of $\\delta E(t)^2$ is shown in Fig.~\\ref{fig:1}. The\nparameters except for $\\omega$ are fixed. The larger $\\omega$ is, the faster\nthe energy diffusion grows, which is consistent with our expectations. The\ndetails will be explained in Sec.~\\ref{sec:diffusion}.\nFor wide parameter values of the next-nearest-neighbor (NNN) coupling $J_2$\nand exchange anisotropy $\\Delta$, the early stage of quantum dynamics\nbecomes to show the normal diffusion in energy space, i.e., a linear growth of \n$\\delta E(t)^2$ in time.\nWhile we proceed to investigate this normal diffusion process, \nenergy variances will\nfinally saturate because the system size we consider is finite. On the\nother hand, energy variances can also saturate because of another reason,\ni.e., the dynamical localization effect associated with a periodic\nperturbation. \n\nDuring the first period, $\\delta E(t)^2$ shows a linear\ngrowth in time as shown in Fig.~\\ref{fig:1} (a). The range of the linear\ngrowth is not sufficiently wide because the number of levels is not\nlarge enough for\n$L=10$. However, if the number of levels as well as the system\nsize is increased, the length of a linear region may be elongated. \nIn fact, the linear growth of $\\delta E(t)^2$ during the first\nperiod can be recognized more clearly for $L=14$ than for $L=10$ \n[see Fig.~\\ref{fig:1} (b)].\nThe diffusion coefficient has to be determined much earlier than the \ntime where saturation begins. \nWe determine the diffusion coefficient $D$  from the fitting\n\\begin{equation}\n    \\delta E(t)^2 = Dt +\\mbox{\\rm const.}\n\\label{eq:defD}\n\\end{equation}\nto some data points around the largest slope\nin the first period, where the normal diffusion is expected. \n\n\\section{\\label{sec:diffusion} Diffusion coefficients: dependence on field\nstrength and frequency}\n\nSince the time evolution of our system starts from the ground state,\nwe consider non-adiabatic regions where inter-level transitions\nfrequently occur. In other words, we suppress a near-adiabatic or the\nso-called Landau-Zener (LZ) region\nwhere the driving frequency $\\omega$ is much smaller than the mean level\nspacing divided by Planck constant.  \nBecause of a large energy gap between the ground and first\nexcited states,\nthe near-adiabatic region cannot result in the notable energy diffusion and\nwill be left outside a scope of the present study.\n\nBeyond the LZ region, however, so long as the changing rate $\\dot{X}$  of a\nperturbation\nparameter is not very large,~\\cite{note2} the diffusion coefficient can be calculated\nusing the Kubo formula. We call such a parameter regime ``linear\nresponse'' regime. In the linear response regime, $D\\propto\\dot{X}^2$\n(See, e.g., Refs.~\\onlinecite{WilAus92} and \\onlinecite{WilAus95}).\nWhen $\\dot{X}$ is large, however, the perturbation theory fails. We call\nsuch a parameter regime ``non-perturbative'' regime. In the \nnon-perturbative regime, the diffusion coefficient is smaller than that\npredicted by the Kubo formula.~\\cite{WilAus95,Cohen00D} According to\nRef.~\\onlinecite{WilAus95}, $D \\propto \\dot{X}^{\\gamma}$ with\n$\\gamma \\le 1$ in the non-perturbative regime. We note that\n$\\dot{X}\\propto B_0\\omega$ in this paper since the perturbation is given by\nEq.~(\\ref{eq:jiba}). \nBoth Refs.~\\onlinecite{WilAus95} and \\onlinecite{Cohen00D} are based on\nthe random matrix models, which are utterly different from our\ndeterministic one.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{fig2.eps}\n\\caption{\\label{fig:2} Driving frequency dependence of the diffusion\n    coefficients.  The chained line and the solid line\n are just eye guides for $D\\propto \\omega^{\\beta}$ with $\\beta=1$ and\n $2$, respectively.  \nThe symbols ($\\diamond$) are the average of the  diffusion coefficients\n    calculated for several values of $\\Delta$ ($0.3\\le \\Delta \\le 0.8$). The\n    parameters are the following: $L=10$, $J_1=1.0$; (a) $J_2=1.0$, \n(b) $J_2=0.2$.}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[width=12cm]{fig3.eps}\n\\caption{\\label{fig:3} Dependence of the diffusion\n    coefficients on the product of field strength $B_0$ and driving\n frequency $\\omega$ for (a) $L=10$ and (b) $L=14$. \nThe symbols ($\\diamond$) are the average of the  diffusion coefficient\n    calculated for several values of $\\Delta$ ($0.3\\le \\Delta \\le 0.8$). \nThe parameters are $J_1=J_2=1.0$; for\n the inset, $J_1=1.0$ and $J_2=0.2$. \nThe chained line and the solid line\n are just eye guides for $D\\propto (B_0\\omega)^{\\beta}$ with $\\beta=1$\n and $2$, respectively. Some error bars are too short to see.}\n\\end{figure*}\n\nNumerical results of diffusion coefficients in Fig.~\\ref{fig:2}\n are almost consistent with the argument of Ref.~\\onlinecite{WilAus95}.\nDiffusion coefficients as a function of $\\omega$\nare shown in Fig.~\\ref{fig:2}.\nIn Fig.~\\ref{fig:2}(a), where $J_2=1.0$\n(i.e., the fully-frustrated case), $D$ is larger\nas $B_0$ is larger for a fixed value of $\\omega$. In a small-$\\omega$ regime,\n$D\\propto \\omega^{\\beta}$ with $\\beta=2$, though $\\beta >2$ for small $B_0$. \nThe latter is merely attributed to the fact that the\nperturbation is\ntoo  small to observe a sufficient energy diffusion \nwhen both $\\omega$ and $B_0$ are small.  \nIn a large-$\\omega$ regime, $\\beta=1$.\nNamely, we observe that $\\beta=2$ in the linear response\nregime\nand $\\beta=1$ in the non-perturbative regime.\nIn fact, for a large-$\\omega$ regime, \nthe increase of energy variances per effective\nperiod hardly depend on $\\omega$ by the time when $\\delta E(t)^2$\n starts to decrease.\nThis explains the observation\nthat $D\\propto \\omega^{\\beta}$ with $\\beta=1$ in both\nFig.~\\ref{fig:2}(a) and Fig.~\\ref{fig:2}(b). \nLet us represent the increase of energy variances per\neffective period\nas $\\Delta(\\delta E^2)$. From the definition of $D$,\ni.e. Eq.~(\\ref{eq:defD}), $D\\propto \\Delta(\\delta E^2)\/T'$.\nIf\n$\\Delta(\\delta E^2)$ is constant, $D\\propto \\omega$.\n\nOn the other hand, in Fig.~\\ref{fig:2}(b) where $J_2=0.2$\n(i.e., a weakly-frustrated\n case),  the region with $\\beta=1$ is expanding. \nFor small $B_0$, $\\beta>2$ in a small-$\\omega$ regime is\nthe same as in the\ncase of $J_2=1.0$.\nFor small $B_0$ and around $\\omega\\sim 1$,\n$D$ seems to rather decrease than increase \nespecially in the case of $J_2=0.2$.\nSome kind of localization would have occurred \nin the very early stage of energy diffusion for large\n$\\omega$ and small $B_0$,\nleading to the suppression of $D$.\n\n\nIt is seen more clearly in\nFig.~\\ref{fig:3} how the behavior of $D$ changes between a linear\nresponse regime and a non-perturbative regime. \nThe diffusion coefficient $D$ obeys the power law\n$D\\propto (B_0 \\omega)^{\\beta}$ with its power $\\beta$ being two in  the\nlinear response regime and $\\beta=1$ in the non-perturbative regime.\nFor small $B_0\\omega$, the power law seems to fail because of some\nfinite-size effects. \nThese universal feature is confirmed in systems of larger size.\nActually, $D$ obeys the power law \nbetter for $L=14$ [Fig.~\\ref{fig:3}(b)] than $L=10$\n[Fig.~\\ref{fig:3}(a)]. In addition, error bars are shorter for $L=14$\nthan $L=10$.\nHere, we have used the data of\n$\\omega \\le 1$. We cannot expect meaningful results in a large-$\\omega$\nregime since, as mentioned above, energy diffusion is not normal there.\n\nFigure~\\ref{fig:3} suggests that the strength of frustration should affect the\nrange of the linear response regime.\nThe linear response regime is shorter for $J_2=0.2$ than for $J_2=1.0$,\nwhile the non-perturbative regime is larger for $J_2=0.2$ than for\n$J_2=1.0$. In fact, when $J_2=0$ (i.e. the integrable case), \n$D\\propto (B_0 \\omega)^{\\beta}$ with $\\beta=1$ for almost all the data\nin the same range of $B_0\\omega$ as that of Fig.~\\ref{fig:3}.    \n\n\\section{\\label{sec:compare} Oscillation Of energy diffusion in \nweakly-frustrated cases}\n\nWe shall now proceed to investigate\noscillations of diffusion which occur in the non-perturbative regime of\na weakly-frustrated case.\nFigure~\\ref{fig:4}(a) shows an example of oscillatory diffusion for\n$J_2=0.2$, which is compared with a non-oscillatory\ndiffusion for $J_2=1.0$. \nThe two examples have the same set of parameters except for $J_2$.\nHowever, the cases of $J_2=1.0$ and $J_2=0.2$ are  in the linear response\nregime  and  in the non-perturbative regime, respectively.\nThe variance for both cases\nshows normal diffusion at the very early stage of time evolution.\nFor $J_2=1.0$, the energy variance seems to saturate after a normal\ndiffusion time. On the contrary, the energy variance for $J_2=0.2$ shows\nlarge-amplitude oscillations.\nTo investigate more details, we introduce another definition of energy \nvariance:\n\\begin{equation}\n   \\delta \\tilde{E}(t)^2 =\\langle \\psi (t)|[H(t)-\\langle\\psi (t) |\nH(t)|\\psi(t)\\rangle]^2\n   |\\psi (t)\\rangle .\n\\end{equation}\nThis follows a standard definition of the variance and quantifies\nthe degree of energy diffusion around the \\textit{time-dependent\nexpectation} of the energy Hamiltonian.\nThe time evolutions of $\\delta \\tilde{E}(t)^2$  \ncorresponding to that of $\\delta E(t)^2$ are shown\nin Fig.~\\ref{fig:4}(b). \nIn the fully-frustrated case ($J_2=1.0$), the profile of \n$\\delta\\tilde{E}(t)^2$ is similar to that of $\\delta E(t)^2$.\nThis observation indicates that an occupation probability spread\nover the whole levels after normal diffusion of energy.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{fig4.eps}\n\\caption{\\label{fig:4} Examples for time evolution of energy variances :  \n(a) $\\delta E(t)^2$ and (b)  $\\delta \\tilde{E}(t)^2$ (see text).\nSolid lines are for $J_2=1.0$; Broken lines, $J_2=0.2$.\nThe parameters are\nthe following: $L=10$, $J_1=1.0$, $\\Delta=0.3$, $B_0=1.5$, $\\omega=0.5$.}\n\\end{figure}\n\nOn the contrary, in a weakly-frustrated case ($J_2=0.2$) \nin Fig.~\\ref{fig:4}, $\\delta \\tilde{E}(t)^2$\nshows small-amplitude oscillations reflecting\nthe large-amplitude oscillations of $\\delta E(t)^2$.\nMost part of $\\delta \\tilde{E}(t)^2$ for $J_2=0.2$ is smaller than that\nfor  $J_2=1.0$. Furthermore, minima of $\\delta \\tilde{E}(t)^2$ come just\nbefore minima and maxima of $\\delta E(t)^2$.  \nThese observations indicates the following: an occupation probability,\nwhich is diffusing slowly,\nclustering around the expectation of energy oscillates together with the\nexpectation in the energy space.\nTo make the picture of such behavior clearer, let us\nconsider an occupation probability described by\n\\begin{equation}\n  P_t(E_n)=| \\langle \\phi_n | \\psi(t) \\rangle |^2,\n\\end{equation}\nwhere $|\\phi_n\\rangle$ is the $n$th excited eigenstate of $H_0$:\n\\begin{equation}\n  H_0 |\\phi_n\\rangle =E_n |\\phi_n\\rangle .\n\\end{equation}\nWhen $t=0$ , $P_t(E_n)$ is given by the Kronecker delta:\n$P_0(E_n)=\\delta_{E_n,E_0}$, where $E_0$ is the energy of the ground\nstate. As $t$ increases, $P_t(E_n)$ forms a wave packet in energy space\nand moves to\nhigher levels. When the wave packet reaches some highest level, it\nreflects like a soliton and moves back to lower levels. Such behavior is\nrepeated, although the wave packet of $P_t(E_n)$ broadens slowly.\nWe have actually watched this soliton-like behavior of $P_t(E_n)$ in\na form of  an animation.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{fig5.eps}\n\\caption{\\label{fig:5} Parts of energy spectra depending on\n adiabatically fixed time $t$ with\n   $0\\le t \\le T\/4$. Effective period is $\\omega T'=2\\pi\/10$. The\n    parameters are the following: $L=10$, $J_1=1.0$, $\\Delta=0.3$, $B_0=0.8$;\n   (a) $J_2=1.0$, (b) $J_2=0.2$.}\n\\end{figure}\n\nThe picture discussed above is also supported by the adiabatic energy\nspectra in Fig.~\\ref{fig:5}. \nFigures~\\ref{fig:5}(a) and \\ref{fig:5}(b) correspond to fully- and\nweakly-frustrated cases, respectively.\nMuch more sharp avoided crossings appear in\nFig.~\\ref{fig:5}(b) than Fig.~\\ref{fig:5}(a). Some energy levels appear\nto be crossing, although they are very close and never crossing in\nfact. At a sharp-avoided-crossing point,\nLandau-Zener formula for two adjacent levels is applicable.\nThen the nonadiabatic transition leads to one-way transfer of a\npopulation from a level to its partner, failing to result in the energy\ndiffusion. \nFor small-$J_2$, therefore,\nsuccessive sharp avoided crossings can suppress diffusion of energy.\n\nWe believe that large-amplitude oscillations of $\\delta E(t)^2$ should\nbe one of characteristic features of the non-perturbative regime in this\nfinite frustrated spin system.\nIn fact, similar oscillations of energy variance are seen for large\n$\\omega$ and large $B_0$ even when $J_2=1.0$ though the energy variance\nrapidly converges after one or two periods. How long such oscillations\ncontinue should depend mainly on $J_2$.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{fig6.eps}\n\\caption{\\label{fig:6} (Color online) Level-spacing distributions at $t=\\pi\/4$ \nfor lowest 300 levels from\n  the ground state (about 10\\% of all 3003 levels).  Blue histogram is for\n  $J_2=1.0$; Red bars, $J_2=0.2$; Solid curve, GOE spectral statistics.\nThe other\n    parameters are the following: $L=14$, $S^z_{\\rm tot}=1$, $J_1=1.0$,\n $\\Delta=0.3$,  $B_0=0.8$.\nThe inset is for all levels when $J_2=1.0$. The numerical methods to\n obtain the level-spacing distributions are referred in\n Refs.~\\onlinecite{Kudo03,Kudo04}. }\n\\end{figure}\n\nIt is a notable fact that, common to both $J_2=1.0$ and $J_2=0.2$, the\nlevel-spacing distributions in Fig.~\\ref{fig:6} show GOE behavior. This\nGOE behavior in the adiabatic energy spectra appears for an arbitrary\nfixed time except for special points\nsuch as $t=T=2\\pi\/\\omega$. This fact suggests that dynamics can reveal\nsome various generic features of quantum many-body systems\n which can never be explained by level\nstatistics. The level-spacing distributions in Fig.~\\ref{fig:6} convey\nanother crucial fact: they have been\ncalculated for low energy levels because our interest is in the low\nenergy region around the ground state. We have confirmed that the\nlevel-spacing distributions for all energy levels in the inset is also described by\nGOE spectral statistics. It is typical of this frustrated spin system that GOE level\nstatistics is observed already in the low energy region.~\\cite{Kudo04}  \n\n\\section{\\label{sec:conc}Conclusions}\n\nWe have explored the energy diffusion from the ground state\nin frustrated quantum $XXZ$  spin chains under the applied oscillating\nmagnetic field.\nIn a wide parameter region of next-nearest-neighbor (NNN) coupling $J_2$\nand exchange anisotropy $\\Delta$,\nthe diffusion is normal in the early stage of diffusion.\nDiffusion coefficients $D$ obey the power law\nwith respect to\nboth the field strength and driving frequency with its power being two in  the\nlinear response regime and equal to unity in the\nnon-perturbative regime.\nIn the case of weakened frustrations with small-$J_2$ \nwe find oscillation of energy diffusion,\nwhich is attributed to a\nnon-diffusive and ballistic nature of\nthe underlying energy diffusion.\nIn this way, the energy diffusion reveals generic features of the\nfrustrated quantum spin chains, which cannot be captured by the analysis\nof level statistics.\n\n\n\\begin{acknowledgments}\nThe authors would like to thank T. Deguchi.\n The present study was partially supported by Hayashi Memorial\n Foundation for Female Natural Scientists.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\t\n\tIn contemporary financial option valuation theory, jump-diffusion processes form a principal class of models for the \n\tevolution of the underlying asset prices, see~e.g.~Cont \\& Tankov~\\cite{CT04book} and Schoutens~\\cite{S03book}.\n\tThe first jump-diffusion process was proposed in 1976 by Merton~\\cite{M76}.\n\tIn this classical model, the relative jump sizes are assumed to be lognormally distributed.\n\tA wide variety of jump-diffusion processes, and more generally, exponential L\\'{e}vy processes, for asset prices has been introduced \n\tin the literature since then, for example the familiar variance gamma (VG), normal inverse Gaussian (NIG) and Carr--Geman--Madan--Yor \n\t(CGMY) models~\\cite{CT04book,S03book}.\n\t\n\tIn this paper we consider the jump-diffusion model proposed by Kou~\\cite{K02}.\n\tIn this model, the relative jump sizes are given by a log-double-exponential distribution.\n\tJust like the models mentioned above, Kou's jump-diffusion model has become popular in financial option valuation theory and practice.\n\tIn this paper we are interested in the valuation of European-style options under a direct extension of Kou's single \n\tasset jump-diffusion model~\\cite{K02} to two assets.\n\tHere the (finite activity) jumps in the two asset prices are assumed to occur contemporaneously.\n\tFinancial option valuation theory then yields a two-dimensional time-dependent partial integro-differential equation (PIDE)\n\tthat must be satisfied for the values of European two-asset options. \n\tHere the integral part, which stems from the contribution of the jumps, is nonlocal: it is taken over the whole, two-dimensional asset \n\tprice domain.\n\tIn general, (semi-)closed analytical solutions to this PIDE are not available in the literature.\n\tAccordingly, in the present paper, we investigate the effective numerical solution of the two-dimensional time-dependent Kou PIDE.\n\t\n\tFor the numerical solution, we follow the well-known and general method of lines (MOL) approach.\n\tThe two-dimensional Kou PIDE is first discretized in space by finite differences, and the resulting large, semidiscrete system of \n\tordinary differential equations (ODEs) is subsequently discretized in time by a suitable implicit time stepping scheme.\n\tThe main challenges for the efficient and stable numerical solution are the treatment of the two-dimensional integral part and the \n\ttreatment of the two-dimensional PDE part, which includes a mixed spatial derivative term.\n\t\n\tSpatial discretization of the PIDE leads to a large, dense matrix for the integral part.\n\tIn each time step of the schemes under consideration, products of this matrix with one or more given vectors need to be computed,\n\twhich can form a computational burden.\n\tFor the one-dimensional Kou PIDE, however, Toivanen~\\cite{T08} derived a simple algorithm for evaluating these matrix-vector \n\tproducts that has optimal computational cost.\n\tA key result of our paper is a generalization of Toivanen's algorithm to the two-dimensional Kou PIDE that maintains optimal \n\tcomputational cost.\n\t\n\tFor the temporal discretization of semidiscretized one-dimensional PIDEs arising in financial option valuation, various authors have \n\tproposed operator splitting schemes where the (stiff) PDE part is handled implicitly and the (nonstiff) integral part explicitly.\n\tCont \\& Voltchkova~\\cite{CV05} considered an {\\it implicit-explicit (IMEX)} splitting scheme where the PDE part is handled by \n\tthe backward Euler method and the integral part by the forward Euler method.\n\tThis IMEX Euler scheme is only first-order consistent.\n\tA variety of higher-order IMEX schemes for PIDEs in finance has been studied since, e.g.~by Briani, Natalini \\& Russo~\\cite{B07},\n\tFeng \\& Linetsky~\\cite{FL08}, Kwon \\& Lee~\\cite{KL11} and Salmi \\& Toivanen~\\cite{ST14}.\n\tThe latter authors proposed the IMEX CNAB scheme, where the PDE part is treated by the Crank--Nicolson method and the integral part \n\tby the second-order Adams--Bashforth method.\n\tThe IMEX CNAB scheme has been successfully applied to two-dimensional option valuation PIDEs in e.g.~\\cite{HT16,STS14}. \n\t\n\tTavella \\& Randall~\\cite{TR00book} and d'Halluin, Forsyth \\& Vetzal~\\cite{HFV05} considered an alternative approach where the PDE \n\tpart is treated by the Crank--Nicolson method and a fixed-point iteration on the integral part is performed in each time step. \n\tThis approach has been applied to two-dimensional option valuation PIDEs in Clift \\& Forsyth~\\cite{CF08}, including the \n\ttwo-dimensional Kou PIDE.\n\tWhen the number of fixed-point iterations is frozen, one arrives at a particular IMEX scheme.\n\t\n\tFor the efficient temporal discretization of semidiscrete two-dimensional PIDEs, a subsequent important improvement is obtained \n\tby using, instead of the Crank--Nicolson method, an {\\it alternating direction implicit (ADI)} splitting scheme for the \n\ttwo-dimensional PDE part.\n\tIn the computational finance literature, a range of effective, second-order ADI schemes has been developed and analyzed for \n\tmulti-dimensional PDEs (without integral part), where in each time step the implicit unidirectional stages are combined with \n\texplicit stages involving the mixed derivative terms, see e.g.~\\cite{H17book,HT16}.\n\tIn this paper, we consider the well-established modified Craig--Sneyd (MCS) scheme, introduced by in 't Hout \\& Welfert~\\cite{HW09}, \n\tand the stabilizing correction two-step Adams-type scheme called SC2A, constructed by Hundsdorfer \\& in 't Hout~\\cite{HH18}.\n\t\n\tThe direct adaptation of ADI schemes for PDEs to PIDEs in finance has first been studied by Kaushansky, Lipton \\& Reisinger~\\cite{KLR18} \n\tand next by in 't Hout \\& Toivanen~\\cite{HT18}.\n\tHere the implicit unidirectional stages are blended with explicit stages involving both the mixed derivative terms and the integral \n\tpart, leading again to second-order schemes.\n\tWe note that an efficient, parallel implementation of the schemes developed in \\cite{HT18} has been designed by Ghosh \\& Mishra~\\cite{GM21} \n\twho apply a parallel cyclic reduction algorithm.\n\t\n\tBoen \\& in 't Hout~\\cite{BH21} recently investigated a collection of seven contemporary operator splitting schemes of both the \n\tIMEX and the ADI kind in the application to the two-dimensional Merton PIDE for the values of two-asset options.\n\tHere the numerical evaluation of the integral term has been done by means of a FFT algorithm, following \n\te.g.~\\cite{AO05,AA00,CF08,HFV05,STS14}. \n\tBased on analytical and numerical evidence in \\cite{BH21,HT18} in the case of the two-dimensional Merton and Bates PIDEs, \n\tit is concluded that, among the schemes under consideration, the adaptation of the MCS scheme introduced in \\cite{HT18} \n\tthat deals with the integral part in a two-step Adams--Bashforth fashion is preferable.\n\t\n\tIn the present paper we consider for the two-dimensional Kou PIDE the same collection of operator splitting schemes as in~\\cite{BH21}. \n\tFor the numerical evaluation of the double integral part, a generalization of the algorithm of Toivanen~\\cite{T08} is derived that has \n\toptimal computational cost.\n\tThis algorithm is simple to implement, requires little memory usage and is computationally much faster than the FFT algorithm \n\tmentioned above.\n\tAs a representative example, we consider the approximation of European put-on-the-average option values, together with their Greeks.\n\tAn outline of the rest of this paper is as follows.\n\t\n\tIn Section~\\ref{SecModel} the two-dimensional Kou PIDE is formulated.\n\tSection~\\ref{SecSpatial} deals with its spatial discretization.\n\tFirst, in Subsection~\\ref{SecPDEpart}, the two-dimensional PDE part is considered and a second-order finite difference discretization \n\ton a suitable nonuniform spatial grid is described.  \n\tNext, in Subsection~\\ref{SecIntpart}, we present the first main contribution of this paper.\n\tA common, second-order spatial discretization of the double integral part is employed and for its highly efficient evaluation \n\twe derive an extension of the algorithm proposed by Toivanen~\\cite{T08}.\n\tIt is shown that the computational cost of the extension is directly proportional to the number of spatial grid points, \n\twhich is optimal.\n\tSection~\\ref{SecTime} subsequently concerns the temporal discretization of the obtained semidiscrete two-dimensional Kou PIDE.\n\tHere the seven contemporary operator splitting schemes of the IMEX and ADI kind from~\\cite{BH21} are considered.  \n\tEach of these schemes conveniently treats the integral part in an explicit manner, where its fast evaluation is performed by the \n\talgorithm derived in Subsection~\\ref{SecIntpart}.\n\tIn Section~\\ref{SecResults} ample numerical experiments are presented.\n\tHere European put-on-the-average option values, together with their Greeks Delta and Gamma, are considered and we examine in detail\n\tthe temporal discretization errors of the different operator splitting schemes.\n\tThe final Section~\\ref{SecConc} gives conclusions.\n\t\n\t\n\n\n\n\t\n\t\\section{The two-dimensional Kou PIDE}\\label{SecModel}\n\tUnder the two-asset Kou jump-diffusion model, the value $v = v(s_1,s_2,t)$ of a European-style option with maturity date $T>0$ and $s_i$ ($i=1,2$) \n\trepresenting the price of asset $i$ at time $\\tau = T-t$, satisfies the following PIDE:\n\t\\begin{align}\\label{PIDE2D}\n\t\t\\dud{t} =&~ \\tfrac{1}{2} \\sigma_1^2s_1^2\\dudd{s_1} + \\rho \\sigma_1\\sigma_2s_1s_2\\duddm{s_1}{s_2} + \\tfrac{1}{2} \\sigma_2^2 s_2^2\\dudd{s_2} + \n\t\t(r-\\lambda \\kappa_1) s_1\\dud{s_1} + (r-\\lambda\\kappa_2) s_2 \\dud{s_2} \\nonumber \\\\\n\t\t& -(r+\\lambda)v+\\lambda\\int_0^{\\infty}\\int_0^{\\infty} f(y_1,y_2) v(s_1y_1, s_2y_2,t) \\mathrm{d} y_1 \\mathrm{d} y_2\n\t\\end{align}\n\twhenever $s_1>0$, $s_2>0$, $0<t\\leq T$.\n\tHere $r$ is the risk-free interest rate, $\\sigma_i > 0$ ($i=1,2$) is the instantaneous volatility for asset $i$ conditional \n\ton the event that no jumps occur, and $\\rho$ is the correlation coefficient of the two underlying standard Brownian motions.\n\tNext, $\\lambda$ is the jump intensity of the underlying Poisson arrival process, and $\\kappa_i$ ($i=1,2$) is the expected \n\trelative jump size for asset $i$.\n    The function $f$ is the joint probability density function of two independent random variables possessing log-double-exponential \n\tdistributions~\\cite{K02},\n\t\\begin{equation}\\label{pdf2D}\n\t\tf(y_1,y_2) = \n\t\t\\left\\{\\begin{array}{lll}\n\t\t\tq_1q_2\\eta_{q_1}\\eta_{q_2}y_1^{\\eta_{q_1}-1}y_2^{\\eta_{q_2}-1} & (0 < y_1, y_2 < 1),\\\\\n\t\t\tp_1q_2\\eta_{p_1}\\eta_{q_2}y_1^{-\\eta_{p_1}-1}y_2^{\\eta_{q_2}-1} & (y_1 \\geq 1, 0< y_2 < 1),\\\\\n\t\t\tq_1p_2\\eta_{q_1}\\eta_{p_2}y_1^{\\eta_{q_1}-1}y_2^{-\\eta_{p_2}-1} & (0 < y_1 < 1, y_2 \\geq 1),\\\\\n\t\t\tp_1p_2\\eta_{p_1}\\eta_{p_2}y_1^{-\\eta_{p_1}-1}y_2^{-\\eta_{p_2}-1} & (y_1, y_2 \\geq 1).\n\t\t\\end{array}\\right.\n\t\\end{equation}  \n\tThe parameters $p_i$, $q_i$, $\\eta_{p_i}$, $\\eta_{q_i}$  are all positive constants with $p_i + q_i = 1$ and $\\eta_{p_i} > 1$.\n\tIt holds that\n\t\\begin{equation*}\n\t    \\kappa_i = \\frac{p_i\\eta_{p_i}}{\\eta_{p_i}-1}+\\frac{q_i\\eta_{q_i}}{\\eta_{q_i}+1}-1 \\quad (i=1,2).\n\t\\end{equation*}\n\tFor \\eqref{PIDE2D}, the initial condition is given by\n\t\\[\n\tv(s_1,s_2,0) = \\phi(s_1,s_2),\n\t\\]\n\twhere $\\phi$ denotes the payoff function of the option.\n\tAs a typical example, we consider in this paper a European put-on-the-average option, which has the payoff function\n\t\\begin{equation}\\label{payoff}\n\t\t\\phi(s_1,s_2) = \\textrm{max} \\left(0\\,,\\,K-\\frac{s_1+s_2}{2}\\right)\n\t\\end{equation}\n\twith strike price $K>0$.\n\tIts graph is shown in Figure~\\ref{FigPayoff}.\n\tConcerning the boundary condition, it holds that the PIDE \\eqref{PIDE2D} is itself satisfied on the two sides $s_{1} = 0$ \n\tand $s_{2} = 0$, respectively.\n\t\n\n\t\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{payoff.pdf}\n\t\\caption{Payoff put-on-the-average option with $K=100$.}\n\t\\label{FigPayoff}\n\t\\end{figure}\n\t\n\t\n\n\n\n\t\n\t\\section{Spatial discretization}\\label{SecSpatial}\n\tFor the numerical solution of the initial-boundary value problem for \\eqref{PIDE2D} we employ the popular method of lines (MOL) approach.\n\tThis approach consists of two consecutive steps~\\cite{HV03book}: first the PIDE \\eqref{PIDE2D} is discretized in space and subsequently in time.\n\tThis Section~\\ref{SecSpatial} deals with the spatial discretization. \n\tIn the next Section~\\ref{SecTime} we shall consider the temporal discretization.\n\t\n\n\n\n\t\n\t\\subsection{Convection-diffusion-reaction part}\\label{SecPDEpart}\n\tFor the numerical solution, the spatial domain is truncated to a bounded set $[0,S_{\\rm max}]\\times[0,S_{\\rm max}]$ \n\twith fixed value $S_{\\rm max}$ chosen sufficiently large.\n\tOn the two far sides $s_{1} = S_{\\rm max}$ and $s_{2} = S_{\\rm max}$ a linear boundary condition is taken, which is well-known in finance,\n\t\\begin{equation}\\label{LBC}\n\t\t\\dudd{s_1} = 0~~(\\textrm{if}~s_{1} = S_{\\rm max}) \\quad \\mbox{ and } \\quad  \\dudd{s_2} = 0 ~~(\\textrm{if}~s_{2} = S_{\\rm max}).\n\t\\end{equation}\n\tIn this subsection we describe the finite difference discretization of the convection-diffusion-reaction part of the PIDE \\eqref{PIDE2D}, \n\tspecified by\n\t\\begin{equation*}\\label{lindiffop}\n\t\t\\mathcal{D} v =  \\tfrac{1}{2} \\sigma_1^2s_1^2\\dudd{s_1} + \\rho \\sigma_1\\sigma_2s_1s_2\\duddm{s_1}{s_2} + \\tfrac{1}{2} \\sigma_2^2 s_2^2\\dudd{s_2} \n\t\t+ (r-\\lambda \\kappa_1) s_1\\dud{s_1} + (r-\\lambda\\kappa_2) s_2 \\dud{s_2} -(r+\\lambda)v.\n\t\\end{equation*}\n\tThe semidiscretization of this part is common and similar to that in~e.g.~\\cite{BH21}.\n\t\n\tLet integers $m_1, m_2 \\geq 1$ be given.\n\tThe option value function $v$ is approximated at a nonuniform, Cartesian set of spatial grid points,\n\t\\begin{equation*}\\label{sgrid}\n\t\t(s_{1,i},s_{2,j})\\in [0,S_{\\rm max}]\\times[0,S_{\\rm max}] \\quad (0\\leq i \\leq m_1, \\ 0\\leq j \\leq m_2),\n\t\\end{equation*}\n\twith $s_{1,0} = s_{2,0} = 0$ and $s_{1,m_1} = s_{2,m_2} = S_{\\rm max}$. \n\tThe nonuniform grid in each spatial direction is defined through a smooth transformation of an artificial uniform grid, such that \n\trelatively many grid points are placed in a region of financial and numerical interest.\n\tFigure~\\ref{FigGrid} shows a sample spatial grid if $m_1=m_2=50$, $K=100$ and $S_{\\rm max} = 5K$.\n\t\n\tLet integer $m \\geq 1$ and parameter $d > 0$.\n\tConsider equidistant points $0 = \\xi_0 < \\xi_1 < \\dots < \\xi_{m} = \\xi_{\\rm max}$ where\n\t\\begin{equation*}\n\t\t\\xi_{\\rm max} = \\xi_{\\rm int} + \\sinh^{-1}\\left(\\frac{S_{\\rm max}}{d}-\\xi_{\\rm int}\\right) \n\t\t\\quad \\textrm{and } \\quad\n\t\t\\xi_{\\rm int} = \\frac{2K}{d}.\n\t\\end{equation*}\n\tThen in each spatial direction a nonuniform mesh $0=s_0<s_1<\\cdots<s_m=S_{\\rm max}$ is constructed by the smooth transformation \n\t$s_{i} = \\varphi(\\xi_i)$ $(0\\le i \\le m)$ with\n\t\\begin{equation*}\n\t\t\\varphi(\\xi) = \n\t\t\\left\\{\\begin{array}{lll}\n\t\t\td \\xi &(0 \\leq \\xi \\leq \\xi_{\\rm int}),\\\\\\\\\n\t\t\t2K + d\\sinh(\\xi-\\xi_{\\rm int}) &(\\xi_{\\rm int} < \\xi \\leq \\xi_{\\rm max}).\n\t\t\\end{array}\\right.\n\t\\end{equation*}  \n\tThis mesh for $s$ is uniform inside the interval $[0,2K]$ and nonuniform outside.\n\tThe parameter $d$ controls the fraction of points $s_{i}$ that lie inside.\n\tIn this paper we (heuristically) choose $d = K\/10$, such that the largest fraction is inside $[0,2K]$.\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.6]{sgrid.pdf}\n\t\t\\caption{Sample spatial grid for $m_1=m_2=50$, $K=100$, $S_{\\rm max} = 5K$.}\n\t\t\\label{FigGrid}\n\t\\end{figure}\n\t\n\tDefine mesh widths $h_i = s_i-s_{i-1}$ ($1\\leq i \\leq m$) and let $u: [0,S_{\\rm max}] \\to \\mathbb{R}$ be any given smooth function. \n\tFor approximating the first and second derivatives of $u$, the following second-order central finite difference formulas are used:\n\t\\begin{align*}\\label{semconv}\n\t\tu^\\prime (s_i) \\approx \\omega_{i,-1} u(s_{i-1}) + \\omega_{i,0} u(s_{i}) + \\omega_{i,1} u(s_{i+1}),\n\t\\end{align*}\n\twith \n\t\\[\n\t\\omega_{i,-1} = \\frac{-h_{i+1}}{h_{i}(h_{i}+h_{i+1})},\\quad\n\t\\omega_{i,0} = \\frac{h_{i+1}-h_{i}}{h_{i}h_{i+1}},\\quad \n\t\\omega_{i,1} = \\frac{h_{i}}{h_{i+1}(h_{i}+h_{i+1})}\n\t\\]\n\tand\n\t\\begin{align*}\\label{semdiff}\n\t\tu^{\\prime\\prime} (s_i) \\approx \\omega_{i,-1} u(s_{i-1}) + \\omega_{i,0} u(s_{i}) + \\omega_{i,1} u(s_{i+1}),\n\t\\end{align*}\n\twith\n\t\\[\n\t\\omega_{i,-1} = \\frac{2}{h_{i}(h_{i}+h_{i+1})},\\quad \n\t\\omega_{i,0} = \\frac{-2}{h_{i}h_{i+1}},\\quad \n\t\\omega_{i,1} = \\frac{2}{h_{i+1}(h_{i}+h_{i+1})}\n\t\\]\n\twhenever $1\\le i \\le m-1$.\n\tFor $i=0$ no finite difference formulas are needed, due to the degeneracy of $\\mathcal{D} v$ at the zero boundaries.\n\tFor $i=m$, the first derivative is discretized by the first-order backward finite difference formula and the second \n\tderivative is equal to zero by the linear boundary condition \\eqref{LBC}.\n\tConcerning the mixed derivative term in $\\mathcal{D} v$, this is approximated by successively applying the relevant \n\tfinite difference formulas for the first derivative in the two spatial directions.\n\t\n\tLet $V_{i,j}(t)$ denote the semidiscrete approximation to $v(s_{1,i},s_{2,j},t)$ for $0\\leq i \\leq m_1$, $0\\leq j \\leq m_2$ \n\tand define the vector \n\t\\[\n\tV(t) = (V_{0,0}(t), V_{1,0}(t),\\ldots,V_{m_1-1,m_2}(t),V_{m_1,m_2}(t))^{\\top} \\in \\mathbb{R}^{(m_1+1)(m_2+1)}.\n\t\\] \n\tThe semidiscretized convection-diffusion-reaction part $\\mathcal{D} v$ of PIDE \\eqref{PIDE2D} is then given by\n\t\\[\n\tA^{(D)} V(t)\n\t\\] \n\twith matrix\n\t\\[\n\tA^{(D)} = A^{(M)} + A_1 + A_2,\n\t\\]\n\twhere\n\t\\begin{align*}\n\t\tA^{(M)} &= \\rho\\sigma_1\\sigma_2\\left(X_2D_2^{(1)}\\right)\\otimes\\left(X_1D_1^{(1)}\\right),\\\\\n\t\tA_1 &= I_2 \\otimes \\left(\\tfrac{1}{2}\\sigma_1^2X_1^2D_1^{(2)} + (r-\\lambda\\kappa_1)X_1D_1^{(1)}-\\tfrac{1}{2}(r+\\lambda)I_1\\right),\\\\\n\t\tA_2 &= \\left(\\tfrac{1}{2}\\sigma_2^2X_2^2D_2^{(2)} + (r-\\lambda\\kappa_2)X_2D_2^{(1)}-\\tfrac{1}{2}(r+\\lambda)I_2\\right)\\otimes I_1.\n\t\\end{align*}\n\tHere $I_k$, $X_k$, $D_k^{(l)}$ are given $(m_k+1)\\times (m_k+1)$ matrices for $k,l\\in \\{1,2\\}$ with $I_k$ being the identity matrix, \n\t$X_k$ being the diagonal matrix\n\t\\begin{equation*}\n\t\tX_k = {\\rm diag}(s_{k,0}, s_{k,1},\\ldots, s_{k,m_k})\n\t\\end{equation*}\n\tand $D_k^{(l)}$ the matrix representing numerical differentiation of order $l$ in the $k$-th spatial direction by the relevant \n\tfinite difference formula above.\n\tThe matrix $A^{(M)}$ corresponds to the mixed derivative term in $\\mathcal{D} v$ and $A_k$ corresponds to all derivative terms \n\tin the $k$-th spatial direction ($k=1,2$), where the reaction term has been distributed evenly across $A_1$ and $A_2$.\n\tThese matrices are all sparse. In particular, $A_1$ and $A_2$ are (essentially) tridiagonal.\n\t\n\t\n\n\n\n\t\n\\subsection{Integral part}\\label{SecIntpart}\nIn the following we consider discretization of the double integral\n\\begin{equation}\\label{integral}\n\t\\mathcal{J} = \\lambda\\int_0^{\\infty}\\int_0^{\\infty} f(y_1,y_2)v(s_1y_1, s_2y_2,t) \\mathrm{d} y_1 \\mathrm{d} y_2\n\\end{equation}\non the spatial grid from Subsection~\\ref{SecPDEpart}.\nFor its efficient evaluation, we derive an extension of the algorithm proposed by Toivanen~\\cite{T08} for the special case of the \none-dimensional Kou model.\n\nAssume $s_1, s_2>0$ are given. A change of variables $y_i = z_i\/s_i$ ($i=1,2$) yields\n\\begin{equation*}\n\t\\mathcal{J} = \\lambda\\int_0^{\\infty}\\int_0^{\\infty} f\\bigg(\\frac{z_1}{s_1},\\frac{z_2}{s_2}\\bigg) v(z_1,z_2,t) \\frac{\\mathrm{d} z_1 \\mathrm{d} z_2}{s_1 s_2}.\n\\end{equation*}\nThe density function $f$ is defined on a partition of four sets of the first quadrant in the real plane, see (\\ref{pdf2D}).\nIt follows that $\\mathcal{J}$ is decomposed into four integrals as $\\mathcal{J} = \\mathcal{J}_1 + \\mathcal{J}_2 + \\mathcal{J}_3 + \\mathcal{J}_4$, where\n\\begin{align*}\n\t\\mathcal{J}_1 &= \\lambda q_1q_2\\eta_{q_1}\\eta_{q_2}s_1^{-\\eta_{q_1}}s_2^{-\\eta_{q_2}} \\int_0^{s_2} \\int_0^{s_1} z_1^{\\eta_{q_1}-1}z_2^{\\eta_{q_2}-1} v(z_1,z_2,t) \\mathrm{d} z_1 \\mathrm{d} z_2,\\\\\n\t\\mathcal{J}_2 &= \\lambda p_1q_2\\eta_{p_1}\\eta_{q_2}s_1^{\\eta_{p_1}}s_2^{-\\eta_{q_2}}  \\int_0^{s_2} \\int_{s_1}^{\\infty} z_1^{-\\eta_{p_1}-1}z_2^{\\eta_{q_2}-1} v(z_1,z_2,t) \\mathrm{d} z_1 \\mathrm{d} z_2,\\\\\n\t\\mathcal{J}_3 &= \\lambda q_1p_2\\eta_{q_1}\\eta_{p_2}s_1^{-\\eta_{q_1}}s_2^{\\eta_{p_2}}  \\int_{s_2}^{\\infty} \\int_0^{s_1} z_1^{\\eta_{q_1}-1}z_2^{-\\eta_{p_2}-1} v(z_1,z_2,t) \\mathrm{d} z_1 \\mathrm{d} z_2,\\\\\t\t\n\t\\mathcal{J}_4 &= \\lambda p_1p_2\\eta_{p_1}\\eta_{p_2}s_1^{\\eta_{p_1}}s_2^{\\eta_{p_2}}   \\int_{s_2}^{\\infty} \\int_{s_1}^{\\infty} z_1^{-\\eta_{p_1}-1}z_2^{-\\eta_{p_2}-1} v(z_1,z_2,t) \\mathrm{d} z_1 \\mathrm{d} z_2.\n\\end{align*}\nWe first consider discretization of the integral $\\mathcal{J}_1$. \nUpon writing\n\\begin{equation*}\n\t\\psi_1(s_1,s_2) = \\lambda q_1q_2\\eta_{q_1}\\eta_{q_2}s_1^{-\\eta_{q_1}}s_2^{-\\eta_{q_2}} \n\t\\quad \\textrm{and} \\quad\n\t\\varphi_1(z_1,z_2) = z_1^{\\eta_{q_1}-1}z_2^{\\eta_{q_2}-1},\n\\end{equation*}\nwe have\n\\begin{equation}\n\t\\mathcal{J}_1 = \\psi_1(s_1,s_2)\\int_0^{s_2} \\int_0^{s_1} \\varphi_1(z_1,z_2) v(z_1,z_2,t) \\mathrm{d} z_1 \\mathrm{d} z_2.\n\\end{equation}\nFor $1\\leq i \\leq m_1$, $1\\leq j \\leq m_2$ let\n\\begin{equation*}\n\t\\mathcal{J}_{1,ij} = \\psi_1(s_{1,i},s_{2,j}) \\int_0^{s_{2,j}} \\int_0^{s_{1,i}} \\varphi_1(z_1,z_2) v(z_1,z_2,t) \\mathrm{d} z_1 \\mathrm{d} z_2\n\\end{equation*}\ndenote the value of $\\mathcal{J}_1$ at the spatial grid point $(s_{1,i},s_{2,j})$.\nDefine\n\\begin{equation*}\n\t\\mathcal{G}_{1,kl} = \\int_{s_{2,l-1}}^{s_{2,l}} \\int_{s_{1,k-1}}^{s_{1,k}} \\varphi_1(z_1,z_2) v(z_1,z_2,t) \\mathrm{d} z_1 \\mathrm{d} z_2\n\\end{equation*}\nwhenever $1\\leq k \\leq m_1$, $1\\leq l \\leq m_2$.\nThen the following useful expression for $\\mathcal{J}_{1,ij}$ in terms of a double cumulative sum is obtained,\n\\begin{equation}\\label{cumsum1}\n\t\\mathcal{J}_{1,ij} = \\psi_1(s_{1,i},s_{2,j}) \\sum_{k=1}^i \\sum_{l=1}^j \\mathcal{G}_{1,kl}\n\t\\quad\n\t(1\\leq i \\leq m_1, 1\\leq j \\leq m_2).\n\\end{equation}\nNotice the obvious but important fact that the $\\mathcal{G}_{1,kl}$ are independent of the indices $i$ and $j$.\nHence, if all values $\\mathcal{G}_{1,kl}$ are given, then computing the double cumulative sums in \\eqref{cumsum1} \nfor all $i$, $j$ can be done in, to leading order, just $2m_1m_2$ additions.\n\nWe subsequently construct approximations $G_{1,kl}$ to $\\mathcal{G}_{1,kl}$ ($1\\leq k \\leq m_1$, $1\\leq l \\leq m_2$)\nand define the approximation to $\\mathcal{J}_{1,ij}$ by\n\\begin{equation}\\label{cumsum1_semi}\n\tJ_{1,ij} = \\psi_1(s_{1,i},s_{2,j}) \\sum_{k=1}^i \\sum_{l=1}^j G_{1,kl}\n\t\\quad\n\t(1\\leq i \\leq m_1, 1\\leq j \\leq m_2).\n\\end{equation}\nFor any given $k, l$ with $1\\leq k \\leq m_1$, $1\\leq l \\leq m_2$ consider the natural choice of bilinear interpolation \nto approximate $v(z_1,z_2,t)$ on the $(z_1,z_2)$-domain $[s_{1,k-1}, s_{1,k}] \\times [s_{2,l-1}, s_{2,l}]$:\n\\begin{equation*}\n\t{\\widetilde v}_{kl}(z_1, z_2,t) = \\ell_{kl}^{00}(z_1,z_2) V_{k-1,l-1}(t) + \\ell_{kl}^{10}(z_1,z_2) V_{k,l-1}(t) +\n\t\\ell_{kl}^{01}(z_1,z_2) V_{k-1,l}(t) + \\ell_{kl}^{11}(z_1,z_2) V_{k,l}(t)\n\\end{equation*}\nwith weights\n\\begin{align*}\n\t\\ell_{kl}^{00}(z_1,z_2) &= (s_{1,k}-z_1)(s_{2,l}-z_2)\/\\delta_{kl},\\\\\n\t\\ell_{kl}^{10}(z_1,z_2) &= (z_1-s_{1,k-1})(s_{2,l}-z_2)\/\\delta_{kl},\\\\\n\t\\ell_{kl}^{01}(z_1,z_2) &= (s_{1,k}-z_1)(z_{2}-s_{2,l-1})\/\\delta_{kl},\\\\\n\t\\ell_{kl}^{11}(z_1,z_2) &= (z_1-s_{1,k-1})(z_2-s_{2,l-1})\/\\delta_{kl},\n\\end{align*}\nwhere $\\delta_{kl} = \\Delta s_{1,k}\\Delta s_{2,l}$ and $\\Delta s_{1,k} = s_{1,k}-s_{1,k-1}$, $\\Delta s_{2,l} = s_{2,l}-s_{2,l-1}$.\nThen we define \n\\begin{equation*}\n\tG_{1,kl} = \\int_{s_{2,l-1}}^{s_{2,l}} \\int_{s_{1,k-1}}^{s_{1,k}} \\varphi_1(z_1,z_2) {\\widetilde v}_{kl}(z_1, z_2,t) \\mathrm{d} z_1 \\mathrm{d} z_2.\n\\end{equation*}\nA straightforward calculation yields the simple, convenient formula \n\\begin{equation}\\label{G1kl}\n\tG_{1,kl} = \n\t\\gamma_{1,kl}^{00} V_{k-1,l-1}(t) + \\gamma_{1,kl}^{10} V_{k,l-1}(t) +\n\t\\gamma_{1,kl}^{01} V_{k-1,l}(t)   + \\gamma_{1,kl}^{11} V_{k,l}(t)\n\\end{equation}\nwith\n\\begin{align*}\n\t\\gamma_{1,kl}^{00} &= (s_{1,k}s_{2,l}\\zeta_{1,kl}^{00} - s_{2,l}\\zeta_{1,kl}^{10} - s_{1,k}\\zeta_{1,kl}^{01} + \\zeta_{1,kl}^{11})\/\\delta_{kl},\\\\\n\t\\gamma_{1,kl}^{10} &= (-s_{1,k-1}s_{2,l}\\zeta_{1,kl}^{00} + s_{2,l}\\zeta_{1,kl}^{10} + s_{1,k-1}\\zeta_{1,kl}^{01} - \\zeta_{1,kl}^{11} )\/\\delta_{kl},\\\\\n\t\\gamma_{1,kl}^{01} &= (-s_{1,k}s_{2,l-1}\\zeta_{1,kl}^{00} + s_{2,l-1}\\zeta_{1,kl}^{10} + s_{1,k}\\zeta_{1,kl}^{01} - \\zeta_{1,kl}^{11} )\/\\delta_{kl},\\\\\n\t\\gamma_{1,kl}^{11} &= (s_{1,k-1}s_{2,l-1}\\zeta_{1,kl}^{00} - s_{2,l-1}\\zeta_{1,kl}^{10} - s_{1,k-1}\\zeta_{1,kl}^{01} + \\zeta_{1,kl}^{11})\/\\delta_{kl}\n\\end{align*}\nand\n\\begin{equation*}\n\t\\zeta_{1,kl}^{ab} =  \n\t\\int_{s_{2,l-1}}^{s_{2,l}} \\int_{s_{1,k-1}}^{s_{1,k}} \\varphi_1(z_1,z_2) z_1^a z_2^b \\mathrm{d} z_1 \\mathrm{d} z_2 =\n\t\\frac{\\bigg(s_{1,k}^{a+\\eta_{q_1}}-s_{1,k-1}^{a+\\eta_{q_1}}\\bigg)\\bigg(s_{2,l}^{b+\\eta_{q_2}}-s_{2,l-1}^{b+\\eta_{q_2}}\\bigg)}{\\big(a+\\eta_{q_1}\\big)\\big(b+\\eta_{q_2}\\big)}\n\\end{equation*}\nfor $a,b\\in \\{0,1\\}$.\n\nThe coefficients $\\gamma_{1,kl}^{ab}$ are completely determined by the Kou parameters and the spatial grid. \nSince they are independent of $t$, they can be computed upfront, before the time discretization.\nClearly, for any given vector $V(t)$, the computation of $G_{1,kl}$ by \\eqref{G1kl} for all $k$, $l$ requires \n$3m_1m_2$ additions and $4m_1m_2$ multiplications.\nNoticing that the values of $\\psi_1$ in \\eqref{cumsum1_semi} can also be computed upfront, it follows that \nthe number of basic arithmetic operations to compute all approximations $J_{1,ij}$ ($1\\leq i \\leq m_1$, \n$1\\leq j \\leq m_2$) by \\eqref{cumsum1_semi} is, to leading order, equal to $10m_1m_2$.\n\nThe discretization and efficient evaluation of the other three integrals is done completely analogously.\nIn the relevant derivation, $v$ is approximated by zero outside the spatial domain $[0,S_{\\rm max}]\\times [0,S_{\\rm max}]$.\nWrite\n\\begin{align*}\n\t\\psi_2(s_1,s_2) &= \\lambda p_1q_2\\eta_{p_1}\\eta_{q_2}s_1^{\\eta_{p_1}}s_2^{-\\eta_{q_2}},\\\\\n\t\\psi_3(s_1,s_2) &= \\lambda q_1p_2\\eta_{q_1}\\eta_{p_2}s_1^{-\\eta_{q_1}}s_2^{\\eta_{p_2}},\\\\ \n\t\\psi_4(s_1,s_2) &= \\lambda p_1p_2\\eta_{p_1}\\eta_{p_2}s_1^{\\eta_{p_1}}s_2^{\\eta_{p_2}}.\n\\end{align*}\nLet $1\\leq i \\leq m_1$, $1\\leq j \\leq m_2$. \nThen the approximations of $\\mathcal{J}_2, \\mathcal{J}_3, \\mathcal{J}_4$ \nat the spatial grid point $(s_{1,i},s_{2,j})$ are given by, respectively, the double cumulative sums\n\\begin{align*}\n\tJ_{2,ij} &= \\psi_2(s_{1,i},s_{2,j}) \\sum_{k=i+1}^{m_1} \\sum_{l=1}^j G_{2,kl},\\\\\n\tJ_{3,ij} &= \\psi_3(s_{1,i},s_{2,j}) \\sum_{k=1}^{i}     \\sum_{l=j+1}^{m_2} G_{3,kl},\\\\\n\tJ_{4,ij} &= \\psi_4(s_{1,i},s_{2,j}) \\sum_{k=i+1}^{m_1} \\sum_{l=j+1}^{m_2} G_{4,kl}\n\\end{align*}\nwith the usual convention that empty sums are equal to zero.\nFor $2\\le \\nu \\le 4$, $1\\leq k \\leq m_1$, $1\\leq l \\leq m_2$ we obtain\n\\begin{equation*}\\label{Gnukl}\n\tG_{\\nu,kl} = \n\t\\gamma_{\\nu,kl}^{00} V_{k-1,l-1}(t) + \\gamma_{\\nu,kl}^{10} V_{k,l-1}(t) +\n\t\\gamma_{\\nu,kl}^{01} V_{k-1,l}(t)   + \\gamma_{\\nu,kl}^{11} V_{k,l}(t)\n\\end{equation*}\nwith\n\\begin{align*}\n\t\\gamma_{\\nu,kl}^{00} &= (s_{1,k}s_{2,l}\\zeta_{\\nu,kl}^{00} - s_{2,l}\\zeta_{\\nu,kl}^{10} - s_{1,k}\\zeta_{\\nu,kl}^{01} + \\zeta_{\\nu,kl}^{11})\/\\delta_{kl},\\\\\n\t\\gamma_{\\nu,kl}^{10} &= (-s_{1,k-1}s_{2,l}\\zeta_{\\nu,kl}^{00} + s_{2,l}\\zeta_{\\nu,kl}^{10} + s_{1,k-1}\\zeta_{\\nu,kl}^{01} - \\zeta_{\\nu,kl}^{11} )\/\\delta_{kl},\\\\\n\t\\gamma_{\\nu,kl}^{01} &= (-s_{1,k}s_{2,l-1}\\zeta_{\\nu,kl}^{00} + s_{2,l-1}\\zeta_{\\nu,kl}^{10} + s_{1,k}\\zeta_{\\nu,kl}^{01} - \\zeta_{\\nu,kl}^{11} )\/\\delta_{kl},\\\\\n\t\\gamma_{\\nu,kl}^{11} &= (s_{1,k-1}s_{2,l-1}\\zeta_{\\nu,kl}^{00} - s_{2,l-1}\\zeta_{\\nu,kl}^{10} - s_{1,k-1}\\zeta_{\\nu,kl}^{01} + \\zeta_{\\nu,kl}^{11})\/\\delta_{kl}\n\\end{align*}\nwhere\n\\begin{align*}\n\t\\zeta_{2,kl}^{ab} &=  \n\t\\frac{\\bigg(s_{1,k}^{a-\\eta_{p_1}}-s_{1,k-1}^{a-\\eta_{p_1}}\\bigg)\\bigg(s_{2,l}^{b+\\eta_{q_2}}-s_{2,l-1}^{b+\\eta_{q_2}}\\bigg)}{\\big(a-\\eta_{p_1}\\big)\\big(b+\\eta_{q_2}\\big)},\\\\\n\t\\zeta_{3,kl}^{ab} &=  \n\t\\frac{\\bigg(s_{1,k}^{a+\\eta_{q_1}}-s_{1,k-1}^{a+\\eta_{q_1}}\\bigg)\\bigg(s_{2,l}^{b-\\eta_{p_2}}-s_{2,l-1}^{b-\\eta_{p_2}}\\bigg)}{\\big(a+\\eta_{q_1}\\big)\\big(b-\\eta_{p_2}\\big)},\\\\\n\t\\zeta_{4,kl}^{ab} &=  \n\t\\frac{\\bigg(s_{1,k}^{a-\\eta_{p_1}}-s_{1,k-1}^{a-\\eta_{p_1}}\\bigg)\\bigg(s_{2,l}^{b-\\eta_{p_2}}-s_{2,l-1}^{b-\\eta_{p_2}}\\bigg)}{\\big(a-\\eta_{p_1}\\big)\\big(b-\\eta_{p_2}\\big)}\n\\end{align*}\nfor $a,b\\in \\{0,1\\}$.\n\nOn the boundary part $\\{(s_1,0): s_1>0\\}$ the double integral \\eqref{integral} reduces to the single integral\n\\begin{equation}\\label{integral1_1D}\n\t\\mathcal{J} = \\lambda\\int_0^{\\infty} f_1(y_1)v(s_1y_1,0,t) \\mathrm{d} y_1.\n\\end{equation}\nThe discretization and efficient evaluation of this integral, which arises in the one-dimensional \nKou PIDE, has been constructed by Toivanen~\\cite{T08}.\nFor completeness, we include the result here.\nAssume $s_1>0$. \nThen for \\eqref{integral1_1D} there holds $\\mathcal{J} = \\mathcal{J}_1 + \\mathcal{J}_2$, where\n\\begin{equation*}\n\t\\mathcal{J}_1 = \\lambda \\int_0^{s_1} f_1\\bigg(\\frac{z_1}{s_1}\\bigg) v(z_1,0,t) \\frac{\\mathrm{d} z_1 }{s_1}\n\t\\quad \\textrm{and} \\quad\n\t\\mathcal{J}_2 = \\lambda \\int_{s_1}^{\\infty} f_1\\bigg(\\frac{z_1}{s_1}\\bigg) v(z_1,0,t) \\frac{\\mathrm{d} z_1 }{s_1}.\n\\end{equation*}\nWrite\n\\begin{equation*}\n\t\\psi_1(s_1) = \\lambda q_1\\eta_{q_1}s_1^{-\\eta_{q_1}} \n\t\\quad \\textrm{and} \\quad\n\t\\psi_2(s_1) = \\lambda p_1\\eta_{p_1}s_1^{\\eta_{p_1}}.\n\\end{equation*}\nLet $1\\leq i \\leq m_1$. \nThen the approximations of $\\mathcal{J}_1, \\mathcal{J}_2$ at the spatial grid point $(s_{1,i},0)$ are given by, \nrespectively, the single cumulative sums\n\\begin{equation*}\n\tJ_{1,i} = \\psi_1(s_{1,i}) \\sum_{k=1}^{i} G_{1,k}\n\t\\quad \\textrm{and} \\quad\n\tJ_{2,i} = \\psi_2(s_{1,i}) \\sum_{k=i+1}^{m_1} G_{2,k}.\n\\end{equation*}\nFor $1\\le \\nu \\le 2$, $1\\leq k \\leq m_1$ we obtain, by employing linear interpolation,\n\\begin{equation*}\\label{Gnuk}\n\tG_{\\nu,k} = \n\t\\gamma_{\\nu,k}^{0} V_{k-1,0}(t) + \\gamma_{\\nu,k}^{1} V_{k,0}(t)\n\\end{equation*}\nwith\n\\begin{equation*}\n\t\\gamma_{\\nu,k}^{0} = (s_{1,k}\\zeta_{\\nu,k}^{0} - \\zeta_{\\nu,k}^{1})\/\\Delta s_{1,k}\n\t\\quad \\textrm{and} \\quad\n\t\\gamma_{\\nu,k}^{1} = (-s_{1,k-1}\\zeta_{\\nu,k}^{0} + \\zeta_{\\nu,k}^{1} )\/\\Delta s_{1,k}\n\\end{equation*}\nwhere\n\\begin{equation*}\n\t\\zeta_{1,k}^{a} =  \n\t\\frac{s_{1,k}^{a+\\eta_{q_1}}-s_{1,k-1}^{a+\\eta_{q_1}}}{a+\\eta_{q_1}}\n\t\\quad \\textrm{and} \\quad\n\t\\zeta_{2,k}^{a} =  \n\t\\frac{s_{1,k}^{a-\\eta_{p_1}}-s_{1,k-1}^{a-\\eta_{p_1}}}{a-\\eta_{p_1}}\n\t\\quad (a=0,1).\n\\end{equation*}\n\nOn the boundary part $\\{(0,s_2): s_2>0\\}$ the double integral \\eqref{integral} reduces \nto the single integral\n\\begin{equation}\\label{integral2_1D}\n\t\\mathcal{J} = \\lambda\\int_0^{\\infty} f_2(y_2)v(0,s_2y_2,t) \\mathrm{d} y_2\n\\end{equation}\nand the discretization is performed completely similarly as above. \nFinally, at the spatial grid point $(s_1,s_2)=(0,0)$ it holds that \n$\\mathcal{J} \\approx  \\lambda V_{0,0}(t)$.\n\nFrom the above it follows that the number of basic arithmetic operations to compute, for any given $t$, the approximation to the double \nintegral \\eqref{integral} on the full spatial grid is, to leading order, equal to $40m_1m_2$.\n\n\t\n\n\n\n\t\n\\subsection{Semidiscrete PIDE}\\label{SecSemiPIDE}\nCombining the above semidiscretizations for the convection-diffusion-reaction and integral parts of PIDE \\eqref{PIDE2D}, the following \nlarge system of ODEs is obtained:\n\\begin{align}\\label{ODEs}\n\tV'(t) = AV(t) \\quad (0<t\\leq T),\n\\end{align}\nwhere \n\\begin{equation*}\n\tA = A^{(D)} + A^{(J)} = A^{(M)} + A_1 + A_2 + A^{(J)}.\n\\end{equation*}\nEven though we never actually compute it in this way, the approximation to the double integral \\eqref{integral} \ndefined in Subsection~\\ref{SecIntpart} is formally denoted here by $A^{(J)} V(t)$ with given (full) matrix $A^{(J)}$.\\\\\n\nThe initial vector $V(0)=V^0$ is defined by pointwise evaluation of the payoff function $\\phi$ on the spatial grid, \nexcept at those grid points that lie near to the line segment given by $s_1+s_2 = 2K$ where $\\phi$ is nonsmooth.\nIt is well-known that using pointwise values of $\\phi$ in such a region can lead to a deteriorated (spatial) \nconvergence behaviour, which can be alleviated by applying cell averaging. \nAccordingly, for $k=1,2$ let\n\\begin{eqnarray*}\n\t&s_{k,l+1\/2} = \\tfrac{1}{2}(s_{k,l}+s_{k,l+1})~~~~ &{\\rm if}~~ 0\\le l < m,\\\\\n\t&h_{k,l+1\/2} = s_{k,l+1\/2}-s_{k,l-1\/2} &{\\rm if}~~ 0\\le l \\le m,\n\\end{eqnarray*}\nwith $s_{k,-1\/2} = -s_{k,1\/2}$ and $s_{k,m+1\/2} = S_{\\rm max}$.\nThen we define \\cite{H17book}\n\\begin{equation}\\label{cellave}\n\tV_{i,j}(0) = \\frac{1}{h_{1,i+1\/2}h_{2,j+1\/2}}\\int_{s_{1,i-1\/2}}^{s_{1,i+1\/2}}\\int_{s_{2,j-1\/2}}^{s_{2,j+1\/2}} \\phi(s_1,s_2) \\mathrm{d} s_2 \\mathrm{d} s_1\n\\end{equation}\nwhenever the cell \n\\[\n[s_{1,i-1\/2}, s_{1,i+1\/2})\\times[s_{2,j-1\/2}, s_{2,j+1\/2})\n\\]\nhas a nonempty intersection with the line segment $s_1+s_2 = 2K$. \nNote that the double integral \\eqref{cellave} is easily calculated.\n\n\n\n\\section{Temporal discretization}\\label{SecTime}\nFor the temporal discretization of the semidiscrete PIDE \\eqref{ODEs} we study seven operator splitting schemes of the IMEX and ADI \nkind~\\cite{BH21}.\nLet integer $N\\ge 1$ be given and define step size $\\Delta t = T\/N$ and temporal grid points $t_n = n \\Delta t$ for $n=0,1,2,\\ldots,N$.\nEach of the following schemes generates an approximation $V^{n}$ to $V(t_n)$ successively for $n=1,2,\\ldots,N$.\nHere the integral part of \\eqref{ODEs}, corresponding to the matrix $A^{(J)}$, is always conveniently treated in an explicit manner\nand evaluated by applying the efficient algorithm derived in Subsection~\\ref{SecIntpart}.\n\n\\begin{enumerate}\n\t\n\t\\item \\textit{Crank--Nicolson Forward Euler (CNFE) scheme}:\n\t\\vskip0.1cm\n\tIn this one-step IMEX method, the convection-diffusion-reaction part is handled implicitly by the Crank--Nicolson scheme and the \n\tintegral part is treated in a simple, explicit Euler fashion:\n\t\\begin{align}\\label{CNFE}\n\t\t\\left(I-\\tfrac{1}{2}\\Delta t A^{(D)}\\right)V^{n} = \\left(I + \\tfrac{1}{2}\\Delta t A^{(D)}\\right)V^{n-1} + \\Delta t A^{(J)} V^{n-1}.\n\t\\end{align}\n\tDue to the application of explicit Euler, the order\\footnote{This means the classical order of consistency, i.e., for fixed \n\t\tnonstiff ODEs.} of the CNFE scheme is just equal to one. \n\t\\vskip0.2cm\n\t\n\t\\item \\textit{Crank--Nicolson scheme with fixed-point iteration (CNFI)}:\n\t\\vskip0.1cm\n\tTavella \\& Randall \\cite{TR00book} and d'Halluin, Forsyth \\& Vetzal \\cite{HFV05} proposed to combine the Crank--Nicolson scheme for \n\tthe convection-diffusion-reaction part with a fixed-point iteration on the integral part:\n\t\\begin{align}\\label{CNFI}\n\t\t\\left(I-\\tfrac{1}{2}\\Delta t A^{(D)}\\right)Y_k = \\left(I + \\tfrac{1}{2}\\Delta t A^{(D)}\\right)V^{n-1} +  \\tfrac{1}{2} \\Delta t A^{(J)} (Y_{k-1}+V^{n-1})\n\t\\end{align}\n\tfor $k=1,2,\\ldots,l$ and $V^n = Y_l$ with starting value $Y_0 = V^{n-1}$. \n\tNumerical evidence in \\cite{CF08,HFV05} indicates that in general $l=2$ or $l=3$ iterations suffice.\n\tWe shall consider in this paper a fixed number of $l=2$ iterations.\n\tThen method \\eqref{CNFI} can be viewed as a one-step IMEX scheme, where the integral part is treated by the explicit trapezoidal rule, \n\talso called the modified Euler method. \n\tBy Taylor expansion, it can be seen that its order is equal to two.\n\t\n\t\\item \\textit{Implicit-Explicit Trapezoidal Rule (IETR)}:\n\t\\vskip0.1cm\n\tAnother blend of the implicit trapezoidal rule (Crank--Nicolson) for the convection-diffusion-reaction part and the explicit \n\ttrapezoidal rule for the integral part has been considered in~\\cite{H17book}:\n\t\\begin{equation}\\label{IETR}\n\t\t\\left\\{\\begin{array}{lll}\n\t\t\tY_0 = V^{n-1} + \\Delta t\\, (A^{(D)}+A^{(J)})V^{n-1},\\\\\n\t\t\t\\widehat{Y}_0 = Y_0+\\tfrac{1}{2} \\Delta t A^{(J)} \\big(Y_0-V^{n-1}\\big),\\\\\n\t\t\tY_1 = \\widehat{Y}_0+\\tfrac{1}{2} \\Delta t A^{(D)} \\big(Y_1-V^{n-1}\\big),\\\\\n\t\t\tV^n = Y_1.\n\t\t\\end{array}\\right.\n\t\\end{equation} \n\tMethod~\\eqref{IETR} also forms a one-step IMEX scheme with order equal to two.  \n\tIt is already well-known in the literature on the numerical solution of PDEs (without integral part), see e.g.~Hundsdorfer \\& \n\tVerwer~\\cite{HV03book}.\n\tNotice that the first two internal stages $Y_0$, $\\widehat{Y}_0$ are explicit, whereas the third internal stage $Y_1$ is implicit.\n\t\\vskip0.2cm \n\t\n\t\\item \\textit{Crank--Nicolson Adams--Bashforth (CNAB) scheme}:\n\t\\vskip0.1cm\n\tThe CNAB method has been amply considered in the literature on the numerical solution of PDEs as well, \n\tsee e.g.~\\cite{FHV97,HV03book,H02}.\n\tIt has been investigated for the numerical solution of PIDEs by Salmi \\& Toivanen \\cite{ST14} and Salmi, Toivanen \\& von Sydow \n\t\\cite{STS14}. \n\tThe convection-diffusion-reaction part is again handled by the Crank--Nicolson scheme, but the integral part is now dealt with \n\tin a two-step Adams--Bashforth fashion:\n\t\\begin{align}\\label{CNAB}\n\t\t\\left(I-\\tfrac{1}{2} \\Delta t A^{(D)}\\right)V^{n} = \\left(I+\\tfrac{1}{2} \\Delta t A^{(D)}\\right)V^{n-1} + \\tfrac{1}{2}\\Delta t A^{(J)}(3V^{n-1}-V^{n-2}).\n\t\\end{align}\n\tMethod \\eqref{CNAB} constitutes a two-step IMEX scheme and is of order equal to two. \n\tIt is interesting to remark that \\eqref{CNAB} can also be viewed as obtained from \\eqref{CNFI}, upon taking $l=1$ and \n\timproved (linearly extrapolated) starting value $Y_0 = 2V^{n-1}-V^{n-2}$.\n\t\\vskip0.2cm \n\t\n\\end{enumerate}\n\nAs for the spatial discretization, it is well-known that the nonsmoothness of the payoff function can have an unfavourable \nimpact on the convergence behaviour of the temporal discretization, even if cell averaging is applied, cf.~\\cite{H17book}.\nFor the Crank--Nicolson scheme a common approach to alleviate this has been proposed by Rannacher~\\cite{R84}. \nIt consists of replacing (only) the first time step by two time steps with step size $\\Delta t\/2$ using the implicit Euler method, \nthus defining the approximation $V^1$ to $V(t_1)$.\nIn the present case of two-dimensional PIDEs, the implicit Euler method can be computationally intensive.\nWe therefore consider, for all four IMEX schemes above, the {\\it IMEX Euler scheme} as the starting method:\n\\begin{align*}    \n\t\\left(I-\\tfrac{1}{2}\\Delta t A^{(D)}\\right)V^{\\tfrac{1}{2}} &= V^{0} + \\tfrac{1}{2}\\Delta t A^{(J)} V^{0},\\\\\n\t\\left(I-\\tfrac{1}{2}\\Delta t A^{(D)}\\right)V^{1} &= V^{\\tfrac{1}{2} } + \\tfrac{1}{2}\\Delta t A^{(J)} V^{\\tfrac{1}{2}}.\n\\end{align*} \n\nClearly, in each of the IMEX schemes under consideration, linear systems need to be solved involving the matrix \n$I-\\tfrac{1}{2}\\Delta t A^{(D)}$. \nWhereas this matrix is sparse, with at most nine nonzero entries per row, it possesses a large bandwidth that is directly \nproportional to $m_1$.\nTo obtain a further reduction in computational work, we consider three recent operator splitting schemes for PIDEs that employ \nalso the (directional) splitting $A^{(D)} = A^{(M)} + A_1 + A_2$ of the two-dimensional convection-diffusion-reaction part, such \nthat only linear systems with tridiagonal matrices arise.\n\n\\begin{enumerate} \n\t\n\t\\item[5.] \\textit{One-step adaptation of the modified Craig--Sneyd (MCS) scheme}:\n\t\\vskip0.1cm\n\tThe MCS scheme is an ADI scheme that was introduced by in 't Hout \\& Welfert \\cite{HW09} for the numerical solution of \n\tPDEs containing mixed spatial derivative terms.\n\tThe following, direct adaptation to PIDEs has recently been investigated by in 't Hout \\& Toivanen \\cite{HT18}:\n\t\\begin{equation}\\label{MCS}\n\t\t\\left\\{\\begin{array}{lll}\n\t\t\tY_0 = V^{n-1} + \\Delta t\\, (A^{(D)}+A^{(J)})V^{n-1},\\\\\n\t\t\tY_j = Y_{j-1} + \\theta\\Delta t A_j(Y_j-V^{n-1}) \\quad (j=1,2),\\\\\n\t\t\t\\widehat{Y}_0 = Y_0 + \\theta \\Delta t\\, (A^{(M)}+A^{(J)})(Y_2 - V^{n-1}),\\\\\n\t\t\t\\widetilde{Y}_0 = \\widehat{Y}_0 + (\\tfrac{1}{2}-\\theta)\\Delta t\\, (A^{(D)}+A^{(J)})(Y_2-V^{n-1}),\\\\\n\t\t\t\\widetilde{Y}_j = \\widetilde{Y}_{j-1} + \\theta \\Delta t A_j(\\widetilde{Y}_j - V^{n-1}) \\quad (j=1,2),\\\\\n\t\t\tV^n = \\widetilde{Y}_2,\n\t\t\\end{array}\\right.\n\t\\end{equation}   \n\twhere $\\theta>0$ is a given parameter.\n\tMethod \\eqref{MCS} is of order two for any value~$\\theta$.\n\tHere we make the common choice $\\theta = \\frac{1}{3}$, which is motivated by stability and accuracy results \n\tfor two-dimensional problems, see e.g.~\\cite{HM11,HT18,HW09,HW16}.\n\tIt is easily verified that in \\eqref{MCS} the integral part and mixed derivative term are both treated by the \n\texplicit trapezoidal rule.\n\tWe note that the explicit stages $\\widehat{Y}_0$, $\\widetilde{Y}_0$ can be merged, so that the integral part \n\tis evaluated (just) twice per time step.\n\tThe implicit stages $Y_j$, $\\widetilde{Y}_j$ (for $j=1,2$) are often called stabilizing corrections.\n\tThe four pertinent linear systems for these stages are tridiagonal and can be solved very efficiently \n\tby means of an a priori $LU$ factorization.\n\tWe mention that \\eqref{MCS} is already applied in the first time step, i.e., for defining $V^1$ (thus \n\tIMEX Euler is not used here).\n\t\\vskip0.2cm \n\t\n\t\\item[6.] \\textit{Two-step adaptation of the MCS (MCS2) scheme}:\n\t\\vskip0.1cm\n\tIn \\cite{HT18} a second adaptation of the MCS scheme to PIDEs has been proposed where, instead of the explicit \n\ttrapezoidal rule, the integral part is now handled in a two-step Adams--Bashforth fashion:\n\t\\begin{equation}\\label{MCS2}\n\t\t\\left\\{\\begin{array}{lll}\n\t\t\tX_0 = V^{n-1}+\\Delta t A^{(D)}V^{n-1},\\\\\n\t\t\tY_0 = X_0 + \\tfrac{1}{2}\\Delta t A^{(J)}(3V^{n-1}-V^{n-2}),\\\\\n\t\t\tY_j = Y_{j-1}+\\theta\\Delta t A_j(Y_j-V^{n-1})\\quad (j=1,2),\\\\\n\t\t\t\\widehat{Y}_0 = Y_0 + \\theta\\Delta t A^{(M)}(Y_2-V^{n-1}),\\\\\n\t\t\t\\widetilde{Y}_0 = \\widehat{Y}_0 + (\\tfrac{1}{2}-\\theta)\\Delta t A^{(D)}(Y_2-V^{n-1}),\\\\\n\t\t\t\\widetilde{Y}_j = \\widetilde{Y}_{j-1} + \\theta \\Delta t A_j(\\widetilde{Y}_j-V^{n-1})\\quad (j=1,2),\\\\\n\t\t\tV^n = \\widetilde{Y}_2.\n\t\t\\end{array}\\right.\n\t\\end{equation}   \n\tMethod \\eqref{MCS2} is also of order two for any value~$\\theta$.\n\tWe choose again $\\theta = \\frac{1}{3}$ and, for starting this two-step method, define $V^1$ by \\eqref{MCS}.\n\t\\vskip0.2cm \n\t\n\t\\item[7.] \\textit{Stabilizing correction two-step Adams-type (SC2A) scheme}:\n\t\\vskip0.1cm\n\tHundsdorfer \\& in 't Hout \\cite{HH18} recently studied a novel class of stabilizing correction multistep methods \n\tfor the numerical solution of PDEs.\n\tWe consider here a direct adaptation to PIDEs of a prominent member of this class, the two-step Adams-type scheme \n\tcalled SC2A:\n\t\\begin{equation}\\label{SC2A}\n\t\t\\left\\{\\begin{array}{lll}   \n\t\t\tY_0 = V^{n-1} + \\Delta t\\, (A^{(M)}+A^{(J)})\\sum_{i=1}^2\\widehat{b}_iV^{n-i} + \\Delta t\\, (A_1+A_2)\\sum_{i=1}^2\\widecheck{b}_iV^{n-i},\\\\\n\t\t\tY_j = Y_{j-1} + \\theta \\Delta t A_j(Y_j-V^{n-1})\\quad (j=1,2),\\\\\n\t\t\tV^n = Y_2,\n\t\t\\end{array}\\right.\n\t\\end{equation}   \n\twith coefficients $(\\widehat{b}_1,\\widehat{b}_2) = \\left(\\frac{3}{2},-\\frac{1}{2}\\right)$ and \n\t$(\\widecheck{b}_1,\\widecheck{b}_2) = \\left(\\frac{3}{2}-\\theta, - \\frac{1}{2}+\\theta\\right)$.\n\tThe integral part and mixed derivative term are now both handled by the two-step Adams--Bashforth scheme.\n\tMethod \\eqref{SC2A} is again of order two for any value~$\\theta$. \n\tFollowing \\cite{HH18}, we take $\\theta = \\frac{3}{4}$, which is based upon stability and accuracy results.\n\tFor starting \\eqref{SC2A}, we define $V^1$ again by \\eqref{MCS} with $\\theta = \\frac{1}{3}$\n\t\\vskip0.2cm    \n\t\n\\end{enumerate}\n\n\n\n\n\n\t\n\\section{Numerical study}\\label{SecResults}\nIn this section we present ample numerical experiments for the seven operator splitting schemes formulated in Section~\\ref{SecTime}, \nwhich provides important insight in their convergence behaviour and mutual performance.\nThe experiments involve three different parameter sets for the two-asset Kou model, labelled as 1, 2, and~3. \nThe parameter set~1 is taken from Clift \\& Forsyth \\cite{CF08}. \nSet~2 is a blend of parameters considered for the one-asset Kou model by Almendral \\& Oosterlee~\\cite{AO05} and d'Halluin, Forsyth \n\\& Vetzal~\\cite{HFV05} (see also Toivanen~\\cite{T08}). \nSet~3 has been newly constructed and includes a relatively large value for the product $\\lambda T$, i.e., the expected number of \njumps in $[0,T]$. \nFor the truncated spatial domain, we (heuristically) select $S_{\\rm max} = 20K, 10K, 30K$ for sets 1, 2, 3, respectively.\n\n\\begin{table}[H]\n\t\\caption{Parameter sets for the two-asset Kou jump-diffusion model.}\n\t\\small\n\t\\begin{tabular}{@{}llllllllllllll@{}}\n\t\t\\toprule\n\t\t& $\\sigma_1$ & $\\sigma_2$ & $r$ & $\\rho$ & $\\lambda$ & $p_1$ & $p_2$ & $\\eta_{p_1}$ & $\\eta_{q_1}$ & $\\eta_{p_2}$ & $\\eta_{q_2}$ & $K$ & $T$ \\\\ \\midrule\n\t\tSet 1 & 0.12 & 0.15 & 0.05 & 0.30 & 0.50 & 0.40 & 0.60 & $1\/0.20$ & $1\/0.15$ & $1\/0.18$ & $1\/0.14$ & 100 & 1 \\\\\n\t\tSet 2 & 0.15 & 0.20 & 0.05 & 0.50 & 0.20 & 0.3445 & 0.50 & 3.0465 & 3.0775 & 3 & 2 & 100 & 0.2 \\\\\n\t\tSet 3 & 0.20 & 0.30 & 0.05 & 0.70 & 8 & 0.60 & 0.65 & 5 & 4 & 4 & 3 & 100 & 1 \\\\ \\bottomrule\n\t\\end{tabular}\n\t\\label{paramsets}\n\\end{table}\n\n\n\\subsection{Option values and numerical convergence behaviour}\\label{SecResults1}\nFigure~\\ref{FigSurfacePlots} shows the numerically computed\\footnote{With $m_1=m_2=200$, $N=100$ and the MCS2 scheme.} option value \nsurfaces for the European put-on-the-average option and the three parameter sets from Table~\\ref{paramsets} on the asset price \ndomain $[0,3K]\\times[0,3K]$ and $t=T$.\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[trim = 1.4in 3.7in 1.3in 3.7in, clip, scale=0.5]{surf1.pdf}\t\n\t\\includegraphics[trim = 1.4in 3.7in 1.3in 3.7in, clip, scale=0.5]{surf2.pdf}\t\n\t\\includegraphics[trim = 1.3in 3.7in 1.3in 3.7in, clip, scale=0.5]{surf3.pdf}\n\t\\caption{Option value surfaces for the European put-on-the-average option under the two-asset Kou model in the case of parameter \n\tset 1 (top left), 2 (top right) and 3 (bottom) given in Table~\\ref{paramsets}.}\n\t\\label{FigSurfacePlots}\n\\end{figure}\n\n\\noindent\nFor future reference, accurate approximations to the option values have been \ncomputed\\footnote{Using $m_1=m_2=1000$, $N=500$, the MCS2 scheme and spline interpolation.} \nfor specifically chosen spot prices $S_0^{(1)}$ and $S_0^{(2)}$ of the two assets in a\nneighbourhood of the strike $K$, see Table~\\ref{tablevalues}. \n\\vskip0.3cm\n\n\\begin{table}[H]\n\t\\caption{\\label{tablevalues}European put-on-the-average option value approximations for parameter sets 1, 2 and 3.}\n\t\\vspace{0.5cm}\n\t\\centering\n\tSet 1\\\\\\vspace{0.1cm}\n\t\\begin{tabular}{llrrrrr}\n\t\t\\toprule\n\t\t&  & $S_0^{(1)} = 90$ &  & $S_0^{(1)} = 100$ &  & $S_0^{(1)} = 110$ \\\\\n\t\t&  & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} \\\\\n\t\t$S_0^{(2)} = 90$  &  & 8.9385 &  & 6.0316 &  & 3.8757 \\\\\n\t\t$S_0^{(2)} = 100$ &  & 5.9655 &  & 3.8038 &  & 2.3370 \\\\\n\t\t$S_0^{(2)} = 110$ &  & 3.7641 &  & 2.2978 &  & 1.3771 \\\\ \n\t\t\\bottomrule\n\t\\end{tabular}\\vspace{0.5cm}\n\n\tSet 2\\\\\\vspace{0.1cm}\n\t\\begin{tabular}{llrrrrr}\n\t\t\\toprule\n\t\t&  & $S_0^{(1)} = 90$ &  & $S_0^{(1)} = 100$ &  & $S_0^{(1)} = 110$ \\\\\n\t\t&  & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} \\\\\n\t\t$S_0^{(2)} = 90$  &  & 9.6863 &  & 5.5616 &  & 2.6400 \\\\\n\t\t$S_0^{(2)} = 100$ &  & 5.6162 &  & 2.6929 &  & 1.1264 \\\\\n\t\t$S_0^{(2)} = 110$ &  & 2.7570 &  & 1.1670 &  & 0.5246 \\\\ \n\t\t\\bottomrule\n\t\\end{tabular}\\vspace{0.5cm}\n\n\tSet 3\\\\\\vspace{0.1cm}\n\t\\begin{tabular}{llrrrrr}\n\t\t\\toprule\n\t\t&  & $S_0^{(1)} = 90$ &  & $S_0^{(1)} = 100$ &  & $S_0^{(1)} = 110$ \\\\\n\t\t&  & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} \\\\\n\t\t$S_0^{(2)} = 90$  &  & 32.7459 &  & 31.0984 &  & 29.5758 \\\\\n\t\t$S_0^{(2)} = 100$ &  & 30.5796 &  & 29.0181 &  & 27.5770 \\\\\n\t\t$S_0^{(2)} = 110$ &  & 28.5830 &  & 27.1033 &  & 25.7396 \\\\ \n\t\t\\bottomrule\n\t\\end{tabular}\\vspace{0.5cm}\n\\end{table}\n\nWe next examine, through numerical experiments, the convergence behaviour of the seven operator splitting schemes formulated \nin Section~\\ref{SecTime} for the temporal discretization of the semidiscrete two-dimensional Kou PIDE. \nTake $m_1=m_2=m$.\nConsider a region of financial interest\n\\begin{equation*}\\label{ROI}\n\t\\textrm{ROI} = \\left\\{(s_1,s_2) : \\tfrac{1}{2}K < s_1,s_2 < \\tfrac{3}{2}K\\right\\}\n\\end{equation*}\nand define the {\\it temporal discretization error} on this ROI for $t=T$ by\n\\begin{equation}\\label{EROI}\n\t\\widehat{E}^{\\rm ROI}(m,N) = \\textrm{max} \\left\\{| V_{i,j}(T) - V_{i,j}^{N'} |: \\Delta t = T\/N' ~~{\\rm and}~~ \n\t(s_{1,i},s_{2,j})\\in \\rm ROI\\right\\}.\n\\end{equation}\nClearly, this error is measured in the (important) maximum norm.\nThe MCS2 scheme is applied with $3000$ time steps to obtain a reference value for the exact semidiscrete solution $V(T)$ to \n\\eqref{ODEs}.\nFor each of the seven operator splitting schemes formulated in Section~\\ref{SecTime}, we study the temporal discretization \nerror for a sequence of values $N$, where the approximation $V^{N'}$ to $V(T)$ is computed using $N'$ time steps.\nHere $N'$ is chosen in function of $N$ depending on the specific scheme, so as to arrive at a fair comparison between the \nschemes.\n\nTheoretical as well as experimental evidence indicates that for the four IMEX schemes the process of solving the linear systems, \ninvolving the matrix $I-\\tfrac{1}{2}\\Delta t A^{(D)}$, essentially determines the total computational time in each time step \nwhenever $m$ is sufficiently large ($m\\gtrsim 150$). \nIn particular, the time for evaluating the integral part, by means of the algorithm from Subsection~\\ref{SecIntpart}, forms \nonly a small fraction of the total computational time.\nAccordingly, for a fair comparison between the IMEX schemes, we apply the three schemes (\\ref{CNFE}), (\\ref{IETR}), (\\ref{CNAB}) \nwith $N' = 2N$ time steps and the scheme (\\ref{CNFI}) with just $N'=N$ time steps, since in the latter scheme double the amount \nof linear systems is solved in each time step.\n\nFor the three ADI schemes we find that the evaluation of the two-dimensional integral part takes about the same computational\ntime as the solution of four pertinent tridiagonal linear systems. \nThe scheme (\\ref{MCS}) employs two evaluations of the integral part, whereas (\\ref{MCS2}) and (\\ref{SC2A}) each use only one \nsuch evaluation.\nNext, both schemes (\\ref{MCS}) and (\\ref{MCS2}) require the solution of four tridiagonal systems, whereas (\\ref{SC2A}) involves \nonly two tridiagonal systems.\nAccordingly, for a fair mutual comparison, we apply the ADI schemes (\\ref{MCS}), (\\ref{MCS2}) and (\\ref{SC2A}) with, respectively, \n$N'=N$, $N' = \\lfloor 3N\/2 \\rfloor$ and $N' = 2N$ time steps.\n\nFor each of the seven splitting schemes, the temporal discretization errors $\\widehat{E}^{\\rm ROI}(m,N)$ have been computed for \n$m = 200$ and a sequence of values $N$ between 10 and 1000. \nFigure~\\ref{FigTemperrors} displays the obtained temporal errors for all three parameter sets given in Table~\\ref{paramsets}, \nwhere the left column shows the results for the IMEX schemes and the right column the results for the ADI schemes.\n\nFrom Figure~\\ref{FigTemperrors} we observe the positive result that, for each given scheme and parameter set, the temporal errors \nare bounded from above by a moderate value and decrease monotonically as $N$ increases. \nAs expected, the CNFE scheme (\\ref{CNFE}) shows an order of convergence equal to one, whereas all other six splitting schemes \nreveal a desirable order of convergence equal to two.\nBy repeating the numerical experiments for spatial grids that are finer (e.g.~$m=400$) or coarser (e.g.~$m=50, 100$) \nwe obtain results that are visually identical to those displayed in Figure~\\ref{FigTemperrors}. \nThis indicates that, for each splitting scheme, the temporal errors are essentially independent of the spatial grid \nsize, and hence, its observed convergence order is valid in a stiff sense, which is very favourable.\n\nIt is interesting to remark that the error constants become larger as the intensity of the jumps increases, keeping all\nelse fixed.\nThis has been observed in additional numerical experiments for increasing values of $\\lambda$ between 0 and 10.\n\nAmong the four IMEX schemes, the IETR and CNAB schemes yield the smallest error constants in the numerical experiments.\nAmong the three ADI schemes, the MCS2 scheme has the smallest error constant in all experiments.\nOverall, the MCS2 scheme is preferred.\nIt yields temporal errors that are smaller than or approximately equal to the IETR and CNAB schemes, but is computationally \nmuch faster since the linear systems appearing in this scheme are just tridiagonal.\nAs an indication for actual run times of the MCS2 scheme, we mention that they are approximately 6e-4, 2e-3, 7e-3, \n4.5e-2 CPU seconds per time step if $m=50,100,200,400$, respectively, in the case of our Matlab implementation, \nversion R2020b, on an Intel Core i7-8665U processor at 1.9 GHz with 16 GB memory.\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{TemperrorIMEX_jul8_set1.pdf}\\includegraphics[trim = 1.3in 3.7in 1.6in 3.7in, clip, scale=0.5]{TemperrorADI_jul8_set1.pdf}\\\\\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{TemperrorIMEX_jul8_set2.pdf}\\includegraphics[trim = 1.3in 3.7in 1.6in 3.7in, clip, scale=0.5]{TemperrorADI_jul8_set2.pdf}\\\\\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{TemperrorIMEX_jul8_set3.pdf}\\includegraphics[trim = 1.3in 3.7in 1.6in 3.7in, clip, scale=0.5]{TemperrorADI_jul8_set3.pdf}\n\t\\caption{Temporal errors $\\widehat{E}^{\\rm ROI}(200,N)$ of the seven operator splitting schemes under consideration. IMEX schemes on the left side and ADI \n\tschemes on the right side, with parameter set 1 (top), set 2 (middle) and set 3 (bottom) from Table~\\ref{paramsets}. The CNFI and MCS schemes are applied \n\twith step size $\\Delta t = T\/N$, the MCS2 scheme is applied with $\\Delta t = T\/\\lfloor 3N\/2 \\rfloor$ and the CNFE, IETR, CNAB, SC2A schemes are applied \n\twith $\\Delta t = T\/(2N)$.}\n\t\\label{FigTemperrors}\n\\end{figure}\n\n\\newpage\n\\subsection{The Greeks}\\label{Greeks}\nThe Greeks are mathematical derivatives of the option value with respect to underlying variables and parameters. \nThey constitute a measure for risk that indicates how sensitive an option value is to changes in underlying variables and parameters \nand are crucial for hedging strategies. \nIn this subsection we consider the numerical approximation of the Delta and Gamma Greeks. \nDelta is a measure for the rate of change of the option value with respect to a change in an underlying asset price.\nAs there are two underlying assets, there are two Deltas:\n\\begin{equation*}\n\\Delta_1 = \\frac{\\partial v}{\\partial s_1} \\quad \\text{and} \\quad \\Delta_2 = \\frac{\\partial v}{\\partial s_2} . \n\\end{equation*}\nNext, Gamma measures the rate of change of a Delta with respect to a change in an underlying asset price. \nThere are three different Gammas:\n\\begin{equation*}\n\t\\Gamma_{11} = \\frac{\\partial \\Delta_1}{\\partial s_1} = \\frac{\\partial^2v}{\\partial s_1^2}, \\quad \n\t\\Gamma_{22} = \\frac{\\partial \\Delta_2}{\\partial s_2} = \\frac{\\partial^2v}{\\partial s_2^2} \\quad \\text{and} \\quad \n\t\\Gamma_{12} = \\frac{\\partial \\Delta_1}{\\partial s_2} = \\frac{\\partial^2v}{\\partial s_2 \\partial s_1} = \\frac{\\partial \\Delta_2}{\\partial s_1} = \\Gamma_{21}.\n\\end{equation*}\n\nBy virtue of the finite difference discretization that has been defined in Section \\ref{SecSpatial}, the Delta and Gamma Greeks\ncan directly be approximated, at essentially no computational cost, by applying the second-order central finite \ndifference formulas considered in Subsection \\ref{SecPDEpart} to the option value approximations on the spatial grid.\n\nAs an illustration, Figure~\\ref{FigGreeks} displays the numerically\\footnote{With $m_1=m_2=200$, $N=100$ and the MCS2 scheme.} \nobtained Delta and Gamma surfaces at maturity for the European put-on-the-average option under the two-asset Kou model for \nparameter set~1. \nAs expected, the Delta surfaces are steepest around the line segment $s_1 + s_2 = 2K$ and, correspondingly, the \nGamma surfaces are highest there.\n\nSimilarly to the option value, we study the temporal convergence behaviour of all operator splitting schemes in \nthe case of the five Greeks.\nAkin to (\\ref{EROI}), the temporal discretization error in the case of Delta $\\Delta_k \\, (k = 1, 2)$ is defined by\n\\begin{equation}\\label{EROIGreeks}\n\t\\widehat{E}^{\\rm ROI}_{\\Delta_k}(m,N) = \\textrm{max} \\left\\{| (\\Delta_k)_{i,j}(T) - (\\Delta_k)_{i,j}^{N'} |: \n\t\\Delta t = T\/N' ~~{\\rm and}~~ (s_{1,i},s_{2,j})\\in \\rm ROI \\right\\}.\n\\end{equation}\nHere $\\Delta_k (T)$ denotes the pertinent finite difference matrix for convection applied to the reference value for the exact\nsemidiscrete solution $V(T)$ given in Subsection~\\ref{SecResults1}. \nNext, $\\Delta_k ^{N'}$ is equal to the same finite difference matrix applied to the approximation $V^{N'}$ of $V(T)$ that is \ngenerated by any one of the seven operator splitting schemes. \nAnalogous temporal discretization error definitions hold in the case of $\\Gamma_{11}, \\Gamma_{22}$, $\\Gamma_{12}$. \n\nFigure~\\ref{FigGreeksError} displays the temporal errors in the case of the five Greeks and parameter set 1 for $m = 200$ and \nthe same sequence of values $N$ between 10 and 1000 as before.\nWe arrive at the same conclusions on the temporal convergence behaviour of the seven splitting schemes as obtained \nin Subsection~\\ref{SecResults1} regarding the option value.\nIn particular, besides CNFE, all splitting schemes reveal a stiff order of convergence equal to two, and MCS2 has the best\nperformance among all these schemes.\n\nWe note that the somewhat larger errors that are observed for the CNFI, IETR, CNAB schemes when $N$ is small are attributed\nto the nonsmootness of the initial (payoff) function and could be alleviated by applying four (instead of two) half time steps\nwith IMEX Euler at the start of the time stepping, cf. Section~\\ref{SecTime}.\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{Delta1.pdf}\\includegraphics[trim = 1.3in 3.7in 1.4in 3.7in, clip, scale=0.5]{Delta2.pdf}\\\\\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{Gamma11.pdf}\\includegraphics[trim = 1.3in 3.7in 1.4in 3.7in, clip, scale=0.5]{Gamma22.pdf}\\\\\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{Gamma12.pdf}\n\t\\caption{\n\tFirst-order Greeks $\\Delta_1$ (top left) and $\\Delta_2$ (top right) and second-order Greeks $\\Gamma_{11}$ (middle left), $\\Gamma_{22}$ \n\t(middle right) and $\\Gamma_{12}$ (bottom) for the European put-on-the-average option under the two-asset Kou model in the case of \n\tparameter set 1.}\n\t\\label{FigGreeks}\n\\end{figure}\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{errorDelta1_jul14_set1.pdf}\\includegraphics[trim = 1.3in 3.7in 1.6in 3.7in, clip, scale=0.5]{errorDelta2_jul14_set1.pdf}\\\\\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{errorGamma11_jul14_set1.pdf}\\includegraphics[trim = 1.3in 3.7in 1.6in 3.7in, clip, scale=0.5]{errorGamma22_jul14_set1.pdf}\\\\\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{errorGamma12_jul14_set1.pdf}\n\t\\caption{Temporal errors of the seven operator splitting schemes under consideration in the case of the five Greeks and parameter set 1: \n\t$\\widehat{E}^{\\rm ROI}_{\\Delta_1}$ (top left), $\\widehat{E}^{\\rm ROI}_{\\Delta_2}$ (top right), $\\widehat{E}^{\\rm ROI}_{\\Gamma_{11}}$ \n\t(middle left), $\\widehat{E}^{\\rm ROI}_{\\Gamma_{22}}$ (middle right) and $\\widehat{E}^{\\rm ROI}_{\\Gamma_{12}}$ (bottom). \n\tThe CNFI and MCS schemes are applied with step size $\\Delta t = T\/N$, the MCS2 scheme is applied with $\\Delta t = T\/\\lfloor 3N\/2 \\rfloor$ \n\tand the CNFE, IETR, CNAB, SC2A schemes are applied with $\\Delta t = T\/(2N)$.}\n\t\\label{FigGreeksError}\n\\end{figure}\n\n\n\n\n\n\n\\newpage\n\\section{Conclusions}\\label{SecConc}\nWe have studied the valuation of European options under the two-asset Kou jump-diffusion model via the numerical solution of the \npertinent two-dimensional time-dependent PIDE.\nA first main contribution of our paper is the extension of an algorithm derived by Toivanen~\\cite{T08}, which enables a highly \nefficient numerical evaluation of the nonlocal double integral appearing in this PIDE.\nThe computational cost of the acquired algorithm is optimal: it is directly proportional to the number of grid points in the \nspatial discretization.\nAlso, it is simple to implement and requires little memory usage.\nSubsequently, for the efficient discretization in time of the semidiscretized two-dimensional Kou PIDE, we have investigated \nseven modern operator splitting schemes of the IMEX and the ADI kind.\nEvery splitting scheme conveniently treats the integral term in an explicit fashion. \nThrough extensive numerical experiments, we have examined the stability and convergence behaviour of the splitting schemes as \nwell as their relative performance. \nAll of the considered schemes, except for the first-order CNFE scheme, show a desirable, stiff order of temporal convergence \nequal to two. \nThe MCS2 scheme, successively developed by in 't Hout \\& Welfert \\cite{HW09} for PDEs and in 't Hout \\& Toivanen \\cite{HT18} for PIDEs,\nstood out favourably among the splitting schemes in view of its superior efficiency. \nThis conclusion agrees with the results recently obtained by Boen \\& in 't Hout~\\cite{BH21} in the case of the two-dimensional \nMerton PIDE. \n\n\n\n\n\n\n\\bibliographystyle{plain}\n\n\\section{Introduction}\n\t\n\tIn contemporary financial option valuation theory, jump-diffusion processes form a principal class of models for the \n\tevolution of the underlying asset prices, see~e.g.~Cont \\& Tankov~\\cite{CT04book} and Schoutens~\\cite{S03book}.\n\tThe first jump-diffusion process was proposed in 1976 by Merton~\\cite{M76}.\n\tIn this classical model, the relative jump sizes are assumed to be lognormally distributed.\n\tA wide variety of jump-diffusion processes, and more generally, exponential L\\'{e}vy processes, for asset prices has been introduced \n\tin the literature since then, for example the familiar variance gamma (VG), normal inverse Gaussian (NIG) and Carr--Geman--Madan--Yor \n\t(CGMY) models~\\cite{CT04book,S03book}.\n\t\n\tIn this paper we consider the jump-diffusion model proposed by Kou~\\cite{K02}.\n\tIn this model, the relative jump sizes are given by a log-double-exponential distribution.\n\tJust like the models mentioned above, Kou's jump-diffusion model has become popular in financial option valuation theory and practice.\n\tIn this paper we are interested in the valuation of European-style options under a direct extension of Kou's single \n\tasset jump-diffusion model~\\cite{K02} to two assets.\n\tHere the (finite activity) jumps in the two asset prices are assumed to occur contemporaneously.\n\tFinancial option valuation theory then yields a two-dimensional time-dependent partial integro-differential equation (PIDE)\n\tthat must be satisfied for the values of European two-asset options. \n\tHere the integral part, which stems from the contribution of the jumps, is nonlocal: it is taken over the whole, two-dimensional asset \n\tprice domain.\n\tIn general, (semi-)closed analytical solutions to this PIDE are not available in the literature.\n\tAccordingly, in the present paper, we investigate the effective numerical solution of the two-dimensional time-dependent Kou PIDE.\n\t\n\tFor the numerical solution, we follow the well-known and general method of lines (MOL) approach.\n\tThe two-dimensional Kou PIDE is first discretized in space by finite differences, and the resulting large, semidiscrete system of \n\tordinary differential equations (ODEs) is subsequently discretized in time by a suitable implicit time stepping scheme.\n\tThe main challenges for the efficient and stable numerical solution are the treatment of the two-dimensional integral part and the \n\ttreatment of the two-dimensional PDE part, which includes a mixed spatial derivative term.\n\t\n\tSpatial discretization of the PIDE leads to a large, dense matrix for the integral part.\n\tIn each time step of the schemes under consideration, products of this matrix with one or more given vectors need to be computed,\n\twhich can form a computational burden.\n\tFor the one-dimensional Kou PIDE, however, Toivanen~\\cite{T08} derived a simple algorithm for evaluating these matrix-vector \n\tproducts that has optimal computational cost.\n\tA key result of our paper is a generalization of Toivanen's algorithm to the two-dimensional Kou PIDE that maintains optimal \n\tcomputational cost.\n\t\n\tFor the temporal discretization of semidiscretized one-dimensional PIDEs arising in financial option valuation, various authors have \n\tproposed operator splitting schemes where the (stiff) PDE part is handled implicitly and the (nonstiff) integral part explicitly.\n\tCont \\& Voltchkova~\\cite{CV05} considered an {\\it implicit-explicit (IMEX)} splitting scheme where the PDE part is handled by \n\tthe backward Euler method and the integral part by the forward Euler method.\n\tThis IMEX Euler scheme is only first-order consistent.\n\tA variety of higher-order IMEX schemes for PIDEs in finance has been studied since, e.g.~by Briani, Natalini \\& Russo~\\cite{B07},\n\tFeng \\& Linetsky~\\cite{FL08}, Kwon \\& Lee~\\cite{KL11} and Salmi \\& Toivanen~\\cite{ST14}.\n\tThe latter authors proposed the IMEX CNAB scheme, where the PDE part is treated by the Crank--Nicolson method and the integral part \n\tby the second-order Adams--Bashforth method.\n\tThe IMEX CNAB scheme has been successfully applied to two-dimensional option valuation PIDEs in e.g.~\\cite{HT16,STS14}. \n\t\n\tTavella \\& Randall~\\cite{TR00book} and d'Halluin, Forsyth \\& Vetzal~\\cite{HFV05} considered an alternative approach where the PDE \n\tpart is treated by the Crank--Nicolson method and a fixed-point iteration on the integral part is performed in each time step. \n\tThis approach has been applied to two-dimensional option valuation PIDEs in Clift \\& Forsyth~\\cite{CF08}, including the \n\ttwo-dimensional Kou PIDE.\n\tWhen the number of fixed-point iterations is frozen, one arrives at a particular IMEX scheme.\n\t\n\tFor the efficient temporal discretization of semidiscrete two-dimensional PIDEs, a subsequent important improvement is obtained \n\tby using, instead of the Crank--Nicolson method, an {\\it alternating direction implicit (ADI)} splitting scheme for the \n\ttwo-dimensional PDE part.\n\tIn the computational finance literature, a range of effective, second-order ADI schemes has been developed and analyzed for \n\tmulti-dimensional PDEs (without integral part), where in each time step the implicit unidirectional stages are combined with \n\texplicit stages involving the mixed derivative terms, see e.g.~\\cite{H17book,HT16}.\n\tIn this paper, we consider the well-established modified Craig--Sneyd (MCS) scheme, introduced by in 't Hout \\& Welfert~\\cite{HW09}, \n\tand the stabilizing correction two-step Adams-type scheme called SC2A, constructed by Hundsdorfer \\& in 't Hout~\\cite{HH18}.\n\t\n\tThe direct adaptation of ADI schemes for PDEs to PIDEs in finance has first been studied by Kaushansky, Lipton \\& Reisinger~\\cite{KLR18} \n\tand next by in 't Hout \\& Toivanen~\\cite{HT18}.\n\tHere the implicit unidirectional stages are blended with explicit stages involving both the mixed derivative terms and the integral \n\tpart, leading again to second-order schemes.\n\tWe note that an efficient, parallel implementation of the schemes developed in \\cite{HT18} has been designed by Ghosh \\& Mishra~\\cite{GM21} \n\twho apply a parallel cyclic reduction algorithm.\n\t\n\tBoen \\& in 't Hout~\\cite{BH21} recently investigated a collection of seven contemporary operator splitting schemes of both the \n\tIMEX and the ADI kind in the application to the two-dimensional Merton PIDE for the values of two-asset options.\n\tHere the numerical evaluation of the integral term has been done by means of a FFT algorithm, following \n\te.g.~\\cite{AO05,AA00,CF08,HFV05,STS14}. \n\tBased on analytical and numerical evidence in \\cite{BH21,HT18} in the case of the two-dimensional Merton and Bates PIDEs, \n\tit is concluded that, among the schemes under consideration, the adaptation of the MCS scheme introduced in \\cite{HT18} \n\tthat deals with the integral part in a two-step Adams--Bashforth fashion is preferable.\n\t\n\tIn the present paper we consider for the two-dimensional Kou PIDE the same collection of operator splitting schemes as in~\\cite{BH21}. \n\tFor the numerical evaluation of the double integral part, a generalization of the algorithm of Toivanen~\\cite{T08} is derived that has \n\toptimal computational cost.\n\tThis algorithm is simple to implement, requires little memory usage and is computationally much faster than the FFT algorithm \n\tmentioned above.\n\tAs a representative example, we consider the approximation of European put-on-the-average option values, together with their Greeks.\n\tAn outline of the rest of this paper is as follows.\n\t\n\tIn Section~\\ref{SecModel} the two-dimensional Kou PIDE is formulated.\n\tSection~\\ref{SecSpatial} deals with its spatial discretization.\n\tFirst, in Subsection~\\ref{SecPDEpart}, the two-dimensional PDE part is considered and a second-order finite difference discretization \n\ton a suitable nonuniform spatial grid is described.  \n\tNext, in Subsection~\\ref{SecIntpart}, we present the first main contribution of this paper.\n\tA common, second-order spatial discretization of the double integral part is employed and for its highly efficient evaluation \n\twe derive an extension of the algorithm proposed by Toivanen~\\cite{T08}.\n\tIt is shown that the computational cost of the extension is directly proportional to the number of spatial grid points, \n\twhich is optimal.\n\tSection~\\ref{SecTime} subsequently concerns the temporal discretization of the obtained semidiscrete two-dimensional Kou PIDE.\n\tHere the seven contemporary operator splitting schemes of the IMEX and ADI kind from~\\cite{BH21} are considered.  \n\tEach of these schemes conveniently treats the integral part in an explicit manner, where its fast evaluation is performed by the \n\talgorithm derived in Subsection~\\ref{SecIntpart}.\n\tIn Section~\\ref{SecResults} ample numerical experiments are presented.\n\tHere European put-on-the-average option values, together with their Greeks Delta and Gamma, are considered and we examine in detail\n\tthe temporal discretization errors of the different operator splitting schemes.\n\tThe final Section~\\ref{SecConc} gives conclusions.\n\t\n\t\n\n\n\n\t\n\t\\section{The two-dimensional Kou PIDE}\\label{SecModel}\n\tUnder the two-asset Kou jump-diffusion model, the value $v = v(s_1,s_2,t)$ of a European-style option with maturity date $T>0$ and $s_i$ ($i=1,2$) \n\trepresenting the price of asset $i$ at time $\\tau = T-t$, satisfies the following PIDE:\n\t\\begin{align}\\label{PIDE2D}\n\t\t\\dud{t} =&~ \\tfrac{1}{2} \\sigma_1^2s_1^2\\dudd{s_1} + \\rho \\sigma_1\\sigma_2s_1s_2\\duddm{s_1}{s_2} + \\tfrac{1}{2} \\sigma_2^2 s_2^2\\dudd{s_2} + \n\t\t(r-\\lambda \\kappa_1) s_1\\dud{s_1} + (r-\\lambda\\kappa_2) s_2 \\dud{s_2} \\nonumber \\\\\n\t\t& -(r+\\lambda)v+\\lambda\\int_0^{\\infty}\\int_0^{\\infty} f(y_1,y_2) v(s_1y_1, s_2y_2,t) \\mathrm{d} y_1 \\mathrm{d} y_2\n\t\\end{align}\n\twhenever $s_1>0$, $s_2>0$, $0<t\\leq T$.\n\tHere $r$ is the risk-free interest rate, $\\sigma_i > 0$ ($i=1,2$) is the instantaneous volatility for asset $i$ conditional \n\ton the event that no jumps occur, and $\\rho$ is the correlation coefficient of the two underlying standard Brownian motions.\n\tNext, $\\lambda$ is the jump intensity of the underlying Poisson arrival process, and $\\kappa_i$ ($i=1,2$) is the expected \n\trelative jump size for asset $i$.\n    The function $f$ is the joint probability density function of two independent random variables possessing log-double-exponential \n\tdistributions~\\cite{K02},\n\t\\begin{equation}\\label{pdf2D}\n\t\tf(y_1,y_2) = \n\t\t\\left\\{\\begin{array}{lll}\n\t\t\tq_1q_2\\eta_{q_1}\\eta_{q_2}y_1^{\\eta_{q_1}-1}y_2^{\\eta_{q_2}-1} & (0 < y_1, y_2 < 1),\\\\\n\t\t\tp_1q_2\\eta_{p_1}\\eta_{q_2}y_1^{-\\eta_{p_1}-1}y_2^{\\eta_{q_2}-1} & (y_1 \\geq 1, 0< y_2 < 1),\\\\\n\t\t\tq_1p_2\\eta_{q_1}\\eta_{p_2}y_1^{\\eta_{q_1}-1}y_2^{-\\eta_{p_2}-1} & (0 < y_1 < 1, y_2 \\geq 1),\\\\\n\t\t\tp_1p_2\\eta_{p_1}\\eta_{p_2}y_1^{-\\eta_{p_1}-1}y_2^{-\\eta_{p_2}-1} & (y_1, y_2 \\geq 1).\n\t\t\\end{array}\\right.\n\t\\end{equation}  \n\tThe parameters $p_i$, $q_i$, $\\eta_{p_i}$, $\\eta_{q_i}$  are all positive constants with $p_i + q_i = 1$ and $\\eta_{p_i} > 1$.\n\tIt holds that\n\t\\begin{equation*}\n\t    \\kappa_i = \\frac{p_i\\eta_{p_i}}{\\eta_{p_i}-1}+\\frac{q_i\\eta_{q_i}}{\\eta_{q_i}+1}-1 \\quad (i=1,2).\n\t\\end{equation*}\n\tFor \\eqref{PIDE2D}, the initial condition is given by\n\t\\[\n\tv(s_1,s_2,0) = \\phi(s_1,s_2),\n\t\\]\n\twhere $\\phi$ denotes the payoff function of the option.\n\tAs a typical example, we consider in this paper a European put-on-the-average option, which has the payoff function\n\t\\begin{equation}\\label{payoff}\n\t\t\\phi(s_1,s_2) = \\textrm{max} \\left(0\\,,\\,K-\\frac{s_1+s_2}{2}\\right)\n\t\\end{equation}\n\twith strike price $K>0$.\n\tIts graph is shown in Figure~\\ref{FigPayoff}.\n\tConcerning the boundary condition, it holds that the PIDE \\eqref{PIDE2D} is itself satisfied on the two sides $s_{1} = 0$ \n\tand $s_{2} = 0$, respectively.\n\t\n\n\t\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{payoff.pdf}\n\t\\caption{Payoff put-on-the-average option with $K=100$.}\n\t\\label{FigPayoff}\n\t\\end{figure}\n\t\n\t\n\n\n\n\t\n\t\\section{Spatial discretization}\\label{SecSpatial}\n\tFor the numerical solution of the initial-boundary value problem for \\eqref{PIDE2D} we employ the popular method of lines (MOL) approach.\n\tThis approach consists of two consecutive steps~\\cite{HV03book}: first the PIDE \\eqref{PIDE2D} is discretized in space and subsequently in time.\n\tThis Section~\\ref{SecSpatial} deals with the spatial discretization. \n\tIn the next Section~\\ref{SecTime} we shall consider the temporal discretization.\n\t\n\n\n\n\t\n\t\\subsection{Convection-diffusion-reaction part}\\label{SecPDEpart}\n\tFor the numerical solution, the spatial domain is truncated to a bounded set $[0,S_{\\rm max}]\\times[0,S_{\\rm max}]$ \n\twith fixed value $S_{\\rm max}$ chosen sufficiently large.\n\tOn the two far sides $s_{1} = S_{\\rm max}$ and $s_{2} = S_{\\rm max}$ a linear boundary condition is taken, which is well-known in finance,\n\t\\begin{equation}\\label{LBC}\n\t\t\\dudd{s_1} = 0~~(\\textrm{if}~s_{1} = S_{\\rm max}) \\quad \\mbox{ and } \\quad  \\dudd{s_2} = 0 ~~(\\textrm{if}~s_{2} = S_{\\rm max}).\n\t\\end{equation}\n\tIn this subsection we describe the finite difference discretization of the convection-diffusion-reaction part of the PIDE \\eqref{PIDE2D}, \n\tspecified by\n\t\\begin{equation*}\\label{lindiffop}\n\t\t\\mathcal{D} v =  \\tfrac{1}{2} \\sigma_1^2s_1^2\\dudd{s_1} + \\rho \\sigma_1\\sigma_2s_1s_2\\duddm{s_1}{s_2} + \\tfrac{1}{2} \\sigma_2^2 s_2^2\\dudd{s_2} \n\t\t+ (r-\\lambda \\kappa_1) s_1\\dud{s_1} + (r-\\lambda\\kappa_2) s_2 \\dud{s_2} -(r+\\lambda)v.\n\t\\end{equation*}\n\tThe semidiscretization of this part is common and similar to that in~e.g.~\\cite{BH21}.\n\t\n\tLet integers $m_1, m_2 \\geq 1$ be given.\n\tThe option value function $v$ is approximated at a nonuniform, Cartesian set of spatial grid points,\n\t\\begin{equation*}\\label{sgrid}\n\t\t(s_{1,i},s_{2,j})\\in [0,S_{\\rm max}]\\times[0,S_{\\rm max}] \\quad (0\\leq i \\leq m_1, \\ 0\\leq j \\leq m_2),\n\t\\end{equation*}\n\twith $s_{1,0} = s_{2,0} = 0$ and $s_{1,m_1} = s_{2,m_2} = S_{\\rm max}$. \n\tThe nonuniform grid in each spatial direction is defined through a smooth transformation of an artificial uniform grid, such that \n\trelatively many grid points are placed in a region of financial and numerical interest.\n\tFigure~\\ref{FigGrid} shows a sample spatial grid if $m_1=m_2=50$, $K=100$ and $S_{\\rm max} = 5K$.\n\t\n\tLet integer $m \\geq 1$ and parameter $d > 0$.\n\tConsider equidistant points $0 = \\xi_0 < \\xi_1 < \\dots < \\xi_{m} = \\xi_{\\rm max}$ where\n\t\\begin{equation*}\n\t\t\\xi_{\\rm max} = \\xi_{\\rm int} + \\sinh^{-1}\\left(\\frac{S_{\\rm max}}{d}-\\xi_{\\rm int}\\right) \n\t\t\\quad \\textrm{and } \\quad\n\t\t\\xi_{\\rm int} = \\frac{2K}{d}.\n\t\\end{equation*}\n\tThen in each spatial direction a nonuniform mesh $0=s_0<s_1<\\cdots<s_m=S_{\\rm max}$ is constructed by the smooth transformation \n\t$s_{i} = \\varphi(\\xi_i)$ $(0\\le i \\le m)$ with\n\t\\begin{equation*}\n\t\t\\varphi(\\xi) = \n\t\t\\left\\{\\begin{array}{lll}\n\t\t\td \\xi &(0 \\leq \\xi \\leq \\xi_{\\rm int}),\\\\\\\\\n\t\t\t2K + d\\sinh(\\xi-\\xi_{\\rm int}) &(\\xi_{\\rm int} < \\xi \\leq \\xi_{\\rm max}).\n\t\t\\end{array}\\right.\n\t\\end{equation*}  \n\tThis mesh for $s$ is uniform inside the interval $[0,2K]$ and nonuniform outside.\n\tThe parameter $d$ controls the fraction of points $s_{i}$ that lie inside.\n\tIn this paper we (heuristically) choose $d = K\/10$, such that the largest fraction is inside $[0,2K]$.\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.6]{sgrid.pdf}\n\t\t\\caption{Sample spatial grid for $m_1=m_2=50$, $K=100$, $S_{\\rm max} = 5K$.}\n\t\t\\label{FigGrid}\n\t\\end{figure}\n\t\n\tDefine mesh widths $h_i = s_i-s_{i-1}$ ($1\\leq i \\leq m$) and let $u: [0,S_{\\rm max}] \\to \\mathbb{R}$ be any given smooth function. \n\tFor approximating the first and second derivatives of $u$, the following second-order central finite difference formulas are used:\n\t\\begin{align*}\\label{semconv}\n\t\tu^\\prime (s_i) \\approx \\omega_{i,-1} u(s_{i-1}) + \\omega_{i,0} u(s_{i}) + \\omega_{i,1} u(s_{i+1}),\n\t\\end{align*}\n\twith \n\t\\[\n\t\\omega_{i,-1} = \\frac{-h_{i+1}}{h_{i}(h_{i}+h_{i+1})},\\quad\n\t\\omega_{i,0} = \\frac{h_{i+1}-h_{i}}{h_{i}h_{i+1}},\\quad \n\t\\omega_{i,1} = \\frac{h_{i}}{h_{i+1}(h_{i}+h_{i+1})}\n\t\\]\n\tand\n\t\\begin{align*}\\label{semdiff}\n\t\tu^{\\prime\\prime} (s_i) \\approx \\omega_{i,-1} u(s_{i-1}) + \\omega_{i,0} u(s_{i}) + \\omega_{i,1} u(s_{i+1}),\n\t\\end{align*}\n\twith\n\t\\[\n\t\\omega_{i,-1} = \\frac{2}{h_{i}(h_{i}+h_{i+1})},\\quad \n\t\\omega_{i,0} = \\frac{-2}{h_{i}h_{i+1}},\\quad \n\t\\omega_{i,1} = \\frac{2}{h_{i+1}(h_{i}+h_{i+1})}\n\t\\]\n\twhenever $1\\le i \\le m-1$.\n\tFor $i=0$ no finite difference formulas are needed, due to the degeneracy of $\\mathcal{D} v$ at the zero boundaries.\n\tFor $i=m$, the first derivative is discretized by the first-order backward finite difference formula and the second \n\tderivative is equal to zero by the linear boundary condition \\eqref{LBC}.\n\tConcerning the mixed derivative term in $\\mathcal{D} v$, this is approximated by successively applying the relevant \n\tfinite difference formulas for the first derivative in the two spatial directions.\n\t\n\tLet $V_{i,j}(t)$ denote the semidiscrete approximation to $v(s_{1,i},s_{2,j},t)$ for $0\\leq i \\leq m_1$, $0\\leq j \\leq m_2$ \n\tand define the vector \n\t\\[\n\tV(t) = (V_{0,0}(t), V_{1,0}(t),\\ldots,V_{m_1-1,m_2}(t),V_{m_1,m_2}(t))^{\\top} \\in \\mathbb{R}^{(m_1+1)(m_2+1)}.\n\t\\] \n\tThe semidiscretized convection-diffusion-reaction part $\\mathcal{D} v$ of PIDE \\eqref{PIDE2D} is then given by\n\t\\[\n\tA^{(D)} V(t)\n\t\\] \n\twith matrix\n\t\\[\n\tA^{(D)} = A^{(M)} + A_1 + A_2,\n\t\\]\n\twhere\n\t\\begin{align*}\n\t\tA^{(M)} &= \\rho\\sigma_1\\sigma_2\\left(X_2D_2^{(1)}\\right)\\otimes\\left(X_1D_1^{(1)}\\right),\\\\\n\t\tA_1 &= I_2 \\otimes \\left(\\tfrac{1}{2}\\sigma_1^2X_1^2D_1^{(2)} + (r-\\lambda\\kappa_1)X_1D_1^{(1)}-\\tfrac{1}{2}(r+\\lambda)I_1\\right),\\\\\n\t\tA_2 &= \\left(\\tfrac{1}{2}\\sigma_2^2X_2^2D_2^{(2)} + (r-\\lambda\\kappa_2)X_2D_2^{(1)}-\\tfrac{1}{2}(r+\\lambda)I_2\\right)\\otimes I_1.\n\t\\end{align*}\n\tHere $I_k$, $X_k$, $D_k^{(l)}$ are given $(m_k+1)\\times (m_k+1)$ matrices for $k,l\\in \\{1,2\\}$ with $I_k$ being the identity matrix, \n\t$X_k$ being the diagonal matrix\n\t\\begin{equation*}\n\t\tX_k = {\\rm diag}(s_{k,0}, s_{k,1},\\ldots, s_{k,m_k})\n\t\\end{equation*}\n\tand $D_k^{(l)}$ the matrix representing numerical differentiation of order $l$ in the $k$-th spatial direction by the relevant \n\tfinite difference formula above.\n\tThe matrix $A^{(M)}$ corresponds to the mixed derivative term in $\\mathcal{D} v$ and $A_k$ corresponds to all derivative terms \n\tin the $k$-th spatial direction ($k=1,2$), where the reaction term has been distributed evenly across $A_1$ and $A_2$.\n\tThese matrices are all sparse. In particular, $A_1$ and $A_2$ are (essentially) tridiagonal.\n\t\n\t\n\n\n\n\t\n\\subsection{Integral part}\\label{SecIntpart}\nIn the following we consider discretization of the double integral\n\\begin{equation}\\label{integral}\n\t\\mathcal{J} = \\lambda\\int_0^{\\infty}\\int_0^{\\infty} f(y_1,y_2)v(s_1y_1, s_2y_2,t) \\mathrm{d} y_1 \\mathrm{d} y_2\n\\end{equation}\non the spatial grid from Subsection~\\ref{SecPDEpart}.\nFor its efficient evaluation, we derive an extension of the algorithm proposed by Toivanen~\\cite{T08} for the special case of the \none-dimensional Kou model.\n\nAssume $s_1, s_2>0$ are given. A change of variables $y_i = z_i\/s_i$ ($i=1,2$) yields\n\\begin{equation*}\n\t\\mathcal{J} = \\lambda\\int_0^{\\infty}\\int_0^{\\infty} f\\bigg(\\frac{z_1}{s_1},\\frac{z_2}{s_2}\\bigg) v(z_1,z_2,t) \\frac{\\mathrm{d} z_1 \\mathrm{d} z_2}{s_1 s_2}.\n\\end{equation*}\nThe density function $f$ is defined on a partition of four sets of the first quadrant in the real plane, see (\\ref{pdf2D}).\nIt follows that $\\mathcal{J}$ is decomposed into four integrals as $\\mathcal{J} = \\mathcal{J}_1 + \\mathcal{J}_2 + \\mathcal{J}_3 + \\mathcal{J}_4$, where\n\\begin{align*}\n\t\\mathcal{J}_1 &= \\lambda q_1q_2\\eta_{q_1}\\eta_{q_2}s_1^{-\\eta_{q_1}}s_2^{-\\eta_{q_2}} \\int_0^{s_2} \\int_0^{s_1} z_1^{\\eta_{q_1}-1}z_2^{\\eta_{q_2}-1} v(z_1,z_2,t) \\mathrm{d} z_1 \\mathrm{d} z_2,\\\\\n\t\\mathcal{J}_2 &= \\lambda p_1q_2\\eta_{p_1}\\eta_{q_2}s_1^{\\eta_{p_1}}s_2^{-\\eta_{q_2}}  \\int_0^{s_2} \\int_{s_1}^{\\infty} z_1^{-\\eta_{p_1}-1}z_2^{\\eta_{q_2}-1} v(z_1,z_2,t) \\mathrm{d} z_1 \\mathrm{d} z_2,\\\\\n\t\\mathcal{J}_3 &= \\lambda q_1p_2\\eta_{q_1}\\eta_{p_2}s_1^{-\\eta_{q_1}}s_2^{\\eta_{p_2}}  \\int_{s_2}^{\\infty} \\int_0^{s_1} z_1^{\\eta_{q_1}-1}z_2^{-\\eta_{p_2}-1} v(z_1,z_2,t) \\mathrm{d} z_1 \\mathrm{d} z_2,\\\\\t\t\n\t\\mathcal{J}_4 &= \\lambda p_1p_2\\eta_{p_1}\\eta_{p_2}s_1^{\\eta_{p_1}}s_2^{\\eta_{p_2}}   \\int_{s_2}^{\\infty} \\int_{s_1}^{\\infty} z_1^{-\\eta_{p_1}-1}z_2^{-\\eta_{p_2}-1} v(z_1,z_2,t) \\mathrm{d} z_1 \\mathrm{d} z_2.\n\\end{align*}\nWe first consider discretization of the integral $\\mathcal{J}_1$. \nUpon writing\n\\begin{equation*}\n\t\\psi_1(s_1,s_2) = \\lambda q_1q_2\\eta_{q_1}\\eta_{q_2}s_1^{-\\eta_{q_1}}s_2^{-\\eta_{q_2}} \n\t\\quad \\textrm{and} \\quad\n\t\\varphi_1(z_1,z_2) = z_1^{\\eta_{q_1}-1}z_2^{\\eta_{q_2}-1},\n\\end{equation*}\nwe have\n\\begin{equation}\n\t\\mathcal{J}_1 = \\psi_1(s_1,s_2)\\int_0^{s_2} \\int_0^{s_1} \\varphi_1(z_1,z_2) v(z_1,z_2,t) \\mathrm{d} z_1 \\mathrm{d} z_2.\n\\end{equation}\nFor $1\\leq i \\leq m_1$, $1\\leq j \\leq m_2$ let\n\\begin{equation*}\n\t\\mathcal{J}_{1,ij} = \\psi_1(s_{1,i},s_{2,j}) \\int_0^{s_{2,j}} \\int_0^{s_{1,i}} \\varphi_1(z_1,z_2) v(z_1,z_2,t) \\mathrm{d} z_1 \\mathrm{d} z_2\n\\end{equation*}\ndenote the value of $\\mathcal{J}_1$ at the spatial grid point $(s_{1,i},s_{2,j})$.\nDefine\n\\begin{equation*}\n\t\\mathcal{G}_{1,kl} = \\int_{s_{2,l-1}}^{s_{2,l}} \\int_{s_{1,k-1}}^{s_{1,k}} \\varphi_1(z_1,z_2) v(z_1,z_2,t) \\mathrm{d} z_1 \\mathrm{d} z_2\n\\end{equation*}\nwhenever $1\\leq k \\leq m_1$, $1\\leq l \\leq m_2$.\nThen the following useful expression for $\\mathcal{J}_{1,ij}$ in terms of a double cumulative sum is obtained,\n\\begin{equation}\\label{cumsum1}\n\t\\mathcal{J}_{1,ij} = \\psi_1(s_{1,i},s_{2,j}) \\sum_{k=1}^i \\sum_{l=1}^j \\mathcal{G}_{1,kl}\n\t\\quad\n\t(1\\leq i \\leq m_1, 1\\leq j \\leq m_2).\n\\end{equation}\nNotice the obvious but important fact that the $\\mathcal{G}_{1,kl}$ are independent of the indices $i$ and $j$.\nHence, if all values $\\mathcal{G}_{1,kl}$ are given, then computing the double cumulative sums in \\eqref{cumsum1} \nfor all $i$, $j$ can be done in, to leading order, just $2m_1m_2$ additions.\n\nWe subsequently construct approximations $G_{1,kl}$ to $\\mathcal{G}_{1,kl}$ ($1\\leq k \\leq m_1$, $1\\leq l \\leq m_2$)\nand define the approximation to $\\mathcal{J}_{1,ij}$ by\n\\begin{equation}\\label{cumsum1_semi}\n\tJ_{1,ij} = \\psi_1(s_{1,i},s_{2,j}) \\sum_{k=1}^i \\sum_{l=1}^j G_{1,kl}\n\t\\quad\n\t(1\\leq i \\leq m_1, 1\\leq j \\leq m_2).\n\\end{equation}\nFor any given $k, l$ with $1\\leq k \\leq m_1$, $1\\leq l \\leq m_2$ consider the natural choice of bilinear interpolation \nto approximate $v(z_1,z_2,t)$ on the $(z_1,z_2)$-domain $[s_{1,k-1}, s_{1,k}] \\times [s_{2,l-1}, s_{2,l}]$:\n\\begin{equation*}\n\t{\\widetilde v}_{kl}(z_1, z_2,t) = \\ell_{kl}^{00}(z_1,z_2) V_{k-1,l-1}(t) + \\ell_{kl}^{10}(z_1,z_2) V_{k,l-1}(t) +\n\t\\ell_{kl}^{01}(z_1,z_2) V_{k-1,l}(t) + \\ell_{kl}^{11}(z_1,z_2) V_{k,l}(t)\n\\end{equation*}\nwith weights\n\\begin{align*}\n\t\\ell_{kl}^{00}(z_1,z_2) &= (s_{1,k}-z_1)(s_{2,l}-z_2)\/\\delta_{kl},\\\\\n\t\\ell_{kl}^{10}(z_1,z_2) &= (z_1-s_{1,k-1})(s_{2,l}-z_2)\/\\delta_{kl},\\\\\n\t\\ell_{kl}^{01}(z_1,z_2) &= (s_{1,k}-z_1)(z_{2}-s_{2,l-1})\/\\delta_{kl},\\\\\n\t\\ell_{kl}^{11}(z_1,z_2) &= (z_1-s_{1,k-1})(z_2-s_{2,l-1})\/\\delta_{kl},\n\\end{align*}\nwhere $\\delta_{kl} = \\Delta s_{1,k}\\Delta s_{2,l}$ and $\\Delta s_{1,k} = s_{1,k}-s_{1,k-1}$, $\\Delta s_{2,l} = s_{2,l}-s_{2,l-1}$.\nThen we define \n\\begin{equation*}\n\tG_{1,kl} = \\int_{s_{2,l-1}}^{s_{2,l}} \\int_{s_{1,k-1}}^{s_{1,k}} \\varphi_1(z_1,z_2) {\\widetilde v}_{kl}(z_1, z_2,t) \\mathrm{d} z_1 \\mathrm{d} z_2.\n\\end{equation*}\nA straightforward calculation yields the simple, convenient formula \n\\begin{equation}\\label{G1kl}\n\tG_{1,kl} = \n\t\\gamma_{1,kl}^{00} V_{k-1,l-1}(t) + \\gamma_{1,kl}^{10} V_{k,l-1}(t) +\n\t\\gamma_{1,kl}^{01} V_{k-1,l}(t)   + \\gamma_{1,kl}^{11} V_{k,l}(t)\n\\end{equation}\nwith\n\\begin{align*}\n\t\\gamma_{1,kl}^{00} &= (s_{1,k}s_{2,l}\\zeta_{1,kl}^{00} - s_{2,l}\\zeta_{1,kl}^{10} - s_{1,k}\\zeta_{1,kl}^{01} + \\zeta_{1,kl}^{11})\/\\delta_{kl},\\\\\n\t\\gamma_{1,kl}^{10} &= (-s_{1,k-1}s_{2,l}\\zeta_{1,kl}^{00} + s_{2,l}\\zeta_{1,kl}^{10} + s_{1,k-1}\\zeta_{1,kl}^{01} - \\zeta_{1,kl}^{11} )\/\\delta_{kl},\\\\\n\t\\gamma_{1,kl}^{01} &= (-s_{1,k}s_{2,l-1}\\zeta_{1,kl}^{00} + s_{2,l-1}\\zeta_{1,kl}^{10} + s_{1,k}\\zeta_{1,kl}^{01} - \\zeta_{1,kl}^{11} )\/\\delta_{kl},\\\\\n\t\\gamma_{1,kl}^{11} &= (s_{1,k-1}s_{2,l-1}\\zeta_{1,kl}^{00} - s_{2,l-1}\\zeta_{1,kl}^{10} - s_{1,k-1}\\zeta_{1,kl}^{01} + \\zeta_{1,kl}^{11})\/\\delta_{kl}\n\\end{align*}\nand\n\\begin{equation*}\n\t\\zeta_{1,kl}^{ab} =  \n\t\\int_{s_{2,l-1}}^{s_{2,l}} \\int_{s_{1,k-1}}^{s_{1,k}} \\varphi_1(z_1,z_2) z_1^a z_2^b \\mathrm{d} z_1 \\mathrm{d} z_2 =\n\t\\frac{\\bigg(s_{1,k}^{a+\\eta_{q_1}}-s_{1,k-1}^{a+\\eta_{q_1}}\\bigg)\\bigg(s_{2,l}^{b+\\eta_{q_2}}-s_{2,l-1}^{b+\\eta_{q_2}}\\bigg)}{\\big(a+\\eta_{q_1}\\big)\\big(b+\\eta_{q_2}\\big)}\n\\end{equation*}\nfor $a,b\\in \\{0,1\\}$.\n\nThe coefficients $\\gamma_{1,kl}^{ab}$ are completely determined by the Kou parameters and the spatial grid. \nSince they are independent of $t$, they can be computed upfront, before the time discretization.\nClearly, for any given vector $V(t)$, the computation of $G_{1,kl}$ by \\eqref{G1kl} for all $k$, $l$ requires \n$3m_1m_2$ additions and $4m_1m_2$ multiplications.\nNoticing that the values of $\\psi_1$ in \\eqref{cumsum1_semi} can also be computed upfront, it follows that \nthe number of basic arithmetic operations to compute all approximations $J_{1,ij}$ ($1\\leq i \\leq m_1$, \n$1\\leq j \\leq m_2$) by \\eqref{cumsum1_semi} is, to leading order, equal to $10m_1m_2$.\n\nThe discretization and efficient evaluation of the other three integrals is done completely analogously.\nIn the relevant derivation, $v$ is approximated by zero outside the spatial domain $[0,S_{\\rm max}]\\times [0,S_{\\rm max}]$.\nWrite\n\\begin{align*}\n\t\\psi_2(s_1,s_2) &= \\lambda p_1q_2\\eta_{p_1}\\eta_{q_2}s_1^{\\eta_{p_1}}s_2^{-\\eta_{q_2}},\\\\\n\t\\psi_3(s_1,s_2) &= \\lambda q_1p_2\\eta_{q_1}\\eta_{p_2}s_1^{-\\eta_{q_1}}s_2^{\\eta_{p_2}},\\\\ \n\t\\psi_4(s_1,s_2) &= \\lambda p_1p_2\\eta_{p_1}\\eta_{p_2}s_1^{\\eta_{p_1}}s_2^{\\eta_{p_2}}.\n\\end{align*}\nLet $1\\leq i \\leq m_1$, $1\\leq j \\leq m_2$. \nThen the approximations of $\\mathcal{J}_2, \\mathcal{J}_3, \\mathcal{J}_4$ \nat the spatial grid point $(s_{1,i},s_{2,j})$ are given by, respectively, the double cumulative sums\n\\begin{align*}\n\tJ_{2,ij} &= \\psi_2(s_{1,i},s_{2,j}) \\sum_{k=i+1}^{m_1} \\sum_{l=1}^j G_{2,kl},\\\\\n\tJ_{3,ij} &= \\psi_3(s_{1,i},s_{2,j}) \\sum_{k=1}^{i}     \\sum_{l=j+1}^{m_2} G_{3,kl},\\\\\n\tJ_{4,ij} &= \\psi_4(s_{1,i},s_{2,j}) \\sum_{k=i+1}^{m_1} \\sum_{l=j+1}^{m_2} G_{4,kl}\n\\end{align*}\nwith the usual convention that empty sums are equal to zero.\nFor $2\\le \\nu \\le 4$, $1\\leq k \\leq m_1$, $1\\leq l \\leq m_2$ we obtain\n\\begin{equation*}\\label{Gnukl}\n\tG_{\\nu,kl} = \n\t\\gamma_{\\nu,kl}^{00} V_{k-1,l-1}(t) + \\gamma_{\\nu,kl}^{10} V_{k,l-1}(t) +\n\t\\gamma_{\\nu,kl}^{01} V_{k-1,l}(t)   + \\gamma_{\\nu,kl}^{11} V_{k,l}(t)\n\\end{equation*}\nwith\n\\begin{align*}\n\t\\gamma_{\\nu,kl}^{00} &= (s_{1,k}s_{2,l}\\zeta_{\\nu,kl}^{00} - s_{2,l}\\zeta_{\\nu,kl}^{10} - s_{1,k}\\zeta_{\\nu,kl}^{01} + \\zeta_{\\nu,kl}^{11})\/\\delta_{kl},\\\\\n\t\\gamma_{\\nu,kl}^{10} &= (-s_{1,k-1}s_{2,l}\\zeta_{\\nu,kl}^{00} + s_{2,l}\\zeta_{\\nu,kl}^{10} + s_{1,k-1}\\zeta_{\\nu,kl}^{01} - \\zeta_{\\nu,kl}^{11} )\/\\delta_{kl},\\\\\n\t\\gamma_{\\nu,kl}^{01} &= (-s_{1,k}s_{2,l-1}\\zeta_{\\nu,kl}^{00} + s_{2,l-1}\\zeta_{\\nu,kl}^{10} + s_{1,k}\\zeta_{\\nu,kl}^{01} - \\zeta_{\\nu,kl}^{11} )\/\\delta_{kl},\\\\\n\t\\gamma_{\\nu,kl}^{11} &= (s_{1,k-1}s_{2,l-1}\\zeta_{\\nu,kl}^{00} - s_{2,l-1}\\zeta_{\\nu,kl}^{10} - s_{1,k-1}\\zeta_{\\nu,kl}^{01} + \\zeta_{\\nu,kl}^{11})\/\\delta_{kl}\n\\end{align*}\nwhere\n\\begin{align*}\n\t\\zeta_{2,kl}^{ab} &=  \n\t\\frac{\\bigg(s_{1,k}^{a-\\eta_{p_1}}-s_{1,k-1}^{a-\\eta_{p_1}}\\bigg)\\bigg(s_{2,l}^{b+\\eta_{q_2}}-s_{2,l-1}^{b+\\eta_{q_2}}\\bigg)}{\\big(a-\\eta_{p_1}\\big)\\big(b+\\eta_{q_2}\\big)},\\\\\n\t\\zeta_{3,kl}^{ab} &=  \n\t\\frac{\\bigg(s_{1,k}^{a+\\eta_{q_1}}-s_{1,k-1}^{a+\\eta_{q_1}}\\bigg)\\bigg(s_{2,l}^{b-\\eta_{p_2}}-s_{2,l-1}^{b-\\eta_{p_2}}\\bigg)}{\\big(a+\\eta_{q_1}\\big)\\big(b-\\eta_{p_2}\\big)},\\\\\n\t\\zeta_{4,kl}^{ab} &=  \n\t\\frac{\\bigg(s_{1,k}^{a-\\eta_{p_1}}-s_{1,k-1}^{a-\\eta_{p_1}}\\bigg)\\bigg(s_{2,l}^{b-\\eta_{p_2}}-s_{2,l-1}^{b-\\eta_{p_2}}\\bigg)}{\\big(a-\\eta_{p_1}\\big)\\big(b-\\eta_{p_2}\\big)}\n\\end{align*}\nfor $a,b\\in \\{0,1\\}$.\n\nOn the boundary part $\\{(s_1,0): s_1>0\\}$ the double integral \\eqref{integral} reduces to the single integral\n\\begin{equation}\\label{integral1_1D}\n\t\\mathcal{J} = \\lambda\\int_0^{\\infty} f_1(y_1)v(s_1y_1,0,t) \\mathrm{d} y_1.\n\\end{equation}\nThe discretization and efficient evaluation of this integral, which arises in the one-dimensional \nKou PIDE, has been constructed by Toivanen~\\cite{T08}.\nFor completeness, we include the result here.\nAssume $s_1>0$. \nThen for \\eqref{integral1_1D} there holds $\\mathcal{J} = \\mathcal{J}_1 + \\mathcal{J}_2$, where\n\\begin{equation*}\n\t\\mathcal{J}_1 = \\lambda \\int_0^{s_1} f_1\\bigg(\\frac{z_1}{s_1}\\bigg) v(z_1,0,t) \\frac{\\mathrm{d} z_1 }{s_1}\n\t\\quad \\textrm{and} \\quad\n\t\\mathcal{J}_2 = \\lambda \\int_{s_1}^{\\infty} f_1\\bigg(\\frac{z_1}{s_1}\\bigg) v(z_1,0,t) \\frac{\\mathrm{d} z_1 }{s_1}.\n\\end{equation*}\nWrite\n\\begin{equation*}\n\t\\psi_1(s_1) = \\lambda q_1\\eta_{q_1}s_1^{-\\eta_{q_1}} \n\t\\quad \\textrm{and} \\quad\n\t\\psi_2(s_1) = \\lambda p_1\\eta_{p_1}s_1^{\\eta_{p_1}}.\n\\end{equation*}\nLet $1\\leq i \\leq m_1$. \nThen the approximations of $\\mathcal{J}_1, \\mathcal{J}_2$ at the spatial grid point $(s_{1,i},0)$ are given by, \nrespectively, the single cumulative sums\n\\begin{equation*}\n\tJ_{1,i} = \\psi_1(s_{1,i}) \\sum_{k=1}^{i} G_{1,k}\n\t\\quad \\textrm{and} \\quad\n\tJ_{2,i} = \\psi_2(s_{1,i}) \\sum_{k=i+1}^{m_1} G_{2,k}.\n\\end{equation*}\nFor $1\\le \\nu \\le 2$, $1\\leq k \\leq m_1$ we obtain, by employing linear interpolation,\n\\begin{equation*}\\label{Gnuk}\n\tG_{\\nu,k} = \n\t\\gamma_{\\nu,k}^{0} V_{k-1,0}(t) + \\gamma_{\\nu,k}^{1} V_{k,0}(t)\n\\end{equation*}\nwith\n\\begin{equation*}\n\t\\gamma_{\\nu,k}^{0} = (s_{1,k}\\zeta_{\\nu,k}^{0} - \\zeta_{\\nu,k}^{1})\/\\Delta s_{1,k}\n\t\\quad \\textrm{and} \\quad\n\t\\gamma_{\\nu,k}^{1} = (-s_{1,k-1}\\zeta_{\\nu,k}^{0} + \\zeta_{\\nu,k}^{1} )\/\\Delta s_{1,k}\n\\end{equation*}\nwhere\n\\begin{equation*}\n\t\\zeta_{1,k}^{a} =  \n\t\\frac{s_{1,k}^{a+\\eta_{q_1}}-s_{1,k-1}^{a+\\eta_{q_1}}}{a+\\eta_{q_1}}\n\t\\quad \\textrm{and} \\quad\n\t\\zeta_{2,k}^{a} =  \n\t\\frac{s_{1,k}^{a-\\eta_{p_1}}-s_{1,k-1}^{a-\\eta_{p_1}}}{a-\\eta_{p_1}}\n\t\\quad (a=0,1).\n\\end{equation*}\n\nOn the boundary part $\\{(0,s_2): s_2>0\\}$ the double integral \\eqref{integral} reduces \nto the single integral\n\\begin{equation}\\label{integral2_1D}\n\t\\mathcal{J} = \\lambda\\int_0^{\\infty} f_2(y_2)v(0,s_2y_2,t) \\mathrm{d} y_2\n\\end{equation}\nand the discretization is performed completely similarly as above. \nFinally, at the spatial grid point $(s_1,s_2)=(0,0)$ it holds that \n$\\mathcal{J} \\approx  \\lambda V_{0,0}(t)$.\n\nFrom the above it follows that the number of basic arithmetic operations to compute, for any given $t$, the approximation to the double \nintegral \\eqref{integral} on the full spatial grid is, to leading order, equal to $40m_1m_2$.\n\n\t\n\n\n\n\t\n\\subsection{Semidiscrete PIDE}\\label{SecSemiPIDE}\nCombining the above semidiscretizations for the convection-diffusion-reaction and integral parts of PIDE \\eqref{PIDE2D}, the following \nlarge system of ODEs is obtained:\n\\begin{align}\\label{ODEs}\n\tV'(t) = AV(t) \\quad (0<t\\leq T),\n\\end{align}\nwhere \n\\begin{equation*}\n\tA = A^{(D)} + A^{(J)} = A^{(M)} + A_1 + A_2 + A^{(J)}.\n\\end{equation*}\nEven though we never actually compute it in this way, the approximation to the double integral \\eqref{integral} \ndefined in Subsection~\\ref{SecIntpart} is formally denoted here by $A^{(J)} V(t)$ with given (full) matrix $A^{(J)}$.\\\\\n\nThe initial vector $V(0)=V^0$ is defined by pointwise evaluation of the payoff function $\\phi$ on the spatial grid, \nexcept at those grid points that lie near to the line segment given by $s_1+s_2 = 2K$ where $\\phi$ is nonsmooth.\nIt is well-known that using pointwise values of $\\phi$ in such a region can lead to a deteriorated (spatial) \nconvergence behaviour, which can be alleviated by applying cell averaging. \nAccordingly, for $k=1,2$ let\n\\begin{eqnarray*}\n\t&s_{k,l+1\/2} = \\tfrac{1}{2}(s_{k,l}+s_{k,l+1})~~~~ &{\\rm if}~~ 0\\le l < m,\\\\\n\t&h_{k,l+1\/2} = s_{k,l+1\/2}-s_{k,l-1\/2} &{\\rm if}~~ 0\\le l \\le m,\n\\end{eqnarray*}\nwith $s_{k,-1\/2} = -s_{k,1\/2}$ and $s_{k,m+1\/2} = S_{\\rm max}$.\nThen we define \\cite{H17book}\n\\begin{equation}\\label{cellave}\n\tV_{i,j}(0) = \\frac{1}{h_{1,i+1\/2}h_{2,j+1\/2}}\\int_{s_{1,i-1\/2}}^{s_{1,i+1\/2}}\\int_{s_{2,j-1\/2}}^{s_{2,j+1\/2}} \\phi(s_1,s_2) \\mathrm{d} s_2 \\mathrm{d} s_1\n\\end{equation}\nwhenever the cell \n\\[\n[s_{1,i-1\/2}, s_{1,i+1\/2})\\times[s_{2,j-1\/2}, s_{2,j+1\/2})\n\\]\nhas a nonempty intersection with the line segment $s_1+s_2 = 2K$. \nNote that the double integral \\eqref{cellave} is easily calculated.\n\n\n\n\\section{Temporal discretization}\\label{SecTime}\nFor the temporal discretization of the semidiscrete PIDE \\eqref{ODEs} we study seven operator splitting schemes of the IMEX and ADI \nkind~\\cite{BH21}.\nLet integer $N\\ge 1$ be given and define step size $\\Delta t = T\/N$ and temporal grid points $t_n = n \\Delta t$ for $n=0,1,2,\\ldots,N$.\nEach of the following schemes generates an approximation $V^{n}$ to $V(t_n)$ successively for $n=1,2,\\ldots,N$.\nHere the integral part of \\eqref{ODEs}, corresponding to the matrix $A^{(J)}$, is always conveniently treated in an explicit manner\nand evaluated by applying the efficient algorithm derived in Subsection~\\ref{SecIntpart}.\n\n\\begin{enumerate}\n\t\n\t\\item \\textit{Crank--Nicolson Forward Euler (CNFE) scheme}:\n\t\\vskip0.1cm\n\tIn this one-step IMEX method, the convection-diffusion-reaction part is handled implicitly by the Crank--Nicolson scheme and the \n\tintegral part is treated in a simple, explicit Euler fashion:\n\t\\begin{align}\\label{CNFE}\n\t\t\\left(I-\\tfrac{1}{2}\\Delta t A^{(D)}\\right)V^{n} = \\left(I + \\tfrac{1}{2}\\Delta t A^{(D)}\\right)V^{n-1} + \\Delta t A^{(J)} V^{n-1}.\n\t\\end{align}\n\tDue to the application of explicit Euler, the order\\footnote{This means the classical order of consistency, i.e., for fixed \n\t\tnonstiff ODEs.} of the CNFE scheme is just equal to one. \n\t\\vskip0.2cm\n\t\n\t\\item \\textit{Crank--Nicolson scheme with fixed-point iteration (CNFI)}:\n\t\\vskip0.1cm\n\tTavella \\& Randall \\cite{TR00book} and d'Halluin, Forsyth \\& Vetzal \\cite{HFV05} proposed to combine the Crank--Nicolson scheme for \n\tthe convection-diffusion-reaction part with a fixed-point iteration on the integral part:\n\t\\begin{align}\\label{CNFI}\n\t\t\\left(I-\\tfrac{1}{2}\\Delta t A^{(D)}\\right)Y_k = \\left(I + \\tfrac{1}{2}\\Delta t A^{(D)}\\right)V^{n-1} +  \\tfrac{1}{2} \\Delta t A^{(J)} (Y_{k-1}+V^{n-1})\n\t\\end{align}\n\tfor $k=1,2,\\ldots,l$ and $V^n = Y_l$ with starting value $Y_0 = V^{n-1}$. \n\tNumerical evidence in \\cite{CF08,HFV05} indicates that in general $l=2$ or $l=3$ iterations suffice.\n\tWe shall consider in this paper a fixed number of $l=2$ iterations.\n\tThen method \\eqref{CNFI} can be viewed as a one-step IMEX scheme, where the integral part is treated by the explicit trapezoidal rule, \n\talso called the modified Euler method. \n\tBy Taylor expansion, it can be seen that its order is equal to two.\n\t\n\t\\item \\textit{Implicit-Explicit Trapezoidal Rule (IETR)}:\n\t\\vskip0.1cm\n\tAnother blend of the implicit trapezoidal rule (Crank--Nicolson) for the convection-diffusion-reaction part and the explicit \n\ttrapezoidal rule for the integral part has been considered in~\\cite{H17book}:\n\t\\begin{equation}\\label{IETR}\n\t\t\\left\\{\\begin{array}{lll}\n\t\t\tY_0 = V^{n-1} + \\Delta t\\, (A^{(D)}+A^{(J)})V^{n-1},\\\\\n\t\t\t\\widehat{Y}_0 = Y_0+\\tfrac{1}{2} \\Delta t A^{(J)} \\big(Y_0-V^{n-1}\\big),\\\\\n\t\t\tY_1 = \\widehat{Y}_0+\\tfrac{1}{2} \\Delta t A^{(D)} \\big(Y_1-V^{n-1}\\big),\\\\\n\t\t\tV^n = Y_1.\n\t\t\\end{array}\\right.\n\t\\end{equation} \n\tMethod~\\eqref{IETR} also forms a one-step IMEX scheme with order equal to two.  \n\tIt is already well-known in the literature on the numerical solution of PDEs (without integral part), see e.g.~Hundsdorfer \\& \n\tVerwer~\\cite{HV03book}.\n\tNotice that the first two internal stages $Y_0$, $\\widehat{Y}_0$ are explicit, whereas the third internal stage $Y_1$ is implicit.\n\t\\vskip0.2cm \n\t\n\t\\item \\textit{Crank--Nicolson Adams--Bashforth (CNAB) scheme}:\n\t\\vskip0.1cm\n\tThe CNAB method has been amply considered in the literature on the numerical solution of PDEs as well, \n\tsee e.g.~\\cite{FHV97,HV03book,H02}.\n\tIt has been investigated for the numerical solution of PIDEs by Salmi \\& Toivanen \\cite{ST14} and Salmi, Toivanen \\& von Sydow \n\t\\cite{STS14}. \n\tThe convection-diffusion-reaction part is again handled by the Crank--Nicolson scheme, but the integral part is now dealt with \n\tin a two-step Adams--Bashforth fashion:\n\t\\begin{align}\\label{CNAB}\n\t\t\\left(I-\\tfrac{1}{2} \\Delta t A^{(D)}\\right)V^{n} = \\left(I+\\tfrac{1}{2} \\Delta t A^{(D)}\\right)V^{n-1} + \\tfrac{1}{2}\\Delta t A^{(J)}(3V^{n-1}-V^{n-2}).\n\t\\end{align}\n\tMethod \\eqref{CNAB} constitutes a two-step IMEX scheme and is of order equal to two. \n\tIt is interesting to remark that \\eqref{CNAB} can also be viewed as obtained from \\eqref{CNFI}, upon taking $l=1$ and \n\timproved (linearly extrapolated) starting value $Y_0 = 2V^{n-1}-V^{n-2}$.\n\t\\vskip0.2cm \n\t\n\\end{enumerate}\n\nAs for the spatial discretization, it is well-known that the nonsmoothness of the payoff function can have an unfavourable \nimpact on the convergence behaviour of the temporal discretization, even if cell averaging is applied, cf.~\\cite{H17book}.\nFor the Crank--Nicolson scheme a common approach to alleviate this has been proposed by Rannacher~\\cite{R84}. \nIt consists of replacing (only) the first time step by two time steps with step size $\\Delta t\/2$ using the implicit Euler method, \nthus defining the approximation $V^1$ to $V(t_1)$.\nIn the present case of two-dimensional PIDEs, the implicit Euler method can be computationally intensive.\nWe therefore consider, for all four IMEX schemes above, the {\\it IMEX Euler scheme} as the starting method:\n\\begin{align*}    \n\t\\left(I-\\tfrac{1}{2}\\Delta t A^{(D)}\\right)V^{\\tfrac{1}{2}} &= V^{0} + \\tfrac{1}{2}\\Delta t A^{(J)} V^{0},\\\\\n\t\\left(I-\\tfrac{1}{2}\\Delta t A^{(D)}\\right)V^{1} &= V^{\\tfrac{1}{2} } + \\tfrac{1}{2}\\Delta t A^{(J)} V^{\\tfrac{1}{2}}.\n\\end{align*} \n\nClearly, in each of the IMEX schemes under consideration, linear systems need to be solved involving the matrix \n$I-\\tfrac{1}{2}\\Delta t A^{(D)}$. \nWhereas this matrix is sparse, with at most nine nonzero entries per row, it possesses a large bandwidth that is directly \nproportional to $m_1$.\nTo obtain a further reduction in computational work, we consider three recent operator splitting schemes for PIDEs that employ \nalso the (directional) splitting $A^{(D)} = A^{(M)} + A_1 + A_2$ of the two-dimensional convection-diffusion-reaction part, such \nthat only linear systems with tridiagonal matrices arise.\n\n\\begin{enumerate} \n\t\n\t\\item[5.] \\textit{One-step adaptation of the modified Craig--Sneyd (MCS) scheme}:\n\t\\vskip0.1cm\n\tThe MCS scheme is an ADI scheme that was introduced by in 't Hout \\& Welfert \\cite{HW09} for the numerical solution of \n\tPDEs containing mixed spatial derivative terms.\n\tThe following, direct adaptation to PIDEs has recently been investigated by in 't Hout \\& Toivanen \\cite{HT18}:\n\t\\begin{equation}\\label{MCS}\n\t\t\\left\\{\\begin{array}{lll}\n\t\t\tY_0 = V^{n-1} + \\Delta t\\, (A^{(D)}+A^{(J)})V^{n-1},\\\\\n\t\t\tY_j = Y_{j-1} + \\theta\\Delta t A_j(Y_j-V^{n-1}) \\quad (j=1,2),\\\\\n\t\t\t\\widehat{Y}_0 = Y_0 + \\theta \\Delta t\\, (A^{(M)}+A^{(J)})(Y_2 - V^{n-1}),\\\\\n\t\t\t\\widetilde{Y}_0 = \\widehat{Y}_0 + (\\tfrac{1}{2}-\\theta)\\Delta t\\, (A^{(D)}+A^{(J)})(Y_2-V^{n-1}),\\\\\n\t\t\t\\widetilde{Y}_j = \\widetilde{Y}_{j-1} + \\theta \\Delta t A_j(\\widetilde{Y}_j - V^{n-1}) \\quad (j=1,2),\\\\\n\t\t\tV^n = \\widetilde{Y}_2,\n\t\t\\end{array}\\right.\n\t\\end{equation}   \n\twhere $\\theta>0$ is a given parameter.\n\tMethod \\eqref{MCS} is of order two for any value~$\\theta$.\n\tHere we make the common choice $\\theta = \\frac{1}{3}$, which is motivated by stability and accuracy results \n\tfor two-dimensional problems, see e.g.~\\cite{HM11,HT18,HW09,HW16}.\n\tIt is easily verified that in \\eqref{MCS} the integral part and mixed derivative term are both treated by the \n\texplicit trapezoidal rule.\n\tWe note that the explicit stages $\\widehat{Y}_0$, $\\widetilde{Y}_0$ can be merged, so that the integral part \n\tis evaluated (just) twice per time step.\n\tThe implicit stages $Y_j$, $\\widetilde{Y}_j$ (for $j=1,2$) are often called stabilizing corrections.\n\tThe four pertinent linear systems for these stages are tridiagonal and can be solved very efficiently \n\tby means of an a priori $LU$ factorization.\n\tWe mention that \\eqref{MCS} is already applied in the first time step, i.e., for defining $V^1$ (thus \n\tIMEX Euler is not used here).\n\t\\vskip0.2cm \n\t\n\t\\item[6.] \\textit{Two-step adaptation of the MCS (MCS2) scheme}:\n\t\\vskip0.1cm\n\tIn \\cite{HT18} a second adaptation of the MCS scheme to PIDEs has been proposed where, instead of the explicit \n\ttrapezoidal rule, the integral part is now handled in a two-step Adams--Bashforth fashion:\n\t\\begin{equation}\\label{MCS2}\n\t\t\\left\\{\\begin{array}{lll}\n\t\t\tX_0 = V^{n-1}+\\Delta t A^{(D)}V^{n-1},\\\\\n\t\t\tY_0 = X_0 + \\tfrac{1}{2}\\Delta t A^{(J)}(3V^{n-1}-V^{n-2}),\\\\\n\t\t\tY_j = Y_{j-1}+\\theta\\Delta t A_j(Y_j-V^{n-1})\\quad (j=1,2),\\\\\n\t\t\t\\widehat{Y}_0 = Y_0 + \\theta\\Delta t A^{(M)}(Y_2-V^{n-1}),\\\\\n\t\t\t\\widetilde{Y}_0 = \\widehat{Y}_0 + (\\tfrac{1}{2}-\\theta)\\Delta t A^{(D)}(Y_2-V^{n-1}),\\\\\n\t\t\t\\widetilde{Y}_j = \\widetilde{Y}_{j-1} + \\theta \\Delta t A_j(\\widetilde{Y}_j-V^{n-1})\\quad (j=1,2),\\\\\n\t\t\tV^n = \\widetilde{Y}_2.\n\t\t\\end{array}\\right.\n\t\\end{equation}   \n\tMethod \\eqref{MCS2} is also of order two for any value~$\\theta$.\n\tWe choose again $\\theta = \\frac{1}{3}$ and, for starting this two-step method, define $V^1$ by \\eqref{MCS}.\n\t\\vskip0.2cm \n\t\n\t\\item[7.] \\textit{Stabilizing correction two-step Adams-type (SC2A) scheme}:\n\t\\vskip0.1cm\n\tHundsdorfer \\& in 't Hout \\cite{HH18} recently studied a novel class of stabilizing correction multistep methods \n\tfor the numerical solution of PDEs.\n\tWe consider here a direct adaptation to PIDEs of a prominent member of this class, the two-step Adams-type scheme \n\tcalled SC2A:\n\t\\begin{equation}\\label{SC2A}\n\t\t\\left\\{\\begin{array}{lll}   \n\t\t\tY_0 = V^{n-1} + \\Delta t\\, (A^{(M)}+A^{(J)})\\sum_{i=1}^2\\widehat{b}_iV^{n-i} + \\Delta t\\, (A_1+A_2)\\sum_{i=1}^2\\widecheck{b}_iV^{n-i},\\\\\n\t\t\tY_j = Y_{j-1} + \\theta \\Delta t A_j(Y_j-V^{n-1})\\quad (j=1,2),\\\\\n\t\t\tV^n = Y_2,\n\t\t\\end{array}\\right.\n\t\\end{equation}   \n\twith coefficients $(\\widehat{b}_1,\\widehat{b}_2) = \\left(\\frac{3}{2},-\\frac{1}{2}\\right)$ and \n\t$(\\widecheck{b}_1,\\widecheck{b}_2) = \\left(\\frac{3}{2}-\\theta, - \\frac{1}{2}+\\theta\\right)$.\n\tThe integral part and mixed derivative term are now both handled by the two-step Adams--Bashforth scheme.\n\tMethod \\eqref{SC2A} is again of order two for any value~$\\theta$. \n\tFollowing \\cite{HH18}, we take $\\theta = \\frac{3}{4}$, which is based upon stability and accuracy results.\n\tFor starting \\eqref{SC2A}, we define $V^1$ again by \\eqref{MCS} with $\\theta = \\frac{1}{3}$\n\t\\vskip0.2cm    \n\t\n\\end{enumerate}\n\n\n\n\n\n\t\n\\section{Numerical study}\\label{SecResults}\nIn this section we present ample numerical experiments for the seven operator splitting schemes formulated in Section~\\ref{SecTime}, \nwhich provides important insight in their convergence behaviour and mutual performance.\nThe experiments involve three different parameter sets for the two-asset Kou model, labelled as 1, 2, and~3. \nThe parameter set~1 is taken from Clift \\& Forsyth \\cite{CF08}. \nSet~2 is a blend of parameters considered for the one-asset Kou model by Almendral \\& Oosterlee~\\cite{AO05} and d'Halluin, Forsyth \n\\& Vetzal~\\cite{HFV05} (see also Toivanen~\\cite{T08}). \nSet~3 has been newly constructed and includes a relatively large value for the product $\\lambda T$, i.e., the expected number of \njumps in $[0,T]$. \nFor the truncated spatial domain, we (heuristically) select $S_{\\rm max} = 20K, 10K, 30K$ for sets 1, 2, 3, respectively.\n\n\\begin{table}[H]\n\t\\caption{Parameter sets for the two-asset Kou jump-diffusion model.}\n\t\\small\n\t\\begin{tabular}{@{}llllllllllllll@{}}\n\t\t\\toprule\n\t\t& $\\sigma_1$ & $\\sigma_2$ & $r$ & $\\rho$ & $\\lambda$ & $p_1$ & $p_2$ & $\\eta_{p_1}$ & $\\eta_{q_1}$ & $\\eta_{p_2}$ & $\\eta_{q_2}$ & $K$ & $T$ \\\\ \\midrule\n\t\tSet 1 & 0.12 & 0.15 & 0.05 & 0.30 & 0.50 & 0.40 & 0.60 & $1\/0.20$ & $1\/0.15$ & $1\/0.18$ & $1\/0.14$ & 100 & 1 \\\\\n\t\tSet 2 & 0.15 & 0.20 & 0.05 & 0.50 & 0.20 & 0.3445 & 0.50 & 3.0465 & 3.0775 & 3 & 2 & 100 & 0.2 \\\\\n\t\tSet 3 & 0.20 & 0.30 & 0.05 & 0.70 & 8 & 0.60 & 0.65 & 5 & 4 & 4 & 3 & 100 & 1 \\\\ \\bottomrule\n\t\\end{tabular}\n\t\\label{paramsets}\n\\end{table}\n\n\n\\subsection{Option values and numerical convergence behaviour}\\label{SecResults1}\nFigure~\\ref{FigSurfacePlots} shows the numerically computed\\footnote{With $m_1=m_2=200$, $N=100$ and the MCS2 scheme.} option value \nsurfaces for the European put-on-the-average option and the three parameter sets from Table~\\ref{paramsets} on the asset price \ndomain $[0,3K]\\times[0,3K]$ and $t=T$.\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[trim = 1.4in 3.7in 1.3in 3.7in, clip, scale=0.5]{surf1.pdf}\t\n\t\\includegraphics[trim = 1.4in 3.7in 1.3in 3.7in, clip, scale=0.5]{surf2.pdf}\t\n\t\\includegraphics[trim = 1.3in 3.7in 1.3in 3.7in, clip, scale=0.5]{surf3.pdf}\n\t\\caption{Option value surfaces for the European put-on-the-average option under the two-asset Kou model in the case of parameter \n\tset 1 (top left), 2 (top right) and 3 (bottom) given in Table~\\ref{paramsets}.}\n\t\\label{FigSurfacePlots}\n\\end{figure}\n\n\\noindent\nFor future reference, accurate approximations to the option values have been \ncomputed\\footnote{Using $m_1=m_2=1000$, $N=500$, the MCS2 scheme and spline interpolation.} \nfor specifically chosen spot prices $S_0^{(1)}$ and $S_0^{(2)}$ of the two assets in a\nneighbourhood of the strike $K$, see Table~\\ref{tablevalues}. \n\\vskip0.3cm\n\n\\begin{table}[H]\n\t\\caption{\\label{tablevalues}European put-on-the-average option value approximations for parameter sets 1, 2 and 3.}\n\t\\vspace{0.5cm}\n\t\\centering\n\tSet 1\\\\\\vspace{0.1cm}\n\t\\begin{tabular}{llrrrrr}\n\t\t\\toprule\n\t\t&  & $S_0^{(1)} = 90$ &  & $S_0^{(1)} = 100$ &  & $S_0^{(1)} = 110$ \\\\\n\t\t&  & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} \\\\\n\t\t$S_0^{(2)} = 90$  &  & 8.9385 &  & 6.0316 &  & 3.8757 \\\\\n\t\t$S_0^{(2)} = 100$ &  & 5.9655 &  & 3.8038 &  & 2.3370 \\\\\n\t\t$S_0^{(2)} = 110$ &  & 3.7641 &  & 2.2978 &  & 1.3771 \\\\ \n\t\t\\bottomrule\n\t\\end{tabular}\\vspace{0.5cm}\n\n\tSet 2\\\\\\vspace{0.1cm}\n\t\\begin{tabular}{llrrrrr}\n\t\t\\toprule\n\t\t&  & $S_0^{(1)} = 90$ &  & $S_0^{(1)} = 100$ &  & $S_0^{(1)} = 110$ \\\\\n\t\t&  & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} \\\\\n\t\t$S_0^{(2)} = 90$  &  & 9.6863 &  & 5.5616 &  & 2.6400 \\\\\n\t\t$S_0^{(2)} = 100$ &  & 5.6162 &  & 2.6929 &  & 1.1264 \\\\\n\t\t$S_0^{(2)} = 110$ &  & 2.7570 &  & 1.1670 &  & 0.5246 \\\\ \n\t\t\\bottomrule\n\t\\end{tabular}\\vspace{0.5cm}\n\n\tSet 3\\\\\\vspace{0.1cm}\n\t\\begin{tabular}{llrrrrr}\n\t\t\\toprule\n\t\t&  & $S_0^{(1)} = 90$ &  & $S_0^{(1)} = 100$ &  & $S_0^{(1)} = 110$ \\\\\n\t\t&  & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} \\\\\n\t\t$S_0^{(2)} = 90$  &  & 32.7459 &  & 31.0984 &  & 29.5758 \\\\\n\t\t$S_0^{(2)} = 100$ &  & 30.5796 &  & 29.0181 &  & 27.5770 \\\\\n\t\t$S_0^{(2)} = 110$ &  & 28.5830 &  & 27.1033 &  & 25.7396 \\\\ \n\t\t\\bottomrule\n\t\\end{tabular}\\vspace{0.5cm}\n\\end{table}\n\nWe next examine, through numerical experiments, the convergence behaviour of the seven operator splitting schemes formulated \nin Section~\\ref{SecTime} for the temporal discretization of the semidiscrete two-dimensional Kou PIDE. \nTake $m_1=m_2=m$.\nConsider a region of financial interest\n\\begin{equation*}\\label{ROI}\n\t\\textrm{ROI} = \\left\\{(s_1,s_2) : \\tfrac{1}{2}K < s_1,s_2 < \\tfrac{3}{2}K\\right\\}\n\\end{equation*}\nand define the {\\it temporal discretization error} on this ROI for $t=T$ by\n\\begin{equation}\\label{EROI}\n\t\\widehat{E}^{\\rm ROI}(m,N) = \\textrm{max} \\left\\{| V_{i,j}(T) - V_{i,j}^{N'} |: \\Delta t = T\/N' ~~{\\rm and}~~ \n\t(s_{1,i},s_{2,j})\\in \\rm ROI\\right\\}.\n\\end{equation}\nClearly, this error is measured in the (important) maximum norm.\nThe MCS2 scheme is applied with $3000$ time steps to obtain a reference value for the exact semidiscrete solution $V(T)$ to \n\\eqref{ODEs}.\nFor each of the seven operator splitting schemes formulated in Section~\\ref{SecTime}, we study the temporal discretization \nerror for a sequence of values $N$, where the approximation $V^{N'}$ to $V(T)$ is computed using $N'$ time steps.\nHere $N'$ is chosen in function of $N$ depending on the specific scheme, so as to arrive at a fair comparison between the \nschemes.\n\nTheoretical as well as experimental evidence indicates that for the four IMEX schemes the process of solving the linear systems, \ninvolving the matrix $I-\\tfrac{1}{2}\\Delta t A^{(D)}$, essentially determines the total computational time in each time step \nwhenever $m$ is sufficiently large ($m\\gtrsim 150$). \nIn particular, the time for evaluating the integral part, by means of the algorithm from Subsection~\\ref{SecIntpart}, forms \nonly a small fraction of the total computational time.\nAccordingly, for a fair comparison between the IMEX schemes, we apply the three schemes (\\ref{CNFE}), (\\ref{IETR}), (\\ref{CNAB}) \nwith $N' = 2N$ time steps and the scheme (\\ref{CNFI}) with just $N'=N$ time steps, since in the latter scheme double the amount \nof linear systems is solved in each time step.\n\nFor the three ADI schemes we find that the evaluation of the two-dimensional integral part takes about the same computational\ntime as the solution of four pertinent tridiagonal linear systems. \nThe scheme (\\ref{MCS}) employs two evaluations of the integral part, whereas (\\ref{MCS2}) and (\\ref{SC2A}) each use only one \nsuch evaluation.\nNext, both schemes (\\ref{MCS}) and (\\ref{MCS2}) require the solution of four tridiagonal systems, whereas (\\ref{SC2A}) involves \nonly two tridiagonal systems.\nAccordingly, for a fair mutual comparison, we apply the ADI schemes (\\ref{MCS}), (\\ref{MCS2}) and (\\ref{SC2A}) with, respectively, \n$N'=N$, $N' = \\lfloor 3N\/2 \\rfloor$ and $N' = 2N$ time steps.\n\nFor each of the seven splitting schemes, the temporal discretization errors $\\widehat{E}^{\\rm ROI}(m,N)$ have been computed for \n$m = 200$ and a sequence of values $N$ between 10 and 1000. \nFigure~\\ref{FigTemperrors} displays the obtained temporal errors for all three parameter sets given in Table~\\ref{paramsets}, \nwhere the left column shows the results for the IMEX schemes and the right column the results for the ADI schemes.\n\nFrom Figure~\\ref{FigTemperrors} we observe the positive result that, for each given scheme and parameter set, the temporal errors \nare bounded from above by a moderate value and decrease monotonically as $N$ increases. \nAs expected, the CNFE scheme (\\ref{CNFE}) shows an order of convergence equal to one, whereas all other six splitting schemes \nreveal a desirable order of convergence equal to two.\nBy repeating the numerical experiments for spatial grids that are finer (e.g.~$m=400$) or coarser (e.g.~$m=50, 100$) \nwe obtain results that are visually identical to those displayed in Figure~\\ref{FigTemperrors}. \nThis indicates that, for each splitting scheme, the temporal errors are essentially independent of the spatial grid \nsize, and hence, its observed convergence order is valid in a stiff sense, which is very favourable.\n\nIt is interesting to remark that the error constants become larger as the intensity of the jumps increases, keeping all\nelse fixed.\nThis has been observed in additional numerical experiments for increasing values of $\\lambda$ between 0 and 10.\n\nAmong the four IMEX schemes, the IETR and CNAB schemes yield the smallest error constants in the numerical experiments.\nAmong the three ADI schemes, the MCS2 scheme has the smallest error constant in all experiments.\nOverall, the MCS2 scheme is preferred.\nIt yields temporal errors that are smaller than or approximately equal to the IETR and CNAB schemes, but is computationally \nmuch faster since the linear systems appearing in this scheme are just tridiagonal.\nAs an indication for actual run times of the MCS2 scheme, we mention that they are approximately 6e-4, 2e-3, 7e-3, \n4.5e-2 CPU seconds per time step if $m=50,100,200,400$, respectively, in the case of our Matlab implementation, \nversion R2020b, on an Intel Core i7-8665U processor at 1.9 GHz with 16 GB memory.\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{TemperrorIMEX_jul8_set1.pdf}\\includegraphics[trim = 1.3in 3.7in 1.6in 3.7in, clip, scale=0.5]{TemperrorADI_jul8_set1.pdf}\\\\\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{TemperrorIMEX_jul8_set2.pdf}\\includegraphics[trim = 1.3in 3.7in 1.6in 3.7in, clip, scale=0.5]{TemperrorADI_jul8_set2.pdf}\\\\\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{TemperrorIMEX_jul8_set3.pdf}\\includegraphics[trim = 1.3in 3.7in 1.6in 3.7in, clip, scale=0.5]{TemperrorADI_jul8_set3.pdf}\n\t\\caption{Temporal errors $\\widehat{E}^{\\rm ROI}(200,N)$ of the seven operator splitting schemes under consideration. IMEX schemes on the left side and ADI \n\tschemes on the right side, with parameter set 1 (top), set 2 (middle) and set 3 (bottom) from Table~\\ref{paramsets}. The CNFI and MCS schemes are applied \n\twith step size $\\Delta t = T\/N$, the MCS2 scheme is applied with $\\Delta t = T\/\\lfloor 3N\/2 \\rfloor$ and the CNFE, IETR, CNAB, SC2A schemes are applied \n\twith $\\Delta t = T\/(2N)$.}\n\t\\label{FigTemperrors}\n\\end{figure}\n\n\\newpage\n\\subsection{The Greeks}\\label{Greeks}\nThe Greeks are mathematical derivatives of the option value with respect to underlying variables and parameters. \nThey constitute a measure for risk that indicates how sensitive an option value is to changes in underlying variables and parameters \nand are crucial for hedging strategies. \nIn this subsection we consider the numerical approximation of the Delta and Gamma Greeks. \nDelta is a measure for the rate of change of the option value with respect to a change in an underlying asset price.\nAs there are two underlying assets, there are two Deltas:\n\\begin{equation*}\n\\Delta_1 = \\frac{\\partial v}{\\partial s_1} \\quad \\text{and} \\quad \\Delta_2 = \\frac{\\partial v}{\\partial s_2} . \n\\end{equation*}\nNext, Gamma measures the rate of change of a Delta with respect to a change in an underlying asset price. \nThere are three different Gammas:\n\\begin{equation*}\n\t\\Gamma_{11} = \\frac{\\partial \\Delta_1}{\\partial s_1} = \\frac{\\partial^2v}{\\partial s_1^2}, \\quad \n\t\\Gamma_{22} = \\frac{\\partial \\Delta_2}{\\partial s_2} = \\frac{\\partial^2v}{\\partial s_2^2} \\quad \\text{and} \\quad \n\t\\Gamma_{12} = \\frac{\\partial \\Delta_1}{\\partial s_2} = \\frac{\\partial^2v}{\\partial s_2 \\partial s_1} = \\frac{\\partial \\Delta_2}{\\partial s_1} = \\Gamma_{21}.\n\\end{equation*}\n\nBy virtue of the finite difference discretization that has been defined in Section \\ref{SecSpatial}, the Delta and Gamma Greeks\ncan directly be approximated, at essentially no computational cost, by applying the second-order central finite \ndifference formulas considered in Subsection \\ref{SecPDEpart} to the option value approximations on the spatial grid.\n\nAs an illustration, Figure~\\ref{FigGreeks} displays the numerically\\footnote{With $m_1=m_2=200$, $N=100$ and the MCS2 scheme.} \nobtained Delta and Gamma surfaces at maturity for the European put-on-the-average option under the two-asset Kou model for \nparameter set~1. \nAs expected, the Delta surfaces are steepest around the line segment $s_1 + s_2 = 2K$ and, correspondingly, the \nGamma surfaces are highest there.\n\nSimilarly to the option value, we study the temporal convergence behaviour of all operator splitting schemes in \nthe case of the five Greeks.\nAkin to (\\ref{EROI}), the temporal discretization error in the case of Delta $\\Delta_k \\, (k = 1, 2)$ is defined by\n\\begin{equation}\\label{EROIGreeks}\n\t\\widehat{E}^{\\rm ROI}_{\\Delta_k}(m,N) = \\textrm{max} \\left\\{| (\\Delta_k)_{i,j}(T) - (\\Delta_k)_{i,j}^{N'} |: \n\t\\Delta t = T\/N' ~~{\\rm and}~~ (s_{1,i},s_{2,j})\\in \\rm ROI \\right\\}.\n\\end{equation}\nHere $\\Delta_k (T)$ denotes the pertinent finite difference matrix for convection applied to the reference value for the exact\nsemidiscrete solution $V(T)$ given in Subsection~\\ref{SecResults1}. \nNext, $\\Delta_k ^{N'}$ is equal to the same finite difference matrix applied to the approximation $V^{N'}$ of $V(T)$ that is \ngenerated by any one of the seven operator splitting schemes. \nAnalogous temporal discretization error definitions hold in the case of $\\Gamma_{11}, \\Gamma_{22}$, $\\Gamma_{12}$. \n\nFigure~\\ref{FigGreeksError} displays the temporal errors in the case of the five Greeks and parameter set 1 for $m = 200$ and \nthe same sequence of values $N$ between 10 and 1000 as before.\nWe arrive at the same conclusions on the temporal convergence behaviour of the seven splitting schemes as obtained \nin Subsection~\\ref{SecResults1} regarding the option value.\nIn particular, besides CNFE, all splitting schemes reveal a stiff order of convergence equal to two, and MCS2 has the best\nperformance among all these schemes.\n\nWe note that the somewhat larger errors that are observed for the CNFI, IETR, CNAB schemes when $N$ is small are attributed\nto the nonsmootness of the initial (payoff) function and could be alleviated by applying four (instead of two) half time steps\nwith IMEX Euler at the start of the time stepping, cf. Section~\\ref{SecTime}.\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{Delta1.pdf}\\includegraphics[trim = 1.3in 3.7in 1.4in 3.7in, clip, scale=0.5]{Delta2.pdf}\\\\\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{Gamma11.pdf}\\includegraphics[trim = 1.3in 3.7in 1.4in 3.7in, clip, scale=0.5]{Gamma22.pdf}\\\\\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{Gamma12.pdf}\n\t\\caption{\n\tFirst-order Greeks $\\Delta_1$ (top left) and $\\Delta_2$ (top right) and second-order Greeks $\\Gamma_{11}$ (middle left), $\\Gamma_{22}$ \n\t(middle right) and $\\Gamma_{12}$ (bottom) for the European put-on-the-average option under the two-asset Kou model in the case of \n\tparameter set 1.}\n\t\\label{FigGreeks}\n\\end{figure}\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{errorDelta1_jul14_set1.pdf}\\includegraphics[trim = 1.3in 3.7in 1.6in 3.7in, clip, scale=0.5]{errorDelta2_jul14_set1.pdf}\\\\\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{errorGamma11_jul14_set1.pdf}\\includegraphics[trim = 1.3in 3.7in 1.6in 3.7in, clip, scale=0.5]{errorGamma22_jul14_set1.pdf}\\\\\n\t\\includegraphics[trim = 1.3in 3.7in 1in 3.7in, clip, scale=0.5]{errorGamma12_jul14_set1.pdf}\n\t\\caption{Temporal errors of the seven operator splitting schemes under consideration in the case of the five Greeks and parameter set 1: \n\t$\\widehat{E}^{\\rm ROI}_{\\Delta_1}$ (top left), $\\widehat{E}^{\\rm ROI}_{\\Delta_2}$ (top right), $\\widehat{E}^{\\rm ROI}_{\\Gamma_{11}}$ \n\t(middle left), $\\widehat{E}^{\\rm ROI}_{\\Gamma_{22}}$ (middle right) and $\\widehat{E}^{\\rm ROI}_{\\Gamma_{12}}$ (bottom). \n\tThe CNFI and MCS schemes are applied with step size $\\Delta t = T\/N$, the MCS2 scheme is applied with $\\Delta t = T\/\\lfloor 3N\/2 \\rfloor$ \n\tand the CNFE, IETR, CNAB, SC2A schemes are applied with $\\Delta t = T\/(2N)$.}\n\t\\label{FigGreeksError}\n\\end{figure}\n\n\n\n\n\n\n\\newpage\n\\section{Conclusions}\\label{SecConc}\nWe have studied the valuation of European options under the two-asset Kou jump-diffusion model via the numerical solution of the \npertinent two-dimensional time-dependent PIDE.\nA first main contribution of our paper is the extension of an algorithm derived by Toivanen~\\cite{T08}, which enables a highly \nefficient numerical evaluation of the nonlocal double integral appearing in this PIDE.\nThe computational cost of the acquired algorithm is optimal: it is directly proportional to the number of grid points in the \nspatial discretization.\nAlso, it is simple to implement and requires little memory usage.\nSubsequently, for the efficient discretization in time of the semidiscretized two-dimensional Kou PIDE, we have investigated \nseven modern operator splitting schemes of the IMEX and the ADI kind.\nEvery splitting scheme conveniently treats the integral term in an explicit fashion. \nThrough extensive numerical experiments, we have examined the stability and convergence behaviour of the splitting schemes as \nwell as their relative performance. \nAll of the considered schemes, except for the first-order CNFE scheme, show a desirable, stiff order of temporal convergence \nequal to two. \nThe MCS2 scheme, successively developed by in 't Hout \\& Welfert \\cite{HW09} for PDEs and in 't Hout \\& Toivanen \\cite{HT18} for PIDEs,\nstood out favourably among the splitting schemes in view of its superior efficiency. \nThis conclusion agrees with the results recently obtained by Boen \\& in 't Hout~\\cite{BH21} in the case of the two-dimensional \nMerton PIDE. \n\n\n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nClassical cameras record a scene using three color channels, namely, the red (R), green (G) and blue (B) channel.\nTo achieve this, the incoming light spectrum is filtered by corresponding filters, e.g., a filter for the red component lets pass the wavelengths from 600 nm to 700 nm.\nThe RGB filters are approximating the properties of the different cones of the human eye.\nThus, if such an image is shown to a human on a proper display, all perceivable information is recorded and displayed.\n\nFor many classification tasks, it is useful to record even more channels, for example also in the infrared (IR) area of the spectrum.\nThus, the concept of RGB imaging is extended to \\textit{multispectral} imaging, where typically six to 16 different bands of the scene are recorded.\nUsually, the filters of multispectral cameras are spectrally distributed depending on the application.\nTherefore, there is no structure regarding the shape of filters or the location of filters in the spectrum.\n\nIn contrast, the filters of \\textit{hyperspectral} cameras are typically equidistantly distributed in the wavelength area of interest.\nSince the employed bandpass filters are narrowband as well, the goal of a hyperspectral camera is to sample the light spectrum in the desired wavelength area for each pixel.\nWhile there are no sharp definitions of multispectral and hyperspectral, these definitions are inline with the definitions given by Hagen et al.\\cite{hagen_review_2013}, which state that most authors distinguish between contiguous (hyperspectral) versus spaced spectral bands (multispectral).\n\nHyperspectral imaging has become increasingly popular for diverse classification tasks.\nFor example, different types of plastics can be separated using hyperspectral cameras\\cite{garaba_airborne_2018}, which is an essential building block for recycling pipelines.\nMoreover, these cameras are employed in agriculture to measure the plant health\\cite{williams_classification_2016} in order to distribute the proper amount of water and fertilizer.\nOther applications include forensics\\cite{edelman_hyperspectral_2012} where hyperspectral cameras can analyse the scene in a non-invasive way, object tracking\\cite{xiong_material_2020}, and tumour detection in medicine\\cite{han_vivo_2016}.\nUsually, for most of these applications, it is beneficial if the hyperspectral camera is able to record videos.\n\n\\begin{figure}\n\t\\centering\n\n\t\\includegraphics[]{figures\/paper-figure0.pdf}\n\t\n\t\\caption{A hyperspectral 3D datacube with two spatial dimensions $x$ and $y$ and one spectral dimension $\\lambda$.}\n\t\\label{fig:hs_cube}\n\\end{figure}\n\n\\begin{table*}[t]\n\t\\centering\n\t\\renewcommand{\\arraystretch}{1.5}\n\t\\caption{Overview over different hyperspectral databases. Note that the 1890 images of the proposed database are put together by 7 scenes, each with 30 frames, rendered from a camera array with 9 cameras.}\n\t\\label{tab:databases}\n\t\\begin{tabular}{ccccccccc}\n\t\tDatabase                                        & Spatial res. & \\#Channels & Wavel. area (nm) & \\#Images & Videos & Rerendering & Cam. array & Depth   \\\\ \\hline\n\t\tCAVE\\cite{yasuma_generalized_2010}              & $512 \\times 512$      & 31    & 400 - 700     & 32   & \\xmark    & \\xmark    & \\xmark & \\xmark \\\\\n\t\tUGR\\cite{eckhard_outdoor_2015}                  & $1000 \\times 900$     & 61    & 400 - 1000    & 14   & \\xmark    & \\xmark    & \\xmark & \\xmark \\\\\n\t\tHordley\\cite{hordley_multi-spectral_2004}       & $\\sim 336 \\times 271$    & 31    & 400 - 700     & 23   & \\xmark    & \\xmark    & \\xmark & \\xmark \\\\\n\t\tLe Moan\\cite{larabi_database_2015}              & $500 \\times 500$      & 160   & 410 - 1000    & 9    & \\xmark    & \\xmark    & \\xmark & \\xmark \\\\\n\t\tFoster\\cite{foster_frequency_2006}              & $1018 \\times 1339$    & 33    & 400 - 720     & 8    & \\xmark    & \\xmark    & \\xmark & \\xmark \\\\\n\t\tChakrabarti\\cite{chakrabarti_statistics_2011}   & $1392 \\times 1040$    & 31    & 420 - 720     & 50   & \\xmark    & \\xmark    & \\xmark & \\xmark \\\\\n\t\tBGU\\cite{leibe_sparse_2016}                     & $1392 \\times 1300$    & 244   & 400 - 1000    & 201  & \\xmark    & \\xmark    & \\xmark & \\xmark \\\\\n\t\tMian\\cite{mian_hyperspectral_2012}              & $752 \\times 480$      & 33    & 400 - 720     & 31   & \\cmark    & \\xmark    & \\xmark & \\xmark \\\\\n\t\tProposed                               & $1600 \\times 1200$    & 31    & 400 - 700     & 1890 & \\cmark    & \\cmark    & \\cmark & \\cmark \\\\\n\t\\end{tabular}\n\t\\vspace*{-0.3cm}\n\\end{table*}\n\nThere are several techniques to record a hyperspectral image.\nThe essential challenge is to record a 3D datacube using only 2D sensors.\nThis hyperspectral 3D datacube is depicted in Fig.\\ \\ref{fig:hs_cube}.\nOne can imagine combining different layers or single data elements to fit onto a 2D sensor.\nSome of the following techniques drop some of these data elements and need a reconstruction process afterwards.\nThe first class of recording techniques are scanning devices.\nHere, one dimension of this 3D datacube is unfolded across the time dimension.\nFor example, pushbroom sensors\\cite{gomez-chova_correction_2008} are scanning one spatial dimension and the spectral dimension using a 2D sensor.\nThe second spatial dimension is scanned over time by moving the scene or camera accordingly.\nAnother example of scanning techniques are filter wheels\\cite{koenig_practice_1998}, where the spectral dimension is scanned by spinning a wheel, such that different filters are in front of the sensor.\nThus, this technique unfolds the spectral dimension over time.\nObviously, these techniques are not able to capture hyperspectral videos, since the 3D datacube is scanned over time and thus not fully available at each time instance.\n\nHyperspectral snapshot cameras, on the other hand, are able to capture hyperspectral videos.\nSeveral ideas exist how to build a snapshot hyperspectral camera.\nThere are concepts using dispersion and fiber bundles\\cite{gat_development_2006} or a 2D dispersion pattern within a computed tomography framework\\cite{descour_demonstration_1997}.\nThus, a reconstruction process is mandatory.\nBoth of these techniques suffer from a very low spatial resolution.\nAnother technique is multispectral beamsplitting\\cite{matchett_volume_2007}, where the incoming light beam is split into the corresponding bands using special mirrors.\nThe problem here is, that the amount of photons for other bands is also drastically reduced by each mirror.\nFor a large number of bands, this technique does not work anymore.\nMultispectral filter arrays\\cite{monno_practical_2015} have a similar idea as a classical Bayer pattern by placing filters periodically over each pixel of a sensor.\nSubsequently, a demosaicing algorithm is necessary to yield a full hyperspectral image.\nHowever, depending on the number of bands, the underlying physical resolution for each channel drops significantly using multispectral filter arrays.\nEspecially, when thinking of hyperspectral imaging which can have up to several hundred bands, the pixels with the same filter are positioned far apart.\nMulti-aperture filtered cameras\\cite{shogenji_multispectral_2004} have a similar idea, but place filters over whole regions of the sensor.\nThus, the filters are not periodically repeated.\nHere, also a reconstruction process is necessary.\nImaging techniques based on compressed sensing\\cite{gehm_single-shot_2007} are also discussed.\nFor these cameras, a coded aperture followed by a dispersion element is used to record different spectral mixtures of different pixels.\nAfterwards, compressed sensing is used to reconstruct the datacube.\nFinally, works on camera arrays\\cite{genser_camera_2020} are currently active as well.\nSince the cameras are spatially distributed, a reconstruction pipeline yielding a consistent datacube is essential.\nTwo examples for the necessity of a reconstruction pipeline for high-resolution hyperspectral snapshot cameras are shown in Fig.\\ \\ref{fig:motivation}.\n\n\\begin{figure}\n\t\\centering\n\t\n\n\t\\includegraphics[]{figures\/paper-figure1.pdf}\n\t\n\t\\caption{Two cases of missing ground-truth pixels of snapshot $3 \\times 3$ multispectral\/hyperspectral cameras. On the left, the warped channels of a camera array are depicted, where pixels are missing due to occlusions. On the right, the channels of a multispectral filter array sensor are shown. Missing pixels are shown in white.}\n\t\\label{fig:motivation}\n\\end{figure}\n\nFrom this short overview of hyperspectral cameras, it becomes clear that it is challenging to get ground-truth high-resolution hyperspectral video data.\nGround-truth data has several requirements.\nFirst, the scene itself should be as realistic as possible.\nOf course, when recording a scene in reality, this is fulfilled by definition, while it is more of a challenge for synthetic data.\nFor synthetic data, the world, including all physical concepts like light, needs to be modeled properly to get as realistic data as possible.\nSecond, ground-truth data needs a high resolution in every dimension to be able to sample in any direction.\nWhile scanning techniques can deliver this as long as the scene is static, some hyperspectral snapshot cameras either record a very low spatial or spectral resolution or have  low underlying spatial or spectral resolution.\nThird, the amount of artefacts and postprocessing should be as low as possible.\nThese artefacts include noise, blur, reconstruction errors, exposure problems, chromatic aberrations and many more.\nHere, recording a scene using existing hyperspectral imaging techniques fails to fulfill this requirement.\nFor a non-moving scene a near ground-truth hyperspectral image can be generated using scanning techniques.\nHowever, for example for a filter wheel, due to different behaviours of lens-filter combinations, the focus and exposure time have to be adjusted, which leads to different behaviours regarding optics and noise.\nSnapshot hyperspectral cameras most often need a reconstruction process, which cannot recover the perfect datacube.\nThus, again artefacts may be introduced.\nFourth, boundary conditions should be given as exact as possible.\nThis includes the used camera(s) including all specifications like sensor size and resolution, as well as the aperture and focal length of the lens.\nFurthermore, properties of the bandpass filters and light are also very important.\nSince there are always slight deviations during manufacturing processes and aging phenomena, it is impossible to properly fulfill this requirement when recording a scene hyperspectrally in reality.\n\nAs one can see, there is a strong need for ground-truth hyperspectral video data, for example also as reference data to measure the performance of snapshot reconstruction algorithms.\nThere are a number of successful hyperspectral image databases, which are summarized in Table \\ref{tab:databases}.\nOne of the most famous examples is the CAVE database\\cite{yasuma_generalized_2010}.\nThis database contains 32 scenes depicting different objects like balloons, persons,  pepper and color patches usually with a black background.\nThe images have a resolution of $512 \\times 512$ pixels in the range from 400 nm to 700 nm in 10 nm steps, resulting in 31 hyperspectral channels.\nThe CAVE database does not contain any natural scene.\nThe UGR database\\cite{eckhard_outdoor_2015} contains 14 images with a resolution of $1000 \\times 900$ pixels.\nThis database depicts mostly urban scenes including trees, cars and buildings.\nThe scenes were recorded from 400 nm to 1000 nm in 10 nm steps, resulting in 61 hyperspectral channels.\nUnfortunately, some of the bands are corrupted by artefacts.\nThe database of Hordley et al.\\cite{hordley_multi-spectral_2004} contains 23 images of packagings and color charts and thus does not hold any natural spectra.\nThe resolution of the images differ.\nThe packagings and color charts were recorded from 400 nm to 700 nm in 10 nm steps, resulting in 31 hyperspectral channels.\nLe Moan et al.\\cite{larabi_database_2015} created a database of 9 different textures including textile, wood and skin from 410 nm to 1000 nm in 160 hyperspectral channels.\nThe textures have a resolution of $500 \\times 500$.\nThe next database of Foster et al.\\cite{foster_frequency_2006} contains 8 hyperspectral images of natural and urban scenes.\nThese were recorded in the range from 400 nm to 720 nm in 10 nm steps, resulting in 33 hyperspectral channels.\nThe images have a resolution of $1018 \\times 1339$.\nChakrabarti et al.\\cite{chakrabarti_statistics_2011} recorded a hyperspectral image database of 50 hyperspectral images with a resolution of $1392 \\times 1040$.\nFor each scene, 31 hyperspectral channels were recorded from 420 nm to 720 nm in 10 nm steps.\nThis database mainly contains urban and indoor scenes.\nThe BGU database\\cite{leibe_sparse_2016} of Arad et al. contains over 200 hyperspectral images of natural, urban and indoor scenes at a resolution of $1392 \\times 1300$ pixels.\nThe database was recorded from 400 nm to 1000 nm at 1.25 nm steps.\nThus, the BGU database offers the highest spectral resolution, one of the highest spatial resolutions and the most diversity in scenes due to the sheer amount of different scenes.\nFinally, a hyperspectral video database\\cite{mian_hyperspectral_2012} containing a single video of a lab scene with a spatial resolution of $752 \\times 480$ exists.\nHere, the spectrum from 400 nm to 720 nm was recorded using 10 nm steps, thus resulting in 33 hyperspectral channels.\n\nThis paper introduces a novel synthetic hyperspectral video database rendered using a camera array.\nThis camera array consists of nine cameras arranged in a three times three grid.\nThus, the scenes are rendered from multiple positions for wavelengths between 400 nm and 700 nm in 10 nm steps, resulting in 31 hyperspectral channels for each camera.\nMoreover, the depth map is generated as well for every frame and camera.\nSince a high resolution representation of spatial positions in three dimensions as well as the spectral dimension is available, many applications can be addressed by this database.\nThese applications do not necessarily have to be in a hyperspectral context, but also range to grayscale and RGB image processing tasks as well as fusion problems, especially regarding depth sensors.\nTo the best of our knowledge, this paper is the first to introduce a \\textit{synthetic} hyperspectral \\textit{video} database.\n\nThe remainder of this paper is organized as follows.\nFirst, the techniques and properties of this hyperspectral video database are described in detail in Section \\ref{sec:database}.\nAfterwards, two applications of this database are considered in Section \\ref{sec:applications} by improving the corresponding state of the art.\nFinally, the paper is summarized in Section \\ref{sec:conclusion}.\n\n\\section{Database}\n\\label{sec:database}\nThe hyperspectral video database was rendered using the open-source software Blender\\cite{blender}, which provides tools for RGB rendering.\nTherefore, the first properties and techniques that are described in this section are related to Blender.\nThe basic idea is to render each wavelength for each frame image by image in grayscale.\nFor that, the textures need to be adjusted to the wavelength as well as the light source.\nOf course, the scene can be rendered from multiple cameras within a camera array.\n\n\\subsection{Renderer}\n\n\\begin{figure*}[t]\n\t\\centering\n\n\t\\includegraphics[]{figures\/paper-figure2.pdf}\n\t\n\t\\caption{The pipeline of the hyperspectral renderer. Essentially, all combinations of channel (cnl), frame (frm) and camera (cmr) are traversed and rendered. The red items in the top row indicate an exemplary state, i.e., currently the textures at 640 nm and the animations at frame 10 of 30 viewed from camera 6 are rendered.}\n\t\\label{fig:rendering_pipeline}\n\\end{figure*}\n\nTo extend Blender to a hyperspectral image renderer, each wavelength is rendered as grayscale image.\nSince the wavelengths from 400 nm to 700 nm were rendered in 10 nm steps, one hyperspectral image consists of 31 rendered grayscale images.\nFor each wavelength, the images of materials as well as the intensity of the light source were changed programmatically through Python.\nMore details on textures and light sources will be presented below.\nThis pipeline is explained in detail in Fig.\\ \\ref{fig:rendering_pipeline}.\n\nBlender has two different rendering engines, namely, Eevee and Cycles.\nWhile Eevee is a simple rasterization engine, Cycles implements path tracing\\cite{purcell_ray_2005} and thus yields much more realistic images.\nConsequently, Cycles was chosen as rendering engine with branched path tracing as integrator.\nThe number of samples per pixel was set to 128.\n\nSince path tracing nearly always yields rendering noise in complex scenes, a post-processing denoiser was employed.\nConsidering that Blender directly implements Non-Local Means (NLM) denoising\\cite{buades_non-local_2011}, NLM was used as denoiser after the grayscale images were rendered.\n\nThe bitdepth of the final output was set to eight to allow for a smaller total database size.\nFurthermore, each grayscale image is saved as PNG.\n\n\\subsection{Hyperspectral Textures and Materials}\n\nAn essential part for setting up a realistic scene in Blender are textures to give surfaces of objects color and details without increasing the complexity of geometry.\nTo give a surface the desired part of a texture, coordinates in images are mapped to coordinates of surfaces.\nSince a hyperspectral scene also needs hyperspectral textures, the possibilities are either to record them, which requires a high resolution hyperspectral camera, or extract them from other databases.\nThe latter was picked for this database.\nTo maintain a consistent appearance of the textures, the BGU database\\cite{leibe_sparse_2016} was chosen to deliver hyperspectral textures.\nThough this database contains wavelengths from 400 nm to 1000 nm in 1.25 nm increments, our novel database only contains wavelengths from 400 nm to 700 nm in 10 nm steps, to keep the rendering time feasible and the database size acceptable.\nThe BGU database contains over 200 images of diverse natural, urban, and indoor scenes.\nThus, different textures like grass, wood, flower, leaves, metal, plastic, different types of stones, parts of cars and many more could be extracted from this database.\nThe sky was also extracted from this database using the image \\textit{hill\\_0325-1228}, which shows a huge portion of a cloud-free sky.\nThis texture was then used for the world material in Blender.\nSince Blender is only an RGB renderer natively, hyperspectral textures consist of one grayscale image for each wavelength.\nThus, depending on the wavelength that is currently rendered, the textures need to be exchanged.\n\nThe scenes have 14 to 34 different textures depending on the complexity.\nFor example, an outdoor scene needs rather less texture, since there is mainly dirt, grass, wood and leaves.\nThen, for these types of materials, different textures for leaves and wood are extracted from the BGU database.\nOn the other hand, indoor scenes have typically more textures, since there are more smaller things lying around the scene, which all need a texture.\nOf course, one can reuse some textures for some objects, but an interesting indoor scene also needs diverse textures even for the same object.\n\n\\begin{figure}\n\t\\centering\n\n\t\\includegraphics[]{figures\/paper-figure3.pdf}\n\t\\caption{The procedure of extracting hyperspectral textures. The RGB image is taken to spot good textures. The extraction is done by warping all hyperspectral channels using a calculated homography matrix.}\n\t\\label{fig:hte}\n\\end{figure}\n\nThe extraction procedure is shown in Fig.\\ \\ref{fig:hte}.\nFirst, four points spanning a rectangle are selected.\nAfterwards, a homography matrix is calculated, which translates these world coordinates in the image plane to coordinates $\\left( (0, 0), (0, H), (W, H), (W, 0) \\right)$, where $W$ and $H$ represent the width and the height of the resulting texture.\nThis has to be defined by the user.\nFrom these four pairs of points $(x_i^{\\text{src}}, y_i^{\\text{src}}) \\leftrightarrow (x_i^{\\text{dst}}, y_i^{\\text{dst}})$, the homography matrix\n\\begin{equation}\n\t\\bm{H} =\n\t\\begin{bmatrix}\n\t\th_{11} & h_{12} & h_{13}\\\\\n\t\th_{21} & h_{22} & h_{23}\\\\\n\t\th_{31} & h_{32} & h_{33}\n\t\\end{bmatrix}\n\\end{equation}\n\nis found by solving the least-squares problem\n\n\\begin{equation}\n\t\\begin{split}\n\t\t\\argmin_{\\bm{H}} \\sum_{i=1}^{4} & \\left( x_i^{\\text{src}} - \\frac{h_{11}y_i^{\\text{dst}} + h_{12}x_i^{\\text{dst}} + h_{13}}{h_{31}x_i^{\\text{dst}} + h_{32}y_i^{\\text{dst}} + h_{33}} \\right) +\\\\\n\t\t& \\left( y_i^{\\text{src}} - \\frac{h_{21}y_i^{\\text{dst}} + h_{22}x_i^{\\text{dst}} + h_{23}}{h_{31}x_i^{\\text{dst}} + h_{32}y_i^{\\text{dst}} + h_{33}} \\right).\n\t\\end{split}\n\\end{equation}\n\nSince a homography matrix has eight degrees of freedom, the final matrix is scaled such that $h_{33} = 1$.\nThen, the position of other points in the destination image plane can be calculated by\n\\begin{equation}\n\t\\begin{bmatrix}\n\t\tx^{\\text{dst}}\\\\\n\t\ty^{\\text{dst}}\\\\\n\t\t1\n\t\\end{bmatrix} = \\bm{H}\n\t\\begin{bmatrix}\n\t\tx^{\\text{src}}\\\\\n\t\ty^{\\text{src}}\\\\\n\t\t1\n\t\\end{bmatrix}.\n\\end{equation}\nThis is necessary to warp the texture from the source image plane to the destination image texture plane.\nThis warping procedure is done for every hyperspectral channel.\nAfterwards, the hyperspectral channels are saved individually as lossless grayscale images.\n\nAfter the textures are extracted, corresponding materials can be set up in Blender.\nThe texture of the material is changed depending on which channel is currently rendered.\nMoreover, a lot of properties can be changed in Blender, which influences how light is reflected or absorbed.\nFor example, the roughness can be changed as well as how specular and metallic a material is, e.g., a shiny car paint is typically very little rough but very metallic, while a street is very rough and less shiny.\nMoreover, also the transparency of a texture can be set, which is very important for windows.\n\n\\subsection{Light Sources}\n\nFor all outdoor scenes or scenes with windows, the light source is the sun.\nIn bright daylight the sun can reach a color temperature up to 6400K.\nThe color temperature of a light source defines its emitting light spectrum.\nThe spectrum is based on the radiation of an ideal black body and can be calculated by\nPlanck's law.\nFor the single indoor scene without windows, a lamp was mounted into that room with a color temperature of 3200K, which is a typical value for studio lamps.\nThe spectra of these two light sources is shown in Fig.\\ \\ref{fig:light_spectra} and were calculated by Planck's law.\nFor each wavelength, the intensity of the sun or the indoor lamp in the scene is set according to this plot.\nFurthermore, the hardness and light strength differs from scene to scene and is also dependent on the angle of the sun.\n\n\\begin{figure}\n\t\\centering\n\n\t\\includegraphics[]{figures\/paper-figure4.pdf}\n\t\\caption{Used light spectra for the sun (6400K) and the indoor lamp (3200K).}\n\t\\label{fig:light_spectra}\n\\end{figure}\n\n\\subsection{Cameras and Camera Array}\n\nAll scenes are rendered from nine cameras arranged in a $3 \\times 3$ grid.\nThis mimicks a camera array from Genser et al.\\cite{genser_camera_2020} and is shown in Fig.\\ \\ref{fig:camsi}.\nTherefore, the specifications are very similar.\nThe baseline between neighboring cameras is 40 mm.\nThe cameras itself are very close to common industrial cameras like a Basler acA1600-60gm, which are also used for the camera array in Fig.\\ \\ref{fig:camsi}.\nThus, the image sensor size is set to 7.2 mm times 5.4 mm and the resolution of the images is 1600 pixels times 1200 pixels.\nThe focal length is different from the original camera array and is set to 6 mm focal length to capture a large portion of the scene.\nThe purpose of rendering the scene from this camera array is, for example, to be able to simulate the multispectral camera shown in Fig.\\ \\ref{fig:camsi}.\nSince this multispectral camera array only records one spectral area with a specific transmission curve per camera, one has to simulate the corresponding filters by calculating a weighted sum of hyperspectral images for each camera.\nThus, in contrast to the camera array in Genser et al.\\cite{genser_camera_2020}, the synthetic data contains all 31 hyperspectral channels for every camera.\nIf the application just demands for a single hyperspectral view of the scene, only the center camera can be extracted, since each camera has its own folder in the database.\n\n\\begin{figure}[t]\n\t\\centering\n\n\t\\includegraphics[]{figures\/paper-figure5.pdf}\n\t\\caption{A multispectral camera with nine channels on the left\\cite{genser_camera_2020} with a sketch of dimensions and indices of cameras on the right. Note that this view is from the front, as well.}\n\t\\label{fig:camsi}\n\\end{figure}\n\n\\subsection{Depth Maps}\n\nMany applications also rely on depth data provided by a depth sensor like RADARs or LIDARs.\nMoreover, some algorithms also estimate the depth as intermediate step, for example, to register multiple views on top of each other.\nTherefore, depth data for all scenes, cameras, and frames is provided.\nThe depth maps are stored in the EXR format as 32 bit floats for a high resolution.\nThe depth maps contain the physical depth in meters.\nConsequently, the disparity between cameras is calculated by\n\n\\begin{equation}\n\t\\label{eq:depth_to_disp}\n\td = \\frac{b \\cdot f}{p \\cdot s},\n\\end{equation}\nwhere $d$ is the disparity of corresponding pixels between two cameras, $b$ is the baseline between the cameras, e.g. 40 mm for directly neighboring cameras in the camera array, $f$ is the focal length, thus 6 mm, $p$ is the depth itself, and $s$ is the size of a single pixel and calculates to $s = 7.2 \\ \\text{mm}\/1600 = 5.4 \\ \\text{mm}\/1200 = 4.5 \\cdot 10^{-3} \\ \\text{mm}$.\nNote that the depth is provided in meters while all other properties are given in millimeters.\nDepth maps can be used for example to evaluate depth estimation algorithms or to warp different views of the camera array to different positions.\n\n\\subsection{Motion}\n\nIn all of the scenes, the camera array moves along a specified path.\nDepending on the scene, this is realized using different approaches.\nIn the first method, the camera array is moved along the scene manually and this trajectory is recorded.\nAfter recording this movement, the paths are smoothed out, since a manual camera movement is typically quite shaky.\nIn the end, each frame is a keyframe for the movement and stores the position and rotation of the camera array.\nThe second way to define movement in this database is to define the starting and endpoint position and rotation and smoothly interpolation between them for intermediate frames.\nThus, there are only two keyframes set and the intermediate points are determined using Bezier interpolation.\nThis approach is also performed for moving objects.\n\n\\subsection{Scenes}\n\n\\begin{figure}\n\t\\centering\n\n\t\\includegraphics[]{figures\/paper-figure6.pdf}\n\t\n\t\\caption{The workflow to create a hyperspectral scene. This general workflow can be executed in any rendering framework.}\n\t\\label{fig:workflow}\n\\end{figure}\n\n\\begin{figure*}\n\t\\centering\n\n\t\\includegraphics[]{figures\/paper-figure7.pdf}\n\t\\caption{The center views of the seven different synthetic hyperspectral scenes: frame 0 of \"family house\", frame 0 of \"medieval seaport\", frame 25 of \"city\", frame 0 of \"outdoor\", frame 10 of \"forest\", frame 0 of \"indoor\" and frame 14 of \"lab\". On the left, an RGB image using standard CIE 1931 color curves can be seen. Next to it, all 31 hyperspectral channels of the corresponding image and the depth map are depicted.}\n\t\\label{fig:scenes}\n\\end{figure*}\n\nThe whole workflow for setting up a 3D hyperspectral scene is summarized in Fig.\\ \\ref{fig:workflow}.\nIn total, seven different scenes were rendered.\nThese scenes are depicted in Fig.\\ \\ref{fig:scenes}, where the RGB images are calculated using standard CIE 1931 color curves.\nFurthermore, all 31 hyperspectral channels of one frame are shown as well as the depth maps corresponding to the respective scene and frame.\nThe camera array moves through each scene for 30 frames.\n\nThere are five outdoor scenes. \\textit{Family house} represents a mixture of urban and rural objects, \\textit{medieval seaport} is a view of a more ancient city incorporating a lot of stone and wood, and \\textit{city} is a newer city including a moving car, normal streets, sidewalks, modern houses and trash bins.\n\\textit{Outdoor} represents a rural scene from an agricultural way and \\textit{forest} is a similar scene without any man-made objects and a lot of trees.\n\\textit{Indoor} depicts a room including furniture and decoration elements, but still with the light source being the sun shining through windows.\nFinally, \\textit{lab} is illuminated by a halogen lamp and depicts a laboratory setup with selective elements on a table similar to the real-world image taken in Genser et al.\\cite{genser_camera_2020}.\n\n\\subsection{Limitations}\n\nNote that this database still contains only \\textit{synthetic} hyperspectral data.\nThus, it is nearly clean from any noise or artefacts and ideal cameras are assumed as well as an ideal camera array is assumed.\nAn array-related real-world artefact would be that the cameras are not perfectly aligned to each other as well as the sensors within the camera housing are not perfectly fitted.\nTherefore, in real-world a calibration procedure between different cameras would always be necessary, e.g., to fulfill the epipolar constraint.\nMoreover, a lot of artefacts of the filter-lens-camera combinations can occur.\nFirst, filters have the tendency to change optical paths slightly dependent on the wavelength.\nSecond, lenses introduce artefacts like optical veiling glare\\cite{mccann_veiling_2007}, lens flares\\cite{rosenhauer_image_1968}, wavelength-dependent chromatic aberration\\cite{waller_phase_2010}, star bursts and many more.\nFinally, cameras always introduce different kind of noise, e.g., shot noise, thermal noise and electronic noise\\cite{hytti_noise_2006}.\nAll of these artefacts are not part of this database and most of them are difficult to simulate.\nThus, if one trains or evaluates algorithms based on this data, one has to be careful, since the data of this database deviates from real-world image acquisitions.\nHowever, depending on the desired application, noise or artefacts could be added to clean data.\nMoreover, while the absolute performance is wrongly estimated due to missing artefacts, the relative performance between two algorithms might still be possible assuming that both algorithms are affected by noise and artefacts equally.\n\n\\section{Applications}\n\\label{sec:applications}\nIn this section, an overview over applications that can benefit from this dataset is given.\nFirst, two applications are presented in detail and improved which is verified using the novel database.\nIn the first application, videos from all cameras and depth maps from the center camera are used.\nThe second application uses just the hyperspectral video of the center camera.\nIn the end, several other applications are described briefly.\n\n\\subsection{Cross Spectral Reconstruction}\n\nThe first application that is covered in detail is cross-spectral image reconstruction.\nConsider imaging a scene with the camera array shown in Fig.\\ \\ref{fig:camsi}.\nThe first step to create a consistent multispectral datacube is a cross-spectral disparity estimation, i.e., finding the correspondence of which pixel in one camera belongs to which pixel in the other camera.\nSince the final reconstruction process is evaluated here, the ground-truth depth maps from the database can be used.\nFor that, the depths needs to be converted to disparities using \\eqref{eq:depth_to_disp}.\nThen, this disparity map can be used to warp the peripheral view to the center view.\nDue to occlusions, there are pixels missing which need to be reconstructed.\nFortunately, the center view is always fully available, since it is not warped to any other position.\nThus, the center view can be used as reference channel for all peripheral views.\nHowever, this center view records a different part of the spectrum than the peripheral views.\nNote that the procedure to simulate a multispectral camera array is described in Section \\ref{subsec:csr_eval}.\nTherefore, one has to find a relationship between the center view and the peripheral view that has to be reconstructed.\nThis whole problem is depicted in Fig.\\ \\ref{fig:CSR_problem}.\nThe peripheral view that has to be reconstructed will be called distorted view in the following description.\n\\begin{figure}\n\t\\centering\n\n\t\\includegraphics[]{figures\/paper-figure8.pdf}\n\t\\caption{The basic problem of cross spectral reconstruction. On the left side the fully available center view is depicted, on the right side the warped peripheral view is shown. Due to occlusions, some pixels are missing in the peripheral view, which are shown in red.}\n\t\\label{fig:CSR_problem}\n\\end{figure}\n\n\\subsubsection{State of the Art}\n\nFor a single multispectral image, Non-local Cross-Spectral Reconstruction (NOCS) in \\cite{sippel_spatio-spectral_2021} is a state-of-the-art approach.\nNOCS first finds similar blocks using the fully available reference images.\nBlock means the set of pixels of usually a square area around a center pixel.\nIn the case of the multispectral camera array, this would be just the center view.\nFor that, the $l_2$-norm between two blocks within the reference image is minimized\n\\begin{equation}\n\td_{\\text{NOCS}}(\\bm{x}, \\bm{y}) = ||\\bm{B}^{\\text{R}}(\\bm{x}) - \\bm{B}^{\\text{R}}(\\bm{y})||_2,\n\\end{equation}\nwhere $\\bm{B}^{R}(\\bm{x})$ describes the reference block at coordinate $\\bm{x}$.\nOf course, this whole block matching procedure only has to be done for pixels that are missing in the distorted view.\nFor missing pixel position $\\bm{x}$, the $B$ best matches are used to stack the corresponding pixel values of the reference view on top of each other, resulting in the vector $\\bm{r}(\\bm{x})$.\nThe same procedure is done for the distorted image, resulting in the distorted vector $\\bm{d}(\\bm{x})$.\nOf course, this vector contains missing pixels.\nMoreover, since a block has distance 0 with itself, the pixel to reconstruct is always at the first entry of the vector.\n\nThese two vectors are then used to build a linear regression model\n\\begin{equation}\n\t\\bm{d}(\\bm{x}) = \\alpha(\\bm{x}) \\cdot \\bm{r}(\\bm{x}) + \\beta(\\bm{x}),\n\\end{equation}\nwhere $\\alpha(\\bm{x})$ and $\\beta(\\bm{x})$ are the parameters of the linear regression and thus the variables that need to be estimated for every missing pixel.\nThey are found by minimizing the $l_2$-norm of the difference between known references of the distorted vector $\\tilde{\\bm{d}}(\\bm{x})$ and its prediction from the reference vector $\\tilde{\\bm{r}}(\\bm{x})$ at the corresponding positions\n\\begin{equation}\n\t\\hat{\\alpha}(\\bm{x}), \\hat{\\beta}(\\bm{x}) = \\argmin_{\\alpha(\\bm{x}), \\beta(\\bm{x})} ||\\alpha(\\bm{x}) \\cdot \\tilde{\\bm{r}}(\\bm{x}) + \\beta(\\bm{x}) - \\tilde{\\bm{d}}(\\bm{x})||_2^2.\n\\end{equation}\nThese parameters are found in closed-form\\cite{sippel_spatio-spectral_2021}.\nAfterwards, this model is used to predict the missing pixel, which is the first element in the distorted vector.\nFor that, the value of the first element in the reference vector is put into the model and the resulting value is the predicted reconstructed pixel value\n\\begin{equation}\n\t\\bm{d}_1(\\bm{x}) = \\hat{\\alpha}(\\bm{x}) \\cdot \\bm{r}_1(\\bm{x}) + \\hat{\\beta}(\\bm{x}).\n\\end{equation}\n\nOf course, not all missing pixels are reconstructed at once, but an iterative procedure is carried out.\nAlways 10\\% (but at least one) of the remaining pixels to reconstruct that have the highest number of non-missing pixels of the distorted view in their vector are reconstructed.\nAfterwards, all vectors that contain a newly reconstructed pixel are updated and the loop starts from the beginning.\nThus, reconstructed pixels will influence the remaining pixels that need to be reconstructed.\n\n\\subsubsection{Novel Approach}\n\nWe propose to extend this method by also using the previous frame and therefore exploit temporal correlation to enhance NOCS.\nThis novel method is called Temporal Cross Spectral Reconstruction (TNOCS).\nTo exploit temporal correlation, the block matching procedure is also executed on the previous frame.\nTherefore, the block matching procedure needs to consider that similar blocks might be in different frames\n\\begin{equation}\n\td_{\\text{TNOCS}}(\\bm{x}, \\bm{y}, t_d) = ||\\bm{B}^{\\text{R}}_{t}(\\bm{x}) - \\bm{B}^{\\text{R}}_{t - t_d}(\\bm{y})||_2,\n\\end{equation}\nwhere $\\bm{B}^{R}_{t}(\\bm{x})$ is a block in the reference video at spatial position $\\bm{x}$ and current frame $t$, and $t_d$ is the difference to the respective frame.\nTherefore, when a similar block for $\\bm{x}$ is searched, the position and the frame of the other block can be varied.\nSince two consecutive frames are usually highly correlated but still depict the scene from a slightly different angle, only the previous and the current frame are searched for similar blocks.\nThus, $t_d$ can only be in the set of $\\{0, 1\\}$.\nThe remaining formulas are the same, since the pixel at position $\\bm{x}$ still needs to be reconstructed and the vectors contain pairs of reference pixels and the corresponding pixels from the distorted video.\n\nFor an improvement over NOCS, it is essential that the camera or object in scene moves from one frame to another frame, since otherwise the pixels to be reconstructed would be exactly the same and the temporal correlation can not be properly exploited.\nNote that only non-reconstructed pixels of the previous frame are used for block matching, since the error in reconstruction should not propagate through all frames.\nOtherwise, this would lead to severe artifacts after some frames.\nOf course, the block matching is also executed on the current frame to further exploit spatial correlation.\nThis idea is summarized in the Fig.\\ \\ref{fig:TNOCS}.\n\n\\begin{figure}\n\t\\centering\n\n\t\\includegraphics[]{figures\/paper-figure9.pdf}\n\t\n\t\\caption{The proposed TNOCS searches similar blocks in the current frame as well as the previous frame to build the linear regression model. The blue box depicts the block for which similar blocks are searched, the red blocks are the best matches. The images depict cutouts of two consecutive frames of the \\textit{lab} scene.}\n\t\\label{fig:TNOCS}\n\\end{figure}\n\n\\subsubsection{Evaluation}\n\\label{subsec:csr_eval}\n\nTo validate that this idea works well, our novel hyperspectral video database is employed.\nFor that, multispectral video data has to be created out of the hyperspectral data.\nTo achieve this, an imaging pipeline needs to be simulated.\nA single pixel of the $i$-th channel of a multispectral image is recorded by\\cite{sippel_structure-preserving_2020}\n\n\\begin{equation}\n\t\\bm{c}_i = \\int q(\\lambda)r(\\lambda)\\bm{f}_i(\\lambda)o(\\lambda)m(\\lambda) \\text{d}\\lambda,\n\\end{equation}\nwhere $q(\\lambda)$ is the spectrum of the light source, $r(\\lambda)$ is the reflectance spectrum of the material, $\\bm{f}_i(\\lambda)$ is the transfer function of the $i$-th filter corresponding to the $i$-th multispectral channel, $o(\\lambda)$ the spectral permeability of the camera lens and $m(\\lambda)$ is the spectral response of the camera sensor.\nIn this case, the database already yields a sampled version of the product of the light source and reflectance $s(\\lambda) = q(\\lambda)r(\\lambda)$ as images.\nThe transfer function of the camera lens and camera sensor are assumed to be perfect in the covered wavelength area from 400 nm to 700 nm.\nSince the images are obviously sampled in the spectral dimension, this equation needs to be sampled as well.\nTherefore, a single hyperspectral pixel from our database can be described by $\\bm{s} = [s(\\lambda_1), s(\\lambda_2), \\dots, s(\\lambda_{31})]^\\text{T}$.\nWhen stacking the sampled filters on top of each other into the filter matrix $\\bm{F} = [\\bm{f}_i(\\lambda_1), \\bm{f}_i(\\lambda_2), \\dots, \\bm{f}_i(\\lambda_{31})]$, the imaging process for a single pixel can be written as\n\\begin{equation}\n\t\\bm{c} = \\bm{F} \\bm{s}.\n\\end{equation}\n\nIn this case, the filter matrix contains sampled values of real filters used for the camera array.\nNamely, six bandpass filters with bandwidth 50 nm at wavelengths 425 nm, 450 nm, 500 nm, 550 nm, 600 nm and 650 nm, and a red filter, a green filter and a blue filter to compose an RGB image.\nThen, the data to reconstruct is created by calculating the multispectral data for the corresponding camera.\nThe ground truth is calculated by doing the same calculations with the middle camera.\nMoreover, the warped images are calculated using the ground-truth depth maps provided by the database.\n\n\\begin{table}[t]\n\t\\centering\n\t\\renewcommand{\\arraystretch}{1.5}\n\t\\caption{Evaluation in terms of PSNR of the novel temporal cross spectral reconstruction algorithm (TNOCS) against the version without exploiting temporal correlation (NOCS).}\n\t\\label{tab:eval_reconstruction}\n\t\\begin{tabular}{c|c|c}\n\t\t& NOCS      & TNOCS             \\\\ \\hline\n\t\tFamily house        & 30.83 dB  & \\textbf{36.43 dB} \\\\ \\hline\n\t\tMedieval seaport    & 40.06 dB  & \\textbf{44.87 dB} \\\\ \\hline\n\t\tCity                & 43.61 dB  & \\textbf{45.39 dB} \\\\ \\hline\n\t\tOutdoor             & 28.94 dB  & \\textbf{34.05 dB} \\\\ \\hline\n\t\tForest              & 27.03 dB  & \\textbf{30.67 dB} \\\\ \\hline\n\t\tIndoor              & 39.33 dB  & \\textbf{41.15 dB} \\\\ \\hline\n\t\tLab                 & 35.40 dB  & \\textbf{36.26 dB} \\\\ \\hline \\hline\n\t\tAverage             & 35.03 dB  & \\textbf{38.40 dB} \\\\\n\t\\end{tabular}\n\t\\vspace*{-0.3cm}\n\\end{table}\n\nUsing this multispectral database, the novel TNOCS algorithm can be evaluated against its non-temporal version NOCS.\nThe results are summarized in Table\\ \\ref{tab:eval_reconstruction}.\nFrom the table it gets obvious that exploiting temporal correlation by just searching for similar blocks in the previous frame is enhancing the performance significantly.\nEspecially for scenes containing a lot of leaves, which leads to many small areas that are missing, the difference is enormous.\nFor other scene like \\textit{lab} order \\textit{city}, the improvement is smaller but still noticeable.\n\nWithout our novel hyperspectral video database, it would be impossible to accurately validate the performance gain of exploiting temporal correlation.\nAs shown, it is also possible to create nearly any other data that lies in the wavelength area from 400 nm to 700 nm.\n\n\\subsection{Hyperspectral Video Coding}\n\nThe second application addressed is hyperspectral video coding.\nThe need for hyperspectral image and video coding is basically already motivated by this database.\nWith just seven scenes, nine cameras and 30 frames, this database already has a size bigger than 60 GB saved as single grayscale PNGs.\nWhen looking at raw values, this number increases even more.\nFor this hyperspectral database, saving all images without any compression and eight bits per grayscale pixel results in $7 \\cdot 9 \\cdot 30 \\cdot 31 \\cdot 1600 \\cdot 1200 \\ \\text{bytes} \\approx 105 \\ \\text{GB}$.\nThis enormous amount of data shall be compressed, since there is a lot of correlation.\nOf course, there is temporal correlation, which can be exploited.\nBut also the different hyperspectral channels are highly correlated which can be exploited by a coder.\nIn the following, the term encoding means producing a compressed bitstream out of a (hyperspectral) video.\nIn contrast, the term decoding means transforming this bitstream back to the (hyperspectral) video.\nFor lossy coders, this decoded video deviates from the original video.\n\n\\subsubsection{State of the Art}\n\nThe presented method is based on a multispectral and hyperspectral image coder named Pel-Recursive Inter-Band Prediction (PRIBP)\\cite{meyer_multispectral_2020}.\nBefore presenting the novel inter component of this coder, the intra hyperspectral coder is reviewed first.\nThe basic idea of\\cite{meyer_multispectral_2020} is to first code three channels, in this paper RGB channels, with a standard coder.\nThen, the three decoded channels serve as reference channels for the next channels to code.\nThe decoded channels need to be used as reference channels, because the other end of the communication channel only has the decoded images, which slightly deviates from the original images depending on the quality settings, available.\nThus, to avoid a drift between encoder and decoder, the original channels cannot be used as reference channels.\nFor that, more intra modes have been implemented into the High Efficiency Video Coding (HEVC) codec\\cite{sullivan_overview_2012}.\nThe prediction is done on a transform block level using previously predicted pixels and reconstructed pixels from adjacent blocks.\nTo code the center pixel $b$ of the current block $\\bm{b}$ of the channel to code, first the best reference channel block $\\bm{r}$ out of the current three reference channels is searched by picking the one with the maximum correlation with the already predicted pixels\n\\begin{equation}\n\t\\rho(\\tilde{\\bm{r}}, \\tilde{\\bm{b}}) = \\frac{\\sum_{i=1}^{N} \\left( \\tilde{\\bm{r}}_i - \\mathbb{E}\\{\\tilde{\\bm{r}}\\} \\right)\\left( \\tilde{\\bm{b}}_i - \\mathbb{E}\\{\\tilde{\\bm{b}}\\} \\right)}{\\sqrt {\\sum_{i=1}^{N} \\left( \\tilde{\\bm{r}}_i - \\mathbb{E}\\{\\tilde{\\bm{r}}\\} \\right)^2} \\sqrt{\\left( \\tilde{\\bm{b}}_i - \\mathbb{E}\\{\\tilde{\\bm{b}}\\} \\right)^2}},\n\\end{equation}\nwhere $\\mathbb{E}\\{\\cdot\\}$ is the expectation operator, $N$ is the number of already predicted pixels, and $\\tilde{\\bm{r}}$ and $\\tilde{\\bm{b}}$ are the already predicted pixels of the current block of the reference signal and the channel, respectively.\nSubsequently, this reference is used to build a linear regression model very similar to the one used in cross-spectral reconstruction.\nTherefore, the linear model reads as\n\\begin{equation}\n\t\\tilde{\\bm{b}} = \\alpha \\cdot \\tilde{\\bm{r}} + \\beta\n\\end{equation}\nwith parameters $\\alpha$ and $\\beta$ to be estimated.\nAgain, the parameters are found by minimizing the $l_2$-norm\n\\begin{equation}\n\t\\hat{\\alpha}, \\hat{\\beta} = \\argmin_{\\alpha, \\beta} ||\\alpha \\cdot \\tilde{\\bm{r}} + \\beta - \\tilde{\\bm{b}}||_2^2.\n\\end{equation}\nFinally, this model is employed for the prediction of the pixel\n\\begin{equation}\n\t\\hat{b} = \\hat{\\alpha} \\cdot r + \\hat{\\beta}.\n\\end{equation}\nThe coding order of the remaining channels is found by sorting the different channels according to their structural similarity index with respect to one anchor channel.\nSince there are always three reference channels, the least similar reference channel is then replaced by the channel, which was just coded.\nNote that this method was focused on multispectral image coding rather than hyperspectral image coding.\n\n\\subsubsection{Novel Approach}\n\n\\begin{figure*}\n\t\\centering\n\n\t\\includegraphics[]{figures\/paper-figure10.pdf}\n\t\n\t\\caption{The basic concept of this novel inter hyperspectral video coder. First, motion compensation is executed on the frame $t$ exploiting the neighboring frames. Afterwards, the current channel is spectrally compensated using the last three channels.}\n\t\\label{fig:inter_hs}\n\\end{figure*}\n\nFor hyperspectral image coding, there is implicitly more structure given, since neighboring channels are also neighbors in the spectrum.\nTherefore, since spectra are typically smooth, the correlation between two neighboring channels is the highest.\nThe temporal and spectral compensation of the novel hyperspectral video coder is shown in Fig.\\ \\ref{fig:inter_hs}.\nAssuming the channels are sorted in ascending order according to their wavelength, the proposed hyperspectral video coder compresses the first three channels as an RGB video with a standard video coder using only intra-prediction to ensure a high quality for motion estimation using these channels.\nSince this work is based on PRIBP, the base video coder is again HEVC.\nThus, the first three decoded channels for every time step $t$ read as\n\\begin{equation}\n\t\\hat{C}^t_0, \\hat{C}^t_1, \\hat{C}^t_2 = \\text{D}^{\\text{HEVC}}\\left( \\text{C}^{\\text{HEVC}} \\left( \\left[ C^t_0, C^t_1, C^t_2 \\right] \\right) \\right),\n\\end{equation}\nwhere $\\text{C}^{\\text{HEVC}}\\left( \\cdot \\right)$ and $\\text{D}^{\\text{HEVC}}\\left( \\cdot \\right)$ are the coder and decoder, respectively, and $C^t_i$ is the $i$-th grayscale channel of a hyperspectral image of size $M \\times N$ at time step $t$.\nTo avoid issues with error propagations, every second frame of every other channel is coded by the hyperspectral intra encoder, while for all other frames a motion-compensated residual is compressed by the hyperspectral intra coder.\nOtherwise, the errors during coding would propagate into the next frame leading to even more errors and a higher residual to code, which would then turn into an even lower quality or higher rate.\nThis residual is calculated by exploiting temporal correlation.\nTherefore, the decoded first three channels are used to estimate motion, which is done by PWC-Net\\cite{sun_pwc-net_2018}.\nThis is done using forward and backward motion estimation\n\\begin{equation}\n\t\\label{eq:motion_vectors}\n\t\\begin{split}\n\t\t\\bm{V}^t_{\\text{fw}} &= \\text{PWC}\\left( \\left[ \\hat{C}^{t-1}_0, \\hat{C}^{t-1}_1, \\hat{C}^{t-1}_2 \\right], \\left[ \\hat{C}^{t}_0, \\hat{C}^{t}_1, \\hat{C}^{1}_2 \\right] \\right) \\\\\n\t\t\\bm{V}^t_{\\text{bw}} &= \\text{PWC}\\left( \\left[ \\hat{C}^{t+1}_0, \\hat{C}^{t+1}_1, \\hat{C}^{t+1}_2 \\right], \\left[ \\hat{C}^{t}_0, \\hat{C}^{t}_1, \\hat{C}^{1}_2 \\right] \\right),\n\t\\end{split}\n\\end{equation}\nwhere $\\bm{V}^t$ is an image of motion vectors for time step $t$.\nThen, these motion estimations can be used for motion compensation\n\\begin{equation}\n\t\\label{eq:motion_compensation}\n\t\\begin{split}\n\t\tP^t_{\\text{fw}, i}[m, n] &= \\tilde{C}^{t - 1}_i \\left( m + \\bm{V}^t_{\\text{fw}, 1}[m, n], n + \\bm{V}^t_{\\text{fw}, 2}[m, n] \\right) \\\\\n\t\tP^t_{\\text{bw}, i}[m, n] &= \\tilde{C}^{t + 1}_i \\left( m + \\bm{V}^t_{\\text{bw}, 1}[m, n], n + \\bm{V}^t_{\\text{bw}, 2}[m, n] \\right),\n\t\\end{split}\n\\end{equation}\nwhere $\\tilde{C} \\left( m, n \\right)$ is the interpolated and extended image of $\\hat{C} \\left[ m, n \\right]$.\nMoreover, corresponding masks, when the motion compensation tries to pull a pixel outside the image scope, need to be calculated\n\\begin{equation}\n\t\\label{eq:motion_mask}\n\tM^t_{j}[m, n] = \\left\\{\n\t\\begin{array}{ c l }\n\t\t1, &\\text{if } \\begin{aligned}\n\t\t\t&0 \\leq m + \\bm{V}^t_{j, 1}[m, n] \\leq M - 1 \\ \\land \\\\\n\t\t\t&0 \\leq n + \\bm{V}^t_{j, 2}[m, n] \\leq N - 1\n\t\t\\end{aligned} \\\\\n\t\t0, &\\text{else},\n\t\\end{array}\n\t\\right.\n\\end{equation}\nwhere $j$ is in the set of $\\left\\{ \\text{fw}, \\text{bw} \\right\\}$.\nSince two predictions for the same image are made, there has to be a merge procedure for these.\nIn general, the simple average of these two predictions is taken to merge them.\nHowever, at the border of the image, it occurs that the warping tries to warp pixels outside of the original image plane to the corresponding pixel in the predicted image.\nIn these situations, the merge procedure entirely relies on the other prediction as long as this prediction is valid.\nIf no prediction is valid, again the average of both predictions is taken as best guess.\nThis summarizes to the final prediction\n\\begin{equation}\n\t\\label{eq:final_prediction}\n\t\\begin{split}\n\t\tP^t_i = \\left\\{\n\t\t\\begin{array}{c l}\n\t\t\t\\frac{P^t_{\\text{fw}, i} + P^t_{\\text{bw}, i} }{2}, &\\begin{aligned}\n\t\t\t\t&\\text{if } \\left(M^t_{\\text{fw}}[m, n] = 1 \\land M^t_{\\text{bw}}[m, n] = 1\\right) \\\\\n\t\t\t\t&\\lor \\left(M^t_{\\text{fw}}[m, n] = 0 \\land M^t_{\\text{bw}}[m, n] = 0\\right)\n\t\t\t\\end{aligned}\\\\\n\t\t\tP^t_{\\text{fw}, i}, &\\text{if } M^t_{\\text{fw}}[m, n] = 1 \\land M^t_{\\text{bw}}[m, n] = 0 \\\\\n\t\t\tP^t_{\\text{bw}, i}, &\\text{if } M^t_{\\text{fw}}[m, n] = 0 \\land M^t_{\\text{bw}}[m, n] = 1.\n\t\t\\end{array}\n\t\t\\right.\n\t\\end{split}\n\\end{equation}\nTo calculate the residual for all odd frames, the predicted image is subtracted from the original decoded image\n\\begin{equation}\n\t\\label{eq:residual}\n\tR^t_i = C^t_i - P^t_i \\quad \\forall \\ t \\bmod 2 = 1.\n\\end{equation}\nThis yields three residual images which serve as reference residual images for the hyperspectral intra coder, later.\nMoreover, as in PRIBP, for all-intra coded frames the first three channels serve as reference channel for the first hyperspectral intra coded image.\n\nSubsequently, all other channels are coded iteratively.\nEven frames are coded using all-intra PRIBP\n\\begin{multline}\n\t\\hat{C}^t_i = \\text{D}^{\\text{PRIBP}}\\left( \\text{C}^{\\text{PRIBP}} \\left( C^t_i, \\left[ \\hat{C}^t_{i - 1}, \\hat{C}^t_{i - 2}, \\hat{C}^t_{i - 3} \\right] \\right) \\right) \\\\\n\t\\forall \\ i \\geq 3 \\land t \\bmod 2 = 0,\n\\end{multline}\nwhere $\\text{C}^{\\text{PRIBP}} \\left( C^t_i, \\left[ \\hat{C}^t_{i - 1}, \\hat{C}^t_{i - 2}, \\hat{C}^t_{i - 3} \\right] \\right)$ means that the $i$-th channel is coded using decoded channels $i - 1$, $i -2$ and $i - 3$ as reference channels.\nAfter being decoded, they replace the oldest reference in the reference buffer for that frame and serve as reference channel for the next three channels.\nThis is done, since, as already described, spectra are typically smooth and therefore, the last three channels have the highest correlation with the current channel from a physical point of view.\nThen, the decoded intra frames at timesteps $t - 1$ and $t + 1$ are used for predicting the current frame at time $t$ using the motion vectors estimated from the first three channels in \\eqref{eq:motion_vectors}.\nThe motion-compensated prediction is calculated using \\eqref{eq:motion_compensation}, \\eqref{eq:motion_mask} and \\eqref{eq:final_prediction}.\nAfterwards, the residual with the frame to code is calculated by \\eqref{eq:residual}.\nThis residual is fed into PRIBP, which uses three reference images again\n\\begin{multline}\n\t\\hat{R}^t_i = \\text{D}^{\\text{PRIBP}}\\left( \\text{C}^{\\text{PRIBP}} \\left( R^t_i, \\left[ \\hat{R}^t_{i - 1}, \\hat{R}^t_{i - 2}, \\hat{R}^t_{i - 3} \\right] \\right) \\right) \\\\\n\t\\forall \\ i \\geq 3 \\land t \\bmod 2 = 1.\n\\end{multline}\nIn this case, the reference images are residual signals as well.\nAgain, as soon as this residual is decoded again, it serves as reference residual for the next three channels.\n\n\\subsubsection{Evaluation}\n\nOur novel hyperspectral video database can directly be used to evaluate the gains of the proposed hyperspectral video coder.\nFor the evaluation, only the videos of the center camera are used.\nThe Quantization Parameters (QPs) for evaluation are set to 22, 27, 32 and 37.\nThese QPs are extracted from the common test conditions for HEVC and thus common practice\\cite{hevc_ctc_2018}.\n\nIn general, different coders are evaluated by setting up rate-distortion curves, i.e., a plot that shows the data rate in terms of bitrate on the x-axis and distortion, here in terms of PSNR, on the y-axis.\nThe different aforementioned QPs are datapoints for this curve.\nBetween these data points the curve is interpolated.\nThus, if the curve of a second coder is to the top left compared to the curve of the base coder, it achieves a better PSNR for the same bitrate, or vice versa, a lower bit rate for the same PSNR.\nThis average distance between rate-distortion curves can be measured using the Bjontegaard-Delta (BD) for the rate direction and PSNR direction\\cite{bjontegaard_calculation_2001}.\nThe results for each scene are summarized in Table\\ \\ref{tab:eval_coding}, which shows the BD rate and PSNR.\nThe proposed hyperspectral video coder is first compared to the all-intra coder PRIBP.\nFor six out of seven scenes, the proposed coder outperforms the intra procedure.\nPWC-Net has problems estimating the motion properly for the scene \\textit{forest}, since there are a lot of leaves, which leads to a high frequency change between background and leaves.\nThus, the motion estimation would have to output a motion field with a very high spatial resolution to enhance the coder.\nIn this case, PWC-Net does not achieve this and even has a bad influence on the coding result.\nOn average, the novel hyperspectral coder saves roughly 3\\% in comparison to PRIBP. Note that this average is calculated over all sizes and PSNRs and not over the rate and PSNR values from the table.\nThus, videos with a large file size influences this average more.\n\nMoreover, the proposed coding technique is also compared to the random access mode of High Efficiency Video Coding (HEVC), thus an RGB inter coder.\nFor that, every three consecutive channels were merged into an RGB video.\nThe remaining channels are coded as grayscale videos.\nFor 31 hyperspectral channels, this results in 10 RGB videos and 1 grayscale video to code for the standard HEVC.\nThe results vary much more in comparison, however, the average bit rate saving of the proposed hyperspectral video coder is over 50\\% in comparison the inter mode of HEVC.\nIt is noticeable that HEVC has problems with high frequency content in comparison, since the scenes \\textit{family house}, \\textit{outdoor} and \\textit{forest} are compressed much worse.\nOn the other side, for scenes with mostly homogeneous regions and movements, HEVC even outperforms the proposed hyperspectral video coder.\n\n\\begin{table}[t]\n\t\\centering\n\t\\renewcommand{\\arraystretch}{1.5}\n\t\\caption{Evaluation of our novel inter hyperspectral coding against all-intra hyperspectral coding PRIBP and HEVC inter coding (random access). The values are given in Bjontegaard-Delta rate and PSNR, respectively. Lower is better for rate, while higher is better for PSNR.}\n\t\\label{tab:eval_coding}\n\t\\begin{tabular}{c|c|c|c|c}\n\t\t& \\multicolumn{2}{c|}{PRIBP} & \\multicolumn{2}{c}{HEVC Inter Coding} \\\\\n\t\t& BD Rate   & BD PSNR   & BD Rate   & BD PSNR   \\\\ \\hline\n\t\tFam. house          & -10.41\\%  & 0.59 dB   & -54.55\\%  & 4.02 dB   \\\\ \\hline\n\t\tM. seaport          & -7.15\\%   & 0.35 dB   & -22.82\\%  & 1.16 dB   \\\\ \\hline\n\t\tCity                & -9.29\\%   & 0.44 dB   & -19.36\\%  & 0.92 dB   \\\\ \\hline\n\t\tOutdoor             & -9.22\\%   & 0.47 dB   & -65.81\\%  & 4.82 dB   \\\\ \\hline\n\t\tForest              & 13.24\\%   & -0.59 dB  & -73.49\\%  & 6.09 dB   \\\\ \\hline\n\t\tIndoor              & -6.77\\%   & 0.22 dB   & 12.71\\%   & -0.38 dB  \\\\ \\hline\n\t\tLab                 & -9.21\\%   & 0.57 dB   & 42.61\\%   & -1.85 dB  \\\\ \\hline \\hline\n\t\tAverage             & -3.36\\%   & 0.15 dB   & -52.46\\%  & 3.27 dB\n\t\\end{tabular}\n\t\\vspace*{-0.3cm}\n\\end{table}\n\n\\subsection{Other Applications}\nOther applications include spectral reconstruction\\cite{sippel_structure-preserving_2020}.\nSpectral reconstruction has the goal of reconstructing light spectra from multispectral images or even RGB images.\nThe database can also be used for cross-spectral depth estimation\\cite{genser_deep_2020}, where the disparity between different views, which record different ranges in the spectrum, is estimated, e.g., using the multispectral camera array shown in Fig.\\ \\ref{fig:camsi}.\nFurthermore, one can also evaluate different camera setups like trinocular arrangements\\cite{mozerov_trinocular_2009}.\nMoreover, depth estimation algorithms and image enhancement procedures that rely on grayscale and RGB cameras\\cite{jeon_stereo_2016} can be used.\nIn a similar direction steer studies that investigate proper color to grayscale conversions for these types of algorithms\\cite{benedetti_color_2012}.\nA related topic to stereo matching is scene reconstruction using different cameras\\cite{vedaldi_atlas_2020}.\n\nIn general, different fusion techniques related to depth sensors can be evaluated.\nA very obvious example is depth estimation aided by a depth map sensor, e.g., a LIDAR sensor\\cite{gao_object_2018}.\nFurthermore, a depth sensor can be used to help upsampling\\cite{park_high-quality_2014} as well.\nAn even better fit to this database is the fusion of RGB images, hyperspectral recordings and LIDAR data for urban land cover classification\\cite{hansch_fusion_2021}.\n\nOf course reconstruction algorithms for different types of hyperspectral cameras can be easily evaluated like demosaicing for multispectral filter arrays\\cite{feng_mosaic_2021}.\nCombinations of these techniques also can be evaluated, e.g., a mixture of multispectral filter arrays and camera arrays.\n\nFinally, multispectral and hyperspectral denoising algorithms can be evaluated, e.g., techniques based on sparse matrix decomposition\\cite{xie_hyperspectral_2020}.\nThe noise can be even further reduced exploiting the temporal correlation between frames, which can be evaluated using the proposed database.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nThis paper introduced a novel synthetic hyperspectral video database containing seven scenes.\nAll scenes were rendered using a camera array with nine cameras arranged in a three times three grid.\nThis camera array moves through the scenes for 30 frames.\nSince depth maps are provided for every scene, camera and frame, this database provides a high resolution in all three spatial as well as spectral and temporal dimension of the corresponding scene.\nTherefore, this database can serve as validation database for many applications ranging from spectral reconstruction over diverse stereo matching problems like cross spectral disparity estimation to sensor fusion problem like combining hyperspectral data, RGB images and data from depth sensing devices.\nFinally, two applications were covered in detail by exploiting the temporal dimension of the data.\nThe proposed two novel algorithms outperformed its non-temporal versions.\nFurthermore, it proved that this new database can be used to evaluate algorithms in diverse image processing tasks.\n\n\\begin{backmatter}\n \\bmsection{Funding} The authors gratefully acknowledge that this work has been supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project number 491814627.\n \\bmsection{Disclosures} The authors declare no conflicts of interest.\n \\bmsection{Data availability} Data underlying the results presented in this paper are available in Dataset 1, Ref. \\cite{hyvid}.\n\\end{backmatter}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\nA knowledge graph~\\cite{hogan2020knowledge} (KG) is an abstraction used in knowledge representation to encode knowledge in one or more domains by representing entities like \\texttt{New York City} and \\texttt{United States} (i.e., nodes) and binary relationships that connect these entities; for example, \\texttt{New York City} and \\texttt{United States} are connected by the relationship  \\texttt{country}, i.e., \\texttt{New York City} has \\texttt{United States} as a \\texttt{country}. Most of KGs also contains relationships that connect entities with \\textit{literals}, i.e., values from known data structures such as strings, numbers, dates, and so on; for example a relationship \\texttt{settled} that connects \\texttt{New York City} and the integer \\texttt{1624} describe a property of the entity \\texttt{New York City}. \nMore in general, we can view a KG under a dual perspective: as a \\textit{directed labeled multi-graph}, where nodes represent entities or literals and labeled edges represent specific relationships between entities or between an entity and a literal, and as a set of \\textit{statements}, also referred to as \\textit{facts}, having the form of subject-predicate-object triples, e.g., (\\texttt{New York City}, \\texttt{country}, \\texttt{United States}) and (\\texttt{New York City}, \\texttt{settled}, \\texttt{1624}). In the following, we will use the notation (h, r, t) (head, relation, tail) to identify a statement in KG, as frequent in the literature about KG embeddings.\n\nThe entities described in KGs are commonly organized using a set of \\textit{types}, e.g., \\texttt{City} and \\texttt{Country}, also referred to as concepts, classes or data types (when referred to literals). For example, the statement (\\texttt{New York City}, \\texttt{type}, \\texttt{City}) states that the entity \\texttt{New York City} has type \\texttt{City}. Indeed, this types are often defined in what is generally referred to as the \\textit{ontology}~\\cite{ehrlinger2016towards}. An ontology is a formal specification of the meaning of types and relationships expressed as a set of logical constraints and rules, which support automated reasoning.\nFor example, DBpedia~\\cite{auer2007dbpedia}, a knowledge graph built upon information extracted from Wikipedia, describes more than 4 million entities and has 3 billion statements\\footnote{\\url{https:\/\/wiki.dbpedia.org\/about\/facts-figures}}. \n\nWhile KGs can be described using a graph, a nice and simple way to visualize a knowledge graph is considering it as a 3-order adjacency tensor (i.e., a 3-dimensional tensor describing the structure of the KG). Formally a 3-dimensional adjacency tensor is defined as $T \\in \\mathbb{R}^{N \\times R \\times N}$, where $N$ is the number of entities and $R$ is the number of relationships. Each dimension of the tensor corresponds to (\\texttt{head}, \\texttt{relation}, \\texttt{tail}) respectively.\n\nMore formally, assume we have a KG $\\mathcal{G} = \\{ (e_i, r_j, e_k) \\} \\subseteq \\mathcal{E} \\times \\mathcal{R} \\times \\mathcal{E}$, where $\\mathcal{E}$ and $\\mathcal{R}$ denote the sets of entities and relations in the KG, respectively, with $|\\mathcal{E}| = N$ and $|\\mathcal{E}| = R$.\nThe adjacency tensor $T \\in \\mathbb{R}^{N \\times R \\times N}$ is defined as follows:\n\\begin{equation*}\n    T_{i,j,k} = \\begin{cases}\n1 & \\text{if } (e_i, r_j, e_k) \\in \\mathcal{G}, \\\\ \n0 & \\text{otherwise}.\n\\end{cases}\n\\end{equation*}\nTo visualize this, imagine a simple adjacency matrix that represents a single relation, such as the \\texttt{country} relation: the two dimensions of the matrix correspond to the head entity and the tail entity.\nEach entity corresponds to an unique index: given a triple (\\texttt{New York City}, \\texttt{country}, \\texttt{United States}), we have a 1 in the cell of the matrix corresponding to the intersection between the $i$-th row and the $j$-th column, where $i, j \\in \\mathbb{N}$ are the indices associated with \\texttt{New York City} and \\texttt{United States}, respectively.\nOn the other hand, any cell in the adjacency matrix corresponding to triples not in the KG contains a 0.\nIf we consider more than one relationship and we stack them together, we obtain a 3-dimensional tensor, generally referred to as the binary tensor representation of a KG.\nSee Figure~\\ref{bianchi:kge:tensor} for a simple visualization of this concept. \n\n\\begin{figure}[t]\n\\includegraphics[width=1\\linewidth]{Figs\/bianchi_tensor.pdf}\n\\caption{Binary adjacency representation of a KG.}\n\\label{bianchi:kge:tensor}\n\\end{figure}\n\nThe term ``knowledge graph embeddings'' refers to the generation of vector representations of the elements that form a knowledge graph\\footnote{Note that knowledge graph embeddings are different from Graph Neural Networks (GNNs). KG embedding models are in general shallow and linear models and should be distinguished from GNNs~\\cite{scarselli2008graph}, which are neural networks that take relational structures as inputs.}. Essentially, what most methods do is to create a vector for each entity and each relation; these embeddings are generated in such a way to capture latent properties of the semantics in the knowledge graph: \\textit{similar} entities and \\textit{similar} relationships will be represented with \\textit{similar} vectors. Figure~\\ref{bianchi:kge:embeddings:idea} provides an intuitive example of what a knowledge graph embedding method does. The tensor representation introduced above is frequently used in many KG embedding methods that learn embeddings by using dimensionality reduction techniques over the tensor.\n\nThe elements are generally represented in a vector space with low dimensionality (with values ranging from 100 dimensions to 1000 dimensions) and one key aspect is given by the notion of similarity: in a vector space similarity can be interpreted with the use of vector similarity measures (e.g., cosine similarity, in which two vectors are more similar if the angle between them is small).\n\n\\begin{figure}[t]\n\\includegraphics[width=1\\linewidth]{Figs\/bianchi_kge_chapter_embeddings_idea.pdf}\n\\caption{Starting from a knowledge graph, embedding methods generate representations of the elements of the knowledge graph that are embedded in a vector space. For example, these representations could be vectors. Vectors encode latent properties of the graph and for example similar entities tend to be described with similar vectors.}\n\\label{bianchi:kge:embeddings:idea}\n\\end{figure}\n\nAn important task is to find ways to extend KGs adding new relationships between entities. This task is generally referred to as link prediction or knowledge graph completion. Adding new facts can be done with the use of logical inference. For example, from a triple (\\texttt{Washington D.C.}, \\texttt{capital}, \\texttt{United States}) we can infer (\\texttt{Washington D.C.}, \\texttt{country}, \\texttt{United States}).\nInferring this last fact comes from background knowledge encoded in an axiom that specify that if a city is a capital of a country, it is also part of that country (e.g., as encoded by a first order logic rule such as $\\forall X, Y:  \\text{capital}(X, Y) \\Rightarrow \\text{country}(X, Y)$). Unfortunately, many knowledge graphs have many observed facts and fewer axioms or rules~\\cite{trouillon2019inductive}. \n\nKG embeddings can be used for link prediction, since they show interesting predictive abilities and are not directly constrained by logical rules. This property comes at the cost of not being directly interpretable (i.e., the vector representations now encode the latent meaning of the entity\/relationship). The explainability of this prediction is often difficult because the result comes from the combination of latent factors that are embedded in a vector space and an evaluation of the inductive abilities of these methods is still an open problem~\\cite{trouillon2019inductive}.\n\nKnowledge graph embeddings projected in the vector space tend to show interesting latent properties~\\cite{DBLP:conf\/pkdd\/MinerviniCMNV17};\nfor example, \\textit{similar} entities tend to be close in the vector space. The value of similarity in the latent space is a function that depends on the way knowledge graph embeddings are generated. Similarity is also important under the point of view of explaining the meaning. \nFor instance, we might not know the meaning of the entity \\texttt{New York City}, but it can be inferred from its topic by looking at closest entities in the geometric space (i.e. \\texttt{Washington D.C.} and \\texttt{United States}).\n\n\nThe components of the vectors representing the entities and relations are not explainable themselves, and it can be hard to assign a natural language label that describes the meaning of that component.\nHowever, we can observe how different entities and relationships are related within the graph by analyzing its structure -- which was also used to generate the vector-based representations.\nIn addition, the training is driven by a similarity principle, which can be easily understood. For example, similar entities have similar embedding representations, and the same is true for similar relationships.\nThus, while it is not possible to explain the exact difference between two vectors of two entities, we can refer to this similarity when using the vectors in more complex neural networks that use these vectors and the additional information to enrich the network capabilities.\n\nKnowledge graph embeddings have been used in different contexts including recommendation~\\cite{DBLP:conf\/sigir\/HuangZDWC18,DBLP:conf\/www\/WangZXG18,DBLP:conf\/kdd\/ZhangYLXM16}, visual relationship detection~\\cite{baier2017improving} and knowledge base completion~\\cite{bordes2013translating}. Moreover, knowledge graph embeddings can be used to integrate semantic knowledge inside deep neural networks, thus enriching the explainability of pure black-box neural networks~\\cite{lecue2019role,hitzler2019neural}, but they also come with some limitations.\n\nIn this chapter, we describe how to build embedding representations for knowledge graphs and how to evaluate them. We discuss related work of the field by mentioning the approaches that improved the state-of-the-art results.\nThen, we focus on knowledge graph embeddings to support explainability, i.e. how knowledge graph embeddings can be adopted to provide explanations by describing the relevant state-of-the-art approaches. Similarity comes has a key factor also in the context of explainability, in recommender systems for example, similarity is a key notion to express suggestions to users.\n\n\\subsection{Overview of this Chapter}\nThis chapter provides an overview of the field in which we describe how KG embeddings are generated and which are the most influential approaches in the filed up to date. Moreover, the chapter should also describe which are the possible usages for KG embeddings in the context of explainability. \nIn the recent literature, many approaches for knowledge graph embeddings have been proposed; we summarize the most relevant models by focusing on the key ideas and their impact on the community. \n\nIn Section~\\ref{bianchisec:knowledge:embeddings:surv} we give a more detailed overview related to how a knowledge graph embedding method can be defined and trained. We will describe  TransE~\\cite{bordes2013translating}, one of the most popular models, and then we will briefly explain how information that does not come from the knowledge graph can be used to extend the capabilities of the embedding models. This will be a general introduction that should help the reader understand how the methods introduced in the other sections work.\n\nIn Section~\\ref{bianchisection:stateart}, we describe the approaches we have selected. We summarize what researchers have experimented within the field, giving to the reader the possibility of exploring different possible ways of generating knowledge graph embeddings. Note that it is difficult to describe which is the best model for a specific task because evaluation results are greatly influenced by hyper-parameters (see Section~\\ref{bianchi:limits:kge}). Nevertheless, we think that most of the approaches have laid the basis for further development in the field and are thus worth describing. We then describe how knowledge graph embeddings are evaluated, showing that the main task is link prediction and that the datasets used have changed over the years. Link prediction is a task that requires high explainability, something that in the context of knowledge graph embeddings is often missing. In general, ComplEx~\\cite{trouillon2016complex} is often considered as one of the best performing models~\\cite{baier2017improving} and gives stable results in inductive reasoning tasks~\\cite{trouillon2019inductive}.\n\n\nThen, in Section~\\ref{bianchi:ref:sec:explainability}, we focus on explainability. Explainability is a difficult term to define~\\cite{lipton2018interpretability}. Knowledge graph embeddings are not explainable by default, because they are sub-symbolic representations of entities in which latent factors are encoded. Knowledge graph embeddings can be used for link prediction, but the prediction is the result of the combination of latent factors that are not directly interpretable. However, there is recent literature that explores the usage of embeddings in the context of explainable and logical inferences. \n\nWe conclude this chapter in Section~\\ref{sec:conclusions}, where we summarize our main conclusions and we describe possible future directions for the field.\n\n\n\\paragraph{Additional Resources} Several works that provide an overview of knowledge graph embeddings have been proposed in the literature. We point the reader to~\\cite{gesese2019survey} that contains a nicely written survey of approaches that are meant to support the embedding of knowledge graph literals and to~\\cite{wang2017knowledgesurvey} for another overview on knowledge graph embeddings. As knowledge graph embeddings provide sub-symbolic representations of knowledge there is a recent increasing interest in finding ways to interpret how these representations interact~\\cite{allen2019understanding}. Inductive capabilities of knowledge graph embeddings methods have been recently evaluated~\\cite{trouillon2019inductive}.\n\n\\section{Knowledge Graph Embeddings}\\label{bianchisec:knowledge:embeddings:surv}\n\n\n\\paragraph{A Short Primer}\nIn this first part, we are going to define the general elements that characterize a knowledge graph embedding method. To better illustrate how knowledge graph embeddings are created we focus our explanation on one of the seminal approaches of the field, TransE~\\cite{bordes2013translating}. We will introduce how TransE embeddings can be generated and how a method like TransE can be extended to consider information that is not included in the set of triples. While we will describe TransE-specific concepts, most of what it is explained in this section is still valid for other methods in the state of the art.\n\nNowadays, a plethora of approaches to generate embedded representations of KGs exists~\\cite{bordes2013translating,nickel2011three,wang2014knowledge,lin2015learning,trouillon2016complex}. In 2011, RESCAL~\\cite{nickel2011three} was the first influential model to create embedded representations of entities and relationships from a KG by relying on a tensor factorization approach upon the 3-dimensional tensor generated by considering subject entity, predicate entity and object entity as the 3 dimensions of the tensor.  There are mainly three elements that are used to distinguish a method to generate KGs embedding: (i) the choice of the representations of entities and relationships, in general vector representations of real numbers are used~\\cite{bordes2013translating,wang2014knowledge}, but there are methods that use matrices to represent relationships~\\cite{nickel2011three} and complex vectors to represent entities and relationships~\\cite{trouillon2016complex}; (ii) the so-called scoring function, which we will refer to as $\\phi$. This function is used to aggregate the information combing from a triple, and is generally referred to as the function that estimates the \\textit{likelihood} of the triple; lastly (iii) the loss function, which defines the objective being minimized during the training of the knowledge graph embedding model. \n\nChanges in these three elements is what generally makes one model better than the other (although, see Section~\\ref{bianchi:limits:kge}, where we explain the impact of different hyperparameters on the comparison). Scoring functions can be extended with many different information like, information coming from images~\\cite{wang2019multimodal} or numerical and relational features~\\cite{garcia2017kblrn}, in which the entity vector of a scoring function might be represented with the aggregation of image representations of that entity or textual content, an entity can be represented by aggregating the information contained inside its textual description.  At the same time, loss functions can be extended considering different parameters, e.g., it is possible to extend a loss function by adding regularization. \nThe interaction between the entity vectors and the relationship vectors is modulated by the score function. The score function computes a confidence value of the likelihood of a triple.  \n\nThe learning process requires both positive and negative data in input and KGs contain only positive information. In KG embeddings the generation of negative is generally achieved generating \\textit{corrupted triples} i.e., triples that are false. For example, if in a knowledge graph we have the triple (\\texttt{New York City}, \\texttt{country}, \\texttt{United States}), a simple corrupted triple is (\\texttt{United States}, \\texttt{country}, \\texttt{New York City}). Note that despite these training procedures might have several limitations, different methods have been proposed to optimize the selection of good negative samples. One of the most advanced techniques is KBGAN~\\cite{cai2018kbgan} that proposes an adversarial method to generate effective negative training examples that can improve the representations of the knowledge graph embedding.\n\n\\paragraph{Making Knowledge Graph Embeddings}\n\nTransE~\\cite{bordes2013translating} uses k-dimensional vectors to represent both entities and relationships; the score function that the authors propose as the following form $d(\\mathbf{h} + \\mathbf{r}, \\mathbf{t})$, where the $d$ function can be the L1 or the L2 norm. The driving idea of this score function is that the sum of the subject vector with the predicate vector should generate the vector representation of the object as output (i.e. $\\mathbf{h}  + \\mathbf{r} \\approx \\mathbf{t}$), in general the scoring function can be also defined as $d(\\mathbf{h}+\\mathbf{r},\\mathbf{t}) = \\left\\|\\textbf{h}+ \\textbf{r}-\\textbf{t}\\right\\|$. The loss function defined to learn the representations is instead:\n\\begin{equation*}\n    \\mathcal{L} = \\sum_{h,r,t \\in S}\\sum_{h',r,t' \\in S'_{h,r,t}}[\\gamma + d(\\mathbf{h}+\\mathbf{r},\\mathbf{t}) - d(\\mathbf{h'}+\\mathbf{r},\\mathbf{t'})]_{+},\n\\end{equation*}\n\\noindent where $[x]_{+}$ is the positive part of $x$ and $\\gamma$ is a margin hyper-parameter. And $S'_{h,r,t}$ is the set of corrupted triples. $d(\\mathbf{h}+\\mathbf{r},\\mathbf{t})$ is the score of the true triple while $d(\\mathbf{h'}+\\mathbf{r},\\mathbf{t'})$ is the score of the true triple. This loss function favors low values of $d(\\mathbf{h}+\\mathbf{r},\\mathbf{t})$ with respect to the corrupted triples, in such a way that the function can be effectively minimized. It is possible to optimize the representation through the use of gradient-based techniques that are now common in machine learning. Figure~\\ref{bianchi:kge:embeddings:transE} shows how TransE combine entities and relationships in the scoring function. Through the training process, TransE learns vector representations of entities and relationships.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.6\\linewidth]{Figs\/bianchi_kge_transE.pdf}\n\\caption{Example of how TransE represents and models the interactions between entities and relationships in vector space.}\n\\label{bianchi:kge:embeddings:transE}\n\\end{figure}\n\n\\paragraph{Augmenting Knowledge Graph Embeddings} Knowledge graph embeddings can be generated by considering information that is not included in the graph itself. Different methods have been introduced to extend knowledge graph embeddings by adding novel information outside from the one provided by knowledge graph triples and we will give a more detailed overview in the next section, here we describe a method that extends TransE using textual information; adding elements to the score function allows us to include novel information inside our representations. \n\nDescription-Embodied Knowledge Representation Learning (DKRL)~\\cite{xie2016representationentity} jointly learns a structure-based representation $h_{s}$ (as TransE) and a description-based representation $t_d$ that can be used in an integrated scoring function, thus combining the relative information coming from both text and facts. \nTo extend with additional information a model like TransE, the scoring function can be extended to optimize also other representations. For example, DKRL uses the following scoring function:\n\\begin{equation*}\n\\left\\|\\textbf{h}_{s}+\\textbf{r}-\\textbf{t}_{s}\\right\\|+\\left\\|\\textbf{h}_{d}+\\textbf{r}-\\textbf{t}_{d}\\right\\| + \\left\\|\\textbf{h}_{s}+\\textbf{r}-\\textbf{t}_{d}\\right\\|+\\left\\|\\textbf{h}_{d}+\\textbf{r}-\\textbf{t}_{s}\\right\\|.\n\\end{equation*}\nOptimizing this joint score function allows us to combine the information coming from both text and triples. In detail, DKRL uses convolutional neural networks to generate description based representations for the entities. Different information can be used to extend the embedding such as images, logical rules, and textual information. In general, the process to introduce new information relies on the extension of the scoring function. Often adding more information allows us to extend the capabilities of the model. For example, the use of text-based representations allows us to generate vector representations of entities for which we have a description but that are not present in the KG.\n\n\\section{State-of-the-art Knowledge Graph Embeddings}\\label{bianchisection:stateart}\nIn this section, we review some of the algorithms that have been introduced in the state of the art. Our main objective is to give the reader an overview of the research that has been done until now and which are the key points in the knowledge graph embedding field.\n\n\\subsection{Structure-based Embeddings}\\label{bianchisec:structure}\n\nApproaches that focus on the use of knowledge graph facts have also been called \\textit{fact alone} methods by other authors~\\cite{wang2017knowledgesurvey}. Table~\\ref{bianchi:table:scoring:function} shows the different scoring function that can be used to define different knowledge graph embeddings methods. The two main categories of approaches are the \\emph{translational models} and the \\emph{bilinear models}. Transnational models are often based on learning the translations from the head entity to the tail entity (e.g., TransE) while bilinear models often tend to use a multiplicative approach and to represent the relationships as matrices in the vector space. In general, bilinear models obtain good results in the link prediction tasks~\\cite{kazemi2018simple}. Main models of this category are RESCAL~\\cite{nickel2011three}, DistMult~\\cite{yang2014embedding}, ComplEx~\\cite{trouillon2016complex}. \n\n\n\n\n\n\\begin{table}[t]\n    \\centering\n   \n    \\begin{tabular}{ccc}\n        \\toprule\n        {\\bf Method} & {\\bf Scoring Function} & {\\bf Representation} \\\\ \\midrule\n        RESCAL~\\cite{nickel2011three}, 2011  & $\\textbf{h}^\\intercal \\textbf{W}_r  \\textbf{t} $ & $\\textbf{h},\\textbf{t} \\in \\mathbb{R}^{d}$, $\\textbf{W}_r \\in \\mathbb{R}^{d \\times d}$ \\\\ \n        TransE~\\cite{bordes2013translating}, 2013 & $ - || \\textbf{h} + \\textbf{r} - \\textbf{t}||$  & $\\textbf{h},\\textbf{t},\\textbf{r} \\in \\mathbb{R}^{d}$ \\\\\n        DistMult~\\cite{yang2014embedding}, 2014 & $\\langle \\textbf{h},\\textbf{r},\\textbf{t} \\rangle$ & $\\textbf{h},\\textbf{t},\\textbf{r} \\in \\mathbb{R}^{d}$ \\\\\n        HolE~\\cite{nickel2016holographic}, 2016 & $\\langle  \\textbf{r}, \\textbf{h} \\otimes \\textbf{t}  \\rangle $ & $\\textbf{h},\\textbf{t},\\textbf{r} \\in \\mathbb{R}^{d}$ \\\\\n        ComplEx~\\cite{trouillon2016complex}, 2016 & $\\text{Re}(\\langle \\textbf{h},\\textbf{r},\\overline{\\textbf{t}} \\rangle)$ & $\\textbf{h},\\textbf{t},\\textbf{r} \\in \\mathbb{C}^{d}$ \\\\\n        RotatE~\\cite{sun2019rotate}, 2019 & $ -  || \\textbf{h} \\circ  \\textbf{r} - \\textbf{t} ||^2 $  & $\\textbf{h},\\textbf{t},\\textbf{r} \\in \\mathbb{C}^{d}$, $|r_i| = 1$ \\\\ \\bottomrule\n    \\end{tabular}\n   \n    \\caption{A short list with knowledge graph embedding approaches with the respective scoring functions and the representation space used for entities and relationships. Lowercase elements are vectors, while uppercase elements are matrices, $\\otimes$ is the circular correlation. $\\overline{\\textbf{t}}$ defines the complex conjugate of an $\\textbf{t}$ and $\\text{Re}$ denotes the real part of a complex vector. We sampled these approaches by considering the novelty they introduced at the time they were presented.\n    Score functions are based on those published in~\\cite{sun2019rotate,balazevic2019tucker}.}\n    \\label{bianchi:table:scoring:function}\n\\end{table}\n\n\\paragraph{Translational Models} We have described how TransE behaves in the previous section. Note that TransE does not efficiently learn the representations for 1-to-N relationships in a knowledge graph. This comes from how the scoring function is defined: suppose the existence of the triples (\\texttt{New York City}, \\texttt{locatedIn}, \\texttt{State of New York}), (\\texttt{New York City}, \\texttt{locatedIn}, \\texttt{United States}. Eventually, a scoring function consistent with $\\mathbf{s} + \\mathbf{p} \\approx \\mathbf{o}$, would make the entities \\texttt{State of New York} and \\texttt{United States} similar, since the elements $\\mathbf{s}$ and $\\mathbf{p}$ of the formula are fixed. Novel models in the translational group have been introduced to reduce the effect of this problem; we can cite in this category TransH~\\cite{wang2014knowledge} and TransR~\\cite{lin2015learning}. In general, translational models have the advantages of having a concise definition and getting good performances. In this same category, recent and relevant approaches are RotatE~\\cite{sun2019rotate} and HAKE~\\cite{zhang2019learning}.\n\n\\paragraph{Bilinear Models} RESCAL~\\cite{nickel2011three} is based on the factorization of the tensor (see Figure~\\ref{bianchi:kge:embeddings:idea} and has a high expressive power due to the use of a full rank matrix for each relationship in the score function $\\textbf{h}^\\intercal \\textbf{W}_r  \\textbf{t}$, where the interaction between the elements comes under the form of vector-matrix products. At the same time, the full rank matrix is prone to overfitting~\\cite{zhang2019learning} and thus researchers that studied bilinear models have added some constraints on those representations. Indeed,  DistMult~\\cite{yang2014embedding} interprets the matrix $\\textbf{W}_r$ as a diagonal matrix, not making difference between head entity and tail entity and thus forcing the modeling of symmetric relationships~\\cite{kazemi2018simple,trouillon2019inductive}: $\\phi(h,r,t) = \\phi(t,r,h)$, $\\forall h,t$, that force symmetry even for anti-symmetric relationships (e.g., \\texttt{country}, \\texttt{hypernym}).\n\nAt the same time DistMult was extended by ComplEx that models the vectors in a complex vector space to better account for anti-symmetric relationships. HolE~\\cite{nickel2016holographic} uses circular correlation, a non commutative operation between vectors, that allows us to effectively surpass the $\\phi(h,r,t) = \\phi(t,r,h)$ problem that DistMult had. Note that it has been proved that HolE and ComplEx are isomorphic~\\cite{hayashi-shimbo-2017-equivalence}. ANALOGY~\\cite{liu2017analogical} is a model that extends the scoring function by considering analogical relationships that exist between entities given the relationships. \nIn their paper~\\cite{liu2017analogical}, the authors have shown that DistMult, ComplEx and HolE are special cases of ANALOGY.\n\n\\paragraph{Neural Models}  Another group with a lower number of proposed approaches consists of neural networks-based models; the Neural Tensor Network~\\cite{socher2013reasoning} is an approach for knowledge graph embeddings that uses a score function that contains a tensor multiplication, that depends on the relationship, to relate entity embeddings, this type of operation provides some interesting reasoning capabilities and was also used in later approaches as a support for reasoning using neural networks in a neural-symbolic model~\\cite{serafini2016logic}. Instead, ConvE~\\cite{dettmers2018conve} introduces the use of convolutional layers, thus being closer to deep learning approaches. While effective, this method suffers from limited explainability and more variation given by the number of hyperparameters that increases with the number of layers~\\cite{sun2019rotate}.\n\n\\paragraph{Recent Approaches}\nWe hereby summarize some recent approaches that have been introduced in the literature and that are relevant with respect to the results they obtained and the ideas that stand behind them.\n\n\\begin{itemize}\n    \\item Hierarchy-Aware Knowledge Graph Embedding (HAKE)~\\cite{zhang2019learning} is one of the few models that also consider the fact that elements in the knowledge graph belong to different levels of the hierarchy (e.g., the authors use the triple \\textit{arbor\/cassia\/palm,  hypernym,  tree} as an example of elements at different levels of the hierarchy). Using polar coordinates they are able to distribute the hierarchical knowledge inside the representations.\n    \\item RotatE~\\cite{sun2019rotate} was introduced to provide a method to effectively represent symmetric properties in knowledge graph embeddings. The authors of this paper propose to use rotation in a complex space to support symmetry and other properties. In Figure~\\ref{bianchi:kge:embeddings:rotate} we show how rotation can effectively support the definition of relationships that are symmetric; the rotation allows you to interpret symmetry as a geometric property. Authors prove that their model, implemented inside a complex vector space, can capture properties like symmetry, inversion, and composition.\n    \\item TuckER~\\cite{balazevic2019tucker} is a recent approach that also uses tensor factorization for knowledge graph embeddings obtaining good results over the link prediction task.\n    \\item Another recent approach tries to apply graph convolutional neural networks to generate knowledge graph embeddings, and this might influence a new way of dealing with knowledge graph structures~ \\cite{schlichtkrull2018modeling}.\n    \\item Contextualized Knowledge Graph Embeddings~\\cite{gupta2019care} (COKE) is a method that has been inspired by recent results of contextual representation of words~\\cite{peters2018deep}: using transformers~\\cite{vaswani2017attention}, the authors propose to capture the different meanings an entity can assume in different parts of the knowledge graph. For example, the entity Barack Obama is connected to entities related to politics, but also to the entity that represents members of his family, showing two different \\textit{contextual meanings} of the same entity. The main difference between COKE and other models is that it models the representations based on the context and thus, differently from other methods, it provides representations that are not static.\n    \\item SimplE~\\cite{kazemi2018simple} extends canonical Polyadic tensor decomposition (CP)~\\cite{hitchcock1927expression} to provide good embeddings for link prediction. CP poorly performs on link prediction because it learns two independent embeddings for each entity. SimplE makes use of inverse relationships to jointly learn the two embeddings of each entity.\n    \\item Quantum embeddings~\\cite{garg2019quantum} are a novel method to embed entities and relationships in a vector space and the representations are generated following ideas that come from quantum logic axioms~\\cite{birkhoff1936logic}. These embeddings preserve the logical structure and can be used to do both reasoning and link prediction.\n\\end{itemize}\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.6\\linewidth]{Figs\/bianch_kge_drawings_rotatE.pdf}\n\\caption{Example of how the use of rotation can support the definition of properties that are symmetric in the vector space. Image is adapted from~\\cite{sun2019rotate}.}\n\\label{bianchi:kge:embeddings:rotate}\n\\end{figure}\n\n\n\n\\subsection{Enhanced Knowledge Graph Embeddings}\\label{bianchisection:enached}\nWhile most of the previous approaches rely mainly on the use of the triples present in the knowledge graph to generate the vector representations; additional information (or different information) can be used inside the embeddings to generate vectors that account for a better representation. As noted by~\\cite{wang2017knowledgesurvey} attributes (like gender) need to be model in an efficient way: the attribute \\textit{male} is connected to multiple entities and thus model like TransE might not be adequate to treat this issue; in the literature, there are in fact models that have been proposed to account for better handling of these attributes~\\cite{lin2016attributes}.\n\n\n\\paragraph{Path-based Embeddings}\nWhile the most common approaches use a score function that is based on triples, more recent approaches try to consider also the information that comes from a path on the graph~\\cite{lin2015modeling,guu2015traversing}. There are approaches that focus on the use of Recurrent Neural Networks (RNNs) to tackle the task of multi-hop predictions~\\cite{yin2018recurrent,das2017chains}. \n\n\\paragraph{Distributional Embeddings} An alternative approach to generate embeddings comes from the computational linguistics field and it is represented by those models that view language under a distributional perspective in which the meaning of words in a language can be extracted from the usage of those words in the language. Word2vec~\\cite{mikolov2013distributed} is a model that embeds words in the vector space by eventually putting words that appear in similar contexts in close positions of the vector space. In the same way, on Wikipedia using user-made links~\\cite{basile2016learning} or using entity linking~\\cite{bianchi2018towards} it is possible to generate embeddings of the entities of a knowledge graph using the word2vec algorithm~\\cite{mikolov2013distributed}. For example, Wiki2vec\\footnote{\\url{https:\/\/github.com\/idio\/wiki2vec}} uses word2vec over Wikipedia text and generates the representations for both entities (by looking at links co-occurrence) and words. TEE~\\cite{bianchi2018towards} proposes to use entity linking to first disambiguate text and generate sequences of entities and then use the knowledge graph to replace the sequences of entities with sequences of most specific types; using word2vec one can generate entity and type embeddings based on the distribution in text. Methods that are based on entity linking suffer from low coverage, caused by the entity linking quality. In general, these models do not provide a direct way to embed relationships. Another prominent model in this category is RDF2Vec~\\cite{ristoski2018rdf2vec}: it uses an approach that combines techniques from the word embeddings community with knowledge graphs. It generates embeddings of entities and relationships by first creating a virtual document that contains lexicalized walks over the graph and then use word embeddings algorithm on the virtual document to create the representations. \n\n\\paragraph{Text-Enhanced Embeddings} There instead exists a variety of models that makes use of textual information~\\cite{wang2016text,fang2016entity,wang2014knowledge2,xie2016representationentity,xiao2017ssp,Jameel2016EntityEW,an2018accurate} to enhance the performance of knowledge graph embeddings techniques. These pre-trained representations can be used to initialize knowledge graph embeddings and to generate representations that can, in some cases, outperform other baselines~\\cite{yang2014embedding}. As stated in the previous section, the use of textual information can be useful to generate the representations of the entities even when they are not present in the knowledge base. For example, Text-enhanced Knowledge Embedding~\\cite{wang2016text} (TEKE) focuses on Wikipedia inner links and replaces them with Freebase entities and then constructs a co-occurrence network of entities and words in the text; eventually, this information is used to enrich the contextual representation of the elements of the knowledge graph. Jointly~\\cite{wang2014knowledge2} is an embedding method in which textual knowledge is used to enrich the representation of entities and relationships. In this work, both entities and words are aligned into a common vector space; vectors associated with words and entities that represent a common concept are then forced to be closer in the vector space by combining different loss functions.  Description-Embodied Knowledge Representation Learning (DKRL)~\\cite{xie2016representationentity} includes the description of the entities in the representation. DKRL uses a convolutional layer to encode the description of the entity into a vector representation and use this representation in the loss function. Words vectors coming from the entity description can be initialized with the use of word2vec embeddings. The model learns two representations for each entity, one that is structure-based (i.e., like TransE) and one that is based on the descriptions. One key advantage of DKRL~\\cite{xie2016representationentity} is that it offers the possibility of doing zero-shot learning of entities by using the description of the entities themselves.\n\n\\paragraph{Image Enhanced Embeddings} \nImage-embodied Knowledge Representation Learning~\\cite{xie2017image} (IKRL) provides a method to integrate images inside the scoring function of the knowledge graph embedding model. Essentially, IKRL uses multiple images for each entity and use the AlexNet convolutional neural network~\\cite{krizhevsky2012imagenet} to generate representations for the images; these representations are then selected and combined with the use of attention to be finally projected in the entity space, generating an image specific representation for images. \nRecently, approaches to exploit \\textit{multi-modal learning} on knowledge graph embeddings that combine image features and other information have also been introduced in the state-of-the-art~\\cite{wang2019multimodal,liu2019mmkg}. \n\n\\paragraph{Logic Enhanced Embeddings} There are approaches that account for the combination of logic and facts~\\cite{wang2016learning,guo2016jointly,guo2018knowledge,rocktaschel2015injecting} for knowledge representation. KALE is a model that combines facts and rules using fuzzy logic~\\cite{guo2016jointly}. There are other approaches that try to embed knowledge graphs by keeping the logical structure consistent, we mentioned embedding with quantum axioms in Section~\\ref{bianchisec:structure}, but there are other methods that starts with the objective of doing logical reasoning over embedded representations~\\cite{serafini2016logic,rocktaschel2017end} (we will present more details of these approaches in the Section~\\ref{bianchi:ref:sec:explainability}, where we discuss explainability).\n\nResearchers have shown that it is possible to combine facts and first-order formulae using a joint optimization process.\nIn \\cite{DBLP:conf\/naacl\/RocktaschelSR15}, the authors propose a general approach for incorporating first-order logic formulae in embedding representations.\nDuring training, their approach samples sets of entities, and jointly minimizes the negative likelihood of the data and a loss function measuring to which extent the model violates the given rules with respect to the sampled entities.\nA shortcoming of this approach is that it relies on a sampling procedure, and it provides no guarantees the model will still produce predictions that are consistent with the logic rules for entities that were not observed during training.\nTo overcome this shortcoming, in \\cite{DBLP:conf\/pkdd\/MinerviniCMNV17} authors incorporate equivalency and inversion axioms between relations by only regularizing the relation representations during the training process, where the shape of the regularizers are derived from the axiom and the model formulations.\nA similar idea is followed by \\cite{DBLP:conf\/emnlp\/DemeesterRR16} for incorporating simple implications between two relations.\nIn \\cite{DBLP:conf\/uai\/MinerviniDRR17}, authors propose using adversarial training for incorporating general first-order logic rules in entity and relation representations: during training, an adversary searches for entities where the model violates the given constraints, and the model is regularized in order to correct such violations.\nEntities can be searched either in entity or in entity embedding space; in the latter case, the problem of finding the entity embeddings where the model maximally violates the logic rules can be efficiently solved via gradient-based optimization.\n\n\n\\paragraph{Schema-Aware Embeddings}\nFew models in the state of the art focus on the differences between instances (i.e., entities) of a knowledge graph and concepts (like, \\texttt{Country},  \\texttt{City} and  \\texttt{Place})~\\cite{lv2018differentiating}.\nSchema-rules can be useful to define constraints over score predictions. For example, they have been used to learn predicate specific parameters to decrease, in an adaptive way, the score of relationships that might be conflicting with schema rules~\\cite{DBLP:conf\/sac\/MinervinidFE16}.\n\nTransC~\\cite{lv2018differentiating} proposes an interesting representation for concepts, in which each concept is represented as a sphere and each entity is a vector. An instance-of relationship can be easily verified by checking if the entity is contained inside the sphere. In one of the previous sections, we mentioned HAKE (Hierarchy-Aware Knowledge Embeddings)~\\cite{zhang2019learning} as a recent method that considers the hierarchical topology in the embedding. This aspect is also important in the context of analysis over explainability: modeling ontologies is a needed step to learn how to model logical reasoning and provide justifiable inferences, however, not all methods are capable of modeling rules~\\cite{gutierrez2018knowledge}.\n\nThere are also approaches that considers the fact that the ontology can be used to provide better representations, for example Type-embodied Knowledge Representation Learning (TKRL)~\\cite{xie2016representationtypes}. Given a triple $h,r,t$, the subject $\\textbf{h}$ and the object $\\textbf{t}$ are projected to the type spaces of this relation as $\\textbf{h}_{r}$  and $\\textbf{t}_{r}$, the projection matrices become type-specific. TKRL optimizes the following scoring function: $||\\textbf{h}_{r} + \\textbf{r} - \\textbf{t}_{r}||$. In this group we also include TRESCAL~\\cite{chang2014typed} an extension of RESCAL~\\cite{nickel2011three} that considers types in the tensor decomposition. On the other hand, there do exist approaches that generate the representations of ontology concept by taking in consideration the co-occurrence of types in text~\\cite{bianchi2018towards}. \n\n\\paragraph{Hyperbolic Embeddings}\nMany approaches in the state-of-the-art rely on the use of representations in the Euclidean space. However, when dealing with the representations of tree-like structures (e.g., some ontologies can be interpreted as trees) Euclidean spaces have to rely on many dimensions and are not suited to represent trees. Euclidean geometries rely on Euclid's axiom of the parallel lines, but there do exist other geometries that do not consider it. Hyperbolic geometries allow us to use hyperbolic planes where trees can be effectively encoded.  These approaches have been now widely used to represent tree-like structure ~\\cite{nickel2017poincare,suzuki2019hyperbolic,sala2018representations} and received recognition in natural language processing~\\cite{le2019inferring,tifrea2018poincare,vimercati2019mapping}. In general, these approaches have been applied to ontological trees (e.g., the WordNet hierarchy) and cannot account for knowledge graph structures that are more complex. Recently, embedding in the hyperbolic plane has shown to be effective also for knowledge graphs~\\cite{balazevic2019multi,kolyvakis2019hyperkg} since they can provide better ways to model topological structures~\\cite{kolyvakis2019hyperkg}. \n\n\\paragraph{Temporal Knowledge Graph Embeddings} There are also approaches that are meant to account for temporality in knowledge graph embeddings by considering temporal link prediction (i.e., consider that some predicates, like \\textit{president of}, have values that change over time) and to study the evolution of knowledge graphs over time~\\cite{jiang2016encoding,esteban2016predicting,garcia2018learning}. For example, recurrent neural networks can be used to learn time-aware relation representations~\\cite{garcia2018learning}.\n\n\\subsection{Evaluation and Replication}\\label{bianchi:evaluation}\nEvaluation in knowledge graph embeddings is often based on link prediction. In general, the link prediction task can be defined as the task of finding an entity that can be used to complete the triple $(h,r,?)$; for example, (\\texttt{New York City}, \\texttt{country}, \\texttt{?}), where \\texttt{?} is United States. To compute the answer for the incomplete triple generally the score function is used to estimate the \\textit{likelihood} of the entities. The procedure is the following: for each triple to test, we remove the head and we compute the value of the score function for each of the entities that we have in the dataset and we rank them from higher to lowest. Then we collect the rank of the correct entity. The same is done by replacing the tail of the triple. At the end, the average rank is computed, this measure is called Mean Reciprocal Rank (MRR). Another measure that is often used in the link prediction setting is the HITS@K (with K commonly in ${1,3,10}$). \n\n\\cite{bordes2013translating} uses a \\textit{filtering} setting that has become a standard of the evaluation. The evaluation of the MRR is influenced by the fact that some \\textit{correct} triples share entity and relationship (e.g., (\\texttt{United States}, \\texttt{countryOf}, \\texttt{?}) is true for multiple triples) and they can be ranked one over the other in the ranking list, thus biasing the results. What it is typically done when computing the MRR for a triple in this setting is to filter out the other triples that are true and that are present in the training\/validation\/test set. \n\nFB15k~\\cite{bordes2013translating} is a subset of Freebase while WN18~\\cite{bordes2013translating} is a Word-Net subset. FB15k and WN18 were both introduced in~\\cite{bordes2013translating} and originally come with a training, validation and test split.\n\nThe quality of these two datasets has been argued in more recent work~\\cite{toutanova2015observed,dettmers2018conve}. FB15k originally contained triples in the test set that are the inverse of those present in the training set, for example \\texttt{\/award\/award\\_nominee} and \\texttt{\/award\\_nominee\/award}. While those links are not false, they could bias the results by making the task easier for learning models (i.e., models can just learn that one relationship is the inverse of the other~\\cite{toutanova2015observed}, and models that force symmetry, like DistMult, could perform better just because of the dataset used). The same problem was found in WN18~\\cite{dettmers2018conve}. This brought researchers to introduce two novel datasets, a subset of the original ones, that do not contain easy-to-solve cases. FB15k-237 has been introduced by~\\cite{toutanova2015observed} and WN18RR was introduced by~\\cite{dettmers2018conve}  and they are a subset of FB15K and WN18 respectively. Take into account that the DistMult model favored the symmetry between the relationships.\n \nYAGO3-10~\\cite{mahdisoltani2014yago3,dettmers2018conve} has recently become quite popular, it contains a subset of the YAGO knowledge graph that consists of entities that have more than 10 relationships each. As noted by~\\cite{dettmers2018conve} the triples in this dataset account for descriptive attributes of people (e.g., as citizenship, gender, and profession). Another really important dataset is Countries~\\cite{bouchard2015approximate}, which is often used to evaluate how well knowledge graph embeddings learn long term logical dependencies.  Note that while in general, the datasets used are the ones we described, some papers introduce new datasets when needed. For example, a subset of the YAGO dataset (namely YAGO39K) has been used to evaluate TransC a work that extended embeddings with the use of concepts~\\cite{lv2018differentiating}.\n\nIn Table~\\ref{bianchi:tab:dataset:sizes} we show numerical data related to these datasets. It is important to notice that these datasets are small with respect to the size of knowledge graphs (e.g., DBpedia has more than 4 million entities).\n\n\n\\begin{table}[t]\n    \\centering\n    \\begin{tabular}{cccccc}\n        \\toprule\n        \\textbf{Dataset} & \\textbf{\\# Entities} & \\textbf{\\# Relations} & \\textbf{Train} & \\textbf{Validation} & \\textbf{Test} \\\\\n        \\midrule\n        FB15k & 14,951 & 1,345 & 483,142 & 50,000 & 59,071 \\\\\n        FB15k-237 & 14,505 & 237 & 272,115 & 17,535 & 20,466 \\\\\n        WN18 & 40,943 & 18 & 141,442 & 5,000 & 5,000 \\\\ \n        WN18RR & 40,943 & 11 & 86,835 & 2,824 & 2,924 \\\\\n        YAGO3-10 & 123,182 & 37 & 1,079,040 & 5,000 & 5,000 \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{Number of entities, relationships and training, validation, test triples for the main dataset used in the state-of-the-art.}\n    \\label{bianchi:tab:dataset:sizes}\n\\end{table}{}\n\nLink prediction is not the only task on which knowledge graph embedding are evaluated, often the evaluation takes into account the task of triple classification, that is the task of verifying if a triple is true or false (i.e., it is a binary classification task over input triples).\n\n\\subsection{Open-source Projects on Knowledge Graph Embeddings}\nMany approaches in the literature share code to reproduce the results in the paper, but often code is written in different languages and does not allow efficient comparison between methods and extensions of the methods. However, there are now some libraries that can be used to replicate the results of different knowledge graph embeddings methods. We cite three and currently active repositories that are popularly used. OpenKE\\footnote{\\url{https:\/\/github.com\/thunlp\/OpenKE}}~\\cite{han2018openke}, the main repository contains Pytorch code, but the authors made the code available also in tensorflow. Ampligraph\\footnote{\\url{https:\/\/github.com\/Accenture\/AmpliGraph}}~\\cite{costabello2019ampligraph}, a tensorflow library that introduces high-level APIs to generate embeddings. Finally, PyTorch BigGraph is another interesting library for knowledge graph embeddings that has been recently introduced by Facebook that can scale to billions of entities\\footnote{\\url{https:\/\/github.com\/facebookresearch\/PyTorch-BigGraph}}~\\cite{lerer2019biggraph}.\n\n\n\\subsection{Limitations of Knowledge Graph Embeddings}\\label{bianchi:limits:kge}\n\nDifferent methods have been introduced in literature and all come with different training methods (e.g., different optimizers, different loss functions, different strategies for sampling negatives). Making the comparison between different methods often difficult and in general not directly possible. \n\nThe first hints of these limitations have been outlined in 2017, where a  work has shown that most of the approaches introduced until then could be outperformed by the use of a simple well-tuned DistMult model~\\cite{kadlec2017knowledge}; the other two competitive models were ComplEx~\\cite{trouillon2016complex} and HolE~\\cite{nickel2016holographic}. As stated by the authors, there is the need to focus on different measures for the evaluation of knowledge graph embeddings\\footnote{Note that the work considered experiment over FB15k and WN18.} and for the intensive study of how hyperparameters are selected. Results are sometime more influenced by training epochs than from actual model complexity.\n\nRecent work~\\cite{calibration} shows that KGE models for link prediction are uncalibrated. This is problematic especially for triple classification tasks where users must define relation-specific thresholds, which can be difficult when working with a large number of relation types. \nMoreover, calibrated probabilities are crucial to provide trustworthy and interpretable decisions (e.g. drug-target discovery scenarios). The authors propose a heuristics that adopts Platt scaling or isotonic regression to calibrate KGE models even without ground truth negatives.\n\nA very recent paper~\\cite{anonymous2020you} has provided new evidence over the limitations of the evaluation of knowledge graph embedding approaches. Authors found that the results of the approaches vary significantly across studies and that they are very much dependent on experimental settings including hyperparameters and loss functions. The main result of this paper is that the conclusions drawn in different papers probably need to be revised in light of the results. Note that the paper address only structure-based embeddings (to which they refer to as pure knowledge graph embeddings), but since many of the enhanced models are based on knowledge graph embeddings, the conclusions drawn from them should also be revised. This paper suggests the lack of a predefined ground of comparison for the embeddings that was already hinted by the need of updating the evaluation datasets (see Section ~\\ref{bianchi:evaluation}, where we explained the limitations of some of the state-of-the-art datasets). The same authors propose LibKGE\\footnote{\\url{https:\/\/github.com\/uma-pi1\/kge}} an open-source library for reproducible research on knowledge graph embeddings that might become useful in providing more robust results to the community.\n\n\\section{Knowledge Graph Embeddings and Explainability}\\label{bianchi:ref:sec:explainability}\n\nWhile explainability is a widely used term and its general meaning is intuitive, there is no agreed definition about what explainability in machine learning is~\\cite{lipton2018interpretability}. Explainability in the context of knowledge graph has recently been outlined by~\\cite{lecue2019role,hitzler2019neural}. In relation to knowledge graph embeddings, explainability has a difficult interpretation: while knowledge graphs are open and in general explainable in terms of direct relationships with other entities, knowledge graph embeddings are often referred to as sub-symbolic, since they represent elements in the vector space, thus losing the original interpretability that comes from logic. The difficulty of mapping vector space representations with logic has been outlined in different work~\\cite{gutierrez2018knowledge,kazemi2018simple} and that in general, some rules are impossible to learn with knowledge graph embeddings (i.e., as described by~\\cite{gutierrez2018knowledge} DistMult can only model a restricted class of subsumption hierarchies). Moreover, explainability passes from the definition of methods that support logical reasoning, since logic offers a paradigm that supports reasoning and its inferences are justifiable and verifiable using logical axioms.\n\nThe problem in parts originates from the fact that there is no agreed view upon how to measure how explainable a system is; the quest of explainable artificial intelligence remains \\textit{how to build intelligent systems able to expose explanation in a human-comprehensible way}~\\cite{lecue2019role}. \n\nExplainability in knowledge graph embeddings is also  important because these latent representations are affected by bias, and social biases have to be taken into consideration when using embeddings for prediction. In fact, as word embedding show stereotypical biases in the representation, evidence of bias in knowledge graph embeddings has been found~\\cite{fisher2019measuring}: males are more likely to be bakers while females are more likely to be home-keepers; this fact could greatly bias the link prediction of novel relationships, think for example of a link prediction system that predicts the most suitable person for a job. Explainability is important in the context of KG embedding because we need to be able to explain these inferences. Same requirement is needed by methods that study drugs effects~\\cite{malone2018knowledge}.\n\nWhat is generally missing is a methodology to effectively explain the predictions of knowledge graph embeddings. From their introduction in the state-of-the-art until now, embedding methods have been mainly evaluated and compared by considering only the accuracy on link prediction tasks. As already outlined in literature~\\cite{pezeshkpour2019investigating}, studies should also be conducted to evaluate the interpretability and the reason why link prediction is feasible in knowledge graphs. For example, Completion Robustness and Interpretability via Adversarial Graph Edits (CRIAGE)~\\cite{pezeshkpour2019investigating} explore the robustness of the approaches by seeing how adding and removing facts affects the general performance of the models. CRIAGE can estimate the effect of those modifications and how they influence the predictions; moreover, it can be used to evaluate the sensitivity of the models towards the addition of fake facts.  CRIAGE~\\cite{pezeshkpour2019investigating} can be indeed used to understand and explain knowledge graph embeddings prediction and explore the limitations and the advantages of different models. In this context, it is worth to cite the closely related, but introduced for graph neural networks, GNNExplainer \\cite{ying2019gnnexplainer}. GNNExplainer is the first work that provides an approach to make sense of the predictions of a graph network: it can be used to identify the most important parts and features of the graph neural network that influence the prediction of a particular instance (e.g., new link, new node label). While this model has been applied to graph neural network it might be possible to adapt it to knowledge graph embeddings.\n\n\\cite{lecue2019role} provides an overview of the challenges, the approaches and the limitations of Explainable Artificial Intelligence in different fields, such as machine learning, planning, natural language processing, computer vision, etc. In particular, the author focuses on how knowledge graphs could be used to support explanations in order to overtake the limitations in each field.  \n\nAn advantage of knowledge graph generated representations, with respect to standard representations generated by deep learning algorithms, is that they come with a previous meaning: each entity vector has a connection with the knowledge graph from which it originates; even if the representation is sub-symbolic. Differently from words~\\cite{mikolov2013distributed}, knowledge graph embedding representations do not suffer from inheriting ambiguity and are can be thus be used more effectively to model reasoning and explainable systems. Moreover, knowledge graph embeddings are not ambiguous in contrast to ``pure words'' in sentences that are ambiguous; this last fact can also help in context of explainability, since it's favorable to provide explanations on something that is not ambiguous and that is linked to a knowledge base.\n\nA key combination can come from the usage of knowledge graph embeddings with logical rules, that can provide justification and explainability over inferences. As stated by \\cite{lecue2019role}, knowledge graphs could provide a semantic layer to support tasks like question answering that are generally tackled with brute force approaches on text. Knowledge graphs can provide generalization capabilities using logic as the source of the generalization: the KG representations can, in fact, be used as sources of inputs to deep learning algorithms and can be used to bridge two worlds that are apart. Knowledge graph representations are linked to knowledge graphs and are thus connected to a source that has explicit connections.\n\nIn fact, there has been a recent spike in the interest for knowledge graph embedding used inside recommender systems to enhance the performance and the explainability of recommendation \\cite{DBLP:conf\/sigir\/HuangZDWC18,DBLP:conf\/www\/WangZXG18,DBLP:conf\/kdd\/ZhangYLXM16}. Deep Knowledge-Aware Network~\\cite{DBLP:conf\/www\/WangZXG18} is a deep network that is used to include external knowledge, trough the use of entity embeddings, inside a news recommendation system; the idea behind this model is to use the information in the knowledge graph to recommend to user news that have a high probability of being clicked. Instead of focusing on word-occurrence based method, like topic models, the proposed model search for more latent factors to use in the recommendation trough the use of embeddings. Instead, other researchers have combined embeddings with recurrent neural networks to account for the recommendation of items based on sequences of user interactions~\\cite{DBLP:conf\/sigir\/HuangZDWC18}.\n\nMany methods for recommendation have limitations regarding the explainability when a multi-hop reasoning is required. In attempt to address this shortcoming, a Knowledge-aware Path Recurrent Network (KPRN) is proposed in \\cite{DBLP:conf\/aaai\/WangWX00C19}.  KPRN models the  sequential  dependencies that connect users and items by also considering the entities and the relationships in between. The running example in the paper is as follows if (\\texttt{Alice}, \\texttt{Interact}, \\texttt{Shape of You}) \\& (\\texttt{Shape of You}, \\texttt{SungBy}, \\texttt{Ed Sheeran}) \\&  (\\texttt{Ed Sheeran}, \\texttt{IsSingerOf}, \\texttt{I SeeFire}) then (\\texttt{Alice}, \\texttt{Interact}, \\texttt{I See Fire}). LSTMs are used to model the sequences of entities and relationships and to predict a recommendation. The embedding of entities and relationships is similar to the path-based embeddings introduced in Section~\\ref{bianchisection:enached}.\n\nAt the same time, the field of conversational agents has also taken into consideration the use of knowledge graph embeddings for explainable conversations~\\cite{moon2019opendialkg}. OpenDialKG~\\cite{moon2019opendialkg} is a corpus in which there is a parallel alignment between the knowledge graph and the dialogues. The authors of the paper propose also an attention-based model that can learn knowledge paths from entities mentioned in the dialog contexts and predicts novel entities that are relevant to the contexts of the dialog: paths provide explanations for entity used in reply to a dialog. Initialization of the model is done through the use of knowledge graph embeddings. \n\nThese last models are close to what has been outlined at the start of this section~\\cite{lecue2019role,hitzler2019neural}: knowledge graphs can provide a semantic and explainable layer (i.e., for conversational agents and for recommendation) that can be useful not only to simply solve tasks but also to provide an effective way to interpret the black-box answers given by neural models.\n\n\\subsection{Knowledge Graph Embeddings and Logical Reasoning}\nLogic is the main explanation paradigm for KGs and one important aspect we want KG embeddings to cover is how to account for axiomatic knowledge inside embeddings.\nThrough a standard KG embedding model it is possible to perform several downstream tasks, such as triple validation or subject, object and relationship prediction.\nKnowledge graph embeddings model relational structure under the form of elements in a low dimensional vector space. While originally methods have been introduced to solve link prediction tasks, more recently many researchers have studied and explored the logical properties of knowledge graph embedding methods. The question that they try to answer is related to how can we effectively model logical knowledge inside a vector space.\n\nIn fact, a recent trend in literature is to propose embedding models that can effectively model some specific logical axioms inside the vector space. This attempt is related to the observation that pure translational approaches like TransE~\\cite{bordes2013translating} are not capable of modeling symmetry in the relationships due to the choice of the score function. \n\nComplEx~\\cite{trouillon2016complex} was proposed as an extension of the DistMult approach in the complex space in which it is easier to model properties of relationships like anti-symmetry (remember that DistMult force symmetry between the relationships). \nInstead, as introduced above, RotatE~\\cite{sun2019rotate} use rotations in a complex plane to capture properties like symmetry, antisymmetry, inversion, and composition. In fact, RotatE models each relationship as a rotation from the subject vector to the object vector in a complex hyperplane. Still, a drawback of the approaches that are based on complex Euclidean geometry is that they require a large number of parameters to train.\n\nA promising direction is how to perform complex logical queries using KGE models. For instance, a query ``Predict communities C? in which user u is likely to upvote a post'' might be expressed as $C?. \\exists P : upvote(u, P) \\wedge belong(P, C?)$.\nIn \\cite{DBLP:conf\/nips\/HamiltonBZJL18} the authors propose a method to map and execute conjunctive queries in a vector space represented by KG embeddings, and further extended by \\cite{DBLP:journals\/corr\/abs-2002-05969} to support disjunctions.\nRecent work such as QUERY2BOX~\\cite{Query2box} goes as far as proposing a hybrid query processing framework. The authors propose a KGE-based query engine that addresses both conjunctive and disjunctive queries by modeling queries as bounding boxes in the embedding space. Besides its intrinsic interpretability - grounded in first-order logical queries it supports - this multi-hop reasoning framework shows how the interplay of KG embeddings and logical queries overcome missing information in the graph when delivering an answer.\n\nThere are approaches that try to combine sub-symbolic representations with reasoning systems: Logic Tensor Networks~\\cite{serafini2016logic}, for example, allows us to define a differential fuzzy logic language over data. Essentially, Logic Tensor Networks (LTN) create the representations for logical constants, functions, and predicates by embedding those in a vector space. While Logic Tensor Networks were not used directly to create knowledge graph embeddings, they have been used with good results on semantic image interpretation tasks~\\cite{donadello2017logic,donadello2019compensating}. Integrating embedding approaches with logical reasoning can account for more complex inferences: combining similarity with logical inferences can bring to interesting results in the field; for example, it is possible to use embedding similarity to extend reasoning on unknown entities. For example, ~\\cite{bianchi2019complementing} shows that combining entity embeddings with logical systems like LTNs can be useful to make inferences that are impossible for rule-based systems.\n\nOn a similar note, the Neural Theorem Prover (NTP)~\\cite{rocktaschel2017end} is an extension of the Prolog programming language that uses embeddings in place of the strict unification provided by Prolog. For such a reason, they are also able to provide an explanation in the form of proof paths, for any given prediction.\n\nWhile both LTN and NTP are not directly knowledge graph embedding methods they use or generate embeddings as part of their training procedure (e.g., LTN embeds elements in the vector space to support logical reasoning). The NTPs provide strong reasoning capabilities with the power of the neural network models but are not scalable to large knowledge bases. Indeed, recently NTPs were extended by the Greedy NTPs (GNTPs)~\\cite{minervini2020differentiable} a model that greatly reduces the computational needs of NTPs, making it possible to use it on large knowledge bases by considering to prune reasoning paths that are not likely when doing inference. We mention again in this section the embeddings inspired by quantum physics have been proposed~\\cite{garg2019quantum} that provide methodologies to reason over embedding by preserving the logical structure.\n\n\n\n\n\n    \n    \n    \n    \n    \n\n\n    \n    \n\n\n\n\n\n    \n\n\n\\section{Summary and Future Directions}\\label{sec:conclusions}\n\nIn this chapter, we have summarized the current state-of-the-art of knowledge graph embeddings by describing many different methods and their main properties. We have also outlined the limitations of these methods, that provide dense representations that while not directly interpretable, but are still connected to a knowledge graph and have thus relationships with other elements. In the context of explainability, we saw that some models are tightly related to logic and try to reconstruct it from the embeddings or to use the embedded representation to perform logical reasoning. However, the actual explainability of these methods is still low and approaches that try to account for it are recent.\n\nThe evolution of the methods in the literature has passed trough different dataset and the evaluation is still subject to a lot of variation due to hyperparameters choice and training procedures. Older approaches perform well when trained with new methodologies. There is the need to define a common ground for evaluation that also takes into account the many differences that each model proposes. \n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDissipative spin models are currently attracting wide interest, because they\ndescribe genuine nonequilibrium states of matter and at the same time,\ncan be realized in various platforms from superconducting circuits to Rydberg atoms\n\\cite{puri2017quantum,clerk2020hybrid,xu2020probing,henriet2020quantum,zeiher2017coherent,labuhn2016tunable,saffman2010quantum,angerer2018superradiant,zhang2017observation}.\nThe model consists of an ensemble of spins that are subject to dissipation caused\nby external baths. The dynamics is described by the Lindblad equation,\nobtained by integrating out the baths' degrees of freedom~\\cite{2002theory}.\n\nGreat efforts have been taken in the investigation of various spin models.\nIn the central spin model~\\cite{kessler2012dissipative}, the dissipative phase\ntransition was located according to the closing of the Liouvillian gap.\nThe transverse-field Ising model was thoroughly studied both under\nthe mean-field approximation and beyond~\\cite{lee2011antiferromagnetic,lee2012collective,\nates2012dynamical,hu2013spatial,marcuzzi2014universal,weimer2015variational,weimer2015variational}, in which\nthe bistability of steady states in some region of the parameter space was found\nto be replaced by a first-order phase transition after the spatial correlation\nwas correctly considered. \nAs the couplings between spins are all-to-all and then the system can be\nseen as a huge spin, the semiclassical approach was adopted.\nThe magnetization was found to display an everlasting oscillation\nin the thermodynamic limit, indicating that the time translational symmetry\nis spontaneously broken into a discrete one~\\cite{ie2018boundary,shammah2018open,tucker2018shattered}. If the dissipation acts in the eigenbasis of the transverse\nfield, then a continuous dissipative phase transition manifests itself as\ncontinuous order parameters with discontinuous derivatives~\\cite{overbeck2017multicritical,foss2017solvable,jin2018phase,wang2021dissipative}. The XYZ-Heisenberg model was also studied by different approximation schemes~\\cite{lee2013unconventional,jin2016cluster,rota2017critical,\ncasteels2018gutzwiller,huybrechts2020validity} to clarify its phase diagram.\n\nThese studies focused on the time-independent Liouvillians. But\nmuch less is known as the Liouvillian changes periodically with time~\\cite{zhu2019dicke}.\nThe topic of this work\nis to study the response to a time-periodic Liouvillian.\nOn the other hand, in closed quantum systems, the response to a time-periodic Hamiltonian\nhas been under intensive investigations.\nThe kicked top models were studied both theoretically~\\cite{haake1987classical,wang2021multifractality,\nbhosale2017signatures,bandyopadhyay2004entanglement,\nruebeck2017entanglement,lombardi2011entanglement,wang2004entanglement,iwaniszewski1995chaos} and\nexperimentally\\cite{chaudhury2009quantum,krithika2019nmr,neill2016ergodic}, which\nis known to exhibit a transition between a regular dynamical phase and a chaotic one,\ndepending on the value of the kicking strength. Similar chaotic behavior was\nfound in the Lipkin-Meshkov-Glick model as the parameters change\nperiodically with time~\\cite{engelhardt2013ac,lerose2019prethermal,\ndas2006infinite,russomanno2015thermalization}.\nThe study in this paper can be seen as an investigation\nof the dissipation effect on the chaotic dynamics in the periodically-driven spin models.\n\nAs a concise example, we study the fully-connected Ising model in the presence of\ncollective dissipation. Without dissipation, chaotic behavior\nappears in the presence of a strongly oscillating\nexternal field~\\cite{lerose2019prethermal,das2006infinite,\nrussomanno2015thermalization}. We find that, the chaotic behavior\nis robust against weak dissipation, if the semiclassical approximation is taken.\nAs the oscillating amplitude of field increases, a periodic response\nchanges into a subharmonic oscillation, and then into\na chaotic behavior. But the numerical simulation shows\nthat, beyond the semiclassical approximation,\nonly the periodic response can survive the quantum fluctuation, but\nneither the subharmonic nor the chaotic dynamics can be observed.\nThe semiclassical approximation works only as the oscillating amplitude\nis small in the periodic-response regime,\nbut it fails as the amplitude is large.\n\n\nThe paper is organized as follows. We introduce the model\nin Sec.~\\ref{sec:model}. Section~\\ref{sec:semi} contributes to\nthe discussion of the semiclassical results. The exact numerical\nsimulations of the dynamics of magnetizations\nare discussed in Sec.~\\ref{sec:finite}. The Floquet Liouvillian spectrum\nis studied in Sec.~\\ref{sec:spec}. Finally, Sec.~\\ref{sec:con} summarizes our results.\n\n\n\\section{ THE MODEL}\n\\label{sec:model}\n\nWe consider the transverse field Ising model with all-to-all couplings,\nand a sinusoidal modulation added to the external field. The Hamiltonian is written as\n\\begin{equation}\\label{1}\n\\hat{H} =-Ng(\\hat{J}_{x})^{2}+N\\Gamma(t)\\hat{J}_{z},\t\n\\end{equation}\nwhere $N$ denotes the total number of spins, $g$ denotes the coupling strength, and\n$\\hat{J}_{\\alpha}=\\sum_{j}\\hat{\\sigma}_{j}^{\\alpha}\/N$ denote the collective spin operators\nwith $\\hat{\\sigma}_{j}^{\\alpha}$ being the Pauli matrices of the $j$th spin and $\\alpha=x,y,z$.\n$\\Gamma(t)$ denotes the time-dependent external field, which is supposed to be\n$\\Gamma(t)= \\Gamma_0+A\\ sin(\\omega_0 t)$, where $A$, $\\Gamma_0$ and $\\omega_0$\nare the oscillating amplitude, mean value and frequency, respectively.\nWe set $g = 1$ as the unit of energy throughout the paper.\n\nIn the presence of dissipation, the dynamics of the system is\ndescribed by the Lindblad equation~\\cite{LindbladmasterE}. The density matrix satisfies\n\\begin{equation}\\label{2}\n\\frac{d\\hat{\\rho}}{\\partial t}=-i[\\hat{H},\\hat{\\rho}]+N{\\gamma_{c}}\\left(\n2\\hat{J}_{-}\\hat{\\rho}\\hat{J}_{+}- \\{\\hat{\\rho},\\hat{J}_{+}\\hat{J}_{-}\\}\\right),\n\\end{equation}\nwhere $ \\hat{J}_{\\pm}=\\left(\\hat{J}_x \\pm i \\hat{J}_y\\right)\/2$ are the jump operators,\nand $\\gamma_{c}$ is the dissipation rate. As $A=0$ and then $\\Gamma(t)= \\Gamma_0$\nis a constant, the solution of Eq.~\\eqref{2}\nwas studied previously~\\cite{wang2021dissipative}. The jump operator forces the spins to be\naligned in the negative $z$-direction, while the interaction between spins favors\nan alignment in the $x$-direction. Their interplay results in a steady state, which\nis either ferromagnetic (with nonzero magnetization in the $x$-direction)\nor paramagnetic. Here we extend to the case of $A\\neq 0$, in which\nthe field oscillation prevents a steady state being reached,\nand then we expect a nontrivial dynamical behavior.\n\n\\begin{figure}\n\t\n\t\\subfigure[]{\n\t\t\\begin{minipage}{0.9\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\linewidth]{fig1.pdf}\n\t\t\n\t\t\\end{minipage}\\label{img2a}\n\t}%\n\n\t\\subfigure[]{\n\t\t\\begin{minipage}{0.9\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\linewidth]{fig2.pdf}\n\t\t\n\t\t\\end{minipage}\\label{img2b}\n\t}%\n\t\n\t\\centering\n\t\\caption{The time evolution of $m_x$ for (a) $A=0.1$ with $25$ different initial states,\n\tand (b) $A=1$ with two initial states - $(\\theta_0=0.5\\pi,\\phi_0=0.1\\pi)$\n\tand $(\\theta_1=0.5\\pi+10^{-7},\\phi_1=0.1\\pi)$. Different line colors\n\tare for different initial states. The dissipation strength is set to $\\gamma_c=1$.}\n\t\\label{img2}\n\\end{figure}\n\n\\begin{figure}\n\\subfigure[]{\n\\begin{minipage}{0.9\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{fig3.pdf}\n\\end{minipage}\\label{img1a}\n}\n\\subfigure[]{\n\\begin{minipage}{0.9\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{fig4.pdf}\n\\end{minipage}\\label{img1b}\n}\n\n\\centering\n\\caption{The time evolution of $m_x$ for (a) $A=0.73$ and (b) $A=0.7345$.\nThe red line highlights a complete period. The transient regime\n($t<150$) is omitted.}\n\\label{img1}\n\\end{figure}\n\n\n\\section{Chaotic dynamics in the semiclassical approach}\n\\label{sec:semi}\n\nThe semiclassical approach is frequently employed for solving the fully-connected models.\nWe choose the order parameters to be $m_{\\alpha}=\\left\\langle \\hat{J}_{\\alpha}\\right\\rangle =Tr[\\hat{\\rho}\\hat{J}_{\\alpha}]$. By ignoring the correlations (i.e., setting $\\left\\langle \\hat{J}_{\\alpha}\\hat{J}_{\\beta}\\right\\rangle =\\left\\langle \\hat{J}_{\\alpha}\\right\\rangle \\left\\langle \\hat{J}_{\\beta}\\right\\rangle$)\nin the limit $N\\to\\infty$, we\nobtain a nonlinear system of differential equations, which read\n\\begin{equation}\n\\begin{aligned}\n\\dot{m}_{x}\t&=-2\\Gamma(t)m_{y}+\\gamma_{c}m_{x}m_{z} , \\\\\n\\dot{m}_{y}\t&=4m_{x}m_{z}+2\\Gamma(t)m_{x}+\\gamma_{c}m_{y}m_{z}, \\\\\n\\dot{m}_{z}\t&=-4m_{x}m_{y}-\\gamma_{c}(m_{x}^{2}+m_{y}^{2}).\n\\end{aligned}\\label{3}\n\\end{equation}\nIt is easy to see that $\\left| {\\bf{m}}(t)\\right|  =\\sqrt{m_x^2 + m_y^2 + m_z^2} $\nis a constant of motion, which can be set to unity without loss of generality.\n${\\bf{m}}$ is moving on a Bloch sphere, and the initial state can be\ndescribed by the azimuthal angles $\\left(\\theta, \\phi\\right)$, which are defined\nby $m_x = \\sin\\theta \\cos\\phi$, $m_y=\\sin\\theta\\sin \\phi$ and $m_z=\\cos \\theta$.\n\nSince the coefficient $\\Gamma(t)$ is a periodic function of $t$ with the period $2\\pi\/\\omega_0$,\none may naively think that the solution ${\\bf{m}}(t)$ is also a periodic function. This\nis the case for small $A$, but not true for large $A$. We note that Eq.~\\eqref{3} bears some\nresemblance to the Lorenz equation~\\cite{lorenz1963deterministic},\none famous example of the deterministic chaos in the classical systems.\nIn Eq.~\\eqref{3}, the possibility of chaos comes from the fact that $\\Gamma(t)$\nis time-periodic, even the trajectory is limited on a two-dimensional sphere.\nIndeed, we observed a periodic ${\\bf{m}}(t)$ for small $A$, but a chaotic\n${\\bf{m}}(t)$ for large $A$.\n\nNext we choose $\\Gamma_0=1$ and $\\omega=1$ as an example for the demonstration.\nIn Fig.~\\ref{img2}(a), we display the evolution of\n$m_x $ for small $A$ ($A=0.1$) and 25 initial states that are evenly distributed\non the Bloch sphere. Except for the initial state at the south pole (${\\bf{m}}=(0,0,1)$),\nall the others eventually evolve into one of the two oscillation\nmodes that are symmetric to each other.\nThe two oscillation modes have the same period, which is exactly\nthe driving period ($2\\pi\/\\omega_0$). For small $A$, the long-time response\nis insensitive to a small deviation in the initial state.\nOn the other hand, Fig. \\ref{img2b} shows the evolution of $m_x $\nfor a large $A$ ($A = 1.0$). Till the largest evolution time that is accessible,\nno periodicity is observed. More importantly, the long-time response is\nextremely sensitive to the initial condition. Even for a tiny deviation\nin the initial state ($\\theta_1-\\theta_0=10^{-7}$), $m_x(t)$\ndisplays a significant difference as $t$ is as large as a few hundreds\n(see the lines of different colors in Fig.~\\ref{img2b}).\n\nFor the values of $A$ between $0.1$ and $1.0$, ${{m}_x}(t)$ displays abundant dynamical\nbehaviors. As $A$ increases up to a certain critical value, the time period is doubled.\nIn Fig.~\\ref{img1a}, we plot ${{m}_x}(t)$ for $A=0.73$. It is easy to see that\nthe period is not $2\\pi\/\\omega_0$, instead, it becomes $4\\pi\/\\omega_0$.\nAs $A$ increases further, the period-doubling happens again.\nFor example, for $A=0.7345$, the period becomes $8\\pi\/\\omega_0$ (see Fig.~\\ref{img1b}).\nThe period-doubling bifurcation is well-known in the classical nonlinear systems.\nUsually, a cascade of period-doubling bifurcations lead to chaos~\\cite{galatzer1997understanding}.\nThis explains why we observe a chaotic dynamics as $A$ is as large as $A=1$.\n\nDepending on the values of $A$, the system exhibits the periodic, subharmonic\nor chaotic responses. We plot the trajectory of the vector ${\\bf{m}}(t)$ on the Bloch sphere\nin Appendix~\\ref{sec:app}, which provides more evidence for the existence\nof different dynamical behaviors.\n\nIn our model, the subharmonic response (i.e., doubled period) to a time-periodic Liouvillian\nmust be distinguished from that in the Floquet time-crystals.\nAs will be shown next, the subharmonic response in our model\ncan only be observed in the semiclassical limit. It cannot\nsurvive the quantum fluctuation that is unavoidable at finite $N$.\n\nTo locate the chaotic phase in the parameter space, we calculate the\nLyapunov exponent. The defining property of a chaotic system is\nthe extreme sensitivity of trajectories to the initial condition.\nTwo points that are initially close will drift apart exponentially over time.\nThe Lyapunov exponent~\\cite{oseledets1968multiplicative} provides a quantitative\nmeasure for this, which is defined as\nthe average exponential rate of convergence or divergence\nbetween adjacent trajectories in the phase space.\nEspecially, the largest Lyapunov exponent (LLE) is frequently employed\nfor determining whether a nonlinear system is chaotic.\nIf the LLE is greater than zero, two initial points will depart\nfrom each other exponentially, and then the system is chaotic~\\cite{wolf1985determining},\notherwise, it is not. Figure~\\ref{fig:h0} displays the dependence of the LLE on $A$ and $\\gamma_c$.\nThe area in red or yellow has a positive LLE, while the area in blue or green has a negative LLE. \nThe chaotic phase is clearly distinguishable in the parameter space.\nAn interesting finding is that for a fixed $\\gamma_c$ in the interval $\\left(0,1\\right)$,\nthe dynamics is chaotic only for an intermediate amplitude of oscillating field, but the dynamics\nis regular either if $A$ is too large or too small. In the presence of\nstrong dissipation ($\\gamma_c>1.5$), there is no chaotic dynamics for\nwhatever $A$.\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{fig5.pdf}\n\t\\caption[]{The largest Lyapunov exponent as a function of $\\gamma_c$ and $A$.\n\tThe system displays chaotic dynamics as $LLE > 0$ (see the area in red or yellow).}\n\t\\label{fig:h0}\n\\end{figure}\n\n\n\\section{Periodic behavior at finite $N$}\n\\label{sec:finite}\n\nTo check the validity of semiclassical approximation,\nwe numerically simulate the real-time dynamics at finite $N$, by exploiting\nthe permutation symmetry of fully-connected models.\nThe Dicke basis with maximum angular momentum is defined as~\\cite{sciolla2011dynamical}\n\\begin{equation}\n\\label{4}\n\\ket{M} =\\sqrt{\\frac{1}{C_{N}^{\\frac{N}{2}+M}}}\n\\sum_{{\\sum_{j=1}^{N} \\sigma_{j}=M}}\\ket{\\sigma_{1},\\sigma_{2},\\cdots,\\sigma_{N}} ,\n\\end{equation}\nwhere $C_{N}^{\\frac{N}{2}+M}$ is the binomial coefficient, and $\\sigma_j=\\pm 1\/2$\nrepresents the spin-up and down states, respectively.\nAnd $M= -N\/2, -N\/2+1,\\cdots, N\/2$ is the magnetization in the $z$-direction.\nThe initial state is supposed to be a pure state with all the spins\naligned in the same direction. We use the azimuthal angles $(\\theta,\\phi)$\nto indicate the initial direction, then the initial state can be\nexpressed in the Dicke basis as\n \\begin{equation}\n \\label{key}\n \t\\ket{ \\theta ,\\phi } =\\sum_{M=-\\frac{N}{2} }^{\\frac{N}{2}} \n\tC_{N}^{M+\\frac{N}{2} }\\cos\\frac{\\theta}{2}^{\\frac{N}{2}+M}\n\t\\sin\\frac{\\theta}{2}^{\\frac{N}{2}-M}e^{i(\\frac{N}{2}-M)\\phi}\\left | M  \\right \\rangle .\n\\end{equation} \nEquation~\\eqref{2} has the permutation symmetry, its solution can then be expressed as\n$\\hat{\\rho}(t) = \\sum_{M,M'} \\rho_{M,M'}(t) \\ket{M}\\bra{M'}$, where\n$\\rho_{M,M'}(t)$ is the density matrix in the Dicke basis. Now Eq.~\\eqref{2} changes into\na system of ordinary differential equations for $\\rho_{M,M'}$, which are solved numerically.\nThe permutation symmetry reduces the dimension of Hilbert space from $2^N$\nto $N$, and then allows us to access a large system with the number of spins $N\\sim 10^2-10^3$.\n\nIn Fig.~\\ref{img4a}, we compare the dynamics of magnetizations at\nfinite $N$s and in the semiclassical limit ($N=\\infty$),\nas $A=0.1$ is in the semiclassical periodic regime.\nAt finite $N$s, the magnetizations display periodic oscillations\nin the asymptotic long time.\nThe curve of $m_z(t)$ at $N=50$ is already very close to the\nsemiclassical one. As $N$ increases, $m_z(t)$\ngoes even closer to the semiclassical result (see Fig.~\\ref{img4a} the inset).\nAs $N\\to\\infty$, we expect that the numerical results should\ncoincide with the semiclassical results.\nIf $A$ is small, then the semiclassical approximation\nis good for large enough $N$s.\n\nFigure~\\ref{img4b} shows the comparison as $A=1.0$\nis in the semiclassical chaotic regime. The results\nare significantly different from those at $A=0.1$. Up to $N=200$,\nwe find no similarity between the numerical result and the semiclassical one.\nFor $N$ ranging between $50$ and $200$, the initial transient\nbehavior of $m_z(t)$ always quickly evolves into the asymptotic behavior\n- periodic oscillation. And the $m_z(t)$s for $N=50$ and $N=200$ have the same period,\nwith only their amplitude being different. But in the semiclassical result, $m_z(t)$ is aperiodic\nat arbitrarily long time. These observations suggest that the\nexact numerical solutions do not conserve to the semiclassical one\nin the limit $N\\to\\infty$. If $A$ is large, then the semiclassical approximation is bad,\neven for large $N$s. This seems to be strange for a fully-connected model.\nWe will further discuss this discrepancy in next section.\n\n\n\\begin{figure}\n\t\\subfigure[]{\n\t\t\\begin{minipage}{1\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\linewidth]{fig6.pdf}\n\t\t\n\t\t\\end{minipage}\\label{img4a}\n\t}%\n\n\t\\subfigure[]{\n\t\t\\begin{minipage}{1\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\linewidth]{fig7.pdf}\n\t\t\n\t\t\\end{minipage}\\label{img4b}\n\t}%\n\t\\centering\n\t\\caption{The time evolution of $m_z$ for (a) $A=0.1$ and (b) $A=1$\n\twith different $N$. The black solid lines represent the semiclassical results.\n\tThe initial condition is fixed to be $\\theta_0=0.5\\pi, \\phi_0=0.1\\pi$.}\n\t\\label{img4(2)}\n\\end{figure}\n\n\\begin{figure}\n\t\\subfigure[]{\n\t\t\\begin{minipage}{1\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\linewidth]{fig8.pdf}\n\t\t\n\t\t\\end{minipage}\\label{img5b}\n\t}%\n\t\n\t\\subfigure[]{\n\t\t\\begin{minipage}{1\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\linewidth]{fig9.pdf}\n\t\t\n\t\t\\end{minipage}\\label{img5a}\n\t}%\n\t\n\t\\centering\n\t\\caption{The evolution of $m_x$ and $m_z$ with different $A$.\n\tThe number of spins is chosen to be $N=100$. And the\n\tinitial condition is $\\theta_0=0.5\\pi,\\phi=0.1\\pi$.}\n\t\\label{img5}\n\\end{figure}\nWe also compare the dynamics of magnetizations for different $A$s, as\n$N$ is fixed. Figure~\\ref{img5b} and~\\ref{img5a} display $m_z(t)$s and\n$m_x(t)$s, respectively, for the values of $A$ ranging from $0.1$ to $1.0$.\nThe magnetization in the $z$-direction always displays a periodic oscillation.\nAnd the oscillations for different $A$s are in phase (see Fig.~\\ref{img5b} the\ninset). On the other hand, the magnetization in the $x$-direction displays\ntwo different dynamical modes, depending on the value of $A$.\nAs seen in Fig.~\\ref{img5a}, as $A$ is small ($A=0.1, 0.3$),\n$m_x$ displays an everlasting oscillation with the period $2\\pi\/\\omega_0$.\nBut as $A$ is large ($A=0.73, 1.0$), $m_x$ rapidly decays to zero.\nFor an intermediate $A$ ($A=0.5$), $m_x$ maintains an oscillation for\na relatively long time, but the decay can still be clearly seen.\nThe decay of $m_x$ is a signal of the discrepancy between the exact\nresults and the semiclassical approximations, for the latter have non-decaying\nmagnetizations in all directions (see App.~\\ref{sec:app}).\n\n\\section{Floquet Liouvillian spectrum}\n\\label{sec:spec}\n\nIn the closed systems, it is widely believed that the semiclassical\napproximation becomes exact for the fully-connected models\nif the number of spins goes to infinity.\nIn the open systems, a similar result was obtained~\\cite{carollo2021exactness},\nas the Liouvillian is time-independent. But in this paper, our numerical\nresults suggest that the semiclassical approximation is bad even in the\nlimit $N\\to\\infty$, if there is a big time-periodic term in the Liouvillian.\nSince we can only obtain the exact results at finite $N$, a scaling\nanalysis is helpful for confirming our viewpoint. Next we give a\nscaling analysis of the Floquet Liouvillian spectrum.\n\n\n\\begin{figure}\n\t\n\t\\subfigure[]{\n\t\t\\begin{minipage}{0.9\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\linewidth]{fig10.pdf}\n\t\t\n\t\t\\end{minipage}\\label{fig.6a}\n\t}%\n\n\t\\subfigure[]{\n\t\t\\begin{minipage}{0.9\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\linewidth]{fig11.pdf}\n\t\t\n\t\t\\end{minipage}\\label{fig.6b}\n\t}%\n\t\\centering\n\t\\caption{The Floquet Liouvillian spectrum for (a) $A=0.1$ and (b) $A=1.0$.\n\tThe number of spins is $N=30$.}\n\t\\label{fig.6}\n\\end{figure}\nThe Lindblad equation can be expressed in a vectorized form as\n$d\\hat{\\rho}\/dt = \\hat{\\hat{\\mathcal{L}}}\\left( \\hat{\\rho}\\right)$, where\n$\\hat{\\hat{\\mathcal{L}}}$ is the so-called Liouvillian\nsuperoperator (or Liouvillian in short), which\nis a non-Hermitian linear operator acting on the vector space of density matrices.\nFor the dissipative systems with time-independent\n$\\hat{\\hat{\\mathcal{L}}}$, it is well known that the eigenvalues\nand eigenvectors of $\\hat{\\hat{\\mathcal{L}}}$ determine the dynamics.\nThe eigenvalues of $\\hat{\\hat{\\mathcal{L}}}$ (Liouvillian spectrum) are complex numbers.\nMore important, if the system size $N$ is finite, all the eigenvalues\nmust have negative real part, except one that is zero.\n\nThese notations can be generalized to the case of time-periodic $\\hat{\\hat{\\mathcal{L}}}$.\nFor a Lindblad equation with $\\hat{\\hat{\\mathcal{L}}}(t) = \\hat{\\hat{\\mathcal{L}}}(t+T)$,\nthere exist a complete set of solutions written as $\\hat{\\rho}(t)= e^{\\lambda t}  \\hat{\\varrho}(t)$,\nwhere $\\hat{\\varrho}(t)=\\hat{\\varrho}(t+T)$ is the time-periodic part of density matrix,\naccording to the Floquet theorem. And $\\hat{\\varrho}(t)$ satisfies\n\\begin{equation}\n\\hat{\\hat{\\mathcal{L}}}\\left( \\hat{\\varrho}(t) \\right) - \\frac{d}{dt} \\hat{\\varrho}(t)\n= \\lambda \\hat{\\varrho}(t).\n\\end{equation}\nThen $\\lambda$ can be seen as the eigenvalue of $\\left(\\hat{\\hat{\\mathcal{L}}}(t)- d\/dt\\right)$, which\nis an operator acting on the generalized vector space of density matrices,\njust as $\\left(\\hat{H}-id\/dt\\right)$ is an operator acting\non the generalized Hilbert space (Sambe space)\nfor a closed system with time-periodic Hamiltonian.\nWhile the eigenstates of $\\left(\\hat{H}-id\/dt\\right)$ are called the\nFloquet spectrum, the eigenstates of $\\left(\\hat{\\hat{\\mathcal{L}}}(t)- d\/dt\\right)$\nare called the Floquet Liouvillian spectrum.\n\nCompared to the Floquet spectrum or Liouvillian spectrum, much less is known about the\nFloquet Liouvillian spectrum. For the dissipative systems, we guess that\nthe Floquet Liouvillian spectrum has the same properties as the Liouvillian spectrum,\ni.e., all the complex eigenvalues have negative real parts except for a unique one\nthat is zero. The numerics support our guess. Figure~\\ref{fig.6}\ndisplays the Floquet Liouvillian spectrum at $N=30$. For both\n$A=0.1$ and $A=1.0$, we see that the rightmost eigenvalue\nin the complex plane is zero, which is non-degenerate.\nAnd the others are to the left of zero.\n\nThe Floquet Liouvillian spectrum completely determines whether\nthe dynamics is periodic, subharmonic or chaotic. To see it, we arrange all\nthe eigenvalues as\n\\begin{equation}\n\\lambda_0=0\\geq \\text{Re}\\lambda_1 \\geq \\text{Re}\\lambda_2 \\geq \\cdots,\n\\end{equation}\nwith the corresponding eigenvectors being $\\hat{\\varrho}_0(t)$, $\\hat{\\varrho}_1(t)$,\n$\\hat{\\varrho}_2(t),\\cdots$. For an arbitrary initial state, the\nsolution of Lindblad equation can be formally expressed as\n\\begin{equation}\n\\hat{\\rho}(t) = \\sum_j K_j e^{\\lambda_j t}  \\hat{\\varrho}_j(t),\n\\end{equation}\nwhere the coefficients $K_j$ depend on the initial state.\nIn the asymptotic long time, the terms with $\\text{Re}\\lambda_j<0$ all\ndecay to zero, and $1\/\\left| \\text{Re}\\lambda_j \\right|$ is just the decay time of\nthe $j$-th mode. At finite $N$, all the $\\lambda_j$ with $j>0$ have\nnegative real parts, therefore, the density matrix in the asymptotic long time\nbecomes $\\hat{\\rho}(t) = K_0  \\hat{\\varrho}_0(t)$, which is exactly periodic with\nthe period $T$. We then expect that the asymptotic behavior is always periodic at finite $N$.\nThe situation is more complicated in the thermodynamic limit.\nAs found in the previous studies of time-independent Liouvillians,\nthere exist possibilities that the real parts of some $\\lambda_j$ decrease\nwith increasing $N$ and vanish in the limit $N\\to\\infty$. As a consequence, the asymptotic\ndensity matrix becomes $\\hat{\\rho}(t) = \\sum_{j=0}^L K_j \ne^{i \\text{Im}\\left(\\lambda_j\\right) t}  \\hat{\\varrho}_j(t)$, where $L$ is the number of\neigenvalues with vanishing real parts. Additionally, if these $\\lambda_j$\nhave also nonvanishing imaginary parts, then $\\hat{\\rho}(t)$ possibly display\nsubharmonic or chaotic behaviors, depending on the values of $\\text{Im}\\left(\\lambda_j\\right)$.\nConversely, if the real parts of $\\lambda_j$ with $j>0$ are all\nfinite in the limit $N\\to\\infty$, then subharmonic oscillation or chaotic\nbehavior are both impossible.\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{fig12.pdf}\n\t\\caption[]{The Floquet Liouvillian gap as a function of $1\/N$. The dots\n\trepresent the numerical results, while the lines represent the fitted curves.}\n\t\\label{fig.7}\n\\end{figure}\nAccording to the above argument, we perform a scaling analysis of\nthe Floquet Liouvillian gap, which is defined as $\\Delta = \\lambda_0 - \\text{Re}\\lambda_1$.\nFigure~\\ref{fig.7} plots $\\Delta$ as a function of $1\/N$.\nThe dots are the numerical results, while the lines are the fitted curves.\nThe gap at $A=1.0$ is significantly smaller than the gap at $A=0.1$.\nBut in both cases, we clearly see that the gap does not go\nto zero as $N\\to\\infty$. This indicates that no $\\lambda_j$ with $j>0$\nhas vanishing real parts, therefore, the asymptotic behaviors\nare periodic for both $A=0.1$ and $A=1.0$, even in the limit $N\\to \\infty$. Such a result is\nconsistent with our previous simulation of the real-time dynamics of magnetizations.\n\n\n\\section{CONCLUSIONS}\n\\label{sec:con}\n\nIn summary, we study the fully-connected Ising model with\na time-periodic external field and subject to a dissipation,\nby using both the semiclassical approach and the exact\nnumerical simulation. If the field amplitude is small, both the semiclassical approach\nand the numerical simulation predict a perfect periodic oscillation\nof magnetizations. And the numerical results in the thermodynamic limit\nare consistent with the semiclassical one.\n\nAs the field amplitude increases, the semiclassical approach predicts\nthe period-doublings or subharmonic oscillations, and a series of period-doublings\nfinally lead to the chaotic dynamics of magnetizations, which is confirmed\nby the calculations of Lyapunov exponents. On the contrary, the\nnumerical simulation show that the magnetizations are always\noscillating periodically, whatever the field amplitude is. No subharmonic\nor chaotic dynamics are observed, even we choose the \nnumber of spins to be as large as a few hundreds in the simulation.\nWe then analyze the Floquet Liouvillian gap, which conserves to a finite\nvalue in the thermodynamic limit, for either small or large field amplitude.\nWe argue that a finite gap is another evidence of the periodic oscillations of observables.\n\nFor the fully-connected models, the\nsemiclassical approximation is generally believed to be good\nfor sufficiently large number of spins. But we find that this is\nnot the case if the time-periodic field and the dissipation are both present.\nFor a large field amplitude, the predictions from the semiclassical approach\nand the numerical method are qualitatively different.\nOur finding suggests that one should be more careful when\nusing the semiclassical approach in the case of time-periodic Liouvillians.\n\n\n\\section*{Acknowledgement}\nThis paper is supported by NSFC under Grant Nos.~11774315,~11835011, and~12174346,\nand by the Junior Associates program of the Abdus Salam International Center for Theoretical Physics.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\@startsection{section}{1}{\\z@}%\n    {-21dd plus-4pt minus-4pt}{10.5dd plus 4pt\n     minus4pt}{\\normalsize\\bfseries\\boldmath}}\n\n\\ifx\\ParFact\\undefined\\def\\ParFact{0.4}\\fi\n\\def\\@setparskip{\\parskip=\\ParFact\\baselineskip \\advance\\parskip by 0pt plus 2pt}\n\\@setparskip\n\\parindent=0pt\n\\makeatother\n\n\\usepackage{tikz}\n\\usetikzlibrary{matrix}\n\\usepackage{caption}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{varwidth}\n\\usepackage{enumitem}\n\\setlist[enumerate]{labelindent=--.5in,leftmargin=0pt}\n\\setlist[itemize]{labelindent=--.5in,leftmargin=0pt}\n\\usepackage{nicefrac}\n\\usepackage{euscript}\n\\usepackage{color}\n\\definecolor{green}{RGB}{0,127,0}\n\\definecolor{red}{RGB}{191,0,0}\n\\usepackage[colorlinks=true]{hyperref}\n\n\\theoremstyle{plain}\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{corollary}[theorem]{Corollary}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{proposition}[theorem]{Proposition}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{remark}[theorem]{Remark}\n\\newtheorem{question}[theorem]{Question}\n\\newtheorem{problem}[theorem]{Problem}\n\n\\numberwithin{equation}{section}\n\n\\newcommand{\\B}[1]{\\mathbf{#1}}\n\\newcommand{\\C}[1]{\\mathcal{#1}}\n\\newcommand{\\F}[1]{\\mathfrak{#1}}\n\n\\begin{document}\n\n\\title[Positive definite functions on Coxeter groups\\dots]{Positive definite functions on Coxeter groups\nwith applications to operator spaces and noncommutative\nprobability}\n\\author{Marek {\\sc Bo\\.zejko}}\n\\address{M.~Bo{\\.z}ejko --- \\normalfont Polska Akademia Nauk, ul.~Kopernika 18, 50-001 Wroc{\\l}aw}\n\\email{bozejko@math.uni.wroc.pl}\n\\author{\\'Swiatos{\\l}aw R.~{\\sc Gal}}\n\\address{\\'S.~R.~Gal --- \\normalfont Uniwersytet  Wroc{\\l}awski, pl.~Grunwaldzki 2\/4, 48-300 Wroc{\\l}aw}\n\\email{sgal@math.uni.wroc.pl}\n\\author{Wojciech {\\sc M{\\l}otkowski}}\n\\address{W.~M{\\l}otkowski --- \\normalfont Uniwersytet  Wroc{\\l}awski, pl.~Grunwaldzki 2\/4, 48-300 Wroc{\\l}aw}\n\\email{mlotkow@math.uni.wroc.pl}\n\n\\begin{abstract}\nA new class of positive definite functions\nrelated to colour-length function on arbitrary Coxeter group is introduced.\nExtensions of positive definite functions, called the Riesz-Coxeter product,\nfrom the Riesz product on the Rademacher (Abelian Coxeter) group to arbitrary\nCoxeter group is obtained. Applications to harmonic analysis, operator spaces\nand noncommutative probability is presented. Characterization of radial and\ncolour-radial functions on dihedral groups and infinite permutation group are shown.\n\\end{abstract}\n\n\\maketitle\n\n\\def{\\mathop{\\rm cb}}{{\\mathop{\\rm cb}}}\n\\def\\mathop{\\rm tr}{\\mathop{\\rm tr}}\n\n\\section*{Introduction}\n\nIn 1979 Uffe Haagerup in his seminal paper \\cite{MR520930}, essentially proved\nthe positive definitness, for $0\\leq q\\leq 1$, of the function  $P_q(x) =\nq^{|x|} = \\exp(-t|x|)$, where $|\\cdot|$ is the word lenght on a free Coxeter\ngroup $W=\\B Z\/2*\\dots *\\B Z\/2$. From this he deduced also Khinchine\ntype inequalities. He has shown that the regular $C^*$-algebra of $W$ has\nbounded approximation property and later \\cite{MR784292} the completely bounded approximation\nproperty \\textsc{\\MakeLowercase{(CBAP)}}. These results of Uffe Haagerup have had\nsignificant impact on harmonic analysis on free groups and, more generally, on\nCoxeter groups; they also influenced free probability theory and other\nnoncommutative probability theories.\n\nIn the paper \\cite{MR950825} it was shown that the function $P_{q} (x) = q^{|x|}$\nis positive definite for $q\\in[-1,1]$ and  all Coxeter groups,\nwhere the length $|\\cdot|$ is the natural word length function on a Coxeter group\nwith repect to the set of its Coxeter generators. This fact implies that infinite\nCoxeter groups have the Haagerup property and do not have Kazhdan's\npropery (T).\n\nLater, Januszkiewicz \\cite{MR1925487} and Fendler \\cite{Fendler} showed,\nin the spirit of Haagerup proof, that $w\\mapsto z^{|w|}$\nis a coefficient of a uniformely bounded Hilbert representation of $W$\nfor all $z\\in\\B C$ such that $|z|< 1$.  As shown in a very short\npaper of Valette \\cite{MR1172955}, this implies \\textsc{\\MakeLowercase{CBAP}}.\nSee the book \\cite{MR2391387} for\nfuther extension of Uffe Haagerup results for a big class of\ngroups.\n\nIn the paper \\cite{MR1388006} Bo{\\.z}ejko and Speicher considered the free\nproduct (convolution) of classic normal distribution $N(0,1)$ and the new\nlength function on the permutation group $\\F S_n$ (i.e. the Coxeter group of\ntype \\textsc{\\MakeLowercase{A}}) was introduced, which we shall call the \\textit{colour-length function}\n$\\|\\cdot\\|$. It is defined as follows: for $w\\in \\F S_n$ in the minimal\n(reduced) representations $w = s_1\\dots s_k$, where each $s_{j}$ belong to the set $S$\nof transposions of the form  $(i,i+1)$, we put $\\|w\\| = \\#\\{s_{1}, s_{2},\\dots, s_{k}\\}$.\n\nFor our study one of  the most important results\nof this paper is that the function\ncalled Riesz-Coxeter product $R_{\\B q}$ defined on all Coxeter groups\n$(W,S)$ as\n$$\nR_{\\B q}(s) = q_{s} \\text{, for $s\\in S$, and\\ }\nR_{\\B q}(xy) = R_{\\B q}(x) R_{\\B q} (y) \\text{, if\\ } \\| x y \\| = \\| x \\| + \\|y\\|\n$$\nis positive definite for $0\\leq q_s\\leq 1$.\n\nThis implies, in particular, that in an arbitrary Coxeter group the set\nof its Coxeter generators is a weak Sidon\nset and also it is completely bounded $\\Lambda^{\\mathop{\\rm cb}}_p$-set, see\nTheorems \\ref{T:5.1} and\n\\ref{T:6.1}.  Equivalently, the span of the linear operators $\\{\\lambda(s)|s\\in\nS\\}$ in the noncommutative $L^{p}$-space $L^p(W)$ is completely boundedly isomorphic\nto row and column operator Hilbert space (see Theorem \\ref{T:6.1}).\n\n\nAnother interesting connection between the two length functions $|\\cdot|$ and\n$\\|\\cdot\\|$ appeared in \\cite{MR1388006} in the formula for the moments of\nfree additive convolution power of the Bernoulli law $\\mu_{-1} =\n(\\delta_{-1} + \\delta_1)\/2$ (cf.~Corollary~6 in cited paper):\n$$\nm_{2n}\\left(\\mu_{-1}^{\\boxplus q}\\right)\n=q^n\\smashoperator{\\sum_{\\pi\\in\\C P_2(2n)}}(-1)^{|\\pi|}q^{-\\|\\pi\\|},\n$$\nfor $q\\in\\B N$.  (See also Section 9 of the present paper.)\n\nAlso, in \\cite{MR2764902} the colour-length function on the permutation group\n$\\F S_n$ was studied. Some of its extensions to pairpartitions appeared\nin the presentation of the proof that classical normal law $N(0,1)$ is free\ninfinitely divisible under free additive convolution $\\boxplus$.\n\nSince we have recent extensions of the free probability (which is related to\ntype \\textsc{\\MakeLowercase{A}} Coxeter groups) to the free probability of type \\textsc{\\MakeLowercase{B}} Coxeter groups\n(see \\cite{MR3373433}), it seems to be\ninteresting to determine the role of the colour-length functions for the Coxeter\ngroups of type \\textsc{\\MakeLowercase{B}} and \\textsc{\\MakeLowercase{D}}.\n\nThe plan of the paper is as follows.\n\nIn Section 1 we recall definitions of Coxeter groups and of the length and the\ncolour-length functions.\n\nIn Section 2 we recall the definition of positive definite funcios and discuss\nvarious classes of those, namely radial, colour-radial, and colour-dependant.\n\nIn Section 3 we discuss Abelian Coxeter groups.\n\nIn Section 4 we show the following formula characterizing the radial normalised\npositive definite functions on these Coxeter groups which contain the infinite\nRademacher group $\\bigoplus_{i=1}^\\infty \\B Z\/2$\nas a parabolic subgroup (these include the infinite\npermutation group $\\F S_\\infty$): every radial positive definite function\n$\\varphi$ is of the form\n$$\n\\varphi (w) =\\int_{-1}^1 q^{|w|}\\,\\mu(dq)\n$$\nfor a probability measure $\\mu$.\n\nThat characterisation is a variation on the classical de Finetti theorem.\nA noncommutative version was shown by K\\\"ostler ans Speicher \\cite{MR2530168}\n(see also \\cite{MR2092722}).\n\nWe also show in Theorem \\ref{T:frac}, that the function $\\exp(-t|w|^p)$\nis positive definite for all $t\\geq 0$ if and only if $p\\in[0,1]$.\n\nIn Section 5 we give a short proof of the equivalence of the two known results\nconcerning positive definite functions on finite Coxeter groups.\n\nIn Section 6 we present the main properties of the colour-dependent positive definite\nfunctions on Coxeter groups, in particular we show in Proposition 4.4. that on\n$\\F S_\\infty$ and some other Coxeter groups, the function $w\\mapsto r^{\\|w\\|}$\nis positive definite if and only if $r\\in[0,1]$.\n\nThe Section 7 gives characterization of all colour-length functions on finite and\ninfinite dihedral groups $\\B D_m$, for $m=1,2,\\dots,\\infty$.\n\nIn Section 8 we prove that the set $S$ of Coxeter generators is a weak Sidon\nset in arbitrary Coxeter groups $(W,S)$ with constant 2 and that it is also a\ncompletely bounded $\\Lambda (p)$ set with contants as $C \\sqrt p$, for $p>2$.\n\nIn Section 9 we prove for arbitrary finitely generated Coxeter group an identity\ninvolving both lengths $|\\cdot|$ and $\\|\\cdot\\|$ (see Proposition \\ref{P:tokyo}).\nWe apply it to give a proof of Corollary 7 from \\cite{MR1388006},\n(see Equation (\\ref{eq:p2-nc2})) where the proof, involving probabilistic considerations,\nwas not presented in \\cite{MR1388006}.\n\n\\section{Coxeter groups}\n\nIn this part we recall the basic facts regarding Coxeter groups and introduce notation\nwhich will be used throughout the rest of the paper. For more details we refer to \\cite{MR0240238,MR1066460}.\n\nA group $W$ is called a \\emph{Coxeter group} if it admits the following presentation:\n\\[\nW=\\left\\langle S\\left|\\left\\{(s_1 s_2)^{m(s_1,s_2)}=1:s_1,s_2\\in S,m(s_1,s_2)\\ne\\infty\\right\\}\\right.\\right\\rangle,\n\\]\nwhere $S\\subset W$ is a set and $m$ is a function $m:S\\times S\\to\\{1,2,3,\\ldots,\\infty\\}$\nsuch that $m(s_1,s_2)=m(s_2,s_1)$ for all $s_1,s_2\\in S$ and $m(s_1,s_2)=1$ if and only if $s_1=s_2$.\nThe pair $(W,S)$ is called a \\emph{Coxeter system}.\nIn particular, every generator $s\\in S$ has order two\nand every element $w\\in W$ can be represented as\n\\begin{equation}\\label{areducedrepr}\nw=s_1 s_2\\ldots s_{m}\n\\end{equation}\nfor some $s_1,s_2,\\ldots,s_m\\in S$.\nIf the sequence $(s_1,\\ldots,s_m)\\in S^{m}$ is chosen in such a way that $m$ is minimal\nthen we write $|w|=m$ and call it the \\emph{length} of $w$.\nIn such a case the right hand side of (\\ref{areducedrepr}) is called a \\emph{reduced representation}\nor \\emph{reduced word} of $w$.\nThis is not unique in general, but the set of involved generators is unique \\cite[Ch.~\\textsc{\\MakeLowercase{iv}}, \\S1, Prop.~7]{MR0240238}, i.e.\nif $w=s_1 s_2\\ldots s_m=t_1 t_2\\ldots t_m$ are two reduced representations\nof $w\\in W$ then $\\{s_1,s_2,\\ldots,s_m\\}=\\{t_1,t_2,\\ldots,t_m\\}$.\nThis set $\\{s_1,s_2,\\ldots,s_m\\}\\subseteq S$ will be denoted $S_w$ and called the \\emph{colour} of $w$.\n\nGiven a subset $T\\subset S$ by $W_T$ we denote the subgroup generated by $T$ and call it the \\emph{parabolic\nsubgroup} associated with $T$.\nTo see that $ S_w$ is independent of the reduced representation of $w$ notice that\n\\begin{equation}\n\\label{def:colour}\ns\\in S_w \\iff w\\not\\in W_{S\\smallsetminus\\{s\\}}.\n\\end{equation}\nWe define the \\emph{colour-length} of $w$ putting $\\|w\\|=\\# S_w$ (the cardinality of $S_w$).\nBoth lengths satisfy the triangle inequality and we have $\\|w\\|\\le |w|$.\n\nIn the case of the permutation group the colour-length has the following pictorial\ninterpretation. If $\\sigma$ is a permutation\nin $\\F S_{n+1}$ then $\\|\\sigma\\|$ equals $n$ minus the number\nof connected components of the diagram representing~$\\sigma$.\nNotice, that $|\\sigma|$ equals to the number of crossings in the diagram\n(the number of pairs of chords that cross).\n\n\\begin{tikzpicture}\n\\newcommand{\\perm}[1]{\\draw \\foreach \\x\/\\y in {#1} {(\\x*.75,1)--(\\y*.75,0)};}\n\\newcommand{\\xtext}[1]{\\draw (1.5,.5) node[fill=white] {#1};}\n\\matrix[column sep=0.8cm,row sep=0.5cm]{\n\\xtext {$\\sigma$};&\\xtext {$e$};&\\xtext {$(12)$};&\\xtext {$(12)(23)$};&\\xtext {$(12)(23)(12)$};\\\\\n&\\perm{1\/1,2\/2,3\/3}&\\perm{1\/2,2\/1,3\/3}&\\perm{1\/2,2\/3,3\/1}&\\perm{1\/3,2\/2,3\/1}\\\\\n\\xtext {$|\\sigma|$};&\\xtext {0};&\\xtext {1};&\\xtext {2};&\\xtext {3};\\\\\n\\xtext {$\\|\\sigma\\|$};&\\xtext {0};&\\xtext {1};&\\xtext {2};&\\xtext {2};\\\\};\n\\end{tikzpicture}\n\n\nIt would be convenient to define \n\\begin{equation}\n\\label{def:block-length}\n\\|w\\|_s\\colon=\n\\begin{cases}\n0&\\text{if $s\\not\\in S_w$,}\\\\\n1&\\text{if $s\\in S_w$,}\n\\end{cases}\n\\end{equation}\nthen, clearly, $\\|w\\|=\\sum_{s\\in S}\\|w\\|_s$.\n\n\\def\\mathop{\\rm Rad}\\nolimits{\\mathop{\\rm Rad}\\nolimits}\n\n\\section{Positive defined functions}\n\nA complex function $\\varphi$ on a group $\\Gamma$ is called\n\\emph{positive definite}\nif we have\n\\[\n\\sum_{x,y\\in \\Gamma}\\varphi(y^{-1}x)\\alpha(x)\\overline{\\alpha(y)}\\ge0\n\\]\nfor every finitely supported function $\\alpha\\colon\\Gamma\\to\\B C$.\n\nA positive definite $\\varphi$ function is Hermitian and satisfies $|\\varphi(x)|\\le\\varphi(e)$\nfor all $x\\in \\Gamma$. Usually it is assumed, that $\\varphi$ is \\emph{normalised}, i.e. that $\\varphi(e)=1$.\n\nIn this and the following sections we discuss the \\textit{radial functions} on Coxeter groups.\nThese are functions which depend on $|w|$ rather then on $w$.\n\nWe call a function $\\varphi$ on $(W,S)$ \\emph{colour-dependent} if\n$\\varphi(w)$ depends only on $S_w$.  We call it \\emph{colour-radial}\nif it depends only on $\\|w\\|$.\n\nAn Abelian Coxeter group generated by $S$ is isomorphic to the direct product $\\oplus_{s\\in S}\\B Z\/2$.\nOn these groups the lengths $|\\cdot|$ and $\\|\\cdot\\|$ coincide and all functions are colour dependent.\n\nThe main example of a positive definite function will be the \\emph{Riesz--Coxeter function}.  Given a sequence $\\B q=(q_s)_{s\\in S}$\nwe define $R_{\\B q}(w)=\\prod_{s\\in S}q_s^{\\|w\\|_s}=\\prod_{s\\in  S_w}q_s$.\nWe will abuse notation and denote by $R_{\\B q}$ also the associated operator\n$\\sum_{w\\in W}R_{\\B q}(w)w$.  That is\n$$\nR_{\\B q}=1+\\sum_{s\\in S}q_ss+\\sum_{w: S_w=\\{s_1,s_2\\}}q_{s_1}q_{s_2}w+\\sum_{w: S_w=\\{s_1,s_2,s_3\\}}q_{s_1}q_{s_2}q_{s_3}w+\\dots\n$$\n\nIn the case all $q_s=q$ we get $R_{\\B q}=\\sum q^{\\|w\\|}w$.\n\nThis generalises the classical case of Rademacher--Walsh functions in the\nRademacher group $\\mathop{\\rm Rad}\\nolimits_n$.  If we denote the generator of the $i$-th factor $\\B Z\/2$\nof the latter by the symbol $r_i$ then, by definition, $r_1^2=1$ and $r_ir_j=r_jr_i$ and\n$$\nR_{\\B q}=\\prod\\limits_{i=1}^n(1+q_ir_i).\n$$\n\n\\section{Rademacher groups}\n\n\nIn this section we are going to study positive definite radial functions on the Abelian Coxeter groups,\n$(W,S)=\\mathop{\\rm Rad}\\nolimits_{S}$.\nSince positive definiteness is tested on functions with finite support, we can assume that $S$ is countable.\nIf $\\#S=n$ we will write $\\mathrm{Rad}_{n}$ instead of $\\mathrm{Rad}_{S}$.\nGiven $n\\in\\B N\\cup\\{\\infty\\}$, we denote by $P_n$ the class of all functions $f\\colon\\{0,1,\\dots,n\\}\\to\\B R$ for $n$ finite and $f\\colon\\B N\\to\\B R$ if\n$n=\\infty$ such that\n$\\varphi(w)=f(|w|)$ is a normalised positive definite on $\\mathop{\\rm Rad}\\nolimits_n$.\n\nThe following observation is straightforward.\n\n\\begin{proposition}\nAssume that $1\\leq m< n\\leq\\infty$ and $f\\in P_n$.  Then the restriction of $f$ to $\\{0,\\dots,m\\}$\nbelongs to $P_m$.  A fuction $f$ belogs to $P_\\infty$ if and only if all its restrictions to $\\{0,\\dots,m\\}$\nfor any $m\\in\\B N$ belong to $P_m$.\n\\end{proposition}\n\n\\begin{theorem}\nAssume $n$ is finite.\nThe set $P_n$ form a simplex whose\nvertices (extreme points) are $f^n_l(k)={n\\choose l}^{-1}\\sum_{i=0}^l(-1)^i{k\\choose i}{n-k\\choose l-i}$,\nwhere $0\\leq l\\leq n$.  Equivalently, every normalised radial positive definite function on the group $\\mathop{\\rm Rad}\\nolimits_n$\nis of the form\n$$\n\\varphi(x)=\\sum_{l=0}^n\\lambda_lf^n_l(|x|),\n$$\nwhere the sequence of nonnegative numbers $(\\lambda_l)_{l=0}^n$ is unique and satisfies\n$\\sum_{l=0}^n\\lambda_l=1$.\n\\end{theorem}\n\n\\begin{proof}\nWe can indentify the dual $\\widehat{\\mathop{\\rm Rad}\\nolimits_n}$ group of $\\mathop{\\rm Rad}\\nolimits_n$ with $\\mathop{\\rm Rad}\\nolimits_n$ via the paring\n$(x,y)=(-1)^{\\sum_{i=1}^n x_iy_i}$.  By Bochner's theorem every normalised\npositive definite function $\\varphi$ on $\\mathop{\\rm Rad}\\nolimits_n$ is of the form\n\\[\n\\varphi(x)=\\int_{\\widehat\\mathop{\\rm Rad}\\nolimits_{\\infty}}(x,y)\\,\\mu(dy),\n\\]\nfor some probability measure $\\mu$.  Clearly, such a function\nis radial if and only if $\\mu$ is invariant under the action of $\\F S_n$.\n\nAmong such measures extreme ones are measures $\\mu_l$ for $0\\leq l\\leq n$,\nwhere $\\mu_l$ is equally distributed among elements of length $l$.\nMoreover,\n\\[\n\\varphi(x)=\\int_{\\widehat\\mathop{\\rm Rad}\\nolimits_{\\infty}}(x,y)\\,\\mu_l(dy)\n=f^n_l(|x|)\n\\]\nas claimed.\n\\end{proof}\n\nThe following theorem is a version of the classical de Finetti Theorem\n(see \\cite[p. 223]{MR0270403}) for the infinite Rademacher group.\n\n\\begin{theorem}\nAssume that $\\varphi$ is a radial function on the Rademacher group\n$\\mathop{\\rm Rad}\\nolimits_\\infty$.\nThen $\\varphi$ is a normalised positive definite if and only if there exists\na probability measure $\\mu$ on $[-1,1]$ such that\n$$\n\\varphi(x)=\\int_{-1}^1 q^{|x|}\\mu(dq).\n$$\nThis measure $\\mu$ is unique.\n\\end{theorem}\n\n\\begin{proof}\nSince the function $q^{|x|}$ is normalised positive definite for $q\\in[-1,1]$, the ``if'' implication is obvious.\n\nAssume that $\\varphi$ is normlised positive definite.\nThe group $\\mathop{\\rm Rad}\\nolimits_{\\infty}$ is discrete and Abelian and its dual is the compact group\n$\\widehat{\\mathop{\\rm Rad}\\nolimits_\\infty}=\\prod_{i=1}^{\\infty}\\B Z\/2$.\nBy Bochner's theorem, there exists a probability measure $\\eta$ on $\\widehat\\mathop{\\rm Rad}\\nolimits_{\\infty}$\nsuch that\n\\[\n\\varphi(x)=\\int_{\\widehat\\mathop{\\rm Rad}\\nolimits_{\\infty}}(x,y)\\,d\\eta(y),\n\\]\nwhere for $x=(x_1,x_2,\\ldots)\\in \\mathop{\\rm Rad}\\nolimits_\\infty$, $y=(y_1,y_2,\\ldots)\\in \\widehat\\mathop{\\rm Rad}\\nolimits_{\\infty}$\nwe put $(x,y)=(-1)^{\\sum_{i=1}^\\infty x_i y_i}$.\nThe radiality of $\\varphi$ is equivalent to the fact that for every permutation $\\sigma\\in\\mathfrak{S}_\\infty$\nwe have $\\varphi(x)=\\varphi(\\sigma(x))$, where $\\sigma(x)=(x_{\\sigma(1)},x_{\\sigma(2)},\\ldots)$.\nThis, in turn, implies that $\\eta$ is $\\sigma$-invariant for every $\\sigma\\in\\mathfrak{S}_\\infty$,\ni.e. we have $\\eta(A)=\\eta(\\sigma(A))$ for every Borel subset $A\\subset\\widehat\\mathop{\\rm Rad}\\nolimits_{\\infty}$.\n\nFor a sequence $\\B \\epsilon=(\\epsilon_i)_{i=1}^n\\in\\{0,1\\}^n$ we define\n$C_{n}(\\B \\epsilon)\\subseteq\\widehat\\mathop{\\rm Rad}\\nolimits_{\\infty}$ by\n\\[\nC_{n}(\\B \\epsilon)=\\{y\\in\\widehat\\mathop{\\rm Rad}\\nolimits_{\\infty}|y_i=\\epsilon_i:1\\leq i\\leq n\\},\n\\]\nin particular $C_0(\\varnothing)=\\widehat\\mathop{\\rm Rad}\\nolimits_{\\infty}$.\nThen we have $\\eta(C_{n}(\\epsilon))=\\eta(C_{n}(\\B\\epsilon'))$\nif $\\epsilon'_i=\\epsilon_{\\sigma(i)}$ for some $\\sigma\\in\\mathfrak{S}_n$\nand every $1\\leq i\\leq n$.\nFor $\\varepsilon \\in \\B R$ we put\n\\[\n\\varepsilon^n=\\underbrace{\\varepsilon,\\varepsilon,\\ldots,\\varepsilon}_{n} \n\\]\nand $a_n=\\eta(C_{n}(1^{n}))$. Moreover, for $n,k\\ge0$ we define the difference operators $\\Delta^{k}a_n$ by induction:\n$\\Delta^{0}a_n=a_n$ and $\\Delta^{k+1}a_n=\\Delta^{k}a_{n+1}-\\Delta^{k}a_{n}$.\nWe claim that\n\\begin{equation}\\label{deltakn}\n(-1)^{k}\\Delta^{k}a_{n}=\\eta\\left(C_{n+k}\\left(1^n0^k\\right)\\right).\n\\end{equation}\nDenoting the right hand side of (\\ref{deltakn}) by $c(n,k)$ we note that $c(n,0)=a_n$ and\n\\[\nC_{n+k+1}\\left(1^n0^k0\\right)\\cup\nC_{n+k+1}\\left(1^n0^k1\\right)\n=C_{n+k}\\left(1^n0^k\\right),\n\\]\nis a disjoint union.  This implies\n\\[\nc(n,k+1)=c(n,k)-c(n+1,k).\n\\]\nThis formula, by induction on $k$, leads to (\\ref{deltakn}).\n\nFrom (\\ref{deltakn}) we see that the sequence $(a_n)$ is \\textit{completely monotone},\ni.e. that $(-1)^{k}\\Delta^{k}a_n\\ge0$ for all $n,k\\ge0$.\nBy the celebrated theorem of Hausdorff (see \\cite[S\\\"atze \\textsc{\\MakeLowercase{ii}} und \\textsc{\\MakeLowercase{iii}}]{MR1544467}),\nthere exists a unique probability measure\n$\\rho$ on $[0,1]$ such that\n\\begin{equation}\n\\label{eq-Haus}\n(-1)^{k}\\Delta^{k}a_n=\\int_{0}^{1} u^n(1-u)^k\\,d\\rho(u).\n\\end{equation}\n(Note that Equation (\\ref{eq-Haus}) for arbitrary $k\\geq 0$ follows from the case $k=0$.)\n \nFor $x=\\left(1^n0^\\infty\\right)\\in \\mathop{\\rm Rad}\\nolimits_\\infty$ so that $|x|=n$, we have\n\\begin{align*}\n\\varphi(x)&=\\int_{\\widehat\\mathop{\\rm Rad}\\nolimits_{\\infty}}(x,y)\\,d\\eta(y)\n=\\int_{\\widehat\\mathop{\\rm Rad}\\nolimits_{\\infty}}(-1)^{\\sum_{i=1}^ny_i}\\,d\\eta(y)\\\\\n&=\\sum_{\\epsilon\\in\\{0,1\\}^{n}}(-1)^{\\sum_{i=1}^n\\epsilon_i}\\eta(C_{n}(\\epsilon))\n=\\sum_{k=0}^{n}\\binom{n}{k}(-1)^{k}\\eta\\left(C_n\\left(1^k0^{n-k}\\right)\\right)\\\\\n&=\\sum_{k=0}^{n}\\binom{n}{k}(-1)^{k}\\int_{0}^{1} u^k(1-u)^{n-k}\\,d\\rho(u)\n=\\int_{0}^{1}(1-2u)^{n}\\,d\\rho(u)\\\\\n&=\\int_{-1}^{1}q^n \\,d\\mu(q),\n\\end{align*}\nwhere $\\mu$ is defined by $\\mu(A)=\\rho\\left(\\frac{1}{2}+\\frac{1}{2}A\\right)$ for a Borel set $A\\subseteq[-1,1]$.\n\\end{proof}\n\n\\section{Remarks on radial positive definite functions on some infinitely generated Coxeter groups}\n\nIn this Section we extend the last theorem of the previous section to a certain class of Coxeter groups.\n\n\\begin{theorem}\nAssume that $(W,S)$ is a Coxeter system and that there is an infinite subset $S_0\\subseteq S$\nsuch that $st=ts$ for $s,t\\in S_0$.\nAssume that $\\varphi$ is a radial function on $W$ with $\\varphi(e)=1$.\nThen $\\varphi$ is positive definite if and only if there exists\na probability measure $\\mu$ on $[-1,1]$ such that\n$$\n\\varphi(\\sigma)=\\int_{-1}^1 q^{|\\sigma|}\\mu(dq).\n$$\nThis measure $\\mu$ is unique.\n\\end{theorem}\n\n\\begin{proof}\nIt is sufficient to note that the group generated by $S_0$ is a parabolic subgroup isomorphic with\n$\\mathop{\\rm Rad}\\nolimits_\\infty$.\n\\end{proof}\n\n\\textbf{Example.}\nFor $W=\\mathfrak{S}_\\infty$ we have $S=\\{(n,n+1):n\\in\\B N\\}$.\nThen we can take $S_0=\\{(2n-1,2n):n\\in\\B N\\}$.  Similar\n$S_0$ can be found in infinitely generated groups of type \\textsc{\\MakeLowercase{B}} and \\textsc{\\MakeLowercase{D}}.\n\n\\begin{problem}\nWhen $-1\\leq q\\leq1$, $q\\neq 0$ is the positive definite function\n$q^{|x|}$ on $\\F S_\\infty$ an extreme point in the set of\nnormalised positive definite functions?\n\\end{problem}\n\n\\begin{theorem}\\label{T:frac}\nThe function $\\varphi_p(\\sigma)=e^{-t|\\sigma|^p}$ is positive definite on $\\F S_\\infty$\nif and only if $0\\leq p\\leq1$.\n\\end{theorem}\n\n\\begin{proof}\n{\\em A contrario.}  Assume that for some $p>1$ and $t_0>0$ the function $\\psi_p(\\sigma)=e^{-t_0|\\sigma|^p}$\nis positive definite on $\\F S_\\infty$.\n\nFor $q_0=e^{-t_0}$, choosing $\\sigma$ such that $|\\sigma|=2n$ we have\n$q_0^{(2n)^p}=\\int_{-1}^1q^{2n}\\,d\\mu_0(q)$ for some probability measure $\\mu_0$\non $[-1,1]$.  Since $\\left(\\int_{-1}^1q^{2n}\\,d\\mu_0(q)\\right)^{1\/n}$ tends to $\\max\\{q^2|q\\in\\mathop{\\hbox{\\rm supp}}\\mu_0\\}$\nwhile $\\left(q_0^{(2n)^p}\\right)^{1\/n}\\to 0$, we conclude that $\\mu_0$ is the Dirac measure at $0$,\nwhich is a contradiction.\n\nThe ``if'' part is standard.\nWe need to show that $f(x)=e^{-tx^p}$ is the Laplace transform of some probability measure supported on $[0,\\infty)$,\nso $f$ is a convex combination of functions of the form $e^{-sx}$.\n\nBy characterisation of Laplace transforms (see \\cite[Satz \\textsc{\\MakeLowercase{iii}}]{MR1544467})\nthis is equivalent to \\emph{complete monotonicity}, that is $(-1)^nf^{(n)}>0$ for all $n=0,1,\\dots$.\nAnd indeed, by induction, $(-1)^nf^{(n)}$ is a positive linear\ncombination of positive functions of the form $x^{pj-n}f(x)$ for $1\\leq j\\leq n$.\n\\end{proof}\n\nThe measures with Laplace transforms $e^{-tx^p}$ for $t\\geq0$ and $0\\leq p\\leq 1$\nare studied in detail in \\cite[Ch.~\\textsc{\\MakeLowercase{ix}.11}]{MR617913}\n(see Propositions~1 and 2 there).\n\nLet us note that for such groups like $\\B Z^k$ or $\\B R^k$ with the Euclidean distance $d$\nthe functions $\\exp(-td^p)$ are positive definite for all $t\\geq 0$ and $0\\leq p\\leq 2$\n(the case $p=2$ corresponds to the Gau\\ss ian Law).\n\n\\section{The longest element}\n\nIf a Coxeter group $W$ is finite, then it contains the unique element $\\omega_\\circ$\nwhich has the maximal length with respect to $|\\cdot|$.\n\nFrom the definition it is clear, that a function $\\varphi$ on a group $\\Gamma$ with values in the field\nof complex numbers $\\B C$ is {\\em positive definite}\nif and only $\\sum_{g\\in\\Gamma}\\varphi(g)g$ is a nonnegative (bounded if the group is finite)\noperator on $\\ell^2\\Gamma$.  (We will identify $g\\in \\Gamma$ with $\\lambda(g)\\in B(\\ell^2\\Gamma)$,\nwhere $\\lambda$ is the left regular representation, for short.)\n\nLet $W$ be a finite Coxeter group.  The following two statements are well known.\n\\begin{enumerate}[label={(\\textsc{\\alph*})}]\n\\item The function $q^{|w|}$ is positive definite for any $0\\leq q\\leq 1$.\n\\item The function $\\Delta(w)=|\\omega_\\circ|\/2-|w|$ is positive definite.\n\\end{enumerate}\nThe first one was proven in \\cite{MR950825} (even for infinite Coxeter groups and also for\n$-1\\leq q\\leq 1$) while the second --- in \\cite[Proposition~6]{MR2009841}.\nHere we give a short direct prove of the following.\n\n\\begin{proposition}\\label{P: }\nThe above statements (\\textsc{a}) and (\\textsc{b}) are equivalent.\n\\end{proposition}\n\n\\begin{proof}\nLet $q=e^{-t}$ (with $t\\geq0$, as we assume $q\\leq 1$).\nThe case (\\textsc{a}) is equivalent to $\\Phi_t=\\sum_{w\\in W}e^{t\\Delta(w)}w=e^{t|\\omega_\\circ|\/2}\\sum_{w\\in W}q^{|w|}w$\nbeing nonnegative.\n\nAssume (\\textsc{a}).\nRecall first, that $|\\omega_\\circ w|=|\\omega_\\circ|-|w|=|w\\omega_\\circ|$.  Therefore\n$|\\omega_\\circ|\/2-|\\omega_\\circ w|=-(|\\omega_\\circ|\/2-|w|)$, ie.\n$\\Delta(\\omega_\\circ w)=-\\Delta(w)$ and similarly, $\\Delta(w\\omega_\\circ)=-\\Delta(w)$.\n\nThe equality $\\Delta(\\omega_\\circ w)=\\Delta(w\\omega_\\circ)$ implies that $\\omega_\\circ$\n(and thus $Q=(1-\\omega_\\circ)\/2$) commutes with $\\Delta$ (and thus $\\Phi_t$).\nSince $Q=Q^2$ is nonnegative we conclude that\n$$\nt^{-1}\\Phi_tQ=\\sum_{w\\in W}\\frac{e^{t(|\\omega_\\circ|\/2-|w|)}-e^{t(|\\omega_\\circ|\/2-|w\\omega_\\circ|)}}{2t}w\n=\\sum_{w\\in W}\\frac{\\sinh(t\\Delta(w))}{t}w.\n$$\nis nonnegative.  Therefore, taking the limit as $t\\to0$, we obtain that\n$\\sum_{w\\in W}\\Delta(w)w$ is nonnegative.\nThus (\\textsc{b}).\n\nAssuming (\\textsc{b}) and using the Schur lemma, which says that the (pointwise)\nproduct of positive definite functions is positive definite, we get that\n$$\n\\Phi_t=\\sum_{w\\in W} e^{t\\Delta(w)}w\n=\\sum_{n\\geq 0}\\frac{t^n}{n!}\\left(\\sum_{w\\in W}\\Delta(w)^nw\\right)\n$$\nis nonnegative.\nThus (\\textsc{a}).\n\\end{proof}\n\n\\section{Colour-dependent positive definite functions on Coxeter groups}\n\nThe question which colour-dependant or colour-radial functions\nare positive functions on Coxeter groups is wide open.\nIn this section we provide some sufficient conditions.\nIn the next section we will\nexamine the dihedral groups in full details.\n\n\\begin{lemma}\\label{alemmasubgroup}\nLet $H$ be a subgroup of a group $\\Gamma$ of index $d$.\nThen the function $\\varphi_r$ defined by $\\varphi_{r}(x)=1$ if $x\\in H$\nand $\\varphi_{r}(x)=r$ otherwise is positive definite on $\\Gamma$\nif and only if $r\\in[-1\/(d-1),1]$,\nwith natural convention that if $d=\\infty$ then $-1\/(d-1)=0$.\n\\end{lemma}\n\nNote, that if $H=\\{1\\}$ then $d=|\\Gamma|$.\n\n\\begin{proof}\nFirst assume that $d$ is finite and let us enumerate the left cosets:\n\\[\n\\{gH:g\\in \\Gamma\\}=\\{H_1,H_2,\\dots,H_d\\}.\n\\]\nNote, that for $x\\in H_i, y\\in H_j$ we have $y^{-1}x\\in H$\nif and only if $i=j$. Therefore, for $r_0=-1\/(d-1)$\nand for a finitely supported\ncomplex function $f$ on $\\Gamma$ we have\n\\[\n\\sum_{x,y\\in \\Gamma} \\varphi_{r_0}(y^{-1}x)f(x)\\overline{f(y)}\n=\\frac{1}{d-1}\\sum_{1\\le i<j\\le d}\n\\left|\\sum_{x\\in H_i}f(x)-\\sum_{y\\in H_j}f(y)\\right|^2,\n\\]\nwhich proves that $\\varphi_{r_0}$ is positive definite.\nFor $r\\in[-1\/(d-1),1]$ the function $\\varphi_r$\nis positive definite as a convex combination of $\\varphi_{r_0}$\nand the constant function $\\varphi_1$.\n\n\nOn the other hand, if we choose $x_i\\in H_i$ for each $i\\le d$\nand define\n$f$ as the characteristic function of the set $\\{x_1,\\ldots,x_{d}\\}$\nthen\n\\begin{equation}\\label{apositivedefiniterestriction}\n\\sum_{x,y\\in \\Gamma}\\varphi_r(y^{-1}x)f(x)\\overline{f(y)}\n=d+(d^2-d)r,\n\\end{equation}\nwhich proves that $r\\ge -1\/(d-1)$ is a necessary condition\nfor positive definiteness of $\\varphi_r$.\n\nIf $d=\\infty$ then $r_0=0$ and the function $\\varphi_0$\nis positive definite as the characteristic function of the subgroup $H$.\nFor ``only if\" part we chose an arbitrarily long sequence $x_1,\\ldots,x_{d'}$ of elements\nfrom different left cosets and use (\\ref{apositivedefiniterestriction}) with $d'$ instead of $d$.\n\\end{proof}\n\n\\begin{theorem}\\label{bpropositionproduct}\nAssume that for every $s\\in S$ we are given a number $q_s$,\n\\[\n\\frac{-1}{d_{s}-1}\\le q_s\\le1,\n\\]\nwhere $d_{s}$ denotes the index if the parabolic subgroup generated by $S\\smallsetminus\\{s\\}$ in $W$:\n$d_{s}=[W:W_{S\\smallsetminus\\{s\\}}]$.\nThen the Riesz--Coxeter $R_{\\B q}$ is positive definite on $W$.\n\\end{theorem}\n\n\\begin{proof}\nFrom Lemma~\\ref{alemmasubgroup} the function $w\\mapsto q_{s}^{\\|w\\|_s}$\nis positive definite for $s\\in S$ and $-1\/(d_{s}-1)\\le q_s\\le1$.\nSince the pointwise product of positive definite functions is positive definite,\nthe statement holds.\n\\end{proof}\n\n\\textbf{Example.} Take $W=\\mathfrak{S}_n$, the permutation group on the set $\\{1,2,\\ldots,n\\}$.\nIt is generated by the transpositions $S=\\{s_i=(i,i+1), 1\\le i\\le n-1\\}$.\nFor $1\\le i\\le n-1$ the parabolic subgroup generated by $S\\smallsetminus\\{s_i\\}$\nis isomorphic with $\\mathfrak{S}_{i-1}\\times \\mathfrak{S}_{n-i-1}$, so its index\nis $i{n\\choose i}$.\n\nIt would be interesting to determine for which $r$ the function\n$w\\mapsto r^{\\|w\\|}$ is positive definite. By Proposition~\\ref{bpropositionproduct}\nthis holds for $r\\in[-1\/(d-1),1]$, where $d$ is the maximal index of the parabolic subgroups\nof the form $W_{S\\smallsetminus\\{s\\}}$.\nWe note a necessary condition.\n\n\\begin{proposition}\nAssume that we have distinct generators $s_0,s_1,\\dots,s_n\\in S$\nsuch that $s_0s_k\\ne s_ks_0$ (i.e. $m(s_0,s_k)>2$) for $1\\le k\\le n$.\nIf the function $w\\mapsto r^{\\|w\\|}$ is positive definite on $W$,\nthen $\\sfrac{-1}{(n-1)}\\le r^3\\le1$.\n\nIf there is an element $s_0\\in S$ for which there are infinitely many $s\\in S$\nsuch that $s_0s\\ne ss_0$ then $ r^{\\|w\\|}$ is positive definite on $W$ if and only if\\\/ $0\\le r\\le 1$.\n\\end{proposition}\n\n\\begin{proof}\nConsider elements $w_k=s_0s_ks_0$.\nNote, that for $k\\ne l$ we have $\\|w_l^{-1}w_k\\|=3$.\nIf $\\varphi_r$ is positive definite on $W$\nthen we have\n\\[\n0\\le\\sum_{k,l=1}^{n}\\varphi_r(x_l^{-1}x_k)=\nn+(n^2-n)r^3,\n\\]\nwhich implies $r^3\\ge\\sfrac{-1}{(n-1)}$.\n\\end{proof}\n\n\\begin{corollary}\nThe function $w\\mapsto q^{\\|w\\|}$ on $\\F S_\\infty$ is positive definite if and only if\\\/ $0\\leq q\\leq 1$.\n\\end{corollary}\n\n\\begin{problem}\nThus, it is valid to ask the following.\nIs it true that {\\em every} normalised positive definite colour-lenght-radial\nfunction $\\phi\\colon\\F S_\\infty\\to\\B R$ is of the form $\\phi(\\sigma)=\\int_0^1 q^{\\|\\sigma\\|}\\,d\\mu(q)$\nfor some probability measure $\\mu$ on $[0,1]$?\n\\end{problem}\n\n\\section{Dihedral groups}\n\nIn this part we are going to examine the class of colour-dependent\npositive definite functions on the case the simplest nontrivial Coxeter groups.\nAssume that $W=\\B D_{2n}=\\langle s,t|(st)^n\\rangle$ (i.e.\\ the group of symmetries of a regular\n$n$-gon), and define a colour-dependent function  on $W$:\n\\begin{equation}\\label{functiondichedralpqr}\n\\phi(w)=\n\\begin{cases}\n1&\\text{if $w=e$,}\\\\\np&\\text{if $w=s$,}\\\\\nq&\\text{if $w=t$,}\\\\\nr&\\text{otherwise.}\n\\end{cases}\n\\end{equation}\nIf $p=q$ then $\\phi$ is colour radial.\nWe are going to determine for which parameters $p,q,r$ the function $\\phi$ is positive definite on $W$.\nIt is easy to observe necessary conditions: $p,q,r\\in[-1,1]$.\nMoreover, since $\\left\\langle st\\right\\rangle$\nis a cyclic subgroup of order $n$, Lemma~\\ref{alemmasubgroup},\nimplies a necessary condition: $-1\/(n-1)\\le r\\le1$.\n\n\\subsection*{Finite dihedral groups}\n\nAssume that $W$ is a finite dihedral group, $W=\\B D_{2n}$,\nso that $(st)^n=1$.\nWe will use the following version of Bochner's theorem:\nA function $f$ on a compact group $G$ is positive definite\nif and only if its \\emph{Fourier transform}:\n\\[\\widehat{f}(\\pi)=\\int_{G} f(x)\\pi(x^{-1})dx\n\\]\nis a positive operator for every $\\pi\\in\\widehat{G}$,\nwhere $\\widehat{G}$ denotes the dual object of $G$, i.e. the family of\nall equivalency classes of unitary irreducible representations of $G$, see~\\cite{MR1363490}.\nThen we have\n\\[\nf(x)=\\sum_{\\pi\\in\\widehat{G}}d_{\\pi}\\mathop{\\rm tr}\\left[\\widehat{f}(\\pi)\\pi(x)\\right].\n\\]\n\nTherefore, for every irreducible representation $\\pi$ of $\\B D_{2n}$\nwe are going to find\n\\[\n\\widehat{\\phi}(\\pi)=\\frac{1}{2n}\\sum_{g\\in G}\\phi(g)\\pi(g^{-1}).\n\\]\n\nWe will identify $s$ with $(0,-1)$ and $t$ with $(1,-1)$.\nIf $n$ is odd then $\\B D_{2n}$ possesses two characters:\n$\\chi_{+,+}$ such that $\\chi_{+,+}(w)=1$ for every $w\\in\\B D_{2n}$\nand $\\chi_{-,-}$ such that $\\chi_{-,-}(s)=\\chi_{-,-}(t)=-1$.\nIf $n$ is even then we have two additional characters\n$\\chi_{+,-}$ and $\\chi_{-,+}$\nsuch that $\\chi_{+,-}(s)=\\chi_{-,+}(t)=1$ and $\\chi_{+,-}(t)=\\chi_{-,+}(s)=-1$.\nIt is easy to check that\n\\[\n2n\\widehat{\\phi}(\\chi_{+,+})=1+p+q+(2n-3)r,\n\\]\n\\[\n2n\\widehat{\\phi}(\\chi_{-,-})\n=1-p-q+r,\n\\]\nwhich gives\n\\[\n-1-(2n-3)r\\le p+q\\le 1+r\n\\]\nand, for $n$ even,\n\\[\n2n\\widehat{\\phi}(\\chi_{+,-})=1+p-q-r,\n\\]\n\\[\n2n\\widehat{\\phi}(\\chi_{-,+})=1-p+q-r,\n\\]\nwhich implies\n\\[\n|p-q|\\le 1-r.\n\\]\nWe have also the family of two dimensional representations $U_a$:\n\\begin{align*}\nU_a(k,1)&=\\begin{pmatrix}\ne^{2\\pi ika\/n}&0\\\\\n0&e^{-2\\pi ika\/n}\n\\end{pmatrix},\\\\\nU_a(k,-1)&=\\begin{pmatrix}\n0&e^{2\\pi ika\/n}\\\\e^{-2\\pi ika\/n}&0\n\\end{pmatrix},\n\\end{align*}\nwhere $a=1,2,\\dots,\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor$.\nThen for the function given by (\\ref{functiondichedralpqr}) we have\n\\begin{align*}\n2n\\widehat{\\phi}(U_a)&=(1-r)\\mathrm{Id}+(p-r)U_a(0,-1)+(q-r)U_a(1,-1)\\\\\n&=\\begin{pmatrix}\n1-r&p-r+(q-r)e^{2\\pi ia\/n}\\\\p-r+(q-r)e^{-2\\pi ia\/n}&1-r\n\\end{pmatrix}.\n\\end{align*}\nThis matrix is positive definite if and only if $r\\le 1$\nand\n\\[\n\\left|p-r+(q-r)e^{2\\pi ia\/n}\\right|\\le 1-r.\n\\]\nTherefore we have\n\\begin{proposition}\\label{propositiondihedralpqr}\nThe function $\\phi$ given by (\\ref{functiondichedralpqr}) is positive definite on $\\B D_{2n}$\nif and only if $$1+p+q+(2n-3)r\\ge0,\\qquad 1-p-q+r\\ge0$$\n(plus $$1+p-q-r\\ge0,\\qquad 1-p+q-r\\ge0$$ whenever $n$ is even)\nand\n\\[\n\\left|p-r+(q-r)e^{2\\pi ia\/n}\\right|\\le 1-r.\n\\]\nfor $a=1,2,\\dots,\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor$.\n\\end{proposition}\n\nLet us confine ourselves to colour-radial functions.\n\n\\begin{corollary}\nAssuming that $p=q$, the function $\\phi$ defined by (\\ref{functiondichedralpqr})\nis positive definite on $W=\\B D_{2n}$ if and only if\n\\[\n\\max\\left\\{\\frac{-2p-1}{2n-3},2p-1\\right\\}\\le r\\le \\frac{1+2p\\cos(\\sfrac{\\pi}n)}{1+2\\cos(\\sfrac{\\pi}n)},\n\\]\ni.e. if and only if the point $(p,r)$ belongs to the triangle whose vertices are\n\\[\n\\left(\\frac{1-n-\\cos(\\sfrac{\\pi}n)}{1+(2n-1)\\cos(\\sfrac{\\pi}n)},\\frac{1-\\cos(\\sfrac{\\pi}n)}{1+(2n-1)\\cos(\\sfrac{\\pi}n)}\\right),\\quad\n\\left(\\frac{n-2}{2n-2},\\frac{-1}{n-1}\\right),\\quad(1,1).\n\\]\n\\end{corollary}\n\n\\begin{proof}\nFor $p=q$ the conditions from Proposition~\\ref{propositiondihedralpqr} reduce to \n\\[\n2p-1\\le r,\\quad\n-1-2p\\le (2n-3)r,\\quad\\mbox{ and }\\quad\n2\\cos(\\sfrac{\\pi}n)|p-r|\\le 1-r.\n\\]\nIt is sufficient to note that $2p-1\\le r$ implies\n$2\\cos(\\sfrac{\\pi}n)(p-r)\\le1-r$ for $p\\le1$.\n\\end{proof}\n\n\\textbf{Example.} For $\\B D_4$ we have the positive definiteness of $\\phi$ is equivalent to\n\\[\n-1+|p+q|\\le r\\le 1-|p-q|,\n\\]\nwhich means that the set of all possible $(p,q,r)$ forms a tetrahedron\nwith vertices $(-1,1,-1)$, $(1,-1,-1)$, $(-1,-1,1)$, $(1,1,1)$.\nFor $p=q$ the condition reduces to $2|p|-1\\le r\\le1$.\n\nIn the case of $\\B D_{6}$ Proposition~\\ref{propositiondihedralpqr} leads to the following conditions:\n\\[\n1-p-q+r\\ge0,\\qquad 1+p+q+3r\\ge0,\n\\]\n\\[\n1-r\\ge\\sqrt{p^2+q^2+r^2-pq-pr-qr},\n\\]\nwhich can be expressed as\n\\[\n\\max\\left\\{\\frac{-1-p-q}{3},p+q-1 \\right\\}\\le r\\le\\frac{1-p^2-q^2+pq}{2-p-q}.\n\\]\n\n\\subsection*{The infinite dihedral group}\n\n\nHere we are going to study $W=\\B D_{\\infty}$.\n\n\\begin{proposition}\nThe function $\\phi$ given by (\\ref{functiondichedralpqr}) is positive definite on $W=\\B D_{\\infty}$\nif and only if $0\\le r$ and $|p-r|+|q-r|\\le1-r$, i.e.\n\\begin{equation}\\label{dichedralinfinite}\n\\max\\left\\{0,p+q-1\\right\\}\\le r\\le\\min\\left\\{1-|p-q|,\\frac{1+p+q}{3}\\right\\}.\n\\end{equation}\n\\end{proposition}\n\n\n\\begin{proof}\nFirst we note that the set of $(p,q,r)\\in\\B R^3$ satisfying (\\ref{dichedralinfinite})\nconstitutes a pyramid which is the convex hull of the points $(\\pm1,0,0)$, $(0,\\pm1,0)$ and $(1,1,1)$ (apex).\nFor these particular parameters it is easy to see that $\\phi$ is positive definite: $(1,1,1)$ corresponds\nto the constant function $1$, $(1,0,0)$ to the characteristic function of the subgroup $\\left\\langle s\\right\\rangle=\\{1,s\\}$,\nand $(-1,0,0)$ to the character $\\chi_{-,-}$ times the characteristic function of $\\left\\langle s\\right\\rangle$.\nSimilarly for $(0,\\pm1,0)$. This, by convexity, proves that (\\ref{dichedralinfinite}) is a sufficient condition.\n\nOn the other hand, we know already that $r\\ge0$ is a necessary condition.\nLet us fix $n$ and define $W^+(n)=\\{x\\in W:|sx|<|x|\\le2n\\}$, $W^-(n)=\\{x\\in W:|tx|<|x|\\le2n\\}$ and\n\\[\nf(x)=\n\\begin{cases}\n\\pm1&\\text{if $x\\in W^\\pm(n)$,}\\\\\n0&\\text{otherwise.}\n\\end{cases}\n\\]\nFor $x,y\\in W^+(n)$ we have $S_{y^{-1}x}=\\varnothing$ in $2n$ cases (namely, if $x=y$)\n$S_{y^{-1}x}=\\{s\\}$ in $2n-2$ cases (namely if $|x|=2k$, $|y|=2k+1$ or vice-versa, $k=1,\\ldots,n-1$)\n$S_{y^{-1}x}=\\{t\\}$ in $2n$ cases (namely if $|x|=2k$, $|y|=2k-1$ or vice-versa, $k=1,\\ldots,n$)\nand $S_{y^{-1}x}=\\{s,t\\}$ in all the other $(2n-1)(2n-2)$ cases.\nSimilarly, for $x,y\\in W^-(n)$ we have $S_{y^{-1}x}=\\emptyset$ in $2n$ cases,\n$S_{y^{-1}x}=\\{s\\}$ in $2n$ cases,\n$S_{y^{-1}x}=\\{t\\}$ in $2n-2$ cases\nand $S_{y^{-1}x}=\\{s,t\\}$ in $(2n-1)(2n-2)$ cases.\nIf $x\\in W^+(n)$, $y\\in W^-(n)$ or vice-versa then $S_{y^{-1}x}=\\{s,t\\}$.\nSumming up, we get\n\\begin{align*}\n\\sum_{x,y\\in W}&\\phi(y^{-1}x)f_n(x)f_n(y)\\\\\n&=4n+(4n-2)p+(4n-2)q+(4n-2)(2n-2)r-8n^2 r\\\\\n&=4n+(4n-2)p+(4n-2)q-(12n-4)r.\n\\end{align*}\nTherefore for every $n\\in\\B N$ we have a necessary condition\n\\[\n1+\\left(1-\\frac{1}{2n}\\right)p+\\left(1-\\frac{1}{2n}\\right)q-\\left(3-\\frac{1}{n}\\right)r\\ge0.\n\\]\nLetting $n\\to\\infty$ we get $1+p+q\\ge3r$.\n\nPut $x_k=stst\\ldots$, $|x_k|=k$. Fix $n$ and define\n\\[\ng(x)=\n\\begin{cases}\n\\chi_{-,+}(x)&\\text{if $x=x_k$ for $1\\le k\\le 4n$,}\\\\\n0&\\text{otherwise,}\n\\end{cases}\n\\]\nwhere, as before, $\\chi_{-,+}$ is the character on $W$ for which $\\chi_{-,+}(s)=-1$, $\\chi_{-,+}(t)=1$.\nThen\n\\[\n\\sum_{x,y\\in W}\\phi(y^{-1}x)g(x)g(y)=\\sum_{k,l=1}^{4n}\\phi(x_{l}^{-1}x_{k})g(x_k)g(x_l).\n\\]\nDenote $c_{k,l}=\\phi(x_{l}^{-1}x_{k})g(x_k)g(x_l)$. Then we have $c_{k,k}=1$, $1\\le k\\le 4n$,\n$c_{k,k-1}=q$ if $k$ is even, $c_{k,k-1}=-p$ if $k$ is odd, $2\\le k\\le4n$ and\n$c_{k,l}=c_{l,k}$ for all $1\\le k,l\\le 4n$. If $1\\le k,l\\le 4n$ and $|k-l|\\ge2$\nthen $c_{k,l}=(-1)^{j}r$, where $j$ is the total number of $s$ appearing in $x_k$ and $x_l$.\nNow it is not difficult to check that\n\\[\n\\sum_{l=1}^{4n}c_{k,l}=\n\\begin{cases}\n1+q-2r&\\text{if $k=1$ or $k=4n$,}\\\\\n1-p+q-r&\\text{if $1<k<4n$,}\n\\end{cases}\n\\]\nwhich implies\n\\[\n\\sum_{x,y\\in W}\\phi(y^{-1}x)g(x)g(y)=4n-(4n-2)p+4nq-(4n+2)r\n\\]\nand leads to necessary condition $r\\le 1-p+q$.\nIn a similar manner we get $r\\le1+p-q$.\n\nFinally, define a function $h$ similarly like $g$, but now we use the character $\\chi_{-,-}$:\n\\[\nh(x)=\n\\begin{cases}\n\\chi_{-,-}(x)=(-1)^k&\\text{if $x=x_k$ for $1\\le k\\le 4n$,}\\\\\n0&\\text{otherwise.}\n\\end{cases}\n\\]\nPutting $d_{k,l}=\\phi(x_{l}^{-1}x_{k})h(x_k)h(x_l)$ we have\n$d_{k,k}=1$, $d_{k,k-1}=-p$ if $2\\le k\\le 4n$ is even\nand $d_{k,k-1}=-q$ if $k$ is odd. Moreover, if $|k-l|\\ge2$, $1\\le k,l\\le 4n$\nthen $d_{k,l}=(-1)^{k+l} r$. Now one can check that\n\\[\n\\sum_{l=1}^{4n}d_{k,l}=\n\\begin{cases}\n1-q&\\text{if $k=1$ or $k=4n$,}\\\\\n1-p-q+r&\\text{if $1<k<4n$,}\n\\end{cases}\n\\]\nwhich yields $1-p-q+r\\ge0$\nand completes the proof that the conditions (\\ref{dichedralinfinite}) are necessary.\n\\end{proof}\n\n\\section{Weak Sidon sets and operator Khinchin inequality}\n\nThe aim of this section is to show that the set of Coxeter generators $S$ in an arbitrary\nCoxeter group $W$ is a \\emph{weak Sidon set}, ie. an interpolation set for the Fourier--Stieltjes\nalgebra $B(W)$.\n\n\n\nGiven a group $\\Gamma$, the Fourier--Stieltjes algebra consists of linear combinations\nof positive definite functions on $\\Gamma$, ie. every element of $B(\\Gamma)$ is of the form\n$f=\\varphi_1-\\varphi_2+i(\\varphi_3-\\varphi_4)$ for some  positive definite functions $\\varphi_i$\n($1\\leq i\\leq4$) on $\\Gamma$.  The norm on $B(\\Gamma)$ is defined as\n$$\n\\|f\\|_{B(\\Gamma)}=\\inf\\left\\{\\sum \\varphi_i(e)|\\hbox{\\rm where $f$ decomposes as above}\\right\\}\n$$\n\n\\begin{theorem}\n\\label{T:5.1}\nThe set of Coxeter generators $S$ in an arbitrary\nCoxeter group $W$ is a weak Sidon set, ie. for every bounded function $f\\colon S\\to[-1,1]$\nthere exists positive definite functions $\\varphi_\\pm$, such that $f(s)=\\varphi_+(s)-\\varphi_-(s)$ for any $s\\in S$.\nOne can take $\\varphi_\\pm=R_{\\B q^\\pm}$ for a suitable choice of\\\/ $\\B q^\\pm$.  Moreover\n$$\n\\|\\varphi_+-\\varphi_-\\|_{B(W)}\\leq 2\n$$\n\\end{theorem}\n\n\\begin{proof}\nPut $S_\\pm(f)=\\{s\\in S|\\pm f(s)> 0\\}$.  Set\n$$\nq^\\pm_s=\\begin{cases}\\pm f(s)&\\hbox{for $s\\in S_\\pm(f)$,}\\\\0&\\hbox{otherwise.}\\end{cases}\n$$\nThen $f(s)=R_{\\B q^+}(s)-R_{\\B q^-}(s)$ as claimed.  The rest of the statement hold as\nthe Riesz-Coxeter function at the identity element equals to one.\n\\end{proof}\n\nGiven a matrix $A\\in M_n(\\B C)$ and $p\\geq 1$ the Schatten $p$-class norm $\\|A\\|_{\\C S_p}$\nis defined as $\\|A\\|^p_{\\C S_p}=\\left(\\mathop{\\rm tr}|A|\\right)^{1\/p}$, where $|A|=(A^*A)^{1\/2}$.\n\nLet $\\lambda$ denote the left regular representation of a group $\\Gamma$.  Given a finite sum\n$f=\\sum c_g\\lambda(g)\\in\\B C[\\Gamma]$ we define noncommutative $L^p$-norm\n$$\n\\|f\\|_{L^p(\\Gamma)}^p=\\left(\\tau\\left((f^**f)^{1\/2}\\right)\\right)^{1\/p}\n$$\nwhere $\\tau(f)=c_e$ is the von Neumann trace and $L^p(\\Gamma)$ is a completion\nof $\\B C[\\Gamma]$ with respect to the above norm.\n\nWe recall, that a scalar-valued map $\\varphi$ on a group $\\Gamma$ is called a \\emph{completely bounded Fourier multiplier}\non $L^p(\\Gamma)$ if the associated operator\n$$\nM_\\varphi(\\lambda(g))=\\varphi(g)\\lambda(g),\\qquad g\\in \\Gamma\n$$\nextends to a completely bounded operator on $L^p(\\Gamma)$.\n\nWe let $M_{\\mathop{\\rm cb}}(L^p(\\Gamma))$ to be an algebra of completely bounded Fourier multipliers equipped\nwith the norm\n$$\n\\|\\varphi\\|_{M_{\\mathop{\\rm cb}}(L^p(\\Gamma))}=\\|M_\\varphi\\otimes\\mathop{\\rm id}\\nolimits_{\\C S^p}\\|.\n$$\n\n\\def\\left\\|(a_s)_{s\\in S}\\right\\|_{\\mathop{R}\\cap \\mathop{C}}{\\left\\|(a_s)_{s\\in S}\\right\\|_{\\mathop{R}\\cap \\mathop{C}}}\n\nFollowing Pisier \\cite{MR2006539}, for $a_s\\in M_n(\\B C)$, where $s\\in S$, we define\n$$\n\\left\\|(a_s)_{s\\in S}\\right\\|_{\\mathop{R}\\cap \\mathop{C}}={\\max\\left\\{\\left\\|\\sum_{s\\in S}(a_sa_s^*)^{1\/2}\\right\\|_{\\C S^p},\\left\\|\\sum_{s\\in S}(a_s^*a_s)^{1\/2}\\right\\|_{\\C S^p}\\right\\}}.\n$$ \nFor a set $E\\in \\Gamma$ we define the \\emph{completely bounded} constant $\\Lambda^{\\mathop{\\rm cb}}_p(E)$ as infimum of $C$ \nsuch that\n$$\n\\left\\|\\sum_{s\\in S}a_s\\otimes\\lambda(s)\\right\\|_{L^p(W)}\\leq C\\left\\|(a_s)_{s\\in S}\\right\\|_{\\mathop{R}\\cap \\mathop{C}}\n$$\nfor all matrices $a_s\\in M_n(\\B C)$ and $n\\in\\B N$.\n\n\\begin{theorem}\n\\label{T:6.1}\nIf\\\/ $a_s\\in M_n(\\B C)$, then for all $p\\geq 2$ and any Coxeter system $(W,S)$\nwe have\n$$\n\\left\\|(a_s)_{s\\in S}\\right\\|_{\\mathop{R}\\cap \\mathop{C}}\\leq\\left\\|\\sum_{s\\in S}a_s\\otimes\\lambda(s)\\right\\|_{L^p(W)}\\leq 2A'\\sqrt p\\left\\|(a_s)_{s\\in S}\\right\\|_{\\mathop{R}\\cap \\mathop{C}}.\n$$\n\\end{theorem}\n\n\\begin{proof}\nIt was shown by Harcharras \\cite[Prop.~1.8]{MR859804} that $\\Lambda^{\\mathop{\\rm cb}}_p(E)$ if finite\nif and only if $E$ is an interpolation set for $M_{\\mathop{\\rm cb}}(L^p(\\Gamma))$, i.e.\nevery bounded function on $E$ can be extended to a multiplier, and\n$$\n\\Lambda^{\\mathop{\\rm cb}}_p(E)\\leq \\Lambda^{\\mathop{\\rm cb}}_p(R)\\mu^{\\mathop{\\rm cb}}_p(E),\n$$\nwhere $R$ is the generating set in the Rademacher group $\\mathop{\\rm Rad}\\nolimits_\\infty$\nand $\\mu^{\\mathop{\\rm cb}}_p(E)$ is the interpolation constant.\n\nAs shown by Buchholz \\cite[Thm.~5]{MR2213610} for $p=2n$, and $S$ te standart generating set in $\\mathop{\\rm Rad}\\nolimits_\\infty$,\n$\\Lambda_{2n}^{\\mathop{\\rm cb}}(R)=\\left((2n-1)!!\\right)^{1\/2n}\\leq A\\sqrt p$ for some absolute $A$.  This was extended\nby Pisier \\cite[Thm.~9.8.2]{MR2006539} for any $p\\geq 2$, i.e\n$$\n\\Lambda_p^{\\mathop{\\rm cb}}(R)\\leq A'\\sqrt p,\n$$\nfor an absolute constant $A'$.\n\nWe have shown in Theorem \\ref{T:5.1} that in an arbitrary Coxeter group $W$ its Coxeter generating set $S$ is\na weak Sidon set, i.e. it is interpolation set for the Fourier--Stieltjes algebra $B(W)$.\nSince for $p\\geq 1$,\n$B(\\Gamma)$ is a subalgebra of $M_{\\mathop{\\rm cb}}(L^p(\\Gamma))$ and\n$$\n\\|\\varphi\\|_{M_{\\mathop{\\rm cb}}(L^p(\\Gamma))}\\leq\\|\\varphi\\|_{B(\\Gamma)},\n$$\nwe see that $\\mu^{\\mathop{\\rm cb}}_p(S)\\leq 2$.  Thus $\\Lambda^{\\mathop{\\rm cb}}_p(S)\\leq 2A'\\sqrt p$.\nThis finishes the proof of the right inequality.\n\nThe left inequality\nholds for any group $\\Gamma$ and any $S\\subset \\Gamma$ (see \\cite{MR859804}).\n\\end{proof}\n\n\\begin{remark}\nFendler \\cite{MR1967380} has shown  that if for all $s,t\\in S$, $s\\neq t$, we have $m_{s,t}\\geq 3$, then\n$$\n\\Lambda_p^{\\mathop{\\rm cb}}(S)\\leq 2\\sqrt 2.\n$$\nSee also \\cite{MR0390658} and \\cite{MR1476122} for related results in the case of free Coxeter groups.\nAlso Haagerup and Pisier have shown that $\\Lambda_\\infty^{\\mathop{\\rm cb}}(S)=2$,\nwhere $\\Lambda^{\\mathop{\\rm cb}}_\\infty(E)=\\sup_{p\\geq 2}\\Lambda^{\\mathop{\\rm cb}}_p(E)$ \\cite{MR1240608}.\nSee the paper of Haagerup \\cite{MR654838} where the best constant\nwas calculated for the set of Coxeter generators of the Rademacher group in case when $a_s$ are scalars.\n\\end{remark}\n\n\n\n\\section{Chromatic length function for Coxeter groups and pairpartitions}\n\n\\def\\mathop{\\C{NC}}\\nolimits{\\mathop{\\C{NC}}\\nolimits}\n\\def\\mathop{\\C P}\\nolimits{\\mathop{\\C P}\\nolimits}\n\\def\\r#1{{:}\\mkern-.5\\thinmuskip{#1}\\mkern-.5\\thinmuskip{:}}\n\nLet $[2n]=\\{1,\\dots,2n\\}$.  Let $2^{[2n]}$ denote the set of subsets of $[2n]$.\nBy a partition of $[2n]$  we mean $\\pi\\subset s^{[2n]}$ such that\n$\\bigcup\\pi=[2n]$ and if $\\pi',\\pi''\\in \\pi$ then $\\pi'=\\pi''$ or\n$\\pi'\\cap\\pi''=\\varnothing$.  We say, that partition $\\varrho$ is a \\emph{coarsening}\nof a partition $\\pi$ if for any $\\pi'\\in\\pi$ there exists $\\varrho'\\in\\varrho$\nsuch that $\\pi'\\subset\\varrho'$.\n\nA partition is called \\emph{crossing}\nif there exist $1\\leq a<b<c<d\\leq 2n$ and $\\pi_1,\\pi_2\\in\\pi$ with\n$a,c\\in\\pi_1\\neq\\pi_2\\ni b,d$; otherwise it is called \\emph{noncrossing}.\nFor any partition $\\pi$ there exists th the smallest noncrossing coarsening $\\Phi(\\pi)$ of $\\pi$\n(ie.\\ if $\\varrho$ is a noncrosing coarsening of $\\pi$ then it is a coarsening of $\\Phi(\\pi)$).\nWe define $\\|\\pi\\|=n-\\#\\Phi(\\pi)$.  The notion for the map $\\Phi$ was introduced in \\cite{MR2423121}.\n\nWe say that $\\pi$ is a \\emph{pairpartition} if every member of $\\pi$ has cardinality two.\nThe set of pairpartitions of $[2n]$ is denoted by $\\mathop{\\C P}\\nolimits_2(2n)$.\nGiven $\\pi\\in\\mathop{\\C P}\\nolimits_2(2n)$ we write $|\\pi|$ to denote the number of ordered quadruples\n$1\\leq a<b<c<d\\leq 2n$ such that both $\\{a,c\\}$ and $\\{b,d\\}$ belong to $\\pi$.\nNote, that $|\\pi|=0$ precisely when $\\pi$ is noncrossing.  The set of\nnoncrossing pairpartitions is denoted $\\mathop{\\C{NC}}\\nolimits_2(2n)$.\n\n\\def\\mathop{\\hbox{\\rm inn}}\\nolimits{\\mathop{\\hbox{\\rm inn}}\\nolimits}\nGiven a noncrossing pairpartition $\\varpi$ we call $\\{b,c\\}\\in\\varpi$ \\emph{an inner block} if there exists $\\{a,d\\}\\in\\varpi$\nwith $a<b<c<d$.  The number of inner blocks of $\\varpi$ we denote as $\\mathop{\\hbox{\\rm inn}}\\nolimits(\\varpi)$.\n\nIn \\cite[Cor.~7]{MR1388006} Bo\\.zejko and Speicher observed the following identity.\n\\begin{equation}\n\\label{eq:p2-nc2}\n\\sum_{\\pi\\in\\mathop{\\C P}\\nolimits_2(2n)}(-1)^{|\\pi|}q^{\\|\\pi\\|}=\\sum_{\\varpi\\in\\mathop{\\C{NC}}\\nolimits_2(2n)}(1-q)^{\\mathop{\\hbox{\\rm inn}}\\nolimits(\\varpi)}.\n\\end{equation}\n\nLet $f_n(q)=\\sum_{\\varpi\\in\\mathop{\\C{NC}}\\nolimits_2(2n)}(1-q)^{\\mathop{\\hbox{\\rm inn}}\\nolimits(\\varpi)}$.  It is\nelementary to derive\n$$\nf_n(q)=C_n\\,\\,\\mathstrut_2\\!F_1\\left({n,1-n\\atop n+2}\\right|\\left.\\vphantom{n,1-n\\atop n+2}q\\right),\n$$\nwhere $C_n={2n\\choose n}-{2n\\choose n-1}=\\#\\mathop{\\C{NC}}\\nolimits_2(2n)$ denote the $n$-th Catalan number\nand $\\mathstrut_2\\!F_1$ is the classical hypergeometric funcion.\nIf we write $f(q)=\\sum_{j=0}^{n-1}t^n_jq^j$, then the triangle $(t^n_j)_{0\\leq j<n}$\nappears in \\cite{oeisA062991} as ``sequence''\n\\textsc{\\MakeLowercase{A062991}}).  Since we are not going to use this formula, we leave it as an exercise to the reader.\nFor the expansion of $f_n(1-t)$ and the Delanoy triangle appearing there the reader may consult \\cite[Prop~6.1]{MR1863276}.\n\n\n\n\nIn what follows, we prove a result about an arbitrary finitely generated Coxeter group, which for permutation\ngroups implies the above one.\nGiven a permutation $\\sigma\\in S_n$, we construct a pairpartition $\\overline\\sigma=\\{\\{i,2n+1-\\sigma(i)\\}|1\\leq i\\leq n\\}$.\nNote, that $|\\overline\\sigma|$ is equal to the length of $\\sigma$ with respect to the Coxeter generators $(1,2)$,\n\\dots, $(n-1,n)$ of $S_n$.  Therefore, denoting by $|w|$\nthe Coxeter length of an element $w$ of some Coxeter group $W$ will not lead into any confusion.\n\nIt is also clear that, with respect to the identification of permutations with\na subset of pairpartitions, the two definitions of $\\|\\cdot\\|$ agree\n(see Equations (\\ref{def:block-length}) and (\\ref{def:colour})).\n\nBy $W(t)$ we denote a growth series of a finitely generated Coxeter group $W$\n(length function).  That is, a power series $W(t)=\\sum_{w\\in W}\nt^{|w|}$.  (Note, that the coefficient at $t$ equals to $\\#S$.  This\nexplains why here and in the rest of this section we consider only finitely\ngenerated Coxeter groups.  We will not repeat this assumption for short.)\nMoreover, for $X\\subset W$ we write\n$X(t)=\\sum_{w\\in X} t^{|w|}$.\n\n\nLet us define a multivariable formal power series (\\emph{chromatic length function}.\nFor any $X\\subset W$ define\n$$\nX(t,\\B q)=\\sum_{w\\in X}t^{|w|}\\prod_{s\\in S_w}q_s.\n$$\nIn particular $X(t)=X(t,\\B 1)$, where $\\B 1=(1)_{s\\in S}$.\n\n\\begin{proposition}\\label{P:tokyo}\nThe polynomial (or formal power series, if\\\/ $W$ is infinite) $W(t,\\B q)$ satisfies\n$$W(t,\\B q)=\n\\sum_{T\\subset S}W_T(t)\\prod_{r\\in T}r_s\\prod_{s\\in S\\smallsetminus T}(1-q_s).$$\n\\end{proposition}\n\n\\begin{proof}\nLet\\\/ $W^\\circ_{R}$ denote the set of all elements of\\\/ $W_R$ not\ncontained in any proper parabolic subgroup of\\\/ $W_R$, ie.\n$W^\\circ_{R}\\colon=W_{R}-\\bigcup_{T\\varsubsetneq R}W_T$.\nThen, by inclusion-exclusion principle, $W^\\circ_R(t)=\\sum_{T\\subset R}(-1)^{\\#(R-T)}W_T(t)$.\nTherefore,\n\\begin{align*}\nW(t,\\B q)&=\\sum_{w\\in W}t^{|w|}\\prod_{s\\in S_w}q_s\n=\\sum_{R\\subset S}W^\\circ_{R}(t)\\prod_{r\\in R}q_r\\\\\n&=\\sum_{R\\subset S}\n\\sum_{T\\subset R}W_T(t)(-1)^{\\#(R-T)}\\prod_{r\\in R}q_r\\\\\n&=\\sum_{T\\subset S}\nW_T(t)\\prod_{r\\in T}q_r\\sum_{T\\subset R\\subset S}\\prod_{s\\in R\\smallsetminus T}(-q_s)\\\\\n&=\\sum_{T\\subset S}W_T(t)\\prod_{r\\in T}q_i\\prod_{s\\in S\\smallsetminus T}(1-q_s).\n\\end{align*}\n\\end{proof}\n\n\\begin{corollary}\\label{C:wat-1}\nIf\\\/ $W$ is a finite Coxeter group then\n\\begin{equation}\n\\label{eq:wat-1}\nW(-1,\\B q)=\\prod\\limits_{s\\in S}(1-q_s).\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nChoose $s\\in T$ and put $W_T^{\\{s\\}}=\\{w\\in W:|w|<|ws|\\}$.  Clearly, $W_T=W_T^{\\{s\\}}W_T$\ntherefore $W_T(t)=W_T^{\\{s\\}}(t)W_{\\{s\\}}(t)$.\nSince $W$ is a finite group, $W_T^{\\{s\\}}$ is a polynomial.\nThus $W_T(-1)=0$ if $T$ is nonempty (and $W_\\varnothing(-1)=1$).\n\\end{proof}\n\nIn order to prove Equation (\\ref{eq:p2-nc2}) we define the \\emph{Wick map} $\\mathop{\\C P}\\nolimits_2(2n)\\ni\\pi\\mapsto\\r{\\pi}\\in\\mathop{\\C{NC}}\\nolimits_2(2n)$\n(related to the normal order in quantum field theory).\nGiven a pairpartition $\\pi$ we define $\\r{\\pi}$ by repetitive resolving crossings.\nThat is, we replace repetitively every crossing pair $\\{a,c\\}$ and $\\{b,d\\}$ with $a<b<c<d$ by\n$\\{a,d\\}$ and $\\{b,c\\}$.  In order to see that the result is independent of the\norder of resolution we describe $\\r{\\pi}$ in an equivalent way.\n\nLet $\\Phi(\\pi)$ be the smallest noncrossing coarsening of $\\pi$.  For each block $\\beta$\nof $\\Phi(\\pi)$ define $\\beta^+=\\{y|(\\exists x)\\ x\\in\\beta,\\ y>x,\\ \\{x,y\\}\\in\\pi\\}$ and\n$\\beta_1^-=\\{x|(\\exists y)\\ y\\in\\beta,\\ y>x,\\ \\{x,y\\}\\in\\pi\\}$.  Order $\\beta^+=\\{y_1,\\dots,y_k\\}$\nin increasing way and $\\beta^-=\\{x_1,\\dots,x_k\\}$ in decreasing way.  Then all\npairs $\\{x_i,y_i\\}$ will be parts of $\\r{\\pi}$.\n\nEquation (\\ref{eq:p2-nc2}) will follow from a more refined statement.\n\n\\begin{proposition}\n\\label{P:tokyo2}\nFor every $\\varpi\\in\\mathop{\\C{NC}}\\nolimits_2(2n)$\n\\begin{equation}\n\\label{eq:tokyo2}\n\\sum_{\\pi\\in\\mathop{\\C P}\\nolimits_2(2n)\\atop \\r{\\pi}=\\varpi}(-1)^{|\\pi|}q^{\\|\\pi\\|}=(1-q)^{\\mathop{\\hbox{\\rm inn}}\\nolimits(\\varpi)}.\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nLet us first consider the case of $\\varpi=\\overline1=\\{(i,2n+1-i)|1\\leq i\\leq n\\}$.\nClearly, $\\{\\pi|\\r{\\pi}=\\overline1\\}=\\{\\overline\\sigma|\\sigma\\in\\F S_n\\}$.  And Equation\n(\\ref{eq:tokyo2}) is equivalent to Equation (\\ref{eq:wat-1}) (as all $q_s$ are set to $q$) for\n$W=\\F S_n$.\n\nIn a general case observe, that $\\Phi(\\pi)$ is a coarsening of $\\varpi=\\r{\\pi}$.  Yet, not every coarsening\nmay appear.  The obvious condition is that for each block $\\beta$ of $\\r{\\pi}$ the pair $\\{\\min\\beta,\\max\\beta\\}$\nbelong to $\\varpi$.  For the purpose of this proof we will call such a coarsening\nadmissible.  Its clear, that abmissible coarsenings $\\rho$ are in one to one correspondence with\nsubsets of $\\varpi$ containing all outer (not inner) parts of $\\varpi$ of the form $\\{\\{\\min\\rho',\\max\\rho'\\}|\\rho'\\in\\rho\\}$.\n\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}\n\\matrix[matrix of nodes,column sep=0.8cm,row sep=0.5cm]{\n\\draw plot [smooth] coordinates {(1,0) (5,1.5) (8,0)};\n\\draw plot [smooth] coordinates {(2,0) (3,0.6) (4,0)};\n\\draw plot [smooth] coordinates {(3,0) (4,0.6) (5,0)};\n\\draw plot [smooth] coordinates {(6,0) (7.5,1) (9,0)};\n\\draw plot [smooth] coordinates {(7,0) (8.5,1.0) (10,0)};\n&$\\pi$\\\\\n\\draw plot [smooth] coordinates {(1,0) (3,1) (5,1.5) (7,1) (7.5,1.2) (8,1) (9,1.2) (10,0)};\n\\draw plot [smooth] coordinates {(2,0) (3,.8) (3.5,.55) (4,.8) (5,0)};\n\\draw plot [smooth] coordinates { (3,0) (3.5,0.45) (4,0)};\n\\draw plot [smooth] coordinates { (6,0) (7,0.8) (7.5,0.55) (8,0.8) (9,0)};\n\\draw plot [smooth] coordinates { (7,0) (7.5,.45) (8,0)};\n&$\\r{\\pi}$\\\\\n\\draw plot [smooth] coordinates {(1,0) (3.5,1.5) (6,0)};\n\\draw plot [smooth] coordinates {(6,0) (6.5,.5) (7,0)};\n\\draw plot [smooth] coordinates {(7,0) (7.5,.5) (8,0)};\n\\draw plot [smooth] coordinates {(8,0) (8.5,.5) (9,0)};\n\\draw plot [smooth] coordinates {(9,0) (9.5,.5) (10,0)};\n\\draw plot [smooth] coordinates {(2,0) (2.5,.5) (3,0)};\n\\draw plot [smooth] coordinates {(3,0) (3.5,.5) (4,0)};\n\\draw plot [smooth] coordinates {(4,0) (4.5,.5) (5,0)};\n&$\\Phi(\\pi)$\\\\};\n\\end{tikzpicture}\n\\caption*{\\textsc{Figure.} Examples of $\\pi$, $\\r{\\pi}$, and $\\Phi(\\pi)$.}\n\\end{figure}\n\nLet us refine Equation (\\ref{eq:tokyo2}) further.  For every $\\varpi\\in\\mathop{\\C{NC}}\\nolimits_2(2n)$ and any\nadmissible coarsening $\\eta$ of $\\varpi$ we have\n\\begin{equation}\n\\label{eq:tokyo3}\n\\sum_{\\pi\\in\\mathop{\\C P}\\nolimits_2(2n)\\atop \\r{\\pi}=\\varpi,\\ \\Phi(\\pi)=\\rho}(-1)^{|\\pi|}=(-1)^{\\#\\rho}.\n\\end{equation}\nEquation (\\ref{eq:tokyo2}) follows from (\\ref{eq:tokyo3}) by multiplying by $q^{n-\\#\\eta}$\nand summing over all admissible coarsenings $\\eta$ of $\\varpi$.\n\nEquation (\\ref{eq:tokyo3}) is again equivalent to to Equation (\\ref{eq:wat-1})\n(for all apermutation groups and all $q_s$ set to $q$) as both\nsides factor as a product over blocks of $\\rho$.\n\\end{proof}\n\n\\begin{question}\nWe have proven Equation \\ref{eq:p2-nc2} with the help of an embedding $\\F S_n\\ni\\sigma\\mapsto\\overline\\sigma\\in\\mathop{\\C{NC}}\\nolimits_2(2n)$\n(or several such embeddings, one for each outer block of\\\/ $\\varpi$).  Corollary\n\\ref{C:wat-1} holds for any Coxeter group.  Is there a corresponding formula\nconcerning some generalization of pairpartitions?\n\\end{question}\n\nIn the proof of Proposition \\ref{P:tokyo} we have not assumed that $W$ was finite.\nLet us finish this section with a discussion of infinite Coxeter groups.  Recall,\nthat $-1$ does not lie in the radius of convergence on $W(t)$ if $W$ is not finite.\nNevertheless, $W(t)$ represents a rational function as follows from the following result.\n\\begin{proposition}(\\cite{MR0230728},\\cite[Prop.~26]{MR0385006})\\label{P:Serre}\nLet $(W,S)$ be an an infinite Coxeter system. Then\n\\begin{equation}\n\\label{eq:serre}\n\\frac{1}{W(t)}=\\sum_{T\\in\\C F}\\frac{(-1)^{\\#T}}{W_T(1\/t)}.\n\\end{equation}\nWhere $\\C F$ denote the family of subsets $T\\subset S$, such that\nthe group $W_T$ generated by $T$ is finite.  In particular, $W(t)$ is a series\nof a rational function (i.e. a quotient of polynomials).\n\\end{proposition}\n\nOne may ask a question what is the class of (infinite) Coxeter groups such that\n$W_T(-1)=0$ for any nonempty subset $T$ of generators.\nA~na\\\"ive argument that\n$$W(t)=W_{\\{s\\}}(t)W^{\\{s\\}}(t)=(1+t)W^{\\{s\\}}(t)$$\nshows, that the question if $W(-1)\\neq0$ is equivalent to whether\n$W^{\\{s\\}}(t)$ can have a pole at $t=-1$.\nOn the other hand note, that if $W$ is of type \\textsc{\\MakeLowercase{\\~A}}$\\mathstrut_2$, ie.~$W$\nis given by a presentation $\\langle s_i:1\\leq i\\leq3|s_i^2,(s_isj)^3:1\\leq i<j\\leq3\\rangle$\nthen, by Equation (\\ref{eq:serre}),  $W(t)=\\frac{1+t+t^2}{(1-t)^2}$ and $W(-1)=\\sfrac14$.\n\nMore generally, it is known (\\cite{MR0240238}) that in each coset of $W_T$\nthere exists the unique shortest element.\nLet $W^T$ denote the set of those shortest representatives.\nMoreover if $w=w^Tw_T$ with $w^T\\in W^T$ and $w_T\\in W_T$ then\n$|w|=|w^T|+|w_T|$. Therefore $W^T(t)W_T(t)=W(t).$\nIn particular, $W^T(t)$ represents a rational function,\nand it is legitimate to ask about the value of $W^T(-1)$.\n\nIn the case of finite Coxeter group $W$, Eng \\cite{MR1817702} observed that\n$$W^T(-1)=\\#\\left\\{w\\in W^T|ww_0w\\in W_T\\right\\},$$ where $w_0$ is the longest\nelement in $W$.\n(Eng's proof was case-by-case. Later, a  general classification-free proof of Eng's theorem\nwas given in \\cite{MR2087303}).\n\nSubsequently, Reiner \\cite{MR1923093} has shown that if $W$ is crystallographic\n(ie. the Weyl group in a compact Lie group $G$),\nthen both sides of the above equality compute the signature of the corresponding\nflag variety $G\/Q_T$, where $Q_T$ is a parabolic subgroup associated to $T$.\n\nWhat is the meaning of $W(-1)$ or $W^T(-1)$ for infinite $W$?\n\nWe do not know if it possible for $W(-1)$ to be negative.\nIf one takes $W=\\langle s_i:1\\leq i\\leq 4|s_i^2, (s_is_j)^3:1\\leq i<j\\leq4\\rangle$.\nThen, by Equation (\\ref{eq:serre}),  $W(t)=\\frac{(1+t)(1+t+t^2)}{3t^3-2t^2-2t+1}$ and\n$W^{s_i}(-1)=\\sfrac{-1}{2}$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{Sec:Intro}\nThe decay $\\mbox{$K_{S}$}\\to 3\\mbox{$\\pi^{0}$}$ violates CP invariance and its observation would be the first\nexample of CP violation in $\\mbox{$K_{S}$}$ decays. \nThe parameter $\\eta_{000}$,  the  ratio of \n\\mbox{$K_{S}$}\\ to \\mbox{$K_{L}$}\\ decay amplitudes, is defined as: \n$\\eta_{000} = A(\\mbox{$K_{S}$} \\to 3\\mbox{$\\pi^{0}$})\/A(\\mbox{$K_{L}$} \\to 3 \\mbox{$\\pi^{0}$})=\n\\epsilon + \\epsilon'_{000}$, where $\\epsilon$ indicates\nthe $\\mbox{$K_{S}$}$ CP impurity and $\\epsilon'_{000}$ \nthe contribution of a direct CP-violating term. \nSince we expect $\\epsilon'_{000}\\;\\ll\\;\\epsilon$ \n\\cite{dambrosio}, it follows that $\\eta_{000} \\sim \\epsilon$.  \nIn the Standard Model, therefore, \nBR($\\mbox{$K_{S}$} \\to 3 \\mbox{$\\pi^{0}$} ) \\sim  1.9 \\times 10^{-9}$,  to a relative accuracy better than 1\\% .\nThe observation \nof such decay remains quite a challenge. \n\n\nPrevious searches follow two alternative methods: via a fit to the interference\npattern or via a direct search. The NA48 collaboration~\\cite{NA48} has \nfit the  $\\mbox{$K_{S}$}\/\\mbox{$K_{L}$}\\to 3\\pi^0$ interference pattern at small decay \ntimes  finding \n$\\Re\\,(\\eta_{000})=-0.002\\;\\pm\\;0.011_{\\rm stat}\\;\\pm\\;0.015_{\\rm sys}$ \nand $\\Im\\,(\\eta_{000})=-0.003\\;\\pm\\;0.013_{\\rm stat}\\;\\pm\\;0.017_{\\rm sys}$, \ncorresponding to a limit on BR($\\mbox{$K_{S}$}\\to 3\\pi^0) \\leq 7.4 \\times 10^{-7}$ at 90\\% C.L. \nThe best upper limit on $BR(\\mbox{$K_{S}$}\\to 3\\mbox{$\\pi^{0}$})$ comes from the direct search \nperformed by the KLOE experiment~\\cite{kloe2008} based on\n450~pb$^{-1}$ of collision data collected during 2001-2002. \nKLOE observed 2 candidates, and quoted a  limit on $BR(\\mbox{$K_{S}$}\\to 3\\mbox{$\\pi^{0}$}) \\leq 1.2 \\times 10^{-7}$ at 90\\%\nC.L.~\\cite{KLOE_old}.\nIn this Letter, we present a twofold improvement of this search based\non a four times larger, and independent, data sample collected\nin 2004-2005 and on  improved techniques used for background rejection.\n\\section{The KLOE detector}\nThe KLOE experiment operated from 2000 to 2006 at \\dafne, the Frascati \n$\\phi$-factory. DA$\\Phi$NE~\\cite{dafne} is an $e^+e^-$ collider running at a \ncenter-of-mass energy of $\\sim 1020$~MeV, the mass of the $\\phi$ meson. \nEqual energy positron and electron beams collide at an angle of $\\pi$-25~mrad,\nproducing $\\phi$ mesons nearly at rest.\nThe detector consists of a large cylindrical Drift Chamber (DC)~\\cite{DCH},\nsurrounded by a lead scintillating fiber Electromagnetic Calorimeter\n(EMC)~\\cite{EMC} both immersed in an axial 0.52~T magnetic field produced by a\nsuperconducting coil around the EMC. At the beams interaction point, IP,\nthe spherical beam pipe of 10~cm radius is made of a Beryllium-Aluminum alloy\nof 0.5 mm thickness.\nLow beta quadrupoles are located inside the detector at a distance of \nabout $\\pm$~50~cm  from the interaction region.\nThe drift chamber, 4~m in diameter and 3.3~m long, has 12582\nall stereo drift cells with tungsten sense wires and is a really light\nstructure with an average thickness less than  $0.1 $ X$_0$,\t\nhaving the chamber shell made of carbon fiber-epoxy composite\nwith an internal wall of $\\sim 1$ mm thickness, and filled with\na  gas mixture of 90\\% helium, 10\\% isobutane, \nto minimize $K_S$ regeneration and photon conversion. \nThe spatial resolutions are $\\sigma_{xy} \\sim 150\\ \\mu$m and \n$\\sigma_z \\sim$~2 mm.\nThe momentum resolution is $\\sigma(p_{\\perp})\/p_{\\perp}\\approx 0.4\\%$.\nThe calorimeter covers 98\\% of the solid angle and is\ncomposed by a barrel  and two endcaps, for a total of 88 modules.\nEach module is read out at both ends by photomultipliers for a total\nof 2440 cells arranged in five layers. \nThe energy deposits are obtained from the signal amplitude, while the\narrival times and particles impact points are obtained from the spatial\ncoordinates of the fired cell and the time differences. \nCells close in time and space are grouped into energy clusters. \nThe cluster energy $E$ is calculated as the sum of the cell energies,\nwhile the cluster time $T$ and position $\\vec{R}$ are energy weighted averages. \nEnergy and time resolutions are parametrized as $\\sigma_E\/E = 5.7\\%\/\\sqrt{E\\ {\\rm(GeV)}}$ \nand  $\\sigma_t = 57\\ {\\rm ps}\/\\sqrt{E\\ {\\rm(GeV)}} \\oplus100\\ {\\rm ps}$, \nrespectively.\nThe trigger \\cite{TRG} uses both calorimeter and chamber information.\nIn this analysis events are selected with the calorimeter trigger,\nrequiring two energy deposits with $E>50$ MeV for the barrel and $E>150$\nMeV for the endcaps. \nData are then analyzed by an event classification filter \\cite{NIMOffline},\nwhich selects and streams various categories of events in different\noutput files.\n\nIn this Letter, we refer only to data collected during 2004-2005 for an integrated luminosity\n$\\mathcal{L} = 1.7~\\mathrm{fb}^{-1}$ with the most stable running conditions and the\nbest peak luminosity. \nA total of 5.1 billion $\\phi$ mesons were produced, yielding  1.7~$ \\times~10^{9}$ \\mbox{$K_{S}$}\\mbox{$K_{L}$}\\ pairs. \nAssuming BR($\\mbox{$K_{S}$} \\to 3 \\mbox{$\\pi^{0}$}$) $\\sim  1.9 \\times 10^{-9}$ about 3 signal \nevents are expected to have been produced.\n\\section{\\boldmath Event selection}\n\\label{Sec:DataSample}\nAt \\dafne\\, the mean decay length of $K_L$, $\\lambda_L$, is equal to\n$~\\sim 340$ cm  and about\n50\\% of $K_L$'s reach the calorimeter before \ndecaying. A very clean $K_S$ tag is provided by the $K_L$ interaction in the calorimeter \n($\\mbox{$K_{L}$}$-crash), which is identified by a cluster with polar angle\n$40^\\circ<\\theta_{cr}<140^\\circ$, not associated\nto any track, with energy $E_{cr}>100$~MeV and with \na time corresponding to a $K_L$ velocity in the $\\phi$ rest frame $\\beta^*$\nin the range [0.17,0.28].\nThe average value of the $e^+e^-$ center of mass \nenergy $W$ is obtained with a precision of 20 keV  \nfor each 200 nb$^{-1}$  running period\nusing large angle Bhabha scattering events~\\cite{kloe2008}. The value of $W$ and \nthe $K_L$-crash cluster position allows us to obtain, \nfor each event, the direction of the $K_S$ with an \nangular resolution of 1$^{\\circ}$ and a momentum \nresolution of about 2 MeV.\n\nBecause of its short decay length, $\\lambda_S \\sim 0.6$ cm, the displacement\nof the $K_S$ from the $\\phi$ decay position \nis negligible. We therefore identify as photons from $\\mbox{$K_{S}$}$ decay,\nneutral particles that travel with $\\beta=1$ \nfrom the interaction point to the EMC (``prompt photons\").\nIn order to retain a large control sample for the \nbackground while preserving high efficiency for the \nsignal, we keep all photons satisfying $E_\\gamma>$ 7 ~MeV and\n$|\\cos \\theta|<$ 0.915.\nEach cluster is required to satisfy the condition \n$|t_{\\gamma}-R_{\\gamma}\/c|<{\\rm min}(3.5\\sigma_t, 2\\ {\\rm ns})$, \nwhere $t_{\\gamma}$ is the photon flight time and $R$ the path \nlength; $\\sigma_t$ also includes a contribution from \nthe finite bunch length (2--3 cm), which introduces \na dispersion in the collision time. \nThe photon detection efficiency of the calorimeter amounts to about 90\\% for $E_\\gamma$ = 20 MeV, and reaches \n100\\% above 70 MeV. After tagging the signal sample is selected requiring 6 prompt photons.\nFor normalization we use the $K_S \\to 2\\pi^0$ decay which is selected requiring 4 prompt photons.\\\\\nFor both channels the expected background as well as the detector acceptance and the analysis efficiency\nare estimated using the\nMonte Carlo simulation of the experiment~\\cite{NIMOffline}.\nThe simulation incorporates a detailed geometry and material composition of the KLOE apparatus\nand most of the data taking conditions of the experiment e.g. DA$\\Phi$NE background rates,\nposition of the interaction point and beam parameters. All the processes contributing to\nthe background were simulated with statistics twice larger than\nthe data sample. Moreover, for the acceptance and the analysis efficiency evaluation a dedicated\n$K_{S}\\to 3\\pi^{0}$ signal simulation was performed,  based on a branching\nratio equal to the best known upper limit~\\cite{KLOE_old} increased by a factor of 30 (about 5000 events).\n\\subsection{The six-photon sample}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{Ecr_plbfinal.eps}\n\\includegraphics[width=0.49\\textwidth]{beta_plbfinal.eps}\n\\end{center}\n\\caption{Distributions of the $K_L$ energy deposit in the EMC ($E_{cr}$) and velocity\nin the $\\phi$ center of mass frame ($\\beta^*$) for all events in the six-photon\nsample. Black points represent data, while the MC background simulation is shown\nas red histogram. The same distributions for events rejected\nby the track veto are shown by the black triangles (data) and green filled histograms \n(MC simulation).}\n\\label{fig1}\n\\end{figure}\nThe selection of the $K_{S}\\to 3\\pi^{0}$ decay is performed by asking for a\n$K_L$-crash and by searching six prompt photons from the decay of pions.\nAfter these requirements we count 76689 events. For these events we perform\nfurther discriminant analysis to increase the signal to background ratio.\\\\\nThe first analysis step aims to reject fake $K_S$ tags (about 2.5\\% of the total background).\nThe distributions of $E_{cr}$ and $\\beta^*$ for the selected data sample\nand background simulations are shown in Fig.~\\ref{fig1}. In the $\\beta^*$ distribution,\nthe peak around 0.215 corresponds to genuine $K_L$ interaction in the calorimeter,\nwhile the flat distribution mainly originates from\n$\\phi \\to K_{S}K_{L} \\to (K_S \\to \\pi^+\\pi^-, K_L \\to 3\\pi^0)$ background events.\nIn this case one of the low momentum charged pions spirals in the forward direction and interacts\nin the low-$\\beta$ \nquadrupoles. This interaction produces neutral particles which simulate the signal of $K_L$\ninteraction in the calorimeter (fake $K_L$-crash), while the $K_L$ meson decays close\nenough to the interaction point to produce six prompt photons.\nTo suppress fake $K_L$-crash we first reject events having charged particles produced close to\nthe interaction region (track veto).\nThe distributions of the kinematical variables for the vetoed\nbackground events are shown in Fig.~\\ref{fig1}.\nTaking advantage of the differences in the $\\beta^*$ and $E_{cr}$ distributions between\nthe tagged $K_S$ events and the fake $K_L$-crash, we have tightened\nthe cuts on these variables: $E_{cr} > 150~\\mathrm{MeV}$ and $0.20 <\\beta^* < 0.225$ ($K_L$-crash hard).\nThis improves by a factor 12 the rejection of this background with respect to the previous\nanalysis~\\cite{KLOE_old}.\\\\\nThe second source of background originates from wrongly reconstructed $K_S\\to 2\\pi^0$ decays.\nThe four photons from this decay can be reconstructed as six due to fragmentation of the electromagnetic\nshowers (splitting). These events are characterized by one or two low-energy clusters reconstructed very close to\nthe position of the genuine photon interaction in the calorimeter and constitute about 67.5\\% of the background.\nAdditional clusters come from accidental time coincidence between $\\phi$ decay\nand machine background photons from DA$\\Phi$NE ($\\sim$~30\\% of the background).\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{chi2fit_plb2.eps}\n\\end{center}\n\\caption{\nDistribution of $\\chi^2_{fit}$ for the tagged sixphoton sample for data\n(black points), background simulation (solid histogram), and simulated $K_S \\to 3\\pi^0$ signal\n(dashed histogram).}\n\\label{fig:chi2}\n\\end{figure}\nAfter tagging with the $K_L$-crash hard algorithm and applying the track veto we remain\nwith a  sample of about 50000 six-photon events.\nA kinematic fit with 11 constraints has been performed imposing\nenergy and momentum conservation, the kaon mass and the velocity of the six photons in the final state.\nThe $\\chi^2$ distribution of the fit for data and background simulation, $\\chi^2_{fit}$, is shown in\nFig.~\\ref{fig:chi2} together with the expected distribution for signal events.\nCutting on $\\chi^2_{fit}$ reduces by about 30\\% the remaining background while keeping the signal efficiency\nat 70\\% level.\n\\\\\nIn order to improve rejection of events with split and accidental clusters, we have exploited\nthe correlation between two $\\chi^2$-like variables named $\\zeta_{2\\pi}$ and $\\zeta_{3\\pi}$. \n$\\zeta_{2\\pi}$ is calculated by an algorithm selecting the best four out of six clusters satisfying\nthe kinematic constraints of the two-body decay in the $K_S \\to 2\\pi^0 \\to 4\\gamma$\nhypothesis:\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{chi2pi3data_plb.eps}\n\\includegraphics[width=0.49\\textwidth]{chi2pi3sig_plb.eps}\n\\end{center}\n\\caption{Distributions of events in the $\\zeta_{3\\pi}$-$\\zeta_{2\\pi}$ plane,\nfor six-photon sample tagged by $K_L$-crash for data (left),\nand for the simulated $K_S\\to 3\\pi^0$ decays (right).\nThe boundaries of the background control regions B1, B2, B3, B4, B5 and\nthe signal region S are as specified in the text.}\n\\label{fig3}\n\\end{figure}\n\\begin{align}\n\\zeta_{2\\pi} &=~\\frac{(m_{1 \\gamma\\gamma} - m_{\\pi^0})^2}{\\sigma^{2}_{2\\pi}}\n+ \\frac{(m_{2 \\gamma\\gamma} - m_{\\pi^0})^2}{\\sigma^{2}_{2\\pi}}\n+ \\frac{(\\theta_{\\pi\\pi} - \\pi)^2}{\\sigma^2_{\\theta_{\\pi\\pi}}}\n\\nonumber\n+ \\frac{\\biggl(E_{K_{S}} - \\displaystyle\\sum_{i=1}^4 E_{\\gamma_{i}}\\biggr)^2}{\\sigma^2_{E_{K_{S}}}}\\\\\n&+ \\frac{\\biggl(p_{K_{S}}^x - \\displaystyle\\sum_{i=1}^4 p_{\\gamma_{i}}^x\\biggr)^2}{\\sigma^2_{p_x}}\n+ \\frac{\\biggl(p_{K_{S}}^y - \\displaystyle\\sum_{i=1}^4 p_{\\gamma_{i}}^y\\biggr)^2}{\\sigma^2_{p_y}}\n+ \\frac{\\biggl(p_{K_{S}}^z - \\displaystyle\\sum_{i=1}^4 p_{\\gamma_{i}}^z\\biggr)^2}{\\sigma^2_{p_z}}~,\n\\label{chi2_2pi_def}\n\\end{align}\nwhere $m_{1 \\gamma \\gamma}$ and $m_{2 \\gamma \\gamma}$ are the reconstructed\n$\\gamma \\gamma$ masses for a given cluster pairing, and $\\theta_{\\pi\\pi}$ denotes the opening angle\nof the reconstructed pion directions in the $K_S$ center of mass frame. $E_{K_S}$ and $p_{K_S}$\nstand for the $K_S$ energy and momentum vector determined from the reconstructed four-momentum\nof $K_L$, while $E_{\\gamma_i}$ and $p_{\\gamma_i}$ are energies and momenta of four out\nof six reconstructed photons.\nThe minimization of $\\zeta_{2\\pi}$ gives the best two photon pairs fulfilling the\n$K_S \\to 2\\pi^0 \\to 4\\gamma$ hypothesis.\nThe resolutions used in Eq.~\\ref{chi2_2pi_def} were estimated independently on data\nand MC simulation using a $K_S \\to 2\\pi^0 \\to 4\\gamma$ control sample.\\\\\nThe second $\\chi^2$-like variable, $\\zeta_{3\\pi}$, instead verifies the signal hypothesis $K_S \\to 3\\pi^0$ by looking\nat the reconstructed masses of the three pions. For each pair of clusters\nwe evaluate $\\zeta_{3\\pi}$ as:\n\\begin{equation}\n\\zeta_{3\\pi}~=~\\frac{(m_{1 \\gamma\\gamma} - m_{\\pi^0})^2}{\\sigma^{2}_{3\\pi}}\n+ \\frac{(m_{2 \\gamma\\gamma} - m_{\\pi^0})^2}{\\sigma^{2}_{3\\pi}}\n+ \\frac{(m_{3 \\gamma\\gamma} - m_{\\pi^0})^2}{\\sigma^{2}_{3\\pi}}~.\n\\label{chi2_3pi_def}\n\\end{equation}\nAs the best combination of cluster pairs, we take the configuration minimizing $\\zeta_{3\\pi}$.\nThe resolution on the $\\gamma\\gamma$ invariant mass in the $3\\pi^0$ hypothesis, $\\sigma_{3\\pi}$,\nwas estimated applying the algorithm to the simulated $K_S \\to 3\\pi^0$ events.\\\\\nThe distributions in the $\\zeta_{3\\pi}$-$\\zeta_{2\\pi}$ plane for the data and\n$K_S \\to 3\\pi^0$ simulated signal are shown in Fig.~\\ref{fig3}. Signal events\nare characterized by small values of $\\zeta_{3\\pi}$ and relatively high\n$\\zeta_{2\\pi}$. To compare data and Monte Carlo simulations we have subdivided\nthe $\\zeta_{3\\pi}$-$\\zeta_{2\\pi}$\nplane into six regions B1, B2, B3, B4, B5, and S as indicated in the left panel of Fig.\\ref{fig3}.\nRegion S, with the largest signal-to-background ratio, is the signal box,\nwhile B1--B5 are control regions used to check the reliability of the simulation\nand optimize our description of the experimental data.\\\\\nSimulation does not reproduce accurately the absolute number of events belonging\nto different background categories. However, their kinematical properties\nare reproduced quite well. To determine the background composition,\nand improve the description of experimental data, we have performed a binned\nlikelihood fit of a linear combination \nof simulated $\\zeta_{3\\pi}$-$\\zeta_{2\\pi}$ distributions to the same data distribution\nfor all background categories.\nThe quality of the fit was controlled by comparing inclusive distributions\nof discriminating variables between data and simulation. Examples are presented\nin Fig.~\\ref{fig4}.\\\\\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\n\\textbf{} & \\textbf{SBOX} & \\textbf{B1} & \\textbf{B2} & \\textbf{B3} & \\textbf{B4} & \\textbf{B5}\\\\\n\\hline\n\\textbf{DATA} & 220 $\\pm$ 15 & 5 $\\pm$ 3 & 15179 $\\pm$ 123 & 26491 $\\pm$ 163 & 6931 $\\pm$ 83 & 137 $\\pm$ 12\\\\\n\\hline\n\\textbf{MC} &  239 $\\pm$ 11 & 4 $\\pm$ 3 & 14905 $\\pm$ 116 & 26964 $\\pm$ 169 & 6797 $\\pm$ 76 & 100 $\\pm$ 7\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\n\\label{tab:box1}\nNumber of events populating control regions in the $\\zeta_{3\\pi}$-$\\zeta_{2\\pi}$ plane defined in\nFig.~\\ref{fig3} after tight requirements on $K_L$-crash and track veto.}\n\\end{table}\nTable~\\ref{tab:box1} shows the comparison of observed number of events with the expectations in each\ncontrol region of the $\\zeta_{3\\pi}$-$\\zeta_{2\\pi}$ plane. The agreement is better\nthan 1.5 $\\sigma$ in all regions except region B5 (2.8 $\\sigma$).\\\\\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{chi23pi_plb.eps}\n\\includegraphics[width=0.49\\textwidth]{chi22pi_plb.eps}\n\\end{center}\n\\caption{Inclusive distributions of the $\\zeta_{3\\pi}$\nand $\\zeta_{2\\pi}$ discriminating variables for six-photon events: data\n(black points), background simulations (red curves). The dashed histograms\nrepresents simulated $K_S \\to 3\\pi^0$ events.\n}\n\\label{fig4}\n\\end{figure}\nTo further improve the $K_S \\to 2\\pi^0$ background rejection we cut on the $\\Delta$ variable defined as:\n\\begin{equation}\n\\Delta~=~(m_{\\phi}\/2 - \\sum E_{\\gamma_{i}})\/\\sigma_{E}~,\n\\label{eqdE}\n\\end{equation}\nwhere $\\sum E_{\\gamma_{i}}$ is the sum of energies of the four prompt photons selected by\nthe $\\zeta_{2\\pi}$ algorithm and $\\sigma_E$ stands for the 4$\\gamma$ energy resolution estimated using the\n$K_S \\to 2\\pi^0 \\to 4\\gamma$ control sample. For $K_S \\to 2\\pi^0$ decays with two additional background\nclusters, we expect $\\Delta \\sim$~0, while for $K_S \\to 3\\pi^0$ events $\\Delta~\\sim~m_{\\pi^0}\/\\sigma_E$.\nTo further reject surviving $K_S \\to 2\\pi^0$ events with split clusters,\nwe cut on the minimal distance between centroids of\nreconstructed clusters, $R_{min}$, considering that the distance between split clusters\nis on average smaller than the distance between clusters originating from $\\gamma$'s of\n$K_S \\to 3\\pi^0$ decay.\nDistributions of these two discriminant variables are presented\nin Fig.~\\ref{fig5}.\\\\\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{deks_plb.eps}\n\\includegraphics[width=0.49\\textwidth]{rmin_plb.eps}\n\\end{center}\n\\caption{Distributions of  $\\Delta$\nand $R_{min}$ discriminating variables for six-photon events:\ndata (black points), background simulations (red curves).\nThe dashed histograms represents simulated $K_S \\to 3\\pi^0$ events. \n}\n\\label{fig5}\n\\end{figure}\nBefore opening the signal box, the cuts on the discriminant variables have been refined minimizing\n$f_{cut}(\\chi^2_{fit},\\zeta_{2\\pi},\\zeta_{3\\pi},\\Delta,R_{min}) = N_{up}\/\\epsilon_{3\\pi}$, where\n$\\epsilon_{3\\pi}$ stands for the signal efficiency and\n$N_{up}$ is the mean upper limit (at 90\\% CL) on the expected number of signal events calculated\non the basis of the expected number of background events\n$B_{exp} = B_{exp}(\\chi^2_{fit},\\zeta_{2\\pi},\\zeta_{3\\pi},\\Delta,R_{min})$\nfrom simulation~\\cite{lal92}. The outcome of the optimizing procedure is $\\chi^2_{fit} < 57.2$, \n$\\Delta > 1.88$ and $R_{min} > 65$~cm. The signal box is defined as:\n$4 < \\zeta_{2\\pi} < 84.9$ and $\\zeta_{3\\pi} < 5.2$.\nAt each stage of the analysis we checked that the simulation describes the data within\nstatistical uncertainty.\nDistributions of $\\chi^2_{fit}$, $\\Delta$ and $R_{min}$ variables are presented\nin Fig.~\\ref{fig6} and Fig.~\\ref{fig7} for events in the signal box.\nIn the right panel of Fig.~\\ref{fig7} we present also the $R_{min}$ distribution just before\nthe last cut $R_{min}>65$~cm.\nAccording to the Monte Carlo simulation, these survived events are all $K_S \\to 2\\pi^0$\ndecays with two split clusters (95$\\%$), or one split and one accidental cluster (5$\\%$).\nA total efficiency of $\\epsilon_{3\\pi} = 0.233 \\pm 0.012_{stat}$ has been estimated.  \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{chi2fit_plb1.eps}\n\\includegraphics[width=0.49\\textwidth]{deks_plb2.eps}\n\\end{center}\n\\caption{Distributions of $\\chi^2_{fit}$ for six-photon events in the signal box (left)\nand $\\Delta$ for six-photon events in the signal box applying the $\\chi^2_{fit} <$~57.2 cut (right).\nBlack points are data, background simulation is the red histogram.\nThe dashed histogram represents simulated $K_S \\to 3\\pi^0$ events.}\n\\label{fig6}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{rmin_plb1.eps}\n\\includegraphics[width=0.49\\textwidth]{rmin_plb3.eps}\n\\end{center}\n\\caption{Distributions of $R_{min}$ for six-photon events in the signal box applying\nthe $\\chi^2_{fit} <$~57.2 cut (left), and applying $\\chi^2_{fit} <$~57.2\nand $\\Delta > 1.88$ cuts (right).\nBlack points are data, background simulation is the red histogram.\nThe dashed histogram represents simulated $K_S \\to 3\\pi^0$ events.}\n\\label{fig7}\n\\end{figure}\nAt the end of the analysis we find  zero candidates in data and in the\nsimulated background sample. To assign an error to the Monte Carlo estimate of the background, $N_b$,\nwe have fit the simulated $R_{min}$ distribution of Fig.~7. (right) with a gaussian and a log-gaussian.\nIntegrating the events above the cut we estimated $N_b = 0.04 ^{+ 0.15}_{-0.03}$.\n\\subsection{The normalization sample}\nThe $K_S \\to 2\\pi^0$ normalization sample is selected requiring four prompt photons.\nThe Monte Carlo simulation shows an amount of background of about 0.1$\\%$ of the total.\nThese events are essentially $\\phi \\to K^+K^-$ decays.\nAfter the $K_L$-crash hard tagging\nwe find $N_{2\\pi} = (7.533 \\pm 0.018) \\times 10^7$ events.\nWith the Monte Carlo simulations we have also determined the $K_S \\to 2\\pi^{0} \\to 4\\gamma$\nefficiency: $\\epsilon_{2\\pi} = 0.660 \\pm 0.002_{stat}$. \nThe final number of produced $K_S \\to 2\\pi^0$ events is:\n$ N_{norm} = N_{2\\pi}\/ \\epsilon_{2\\pi} = (1.142 \\pm 0.005) \\times 10^8$.\n\\subsection{Evaluation of systematic uncertainties}\nThe systematic uncertainties are related to the number of \nbackground events and  to the determination of\nthe acceptance and total efficiencies for the signal,\n$\\epsilon_{3\\pi}$, and normalization, $\\epsilon_{2\\pi}$,  samples.\n\nFor the tagged six-photon sample, we have investigated the uncertainties related to\nthe observed background at the end of the analysis.\nA difference of  $\\sim$ 2.4\\%  in the EMC energy scale and resolution has been observed between\ndata and MC simulation and has been  studied using a control sample of $K_S \\to 2\\pi^0$\nevents. To evaluate the related systematic uncertainty on the background, we have\nrepeated the upper limit evaluation with several values of the energy\nscale correction in the range of 2.2\\%-2.6\\%. Similarly, the analysis has been repeated\nmodifying the resolution used in the definition\nof $\\zeta_{2\\pi}$ and $\\zeta_{3\\pi}$.\nMoreover, we have varied of 1 $\\sigma$ the resolution used in the $\\Delta$ variable calculation\nand removed a data--MC shift correction on $R_{min}$. These variations correspond to a cut change\nof 5\\% and 6\\%, respectively. Similarly, we have removed the data--MC scale correction for $E_{cr}$\nand the additional gaussian smearing in the MC $\\beta^*$ distribution, both\ncorresponding to a 5\\% variation of the cuts.\nThe full analysis was repeated in total twenty times applying each\ntime one of the changes mentioned above. For all of these\nchecks, we have observed no variation in the number of simulated background.\n\nFor the acceptance of  both the signal and normalization\nsamples, we have evaluated the systematic uncertainty on the photon counting\nby comparing data and simulation splitting, accidental probabilities and cluster\nreconstruction efficiency.\nTo determine the probabilities of one, $P_{A1}$, or two, $P_{A2}$,  accidental\nclusters in the event we have used out of time clusters originated from earlier bunch\ncrossing. To estimate the probability of generating one, $P_{S1}$, or more fragments,\n$P_{S2}$, per cluster, we have fit the photon multiplicities observed in data using\nthe experimental values of $P_{A1}$ and $P_{A2}$, and the photon\nmultiplicities obtained by the simulation~\\cite{silarskiPHD,note201}.\nResults of these fits are reported in Tab.~\\ref{tabprob}.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n {}& $\\boldsymbol{P_{A1}}$ [\\%] & $\\boldsymbol{P_{A2}}$ [\\%] & $\\boldsymbol{P_{S1}}$ [\\%] & $\\boldsymbol{P_{S2}}$ [\\%]\\\\\n\\hline\n\\textbf{DATA} & 0.378 $\\pm$ 0.004 & 0.025 $\\pm$ 0.001 & 0.30 $\\pm$ 0.01 & 0.0103 $\\pm$ 0.0001\\\\\n\\hline\n\\textbf{MC}& 0.492 $\\pm$ 0.004& 0.027 $\\pm$ 0.001 & 0.31 $\\pm$ 0.01 & 0.0156 $\\pm$ 0.0002\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\n\\label{tabprob}\nThe probabilities to find one ( $P_{A1}$ ) or two ( $P_{A2}$ ) accidental clusters\nand to reconstruct one ( $P_{S1}$ ) or more ( $P_{S2}$ ) split clusters\nestimated using out of time clusters and fit to the photon multiplicities,\nas described in the text.}\n\\end{table}\nThe photon reconstruction efficiency, for both data and MC,\nwas evaluated using a control sample of $\\phi \\to \\pi^+\\pi^-\\pi^0$ events. The momentum\nof one of the photons is estimated from tracking information and position\nof the other cluster. The candidate photon is then searched for within a search cone.\nThe systematic error related to the cluster efficiency has been estimated by\nremoving the data\/MC efficiency correction. The total systematic\nuncertainty on  the acceptance for both\nmeasured samples are listed in Tab.~\\ref{tabsys}.\nAnother source of systematic uncertainties originates from the offline filter FILFO~\\cite{memo288}\nused, during data reconstruction, to reject cosmic rays and machine\nbackground events before starting the track reconstruction. The FILFO efficiency, for both\nnormalization and signal samples, has been\nestimated using the simulation and is very close to 100\\%~\\cite{silarskiPHD}. \nWe have conservatively assigned as systematic uncertainty in data  half of\nthe difference between the MC evaluated efficiency and 100\\%.\nWe consider completely negligible the influence of trigger efficiency for\nboth samples, since in~\\cite{KLOE_old} it was about 99.5\\%\nand the $K_L$-crash hard tagging requires a\nlarger energy release in the calorimeter, which translates in a \nlarger trigger efficiency.\n\nThe observed difference in the EMC energy scale and resolution between data and simulation\nenters also in the $\\epsilon_{3\\pi}$ evaluation. The effects have been estimated as\n$\\Delta \\epsilon_{3\\pi}\/\\epsilon_{3\\pi} = 1.0 $\\% from the \tenergy scale,\nand $\\Delta \\epsilon_{3\\pi}\/\\epsilon_{3\\pi} = 1.1 $\\% from the resolution. \nThe effect of the cut on $\\chi^2_{fit}$ has been tested constructing the ratio between\nthe cumulative distributions for experimental data and simulation\nwhich leads to a systematics of $\\Delta \\epsilon_{3\\pi}\/\\epsilon_{3\\pi} = 1.46 $\\%.\nFinally, we have investigated the systematic effect related to\nthe $R_{min}$ cut by varying its value by 6\\%, and estimated its contribution\nto be $\\Delta \\epsilon_{3\\pi}\/\\epsilon_{3\\pi} =0.9$\\%.\n \nAll the contributions to the systematic uncertainty are summarized\nin Tab.~\\ref{tabsys}, with the total systematic uncertainty evaluated adding\nall effects in quadrature.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n\\textbf{Source} & $\\boldsymbol{\\Delta \\epsilon_{2\\pi}\/\\epsilon_{2\\pi}~[\\%]}$ & $\\boldsymbol{\\Delta \\epsilon_{3\\pi}\/\\epsilon_{3\\pi}~[\\%]}$\\\\\n\\hline\nAcceptance & 1.60 & 0.21\\\\\n\\hline\nOffline filter &0.46 & 0.30\\\\\n\\hline\nCalorimeter energy scale & -- & 1.00\\\\\n\\hline\nCalorimeter energy resolution & -- & 1.10\\\\\n\\hline\n$\\chi^2_{fit}$ cut & -- & 1.46\\\\\n\\hline\n$R_{min}$ cut & -- & 0.90\\\\\n\\hline\n\\bf{TOTAL} & \\textbf{1.65} & \\textbf{2.30}\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\n\\label{tabsys}\nSummary table of the systematic uncertainties on the total\nefficiencies for the signal, $\\epsilon_{3\\pi}$,\nand  normalization samples, $\\epsilon_{2\\pi}$.}\n\\end{table}\n\\section{Results}\nNo events were observed on data in the signal region. \nEqually, no background events are found in the MC simulation based on twice the data statistics.\nIn the conservative assumption of no background, we estimate an upper\nlimit on the expected number of signal events UL(Nev$(\\mbox{$K_{S}$} \\to  3\n\\mbox{$\\pi^{0}$})$) = 2.3  at 90\\% ~C.L.,   with a signal efficiency of\n$\\epsilon_{3\\pi} = 0.233 \\pm 0.012_{stat} \\pm 0.006_{sys}$.\nIn the same tagged sample we count $N_{norm} = (1.142 \\pm 0.005) \\times 10^8$\n$\\mbox{$K_{S}$} \\to 2 \\mbox{$\\pi^{0}$}$ events.\\\\\nSystematic uncertainties on background  determination, as well as on\nthe efficiency evaluation for the signal and normalization samples,\nare  negligible in the calculation of the limit.\n\nUsing the value BR($\\mbox{$K_{S}$} \\to 2 \\mbox{$\\pi^{0}$})= 0.3069 \\pm 0.0005$~\\cite{pdg2012} we obtain:\n\\begin{equation}\nBR(\\mbox{$K_{S}$}\\to 3 \\mbox{$\\pi^{0}$}) \\leq 2.6 \\times 10^{-8}\\; \\;\\;\\; {\\rm at}\\;\\; \\; 90\\%\\;\\;\\; {\\rm C.L.}\n\\end{equation}\nwhich represents the best limit on this decay, improving \nby a factor of $\\sim$~5 previous result \\cite{KLOE_old}.\\\\\nThis result can be translated into a limit on $|\\eta_{000}|$: \n\\begin{equation}\n|\\eta_{000}| = \\left|\\frac{A(K_S \\to 3\\pi^0)}{A(K_L \\to 3\\pi^0)}\\right| = \\sqrt{\\frac{\\tau_L}{\\tau_S}\n\\frac{BR(K_S \\to 3\\pi^0)}{BR(K_L \\to 3\\pi^0)}} \\leq 0.0088\\;\\;\\; {\\rm at}\\;\\; \\; 90\\%\\;\\;\\; {\\rm C.L.}\n\\end{equation}\n\nThis describes a circle of radius 0.0088 centered at zero in the $\\Re({\\eta_{000}})$, $\\Im{(\\eta_{000})}$\nplane and  represents a limit two times smaller than previous result~\\cite{KLOE_old}.\n\\section*{Acknowledgments}\nWe warmly thank our former KLOE colleagues for the access to the data\ncollected during \nthe KLOE data taking campaign.\nWe thank the DA$\\Phi$NE team for their efforts in maintaining low\nbackground running conditions and their \ncollaboration during all data taking. We want to thank our technical staff: \nG.F. Fortugno and F. Sborzacchi for their dedication in ensuring efficient operation of the KLOE computing facilities; \nM. Anelli for his continuous attention to the gas system and detector safety; \nA. Balla, M. Gatta, G. Corradi and G. Papalino for electronics maintenance; \nM. Santoni, G. Paoluzzi and R. Rosellini for general detector support; \nC. Piscitelli for his help during major maintenance periods.\\\\\nWe acknowledge the support of the European Community-Research\nInfrastructure Integrating Activity `Study of Strongly Interacting Matter'\n(acronym HadronPhysics2, Grant Agreement No. 227431) under the Seventh Framework\nProgramme of EU.\nThis work was supported also in part by the EU Integrated Infrastructure\nInitiative Hadron Physics \nProject under contract number RII3-CT-2004-506078; by the European\nCommission under the 7th \nFramework Programme through the `Research Infrastructures' action of\nthe `Capacities' Programme, Call: \nFP7-INFRASTRUCTURES-2008-1, Grant Agreement No. 283286; by the Polish\nNational Science Centre through the \nGrants Nos. 0469\/B\/H03\/2009\/37, 0309\/B\/H03\/2011\/40, \nDEC-2011\/03\/N\/ST2\/02641, \\\\2011\/01\/D\/ST2\/00748, 2011\/03\/N\/ST2\/02652,\n2011\/03\/N\/ST2\/02641 and by the Foundation for Polish Science through the MPD programme \nand the project HOMING PLUS BIS\/2011-4\/3.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n  Many electronic-structure methods use basis sets made of states\nthat move or change with atomic positions. \n  A very popular subset of these methods (most quantum-chemical\\cite{Gaussian,\nTurbomol,Gamess,ADF} and a significant fraction of solid-state \nmethods\\cite{Siesta,CP2K,Crystal,OpenMX,FHI-AIMS}) use atomic-like basis \nfunctions, \ncomposed\nof the product of a radial function and a spherical harmonic, normally \ncentered around atoms.\n  In other cases, the localised basis is obtained dynamically, using a finer\nauxiliary basis.\\cite{Onetep,Conquest,BigDFT}\n  The detail of the kind of functions is not important in this work,\nwhat matters here is that such a basis is generally not orthonormal,\nit spans a subspace of the Hilbert space (a finite basis is always used), \nand both the basis and the subspace change with the evolution of a set \nof external parameters such as the atomic positions.\n\n  Non-orthogonal basis sets have been used since the early times of \nquantum mechanics, especially in the context of quantum chemistry.\\cite{Lowdin1950} \n  The matrix representation of Schr\\\"odinger's equation (using Dirac notation)\n$H | \\psi \\rangle = E | \\psi \\rangle$ in a basis $\\{ | e_{\\mu} \\rangle , \\mu = 1 \\dots \\cal{N} \\} $\ngives\n\\begin{equation*}  \n\\sum_{\\nu} H_{\\mu \\nu} C_{\\nu} = E \\sum_{\\nu} S_{\\mu\\nu} C_{\\nu} \\; ,\n\\end{equation*}\nwhere \n\\begin{equation}\n\\label{coeff-expansion}\n| \\psi \\rangle = \\sum_{\\mu} | e_{\\mu} \\rangle \\, C_{\\mu} \\,  ,\n\\end{equation}\n$H_{\\mu \\nu}= \\langle e_{\\mu} | H | e_{\\nu} \\rangle$ and \n$S_{\\mu \\nu}= \\langle e_{\\mu} | e_{ \\nu} \\rangle$, the latter being the\noverlap matrix.\n \n  Similarly, there are electronic structure methods based on \nthe integration of time-evolving quantum problems, most prominently\nbased on time-dependent density-functional \ntheory.\\cite{RungeGross,Yabana,Tsolakidis2002,Octopus,ONTDDFT,Bowler} \n  The time-dependent Kohn-Sham equation arising is \nanalogous to the time dependent Schr\\\"odinger equation \n$H | \\psi \\rangle = i \\,  \\partial_t | \\psi \\rangle$ (using \n$\\hbar = m_e = e = 1$), which, for a non-orthogonal \nbasis set, becomes\n\\begin{equation*}\n\\sum_{\\nu} H_{\\mu \\nu}  C_{\\nu} = \ni \\, \\sum_{\\nu} S_{\\mu \\nu} \\, \\partial_t C_{\\nu} \\; ,\n\\end{equation*}\nin a situation in which the basis set is fixed.\\cite{Tsolakidis2002}\n  If the basis set moves in time (e.g., related to nuclear motion)\nthe equation becomes rather\n\\begin{equation}\n\\label{old-td-eq}\n\\sum_{\\nu} (H_{\\mu \\nu} - i \\, D_{\\mu \\nu} ) C_{\\nu} = \ni \\, \\sum_{\\nu} S_{\\mu \\nu} \\, \\partial_t C_{\\nu} \\; ,\n\\end{equation}\nin which the new terms $D_{\\mu \\nu}= \\langle e_{\\mu} | \\partial_t | e_{\\nu} \\rangle\n= \\langle e_{\\mu} | \\partial_t  e_{\\nu} \\rangle$ appear related to the\nbasis set evolution,\\cite{Todorov2001,Rudiger2002,Kaxiras2015} \nalthough at sufficiently low nuclear velocities these terms can be\nneglected.\\cite{Kaxiras}\n  Similar objects to the $D_{\\mu\\nu}$ matrix in Eq.~\\ref{old-td-eq} can also \nbe found in time-dependent methods using localised molecular \norbitals.\\cite{WeitaoYang2010}\n  Extra terms related to derivatives also appear when calculating\nthe forces on atoms for geometry relaxation, ab initio molecular\ndynamics calculations, or Ehrenfest dynamics simulations.\\cite{Todorov2001,Rudiger2002} \n   There are terms arising, called Pulay forces,\\cite{Pulay1969} which again involve \nbasis vector derivatives. \n\n  The matrix representation of the quantum formalism used above has its limitations,\nhowever, and a more general formalisation was introduced\\cite{Vanderbilt1984,\nBallantine1986, Artacho1991, Head-Gordon1993} based on\ntensors, which offers a better suited and more flexible framework for non-orthogonal\nbasis sets, corresponding to a description of magnitudes in an Euclidean space with\noblique axes. \n  It allowed, for instance, the formalisation of second quantisation based on \nnon-orthogonal bases,\\cite{Artacho1991} which was then used to\nformulate many-body theories using such bases,\\cite{Head-Gordon1993,Holomorphic}\nand to formulate corrective methods such as DFT+$U$ for the non-orthogonal \ncase.\\cite{ORegan2011,Palacios2014,Jacob2015}\n  It also connected naturally with non-Hermitian representations proposed\nearlier for the better exploitation of localisation.\\cite{Weeks,Bullett,Anderson}\n\n  In this paper we use \nconcepts of differential geometry \nto extend that oblique-axis formalism to the calculation of derivatives \nwhen the basis and the Hilbert space it spans change with parameters \nsuch as atomic positions or time. \n  This offers insights into the geometric interpretation of the dynamical equations \narising with moving basis sets.\n  In particular, the affine connection defined for the changing basis allows the\nproposal of optimised propagators for the numerical integration of quantum \ntime-evolving problems. \n  It should be noted that there have been previous works on derivatives \nin the tensorial formalism for non-orthogonal bases in electronic \nstructure,\\cite{MHG-gradients} including relaxations with curvy steps.\\cite{MHG-curvy}\n  They were, however, always addressing \nderivatives of a scalar, the total energy, \na special case which allows for circumvention of the key concepts in this work, \nthe affine connection and the covariant derivative.\n  An alternative way of using differential geometry in electronic \nstructure was initially explored in \nRef.~\\onlinecite{ORegan}\nwhen calculating derivatives with respect to the basis functions themselves,\ninstead of external parameters as in this work.\n  This is beyond the scope of the present paper.\n  \n  The ideas in this paper should be useful for time-dependent \n(or parameter dependent) methods involving basis functions, \nauxiliary support functions, or any kind of states, which move\nduring simulations, including atomic-like basis orbitals, support functions  \nor generalised Wannier functions in large scale electronic-structure methods, or\neven the projectors for the core-electron description in projected-augmented-wave \n(PAW) methods.  \n  The connection is also made with the Berry formalism \nof geometric phases.\n  In Section II the general formalism is presented, while Section III shows its\napplication in several contexts.\n  Many derivations have been pushed to appendices \nwith a view to attaining\na more concise exposition of the relevant ideas in the main text, \nwhile preserving a reasonably self-contained paper.  \n\n\n\n\n\n\\section{Formulation}\n\n\\subsection{Tensorial representations}\n\\label{SectForm}\n\n  In this work we will use tensorial representations as used \nin Ref.~\\onlinecite{Artacho1991}.\n  Here are the essentials before we get into derivatives.\n  Consider a basis consisting on a set of linearly-independent, non-orthogonal\nstates,\n\\begin{equation*}\n\\{ | e_{\\mu} \\rangle, \\; \\mu = 1 \\dots \\cal{N} \\}\n\\end{equation*}\nspanning a subspace $\\Omega$ of the relevant Hilbert space $\\cal{H}$\nfor our quantum problem.\n  We will use here the tensorial notation for oblique \nangles.\\cite{Vanderbilt1984, Ballantine1986, Artacho1991, Head-Gordon1993}\n  For this, the dual basis\\cite{Algebra,Artacho1991} is defined as the set of vectors \n$\\{ | e^{\\mu} \\rangle, \\; \\mu = 1 \\dots \\cal{N} \\}$ in the same space $\\Omega$ that \nfulfil\n\\begin{equation*}\n\\langle e^{\\mu} | e_{\\nu} \\rangle = \\delta^{\\mu}_{\\phantom{e}\\nu} = \n\\langle e_{\\nu} | e^{\\mu} \\rangle = \\delta_{\\nu}^{\\phantom{e}\\mu} \\; ,\n\\end{equation*}\nwhere $\\delta^{\\mu}_{\\phantom{e}\\nu}=\\delta_{\\nu}^{\\phantom{e}\\mu}=\n\\delta^{\\mu}_{\\nu}$ is Kronecker's delta.\n  They also fulfil\\cite{Artacho1991}\n\\begin{equation*}\n\\sum_{\\mu} | e_{\\mu} \\rangle \\langle e^{\\mu} | = \\sum_{\\mu} | e^{\\mu} \\rangle \\langle \ne_{\\mu} | = P_{\\Omega} \\; ,\n\\end{equation*}\nwhere $P_{\\Omega}$ is the projector onto the $\\Omega$ Hilbert space.\n  The metric tensors are given by the overlap, \n$S_{\\mu\\nu} = \\langle e_{\\mu} | e_{\\nu} \\rangle$, and its upper-indices counterpart \n$S^{\\mu\\nu} = \\langle e^{\\mu} | e^{\\nu} \\rangle$, which is the inverse\n(in the matrix sense) of the overlap matrix. \n\n  In this paper we will mostly (always, unless explicitly stated) use the\n{\\it natural representation} as defined in Ref.~\\onlinecite{Artacho1991}.\n  A state $| \\psi \\rangle \\in \\Omega$ is represented by the contravariant \nfirst-rank tensor \n\\begin{equation*}\n\\psi^{\\mu}= \\langle e^{\\mu} | \\psi \\rangle \\; ,\n\\end{equation*}\nwhich corresponds to the coefficients $C_{\\mu}$ of the vector expansion\nin Eq.~\\ref{coeff-expansion}, since, \n\\begin{equation*}\n| \\psi \\rangle = P_{\\Omega} | \\psi \\rangle\n= \\sum_{\\mu} | e_{\\mu} \\rangle \\langle e^{\\mu} | \\psi \\rangle\n= \\sum_{\\mu} | e_{\\mu} \\rangle \\psi^{\\mu}\n= | e_{\\mu} \\rangle \\psi^{\\mu} \\; .\n\\end{equation*}\n  The last identity just reflects the fact that from now on we will use \nEinstein's convention for tensors,\\cite{Einstein1916} by which repeated \nindices imply a sum.\n  A bra $\\langle \\psi | \\in \\Omega^\\dagger$ will be represented by the \nequivalent covariant tensor,\n\\begin{equation*}\n\\psi_{\\mu} = \\langle \\psi | e_{\\mu} \\rangle \\; ,\n\\end{equation*}\ncoming from \n\\begin{equation*}\n\\langle \\psi | = \\langle \\psi | e_{\\mu} \\rangle \\langle e^{\\mu} | = \n\\psi_{\\mu} \\langle e^{\\mu} |  \\; .\n\\end{equation*}\n  Note that the representation of the bra is not the complex conjugate of\nthe representation of the ket (see Appendix~\\ref{NotationAppendix}).\n  The covariant and contravariant character of a tensor relates to the way\nthey transform under basis change (see Appendix~\\ref{BasisChangeAppendix}\nfor the definition).\n  \n  An operator acting in $\\Omega$ is represented by the second-rank tensor given by\n\\begin{equation*}\nH^{\\mu}_{\\phantom{e}\\nu} = \\langle e^{\\mu} | H | e_{\\nu} \\rangle \\; ,\n\\end{equation*}\nsince\n\\begin{equation*}\nP_{\\Omega} H P_{\\Omega} = \\left ( |e_{\\mu} \\rangle \\langle e^{\\mu} | \\right ) \\, H \\, \n\\left ( | e_{\\nu} \\rangle \\langle e^{\\nu} | \\right ) = |e_{\\mu} \\rangle \nH^{\\mu}_{\\phantom{e}\\nu} \\langle e^{\\nu} | \\; .\n\\end{equation*}\n  Schr\\\"odinger's equation $H | \\psi \\rangle = E | \\psi \\rangle$ in this representation \nthen becomes\n\\begin{equation*}\nH^{\\mu}_{\\phantom{e}\\nu} \\, \\psi^{\\nu} = E \\, \\psi^{\\mu}  \\; .\n\\end{equation*}\n\n  It should be noted at this point that the formalism described here is equally \nvalid for $| \\psi \\rangle$ being a single-particle or a many-particle state.\n  It just requires the $| e_{\\mu} \\rangle$ basis states (and their duals) to \nrepresent the same number of particles as $|\\psi \\rangle$.\n  One can also use this formalism for many-particle systems building\non single-particle non-orthogonal basis states by using \nnon-orthogonal second quantization.\\cite{Artacho1991,Head-Gordon1993}\n  In some parts below we will refer to single-particle (mean-field-like)\nsituations since they are frequently found in different contexts, such as\nKohn-Sham density-functional theory,\\cite{Kohn-Sham} but the present\nformalism is not limited to such situations.\n\n  The other representation to be considered in this work is the traditional \nquantum-chemical representation (henceforth called the {\\it matrix representation}), \nwhich uses $\\psi^{\\mu}$ and $H_{\\mu\\nu}= \\langle e_{\\mu} | H | e_{\\nu} \\rangle$ \nfor the representation of states and operators, respectively, the Schr\\\"odinger \nequation now reading\n\\begin{equation*}\nH_{\\mu\\nu} \\, \\psi^{\\nu} = E \\, S_{\\mu\\nu} \\, \\psi^{\\nu} \\; ,\n\\end{equation*}\nas detailed in Ref.~\\onlinecite{Artacho1991}.\n\n  An Hermitian operator in the natural representation would fulfil\n\\begin{equation*}\nH^{\\mu}_{\\phantom{e}\\nu} = \\left ( H_{\\nu}^{\\phantom{e}\\mu} \\right )^*\n= \\left ( S_{\\nu\\lambda} H^{\\lambda}_{\\phantom{e}\\sigma} S^{\\sigma\\mu} \\right )^* \\; ,\n\\end{equation*}\nwhere $H_{\\nu}^{\\phantom{e}\\mu} = \\langle e_{\\nu} | H | e^{\\mu}\\rangle$. \n  In the matrix representation hemiticity is reflected just by \n$H_{\\mu\\nu}= \\left ( H_{\\nu\\mu} \\right )^*$.\n  Further details on the tensorial notation used in this work are found in \nAppendix~\\ref{NotationAppendix}.\n\n  All of these magnitudes are tensors in the sense that they represent abstract \nobjects that are defined independently of the basis set.\n  Tensor components transform in a well defined fashion when\nchanging the basis set.\n  Transformations under basis change of the different tensors in this paper \nare discussed in Appendix~\\ref{BasisChangeAppendix}.\n  \n  \n  \n\n \n\\subsection{Parameter vector space}\n\\label{ParameterSpace}\n\n  In this work we provide a comprehensive formalisation of derivatives of\nthe quantities just defined with respect to any parameters that the basis\nmay depend on, including both the basis change within $\\Omega$ and\nthe evolution of $\\Omega$ itself.\n  Such parameters will normally be nuclear positions as in a molecular \ndynamics or Ehrenfest simulations, or time, as when following the\ndynamics governed by time-dependent Schr\\\"odinger equation (or the\ntime-dependent Kohn-Sham equation in time-dependent density-functional\ntheory, or analogous mean-field-like equation).\n\n  We will then consider those parameters as defining a vector space $\\Theta$\nof dimension $N$, spanned by the basis\n\\begin{equation*}\n\\{ \\mathbf{u}_i, \\; i = 1 \\dots N \\} \\; ,\n\\end{equation*}\nsuch that any vector $\\mathbf{R} \\in \\Theta$ is expanded as\n\\begin{equation*}\n{\\mathbf R} = R^i {\\mathbf u}_i \\: .\n\\end{equation*} \n  Please keep in mind that these $R^i$ variables represent any parameters\nthat a particular quantum problem may depend on, not necessarily nuclear\npositions (in the Applications section below there will be examples for \nnuclear positions, but also for time as a single parameter in a one-dimensional\n$\\Theta$).\n  We keep the tensor notation for the vectors in $\\Theta$ for convenience.\nThis allows for oblique angles in this space as well if ever wanted.\n  We will always use Greek letters as indices for the quantum (electronic)\ncomponents and Latin for the components of vectors in parameter space.\n  Our electronic basis set and space do then depend on $\\mathbf{R}$, i.e.\n$\\Omega=\\Omega(\\mathbf{R})$, $| e_{\\mu} \\rangle = | e_{\\mu} (\\mathbf{R}) \\rangle$,\nand so will the projector $P_{\\Omega}$ and all the tensors defined above. \n\n  It is important to note that, although we will exploit analogies with the \ndifferential geometry defined for curved spaces as in general relativity, \nthe situation described here may be discussed  more formally using the \nlanguage of fibre bundles,\\cite{FibreBundle} with $\\Theta$ as the base space, \nand $\\Omega(\\mathbf{R})$ as a fibre for each {\\bf R}. \n  When moving in the base space, the Hilbert space associated to each fibre\nturns within the ambient Hilbert space ${\\cal H}$, very much as the tangent space \nwould turn for a curved space.\n  Although both the base space $\\Theta$ and each fiber $\\Omega(\\mathbf{R})$ \nare each flat Euclidean spaces, the overall bundle is curved.\n\n  The parameter-dependent Hilbert space $\\Omega (\\mathbf{R})$ and its turning\nis also the basis of the Berry formalism of geometric phases in quantum \nmechanics.\\cite{Berry-orig, Berry} \n  In this case the relevant $\\Omega$ space would \nbe the one associated with the ground state, or, in a mean-field-like \nsetting (as in e.g. density-functional theory), the space spanned by\nthe occupied single-particle states (occupied space),\nalthough also larger spaces are considered, e.g. for\nmetallic systems, or disentangling bands.\\cite{Marzari1997}\n  We relate this work to the Berry formalism below.\n  Finally, the change of basis between fibres can be regarded as a gauge transformation\ngiven in principle by any $\\cal{N} \\times \\cal{N}$ invertible matrix of complex numbers,\ni.e., belonging to the general linear group GL$(\\cal{N},\\mathbb{C})$.\n  In this paper, however, we prefer to introduce the formalism in an accessible, \nself-contained manner, using basic quantum mechanics, tensors, and simple \nmanipulations therein.\n\n   \n   In what follows, we will investigate the rate of change of quantum states, \nand operators acting upon them, with respect to the evolving basis vectors as we \nnavigate the parameter vector space. \n   We will find that the components of such change due to space preserving and \nspace non-preserving basis function evolution must be separately considered, \nas follows.\n\n\n\n\n\n\n\n\\subsection{Differential geometry}\n\n\\subsubsection{Covariant derivative}\n\n  The derivative of the $\\psi^{\\mu}$ components of a quantum state $| \\psi \\rangle$\nwith respect to $R^i$ will be indicated by \n\\begin{equation*}\n\\partial_i \\psi^{\\mu} = {\\partial \\psi^{\\mu} \\over \\partial R^i} \\; .\n\\end{equation*}\n  It is easy to show (see Appendix~\\ref{BasisChangeAppendix}) that\nsuch a derivative does not transform as a tensor under basis change.\n  Using the conventional nomenclature: it is a non-tensor.\n  \n  Let us then define the covariant derivative as one that transforms \nas a tensor, which we can easily do as follows:\n\\begin{equation}\n\\label{covdevdef}\n\\text{\\dh}_i \\psi^{\\mu} \\equiv \\langle e^{\\mu} | P_{\\Omega} \\partial_i \n\\left \\{ P_{\\Omega} | \\psi \\rangle \\right \\} =\n\\langle e^{\\mu} | \\partial_i \\left \\{ P_{\\Omega} | \\psi \\rangle \\right \\} \\; .\n\\end{equation}\n  The projector directly acting on $| \\psi \\rangle$ might appear redundant\nif starting with $| \\psi \\rangle \\in \\Omega$.\n  It is not so, however, the important point being that the derivative is calculated\nfor the state being projected on the {\\it varying} $\\Omega$ space, \nand therefore, even though $| \\psi \\rangle$ and $P_{\\Omega} |\\psi \\rangle$ \nare equal at {\\bf R}, they are not necessarily equal at any nearby point, \n$\\mathbf{R} + \\mathrm{d} R^i \\mathbf{u}_i$.\n  Put another way, the covariant derivative must be applied before \nprojection onto $\\Omega$, since both $|\\psi \\rangle$ and $P_{\\Omega}$ \nmay evolve in time.\n\n  This definition gives a well-behaved tensor since\nthe defined $\\text{\\dh}_i \\psi^{\\mu}$ is the tensor representation of the vector\n$P_{\\Omega} \\partial_i \\left \\{ P_{\\Omega} | \\psi \\rangle \\right \\}  \\in \\Omega$\n(see also Appendix~\\ref{BasisChangeAppendix}).\n  The justification for this particular definition will become clear throughout \nthis section. \n  We can already point to the fact that for $| \\psi \\rangle \\in \\Omega$\nsuch that neither $|\\psi \\rangle$ nor $\\Omega$ change with {\\bf R},\nand for a basis set that does change, the proposed covariant derivative is\nzero (while the usual derivative is not), thereby indicating that it is the \nintrinsic rate of change of the state what is being measured, excluding\nthe basis set change. \n  This point will be proven more generally below.\n  It is analogous to the definition of covariant derivatives in gauge-dependent\ntheories, as the physical derivative independent of change of local gauge.\n\n  The \nrelationship between\n$\\text{\\dh}_i \\psi^{\\mu}$ and $\\partial_i \\psi^{\\mu}$ is\nobtained as follows:\n\\begin{align*}\n\\text{\\dh}_i \\psi^{\\mu} &= \\langle e^{\\mu} | \\partial_i \\left \\{ P_{\\Omega} | \\psi \\rangle \\right \\} =\n\\langle e^{\\mu} | \\partial_i \\left \\{  | e_{\\nu} \\rangle \\langle e^{\\nu} | \\psi \\rangle \\right \\} \\\\\n&= \\langle e^{\\mu} | \\partial_i \\left \\{  | e_{\\nu} \\rangle \\psi^{\\nu} \\right \\} \n= \\langle e^{\\mu} | \\partial_i | e_{\\nu} \\rangle \\psi^{\\nu} + \\partial_i \\psi^{\\mu} \\; .\n\\end{align*}\n  This gives us an alternative definition of the covariant derivative,\nexpressed in objects all defined within $\\Omega$, namely,\n\\begin{equation}\n\\label{covdevdef2}\n\\text{\\dh}_i \\psi^{\\mu} = \\partial_i \\psi^{\\mu} + D^{\\mu}_{\\phantom{i} \\nu i} \\psi^{\\nu} \\; ,\n\\end{equation}\nwhere we have used the following definition\n\\begin{equation}\n\\label{christoffel}\nD^{\\mu}_{\\phantom{i} \\nu i}  \\equiv  \\langle e^{\\mu} | \\partial_i | e_{\\nu} \\rangle\n= \\langle e^{\\mu} | \\partial_i e_{\\nu} \\rangle \\; .\n\\end{equation}\n  The $i$ index is located after the index of the state being differentiated.\n  This choice can be remembered by thinking of it as $\\partial_i  | e_{\\mu} \\rangle = \n\\partial | e_{\\mu} \\rangle \/ \\partial R^i$.\n  This second expression of the covariant derivative (Eq.~\\ref{covdevdef2})\ncan be shown to give a tensor within this \nformalism (see Appendix~\\ref{BasisChangeAppendix}).\n\n\n\n\n\n\\subsubsection{Affine connection}\n\n  The quantity defined in Eq.~\\ref{christoffel} is also a non-tensor and \nplays the role of the Christoffel symbols of the second kind in the Levi-Civita \nconnection of conventional differential geometry.\n  Keeping the nomenclature, our Christoffel symbols as defined in \nEq.~\\ref{christoffel} thus define the affine connection relevant to our problem.\n  Remember, however, that this is not differential geometry for curved\nspaces, where the tangent space at one point directly relates to the \noverall manifold, but \nrather\nfor a rotating Hilbert space $\\Omega$ within ${\\cal H}$ \nwhen moving in parameter space $\\Theta$.\n  In the former, there is one metric tensor at every point, while in the latter\nthere are still two metrics, one for $\\Omega$ and one for $\\Theta$.\n  Hence, \nwe do not establish a relationship\nbetween the defined Christoffel symbols \nand the electronic metric and its derivatives.\n\n  The defined Christoffel symbols \nexhibit\nother expected properties.\n  They give, for instance, the expansion coefficients in $\\Omega$ of the \nderivative of a basis vector:\n\\begin{equation*}\nP_{\\Omega} \\, \\partial_i | e_{\\nu} \\rangle = | e_{\\mu} \\rangle \\langle e^{\\mu} \n| \\partial_i | e_{\\nu} \\rangle = | e_{\\mu} \\rangle D^{\\mu}_{\\phantom{i} \\nu i} \\, ,\n\\end{equation*}\nso that when moving from one point $\\mathbf{R} \\in \\Theta$ to another\ninfinitesimally close to it $\\mathbf{R} + \\mathrm{d}\\mathbf{R}$, the basis\nvectors transform as\n\\begin{equation}\n\\label{BasisProp}\n| e_{\\nu} (\\mathbf{R} + \\mathrm{d}\\mathbf{R}) \\rangle =\n| e_{\\nu} (\\mathbf{R}) \\rangle + | e_{\\mu} (\\mathbf{R}) \\rangle \nD^{\\mu}_{\\phantom{e}\\nu i} \\mathrm{d} R^i \\; ,\n\\end{equation}\nto linear order for infinitesimal $\\mathrm{d} \\mathbf{R}$.\n  The second term of the right hand side accounts for the fact that\nnot only the space is turning, but the basis set itself is changing when\ndisplacing in $\\Theta$.\n\n  The turning of the Hilbert space $\\Omega$ as we move in parameter\nspace $\\Theta$ demands the definition of\nthe way the vector $|\\psi (\\mathbf{R}) \\rangle$ in $\\Omega (\\mathbf{R})$\npropagates when moving to the neighbouring point $\\mathbf{R} + \\mathrm{d}\n\\mathbf{R}$ into the slightly turned Hilbert space $\\Omega(\\mathbf{R}+\\mathrm{d}\n\\mathbf{R})$. \n   The required propagation is given by \n\\begin{equation}\n\\label{prop}\n\\psi^{\\mu} (\\mathbf{R} + \\mathrm{d}\\mathbf{R}) = \n\\psi^{\\mu} (\\mathbf{R}) + \\partial_i \\psi^{\\mu} (\\mathbf{R}) \\mathrm{d} R^i \\; ,\n\\end{equation}\nagain for infinitesimal $\\mathrm{d} \\mathbf{R}$.\n  This means that the vector is first propagated in $\\Omega (\\mathbf{R})$\nand then projected into $\\Omega (\\mathbf{R} + \\mathrm{d}\\mathbf{R})$ by \nretaining the same vector components. \n  This is consistent \nto linear order\nwith the projective propagation of the state $|\\psi\\rangle$ when\nmoving from $\\mathbf{R}$ to $\\mathbf{R} + \\mathrm{d}\\mathbf{R}$, \n\\begin{multline}\n\\label{abstractprop}\n|\\psi (\\mathbf{R} + \\mathrm{d}\\mathbf{R}) \\rangle =\nP_{\\Omega (\\mathbf{R} + \\mathrm{d}\\mathbf{R})} \\{\nP_{\\Omega (\\mathbf{R})} |\\psi (\\mathbf{R}) \\rangle \\\\ +\nP_{\\Omega (\\mathbf{R})} \\partial_i [ P_{\\Omega (\\mathbf{R})}\n| \\psi (\\mathbf{R})\\rangle ] \\mathrm{d} R^i \\} \\, .\n\\end{multline}\n  This last expression shows the propagation of a vector when moving in\n$\\Theta$, irrespective of basis set and basis set transformation.\n  It involves the covariant derivative in its final term.\n  The equivalence between Eqs.~\\ref{prop} and \\ref{abstractprop}\nis shown in Appendix~\\ref{PropAppendix}.\n\n Differently from canonical differential geometry of curved spaces,\nin the formalization presented here the proposed Christoffel symbols\ndo not necessarily contract with the metric tensor following conventional\nrules \nfor lowering or raising indices. \n  The required equalities \namong these objects are presented in Appendix~\\ref{ChristAppendix}.\n\n\n\n\n\\subsubsection{Covariant derivative for the bra representation}\n\n  The natural representation of the bra of any state, $\\langle \\psi | $,\nis $\\psi_{\\mu} = \\langle \\psi | e_{\\mu} \\rangle$.\n  Its covariant derivative is defined in analogy to Eq.~\\ref{covdevdef},\nnamely,\n\\begin{equation}\n\\label{covdevbra}\n\\text{\\dh}_i \\psi{\\mu} \\equiv \\, \\partial_i \\left \\{ \\langle \\psi | P_{\\Omega} \\, \\right \\} \n| e_{\\mu} \\rangle \\, ,\n\\end{equation}\nwhich, again working in analogy with that which was\ndone for Eq.~\\ref{covdevdef2}, gives\n\\begin{equation}\n\\label{covdevbra2}\n\\text{\\dh}_i \\psi_{\\mu} = \\partial_i \\psi_{\\mu} + \\psi_{\\nu} D^{\\nu}_{\\phantom{i} i \\mu} \\; ,\n\\end{equation}\nwhere we have defined the corresponding Christoffel symbol\n\\begin{equation}\n\\label{christoffelbra}\nD^{\\nu}_{\\phantom{i} i \\mu} \\equiv  \\langle \\partial_i e^{\\nu} | e_{\\mu} \\rangle \n\\end{equation}\n(see the different order of indices in the symbol as compared to the \ndefinition in Eq.~\\ref{christoffel}).\n\n  The Christoffel symbols of Eqs.~\\ref{christoffel} and \\ref{christoffelbra}\nare easy to interrelate, since \n$\\langle e^{\\mu} | e_{\\nu} \\rangle = \\delta^{\\mu}_{\\phantom{i}\\nu} $ for all\n$\\mathbf{R} \\in \\Theta$, and, therefore,\n\\begin{equation*}\n\\partial_i \\delta^{\\mu}_{\\phantom{i}\\nu} = \\langle \\partial_i e^{\\mu} | e_{\\nu} \\rangle +\n\\langle e^{\\mu} | \\partial_i e_{\\nu} \\rangle = D^{\\mu}_{\\phantom{i} i \\nu} +\nD^{\\mu}_{\\phantom{i} \\nu i} = 0 \\; ,\n\\end{equation*}\nwhich leads to\n\\begin{equation}\n\\label{christoffelrelation}\nD^{\\mu}_{\\phantom{i} i \\nu} = - D^{\\mu}_{\\phantom{i} \\nu i} \\; \n\\end{equation}\n(general relations among Christoffel symbols can be found in \nAppendix~\\ref{ChristAppendix}).\n  The covariant derivative for the bra can then be written as\n\\begin{equation}\n\\label{covdevbra3}\n\\text{\\dh}_i \\psi_{\\mu} = \\partial_i \\psi_{\\mu} - \\psi_{\\nu} D^{\\nu}_{\\phantom{i} \\mu i}  \\; .\n\\end{equation}\n\n  This derivative transforms as the corresponding tensor, as can be easily checked\nfollowing any of the two procedures used above (and in \nAppendix~\\ref{BasisChangeAppendix}) for the case of the ket representation.\n\n\n\n\n\n\n\\subsubsection{Covariant derivative for operators}\n\n  Following the spirit of Eqs.~\\ref{covdevdef} and \\ref{covdevbra}, let us\ndefine the covariant derivative of an operator $H$ in its natural representation\n$H^{\\mu}_{\\phantom{e}\\nu}$ as\n\\begin{equation}\n\\label{covdevop}\n\\text{\\dh}_i H^{\\mu}_{\\phantom{e}\\nu} \\equiv \\langle e^{\\mu} |  \\partial_i \\left \\{\nP_{\\Omega} H P_{\\Omega} \\right \\} | e_{\\nu} \\rangle \\;,\n\\end{equation}\nwhich becomes\n\\begin{equation}\n\\label{covdevop2}\n\\text{\\dh}_i H^{\\mu}_{\\phantom{e}\\nu} = \\partial_i H^{\\mu}_{\\phantom{e}\\nu} + \nD^{\\mu}_{\\phantom{e}\\lambda i} H^{\\lambda}_{\\phantom{e}\\nu} - \nH^{\\mu}_{\\phantom{e}\\lambda} D^{\\lambda}_{\\phantom{e}\\nu i} \\; .\n\\end{equation}\n  For the last term we have used Eq.~\\ref{christoffelrelation}.\n  This last expression coincides with the usual definition of the \ncovariant derivative of a second-rank tensor in other differential\ngeometry contexts, including general relativity, once the \nconnection (the Christoffel symbols) has been defined.\n  It can also be written as \n\\begin{equation*}\n\\text{\\dh}_i H^{\\mu}_{\\phantom{e}\\nu} = \\partial_i H^{\\mu}_{\\phantom{e}\\nu} \n+  [D,H]^{\\mu}_{\\phantom{e}\\nu i} \\; ,\n\\end{equation*}\nwhere the conmutator in the last term is defined as the two last terms\nin Eq.~\\ref{covdevop2}.\n\n  This definition gives a well-behaved tensor, transforming as such\nunder a basis set change. \n  It can be straightforwardly checked following either of\nthe two procedures used before for $\\text{\\dh}_i \\psi{^\\mu}$: by noticing\nfrom Eq.~\\ref{covdevop} that it is a tensor representation of an operator \nwithin $\\Omega$, and by following Appendix~\\ref{BasisChangeAppendix}. \n  It is also a definition consistent with the previous ones. \n  The following Leibniz chain rule for a vector\n\\begin{equation}\n\\label{chain}\n\\text{\\dh}_i \\left ( H^{\\mu}_{\\phantom{e}\\nu} \\psi^{\\nu} \\right ) =\n\\left ( \\text{\\dh}_i  H^{\\mu}_{\\phantom{e}\\nu} \\right ) \\psi^{\\nu} +\nH^{\\mu}_{\\phantom{e}\\nu} \\left ( \\text{\\dh}_i \\psi^{\\nu} \\right ) \\; ,\n\\end{equation}\nand the expected behavior of a scalar (zero-rank tensor)\n\\begin{equation}\n\\label{e_deriv}\n\\text{\\dh}_i E = \n\\text{\\dh}_i \\left ( \\psi_{\\mu} H^{\\mu}_{\\phantom{e}\\nu} \\psi^{\\nu} \\right ) = \n\\partial_i \\left ( \\psi_{\\mu} H^{\\mu}_{\\phantom{e}\\nu} \\psi^{\\nu} \\right ) = \\partial_i E \\; ,\n\\end{equation}\nare proved in Appendix~\\ref{ChainRuleAppendix}.\n\n\n\n\n\n\\subsubsection{Matrix representation}\n\n  Let us extend the previous definitions to the matrix representation, \nwhich will be useful in the next section.  \n  Since the ket is equally represented in the natural and matrix representations,\nthe covariant derivative of a ket is also equally defined, as in Eq.~\\ref{covdevdef2}.\n  The bra is, however, different, since $\\langle \\psi |$ is represented by\n$\\psi^{\\mu *} = \\langle \\psi | e^{\\mu} \\rangle$ (see details of present notation\nin Appendix~\\ref{NotationAppendix}).\n  Its covariant derivative is then defined as\n\\begin{equation*}\n\\text{\\dh}_i \\psi^{\\mu *} = \\partial_i  \\left \\{ \\langle \\psi | P_{\\Omega} \\right \\} | \ne^{\\mu} \\rangle \\; ,\n\\end{equation*}\nand, therefore, without resorting to the ambient Hilbert space ${\\cal H}$, \nit is expressed as\n\\begin{equation*}\n\\text{\\dh}_i \\psi^{\\mu *} = \\partial_i \\psi^{\\mu *} + \\psi^{\\nu *} \nD_{\\nu i}^{\\phantom{ei}\\mu} \\; ,\n\\end{equation*}\nwhich is just the Hermitian conjugate of Eq.~\\ref{covdevdef2}\n(see Appendix~\\ref{ChristAppendix} for relations among Christoffel symbols).\n  For an operator in this representation, the covariant derivative is defined as\n\\begin{equation*}\n\\text{\\dh}_i H_{\\mu\\nu} = \\langle e_{\\mu} | \\partial_i \\left \\{ P_{\\Omega} H\nP_{\\Omega} \\right \\} | e_{\\nu} \\rangle \\; ,\n\\end{equation*}\nwhich becomes\n\\begin{equation}\n\\label{covdevHmatrix}\n\\text{\\dh}_i H_{\\mu\\nu} = \\partial_i H_{\\mu\\nu} +\nD_{\\mu\\phantom{e} i}^{\\phantom{e}\\sigma} H_{\\sigma\\nu} +\nH_{\\mu\\sigma} D_{\\phantom{e} i \\nu}^{\\sigma} \\; .\n\\end{equation}\n\n\n\n\n\n\n\\subsubsection{Geometric interpretation of the affine connection}\n\n  Eqs.~\\ref{prop} and \\ref{abstractprop} give the basis for a clearer geometric \ninterpretation of the affine connection defined above, and the\ncorresponding covariant derivative.\n  Closing Eq.~\\ref{abstractprop} from the left with $\\langle e^{\\mu} (\n\\mathbf{R} + \\mathrm{d}\\mathbf{R}) |$, one obtains\n\\begin{equation}\n\\label{affinerot}\n\\psi^{\\mu} ( \\mathbf{R} + \\mathrm{d}\\mathbf{R})  =\nA^{\\mu}_{\\phantom{e}\\nu} (\\mathbf{R} + \\mathrm{d}\\mathbf{R} : \\mathbf{R})\n\\big \\{ \\psi^{\\nu} + ( \\text{\\dh}_i \\psi^{\\nu} ) \\mathrm{d} R^i \\big \\} \\, ,\n\\end{equation}\nwhere the terms within the curly brackets are defined at {\\bf R},\nand \n\\begin{equation}\n\\label{TimeOverlap}\nA^{\\mu}_{\\phantom{e}\\nu} (\\mathbf{R} + \\mathrm{d}\\mathbf{R} : \\mathbf{R})\n\\equiv \\langle e^{\\mu} (\\mathbf{R} + \\mathrm{d}\\mathbf{R}) | \ne_{\\nu} (\\mathbf{R}) \\rangle \\, ,\n\\end{equation}\ndefines the basis transformation (as in Appendix~\\ref{BasisChangeAppendix})\nwhen moving from {\\bf R} to $\\mathbf{R} + \\mathrm{d}\\mathbf{R}$ \n(the linear equivalence between Eqs.~\\ref{affinerot} and \\ref{prop} is shown in \nAppendix~\\ref{Geometric-Appendix}).\n  Eq.~\\ref{TimeOverlap} represents the local gauge transformation.\nIn principle any invertible matrix with a smooth behaviour with respect to {\\bf R}\nis allowed.\n\n  In Eq.~\\ref{affinerot}, the basis set transformation information is now carried by\nthe $A$ tensor instead of the Christoffel symbols.\n  The Christoffel symbols would reappear in Eq.~\\ref{affinerot}\nif replacing the covariant derivative by its definition in terms of \nthe regular derivative, Eq.~\\ref{covdevdef2}.\n  That would indeed defeat the purpose of Eq.~\\ref{affinerot} (and would\nbring us back to Eq.~\\ref{prop}, as shown in the mentioned \nAppendix~\\ref{Geometric-Appendix}). \n  The usefulness of  expressing the propagation as in Eq.~\\ref{affinerot}\nbecomes evident when using it, for instance, for finite-differences time integrators \nfor solving the time-dependent Schr\\\"odinger equation, as we will do in \nSection~\\ref{quantumevolution} and Appendix~\\ref{ModifiedCK-Appendix}, \nwhere we will essentially replace \n$\\text{\\dh}_t \\psi^{\\mu}$ by $-i H^{\\mu}_{\\phantom{e}\\nu} \\psi^{\\nu}$. \n\n  In addition to its utility for integrators, it is presented here because it conveys \nthe geometric meaning of the affine connection quite clearly: the covariant derivative \nis the intrinsic one, independent of the basis set change, accounting for both the \nphysical variation of the state and the turning of the Hilbert space $\\Omega$, \nwhile the Christoffel symbols linearly account for the basis set transformation.\n\n\n\n\\subsubsection{Rotation versus deformation}\n\n  So far we have talked about basis change or transformation in general.\n  We make here the distinction between pure rotations of\nthe basis and what we will call basis {\\it deformation} (in analogy with \nelasticity theory).\n  They are defined as the ones for anti-Hermitian and Hermitian $D_{\\mu\\nu i}$,\nrepectively.\n   A small arbitrary transformation will thus have a rotation and a deformation\n component, that can be obtained from $( D_{\\mu\\nu i} - D^{*}_{\\nu\\mu i}) \/ 2$,\n and $( D_{\\mu\\nu i} + D^{*}_{\\nu\\mu i} ) \/ 2$, respectively.\n\n   Appendix~\\ref{rotation-appendix} shows how a small unitary transformation of\n the basis, defined as one that keeps constant overlap (and therefore\ncorresponding to basis rotations in $\\Omega$) has an associated anti-Hermitian\n$D_{\\mu\\nu i}$ tensor.\n  This consideration will be relevant in the Applications section below, in the\ncontext of finite-differences integrators.\n \n\n\n\\subsubsection{Parallel transport and unitary propagation}\n\n\\label{ParallelTransport}\n\n  The analog of the parallel transport in curved spaces would be the \npropagation of a state vector $|\\psi \\rangle \\in \\Omega (\\mathbf{R})$, \nwhich, in itself, would not vary (it would be constant if $\\Omega={\\cal H}$) \nwhen moving in $\\Theta$ away from point {\\bf R}. \n  Such intrinsic constancy is reflected by a null covariant derivative, \n\\begin{equation*}\n\\text{\\dh}_i \\psi^{\\mu} = \\partial_i \\psi^{\\mu} + \nD^{\\mu}_{\\phantom{e}\\nu i} \\psi^{\\nu} = 0 \\, .\n\\end{equation*}\n  Therefore, parallel transport of any vector along any line in $\\Theta$\nis then obtained from Eq.~\\ref{prop}, propagating \n\\begin{equation*}\n\\psi^{\\mu} (\\mathbf{R} + \\mathrm{d}\\mathbf{R}) = \n\\psi^{\\mu} (\\mathbf{R}) - D^{\\mu}_{\\phantom{e}\\nu i} (\\mathbf{R}) \n\\psi^{\\nu} (\\mathbf{R}) \\mathrm{d} R^i \n\\end{equation*}\n(for infinitesimal d{\\bf R})\nalong the corresponding line.\n\n  Vectors that are orthogonal to each other at a given point would\npropagate keeping their orthogonality. \n  Appendix~\\ref{UnitaryAppendix} shows that any unitary propagation\n(one such the condition $\\text{\\dh}_i \\{ \\psi_{n\\mu} \\psi^{\\mu}_{\\phantom{e}m} \\} \n= 0$ is preserved)\nmaintains the orthogonality of propagated vectors.\n  Parallel transport is a special case, since $\\text{\\dh}_i \\psi^{\\nu}_m\n= \\text{\\dh}_i \\psi_{\\mu n} = 0$.\n\n  More generally, a set of vectors in parallel transport keep their mutual \nscalar products (this can also be seen following the reasoning of \nAppendix~\\ref{UnitaryAppendix}).\n  This also applies to the two metric tensors, which are composed \nprecisely of the scalar products of basis vectors.\n  Since such scalar products are preserved under parallel transport, \nthen\n\\begin{equation*}\n\\text{\\dh}_i S_{\\mu\\nu}= \\text{\\dh}_i S^{\\mu\\nu} = 0 \\, ,\n\\end{equation*}\nwhich reflects a fundamental property of the theory, that the covariant derivative \nconserves the metric.\n  This result is also derived explicitly in Appendix~\\ref{ChristAppendix}.\n\n\n\n\n\n\n\\subsubsection{Curvature}\n\n  The Riemann-Christoffel curvature for a curved space characterises the\nfact that if you take one vector in a vector field using the Levi-Civita connection \nalong one direction, and then along another direction to reach a certain point, \nit gives a different result than if changing the order of directions in which you \narrive to the same point (or analogously, if following a closed loop).\n  This is locally quantified with the difference in changing the order of\nsecond derivatives, by defining the curvature tensor\n$R^{\\mu}_{\\phantom{e}i\\nu j}$ such that\n\\begin{equation}\n\\label{curvdef}\nR^{\\mu}_{\\phantom{e}i \\nu j} \\psi^{\\nu} = \n\\text{\\dh}_i \\text{\\dh}_j \\psi^{\\mu} - \\text{\\dh}_j \\text{\\dh}_i \\psi^{\\mu} \\; .\n\\end{equation}\n  Using the covariant derivatives defined above, one obtains:\n\\begin{equation}\n\\label{curvexp}\nR^{\\mu}_{\\phantom{e} i \\nu j} = \n\\partial_i D^{\\mu}_{\\phantom{e}\\nu j} - \\partial_j D^{\\mu}_{\\phantom{e}\\nu i} + \nD^{\\mu}_{\\phantom{e}\\lambda i} D^{\\lambda}_{\\phantom{e}\\nu j} -\nD^{\\mu}_{\\phantom{e}\\lambda j} D^{\\lambda}_{\\phantom{e}\\nu i} \\, ,\n\\end{equation}\nagain in perfect analogy to the expression for curved spaces.\n\n\n\n\n\\subsubsection{Relation to Berry connection and curvature}\n\n  In Subsection~\\ref{ParameterSpace} the analogy with Berry's geometric phase\nformalism\\cite{Berry} was mentioned. \n  Indeed, for a quantum mechanical (single- or many-particle) state \n$| \\Psi ( \\mathbf{R} ) \\rangle$,\nthe Berry connection is normally defined as \n\\begin{equation*}\n{\\cal A}_{j} = i \\, \\langle \\Psi | \\partial_j \\Psi \\rangle\n\\end{equation*}\n($i=\\sqrt{-1}$),\nwhich is nothing but ($i$ times) the connection defined in this work \n(Eq.~\\ref{christoffel}) for a space $\\Omega$ spanned by a single state.\n  It generalises to the trace\n\\begin{equation}\n\\label{BerryConnection}\n{\\cal A}_{j} = i \\, \\sum_n^{occ} \\langle \\psi_n | \\partial_j \\psi_n \\rangle \\; ,\n\\end{equation}  \nfor a set of single-particle states $| \\psi_n \\rangle$ spanning the occupied space\nin the context of a mean-field-like approach to the many-particle problem (as is the case \nfor the Kohn-Sham states in density-functional theory).\n  As expected from Berry's work, it is easy to see that \n\\begin{equation}\n\\label{TraceConnection}\n{\\cal A}_j = i \\, D^{\\mu}_{\\phantom{e}\\mu j} \\; ,\n\\end{equation}\nwhere $D^{\\mu}_{\\phantom{e}\\mu j}$ is \nthe trace of the connection defined in \nEq.~\\ref{christoffel}\n(bear in mind that the $| \\psi_n \\rangle$ states in this context play\nthe role of the basis of the relevant space, i.e., the $| e_{\\mu}\\rangle$ states\nof the previous sections, and $\\Omega$ refers here to the occupied space).\n  The expression in Eq.~\\ref{BerryConnection} assumes orthonormal states,\nwhereas Eq.~\\ref{TraceConnection} is valid for any non-orthogonal basis of \nthe relevant $\\Omega$ space.\n  More generally, the Berry connection matrix \n${\\cal A}_{mnj} = i \\, \\langle \\psi_m | \\partial_j \\psi_n \\rangle$ corresponds to\n($i$ times) this work's $D_{mn j}$ connection in the matrix representation, which\ncan be transformed to any other tensorial representation for\nnon-orthogonal states.\n\n  Similarly to what happens to the connection, the curvature of this work and that \nof Berry are closely related.\n  The Berry curvature is usually defined as \n\\begin{equation*}\n{\\cal R}_{ij} = - 2 \\; \\mathrm{Im} \\left \\{ \\langle \\partial_i \\Psi | \\partial_j \\Psi \\rangle \\right \\} \\; ,\n\\end{equation*}\nfor a quantum mechanical state $| \\Psi \\rangle$,\nwhich generalises to \n\\begin{equation}\n\\label{BerryCurv}\n{\\cal R}_{ij} = - 2 \\; \\mathrm{Im} \\left \\{ \\sum_n^{occ} \\langle \\partial_i \\psi_n | \\partial_j \\psi_n \n\\rangle \\right \\} \n\\end{equation}\nfor a set of single-particle states $| \\psi_n \\rangle$ spanning the occupied space.\n  The curvatures in Eqs.~\\ref{curvexp} and \\ref{BerryCurv} are very closely \ninterrelated when considering our $\\Omega$ Hilbert space as the occupied space\n(or any specific subspace that we are computing the curvature for).\n  Starting with \n\\begin{equation*}\n\\partial_i D^{\\mu}_{\\phantom{e}\\nu j} = \\langle \\partial_i e^{\\mu} | \\partial_j e_{\\nu} \\rangle\n- \\langle e^{\\mu} | \\partial_i \\partial_j e_{\\nu} \\rangle \\; ,\n\\end{equation*}\nwe can easily see that\n\\begin{equation*}\n\\partial_i D^{\\mu}_{\\phantom{e}\\nu j} - \\partial_j D^{\\mu}_{\\phantom{e}\\nu i} = \n\\langle \\partial_i e^{\\mu} | \\partial_j e_{\\nu} \\rangle -\n\\langle \\partial_j e^{\\mu} | \\partial_i e_{\\nu} \\rangle \\; .\n\\end{equation*}\nIf we now trace over the quantum variables, in analogy with the Ricci curvature\n\\begin{equation}\n\\label{RicciBerry}\n{\\cal R}_{ij} = R^{\\mu}_{\\phantom{e} i \\mu j} = \n\\langle \\partial_i e^{\\mu} | \\partial_j e_{\\mu} \\rangle -\n\\langle \\partial_j e^{\\mu} | \\partial_i e_{\\mu} \\rangle \\; ,\n\\end{equation} \nsince the following traces annihilate:\n\\begin{equation*}\nD^{\\mu}_{\\phantom{e}\\lambda i} D^{\\lambda}_{\\phantom{e}\\mu j} -\nD^{\\mu}_{\\phantom{e}\\lambda j} D^{\\lambda}_{\\phantom{e}\\mu i} = 0 \\; .\n\\end{equation*}\n\n  If the basis $\\{ | e_{\\mu} \\rangle \\}$ is invariably orthonormal, then\n$\\langle e^{\\mu} | = \\langle e_{\\mu} |$, for any {\\bf R}, and thus\n$\\langle \\partial_i e^{\\mu} | = \\langle \\partial_i e_{\\mu} |$, and\n\\begin{equation*}\n\\langle \\partial_j e_{\\mu} | \\partial_i e_{\\mu} \\rangle =\n\\langle \\partial_i e_{\\mu} | \\partial_j e_{\\mu} \\rangle ^* \\; ,\n\\end{equation*}\nand, therefore, \n\\begin{equation}\n\\label{BerryRicci}\n{\\cal R}_{ij} =   2 i \\; \\mathrm{Im} \\left \\{\n\\langle \\partial_i e_{\\mu} | \\partial_j e_{\\mu} \\rangle \\right \\} \\; ,\n\\end{equation} \nwhich is nothing but Eq.~\\ref{BerryCurv} (times $i$) for the \n$| \\psi_n \\rangle$ states taken as an orthonormal basis of \noccupied space $\\Omega$.\n  Therefore, Berry's curvature is nothing but the Ricci curvature of \nour turning occupied space.\n\n  This result is directly generalizable to any other orthonormal basis\nof occupied space, e.g., a basis of Wannier functions, still \nunder Eq.~\\ref{BerryRicci}.\n  If the basis is non-orthogonal (non-orthogonal Wannier functions), \nEq.~\\ref{RicciBerry} is then the relevant definition.\n  If seeking an expression closer to Eq.~\\ref{BerryRicci}, the definition in \nEq.~\\ref{RicciBerry} can also be re-expressed as\n\\begin{align}\n\\label{BerryRicciNon}\n{\\cal R}_{ij} =   & 2 i \\; \\mathrm{Im}  \\left \\{ S^{\\mu\\nu}\n\\langle \\partial_i e_{\\nu} | \\partial_j e_{\\mu} \\rangle \\right \\}  \\notag \\\\ \n& + ( \\partial_i S^{\\mu\\nu} ) D_{\\nu\\mu j} - ( \\partial_j S^{\\mu\\nu} ) D_{\\nu\\mu i} \\; ,\n\\end{align}\nwhich, in addition to the expected redefinition of the trace with the metric tensor \nin the first term, includes two additional terms related to the variation \nof the metric itself.\n\n\n\n\\subsubsection{The topology of $\\Omega (\\mathbf{R})$}\n\n   When solving a quantum-mechanical problem using a finite basis\nset that changes in parameter space, we will then have two relevant\nfibre bundles, one within the other: the one spanned by the basis, $\\Omega$,\nand the one spanned by the occupied states. \n  The curvature and topology of $\\Omega$({\\bf R})\nand its relation with the ones corresponding to the occupied space could have \nimplications on the effect of the approximation implied by the basis.\n  We will not explore this point further in this paper. \n  We can however, point to Ref.~\\onlinecite{Mead1992} for the implications\nof Berry concepts on the dependence of the occupied space on atomic positions\nin molecular systems, including the effect of conical intersections, for instance.\n  Rethinking such ideas considering a wider-than-occupied space could be\nan avenue for future investigation.\n\n\n\n\\section{Applications}\n\n\n\n\n\\subsection{Quantum time evolution}\n\n\\label{quantumevolution}\n\n\\subsubsection{Basic equations}\n  \n  Using the present formalism, we can consider a one-dimensional \nparameter space with time as the only variable. \n In the natural representation, the time dependent Schr\\\"odinger \nequation $H | \\psi \\rangle = i \\partial_t | \\psi \\rangle$ becomes \nsimply\n\\begin{equation}\n\\label{td}\nH^{\\mu}_{\\phantom{e}\\nu} \\psi^{\\nu} = i \\, \\text{\\dh}_t \\, \\psi^{\\mu} \\, ,\n\\end{equation}\nwhere the time covariant derivative is defined as \n\\begin{equation*}\n\\text{\\dh}_t    \\, \\psi^{\\mu} = \\partial_t  \\, \\psi^{\\mu} + D^{\\mu}_{\\phantom{e}\\nu t} \n\\, \\psi^{\\nu} \\, ,\n\\end{equation*}\nand the corresponding temporal Christoffel symbol as\n\\begin{equation*}\nD^{\\mu}_{\\phantom{e}\\nu t}  = \\langle e^{\\mu} | \\partial_t | e_{\\nu} \\rangle\n= \\langle e^{\\mu} | \\partial_t  e_{\\nu} \\rangle \\, .\n\\end{equation*}\n   Equation \\ref{td} reflects the physics in a basis-set independent form, \nin the sense that the well-behaved tensors in the equation\nall transform as in Appendix~\\ref{BasisChangeAppendix}, \nand transmit the physics of the original Schr\\\"odinger\nequation for an evolving basis set and Hilbert space.\n  Eq.~\\ref{td} can be obtained by minimising the action\nfor the following Lagrangian: \n\\begin{equation*}\nL = i \\, \\psi_{\\mu} \\text{\\dh}_t \\psi^{\\mu} - \\psi_{\\mu} H^{\\mu}_{\\phantom{e}\\nu}\n\\psi^{\\nu} \\; ,\n\\end{equation*}\nwhich is easily obtained using the ideas above from the standard\n$L = i \\, \\langle \\psi | \\partial_t \\psi \\rangle - \\langle \\psi | H | \\psi \\rangle$.\n  \n  The matrix representation also allows a concise representation\nof the physical equation, albeit less elegantly, carrying around\nthe metric tensors, as follows.\n\\begin{equation}\n\\label{matschroe}\nH_{\\mu\\nu} \\psi^{\\nu} = i \\, S_{\\mu\\nu} \\,\\text{\\dh}_t \\, \\psi^{\\nu} \\, ,\n\\end{equation}\nwhich corresponds to Eq.~\\ref{old-td-eq} of the Introduction, or\n\\begin{equation*}\nS^{\\mu\\sigma} H_{\\sigma\\nu} \\psi^{\\nu} = i \\,\\text{\\dh}_t \\, \\psi^{\\mu} \\, .\n\\end{equation*}\n\n  If following the propagation of the density matrix\ninstead of that of the wave functions, the dynamics is\ndefined by the Liouville - Von Neumann equation,\n\\begin{equation*}\ni \\, \\partial_t \\rho = [ H, \\rho]  \\; ,\n\\end{equation*}\nwhere the density operator $\\rho$ would be\n\\begin{equation*}\n\\rho (t) = | \\Psi (t) \\rangle \\langle \\Psi (t) |\n\\end{equation*}\nfor a pure quantum state, or \n\\begin{equation*}\n\\rho (t) = \\sum_n^{occ} | \\psi_n (t) \\rangle \\langle \\psi_n (t) |\n\\end{equation*}\nfor the set of occupied states in a mean-field setting. \n  (It can also be generalised to statistical mixtures in general,\nincluding thermal). \n\n   Again, for the evolving Hilbert space $\\Omega$, the\nexpression of this equation in terms of the corresponding\ntensors is obtained by closing it with $\\langle e^{\\mu} |$ \nfrom the left and $ | e_{\\nu} \\rangle $ from the right, and\nsubstituting $\\rho$ and $H$ by \n$P_{\\Omega} \\rho P_{\\Omega}$ and \n$P_{\\Omega} H P_{\\Omega}$, giving\n\\begin{equation*}\ni \\, \\text{\\dh}_t \\rho^{\\mu}_{\\phantom{e}\\nu} = \n[ H, \\rho]^{\\mu}_{\\phantom{e}\\nu} = \nH^{\\mu}_{\\phantom{e}\\sigma} \\rho^{\\sigma}_{\\phantom{e}\\nu} -\n\\rho^{\\mu}_{\\phantom{e}\\sigma} H^{\\sigma}_{\\phantom{e}\\nu}  \\; \n\\end{equation*}\nin its natural representation.\n  The matrix representation of this equation\nis much less elegant.\n  The conventional definition of  density matrix in a typical quantum chemistry\nsetting is $\\sum_n^{occ} C_{\\mu n} C_{\\nu n}^*$, in the language \nof Eq.~\\ref{coeff-expansion} of the Introduction. \n  This is nothing but  \n\\begin{equation*}\n\\rho^{\\mu\\nu} = \\langle e^{\\mu} | \\, \\rho \\, | e^{\\nu} \\rangle \\; .\n\\end{equation*}  \n  In this representation, the Louiville - Von Neumann equation becomes\n\\begin{equation*}\ni \\, \\text{\\dh}_t \\rho^{\\mu\\nu} = \nS^{\\mu\\sigma} H_{\\sigma\\kappa} \\rho^{\\kappa\\nu} -\n\\rho^{\\mu\\sigma} H_{\\sigma\\kappa} S^{\\kappa\\nu} \\; .\n\\end{equation*}\n\n  For any of the former equations, the time dependence of the basis \norbitals may be due to the variation in time of other parameters like\natomic positions $R^i$.\n  In such cases, the Christoffel symbols in the covariant derivatives would\nsatisfy\n\\begin{equation*}\nD^{\\mu}_{\\phantom{e} \\nu t} = v^i D^{\\mu}_{\\phantom{e} \\nu i} \\; ,\n\\end{equation*} \n(or equivalent representations) where $v^i = \\partial R^i \/ \\partial t$ \nare the corresponding nuclear velocities.\n\n\n\n\n\n\\subsubsection{Crank-Nicholson integrator}\n\n  In various contexts a time-dependent Schr\\\"odinger-like equation \nis numerically solved by discretizing time, using adequate\nintegrator algorithms (for a comparison of the performance and\nstability of different options see Refs.~\\onlinecite{Koslov,Correa}).\n  In the context of non-orthogonal basis sets \nand within mean-field-like\ntheories for which matrix inversion is affordable, \nthe Crank-Nicholson algorithm has been used quite \nsuccessfully.\\cite{Tsolakidis2002}  \n  The generalization of that procedure to a moving basis set was\nachieved by incorporating a L\\\"owdin orthonormalisation step, \nan idea due to Sankey and collaborators,\\cite{Sankey} which will \nbe discussed below. \n  The Crank-Nicholson-L\\\"owdin procedure proved quite successful\nin the integration of the Kohn-Sham equations for several studies\nof electronic stopping power for ionic projectiles shooting through \nvaried materials.\\cite{Zeb2013,Correa2013,Ullah2015} \n  Here we define new integrators based on the Crank-Nicholson idea,\ninspired by the affine connection defined above, and we compare\nthem with the Crank-Nicholson-L\\\"owdin procedure.\n\n  Let us first revise the Crank-Nicholson method in this context\nfor non-moving bases.\\cite{Tsolakidis2002}\n  The basics: a state $| \\psi \\rangle$ evolving according to\n$H | \\psi \\rangle = i \\partial_t | \\psi \\rangle$ can be propagated from $t$ to \n$t+\\mathrm{d}t$ by considering the backwards and forwards evolution \nfrom each of those time points to the one in the middle, as follows:\n\\begin{align}\n\\label{CK-abstract}\n| \\psi (t+\\mathrm{d}t\/2 ) \\rangle \n&= \\Big \\{ 1 - i \\frac{\\mathrm{d}t}{2} H (t) \\Big \\} | \\psi (t) \\rangle \\notag \\\\\n| \\psi (t+\\mathrm{d}t\/2 ) \\rangle \n&= \\Big \\{ 1 + i \\frac{\\mathrm{d}t}{2} H (t+\\mathrm{d}t) \\Big \\} \n| \\psi (t+\\mathrm{d}t) \\rangle \\, ,\n\\end{align}\nwhich, by eliminating the middle point becomes\n\\begin{align*}\n| \\psi (t+\\mathrm{d}t) \\rangle = \\Big [ 1 + i \\frac{\\mathrm{d}t}{2} \nH (t+\\mathrm{d}t) \\Big ]^{-1}\n\\Big \\{ 1 - i \\frac{\\mathrm{d}t}{2} H (t) \\Big \\} | \\psi (t) \\rangle\n\\end{align*}\nthereby ensuring, by construction, that the algorithm respects\ninvariance under time reversal.\n\n  The use of this algorithm is complicated by the dependence\non $H (t+\\mathrm{d}t)$, which requires the use of an \niterative self-consistency procedure.\\cite{Kaxiras}\n  For practical purposes, however, in many implementations the \nalgorithm is simplified by using $H(t)$ in both factors, \n\\begin{align}\n\\label{CK-kets}\n| \\psi (t+\\mathrm{d}t) \\rangle = \\Big [ 1 + i \\frac{\\mathrm{d}t}{2} \nH (t) \\Big ]^{-1} \\Big \\{ 1 - i \\frac{\\mathrm{d}t}{2} H (t) \\Big \\} | \\psi (t) \\rangle\n\\end{align}\nwhich allows a direct evaluation of $| \\psi (t+\\mathrm{d}t) \\rangle$\nfrom information of the previous time step.\n  This change is of course of no consequence if the Hamiltonian does\nnot change with time, but this is hardly the case for any mean-field-like\nHamiltonian (such as that of Kohn and Sham), given the dependence of the \nHamiltonian on the evolving electron density or wave-functions. \n\n  For a fixed, not evolving basis set, this is used\\cite{Tsolakidis2002} as\n\\begin{multline}\n\\label{CK-natural}\n\\psi^{\\mu} (t+\\mathrm{d}t) = \\\\  \\Big [ \\delta_{\\mu}^{\\phantom{e}\\sigma} + \ni \\frac{\\mathrm{d}t}{2} H_{\\mu}^{\\phantom{e}\\sigma} (t) \\Big ]^{-1}\n\\Big \\{ \\delta^{\\sigma}_{\\phantom{e}\\nu}  - i \\frac{\\mathrm{d}t}{2} \nH^{\\sigma}_{\\phantom{e}\\nu}  (t) \\Big \\}  \\psi^{\\nu}(t) \\, \n\\end{multline}\n(please note that the first factor does not indicate an inversion of the\nparticular $(\\mu,\\sigma)$ element, but of the tensor as a whole; see \nAppendix~\\ref{InverseAppendix} for the notation and relevant definitions \nused in this paper for inverse second-rank tensors).\n  In the matrix representation it becomes\n\\begin{multline}\n\\label{CK-matrix}\n\\psi^{\\mu} (t+\\mathrm{d}t) = \\\\ \\Big [ S_{\\mu\\sigma} + i \\frac{\\mathrm{d}t}{2} \nH_{\\mu\\sigma} (t) \\Big ]^{-1}  \n\\Big \\{ S_{\\sigma\\nu}   - i \\frac{\\mathrm{d}t}{2} H_{\\sigma\\nu}  (t) \\Big \\}\n\\psi^{\\nu}(t) \\, ,\n\\end{multline}\nkeeping in mind that the lower indices within the inverted tensor become \nupper indices (see Appendix~\\ref{InverseAppendix}).\n\n  For fixed bases this algorithm has the enormous virtue of \nbeing strictly unitary,\\cite{Tsolakidis2002}  in the sense that when \npropagating an orthonormal set of states $\\psi^{\\mu}_m$, the set \nremains orthonormal at $t+\\mathrm{d}t$ regardless of the size of d$t$.\n\n\n\n\n\\subsubsection{Revising Crank-Nicholson for a moving basis}\n\n  For a moving basis set, the algorithm has to be generalized.\n  One straightforward generalisation is achieved by replacing\n$H_{\\mu\\sigma}(t)$ in Eq.~\\ref{CK-matrix} by $H_{\\mu\\sigma}(t) \n- i D_{\\mu\\sigma t}(t)$ (in the matrix representation, for instance), giving\n\\begin{multline}\n\\label{CKmov-matrix}\n\\psi^{\\mu} (t+\\mathrm{d}t) =  \\Big [ S_{\\mu\\sigma}  + i \\frac{\\mathrm{d}t}{2} \n( H_{\\mu\\sigma}  - i D_{\\mu\\sigma t} )  \\Big ]^{-1}  \\\\ \\times\n\\Big \\{ S_{\\sigma\\nu} - i \\frac{\\mathrm{d}t}{2} ( H_{\\sigma\\nu} - i D_{\\sigma\\nu t} ) \\Big \\} \n\\psi^{\\nu}(t)\n\\end{multline}\n(where we have dropped the $t$-dependence of the tensors for\nclarity).\nHere, $D_{\\mu\\sigma t} = \\langle e_{\\mu} | \\partial_t e_{\\sigma} \\rangle$\nis the required temporal Christoffel symbol.\n  Eq.~\\ref{CKmov-matrix} can be derived from Eq.~\\ref{prop} using the \ntensorial time-dependent Schr\\\"odinger equation in Eq.~\\ref{td} and the \ndefinition of the covariant derivative, Eq.~\\ref{covdevdef2}.\n  This would be only linearly correct in d$t$ in what concerns both\nHilbert space turning and basis set transformation, \nand therefore the nice \nunitary propagation feature for arbitrary d$t$ is lost.\n  Indeed, the loss of hermiticity of the propagated effective Hamiltonian\nmakes this problem apparent,\nalbeit that the propagation is correct and well-behaved in the limit of\nsmall d$t$.\n\n   A much more promising  approach is obtained by building on \nEq.~\\ref{affinerot} instead of Eq.~\\ref{prop}, \nwhich exactly accounts \nfor basis set change if $\\Omega(t+\\mathrm{d}t)=\\Omega(t)$.\n  Within the matrix representation, one obtains\n\\begin{multline}\n\\label{CKmov-matrix2}\n\\psi^{\\mu} (t+\\mathrm{d}t) = \n\\Big [ S_{\\mu\\lambda}(t+\\mathrm{d}t)  + i \\frac{\\mathrm{d}t}{2} \nH_{\\mu\\lambda}  (t+\\mathrm{d}t) \\Big ]^{-1} \\\\ \\times\nA_{\\lambda}^{\\phantom{e}\\sigma} (t +\\mathrm{d}t : t) \n\\Big \\{ S_{\\sigma\\nu}(t) - i \\frac{\\mathrm{d}t}{2} H_{\\sigma\\nu} (t) \\Big \\} \n\\psi^{\\nu}(t) \\; ,\n\\end{multline}\nwhere $A_{\\lambda}^{\\phantom{e}\\sigma} (t +\\mathrm{d}t : t) =\n\\langle e_{\\lambda} (t +\\mathrm{d}t) | e^{\\sigma} (t) \\rangle$ is defined \nanalogously to Eq.~\\ref{TimeOverlap}.\n  The derivation of Eq.~\\ref{CKmov-matrix2} is given in \nAppendix~\\ref{ModifiedCK-Appendix}.\n  The algorithm given by Eq.~\\ref{CKmov-matrix2} would again require an\niterative self-consistency procedure for every time step.\n  An analog to Eq.~\\ref{CK-kets} can also be considered \nwith a view to\nremoving the dependence on $H_{\\mu\\nu} (t+\\mathrm{d}t)$ and\n$S_{\\mu\\nu} (t+\\mathrm{d}t)$.\n  This is achieved with the following Ansatz:\n\\begin{multline}\n\\label{CKmov-matrix3}\n\\psi^{\\mu} (t+\\mathrm{d}t) = A^{\\mu}_{\\phantom{e}\\lambda} (t +\\mathrm{d}t : t) \n\\\\ \\times \\Big [ S_{\\lambda\\sigma} (t)  + i \\frac{\\mathrm{d}t}{2} \nH_{\\lambda\\sigma} (t)  \\Big ]^{-1}\n\\Big \\{ S_{\\sigma\\nu} (t) - i \\frac{\\mathrm{d}t}{2} H_{\\sigma\\nu} (t)  \\Big \\} \n\\psi^{\\nu}(t) \\, .\n\\end{multline}\n  The integrator in Eq.~\\ref{CKmov-matrix3} keeps the strict unitary character \nof the algorithm for a transforming basis set in a fixed $\\Omega$ space. \n  This can be seen by noticing that the last two transformations in \nEq.~\\ref{CKmov-matrix3} (the ones in curly and square brackets) correspond\nto the original Crank-Nicholson scheme for a fixed Hamiltonian, and that\nthe $A$ tensor transformation is nothing but a change of basis. \n  The latter, however, is only linearly correct in d$t$ if $\\Omega$ does turn. \n  Indeed, for an arbitrarily large d$t$, the set \n$P_{\\Omega(t + \\mathrm{d}t)} | \\psi_m \\rangle$ \nis not necessarily orthonormal even if the set $P_{\\Omega(t)} | \n\\psi_m \\rangle$ was.\n  The advantage of Eq.~\\ref{CKmov-matrix3} with respect to \nEq.~\\ref{CKmov-matrix} is important, however, since the space \nturning diminishes with better converged bases, while the basis \nchange does not necessarily diminish (think of the perfectly converged limit of \n$\\Omega={\\cal H}$ for a moving basis: the basis still changes and the \nHilbert space does not).\n\n  In practice, the tensor $A^{\\mu}_{\\phantom{e}\\lambda} (t +\\mathrm{d}t : t)$\nof Eq.~\\ref{CKmov-matrix3} is somewhat inconvenient for calculations.\n  It can be replaced by\n\\begin{multline}\n\\label{CKmov-matrix4}\n\\psi^{\\mu} (t+\\mathrm{d}t) = S^{\\mu\\kappa}(t+\\mathrm{d}t) A_{\\kappa\\lambda} \n(t +\\mathrm{d}t : t) \\\\ \\times \\Big [ S_{\\lambda\\sigma} (t) + i \\frac{\\mathrm{d}t}{2} \nH_{\\lambda\\sigma} (t)  \\Big ]^{-1}\n\\Big \\{ S_{\\sigma\\nu} (t) - i \\frac{\\mathrm{d}t}{2} H_{\\sigma\\nu} (t) \\Big \\} \n\\psi^{\\nu}(t) \\, ,\n\\end{multline}  \nwhich involves the inversion of the overlap matrix at $t+\\mathrm{d}t$\nand the calculation of the overlap between the $t$ basis vectors and\nthe ones at $t+\\mathrm{d}t$ (in addition to the Crank-Nicholson operations at $t$).\n  Similarly, the practical implementation of the self-consistent procedure\nimplied by Eq.~\\ref{CKmov-matrix2} would actually imply using the\nfollowing self-consistent integrator instead:\n\\begin{multline}\n\\label{CKmov-matrix5}\n\\psi^{\\mu} (t+\\mathrm{d}t) = \n\\Big [ S_{\\mu\\lambda}(t+\\mathrm{d}t)  + i \\frac{\\mathrm{d}t}{2} \nH_{\\mu\\lambda}  (t+\\mathrm{d}t) \\Big ]^{-1} \\\\ \\times\nA_{\\lambda\\kappa} (t +\\mathrm{d}t \\! : \\! t) \\, \\Big \\{ \\delta^{\\kappa}_{\\phantom{e}\\nu} \n- i \\frac{\\mathrm{d}t}{2} S^{\\kappa\\sigma} (t) H_{\\sigma\\nu} (t) \\Big \\} \n\\psi^{\\nu}(t) \\, .\n\\end{multline}\n\n\n\n\n\n\n\n\\subsubsection{Procedure based on L\\\"owdin orthonormalization}\n\\label{secSankey}\n \n  An alternative integrator was proposed\\cite{Sankey} and has been \nused\\cite{Zeb2013,Correa2013,Ullah2015} for strict unitary propagation\nfor arbitrary d$t$.\n  Following the previous notation, this propagator can be written as\n\\begin{multline}\n\\label{Sankey}\n\\psi^{\\mu} (t+\\mathrm{d}t) = \\{ S^{-1\/2} (t+\\mathrm{d}t) \\}^{\\mu l} \n\\{ S^{1\/2} (t) \\}_{l\\lambda}  \n\\\\ \\times \\Big [ S_{\\lambda\\sigma} (t) + i \\frac{\\mathrm{d}t}{2} \nH_{\\lambda\\sigma} (t)  \\Big ]^{-1}\n\\Big \\{ S_{\\sigma\\nu} (t) - i \\frac{\\mathrm{d}t}{2} H_{\\sigma\\nu} (t) \\Big \\} \n\\psi^{\\nu}(t) \\, .\n\\end{multline}  \n  This is analogous to Eq.~\\ref{CKmov-matrix4} inasmuch as it first does the\nphysical (Crank-Nicholson) propagation for the basis at $t$ and within \n$\\Omega(t)$, and then \nit transforms to conform to the basis at time $t+\\mathrm{d}t$.\n  The basis transformation is done differently, however. It can be seen as the\nfollowing process: ($i$) It first changes basis \nwithin $\\Omega(t)$ to a L\\\"owdin orthonormal basis.\n  ($ii$) It then assumes that the coefficients in this basis do not change\nwhen changing to the L\\\"owdin basis of $t+\\mathrm{d}t$.   \n  ($iii$) It then undoes the L\\\"owdin transformation in $t+\\mathrm{d}t$\nobtaining the sought coefficients in the non-orthogonal basis\nat $t+\\mathrm{d}t$.\n\n  This procedure has the advantage that\nthe propagation is now strictly unitary by construction for any d$t$,\neven considering the turning of the Hilbert space, which should give\nlarger stability to the method for relatively large values of d$t$.\n  The propagated vectors are guaranteed to be orthonormal.\n  \n  However, the vectors propagated following this L\\\"owdin procedure\n(Eq.~\\ref{Sankey}) do not constitute a fair representation of what the \nevolution of the corresponding vectors would be in ${\\cal H}$ in the sense\nthat it does not properly counter the effect of the transforming basis.\n  In order to see this, consider the case of a set of vectors at $t=0$, \n$ \\{ | \\psi_n \\rangle \\}$, all within $\\Omega$ and all initially orthonormal, \n$ {\\cal S}_{nm} = \\langle \\psi_n | \\psi_m \\rangle = \\delta_{nm}$. \n  Consider, as well, that the dynamics is such that the vectors do not \nrotate with time, changing only in phase, as would be the case for\neigenstates of a time-independent Hamiltonian, for instance.\n  Take now a basis set for $\\Omega$ that does rotate with time, but\nkeeps orthonormality at all times.\n  In this scenario it is clear that the coefficients $\\psi^{\\mu}_n$ should\ntransform with time to capture the fact that the non-rotating eigenvectors \nare described in the frame of a rotating basis.\n  This is properly taken care of in the integrator proposed in \nEq.~\\ref{CKmov-matrix4} by the $A_{\\kappa\\lambda} \n(t +\\mathrm{d}t : t)$ tensor, which does the needed basis transformation.\n   \n   For the L\\\"owdin procedure, however, the coefficients do not change,\n except for the global phase dictated by the Hamiltonian evolution, i.e., \n \\begin{equation*}\n \\psi^{\\mu}_{\\phantom{e}n} (t + \\mathrm{d} t) = e^{-i \\omega_n \n \\mathrm{d} t}  \\psi^{\\mu}_{\\phantom{e}n} (t), \\, ,\n \\end{equation*}\nas is obvious from the fact that if $S_{\\mu\\nu} = \\delta_{\\mu\\nu}$ at all\ntimes, and therefore\n\\begin{equation*}\nS^{-1}_{\\mu\\nu} = S^{-1\/2 \\, \\mu\\nu} = S^{1\/2}_{\\mu\\nu} = \\delta_{\\mu\\nu}\n\\end{equation*} \nat all times. \n  This simple situation clearly illustrates the above assertion on the \nunfair representation by Eq.~\\ref{CKmov-matrix4} of the \nstates' evolution in a generic situation.\n  It is not clear however how significant such discrepancies can be.\n  In particular, the case made above is for pure rotations of the basis set.\n  We explore this point in more depth in Appendix~\\ref{SankeyAppendix}, \nfinding interesting dependencies on the rotating versus deforming basis sets.\n  In particular, for fixed-shape atomic-like orbitals as basis functions, \nsuch basis rotations correspond to Galileo transforms in parameter space,\nwhich would suggest no physical significance to the discrepancies\nbetween the solvers in Eqs.~\\ref{Sankey} and \\ref{CKmov-matrix4} for such \nrotations.\n  This could be behind the apparent success of Eq.~\\ref{Sankey}.\n  A more quantitative analysis should be the focus of further work.\n  \n\n\n\n\n\n\\subsubsection{Strictly unitary propagation}\n\n  If not using Eq.~\\ref{Sankey} for propagation, we are left with \nEq.~\\ref{CKmov-matrix4} (or Eq.~\\ref{CKmov-matrix5} if using\nself-consistency), which does not strictly conserve the orthonormality\nof propagated states for finite d$t$ if the space $\\Omega$ turns within\n${\\cal H}$.\n  The propagator can still be used as long as d$t$ is small enough such\nthat the unitary propagation is preserved within a desired tolerance.\n  An alternative, however, is re-orthonormalising the states. \n  This can be done with any orthonormalisation procedure, e.g.,\nGram-Schmidt or the iterative procedure used when finding eigenstates\nby minimization in electronic structure (see, e.g., \nRef~\\onlinecite{Payne1990}).\n\n  The L\\\"owdin orthonormalization method described above can\nbe used for this as well, with the advantage that the orthonormal states\nkeep closest to the states prior to orthonormalization (we need to remember\nwe are not just after an orthonormal basis of the evolved occupied space, \nbut actually following the evolution of separate states).\n\n  Consider a set of $M$ states, $\\{ | \\psi_n \\rangle , n=1, ... , M \\}$, where \n$M < {\\cal N}$, being ${\\cal N}$ the dimension of $\\Omega$, and which are\nrepresented by $\\{ \\psi^{\\mu}_{\\phantom{e}n}, n= 1 ... M, \\mu = 1, ... , {\\cal N} \\}$, \nand are all, therefore, within $\\Omega$.\n  Consider they have been propagated by Eq.~\\ref{CKmov-matrix4} and\nare not strictly orthonormal, i.e.,\n\\begin{equation*}\n{\\cal S}_{nm} = \\langle \\psi_n | \\psi_m \\rangle  \\ne \\delta_{nm} \\; ,\n\\end{equation*}\nwhere \n\\begin{equation*}\n{\\cal S}_{nm} = \\psi_{\\mu n} \\psi^{\\mu}_{\\phantom{e}m}\n\\end{equation*}\nis an $M\\times M$ matrix. \n  By diagonalising ${\\cal S}_{mn}$, one can obtain ${\\cal S}^{-1\/2\\phantom{i}nm}$,\nwhereupon the strictly unitary propagator version of Eq.~\\ref{CKmov-matrix4}\nbecomes\n\\begin{multline}\n\\label{CKmov-unitary1}\n\\psi^{\\mu}_{\\phantom{e}n} (t+\\mathrm{d}t) = {\\cal S}^{-1\/2 \\phantom{i} nm} \n(t+\\mathrm{d}t) S^{\\mu\\kappa}(t+\\mathrm{d}t) A_{\\kappa\\lambda} \n(t +\\mathrm{d}t : t) \\\\ \\times \\Big [ S_{\\lambda\\sigma} (t) + i \\frac{\\mathrm{d}t}{2} \nH_{\\lambda\\sigma} (t)  \\Big ]^{-1}\n\\Big \\{ S_{\\sigma\\nu} (t) - i \\frac{\\mathrm{d}t}{2} H_{\\sigma\\nu} (t) \\Big \\} \n\\psi^{\\nu}_{\\phantom{e}m} (t) \\, .\n\\end{multline}  \n  The self-consistent alternative orthonormalising Eq.~\\ref{CKmov-matrix5}\nwould read:\n\\begin{multline}\n\\label{CKmov-unitary2}\n\\psi^{\\mu}_{\\phantom{e}n} (t+\\mathrm{d}t) = \n{\\cal S}^{-1\/2 \\phantom{i} nm} (t+\\mathrm{d}t) \\\\ \\times\n\\Big [ S_{\\mu\\lambda}(t+\\mathrm{d}t)  + i \\frac{\\mathrm{d}t}{2} \nH_{\\mu\\lambda}  (t+\\mathrm{d}t) \\Big ]^{-1} A_{\\lambda\\kappa} (t +\\mathrm{d}t \\! : \\! t) \n\\, \\\\ \\times \\Big \\{ \\delta^{\\kappa}_{\\phantom{e}\\nu} - i \\frac{\\mathrm{d}t}{2} \nS^{\\kappa\\sigma} (t) H_{\\sigma\\nu} (t) \\Big \\} \\psi^{\\nu}_{\\phantom{e}m} (t) \\, .\n\\end{multline}\n\n  This orthonormalisation step can be done at every evolution step, \nafter a determined number of them, or when the deviation from orthonormality \nreaches some tolerance.\n\n\n\n\\subsection{Forces}\n\n  Additional terms related to derivatives also appear when calculating\nthe forces on atoms for geometry relaxation or ab initio molecular\ndynamics calculations.\n  In the following we restrict the discussion to adiabatic forces, leaving for \nfurther work the consideration of additional terms that appear for moving \nbasis sets in non-adiabatic settings.\\cite{Todorov2001}\n  Considering for simplicity one single state $| \\psi \\rangle$, and using \nthe language of the Introduction, one calculates quantities like\n\\begin{align}\n\\label{PulayOld}\n& \\frac{\\partial E} {\\partial R_i} = \\partial_i \\langle \\psi | H | \\psi \\rangle =\n\\partial_i \\left \\{ \\sum_{\\mu\\nu} C_{\\mu}^* \\langle e_{\\mu} | H | e_{\\nu} \n\\rangle C_{\\nu} \\right \\} \\notag \\\\\n&= \\sum_{\\mu\\nu} \\bigg \\{ C_{\\mu}^* \\langle e_{\\mu} | \\partial_iH | e_{\\nu} \n\\rangle C_{\\nu}  \\notag \\\\\n& +(\\partial_i C_{\\mu}^*) \\langle e_{\\mu} | H | e_{\\nu} \\rangle C_{\\nu} \n+ C_{\\mu}^* \\langle e_{\\mu} | H | e_{\\nu} \\rangle (\\partial_i C_{\\nu}) \\notag \\\\ \n& + C_{\\mu}^* \\langle \\partial_i e_{\\mu} | H | e_{\\nu} \\rangle C_{\\nu}\n+ C_{\\mu}^* \\langle e_{\\mu} | H | \\partial_i e_{\\nu} \\rangle C_{\\nu} \\bigg \\}  \\, .\n\\end{align}  \n  The last two terms, called Pulay forces,\\cite{Pulay1969} involve again \nbasis vector derivatives. \n  We discuss here the relevance of the present formalism for\nthese forces and related concepts.\n\n\n\n\n\n\\subsubsection{Hellmann-Feynman theorem}\n\n  The Hellman-Feynman theorem states that, given \n$E = \\langle \\psi | H | \\psi \\rangle$ (for a normalised $| \\psi \\rangle$),\nthe derivative of $E$ with respect to $R^i$ fulfils\n\\begin{equation}\n\\label{HF-general}\n\\partial_i E = \\langle \\psi | \\partial_i H | \\psi \\rangle\n\\end{equation}\nif $H | \\psi \\rangle = E | \\psi \\rangle$ (and $\\langle \\psi | H = \\langle \\psi | E$).  \nIt is easy to see that, using the latter equations in $\\Omega$, \n\\begin{equation}\n\\label{schroe}\nH^{\\mu}_{\\phantom{e}\\nu} \\psi^{\\nu} = E \\psi^{\\mu} \\; \\, {\\rm and} \\; \\; \n\\psi_{\\mu} H^{\\mu}_{\\phantom{e}\\nu} = \\psi_{\\nu} E \\; ,\n\\end{equation}\nand the Hellman-Feynman theorem then reads,\n\\begin{equation*}\n\\partial_i E = \\psi_{\\mu} \\left ( \\partial_i \nH^{\\mu}_{\\phantom{e}\\nu} \\right ) \\psi^{\\nu} \\; .\n\\end{equation*}\n\n  In Eq.~\\ref{e_deriv} we saw that the derivative of a scalar\nneeds no correction. Actually, \n\\begin{equation}\n\\label{forces}\n\\text{\\dh}_i E = \\partial_i E = \\psi_{\\mu} \\left ( \\partial_i \nH^{\\mu}_{\\phantom{e}\\nu} \\right ) \\psi^{\\nu} \n= \\psi_{\\mu} \\left ( \\text{\\dh}_i \nH^{\\mu}_{\\phantom{e}\\nu} \\right ) \\psi^{\\nu}  \\; ,\n\\end{equation}\nthe last identity being easy to check by using \nEqs.~\\ref{covdevop2} and \\ref{schroe}.\n  This is a very transparent expression of the theorem in \nits general quantum mechanical form in Eq.~\\ref{HF-general}.\n\n\n\n\n\n\n\\subsubsection{Hellman-Feynman theorem in matrix representation}\n\n  The matrix representation gives a less concise expression of the same \ntheorem, except when using the covariant derivative of $H$. \n  Starting with the Schr\\\"odinger equation, instead of Eq.~\\ref{schroe}, we have \n\\begin{equation}\n\\label{schroematrix}\nH_{\\mu\\nu} \\psi^{\\nu} = E S_{\\mu\\nu} \\psi^{\\nu} \\; \\, {\\rm and} \\; \\; \n\\psi^{\\mu *} H_{\\mu\\nu} = \\psi^{\\mu *} S_{\\mu\\nu} E \\; .\n\\end{equation}\nExpanding the derivative $\\partial_i E=\n\\partial_i \\left ( \\psi^{\\mu *} H_{\\mu\\nu}\\psi^{\\nu} \\right )$ \nand using the fact that\n$\\langle \\psi | \\psi \\rangle = 1 = \\psi^{\\mu *} S_{\\mu\\nu} \\psi^{\\nu}$,\nthe Hellman-Feynman theorem is obtained in this representation as\n\\begin{equation*}\n\\partial_i E = \\psi^{\\mu *} \\left [ \\partial_i  H_{\\mu\\nu} - \nE \\, \\partial_i S_{\\mu\\nu}  \\right ] \\psi^{\\nu} \\; ,\n\\end{equation*}\nwhich can also be expressed as\n\\begin{equation}\n\\label{HFmatrix}\n\\partial_i E = \\psi^{\\mu *} \\left [ \\partial_i  H_{\\mu\\nu}  - \nE ( D_{\\mu i\\nu} + D_{\\mu\\nu i} ) \\right ] \\psi^{\\nu} \\; .\n\\end{equation}\n\n  Let us see how it looks using the covariant derivative.\n  Introducing its definition for the matrix\nrepresentation of $H$ (Eq.~\\ref{covdevHmatrix}) into the previous\nexpression (Eq.~\\ref{HFmatrix}), \n\\begin{multline}\n\\label{stepHFmatrixcov}\n\\partial_i E =  \\psi^{\\mu *}  [ \\text{\\dh}_i  H_{\\mu\\nu}   \n- D_{\\mu\\phantom{e}i}^{\\phantom{e}\\sigma} H_{\\sigma\\nu}\n- H_{\\mu\\sigma} D^{\\sigma}_{\\phantom{e}i\\nu} \\\\\n- E ( D_{\\mu i\\nu} + D_{\\mu\\nu i} )  ] \\psi^{\\nu} \\; .\n\\end{multline}\n  Using now the Schr\\\"odinger equation again (Eq.~\\ref{schroematrix}),\nthe following elements of the previous equation become:\n\\begin{align*}\n\\psi^{\\mu *} D_{\\mu\\phantom{e}i}^{\\phantom{e}\\sigma} H_{\\sigma\\nu} \\psi^{\\nu} &=\nE \\, \\psi^{\\mu *} D_{\\mu\\phantom{e}i}^{\\phantom{e}\\sigma} S_{\\sigma\\nu} \\psi^{\\nu} \n\\, , \\quad \\mbox{and} \\\\\n\\psi^{\\mu *} H_{\\mu\\sigma} D_{\\phantom{e}i\\nu}^{\\sigma} \\psi^{\\nu} &=\nE \\psi^{\\mu *} S_{\\mu\\sigma} D_{\\phantom{e}i\\nu}^{\\sigma} \\psi^{\\nu} \\; ,\n\\end{align*}\nwhereupon Eq.~\\ref{stepHFmatrixcov} becomes\n\\begin{multline}\n\\label{step2HFmatrixcov}\n\\partial_i E =  \\psi^{\\mu *}  [ \\text{\\dh}_i  H_{\\mu\\nu}   \n- E ( D_{\\mu\\phantom{e}i}^{\\phantom{e}\\sigma} S_{\\sigma\\nu}\n+ S_{\\mu\\sigma} D^{\\sigma}_{\\phantom{e}i\\nu} \\\\\n+ D_{\\mu i\\nu} + D_{\\mu\\nu i} )  ] \\psi^{\\nu} \\; .\n\\end{multline}\n  Using now the relations between Christoffel symbols in \nEqs.~\\ref{ChristRelations}, we find that\n\\begin{align*}\nD_{\\mu\\phantom{e}i}^{\\phantom{e}\\sigma} S_{\\sigma\\nu} &= \n- D_{\\mu i}^{\\phantom{ee}\\sigma} S_{\\sigma\\nu} = \n- D_{\\mu i\\nu} \\\\\nS_{\\mu\\sigma} D^{\\sigma}_{\\phantom{e}i\\nu} &=\n- S_{\\mu\\sigma} D^{\\sigma}_{\\phantom{e}\\nu i} =\n- D_{\\mu \\nu i} \\; ,\n\\end{align*}\nand introducing these in Eq.~\\ref{step2HFmatrixcov} gives a\nquite simple final result for the Hellman-Feynman theorem in the matrix\nrepresentation in terms of the Hamiltonian's covariant derivative,\nnamely,\n\\begin{equation}\n\\label{HFmatrixcov}\n\\partial_i E = \\text{\\dh}_i E = \\psi^{\\mu *} ( \\text{\\dh}_i H_{\\mu\\nu} ) \\psi^{\\nu} \\; .\n\\end{equation}\n\n  This last equation can also be derived directly from Eq.~\\ref{forces} using the\nfact that the covariant derivative conserves the metric, as discussed in\nsection~\\ref{ParallelTransport}, i.e., \n$\\text{\\dh}_i S_{\\mu\\nu} = \\text{\\dh}_i S^{\\mu\\nu} = 0$.\n  From Eq.~\\ref{forces}, we have\n\\begin{align*}\n\\text{\\dh}_i E &= \\psi_{\\mu} \\left ( \\text{\\dh}_i \nH^{\\mu}_{\\phantom{e}\\nu} \\right ) \\psi^{\\nu} =\n\\psi^{\\mu *} S_{\\mu\\lambda} (\\text{\\dh}_i S^{\\lambda\\sigma} H_{\\sigma\\nu} )\n\\psi^{\\nu} \\\\\n&= \\psi^{\\mu *} S_{\\mu\\lambda} \\big \\{ S^{\\lambda\\sigma} \n(\\text{\\dh}_i  H_{\\sigma\\nu} ) + (\\text{\\dh}_i S^{\\lambda\\sigma} )\nH_{\\sigma\\nu} \\big \\} \\psi^{\\nu} \\\\\n&= \\psi^{\\mu *} \\delta_{\\mu}^{\\phantom{e}\\sigma}\n(\\text{\\dh}_i H_{\\sigma\\nu} ) \\psi^{\\nu} +\n\\psi^{\\mu *} S_{\\mu\\lambda} \\, 0 \\,\nH_{\\sigma\\nu} \\psi^{\\nu} \\\\\n&= \\psi^{\\mu *}  (\\text{\\dh}_i  H_{\\mu\\nu} ) \\psi^{\\nu} \\;  ,\n\\end{align*}  \nwhich is nothing but Eq.~\\ref{HFmatrixcov}.  \n\n\n\\subsubsection{Pulay forces}\n\n  When facing the problem of calculating the forces, $\\partial_i E$, \none still needs to calculate $\\partial_i H^{\\mu}_{\\phantom{e}\\nu}$.\n  For the time-dependent Schr\\\"odinger or the von Neumann\nequations above, the relevant derivatives were obtained by solving\nequations defined in (an evolving) $\\Omega$.\n  In this case, however, the calculation of  $\\partial_i H^{\\mu}_{\\phantom{e}\\nu}$ \nis done by integration, effectively using an auxiliary basis set in $\\cal{H}$ \n(analytically with Gaussians, on a real-space grid, etc.).\n  The usual procedure follows\n\\begin{align}\n\\label{pulay}\n\\partial_i & H^{\\mu}_{\\phantom{e}\\nu} = \\partial_i \\langle \\psi^{\\mu} |\nH | \\psi_{\\nu} \\rangle = \\\\ &= \n\\langle e^{\\mu} | \\partial_i H | e_{\\nu} \\rangle +\n\\langle \\partial_i e^{\\mu} | H | e_{\\nu} \\rangle + \n\\langle e^{\\mu} | H | \\partial_i e_{\\nu} \\rangle \\; . \\notag\n\\end{align}\n  The last two terms give rise to the so-called Pulay terms,\\cite{Pulay1969}\nas already presented in Eq.~\\ref{PulayOld}.\n  There is nothing substantially new offered by differential geometry \nin this context.\n\n  It is interesting, however, to separate the terms residing fully within $\\Omega$ from\nthe contributions outside it. \n  Defining $Q_{\\Omega}$ as  $P_{\\Omega}$'s complement, i.e.,\n$P_{\\Omega} + Q_{\\Omega} = \\mathbb{1}$ (the identity operator in $\\cal{H}$), \nwe can expand\n\\begin{align*}\n\\langle \\partial_i e^{\\mu} | H | e_{\\nu} \\rangle &=\n\\langle \\partial_i e^{\\mu} | ( P_{\\Omega} + Q_{\\Omega}) H | e_{\\nu} \\rangle \\\\ &=\n\\langle \\partial_i e^{\\mu} | e_{\\sigma} \\rangle \\langle e^{\\sigma} | H | e_{\\nu} \\rangle \n+ \\langle \\partial_i e^{\\mu} | Q_{\\Omega} H | e_{\\nu} \\rangle  \\\\ &=\nD^{\\mu}_{\\phantom{e}i\\sigma} H^{\\sigma}_{\\phantom{e}\\nu} \n+ \\langle \\partial_i e^{\\mu} | Q_{\\Omega} H | e_{\\nu} \\rangle \\; .\n\\end{align*}\n  Doing this for both Pulay terms and using the definition of the covariant\nderivative $\\text{\\dh}_i H^{\\mu}_{\\phantom{e}\\nu}$, one arrives at\n\\begin{equation*}\n\\text{\\dh}_i H^{\\mu}_{\\phantom{e}\\nu} = \n\\langle e^{\\mu} |  \\partial_i H | e_{\\nu} \\rangle +\n\\langle \\partial_i e^{\\mu} | Q_{\\Omega} H | e_{\\nu} \\rangle +\n\\langle e^{\\mu} | H Q_{\\Omega} | \\partial_i e_{\\nu} \\rangle \n\\end{equation*}\nthe last two terms being explicitly built from the components out of\n$\\Omega$ of both vectors $H | e_{\\nu} \\rangle$ and \n$| \\partial_i e_{\\nu} \\rangle$ (and their duals\/bras).\n  Indeed, if neglecting out-of-space components, then\n\\begin{equation*}\n\\text{\\dh}_i  H^{\\mu}_{\\phantom{e}\\nu}  \\simeq\n\\langle e^{\\mu} | \\partial_i H | e_{\\nu} \\rangle \\; ,\n\\end{equation*}\nand considering Eq.~\\ref{forces}, we arrive upon\n\\begin{equation*}\n\\partial_i E \\simeq  \\psi_{\\mu} \\langle e^{\\mu} | \\partial_i H | e_{\\nu} \\rangle\n\\psi^{\\nu} \\; ,\n\\end{equation*}\nor, in the matrix representation,\n\\begin{equation*}\n\\partial_i E \\simeq  \\psi^{\\mu *} \\langle e_{\\mu} | \\partial_i H | e_{\\nu} \\rangle\n\\psi^{\\nu} \\; ,\n\\end{equation*}\nwhere the Pulay terms have disappeared. \n  Neglecting those terms, however, spoil the correspondence between $E$\nand $\\partial_i E$.\n  This just reflects the fact that, in the time-dependent equations above,\nthe derivatives of the basis vectors were naturally projected onto $\\Omega$,\nwhich is not the case here.\n\n\n\n\n\n\n\\section{Conclusions}\n\n  Covariant derivatives are defined for derivatives of quantum mechanical states\nin situations of basis sets varying as a function of external parameters.\n  The concepts from differential geometry used to re-formalize dynamical equations\nallow for better insights into the meaning of connection terms appearing in\ndynamical equations. \n  In addition to relating to the Berry-phase and gauge formalisms, \nthese geometrical insights enable the evaluation of existing, and proposal of new,\nfinite-differences propagators for time evolution equations.\n\n\n\n\n\n\n\\begin{acknowledgments}\n  We would like to thank \nDaniel Sanchez-Portal and Jorge Kohanoff for interesting and intense discussions\non the problem of integrating time-dependent Kohn-Sham equations, Ivo\nSouza for discussions on the relation of this work with the Berry formalism,\nand Jonathan M. Evans for suggestions on the mathematics.\n  D. D. O'R. would like to thank S. M.-M. Dubois, A. A. Mostofi, C.-K. Skylaris, and \nM. C. Payne for helpful comments an early stage of this work.\n  Both authors would like to thank the anonymous reviewers for the careful \nreading of the manuscript and for their constructive comments, which have\nhelped to improve the paper noticeably. \n  E. A. acknowledges funding from the EU through the ElectronStopping Grant \nNumber 333813, within the Marie-Curie CIG program, and from MINECO, Spain, \nthrough Grant Number FIS2015-64886-C5-1-P.\n  D. D. O'R. gratefully acknowledges the support of the National University of Ireland \nand the School of Physics at Trinity College Dublin.\n\\end{acknowledgments}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\subsection{Dataset}\nThe dataset given the VLSP 2020's Organizer\\footnote{https:\/\/vlsp.org.vn\/vlsp2020\/eval} contains 4,372 training examples and about 1,642 examples for each public test set and private test set. Each example includes some information such as the encoded id of the owner, the text content of the post, the number of likes, shares, comments, and some photos. This dataset is really imbalanced. The unreliable class accounts for about 17\\% of the training set, the reliable class dominates with about 83\\%. This dataset also contains nearly 10\\% long texts with more than 500 tokens.\n\n\\subsection{Experimental Setup}\nFor training, we use Adam optimizer with a learning rate of 3e-5 and a weight decay of 0.01. Initially we freeze all layers of Transformer-based model to make sure the gradients are not calculated in the first epoch. We initialize each of the models described in Section \\ref{singmodel} with 10 different random states, then we train them for 20 epochs and select the models with the highest ROC-AUC score in the validation set for predicting.\n\n\\subsection{Results}\nIn public test set, all our models trained on lower-case texts. Some models fail in the validation set. Thus, we do not use them for predicting. As can be seen from Table \\ref{table:1}, the monolingual models such as viBERT, viELECTRA, and PhoBERT achieve better performance than the multilingual models. The effectiveness of length is also different for each model architecture. Specifically, viBERT with the length of 512 is worse than 256-length. Meanwhile, viELECTRA with the length of 512 significantly outperforms 256-length viELECTRA, about 3.21\\% improvement. 512-length viELECTRA also overcomes all single models, it achieves the highest score in both public test and private test. \n\nIn order to leverage the ability of the ensemble method, we select three best models with different styles including 256-length viBERT, 512-length viELECTRA, and 256-length PhoBERT to average their output probabilities, we refer this as 3-Ensemble model. In addition, we also denote the top six models with 6-Ensemble. This experiment result demonstrates the ensemble method can further boost the performance.\n\\input{result_table}\n\nAfter reviewing the data, arbitrary capitalization makes the text content unprofessional. In private test set, we investigate the influence of letter case on model performance. viELECTRA with the length of 512 and PhoBERT with the length of 256 achieve the highest score in the public test set, so we decide to train them on the raw texts which remain the upper-case and newline character. The results are described in Table \\ref{table:2} shows that the cased models significantly improve the uncased models. The ROC-AUC values being 1.49\\% and 1.02\\% above the uncased models for viELECTRA and PhoBERT. If only based on text content, it is difficult to identify fake news. In common, the news sources will be an important factor to distinguish reliable information. Because of that, we deliver the username feature into our models. The usernames with high unreliable examples will be penalized, which reduces their output probabilities by a certain value and vise versa. This intuition brings a significant improvement to 512-length cased viELECTRA, from 93.09\\% to 93.78\\%.\n\n\n\\section{Introduction}\n\\input{introduction}\n\n\\section{Methodology}\n\\input{methodology}\n\n\\section{Experiments}\n\\input{experiments}\n\n\\section{Conclusion and Future work}\n\\input{conclusion}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\printbibliography\n\\end{document}\n\n\\subsection{Data Pre-processing}\nWe process the text contents in two phases. First, we tokenize the contents using TweetTokenizer from the NLTK toolkit\\footnote{https:\/\/www.nltk.org\/} and use the emoji\\footnote{https:\/\/pypi.org\/project\/emoji\/} package to translate icons into text strings. Word segmentation is required in some pre-trained models, we use VnCoreNLP \\cite{vu-etal-2018-vncorenlp} to tokenize the input. In the text content, several features can affect model performance. To clarify this, we duplicate the texts into two versions, one is lower-case texts and no newline characters, the other is raw texts.\n\nIn the second phase, to fine-tune the Transformer-based models such as BERT Multilingual, viBERT, and viELECTRA, we must insert [CLS] special token and [SEP] special token to the input. The [CLS] is encoded including all representative information of the whole input sentence. Meanwhile, the use of [SEP] token is simply to separate different sentences of an input. In our method, we only need to insert the [SEP] token to the end of every input. Finally, we convert the new input to the sequence of indices with the same length. We will pad the sequences with the [PAD] token if their length is less than the specified length. In PhoBERT and XLM-RoBERTa, instead of [CLS], [SEP], and [PAD], we use $<$s$>$, $<$\/s$>$, $<$pad$>$ respectively.\n\n\\subsection{Models}\n\\subsubsection{Single Models}\\label{singmodel}\nWe leverage the ability of pre-trained Transformer-based models, which are trained on large-scale datasets by a variety of unsupervised learning methods. In this task, we list some popular models in the Vietnamese language. They divide into two categories following:\n\\begin{itemize}\n    \\item Multilingual: BERT Multilingual employs masked language model and next sentence prediction for pre-training. Meanwhile, XLM-RoBERTa relies on masked language modeling objective and cross-lingual language modeling objective without next sentence pre-training objective. Both models trained on a large-scale dataset with multiple languages.\n    \\item Monolingual: viBERT uses the same architecture as BERT Multilingual. However, it just trained on Vietnamese dataset. viELECTRA is released with viBERT. viELECTRA uses a new pre-training task, called replaced token detection. PhoBERT is also masked language model, which trained on 20GB Vietnamese corpus.\n\\end{itemize}\n\nWe examine the diversity of input lengths for each of the above models including 256-length, 512-length, and multiple 512-length as a long document. Handling long document is inspired by the research in \\cite{9003958}, we segment the input into multiple chunks with the length of 512 and feed them into the Transformer-based model to obtain contextual representations. Then we propagate each output through a Bi-LSTM layer to get document embedding. Finally, we perform the final classification in a linear layer. \n\n\nWe stack a linear classifier on the top of output representations. Instead of just using the last hidden layer, based on the result in \\cite{devlin2018bert}, we concatenate the last four hidden layers as the input of linear classifier, this modification might slightly improve model performance. For more clearly, Figure \\ref{fig:model} illustrates the architecture of our model in detail.\n\n\\subsubsection{Ensemble Method}\nTo utilize the robustness of the different Transformer-based models such as viBERT, viELECTRA, PhoBERT. We select three different Transformer-based models with the highest ROC-AUC score in the validation set. Then, we average their label probabilities to get final probabilities. \n\n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=0.5\\textwidth,height=8.5cm]{classification_model.pdf}\n    \\caption{Proposed Neural Network Architecture}\n    \\label{fig:model}\n\n\\end{figure}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe study of quantum systems interacting with their environment, namely open quantum systems, has gained immense importance lately. This is both due to practical reasons as well as the relevance of open quantum systems to the foundations of quantum mechanics \\cite{BPbook}. In particular, precise quantum coherent control and state preparation necessary for quantum computation and information is only possibly by taking the interaction of a quantum system with its environment, while the process of decoherence sheds light on the so-called measurement problem. Amongst the variety of tools and techniques developed to tackle the dynamics of open quantum systems, the most popular approach is the quantum master equation approach whereby a differential equation for the time evolution of the system state is obtained and solved. The basic approach is that an initial state of the system and the environment is assumed, the total system-environment Hamiltonian is written down, and then using this total Hamiltonian, the time evolution of the total system-environment state is examined. Since we are typically interested only in the system dynamics, the environment is traced out to obtain the master equation. Unfortunately, performing this process to obtain the master equation for most realistic system-environment models involves making a series of approximations. For example, the system-environment coupling strength is assumed to be weak so that the joint time evolution operator can be found perturbatively \\cite{BPbook,Weissbook}. The environment is assumed to have a short `memory time' (the Markovian approximation) - this means that the environment loses information about the system very quickly \\cite{deVegaRMP,BreuerRMP}. Finally, the initial system-environment state is assumed to a simple product state, with the system and the environment independent of one another \\cite{deVegaRMP}. \n\nWith increasingly sophisticated quantum technologies, each of the assumptions typically made in the derivation of master equations has come under renewed reexamination. Master equations that allow one to deal with stronger system-environment coupling strengths have been formulated (see, for example, Ref.~\\cite{MoussaPRA2014}). Measures of non-Markovianity have also been put forward \\cite{BreuerRMP}. Most pertinent for us, the role of the initial system-environments - the correlations present in the total system-environment state at the initial state - has been investigated widely \\cite{HakimPRA1985, HaakePRA1985, Grabert1988, SmithPRA1990, GrabertPRE1997, PazPRA1997, LutzPRA2003, BanerjeePRE2003, vanKampen2004, BanPRA2009, HanggiPRL2009, UchiyamaPRA2010, TanimuraPRL2010, SmirnePRA2010, DajkaPRA2010, ZhangPRA2010,TanPRA2011, CKLeePRE2012,MorozovPRA2012, SeminPRA2012,  ChaudhryPRA2013a,ChaudhryPRA2013b,ChaudhryCJC2013,FanchiniSciRep2014,FanSciRep2015correlation,ChenPRA2016,VegaRMP2017,VegaPRA2017,ShibataJPhysA2017,CaoPRA2017,MehwishEurPhysJD2019}. Such studies have generally been performed using exactly solvable models such as the pure dephasing model of a two-level system interacting with a collection of harmonic oscillators \\cite{MorozovPRA2012,ChaudhryPRA2013a}. Some efforts have nevertheless been made to consider master equations that include the effect of the initial correlations. For example, in Ref.~\\cite{ChaudhryPRA2013b}, the system and its environment were allowed to come to thermal equilibrium and thereafter a projective measurement was performed on the system to prepare the desired initial system state. It was shown that the effect of the initial correlations appears as an additional term in the second-order master equation, similar in form to the first term in the master equation that describes the effect of the free system-Hamiltonian. This approach was later generalized to higher order system-environment coupling strengths \\cite{ShibataJPhysA2017}. Along similar lines, in this paper, we consider the quantum system and its environment to reach a joint equilibrium state. A unitary operator is then performed on the system to prepare (approximately) a desired initial system state, and a time-local master equation, correct to second-order in the system-environment coupling strength, that describes the ensuing dynamics of the system is derived. In fact, the system Hamiltonian before the application of the system Hamiltonian can be different from the system Hamiltonian after the unitary operator - the former plays a role in the initial state preparation, while the latter plays a role in the dynamics thereafter. We show that the effect of the initial correlations is now again contained in an additional term in the master equation, but now the form of this additional term is similar to that of the second term in the usual master equation that describes relaxation and decoherence of open quantum systems. We then apply our master equation to a collection of two-level systems interacting with a common environment of harmonic oscillators. We work out the additional term in the master equation, and perform numerical simulations to show that the effect of the initial correlations increases as the number of two-level systems increases. Along similar lines, we also apply our master equation to analyze the effect of initial correlations for a collection of two-level systems interacting with a spin environment.\n\n\nThis paper is organized as follows. In Sec.~II, we derive our general time-local second-order master equation. Sec.~III discusses the application of this master equation to the large spin-boson model, while Sec.~IV applies the master equation to a collection of two-level systems interacting with a spin environment. We then conclude in Sec.~V. The appendices consist of some technical details regarding the derivation of the usual relaxation term in the master equation, as well as the details of the exactly solvable limit of the large spin-boson model.\n\n\\section{The formalism}\nWe start by briefly discussing the problem we wish to solve. We are given a quantum system which is interacting with its environment and has reached a joint equilibrium state. At the initial time, a unitary operation is performed on the system alone, and the system Hamiltonian itself may also be changed. After all, one can, for example, apply a large magnetic field in order to prepare the initial state, and thereafter alter the value of this magnetic field to generate some desired system dynamics. Our problem is to derive the master equation, correct to second-order in the system-environment coupling strength, that describes the system dynamics. We then write the system-environment Hamiltonian as \n\\begin{align}\n    H_{tot} =\n    \\begin{cases}\n      H_{S0} + H_B +\\alpha V & t\\le 0,\\\\\n      H_{S} + H_B +\\alpha V & t\\ge 0.\\\\\n    \\end{cases}   \n\\end{align}\nHere $H_S$ is the system Hamiltonian corresponding to coherent evolution of the system only after the initial time $t = 0$ at which the system state is prepared. $H_\\text{S0}$ is similar to $H_S$\nbut with possibly different parameters in order to prepare the initial system state. $H_B$ is the Hamiltonian of the environment, and $V$ corresponds to the system-environment coupling. $\\alpha$ is simply a dimensionless parameter introduced to keep track of the perturbation order; at the end of the calculation, we will set $\\alpha = 1$. Let us now discuss in detail the initial state preparation.\n\n\\subsection{Initial state preparation}\nWe let our system to come to a joint equilibrium state with the environment. What we mean by this is that the equilibrium state of the system is not simply proportional to $e^{-\\beta H_{S0}}$ - there are corrections due to the finite system-environment coupling strength. We instead consider the system and the environment together in the thermal equilibrium state proportional to  $e^{-\\beta H_{\\text{tot}}}$ with $H_{\\text{tot}} =  H_{S0} + H_B +\\alpha V$; the system state can then be obtained by simply tracing out the environment. A unitary operator $\\Omega$ is then applied to the system. The initial system-environment state is then \n\\begin{align}\n    \\rho_\\text{tot}(0)\n    &=\\frac{\\Omega e^{-\\beta H_\\text{tot}}\\Omega^\\dagger}{Z_\\text{tot}}, \\label{ref11}\n\\end{align}\nwith $Z_\\text{tot}=\\text{Tr}_\\text{S,B}\\left[e^{-\\beta H_\\text{tot}}\\right]$ the partition function and $\\text{Tr}_{S,B}$ denotes the trace over the system and the environment. Now, assuming the system-environment coupling strength to be weak, we can use the Kubo identity to expand the joint state given by Eq.~\\eqref{ref11}. The Kubo identity tells us that for any two arbitrary operators $X$ and $Y$,\n\\begin{align}\n    e^{\\beta(X+Y)}&=e^{\\beta X}\\left[1+\\alpha\\int_{0}^{\\beta} e^{-\\lambda X}Y^{\\lambda(X+Y)}d\\lambda\\right]. \\label{ref12}\n\\end{align}\nBy setting $X=-(H_\\text{S0}+H_B)$  and  $Y=-V$, and using the Kubo identity twice, we find that to second order in the system-environment coupling strength\n\\begin{align}\n    &e^{-\\beta(H_\\text{S0}+H_B+V)}=e^{-\\beta(H_\\text{S0}+H_B)}\\nonumber \\\\& -\\alpha e^{-\\beta(H_\\text{S0}+H_B)}\\int_{0}^{\\beta} e^{\\lambda(H_\\text{S0}+H_B)}V  \n    \\times e^{-\\lambda(H_\\text{S0}+H_B)}d\\lambda \\nonumber \\\\ &+\\alpha^2e^{-\\beta(H_\\text{S0}+H_B)}\\int_{0}^{\\beta}d\\lambda e^{\\lambda(H_\\text{S0}+H_B)}V  \\nonumber \\\\\n    &\\times e^{-\\lambda(H_\\text{S0}+H_B)}\\int_{0}^{\\lambda} e^{\\lambda'(H_\\text{S0}+H_B)} Ve^{-\\lambda'(H_\\text{S0}+H_B)}d\\lambda'.\\label{ref14}\n\\end{align}\nWe now write the system environment coupling $V$ as $F \\otimes B$, where $F$ and $B$ are operators living in the system and environment Hilbert space respectively. The extension to the more general case where $V = \\sum_\\alpha F_\\alpha \\otimes B_\\alpha$ is straightforward. Eq.~\\eqref{ref14} can then be simplified as\n\\begin{align} \n    &e^{-\\beta(H_\\text{S0}+H_B+V)}=\n    e^{-\\beta(H_\\text{S0}+H_B)} -\\alpha e^{-\\beta(H_\\text{S0}+H_B)}\\nonumber \\\\ &\\times \\int_{0}^{\\beta} F(\\lambda)\\otimes B(\\lambda)d\\lambda + \\alpha^2 e^{-\\beta(H_\\text{S0}+H_B)}\\nonumber\\\\\n   &\\times \\int_{0}^{\\beta}d\\lambda F(\\lambda)\\otimes B(\\lambda)\\int_{0}^{\\lambda} F(\\lambda')\\otimes B(\\lambda')d\\lambda'.\n\\end{align}\nwhere $F\\left(\\lambda\\right)=e^{\\lambda H_\\text{S0}}Fe^{-\\lambda H_\\text{S0}}$ and $B\\left(\\lambda\\right)=e^{\\lambda H_B}Be^{-\\lambda H_B}$.\nWe now use this in Eq.~\\eqref{ref11} and thereafter take the trace over the environment to find the initial system state correct to second order in the system-environment coupling strength. This is important because our aim is to derive a master equation correct to second-order in the system-environment strength. For consistency, the initial system state used to solve this master equation should also be accurate to second-order in the system-environment coupling strength. For ease of notation, we write the initial system state as\n\\begin{align}\n    \\rho(0)=\\rho^{(0)}(0)+\\rho^{(1)}(0)+\\rho^{(2)}(0),\n    \\label{sysdminitial}\n\\end{align}\nwhere,\n\\begin{align}\n    &\\rho^{(0)}(0)\n    =\\frac{\\text{Tr}_B\\left[\\Omega\\left( e^{-\\beta(H_\\text{S0}+H_B)}\\right)\\Omega^\\dagger\\right]}{Z_\\text{tot}}\\\\\n    &\\rho^{(1)}(0) \\nonumber \\\\\n    &=\\frac{\\text{Tr}_B\\left[-\\alpha \\Omega\\left(e^{-\\beta(H_\\text{S0}+H_B)}\\int_{0}^{\\beta} F(\\lambda)\\otimes B(\\lambda)d\\lambda\\right)\\Omega^\\dagger\\right]}{Z_\\text{tot}}\\\\\n    &\\rho^{(2)}(0)= \\frac{1}{{Z_\\text{tot}}}\\times \\text{Tr}_B\\Bigg[\\alpha^2 \\Omega\\Big(e^{-\\beta(H_\\text{S0}+H_B)}\\nonumber \\\\\n    & \\times \\int_{0}^{\\beta}d\\lambda F(\\lambda)\\otimes B(\\lambda) \\int_{0}^{\\lambda}F(\\lambda')\\otimes B(\\lambda')d\\lambda'\\Big)\\Omega^\\dagger\\Bigg]. \n\\end{align}\nLet us simplify these relations one by one. $\\rho^{(0)}(0)$ can be simplified as,\n\\begin{align*}\n   \\rho^{(0)}(0)=\\frac{e^{-\\beta \\widetilde{H}_\\text{S0}} Z_B}{Z_\\text{tot}},\n\\end{align*}\nwhere $\\widetilde{H}_{\\text{S0}} = \\Omega H_{\\text{S0}}\\Omega^\\dagger$ and $Z_B= \\text{Tr}_B\\left[ e^{-\\beta H_B}\\right]$. As for $\\rho^{(1)}(0)$, we can write\n\\begin{align*}\n    \\rho^{(1)}(0)= \\frac{-\\alpha Z_B\\int_{o}^{\\beta}\\Omega e^{-\\beta H_\\text{S0}}F(\\lambda)\\Omega^{\\dagger} \\big\\langle B(\\lambda)\\big\\rangle_Bd\\lambda}{Z_\\text{tot}},\n\\end{align*}\nwhere $\\langle \\hdots \\rangle_B = \\text{Tr}_B [e^{-\\beta H_B} (\\hdots)\/Z_B]$. Since $\\langle B(\\lambda)\\rangle_B$ is zero for most system-environment models, we simply get that $\\rho^{(1)}(0)=0$. Carrying on, $ \\rho^{(2)}(0)$ can be simplified as\n\\begin{align*}\n    &\\rho^{(2)}(0) = \\nonumber\\\\\n    &\\frac{\\alpha^2 Z_B\\Omega e^{-\\beta H_\\text{S0}}\\int_{0}^{\\beta}\\int_{0}^{\\lambda} F(\\lambda) F(\\lambda') \\Omega^{\\dagger} \\big\\langle B(\\lambda)  B(\\lambda')\\big\\rangle_Bd\\lambda'd\\lambda}{Z_\\text{tot}}. \n\\end{align*}\nTo proceed further, we evaluate the partition function $Z_{\\text{tot}}$. We note that $Z_{\\text{tot}}$ has to be such that the trace of the system state $\\rho(0)$ in Eq.~\\eqref{sysdminitial} is one. It is then clear that \n\\begin{align}\n    Z_\\text{tot}\n    =&Z_B\\text{Tr}_\\text{S}\\left[e^{-\\beta H_\\text{S0}} \\right] +\\alpha^2Z_B\\text{Tr}_\\text{S}\\Bigg[\\Omega e^{-\\beta H_\\text{S0}}  \\nonumber \\\\\n    &\\times\\int_{0}^{\\beta}\\int_{0}^{\\lambda} F(\\lambda) F(\\lambda') \\Omega^{\\dagger} \\big\\langle B(\\lambda)  B(\\lambda')\\big\\rangle_Bd\\lambda'd\\lambda\\Bigg]\\notag.\n\\end{align}\nIt is then clear that the initial system density matrix can be written as \n\\begin{align}\n    \\rho(0)\n    &=\\frac{e^{-\\beta \\widetilde{H}_{S0}}}{Z_{S0}Z'}\\Big[\\mathds{1}+\\int_{0}^{\\beta}\\int_{0}^{\\lambda} F^R(\\lambda)F^R(\\lambda')  \\nonumber \\\\ \n    &\\times\\big\\langle B(\\lambda)  B(\\lambda')\\big\\rangle_Bd\\lambda'd\\lambda\\Big]\n\\end{align}\nwhere $F^R(\\lambda) = \\Omega F(\\lambda)\\Omega^\\dagger$, $Z_{S0} = \\text{Tr}_S[e^{-\\beta H_{S0}}]$, and \n\\begin{align}\n    Z'=1 + \\int_{0}^{\\beta}\\int_{0}^{\\lambda} \\langle F(\\lambda) F(\\lambda') \\rangle_S  \n   \\langle B(\\lambda)  B(\\lambda')\\rangle_B d\\lambda'd\\lambda,\n\\end{align}\nwith $\\langle \\hdots \\rangle_S = \\text{Tr}_S [ e^{-\\beta H_{S0}}(\\hdots)\/Z_{S0}]$. With the initial system state, correct to second order in the system-environment coupling strength available, we now turn our attention to deriving the second-order master equation.\n\n\n\n\\subsection{Derivation of the master equation}\nWe now derive a master equation that describes the time evolution of the system for $t > 0$. The system-environment Hamiltonian is \n\\begin{align*}\n    H_\\text{tot} = H_\\text{S}+H_\\text{B}+\\alpha V \\equiv H_0 + \\alpha V.\n\\end{align*}\nNote that the system Hamiltonian $H_S$ can be different from the previous system Hamiltonian $H_{S0}$. Using perturbation theory, the unitary time evolution for such an Hamiltonian can be written as,\n\\begin{align}\n    U (t) \\approx U_0\\left(t\\right) \\left[ 1 - \\alpha \\int_0^t U_0^\\dagger (s) V U_0(s)\\,ds \\right] \\label{ref15}\n\\end{align}\nwhere, $U_0(t) \\equiv U_S(t) \\otimes U_B(t)$ is the free unitary time evolution corresponding to $H_0.$ The matrix elements of the system density matrix can be written as $\\rho_{mn}\\left(t\\right) = \\text{Tr}_S\\left[\\ket{n}\\bra{m} \\rho \\left(t\\right) \\right]$, where $\\ket{n}$ and $\\ket{m}$ are some basis states of the system. Since $\\rho(t) = \\text{Tr}_B \\rho_{\\text{tot}}(t)$, we can alternatively write\n\\begin{align*}\n    \\rho_{mn}(t) = \\text{Tr}_\\text{S,B}\\left[X_{nm}^H\\left(t\\right) \\rho_\\text{tot} (0) \\right],\n\\end{align*}\nwhere $X_{nm}^H(t) = U^\\dagger(t) (\\ket{n}\\bra{m} \\otimes \\mathds{1}_B)U(t)$. The master equation can then be written in the general form \n\\begin{align}\n    \\frac{d}{dt}\\rho_{mn}(t) &=\\text{Tr}_\\text{S,B}\\left[\\rho_\\text{tot}(0)\\frac{d}{dt}X_{nm}^H(t)\\right].\n    \\label{mastereqgen}\n\\end{align}\nTo make further progress, we note that $X_{nm}^H(t)$ is a Heisenberg picture operator. Using the Heisenberg equation of motion and Eq.~\\eqref{ref15}, it can be shown that, correct to second order in the system-environment coupling strength, \n\\begin{align} \\label{ref16}\n    \\frac{d}{dt}X_{nm}^H(t) &=i\\left[H_0^H(t),X_{nm}^H(t)\\right]+i\\alpha\\left[\\Tilde{V}(t),\\Tilde{X}_{nm}(t)\\right] \\nonumber \\\\\n    &+ \\alpha^2\\int_{0}^{t}ds\\left[\\left[\\Tilde{V}(t),\\Tilde{X}_{nm}(t)\\right],\\Tilde{V}(s)\\right],\n\\end{align}\nwhere the tildes denote time evolution under the `free' unitary operator $U_0 \\left(t\\right)$ while the superscript `$H$' denotes time evolution with the full time evolution operator. Using Eq.~\\eqref{ref16} and given the initial system-environment state $\\rho_\\text{tot}(0)$ in Eq.~\\eqref{sysdminitial}, we can derive our master equation that describes the time evolution of the quantum system by simplifying Eq.~\\eqref{mastereqgen}. The first term is very straightforward. We simply have that \n\\begin{align}\n    &\\text{Tr}_\\text{S,B}\\left[\\rho_\\text{tot}(0)i\\left[H_0^H(t),X_{nm}^H(t)\\right]\\right] \\nonumber\\\\\n    &=i\\text{Tr}_\\text{S,B}\\left[\\rho_\\text{tot}(t)\\left[H_S + H_B,(\\ket{n}\\bra{m}\\otimes \\mathds{1}_B)\\right]\\right]\\nonumber\\\\\n    &=i\\text{Tr}_\\text{S}\\left[\\rho(t)\\left[H_\\text{S},\\ket{n}\\bra{m}\\right]\\right]\\nonumber\\\\\n    &=i\\bra{m}\\left[\\rho(t),H_\\text{S}\\right]\\ket{n}. \\label{ref20}\n\\end{align}\nThis term simply tells us about free system evolution corresponding to the system Hamiltonian $H_\\text{S}$.\n\nTo calculate the next term in our master equation, that is,\n \\begin{align*}\n    i\\alpha \\text{Tr}_\\text{S,B}\\left[\\rho_\\text{tot}(0)\\left[V(t),X_{nm}(t)\\right]\\right],\n\\end{align*}\nit is useful to write the initial system-environment state as $\\rho_{\\text{tot}}(0) = \\rho_{\\text{tot}}^{(0)} + \\rho_{\\text{tot}}^{(1)}$, where, using Eq.~\\eqref{ref14}, it should be clear that \n\\begin{align}\n    \\rho_\\text{tot}^{(0)}(0)\n    &=\\frac{\\Omega e^{-\\beta(H_\\text{S0}+H_B)}\\Omega^\\dagger}{Z_\\text{tot}}  = \\widetilde{\\rho}_{S0} \\otimes \\rho_B \\label{ref18}\\\\\n    \\rho_\\text{tot}^{(1)}(0)\n    &=\\frac{-\\alpha \\Omega e^{-\\beta(H_\\text{S0}+H_B)}Q_\\text{SB}(\\beta)\\Omega^\\dagger}{Z_\\text{tot}}, \\label{ref19}\n\\end{align}\nwith $\\widetilde{\\rho}_{S0} = e^{-\\beta \\widetilde{H}_{S0}}\/Z_{S0}$, $\\rho_B = e^{-\\beta H_B}\/Z_B$, the partition function $\\tot{Z} = Z_{S0} Z_B$ and $Q_{\\text{SB}}(\\beta) = \\int_0^\\beta d\\lambda F(\\lambda)\\otimes B(\\lambda)$. We do not need the higher order terms since there is already a factor of $\\alpha$ in $i\\alpha\\left[V(t),X_{nm}(t)\\right]$. Now, the contribution of $\\rho_\\text{tot}^{(0)}$ is\n\\begin{align*}\n    &i\\alpha \\text{Tr}_\\text{S,B}\\left [\\rho_\\text{tot}^{(0)}(0)\\left[U_0^\\dagger(t)VU_0(t),U_0^\\dagger(t)X_{nm}U_0(t)\\right]\\right]\\\\\n    &=i\\alpha \\text{Tr}_\\text{S,B}\\left[\\widetilde{\\rho}_{S0}\\otimes\\rho_B U_0^\\dagger(t) \\left[F\\otimes B,\\ket{n}\\bra{m}\\otimes \\mathds{1}_B\\right] U_0(t)\\right]\\\\\n    &=i\\alpha \\text{Tr}_\\text{S}\\left [\\Tilde{\\rho}_{S0}U^\\dagger(t)\\left[F,Y_{nm}\\right]U(t)\\right ]\\times \\langle B(t)\\rangle_B.\n\\end{align*}\nAgain, since $\\langle B(t)\\rangle_B$ is usually zero for most system-environment models, this contribution turns out to be zero. \n\nThe most interesting contribution is due to $\\rho_\\text{tot}^{(1)}(0)$. This is\n\\begin{align}\n    &i\\alpha \\text{Tr}_\\text{S,B}\\Big[\\rho_\\text{tot}^{(1)}(0)\\left[U_0^\\dagger(t)VU_0(t),U_0^\\dagger(t)X_{nm}U_0(t)\\right]\\Big]\\nonumber\\\\\n    &=\\frac{-i\\alpha^2}{Z_\\text{S0}}\\int_{0}^{\\beta}\\text{Tr}_\\text{S,B}\\Big[\\rho_B{\\Omega e^{-\\beta H_\\text{S0}} F(\\lambda)\\Omega^\\dagger\\otimes  B(\\lambda)}U_S^\\dagger(t)  \\nonumber \\\\\n    & \\times\\left[F,\\ket{n}\\bra{m}\\right]U_S(t)U_B^\\dagger(t)BU_B(t)\\Big]d\\lambda\\nonumber\\\\\n    &=\\frac{-i\\alpha^2}{Z_\\text{S0}}\\int_{0}^{\\beta}\\text{Tr}_\\text{S}\\Big[{\\Omega e^{-\\beta H_\\text{S0}} F(\\lambda)\\Omega^\\dagger}U_S^\\dagger(t)\\left[F,\\ket{n}\\bra{m}\\right] \\nonumber \\\\\n    &\\times U_S(t)\\Big] \\text{Tr}_B\\Big[\\rho_B B(\\lambda)B(t)\\Big]d\\lambda\\nonumber\\\\\n    &=\\frac{-i\\alpha^2}{Z_\\text{S0}}\\int_{0}^{\\beta}\\bra{m}\\left[U_S(t)\\Omega e^{-\\beta H_\\text{S0}} F(\\lambda)\\Omega^\\dagger U_S^\\dagger(t),F\\right]\\ket{n}  \\nonumber \\\\\n    & \\times B_{\\text{corr}}(\\lambda,t)\\, d\\lambda, \\label{ref21}.\n\\end{align}\nwhere we have defined $B(t) = U_B^\\dagger (t) B U_B(t)$ and $B_{\\text{corr}}(\\lambda,t) = \\text{Tr}_B \\Big[\\rho_B B(\\lambda)B(t)\\Big]$. This is additional term in the master equation that takes into account the effect of the initial correlations, correct to second order in the system-environment coupling strength. In basis-independent form, we can write this term as \n\\begin{align}\n    -i\\left[ \\widetilde{\\rho}(t)J^R_\\text{corr}(\\beta,t),F\\right],\n\\end{align}\nwhere we have defined \n\\begin{align} \\label{ref23}\n    J^R_\\text{corr}(\\beta,t)\n    &=\\int_{0}^{\\beta}  \\overleftarrow{F}^R(\\lambda,t)\\text{B}_\\text{corr}(\\lambda,t)d\\lambda,\\\\\n    \\overleftarrow{F}^R(\\lambda,t)\n    &= U_S(t)\\Omega e^{\\lambda H_\\text{S0}}Fe^{-\\lambda H_\\text{S0}}\\Omega^\\dagger U_S^\\dagger(t).\n\\end{align}\nWe now write \n\\begin{align}\n    -i\\left[ \\widetilde{\\rho}(t)J^R_\\text{corr}(\\beta,t),F\\right] = -\\frac{i}{2}\\left(\\left[ \\widetilde{\\rho}(t)J^R_\\text{corr}(\\beta,t),F\\right] - \\text{h.c.}\\right).\n\\end{align}\nThis is permitted because $ -i\\left[ \\widetilde{\\rho}(t)J^R_\\text{corr}(\\beta,t),F\\right]$ is hermitian, so \n$\\left[ \\widetilde{\\rho}(t)J^R_\\text{corr}(\\beta,t),F\\right]$ is anti-hermitian. We now replace $\\widetilde{\\rho}(t)$ by $\\rho(t)$. This step is permitted since the corrections are of order higher than the second-order master equation that we are considering. Consequently, the term in the master equation that takes into account the initial correlations is $-\\frac{i}{2}\\left(\\left[ \\rho(t)J^R_\\text{corr}(\\beta,t),F\\right] - \\text{h.c.}\\right)$.\n\n\tWe next simplify the contribution of the third term in Eq.~\\eqref{ref16}. It is clear that now only $\\rho_{\\text{tot}}^{(0)}(0)$ contributes. Similar manipulations to those performed above lead to (see the appendix for details) \n$$ \\alpha^2\\int_{0}^{t}\\bra{m}\\Big(\\left[\\Bar{F}(t,s)\\widetilde{\\rho}_{S}(t),F\\right]C_{ts}+\\text{h.~c.}\\Big)\\ket{n}\\,ds, $$\nwhere the environment correlation function is $C_{ts} = \\langle B(t)B(s) \\rangle_B$, $\\widetilde{\\rho}_S(t) = U_S(t) \\widetilde{\\rho}_{S0} U_S^\\dagger (t)$, $\\Bar{F}(t,s)=U_S^\\dagger(t,s)FU_S(t,s)$, and $\\text{h.~c.}$ denotes the hermitian conjugate. We can further replace $\\widetilde{\\rho}_S(t)$ by $\\rho(t)$ to get \n$$ \\alpha^2\\int_{0}^{t}\\bra{m}\\Big(\\left[\\Bar{F}(t,s)\\rho(t),F\\right]C_{ts}+\\text{h.~c.}\\Big)\\ket{n}\\,ds. $$\nOnce again, this is permitted since the corrections lead to terms of higher order in the master equation (compared to the second order master equation that we are considering). We now put all the terms together to arrive at the general basis-independent form of the master equation given by \n\n\\begin{align}\n    &\\frac{d}{dt}\\rho(t)\n    =i\\left[\\rho(t),H_S\\right]-\\frac{i}{2}\\left(\\left[ \\rho(t)J^R_\\text{corr}(\\beta,t),F\\right] - \\text{h.c.}\\right)\\nonumber\\\\\n    &+\\int_{0}^{t}\\left(\\left[\\Bar{F}(t,s)\\rho(t),F\\right]C_{ts}+\\text{h.c.}\\right)\\,ds.\n     \\label{finalme}\n\\end{align}\n\n\n\n\n\n\n\n\\begin{comment}\n\\subsubsection{$\\Omega$ as a mixed state projection operator $\\ket{\\psi_i}\\bra{\\psi_i}$}\nFirst and third term will be totally unchanged and while second term will have some modifications, Eq. \\eqref{ref21} under the action projection operator and after some those little modifications, the second term turns out to be.\n\\begin{align*}\n    &-i\\int_{o}^{\\beta}\\frac{1}{\\mathcal{Z'}}\\sum_{i=1}^{n}P_i\\bra{\\psi_i} e^{-\\beta H_\\text{S0}} F(\\lambda)\\ket{\\psi_i} \\text{B}_{\\text{corr}}(\\lambda,t)d\\lambda\\\\\n    &=-i\\bra{m}\\left[\\rho(t),F\\right]\\ket{n}\\mathcal{F}_\\text{corr},\n    \\end{align*}\nhence overall master equation in operator form become\n\\begin{align}\n    &\\frac{d}{dt}\\rho(t)\n    =i\\left[\\rho(t),H_S\\right]-i\\left[\\rho(t),F\\right]\\mathcal{F}_\\text{corr}\\nonumber\\\\\n    &+\\int_{0}^{t}\\left(\\left[\\Bar{F}(t,s)\\rho(t),F\\right]C_\\text{ts}+\\text{hc}\\right)ds\n\\end{align}\nwith\n\\begin{align}\n    \\mathcal{F}_\\text{corr}(t)\n    &=\\int_{o}^{\\beta}\\sum_{i=1}^{n}\\frac{P_i\\bra{\\psi_i} e^{-\\beta H_\\text{S0}} F(\\lambda)\\ket{\\psi_i} \\text{B}_{\\text{corr}}(\\lambda,t)d\\lambda}{\\mathcal{Z'}}\\\\\n    \\mathcal{Z'} \n    &= \\bra{\\psi_i} e^{-\\beta H_\\text{S0}} \\ket{\\psi_i}\n    \\label{finalme}\n\\end{align}\n\\end{comment}\n\n\n\n\n\n\n\\section{Application to the large spin-boson model}\nWe will now apply our derived master equation to a variant of the paradigmatic spin-boson model \\cite{BPbook} with a large number of two-level systems interacting with a common environment of harmonic oscillators \\cite{KurizkiPRL20112,ChaudhryPRA2013a,ChaudhryPRA2013b}. Recall that the total system-environment Hamiltonian is given by $H_\\text{tot} = H_\\text{S0}+H_\\text{B}+V$ for $t < 0$, while $H_{\\text{tot}} = H_S+H_B+V$ for $t \\geq 0$. For the large spin-boson model, we write\n\\begin{align}\n    H_\\text{S0}\n    &= \\varepsilon_0 J_z + \\Delta_0 J_x\\\\\n    H_S&= \\varepsilon J_z + \\Delta J_x\\\\\n    H_B&= \\sum_{k} \\omega_k b_{k}^{\\dagger} b_k,\\\\\n    V&=J_z \\sum_k \\big( g_k^* b_k + g_k b_k^\\dagger \\big)\n\\end{align}\nwhere $J_x,J_y,J_z$ are the collective spin operators with $J^2 =J_x^2+J_y^2+J_z^2, \\varepsilon$ is energy bias, $\\Delta$ is the tunneling amplitude, $H_B$ is the bath of harmonic oscillators (we are ignoring zero point energy), while $V$ describes the interaction between the common harmonic oscillator bath and the spin system. We have set $ \\hbar =1 $ throughout and the values of other parameters will be in dimensionless units. Note that the system operator $F=J_z,$ and the bath operator $B=\\sum_k \\big( g_k^* b_k + g_k b_k^\\dagger \\big). $ One imagine that the large-spin system has been interacting with the environment for a long time with a relatively large value of $\\varepsilon_0$ and a small value of $\\Delta_0$. In such a situation, the state of the system will be approximately corresponding to the state with all spins down in the $z$-direction. At time $t = 0$, we then apply a unitary operator to prepare the desired initial state. For example, if the desired initial state is one with all spins in the $x$-direction. then the unitary operator that should be applied is $U_R = e^{i\\pi J_y\/2}$. In other words, a $\\frac{\\pi}{2}$-pulse is used to prepare the initial system state, with the assumption that this pulse takes a very short time to be applied. With the initial state approximately prepared, we can then change the parameters to the system Hamiltonian to whatever values we desire to generate any required system evolution. Let us look at how the initial system-environment correlations appear in this system evolution using our general master equation. \n\nOur objective is to calculate the operator $J^R_{\\text{corr}}$. To do so, we first find \n\\begin{align*}\n    &\\overleftarrow{F}^R(\\lambda,t)=U(t)\\left[U_R\\left(e^{\\lambda H_{S0}}Fe^{-\\lambda H_{S0}}\\right)U^\\dagger_R\\right]U^\\dagger(t)\\\\\n    &=J_x\\left[a_xd_x+a_yc_x-a_z b_x\\right]+J_y\\left[a_xd_y+a_yc_y-a_z b_y\\right]\\\\\n    &+J_z\\left[a_xd_z+a_yc_z-a_zb_z\\right],\n\\end{align*}\nwith \n\\begin{align*}\n       a_x\n    &=\\frac{\\varepsilon_0\\Delta_0}{\\Delta'^2} \\left\\{1-\\cosh\\left({\\lambda\\Delta'} \\right)\\right\\},\\\\\n    a_y\n    &=\\frac{-i\\Delta_0}{\\Delta'}\\sinh \\left(\\lambda\\Delta'\\right),\\\\\n    a_z\n    &= \\frac{\\varepsilon_0^2 + \\Delta^2_0 \\cosh \\left({\\lambda\\Delta'}\\right)}{\\Delta'^2},\\\\\n     b_x\n    &= \\frac{\\Delta^2 + \\varepsilon^2 \\cos \\left({\\widetilde{\\Delta}t}\\right)}{\\widetilde{\\Delta}^2},\\\\\n    b_y\n    &= \\frac{ \\varepsilon}{\\widetilde{\\Delta}}\\sin \\left( \\widetilde{\\Delta}t\\right),\\\\\n    b_z\n    &= \\frac{\\varepsilon \\Delta}{\\widetilde{\\Delta}^2} \\left\\{ 1 - \\cos\\left({\\widetilde{\\Delta}t} \\right)\\right\\},\\\\\n    c_x \n    &= -\\frac{\\varepsilon }{\\widetilde{\\Delta}}\\sin{\\left(\\widetilde{\\Delta}t\\right)},\\\\\n    c_y\n    &= \\cos{\\left(\\widetilde{\\Delta}t\\right)},\\\\\n    c_z\n    &= \\frac{\\Delta }{\\widetilde{\\Delta}}\\sin{\\left(\\widetilde{\\Delta}t\\right)},\\\\\n     d_x &= \\frac{\\varepsilon\\Delta }{\\tilde{\\Delta}^2}\\left\\{1-\\cos{\\left(\\widetilde{\\Delta}t\\right)}\\right\\},\\\\\n    d_y &=-\\frac{\\Delta }{\\widetilde{\\Delta}}\\sin{\\left(\\widetilde{\\Delta}t\\right)},\\\\\n    d_z &= 1+\\frac{\\Delta^2 }{\\tilde{\\Delta}^2}\\left\\{\\cos{\\left(\\widetilde{\\Delta}t\\right)}-1\\right\\}.\n\\end{align*}\nHere $\\Delta'^2 = \\varepsilon_0^2 + \\Delta_0^2$ and $\\widetilde{\\Delta}^2 = \\varepsilon^2 + \\Delta^2$. In short, \n\\begin{align} \\label{ref24}\n    \\overleftarrow{F}^R(\\lambda,t)=\\alpha_1(\\lambda,t)J_x+\\alpha_2(\\lambda,t)J_y+\\alpha_3(\\lambda,t)J_z,\n\\end{align}\nwhere,\n\\begin{align*}\n    \\alpha_1(\\lambda,t)\n    &=a_xd_x+a_yc_x-a_z b_x,\\\\\n    \\alpha_2(\\lambda,t)\n    &=a_xd_y+a_yc_y-a_z b_y,\\\\\n    \\alpha_3(\\lambda,t)\n    &=a_xd_z+a_yc_z-a_zb_z.\n\\end{align*}\nIt then follows that \n\\begin{align}\n\\label{initialcorrelations}\n    J_\\text{corr}^R(\\beta,t)\n    &=P(\\beta,t)J_x+Q(\\beta,t)J_y+R(\\beta,t)J_z,\n\\end{align}\nwith,\n\\begin{align*}\n    P(\\beta,t)\n    &=\\int_{0}^{\\beta}  \\alpha_1(\\lambda,t)B_{\\text{corr}}(\\lambda,t)d\\lambda\\\\\n    Q(\\beta,t)\n    &=\\int_{0}^{\\beta}  \\alpha_2(\\lambda,t)B_{\\text{corr}}(\\lambda,t)d\\lambda\\\\\n    R(\\beta,t)\n    &=\\int_{0}^{\\beta}  \\alpha_3(\\lambda,t)B_{\\text{corr}}(\\lambda,t)d\\lambda.\n\\end{align*}\nWe now calculate bath correlation term $B_{\\text{corr}}(\\lambda,t)$. First, \n\\begin{align}\n    B(t)=\\sum_k\\left(g_k^*e^{-i\\omega_{k}t}b_k+g_ke^{i\\omega_{k}t}b_k^\\dagger\\right),\n\\end{align}\nand,\n\\begin{align}\n    B(\\lambda)=\\sum_k\\left(g_k^*e^{-\\lambda\\omega_{k}}b_k+g_ke^{\\lambda\\omega_{k}}b_k^\\dagger\\right).\n\\end{align}\nSince $B_{\\text{corr}}(\\lambda,t) =\\text{Tr}[\\rho_B B(\\lambda) B(t)]$, we find \n\\begin{align}\n    B_{\\text{corr}}(\\lambda,t)&=\\sum_k  |g_k|^2\\Big\\{e^{-\\omega_{k}\\left(\\lambda-it\\right)}\\nonumber \\\\& +2n_k\\cosh{\\left(\\lambda\\omega_{k}-i\\omega_{k}t\\right)}\\Big\\},\\label{ref26}\n\\end{align}\nwith $n_k$ given by Bose-Einstein statistics as \n\\begin{align}\n    n_k&=\\frac{1}{2}\\left\\{\\coth{\\left(\\frac{\\beta\\omega_k}{2}\\right)}-1\\right\\}\\nonumber.\n\\end{align}\nTo perform the sum over the environment modes, we use the spectral density $J(\\omega)$ via $\\sum_k |g_k|^2 (\\hdots) \\rightarrow \\int_0^\\infty d\\omega \\, J(\\omega) (\\hdots)$. We generally use an Ohmic spectral density of the form $J(\\omega) = G\\omega e^{-\\omega\/\\omega_c}$. The integrals are performed numerically to find $J_{\\text{corr}}(\\beta,t)$, and the results are incorporated in the numerical simulations of the master equation. We start by first looking at the pure dephasing case ($\\Delta=\\Delta_0=0$), since this case can be solved exactly and serves as a useful benchmark (see the appendix for details regarding the exact solution). We illustrate our results in Fig.~\\ref{Puredephasing-N=1-unitary} for $N = 1$. Two points should be noted. First, the role played by initial correlations is very small in this case. Second, our master equation reproduces the exact results very well. Since we expect that the role of the initial correlations increases with increasing $N$, we next look at $N = 4$ and $N = 10$. Results are shown in Figs.~\\ref{Puredephasing-N=4-unitary} and \\ref{Puredephasing-N=10-unitary}. It is clear that as $N$ increases, the initial correlations play a larger and larger role. Moreover, the extra term in our master equation is able to take into account the effect of the initial correlations very well.\n\n\n\\begin{figure}[htp]\n  \\centering \n  \\includegraphics[width=0.95\\linewidth]{Puredephasing-N=1-unitary.eps}\n\\caption{Behavior of $j_x$ versus $t$ for N = 1 using the exact solution with (blue circled dot) and without (purple squares) initial correlations, as well as using the master\nequation with (solid, black line) and without (dashed, red line)initial correlations. We have used $\\varepsilon=\\varepsilon_0=4$, $G = 0.05$, $\\beta=1$ and $\\omega_c=5$. Here and in all other figures, the plotted variables are all in dimensionless units} \\label{Puredephasing-N=1-unitary}\n\\end{figure} \n \\begin{figure}[htp]\n  \\centering \n  \\includegraphics[width=0.95\\linewidth]{Puredephasing-N=4-unitary.eps}\n\\caption{Same as Fig. \\ref{Puredephasing-N=1-unitary}, except that we now have $N = 4$.}\n\\label{Puredephasing-N=4-unitary}\n\\end{figure}\n\\begin{figure}[htp]\n  \\centering \n  \\includegraphics[width=0.95\\linewidth]{Puredephasing-N=10-unitary.eps}\n\\caption{Same as Fig. \\ref{Puredephasing-N=4-unitary}, except that we now have $N = 10$.} \\label{Puredephasing-N=10-unitary}\n\\end{figure}\n\nHaving shown that our master equation is able to reproduce results for the pure dephasing model, we are now in a position to go beyond the pure dephasing model and see the effects of the initial correlations. In Fig.~\\ref{Beyond-PD-N=2}, we have shown the dynamics of $j_x$ with a non-zero value of the tunneling amplitude for $N = 2$. It is clear that the initial correlations do have a small influence on the dynamics. This effect becomes more pronounced as we increase $N$ (see Figs.~\\ref{Beyond-PD-N=4} and \\ref{Beyond-PD-N=10}). We have also looked at how the role played by the initial correlations changes as the temperature changes. To this end, we compare Fig.~\\ref{Beyond-PD-N=10}, where the inverse temperature is $\\beta = 1$, with Fig.~\\ref{Beta=0.5} where $\\beta = 0.5$ and Fig.~\\ref{Beta=1.5} where $\\beta = 1.5$. As expected, at higher temperatures, the effect of the initial correlations decreases, while at lower temperatures, the effect of the initial correlations increases. \\\\\n\\begin{figure}[htp]\n  \\centering \n  \\includegraphics[width=0.95\\linewidth]{Beyond-PD-N=2new.eps}\n\\caption{Behavior of $j_x$ against $t$ for $N=2$ with (black, solid) and without (dashed, red) taking into\naccount initial correlations. Here we have used $\\varepsilon_0 = 4$, $\\varepsilon=2.5$ and $\\Delta=\\Delta_0=0.5$, while the rest of the parameters used are the same as Fig. \\ref{Puredephasing-N=1-unitary}.} \\label{Beyond-PD-N=2}\n\\end{figure}\n\n\\begin{figure}[htp]\n  \\centering \n  \\includegraphics[width=0.95\\linewidth]{Beyond-PD-N=4new.eps}\n\\caption{Same as Fig. \\ref{Beyond-PD-N=2}, except that we now have $N = 4$.} \\label{Beyond-PD-N=4}\n\\end{figure}\n\n\\begin{figure}[htp]\n  \\centering \n  \\includegraphics[width=0.95\\linewidth]{Beyond-PD-N=10.eps}\n\\caption{Same as Fig. \\ref{Beyond-PD-N=4}, except that we now have $N = 10$.} \\label{Beyond-PD-N=10}\n\\end{figure}\n\\begin{figure}[htp]\n  \\centering \n  \\includegraphics[width=0.95\\linewidth]{fig6-beta=0.5new.eps}\n\\caption{Behavior of $j_x$ against $t$ for $N=10$ with (black, solid) and without (dashed, red) taking into account initial correlations. Here we have used the same parameters as Fig.~\\ref{Beyond-PD-N=10}, except that $\\beta = 0.5$.} \\label{Beta=0.5}\n\\end{figure}\n\\begin{figure}[htp]\n  \\centering \n  \\includegraphics[width=0.95\\linewidth]{fig6beta=1.5new.eps}\n\\caption{Same as Figs.~\\ref{Beyond-PD-N=10} and \\ref{Beta=0.5}, except that we now have $\\beta=1.5$.} \\label{Beta=1.5}\n\\end{figure}\n\nWe now show that the effect of initial correlations are not manifested in the dynamics of $j_x$ alone, we show in Figs.~\\ref{Jx^2-plot-for-N=4} and \\ref{Jx^2-plot-for-N=10} the dynamics of $j_x^{(2)}\\equiv 4\\expval{J_x^2}\/N^2$. Such an observable is relevant in the study of spin squeezing and entanglement. Once again, the initial correlations can affect the dynamics significantly.\n\\begin{figure}[htp]\n  \\centering \n  \\includegraphics[width=0.95\\linewidth]{Jx_2-plot-for-N=4.eps}\n\\caption{Behavior of $j_x^{(2)}$ against $t$ for $N = 4$\nwith (black, solid) and without (dashed, red) taking into account initial correlations. The rest of the parameters used are the same as those in Fig.~\\ref{Beyond-PD-N=2}.} \\label{Jx^2-plot-for-N=4}\n\\end{figure}\n\\begin{figure}[htp]\n  \\centering \n  \\includegraphics[width=0.95\\linewidth]{Jx_2-plot-for-N=10.eps}\n\\caption{Same as Fig. \\ref{Jx^2-plot-for-N=4}, except that we now have $N=10$.} \\label{Jx^2-plot-for-N=10}\n\\end{figure}\n\nFinally, let us also demonstrate the effect of the initial correlations with sub-Ohmic environment, that is, $J(\\omega) = G\\omega^s \\omega_c^{1 - s} e^{-\\omega\/\\omega_c}$. Since these environments have longer correlation times, we expect that the effect of the initial correlations will be greater as well. This is indeed the case, as can be seen by comparing Figures \\ref{subOhmic1} and \\ref{subOhmic2} with Figs.~\\ref{Beyond-PD-N=4} and Figs.~\\ref{Beyond-PD-N=10} where an Ohmic environment had been used. \n\n\\begin{figure}[htp]\n  \\centering \n  \\includegraphics[width=0.95\\linewidth]{subOhmic-S=0.5,J=2.eps}\n\\caption{Behavior of $j_x$ against $t$ for $N=4$ with (black, solid) and without (dashed, red) taking into\naccount initial correlations. Here we have used a sub-Ohmic environment with $s = 0.5$. We also have $\\varepsilon_0 = 4$, $\\varepsilon=2.5$ and $\\Delta=\\Delta_0=0.5$, while the rest of the parameters used are the same as Fig. \\ref{Puredephasing-N=1-unitary}.} \\label{subOhmic1}\n\\end{figure}\n\n\\begin{figure}[htp]\n  \\centering \n  \\includegraphics[width=0.95\\linewidth]{subOhmic-S=0.5,J=5.eps}\n\\caption{Same as Fig.~\\ref{subOhmic1}, except that now $N = 10$.} \\label{subOhmic2}\n\\end{figure}\n\n\n\n\\section{Application to the spin-spin environment model}\n\nWe consider now a collection of identical two-level systems interacting with an environment consisting of two-level systems \\cite{cucchietti2005decoherence,CamaletPRB2007, Schlosshauerbook,VillarPhysLettA2009,SegalJCP2014}. We then have\n\\begin{align*}\n    H_\\text{S0}\n    &= \\varepsilon_0 J_z + \\Delta_0 J_x\\\\\n    H_S&= \\varepsilon J_z + \\Delta J_x\\\\\n    H_B&= \\sum_{k} \\frac{\\omega_k}{2} \\sigma_{x}^{(k)},\\\\\n    V&=J_x \\otimes \\sum_k g_k \\sigma_{z}^{(k)}\n\\end{align*}\nwhere $\\sigma_{z}^{(k)}$ and $ \\sigma_{x}^{(k)}$ are the Pauli $z$-spin and $x$-spin operators\nof the $k$th environment spin respectively, $\\omega_k$ denotes the tunneling matrix element for the $k^{\\text{th}}$ environment spin and $g_k$ quantifies the coupling strength.\\\\\nThe different environment leads to a different correlation function $C_{ts}$ as well as a different factor $J^R_\\text{corr}(\\beta,t)$ that takes into account the effect of the initial system-environment correlations. We first note that   \n\\begin{align}\n    B_{\\text{corr}}(\\lambda,t)&=\\sum_k  |g_k|^2\\Big\\{\\tanh{\\left(\\frac{\\beta \\omega_k}{2}\\right)}e^{-\\omega_{k}\\left(\\lambda-it\\right)}\\nonumber\\\\\n    &+2n_k\\sinh{\\left(\\lambda\\omega_{k}-i\\omega_{k}t\\right)}\\Big\\},\n\\end{align}\nwith, \n\\begin{align}\n    n_k=\\frac{1}{2}\\left\\{\\coth{\\left(\\frac{\\beta\\omega_k}{2}\\right)}-1\\right\\}\\nonumber.\n\\end{align}\nSince the factors $\\alpha_1(\\lambda,t)$, $\\alpha_2(\\lambda,t)$, and $\\alpha_3(\\lambda,t)$ are the same as before, this allows us to work out the role of the initial correlations [see Eq.~\\eqref{initialcorrelations}]. In a similar manner, the environment correlation function can also be worked out. As before, to perform the sum over the environment modes, we\nuse the same procedure as mentioned earlier, that is, $\\sum_k |g_k|^2 (...) \\rightarrow \\int_0^{\\infty}d\\omega J(\\omega) (...)$. Results are shown in Figs.~\\ref{spinenv1} and \\ref{spinenv2}. Once again, the role of the initial correlations is relatively small for a smaller value of $N$. However, as $N$ increases, it is clear that we need to take into account the role of the initial correlations to obtain an accurate picture of the system dynamics.\n\n\\begin{figure}[htp]\n  \\centering \n  \\includegraphics[width=0.95\\linewidth]{Spin-E-Ohmic-J=2.eps}\n\\caption{Behavior of $j_x$ against $t$ for $N = 4$\nwith (black, solid) and without (dashed, red) taking into account initial correlations. Here we have $\\varepsilon_0 = 4$, $\\varepsilon=2.5$, $\\Delta=\\Delta_0=0.5$, $G = 0.05$, $\\beta=1$ and $\\omega_c=5$.} \\label{spinenv1}\n\\end{figure}\n\n\\begin{figure}[htp]\n  \\centering \n  \\includegraphics[width=0.95\\linewidth]{Spin-E-Ohmic-J=5.eps}\n\\caption{Same as Fig.~\\ref{spinenv1}, except that now $N = 10$.} \\label{spinenv2}\n\\end{figure}\n\n\n\\section{Conclusion}\nTo conclude, we have shown that if we start from the joint thermal equilibrium state of a quantum system and its environment, and then apply a unitary operation to the system to prepare the system quantum state, the initial correlations that exist in the joint thermal equilibrium state influence the subsequent dynamics of the system. We have derived a time-local master equation, correct to second-order in the system-environment coupling strength, that takes into account the effect of these correlations, showing therefore that one need not necessarily be in the strong system-environment coupling regime to observe the effects of the initial correlations. The structure of this master equation is very interesting, as the form of the term that takes into account the initial correlations is the same as the relaxation and dephasing term. In this sense, one can say that the initial correlations affect the decoherence and dephasing rates, a fact which has already been pointed out in studies of the role of initial correlations in pure dephasing models \\cite{MorozovPRA2012}. Finally, we actually applied our master equation to the large spin-boson model as well as to a collection of two-level systems interacting with a spin environment to quantitatively investigate the role of the initial correlations. We found that when the number of spins is small, then the initial correlations do not play a significant role. However, for a larger number of spins, the initial correlations must be accounted for in order to explain the dynamics accurately. \n\n\n\n\n\\section*{acknowledgements}\nA.~R.~M, M.~Z.~and A.~Z.~C. are grateful for support from HEC under grant No 5917\/Punjab\/NRPU\/R\\&D\/HEC\/2016. Support from the LUMS Faculty Initiative Fund is also acknowledged.\n\n\n\n\\begin{comment}\n\\subsection{State preparation via projection operator}\nFor an initial state prepared by a pure state mixed operator, work has been already done by [citation.]. As mentioned earlier, the initial state prepared using a pure state projection operator is given by Eq.~\\eqref{pure-projection-state.}.\nWe are considering the work done by [citation] as a benchmark and therefore, we are using the similar Hamiltonian to verify the results in this and the next sub-section. The total system-environment Hamiltonian can be written as:\n\\begin{align*}\n    H=H_S + H_B +V.\n\\end{align*}\nwhere, $H_S$ and $H_B$ are same as used previously, whereas, \n\\begin{align*}\n    V= J_x \\sum_k \\left( g_k^* b_k + g_k b_k^\\dagger \\right)\n\\end{align*}\nBy choosing an initial state $\\ket{\\psi}$, such that $J_Z\\ket{\\psi}= -\\frac{N}{2}\\ket{\\psi}$, it can be shown that,\n\\begin{align*}\n    &f_\\text{corr}(t)= N \\sum_k \\abs{g_k}^2 \\cos\\left(\\omega_k t\\right) \\times \\nonumber \\\\\n    &\\left[ \\frac{A}{\\omega_k} + \\frac{D}{\\tilde{\\Delta}^2-\\omega_k^2}\\left\\{\\tilde{\\Delta} \\coth\\left(\\frac{\\beta \\omega_k}{2} \\right)-\\omega_k \\coth\\left(\\frac{\\beta\\tilde{\\Delta}}{2}\\right)\\right\\} \\right],\n\\end{align*}\nwith,\n\\begin{align*}\n    A &= -\\frac{\\Delta}{\\tilde{\\Delta}} \\frac{\\tilde{\\Delta} + \\varepsilon \\coth\\left(\\beta\\tilde{\\Delta}\/2\\right)}{\\tilde{\\Delta} \\coth\\left(\\beta\\tilde{\\Delta}\/2\\right) + \\varepsilon},\\\\\n    D&= \\frac{\\varepsilon \\Delta \\tilde{\\Delta}}{\\tilde{\\Delta} \\coth\\left(\\beta\\tilde{\\Delta}\/2\\right) + \\varepsilon }.\n\\end{align*}\nIt can also be shown that, $f_\\text{corr}\\left( t\\right)$ is real.\n\\subsection{State preparation via mixed state projection operator}\nThis is a more generalized case of what we have done in previous sub-section. By choosing an initial state $\\ket{\\psi}= P_1 \\ket{\\psi_1} + P_2 \\ket{\\psi_2} $, such that $J_Z\\ket{\\psi_1}= -\\frac{N}{2}\\ket{\\psi_1}$ and $J_Z\\ket{\\psi_2}= \\frac{N}{2}\\ket{\\psi_2}$. By generalizing the previous results, it can be shown that,\n\\begin{align}\n    f_\\text{corr}(t) = f_{\\text{corr},1} +  f_{\\text{corr},2}\n\\end{align}\nwhere,\n\\begin{align}\n    &f_{\\text{corr},1}= P_1 N \\sum_k \\abs{g_k}^2 \\cos\\left(\\omega_k t\\right) \\times \\nonumber \\\\\n    &\\left[ \\frac{A_1}{\\omega_k} + \\frac{D_1}{\\tilde{\\Delta}^2-\\omega_k^2}\\left\\{\\tilde{\\Delta} \\coth\\left(\\frac{\\beta \\omega_k}{2} \\right)-\\omega_k \\coth\\left(\\frac{\\beta\\tilde{\\Delta}}{2}\\right)\\right\\} \\right],\n\\end{align}\n\\begin{align}\n    &f_{\\text{corr},2}= P_2 N \\sum_k \\abs{g_k}^2 \\cos\\left(\\omega_k t\\right) \\times \\nonumber \\\\\n    &\\left[ \\frac{A_2}{\\omega_k} + \\frac{D_2}{\\tilde{\\Delta}^2-\\omega_k^2}\\left\\{\\tilde{\\Delta} \\coth\\left(\\frac{\\beta \\omega_k}{2} \\right)-\\omega_k \\coth\\left(\\frac{\\beta\\tilde{\\Delta}}{2}\\right)\\right\\} \\right],\n\\end{align}\nwith,\n\\begin{align}\n    A_1 &= -\\frac{\\Delta}{\\tilde{\\Delta}} \\frac{\\tilde{\\Delta} + \\varepsilon \\coth\\left(\\beta\\tilde{\\Delta}\/2\\right)}{\\tilde{\\Delta} \\coth\\left(\\beta\\tilde{\\Delta}\/2\\right) + \\varepsilon},\\\\\n    D_1&= \\frac{\\varepsilon \\Delta \\tilde{\\Delta}}{\\tilde{\\Delta} \\coth\\left(\\beta\\tilde{\\Delta}\/2\\right) + \\varepsilon },\\\\\n    A_2 &= -\\frac{\\Delta}{\\tilde{\\Delta}} \\frac{\\tilde{\\Delta} - \\varepsilon \\coth\\left(\\beta\\tilde{\\Delta}\/2\\right)}{\\tilde{\\Delta} \\coth\\left(\\beta\\tilde{\\Delta}\/2\\right) - \\varepsilon},\\\\\n    D_2&= -\\frac{\\varepsilon \\Delta \\tilde{\\Delta}}{\\tilde{\\Delta} \\coth\\left(\\beta\\tilde{\\Delta}\/2\\right) - \\varepsilon }.    \n\\end{align}\nIt can also be shown that, $f_\\text{corr}\\left( t\\right)= f_{\\text{corr},1} +  f_{\\text{corr},2}$ is real.\n\n\\end{comment}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{}\n\\begin{abstract}\nThe study of non-linearity (linearity) of Boolean function was initiated by Rothaus in 1976. The classical non-linearity of a Boolean function is the minimum Hamming distance of its truth table to that  of affine functions. \nIn this note we introduce new \"multidimensional\"  non-linearity parameters $(N_f,H_f)$ for conventional and vectorial Boolean functions $f$ with $m$ coordinates in $n$ variables.\n The classical non-linearity may be treated as a 1-dimensional parameter in the new definition.  $r$-dimensional parameters for $r\\geq 2$ are relevant to possible multidimensional extensions of the Fast Correlation Attack in stream ciphers and Linear Cryptanalysis in block ciphers. Besides we introduce a notion of  optimal vectorial Boolean functions relevant to the new parameters. For $r=1$ and even $n\\geq 2m$ optimal Boolean functions  are exactly  perfect nonlinear functions (generalizations of Rothaus' bent functions) defined by Nyberg in 1991. By a computer search we find that this property  holds for $r=2, m=1, n=4$ too. That is an open problem for larger $n,m$ and $r\\geq 2$. The definitions may be easily  extended to $q$-ary functions.       \n\\end{abstract}\n\\section{Conventional Boolean Functions}\\label{Boolean}\nLet $f(x)=f(x_1,\\ldots,x_n)$ be a Boolean function (takes $0,1$-values) in $n$ Boolean variables $x=(x_1,\\ldots,x_n)$. Let   $v$ denote its weight (the number of values $1$ in the truth table). One constructs a probability distribution $p$ on binary $n$-strings, such that $p_x=1\/v$ if $f(x)=1$ and $p_x=0$ otherwise.\n\nLet $r$ be a fixed number $1\\leq r\\leq n$ and  $U$  an $r\\times n$ binary matrix of rank $r$. The matrix defines a linear transform from the space of $n$-bit strings to the space of $r$-bit strings. That induces  \n  a probability distribution $q$ on binary $r$-strings. \n Namely, $q_y=\\sum_{y=Ux}p_x$, where the sum is over  $x$ such that $y=Ux$.  The distribution $q$ depends on $U$. For $r=n$ the distribution $q$ is a permuted distribution $p$ with a linear permutation defined by $U$.\n \n  For $r=1$ the distributions (in a slightly different form) are used in  Correlation and Fast Correlation Attacks in stream ciphers, see \\cite{MS}.\n  The efficiency of  Correlation  \nAttacks,  e.g., for a Filter Generator with a filtering function $f$, depends on the probability\n$\\mathbf{Pr}(Ux=f(x)).$ By the definition of $q=(q_0,q_1)$, one gets $\\mathbf{Pr}(Ux=1,f(x)=1)=vq_1\/2^n$ and $\\mathbf{Pr}(Ux=0,f(x)=0)=1\/2-vq_0\/2^n$. \nSo \n\\begin{eqnarray}\n\\mathbf{Pr}(Ux=f(x))&=&\\mathbf{Pr}(Ux=1,f(x)=1)+\\mathbf{Pr}(Ux=0,f(x)=0)=\\frac{1}{2}+\\frac{v(q_1-q_0)}{2^n}.\\nonumber\\end{eqnarray}\n For $r\\geq 2$ the distribution $q$ may potentially    \n    be used in multidimensional extensions of Correlation Attacks. \n    \n In cryptanalysis one may want to distinguish non-uniform distributions from uniform. The number of zero values of $q_y$ denoted $N_q$ and the entropy of $q$ on its support denoted $H_q$ are relevant parameters. \n  For a fixed $r$ the distributions $q$ may be partitioned into  classes by equivalence, where $q_1$ and $q_2$ are equivalent if $N_{q_1}=N_{q_2}$ and $H_{q_1}=H_{q_2}$. Obviously, the number of zero values provides with a stronger distinguisher than the entropy. So we define\n   an order on classes $\\{q\\}$ induced by the relation  \n    $\\{q_1\\}>\\{q_2\\}$ which holds  if $N_{q_1}>N_{q_2}$ or if $N_{q_1}=N_{q_2}$ and $H_{q_1}<H_{q_2}$. The parameters $(N_q,H_q)$ of the largest (according to $>$) class we call $r$-dimensional non-linearity of $f$. They are  denoted $(N_f,H_f)$.  Let, for instance,  $r=n$ then $(N_f,H_f)=(2^n-v,\\ln(v))$.   It is easy to see that $r$-dimensional non-linearity of $f$ is invariant under  affine change of variables in $f$. \n      \n       Let $r=1$, then $Ux$ is a non-zero linear  function and \n  \\begin{eqnarray}\nq_0-q_1&=&\\sum_xp_x\\,(-1)^{Ux}=\\frac{1}{v}\\sum_{x}f(x)\\,(-1)^{Ux}\\nonumber\\\\\n&=&\\frac{1}{v}\\sum_{x}\\frac{1-(-1)^{f(x)}}{2}\\,(-1)^{Ux}=\\frac{-2^{n-1}}{v}W_U\n.\\nonumber\n\\end{eqnarray}\nThe numbers\n$W_a=\\frac{1}{2^n}\\sum_{x}(-1)^{f(x)+ax}$, where $a$ are encoded by binary $n$-strings, may be called Walsh-Hadamard spectrum of $f$. Also $(-1)^{f(x)}=\\sum_{a}W_a\\,(-1)^{ax}$. \n\n It is well known and easy to prove that  $W_a=\\frac{v_a-2^{n-1}}{2^{n-1}}$, where $v_a$ is the number of $x$ such that $f(x)= ax$. \n  Minimum distance of $f$ to  affine functions (classical non-linearity of $f$)  is defined by\n $$d_f=\\min_{a}(v_a,2^n-v_a)=2^{n-1}(1-\\max_{a }|W_a|).$$  \n \nTo construct $U$, where the distribution $q_0,q_1$ has the smallest entropy (largest bias $|q_0-q_1|$), one computes  Walsh-Hadamard spectrum of $f$ and chooses $U$ such that $W_U$ is the largest in absolute value. The computation takes $n2^n$ integer additions and subtractions.\nSo the largest   Walsh-Hadamard spectrum value $|W_a|,a\\ne 0$ is a $1$-dimensional parameter of the Boolean function $f$. For balanced Boolean functions $W_0=0$. So $1$-dimensional non-linearity parameter for a balanced Boolean function is also defined  by the classical non-linearity of  $f$.\n\nIn order to find $r$-dimensional parameters for $r\\geq 2$ one can brute force all matrices $U$ (up to an  equivalence by row operations), calculate the distribution $q$, its entropy and the number of its zero values. The number of inequivalent matrices grows fast with $r$, so the calculation is infeasible even for moderate $n$. \n    \n    \n    \n We consider an example.       For the Boolean function $$f(x_1,x_2,x_3,x_4,x_5)=x_1x_2x_3 +x_1x_2x_4 +x_1x_2x_5 +x_1x_4 +x_2x_5 +x_3 +x_4 +x_5$$\n$r$-dimensional  parameters are shown in Table \\ref{Boolean} for $r=1,2,3,4$, where $u$ is the  number of $r\\times n$-matrices  $U$ up to a row equivalence, $c$ is the number of classes of equivalent distributions.   \nAlso the table contains a  representative $q$ of the largest (according to the order above)  class of the distributions, a matrix $U_q$ and   \nthe number $T_q$ of the distributions in that class, and the parameters $N_f,H_f$.  The linear transform $U$ is represented by its coordinate linear functions.        \n\\begin{table}[htp]\n\\caption{$r$-dimensional non-linearity parameters of $f$}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\\hline\n\n$r$& $u$&$c$& $U_q$ & $q$&  $N_f$&  $H_f$& $T_q$\\\\ \\hline\n&&&  & & && \\\\\n$1$&31&2& $x_4+x_5$ & $\\frac{3}{8},\\frac{5}{8}$& 0&$0.95441$& $16$\\\\\n&  & & && &&\\\\ \\hline\n&&&  & & && \\\\\n$2$&155&5 &$x_3+x_5$ & $\\frac{5}{16},\\frac{5}{16},\\frac{5}{16},\\frac{1}{16}$& 0&$1.82320$& $8$\\\\\n& & &$x_4+x_5$ & & &&  \\\\\n \\hline\n $3$&155&7& $x_2$ & & 1&$2.65563$& $12$\\\\\n &&& $x_3+x_5$ & $\\frac{1}{16},\\frac{1}{16},\\frac{3}{16},\\frac{3}{16},0,\\frac{1}{4},\\frac{1}{8},\\frac{1}{8}$ & && \\\\\n&&& $x_4+x_5$ & & && \\\\\n \\hline\n  $4$&31&3& $x_1$ & & 6&$ 3.2500$& $1$\\\\\n  && & $x_2$ &$\\frac{1}{16},\\frac{1}{16},\\frac{1}{16},\\frac{1}{16},0,\\frac{1}{8},\\frac{1}{8},0$ & && \\\\\n &&& $x_3+x_5$ & & && \\\\\n&&& $x_4+x_5$ & $0,0,\\frac{1}{8},\\frac{1}{8},0,\\frac{1}{8},0,\\frac{1}{8}$ & && \\\\\n&&&  & & && \\\\\n \\hline\n\\end{tabular}\n\\end{center}\n\\label{Boolean}\n\\end{table}%\n \n\n\n\n   \n   \n\\section{Vectorial Boolean Functions}\nA variation of the above definition may be extended to vectorial Boolean functions. Let $y=(y_1,\\ldots,y_m)=f(x_1,\\ldots,x_n)$ be a vectorial Boolean function in $n$ variables $x=(x_1,\\ldots,x_n)$. One defines a probability distribution on $(n+m)$-binary vectors $p_{x,y}=1\/2^n$ if $y=f(x)$ and $p_{x,y}=0$ otherwise. Let $U$ be an $r\\times (n+m)$ binary matrix of rank $r$. That matrix defines a probability distribution $q$ on binary $r$-strings as\n$q_z=\\sum_{z=U(x,y)}p_{x,y}$, where the sum is computed over  $x,y$ such that $z=U(x,y)$, and $(x,y)$ is a column vector of length $n+m$.\nHow to  find efficiently $U$ such that the distribution $q$ is far away from the uniform? For $r=1$ the distribution is $q=(q_0,q_1)$ and the function $U(x,y)=ax+by$ is a conventional linear approximation used in Matsui's Linear Cryptanalysis of block ciphers, see \\cite{Matsui}.  The best $ax+by$ is found after  applying Walsh-Hadamard transform to the distribution $p$ as\n$$q_0-q_1=\\sum_{x,y}p_{x,y}(-1)^{ax+by}.$$\nThe computation takes $(n+m)2^{n+m}$ arithmetic operations. Let $r=m=1$.  We set $b=1$, otherwise the distribution $q$ is uniform. Then\n$$q_0-q_1=\\sum_{x,y}p_{x,y}(-1)^{ax+y}=\\frac{1}{2^n}\\sum_{x}(-1)^{ax+f(x)}=W_a.$$\nOne takes $a$ with the largest $|W_a|$ and constructs the  distribution $q$ with the smallest entropy by using $U(x,y)=ax+y$.\n\n Similar to Section \\ref{Boolean}, we define  $r$-dimensional non-linearity parameters $(N_f,H_f)$ for $f$.\n in case $r\\geq 2$  efficient method to compute those parameters  is unknown. However for small $n$ one can  brute force all matrices $U$.   \nFor instance, let $n=m=4$ and $f(x_1,x_2,x_3,x_4)=(y_1,y_2,y_3,y_4)$, where \n\\begin{eqnarray}\n(x_1\\alpha^3+x_2\\alpha^2+x_3\\alpha+x_4)^{-1}=(y_1\\alpha^3+y_2\\alpha^2+y_3\\alpha+y_4)\\quad \\hbox{mod} \\quad\\alpha^4+\\alpha+1\n\\end{eqnarray}\nif $(x_1,x_2,x_3,x_4)\\ne (0,0,0,0)$ and $f(0,0,0,0)=(0,0,0,0).$  $r$-dimensional non-linearity parameters for $f$ for $r=1,\\ldots,7$ are in Table \\ref{Vectorial}.\n\n\\begin{table}[htp]\n\\caption{$r$-dimensional non-linearity parameters of the vectorial $f$}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\\hline\n\n$r$& $u$&$c$& $U_q$ & $q$&  $N_f$&  $H_f$& $T_q$\\\\ \\hline\n&&&  & & && \\\\\n$1$&255&3& $x_1+x_2+x_3+y_1$ & $\\frac{3}{4},\\frac{1}{4}$& 0&$0.8112$& $30$\\\\\n&  & & && &&\\\\ \\hline\n&&&  & & && \\\\\n$2$&10795&12 &$x_1+x_3+x_4+y_4,$ & $\\frac{1}{2},\\frac{1}{4},0,\\frac{1}{4}$& 1&$1.5$& $135$\\\\\n& & &$x_2$ & & &&  \\\\\n \\hline\n $3$&97155&35& $x_1+y_1,$ & & 3&$2$& $15$\\\\\n &&& $x_2+y_1+y_2,$ & $\\frac{1}{2},0,0,0,\\frac{1}{8},\\frac{1}{8},\\frac{1}{8},\\frac{1}{8}$ & && \\\\\n&&& $x_3+y_1+y_3$ & & && \\\\\n \\hline\n  $4$&200787&49& $x_1+y_2+y_3,$ & & 10&$ 2.4056$& $3$\\\\\n  && & $x_2+y_1+y_2+y_4,$ &$\\frac{3}{8},0,\\frac{1}{8},\\frac{1}{8},0,0,0,\\frac{1}{8}$ & && \\\\\n &&& $x_3+y_3+y_4,$ & & && \\\\\n&&& $x_4+y_2$ & $0,0,0,\\frac{1}{8},0,\\frac{1}{8},0,0$ & && \\\\\n&&&  & & && \\\\\n \\hline\n $5$&97155&21& $x_1+y_2+y_3,$ &$\\frac{1}{4},0,0,0,\\frac{1}{16},0,\\frac{1}{16},0$ & 23&$ 3$& $30$\\\\\n  && & $x_2+y_2+y_4,$ &$0,\\frac{1}{8},0,0,0,\\frac{1}{16},\\frac{1}{8},\\frac{1}{16}$ & && \\\\\n &&& $x_3+y_3+y_4,$ &$0,0,0,\\frac{1}{8},0,0,0,0$ & && \\\\\n&&& $x_4+y_2$ & $0,0,0,0,0,0,0,\\frac{1}{8}$ & && \\\\\n&&& $y_1$ & & && \\\\\n \\hline\n $6$&10795&9& $x_1+y_3,x_2,x_3,$ &$\\frac{3}{16},0,0,\\frac{1}{16},0,\\ldots,0$ & 52&$ 3.4528$& $90$\\\\\n&&& $x_4+y_3+y_4,y_1,y_2$ &  & && \\\\\\hline\n$7$&255&3& $x_1,x_2,x_3,$ &$\\frac{1}{8},0,0,,\\ldots,\\frac{1}{16},0,0,0$ & 114&$ 3.75$& $15$\\\\\n&&& $x_4+y_4,y_1,y_2,y_3$ &  & && \\\\\n \\hline\n\n\n\\end{tabular}\n\\end{center}\n\\label{Vectorial}\n\\end{table}%\nOne can construct an extension to the Linear Cryptanalysis based on $r$-dimensional parameters for  $r\\geq 2$. \n\\section{Optimal Boolean Functions}\\label{optimal}\nLet $y=f(x_1,\\ldots,x_n)$ be a vectorial Boolean function with $m$ coordinates and in $n$ variables and let $1\\leq r\\leq n$. \nOne can split Boolean functions with the same $n,m$ into classes of equivalence and define an order on the equivalence classes.\n\n Let $f'$ be another Boolean function with the same parameters $n,m$. One says $f,f'$ are equivalent (belong to the same class denoted $\\{f\\}$) if $(N_f,H_f)=(N_{f'},H_{f'})$. One now defines an order on the classes by \n$\\{f\\}<\\{f'\\}$ if $N_{f}<N_{f'}$ or if $N_{f}=N_{f'}$, then $H_{f}>H_{f'}$. Boolean functions from the smallest (according to $<$) class are called optimal. \n\n Then it is easy to show that for  $r=1,m=1$ and even $n$  optimal Boolean functions     are exactly  Boolean bent functions introduced by Rothaus in \\cite{R76}. For $m\\geq 1$ and  even $n\\geq 2m$ optimal Boolean functions are perfect nonlinear according to \\cite{N91} and vice versa. By a computer search we find that the property  holds for $r=2, m=1, n=4$. For larger $n,m$ and $r\\geq 2$ this is an open problem. \n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{section:litReview}\n\nInsurance is based on the idea that society asks for protection against unforeseeable events which may cause serious financial damage. Insurance companies offer a financial protection against these events. The general idea is to build a community where everybody contributes a certain amount and those who are exposed to the damage receive financial reimbursement \\cite{wuthrich2019non} . \n\\smallskip\n\nWhen (non-life) insurers set premium prices they usually start by finding the so-called pure premium, which is the expected value of the total claims that will occur in one time unit. However, when pricing insurance policies, insurers must take into account the risk associated with the policy as well as additional costs (e.g. operational cost, capital cost, etc.). Therefore, a so-called security loading is added to cover the risk and additional costs. The security loading is often calculated using some premium calculation principle,\nand the insurance premium is obtained once the security loading has been determined and added to the pure premium. The main concerns are usually whether the loading is an appropriate measure of the risk and which premium principle to choose. The higher the loading the higher the premium and consequently, the underwriting risk will be lower. However, if the premium price is too high then the exposure will be too low due to competition, and the operational cost of the insurer will engulf the premium income resulting in financial instability. Therefore, insurers usually require sophisticated premium calculations in order to secure stability.\n\\smallskip\n\nCollective risk models are fundamental in actuarial science to model the aggregate claim amount of a line of business in an insurance company. The collective risk model has two random components, the number of claims and the severity of claims, and is usually modelled with a compound process \\cite[Chapter~3]{kaas2008modern}. The classical Lundberg risk process has been studied extensively and there exist many variations, for example including reinsurance or investments \\cite{hipp2000optimal}. It assumes that premia come in a continuous stream while claims happen at discrete times according to a Poisson distribution.\n\\smallskip\n\nAnother common assumption is that the risk can be divided into groups of homogeneous risks such that the pure premia and security loadings can be estimated separately for each risk group. The pure premia of these individual groups are usually modelled with generalized linear models (GLM). GLM's have been applied extensively in actuarial work and a good overview is provided in \\cite{ohlsson2010glm, yao2013generalized}. Traditional risk theory has usually assumed independence between risks due to its convenience, but it is generally not very realistic. Claims in an insurer's risk portfolio are correlated as they are subject to the same event causes \\cite{campana2005distortion}. Completely homogeneous risk groups are extremely rare and dependence among risks has become a flourishing topic in actuarial literature \\cite{Copulas_Barning}. Dependence has mostly been measured through linear correlation coefficients \\cite{britt2005linear}. The popularity of linear coefficient is mainly due to the ease with which dependencies can be parameterized, in terms of correlation matrices. Most random variables, however, are not jointly elliptically distributed and it could be very misleading to use linear coefficients \\cite{straumann2001correlation}. This motivated the use of concordance measures. Two random variables are concordant when large values of one go with large values of the other \\cite{nelsen2007introductionCopulas}. The Lundberg risk model is a L\u00e9vy jump process, \\cite{ContTankov2004Jumps} which means that the dependency of two claim processes is best explained through their L\u00e9vy measure \\cite{tankov2016levy}. This study will not go into details about L\u00e9vy processes, but both \\cite{ContTankov2004Jumps} and \\cite{papapantoleon2008introduction} provide a very good introduction, and \\cite{Copulas_Barning, BAUERLE2011398, sato1999levy, van2012parameter, avanzi_cassar_wong_2011} are examples of applications of L\u00e9vy copulas to risk processes. For example, van Velesen \\cite{van2012parameter} showed how L\u00e9vy copulas can be used in operational modelling and discussed how dependence is implied by the L\u00e9vy copula. In this work we consider bivariate claim processes, but the presented theory can be straightforwardly extended to multiple claim processes.\n\\smallskip\n\nRuin probability is a classical measure of risk and has been extensively studied \\cite{kaas2008modern, hipp2000optimal,kasumo2018minimizing, trufin2011properties}. Although there is no absolute meaning to the probability of ruin, it still measures the stability of insurance companies. A high ruin probability indicates instability, and risk mitigation techniques should be used, like reinsurance or raising premia \\cite{kaas2008modern}. Most non-life insurance products have a term of one year and therefore it can be argued that the one year ruin probability should be used. The one year ruin probability is the probability that the capital of an insurance company will hit zero within one year. However, the appropriateness of risk measures defined over fixed time horizons can be questioned, since ruin in a given time span can be minimized by increasing the probability of ruin in the aftermath of that period. Lundberg concluded that the actual assumptions behind the classical collective risk model are in fact less restrictive when time-invariant quantities like the infinite time ruin probability are considered \\cite{trufin2011properties}. Therefore, we focus on the infinite time ruin probability in this paper.\n\\smallskip\n\nIn this work, the optimal loadings based on two strategies are derived, and compared. One strategy maximizes the profit and the other minimizes the ruin probability. We show that the two loading strategies give different results. Furthermore, we show how the optimal loading with respect to the ruin probability can be found and compare it to the one obtained when the expected profit is maximized. We consider dependencies and illustrate how L\u00e9vy copulas can be used to model claim process dependencies and how dependencies can affect the riskiness of the insurance portfolio. We take this idea further and consider dependency between the acquisition of insurance for different risks by policyholders. This is a realistic assumption as policyholders usually buy multiple insurance products from the same insurance company. We also take into account the fact that the market risk process and the company's risk process are not the same, and how the company's risk process depends on its exposure to the market. This is, to our knowledge, the first analysis of the interplay of the ruin probability, the dependency structure of claim, and the dependency structure of acquisition of insurance. We demonstrate that even if there is a strong dependency between insurance products within the market, small insurance companies have less dependency and therefore less risk than bigger insurance companies, provided the dependency between acquisition of insurance for different risks is not too strong. \n\\medskip\n\nThe paper is organized as follows: Section \\ref{section:min_ruin} contains some background material about ruin probabilities in the Lundberg process and aggregation of compound Poisson processes. Section \\ref{section:demand} deals with the single-risk case. We characterize the optimal loading and compare it with the loading maximizing the expected profit.  Section \\ref{section:companyRisk} handles the multiple risks case. We show how the dependency structure existing in the market (i.e. the general population) translates into the risk exposure of the company through its market shares on different risks and the likelihood that clients acquire insurance for more than one risk. Section \\ref{seq:numerical} contains a numerical illustration. A numerical scheme to compute the ruin probabilities is given in the appendix. \n\n\n\n\n\n\\section{Preliminaries}\n\\label{section:min_ruin}\n\n\n\\subsection{Claim and Surplus Processes}\n\nThe Lundberg risk model describes the evolution of the capital of an insurance company and assumes that exposure is constant in time, losses follow a compound Poisson process, and premia arrive at a fixed continuous rate:\n\n$$ X_t = u + ct-\\sum_{i=0}^{N_t} Y_i = u + ct - S_t, \\qquad Y_0 \\coloneqq 0  ,$$\n\nwhere $u$ is the initial surplus, $c$ is the risk premium rate, $N_t$ is a time homogeneous Poisson process with intensity parameter $\\lambda$, and  $Y_i$ are i.i.d. random variables representing the severity of claim $i$, $i = 0,\\dots, N_t$. Here it is assumed that $Y_i$ are positive. In the following sections, $Y$ denotes an arbitrary random variable with the same distribution as any $Y_i$. The severity distribution is denoted as $F(x)$ and the severity survival distribution as $\\overline{F}(x)$. $S_t$ is a compound Poisson process and thus $X_t$ is a stochastic process (sometimes called the surplus process) representing the insurance wealth at time $t$. $X_t$ increases because of earned premia and decreases when claims occur. When the capital of an insurance company hits zero, the insurance company is said to be ruined. Formally, the ruin probability is defined as follows.\n\n\\begin{definition}[Probability of Ruin]\n\\label{SimpleRuin}\nLet $(\\Omega, \\mathcal{F}, \\{\\mathcal{F}_t\\}_{t\\geq0}, \\mathbbm{P})$ be a filtered probability space and $X = (X_t)_{t \\in [0, \\infty[}$ a surplus process which is adapted and Markov with respect to the filtration. The state space is $(\\mathbbm{R}, \\mathcal{B}(\\mathbbm{R}))$. If X is time homogeneous, the infinite time ruin probability is the function $V: \\mathbbm{R} \\mapsto [0,1]$ such that\n\n$$V(x) = \\Prob \\big( \\exists s\\in[0,+\\infty[:X_s \\leq 0  \\given[\\big]   X_0 = x \\big), \\quad x \\in \\mathbbm{R} .$$\n\n\\end{definition}\n\nSometimes it is useful to use the survival (non-ruin) probability, defined as $\\overline{V}(x) = 1-V(x)$. The ruin probability can be calculated using the following integro-differential equation \\cite{grandell2012aspects}.\n\n\\begin{proposition}\n\\label{prop:Ruin_lundberg_inf}\nAssume that $X_t$ is defined as above and the premium rate satisfies $c > \\lambda \\E[Y]$. If $V \\in C^1(]0, \\infty[),$ then the probability of ruin with infinite time horizon satisfies the following equation:\n\\begin{align}\n\\label{eq:Ruin_lundberg_inf}\n    0 =  c \\deriv{x}V(x) + \\lambda \\bigg( \\int_{0}^{x} V(x-y)dF(y) - V(x) + 1-F(x) \\bigg), \\quad x>0,\n\\end{align}\nwith the following boundary condition:\n\\[\n\\begin{cases}\n   V(x) = 1  & x\\leq 0,\\\\\n  \\lim_{x \\to 0^+}V(x) = \\frac{\\lambda}{c}\\E[Y] . \\\\\n\\end{cases}\n\\]\nFurthermore, the probability of non-ruin satisfies the following equation:\n\\begin{equation}\n\\label{eq:Ruin_lundberg_inf_survival}\n    \\overline{V}(x)-\\overline{V}(\\epsilon) = \\frac{\\lambda}{c}\\int_\\epsilon^x\\overline{V}(x-y)\\overline{F}(y)dy\n\\end{equation}\nfor $0 < \\epsilon \\leq x <  + \\infty$  with the following boundary condition:\n\\[\n\\begin{cases}\n   \\overline{V}(x) = 0  & x\\leq 0,\\\\\n  \\lim_{x \\to 0^+} \\overline{V}(x) =1- \\frac{\\lambda}{c}\\E[Y] . \\\\\n\\end{cases}\n\\]\n\\end{proposition}\n\\medskip\n\nA numerical scheme solving equation \\eqref{eq:Ruin_lundberg_inf} can be found in Appendix \\ref{appendix:NA_ez_Poi_Exp_inf}. \n\n\n\\subsection{Accounting for Claim Dependencies}\n\nConsider the surplus process $\\bm{X} = (X_t^{(1)},...,X_t^{(n)}) $ where\n\\begin{equation}\n\\label{multiclaims}\n    \\begin{split}\n        X_t^{(1)} &= u^{(1)}+ c^{(1)}t - \\sum_{i = 0}^{N_t^{(1)}}Y_i^{(1)} \\\\\n        \\vdots &\\\\\n        X_t^{(n)} &= u^{(n)}+ c^{(n)}t - \\sum_{i = 0}^{N_t^{(n)}}Y_i^{(n)}  \\\\\n    \\end{split}\n\\end{equation}\n\nIf these processes are independent, it is relatively easy to combine them into a single process using the aggregation property of compound Poisson processes as described in W\\\"{u}trich \\cite{wuthrich2019non}. The aggregation property allows the combination of multiple surplus processes into a single risk process as follows:\n\\begin{equation*}\n    X_t = \\sum_{j= 1}^n u^{(j)} +\\sum_{j = 1}^n c^{(j)}t -\\sum_{i = 0}^{N_t}Y_i,\n\\end{equation*}\nwhere $N_t$ is a Poisson r.v. with $\\lambda = \\lambda_{1} + ...+ \\lambda_{n}$ and $Y_i$ are i.i.d. random variables, which follow the severity distribution $F(x) = \\sum_{j = 1}^n \\frac{\\lambda_j}{\\lambda} F_j(x)$. This aggregation property allows us to use the integro-differential equation \\eqref{eq:Ruin_lundberg_inf} to calculate the ruin of multiple surplus processes.\n\\smallskip\n\nIf the risks are not independent, then we can use the fact that compound Poisson processes are characterized by their L\u00e9vy measure to decompose the claim process into independent processes to which the aggregation property can be applied.\nIn particular, for $n = 2$ risks, we obtain the decomposition:\n\\begin{equation*}\n    \\begin{split}\n          X_t = X_t^{(1)} + X_t^{(2)} = u + ct - S_t^{1\\perp} - S_t^{2\\perp} -S_t^\\parallel.\n    \\end{split}\n\\end{equation*}\nwhere $S^{1\\perp}$ and $S^{2\\perp}$ are compound Poisson processes accounting for events concerning only risk 1 and risk 2, respectively. $S^{\\parallel}$ is a compound Poisson process accounting for events concerning both risks simultaneously. Furthermore, $S^{1\\perp}$, $S^{2\\perp}$ and $S^{\\parallel}$ are mutually independent. \\smallskip\n\nIn this section, we briefly explain how this can be achieved. Further details can be found in \\cite{tankov2016levy}. We will use the following definitions:\n\n\\begin{definition}\nThe tail integral of a L\u00e9vy measure $\\nu$ on $[0, \\infty]^2$ is given by a function $U:[0, \\infty]^2 \\mapsto [0, \\infty]$\n\\begin{equation}\n\\label{eq:tail_int}\n    \\begin{split}\n        & U(x_1,x_2) = 0 \\quad if \\quad x_1 = \\infty \\quad or \\quad x_2 = \\infty ,\\\\\n        & U(x_1,x_2) = \\nu\\big([x_1, \\infty[ \\times [x_2, \\infty[ \\big) \\quad for \\quad (x_1,x_2) \\in ]0, \\infty[^2 , \\\\\n        & U(0,0) = \\infty.\n    \\end{split}\n    \\end{equation}\n\\end{definition}\n\n\\begin{definition}[L\u00e9vy Copula for Processes with Positive Jumps]\nA two-dimensional L\u00e9vy copula for L\u00e9vy processes with positive jumps, or for short, a positive L\u00e9vy copula, is a 2-increasing grounded function $\\mathcal{C}: [0,\\infty]^2 \\to [0, \\infty]$ with uniform margins, that is, $\\mathcal{C}(x,\\infty) = \\mathcal{C}(\\infty,x) = x$.\n\\end{definition}\n\nSimilarly to Sklar's theorem for ordinary copulas \\cite{nelsen2007introductionCopulas}, it has been shown that the dependency structure of $(X_t^{(1)},X_t^{(2)})$ can be characterized by a Levy copula $\\mathcal{C}$ such that $\\mathcal{C}(U_1(x_1), U_2(x_2))$ where $U_1$ and $U_2$ are the marginal tail integrals for $X_t^{(1)}$ and $X_t^{(2)}$. If $U_1$ and $U_2$ are absolutely continuous, this L\u00e9vy copula is unique, otherwise it is unique on $Range(U_1)\\times Range(U_2)$, the product of ranges of one-dimensional tail integrals, \\cite[Theorem~5.4]{ContTankov2004Jumps}\n\\smallskip\n\nConsider a two dimensional claim process:\n\\begin{equation}\n    \\label{eq:2DimClaimProcess}\n    S_t = (S_t^{(1)}, S_t^{(2)}) = \\sum_{i = 0}^{N_t} (Y_i^{(1)},Y_i^{(2)}),\n\\end{equation}\nwhere $N_t$ is a Poisson process with intensity $\\lambda$ and $Y_i = (Y_i^{(1)},Y_i^{(2)})$, $i \\in \\mathbbm{N}$ are independent random variables with common joint distribution $F_Y$. The components of $S$, $S^{(1)}$ and $S^{(2)}$, are one-dimensional compound Poisson processes with intensities $\\lambda_1$ and $\\lambda_2$ and severity distributions $F_{Y^{(1)}}$ and $F_{Y^{(2)}}$, respectively. \nWe wish to obtain a decomposition:\n\\begin{equation}\n\\label{eq:bracketSum}\n     (S_t^{(1)}, S_t^{(2)}) = \\sum_{i = 0}^{N_t^{1\\perp}}  (Y_i^{(1\\perp)}, 0) + \\sum_{i = 0}^{N_t^{2\\perp}}  (0, Y_i^{(2\\perp)}) + \\sum_{i = 0}^{N_t^{\\parallel}}(Y_i^{(1\\parallel)}, Y_i^{(2\\parallel)}),\n\\end{equation}\nwhere $\\sum_{i = 0}^{N_t^{1\\perp}}  Y_i^{(1\\perp)}$,$\\sum_{i = 0}^{N_t^{2\\perp}} Y_i^{(2\\perp)}$ and $\\sum_{i = 0}^{N_t^{\\parallel}}(Y_i^{(1\\parallel)}, Y_i^{(2\\parallel)})$ are independent compound Poisson processes with intensities $\\lambda_1^\\perp$, $\\lambda_2^\\perp$, $\\lambda^\\parallel$ and severity distributions $F_{Y^{1\\perp}}$,$F_{Y^{2\\perp}}$,$F_{Y^{\\parallel}}$, respectively. In the above setting, we consider \\begin{equation}\n\\label{eq:AllDistZero}\n    F_Y(0,0) = F_{Y^{\\parallel}}(0,0) = 0, \\quad F_{Y^{(1)}}(0) = F_{Y^{1\\perp}}(0) = F_{Y^{(2)}}(0) = F_{Y^{2\\perp}}(0) \n\\end{equation}\n\nA compound Poisson process, $S$, is a L\u00e9vy process with L\u00e9vy measure $\\nu(dx) = \\lambda dF(x)$, with tail integral\n\\begin{equation*}\n    U(x_1,x_2) = \n    \\begin{cases}\n    \\lambda \\Prob \\big(Y^{(1)} \\geq x_1, Y^{(2)} \\geq x_2 \\big) & \\textrm{if } x_1 >0 \\textrm{ or } x_2 >0 \\\\\n    +\\infty & \\textrm{if } x_1 = x_2 = 0 .\n    \\end{cases}\n\\end{equation*}\nThe components $S^{(1)}$ and $S^{(2)}$ are independent if and only if $U(x_1, x_2) = 0$ for every $(x_1, x_2) \\in ]0, + \\infty[$, i.e., if and only if $\\lim_{x_1 \\to 0^{+}, x_2 \\to 0^{+}} U(x_1, x_2) = 0 $.\n\\smallskip\n\nThe L\u00e9vy measure of the processes $S^{(i)}$, $i = 1,2$, have tail integrals\n\\begin{equation*}\n    \\begin{split}\n        U_1(x_1) &= \\lambda_1 \\Prob \\big( Y^{(1)} \\geq x_1 \\big) = U(x_1,0) \\\\\n        U_2(x_2) &= \\lambda_2 \\Prob \\big( Y^{(2)} \\geq x_2 \\big) = U(0,x_2)\n    \\end{split}\n\\end{equation*}\nTaking equation \\eqref{eq:AllDistZero} into account, one obtains \n\\begin{equation*}\n     \\lambda_i = \\lim_{x_i \\to 0^{+}} U_i(x_i), \\quad i = 1,2 \n\\end{equation*}\n\\begin{equation*}\n     \\lambda =\\lim_{x_1, x_2 \\to 0^{+}} \\big(U_1(x_1) + U_2(x_2) - U(x_1,x_2) \\big) \n     = \\lambda_1 + \\lambda_2 - \\lambda^\\parallel\n\\end{equation*}\n\\begin{equation*}\n    \\lambda^\\parallel =\\lim_{x_1, x_2 \\to 0^{+}} U(x_1,x_2) \n\\end{equation*}\n\\begin{equation*}\n     \\lambda_i^\\perp = \\lambda_i - \\lambda^\\parallel, \\quad i = 1,2 .\n\\end{equation*}\nThe severity distributions $F_{Y^{1\\perp}}$, $F_{Y^{2\\perp}}$, and $F_{Y^{\\parallel}}$ can be recovered from the tail integrals:\n\\begin{equation*}\n    \\Prob \\big(Y^{1\\perp} \\geq x_1 \\big) = \\frac{1}{\\lambda_1^\\perp} \\lim_{x_2 \\to 0^{+}} \\Big( U_1(x_1) - U(x_1,x_2) \\Big)\n\\end{equation*}\n\\begin{equation*}\n    \\Prob \\big(Y^{2\\perp} \\geq x_2 \\big) = \\frac{1}{\\lambda_2^\\perp} \\lim_{x_1 \\to 0^{+}} \\Big( U_2(x_2) - U(x_1,x_2) \\Big)\n\\end{equation*}\n\\begin{equation*}\n    \\Prob \\big(Y^{1\\perp} \\geq x_1, Y^{2\\perp} \\geq x_2 \\big) = \\frac{1}{\\lambda^\\parallel}  U(x_1,x_2) .\n\\end{equation*}\n\nIf the dependency between $S^{(1)}$ and $S^{(2)}$ is characterized by a L\u00e9vy copula, $\\mathcal{C}$, i.e. $U(x_1, x_2) = \\mathcal{C}(U_1(x_1), U_2(x_2))$ for $(x_1, x_2) \\in [0, +\\infty[^2$, then the relations above can be written using the L\u00e9vy copula and one-dimensional tail integrals:\n\\begin{equation*}\n\\lambda^\\parallel = \\lim_{u_1 \\to \\lambda_1^{+}, u_2 \\to \\lambda_2^{+}} \\mathcal{C}(u_1, u_2)\n\\end{equation*}\n\\begin{equation*}\n    \\Prob \\big(Y^{1\\perp} \\geq x_1 \\big) = \\frac{1}{\\lambda_1^\\perp} \\lim_{u_2 \\to \\lambda_2^{+}} \\Big( U_1(x_1) - \\mathcal{C}(U_1(x_1),u_2) \\Big)\n\\end{equation*}\n\\begin{equation*}\n    \\Prob \\big(Y^{2\\perp} \\geq x_2 \\big) = \\frac{1}{\\lambda_2^\\perp} \\lim_{u_1 \\to \\lambda_1^{+}} \\Big( U_2(x_2) - \\mathcal{C}(u_1, U_2(x_2)) \\Big)\n\\end{equation*}\n\\begin{equation*}\n    \\Prob \\big(Y^{1\\perp} \\geq x_1, Y^{2\\perp} \\geq x_2 \\big) = \\frac{1}{\\lambda^\\parallel}  \\mathcal{C}(U_1(x_1), U_2(x_2)) .\n\\end{equation*}\n\nUsing the above methodology, the surplus process\ncan be represented as\n\\begin{equation*}\n        X_t = u + ct - \\sum_{i = 0}^{N_t^{1\\perp}} Y_i^{1\\perp} - \\sum_{i = 0}^{N_t^{2\\perp}} Y_i^{2\\perp} -  \\sum_{i = 0}^{N_t^{\\parallel}} ( Y_i^{1\\parallel} + Y_i ^{2 \\parallel} ) =\n        u +ct - \\sum_{i = 1}^{N_t^{*}} Y_i^{*} ,\n\\end{equation*}\nwhere $u = u_1 + u_2$, $c = c_1 + c_2$, $N^{*}$ is a Poisson process with intensity $\\lambda = \\lambda_1^\\perp + \\lambda_2^\\perp + \\lambda^\\parallel$ and $Y_i^{*}$ are i.i.d. random variables with distribution:\n\\begin{equation*}\n    F^{*} = \\frac{\\lambda_1^\\perp}{\\lambda} F_{Y^{1\\perp}} + \\frac{\\lambda_2^\\perp}{\\lambda} F_{Y^{2\\perp}} + \\frac{\\lambda^\\parallel}{\\lambda} F_{Y^{1\\parallel} + Y^{2\\parallel}} ,\n\\end{equation*}\nwhere\n\\begin{equation*}\n    F_{Y^{1\\parallel} + Y^{2\\parallel}}(x)  = \\int_{x_1 + x_2 \\leq x} dF_{Y^\\parallel}(x_1,x_2).\n\\end{equation*}\n\n\n\\section{The Optimal Loading for a Single Risk}\n\\label{section:demand}\n\nAn insurer can control the volume of its business through the premium loading $\\theta$. A reasonable assumption is that the higher the loading, the smaller the number of contracts in its portfolio, which means that the claim intensity (or business volume) will decrease. Therefore, both the claim intensity $\\E^\\theta[N_1]$, and the premium rate $c(\\theta)$, will depend on $\\theta$. It is reasonable to assume that $\\E^\\infty[N_1] = 0$, as abnormal premium rates will not attract customers \\cite{hipp2004insurancecontrol}. To capture these concepts let $\\E^\\theta[N_1] = \\lambda p(\\theta)$. Here $\\lambda$ is the average number of claims per unit of time for the whole market, and $p(\\theta)$ is the probability that a potential claim is filed as an actual claim to the particular insurer under consideration. In other words, $p(\\theta)$ reflects the demand or the market share sensitivity to the loading parameter $\\theta$. $p(\\theta)$ can be interpreted as a probability that a customer buys an insurance product. For example, we may assume that demand of insurance contracts is described by a logit glm model as in Hardin and Tabari \\cite{hardin2017renewal}. Thus, $p(\\theta)$, will be :\n\\begin{equation}\n\\label{eq:logit}\n    p(\\theta) = \\frac{1}{1+e^{\\beta_0 + \\beta_1 \\theta}},\n\\end{equation}\nwhere $\\beta_0$ and $\\beta_1$ are determined from the glm and $\\theta$ is the loading parameter. $\\beta_1$ will be a positive number so $p\\to 0$ when $\\theta \\to \\infty$ and $p\\to 1$ when $\\theta \\to -\\infty$. Assuming that the company has some fixed costs, independent of the risk exposure, denoted by $r > 0$, the expression for the net premium income becomes:\n\\begin{equation*}\n    c(\\theta) = (1 + \\theta)\\E^\\theta[N_1]\\E[Y] -r.\n\\end{equation*}\n\nThe following proposition characterizes the behaviour of the solution of equation \\eqref{eq:Ruin_lundberg_inf} with respect to the loading $\\theta$.\n\n\\begin{proposition}\n\\label{prop:alpha_proof}\nIf $V(x,\\theta)$ satisfies equation \\eqref{eq:Ruin_lundberg_inf} then $V(x,\\theta)$ is strictly increasing with respect to the parameter $\\alpha= \\frac{ \\E^\\theta [N_1]}{c(\\theta)}$.\n\\end{proposition}\n\n\\begin{proof}\nIt is possible to integrate Equation \\eqref{eq:Ruin_lundberg_inf} on the interval $]0, x]$ to obtain:\n\\begin{equation}\n\\label{eq:omega_prove_1}\n    \\begin{split}\n        V(x, \\theta) = \\frac{ \\E^\\theta [N_1]}{c(\\theta)} \\Bigg( \\E[Y] + \\int_0^x \\Big( V(z, \\theta) - \\int_0^z V(z-y)dF(y) + F(z) -1 \\Big)dz \\Bigg) .\n    \\end{split}\n\\end{equation}\n\nTo prove the proposition, we will study equations of the general form:\n\\begin{equation}\n\\label{eq:alpha_prove_2}\n    \\begin{split}\n        u(x) = \\alpha \\Bigg( g(x) + \\int_0^x \\Big( u(z) - \\int_0^z u(z-y)dF(y) \\Big)dz \\Bigg) .\n    \\end{split}\n\\end{equation}\n\nWe introduce the operator $\\Psi$, acting on measurable locally bounded functions $h:[0, +\\infty] \\mapsto \\mathbbm{R}$, as:\n\\begin{equation}\n\\label{eq:IntegralOperator}\n(\\Psi h)(x) = \\int_0^x \\Big( h(z) - \\int_0^z h(z-y)dF(y) \\Big)dz, \\quad x \\geq 0 \n\\end{equation}\nNotice that the transformation $h \\mapsto \\Psi h$ is linear and for every $h$, $\\Psi h : [0, + \\infty[ \\mapsto \\mathbbm{R}$ is continuous, hence measurable and locally bounded. Thus, powers of the operator $\\Psi$ are defined in the usual way.\n\\begin{equation*}\n\\Psi^0h = h, \\qquad \\Psi^nh = \\Psi (\\Psi^{n-1}h), \\quad  n \\in \\mathbb N.\n\\end{equation*}\n\nLet, $\\norm{h}_{[0,x]} = \\sup_{z \\in [0,x]} |h(z)| $. Then:\n\\begin{equation*}\n    \\begin{split}\n    | (\\Psi h)(x) |  &\\leq \\int_0^x \\Big( |h(z)| +\\int_0^z |h(z-y)|dF(y) \\Big)dz \\leq 2x \\norm{h}_{[0,x]}.\n    \\end{split} \n\\end{equation*}\nIf the inequality \n\\begin{equation}\n\\label{Eq bound Psi n}\n    \\norm{(\\Psi^n h)}_{[0,x]} \\leq \\frac{2^n x^n}{n!} \\norm{h}_{[0,x]}\n\\end{equation}\nholds, for some $n \\in \\mathbbm{N}$, then\n\\begin{equation*}\n    \\begin{split}\n    | (\\Psi^{n+1} h)(x) |  &\\leq \\int_0^x \\Big( | (\\Psi^{n}h)(z)| +\\int_0^z |(\\Psi^{n}h)(z-y)|dF(y) \\Big)dz \\\\\n     &\\leq \\int_0^x 2 \\frac{2^n z^n}{n!} \\norm{h}_{[0,x]} dz = \\frac{2^{n+1} x^{n+1}}{(n+1)!} \\norm{h}_{[0,x]}.\n    \\end{split} \n\\end{equation*}\nThus, by induction, \\eqref{Eq bound Psi n} holds for every $n \\in \\mathbb N$.\nTherefore, for every $x \\in [0, \\infty[$, fixed, there is some $n \\in \\mathbb N$ such that $\\Psi^n$ is a contraction in the space of measureable and bounded functions $h:[0,x] \\mapsto \\mathbbm{R}$. It follows from the contraction principle that equation \\eqref{eq:omega_prove_1} has one unique solution. Further, $\\lim_{n \\to \\infty} (\\alpha^n \\Psi^n) h = 0$, uniformly in $[0,x]$ for any given $h$ and any fixed $x \\in [0, +\\infty[$. \n\nLet $u_{\\alpha, g}$ be the solution of equation \\eqref{eq:alpha_prove_2}  for given $g$ and $\\alpha$. Then,\n\\begin{equation*}\n    \\begin{split}\n        u_{\\alpha,g} &= \\alpha (g + \\Psi u_{\\alpha,g}) = \\alpha g + \\alpha \\Psi(\\alpha(g + \\Psi u_{\\alpha,g}))\n        = \\alpha g + \\alpha^2 \\Psi g + \\alpha^2 \\Psi u_{\\alpha,g} \\\\ \n        &= \\alpha g + \\alpha^2 \\Psi g + \\dots +\\alpha^{n+1} \\Psi^n g  +\\alpha^{n+2} \\Psi^{n +1}  u_{\\alpha,g} .\n    \\end{split}\n\\end{equation*}\nSince $\\lim_{n \\to \\infty} \\alpha^n \\Psi^n u_{\\alpha,g}(x) = 0$, this shows that $u_{\\alpha,g}$ admits the series representation:\n\\begin{equation*}\n    \\begin{split}\n        u_{\\alpha,g} &= \\sum_{n=0}^\\infty\\alpha^{n+1} \\Psi^n g ,\n    \\end{split}\n\\end{equation*}\nwhich converges uniformly with respect to $\\alpha$ on compact intervals. \nThus, we can differentiate term by term and obtain\n\\begin{equation*}\n    \\begin{split}\n        \\frac{d}{d\\alpha} u_{\\alpha,g}(x) &= \\sum_{n=0}^\\infty (n+1)\\alpha^n(\\Psi^n g)(x)\\\\\n        &= \\sum_{n = 0}^\\infty \\alpha^n (\\Psi^n g)(x) + \\sum_{n = 1}^\\infty n \\alpha^n (\\Psi^n g)(x) \\\\\n        &= \\frac{1}{\\alpha}u_{\\alpha,g} + \\sum_{n=1}^\\infty \\alpha^n (\\Psi^n g)(x) + \\sum_{n=2}^\\infty (n-1) \\alpha^n (\\Psi^n g)(x)\\\\\n        &= \\frac{1}{\\alpha}u_{\\alpha,g} + \\sum_{n=0}^\\infty \\alpha^{n+1} (\\Psi^{n+1} g)(x) + \\sum_{n=1}^\\infty n \\alpha^{n+1} (\\Psi^{n+1} g)(x) \\\\\n        &= \\frac{1}{\\alpha} u_{\\alpha,g} + (\\Psi u_{\\alpha,g})(x) + \\sum_{n=1}^\\infty \\alpha^{n+1} (\\Psi^{n+1} g)(x) + \\sum_{n=2}^\\infty (n-1) \\alpha^{n+1} (\\Psi^{n+1} g)(x)  \\\\\n        &= \\frac{1}{\\alpha} u{\\alpha,g} + (\\Psi u_{\\alpha,g})(x) + \\sum_{n=0}^\\infty \\alpha^{n+2} (\\Psi^{n+2} g)(x) + \\sum_{n=1}^\\infty n \\alpha^{n+2} (\\Psi^{n+2} g)(x)  \\\\\n        &= \\frac{1}{\\alpha} u_{\\alpha,g} + (\\Psi u_{\\alpha,g})(x) + ( \\alpha \\Psi^2 u_{\\alpha,g})(x) + \\dots + ( \\alpha^{k-1} \\Psi^k u_{\\alpha,g})(x) + \\sum_{n = 1}^\\infty n \\alpha^{n+k} (\\Psi^{n+k} g)(x) \\\\\n        &= \\sum_{n=0}^\\infty \\alpha^{n-1} (\\Psi^n u_{\\alpha,g})(x) = \\frac{1}{\\alpha^2} u_{\\alpha,u_{\\alpha,g}}.\n    \\end{split}\n\\end{equation*}\n\nFor any $h:[0,x] \\mapsto \\mathbbm{R}$ locally absolutely continuous function:\n\\begin{equation*}\n    \\begin{split}\n        (\\Psi h)(x) &= \\int_0^x \\Big( h(z) - \\int_0^z h(z-y)dF(y) \\Big)dz \\\\\n        &= \\int_0^x \\Big(h(z) - [h(z-y)F(y)]_{y = 0}^{y = z} - \\int_0^z h'(z-y)F(y)dy \\Big)dz \\\\\n        &= \\int_0^x (h(z) - h(0)F(z))dz - \\int_0^x \\int_y^x h'(z-y)F(y)dzdy \\\\\n        &= \\int_0^x (h(z) - h(0)F(z))dz - \\int_0^x \\big(h(x-y) - h(0) \\big)F(y)dy \\\\\n        &= \\int_0^x h(z)dz - \\int_0^x h(x-y)F(y)dy = \\int_0^x h(z) (1+F(x-z))dz .\\\\\n    \\end{split}\n\\end{equation*}\nThus, $h>0$ implies $(\\Psi h)>0$, which implies $(\\Psi^n h)>0$, $\\forall n \\in \\mathbbm{N}$, and therefore $u_{\\alpha,h}>0$ for any $\\alpha >0$. This argument shows that $\\frac{d}{d\\alpha} V =\\frac{1}{\\alpha^2} u_{\\alpha,V} >0$ as $V >0$. Therefore $V$ is strictly increasing with $\\alpha$.\n\\end{proof}\n\nAccording to Proposition \\ref{prop:alpha_proof}, in order to find $\\theta$ minimizing the probability of ruin, it is sufficient to find $\\theta$ minimizing $\\frac{\\mathbb E^\\theta[N_1]}{c(\\theta)}$. For example, using the logit demand model \\eqref{eq:logit}, the optimal loading is found with direct differentiation of $\\alpha$ and is given by:\n\\begin{equation}\n\\label{eq:alpha_direct}\n    \\begin{split}\n\\theta_{ruin} = \\frac{1}{\\beta_1} \\Big( \\ln \\big( \\frac{\\lambda \\E[Y]}{r \\beta_1}  \\big) - \\beta_0 \\Big).\n    \\end{split}\n\\end{equation}\nHowever, the loading that maximizes the expected profit is:\n\\begin{equation*}\n    \\begin{split}\n    \\theta_{profit} = \\argmax_\\theta \n        \\E^\\theta[X_1 \\given[] X_0 = x] = \\argmax_\\theta \\{ \\theta\\E^\\theta[N_1]\\E[Y] -r\\},\n    \\end{split}\n\\end{equation*} \nwhich is, in the case of logit demand \\eqref{eq:logit}, the unique solution of:\n\\begin{equation}\n\\label{eq:profit_equation}\n    \\begin{split}\n    1 + e^{\\beta_0 + \\beta_1 \\theta} - \\beta_1 \\theta e^{\\beta_0 + \\beta_1 \\theta} = 0.\n    \\end{split}\n\\end{equation}\nThus, in general, $\\theta_{ruin}$ does not coincide with $\\theta_{profit}$.\n\n\n\\section{The Multiple Risk Case }\n\\label{section:companyRisk}\n\nIn this section, we explore how dependencies between risks available in an insurance market translate into risk exposure for a company through its market shares on the different risks. It turns out that this mechanism is non trivial when the risks are dependent. For the sake of simplicity, we assume that the company offers insurance for two risks in a market constituted by identical individuals, all of them exposed to both risks. Using the notation in equations \\eqref{eq:2DimClaimProcess} and \\eqref{eq:bracketSum} to denote the market claim process, $S_t = (S_t^{(1)}, S_t^{(2)})$ is the vector of the total (accumulated) amount of claims of each risk that occurred in the market, up to time $t$. The marginal distributions of $S^{(1)}$ and $S^{(2)}$ are characterized by claim intensities $\\lambda_1$ and $\\lambda_2$ and the severity distributions $F_{Y^{(1)}}$, $F_{Y^{(2)}}$ and their dependency structure is characterized by a parameter $\\lambda^\\parallel \\in [0, \\min(\\lambda_1, \\lambda_2)]$ and a joint distribution $F_{(Y^{1\\parallel},Y^{2\\parallel})}$, as explained in Section  $\\ref{section:min_ruin}$. \n\n\n\\subsection{Risk Exposure as a Function of Market Shares}\n\\label{section:sub4.1}\n\nTo extend the demand model outlined in Section \\ref{section:demand} to a market with multiple risks where the acquisition of insurance for different risks may not be independent, we propose the following interpretation for the function $p$.\n\\smallskip\n\nLet $(\\theta_1, \\theta_2)$ be the loadings charged by the company for each risk. We assume that every  individual in the market (a potential client) is provided with a vector of bid prices ($b_1$, $b_2$). The client acquires the insurance for risk $i$ if $b_i \\geq \\theta_i$ (for convenience, we consider prices net of the pure premium). The distribution of the price vectors in the market is modelled by a random vector $B = (B_1, B_2)$. Thus, $p_i(\\theta) = p_i(\\theta_i) = \\Prob \\big( B_i \\geq \\theta_i \\big)$ is the company's market share for the insurance of risk $i$ at equilibrium, given the loadings $\\theta = ( \\theta_1, \\theta_2)$. Let $p^{(1,0)}$ be the proportion of individuals in the market holding a policy for risk 1 and no policy for risk 2. Similarly, $p^{(0,1)}(\\theta)$ and $p^{(1,1)}(\\theta)$ denote the proportion of individuals holding a policy only for risk 2 and for both risks, respectively. If the acquisition of polices for different risks is independent, then:\n\\begin{equation}\n\\label{eq:AcquisitionProbabilities}\n    p^{(1,1)}(\\theta) = p_1(\\theta_1)p_2(\\theta_2), \\quad p^{(1,0)}(\\theta) = p_1(\\theta_1)(1-p_2(\\theta_2)), \\quad p^{(0,1)}(\\theta) = p_2(\\theta_2)(1-p_1(\\theta_1)).\n\\end{equation}\n\nDependency between the acquisition of different risks can be introduced by considering dependent bid prices $B = (B_1, B_2)$. In particular, if the joint distribution of $B$ is characterized by an ordinary copula $C: [0,1]^2 \\mapsto [0,1]$, then, according to Sklar's theorem $F_B(\\theta_1, \\theta_2) = C(F_{B_1}(\\theta_1), F_{B_2}(\\theta_2))$ \\cite{nelsen2007introductionCopulas}. This gives:\n\\begin{equation*}\n\\label{eq:EQ8}\n    \\begin{split}\n        p^{(1,0)} &= F_{B_2}(\\theta_2^{-}) - C(F_{B_1}(\\theta_1^{-}), F_{B_2}(\\theta_2^{-})), \\\\\n        p^{(0,1)} &= F_{B_1}(\\theta_1^{-}) - C(F_{B_1}(\\theta_1^{-}), F_{B_2}(\\theta_2^{-})), \\\\\n        p^{(1,1)} &=  1 -F_{B_1}(\\theta_1^{-})- F_{B_2}(\\theta_2^{-}) + C(F_{B_1}(\\theta_1^{-}), F_{B_2}(\\theta_2^{-})). \\\\\n    \\end{split}\n\\end{equation*}\n\nUnder this model, the company's surplus process is:\n\\begin{equation}\n\\label{eq:agg_company_claims}\n    \\Tilde{X}_t = u^{(1)} + u^{(2)} + \\big( c^{(1)}(\\theta_1) + c^{(2)}(\\theta_2) \\big)t - \\sum_{i = 0}^{\\Tilde{N}_t^{1\\perp}} \\Tilde{Y}_i^{1\\perp} - \\sum_{i = 0}^{\\Tilde{N}_t^{2\\perp}} \\Tilde{Y}_i^{2\\perp} - \\sum_{i = 0}^{\\Tilde{N}_t^{\\parallel}} \\Big(Y_i^{1\\parallel} + Y_i^{2\\parallel}  \\Big),\n\\end{equation}\nwhere $\\Tilde{N}_t^{1\\perp}$, $\\Tilde{N}_t^{2\\perp}$ , and $\\Tilde{N}_t^{\\parallel}$ count the number of claims received by the company concerning only risk 1, only risk 2, and both risks, respectively. Their intensities are, respectively,\n\\begin{equation*}\n    \\begin{split}\n        \\Tilde{\\lambda}_1^\\perp &= p^{(1,0)}(\\theta) \\big( \\lambda_1^\\perp + \\lambda^\\parallel \\big) + p^{(1,1)}(\\theta) \\lambda_1^\\perp = p_1(\\theta_1) \\lambda_1^\\perp + p^{(1,0)}(\\theta) \\lambda^\\parallel, \\\\\n        \\Tilde{\\lambda}_2^\\perp &= p_2(\\theta_2) \\lambda_2^\\perp + p^{(0,1)}(\\theta) \\lambda^\\parallel, \\\\\n        \\Tilde{\\lambda}^\\parallel &= p^{(1,1)}(\\theta) \\lambda^\\parallel.\n    \\end{split}\n\\end{equation*}\n\nThe distribution of the single risk claim amounts $\\Tilde{Y}^{1\\perp}$ (resp., $\\Tilde{Y}^{2\\perp}$) is a mixture of the distributions $Y^{1\\perp}$ and $Y^{1\\parallel}$ (resp., $Y^{2\\perp}$ and $Y^{2\\parallel}$):\n\\begin{equation*}\n    \\begin{split}\n     &F_{\\Tilde{Y}^{1\\perp}} = \\frac{p_1\\lambda_1^\\perp }{p_1 \\lambda_1^\\perp + p^{(1,0)} \\lambda^\\parallel } F_{Y^{1\\perp}} + \\frac{p^{(1,0)} \\lambda^\\parallel }{p_1 \\lambda_1^\\perp + p^{(1,0)} \\lambda^\\parallel } F_{Y^{1\\parallel}} \\\\\n     &F_{\\Tilde{Y}^{2\\perp}} = \\frac{p_2 \\lambda_2^\\perp }{p_2 \\lambda_2^\\perp + p^{(0,1)} \\lambda^\\parallel } F_{Y^{2\\perp}} + \\frac{p^{(0,1)} \\lambda^\\parallel }{p_2 \\lambda_2^\\perp + p^{(0,1)} \\lambda^\\parallel } F_{Y^{2\\parallel}}\n    \\end{split}\n\\end{equation*}\nThis is because some customers insure risk 1, but not risk 2 and vice-versa. Therefore, the aggregate process for the insurer is\n\\begin{equation}\n\\label{eq:indp_company_claim}\n    \\Tilde{X}_t = u^{(1)} + u^{(2)} + \\big( c^{(1)}(\\theta_1) + c^{(2)}(\\theta_2) \\big)t - \\sum_{i = 0}^{\\Tilde{N}_t} \\Tilde{Y}_i,\n\\end{equation}\nwhere $\\Tilde{N}_t$ is a Poisson process with intensity\n\\begin{equation}\n\\label{eq:lambda_tilde}\n    \\Tilde{\\lambda} = p_1 \\lambda_1^\\perp + p_2 \\lambda_2^\\perp + \\big( p^{(1,0)} + p^{(0,1)}+ p^{(1,1)} \\big)\\lambda^\\parallel = p_1 \\lambda_1 + p_2 \\lambda_2 - p^{(1,1)}\\lambda^\\parallel ,\n\\end{equation}\nand $\\Tilde{Y}_i$, $i \\in \\mathbb N$ are i.i.d random variables with distribution\n\\begin{equation}\n\\label{eq:DepDist}\n\\begin{split}\n    F_{\\Tilde{Y}} &= \\frac{p_1 \\lambda_1^{\\perp} }{\\Tilde{\\lambda}} F_{Y^{1 \\perp}} + \\frac{p_2 \\lambda_2^{\\perp} }{\\Tilde{\\lambda}} F_{Y^{2 \\perp}} \n    + \\frac{p^{(1,0)} \\lambda^{\\parallel} }{\\Tilde{\\lambda}} F_{Y^{1 \\parallel}}\n    + \\frac{p^{(0,1)} \\lambda^{\\parallel} }{\\Tilde{\\lambda}} F_{Y^{2 \\parallel}}\n    + \\frac{p^{(1,1)} \\lambda^{\\parallel} }{\\Tilde{\\lambda}} F_{Y^{1 \\parallel}+ Y^{2 \\parallel}} \\\\\n    &=\\frac{1}{p_1 \\lambda_1 + p_2 \\lambda_2 - p^{(1,1)} \\lambda^\\parallel} \\bigg( p_1 \\lambda_1 F_{Y^{1}} + p_2 \\lambda_2 F_{Y^{2}} + p^{(1,1)}\\lambda^\\parallel \\Big( F_{Y^{1\\parallel} + Y^{2\\parallel}} - F_{Y^{1\\parallel}} - F_{Y^{2\\parallel}} \\Big) \\bigg). \n    \\end{split}\n\\end{equation}\n\nThus, if the risks in the market are independent (i.e. if $\\lambda^\\parallel = 0$), then the risk in the company's portfolio is just a sum of the risks $S^{(1)}$ and $S^{(2)}$, weighted by the respective market shares, $p_1$ and $p_2$, irrespective of any dependency between sales of policies for different risks. However, if the risks in the market are dependent ( $\\lambda^\\parallel \\neq 0$), then the company's risk is not, in general, a weighted sum of $S^{(1)}$ and $S^{(2)}$. Further, this effect persists even in the case where sales of different policies are independent (i.e., $p^{(1,1,)} = p_1 p_2$).  On the other hand, equalities \\eqref{eq:lambda_tilde} and \\eqref{eq:DepDist} show that in the (unlikely) situation where clients always buy insurance for only one risk, the risk exposure of the insurer is accurately computed using only the marginal distributions of each risk (i.e. assuming that the risks are independent). This is due to the static nature of our model. For example, it does not take into account the possibility of external factors changing the frequency of claim events in both risks simultaneously.\n\n\n\\subsection{The Impact of Dependencies on Ruin Probability}\n\nFrom the discussion above and Proposition \\ref{prop:alpha_proof}, it follows that the ruin probability of a company with market shares ($p_1$, $p_2$, $p^{(1,1)}$) solves the equation \n\\begin{align}\n\\label{eq:EQ3}\n\\frac{dV(x)}{dx} = &\n\\frac{\\Tilde{\\lambda}}{c^{(1)} + c^{(2)}} \\Big( V(x) - \\int_0^x V(x-y)dF_{\\Tilde{Y}}(y) + F_{\\Tilde{Y}}(x) -1 \\Big),\n\\\\ \\label{eq:EQ4}\n    V(0^{+}) =&\n    \\frac{\\Tilde{\\lambda}}{c^{(1)} + c^{(2)}} \\E[\\Tilde{Y}] ,\n\\end{align}\nwith $\\Tilde{\\lambda}$ and $F_{\\Tilde{Y}}$ given by equations \\eqref{eq:lambda_tilde} and \\eqref{eq:DepDist}.\n\\smallskip\n\nSince estimating the dependency structure may pose substantial difficulties, we may wish to have an a-priori bound for the error introduced by neglecting dependencies, that is, by substituting the probability $V_{ind}(x)$ for $V(x)$, where $V_{ind}(x)$ solves the equation.\n\\begin{equation}\n    \\label{eq:EQ5}\n    \\frac{dV(x)}{dx} = \\frac{\\hat{\\lambda}}{c^{(1)} + c^{(2)}} \\Big( V(x) - \\int_0^x V(x-y)dF_{\\hat{Y}}(y) + F_{\\hat{Y}}(x) -1 \\Big) ,\n\\end{equation}\nwhere $\\hat{\\lambda} = \\lambda_1p_1 + \\lambda_2p_2$ and $F_{\\hat{Y}}(x) = \\frac{\\lambda_1 p_1 F_{Y^{(1)}} + \\lambda_2 p_2 F_{Y^{(2)}}}{\\hat{\\lambda}}$. Notice that $\\hat{\\lambda} \\E[\\hat{Y}] = \\Tilde{\\lambda} \\E[\\Tilde{\\lambda}]$ and therefore the boundary condition for \\eqref{eq:EQ5} is again \\eqref{eq:EQ4}.\n\\smallskip\n\nThe discussion in Subsection \\ref{section:sub4.1} shows that the difference $V(x) - V_{ind}(x)$ is expected to be small when $p^{(1,1)}$ is small compared to $p_1 + p_2$. The following proposition gives a precise meaning for this statement.\n\n\\begin{proposition}\n\\label{prop:P1}\nWith the notation above:\n\\begin{equation*}\n    |V(x) - V_{ind}(x)| \\leq p^{(1,1)} \\lambda^\\parallel \\frac{e^{\\frac{2\\Tilde{\\lambda}x}{c^{(1)} + c^{(2)}}} - 1}{\\Tilde{\\lambda}}\n\\end{equation*}\nfor every amount of initial reserve $x \\geq 0$.\n\\end{proposition}\n\n\\begin{proof}\nFrom equalities \\eqref{eq:EQ3}, \\eqref{eq:EQ4} and \\eqref{eq:EQ5}, straightforward computations yield:\n\\begin{equation}\n\\label{eq:EQ6}\n    \\begin{split}\n        V(x) -V_{ind}(x) &= \\frac{p^{(1,1)} \\lambda^\\parallel}{c^{(1)} + c^{(1)}} \\Bigg(  \\int_0^x V_{ind}(z) - \\int_0^z V_{ind}(z-y)dF_{Y^{1\\parallel} + Y^{2\\parallel}}(y) + F_{Y^{1\\parallel} + Y^{2\\parallel}}(z) -1 dz -  \\\\\n        & \\quad \\int_0^x V_{ind}(z) - \\int_0^z V_{ind}(z-y)dF_{Y^{1\\parallel}}(y) + F_{Y^{1\\parallel} }(z) -1 dz - \\\\\n        & \\quad \\int_0^x V_{ind}(z) - \\int_0^z V_{ind}(z-y)dF_{Y^{2\\parallel}}(y) + F_{Y^{2\\parallel}}(z) -1 dz -\\Bigg)  \\\\\n        & \\quad + \\frac{\\Tilde{\\lambda}}{c^{(1)} + c^{(2)}} \\int_0^x (V - V_{ind})(z) - \\int_0^z (V - V_{ind})(z-y)dF_{\\Tilde{Y}}(y)dz\n    \\end{split}\n\\end{equation}\n\nIt can be checked that for every distribution function $G:[0, +\\infty[ \\mapsto [0,1]$,\n\\begin{equation*}\n    -x \\leq \\int_0^x V_{ind}(z) -\\int_0^z V_{ind}(z-y)dG(y) + G(z) -1dz \\leq 0\n\\end{equation*}\n\nTherefore, \\eqref{eq:EQ6} implies:\n\\begin{equation*}\n    \\max_{y \\in [0,x]} |V(x) - V_{ind}(x)| \\leq \\frac{ p^{(1,1)} \\lambda^\\parallel}{c^{(1)} + c^{(2)}}2x + \\frac{\\Tilde{\\lambda}}{c^{(1)} + c^{(2)}} \\int_0^x 2  \\max_{y \\in [0,z]} |V(x) - V_{ind}(y)|dz\n\\end{equation*}\nThus, the result follows by Gr\u00f6nwall's inequality \\cite{dragomir2003some}.\n\\end{proof}\n\n\n\\subsection{The Impact of Dependencies on Small Companies}\n\nNow, we proceed with the argument above to explore how dependencies affect companies of different size. We measure the size of the company by it's expected total value of claims, $\\Tilde{\\lambda} \\E[\\Tilde{Y}]$ and, to make comparisons meaningful, we consider that the total revenue is proportional to the company's size, i.e.\n\\begin{equation*}\n    c^{(1)} + c^{(2)} = (1 + \\theta) \\Tilde{\\lambda} \\E [\\Tilde{Y}], \\quad \\textrm{with } \\theta>0 \\textrm{ constant.}\n\\end{equation*}\nSimilarly, we consider the initial reserve to be proportional to size, i.e.:\n\\begin{equation*}\n    x = x_o  \\Tilde{\\lambda} \\E [\\Tilde{Y}], \\quad \\textrm{with } x_0>0 \\textrm{ constant.}\n    \\end{equation*}\nNotice that, due to equations \\eqref{eq:EQ3}, \\eqref{eq:EQ4} and \\eqref{eq:EQ5}, the effect of dependencies must be bounded in the sense that\n\\begin{equation*}\n    |V(x_0\\Tilde{\\lambda}\\E[\\Tilde{Y}]) - V_{ind}(x_0\\Tilde{\\lambda}\\E[\\Tilde{Y}])| \\leq K_1 x_0\\Tilde{\\lambda}\\E[\\Tilde{Y}] \\leq K_2(p_1 + p_2) ,\n\\end{equation*}\nfor some constants $K_1, K_2 < +\\infty$. However, we can use Proposition \\ref{prop:P1} to obtain a better estimate:\n\\begin{equation}\n    \\label{eq:EQ7}\n    |V(x_0\\Tilde{\\lambda}\\E[\\Tilde{Y}]) - V_{ind}(x_0\\Tilde{\\lambda}\\E[\\Tilde{Y}])| \\leq p^{(1,1)} \\lambda^\\parallel \\frac{e^{\\frac{2\\Tilde{\\lambda}\\frac{x_0}{1 + \\theta}}{c^{(1)} + c^{(2)}}} - 1}{\\Tilde{\\lambda}}.\n\\end{equation}\n\nNotice that the right-hand side of \\eqref{eq:EQ7} has the same asymptotic behaviour as \n\\begin{equation*}\n    p^{(1,1)} \\lambda^\\parallel \\frac{x_0}{1 + \\theta}, \\quad \\textrm{when } p_1 + p_2 \\to 0.\n\\end{equation*}\nFurther, if the sales of policies for different risks to the same individual are independent, then $p^{(1,1)} = p_1 p_2$ goes to zero faster than $\\Tilde{\\lambda} \\E[\\Tilde{Y}] = p_1 \\lambda_1 \\E[Y^{(1)}] + p_2 \\lambda_2 \\E[Y^{(2)}]$, when $p_1 + p_2 \\to 0$. Thus, a small company selling policies for different risks independently is relatively immune to the effects of dependencies between the risks, contrary to a large company (it is obvious that a monopolistic company is fully exposed to the dependencies between risks). \nThis immunity to risk's dependencies may persist even when sales of policies for different risks are not independent, provided the dependency in sales is  sufficiently mild. For example, $\\lim_{p_1 + p_2 \\to 0} \\frac{p^{(1,1)}}{p_1 + p_2} = 0$ if the dependency between sales is modelled by a Clayton or a Frank copula in \\eqref{eq:EQ8}. However, small companies are not specially protected from risk dependencies if the dependency between sales is modelled by a Pareto or a Gumbel copula.\n\n\\subsection{Optimal Loadings and Market Shares}\n\nSince the right-hand sides of equalities \\eqref{eq:EQ3} and \\eqref{eq:EQ4} depend on the loadings through both $\\frac{\\E[\\Tilde{N}_1}{c^{(1)} + c^{(2)}}$ and $F_{\\Tilde{Y}}$, Proposition \\ref{prop:alpha_proof} can not be generalized to models with multiple risks. However, it is possible to provide optimality conditions for the loadings $\\theta = (\\theta_1, \\theta_2)$ minimizing the ruin probability.\n\\smallskip\n\nTo do this, we extend the notation introduced in the proof of Proposition \\ref{prop:alpha_proof}. For any distribution function $G:[0, +\\infty[ \\mapsto [0,1]$, we consider the compounding operator of type \\eqref{eq:IntegralOperator}\n\\begin{equation*}\n    (\\Psi_G h)(x) = \\int_0^x \\Big(  h(z) - \\int_0^z h(z-y) d\\theta(y)\\Big)dz, \\quad x \\geq 0.\n\\end{equation*}\nThus, the 2-risk version of equation \\eqref{eq:omega_prove_1} can be written as \n\\begin{equation}\n\\label{eq:EQ10}\n    V_\\theta(x) = \\frac{\\Tilde{\\lambda}_\\theta}{c(\\theta)} \\bigg( \\int_x^\\infty 1- F_\\theta(z)dz + \\Big(\\Psi_{F_\\theta} V_\\theta \\Big)(x)  \\bigg) ,\n\\end{equation}\nwhere\n\\begin{equation*}\n    F_\\theta = \\frac{\\lambda_1^\\perp}{\\Tilde{\\lambda}_\\theta} p_1(\\theta) F_{Y^{1\\perp}} + \\frac{\\lambda_2^\\perp}{\\Tilde{\\lambda}_\\theta} p_1(\\theta) F_{Y^{2\\perp}} + \\frac{\\lambda^\\parallel}{\\Tilde{\\lambda}_\\theta} p^{(1,0)}(\\theta) F_{Y^{1\\parallel}} + \\frac{\\lambda^\\parallel}{\\Tilde{\\lambda}_\\theta} p^{(0,1)}(\\theta) F_{Y^{2\\parallel}} + \\frac{\\lambda^\\parallel}{\\Tilde{\\lambda}_\\theta} p^{(1,1)}(\\theta) F_{Y^{1\\parallel} + Y^{2\\parallel}} .\n\\end{equation*}\nSince $\\frac{\\lambda_1^\\perp}{\\Tilde{\\lambda}_\\theta} p_1(\\theta) + \\frac{\\lambda_2^\\perp}{\\Tilde{\\lambda}_\\theta}p_2(\\theta) + \\frac{\\lambda^\\parallel}{\\Tilde{\\lambda}_\\theta} p^{(1,0)}(\\theta) + \\frac{\\lambda^\\parallel}{\\Tilde{\\lambda}_\\theta} p^{(0,1)}(\\theta) + \\frac{\\lambda^\\parallel}{\\Tilde{\\lambda}_\\theta} p^{(1,1)}(\\theta) = 1$, \\eqref{eq:EQ10} becomes\n\\begin{equation*}\n    \\begin{split}\n        V_{\\theta}(x) &= \\frac{\\lambda_1^\\perp p_1(\\theta)}{c(\\theta)} \\bigg( \\int_x^\\infty 1- F_{Y^{1\\perp}}(z)dz + (\\Psi_{F_{Y^{1\\perp}}}V_\\theta)(x) \\bigg) \\\\\n        & \\quad + \\frac{\\lambda_2^\\perp p_2(\\theta)}{c(\\theta)} \\bigg( \\int_x^\\infty 1- F_{Y^{2\\perp}}(z)dz + (\\Psi_{F_{Y^{2\\perp}}}V_\\theta)(x) \\bigg) \\\\\n        & \\quad + \\frac{\\lambda^\\parallel p^{(1,0)}(\\theta)}{c(\\theta)} \\bigg( \\int_x^\\infty 1- F_{Y^{1\\parallel}}(z)dz + (\\Psi_{F_{Y^{1\\parallel}}}V_\\theta)(x) \\bigg) \\\\\n        & \\quad + \\frac{\\lambda^\\parallel p^{(0,1)}(\\theta)}{c(\\theta)} \\bigg( \\int_x^\\infty 1- F_{Y^{2\\parallel}}(z)dz + (\\Psi_{F_{Y^{2\\parallel}}}V_\\theta)(x) \\bigg) \\\\\n        & \\quad + \\frac{\\lambda^\\parallel p^{(1,1)}(\\theta)}{c(\\theta)} \\bigg( \\int_x^\\infty 1- F_{Y^{1\\parallel} +Y^{2\\parallel}}(z)dz + (\\Psi_{F_{Y^{1\\parallel} + Y^{2\\parallel}}}V_\\theta)(x) \\bigg) .\n    \\end{split}\n\\end{equation*}\n\nWe write this in abbreviated form:\n\\begin{equation*}\n    V_{\\theta}(x) = <\\alpha(\\theta), \\Gamma(x)> + (<\\alpha(\\theta), \\Psi>V_\\theta)(x) ,\n\\end{equation*}\nwhere $\\alpha(\\theta)$ is the vector\n\\begin{equation*}\n    \\alpha(\\theta) = \\frac{1}{c(\\theta)} \\Big(\\lambda_1^\\perp p_1(\\theta), \\lambda_2^\\perp p_2(\\theta), \\lambda^\\parallel p^{(1,0)}, \\lambda^\\parallel p^{(0,1)}, \\lambda^\\parallel p^{(1,1)} \\Big),\n\\end{equation*}\n$\\Gamma(x)$ is the vector function\n\\begin{equation*}\n    \\Gamma(x) = \\Big(\\int_x^\\infty 1- F_{Y^{1\\perp}}(z)dz, \\int_x^\\infty 1- F_{Y^{2\\perp}}(z)dz, \\int_x^\\infty 1- F_{Y^{1\\parallel}}(z)dz, \\int_x^\\infty 1- F_{Y^{2\\parallel}}(z)dz, \\int_x^\\infty 1- F_{Y^{1\\parallel} + Y^{2\\parallel}}(z)dz \\Big),\n\\end{equation*}\n$\\Psi$ is the vector of operators\n\\begin{equation*}\n    \\Psi = \\Big(\\Psi_{F_{Y^{1\\perp}}}, \\Psi_{F_{Y^{2\\perp}}}, \\Psi_{F_{Y^{1\\parallel}}}, \\Psi_{F_{Y^{2\\parallel}}}, \\Psi_{F_{Y^{1\\parallel}} + Y^{2\\parallel}} \\Big) ,\n\\end{equation*}\nand $<\\cdot, \\cdot>$ is the usual inner product in $\\mathbbm{R}^5$. \n\\smallskip\n\nUsing the argument in the proof of Proposition \\ref{prop:alpha_proof}, we see  that $V_\\theta$ admits the series representation\n\\begin{equation*}\n    V_\\theta(x) = \\sum_{n=0}^\\infty \\Big(<\\alpha(\\theta), \\Psi>^n <\\alpha(\\theta), \\Gamma> \\Big)(x) .\n\\end{equation*}\nSimilarly, any vector $\\gamma \\in \\mathbbm{R}^5$ and any bounded measurable function $g:[0, +\\infty[ \\mapsto \\mathbbm{R}$ define one unique function\n\\begin{equation*}\n    u_{\\gamma, g} (x) = \\sum_{n = 0}^\\infty \\Big(<\\gamma, \\Psi>^n g\\Big)(x) .\n\\end{equation*}\nThis function is analytic with respect to $\\gamma$, with partial derivatives\n\\begin{equation*}\n    \\frac{\\partial{u_{\\gamma, g}}}{\\partial{\\gamma_i}} = \\sum_{n = 0}^\\infty <\\gamma, \\Psi>^n\\big( \\Psi_i u_{\\gamma,g}\\big) = u_{\\gamma, \\Psi_i u_{\\gamma, g}}, \\quad i = 1, \\dots, 5 .\n\\end{equation*}\n\nTaking into account the chain rule for derivatives, this proves the following proposition.\n\n\\begin{proposition}\n\\label{prop:P2}\nIf $\\theta \\mapsto \\alpha(\\theta)$ is differentiable, then $\\theta \\mapsto V_\\theta(x)$ is differentiable for every $x \\geq 0$ and\n\n$$\\frac{\\partial{}}{\\partial{\\theta_i}} V_\\theta(x) = \\sum_{j = 1}^5 u_{\\alpha(\\theta),(\\Gamma_j + \\Psi_j V_\\theta)} \\frac{\\partial{\\alpha_j(\\theta)}}{\\partial{\\theta_i}}, \\quad i = 1,2.$$\n\\end{proposition}\n\nBy Proposition \\ref{prop:P2}, the optimal loadings satisfy the equation \n\\begin{equation}\n    \\label{eq:EQ11}\n     \\sum_{j = 1}^5 u_{\\alpha(\\theta),(\\Gamma_j + \\Psi_j V_\\theta)} \\frac{\\partial{\\alpha_j(\\theta)}}{\\partial{\\theta_i}} = 0, \\quad i = 1,2\n\\end{equation}\n\nContrary to the single-risk case, the odds of finding explicit solutions for this equation seem very low, even in simple cases. \nHowever, \\eqref{eq:EQ11} can be numerically solved by Newton's algorithm, the second-order partial derivatives being\n\\begin{equation*}\n    \\frac{\\partial^2}{\\partial{\\theta_i}\\partial{\\theta_j}} V_\\theta(x) = \\sum_{k = 1}^5 u_{\\alpha(\\theta),(\\Gamma_k + \\Psi_k V_\\theta)} \\frac{\\partial^2{\\alpha_k(\\theta)}}{\\partial{\\theta_i}\\partial{\\theta_j}} + \\sum_{k = 1}^5\\sum_{l = 1}^5 u_{\\alpha(\\theta),\\Psi_k} u_{\\alpha(\\theta),(\\Gamma_l + \\Psi_l V_\\theta)} \\frac{\\partial{\\alpha_k(\\theta)}}{\\partial{\\theta_i}}  \\frac{\\partial{\\alpha_l(\\theta)}}{\\partial{\\theta_j}} .\n\\end{equation*}\n\nNotice that the expected profit is\n\\begin{equation*}\n    c^{(1)}(\\theta) + c^{(2)}(\\theta) - \\Tilde{\\lambda} \\E[\\Tilde{Y}] = \\theta_1 p_1(\\theta_1)\\lambda_1\\E[Y^{(1)}] + \\theta_2 p_2(\\theta_2)\\lambda_2\\E[Y^{(2)}].\n\\end{equation*}\nThus, it depends only on the marginal distribution of the claim processes $S^{(1)}$, $S^{(2)}$, being independent of the dependency structure. It follows that the loadings minimizing the joint profit coincide with the loadings minimizing the profit on each risk, separately.  That is, a pricing strategy that completely focus on expected profit completely fails to take both dependencies between risks and dependencies between sales of policies into account.\n\n\n\n\n\n\\section{Numerical Results}\n\\label{seq:numerical}\n\nThroughout this section, $Y_i^{(i)}$ are assumed to be i.i.d gamma distributed random variables with shape parameter, $a^{(i)}$, and scale parameters, $k^{(i)}$, which means that the mean is, $\\E[Y^{(i)}] = a^{(i)}k^{(i)}$, for $i = 1,2$. In the following numerical analysis let $a^{(1)} = a^{(2)} = 2$, $k^{(1)} = k^{(2)} = 500$, $\\lambda^{(1)} = \\lambda^{(2)} = 800$, $\\beta_0^{(1)} = \\beta_0^{(2)} = -0.5$, $\\beta_1^{(2)} = 4$ and $\\beta_1^{(1)} = 4.5$. That is, the difference stems from surplus process 2 being more sensitive to the loading via the parameter $\\beta_1^{(2)}$.  $r^{(i)}$ is taken to be $20\\%$ of the pure premium if the exposure was $40\\%$, that is $r^{(i)} = 0.4 * 0.2 k^{(i)} a^{(i)} N^{(i)}$. The operational cost is therefore $8\\%$ of the expected total amount of claims in the market. The Clayton L\u00e9vy copula is considered for positive dependence and the parameter is set to $\\omega = 1$. Finally, let $\\theta_{ruin}^*$ and $\\theta_{profit}^*$ denote the optimal loading when the ruin probability and expected profit criterion is used, respectively. The programming language R was used for every calculation. \\\\\n\n\n\\subsection{Single Surplus Process}\n\nThe surplus processes are first considered separately. The ruin probability and the expected profit is plotted as a function of $\\theta$ for the two processes in Figures \\ref{fig:logit_1} and \\ref{fig:logit_2}. $\\theta_{ruin}^*$ was found by minimizing $\\alpha$.\n\n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[scale = 0.4]{.\/logit_1.png}\n  \\caption[Optimal loading parameter of surplus process 1.]{Surplus process 1. The blue lines show the ruin probability as a function of $\\theta$ for a given surplus x. The black line shows the expected profit per time unit as a function of $\\theta$. The blue dots show the minimum ruin probability for each surplus. $\\theta_{profit}^*$ and $\\theta_{ruin}^*$ denote the optimal security loading parameter for the expected profit and for the probability of ruin, respectively. }\n  \\label{fig:logit_1}\n\\end{figure}\n\n\nFrom Figure \\ref{fig:logit_1} it can be seen that the optimal security loading parameter for the ruin probability is, $\\theta_{ruin}^* = 0.435$, while the $\\theta$ that maximizes the expected profit is lower, $\\theta_{profit}^* = 0.359$. Moreover, in this example, the maximum expected profit is 22.843 units and is given at $\\theta_{profit}^*$. The expected profit taken at the point $\\theta_{ruin}^*$ is lower, close to 20.000 units. \n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[scale = 0.4]{.\/logit_2.png}\n  \\caption[Optimal loading parameter of surplus process 2.]{Surplus process 2. The blue lines show the ruin probability as a function of $\\theta$ for a given surplus x. The black line shows the expected profit per time unit as a function of $\\theta$.  The blue dots show the minimum ruin probability for each surplus. $\\theta_{profit}^*$ and $\\theta_{ruin}^*$ denote the optimal security loading parameter for the expected profit and for the probability of ruin, respectively.}\n  \\label{fig:logit_2}\n\\end{figure}\n\n\nFrom Figure \\ref{fig:logit_2} it can be seen that the optimal security loading parameter for the ruin probability is $\\theta_{ruin}^* = 0.358$, while the $\\theta$ that maximizes the expected profit is again lower or $\\theta_{profit}^*= 0.319$.  \\\\\n\nObviously, for both processes, the ruin probability decreases with increasing surplus. Moreover, it can be seen that surplus process $X_2$ has higher probability of ruin than surplus process $X_1$ for the same amount of surplus. The sensitivity of the demand curve affects the ruin probability and $\\theta_{ruin}^*$ greatly. The more sensitive to the exposure the demand curve is, the closer the $\\theta_{profit}^*$ and $\\theta_{ruin}^*$ are. This more sensitive curve also has higher probability of ruin for a given surplus, which indicates that more competitive insurance products are riskier. These effects can be seen if the two Figures (\\ref{fig:logit_1} and \\ref{fig:logit_2}) are compared. Conversely, if the demand curve is not sensitive to the price, then the gap between $\\theta_{profit}^*$ and $\\theta_{ruin}^*$ can become quite large. Additionally, it can be seen from the curve at surplus = 100 that the ruin probability for $\\theta_{profit}^*$ and $\\theta_{ruin}^*$ are similar but as the surplus grows the values start to differ and once the surplus is great enough the two values $\\theta_{profit}^*$ and $\\theta_{ruin}^*$ result in similar ruin probabilities again. This means that if the insurance firm has high enough surplus then they can choose arbitrary $\\theta$ without risking the chance of ruin. If the surplus is great enough then the value of $\\theta$ does not matter as much. However, having too much reserves can be bad for insurance companies as it can be seen as a negative leverage. The bowl shape of the blue curves in the two Figures (\\ref{fig:logit_1} and \\ref{fig:logit_2})  is because of the interplay between the fixed cost and the demand curve. \\\\\n\n$\\theta_{ruin}^*$ should give the minimum ruin probability at all surplus values. This can be tested by graphing multiple ruin probability curves and compare it with the one obtained by $\\theta_{ruin}^*$. Figure \\ref{fig:test_many_thetas} shows that $\\theta_{ruin}^*$ gives the minimum ruin probability indeed.\n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[scale = 0.4]{.\/test_many_thetas.png}\n  \\caption[Optimal value function of surplus process, $X_1$.]{The figure compares the optimal value function of surplus process, $X_1$ (blue line) to other cost functions (grey lines). The blue line is achieved by setting $\\theta = \\theta_{ruin}^*$.  All the cost functions lie above the optimal value function, as is expected. }\n  \\label{fig:test_many_thetas}\n\\end{figure}\n\n \n\n\\subsection{Two Aggregated Surplus Processes with Common Loading}\n\nNext, the two surplus processes, $X_1$ and $X_2$ are aggregated, both when the claims are independent and dependent. The acquisition is independent in this subsection.\n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[scale = 0.38]{.\/dep_vs_indp.png}\n  \\caption[Surplus processes 1 and 2 aggregated. Showing the ruin probability and optimal $\\theta$ both in the case of independence and dependence. ]{Ruin probability when $X_1$ and $X_2$ are aggregated as a function of the security loading parameter, $\\theta$, both when they are independent and dependent via Clayton L\u00e9vy copula with $\\omega = 0.5$. The blue curves show the ruin probability when the two processes are independent for different values of the surplus and the red curves show the same for the dependent case. The curves have similar shapes, but the ruin probability is higher in the case of dependence, for the same surplus. The values in the legend show the minimum ruin probability for a given surplus (surplus $\\rightarrow$ probability).}\n  \\label{fig:logit_dep_vs_indp}\n\n\\end{figure}\n\nFigure \\ref{fig:logit_dep_vs_indp} shows the ruin probability of the aggregated surplus process as a function of the security loading parameter, $\\theta$, both when they are independent and dependent via Clayton L\u00e9vy copula. The red curves represent dependence while the blue curves represent independence. \\\\\n\nFirstly, it can be seen that the expected profit is the same for dependence and independence and from the figure, $\\theta^*_{profit} \\approx 0.34$. The reason is that the claim mean and the claim frequency is almost the same (numerically) for dependence and independence. \\\\\n\nSecondly, the dependent case has a higher probability of ruin than the independent case for the same amount of surplus. However, the ruin probability is almost the same for small surplus values as can be seen from the figure. Interestingly, the optimal loading for dependence and independence seem to be the same and numerically the values are $\\theta^*_{ruin,dep}  = 0.4  = \\theta^*_{ruin,indp}$.  The surplus value does not change the optimal loading $\\theta^*$, as expected. The reason why the ruin probability difference between the dependent and independent cases is relatively small is because of the probability $p^{(0,1)}(\\theta)$. The fact that the insurance company does not always have the both claims $Y^{1\\parallel}$ and $Y^{1\\parallel}$ when a common jump occurs reduces the risk.\n\\\\\n\n\nFinally, the difference of the two ruin probability curves (red and blue) for a given surplus seems to be increasing with increasing surplus, meaning that the ruin probability in the independent case decreases more rapidly with increasing surplus then for the dependent case. Therefore, it is clear that the positive dependent case is riskier. \\\\\\\n\n\nNote that   $\\theta_{ruin}^* \\approx 0.4$, which is very close to the weighted average of the optimal loading parameter of the isolated surplus processes where the weight is the exposure ratio of each surplus process, that is\n\n\\begin{equation*}\n    \\theta_{weighted} = \\frac{0.435 \\frac{1}{1 + \\exp(-0.6 + 4*0.4)} + 0.358\\frac{1}{1 + \\exp(-0.6 + 4.5*0.4)}}{\\frac{1}{1 + \\exp(-0.6 + 4*0.4)} + \\frac{1}{1 + \\exp(-0.6 + 4.5*0.4)}} \\approx 0.4,\n\\end{equation*}\n\nwhich strongly indicates that the optimal value, $\\theta_{ruin}^*$, is simply the weighted average. \\\\\n\n\\subsubsection{Two Aggregated Surplus Processes with Separate Loadings}\n\nIt is more realistic to consider $\\theta$ as a vector so that the loading parameter can be different for each surplus process separately, to spread the total premium over the policies in an optimal way. The two surplus processes, $X_1$ and $X_2$, are aggregated as before and the constants are the same, but let $\\theta = (\\theta^{(1)}, \\theta^{(2)})$. \\\\\n\n\\begin{figure}[ht]\n  \\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=.9\\linewidth]{.\/indp_multi_profit.png}  \n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=.9\\linewidth]{.\/indp_multi_ruin.png}  \n\\end{subfigure}\n  \\caption[Two security loading parameters of two aggregated independent surplus processes. ]{Expected profit (left) and the ruin probability (right) when $X_1$ and $X_2$ are aggregated, as a function of the security loading parameters, $\\theta^{(1)}$ and $\\theta^{(2)}$. The processes are assumed to be independent and the surplus is fixed at $x = 5000$. The parenthesis in the right figure shows the optimal values of $\\theta^{(1)}$ and $\\theta^{(2)}$ with the ruin probability as a criterion. The arrow indicates which values $\\theta^{(1)}$ and $\\theta^{(2)}$ are mapped into, thus showing the minimum ruin probability. The parenthesis in the left figure shows the same for the expected profit. The shape of the contour plot is due to the fact that the $\\theta$ grid considered is sparser for values that give high ruin probability.}\n  \\label{fig:both_two_indp_15000}\n\\end{figure}\n\nFigure \\ref{fig:both_two_indp_15000} shows the expected profit (left) and the ruin probability (right), when $X_1$ and $X_2$ are assumed to be independent and aggregated, as a function of the security loading parameters, $\\theta^{(1)}$ and $\\theta^{(2)}$. The surplus is fixed at $x = 5000$ and the optimal values are shown. It should be noted that many surplus values were tested and they all gave the same value for $\\theta_{ruin}^{*(1)}$, $\\theta_{ruin}^{*(2)}$, $\\theta_{profit}^{*(2)}$, and $\\theta_{profit}^{*(2)}$ as shown, only the ruin probability level changed. Note that the optimal loading parameters for the expected profit are the same as those for the individual surplus processes. However, the optimal loading parameters for the ruin probability change when compared to the individual one (compare it with Figures \\ref{fig:logit_1} and \\ref{fig:logit_2}). When compared to the optimal loading parameter for the individual surplus process, $\\theta^{(1)}$ decreases from $0.435$ to $0.42$ and $\\theta^{(2)}$ increases from 0.358 to 0.38. Therefore, the optimal security loading parameter decision is to decrease the loading parameter of the less sensitive surplus process while increasing the loading parameter of the more sensitive surplus process. Additionally, when compared to Figure \\ref{fig:logit_dep_vs_indp}, the minimum ruin probability for one shared loading is $0.57$ while the ruin for two loadings is $0.56$, showing only a marginal difference. When the same is done for other surplus values a similar difference is found. The expected profit is marginally higher. \\\\\n\nLastly, consider the case when the surplus processes are assumed to be dependent via L\u00e9vy Clayton copula and the loadings can be different for each surplus process separately. Figure \\ref{fig:both_two_dep_15000} shows the ruin probability when $X_1$ and $X_2$ are aggregated as a function of the security loading parameters, $\\theta^{(1)}$ and $\\theta^{(2)}$. The shape of the contour plots is due to the fact that the $\\theta$ grid considered is sparser for values that give high ruin probability. The surplus is fixed at $x = 5000$. \\\\\n\n\\begin{figure}[ht]\n  \\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=.96\\linewidth]{.\/dep_multi_profit.png}  \n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=.96\\linewidth]{.\/dep_multi_ruin.png}  \n\\end{subfigure}\n  \\caption[Two security loading parameters of two aggregated dependent surplus processes.]{Expected profit (left) and the ruin probability (right) when $X_1$ and $X_2$ are aggregated, as a function of the security loading parameters, $\\theta^{(1)}$ and $\\theta^{(2)}$. The processes are assumed to be dependent via Clayton L\u00e9vy copula and the surplus is fixed at $x = 5000$. The parenthesis in the right figure shows the optimal values of $\\theta^{(1)}$ and $\\theta^{(2)}$ with the ruin probability as a criterion. The arrow shows which values $\\theta^{(1)}$ and $\\theta^{(2)}$ are mapped into, thus showing the minimum ruin probability. The parenthesis in the left figure shows the same except for the expected profit. The shape of the contour plot is due to the fact that the $\\theta$ grid considered is sparser for values that give high ruin probability.}\n  \\label{fig:both_two_dep_15000}\n\\end{figure}\n\n\n\nIt can be seen that the optimal loadings $\\theta^{(1)}$ and $\\theta^{(2)}$ are the same as the ones in the case of independence and the minimum ruin probability is higher (compared to the case in Figure \\ref{fig:both_two_indp_15000}). Both the values and the optimal loadings of the expected profit are the same as the independent case. \nAgain, the optimal security loading parameter decision is to decrease the loading parameter of the less sensitive surplus process while increasing the loading parameter of the more sensitive surplus process. The difference between the ruin probability in Figure \\ref{fig:both_two_dep_15000} vs Figure \\ref{fig:both_two_indp_15000} is only $0.03$ but in this case the surplus is low compared to the expected profit. If the surplus would be increased to $\\approx 20.000$ the difference would become greater. The difference would then decrease again if the surplus were increased to $\\approx 40.000$. \\\\\n\nAdditionally, when compared to Figure \\ref{fig:logit_dep_vs_indp}, the minimum ruin probability for one common loading is $0.59$, which is the same as the ruin probability for separate loading selections, therefore the difference is only marginal. \\\\\n\n\\subsection{Dependent Claims and Dependent Acquisition}\n\nIt is time to look at the case when we have dependent claims and dependent acquisition. Note that the case when we have independent claims and dependent acquisition is the same as the total independence case. We will look both at the case when the acquisition is modelled with a Gumbel and Clayton dependency structure. To compare these two structures we use Kendell's tau. The following equations relate the copula parameters, $\\omega_{clayton}$ and $\\omega_{gumbel}$ to kendell's tau, $\\tau$. \n\n\\begin{equation*}\n  \\omega_{clayton} = \\frac{2 \\tau}{1-\\tau}, \\quad   \\omega_{gumbel} = \\frac{1}{1-\\tau}.\n\\end{equation*}\n\nWe know that the expected profit is the same as before for all values of $\\tau$. Therefore, we analyze the ruin probability. \\\\\n\n\\begin{figure}[ht]\n  \\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=.9\\linewidth]{.\/Clayton.png}  \n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=.9\\linewidth]{.\/Gumbel.png}  \n\\end{subfigure}\n  \\caption[Acquisition dependency.]{The ruin probability when the acquisition is modelled with a Clayton (left) and a Gumbel (right) dependency structure. The surplus is constant. In both cases, the ruin probability is higher for higher kendell's tau. The Gumbel case is more riskier. The surplus is fixed at 5000 units.}\n  \\label{fig:acquisition}\n\\end{figure}\n\nIn Figure \\ref{fig:acquisition} we can see the ruin probability for different dependency values when the surplus is fixed at 5000 units. We can see that the ruin probability is higher for more dependent acquisition, as we expected. Also, we can see that the Gumbel acquisition model gives higher ruin probabilities than the Clayton model for the same Kendell's tau value. They give the same optimal loading parameter. It can also be seen that when kendell's tau is $0.05$ (close to $0$) the ruin probability is close to the case of independent acquisition, as expected. The optimal loading parameter is the same for all dependency levels. \n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{}\n\n\n\nUntil recently, the nuclear magic numbers have been considered to be robust \nand durable with respect to the shell structure of atomic nuclei.\nThe nuclear shell model initiated by Mayer and Jensen \\cite{MayJen}\nsuccessfully accounts for the shell structure of nuclei on and near \nthe $\\beta$-stability line. \nHowever, recent experimental studies on neutron-rich nuclei \nfar from the $\\beta$-stability line have revealed \ndisappearance of magic numbers \n(e.g., N $=$ 8 \\cite{be12a,be12b,be12c,be12d}, 20 \\cite{mg32}) \nand appearance of new magic numbers (e.g., N $=$ 16 \\cite{n16magic}).\nThe new N $=$ 16 magic number comes from the reduction of the N $=$ 20\nshell gap. \nOne theoretical explanation \\cite{otsuka1, utsuno1} for this phenomenon \nis that the attractive \nmonopole part of the tensor force acting between the $\\pi$d$_{5\/2}$ and \n$\\nu$d$_{3\/2}$ orbitals \nreduces the N $=$ 20 gap in nuclei with high N\/Z ratios. \n\nAnother anomalous phenomenon discovered in neutron-rich nuclei is \nthe occurrence of a {\\it strongly deformed} ground state on and near \nthe N $=$ 20 magic nuclei in the Z $\\sim$ 12 region, \nthe so-called 'island of inversion' \\cite{inversion}. \nHere, the intruder two-particle, two-hole (2p--2h) configuration occupying \nthe {\\it fp} shell and the normal 0p--0h configuration in the {\\it sd} shell \nare inverted in energy, and the 2p--2h configuration dominates in the \nground state. \nThis is a new challenge to the nuclear shell model. \nThe most convincing evidence for the large quadrupole collectivity of the nuclei \nin the 'island of inversion' was the measured low E(2$_{1}^{+}$) energy \\cite{mg32a} \nand the high B(E2) strength \\cite{mg32} as well as the enhancement of the binding \nenergy \\cite{mg32mass}. \nThe experimentally deduced large B(E2; 0$_{g.s.}^{+}\\rightarrow$2$_{1}^{+}$) value \nindicates large deformation ($\\beta_{2}=$ 0.512(44)) in $^{32}$Mg \\cite{mg32}.\n$^{34}$Mg (N $=$ 22) shows an even larger deformation ($\\beta_{2}=$ 0.58(6)) \n\\cite{mg34}.\nMonte Carlo shell model calculations \\cite{mcsm} that include the effects of \nthe narrowed shell gap at N $=$ 20 and enhanced 2p--2h excitations reproduce \nthe experimental values quite well \\cite{mg34}. \n\nOn the other hand, such a multiparticle--multihole (mp--mh) excitation appears \nin the excited states of nuclei near the $\\beta$-stability line.\nThese states can be studied by the heavy-ion induced fusion-evaporation reaction\nof stable isotopes. \nIn fact, an mp--mh excitation from the {\\it sd} to {\\it fp} shell produces \nsuperdeformation (SD) in N $=$ Z light mass nuclei, \nsuch as $^{36}$Ar \\cite{ar36}, $^{40}$Ca \\cite{ca40}, and $^{44}$Ti \\cite{ti44}.\nThese SD nuclei exhibit a large deformation of $\\beta_{2}\\sim$ 0.5, which is about \nthe same magnitude as the ground state deformation in the 'island of inversion'.\nThe presence of SD in the three abovementioned nuclei can be also understood \nin terms of the SD shell gaps of N $=$ Z $=$ 18, 20, and 22, respectively \\cite{ca40}. \nIn this region, the spherical and SD magic numbers occur at similar \nparticle numbers, which results in shape coexistence. \nHowever, the existence of a SD shell structure in neutron-rich nuclei \nhas not been experimentally confirmed yet.\n\nIn order to access the currently reachable neutron-richest SD states with asymmetric \nSD magic numbers, especially the nucleus with N $=$ 22 corresponding to $^{34}$Mg, \nwe employed a proton emission channel (2p2n) in the fusion-evaporation reaction using \nthe neutron-richest beam and target combination of stable isotopes obtained so far. \nConsequently, we successfully populated the high-spin states of the SD double magic\nZ $=$ 18 and N $=$ 22 nucleus, $^{40}$Ar. \nIn this Letter, we report experimental results on the SD states in $^{40}$Ar \nassociated with the mp--mh excitation between the {\\it sd} and {\\it fp} shells. \n\n\nHigh-spin states in $^{40}$Ar have previously been studied by proton-$\\gamma$ \ncoincidence measurements using the $^{37}$Cl($\\alpha$,~p$\\gamma$) reaction \n\\cite{old40ar}.\nHigh-spin levels below 6.8~MeV were identified up to (8$^{+}$) and spin-parity \nassignments up to the 6$^{+}$ state were obtained from the particle-$\\gamma$ \nangular correlations. \nThe parity of the 5$^{-}$ state at 4.494~MeV was determined by the linear \npolarization of the 5$^{-}\\rightarrow$4$^{+}$ transition at 1602~keV.\nThe lifetimes of low-lying levels were measured by the Doppler-shift attenuation \nmethod. \nThe high E2 strengths of the 6$_{2}^{+}\\rightarrow$4$_{2}^{+}$ \nand 4$_{2}^{+}\\rightarrow$2$_{2}^{+}$ transitions are respectively deduced to be \n67$_{-19}^{+38}$ and 46$_{-10}^{+15}$ in Weisskopf units, \nwhich indicates the large collectivity of the band. \nHowever, the (8$^{+}$) assignment was based solely on the similarity of the\nlevel structure to that in $^{42}$Ca, but the $\\gamma-\\gamma$ coincidence\nrelations of the in-band transitions were not examined and the presence of the \nband structure was not unambiguously identified by experiment. \nTherefore, it is essential to find the higher-spin members of the rotational band \nand to confirm the coincidence relations between the in-band $\\gamma$ transitions.\n\n\nHigh-spin states in $^{40}$Ar were populated via the \n$^{26}$Mg($^{18}$O, 2p2n)$^{40}$Ar reaction with a 70-MeV $^{18}$O beam\nprovided by the tandem accelerator facility at the Japan Atomic Energy Agency.\nTwo stacked self-supporting foils of $^{26}$Mg enriched isotopes with \nthicknesses of 0.47 and 0.43 mg\/cm$^{2}$ were used. \nThe mean beam energy of the $^{18}$O beam used to irradiate the $^{26}$Mg foils \nwas 69.0~MeV.\nGamma rays were measured by the GEMINI-II array \\cite{gemini} consisting of \n16 HPGe detectors with BGO Compton suppression shields, in coincidence with \ncharged particles detected by the Si-Ball \\cite{siball}, a 4 $\\pi$ array \nconsisting of 11 $\\Delta$E Si detectors that were 170~$\\mu$m thick. \nThe most forward placed Si detector was segmented into five sections and \nthe other detectors were segmented into two sections each, giving a total of \n25 channels that were used to enhance the selectivity of multi charged-particle \nevents.\nWith a trigger condition of more than two Compton-suppressed Ge detectors\nfiring in coincidence with charged particles, a total number of \n6.6$\\times$10$^{8}$ events were collected.\n\nBased on the number of hits in the charged particle detectors, events were \nsorted into three types of E$_{\\gamma}-$E$_{\\gamma}$ coincidence matrices \nfor each evaporation channel.\nA symmetrized matrix was created and the RADWARE program ESCL8R \\cite{radware}\nwas used to examine the coincidence relations of $\\gamma$ rays. \nBy gating on the previously reported $\\gamma$ rays, high-spin states\nin $^{40}$Ar were investigated.\n\n\n\nBy gating on the known 1461, 1432, and 571~keV peaks of the \n2$^{+}\\rightarrow $0$^{+}$, \n4$^{+}\\rightarrow $2$^{+}$, and \n6$^{+}\\rightarrow $4$^{+}$ transitions respectively, \nseveral new levels were identified above the 5$^{-}$ states at 4.49~MeV by \nconnecting with high-energy $\\gamma$ transitions of $\\geq$2.5~MeV.\nThe previously assigned deformed band members of 2$_{2}^{+}$, 4$_{2}^{+}$, \nand 6$_{2}^{+}$ states were confirmed at 2.522, 3.515, and 4.960~MeV, \nrespectively.\nIn addition, two $\\gamma$-ray cascade transitions of 2269 and 2699~keV \nwere identified in coincidence with the 993, 1445, and 1841~keV transitions, \nwhich form a rotational band up to the (12$^{+}$) state at 11.769~MeV. \nLinking $\\gamma$ transitions were also observed between the excited \n2$_{2}^{+}$, 4$_{2}^{+}$, and 6$_{2}^{+}$ states and the low-lying \n2$_{1}^{+}$ and 4$_{1}^{+}$ levels, which establishes the excitation energies \nand the spin-parity assignment of the band. \n\n\nSpins of the observed levels are assigned on the basis of the \nDCO (Directional Correlations from Oriented states) ratios of $\\gamma$ rays \nby analyzing an asymmetric angular correlation matrix. \nThe multipolarities of the in-band transitions of the band and \nthe linking transitions of 4$_{2}^{+}\\rightarrow $2$_{1}^{+}$ and \n6$_{2}^{+}\\rightarrow $4$_{1}^{+}$ are consistent with a stretched quadrupole \ncharacter. \nAssuming E2 multipolarity for the stretched quadrupole transition, the parity \nof the band was assigned to be positive.\nThe multipolarity of the 2699~keV transition could not be determined due to \nthe lack of statistics, but it was in coincidence with other $\\gamma$ \ntransitions in the band and assigned as E2. \n\nTo determine the deformation of the band, the transition quadrupole moment Q$_{t}$ \nwas deduced. \nLifetimes were estimated by the \\cite{DSAM} technique, which is based on the \nresidual Doppler shift of the $\\gamma$-ray energies emitted from the deceleration \nof recoiling nuclei in a thin target. \nThe average recoil velocity $<\\beta>$ is expressed as a function of the \ninitial recoil velocity to obtain $F(\\tau) \\equiv <\\beta>\/\\beta_{0}$.\nIn Fig.~3, the fractional Doppler shift $F(\\tau)$ is plotted against the \n$\\gamma$-ray energy. \nThe experimental $F(\\tau)$ values are compared with the calculated values\nbased on known stopping powers \\cite{srim2003}. \nIn this calculation, the side feeding into each state is assumed to consist\nof a cascade of two transitions having the same lifetime as the in-band\ntransitions.\nThe intensities of the side-feeding transitions were modeled to reproduce\nthe observed intensity profile.\nThe data are best fitted with a transition quadrupole moment \n$Q_{t} = 1.45_{-0.31}^{+0.49} e$b, which corresponds to a quadrupole \ndeformation of $\\beta_{2}=0.53_{-0.10}^{+0.15}$.\nThis result is consistent with a superdeformed shape of the band.\n\n\n\nIn order to compare the high-spin behavior of the rotational band in $^{40}$Ar\nwith the SD bands in $^{36}$Ar and $^{40}$Ca, \nthe so-called 'backbending' plot of the SD bands is displayed in Fig.~\\ref{fig4}.\nThe gradients of the plots correspond to the kinematic moments of inertia. \nBecause $^{40}$Ar has a similar gradient to $^{36}$Ar and $^{40}$Ca, \nthe deformation size of the $^{40}$Ar rotational band is expected to be as \nlarge as the deformation of the SD bands in $^{36}$Ar and $^{40}$Ca. \nUnlike $^{36}$Ar, no backbending was observed in $^{40}$Ar.\nIts behavior is rather similar to that of $^{40}$Ca.\nMany theoretical models, including the shell model \\cite{ar36,caurier,poves,sun} \nand various mean-field models \\cite{inakura,bender,HFB}, have been used to analyze \n$^{36}$Ar.\nAll the calculations reveal that the strong backbending in \n$^{36}$Ar is due to the simultaneous alignment of protons and neutrons \nin the f$_{7\/2}$ orbital.\n\n\nThe pronounced difference in the high-spin behaviors of $^{36}$Ar and \n$^{40}$Ar implies that the addition of four neutrons to $^{36}$Ar gives rise to  \na dramatic effect on its structure. \nIn order to understand this structural change theoretically, cranked \nHartree--Fock--Bogoliubov (HFB) calculations with the P+QQ force \\cite{HFB} \nwere conducted.\nThe evolution of the nuclear shape was treated in a fully self-consistent manner \nand the model space of the full {\\it sd-fp} shell plus the g$_{9\/2}$ orbital \nwas employed.\nThe calculation shows that $\\beta_2 = 0.57$ at $J = 0 \\hbar$ and that the deformation \ngradually decreases to 0.45 at $J = 12\\hbar$. \nTriaxiality is found to be almost zero ($\\gamma\\simeq 0$) throughout this \nangular momentum range. \nThis result agrees with the experimental $Q_t$ value within the error bars.\n\n\n\nThe occupation number of each orbital was also calculated (Table \\ref{occupation}).\nThe ground-state configuration is expressed as \n({\\it sd})$^{-2}$({\\it fp})$^{2}$ relative to the ground-state configuration of \n$^{40}$Ca, where the Fermi levels for protons and neutrons lie at d$_{3\/2}$\nand f$_{7\/2}$, respectively.\nThe self-consistently calculated second $0^+$ state has \nthe ({\\it sd})$^{-6}$({\\it fp})$^{6}$ configuration.\nHere, the {\\it fp} shell is occupied by two protons and four neutrons,\nwhile the {\\it sd} shell has four proton holes and two neutron holes.\nConsidering the rise of the neutron Fermi level relative to that in $^{36}$Ar,\nthis excited configuration is essentially equivalent to the 4p--4h superdeformed \nconfiguration in $^{36}$Ar.\n\nCranking is then performed to study high-spin states. \nIn the proton sector, the behaviors of single-particle orbitals are similar to \nthose of $^{36}$Ar \\cite{HFB}. \nFor example, the occupation numbers in the $\\pi$p$_{3\/2}$ orbital monotonically\ndecrease up to $J=16 \\hbar$ while the $\\pi$f$_{7\/2}$ orbital starts to increase \nat $J=$ 8 $\\hbar$ due to the disappearance of the pairing gap energy.\n\nIn the neutron sector, clear differences are observed from the $^{36}$Ar case.\nThe occupation number in the $\\nu$f$_{7\/2}$ orbital is almost constant ($\\sim$3) \nagainst the total angular momentum; it is about 1.5 times larger than that for \n$^{36}$Ar. \nThe $\\nu$d$_{5\/2}$ orbitals are almost fully occupied ($\\simeq 5.5$) \nfrom the low- to the high-spin regions.\nIn the case of $^{36}$Ar, the structural change involving the sharp backbending\nis caused by a particle feeding from the p$_{3\/2}$ orbital to the f$_{7\/2}$ \norbital for both protons and neutrons.\nIn the neutron sector of $^{40}$Ar, this feeding happens from the p$_{3\/2}$\nto the many other single-particle orbitals.\nThis is because the rise of the neutron Fermi level enhances the occupation \nnumbers of the single-particle orbitals, particularly at the bottom end of \nthe {\\it fp} shell. \nFor example, the f$_{7\/2}$ is well occupied by $\\simeq 40\\%$. \nThis high occupation influences the response of the system to the Coriolis \nforce. \nIn general, low-$\\Omega$ states tend to come down energetically lower, so that \nsuch states are the first to be \"submerged\" when the Fermi level rises. \nAs a result, neutron states near the Fermi level in $^{40}$Ar possess a higher \n$\\Omega$ value and the rotational alignment is suppressed. \nIn $^{36}$Ar, many $\\Omega = 1\/2$ states are vacant in the {\\it fp} shell, \nso that it is possible to place particles in the $\\Omega = 1\/2$ states during \nthe feeding from the p$_{3\/2}$ orbital to the f$_{7\/2}$ orbital.\nHowever, in $^{40}$Ar, such $\\Omega = 1\/2$ states are filled due to the\nrise of the neutron Fermi level. \nIt is thus necessary to place neutrons in the $\\Omega$ = 3\/2 or 5\/2 levels \nin order to generate angular momentum. \nBut this way of feeding weakens the rotational alignment.\nThis ``Pauli blocking effect'' is one of the reasons why $^{40}$Ar does not \nbackbend (at least, not in the spin region so far observed). \nIt is also worth mentioning that because of the rise of the neutron Fermi level \nin $^{40}$Ar, angular momentum generation is spread among the extra f$_{7\/2}$ \nneutrons, in comparison with $^{36}$Ar. \nThis means that, unlike their neutron counterparts, the f$_{7\/2}$ protons do not \nneed to \"work hard\" to generate angular momentum. \nAs a result, simultaneous alignment of the f$_{7\/2}$ protons and neutrons \ndoes not occur in $^{40}$Ar. \nOur calculation confirms this picture. \n\n\nIn summary, a discrete-line superdeformed band has been identified in $^{40}$Ar. \nThe observed large transition quadrupole moment ($Q_{t}= 1.45_{-0.31}^{+0.49} e$b)\nsupports its SD character. \nThe properties of the SD band could be reasonably well explained by cranked HFB \ncalculations with the P+QQ force. \nThis finding of the SD band in $^{40}$Ar is similar to those observed in $^{36}$Ar \n\\cite{ar36} and $^{40}$Ca \\cite{ca40}, indicating the persistence of the SD shell \nstructure in the neutron-rich A $=$ 30 $\\sim$ 40 nuclei and possibly implying that \n$^{40}$Ar is a doubly SD magic nucleus with Z $=$ 18 and N $=$ 22.\nThe observed SD structure with a deformation of $\\beta_2 \\sim$ 0.5 caused by the \nmp--mh excitation across the {\\it sd--fp} shell gap might explain\nthe origin of the strongly deformed ground state in the 'island of inversion'.\n\n\nThe authors thank the staff at the JAEA tandem accelerator for providing \nthe $^{18}$O beam. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn his seminal paper, published in 1870, about the solution of a nonlinear equation in a single unknown, \n\\begin{equation} \nf(z)=0,\n\\end{equation}\\label{eq1}\nSchr\\\"oder deals with the problem of characterizing general iterative algorithms to solve~\\eqref{eq1} with a prefixed order of convergence $\\omega\\ge 2$ (see the original paper \\cite{Sch} or the commented English translation \\cite{Ste}). The main core of Schr\\\"oder's work studies two families of iterative processes, the well-known families of first and second kind \\cite{Pet1}, \\cite{Pet2}. The $\\omega$-th member of  these families is an iterative method that converges with order $\\omega$ to a solution of~\\eqref{eq1}. In this way, the second method of both families is Newton's method. The third method of the first family is Chebyshev's method. The third method of the second family is Halley's method. The rest of the methods in both families (with order of convergence $\\omega\\ge 4$) are not so well known. \n\nNote that Newton's, Chebyshev's and Halley's method are also members of another famous family of iterative methods, the known as Chebyshev-Halley family of methods (introduced by Werner \\cite{Wer} and reported by may other authors such as \\cite{Amat} or \\cite{Dub}):\n \\begin{equation}\\label{eq2}\nz_{k+1}=z_k-\\left(1-\\frac{1}{2}\\frac{L_f(z_k)}{1-\\alpha L_f(z_k)}\\frac{f(z_k)}{f'(zk+1)}\\right), \\quad \\alpha\\in \\mathbb{R},\\quad n\\ge 0, \\quad z_0\\in\\mathbb{C},\n\\end{equation}\nwhere we have used the notation\n \\begin{equation}\\label{eq3}\n L_f(z)=\\frac{f(z)f''(z)}{f'(z)^2}.\n\\end{equation}\nIn fact, Chebyshev's method is obtained for $\\alpha=0$, Halley's method appears for $\\alpha=1\/2$ and Newton's method can be obtained as a limit case when $|\\alpha| \\to \\infty$. Except for the limit case of Newton's method, all the methods in the family have third order of convergence.\n\nIn this general context of families of iterative methods for solving nonlinear equations, we would like to highlight a detail that appears in the aforementioned paper by Schr\\\"oder~\\cite{Sch}.  Actually, in the third section of this article, Schr\\\"oder constructs an algorithm by applying Newton's method to the equation \n$$\n\\frac{f(z)}{f'(z)}=0.\n$$\nThe resulting iterative scheme can be written as\n$$\nz_{k+1}=z_k-\\frac{f(z_k)f'(z_k)}{f'(z_k)^2-f(z_k)f''(z_k)}, \\quad n\\ge 0, \\quad z_0\\in\\mathbb{C},\n$$\nas it is known as \\emph{Schr\\\"oder's method} by many authors (see for instance \\cite{Proinov} or  \\cite{Scavo}).\n\nFor our convenience, we denote by $S_f(z)$ the iteration function of Schr\\\"oder's method. Note that it can be written in terms of the function $L_f(z)$ introduced in~\\eqref{eq3} in the following way\n \\begin{equation}\\label{eq4}\nz_{k+1}=S_f(z_k)=z_k-\\frac{1}{1-L_f(z_k)}\\frac{f(z_k)}{f'(z_k)} \\quad n\\ge 0, \\quad z_0\\in\\mathbb{C}.\n\\end{equation}\nThe same Schr\\\"oder compares the resulting algorithm~\\eqref{eq4} with Newton's method and says: \n\\begin{quote}\n``It is an equally worthy algorithm which to my knowledge has not been previously considered. Besides, being almost as simple, this latter algorithm has the advantage that it converges quadratically even for multiple roots''.\n\\end{quote}\n\nCuriously, Schr\\\"oder's method~\\eqref{eq4} does not belong neither to the Schr\\\"oder's families of first and second kind nor the Chebyshev-Halley family~\\eqref{eq2}. It has very interesting numerical properties, such as the quadratic convergence even for multiple roots, but the fact of having a high computational cost (equivalent to the the third order methods in~\\eqref{eq2}) could be an important handicap for practical purposes.\n\nIn this paper we present a first  approach to the dynamical behavior of Schr\\\"oder's method. So we show that for polynomials with two different roots and different multiplicities, it is possible to characterize the basins of attraction and the corresponding Julia set. We can appreciate the influence of the multiplicities in such sets.\n\n\\section{Preliminaries}\n\nIn the 13th section of Schr\\\"oder's work~\\cite{Sch}, that has the title \\emph{The Principal Algorithms Appllied to very Simple Examples}, we can find the first dynamical study of a couple of rootfinding methods. Actually, Schr\\\"oder's considers those who, in his opinion, are the two most useful methods: Newton's method, defined by the iterative scheme\n\\begin{equation}\\label{eq5}\nz_{k+1}=N_f(z_k)=z_k-\\frac{f(z_k)}{f'(z_k)} \\quad n\\ge 0, \\quad z_0\\in\\mathbb{C},\n\\end{equation}\nand the method $z_{k+1}=S_f(z_k)$ given in~\\eqref{eq4}.\n\nIn the simplest case, namely equations with only one root,  we can assume without loss of generality that $f(z)=z^n$. It is easy to see that\n$$\nN_f(z)=\\frac{n-1}{n}z,\\quad S_f(z)=0.\n$$\nSo Schr\\\"oder's method gives the correct root ($z=0$) of the equation in just one step, whereas Newton's method converges to this root with lineal convergence:\n$$\nz_k=\\left(\\frac{n-1}{n}\\right)^k z_0.\n$$\nConsequently, for equations with s single root Schr\\\"oder concludes that the convergence regions of these two methods is the entire complex plane.\n\nThe next simple case considered by Schr\\\"oder is the quadratic equation. Again, without loss of generality he asumes \n$f(z)=(z-1)(z+1)$. After a series of cumbersome calculus, he estates that in this case and for both methods, the entire complex plane decomposes into two regions separated by the imaginary axis. A few years later, Cayley~\\cite{Cay} addresses the same problem, only for Newton's method. In a very elegant way, Cayley proves that for polynomials\n\\begin{equation}\\label{eq6}\nf(z)=(z-a)(z-b),\\quad a,b\\in \\mathbb{C}, \\quad a\\ne b,\n\\end{equation}\nNewton's iterates converge to the root $a$ if $|z_0-a|<|z_0-b|$ and to the  root $b$ if $|z_0-b|<|z_0-a|$. The Julia set is the equidistant line between the two roots. The key to prove this result is to check that Newton iteration function~\\eqref{eq5} applied to polynomials~\\eqref{eq6} is conjugate via the M\\\"obius map\n\\begin{equation}\\label{eq7}\nM(z)=\\frac{z-a}{z-b}\n\\end{equation}\nwith the function $R(z)=z^2$, that is, $R(z)=M\\circ N_f\\circ M^{-1}(z)$. The unit circle $S^1=\\{z\\in\\mathbb{C}; |z|=1\\}$ is invariant by $R$. Its anti-image by $R$ is the bisector between the roots $a$ and $b$.\n\nTwo functions $f,g: \\mathbb{C} \\to  \\mathbb{C} $ are said topologically conjugate if  there exists a homeomorphism $\\varphi$ such that \n$$\n\\varphi\\circ g=f\\circ \\varphi.\n$$\nTopological conjugation is a very useful tool in dynamical systems (see \\cite{Dev} for more details)\nbecause two conjugate functions share the same dynamics properties, from the topological viewpoint. For instance, the fixed points of one function are mapped into the fixed points of the other, the periodic points of one function are mapped into the periodic points of the other function, and so on. Speaking informally, we can say that the two functions are the same from a dynamical point of view. As we have just seen, in some cases one of the functions in a conjugation could be much simpler than the other. In the case of Cayley's problem $R(z)=z^2$ is topologically conjugate (and much simpler) to\n$$\nN_f(z)=\\frac{a b-z^2}{a+b-2 z}.\n$$\n\n\nIn the same way, we have that Schr\\\"oder's method~\\eqref{eq4} applied to polynomials~\\eqref{eq6} \n$$\nS_f(z)=\\frac{z^2 (a+b)-4 a b z+a b (a+b)}{a^2-2 z (a+b)+b^2+2 z^2}\n$$is conjugated with $-R(z)$ via the M\\\"obius map defined in~\\eqref{eq7},  that is $M\\circ S_f\\circ M^{-1}(z)=-z^2$. Consequently, the dynamical behavior of Schr\\\"oder's method for quadratic polynomials mimics the behavior of Newton's method: the Julia set is the bisector between the two roots and the basins of attraction are the corresponding half-planes.\n\n\\section{Main results}\n\nNow we consider the case of polynomials with two roots, but with different multiplicities, $m\\ge n\\ge 1$:\n\\begin{equation}\\label{eq8}\nf(z)=(z-a)^m(z-b)^n,\\quad a,b\\in \\mathbb{C}, \\quad a\\ne b.\n\\end{equation}\nFor our simplicity, and to better appreciate symmetries, we move the roots $a$ and $b$ to $1$ and $-1$. To do this, we conjugate with the affine map\n\\begin{equation}\\label{eq9}\nA(z)=1+2\\frac{z-a}{a-b}\n\\end{equation}\nto obtain a simpler function that does not depend on the roots $a$ and $b$. Let $T_{m,n}(z)$ be the corresponding conjugate map:\n\\begin{equation}\\label{eq10}\nT_{m,n}(z)=A\\circ S_f\\circ A^{-1}(z)=\\frac{(m-n) z^2 +2 (m+n) z +m-n}{(m+n) z^2 +2(m-n)z +m+n}.\n\\end{equation}\nA new conjugation of $T_{m,n}(z)$ with the M\\\"obius map \\eqref{eq7}, when $a=1$ and $b=-1$ provides us a new rational function whose dynamics are extremely simple. Actually:\n\\begin{equation}\\label{eq11}\nR_{m,n}(z)=M\\circ T_{m,n}\\circ M^{-1}(z)=-\\frac{nz^2}{m}.\n\\end{equation}\n\nNote that the circle $C_{m,n}=\\{z\\in\\mathbb{C}; |z|=m\/n\\}$ is invariant by $R_{m,n}(z)$. After iteration by $R_{m,n}(z)$, the orbits of the points $z_0$ with  $|z_0|<m\/n$ go to 0, whereas the orbits  of the points $z_0$ with  $|z_0|>m\/n$ go to $\\infty$. Consequently, $C_{m,n}$ is the Julia set of the map $R_{m,n}(z)$.\n\n\n\\begin{Theorem}\\label{teo1}\nLet $T_{m,n}(z)$ be the rational map defined by~\\eqref{eq10} and let us denote by  $J_{m,n}$ its Julia set. Then we have:\n\\begin{enumerate}\n\\item   If $m=n$,  then $J_{m,m}$ is the imaginary axis.\n\\item   If $m>n\\ge 1$, then $J_{m,n}$  is the circle\n$$J_{m,n}=\\left\\{z\\in\\mathbb{C}; \\left|z+\\frac{m^2+n^2}{m^2-n^2}\\right|=\\frac{2mn}{m^2-n^2}\\right\\}.$$\n\\end{enumerate}\n\\end{Theorem}\n\n\\begin{proof\nThe proof follows immediately, just by taking into account that $J_{m,n}$ is the pre-image by $M(z)=(z-1)\/(z+1)$ of the circle $C_{m,n}$ and by distinguishing the two situations indicated in the statement of the theorem.\n\\end{proof}\n\n\\begin{Theorem}\\label{teo2}\nLet $S_f(z)$ be the rational map defined by applying Schr\\\"oder's method to polynomials~\\eqref{eq8} and let us denote by  $J_{m,n,a,b}$ its Julia set. Then we have:\n\\begin{enumerate}\n\\item   If $m=n$,  $J_{m,m,a,b}$ is the equidistant line between the points $a$ and $b$.\n\\item   If $m>n\\ge 1$, then $J_{m,n,a,b}$  is the circle\n$$J_{m,n,a,b}=\\left\\{z\\in\\mathbb{C}; \\left|z+\\frac{b m^2-a n^2}{m^2-n^2}\\right|=\\frac{mn |a-b|}{m^2-n^2}\\right\\}.$$\n\\end{enumerate}\n\\end{Theorem}\n\n\\begin{proof\nNow we deduce this result  by calculating the pre-images of $J_{m,n}$ by the affine map $A(z)$ defined in~\\eqref{eq9}  in the two situations indicated in the previous theorem.\n\\end{proof}\n\n\n\n\\section{Conclusions and further work}\n\n\nWe have studied the behavior of Schr\\\"oder's method for polynomials with two different complex roots and with different multiplicities~\\ref{eq8}. Actually, we have proved that the Julia set of the corresponding rational functions obtained in this case is a circle given in Theorem~\\ref{teo2}.\n\nIn addition, Theorem~\\ref{teo1} gives us a universal result that characterizes the behavior of Schr\\\"oder's method in a very simplified form, depending only of the values of the multiplicities $m$ and $n$. The influence of the roots $a$ and $b$ is revealed in Theorem~\\ref{teo2}, and is just an affine transformation of the situation given in Theorem~\\ref{teo1}.\n\n\nLet us consider the points $(x,y)\\in\\mathbb{R}^2$ given by the centers and radius of the circles defined in Theorem~\\ref{teo1}, that is\n$$\n(x,y)=\\left( \\frac{m^2+n^2}{m^2-n^2}, \\frac{2mn}{m^2-n^2}\\right).\n$$\nThese points belong to the hyperbola $x^2-y^2=1$ in the real plane $\\mathbb{R}^2$.\n\nIn addition, we appreciate that are polynomials for which Schr\\\"oder's method  has the same dynamical behavior. Actually, if we introduce the new parameter\n$$\np=\\frac{m}{n},\n$$\nwe have that the circles $J_{m,n}$ defined in Theorem~\\ref{teo1} can be expressed as\n$$J_{p}=\\left\\{z\\in\\mathbb{C}; \\left|z+\\frac{p^2+1}{p^2-1}\\right|=\\frac{2p}{p^2-1}\\right\\}.$$\nTherefore Schr\\\"oder's method applied to polynomials with couples of multiplicities $(m,n)$ having the same quotient $p$ have the same Julia set.\n\nWe can schematize of the dynamics of Schr\\\"oder's method applied to polynomials $(z-1)^m(z+1)^n$, $m>n$ in the following way:\n\\begin{itemize}\n\\item When $p=m\/n\\to \\infty$, the Julia sets $J_p$ are circles that tends to collapse in the point $z=-1$.\n\\item When $p=m\/n\\to 1^+$, the Julia sets $J_p$ are circles with centers in the negative real line. Note\nthat centers\n$$\n-\\frac{p^2+1}{p^2-1}\\to -\\infty \\text{ when } p\\to 1^+\n$$\nand radius\n$$\n\\frac{2p}{p^2-1}\\to \\infty \\text{ when } p\\to 1^+.\n$$\nSo when $p\\to 1^+$ the Julia set are circles getting bigger and tending to ``explode'' into the limit case, given by the imaginary axis when $p=1$.\n\\end{itemize}\n\nIf we consider the presence of the roots $a$ and $b$, the dynamics of Schr\\\"oder's method applied to polynomials $(z-a)^m(z-b)^n$, $m>n$ can be summarized in a ``travel'' from a circle concentrated in the root with the smallest multiplicity, $b$ to circles with the center in the line connecting the roots $a$ and $b$ and radius tending to infinity until the ``explosion'' into the limit case, given by the bisector of the two roots,  when $p=1$.\n\nIn Figures~\\ref{fig1}--\\ref{fig3} we show some graphics of different Julia sets obtained when Schr\\\"oder's method is applied to polynomials $(z-a)^m(z-b)^n$, $m\\ge n\\ge 1$. We compare these dynamical planes with the ones obtained for Newton's.   For instance, in Figure~\\ref{fig1} we show the behavior when $p=m\/n$ is increasing. We appreciate how the Julia set for Schr\\\"oder's method (a circle) tends to to collapse in the point $z=-1$  that in this case is the simple root. In the case of Newton's method, the Julia set is a kind of ``deformed parabola'', whose ``axis of symmetry''  is the real line, it is open to the left, the ``vertex'' tends to the simple root $z=-1$ and the ``latus rectum''  tends to zero. We see how the basin of attraction of the multiple root $z=1$ invades more and more the basin of the simple root $z=-1$, as it was pointed out by Hern\\'andez-Paricio et al. \\cite{Gut}.\n\nIn Figure~\\ref{fig2} we see what happens when $p=m\/n\\approx 1$. The Julia set for Schr\\\"oder's method are circles getting bigger as $p$ approaches the value 1 and exploding into a half-plane limited by the imaginary axis when $p=1$. In the case of Newton's method, the Julia set is again a ``deformed parabola'' with  the real line as   ``axis of symmetry''  and open to the left. However as $p$ goes to 1, the ``vertex'' tends to  $z=0$ and the ``latus rectum''  tends to infinity. As a limit case, when $p=1$ this ``deformed parabola'' becomes a straight line, actually the imaginary axis.\n\n\nFigure~\\ref{fig3} shows the circle corresponding to the Julia of Schr\\\"oder's method applied to polynomials $(z-1)^m(z+1)^n$ with $p=m\/n=2$. We can also see the Julia set of Newton's method applied to such polynomials. In the case of Newton's method we observe that the behavior is not the same fo values of $m$ and $n$ such that $p=m\/n=2$. The corresponding ``deformed parabola'' tends to be smoother when the values of $m$ and $n$ increase.\n\nFinally in Figure~\\ref{fig4} we show the Julia set $J_{m,n,a,b}$ defined in Theorem~\\ref{teo2} in the case $m=2$, $n=1$, $a=1$, $b=i$ together with the corresponding Julia set for Newton's method. In these figures we appreciate the loss of symmetry respect to the imaginary axis. This role is now played by the equidistant line between the roots $a$ and $b$.\n\nAs a further work, we would like to explore the  influence of the multiplicity in the Julia set of Newton's method applied to polynomials $(z-a)^m(z-b)^n$, $m\\ge n\\ge 1$ and its possible relationship between the study of Schr\\\"oder's method. In particular, we are interested in characterize the main properties of the ``deformed parabolas'' that appear in the case of Newton's method.\n\n\n\n\n\n\\begin{figure}[H]\n\\centering \n\\twofractals{imagenes\/sch_m_2n_1.jpg}{Schr\\\"oder $m=2, \\, n=1.$}{imagenes\/newton_m_2n_1}{Newton $m=2, \\, n=1.$}\n\n\\twofractals{imagenes\/sch_m_5n_1}{Schr\\\"oder $m=5, \\, n=1.$}{imagenes\/newton_m_5n_1}{Newton $m=5, \\, n=1.$}\n\n\\twofractals{imagenes\/sch_m_8n_1}{Schr\\\"oder $m=8, \\, n=1.$}{imagenes\/newton_m_8n_1}{Newton $m=8, \\, n=1.$}\n\\caption{Basins of attraction of Schr\\\"oder's  and Newton's methods applied to polynomials $(z-1)^n(z+1)^m$ for $n=1$, $m=2, 5, 8$.}\n\\label{fig1}\n\\end{figure}  \n\n\\begin{figure}[H]\n\\centering \n\\twofractals{imagenes\/sch_n_6_m_6}{Schr\\\"oder $m=6, \\, n=6.$}{imagenes\/newton_n_6_m_6}{Newton $m=6, \\, n=6.$}\n\n\\twofractals{imagenes\/sch_m_7n_6}{Schr\\\"oder $m=7, \\, n=6.$}{imagenes\/newton_m_7n_6}{Newton $m=7, \\, n=6.$}\n\n\\twofractals{imagenes\/sch_m_8n_6}{Schr\\\"oder $m=8, \\, n=6.$}{imagenes\/newton_m_8n_6}{Newton $m=8, \\, n=6.$}\n\\caption{Basins of attraction of Schr\\\"oder's  and Newton's methods applied to polynomials $(z-1)^n(z+1)^m$ for $n=6$, $m=6, 7, 8$.}\n\\label{fig2}\n\\end{figure}  \n\n\n\\begin{figure}[H]\n\\centering \n\\imagen{imagenes\/sch_m_4n_2}{Schr\\\"oder $p=m\/n=2.$}{imagenes\/newton_m_4n_2}{Newton $m=4, \\, n=2.$}\n\n\\imagen{imagenes\/newton_m_6n_3}{Newton $m=6, \\, n=3.$}{imagenes\/newton_m_8n_4}{Newton $m=8, \\, n=4.$}\n\n\\caption{The first graphic shows the basin of attraction of Schr\\\"oder's method applied to polynomials $(z-1)^n(z+1)^m$ with $p=m\/n=2$. The other graphics show the basins of attraction of  Newton's method applied to the same polynomials  for different values of $m$ and $n$ with $p=m\/n=2$.}\n\\label{fig3}\n\\end{figure}  \n\n\\begin{figure}[H]\n\\centering \n\\includegraphics[width=0.4\\textwidth]{imagenes\/sch-i_m2_n1.jpg}\\quad \n\\includegraphics[width=0.4\\textwidth]{imagenes\/newton1-i_m2_n1.jpg}\n\n\n\\caption{Basin of attraction of Schr\\\"oder's and Newton's methods applied to polynomials $(z-1)^2(z+i)$.}\n\\label{fig4}\n\\end{figure}  \n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRayleigh-B\\'enard convection of a horizontal fluid layer heated from\nbelow is a paradigmatic system for the study of complex spatial and\nspatio-temporal patterns \\C{BoPe00}. Usually, the system parameters\nare chosen such that variations of the fluid properties across the\nlayer are very small. In the theoretical description this allows the\nOberbeck-Boussinesq (OB) approximation, in which the temperature\ndependence is neglected in all fluid properties except for the density\nterm that is responsible for the buoyancy. For large temperature\ndifferences across the layer the variation of the fluid properties\nwith the temperature become significant. These non-Oberbeck-Boussinesq\n(NB) effects  break the up-down symmetry that is characteristic of the\nOB approximation and, thus, allow a resonant triad interaction among\nthree Fourier modes whose wavevectors form a hexagonal planform. Due\nto the resonant interaction the primary bifurcation to the hexagons is\ntranscritical, and hexagons are preferred over rolls in the immediate\nvicinity of onset \\cite{Bu67}. \n\nWithin the framework of the leading-order weakly nonlinear analysis,\nhexagons typically become unstable to rolls further above threshold,\nwhere the amplitudes are larger and the resonant-triad interaction\nloses significance compared to interactions involving four modes\n\\cite{Pa60,SeSt62,Se65,Bu67,PaEl67,DaSe68}. This scenario of a\ntransition from hexagons to rolls has been confirmed in quite a number\nof experimental investigations\n\\cite{SoDo70,DuBe78,Ri78,WaAh81,BoBr91,PaPe92}, and quite commonly it\nhas been assumed that hexagon patterns in NB convection are confined\nto the regime close to onset.\n\nTwo convection experiments using SF$_6$ as the working fluid\n\\C{AsSt96,RoSt02} have shown, however, that even in the strongly\nnonlinear regime stable hexagon patterns can be observed. Under OB\nconditions \\cite{AsSt96} hexagons were found at relatively high\nRayleigh numbers, $\\epsilon \\equiv (R-R_c)\/R_c \\approx 3.5$ \n\\C{AsSt96}. Due to the mid-plane symmetry of OB convection hexagons\nwith up-flow in the center coexist stably with down-flow hexagons  in\nthis regime. In experiments using SF$_6$ under NB conditions\n\\cite{RoSt02} the hexagons that appear at threshold were replaced by\nrolls for somewhat larger heating and then reappeared in the strongly\nnon-linear regime near $\\epsilon ={\\mathcal O}(1)$. The\nrestabilization was attributed to the large compressibility of SF$_6$\nnear its critical point \\cite{RoSt02}. The hexagons that regain\nstability were termed {\\em reentrant hexagons}.\n\nRecent numerical computations \\cite{MaRi05} have demonstrated that \nhexagons can restabilize in NB convection even if the fluid is {\\em\nincompressible}.  For instance, in water hexagons that are unstable at\n$\\epsilon=(R-R_c)\/R_c =0.15$ can already restabilize at\n$\\epsilon=0.2$. The origin of this reentrance was traced back to the\nexistence of stable hexagons in OB convection at large Rayleigh\nnumbers, and the dependence of the NB effects on the Rayleigh number.\n\nAt low Prandtl numbers ($Pr\\simeq 1$) and further above onset OB\nconvection exhibits a new state: {\\em spiral defect chaos} (SDC). It\nwas first found experimentally  \\C{MoBo93,MoBo96,LiAh96} and then\ninvestigated theoretically using simple models \\cite{XiGu93,CrTu95} as\nwell as simulations of the full fluid equations \\cite{DePe94a,ChPa03}.\nThis fascinating state of spatio-temporal chaos is characterized by\nrotating spirals with varying numbers of arms and of different size,\nwhich appear and disappear irregularly and interact with each other\nand with other defects. SDC arises from the roll state at a threshold \nthat can be as low as $\\epsilon=0.1$ in the limit of small $Pr$.  It\nis driven by large-scale flows that are induced by roll curvature and\nhave a strength that is proportional to $Pr^{-1}$ \\cite{ChPa03}.\n\nSo far, strongly non-linear NB convection has been studied mostly for\nlarge Prandtl numbers \\cite{YoRi03b,MaPe04,MaRi05}, but little is\nknown for small ( $Pr\\simeq 1$) or very small Prandtl numbers ($Pr\\ll \n1$). In particular, whether reentrant hexagons exist at large\n$\\epsilon$  in the presence of the large-scale flows that arise at low\n$Pr$, and how NB effects impact spiral defect chaos are interesting\nquestions, which we address in this paper. \n\nHere we study NB convection in gases with small Prandtl numbers.\nSpecifically, we consider parameters corresponding to convection in\nCO$_2$ and SF$_6$ ($Pr\\simeq 0.8$) and in a H$_2$-Xe mixture\n($Pr=0.17$). We show that reentrant hexagons are possible in\nconvection in CO$_2$. For spiral defect chaos in SF$_6$ we find that\nNB effects promote small convection cells (`bubbles'). In H$_2$-Xe,\ncloser to threshold, roll-like structures dominate. In both cases the\nNB effects reduce the spiral character of the pattern. We quantify the\nimpact of the NB effects on spiral defect chaos using geometric\ndiagnostics that we have proposed recently \\cite{RiMa06}.  \n\nThe paper is organized as follows. In Sec.\\ref{sec:basicequations} we\nbriefly review the basic equations, emphasizing how our computations\nfocus on weakly non-Boussinesq effects, but strongly nonlinear\nconvection. In Sec.\\ref{sec:co2stability} we present the results for\nthe linear stability of hexagons and rolls in CO$_2$ for a range of\nparameters accessible experimentally. To compare with experiments, we\nstudy the influence of different lateral boundary conditions on the\ntransition from hexagons to rolls in Sec. \\ref{sec:sim-co2}.  In Sec.\n\\ref{sec:sim-sf6} we discuss  spiral defect chaos in SF$_6$ under NB\nconditions.  The stability of hexagons and spiral defect chaos in\nfluids with very low Prandlt number ($Pr=0.17$) is studied in a \nmixture of $H_2$ and $Xe$ in Sec. \\ref{sec:h2xe}. Finally, conclusions\nare drawn in Sec.\\ref{sec:conclusions}.\n\n\n\\section{Basic equations \\LB{sec:basicequations} }\n\nThe basic equations that we use for the description of NB\nconvection have been discussed in detail previously\n\\cite{YoRi03b,MaRi05}. We therefore give here only a brief summary. \nWe consider a horizontal fluid layer of thickness $d$, density $\\rho$,\nkinematic viscosity $\\nu$, heat conductivity $\\lambda$, thermal\ndiffusivity $\\kappa$, and specific heat $c_p$. The system is heated\nfrom below (at  temperature $T_1$) and cooled from above (at\ntemperature $T_2 < T_1$). \n\nTo render the governing equations and boundary conditions\ndimensionless we choose the length $d$, the time $d^{2}\/\\kappa_0$, the\nvelocity $\\kappa_0\/d$, the pressure $\\rho_0\\nu_0 \\kappa_0\/d^{2}$, and\nthe temperature $T_s=\\nu_0 \\kappa_0\/\\alpha_0 g d^3$ as the respective\nscales. The subscripted quantities refer to the respective values at\nthe middle of the fluid layer in the conductive state. The\nnon-dimensionalization gives rise to two dimensionless quantities: the\nPrandtl number $Pr=\\nu_0\/\\kappa_0$, and  the Rayleigh number\n$R=\\alpha_0 \\Delta T g d^3\/\\nu_0 \\kappa_0$.  Furthermore, we write the\nequations in terms of the dimensionless momentum density $v_i=\\rho d\nu_i\/\\rho_0 \\kappa_0$ instead of the velocities $u_i$. The\ndimensionless form of the temperature $\\hat T =T\/T_s$, heat\nconductivity $\\hat \\lambda =\\lambda\/\\lambda_0$, density  $\\hat \\rho\n=\\rho\/\\rho_0$, kinematic viscosity $\\hat \\nu =\\nu\/\\nu_0$, and specific\nheat $\\hat c_p =c_p\/c_{p0}$ will be used in the  ensuing equations and\nthe hats dropped for clarity.  In dimensionless form the equations for\nthe momentum, mass conservation and heat are then given, respectively,\nby\n\\begin{eqnarray}\n\\frac{1}{Pr}\\left(\\partial_tv_i+v_j\\partial_j\\left(\\frac{v_i}{\\rho}\\right)\\right)&=&-\\partial_i\np  \\\\\n&&+\\delta_{i3}\\left(1+\\gamma_1(-2 z+\\frac{\\Theta}{R})\\right)\\Theta\\nonumber \\\\\n&&+\\partial_j\\left[\\nu\\rho\\left(\\partial_i(\\frac{v_j}{\\rho})+\\partial_j(\\frac{v_i}{\\rho})\\right)\\right]\n\\nonumber \\LB{e:v}\\\\\n\\partial_jv_j&=&0, \\LB{e:cont}\\\\\n\\partial_t\\Theta+\\frac{v_j}{\\rho}\\partial_j\\Theta\n& =&\\frac{1}{\\rho\nc_p}\\partial_j(\\lambda\\partial_j\\Theta)-\\gamma_3\\partial_z\\Theta-\\nonumber \\\\\n&& R\\frac{v_z}{\\rho}(1+\\gamma_3z).\\LB{e:T}\n\\end{eqnarray}\nwith the dimensionless boundary conditions \n\\begin{eqnarray}\n\\vec{v}(x,y,z,t)=\\Theta(x,y,z,t)=0  \\mbox{ at } z= \\pm \\frac{1}{2}.\\LB{e:bc}\n\\end{eqnarray}\nHere $\\Theta$ is the deviation of the temperature field from the basic\nconductive profile. Summation over repeated indices is assumed.\n\nWe consider the NB effects to be weak and retain in a\nTaylor expansion of all material properties only the leading-order\ntemperature dependence {\\it beyond} the OB approximation. For\nthe density this implies also a quadratic term with coefficient\n$\\gamma_1$. It contributes, however, only to the buoyancy term in\n(\\ref{e:v}); in all other expressions it would constitute only a\nquadratic correction to the leading-order NB effect. Thus,\nthe remaining temperature dependence of the fluid parameters $\\rho$,\n$\\nu$, $\\lambda$, and $c_p$ in (\\ref{e:v},\\ref{e:cont},\\ref{e:T}) is\ntaken to be linear\n\\begin{eqnarray}\n\\rho(\\Theta)&=&1-\\gamma_0(-z+\\frac{\\Theta}{R}),\\LB{e:rhoTh}\\\\\n\\nu(\\Theta)&=& 1+\\gamma_2(-z+\\frac{\\Theta}{R}),\\LB{e:nuTh}\\\\\n\\lambda(\\Theta)&=&1+\\gamma_3(-z+\\frac{\\Theta}{R}),\\LB{e:lambdaTh}\\\\\nc_p(\\Theta)&=&1+\\gamma_4(-z+\\frac{\\Theta}{R}).\\LB{e:cpTh}\n\\end{eqnarray}\n\nThe coefficients $\\gamma_i$ give the difference of the respective\nfluid properties across the layer. They depend therefore linearly on\nthe Rayleigh number, \n\\begin{eqnarray}\n\\gamma_i(\\Delta T)=\\gamma_i^{c}\\,\\left(\\frac{R}{R_c} \\right) \n=\\gamma_i^{c} \\, (1+\\epsilon) ,\n\\end{eqnarray}\nwhere $\\gamma_i^{c}$ is the value of $\\gamma_i$ at the onset of convection\nand $\\epsilon\\equiv (R-R_c(\\gamma_i^{c}))\/R_c(\\gamma_i^{c})$ is the\nreduced Rayleigh number. \n\nIn analogy to \\cite{Bu67}, we further omit NB terms that\ncontain cubic nonlinearities in the amplitudes $v_i$ or $\\Theta$, as\nthey arise from the expansion of the advection terms $v_j\n\\partial_j(v_i\/\\rho)$ and $(v_j\/\\rho)\\partial_j \\Theta$ when the\ntemperature-dependence of the density is taken into account.  Since we\nwill be considering Rayleigh numbers up to twice the critical value,\nwhich implies enhanced NB effects, these approximations\nmay lead to quantitative differences compared to the fully\nNB system, even though the temperature-dependence of the\nmaterial properties themselves is in most situations well described by\nthe linear (or quadratic in the case of the density) approximation.  \n\nTo quantify the overall strength of the NB-effects we use Busse's\nNB parameter $Q$, which is given by\n\\begin{eqnarray}\nQ = \\sum_{i=0}^{4}\\gamma_i^{c} {\\cal P}_i,\\LB{e:busseq}\n\\end{eqnarray}\nwhere the quantities ${\\cal P}_i$ are linear functions of $Pr^{-1}$.\nThe NB parameter $Q$ quantifies the breaking of the up-down symmetry,\nwhich renders at most one of the two types of hexagons stable. Gases\nhave a positive value of $Q$ and exhibit hexagons with down-flow in\nthe center ($g$-hexagons), whereas  liquids have negative $Q$ and show\nhexagons with up-flow ($l$-hexagons).\n\nWe focus in this paper on the stability properties of patterns in the\nstrongly nonlinear regime. They are determined by a Galerkin method\n(e.g. \\cite{BuCl79a}). We use a Fourier expansion on a hexagonal\nlattice in the lateral directions.  The Fourier wave vectors ${\\bf q}$\nare constructed as linear combinations  of the hexagon basis vectors\n${\\bf b}_1 =q(1,0)$ and  ${\\bf b}_2 =q(1\/2, \\sqrt{3}\/2)$ with  ${\\bf\nq} = m {\\bf b}_1 + n {\\bf b}_2$ where the integers $m$ and $n$ are in\nthe range  $|m {\\bf b}_1+n{\\bf b}_2 | \\le n_q q$.  The largest\nwavenumber is then $n_q q$ and the number of Fourier modes retained is\ngiven by $1+6\\sum_{j=1}^{n_q}j$. Typically we use $n_q =3 $. The top\nand bottom boundary conditions are satisfied by using appropriate\ncombinations of trigonometric and Chandrasekhar functions in $z$\n(\\cite{Ch61,Bu67}). In most of the computations we use $n_z=6$ modes\nfor each field in Eq.(\\ref{e:v},\\ref{e:cont},\\ref{e:T}). The linear\nanalysis yields the critical Rayleigh number $R_c$ as well as the\ncritical wavenumber $q_c$. Both depend on the NB-coefficients\n$\\gamma_i^{c}$ which in turn depend on $R_c$. Thus, in principle one\nobtains an implicit equation for the $\\gamma_i^{c}$. The shift in the\ncritical Rayleigh number away from the classical value $R_c=1708$ due\nto the NB-effects is, however, quite small (less than 1 percent) and\ntherefore the resulting change in the $\\gamma_i^{c}$ is negligible. In\nthis paper we therefore choose the $\\gamma_i^{c}$ corresponding to\n$R_c=1708$. \n\n\nTo investigate the nonlinear hexagon solutions, we start with\nthe standard weakly nonlinear analysis to determine the\ncoefficients of the three coupled amplitude equations for the\nmodes making up the hexagon pattern.  To obtain the fully\nnonlinear solutions requires the solution of a set of nonlinear\nalgebraic equations for the expansion coefficients with respect\nto the Galerkin modes. This is achieved with a Newton solver for\nwhich the weakly nonlinear solutions serve as convenient\nstarting solutions. In the Galerkin code amplitude instabilities\nare tested by linear perturbations of the expansion\ncoefficients. In addition,  modulational instabilities are considered, which involve the introduction of Floquet\nmultipliers  $\\exp (i {\\bf s}\\cdot (x,y))$ in the Fourier ansatz  for the linear perturbations \nof the Galerkin solutions. \n\nWe also study the temporal evolution of the system. For that we employ\na Fourier spectral code on a rectangular grid $(i\\,dq_x,j\\,dq_y)$,\n$i,j=1...N$, with $dq_y\/dq_x=\\sqrt{3}\/2$ to accomodate perfect\nhexagonal patterns. The same vertical modes are used as in the\nGalerkin stability code \\cite{DePe94a}. To solve for the time\ndependence a fully implicit scheme is used for the linear terms,\nwhereas the nonlinear parts are treated explicitly (second order\nAdams-Bashforth method). The time step is typically taken to be\n$t_v\/500$, where $t_v$ is the vertical diffusion time. We have tested\nthat the stability regimes obtained from the Galerkin analysis are\nconsistent with the direct numerical simulations. Both codes employed\nin this paper have been kindly provided by W. Pesch\n\\cite{DePe94a,Pe96}. \n\n\\section{Reentrant hexagons in CO$_2$ \\LB{sec:co2stability}}\n\nIn this paper we investigate specific scenarios for convection in\ngases that should be experimentally realizable. We focus in this\nsection on CO$_2$ in a conventional range for the layer thickness,\npressure, and temperature. Table \\ref{t:gamma-co2} provides the NB\ncoefficients and the $Q$-value at the onset of convection for a\nrepresentative range of the mean temperature $T_0$ in a layer of\nthickness $d=0.08\\,cm$ \\footnote{These values were obtained with a\ncode kindly provided by G. Ahlers.}. \n\n\\begin{table}\n\\begin{tabular}{|c|cccccccc|}\\hline \n$T_0$ & $\\Delta T_c$ & $Pr$ & $\\gamma_0^{c}$ &\n$\\gamma_1^{c}$ & $\\gamma_2^{c} $ & $\\gamma_3^{c}$ &\n$\\gamma_4^{c}$ & $Q$ \\\\ \\hline \n20 & 9.43  & 0.87  & 0.0486 & -0.0669 & 0.0779 &0.0236 & -0.0251&  1.199        \\\\ \\hline   \n40 & 15.52 & 0.84  & 0.0685 & -0.0883 & 0.1132 &0.0508 & -0.0184& 1.724       \\\\ \\hline   \n60 & 23.80 & 0.82  & 0.0931 & -0.1148 & 0.1566 &0.0919 & -0.0074 & 2.430       \\\\ \\hline   \n\\end{tabular}  \n\\caption{Values for the critical temperature (in $^\\circ C$), the \nPrandtl number $Pr$, NB coefficients\n$\\gamma_i^{c}$,  and Busse's parameter $Q$ for CO$_2$ at the\nonset of convection for three values of the mean temperature (in\n$^\\circ C$). The values correspond to a layer thickness of \n$d=0.08\\,cm$ and a pressure of $P=300$\\,psi.  \\LB{t:gamma-co2} } \n\\end{table} \n\n\n\\subsection{Amplitude instabilities}\n\nIn our analysis we first concentrate on spatially periodic solutions\nwith their wavenumber being fixed at the critical wavenumber and\ndiscuss their domains of existence and their stability with respect to\namplitude instabilities, which do not change the wavenumber.   For\nthree different cells with thicknesses $d=0.07,\\, 0.08, \\mbox{ and }\n0.09$\\, cm, respectively, we consider the range $0 < \\epsilon < 1$ at\na pressure of $P=300$\\,psi. \n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth,angle=0]{arti-co2-arti-3.eps}\n\\caption{Stability regions for hexagons and rolls in CO$_2$ with respect to amplitude \nperturbations for three fluid depths: $d=0.07\\, cm$ (dotted\nlines), $d=0.08\\, cm$ (full lines), $d=0.09\\, cm$ (dot-dashed line).\nPressure is kept at $P=300$\\,psi.\nThick lines: stability boundaries for hexagons. Thin lines: stability\nboundaries for rolls. For a given depth, rolls are stable above the thin \nline, and hexagons are unstable in the inner region of the thick line.  \\LB{fig:co2-ampli3d}}\n\\end{figure}\n\\end{center}\n\nThe results of the stability analysis for hexagons and rolls are shown\nin Fig.\\ref{fig:co2-ampli3d}. The hexagons are linearly stable for\nvery small $\\epsilon$. For a given layer thickness and not too high\nmean temperature $T_0$  the hexagons become unstable as  the control\nparameter is increased. The hexagon patterns then undergo  a second\nsteady bifurcation as the control parameter is increased further and\nbecome stable again. Such restabilized hexagons have been termed {\\em\nreentrant hexagons} \\cite{AsSt96,RoSt02,MaRi05}. As the mean\ntemperature is increased or the layer thickness is decreased the\ncritical heating and with it the NB effects increase. This shifts the\npoint of reentrance to lower $\\epsilon$ and the lower stability limit\nto higher $\\epsilon$, decreasing the $\\epsilon$-range over which the\nhexagons are unstable, until the two limits merge at a temperature\n$T_m$. For $T_0>T_m$ the hexagons are amplitude-stable over the whole\nrange of $\\epsilon$ considered ($0 \\le \\epsilon \\le 1$).\n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth,angle=270]{artico2-To40.ps}   \n\\caption{ Stability regions for hexagons in CO$_2$ with respect\nto amplitude  (dashed line) and side-band perturbations \n(solid line) for layer depth $d=0.08\\,\ncm$, pressure $P=300$\\,psi, mean temperature $T_0=40\\,^oC$, and Prandtl number $Pr=0.84$.\nThe NB coefficients $\\gamma_i^{c}$ are reported in  Tab. \\ref{t:gamma-co2}.\nHexagons are stable with respect to amplitude perturbations outside the\ndashed-line region, and stable with respect to side-band\nperturbation inside the solid-line region.\\LB{fig:side-co2}}\n\\end{figure}\n\\end{center}\n\nWe have also computed the stability of rolls with respect to amplitude\nperturbations. The corresponding stability limits are indicated  in\nFig.\\ref{fig:co2-ampli3d} by thin lines. Rolls are stable above these\nlines. As the NB effects become stronger the stabilization\nof rolls is shifted to larger $\\epsilon$. In contrast to the\nhexagons,  the rolls do not undergo a second bifurcation within the\nparameter regime investigated and remain amplitude-stable beyond\n$\\epsilon =1.0$.  For strong NB\neffects one has therefore a large range of parameters over which the\ncompeting rolls and hexagons are both linearly amplitude-stable. \n\nThe amplitude-stability limits of the hexagons and rolls depend on\ntheir wavenumber.  This is illustrated for the hexagons in Fig.\n\\ref{fig:side-co2} for a mean temperature of $T_0=40\\,^o C$. The\ninstability region with respect to amplitude perturbations forms a\nbubble-like closed curve, inside of which the hexagons are unstable\nwith respect to amplitude perturbations. \n\nIt is worth mentioning that the stability limits for hexagons in\nCO$_2$ are quite similar to those of NB convection in\nwater, except that in CO$_2$ the NB effects increase\nrather than decrease with increasing mean temperature \\cite{MaRi05}.\n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth,angle=270]{side-to27-co2.ps}\n\\caption{ Stability regions for hexagons in CO$_2$ with respect to \nside-band perturbations for layer thickness $d=0.052\\, cm$, pressure\n$P=300$\\,psi, mean temperature $T_0=27^o C$, and Prandtl $Pr=0.86$. The NB coefficients\nare $\\gamma_0^{c}=0.2053$,  $\\gamma_1^{c}= -0.2877$,  $\\gamma_2^{c}=0.3267$, $\\gamma_3^{c}=0.1152$, and $\\gamma_4^{c}=-0.086$.\nThe hexagons are stable with respect to side-band perturbations in the region \ninside the solid lines and unstable outside. The dashed line corresponds to\nthe neutral curve. \\LB{fig:side-to27}} \n\\end{figure}\n\\end{center}\n\n\n\\subsection{Side-Band Instabilities}\n\nIn systems with sufficently large aspect ratio side-band instabilities\ncan be the most relevant instabilities. Using the Galerkin method, we\nhave studied the stability of the hexagons with respect to long- and\nshort-wave side-band perturbations for $T_0=40^oC$. The results are\nshown in Fig.\\ref{fig:side-co2}. We find that over the whole range\n$0\\le \\epsilon \\le 1$ the only relevant side-band perturbations are\nlong-wave and steady, as is the case in the weakly nonlinear regime.\nThe same is true also in water with higher Prandtl number \\C{MaRi05}.\nIn this parameter regime the stability region consists of two\ndisconnected domains, reflecting the reentrant nature of the hexagons.\nThe stability domain near onset is very small and closes up as the\namplitude stability limit is reached. In the reentrant regime the\nstable domain opens up again in an analogous fashion  when the\namplitude-stability limit is crossed. Note that the stability\nboundaries are leaning toward lower wavenumbers. Thus, stable\nreentrant hexagon patterns are expected to have wavenumbers below\n$q_c$. This is related to the fact that in the OB case hexagons can be\nstable for large $\\epsilon$, but they are side-band stable only for\nsmall wavenumbers \\cite{ClBu96}.\n\nAs the mean temperature is increased the bubble of the amplitude\ninstability shrinks and eventually disappears, as shown in\n\\F{fig:side-to27} for a cell with thickness $d=0.052\\, cm$ and mean\ntemperature $T_0 =27\\,^o C$. As before, the relevant side-band\ninstabilities are long-wave all along the stability limits and with\nincreasing $\\epsilon$ the wavenumber range over which the hexagons are\nstable shifts towards smaller wavenumbers. For these parameters the\nregion of side-band-stable hexagons reaches without interruption from\nthe strongly nonlinear regime all the way down to threshold. For yet\nstronger NB effects the range of stable wavenumbers widens. \n\n\\section{Comparison with experiments in CO$_2$ \\LB{sec:sim-co2}}\n\nBodenschatz {\\em et al.} \\cite{BoBr91} carried out a set of experiments on\nconvection in CO$_2$ in a cylindrical cell  with  aspect ratio\n$\\Gamma\\sim172$, thickness $d=0.052\\, cm$, and pressure $P=300$\\, psi.\nUnder these conditions NB effects are relevant. In the experiments a\nweakly hysteretic transition from hexagons to rolls was found near\n$\\epsilon=0.1$. Noting that this transition point was below the\namplitude instability of hexagons to rolls as predicted by weakly\nnonlinear theory, the authors interpreted their results in terms of\nthe heterogeneous nucleation of rolls by the sidewalls. They found\nthat for small $\\epsilon$ the concentric rolls induced by the sidewall\nheating remained confined to the inmediate vicinity of the sidewalls;\nhowever, as $\\epsilon$ was increased the rolls invaded the hexagons\nand filled the whole cell, inducing a transition from hexagons to\nrolls. \n\nA comparison of the experimental findings with the stability results\nshown in Fig. \\ref{fig:side-to27} shows that indeed the transition\ncannot be due to an amplitude instability of the hexagons. In fact, in\nthis regime the NB effects are so strong that, in contrast to the\npredictions of the weakly nonlinear theory, the hexagons do not\nundergo an amplitude instability at all. To clarify the influence of\nthe sidewalls and to assess the significance of the side-band\ninstabilities for the transition from hexagons to rolls, we perform\ndirect simulations of the Navier-Stokes equations\n(\\ref{e:v},\\ref{e:cont},\\ref{e:T},\\ref{e:bc}) for two different sets\nof boundary conditions \\footnote{In the experimental setup the top\ntemperature is held constant at $T=12.84 ^oC$, and therefore the mean\ntemperature changes as $\\epsilon$ is increased. In our computations,\nhowever, we keep $T_0$ fixed. Since the transition occurs quite close\nto threshold this is a good aproximation to the experimental\nprocedure.}: \n\ni) periodic boundary conditions,\n\nii) concentric rolls as boundary conditions. In our computations this\ntype of boundary condition is generated by a suitably patterned \nheating in the interior of the fluid.  In the experiments concentric\nrolls near the side walls were generated by a side-wall heating\n\\cite{BoBr91}. \n\n\ni) according to Fig. \\ref{fig:side-to27}, for $\\epsilon=0.3$ hexagons\nwith wavenumbers larger than $q>2.98$, which includes the critical\nwavenumber, are unstable with respect to side-band instabilities. To\ntest whether these side-band instabilities trigger a transition to\nrolls we perform numerical simulations with periodic boundary\nconditions  and hexagons as initial conditions. Fig.\n\\ref{fig:perio-co2} presents some snapshots of the ensuing temporal\nevoluation in a cell of size $L=8\\cdot 2\\,\\pi\/q_c=16.11$ using\n$128\\times 128$ Fourier modes. More precisely, to allow perfect\nhexagons in a rectangular container the container cannot be square. In\nour simulations we use $L_x=L$ and $L_y=\\sqrt{3}L\/2$. The sideband\ninstability of the initially almost perfect hexagon pattern (cf.\n\\F{fig:perio-co2}(a)) induces a shearing of the pattern (cf.\n\\F{fig:perio-co2}(b)). At the same time a few  penta-hepta defects\narise and some hexagonal convection cells tend to connect with each\nother forming short roll-like structures. However, as time evolves the\nsystem becomes progressively more ordered again and eventually after\nlosing a number of convection cells a defect-free hexagon pattern with\na smaller wavenumber is established (cf. \\F{fig:perio-co2}(c) at \n$t\\simeq 60t_v$).  \n\nThus, while roll-like features appear for this value of $\\epsilon$,\nwith periodic boundary conditions no transition to rolls occurs and\nthe system relaxes to a new ordered hexagon pattern. Only for yet\nlarger values of $\\epsilon$ the roll-like structures that arise in the\nintermediate, disordered state take over and lead to a transition to\nrolls induced by the sideband instabilities.\n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{R2193-a-b-c-2.eps}\n\\caption{ Numerical simulation corresponding to a layer of \nCO$_2$ with thickness $d=0.052\\,cm$, pressure $P=316$\\,psi, \nmean temperature $T_0=27.3^o C$,  Prandtl number $Pr=0.87$, and control parameter  $\\epsilon=0.3$\n(cf. Fig.\\ref{fig:side-to27}). The size of the integration domain is\n$L=8\\cdot 2\\,\\pi\/q_c=16.11$ and  the boundary conditions are\nperiodic.  As initial condition a perfect hexagon lattice with \nwavenumber $q_c=3.12$ has been used.  a) corresponds to $t=0$, \nb) to  $t=18t_v$, and c) to $t=60t_v$.\\\\ \\LB{fig:perio-co2}} \n\\end{figure}\n\\end{center}\n\nii) clearly, the simulations for periodic boundary conditions do not match\nthe experimental results described above, where a transition from\nhexagons to rolls occurs already for $\\epsilon \\gtrsim 0.11$. To\naddress this disagreement we take into account the fact that in the\nexperiments the side walls promote the nucleation of rolls\n\\cite{BoBr91}.\n\nStrictly speaking, the code we are using does not allow non-periodic\nboundary conditions. To mimic the experimentally used clylindrical\ncell we employ a step ramp in $\\epsilon$ that reduces $\\epsilon$ to\nvalues well below $\\epsilon=0$ outside a circle of radius $r=0.45L$\nwith $L=16\\cdot 2\\pi\/q_c=32.22$ \\C{DePe94a}. To induce concentric\nrolls near the sidewalls we introduce for $r> 0.45L$ an additional\nheating in the interior of the fluid in the form of concentric rings.\nUsing hexagonal initial conditions in the bulk, this leads to an\ninitial state as shown in Fig. \\ref{fig:boun-co2}a.\n\nFig. \\ref{fig:boun-co2}a,b shows two snapshots at $t=t_v$ and\n$t=158t_v$ demonstrating how the rolls induced by the side walls\ninvade the carefully prepared hexagonal state in the bulk already for\n$\\epsilon=0.2$. This is well below the $\\epsilon$-value for which with\nperiodic boundary conditions the hexagons persisted even through the\nside-band instability. The final steady state consists of concentric\nrolls as observed in the experiments  (cf. Fig. 5 in \\cite{BoBr91}).\nFor lower $\\epsilon$, however, the experimentally observed final state\nconsists of hexagons in the bulk of the system surrounded by\nwall-induced concentric rolls (cf. Fig. 4 in \\cite{BoBr91}). We find\nthis also in our numerical simulations, as shown in\n\\F{fig:boun-co2}(c-d). There the forcing of rolls is identical to that\nin \\F{fig:boun-co2}(a-b) but $\\epsilon=0.05$. Starting with random\ninitial conditions, the forcing gives rise to a ring contained in the\nsquare integration domain.  At the beginnig of the simulation\n(\\F{fig:boun-co2}(c)) the rolls created by the forcing invade the\ninterior of this small system. However, as time progresses the rolls\npull out of the central region of the cell,  and the  final steady\nstate  (for $t \\gtrsim  165\\,t_v$)  consists of stable hexagons\nsurrounded by a couple of concentric rolls in addition to those \ninduced by the forcing. \n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth]{ergT.000005-ergT.000790-co2-to27-qc-eps0.2-forcing.rolls.eps}\\\\\n\\vspace{0.5cm}\n\\includegraphics[width=0.4\\textwidth]{ergT-eps0.05-for.rolls-init.random.eps}\n\\caption{ Numerical simulation in a cell of CO$_2$ with thickness\n$d=0.052\\,cm$, pressure $P=300$\\,psi, mean temperature $T_0=27^o\nC$, and  Prandtl number $Pr=0.86$. The size of the integration domain\nis $L=16\\cdot 2\\,\\pi\/q_c=32.22$, and the boundary conditions are rings\nof concentric rolls  generated with and external forcing. \\\\ For\n$\\epsilon=0.2$ (upper snapshots) hexagon inititial conditions have\nbeen used, with $t=t_v$ (left snapshot) and $t=158\\,t_v$ (right). The\nlower snapshots correspond to a simulation with $\\epsilon=0.05$ and\nrandom initial conditions, at $t=40\\,t_v$ \n(left) and $t=158\\,t_v$ (right).\\LB{fig:boun-co2}} \n\\end{figure}\n\\end{center}\n\nThus, our simulations suggest that the experimentally observed\ntransition from hexagons to rolls is neither due to amplitude\ninstabilities nor to side-band instabilities. Rather, there is a large\nrange of parameters in which hexagons and rolls are both linearly\nstable and the final state is selected by one type of pattern invading\nthe other. The transition to rolls at these low values of $\\epsilon$\nis made possible by the boundaries, which provide a seed for the\nnucleation of rolls. We expect that by applying a forcing that is\nconfined to the region near the boundaries and that replaces the rolls\nby hexagons the transition to rolls could be shifted to substantially\nlarger values of $\\epsilon$. Such a forcing could be achieved by a\npatterned heating of the interior of the fluid \\cite{SeSc02} or\npossibly by a suitable geometric patterning of the bottom plate\n\\cite{Bopriv}. \n\n\n\\section{NB Spiral Defect Chaos in SF$_6$\\LB{sec:sim-sf6}}\n\nA fascinating state observed in convection at low Prandtl numbers is\nspiral defect chaos. It is characterized by self-sustained chaotic\ndynamics of rotating spirals, as well as dislocations and\ndisclinations, and arises in fluids  with $Pr\\lesssim 1$\n\\C{MoBo93,XiGu93,DePe94a,CrTu95,LiAh96} in a parameter range where\nstraight rolls are linearly stable. Spiral defect chaos has so far\npredominantly been investigated under OB conditions, in which\nup-flows and down-flows are equivalent. \n\nAs mentioned before, NB effects break the up-down symmetry and\ndifferent flow structures may be predominantly associated with up-flow\nand down-flow, respectively. Moreover, in the absence of the OB\nsymmetry a resonant triad interaction is allowed. If it is strong\nenough it leads to the formation of hexagons. For weaker interaction\none may still expect an enhancement of cellular rather than roll-like\nstructures.\n\nTo investigate the impact of NB effects on spiral defect chaos we\nconsider convection in a layer of SF$_6$. This gas has been used\npreviously in experimental convection studies under usual laboratory\nconditions \\C{BoCa92}, and near the thermodynamical critical point\n\\C{AsSt94,AsSt96,RoSt02}. In Fig. \\ref{fig:sf6-ampli}(a) we present\nthe stability diagram for hexagons and rolls with respect to amplitude\nperturbations in a layer of SF$_6$ of  thickness $d=0.0542$\\,cm,\npressure $P=140\\,$psi, and a range of temperatures that is\nexperimentally accessible. Hexagons are amplitude-stable to the right\nof the solid line and rolls above the dashed line.  As in the case of\nCO$_2$ the NB effects increase with increasing mean temperature $T_0$\nand above a certain value of $T_0$ hexagons are linearly\namplitude-stable over the whole range of $\\epsilon$ investigated. Here\nwe focus on relatively strong NB effects. We therefore show in\nFig.\\ref{fig:sf6-ampli}b the stability limits with respect to\nside-band perturbations for a relatively large mean temperature,\n$T_0=80^\\circ C$. As in the case of CO$_2$, the wavenumber range over\nwhich the hexagons are stable is leaning towards smaller wavenumbers.\nOverall, amplitude and side-band stability limits are qualitatively\nsimilar to those of convection in CO$_2$ (cf. Fig.\n\\ref{fig:co2-ampli3d}).\n\n\\begin{center}\n\\begin{figure}\n\\begin{minipage}{0.35\\textwidth}\n\\includegraphics[width=\\textwidth,angle=0]{sf6.eps}\n\\end{minipage} \n\\hspace{0.4cm}  \n\\begin{minipage}{0.35\\textwidth}\n\\includegraphics[width=\\textwidth,angle=270]{sf6-to80-side2.ps}\n\\end{minipage}\n\\caption{ \nStability regions of hexagons and rolls in a layer of  SF$_6$ \nof thickness $d=0.0542\\,cm$,  pressure $P=140$\\,psi, and Prandtl number $Pr=0.8$. \\\\\na)  Stability regions with respect to amplitude perturbations. \nContinous line: stability boundary for hexagons. Dashed line: stability boundary for\nrolls. Stability limits obtained for the critical wavenumber $q_c$.\\\\\nb)  Stability regions with respect to side-band perturbations for the above\nlayer with a mean temperature $T_0=80\\,^oC$ The corresponding NB coefficients\nare $\\gamma_0^{c}=0.1714$,  $\\gamma_1^{c}= -0.2118$, $\\gamma_2^{c}=0.2836$, $\\gamma_3^{c}=0.1905 $, and\n$\\gamma_4^{c}=0.0624$ corresponding to $Q=4.2$.. The dashed line corresponds to\nthe neutral curve. \n\\LB{fig:sf6-ampli}}\n\\end{figure}\n\\end{center}\n\nFig.\\ref{fig:sf6-nb-ob} shows two snapshots obtained by direct \nnumerical simulations of the Navier-Stokes equations corresponding to\nconvection in SF$_6$ for $T_0=80\\,^oC$ and $\\epsilon=1.4$ in a\nconvective cell of thickness $d=0.0542\\,cm$ and horizontal size\n$L=16\\cdot 2 \\,\\pi\/q_c=32.22$. Periodic boundary conditions are used\nwith $128\\times 128$ Fourier modes and 6 vertical modes.  Both states\nare obtained after an integration time of $160\\,t_v$, starting from\nrandom initial conditions. While in\nFig.\\ref{fig:sf6-nb-ob}b all NB are retained, in\nFig.\\ref{fig:sf6-nb-ob}a the same values are used for $Pr$ and\n$\\epsilon$, but all NB parameters $\\gamma_i^c$ are set to 0, i.e. the\nsystem is treated as if it was Boussinesq. \n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth]{sf6-nb-ob-4-sqrt.eps\n\\caption{ \nDirect numerical simulation of equations (\\ref{e:v}-\\ref{e:bc}) for SF$_6$ in a\ncell of thickness $d=0.0542\\,cm$, pressure $P=140$\\,psi, mean temperature \n$T_0=80^oC$, Prandtl number $Pr=0.8$, and control parameter $\\epsilon=1.4$. \nThe cell has size $L=16\\cdot 2 \\,\\pi\/q_c=32.22$ with periodic boundary conditions. \nStarting from random initial conditions both snapshots are taken at\n$t=160\\,t_v$. Left panels for OB conditions ($\\gamma_i$=0, $i=0..4$, left), \nright panels for NB conditions appropriate for $T_0=80^oC$ \n(cf. Fig\\ref{fig:sf6-ampli}b). Bottom panels give the corresponding contour lines used\nfor the pattern diagnostics.\n\\LB{fig:sf6-nb-ob}}\n\\end{figure}\n\\end{center}\n\nThe snapshots depicted in Fig.\\ref{fig:sf6-nb-ob} show, as expected,\nthat due to the NB effects down-flow convection cells, which are white\nin Fig.\\ref{fig:sf6-nb-ob}, outnumber cells with up-flow (black).\nMoreover, in this regime the NB effects enhance the overall cellular\nrather than roll-like character of SDC.  This manifests itself in the\nappearance of numerous small down-flow convection cells (white\n`bubbles') and in the appearance of quite noticeable bulges on the NB\nconvection rolls. To quantify these and other differences  we analyse\na long sequence of snapshots with a recently introduced geometric\napproach \\cite{RiMa06}. It is based on the contour lines corresponding\nto the intensity half-way between the minimal and maximal intensity of\nall snapshots in a given run.  The contour lines corresponding to the\ntemperature field of snapshots Fig.\\ref{fig:sf6-nb-ob}(a,b) are shown in\nFig.\\ref{fig:sf6-nb-ob}(c,d). In the following we\npresent various statistics of these contour lines.\n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth,angle=0]{NOB-OB-ratio0.8-2.eps} \\\\%{NOB-OB-2.ps}\n\\caption{ \nNumber of black and white closed contours  as a function of time\nfor NB and OB conditions for the simulations corresponding to the\nsnapshots shown in \\F{fig:sf6-nb-ob}. \n\\LB{fig:num-bubbles}}\n\\end{figure}\n\\end{center}\n\nThe most striking difference between the OB and the NB case is the\nasymmetry that is induced by the NB effects between `black' and\n`white' components, i.e. between closed contours that enclose up- and\ndown-flow regiones, respectively. To quantify this asymmetry we\nmeasure the number of white and black components. These two\ntopological measures correspond, respectively, to the Betti numbers of\norder 1 and 2 of the pattern defined by the white components\n\\cite{GaMi04}. Fig.\\ref{fig:num-bubbles} shows these two quantities as\na function of time in the OB and the NB case. As expected, in the OB\ncase the number of black and white components is essentially the same\nat all times, whereas in the NB case the white components\nsignificantly outnumber the black ones. The ratio of white to black\ncomponents is therefore a sensitive indicator for the significance of\nNB effects. Fig.\\ref{fig:num-bubbles} also illustrates how much the\nnumber of components fluctuates during these runs. Recently, the two\nBetti numbers have also been measured based on patterns obtained in\nexperiments on SDC in convection in CO$_2$. Scanning $\\epsilon$ in\nvery small steps the authors report steps in the Betti numbers\nindicative of transitions between different chaotic, disordered states\n\\cite{KrGaunpub}.\n\nFig.\\ref{fig:num-bubbles} shows that the total number of components\n(closed contours) is considerably larger in the NB case than in the OB\ncase. This is presented in more quantitative detail in\nFig.\\ref{fig:num-bubbles-eps}, which gives the mean value of the total\nnumber of components, i.e. the sum of black and white components, as a\nfunction of $\\epsilon$ for OB as well as NB conditions. In the NB case\nthe total number of components is up to four times larger than in the\nOB case. We attribute this difference to the resonant triad\ninteraction that is made possible by the breaking of the OB symmetry\nand which tends to enhance cellular rather than filamentary roll-like\nstructures.\n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth,angle=0]{meanNP-NPratio0.8-2.eps}\n\\caption{ \nTotal number of closed contours as a function of the control parameter \n$\\epsilon$ for NB (circles) and OB (squares) conditions. \n\\LB{fig:num-bubbles-eps}}\n\\end{figure}\n\\end{center}\n \nTo characterize the components better and to distinguish cellular and\nroll-like structures we introduce the `compactness' ${\\mathcal C}$ of\ncomponents \\cite{RiMa06},\n\\begin{eqnarray}\n{\\mathcal C}=4\\pi \\frac{{\\mathcal A}}{{\\mathcal P}^2}.  \\LB{e:compact}\n\\end{eqnarray} \nHere ${\\mathcal A}$ is the area inside a closed contour and $P$ its\nperimeter. With the normalization used in (\\ref{e:compact}) compact,\ncellular structures are charecterized by ${\\mathcal C}\\lesssim 1$,\nwhereas filamentary, roll-like structures have ${\\mathcal C} \\ll 1$.\n\n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth,angle=0]{histo_compact_SF.eps}\n\\caption{ \nMean number of closed contours per snapshot with a \ngiven compactness ${\\mathcal C}$ for convection in  SF$_6$ at\n$\\epsilon=1.4$ under NB (circles) and under OB (squares) conditions \n(cf. \\F{fig:sf6-nb-ob}).\n\\LB{fig:hist-compact}}\n\\end{figure}\n\\end{center}\n\n\\F{fig:hist-compact} shows the mean number of closed contours per\nsnapshot for a given compactness $\\mathcal C$ for the NB and the OB\nsimulation at $\\epsilon=1.4$ over the duration $t=360\\,t_v$. As\nexpected, in the NB case the number of white components is much larger\nthan that of black components, whereas in the OB case both are about\nthe same. The total number of components is noticeably larger in the\nNB case, which also shows  an increase in the number of white,\nfilamentary contours with small compactness. \n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth,angle=0]{histo_compact_smoothed_normalized.eps}\n\\caption{ \nDistribution function for the compactness ${\\mathcal C}$ for NB\n(circles) and OB (squares) conditions (cf. \\F{fig:sf6-nb-ob}).\n\\LB{fig:hist-compact-normalized}}\n\\end{figure}\n\\end{center}\n\nThe relative distribution among components with different compactness\nis more clearly visible in the {\\it relative} frequency of contours as\na function of the compactness, which is shown in\nFig.\\ref{fig:hist-compact-normalized}. More precisely, its data result\nfrom running averages over adjacent 10 points of the corresponding\ndata shown in Fig.\\ref{fig:hist-compact}, which are then normalized.\nThe normalized data show that the increase in the relative frequency\nof white filamentary components (${\\mathcal C}\\ll 1$) is essentially\nthe same in the NB case and in the OB case. However, only very few\nblack filamentary components arise in the NB case. \n\nA feature of Fig.\\ref{fig:hist-compact-normalized} that is surprising\nat first sight is the essentially equal height of the NB peak and the\nOB peak for components with ${\\mathcal C}\\sim 1$. Visually, the NB\nsnapshot exhibits many more small compact `bubbles' than the OB run.\nAn explanation for this observation can be obtained by correlating the\ncompactness of the closed contours with their length ${\\mathcal P}$.\nThe joint distribution function for these two quantities is shown in\nFig.\\ref{fig:corr-cont-comp} using logarithmic scales for ${\\mathcal\nP}$ and ${\\mathcal C}$. Focussing on the compact objects with\n${\\mathcal C}\\lesssim 1$ one recognizes in the OB case a second peak\nat somewhat larger contourlength. We associate this peak with the\nappearance of target-like structures, i.e. with a second contourline\nencircling a smaller compact, almost circular contourline. In the NB\ncase this second peak is barely visible. Instead, the shoulder of the\nmain peak is extended significantly towards smaller contourlength\n${\\mathcal P}$. It signifies the appearance of compact objects that\nare smaller than a typical wavelength, which we associate with the\nsmall `bubbles' that are easily recognized in the snapshot of the NB\ncase Fig. \\ref{fig:sf6-nb-ob}b. Thus, the comparable relative\nfrequency of small components shown in\nFig.\\ref{fig:hist-compact-normalized} has a different origin in the OB\nand the NB case. Whereas in the NB case it is mostly due to small\nbubbles, it seems to originate from target-like structures in the OB\ncase. \n\nIn the logarithmic scaling used in Fig.\\ref{fig:corr-cont-comp} a\nstraight ridge arises in the distribution function for large\ncontourlengths. It is characteristic for filamentary structures with a\ntypical width, which corresponds here to half a wavelength $\\lambda$\nof the convection rolls,\n\\begin{eqnarray}\n{\\mathcal C}\\sim 4\\pi\n\\frac{\\lambda \\, {\\mathcal P}}{4{\\mathcal P}^2} \\propto {\\mathcal\nP}^{-1}.\n\\end{eqnarray}\nIn the OB case one can discern deviations from this scaling. They are\nconfined to larger rather than smaller compactness values for a given\ncontourlength, indicating that the long rolls can be wider but not\nnarrower than a certain thickness. In the NB case these deviations are\nmuch smaller. \n\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{corr-cont-comp.MaRi05a.f12a-4.eps}\n\\includegraphics[width=0.48\\textwidth]{corr-cont-comp.MaRi05a.f12b-2.eps}\n\\caption{Distribution function for contour length and \ncompactness for OB (a) and NB conditions (b)\n$Pr=0.8$ and $\\epsilon=1.4$.\n\\LB{fig:corr-cont-comp}\n} \n\\end{figure}\n\nTo identify spiral components in the pattern directly we also measure\nthe winding number of the components \\cite{RiMa06}. It is defined via\nthe angle $\\theta$ by which the (spiral) arm of a pattern component is\nrotated from its tip to its end at the vertex at which it merges with\nthe rest of the component. In cases in which a component has no\nvertices we split it into two arms at the location of minimal\ncurvature \\cite{RiMa06}. The winding number is then defined as\n$|{\\mathcal W}|=\\theta\/2\\pi$. To assess the impact of the NB effects\non the spiral character of the pattern we measure the number of\nspirals in each snapshot and show the resulting histogram over the\nwhole run in Fig.\\ref{fig:num-spiral}. We use three different\nthresholds ${\\mathcal W}_{min}$ for the identification of spirals,\n${\\mathcal W}_{min}= 1$, ${\\mathcal W}_{min}= 1\/2$, and ${\\mathcal\nW}_{min}= 1\/4$. As  Fig.\\ref{fig:num-spiral} shows, the number of\nsmall spirals with $|{\\mathcal W}|\\gtrsim 1\/4$ is quite similar in the\nOB and the NB case. However, larger spirals with  $|{\\mathcal W}|\\ge\n1\/2$ or even $|{\\mathcal W}|\\ge 1$ are much more rare in the NB case;\nin fact, for the system size $L=16 \\cdot 2\\pi\/q_c=32.22$ that we have\nused in these simulations there was at most one spiral with \n$|{\\mathcal W}| \\ge 1$ at any given time. \n\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{histo_num_spiral.OB_NB.1.eps}\n\\caption{Distribution function for the number of spirals in spiral\ndefect chaos under OB and NB conditions for three values of the\nthreshold, ${\\mathcal W}_{min}=1\/4$ (circles),${\\mathcal W}_{min}=1\/2$\n(squares), and ${\\mathcal W}_{min}=1$ (diamonds).  \n\\LB{fig:num-spiral}\n} \n\\end{figure}\n\nThe reduced spiral character of NB spiral defect chaos is also quite\napparent in Fig.\\ref{fig:corr-arc-winding}, which shows the\ncorrelation between the winding number and the arclength of the spiral\narm. More precisely, each dot marks the occurrence of one spiral arm\nin a snapshot. In the OB case one can see quite clearly a maximal\nwinding number for any given arclength, which is consistent with an\nArchimedean shape of the spiral \\cite{RiMa06}\\footnote{Note that\ndetailed analyses of large spirals show deviations from the\nArchimedean shape due to the dislocations that accompany finite\nspirals \\cite{Pl97}}. In the NB case, however, only components with\nvery small contourlength reach the Archimedean limit and most\ncomponents have winding numbers that are much smaller, i.e. the\ncomponents are quite straight.\n\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{corr-arc-winding_SF6-OB-NB.eps}\n\\caption{Correlation between the arclength and the winding number of\nspiral arms under OB (black squares) and NB conditions (red circles).\n\\LB{fig:corr-arc-winding}} \n\\end{figure}\n\n\\section{Hexagons and Spiral Defect Chaos at Very Low Prandtl numbers. H$_2$-Xe\nMixtures \\LB{sec:h2xe}}\n\nAs mentioned above, the restabilization of NB hexagons at larger\nRayleigh number is related to the existence of stable OB hexagons at\nlarge Rayleigh numbers. The wavenumber range over which the OB\nhexagons are stable shrinks with decreasing Prandtl number and for $Pr\n< 1.2$ the OB hexagons are side-band unstable at {\\it all} wavenumbers\n\\cite{ClBu96}. However, as seen in the case of CO$_2$ and SF$_6$, NB\nhexagons can be side-band stable at large $\\epsilon$ even below\n$Pr=1.2$ due to the additional stabilizing effect of the resonant\ntriad interaction. It is therefore of interest to investigate whether\nthe NB effects can be sufficient to stabilize strongly nonlinear\nhexagons even for Prandtl numbers significantly below $Pr=1$. \n\nPrandtl numbers well below $Pr=1$ can be reached by using a mixture of\na heavy and a light gas. An experimentally investigated case is a\nmixture of H$_2$ and Xe \\cite{LiAh97}. With a mole fraction of\n$\\chi=0.4$, one can reach Prandtl numbers as small as $Pr=0.17$. The\nLewis number of such a mixture is close to one \\cite{LiAh96,Ah05}. \nTherefore such mixtures are expected to behave essentially like a pure\nfluid with the same Prandtl number \\cite{BoPe00}.\n\nWe investigate the stability of hexagon and roll convection in a\nH$_2$-Xe mixture with mole fraction $\\chi=0.4$ at a pressure of $300$\n\\,psi and a layer thickness of $d=0.1$cm. With respect to amplitude\ninstabilities the stability diagram is very similar to that of\nconvection in CO$_2$ and SF$_6$ with hexagons becoming reentrant at\n$\\epsilon$-values as low as $\\epsilon=0.14$ for $T_0=20\\,^oC$. \n\nFocusing on strong NB effects we perform a detailed stability analysis\nwith respect to sideband perturbations at a mean temperature of\n$T_0=80^oC$ using the same layer thickness of $d=0.1$cm.  For these\nlow Prandtl numbers the numerical resolution has to be increased to\nobtain sufficiently well resolved flows. While for the stability\nanlyses of hexagons in CO$_2$ and SF$_6$ it is sufficient to use  \n$n_z=6$ and $n_q=3$ in the Galerkin expansion, for the H$_2$-Xe\nmixture with $P_r\\simeq0.17$ at least $n_q=5$ and $n_z=6$ are\nrequired. \\F{fig:side-h2xe}  depicts the resulting stability diagram. It shows\nthat the region of side-band stable hexagons is not contiguous but\nconsists of the usual region immediately above threshold and an\nadditional, disconnected region at larger Rayleigh numbers. We could\nnot follow the stability limits to smaller values of $q$ than shown in\nFig.\\ref{fig:side-h2xe} due to numerical convergence problems.\nPresumably, these arise due to bifurcations involving additional,\nresonant wavevectors \\cite{Mo04a}, somewhat similar to the resonances\nstudied in Taylor vortex flow \\cite{RiPa86,PaRi90}. The fact that the\nregion of stability is disconnected is remarkable since the region of\namplitude-stability (not shown) is actually contiguous. This is in\ncontrast to the behavior found in CO$_2$ and SF$_6$ where the two\nside-band stable regions become connected when the bubble-like region\nof amplitude instability disappears (cf.\nFig.\\ref{fig:side-co2},\\ref{fig:side-to27}). The comparison of the\nstability limits with those of CO$_2$ and of SF$_6$ \n(Fig.\\ref{fig:sf6-ampli}b) shows further that the maximal wavenumber\n$q$ at which the hexagons are stable with respect to side-band\nperturbations decreases with decreasing Prandtl number and as a result\nthe over-all stability region shrinks as well. \n\n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth,angle=270]{h2xe-to80.ps\n\\caption{ \nStability limits with respect to sideband perturbations  in  a mixture\nof $H_2-Xe$ with molar fraction $\\chi=0.4$, thickness $d=0.1 cm$, pressure $P=300$\\,psi, \nmean temperature $T_0=80\\,^oC$,  and Prandtl number $Pr=0.17$. The NB coefficients are \n$\\gamma_0=0.5535$, $\\gamma_1=-0.6421$, $\\gamma_2=0.9224$, $\\gamma_3=0.3647$, \n$\\gamma_4=-0.0712$ resulting in $Q=13.8$. Hexagons are stable in the\nregions enclosed by the solid lines and unstable outside. The dashed line represents \nthe neutral curve. The results for H$_2$-Xe are obtained with $n_z=6$ and $n_q=5$  \n\\LB{fig:side-h2xe}.}\n\\end{figure}\n\\end{center}\n\n\nThe stability analysis of the H$_2$-Xe mixture also reveals an\noscillatory instability of the hexagon patterns at $\\epsilon \\sim 1$.\nWithin the $\\epsilon$-range investigated, no such oscillatory\ninstability was found at the larger Prandtl numbers relevant for\nCO$_2$ and SF$_6$. Unfortunately, it turns out that before the onset\nof the oscillatory instability the hexagons already become unstable to\na side-band instability at half the hexagon wavelength, which will\nalways preempt the oscillatory instability. For the rolls we also find\nan oscillatory instability. It is presumably related to the well-known\noscillatory instability of Boussinesq rolls \\cite{CrWi89}. \n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.3\\textwidth,angle=0]{h2xe.snap.001401-sqrt.eps}\n\\caption{ \nTypical pattern for convection in H$_2$-Xe at $\\epsilon=0.3$.\n\\LB{fig:pattern-h2xe}.  }\n\\end{figure}\n\\end{center}\n\nFor not too small $\\epsilon$ generic initial conditions will not lead\nto hexagonal patterns but rather to spiral defect chaos. A typical\nsnapshot of a pattern in this state is shown in\nFig.\\ref{fig:pattern-h2xe} for $\\epsilon=0.3$. Compared to the\npatterns obtained in NB convection in SF$_6$ (cf.\nFig.\\ref{fig:sf6-nb-ob}b) the patterns in H$_2$-Xe are less cellular\nand do not show a large number of small bubbles. To quantify these and\nother characteristics of the patterns we again apply the geometric\ndiagnostics introduced earlier \\cite{RiMa06}.  \n\nIn Fig.\\ref{fig:compact-h2xe} we show the normalized distribution\nfunctions for the compactness of white and black components (cf.\nFig.\\ref{fig:hist-compact-normalized}). Since there are only very few\nblack components their distribution function exhibits large\nstatistical fluctuations. Of particular interest is the distribution\nfunction for the white components. It confirms the visual impression\nthat the number of compact components is significantly reduced\ncompared to the case of SF$_6$; in fact, while in SF$_6$ the maximum\nof the distribution function is close to ${\\mathcal C}=1$, in H$_2$-Xe\nthe absolute maximum is at ${\\mathcal C}\\sim 0.1$, which corresponds\nto filamentary structures.\n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth,angle=0]{histo_compact.h2xe.eps}\n\\caption{ \nDistribution function of the compactness of closed contours \nfor H$_2$-Xe ($Pr=0.17$, $\\epsilon=0.3$)\n\\LB{fig:compact-h2xe}.  }\n\\end{figure}\n\\end{center}\n\nThe lack of small bubbles is demonstrated in more detail in the joint\ndistribution function for the contourlength and the compactness, which\nis shown in Fig.\\ref{fig:corr-cont-compt-h2xe}. Note that the view is\nrotated compared to Fig.\\ref{fig:corr-cont-comp}. The distribution\nfunction is lacking the broad shoulder seen in NB convection in SF$_6$\n(see Fig.\\ref{fig:corr-cont-comp}b). Instead, the decay of the\ndistribution function towards long filamentary contours is quite\nslow. \n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth,angle=0]{corr-cont-comp.MaRi05a.f18-d.eps\n\\caption{ \nJoint distribution function for the contour length and \nthe compactness of closed contours for H$_2$-Xe ($Pr=0.17$, $\\epsilon=0.3$).\n\\LB{fig:corr-cont-compt-h2xe}.}\n\\end{figure}\n\\end{center}\n\nTo assess the spiral character of NB spiral defect chaos at these low\nPrandtl numbers we show in Fig.\\ref{fig:histo-winding} the\ndistribution function for the absolute value $|{\\mathcal W}|$ of the\nwinding number for NB convection in H$_2$-Xe (at $\\epsilon=0.3$ and\n$PR=0.17$) as well as for Boussinesq and non-Boussinesq convection in\nSF$_6$ (at $\\epsilon=1.4$ and $Pr=0.8$). As had been noted\npreviously in the Boussinesq case \\cite{EcHu97,RiMa06} the\ndistribution function is roughly consistent with exponential behavior.\nIn the NB case the exponential decays substantially faster than in the\nBoussinesq case and spirals with winding numbers above ${|\\mathcal\nW}|=1$ are rare. In the Boussinesq case we had found that the decay\nrate depends mostly on the Prandtl number, but only very little on\n$\\epsilon$ \\cite{RiMa06}. Unfortunately, we do not have enough\nnon-Boussinesq data to investigate such trends in the $\\epsilon$- and\n$Pr$-dependence. However, it is worth noting that in the two\nnon-Boussinesq cases shown in Fig.\\ref{fig:histo-winding} the decay\nrates are essentially the same despite their substantial difference in\nPrandtl number and both decays are much faster than that in the\nBoussinesq case. Thus, possibly the impact of NB effects dominates\nthe dependence on the Prandtl number.\n\n\\begin{center}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth,angle=0]{histo_winkel.H2Xe_SF6.eps}\n\\caption{ \nDistribution function for the absolute value of the winding number for\nNB convection in H$_2$-Xe in comparison with convection in \nSF$_6$ in the NB and the OB case.\n\\LB{fig:histo-winding}.  }\n\\end{figure}\n\\end{center}\n\nIn NB convection in H$_2$-Xe, the distribution function for the number\nof spirals is qualitatively similar to that in SF$_6$, which is shown\nin Fig.\\ref{fig:num-spiral} above. In particular, there are almost no\nspirals with a winding number $|{\\mathcal W}|>1$. Similarly, the\ncorrelation between the arclength and the winding number of spirals\nreveals that in NB convection in H$_2$-Xe there is no significant\ntrend towards Archimedean spirals. \n\n\\section{Conclusion}\n\\LB{sec:conclusions}\n\nIn this paper we have studied non-Boussinesq convection in gases\n(CO$_2$, SF$_6$, H$_2$-Xe) with Prandtl numbers ranging from $Pr\\sim\n1$ down to $Pr=0.17$ in experimentally relevant parameter regimes. We\nhave complemented a Galerkin stability analysis of hexagon patterns\nwith direct numerical simulations of the fluid equations to study\ntransitions between different hexagon patterns and to quantify the\nimpact of non-Boussinesq effects on spiral defect chaos.\n\nWe find that the reentrance of hexagons that we have identified\npreviously in non-Boussinesq convection in water \\cite{MaRi05} also\noccurs at low Prandtl numbers. As was the case at large Prandtl\nnumbers, compressibility is not neccessary for reentrance. Since, in\naddition, the range of wavenumbers for which the reentrant hexagons\nare stable differs significantly from that of the reentrant hexagons\nobserved in experiments on convection in SF$_6$ near the thermodynamic\ncritical point \\cite{RoSt02}, the mechanisms underlying the two types\nof restabilization of the hexagons is most likely different.\nReflecting the fact that in gases the non-Boussinesq effects increase\nwith increasing temperature the reentrance is shifted to lower values\nof the Rayleigh number when the mean temperature is increased, opposite to\nthe behavior in water \\cite{MaRi05}. In convection in water the\nreentrant hexagons are stable only for wavenumbers below the critical\nwavenumber. This trend becomes more pronounced with decreasing Prandtl\nnumber. In fact, for the gas mixture with $Pr=0.17$ the wavenumber at\nthe stability limit of the hexagons decreases so rapidly with\nincreasing Rayleigh number that the range in Rayleigh number over\nwhich the hexagons are stable becomes quite small. \n\nThe comparison of our stability results with experiments on the\ntransition between hexagons and rolls in CO$_2$ \\cite{BoBr91} shows\nthat this transition is not due to an amplitude or a side-band\ninstability. As a matter of fact, for the parameters of the\nexperimental system the hexagons do not undergo any linear amplitude\ninstability to rolls, contrary to the prediction of the weakly\nnonlinear theory \\cite{BoBr91}. We have performed detailed numerical\nsimulations with various lateral boundary conditions and confirm that\nthe transition is the result of the heterogeneous nucleation of rolls\nat the side walls of the container, which then invade the whole system\nif the Rayleigh number is sufficiently high. Our simulations suggest\nthat hexagons could be stabilized  well beyond the experimentally\nobserved transition point if the influence of the lateral walls can be\nreduced by applying a spatially patterned forcing that drives hexagons\nat the wall. Such a forcing can be achieved by localized heating\n\\cite{SeSc02} or by geometric patterning of the bottom plate\n\\cite{Bopriv}. Of course, the wavenumber of the forced hexagons would\nhave to be adjusted to lie in the stable range.\n\nWe have also investigated the stability of hexagons in H$_2$-Xe\nmixtures with very small Prandlt number ($Pr=0.17$). There also stable\nreentrant hexagons are possible, but they are restricted to a small\nrange in wavenumber ($q<q_c$) and Rayleigh number. Since for such\nsmall Prandtl numbers Boussinesq hexagons are always side-band\nunstable \\cite{ClBu96}, the stability of the hexagons is a result of\nthe non-Boussinesq effects. \n\nA fascinating state that is characteristic for convection at low\nPrandtl numbers is spiral defect chaos \\cite{MoBo93}. We have studied\nthe influence of NB effects on spiral defect chaos in a set-up\ncorresponding to convection in SF$_6$ with a Prandtl number of $Pr=0.8$\nand in H$_2$-Xe with a Prandtl number of $Pr=0.17$. To quantify the\ndifferences between Boussinesq and non-Boussinesq spiral defect chaos\nwe have employed a recently suggested set of geometric diagnostics of\nthe patterns \\cite{RiMa06}. \n\nAs expected, in SF$_6$ and in H$_2$-Xe the non-Boussinesq effects\nbreak the equivalence of up- and down-flows and, consequently,  the\nmean number of pattern components (closed contours) corresponding to\ndown-flow differs significantly from that of the up-flow components.\nMore interesting is our finding that in SF$_6$ the total number of\ncomponents (Betti number \\cite{GaMi04}) is more than twice as large in\nthe non-Boussinesq case than in the Boussinesq case for otherwise\nequal parameters. We attribute this enhancement of cellular rather\nthan roll-like structures to the resonant triad interaction that is\nintroduced by the non-Boussinesq effects. \n\nAnother striking difference between Boussinesq and non-Boussinesq\nspiral defect chaos in SF$_6$ becomes apparent in the joint\ndistribution function for the contourlength of the components and\ntheir compactness. While the Boussinesq case exhibits a strong\nsignature of target-like structures, the non-Boussinesq case displays\ninstead a marked increase in the number of small compact components\n(`bubbles'). This trend towards a more cellular structure is also\napparent in the analysis of spirals. The number of spirals with a\nwinding number above 1\/2 is much smaller in the non-Boussinesq than in\nthe Boussinesq case. In the correlations between the winding number\nand the arclength of the components this is reflected by the missing\nof large Archimedean spirals in the non-Boussinesq case. \n\nInterestingly, we do not find many small bubbles in our simulations of\nH$_2$-Xe at $Pr=0.17$. In fact, the maximum of the distribution\nfunction for the compactness of pattern components is shifted away\nfrom ${\\mathcal C}\\lesssim 1$ to ${\\mathcal C}\\sim 0.1$, i.e. to\nfilamentary structures. We do not know whether this implies that the\ntendency towards small bubbles is maximal at moderately small Prandtl\nnumbers, i.e. for $Pr \\sim 1$, or whether it is due to the fact that for\ncomputational reasons the simulations in H$_2$-Xe were performed at a\nlower Rayleigh number ($\\epsilon=0.3$) than in SF$_6$ ($\\epsilon=1.4$). \n\nDue to computational limitations we have not been able to investigate\nthe transition from hexagons to spiral defect chaos at very small\nPrandtl numbers. Since for very small Prandtl numbers there are\nessentially no large spirals, the transition from hexagons to spiral\ndefect chaos may follow a different path than that observed\nexperimentally in CO$_2$, where large spirals were reported to arise\nin the transition from hexagons to rolls \\cite{BoBr91}. We surmise\nthat the geometric diagnostics that we have employed to characterize\nthe well developed spiral defect chaos would provide also valuable\ninsight into this transition. Since the diagnostics require a\nsubstantial amount of data the transition would be best investigated\nexperimentally.\n\nWe deeply appreciate the support by W. Pesch and his students, who\nhave developed the codes we have used in this study \\cite{DePe94a}. We\nthank G. Ahlers for providing us with the code to determine the NB\ncoefficient. We have benefitted from discussions with G. Ahlers, G.\nGunaratne, K. Krishan, K. Mischaikow, W. Pesch, and M. Schatz. Support\nfrom the Department of Energy (DE-FG02-92ER14303) and NSF\n(DMS-9804673) is gratefully acknowledged.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAfter the Syrian Civil War started in 2011-12, civilians in increasing numbers sought refuge in neighboring countries. By May 2017, Turkey had received over 3 million refugees \u2014 the largest refugee population in the world. About 30\\% of them live in government-run camps near the Syrian border. Many have moved to cities looking for work and better living conditions. They face problems of integration, income, welfare, employment, health, education, language, social tension, and discrimination.\\footnote{This is the preprint of the D4R Challenge description paper. Please check the Challenge website for the final version, and the correct citation. Author affiliations: AAS (corresponding author, salah@boun.edu.tr) - Bo\u011fazi\u00e7i University and Nagoya University; AP - MIT; BL - Fondazione Bruno Kessler; EL - MIT and DataPop Alliance; PV - Harvard University; Y-AdM - Imperial College London; XD - University of Oxford; \u00d6D - T\u00fcrk Telekom} \nThe Data for Refugees (D4R) Challenge\\footnote{\\url{http:\/\/d4r.turktelekom.com.tr}} is a non-profit project to ultimately improve the conditions of the Syrian refugees in Turkey by providing a special database to scientific community for enabling research on some urgent problems. The challenge datasets opened to the community are based on anonymised mobile Call Detail Record (CDR) of phone calls and SMS messages of T\u00fcrk Telekom customers. It indicates broad activity and mobility patterns in Turkey for one year. The D4R Challenge, called the Challenge hereafter, opens the data to research groups submitting project proposals, after an evaluation, and on strictly regulated terms. The five focus themes of the challenge are health, education, unemployment, safety, and social integration, respectively. The project is initiated by T\u00fcrk Telekom, in partnership with the Turkish Academic and Research Council (T\u00dcB\u0130TAK) and Bo\u011fazi\u00e7i University and in collaboration with several academic and non governmental organizations, including UNHCR Turkey, UNICEF, and International Organization for Migration.\n\nA scientific committee of international experts guides its execution. A Project Evaluation Committee (PEC)\\footnote{\\url{http:\/\/d4r.turktelekom.com.tr\/presentation\/project-evaluation-committee}} is formed with representatives from academia, government (i.e. ministries related to the challenge), and NGOs working in this area. This committee represents refugee interests, and its job is to ensure that the submitted research projects that are granted access to the data have clear goals, with foreseen benefits to the refugee population in Turkey and elsewhere. Access to D4R data requires PEC approval.\n\nThe general aims of the Data for Refugees (D4R) project are to:\n\\begin{enumerate}\n    \\item Contribute to the welfare of the refugee populations,\n    \\item Gain insights on key issues, including safety and security, health, education, unemployment, social integration and segregation, mobility, and distribution of resources and infrastructure,\n    \\item Help governments and international bodies model the dynamics of the refugee populations and to discover vulnerabilities (socio-economic vulnerabilities, gaps in education and services, etc.),\n    \\item Seed further projects, co-created with refugees, resulting in new applications, services, and innovative solutions for refugees in Turkey and elsewhere.\n\\end{enumerate}\n\nThe lack of data on refugee mobility is a very important hurdle to the proper functioning of government services and international aid bodies. Innovative approaches attempted to overcome this included the use of satellite imagery to obtain information from the regions in crisis, with limited success~\\cite{machado2015analyzing}. This project will, for the first time, allow the analysis of a large-scale mobile CDR database on refugees. The D4D Challenges~\\cite{blondel2012data,de2014d4d} have illustrated the usefulness of such data, and the numerous projects completed on these challenges provided insights for social data science. We believe the challenge will be instrumental in understanding refugee mobility, optimizing infrastructure and humanitarian help, for better understanding the effects of interventions in health, education, and integration. Additionally the Challenge, by involving research groups from all around the globe, aims to raise awareness for the refugee issues on a grand scale.\n\nThe possibilities that mobile CDR data affords for analysis of a broad set of problems are surveyed in~\\cite{blondel2015survey}. Examples of projects conducted with similar data include analysis of disaster resilience~\\cite{tomaszewski2014geographic}, infrastructure planning~\\cite{martinez2015using}, quantifying mobility effects on the spread of infectious diseases~\\cite{baldo2013disease,mari2017big}, developing agent based models for disease migrations~\\cite{tompkins2016migration}, disease containment~\\cite{lima2015disease}, analysis of community structures and socio-demographic indicators~\\cite{trestian2017towards}, detection of unusual events~\\cite{gundogdu2016countrywide}, poverty analysis~\\cite{pokhriyal2017combining}, mobility during holidays and religious festivals~\\cite{scharff2015human}, to name a few. \n\nThe D4R Challenge has a distinct focus around refugee problems, and aims to enable evaluation of refugee related interventions and activities, including, but not limited to, educational activities, social gatherings, NGO actions, government infrastructure investments, etc. It also has the potential to bring insights to the analysis of residential segregation, population structures for specific geographical locations, and factors on social integration~\\cite{silm2014ethnic}.\n\n\\section{Description of D4R Data}\n\\subsection{Brief summary}\nThe D4R Challenge is based on the successful Data for Development (D4D) Challenge series~\\cite{blondel2012data}. Three datasets are made available to the challenge participants, along with external helper files. The main difference from D4D is that D4R has a ``Refugee\" flag, which indicates (with a high probability) that the CDR belongs to a refugee customer. This flag is given to customers in the database that 1) have ID numbers given to refugees and foreigners in Turkey, 2) registered with Syrian passports, 3) use special tariffs reserved for refugees. None of these groups are guaranteed to include only and exclusively refugees, which serves as a layer of protection: It is not possible to say with certainty that a particular CDR belongs to a refugee or not, but it is only possible to derive patterns from aggregated records. Initially, we have planned to perturb these labels for adding such a protection, but the collection is already noisy, and no further noise is introduced. We list the datasets contained in the Challenge in individual subsections.\n\nWe acknowledge that the term ``refugee\" is used as a blanket term in the dataset, and includes migrants, asylum seekers, and even foreigners who have acquired a ``temporarily protected foreign individual\" ID number in Turkey (i.e. starting with 98 or 99). The dataset needs to be approached with these reservations in mind, and the analysis should carefully consider such biases in the data.\n\nThe D4R dataset is collected from 992.457 customers of T\u00fcrk Telekom, of which 184.949 are tagged as ``refugees'', and 807.508 as Turkish citizens. A total of 1.211.839 subscriptions are included. Of these, 980.697 belong to Turkish citizens, and 231.142 belong to refugees (we refer to these customers as refugees, but as mentioned before, there is some noise in this indicator). Some of the customers had multiple phone lines; each line corresponds to a single subscription. \n\n75\\% of the refugee-tagged customers are recorded as ``male\", and 25\\% as ``female\". There is clearly a gender bias in the ownership of the phone line. This does not mean that 75\\% of the phone lines are used by men, however. We have sampled the 807K Turkish customers with the same gender distribution. \n\n45\\% of the refugee customers were registered in Istanbul. Other major cities with refugee presence were Gaziantep, \u0130zmir, \u015eanl\u0131urfa, and Mersin. We have sampled the Turkish citizen customers mainly from the cities with registered refugee presence, to simplify comparisons.  Table~\\ref{table:refugee_distrib} and Table~\\ref{table:tc_distrib} show the distribution of customers and their registered cities. Only the top locations are shown. The distribution over all the cities of the country is provided to the participants in the file ``Refugees per city (March 2017).xls\". This file shows the official number of refugees registered per city, the official city population in 2017 (excluding refugees), and the percentage of refugees with respect to the city population. Additionally, it shows the number of T\u00fcrk Telekom customers used for the entire D4R data collection per city, broken into ``refugee\" and ``citizen\" counts.\nThe numbers of registered refugees and asylum seekers in Turkey according to registration dates can be obtained from UNHCR websites\\footnote{See \\url{https:\/\/data2.unhcr.org\/en\/situations\/syria\/location\/113}}. Another useful source of data is the TUIK census estimates for Turkish cities, according to years\\footnote{See \\url{http:\/\/www.tuik.gov.tr\/PreIstatistikTablo.do?istab_id=1590}}. This source indicates the population size and growth of each city between 2000-2017.\n\n\\begin{table}[tb]\n    \\centering\n    \\begin{tabular}{|c|c|c|}\n         \\hline\n         Location & Number of Customers & Percentage \\\\ \\hline\n\u0130stanbul & 84.173 & 45,511 \\\\\nGaziantep &14.898 & 8,055\\\\\n\u0130zmir & 10.425 & 5,637\\\\\n\u015eanl\u0131urfa & 9.701 & 5,245\\\\\nMersin & 9.660 & 5,223\\\\\nHatay & 7.024 & 3,798\\\\\nAnkara & 5.580 & 3,017\\\\\nKonya & 4.718 & 2,551\\\\\nBursa & 3.479 & 1,881\\\\\nOutside Turkey & 2.902 & 1,569\\\\\nOther & 32.440 & 17,540 \\\\\n\\hline\n    \\end{tabular}\n    \\caption{The distribution of customers tagged as refugees and their registered locations. Numbers rounded to the third significant digit.}\n    \\label{table:refugee_distrib}\n\\end{table}\n\n\\begin{table}[tb]\n    \\centering\n    \\begin{tabular}{|c|c|c|}\n         \\hline\n         Location & Number of Customers & Percentage \\\\ \\hline\n\u0130stanbul & 363.334 & 44,994 \\\\\nGaziantep &80.655 & 9,988\\\\\n\u0130zmir & 40.501 & 5,016\\\\\nAnkara & 40.443 & 5,009\\\\\nAdana & 40.415 & 5,005\\\\\nHatay & 40.394 & 5,002\\\\\nKonya & 40.388 & 5,002\\\\\nAntalya &40.367 & 4,999\\\\\nBursa &40.359& 4,998\\\\\n\u015eanl\u0131urfa & 40.321 & 4,993\\\\\nMersin & 40.242 & 4,983\\\\\n\\hline\n    \\end{tabular}\n    \\caption{The distribution of customers tagged as Turkish citizens and their registered locations. Numbers rounded to the third significant digit.}\n    \\label{table:tc_distrib}\n\\end{table}\n\nThe usage of the D4R data requires caution in interpreting the representativeness of the data for the refugee population in Turkey. At the end of March 2017, there were 75.724.413 mobile customers in Turkey across all operators (\\%94,9 penetration rate)~\\cite{btk17}. Excluding machine to machine (M2M) and population of 0-9 age range, the mobile penetration was \\%107. According to data from the first three months of 2017, the mobile market share of T\u00fcrk Telekom (Avea), from which the challenge data is collected, was \\%24,7~\\cite{btk17}. \n\nWe have used the entire refugee customer base (with the filtering conditions described before), but the market share of T\u00fcrk Telekom also shows fluctuations according to the individual cities. Therefore, it is useful to look at official numbers of refugees distributed over the country. This is partly depicted in Fig.~\\ref{fig:distribution} as per Ministry of Interior, Directorate General of Migration Management figures\\footnote{Only the top ten cities are shown, more detailed information can be obtained from \\url{http:\/\/www.goc.gov.tr\/icerik\/migration-statistics_915_1024}.}. However, these figures are from 2018. We have supplied the official figures from March 2017, as mentioned before.\n\nWe now describe the contents of the dataset in more detail. \n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=10cm]{figures\/distribution-of-refugees.jpg}\n\\caption{The distribution of refugees in the country according to data from Ministry of Interior, Directorate General of Migration Management.}\n\\label{fig:distribution}\n\\end{figure}\n\n\\subsection{Cell tower locations}\nThe cell tower (i.e. base station) locations are provided in the file ``Base\\_Station\\_Location.txt\". The file contains the following fields:\n\\begin{itemize}\n    \\item BTS\\_ID: The identifier of the base tower.\n    \\item MX\\_LAT1,MX\\_LAT2,MX\\_LAT3: DMS latitude of the base tower.\n    \\item MX\\_LONG1,MX\\_LONG2,MX\\_LONG3: DMS longitude of the base tower.\n    \\item MX\\_SAHAIL: The registered city of the base tower.\n    \\item MX\\_SAHAILCE: The registered district of the base tower.\n    \\item MX\\_POPAREA: An inofficial note about the population type around the base tower, used internally in T\u00fcrk Telekom. It takes values of RURAL, SUB\\_URBAN, INDUSTRIAL, SEASONAL AREAS, DENSE\\_URBAN, HOT SPOT, OPEN IN URBAN, AIRPORT, SUBURBANLOW, POPRURAL, and INDOOR.\n\\end{itemize}\nIn some rare cases, the precise location information of the base station is missing, only the city is indicated.\nThe interpretation of the latitude and longitude follows degree, minutes, seconds (DMS) syntax. For example, the district of BARTIN in the city of ZONGULDAK is represented by these six numbers as follows: (41 25 43.1184 32 4 37.9344). This corresponds to $41^\\circ 25' 43.1184''$ N DMS latitude, and $32^\\circ 4' 37.9344''$ E longitude.\n\\subsection{District locations}\nTo disambiguate the base stations, we provide a file that contains district coordinates. This file, named ``district\\_coordinates.csv\", has the following fields:\n\\begin{itemize}\n    \\item CITY: Name of the city.\n    \\item DISTRICT: Name of the district.\n    \\item POPULATION\\_2014: The official census population of the district in 2014.\n    \\item LATITUDE, LONGITUDE: The 2D coordinates of the district.\n    \\item MX\\_LAT1, MX\\_LAT2, MX\\_LAT3, MX\\_LONG1, MX\\_LONG2, MX\\_LONG3: The DMS coordinates of the district.\n\\end{itemize}\n\nA conversion script (such as \\url{https:\/\/www.latlong.net\/lat-long-dms.html}) can be used to convert the (latitude, longitude) variables into DMS. For example, the 2D coordinates of (41.428644\t32.077204) for Zonguldak, Bart\u0131n, translate into the DMS coordinates given in the previous subsection.\n\\subsection{Dataset 1: Antenna Traffic}\nThe first database we provide includes one year site-to-site traffic on an hourly basis. This dataset contains the traffic between each site for a year. Calls between T\u00fcrk Telekom (TT) customers and other service providers (SP) only have information about the TT side. For each record, total number and duration of calls are recorded in an aggregated fashion. \n\nThe database is split into voice and SMS partitions. For the voice partition, the file ``Dataset 1\\_2017XX.txt'' contains the data for month XX, and there are 12 such files. Each file contains the following fields:\n\\begin{itemize}\n\\item TIMESTAMP: Day and hour considered in format DD-MM-YYYY HH (24 hours format).\n\\item OUTGOING\\_SITE\\_ID: ID of the site the call originated from. Unknown stations are denoted as ``-99'' or ``9999''.\n\\item INCOMING\\_SITE\\_ID: ID of the site receiving the call.\n\\item NUMBER\\_OF\\_CALLS: The number of calls in this 1 hour slot.\n\\item NUMBER\\_OF\\_REFUGEE\\_CALLS: The number of calls involving numbers tagged as ``refugee''.\n\\item NUMBER\\_OF\\_TOTAL\\_CALL\\_DURATION: The total call duration from all calls.\n\\item REFUGEE\\_CALL\\_DURATION: The total call duration from calls involving numbers tagged as ``refugee''.\n\\end{itemize}\n\nFor the SMS partition, the file ``Dataset 1\\_SMS\\_2017XX.txt'' contains the data for month XX, and there are 12 such files. Each file contains the following fields:\n\\begin{itemize}\n\\item TIMESTAMP: Day and hour considered in format DD-MM-YYYY HH (24 hours format).\n\\item OUTGOING\\_SITE\\_ID: ID of site the SMS originated from. Unknown stations are denoted as ``-99'' or ``9999''.\n\\item INCOMING\\_SITE\\_ID: ID of the site receiving the SMS.\n\\item NUMBER\\_OF\\_SMS: The number of SMS messages in this 1 hour slot.\n\\item NUMBER\\_OF\\_REFUGEE\\_SMS: The number of SMS messages involving numbers tagged as ``refugee''.\n\\end{itemize}\n\\subsection{Dataset 2: Fine Grained Mobility}\nThe dataset contains cell tower identifiers used by a group of randomly chosen active users to make phone calls and send texts. The data are timestamped and a particular group of users are observed for a period of 2 weeks. At the end of the two-week period, a fresh sample of active users are drawn at random. We provide data for the entire 1-year sampling period. The users are represented by random numbers in the dataset, and no personal information is stored. These numbers start with 1 for refugees, 2 for non-refugees, 3 for unknown. However, this indicator should be considered to be somewhat noisy. Among the users who are marked as refugees, there may be customers who are not refugees, and vice versa. Consequently, it will not be possible to say with 100\\% certainty whether an invitation CDR belongs to a refugee or not. (This numbering scheme is also used for Dataset 3.) There is no identifying information about the other party of the call; only the callee prefix (1: refugee, 2: not refugee, 3: unknown) is given. \n\nTo protect privacy, new random identifiers are chosen for every two-week period, and if a user is sampled in more than one period, these records cannot be associated with each other. For missing antenna locations, a code of -99 or 9999 is assigned. This dataset is also separated into voice and SMS partitions. Furthermore, to deal with large file sizes, it is further divided into files containing incoming (IN) and outgoing (OUT) calls, resulting in four files per 15-day period.\n\nThe files ``Dataset 2\\_2017XXW\\_In.txt,\" ``Da-taset 2\\_2017XXW\\_Out.txt,\" ``Dataset 2\\_2017XXW\\_SMS\\_In.txt,\" and ``Dataset 2\\_2017XXW\\_SMS\\_Out.txt\" all have similar structure, where XX ranges from 01 to 26, and denotes a 15-day period, starting from 1-15 January 2017 in the first file. The files have the following fields: \n\\begin{itemize}\n    \\item CALLER\\_ID: Random number for a user generated specifically for this 15-day period. Note that the user is not necessarily the initiator of the call, that is determined by the CALL\\_TYPE flag. The first digit denotes ``refugee'' (1), ``non-refugee'' (2), ``unknown'' (3). \n    \\item TIMESTAMP: Day and hour considered in format DD-MM-YYYY HH (24 hours format).\n    \\item CALLEE\\_PREFIX: A value for the other party of the call that denotes ``refugee'' (1), ``non-refugee'' (2), ``unknown'' (3).\n    \\item SITE\\_ID: The ID of the base station.\n    \\item CALL\\_TYPE: The Call type is either outgoing (1) or incoming (2).\n\\end{itemize}\n\n\\subsection{Dataset 3: Coarse Grained Mobility}\nIn this dataset, the trajectories of a randomly selected subset of users are provided for the entire observation period, but with reduced spatial resolution. We divide the entire country into the electoral prefectures (or districts), and for each call record, only the prefecture information is provided. The IDs are randomly assigned, and two different users may have the same ID in Dataset 2 and Dataset 3. The database is split into incoming (IN) and outgoing (OUT) calls to deal with large files. The files ``Dataset 3\\_2017XX\\_In.txt\" and ``Dataset 3\\_2017XX\\_Out.txt\" have a similar structure, and contain the fields:\n\\begin{itemize}\n    \\item CALLER\\_ID: The randomly assigned ID of the user (different from Dataset 2). Similarly to Dataset 2, the call initiator is determined by CALL\\_TYPE.\n    \\item TIMESTAMP: Day and time considered in format DD-MM-YYYY HH:MM (24 hours format).\n    \\item ID: The ID of the district.\n    \\item CITY\\_ID: The ID of the city.\n\\end{itemize}\n\nIn order to achieve the mapping to the cities and prefectures, two additional files are provided. In ``Dataset 3\\_City Mapping.txt,'' the CITY\\_ID is followed by CITY\\_DESC, which is the name of the city. 81 cities are included. In ``Dataset 3\\_District Mapping.txt,'' the ID field represents the district ID, as used in the Dataset 3, and the BTS\\_DISTRICT field gives the name of the district. There are a total of 1025 districts.\n\\section{Ethical and privacy issues}\nIn this section, we briefly discuss the ethical and privacy issues regarding the Challenge data. The collection, storage and protection of data in the D4R Challenge complies with European Union requirements regarding the protection of personal data and the protection of privacy in the electronic communications sector. Furthermore, research on the previously conducted D4D Challenges established that the data such as offered in this project does not allow identification of individuals~\\cite{al2015literature,cecaj2016re,gambs2014anonymization,taylor2016no}. Sharad and Danezis note that providing aggregated data such as antenna traffic results in ``little scope of privacy breach\\ldots since it contains no personally identifiable information about the users. It could be used to study traffic patterns during the entire period but reveals no information pertaining to the users.\"~\\cite{sharad2013anonymizing}.\n\n\\textbf{Definitions:} Personal data means any information relating to an identified or identifiable natural person. Personal data does not include anonymous information, that is, information that does not relate to an identified or identifiable natural person or to data rendered anonymous in such a way that the Data Subject is not or is no longer identifiable. Data Subject means a natural person (i.e. an individual) who can be identified directly or indirectly, in particular by reference to Personal Data.\n\n\\textbf{Consent, legitimate and fair processing:} The data in the Database comes exclusively from T\u00fcrk Telekom customers, who has consented to its anonymised use for research purposes through the mandatory user agreement at the time of the purchase of the phone line. The content of phone activity, actual phone numbers, identities, addresses, or similar personal information are neither stored, nor distributed with the Database. Subsequently, it is not possible to identify natural persons with the Database.\n\n\\textbf{Transparent processing:} The nature of the data, the assurance of its anonymity, as well as the ethical precautions to ensure its proper use are (at the time of the start of the Challenge) documented openly on the Challenge website. Accessible and plain language is used, and further contact information is supplied to respond to questions about the data usage.\n\n\\textbf{Project Evaluation Committee (PEC):} The PEC is formed with representatives from academia, government, and related NGOs. Its aim is to represent refugee interests in the Challenge, and all project proposals are pre-screened by the Scientific Committee and by PEC. The proposals that pass the initial screening are granted access to the dataset, upon submitting the signed User Agreement form.\nCriteria for passing the initial screening were:\n\\begin{itemize}\n\\item A project proposal is submitted (in English, which is the common language for the international PEC), and all the team members who will access the data are identified.\n\\item The project PI has a permanent affiliation. \n\\item The project uses the D4R data meaningfully. \n\\item The project aims are aligned with the goals described in the previous section, do not represent a commercial interest, and do not endanger the privacy or well-being of individuals or groups.\n\\end{itemize}\n\nData access for the Database is granted to participants during the designated Challenge period, by a mandatory user agreement prepared by T\u00fcrk Telekom lawyers, and approved by ADIEK, the Ethical Conduct committee of Bo\u011fazi\u00e7i University. The agreement permits third parties to analyze the anonymised and aggregated data, summarized previously, to submit a research report at the end of the Challenge and to present the results at a special workshop. A white paper is be prepared to inform the related government bodies and NGOs about the results of the projects. The project reports are published publicly on the project website, after evaluation by the D4R PEC and D4R SC. There are three possible outcomes for submitted reports: (1) Normal publication, for papers that treat the ethical issues correctly. (2) Ask to consider adjustments, for papers that require amendments and removal of sensitive material before publication. We ask the projects to be careful not to include statements that may harm the refugee population in any way, or may promote negative sentiments about the refugee populations. (3) Do Not Publish, either because the report is on a sensitive issue, or because it is not scientifically rigorous and its conclusions are not warranted. Sensitive reports may be shared directly with related institutions or authorities. The PEC and the project proposers jointly decide on this, on a case by case basis. \n\n\\textbf{Retention, destruction and archiving:} The Challenge mandates that all participants destroy the Database upon completing the challenge. Any publication based on the Database requires the pre-approval of the Project Evaluation Committee. Any further use of the Database (for instance to complete numerical experiments for a publication under review) will be regulated by PEC, and extensions are to be conditionally granted for specific purposes on a case by case basis.\n\n\\textbf{Information:} Every care has been given to ensure that the information provided in the Database does not cause any harm, prejudice, or distress to customers, regardless of their refugee status. PEC provides an additional layer of control, and will examine project reports confirm to this maxim.\n\n\\textbf{Access, correction, erasure, objection:} The Database does not contain personal information, and it is not possible for individuals to request access to personal data. The data are anonymised and aggregated in a way to prevent identification of persons. For the same reason, correction, erasure, and objection do not apply for the Database. We note that this is a stricter protection condition than most envisioned CDR usage scenarios. Furthermore, the Database does not contain children's data, as each registered customer has to be over 18 years of age, and thus legally permitted to own and use a mobile phone line. Personal profiling (such as used for CRM applications) is not possible with the Database.\n\n\\textbf{Responsibility and accountability:} The responsibilities of all parties concerned are defined clearly, and set out in the data agreement prepared by TT lawyers.\n\n\\textbf{Data protection by design and default:} Data collection follows this principle, where any names, real phone numbers, or other identifying information is excluded from the design of the Database. The pseudo-random numbers representing customers are not stored anywhere along with actual phone numbers. Subsequently, the anonymisation works only one way. Refugee status is indicated by purposefully noisy indicators, and no effort is spent to ensure its validity. Subsequently, only aggregate-level conclusions can be drawn from the Database. It is not possible to use the Database for the surveillance and tracking of individuals. \n\n\\textbf{Limitations:} Data access is not provided to institutions in a blanket permission, but to specific individuals within institutions, whose names and roles in the proposed research project is clearly defined in the data agreement.\n\\section{Acknowledgments}\nWe thank Duhan Can \u00c7ak\u0131 (T\u00fcrk Telekom), Salim Y\u0131lmaz (T\u00fcrk Telekom), Oktay Namver (T\u00fcrk Telekom), Ali G\u00f6r\u00e7in (T\u00dcB\u0130TAK B\u0130LGEM), Merve Astekin (T\u00dcB\u0130TAK B\u0130LGEM), and the members of the D4R PEC for their contributions. This work is partially supported by a MIT MISTI grant to Alex Pentland and Albert Ali Salah.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nRadio interferometers, which operate by correlating the signals from two or\nmore antennas, have many advantages over traditional single-dish telescopes,\nincluding greater scalability, independent control of aperture size and\ncollecting area, and self-calibration.  Since the first digital correlator\nbuilt by Weinreb \\citep{weinreb_1961}, the processing power of\nthese systems has been tracking the Moore's Law growth of digital electronics.\nThe decreasing cost per performance of these systems has influenced the design\nof many new radio antenna array telescopes.  Some\nnext-generation array telescopes at meter, centimeter and millimeter\nwavelengths are: \nthe LOw Frequency ARray (LOFAR), \nthe Precision Array for Probing the Epoch of Reionization (PAPER), \nthe Murchison Widefield Array (MWA), \nthe Long Wavelength Array (LWA),\nthe Expanded Very Large Array (EVLA), \nthe Allen Telescope Array (ATA), \nthe Karoo Array Telescope (MeerKAT), \nthe Australian Square Kilometer Array Demonstrator (ASKAP),\nthe Atacama Large Millimeter Array (ALMA).\nand the Combined Array for Research Millimeter-wave Astronomy (CARMA). \nThis paper presents a novel approach to the intense digital signal \nprocessing requirements of these instruments that has many other applications\nto astronomy signal processing.\n\nWhile each generation of electronics has brought new commodity data processing\nsolutions, the need for high-bandwidth communication between processing nodes\nhas historically lead to specialized system designs.  This communication\nproblem is particularly germane for correlators, where the number of\nconnections between nodes scales with the square of the number of antennas.\nSolutions to date have typically consisted of specialized processing boards\ncommunicating over custom backplanes using non-standard protocols.  However,\nsuch solutions have the disadvantage that each new generation of digital\nelectronics requires expensive and time-consuming investments of engineering\ntime to re-solve the same connectivity problem.  Redesign is driven by the same\nMoore's Law that makes digital interferometry attractive, and is not unique to\nthe interconnect problem; processors such as Application-Specific Integrated\nCircuits (ASICs) and Field Programmable Gate Arrays (FPGAs) also require\nredesign, as do the boards bearing them, and the signal processing algorithms\ntargeting their architectures.\n\nOur research is aimed at reducing the time and cost of correlator design and\nimplementation.  We do this, firstly, by developing a packetized communication\narchitecture relying on industry-standard Ethernet switches and protocols to\navoid redesigning backplanes, connectors, and communication protocols.\nSecondly, we develop flexible processing modules that allow identical boards to\nbe used for a multitude of different processing tasks.  These boards are\napplicable to general signal processing problems that go beyond \ncorrelators and even radio science to include, e.g., ASIC design and\nsimulation, genomics, and research into parallel processor architectures.  \nGeneral\npurpose hardware reduces the number of boards that have to be redesigned and\ntested with each new generation of electronics.  Thirdly, we create\nparametrized signal processing libraries that can easily be recompiled and\nscaled for each generation of processor.  This allows signal processing systems\nto quickly take advantage of the capabilities of new hardware.  Finally, we\nemploy an extension of a Linux kernel to interface between CPUs and FPGAs for\nthe purposes of testing and control, presenting a standard file interface\nfor interacting with FPGA hardware.\n\nThis paper begins with a presentation of the new correlator\ndesign architecture in \\S\\ref{sec:architecture}. The hardware to \nimplement this architecture follows in \\S\\ref{sec:hardware}, and\nthe FPGA gateware used in the hardware is summarized in \\S\\ref{sec:gateware}.\nIssues concerning system integration are given in \\S\\ref{sec:integration},\nand performance characterization of subsystems are given in \n\\S\\ref{sec:characterization}. Results from our first deployments of\nthe packetized correlator are displayed in \\S\\ref{sec:deployments}.\nOur final section summarizes our progress and points to a number\nof directions we are pursuing for the next generation of scalable\ncorrelators based on modular hardware, reuseable gateware and\ndata packetization. An appendix gives a glossary of technical\nacronyms since this paper makes heavy use of abbreviated terms.\n\n\\section{A Scalable, Asynchronous, Packetized FX Correlator Architecture}\n\\label{sec:architecture}\n\nCorrelators integrate the pairwise correlation between complex voltage samples \nfrom polarization channels of array antenna receivers at a set of\nfrequencies.\nOnce instrumental effects have been calibrated and removed, the resultant \ncorrelations (called visibilities) represent the self-convolved electric field\nacross an aperture sampled at locations \ncorresponding to separations between antennas.  These visibilities can be \nused to reconstruct an image of the sky by inverting the interferometric \nmeasurement equation:\n\\begin{equation}\nV_{\\nu}(u,v)=\\int\\!\\!\\!\\!\\int{G_{i,\\nu}G_{j,\\nu}^*I_\\nu(\\ell,m)}\n{e^{-2\\pi i(u\\ell+vm+w(\\sqrt{1-\\ell^2-m^2}-1))}d\\ell dm}\n\\label{eq:vis}\n\\end{equation}\n$I_\\nu$ represents the sky brightness in angular coordinates $(\\ell,m)$, and \n$(u,v,w)$ correspond to the separation in wavelengths of an antenna pair \nrelative to a pointing direction.\nFor antennas with separate polarization feeds, cross-correlation\nof polarizations yields components of the four Stokes parameters that\ncharacterize polarized radiation, here defined in terms of linear\npolarizations ($\\|,\\perp$) for all pairs of antennas $A$ and $B$ \n\\citep{rybicki_lightman1979}:\n\\begin{equation}\n\\begin{array}{ll}\n\\displaystyle I=A_\\| B_\\|^*+A_\\perp B_\\perp^* &\\ \\ \\ \nQ=A_\\| B_\\|^*-A_\\perp B_\\perp^* \\nonumber \\\\\n\\displaystyle U=A_\\| B_\\perp^*+A_\\perp B_\\|^* &\\ \\ \\ \nV=A_\\| B_\\perp^*-A_\\perp B_\\|^* \n\\label{eq:pol}\n\\end{array}\n\\end{equation}\nI measures total intensity, V measures the degree of circular polarization,\nand Q and U measure the amplitude and orientation of linear polarization.\n\nThe problem of computing pairwise correlation as a function of frequency can be\ndecomposed two mathematically equivalent but architecturally distinct ways.\nThe first architecture is known as ``XF'' correlation because it first\ncross-correlates antennas (the ``X'' operation) using a time-domain ``lag''\nconvolution, and then computes the spectrum (the ``F'' operation) for each resulting\nbaseline using a Discrete Fourier Transform (DFT).  An alternate architecture\ntakes advantage of the fact that convolution is equivalent to multiplication in\nFourier domain.  This second architecture, called ``FX'' correlation, first\ncomputes the spectrum for each individual antenna (the F operation), and then\nmultiplies pairwise all antennas for each spectral channel (the X operation).  \nAn FX correlator has an advantage over XF\ncorrelators in that the operation that scales as $O(N^2)$ with the\nnumber of antennas, N, is a complex multiplication as opposed to a full \nconvolution in an XF correlator \\citep{daddario2001,yen1974}.\n\nThough there are mitigating factors (such as bit-growth for representing the\nhigher dynamic range of frequency-domain data) that favor XF correlators for\nsmall numbers of antennas \\citep{thompson_et_al2001}, FX correlators are more\nefficient for larger arrays.  Since scalability to large numbers of antennas is\none of the primary motivations of our correlator architecture, we have chosen\nto develop FX architectures exclusively.\n\n\\subsection{Scalability With Number of Antennas and Bandwidth}\n\\label{sec:scalability}\n\nThe challenge of creating a scalable FX correlator is in designing a\nscalable architecture for factoring the total computation into manageable \npieces and efficiently bringing together data in each piece for computation.\nTraditionally, the spectral decomposition (in F engines) has been scaled to\narbitrary bandwidths by using analog mixers and filters to divide the operating\nband of each antenna into the widest subbands that can be processed digitally\nusing existing technology.  Within correlation of a given subband,\nthe complexities of computation and of data distribution both scale\nlinearly with bandwidth and quadratically with the number of antennas. It is\nimperative that the arrangement of cross-multiplication engines (hereafter\nreferred to as X engines) minimize data replication\/retransmission, even as X\nengines expand to encompass many boards.  Fortunately, each frequency channel\nof an FX correlator is computationally independent, providing a natural\nboundary for dividing computation among processing nodes.\n\n\\placefigure{fig:corr_arch1}\n\nFigure \\ref{fig:corr_arch1} illustrates a simplistic architecture for an FX\ncorrelator that takes advantage of the computational independence of channels\nto avoid unnecessary data transmission;\nthe total X computation has been factored into X engines that cross-multiply \nall antenna pairs for a single frequency channel.\nThis architecture is overly\nsimplistic, since an X engine's performance can be equated to an aggregate \ninput bandwidth that it can handle.  For the sake of efficiency, an X engine \nprocessor \nshould receive as many channels as it has capacity to process.  In this case, \nthe number of X engines is given by:\n\\begin{equation}\n\\#\\ {\\rm X\\ Engines} = \\frac{({\\rm Antenna\\ Bandwidth})\\times \n  (\\#\\ {\\rm Antennas})}{ {\\rm X\\  Engine\\ Processing\\ Bandwidth}}\n\\end{equation}\nMultiplexing channels into X engines makes cross-multiplication\ncomplexity independent of the number of channels.  There are three\npotential bottlenecks for scaling this architecture: the complexity of\ninterconnecting F engines and X engines, the bandwidth into individual X\nengines, and the amount of computation in an X engine relative to the size of a\nprocessing chip\/board\/system.  Each of these bottlenecks warrants further\ndiscussion.\n\nThe potential bottleneck of connecting $N$ antenna-based F engines to $M$\nchannel-based X engines is highlighted by the criss-crossed lines in Figure\n\\ref{fig:corr_arch1}.  Historically, this bottleneck has been addressed with\ncustom backplanes and transmission protocols.  However, our group has taken the\nnovel approach of using high-performance, commercially available, \n10-Gbit\/s Ethernet (10GbE) switches to solve this problem.  \nAs will be discussed, these switches currently have the bandwidth and switching\ncapacity to handle large correlators, and represent a negligible fraction of\nthe total cost of correlator hardware.  Furthermore, switching technology is\ndriven by commercial applications and by Moore's Law, making it likely that\nfuture switches will continue increasing in number of ports and bandwidth per\nport.\n\nA second potential bottleneck concerns how data rates and \nnumbers of X engines scale with antenna bandwidth.  It is important that\nwe consider various bandwidth cases, owing to the variety of science\napplications driving large, next-generation systems.  For example, correlators\nfor large arrays of low-bandwidth antennas will need to multiplex data into\nhigher bandwidth processors, while arrays with larger bandwidths will face the\nopposite problem. In our architecture, we make the reasonable assumption that\nthe number of frequency channels always exceeds the number of antennas.  \nThis assumption\nensures that the per-port bandwidth into an X engine never exceeds what is\ntransmitted per antenna.  Multiple channels may then be mapped into an X engine\nup to its computational capacity (allowing efficient resource utilization for\nlow-bandwidth arrays), and additional X engines may be added for high-bandwidth\napplications.  Antenna bandwidths requiring transmission above 10 Gbits\/s can\nbe accommodated by connecting F engines to multiple 10GbE ports.\nFrequency channels are then assigned to each port, which connect separate\nswitches and sub-networks of X engines.  In this way, bandwidths may be scaled\nup to the transmission capability of an F processor by increasing the number of\nsubnets, and not switch complexity.\n\nThe third and final potential bottleneck concerns how the sizes of individual X\nengines scales with the number of antennas.  Both large and small numbers of\nantennas pose scaling problems.  The size of an X engine responsible for\ncomputing all baseline cross-multiples with a fixed input data rate\nscales as $O(N)$, while\nthe number of X engines required to accommodate the expanding data bandwidth\nwith increasing numbers of antennas also scales as $O(N)$,\naccounting for the $O(N^2)$ scaling of computing in a correlator.  For\nsufficiently large $N$, the size of an X engine can exceed the size of any\nprocessing chip or board.  Our solution has been to develop an X engine whose\npipelined architecture allows it to be split across multiple processors with\nsimple point-to-point connectivity.  This allows many processors to be chained\ntogether from a switch port to meet the computational demands of an X engine.\nScaling to small $N$ is equally challenging, because the aggregate correlator\nbandwidth decreases as $O(N)$, while computational complexity scales down as\n$O(N^2)$.  As a result, we can find that the number of X engines that\nfit onto a chip\/board exceeds the rate at which data can be received.  The\nthreshold where this problem is encountered can be changed by designing\nprocessors with greater connectivity, but once hardware is fixed, there is no\nother recourse but to accept a certain inefficiency for low numbers of\nantennas.  While this is a fundamental limitation of our architecture, \nthe cost of small correlators is typically dominated by development\n(not hardware), so a certain architectural inefficiency can be accommodated for\nthe savings it affords in development time.\n\n\\subsection{Globally Asynchronous Locally Synchronous Systems}\n\\label{sec:gals}\n\nPacketized data transmission simplifies the cross-connect problem inherent to\ncorrelators, but this comes at the price of global synchronicity.  Packetized\ncommunication is fundamentally asynchronous: data can arrive scrambled,\ndelayed, or not at all.  Locally-synchronous X engine processing must therefore\ntransition from being timing-driven (with throughput tied to an FPGA clock, for\nexample) to being asynchronously data-driven.  Though data buffers and control\nsignals complicate development, Globally Asynchronous Locally Synchronous\n(GALS) design facilitates system integration and leads to robust design\n\\citep{chapiro1984,luis_et_al2007}.  Processors run at clock rates above the\ndata rate, using local oscillators that can drift with temperature.  By\nallowing for non-transmission of data, individual components can fail without\ncausing global failure--an important feature for large systems where\ncomponents may fail regularly during operation.  GALS design also insulates\nprocessing architectures from decisions regarding sample rates and antenna\nbandwidths, allowing for greater operational flexibility.  Finally, individual\nprocessing elements may be redesigned and upgraded in a GALS system without\naffecting the overall architecture, facilitating early adoption of new\ntechnology.\n\nData-driven processing on locally synchronous processors like FPGAs requires\ncontrolling propagation through the processing pipeline.  However, routing\ncontrol signals to every multiplier, accumulator, and logic element in a\npipeline can lead to excessive routing and gating demands.  To avoid this, we\nhave implemented a window-based processing architecture for algorithms where\nthe results derived from one set of data samples are computationally\nindependent from the next.  In this architecture, processing elements are\nallowed to run freely at their native rate without being enabled\/disabled, but\nare only provided data when an entire window of data has been buffered.  These\nwindows of data are provided synchronously with the inherent window boundaries\nof the processing element, and an entire output window is flagged as valid.\nInternally, a processor processes both valid and invalid data--it is only the\nexternal buffering system that keeps track of data validity.  This technique is\napplicable to many common operators such as cross-multipliers, DFTs, and\naccumulators.  Finite Impulse Response (FIR) filtering is an\noperation notable for not being window-based.\n\n\\subsection{Example Applications}\n\\label{sec:example}\n\n\\placefigure{fig:ex_app1}\n\nPerhaps the best method for demonstrating the flexibility and scalability of\nour correlator architecture is through example applications.  To illustrate\ntechniques for using hardware and ports efficiently, we will map processing\ninto fictitious hardware that corresponds roughly in capability to the\nCASPER (Center for Astronomy Signal Processing \nand Engineering Research)\\footnote{http:\/\/casper.berkeley.edu}\nhardware discussed in Section \\ref{sec:hardware}.\n\nOur first example (Fig. \\ref{fig:ex_app1}) illustrates an antenna signal\nbandwidth sufficiently low so that data from 2 polarization channels of 2\nantennas can be transmitted over one 10GbE connection.  Assuming that the\nnumber of antennas evenly divides the number of frequency channels, and that\nthe processing bandwidth of an X engine matches the data bandwidth of one\nantenna, there will be the same number of X engines as F engines, and each X\nengine will receive 1\/N$^{\\rm th}$ of the total bandwidth, where N is\nthe number of antennas.  F engine\ntransmission and X engine reception are combined on a single port to make use\nof the bi-directionality of 10GbE.  This optimization halves the size of the\nswitch needed.  Multiple X processors can be chained together from a single\n10GbE port using point-to-point connections.  For cases where the number of\nantennas does not evenly divide the number of frequency channels, one can adjust\npacket transmission to drop remainder channels so that the band may be equally\ndivided among X engines.\n\n\\placefigure{fig:ex_app2}\n\nA second example (Fig. \\ref{fig:ex_app2}) illustrates a case where the\nbandwidth from a single F engine exceeds the transmission capacity of a 10GbE\nlink.  Here, data can be split by frequency channel across two\nports.  Since different channels are never cross-multiplied, each of these\nlinks goes to a separate subnet of switched X engines.  Thus,\ntwo smaller (and often less expensive per port) switches may be \nsubstituted for one large\none.  Each X engine still receives the same bandwidth as in the previous\nexample, although this now represents a smaller fraction of the total\nbandwidth.  Note that the same X processor used in the first example functions\nhere without modification.  Only the number of X engines and the transmission\npattern has changed.\n\n\\placefigure{fig:ex_app3}\n\nA final example (Fig. \\ref{fig:ex_app3}) explores the case where the capacity\nof an X processor and a 10GbE link both exceed the data bandwidth.  In this\ncase, multiple F engines can (but do not have to) be chained together to\nminimize the number of switched ports.  As should be the case, only half as\nmany X engines (as compared to Fig. \\ref{fig:ex_app1}) are necessary for a\ngiven number of antennas.  X processors operate in the same configuration as\nbefore, oblivious to changes in F engines.\n\nThese examples highlight the flexibility of the hardware and gateware for\ntargeting a number of applications. One shortcoming they also illustrate is\nhow the cabling between components differs for different bandwidths.\nTherefore the different bandwidth operations are not as easily reconfigured as\nmight be desired for varying science goals on a given telescope. Research is\nongoing to improve the rapid reconfigurability that is an essential\nspecification for the most general radio interferometer array applications.\n\n\\section{Modular, FPGA-based Processing Hardware}\n\\label{sec:hardware}\n\nA flexible and scalable correlator architecture is of limited use without\nequally dynamic processing hardware that can support a variety of\nconfigurations.  FPGAs provide a unique combination of flexibility and\nperformance that make them well-suited for moderate-scale signal processing\napplications such as correlators and spectrometers \\citep{parsons_et_al2006}.\nA primary goal of the CASPER group has been development of\nmultipurpose processing modules that can be of general use to the astronomy\nsignal processing community, and beyond.  We seek to\nminimize the effort of redesigning and upgrading hardware by modularizing\nprocessing hardware, by minimizing the number of different modules \nin a system, and by employing industry-standard interconnection protocols.\n\nHardware modularity is the idea that boards should have consistent interfaces\nin order to be connectible with an arbitrary number of heterogeneous components\nto meet the computing needs of an application (``computing by the yard''), and\nthat upgrading\/revising a component does not change the way in which components\nare combined in the system.\nMinimization of hardware reproduction costs is often used to motivate the\ndesign of specialized hardware for large-scale correlators.  However, \nthe longer development times inherent to such solutions, and\nthe necessity of targeting specific components from the outset,\nsuggest that a modular solution, initiated nearer to the deployment date,\nwill employ newer technology that costs less and uses less\npower per operation.  The predicted economy of mass-producing\nspecially-designed hardware must be tempered by its expected devaluation\nby Moore's Law over the course of correlator development.  This devaluation\nmakes the argument that hardware modularity can reduce the overall system\ncost, even for large-scale systems, by reducing development time.\n\nIn current correlator systems, we rely on two\nCASPER FPGA-based processing boards; Internet Break-Out Boards (IBOBs) are\ngenerally used for implementing per-antenna F engine processing, and\nsecond-generation Berkeley Emulation Engines (BEE2s) implement X engine\nprocessing.  Work is progressing on a new board, the Reconfigurable Open\nArchitecture for Computing Hardware (ROACH), that will provide a single-board\nsolution to both F and X processing.\n\n\\placefigure{fig:ibobadcbee2}\n\nIBOBs (Fig. \\ref{fig:ibobadcbee2}) can interface to two\nAnalog-to-Digital Converter (ADC) boards, each capable of digitizing two\nstreams at 1 Gsamples\/sec or a single stream at 2 Gsamples\/sec using an Atmel\nAT84AD001B dual 8-bit ADC chip.  This data is processed by a Xilinx XC2VP50\nFPGA containing 232 18$\\times$18-bit multipliers, two PowerPC CPU cores, and\nover 53,000 logic cells.  Two ZBT SRAM chips provide 36 Mbits of extra\nbuffering, and two 10GbE-compatible CX4 connectors provide a standard interface\nfor connecting to other boards, switches, and computers.  A detailed discussion\nof ADC signal fidelity is presented in Section \\ref{sec:characterization}.\nWe are developing a second ADC board that allows four signal sampling at\n200 Msample\/sec. \n\nThe BEE2 board \\cite{chang_et_al2005} (Fig. \\ref{fig:ibobadcbee2}) was\noriginally designed for high-end reconfigurable computing applications such as\nASIC design, but has been conscripted for astronomy applications in a\ncollaboration between the BWRC\\footnote{Berkeley Wireless Research Center\nhttp:\/\/bwrc.eecs.berkeley.edu}, \nthe UC Berkeley Radio Astronomy Laboratory, and the UC Berkeley SETI group. \nThe 500\nGops\/sec of computational power in the BEE2\nis provided by 5 Xilinx XC2VP70 Virtex-II Pro\nFPGAs, each containing 328 multipliers, two PowerPC CPU cores capable of\nrunning Linux, and over 74,000 configurable logic cells.  Each FPGA connects to\n4 GB of DDR2-SDRAM, and four 10GbE-compatible CX4 connectors, and all FPGAs\nshare a 100-Mbps Ethernet port.  The size and connectivity of the\nBEE2 board make it suitable for implementing X engine processing in our\ncorrelator architecture.\n\nThe ROACH board is being developed in collaboration with MeerKAT and \nNRAO,\\footnote{The National Radio Astronomy\nObservatory (NRAO) is owned and operated by Associated Universities, Inc. with\nfunding from the National Science Foundation} \nand is scheduled for release in the third quarter of 2008.  It is intended as a\nreplacement for both IBOB and BEE2 boards.  A single Xilinx Virtex-5 XC5VSX95T\nFPGA containing 94,000 logic cells and 640 multiplier\/accumulators provides 400\nGops\/sec of processing power and is connected to a separate PowerPC 440EPx\nprocessor with a 1 GbE network connection.  The board contains 4 GB of DDR2\nDRAM and two 36Mbit QDR SRAMs, four 10GbE-compatible CX4 connectors, and two\ninterfaces that allow the use of the current ADC boards, or a new 3\nGsamples\/sec (6 Gsamples\/sec dual-board interleaved) ADC.  The scale, economy,\nand peripheral interfaces of this board will make it appropriate for both F and\nX engine processing, and will enable a single-board correlator architecture.\n\n\\placetable{tab:hardware_price}\n\n\\section{Gateware}\n\\label{sec:gateware}\n\nEfficient, customizable signal processing libraries are another important\ncomponent of a flexible and scalable correlator architecture.  Towards this\ngoal, our group has designed a set of open-source libraries\\footnote{Available\nat http:\/\/casper.berkeley.edu} for the Simulink\/Xilinx System Generator FPGA\nprogramming language.  These libraries abstract chip-specific components to\nprovide high-level interfaces targeting a wide variety of devices.  Signal\nprocessing blocks in these libraries are parametrized to scale up and down to\narbitrary sizes, and to have selectable bit widths, latencies, and scaling.\nThough the design principles of parametrization and scalability have added\ncomplexity to the initial design of these libraries, it dramatically enhances\ntheir applicability and potential for longevity as hardware evolves.  It also\ndecreases testing time by allowing developers to debug scale models of systems\nthat derive from the same parametrization code and are behaviorally similar to\nlarger systems.  In this section, we present several components of our\nlibraries vital to the design of flexible correlators.\n\n\\subsection{A Digital Down-Converter}\n\\label{sec:downconverter}\n\nThe rising speed of ADCs has enabled digitization to occur increasingly early\nin the antenna receiver chain.  We are thus replacing analog electronics\ncommonly known as intermediate frequency processor (gain, band definition)\nand baseband mixer (conversion to zero frequency and filtering).\nThere are numerous advantages to doing this.\nDigital mixing allows dynamically selecting an operating frequency within the\ndigitized band while ensuring perfect sine-cosine phasing in the local\noscillator (LO) mixing frequency. \nDigitizing a wider bandwidth than will be ultimately processed makes analog\nfiltering less critical; inexpensive filters with slow roll-offs can be\nused, and passband rippling can be corrected.  Finally, digital filtering\nallows flexibility and control in selecting passband shapes and adjusting fine\ndelays.  One can even split out several bands from the same signal.\nThe issue of quantization levels and other digital artifacts needs to be\ncarefully addressed.\n\nOur library provides a digital down-conversion core with a runtime-selectable\nmixing frequency.  Using a discretely sampled sine wave in an addressable\nlookup table, we can approximate nearly any mixing frequency by rounding a wide\naccumulation register (incremented every clock) to the nearest address in the\nlookup table.  Digital sine waves have an accuracy dictated by the number of\nbits used to represent a value; a lookup table need only have enough samples to\nachieve comparable accuracy.  The fact that the derivative of $\\sin(x)$ reaches \na maximum magnitude of 1 allows the sampling interval of a sine wave to be\nsimply equated to the accuracy of a coefficient over that time interval.\nAs a result, a lookup table only need be addressed with the same\nbit-width as the sample width to implement an arbitrary mixing frequency.\n\n\\placefigure{fig:ddc_passband}\n\nOur library also contains a decimating FIR filter.  Digital filters have\nadvantages over analog filters by being reprogrammable and by providing exact,\ncalculable passbands.  This filter is often used for suppressing harmonics of\nthe mixing frequency and for steepening the rolloff of cheaper analog filters,\nbut it has also been relied upon for implementing IF sub-band selection\ndigitally.  In practice, one must weigh the need for performance and\nflexibility against the cost of FPGA resources compared to analog filters.  As\nan example, the response of the FIR filter used in various correlator designs\nis shown in Figure \\ref{fig:ddc_passband}.  Since the exact shape \nof this filter can be calculated, it is possible to remove passband\nripple post-channelization because of the large dynamic range available in\noutput of our FFT core.\n\n\\subsection{A Polyphase Filter Bank Front-End}\n\\label{sec:pfb}\n\nThe Polyphase Filter Bank (PFB) \\citep{crochiere+rabiner1983, vaidyanathan1990}\nis an efficient implementation of a bank of evenly spaced, decimating FIR\nfilters.  The PFB algorithm decomposes these filters into a single polyphase\nconvolution followed by a DFT.  Since DFTs have been highly optimized\nalgorithmically, this results in an extremely efficient implementation.\nEquivalently, the PFB may be regarded as an improvement on the Fast Fourier\nTransform (FFT) that uses a front-end polyphase FIR filter to improve the\nfrequency response of each spectral channel (Fig. \\ref{fig:pfb_bin_resp}).\nThis improvement comes at the cost of buffering an additional window of samples\nand adding a complex cross-multiplication for each additional tap in the\npolyphase FIR.  This PFB implementation has seen widespread use in the astronomy\ncommunity in 21 cm hydrogen surveys \\citep{heiles_et_al2004}, pulsar surveys\n\\citep{demorest_et_al2004}, antenna arrays \\citep{bradley_et_al2005}, Very Long\nBaseline Interferometry, and other applications.\n\n\\placefigure{fig:pfb_bin_resp}\n\nOur core is parametrized to use selectable windowing functions, allowing\nadjustment of the out-of-band rejection and passband ripple\/rolloff.  Blackman\nand Tukey \\citep{blackman_tukey1958} provides a summary of the characteristics\nand trade-offs of various windows.  Each polyphase FIR tap, at the cost of\nincreased buffering and additional multipliers, increases filter steepness by\nadding samples (in increments of the number of channels) to the time window\nused in the PFB.  For fixed-point implementations, a practical upper limit to\nthe number of PFB taps is set by the number of bits used to represent filter\ncoefficients; the sinc function's 1\/x tapering ceases to be representable when\n$\\pi T > \\pi + 2^{B+1}$ where $T$ is the number of taps, and $B$ is the\ncoefficient bit width.  Finally, the width of a PFB channel is tunable by\nadjusting the period of the sinc function, forcing adjacent bandpass filters to\noverlap at a point other than the -3 dB point.  Note that this causes\npower to no longer be conserved in the Fourier transform operation.\n\n\\subsection{A Bandwidth-Agile Fast Fourier Transform}\n\\label{sec:fft}\n\nThe computational core of our FFT library is an implementation of a radix-2\nbiplex pipelined FFT \\citep{rabiner_gold1975} capable of analyzing two\nindependent, complex data streams using a fraction of the FPGA resources of\ncommercial designs \\citep{dick2000}.  This architecture takes advantage of the\nstreaming nature of ADC samples by multiplexing the butterfly computations of\neach FFT stage into a single physical butterfly core.  When used to analyze two\nindependent streams, every butterfly in this biplex core outputs valid data\nevery clock for 100\\% utilization efficiency.\n\nThe need to analyze bandwidths higher than the native clock rate of an FPGA led\nus to create a second core that combines multiple biplex cores with additional\nbutterfly cores to create an FFT that is parametrized to handle $2^P$ samples\nin parallel \\citep{parsons2008}.  This FFT architecture uses only 25\\% more\nbuffering than the theoretical minimum, and still achieves 100\\% butterfly\nutilization efficiency.  This feat is achieved by decomposing a $2^N$\nchannel FFT into $2^P$ parallel biplex FFTs of length $2^{N-P}$, followed by a\n$2^P$ channel parallel FFT core using time-multiplexed twiddle-factor\ncoefficients.\n\nFinally, we have written modules for performing two real FFTs with each half of\na biplex FFT using Hermitian conjugation.  Mirroring and\nconjugating the output spectra to reconstitute the negative frequencies, this\nmodule effects a 4-in-1 real biplex FFT that can then be substituted for the\nequivalent number of biplex cores in a high-bandwidth FFT.  Thus, our real FFT\nmodule has the same bandwidth flexibility as our standard complex FFT.\n\nDynamic range inside fixed-point FFTs requires careful consideration.  Tones\nare folded into half as many samples through each FFT stage, causing magnitudes\nto grow by a factor of 2 for narrow-band signals, and $\\sqrt{2}$ for random \nnoise.  To\navoid overflow and spectrum corruption, our cores contain optional downshifts\nat each stage.  In an interference-heavy environment, one must balance loss of\nSNR from downshifting signal levels against loss of integration time due to\noverflows.  A good practice is to place time-domain input into the\nmost-significant bits of the FFT and downshift as often as possible to\navoid overflow and minimize rounding error in each butterfly stage.  However,\nit is also best to avoid using the top 2 bits on input since the first \n2 butterfly\nstages can be implemented using negation instead of complex multiplies, but the\nasymmetric range of 2's complement arithmetic can allow this negation to\noverflow.\n\n\\subsection{A Cross-Multiplication\/Accumulation (X) Engine}\n\\label{sec:x_engine_arch}\n\n\\placefigure{fig:x_engine_schem}\n\nOur FX correlator architecture employs\nX engines to compute all antenna cross-multiples within a frequency\nchannel, and multiple frequencies are multiplexed into the core as dictated by\nprocessor bandwidth; the complex visibility $V_{ij}$ (Eq. \\ref{eq:vis})\nis the average of the product of complex voltage samples from antenna $i$ and\nantenna $j$ with the convention that the voltage $j>i$ is conjugated prior to\nforming product.\nIn collaboration with Lynn Urry of UC Berkeley's Radio\nAstronomy Lab we have implemented a parametrized module (Fig.\n\\ref{fig:x_engine_schem}) for computing and accumulating all visibilities for a\nspecified number of antennas.  An X engine operates by receiving $N_{ant}$ data\nblocks in series, each containing $T_{acc}$ data samples from one frequency\nchannel of one antenna.  The first samples of all blocks are\ncross-multiplied, and the $N_{ant}(N_{ant}+1)\/2$ results are added to the\nresults from the second samples, and so on, until all $T_{acc}$ samples have\nbeen exhausted.  Accumulation prevents the data rate out of a\ncross-multiplier from exceeding the input data rate.  An X engine is divided\ninto stages, each responsible for pairing two different data blocks\ntogether: the zeroth stage pairs adjacent blocks, the first stage pairs blocks\nseparated by one, and so on.  As the final accumulated results become available,\nthey are loaded onto a shift register and output from the X engine.\n\nHowever, as a new window of $N_{ant}\\times T_{acc}$ samples arrives, some\nstages, behaving as described above, would compute invalid results using\ndata from two different windows.  To avoid this, each stage switches between\ncross-multiplying separations of $S$ to separations of $N_{ant}-S$, which\nhappen to be valid precisely when separations of $S$ would be invalid.  As a\nresult, there need be only $floor({N_{ant}\/2}+1)$ stages in an X engine.  Every\n$T_{acc}$ samples, each stage outputs a valid result, yielding $N_{ant}\\times\nfloor({N_{ant}\/2}+1)$ total accumulations; for even values of $N_{ant}$,\n$N_{ant}\/2$ of the results from the last stage are redundant.\nAll other multiplier\/accumulators are 100\\% utilized.  Each stage\nalso computes all polarization cross-multiples (Eq. \\ref{eq:pol})\nusing parallel multipliers.\n\nWhen one X engine no longer fits on a single FPGA, it may be divided across\nchips at any stage boundary at the cost of a moderate amount of bidirectional\ninterconnect.  The output shift register need not be carried between chips;\neach FPGA can accumulate and store the results computed locally.  In order for\nthe output shift register's $floor({N_{ant}\/2}+1)$ stages to clear before the\nnext accumulation is ready, an X engine requires a minimum integration length\nof: $T_{acc}>floor({N_{ant}\/2}+1)$.  In current hardware, a practical upper\nlimit on $T_{acc}$ is set by the 2$\\times$4 Mbit of SRAM storage available on\nthe IBOB.  For 2048 channels with 4-bit samples, and double buffering for 2\nantennas, 2 polarizations, this limit is $T_{acc}\\le 128$.  Longer integration\nrequires an accumulator capable of buffering an entire vector of visibility\ndata, and typically occurs in off-chip DRAM.  The maximum theoretical\naccumulation length in correlator is determined by the fringe rate of sources\nmoving across the sky, and is a function of observing frequency, maximum\nantenna separation, and (for correlators with internal fringe rotation)\nfield-of-view across the primary beam.\n\nCross-multiplication comes to dominate the total correlator processing budget\nfor large numbers of antennas.  As a result, care must be taken both to reduce\nthe footprint of a complex multiplier\/accumulator and to make full and\nefficient use of the resources on an FPGA processor.  The number of bits used\nto carry a signal should be minimized while retaining sufficient dynamic range\nto distinguish signal from noise.  We have chosen to focus on 4-bit multipliers\nin current applications, and the subjects of dynamic equalization and Van Vleck\ncorrection generalized to 4 bits are explored in Section\n\\ref{sec:characterization} for optimizing signal-to-noise ratios (SNR) in our\ncorrelators.  To make full use of FPGA resources, we construct\n4-bit complex multipliers using distributed logic, dedicated multiplier cores, \nand look-up tables implemented in Block RAMs.  \n\nIt is possible to perform the bulk of an $N$-bit complex multiply in an $M$-bit\nmultiplier core by sign-extending numbers to $2N$ bits and combining them into\ntwo $M$-bit, unsigned numbers.  Multiplying $(a+bi)(c+di)$, these\nrepresentations are $(2^{M-2N}a_s+b_s)$ and $(2^{M-2N}c_s+d_s)$, where\n$n_s=2^{2N}+n$.  The bits corresponding to $ac, ad+bc, bd$ may be selected from\nthe product, provided that the\nsign-extension to $2N$ bits shifts $a+d$ beyond the bits occupied by $ad$.\nThis yields the constraint: \n\\begin{equation} 6N-1 < M \\end{equation} \nThe 18-bit multipliers in current Xilinx \nFPGAs can efficiently perform 3-bit complex\nmultiplies, but fall short of 4 bits.\n\n\\section{System Integration}\n\\label{sec:integration}\n\n\\subsection{F Engine Synchronization}\n\\label{sec:F_synch}\n\n\\placefigure{fig:corr_vs_dly}\n\nThough we have touted GALS design principles for X engine processing,\ndigitization and spectral processing within F engines must be synchronized to a\ntime interval much smaller than a spectral window to avoid severe degradation\nof correlation response (Fig. \\ref{fig:corr_vs_dly}).  This attenuation effect,\nresulting from the changing degree of overlap of correlated signals within a\nspectral window, can be caused by systematic signal delay between antennas, as\nwell as by source-dependent geometric delay; FX correlators with insufficient\nchannel resolution experience a narrowing of the field of view related to\nchannel bandwidth.  This effect has been well explored for FX correlators\nemploying DFTs (see Chapter 8 of \\citet{thompson_et_al2001}), but Polyphase\nFilter Banks show a different response owing to a weighting function that\nextends well beyond the number of samples used in a DFT. \nGiven a standard form for PFB sample weighting of\n${\\rm sinc}\\left(\\frac{\\pi t}{N\\tau_s}\\right)\nW\\left(\\frac{t}{2TN\\tau_s}\\right)$, \nwhere $N$ is the number of output channels,\n$T$ is the number of PFB taps, $\\tau_s$ is the delay between time-domain\nsamples, and $W$ is an arbitrary windowing function that tapers to 0 at\n$\\pm1$, the gain versus delay $G(\\tau)$ of a PFB-based FX correlator is\ngiven by:\n\\begin{displaymath}\nG(\\tau)=\\int_{-\\infty}^{\\infty}{\n\\left[{\\rm sinc}\\left(\\frac{\\pi t}{N\\tau_s}\\right)\nW\\left(\\frac{t}{2TN\\tau_s}\\right)\\right] \\times\n\\left[{\\rm sinc}\\left(\\frac{\\pi (t-\\tau)}{N\\tau_s}\\right)\nW\\left(\\frac{t-\\tau}{2TN\\tau_s}\\right)\\right]\\ dt\n}\n\\end{displaymath}\n\nFor the purpose of F Engine synchronization, we\nrely on a one-pulse-per-second (1PPS) signal with a fast edge-rate provided\nsynchronously to a bank of F processors running off identical system clocks.\nThis signal is sampled by the system clock on each processor, and provided\nalongside ADC data.  A slower, asynchronous ``arm'' signal is sent from\na central node to each F engine at the half second phase \nto indicate that the next 1PPS signal should be\nused to generate the reset event that synchronizes spectral windows and packet\ncounters.  This ensures that samples from different antennas entering X engines\ntogether were acquired within one or two system clocks of one another.  The\ndegree of synchronization is determined by the difference in path lengths of\n1PPS and the system clock from their generators to each F engine.  This path\nlength can be determined from celestial source observations\nusing self-calibration, and barring temperature\neffects, will be constant for a correlator configuration following power-up.\n\n\\subsection{Asynchronous, Packetized ``Corner Turner''}\n\\label{sec:packetization}\n\nThe choice of the accumulation length $T_{acc}$ in X engines \ndetermines the natural size of UDP packets in our\npacket-switched correlator architecture.  For current CASPER hardware where\nchannel-ordering occurs in IBOB SRAM, $T_{acc}$ is constrained by the available\nmemory to an upper limit of 128 samples for 2048-channel dual-polarization, \n4-bit,\ncomplex data, yielding a packet payload of 256 bytes.  A header containing\n2 bytes of antenna index and 6 bytes of frequency\/time index is added to each\npacket to enable packet unscrambling on the receive side.  The frequency\/time\nindex (hereafter referred to as the master counter, or MCNT) is a counter that\nis incremented every packet transmission.  The lower bits count frequencies\nwithin a spectrum, and the rest count time.  Combined with the antenna\nindex, MCNT completely determines the time, frequency, source, and destination\nof each packet; MCNT maps uniquely to a destination IP address.\n\n\\placefigure{fig:packet_rx}\n\nPacket reception (Fig. \\ref{fig:packet_rx}) is complicated by the realities of\npacket scrambling, loss, and interference.  A circular buffer holding $N_{win}$\nwindows worth of X engine data stores packet data as they arrive.  The lower\nbits of MCNT act as an address for placing payloads into the the correct\nwindow, and the antenna index addresses the position within that window.  When\ndata arrives $N_{win}\/2$ windows ahead of a buffered window, that window is\nflagged for readout, and is processed contiguously on the next window boundary\nof the free-running X engine.  Using packet arrival to determine when a window\nis processed allows a data-rate dependent time interval for all packets to\narrive, but pushes data through the buffer in the event of packet loss.  On\nreadout, the buffer is zeroed to ensure that packet loss results in loss of\nsignal, rather than the introduction of noise.  F engines can be intentionally\ndisconnected from transmission without compromising the correlation of\nthose remaining.\n\nPacket interference occurs when a well-formed packet contains an invalid MCNT\nas a result of switch latency, unsynchronized F engines, or system\nmisconfiguration.  Such packets must be prevented from entering the receive\nbuffer, since they can lead to data corruption; one would prefer that a\nmisconfigured F engine antenna result in data loss for that antenna, rather\nthan data loss for the entire system.  To ensure this behavior, incoming\npackets face a sliding filter based on currently active MCNTs.  Packets are\nonly accepted if their MCNT falls within the range of what can currently be\nheld in the circular buffer.  As higher MCNTs are received and accepted, old\nwindows are flagged for read out, freeing up buffer space for still\nhigher MCNTs.  This system forces MCNTs to advance by small increments and\nprevents the large discontinuities indicative of packet\ninterference.  In the eventuality that a receive buffer accidentally locks onto\nan invalid MCNT from the outset, a time-out clause causes the currently active\nMCNT to be abandoned for a new one if no new data is accepted into the receive\nbuffer.\n\nA final complication comes when implementing a bidirectional 10GbE transmission\narchitecture such as the one outlined in Figure \\ref{fig:ex_app1}.\nCommercial switches do not support\nself-addressed packet transmission; they assume that the transmitter\n(usually a CPU) intercepts these packets and transfers them to the receive\nbuffer.  On FPGAs, this requires an extra buffer for holding ``loopback'', and\na multiplexer for inserting these packets into the processing stream.  A simple\nmethod for this insertion would be to always insert loopback packets, if\navailable, and otherwise to insert packets from the 10GbE\ninterface.  However, there is a maximum interval over which packets with\nidentical MCNTs can be scrambled before the receive system rejects\npackets for being outside of its buffer.  This simple method has the\nundesirable effect of including switch latency in the time interval over which\npackets are scrambled, causing unnecessary packet loss.  Our solution is to\npull loopback packets only after packets with the same MCNT \narrive through the switch.\n\n\\subsection{Monitor, Control, and Data Acquisition}\n\\label{sec:data_aq}\n\nThe toolflow we have developed for CASPER hardware provides convenient\nabstractions for interfacing to hardware components such as ADCs, DRAM, and 10\nGbE transceivers, and allows specified registers and BRAMs to be automatically\nconnected to CPU-accessible buses.  On top of this framework, we run BORPH--an\nextension of the Linux operating system that provides kernel support for FPGA\nresources \\citep{so_broderson2006,so2007}.  This system allows FPGA\nconfigurations to be run in the same fashion as software processes, and creates\na virtual file system representing the memories and registers defined on the\nFPGA.  Every design compiled with this toolflow comes equipped with this\nreal-time interface for low- to moderate-bandwidth data I\/O.  By emulating\nstandard file-I\/O interfaces, BORPH allows programmers to use standard\nlanguages for writing control software.  The majority of the monitor, control,\nand data acquisition routines in our correlators are written in C\nand Python.  For 8-16 antenna correlators, the bandwidth through BORPH on a\nBEE2 board is sufficient to support the output of visibility data with 5-10s\nintegrations.\n\nFor correlators with more antennas or shorter integration times, the bandwidth\nof the CPU\/FPGA interface is incapable of maintaining the full correlator\noutput.  This limitation is being overcome by transmitting the final correlator\noutput using a small amount of the extra bandwidth on the 10GbE ports already\nattached to each X engine.  After accumulation in DRAM, correlator output is\nmultiplexed onto the 10GbE interface and transmitted to one or more Data\nAcquisition (DA) systems attached to the central 10GbE switch.  These systems\ncollect and store the final correlator output.  With a capable DA system, the\nadded bandwidth through this output pathway can be used to attain millisecond\nintegration times, opening up opportunities for exploring transient events and\nincreasing time resolution for removing interference-dominated data. \n\nThe capabilities of correlators made possible by our research are placing\nnew challenges on DA systems \\citep{wright2005}.  There is a severe (factor of\n100) mismatch between the data rates in the on-line correlator hardware and\nthose supported by the off-line processing.  Members of our team are currently\npursuing research on how this can be resolved both for correlators and for\ngeneric signal processing systems using commercially available compute\nclusters.  For correlators, our group is currently exploring how to implement\ncalibration and imaging in real-time to reduce the burden of\nexpert data reduction on the end user, and to make best use of both telescope\nand human resources.\n\n\n\\section{Characterization}\n\\label{sec:characterization}\n\n\\subsection{ADC Crosstalk}\n\\label{sec:crosstalk}\n\n\\placefigure{fig:crosstalk}\n\nCrosstalk is an undesirable but prevalent characteristic of analog systems\nwherein a signal is coupled at a low level into other pathways.  This can pose\na major threat to sensitivity in systems that integrate noise-dominated data to\nreveal low-level correlation.  For CASPER hardware, we have examined crosstalk\nlevels between signal inputs sharing an ADC chip, and between different ADC\nboards on the same IBOB.  Figure \\ref{fig:crosstalk} illustrates a one-hour\nintegration of uncorrelated noise of various bandwidths input to the ``Pocket\nCorrelator'' system (see Section \\ref{sec:deployments}).  Between inputs \nof the same ADC board, a coupling coefficient of $\\sim0.0016$ indicates\ncrosstalk at approximately $-28$ dB.  This coupling is a factor of $5$ higher\nthan the $-35$ dB isolation advertised by the Atmel ADC chip, and is most\nlikely the result of board geometry and shared power supplies.  Crosstalk\nbetween inputs on different ADCs also peaks at the $-28$ dB level, but shows\nmore frequency-dependent structure.\n\n\\placefigure{fig:crosstalk_stability}\n\nCrosstalk may be characterized and removed, provided that its timescale for\nvariation is much longer than the calibration interval.  Figure\n\\ref{fig:crosstalk_stability} demonstrates that for integration intervals\nranging from 7.15 seconds to approximately 1 day (the limit of our testing),\ncrosstalk amplitudes and phases vary around stable values in a\nlab test that, when\nsubtracted, yield noise that integrates down with time.  Even\nthough crosstalk is encountered at the $-28$ dB level, its stability allows\nsuppression to at least $-62$ dB.  This stability has allowed crosstalk\nto be removed post-correlation, and we have until recently deferred\nadding phase switching.  Developments along this line are proceeding by\nintroducing an invertible mixer (controlled via a Walsh counter on an IBOB)\nearly in the analog signal path, and removing this inversion after\ndigitization.  Phase switching must be coupled with data blanking near \nboundaries when the\ninversion state is uncertain.  Blanking will be most easily implemented by\nintentionally dropping packets of data from F engine transmission, and by\nproviding a count of results accumulated in each integration for normalization\npurposes.\n\n\\subsection{XAUI Fidelity and Switch Throughput}\n\\label{sec:10gbe_sw}\n\nCASPER boards are currently configured to transmit XAUI protocol over CX4 ports\nas a point-to-point communication protocol and as the physical layer of 10GbE\ntransmission.  Because the Virtex-II FPGAs used in current CASPER hardware do\nnot fully support XAUI transmission standards \\cite{xilinx_ug024,xilinx_ds083}, \ncurrent devices can have\nsub-optimal performance for certain cable lengths.  We expect the new ROACH\nboard, which employs Virtex-5 FPGAs, to have better\nperformance in this regard.  For cable lengths supported in current hardware,\nwe tested XAUI transmission fidelity using matched Linear Feedback Shift\nRegisters (LFSRs) on transmit and receive.  Error detection was verified using\nprogrammable bit-flips following transmitting LFSRs.  Over a period of 16\nhours, 573 Tb of data were transmitted and received on each of 8 XAUI\nlinks.  During this time, no errors were detected, resulting in an estimated\nbit-error rate of $2.2\\cdot 10^{-16}$ Hz.  We also tested the capability of two\nFujitsu switches (the XG700 and the XG2000) for performing the full\ncross-connect packet switching required in our FX correlator architecture.  By\ntuning the sample rate inside F engines of an 8-antenna (4-IBOB) packetized\ncorrelator, we controlled the transmission rate per switch port over a range of\n5.96 to 8.94 Gb\/s.  In 10-minute tests, packet loss was zero for both\nswitches in all but the 8.94 Gb\/s case.  Packet loss in this final case was\ntraced to intermittent XAUI failure as a result of imperfect compliance with\nthe XAUI standard, as described previously. Overheating of FPGA chips in the\nfield has also been reported as a source of intermittent operation.\n\n\\subsection{Equalization and 4-Bit Requantization}\n\\label{sec:equalization}\n\n\\placefigure{fig:4_bit_quant}\n\nCorrelator processing resources can be reduced by limiting the bit width of\nfrequency-domain antenna data before cross-multiplication.  However, digital\nquantization requires careful setting of signal levels for optimum\nSNR and subsequent calibration to a linear power scale \n\\citep{thompson_et_al2001,jenet_anderson1998}.  Correlators using 4 bits \nrepresent\nan improvement over their 1 and 2 bit predecessors, but there are still\nquantization issues to consider.  The total power of a 4-bit quantizer has a\nnon-linear response with respect to input level as shown in Figure\n\\ref{fig:4_bit_quant}.  In currently deployed correlators, we perform\nequalization (per channel scaling) to control the RMS channel values before\nrequantizing from 18 bits to 4 bits.  This operation saturates RFI and flattens\nthe passband to reduce dynamic range and to hold the passband in\nthe linear regime of the 4-bit quantization power curve.  Equalization is\nimplemented as a scalar multiplication on the output of each PFB using 18-bit\ncoefficients from a dynamically updateable memory.  These coefficients allow\nfor automatic gain control to maintain quantization fidelity through changing\nsystem temperatures.\n\n\\section{Deployments and Results}\n\\label{sec:deployments}\n\n\\subsection{A Pocket Correlator}\n\\label{sec:pocket_corr}\n\n\\placefigure{fig:f_engine}\n\nThe ``Pocket Correlator'' (Fig. \\ref{fig:f_engine}) is a single IBOB system\nthat includes F and X engines on a single board for correlating and\naccumulating 4 input signals.  Each input is sampled at 4 times the FPGA clock\nrate (which runs up to 250 MHz), and a down-converter extracts half of the\ndigitized band.  This subband is decomposed into 2048 channels by an 8-tap PFB,\nequalized, and requantized to 4 bits.  With all input signals on one chip, X\nprocessing can be implemented directly as multipliers and vector accumulators,\nrather than as X engines.  Limited buffer space on the IBOB permits only 1024\nchannels (selectable from within the 2048) to be accumulated.  Output occurs\neither via serial connection (with a minimum integration time of 5\nseconds) or via 100-Mbit UDP transmission (with a minimum integration time in\nthe millisecond range).  This system can act as a 2-antenna, full Stokes\ncorrelator, or as a 4-antenna single polarization correlator.\n\n\\placefigure{fig:skymap}\n\nThe Pocket Correlator is valuable as a simple, stand-alone instrument, and for\nboard verification in larger packetized systems.  It is being applied as a\nstand-alone instrument in PAPER, the ATA, and the UNC PARI observatory. A\n4-antenna, single polarization deployment of the PAPER experiment in Western\nAustralia in 2007 used the Pocket Correlator to collect the data used to\nproduce a 150 MHz all-sky map illustrated in Figure \\ref{fig:skymap}.  In\naddition to demonstrating the feasibility of post-correlation crosstalk\nremoval, this map (specifically, the imperfectly removed sidelobes of sources)\nillustrates a problem that will require real-time imaging to solve for large\nnumbers of antennas.\n\n\\subsection{An 8-Antenna, 2-Stokes, Synchronous Correlator}\n\\label{sec:8_ant_corr}\n\nThis first generation multi-board correlator demonstrated the functionality\nof signal processing algorithms and CASPER hardware, but preempted the\ncurrent packetized architecture--it operated synchronously.  This version of\nthe correlator was most heavily limited by X engine resources, all of which\nwere implemented on a single FPGA to simplify interconnection. The\ntotal number of complex multipliers in the X engines of an $N_{ant}$ antenna\narray is: $N_{cmac} = floor({N_{ant}\/2}+1)\\times N_{ant}\\times N_{pol}$; the\nlimited number of multipliers on a BEE2 FPGA only allowed for supporting half\nthe polarization cross-multiples.  This system was an\nimportant demonstration of the basic capabilities of our hardware and software,\nand provided a starting point for evolving a more sophisticated system.  \nDeployments of this\nsystem at the NRAO site in Green Bank as part of the PAPER\nexperiment, and briefly\nat the Hat Creek Radio Observatory for the Allen Telescope Array,\nare being supersede by the packetized correlator presented in the next\nsection.\n\n\\subsection{A 16-Antenna, Full-Stokes, Packetized Correlator}\n\\label{sec:packet_deploy}\n\nThis packetized FX correlator is a realization of the architecture outlined in\nFigure \\ref{fig:ex_app1}, with F processing for 2 antennas implemented on each\nIBOB, and matching X processors implemented on each corner FPGA of two BEE2s.\nEach F processor is identical to a Pocket Correlator (Fig. \\ref{fig:f_engine}),\nbut branches data from the equalization module to a matrix transposer in IBOB\nSRAM to form frequency-based packets.  Packet data for each antenna are\nmultiplexed through a point-to-point XAUI connection to a BEE2-based X\nprocessor, and then relayed in 10GbE format to the switch.  The number of\nchannels in this system is limited to 2048 by memory in IBOB SRAM for\ntransposing the 128 spectra needed to meet bandwidth restrictions between X\nengines and DRAM-based vector accumulators.\n\n\\placefigure{fig:x_processor}\n\nThe X processor in this packetized correlator implements the transmit and\nreceive architecture illustrated in Figure \\ref{fig:x_processor} \nfor two X engines sharing the same 10GbE link.\nEach X engine's data processing rate is\ndetermined by packets arriving in its own receive buffer, and results are\naccumulated in separate DRAM DIMMs.  The accumulated output of each X processor\nis read out of DRAM at a low bandwidth and transmitted via 10GbE packets to\na CPU-based server where\nall visibility data is collected and\nwritten to disk in MIRIAD format\n\\citep{sault_et_al1995} using interfaces from the Astronomical Interferometry\nin PYthon (AIPY) package\\footnote{http:\/\/pypi.python.org\/pypi\/aipy}.\n\nThe clocks for the BEE2 FPGAs are asynchronous 200-MHz oscillators, and IBOBs\nrun synchronously at any rate lower than this.  Packet transmission is\nstatically addressed so that all each X engine processes every 16th channel.\nWe use 8 ports of a Fujitsu XG700 switch to route data.  This system is is\nscalable to 32 antennas before two X engines no longer fit on a single FPGA.\nFor larger systems, the number of BEE2s will scale as the square of the number\nof antennas, and the number of IBOBs will scale linearly.  A 32-antenna,\n200-MHz correlator on 16 IBOBs and 4 BEE2s is now working in the lab, and a\n16-antenna version using 8 IBOBs and 2 BEE2s has been deployed to the NRAO site\nin Green Bank with the PAPER experiment.  \n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nBy decreasing the time and engineering costs of building and upgrading\ncorrelators, we aim to reduce the total cost of correlators for a wide range of\nscales.  Small- and medium-scale correlators with total cost dominated by\ndevelopment clearly stand to benefit from our research.  It is less clear if\nthe cost of large-scale correlators can be reduced by the general-purpose\nhardware used in our architecture.  Though minimization of replication cost\nfavors the development of specialized parts, there are two factors\nthat can make a generic, modular solution cost less.\n\nThe first factor to consider is time to deployment.  Even if the monetary cost\nof development is negligible in the budget of a large correlator, the cost of\ndevelopment time can be significant.  If a custom solution takes several years\nto go from design to implementation, the hardware that is deployed will be out\nof date.  Moore's Law suggests that when a custom solution taking 3 years to\ndevelop is deployed, there will exist processors 4 times more powerful, or 4\ntimes less expensive for the equivalent system.  The cost of a generic, modular\nsystem has to be tempered by the expected savings of committing to hardware\ncloser to the ultimate deployment date.\n\nThe second factor is the cost of upgrade.  Many facilities (including the ATA)\nare beginning to appreciate the advantages of designing arrays with wider\nbandwidths and larger numbers of antennas than can be handled by current\ntechnology.  Correlators may then be implemented inexpensively on scales\nsuited to current processors, and upgraded as more powerful processors\nbecome available.  Modular solutions facilitate this methodology.\n\n\n\\acknowledgments\n\nThis and other CASPER research are supported by the National Science Foundation\nGrant No. 0619596 for Low Cost, Rapid Development Instrumentation for Radio\nTelescopes.  We would like to acknowledge the students, faculty and sponsors of\nthe Berkeley Wireless Research Center, and the National Science Foundation\nInfrastructure Grant No.  0403427.  Correlator development for the PAPER\nproject is supported by NSF grant AST-0505354, and for the ATA project by NSF\ngrant AST-0321309 as well as the Paul G. Allen Foundation.  Chips and software\nwere generously provided by Xilinx, Inc.  JM and PM gratefully acknowledge\nfinancial support from the MeerKAT project and South Africa's National Research\nFoundation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAn accurate and scalable multi-target tracking solution is a critical component of many wide-area urban surveillance systems.\nFor example, human and vehicle detection with closed-circuit television (CCTV) networks leverages multiple bearing-only sensors to uniquely track targets throughout a city \\cite{Chen2015,Hou2016,Anuj2017,Yang2009}.\nAnother important area involves tracking unauthorized unmanned aerial vehicles (UAVs) using heterogeneous and spatially distributed sensors \\cite{Poullin2018,Jovanoska2018,Ganti2016}. \nFor commercial UAVs that stream video and telemetry data, passive RF detection mechanisms have also been suggested \\cite{Watson2017,Watson2016,Scerri2007}.\nAcross all of these applications, the deployment and positioning of the sensors over time has a major impact on multi-target tracking performance. \nThis is especially true when tracking with passive sensors, which requires fusing multiple sensors to unambiguously resolve target position and velocity.\nExamples of these passive sensor types include received signal strength indicator (RSSI), time difference of arrival (TDOA), frequency difference of arrival (FDOA), and angle of arrival (AOA).\n\nSensor deployment and path planning for multi-target tracking falls under the broad research area of \\emph{sensor control}.\nSensor control began receiving considerable attention from the information fusion community in the late 1990s \\cite{Hero2011, Ng2000}.\nMany of the techniques in the area initially focused on dynamic reconfiguration of individual sensors in order to maintain strong target tracking performance (e.g., beam scheduling \\cite{Krishnamurthy2001}, waveform selection \\cite{Kershaw1994,Sira2009}). \nIn the early 2000s, however, the focus shifted to include sensor selection for wireless sensor network (WSN) applications \\cite{Ramya2012, Cao2016}.\nThe size, weight, and power (SWaP) requirements of these systems necessitated sensor control techniques that could balance tracking performance with the energy cost of obtaining and communicating sensor measurements across the network \\cite{Guo2004,Zuo2008,Masazade2013,Niu2018}. \nDecentralized multi-target tracking techniques were proposed to maintain communication bandwidth scalability and resilience to sensor failure \\cite{Hlinka2013,Hlinka2014,Meyer2018,Msechu2008,Ribeiro2006,Zuo2010}. \nThe majority of WSN applications focused on stationary installations, allowing \\emph{offline} solutions to the problem of sensor deployment optimization.\nA number of WSN deployment optimization solvers were proposed by drawing analogies to the NP-hard art gallery problem from computational geometry \\cite{Efrat2005}. \nMeta heuristics formed the basis for many of these solvers, including techniques such as particle swarm optimization and genetic algorithms \\cite{Bojkovic2008,Kulkarni2011,Perez2015,Hu2008}.\nThe sensor control problem for \\emph{online} path planning in the context of multi-target tracking, however, is significantly more challenging and less studied.\nAs opposed to existing offline techniques for path planning in mobile sensor networks \\cite{Singh2009}, the multi-target tracking variation of the problem necessitates an online solution due to the lack of \\emph{a priori} information on target trajectories.\n\nVery few online mobile sensor control techniques in the current state-of-the-art are capable of addressing the unique challenges associated with non-cooperative target surveillance in urban environments.\nThis is primarily because the urban environment highly constrains sensor coverage and maneuverability based on terrain elevation and building geometries. \nThe same shadowing issues that make target sensing difficult also introduce challenges in maintaining end-to-end network connectivity, thus rendering centralized fusion approaches impractical.\nStrong performance and safe operation of a mobile sensor network in this scenario requires an understanding of how the terrain impacts the relevant tracking and sensor control algorithms, all while maintaining decentralized operation.\n\nThe goal of this paper is to provide a brief summary of the current multi-target multi-sensor tracking approaches using a mobile sensor network.\nWe use this summary to highlight the main limitations that prevent immediate application of these architectures to the urban environment.\nIn the sections that follow, we briefly summarize the model generally assumed for the mobile sensor network control problem. \nFollowing this, we provide a brief literature review of the current-state-of-the art in multi-target tracking with mobile sensor networks in non-urban environments.\nWe then conclude by discussing three open challenges related to urban surveillance with commercial-off-the-shelf (COTS) UAVs, or more specifically, quadcopters.\n\n\n\\section{Problem Overview}\n\\subsection{Integrated Sensing and Control Architecture}\nFigure~\\ref{fig:sensingArch} shows a typical architecture used for decentralized target tracking and mobile sensor control for a single platform. \nA sensor interface provides derived target measurements, such as time of arrival, Doppler shift, TDOA\/FDOA, RSSI, or AOA.\nThe measurements are usually obtained under measurement origin uncertainty.\nThat is, it is not known \\emph{a priori} which sensor measurements correspond to clutter and which to existing targets.\nIn addition, measurements obtained from targets may be miss-detected at a given time step.\nThe posterior distributions for each target from the previous time step are propagated forward in time under known target birth and survival dynamics.\nThe mechanism for performing this forward prediction is usually a variant of the Chapman-Kolmogorov integral \\cite{Sheldon2014,Chen2003}.\nA data association process uses these predicted posterior distributions to generate a mapping from the measurements to newborn and persisting targets.\nThe association map, sensor measurements, and platform telemetry are then used to apply the Bayes update for each target's predicted posterior distribution.\nIf the tracker update step is decentralized, a consensus process is used to jointly process the sensor log likelihood messages over the network with one-hop neighbors.\nThe updated posterior distribution per target is used to perform state extraction, which generates state estimates and covariance ellipses.\nThe sensor control policy finally uses the updated posterior distribution to determine which platform control actions (e.g., heading, acceleration, or waypoint) to use for the next time step.\nA separate consensus procedure may also occur to synchronize agent control actions.\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=0.75\\textwidth]{fig1.png}}\n\\caption{Typical estimation and control architecture used for multi-target tracking. Network communication interfaces shown for decentralized operation. Interfaces shown as gray boxes. Algorithms shown as blue boxes.}\n\\label{fig:sensingArch}\n\\end{figure*}\n\n\n\\subsection{POMDP Formulation for Mobile Sensor Control}\nControl of mobile sensor networks for multi-target tracking typically follows a \\emph{partially observable Markov decision process} (POMDP) formulation \\cite[Chapter 5.6]{Bertsekas2012}.\nIn a POMDP, the target states (i.e., position and velocities) evolve according to a Markov process and are observed indirectly through sensor measurements.\nSensor states also evolve according to a Markov process based on the control action applied at the current time step.\nDepending on the inertial sensor and kinematic models, the corresponding relationship between platform state dynamics and control may be deterministic or stochastic with directly or partially observable states.\nThe relationship between the sensor measurements and the target and platform states at the current time step is given by a set of likelihood functions.\nThe reward function is designed to capture target tracking goals (e.g., minimized cumulative uncertainty in state estimates), obstacle and inter-agent collision avoidance requirements, and constraints on platform control actions.\nGiven this reward function defined over target states, platform states, and platform actions, the goal is to construct a closed-loop control policy that maximizes the infinite horizon expected cost-to-go (i.e., discounted cumulative reward).\nThe information available for a control policy at the current time step is the measurement history of all target and platform states and the control inputs used at each platform.\nFor simplicity, the following discussion will assume that the sensor platform states are completely observable.\n\nTo prevent the growth of the control and state space dimensionality as new measurements are obtained, the POMDP model is typically reformulated into an equivalent Markov decision process (MDP).\nThis is accomplished using a sufficient statistic that subsumes the measurements up until the current time step \\cite[Chapter 4.3]{Bertsekas2017}.\nThe corresponding sufficient statistic is termed the \\emph{belief state} of the system.\nThe belief state for the sensor control application here is the posterior probability of the target states given the observed measurements up until the current time step.\nIn general, the belief state per target is estimated using application-specific variations of the recursive Bayes filter \\cite{Chen2003}.\nFor the multi-target case, extensions exist for the joint target probability under soft associations \\cite{Barshalom1988} or for multi-target probability distribution under the random finite sets (RFS) formalism \\cite{Mahler2014}.\nThe sensor control reward function at each time step is then mapped to the belief states through an information theoretic measure of the quality of the current target state estimate.\nThe most commonly used measure is the \\emph{mutual information} \\cite{Cover2006} between the future states per target and the predicted sensor measurements obtained over a finite horizon lookahead window.\nEach plausible control action affects the locations of the platforms at future time steps, which in turn affects the posterior belief state for each target.\nThe core idea is that the mutual information metric quantifies how the the sharpness of the belief state per target changes over a finite horizon lookahead under each control action.\n\nDespite this simplification, the resulting \\emph{belief MDP} state space is a subset of the space of multivariate probability distributions.\nThese belief states are always continuous, even if the partially observable states are discrete.\nAs a result, very few closed-form optimal policies for belief MDPs exist.\nThe most well known solution is for the case of a linear Gaussian POMDP with quadratic cost. \nHere, the optimal solution reduces to a Kalman update of the belief state, and the control solution results from solving the discrete-time algebraic Riccati equation \\cite{Georgiou2013,Aastrom2012}.\nIn all other cases, the policies must be determined using approximate online dynamic programming techniques for infinite dimensional state spaces, such as model predictive control (MPC) or stochastic rollout \\cite[Chapter 6.4-5]{Bertsekas2017}.\nFor these techniques, achieving real-time implementation of the control policy involves an application-specific treatment of computational complexity.\n\n\n\n\\section{Related Work in Non-Urban Environments}\n\\subsection{Myopic Control of Mobile Sensors}\nRistic and Vo in \\cite{Ristic2010} used the RFS formalism \\cite{Mahler2014} to derive a myopic sensor control policy for a single integrator (i.e., velocity controlled) plant. \nThe tracking algorithm was a particle filter approximation of the multi-target Bayes filter.\nThis controller used the Renyi divergence between the predicted and the future expected multi-object posterior after obtaining measurements from range-only mobile sensors.\nRistic et al. provided a similar myopic scheme in \\cite{Ristic2011} for range-only tracking, but specialized the multi-object Renyi divergence of \\cite{Ristic2010} to the more computationally tractable probability hypothesis density (PHD) filter \\cite{Vo2005}.\n\nGostar et al. in \\cite{Gostar2017} leveraged the Cauchy-Schwarz divergence for Poisson point processes \\cite{Hoang2015} to derive myopic sensor control policies for a sequential Monte Carlo implementation of the labeled multi-Bernoulli (LMB) filter.\nTo maximize computational efficiency, the Cauchy-Schwarz divergence between the PHDs of the LMB filter's predict and update steps (per control action) was used as the reward function.\nThis reward was efficiently realized by evaluating the difference between each predicted and updated particle systems' weights, scaled by each target's probability of existence.\nA similar approach was applied by Gostar et al. to the cardinality-balanced multi-target multi-Bernoulli (CB-MeMBer) filter in \\cite{Gostar2016}.\nTo further accelerate computation, both \\cite{Gostar2017} and \\cite{Gostar2016} applied certainty equivalent (i.e., noiseless) measurement models when predicting future filter states per control action.\n\nKoohifar et al. in \\cite{Koohifar2018} provided a single sensor myopic control policy based on the steepest descent direction of the predicted posterior Cramer-Rao lower bound (PCRLB). \nThis policy generalized their previous work in \\cite{Koohifar2017} by deriving the sensor likelihoods for an RSSI-only measurement.\nThe RSSI measurement likelihood further modeled packet transmission statistics via a Bernoulli process.\nThe plant model was assumed velocity controllable, where the heading at each control step was chosen from a fixed quantized set.\n\nHoffman and Tomlin in \\cite{Hoffman2010} leveraged a particle filtering solution and distributed myopic control policy for a constrained double integrator (i.e., acceleration controlled) plant using bearing-only or range-only measurements. \nThe plant was designed to model the STARMAC quadcopter \\cite{Hoffmann2007} moving at slow speeds. \nThe reward function was the mutual information between the predicted future target states and measurements.\nTo maintain computational tractability, mutual information was evaluated using either local contributions per node, or limited pair-wise contributions between nodes.\n\nDames and Kumar in \\cite{Dames2013} leveraged the PHD filter to construct a distributed sensor control scheme for indoor unmanned ground vehicles (UGV).\nIn contrast to \\cite{Hoffman2010}, the policy used the mutual information between the predicted target states and the binary event of an agent observing an empty measurement set at subsequent time steps. \nDecentralized estimation and control was achieved by transitioning sensors through operating modes, and organizing them into smaller sub-groups where their PHD states were directly synchronized.\n\nMeyer et al. in \\cite{Meyer2015} proposed a myopic gradient descent algorithm for a class of decentralized multi-target tracking algorithms based on loopy belief propagation (BP). \nThe cost function used was the conditional entropy of the target states at the next time step given the expected sensor measurements at the next time step. \nThe relevant BP messages and conditional entropy gradient were approximated using multiple particle systems under perfect knowledge of the number of targets and the target-to-measurement association.\nAlthough the simulations presented in \\cite{Meyer2015} were for single integrator dynamics, the corresponding technique is general enough to accommodate non-linear sensor and target dynamic models.\n\nChung et al. in \\cite{Chung2006} proposed a decentralized myopic control strategy based on memoryless, zero cross-covariance, track-to-track fusion. \nThe individual sensors provided range-bearing measurements which were fused decentrally using the Kalman equations.\nThe reward function was the determinant of the fused covariance matrix, which is directly related to the entropy of the fused target state estimate.\nThe reward gradient was derived, and consisted of a sum of per-sensor reward gradients.\nAs a result, the control policy was the same for each sensor based on its state, sensor model, and fused target covariance estimates.\nA similar controller was derived for the case where imperfect communications contribute to additional errors in the fused estimates.\n\n\\subsection{Non-myopic Control of Mobile Sensors}\nBeard et al. in \\cite{Beard2017} used the generalized labeled multi-Bernoulli (GLMB) filter to apply the Cauchy-Schwarz divergence for Poisson point processes \\cite{Hoang2015} as the sensor control reward function. \nThe authors also proposed the use of RFS void probabilities to achieve collision avoidance with targets.\nAn example of controlling of a single range-bearing sensor tracking multiple targets under measurement origin uncertainty was presented.\nA finite horizon controller was simulated that assumed a constant velocity plant with instantaneously controllable heading.\nClosed form equations of the Cauchy-Schwarz divergence for the case when the GLMB single object posterior densities are modeled as Gaussian mixtures were provided.\nThe derivations in \\cite{Beard2017} for the GLMB can also be applied to the LMB filter as a special case, but exhibit higher complexity than those described by Gostar in \\cite{Gostar2016,Gostar2017}.\nImplementation details of these control techniques, including pseudo-code, can be found in \\cite{Beard2016}\n\nDames and Kumar in \\cite{Dames2014} demonstrated a non-myopic tracking and control solution on real UGVs.\nThe tracking and control algorithms included a particle filter implementation of the PHD filter and an online estimate of mutual information. \nSimilar to \\cite{Dames2013}, the non-myopic policy was achieved by evaluating the mutual information against the potential of observing empty measurement sets.\nReceding horizon control was achieved through a combination of efficient action-set generation and adaptive sequential information planning \\cite{Charrow2014}.\n\nAtanasov et al. in \\cite{Atanasov2014} proposed a reduced value iteration (RVI) algorithm demonstrated on a target-linearized range-bearing measurement  model for a single sensor. \nAn important distinction was that the relationship between platform state and the observed measurements remained non-linear.\nThe specific technique did not require linearized sensor platform dynamics, and as such, it was demonstrated in simulation for a single target tracking scenario under differential drive dynamics.\nThis RVI algorithm was later generalized by Schlotfeldt et al. in \\cite{Schlotfeldt2018}  to an anytime planning algorithm.\nThe resulting technique, denoted Anytime-RVI (ARVI) was decentralized and tested on a set of quadcopters attempting to localize ground-based robots using range and bearing estimates.\n\nRagi and Chong in \\cite{Ragi2013} assumed linear-Gaussian state and measurement dynamics and applied a joint-probabilistic data association (JPDA) tracker \\cite{Barshalom1988}.\nThe sensor control technique used in this paper is known as nominal belief-state optimization (NBO) \\cite{Miller2009}.\nNBO is a POMDP approach that assumes the associated belief-states (i.e., per target posteriors) are completely characterized by a normal distribution (presumably through a Kalman update).\nA certainty-equivalent principle was applied to remove the expectation across belief states.\nBoth single and multi-step lookahead rollout approaches were provided.\nThe approach in \\cite{Ragi2013} additionally considered forward acceleration thrust and heading dynamics for the platform under wind force disturbances.\nInter-agent collision and obstacle avoidance constraints were considered by including a scaled regularization parameter in the cost-to-go function.\n\nGrocholsky et al. in \\cite{Grocholsky2003} assumed a fixed wing aircraft with constant forward velocity and controllable yaw rate to implement a decentralized control rule for bearing-only sensors.\nDecentralized data fusion was achieved by leveraging the information form of the Kalman filter \\cite{Manyika1994}.\nThe control law used the expected mutual information gain of the information matrix at the beginning and end of a finite lookahead window.\nThis law was made computationally feasible by linearizing the measurement and sensor state evolution dynamics and solving the resulting linear-quadratic-Gaussian (LQG) optimal control problem. \n\n\n\\section{Challenges Specific to Urban Surveillance}\n\\subsection{Terrain-aware Tracking and Sensor Control}\\label{sec:terrain}\nThe primary sensor control challenge in urban surveillance involves understanding how the terrain and building geometries affect tracking performance. \nIn a camera based solution, for example, the observed measurements are AOAs where the probability of detection is dependent on the platform's ability to maintain line-of-sight (LOS) on targets. \nA similar argument can be used for passive RF observations from low power transmitters, where the detectability of multi-path effects is negligible\\footnote{Examples of localization in multi-path rich environments have been proposed based off of pattern recognition techniques \\cite{Tsalolikhin2011}. These are outside of the scope of this review.}.\nIf targets maneuver into a non-line-of-sight (NLOS) region to all sensors, the uncertainty on the target's position and velocity increases due to the lack of measurement updates.\nThus, platform maneuvers that keep as many targets within LOS to their corresponding sensors will lead to an increase in the mutual information between predicted future target states and measurements.\nConsequently, the number of sensors that have LOS to a given target and their sensing geometry in the LOS region is also important.\n\nThere are a number of related studies that provide tracking functionality for targets constrained to road networks.\nUlmke and Koch in \\cite{Ulmke2006} describe a particle filtering technique for tracking a single target maneuvering on partially obstructed road networks.\nThe authors showed that improved tracking performance results when conditioning the measurement detection process on LOS\/NLOS information.\nUlmke et al. extended their work in \\cite{Ulmke2006} to the RFS formalism in \\cite{Ulmke2010} using the Gaussian-mixture cardinalized PHD filter \\cite{Vo2007}.\nWithin these efforts, the detectable regions were constrained by the road network as observed from an overhead sensor.\nIn the general urban surveillance case, the sensors may not necessarily be overhead.\nFurthermore, many practical target types will not be constrained to road networks (e.g., unauthorized UAVs).\n\nThe conditioning of the sensor control policy on LOS\/NLOS sensing regions as described above necessitates a non-parametric approximation of the belief state per target. \nAcross all applications, these approximation techniques are computationally complex and make implementing non-myopic policies at high update rates very challenging. \nA number of point-based value iteration (PBVI) approaches \\cite{Kurniawati2008,Smith2005,Kochenderfer2015} have been proposed to solve loosely related target surveillance problems \\cite{Egorov2016,Hsu2008,He2010}.\nThis computational complexity is made worse by the requirement to perform moderate to high fidelity ray-casting under each sensor action to identify the LOS\/NLOS regions.\nIn order to maintain the computational complexity of the PBVI approaches, these shadowing computations necessitate some form of GPU-based acceleration from the computational geometry literature \\cite{Tomczak2012}. \n\nAnother important consideration is the incorporation of safety-guaranteed operation with respect to inter-agent and obstacle collision avoidance. \nSome studies such as \\cite{Ragi2013} attempt to address the collision avoidance constraints for sensor control by directly penalizing the reward function estimate when targets are too close to other agents or other obstacles. \nA central issue, however, is how the safety constraint penalization term should be weighted when estimating the discounted cost-to-go.\nA regularization weight that is too small may not be capable of preventing a collision under the assumed dynamics of the platform. \nConversely, a penalization that is too large may over constrain the system and unnecessarily degrade the optimality of the policy. \nA better solution would be to select a POMDP solver that is capable of guaranteeing satisfiability of the safety constraints given an accurate map of the environment and agent positions. \nMinimum-norm controllers that modify the planned action from sensor control using safety barrier functions are one option \\cite{Ames2016,Freeman1996}, but can potentially over-compensate for safety when the optimization reward is not quantifiable by a control Lyapunov function. \nComputationally tractable implementations of this technique also require a plant model that is affine in the control actions. \nPath planning and graph traversal techniques, such as A* \\cite{Hart1968} and RRT \\cite{LaValle2001} provide another option, but require a discretization of the platform state space that may not be kinematically feasible. \nRelevant work by the controls community applying such graph search techniques to the trajectory planning problem is given in \\cite{Liu2018,Fink2012,Fink2013}.\n\nWhen digital surface models (DSM) of the terrain and buildings are not available \\emph{a priori}, an online estimate is usually computed via a simultaneous localization and mapping (SLAM) technique. \nThe use of online estimated map data necessitates sensor control robustness under uncertainty. \nThat is, the LOS\/NLOS sensor control techniques should be designed to maintain strong performance up to a pre-specified level of error in the estimated map data. \nSimilarly, the obstacle avoidance techniques should guarantee collision avoidance up to the same pre-specified level of map error.\n\n\n\\subsection{Control Space Fidelity for Quadcopters}\nA major contributing factor to the current interest in urban surveillance with mobile sensor networks is the abundance of commercially available UAVs. \nQuadcopters, for example, provide vertical takeoff and landing functionality in addition to high agility maneuvers.\nThese platforms and their flexible APIs for flight control tasking \\cite{Mavlink,DJIsdk,ROS} make real-time experimentation of mobile sensor network applications very attractive. \nThe sensor control methods that exist in the current state-of-the-art, however, make overly simplifying assumptions in the platform kinematics to further reduce computational complexity (e.g., first or second order integrator dynamics).\nAs a result, the platforms are forced to maneuver at slower velocities so that the actions generated by the sensor control algorithms are representative of the POMDP state dynamics. \nA more critical flaw in this approach is that, under the dynamics of the urban environment, platforms may attempt to delay necessary actions for maintaining collision-free flight until it is too late.\n\nQuadcopter platform dynamics have been studied extensively by the controls and aeronautics communities \\cite{Michael2010}.\nThe quadcopter is a six degree-of-freedom system consisting of position and orientation in 3D Euclidean space. \nHowever, it provides only four actuation points consisting of the total upward thrust force and the roll, pitch, and yaw moments. \nThis makes the quadcopter \\emph{underactuated}, implying that its position and orientation can not be accelerated in any arbitrary direction. \nInstead, translational and rotational acceleration are achieved by applying time-varying attitude control.\nA naive incorporation of these plant state and action dynamics under a fixed discretization in a POMDP-based sensor control algorithm is not computationally feasible.\n\nEarly attempts at quadcopter control applied small-angle approximation techniques to linearize the flight dynamics around the hover state \\cite{Hoffmann2008}.  \nAn important finding was made by Mellinger and Kumar in \\cite{Mellinger2011,Mellinger2012}, where the quadcopter was determined to be \\emph{differentially flat} in terms of its 3D Euclidean position and yaw angle. \nDifferential flatness of a system implies that the original states and inputs can be rewritten as algebraic functions of (potentially fewer) state variables and their derivatives.\nThese algebraic functions define a diffeomorphism that ensures any trajectories of sufficient smoothness in the flat outputs will be sufficiently smooth in the original state and control space. \n\nFor the quadcopter, the highest degree derivative of the flat position outputs in their expressions for the original control inputs is four (i.e., trajectory snap). \nSimilarly, the highest degree derivative of the flat yaw output in the expressions for the original control inputs is two (i.e., yaw acceleration).\nUsing this insight, Mellinger and Kumar \\cite{Mellinger2011,Mellinger2012} provided a series of waypoint-based quadcopter trajectory generation techniques that minimize the control effort under trajectory snap and yaw acceleration (i.e., minimum snap trajectories).\nThese waypoint generation methods assume a concatenation of piecewise polynomial functions that pass through pre-defined waypoints.\nSolving for the trajectory polynomial coefficients is done by solving a computationally tractable quadratic program (QP).\nRegulating the original state dynamics of the quadcopter according to this trajectory can then be achieved through the use of a backstepping controller \\cite{Michael2010,Lee2010}.\n\nThe key takeaway from the above discussion is that the accuracy of the quadcopter control space in a mobile sensor control algorithm can be maintained provided that the actions commanded to the platform generate smooth trajectories up until the fourth derivative of position and second derivative of yaw rate. \nFor sensor control with a quadcopter platform, a natural solution is to assume that the plant consists of a fourth order differential equation on the flat outputs, with an input consisting of the trajectory snap at each time step.\nThis trajectory snap input is termed a \\emph{motion primitive} \\cite{Liu2017,Liu2018}.\nUnder polynomial trajectories, these motion primitives induce a resolution-complete discretization in the flat outputs. \nSikang et al. in \\cite{Liu2017} derive this discretization and suggest optimal search techniques for trajectory generation between waypoints using A* \\cite{Hart1968}. \nFor dynamic environments, Sikang et al. in \\cite{Liu2017} proposed a receding horizon control technique based on Lifelong Planning A* (LPA*).\nThe techniques presented in \\cite{Liu2019} were shown to provide collision avoidance guarantees between static and dynamic obstacles, and generate robust paths with respect to random platform disturbances. \n\n\n\\subsection{Tracking and Control Algorithm Decentralization}\nDecentralized operation is a critical requirement of an urban surveillance system.\nAs discussed in Section~\\ref{sec:terrain}, the terrain and building geometries present a strongly RF shadowed propagation environment.\nThis poses a significant challenge for inter-agent communication, and thus renders centralized tracking and sensor control techniques impractical.\nIn general, decentralization of multi-target tracking and sensor control algorithms is very challenging\nThe BP tracking approaches discussed by Meyer et al. in \\cite{Meyer2018} provide an intuitive framework for performing average consensus on the relevant belief state parameters with one-hop neighbors.\nFor particle filtering approaches to the multi-target tracking problem, the BP approaches are decentralized using a consensus-over-weights approach \\cite{Farahmand2011}.\nConsensus over-weights assumes that the particle systems sampled at each agent are identical, which necessitates the use of synchronized random number generators.\nLikelihood consensus \\cite{Hlinka2012} is a slightly less restrictive approach that overcomes the need for synchronized random number generators by projecting the sensor likelihood functions onto a common set of basis functions. \nOther alternatives to the consensus-over-weights scheme include fusion via Gaussian mixture approximations \\cite{Li2018} and kernel-based methods \\cite{Tslil2018}. \n\nAlthough these techniques work well when decentralizing target tracking algorithms, it is difficult to apply them to mobile sensor control policies for the urban environment. \nSince the techniques suggested in Section~\\ref{sec:terrain} necessitate an online simulation-based approach, it is not immediately clear how a consensus algorithm should be constructed.\nOne approach to circumvent this challenge is to implement the centralized mobile sensor control policy in a high-fidelity simulation and perform \\emph{imitation learning} to generate a decentralized policy. \nImitation learning is a variation of reinforcement learning, where the goal is to make observations on a set of oracle control decisions and determine a non-parametric representation of the policy \\cite{Schaal2003}.\nThis type of learning has been applied regularly in robotics to learn specific robotic manipulator movements via kinesthetic examples \\cite{Kober2010,Pervez2017,Zhang2015}. In these efforts, a convolutional neural network (CNN) is commonly used as approximate architecture for the state-action value function.\n\nA recent study by Gama et al. in \\cite{Gama2019} has shown how the convolution and pooling operations used in CNNs can be generalized to support learning with signals supported over graphs. \nThe resulting learning architecture, titled a graph neural network (GNN), may be capable of supporting an imitation learning procedure where the one-hop features that may be relevant to consensus are analogous to those signals supported over a communication network graph.\nA more thorough investigation of imitation learning of decentralized policies from centralized ones using GNNs is an ongoing and open area of research.\n\n\n\\section{Conclusion}\nIn this paper, we presented an overview of the mobile sensor control problem for multi-target tracking with a specific emphasis on urban surveillance problems. \nIn addition to providing a brief background on the sensor control POMDP formulation, we provided a detailed literature review of the current state-of-the-art and suggested three challenge areas that have yet to receive considerable attention by the community\nThese three areas were terrain-aware tracking and sensor control, control space fidelity for quadcopters, and joint estimation and control algorithm decentralization.\nA number of these challenges are addressed separately in the information fusion, tracking, and control communities.\nWe suggest a coordinated effort amongst these communities in order to arrive at solutions that are capable of addressing these challenges together in a computationally tractable and bandwidth efficient manner.\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nLet $K$ be an infinite field, $A$ be a Noetherian $K$-algebra and $\\mathbb P^n_A$ the $n$-dimensional projective space over $A$. A classical approach to investigate a closed projective scheme $W$ consists of \nconsidering a general hyperplane section of $W$, because many properties of $W$  \nare preserved under general hyperplane sections and can be easier recognized in subschemes of lower dimension. \n\nThe inverse problem that consists in finding a scheme $W$ starting from a possible hyperplane section is called a \\emph{lifting problem} and investigations in this topic can produce methods to obtain affine or projective schemes with specific properties. \n\nTo address the question, we recall that in \\citep{Macaulay} Macaulay lifted monomial ideals to obtain sets of points with a given Hilbert functions, in \\citep{H66} Hartshorne proved the connectedness of a Hilbert scheme by lifting Borel ideals and in \\citep{RA} Reeves computed the radius of a Hilbert scheme with an analogous procedure (distractions). Moreover, by $t$-liftings and pseudo-liftings in \\citep{MiNa} Migliore and Nagel obtained special configurations of linear varieties (stick figures). Much interest has been in particular given to the study of the $x_n$-liftings, which can be defined in terms of ideals (see \\citep{GGR, Roitman}) or equivalently in terms of $K$-algebras, like proposed by Grothendieck (see \\citep{BuchEis,Roitman} and the references therein). \nStarting from the papers \\citep{Laudal} and \\citep{Strano}, many authors also give significant contributions to find conditions on some invariants of a variety $W$ so that a degree $d$ hypersurface containing a hyperplane section of $W$ lifts to a hypersurface containing $W$.\n\nIn this paper, we consider the following lifting problem: given a closed subscheme $Y\\subset \\mathbb P^{n-1}_K$, explicitly describe all closed subschemes $W\\subset \\mathbb P^n_A$ such that $Y$ is a general hyperplane section of $W$, up to an extension of scalars. \nEvery such scheme $W$ is called a {\\em lifting} of $Y$ over $A$ (Definition \\ref{def:geometric lifting}) and the saturated defining ideal $I$ of $W$ is called a {\\em lifting} of the saturated defining ideal $I'$ of $Y$ (see Definition \\ref{def:geometric lifting per gli ideali} and Proposition \\ref{prop:h generico}). By the definition, we almost suddenly obtain that $W$ is smooth at every point at which $Y$ is smooth (see Proposition \\ref{prop:smooth points}).\n\nIn order to have a better understanding of the issue we have in mind, let us consider a double point $P\\in \\mathbb P^n_K$. It is very well known that, for any negative integer $g$, there exist double lines $C_g\\subset \\mathbb P^3_K$ with arithmetic genus $g$ and having $P$ as general hyperplane section. So, even when $Y\\subset \\mathbb P^{n-1}_K$ is let to vary in a concrete quasi-projective scheme, we cannot hope that the liftings of $Y$ are parameterized \nby a quasi-projective scheme. In view of this we are led to set the Hilbert polynomial $p_Y(t)$ of $Y$ and look for a functor describing all the liftings of $Y$ with a given Hilbert polynomial $p(t)$ such that its first difference $\\Delta p(t):=p(t)-p(t-1)$\nis equal to $p_Y(t)$.\n\nThe framework of the present paper is both functorial and constructive. \nWe use computational methods that are borrowed from Gr\\\"obner and marked bases theories and which involve quasi-stable ideals (see \\cite{LR,LR2,BCR2}).\nThroughout the paper, a functor $F:{\\text{\\rm Noeth-}K\\text{\\rm -Alg}}\\rightarrow {\\mathrm{Sets}}$ will be said \\emph{representable} if there is a scheme $X$ and a natural isomorphism between $F$ and the functor of points $h_X=\\mathrm{Hom}( - ,X)$ applied on affine schemes.\nIf $X$ is an affine scheme, this definition coincides with the usual one in category theory (see, for instance, \\cite[Definition 2.1]{Vistoli}).\n\nDenoting by $\\mathrm{Hilb}_{p(t)}^n$ the Hilbert scheme parameterizing all the subschemes of $\\mathbb P^n_K$ with Hilbert polynomial $p(t)$, our main results, which are collected in Theorems \\ref{th:schema che rappresenta} and \\ref{th:funtore di punti}, can be summarized in the following way.\n\\vskip 1mm\n\\noindent{\\bf Theorem A.} {\\em  Let $p(t)$ be the Hilbert polynomial of a lifting of $Y$.\n\\begin{itemize}\n\\item[(i)] If $Y$ is equidimensional, then the family of the equidimensional liftings of $Y$ with Hilbert polynomial $p(t)$ is parameterized by a locally closed subscheme of $\\mathrm{Hilb}_{p(t)}^n$.\n\\item[(ii)] The family of the liftings of $Y$ with Hilbert polynomial $p(t)$ is parameterized by a subscheme of $\\mathrm{Hilb}_{p(t)}^n$ which can be explicitly constructed.\n\\end{itemize}\n}\nThis paper has been motivated by the investigation of $x_n$-liftings of a homogeneous polynomial ideal (see Definition \\ref{def:lifting}) that has been faced in \\citep{BCGR} from a functorial point of view. However, the question treated here is different and presents new problems to be solved.\nWe now give a detailed outline of the contents of the paper.\n\nIn what follows, we consider $K[x_0,\\dots,x_{n-1}]$ as a subring of $A[x_0,\\dots,x_{n-1},x_n]$ and the variable $x_n$ is \\emph{generic} for every ideal in $A[x_0,\\dots,x_n]$ we take. \nThis assumption allows us to exploit the behavior of Gr\\\"obner bases with respect to the degree reverse lexicographic order when we need the saturation of homogeneous ideals and of their initial ideals (see Remark \\ref{rem:generic and degrevlex}). The study of $x_n$-lifting presented in \\citep{BCGR} needed weaker hypotheses on the term order that was indeed chosen in a more general class of term orders (see Definition 4.4.1 in \\citep{KR2}, and \\citep{erdos} for a first generic classification of term orderings). \n\nHere, we only consider the degrevlex term order. In this setting, we recall the notion of Gr\\\"obner functor and Gr\\\"obner stratum (see Definition \\ref{def:GrStratum}) \ntogether with the functor of $x_n$-liftings. Then, we prove that our Definition \\ref{def:geometric lifting} of lifting of a projective scheme is equivalent to Definition \\ref{def:geometric lifting per gli ideali} of lifting of a saturated polynomial ideal (see Proposition \\ref{prop:h generico}). This allows to observe that \nthe algebraic counterpart of our lifting problem consists in identifying the liftings $W$ of $Y$ with the $x_n$-liftings of ideals having saturation equal to $I'$ (see Proposition \\ref{prop:geometric lifting}). This characterization implies that all the liftings with a given Hilbert polynomial $p(t)$ can be identified with the points of a disjoint union of locally closed subschemes in the Hilbert scheme $\\mathrm{Hilb}_{p(t)}^n$ via suitable Gr\\\"obner strata (see  \nTheorem \\ref{th:parametrizzazione}).\n\nMoreover, in the further non-restrictive hypothesis that the variable $x_{n-1}$ is generic for $I'$, we prove that a subscheme $W$ is a lifting of $Y$ only if $I$ belongs to a Gr\\\"obner stratum over a monomial ideal $J$ which is a lifting of the initial ideal of $I'$ (see Theorem \\ref{th:dove}). The proofs of these results are constructive and then produce a method for the computation of the locally closed subschemes in a Hilbert scheme whose disjoint union corresponds to all liftings with a given Hilbert polynomial via Gr\\\"obner strata (see Algorithm \\ref{alg:computation in GS}). \n\nSo, we obtain embeddings of liftings with a given Hilbert polynomial in Gr\\\"obner strata and, hence, in a Hilbert scheme. Then, it is natural to look at liftings from a functorial point of view. Thanks to the constructive characterization of liftings given in Section \\ref{sec:Construction}, we are finally able to define functors related to our liftings as subfunctors of a Hilbert functor (Definitions \\ref{def:funtore lifting} and \\ref{def:funtore lifting eq}). Given a Hilbert polynomial $p(t)$, we prove that the functor $\\underline{\\mathrm{L}}_Y^{p(t)}$ of liftings of $Y$ and the functor $\\underline{\\mathrm{L}}_Y^{p(t),e}$ of equidimensional liftings with Hilbert polynomial $p(t)$ are representable. \n\nFor what concerns the functor $\\underline{\\mathrm{L}}_Y^{p(t),e}$, we adapt to our situation classical arguments of algebraic geometry, like the upper semicontinuity of the dimension of the fibers of a dominant map (Theorem \\ref{th:schema che rappresenta}). \n\nFor what concerns the functor $\\underline{\\mathrm{L}}_Y^{p(t)}$, we observe that the locally closed subfunctors that are represented by the locally closed schemes above introduced in suitable Gr\\\"obner strata are not necessarily open subfunctors of $\\underline{\\mathrm{L}}_Y^{p(t)}$ and so are not  expected to be suitable to give a unique scheme representing $\\underline{\\mathrm{L}}_Y^{p(t)}$. At this point our constructive approach is pushed forward by the following fact. Given the degrevlex term order, if the initial ideal of the saturated ideal $I'$ defining $Y$ is quasi-stable then the initial ideal of the saturated ideal $I$ defining a lifting of $Y$ is quasi-stable (see Theorem \\ref{th:lifting}). This result, which is achieved in Section \\ref{sec:quasi-stable}, allows us to study liftings in marked schemes over truncations of quasi-stable ideals, which are open subschemes in a Hilbert scheme (see Definition \\ref{def:markedscheme}). Then, we are able to replace the disjoint locally closed subschemes described by means of Gr\\\"obner strata by open subschemes that we describe by means of marked schemes.\nIndeed, we exploit the features of marked schemes in order to construct an open covering of the functor  $\\underline{\\mathrm{L}}_Y^{p(t)}$ that provides a scheme representing  $\\underline{\\mathrm{L}}_Y^{p(t)}$ by a gluing procedure (see Lemma \\ref{lemma:sezioni ideali marcati}, Theorems \\ref{th:aperti} and \\ref{th:funtore di punti} and Remark \\ref{rem:stromme}). \nThe proof of this last result is constructive and gives rise to Algorithm~\\ref{alg:computation in MS}. In Example \\ref{ex:esempio bello} we exhibit an application of this construction. \n\n\n\n\\section{Setting}\n\nWe consider commutative unitary rings and morphisms that preserve the unit. \nGiven sets of variables $\\mathbf x=\\lbrace x_0,\\dots,x_{n-1}\\rbrace$ and $\\mathbf x,x_n=\\lbrace x_0,\\dots,x_{n}\\rbrace$, we assume they are ordered as $x_0>\\cdots>x_n$. \nFor a term $x^\\alpha:=x_0^{\\alpha_0}\\cdots x_n^{\\alpha_n}$\nother than $1$, we denote by $\\min(x^\\alpha)$ the smallest variable appearing in $x^\\alpha$ with a non-zero exponent\nand by $\\deg(x^\\alpha)=\\vert\\alpha\\vert:=\\sum_i \\alpha_i$ the degree of $x^\\alpha$.\n\nLet $K$ be an infinite field. We denote the polynomial ring $K[x_0,\\dots,$ $x_{n-1}]$ by $K[\\mathbf x]$ and the polynomial ring $K[x_0,\\dots,x_n]$ \nby $K[\\mathbf x,x_n]$. For any (Noetherian) $K$-algebra $A$, $A[\\mathbf x]$ denotes the polynomial ring $K[\\mathbf x]\\otimes_K A$ and $A[\\mathbf x,x_n]$ denotes \n$K[\\mathbf x,x_n]\\otimes_K A$ as \\emph{standard graded algebras}. \nObviously, $A[\\mathbf x]$ is a subring of $A[\\mathbf x,x_n]$, hence the following notations and assumptions will be stated for $A[\\mathbf x,x_n]$ but will hold for $A[\\mathbf x]$ too. \n\nFor any non-zero homogeneous polynomial $f \\in A[\\mathbf x,x_n]$, the {\\em support} $\\mathrm{supp}(f)$ of $f$ is the set of terms in the variables $\\mathbf x,x_n$ that appear in $f$ with a non-zero coefficient.\nWe denote by $\\mathrm{coeff}(f)\\subset A$ the set of the coefficients in $f$ of the terms of $\\mathrm{supp}(f)$. For any subset $\\Gamma\\subseteq A[\\mathbf x,x_n]$, $\\Gamma_t$ is the set of homogeneous polynomials of $\\Gamma$ of degree $t$. Furthermore, we denote by $\\langle \\Gamma\\rangle$ the $A$-module generated by $\\Gamma$. \nWhen $\\Gamma$ is a homogeneous ideal, we denote by $\\Gamma_{\\geq t}$ the ideal  containing\nthe homogeneous polynomials of $\\Gamma$ of degree $\\geq t$. \n\nLet $I$ be a homogeneous ideal of $A[\\mathbf x,x_n]$. \nThe {\\em saturation} of $I$ is $I^{\\textnormal{sat}}=\\{f\\in A[\\mathbf x,x_n] \\ \\vert \\ \\forall \\ i=0,\\dots n, \\exists \\ k_i : x_i^{k_i} f \\in I\\}$. The ideal $I$ is {\\em saturated} if $I=I^{{\\textnormal{sat}}}$ and is {\\em $m$-saturated} if $I_t=(I^{\\textnormal{sat}})_t$ for every $t\\geq m$. The \\emph{satiety} of $I$, denoted by ${\\textnormal{sat}}(I)$, is the smallest $m$ for which $I$ is $m$-saturated.\nA linear form $h\\in A[\\mathbf x,x_n]$ is said \\emph{generic} for $I$ if $h$ is not a zero-divisor in $A[\\mathbf x,x_n]\/I^{\\textnormal{sat}}$.\n\nA {\\em monomial ideal} $J$ of $A[\\mathbf x,x_n]$ is an ideal generated by terms. We denote by $B_J$ the minimal set of terms generating $J$ and by $\\mathcal N(J)$ the sous-escalier of $J$, i.e.~the set of terms outside $J$. \n\n\\begin{definition}\\label{def:stable} \nA monomial ideal $J\\subset A[\\mathbf x,x_n]$ is \\emph{quasi-stable} if for every $x^\\alpha \\in J$ and $x_j>\\min(x^\\alpha)$, there is $t\\geq 0$ such that $\\frac{x_j^t x^\\alpha}{\\min(x^\\alpha)}$ belongs to $J$.\n\\end{definition}\n\nGiven a monomial ideal $J$ and an ideal $I$, a {\\em $J$-reduced form modulo $I$} of a polynomial $f$ is a polynomial $\\bar f$ such that $f-\\bar f$ belongs to $I$ and $\\mathrm{supp}(\\bar f)$ is contained in $\\mathcal N(J)$.\nIf $\\bar f$ is the unique possible $J$-reduced form modulo $I$ of $f$, then it is called the {\\em $J$-normal form modulo $I$} of $f$ and is denoted by $\\mathrm{Nf}(f)$.\n\nWith the usual language of Gr\\\"obner bases theory, {\\em from now,} we consider the degree reverse lexicographic term order $\\succ$ (degrevlex, for short) and, for every non-zero polynomial $f\\in A[\\mathbf x,x_n]$, denote by $\\mathrm{in}(f)=\\max_{\\succ}\\mathrm{supp}(f)$ its initial term and by $c_f$ the coefficient of $\\mathrm{in}(f)$ in $f$. For every ideal $I\\subset A[\\mathbf x,x_n]$ let $\\mathrm{in}(I)=(c_f \\mathrm{in}(f) : f\\in I)$ be its initial ideal. A set $\\{f_1,\\dots,f_t\\}\\subset I$ is a Gr\\\"obner basis of $I$ if $\\{c_{f_1}\\mathrm{in}(f_1),\\dots,c_{f_t}\\mathrm{in}(f_t)\\}$ generates $I$. \nIn general, Gr\\\"obner bases theory over rings \\citep{CM,Z,M} is more intricate than over fields (see for instance the detailed discussion in \\citep{Lederer2011}), and the possibly non-invertible leading coefficients make Gr\\\"obner bases over rings not well-suited for functorial constructions.  This is the reason why\nin this paper we consider homogeneous ideals $I$ generated by either Gr\\\"obner bases in $K[\\mathbf x,x_n]$ or \\emph{monic} Gr\\\"obner bases in $A[\\mathbf x,x_n]$ or \\emph{marked bases} over quasi-stable ideals in $K[\\mathbf x,x_n]$ and $A[\\mathbf x,x_n]$ (see Section \\ref{sec:backgroundMarkedBases} for the definition of marked basis).\nSo, the quotients $A[\\mathbf x,x_n]\/I$ are free graded $A$-modules and \nthis is a very key point for the use of functors we will introduce (see \\citep{BGS91}, \\citep[Lemma 6.1]{BCR2}, \\citep[Section 5]{LR2}).\nThe crucial fact is that, if $\\varphi \\colon A \\rightarrow B$ is a morphism of $K$-algebras and $I$ is an ideal in $A[\\mathbf x]$ (or $A[\\mathbf x,x_n]$) generated by a {\\em monic} Gr\\\"obner basis (resp.~a marked basis over a quasi-stable ideal) $G_I$, then $I\\otimes_A B$ is generated by $\\varphi(G_I)$ which is again a {\\em monic} Gr\\\"obner basis (resp.~a marked basis over a quasi-stable ideal) \n(see Proposition~\\ref{prop:funtore}). \n\nIn this setting, for a homogeneous ideal $I$ of $A[\\mathbf x,x_n]$ we can consider the {\\it Hilbert function} $h_{A[\\mathbf x,x_n]\/I}$ of the free graded $A$-module $A[\\mathbf x,x_n]\/I$ and its {\\it Hilbert polynomial} $p_{A[\\mathbf x,x_n]\/I}(t)$ like the Hilbert function and the Hilbert polynomial of $K[\\mathbf x, x_n]\/\\mathrm{in}(I)$. When we say \\lq\\lq ideal $I$ with Hilbert polynomial $p(t)$\\rq\\rq \\ we mean $p_{A[\\mathbf x,x_n]\/I}(t)=p(t)$. For the geometric definition of Hilbert polynomial of a projective scheme over a field we refer to \\citep[Chapter III, Exercise 5.2]{H}.\n\n\\begin{remark}\\label{rem:generic and degrevlex} \n\\\n\\begin{enumerate}\n\\item If $h$ is a generic linear form for $I$, then $I^{\\textnormal{sat}}=(I : h^\\infty):=\\{f\\in A[\\mathbf x,x_n] \\ \\vert \\ \\exists t\\geq 0 : h^tf\\in I\\}$ (see \\citep{BS}).\n\\item \\label{item:remgeneric} \nRecall that the smallest variable is generic for a homogeneous polynomial ideal $I\\subset A[\\mathbf x,x_n]$ generated by a monic Gr\\\"obner basis if and only if it is generic for $\\mathrm{in}(I)$. Indeed, the initial\nterm  with respect to degrevlex of a homogeneous  polynomial $f$ is divisible by $x_n^r$ if and only if $f$ is divisible by $x_n^r$.\n\\item \\label{item:estensioni e contrazioni}\nIf $L$ is a homogeneous ideal of $K[\\mathbf x,x_n]$ then\n$L A[\\mathbf x,x_n]=L\\otimes_K A$, \\ $L = L A[\\mathbf x,x_n] \\cap~K[\\mathbf x,x_n]$, \\ $(L A[\\mathbf x,x_n])^{\\textnormal{sat}} = L^{\\textnormal{sat}} A[\\mathbf x,x_n]$, and if $I$ is a homogeneous ideal of $A[\\mathbf x,x_n]$ then $I^{\\textnormal{sat}} \\cap K[\\mathbf x,x_n] = (I \\cap K[\\mathbf x,x_n])^{\\textnormal{sat}}$.\n\\end{enumerate} \n\\end{remark}\n\n\n\n\\section{Background I: Gr\\\"obner functor and functor of $x_n$-liftings}\n\\label{sec:Groebner strata and $x_n$-liftings}\n\nIn this section, referring to \\citep{LR,LR2} and to \\citep{BCGR}, we collect some known information about Gr\\\"obner functor and functor of $x_n$-liftings.\nBoth these functors are subfunctors of a Hilbert functor, for which we refer to \\citep{GroHilb,Nit}. We only recall that the Hilbert functor $\\underline{\\mathrm{Hilb}}^n$ associates to a locally Noetherian $K$-scheme $S$ the set $\n\\underline{\\mathrm{Hilb}}^n(S) = \\Bigl\\{W \\subset \\mathbb P_S^n \\ \\vert \\ W \\text{ is flat over }S\\Bigr\\}\n$ of flat families of subschemes of $\\mathbb P^n_S=\\mathbb P^n_K\\times_{\\mathrm{Spec}(K)} S$ parameterized by $S$, \nand to any morphism $\\phi: T \\rightarrow S$ of locally Noetherian $K$-schemes the map\n$$\\begin{array}{lccc}\\underline{\\mathrm{Hilb}}^n(\\phi): &\\underline{\\mathrm{Hilb}}^n(S) &\\longrightarrow &\\underline{\\mathrm{Hilb}}^n(T)\\\\\n& W &\\longmapsto &W \\times_S T.\n\\end{array}\n$$\nGrothendieck shows that the functor  $\\underline{\\mathrm{Hilb}}^n$ is representable by a locally Noetherian scheme $\\mathrm{Hilb}^n$, called Hilbert scheme (see \\citep{GroHilb}).\nGiven the subfunctor\n\\[\n\\underline{\\mathrm{Hilb}}^n_{p(t)}(S):=\\Bigl\\{W \\subset \\mathbb P_S^n \\ \\vert \\ \\ W \\text{ is flat over }S\\text{ and has fibers with Hilbert polynomial } p(t)\\Bigr\\},\n\\]\nthen $\\underline{\\mathrm{Hilb}}^n$ decomposes as co-product:\n\\begin{equation}\\label{eq:co-product}\n\\underline{\\mathrm{Hilb}}^n=\\coprod_{p(t)\\text{ admissible for schemes in } \\mathbb P_K^n}\\underline{\\mathrm{Hilb}}^n_{p(t)}.\n\\end{equation}\nFor every admissible polynomial $p(t)$ of $\\mathbb P_K^n$, $\\underline{\\mathrm{Hilb}}^n_{p(t)}$ is represented by a projective scheme $\\mathrm{Hilb}_{p(t)}^n$.\nThe fact that $\\mathrm{Hilb}^n$ and $\\mathrm{Hilb}_{p(t)}^n$ are locally Noetherian allows to consider the restriction of the functors $\\underline{\\mathrm{Hilb}}^n$ and $\\underline{\\mathrm{Hilb}}^n_{p(t)}$ to the category of Noetherian $K$-algebras (e.g.~\\cite[Proposition VI-2 and Exercise VI-3]{EH}). \nEvery $K$-point of a Hilbert scheme is identified with the saturated ideal $I\\subseteq K[x_0,\\dots,x_n]$ that defines the corresponding fiber in $\\mathbb P^n_K$. \n\nSince in this paper we only consider the degrevlex order, we now recall the notion of Gr\\\"obner functor in this particular setting.\n\n\\begin{definition}\\label{def:GrStratum} {\\rm \\cite[Section 5 and  Theorem 5.3]{LR2}} \nGiven a monomial ideal $J\\subset K[\\mathbf x,x_n]$, the {\\em Gr\\\"obner functor} $\\underline{\\mathrm{St}}_J: \\text{Noeth-}K\\text{-Alg} \\rightarrow \\text{Sets}$ associates to any Noetherian $K$-algebra $A$ the set $\\underline{\\mathrm{St}}_J(A):=\\{I\\subset A[\\mathbf x,x_n] : \\mathrm{in}(I)=J\\otimes_K A\\}$ and to any $K$-algebra morphism $\\phi: A\\rightarrow B$ the function $\\underline{\\mathrm{St}}_J(\\phi): \\underline{\\mathrm{St}}_J(A) \\rightarrow \\underline{\\mathrm{St}}_J(B)$ such that the image of an ideal $I$ is $I\\otimes_A B$. The affine scheme $\\mathrm{St}_J$ representing the Gr\\\"obner functor is called {\\em Gr\\\"obner stratum}. \n\\end{definition}\n\nIn order to briefly recall the construction of a Gr\\\"obner stratum $\\mathrm{St}_J$, take the set of polynomials\n\\begin{equation}\\label{JbaseC}\nG:= \\left\\{ f_\\alpha=x^\\alpha +  \\sum_{{red}x^\\alpha\\succ x^\\gamma\\in\\mathcal N(J)_{\\vert \\alpha\\vert}} C_{\\alpha\\gamma} x^\\gamma \\ : \\ \n\\mathrm{in}(f_\\alpha) = x^\\alpha\\in B_J \\right\\} \\subset K[C_J][\\mathbf x,x_n]\n\\end{equation}\nwhere $C_J$ denotes the set of the new variables $C_{\\alpha \\gamma}$.\nLet $\\mathfrak{a}_J$ be the ideal in $K[C_J]$ generated by the coefficients  (w.r.t.~variables $\\mathbf x,x_n$) of the terms \n in $J$-reduced forms $\\overline{S(f_\\alpha,f_{\\beta})}$ of the $S$-polynomials $S(f_\\alpha,f_{\\beta})$ modulo $(G)$. Due to \\cite[Proposition 3.5]{LR} the ideal $\\mathfrak{a}_J$ depends only on $J$ and on the given term order, which here is supposed to be the degrevlex. Hence, the ideal $\\mathfrak{a}_J$ defines the affine scheme $\\mathrm{St}_J=\\mathrm{Spec}(K[C_J]\/\\mathfrak{a}_J)$.\n\n\\begin{theorem} \\label{th:main features} \\rm{(\\cite[Lemma 5.2]{LR2},  \\cite[Theorem 2.2]{BCGR})} \nLet $J\\subset A[\\mathbf x,x_n]$ be a monomial ideal and $p(t)$ the Hilbert polynomial of $A[\\mathbf x,x_n]\/J$. With the degrevlex term order, \n\\begin{itemize}\n\\item[(i)] $\\underline{\\mathrm{St}}_{J}$ is a Zariski sheaf. \n\\item[(ii)] If the terms in $B_J$ are not divisible by $x_n$, then \n$\\mathrm{St}_J\\cong \\mathrm{St}_{J_{\\geq m}}, \\text{ for every integer } m$, \nand $\\underline{\\mathrm{St}}_J$ is a locally closed subfunctor of the Hilbert functor $\\underline{\\mathrm{Hilb}}^n_{p(t)}$. \n\\end{itemize}\n\\end{theorem}\n\n\\begin{definition}\\label{def:lifting} \\citep{GGR,Roitman}\nA homogeneous ideal $I$ of $A[\\mathbf x,x_n]$ is called a {\\em $x_n$-lifting} of a homogeneous ideal $H$ of $K[\\mathbf x]$ if \n\\begin{enumerate}\n\\item[(a)] the indeterminate $x_n$ is a non-zero divisor in $A[\\mathbf x,x_n]\/I$;\n\\item[(b)] $(I,x_n)\/(x_n)\\simeq H A[\\mathbf x]$ under the canonical isomorphism\n $A[\\mathbf x,x_n]\/(x_n)\\simeq A[\\mathbf x]$; \n\nor, equivalently,\n\\item[(b$'$)] $\\{ g(x_0,x_1,\\ldots,x_{n-1},0) : g\\in I\\} = H A[\\mathbf x]$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{theorem} {\\rm(\\cite[Theorem 2.5]{CFRo}, \\cite[Proposition 6.2.6]{KR2},  \\cite[Theorem 3.2 and Corollary 3.3]{BCGR})}\\label{FM}\nLet $H$ be a homogeneous ideal of $K[\\mathbf x]$. A homogeneous ideal $I$ of $A[\\mathbf x,x_n]$ is a $x_n$-lifting of $H$ if and only if the reduced Gr\\\"obner basis of $I$ is of type $\\lbrace f_\\alpha+g_\\alpha\\rbrace_\\alpha$, \nwhere $\\lbrace f_\\alpha\\rbrace_\\alpha$ is the reduced Gr\\\"obner basis of $H$ and $g_\\alpha \\in (x_n)A[\\mathbf x,x_n]$.\nIf $I$ is a $x_n$-lifting of $H$, then   $\\mathrm{in}(I)$ is generated by the same terms as $\\mathrm{in}(H)$.\n\\end{theorem}\n\n\\begin{definition}\\label{def:xn-liftings} \\cite[Definition 3.4]{BCGR}\nThe {\\em functor} \\ $\\underline{\\mathrm{L_{H}}}: \\underline{\\text{Noeth-}K\\text{-Alg}}\\rightarrow \\underline{\\mathrm{Sets}}$ {\\em of $x_n$-liftings} of a homogeneous ideal $H$ of $K[\\mathbf x]$\nassociates to every Noetherian $K$-algebra $A$ the set $\n\\underline{\\mathrm{L_{H}}}(A)=\\lbrace I\\subseteq A[\\mathbf x,x_n] : I \\text{ is a $x_n$-lifting of }H \\rbrace$ \nand to every morphism of $K$-algebras $\\phi:A \\rightarrow B$ the map\n\\[\\begin{array}{rcl}\n\\underline{\\mathrm{L_{H}}}(\\phi): \\underline{\\mathrm{L_{H}}}(A)&\\rightarrow& \\underline{\\mathrm{L_{H}}}(B)\\\\\n I&\\mapsto& I\\otimes_A B.\n\\end{array}\\]\n\\end{definition}\nWith the notation of Definition \\ref{def:xn-liftings}, let $J:=\\mathrm{in}(H)A[\\mathbf x,x_n]$ and $p(t):=p_{A[\\mathbf x,x_n]\/J}$. The functor $\\underline{\\mathrm{L_{H}}}$ is a closed subfunctor of $\\underline{\\mathrm{St}}_J$ represented by a closed affine subscheme $\\mathrm{L_H}$ of $\\mathrm{St}_J$ and, hence, a locally closed subscheme of the Hilbert scheme $\\mathrm{Hilb}_{p(t)}^n$ thanks to Theorem \\ref{th:main features} \\cite[Theorem 4.3 and Proposition 6.1]{BCGR}. Moreover, $\\underline{\\mathrm{L_H}}$ is a Zariski sheaf.\n\n\n\\section{Liftings of projective schemes}\n\\label{sec:Liftings of projective schemes}\n\n\\begin{definition}\\label{def:geometric lifting}\nLet $\\mathcal{H}=\\mathbb P^{n-1}_K$ be the hyperplane of $\\mathbb P^n_K$ defined by  the ideal $(x_n)$, $Y$ be a closed subscheme of $ \\mathbb P^{n-1}_K$ with Hilbert polynomial $p_Y(t)$   and $A$ be a $K$-algebra. A {\\em lifting of $Y$ over $A$}   is a closed subscheme $W\\in \\underline{\\mathrm{Hilb}}^n(A)$ of $\\mathbb P^n_A=\\mathbb P^n_K\\times_{\\mathrm{Spec}(K)} \\mathrm{Spec}(A)$, such that:\n\\begin{itemize}\n\\item[(i)] $\\Delta p_W(t):=p_W(t)-p_W(t-1)=p_Y(t)$;\n\\item[(ii)] $W\\cap (\\mathcal{H} \\times_{\\mathrm{Spec}(K)} \\mathrm{Spec}(A)) = Y\\times_{\\mathrm{Spec}(K)} \\mathrm{Spec}(A)$. \n\\end{itemize}\n\\end{definition}\n\nObserve that in Definition \\ref{def:geometric lifting} we assume that the scheme $Y$ is contained in the hyperplane defined by the ideal $(x_n)$. This assumption is not restrictive because, given a linear form $h\\in K[\\mathbf x,x_n]$, we can always replace $h$ by the smallest variable $x_n$ thanks to a suitable (deterministic) change of coordinates. \n\n\\begin{definition}\\label{def:geometric lifting per gli ideali}\nLet $I'$ be a homogeneous saturated ideal of $K[\\mathbf x]$. A homogeneous saturated ideal $I$ of $A[\\mathbf x,x_n]$ is called a {\\em lifting of $I'$} if the following conditions are satisfied:\n\\begin{enumerate}\n\\item[(a)] the indeterminate $x_n$ is generic for $I$;\n\\item[(b)] $\\Bigl((I,x_n)\/(x_n)\\Bigr)^{{\\textnormal{sat}}}\\simeq I' A[\\mathbf x]$ under the canonical isomorphism\n $A[\\mathbf x,x_n]\/(x_n)\\simeq A[\\mathbf x]$; \n\nor, equivalently,\n\\item[(b$'$)] $(\\{g(x_0,x_1,\\ldots,x_{n-1},0) : g\\in I\\})^{{\\textnormal{sat}}} = I' A[\\mathbf x]$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{proposition}\\label{prop:h generico}\nLet $Y$ be a closed subscheme of $\\mathbb P^{n-1}_K$ defined by the homogeneous saturated ideal $I'\\subset K[\\mathbf x]$ and $W\\in \\underline{\\mathrm{Hilb}}^n(A)$ be a closed subscheme of $\\mathbb P^n_A$ defined by the homogeneous saturated ideal $I\\subset  A[\\mathbf x,x_n]$. Then, $W$ is a lifting of $Y$ if and only if $I$ is a lifting of $I'$.\n\\end{proposition}\n\n\\begin{proof}\nBy condition (ii) of Definition \\ref{def:geometric lifting}, we have $\\Bigl(\\frac{(I,x_n)}{(x_n)}\\Bigr)^{{\\textnormal{sat}}}=I' A[\\mathbf x]=I'\\otimes_K A$, in particular the quotient $A[\\mathbf x,x_n]\/(I,x_n)$ has Hilbert polynomial $p_Y(t)$. Thus, by the following short exact sequence\n\\begin{equation}\\label{short exact sequence zero-divisor}\n0 \\rightarrow (A[\\mathbf x,x_n]\/(I:x_n))_{t-1} \\xrightarrow{\\cdot x_n} (A[\\mathbf x,x_n]\/I)_t \\rightarrow (A[\\mathbf x,x_n]\/(I,x_n))_t \\rightarrow 0\n\\end{equation}\ncondition (i) of Definition \\ref{def:geometric lifting}, i.e.~$\\Delta p_W(t)= p_Y(t)$, implies that the quotient $A[\\mathbf x,x_n]\/(I:x_n)$ has the same Hilbert polynomial $p_W(t)$ as $A[\\mathbf x,x_n]\/I$. Hence, we obtain $(I:x_n)^{{\\textnormal{sat}}}=I^{{\\textnormal{sat}}}=I$ because $(I:x_n)\\supseteq I$. In conclusion, we have $I^{{\\textnormal{sat}}}\\supseteq(I:x_n)^{{\\textnormal{sat}}}\\supseteq (I:x_n) \\supseteq I=I^{{\\textnormal{sat}}}$, \nnamely $(I:x_n)=I$, which is possible only if $x_n$ is generic for $I$.\n\nConversely, if $I$ is a lifting of $I'$ then it is quite immediate that $W$ is a lifting of $Y$. Indeed, if $x_n$ is generic for $I$, then $(I:x_n)=I=I^{\\textnormal{sat}}$. Hence, $A[\\mathbf x, x_n]\/(I:x_n)$ and $ A[\\mathbf x, x_n]\/I$ have the same Hilbert polynomial. From \\eqref{short exact sequence zero-divisor}, we obtain $\\Delta p(t)=p_{Y}(t)$ and condition (ii) of Definition~\\ref{def:geometric lifting} also follows.\n\\end{proof}\n\nWith the notation we have already introduced in Definition \\ref{def:geometric lifting}, we consider a closed subscheme $Y\\subset \\mathcal H \\subset \\mathbb P^{n}_K$, where $\\mathcal H$ is defined by the ideal $(x_n)$. If $W\\subset \\mathbb P^n_A$ is a lifting of $Y$, then $\\deg W=\\deg Y$ and $\\dim W = \\dim Y +1$, so there are natural restrictions on the Hilbert polynomial $p(t)$ of $W$ because $\\Delta p(t)$ must be the Hilbert polynomial $p_Y(t)$ of $Y$. Hence, the non-constant part of the Hilbert polynomial of $W$ is determined by the Hilbert polynomial of $Y$. However, in general\nthere are no limits on the constant term of the Hilbert polynomial of $W$, even \nif we only consider liftings without zero-dimensional components, as shown by the following example.\n\n\\begin{example}\\label{ex:primoes}For every positive integer $k$, consider the double line $W_k \\subset \\mathbb P^3_K$ defined by the ideal $I=(x_0^2,x_0x_1,x_1^2,x_0x_2^k-x_1x_3^k)\\subseteq K[x_0,x_1,x_2,x_3]$. The Hilbert polynomial of $W_k$ is $p(t)=2t+k+1$ and $W_k$ is a lifting of the double point $Y\\subset \\mathbb P^2_K$ defined by the ideal $I'=(x_0, x_1^2)\\subseteq K[x_0,x_1,x_2]$. In conclusion, for every positive integer $k$, we find a lifting of $Y$ with Hilbert polynomial $2z+k+1$ and without zero-dimensional components.\n\\end{example}\n\nWe conclude this section highlighting a geometric feature of liftings.\n \n\\begin{proposition} \\label{prop:smooth points}\nLet $W$ be a lifting of a scheme $Y$. If $Y$ is smooth on a point $P$ then also $W$ is smooth on $P$. \n\\end{proposition}\n\n\\begin{proof}\nBy definition of lifting, $Y$ is a Cartier divisor in $W$. Then the dimension of the Zariski tangent space of a point $y$ in $Y$ is not lower than the dimension of the Zariski tangent space of the point $y$ in $W$ minus $1$. Indeed, if $\\mathfrak m$ is the local ring at $y$ in $Y$ and $M$ is the local ring at $y$ in $W$, we have $\\dim_K \\frac{\\mathfrak m}{\\mathfrak m^2} \\geq \\dim_K \\frac{M}{M^2}-1$. Hence, if we had $\\dim_K \\frac{M}{M^2} > \\dim(W)$ that we would obtain $\\dim_K \\frac{\\mathfrak m}{\\mathfrak m^2}\\geq \\dim_K \\frac{M}{M^2}-1 > \\dim(W)-1=\\dim(Y)$.\n\\end{proof}\n\n\n\\section{Construction of liftings of projective schemes}\n\\label{sec:Construction}\n\nIn this section, we obtain a constructive characterization of liftings of projective schemes by investigating relations between the notion of lifting of a saturated homogeneous ideal in $K[\\mathbf x]$ (Definition \\ref{def:geometric lifting per gli ideali}) and that of $x_n$-lifting of a homogeneous ideal in $K[\\mathbf x]$ (Definition \\ref{def:lifting}). \n\nIn general a lifting is not a $x_n$-lifting, as shown by the following easy example. Nevertheless, we will show how to recover every lifting of a given saturated ideal by constructing the $x_n$-liftings of suitable families of ideals.\n\n\\begin{example}\nThe ideal \n$I=(x_0^2+x_3^2,x_0x_1,x_0x_2,x_1^2,x_1x_2)\\subset K[x_0,x_1,x_2,x_3]$ is a lifting of $I'=(x_0,x_1)\\subseteq K[x_0,x_1,x_2]$ but is not a $x_3$-lifting of $I'$, as one can easily verify using Theorem \\ref{FM}.\n\\end{example}\n\n\\begin{lemma} \\label{lemma:sezione}\nLet $I'\\subset K[\\mathbf x]$ be a saturated ideal. If $I\\subset A[\\mathbf x,x_n]$ is a lifting of $I'$ then $I'=\\Bigl(\\frac{(I,x_n)}{(x_n)}\\Bigr)^{{\\textnormal{sat}}}\\cap K[\\mathbf x]=\\Bigl(\\frac{(I,x_n)}{(x_n)} \\cap K[\\mathbf x]\\Bigr)^{{\\textnormal{sat}}}$. \n\\end{lemma}\n\n\\begin{proof}\nDefinition \\ref{def:geometric lifting per gli ideali}\nof lifting immediately implies the thesis by Remark \\ref{rem:generic and degrevlex}\\eqref{item:estensioni e contrazioni}. \n\\end{proof}\n\n\\begin{proposition}\\label{prop:geometric lifting}\nLet $I'\\subseteq K[\\mathbf x]$ be a homogeneous saturated ideal. A homogeneous ideal $I\\subseteq A[\\mathbf x,x_n]$ is a lifting of $I'$ if and only if there exists a homogeneous ideal $H\\subseteq K[\\mathbf x]$ such that $H^{{\\textnormal{sat}}}=I'$ and $I$ is a $x_n$-lifting of $H$.\n\\end{proposition}\n\n\\begin{proof}\nFirst assume that $I$ is a lifting of $I'$ and take $H:=\\frac{(I,x_n)}{(x_n)}\\cap K[\\mathbf x]$. Then, we have $H^{{\\textnormal{sat}}}=I'$ by Lemma \\ref{lemma:sezione}. Furthermore, it is immediate that $I$ is a $x_n$-lifting of $H$,  because $HA[\\mathbf x]=\\frac{(I,x_n)}{(x_n)}$ by Remark \\ref{rem:generic and degrevlex}\\eqref{item:estensioni e contrazioni} and $x_n$ is a non-zero divisor in $A[\\mathbf x,x_n]\/I$ by definition.\n\nConversely, let $I\\subset A[\\mathbf x,x_n]$ be an ideal which is a $x_n$-lifting of a homogeneous ideal $H\\subseteq K[\\mathbf x]$ such that $H^{{\\textnormal{sat}}}=I'$. Then, $I$ is a lifting of $I'$, because $x_n$ is generic for $I$, so $I$ is saturated, and $\\frac{(I,x_n)}{(x_n)} \\simeq H\\otimes_K A$ implies $((I,x_n)\/(x_n))^{\\textnormal{sat}}=I'A[\\mathbf x,x_n]$, by Remark \\ref{rem:generic and degrevlex}\\eqref{item:estensioni e contrazioni}.\n\\end{proof}\n\n\\begin{theorem} \\label{th:parametrizzazione}\nLet $I'\\subseteq A[\\mathbf x]$ be a homogeneous saturated ideal and $I\\subseteq A[\\mathbf x,x_n]$ a lifting of $I'$. The locus of liftings of $I'$ in $\\underline{\\mathrm{St}}_{\\mathrm{in}(I)}(A)$ is parameterized by an affine scheme obtained by linear sections of ${\\mathrm{St}}_{\\mathrm{in}(I)}$.\n\\end{theorem}\n\n\\begin{proof}\nLet $J':=\\mathrm{in}(I')\\subset K[\\mathbf x]$ and $J:=\\mathrm{in}(I)\\subset A[\\mathbf x,x_n]$. \nBy Proposition \\ref{prop:geometric lifting}, $I$ is a $x_n$-lifting of a homogeneous ideal $H\\subset K[\\mathbf x]$ with $H^{{\\textnormal{sat}}}=I'$. Hence, by Theorem \\ref{FM}, $\\mathrm{in}(H)$ has the same generators as $J$, so $\\frac{(J,x_n)}{(x_n)}=\\mathrm{in}(H)\\otimes_K A$ and $\\mathrm{in}(H)=J\\cap K[\\mathbf x]$. \n\nLet $\\mathfrak a_J \\subset A[C_J]$ be the defining ideal of the Gr\\\"obner stratum  $\\mathrm{St}_{J}$ and $\\mathfrak a_{J\\cap K[\\mathbf x]} \\subset K[C_{J\\cap K[\\mathbf x]}]$ the defining ideal of the Gr\\\"obner stratum $\\mathrm{St}_{J\\cap K[\\mathbf x]}$, where $C_{J\\cap K[\\mathbf x]}\\subseteq C_J$.\n\nThe ideal $H\\subset K[\\mathbf x]$ is characterized by the following two conditions. The first condition is that $H$ belongs to the family $\\underline{\\mathrm{St}}_{J\\cap K[\\mathbf x]}(K)$, so the reduced Gr\\\"obner basis of $H$ consists of polynomials of the following type\n\\begin{equation}\\label{base di H}\nf_\\beta = x^\\beta +\\sum_{x^\\beta >x^\\gamma \\in \\mathcal N(\\mathrm{in}(H))_{\\vert \\beta\\vert}} C_{\\beta\\gamma} x^\\gamma, \\quad f_\\beta  \\in K[C_{J\\cap K[\\mathbf x]}][\\mathbf x],\n\\end{equation}\nfor every term $x^\\beta$ minimal generator of $J$. The second condition is that the polynomials $f_\\beta$ belong to $I'$, because $H$ is contained in $I'$. This second condition implies that the saturation of $H$ is $I'$ because $K[\\mathbf x]\/I'$ and $K[\\mathbf x]\/H$ have the same Hilbert polynomial, by the first condition.\n\nBy construction, $\\mathrm{in}(H)$ is contained in $J'$. Hence, we have $\\mathcal N(J')\\subseteq\\mathcal N(\\mathrm{in}(H))$ and can obtain a $J$-reduced form $\\overline{f_\\beta}$ modulo $I'$ of the polynomial $f_\\beta$ using the reduced Gr\\\"obner basis of $I'$. Imposing that $\\overline{f_\\beta}$ is zero we obtain that $H$ is contained in $I'$ and collect some new constraints on the coefficients in the parameters $C_{\\beta\\gamma}$ in $C_{J\\cap K[\\mathbf x]}$. Let $\\mathfrak b_{J\\cap K[\\mathbf x]} \\subset K[C_{J\\cap K[\\mathbf x]}]$ be the ideal generated by these constraints, for every $x^\\beta\\in B_J$. The ideal $\\mathfrak a_{J\\cap K[\\mathbf x]} + \\mathfrak b_{J\\cap K[\\mathbf x]} \\subset K[C_{J\\cap K[\\mathbf x]}]$, hence the affine scheme $\\textnormal{Spec}\\,\\left(\\frac{ K[C_{J\\cap K[\\mathbf x]}]}{\\mathfrak a_{J\\cap K[\\mathbf x]} + \\mathfrak b_{J\\cap K[\\mathbf x]}}\\right)$, parameterizes the locus in $\\underline{\\mathrm{St}}_{J\\cap K[\\mathbf x]}(K)$ of all the ideals $H\\subset K[\\mathbf x]$ such that $H^{\\textnormal{sat}}=I'$. \n\nFinally, we apply Theorem \\ref{FM} and hence consider \n\\begin{equation}\\label{base di I}\ng_\\beta :=f_\\beta + \\sum_{x^\\delta \\in \\mathcal N(J)_{\\vert\\beta\\vert-1}} C_{\\beta\\delta} x_n x^\\delta, \\quad g_\\beta  \\in A[C_J][\\mathbf x,x_n],\n\\end{equation}\nfor every term $x^\\beta$ minimal generator of $J$. The set $\\{g_\\beta\\}_{x^\\beta \\in B_J}$ is a Gr\\\"obner basis with initial ideal $J$ modulo the ideal $\\mathfrak a_J\\subset A[C_J]$ which defines the Gr\\\"obner stratum $\\mathrm{St}_{J}$.\n\nWe now observe that if the set of polynomials $g_\\beta$ is a Gr\\\"obner basis then also the set of polynomials $f_\\beta$ is a Gr\\\"obner basis, due to the hypothesis on $J$ and $\\mathrm{in}(H)$. This fact means that the ideal $\\mathfrak a_{J\\cap K[\\mathbf x]} A[C_J]$ is contained in $\\mathfrak a_J$. Then, the ideal $\\mathfrak a_J+\\mathfrak b_{J\\cap K[\\mathbf x]}  A[C_J]$, hence the affine scheme $\\textnormal{Spec}\\,\\left(\\frac{A[C_J]}{\\mathfrak a_J+\\mathfrak b_{J\\cap K[\\mathbf x]}  A[C_J]}\\right)$, parameterizes the locus of the liftings of $I'$ in the family $\\underline{\\mathrm{St}}_{J}(A)$. \n\nIt remains to show that the constraints we obtain by rewriting the polynomials $f_\\beta$ of \\eqref{base di H} by the reduced Gr\\\"obner basis of $I'$ are linear, i.e.~the ideal $\\mathfrak b_{J\\cap K[\\mathbf x]}$ has linear generators. This fact is immediate, because the coefficients of the polynomials of the reduced Gr\\\"obner basis of $I'$ belong to the field $K$.\n\\end{proof}\n\nTheorem \\ref{th:parametrizzazione} describes the locus of liftings of a homogeneous saturated ideal $I'\\subset K[\\mathbf x]$ in a given Gr\\\"obner stratum. In the further hypothesis that the variable $x_{n-1}$ is generic for $I'$,  we can recognize what Gr\\\"obner strata are candidate to contain these liftings.\n\n\\begin{theorem}\\label{th:dove}\nLet $I'\\subseteq K[\\mathbf x]$ be a homogeneous saturated ideal. If $x_{n-1}$ is generic for $I'$, then the liftings of $I'$ belong to the Gr\\\"obner stratum over a monomial lifting $J\\subset A[\\mathbf x,x_n]$ of \\ $\\mathrm{in}(I')$.\n\\end{theorem}\n\n\\begin{proof}\nIt is enough to prove that if $I\\subset A[\\mathbf x,x_n]$ is a lifting of $I'$ and $x_{n-1}$ is generic for $I'$, then $\\Bigl(\\frac{(\\mathrm{in}(I),x_n)}{(x_n)}\\Bigr)^{{\\textnormal{sat}}}=\\mathrm{in}(I')\\otimes_K A$. Hence, we will conclude by applying Theorem \\ref{th:parametrizzazione} and observing that if $x_n$ is generic for $I$ then it is a non-zero divisor for $\\mathrm{in}(I)$.\n\nLet $J':=\\mathrm{in}(I')$ and $J:=\\mathrm{in}(I)$. By definition of lifting we have $\\Bigl(\\frac{(I,x_n)}{(x_n)}\\Bigr)^{{\\textnormal{sat}}}=I'\\otimes_K A$. Hence, there exists an integer $s\\geq 0$ such that\n$\\frac{(I_{\\geq s},x_n)}{(x_n)}=\\Bigl(\\frac{(I,x_n)}{(x_n)}\\Bigr)_{\\geq s}=I'_{\\geq s}\\otimes_K A$. \nBy \\cite[Lemma 2.2]{BS}, we have $\\mathrm{in}(I_{\\geq s},x_n)=(\\mathrm{in}(I_{\\geq s}),x_n)$ and obtain\n\\begin{equation}\\label{eq:saturation}\n\\frac{(J_{\\geq s},x_n)}{(x_n)}={J'}_{\\geq s}\\otimes_K A\n\\end{equation}\nbecause $\\mathrm{in}(I_{\\geq s})=\\mathrm{in}(I)_{\\geq s}= J_{\\geq s}$ and $\\mathrm{in}(I'_{\\geq s})=\\mathrm{in}(I')_{\\geq s}= J'_{\\geq s}$.\nNow, it is enough to recall that $J'$ is saturated because $I'$ is saturated and $x_{n-1}$ is not a zero-divisor in $K[\\mathbf x]\/I'$.\n\\end{proof}\n\n\\begin{remark} \nThe condition that $x_{n-1}$ is generic for $I'$ is not restrictive because it can always be obtained up to a suitable change of variables. On the other hand, the result of Theorem \\ref{th:dove} does not hold without this hypothesis: for example, for the saturated ideal $I'=(x_0^2,x_1x_0+x_2^2)\\subset  K[x_0,x_1,x_2]$ we obtain $\\mathrm{in}(I')=(x_0^2, x_1x_0,x_2^2x_0,x_2^4)$, that is not saturated.\n\\end{remark}\n\nThanks to Theorems \\ref{th:parametrizzazione} and \\ref{th:dove}, we know {\\em where} the liftings of a given saturated polynomial ideal $I'\\subseteq A[\\mathbf x]$ are located and {\\em how} they can be constructed, obtaining Algorithm \\ref{alg:computation in GS}.\n\n\\begin{algorithm}[!ht]\n\\caption{\\label{alg:computation in GS} Algorithm for computing parameter schemes for the liftings of a saturated homogeneous ideal $I'\\subset K[\\mathbf x]$ over a Noetherian $K$-algebra $A$ in $\\mathrm{Hilb}_{p(t)}^n$ by means of Gr\\\"obner strata.}\n\\begin{algorithmic}[1]\n\\STATE $\\textsc{LiftingGS}\\big(I',p(t)\\big)$\n\\REQUIRE $I'\\subset K[\\mathbf x]$ a saturated polynomial ideal such that $x_{n-1}$ is generic for $I'$.\n\\REQUIRE  $p(t)$ a Hilbert polynomial such that $\\Delta p(t)= p_Y(t)$, where $p_Y(t)$ is the Hilbert polynomial of the scheme $Y$ defined by $I'$.\n\\ENSURE  A set $\\mathfrak B$ containing parameter schemes for the liftings of $I'$ in the Gr\\\"obner stratum over $J$, for every monomial lifting $J\\subset A[\\mathbf x,x_n]$ of $\\mathrm{in}(I')$ with Hilbert polynomial $p(t)$.\n\\STATE $\\mathcal L:=\\{ J \\subset A[\\mathbf x,x_n] \\ \\vert \\ J \\text{ monomial lifting of } \\mathrm{in}(I') \\text{ with Hilbert polynomial } p(t)\\}$; \n\\STATE $\\mathfrak B=\\emptyset$;\n\\FOR{$J \\in \\mathcal L$}\n\\STATE let $\\mathfrak a_J \\subset A[C_J]$ be the defining ideal of the Gr\\\"obner stratum $\\mathrm{St}_{J}$;\n\\STATE $\\mathfrak b_{J\\cap K[\\mathbf x]}:=(0)$;\n\\FOR{$x^\\beta\\in B_J$}\n\\STATE construct the polynomial $f_\\beta$ as in \\eqref{base di H};\n\\STATE $\\mathfrak b_{J\\cap K[\\mathbf x]}:= \\mathfrak b_{J\\cap K[\\mathbf x]}+(\\mathrm{coeff}(\\overline{f_\\beta}))$, where $\\overline{f_\\beta}$ is the $J$-normal form of $f_\\beta$ modulo $I'$; \n\\ENDFOR\n\\STATE $\\mathfrak B:=\\mathfrak B\\cup \\{\\mathfrak a_J+\\mathfrak b_{J\\cap K[\\mathbf x]}A[\\mathbf x,x_n]\\}$;\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\\section{Functors of liftings}\n\nFrom Theorem \\ref{th:parametrizzazione} we obtain the following generalization of \\cite[Corollary~3.3]{BCGR}\nin which, if $\\phi:A\\rightarrow B$ is a $K$-algebra morphism, we also denote by $\\phi$ the natural extension\nof $\\phi$ to $A[\\mathbf x,x_n]$. Recall that the image under $\\phi$ of every ideal $I$ in $A[\\mathbf x,x_n]$ generates the extension  $I^e = IB[\\mathbf x,x_n]=I \\otimes_A B$ (see \\citep{BGS91}). \n\n\\begin{proposition} \\label{prop:funtore}\nLet $I'\\subset K[\\mathbf x]$ be a saturated ideal and $\\phi:A\\rightarrow B$ a $K$-algebra morphism. For every lifting $I \\subset A[\\mathbf x,x_n]$ of $I'$ over $A$ the ideal $I\\otimes_A B$ is a lifting of $I'$ over $B$, with the same Hilbert polynomial as $I$. \n\\end{proposition}\n\n\\begin{proof} Let $G_I$ be the reduced monic Gr\\\"obner basis of $I$ and $J=\\mathrm{in}(I)$. Then, $\\phi(G_I)$ is a monic Gr\\\"obner basis of $I\\otimes_A B$ with the same  initial\nterms of $G_I$ because $G_I$ is monic. Let $H\\subset K[\\mathbf x]$ be the ideal such that $I$ is a $x_n$-lifting of $H$ and $H^{\\textnormal{sat}}=I'$. Let $G_H=\\{f_\\beta\\}_{x^\\beta\\in B_J}$ be the reduced (monic) Gr\\\"obner basis of $H$. By Theorems \\ref{th:parametrizzazione} and \\ref{FM} we have that $G_I$ is of the following type\n\\[\nG_I=\\lbrace f_\\beta + \\sum_{x^\\delta \\in \\mathcal N(J)_{\\vert \\beta\\vert-1}} {c}_{\\beta\\delta} x_n x^\\delta \\rbrace_\\beta, \\quad c_{\\beta\\delta} \\in A.\n\\]\nThe ideal $I\\otimes_A B$ is then generated by $\\phi(G_I)=\\lbrace f_\\beta - \\sum_{x^\\delta \\in \\mathcal N(J)_{\\vert \\beta\\vert-1}} \\phi({c}_{\\beta\\delta}) x_n x^\\delta \\rbrace_{x^\\beta\\in B_J}$, which is still a reduced Gr\\\"obner basis  because the polynomials of $G_I$ are monic. Hence, $I\\otimes_A B$ is a $x_n$-lifting of $H$ by Theorem \\ref{FM} \nand a lifting of $I'$ by Proposition  \\ref{prop:geometric lifting}.\n\nFor the statement concerning \nthe Hilbert polynomial, it is sufficient to observe that $I$ and $I\\otimes_K B$ have the same initial ideal, hence the same Hilbert polynomial.\n\\end{proof}\n\nGiven a scheme $Y\\subseteq \\mathbb P^{n-1}_K$, thanks to Proposition \\ref{prop:funtore} we can now easily define some functors concerning the liftings of $Y$. \n\n\\begin{definition}\\label{def:funtore lifting}\nLet $Y=\\textnormal{Proj}\\,(K[\\mathbf x]\/I')$ be a closed subscheme of $\\mathbb P^{n-1}_K$ with Hilbert polynomial $p_Y(t)$ and $p(t)$ be a Hilbert polynomial such that $\\Delta p(t)=p_Y(t)$. \n\\begin{itemize}\n\\item[(a)] The {\\em functor of liftings of $Y$},  $\\underline{\\mathrm{L}}_{Y}:{\\text{Noeth-}K\\text{-Alg}}\\rightarrow {\\mathrm{Sets}}$, \nassociates to every Noetherian $K$-algebra $A$ the set \n$$\\underline{\\mathrm{L}}_{Y}(A):=\\{I \\subset A[\\mathbf x,x_n] : I \\text{ lifting of } I' \\}$$\nand to every morphism of $K$-algebras $\\phi: A \\rightarrow B$ the map\n\\[\\begin{array}{rcl}\n\\underline{\\mathrm{L}}_{Y}(\\phi): \\underline{\\mathrm{L}}_{Y}(A)&\\rightarrow& \\underline{\\mathrm{L}}_{Y}(B)\\\\\n I&\\mapsto& I\\otimes_A B.\n\\end{array}\\]\n\n\\item[(b)] The {\\em functor of liftings of $Y$ with Hilbert polynomial $p(t)$}, $\\underline{\\mathrm{L}}_Y^{p(t)}: {\\text{Noeth-}K\\text{-Alg}}\\rightarrow {\\mathrm{Sets}}$, associates to every Noetherian $K$-algebra $A$ the set \n$$\\underline{\\mathrm{L}}_{Y}^{p(t)}(A):=\\{I \\subset A[\\mathbf x,x_n]  : I \\text{ lifting of } I' \\text{ with Hilbert polynomial } p(t)\\}$$\nand to every morphism of $K$-algebras $\\phi: A \\rightarrow B$ the map\n\\[\\begin{array}{rcl}\n\\underline{\\mathrm{L}}_{Y}^{p(t)}(\\phi): \\underline{\\mathrm{L}}_{Y}^{p(t)}(A)&\\rightarrow& \\underline{\\mathrm{L}}_{Y}^{p(t)}(B)\\\\\n I&\\mapsto& I\\otimes_A B.\n\\end{array}\\]\n\\end{itemize}\n\\end{definition}\n\nIt is immediate that $\\underline{\\mathrm{L}}_{Y}^{p(t)}$ is a subfunctor of $\\underline{\\mathrm{L}}_{Y}$, and furthermore $\\underline{\\mathrm{L}}_{Y}$ (resp.~$\\underline{\\mathrm{L}}_{Y}^{p(t)}$) is a subfunctor of $\\underline{\\mathrm{Hilb}}^n$ (resp.~$\\underline{\\mathrm{Hilb}}^n_{p(t)}$). In fact, recall that the functors $\\underline{\\mathrm{Hilb}}^n$ and $\\underline{\\mathrm{Hilb}}^n_{p(t)}$ are both representable by locally Noetherian schemes, hence it is enough to consider their restrictions to the category of Noetherian $K$-algebras.\n\nUsing the same arguments on Hilbert schemes that give \\eqref{eq:co-product}, we obtain that the functor of litings of $Y$ decomposes as a co-product of the above subfunctors \n\\begin{equation}\\label{eq:co-product lifting}\n\\underline{\\mathrm{L}}_{Y}=\\coprod_{p(t)\\text{ admissible for liftings of $Y$ in } \\mathbb P_K^n} \\underline{\\mathrm{L}}_{Y}^{p(t)}.\n\\end{equation}\n\n\\begin{proposition}\\label{prop:Zariski sheaf}\nThe functor $\\underline{\\mathrm{L}}_Y^{p(t)}$ is a Zariski sheaf.\n\\end{proposition}\n\n\\begin{proof}\nLet $A$ be a Noetherian $K$-algebra and $\\{U_i=\\textnormal{Spec}\\,(A_{a_i})\\}_{i=1,\\dots,r}$ be an open covering of $\\textnormal{Spec}\\,(A)$. This is equivalent to the fact $(a_1,\\dots,a_r)=A$. Consider a set of ideals $I_i\\in \\underline{\\mathrm{L}}_Y^{p(t)}(A_{a_i})$ such that for any pair of indexes $i\\not= j$ we have\n\\begin{equation}\\label{eq:Zariski sheaf}\nI_{ij}:=I_i\\otimes_{A_{a_i}} A_{a_ia_j} = I_j\\otimes_{A_{a_j}} A_{a_ia_j} \\in \\underline{\\mathrm{L}}_Y^{p(t)}(A_{a_ia_j}).\n\\end{equation}\nWe need to show that there is a unique ideal $I\\in \\underline{\\mathrm{L}}_Y^{p(t)}(A)$ such that $I_i=I\\otimes_A A_{a_i}$ for every $i$. \n \nBy Proposition \\ref{prop:geometric lifting}, there are $H_i$ and $H_j$ ideals in $K[\\mathbf x]$ such that $H_i^{\\textnormal{sat}}=H_j^{\\textnormal{sat}}=I'$ and $I_i$ is a $x_n$-lifting of $H_i$, while $I_j$ is a $x_n$-lifting $H_j$. By Theorem \\ref{FM} and assumption \\eqref{eq:Zariski sheaf}, $H_i=H_j\\subset K[\\mathbf x]$, hence $I_i$ belongs to $\\underline{\\mathrm L_H}(A_i)$ and $I_j$ belongs to $\\underline{\\mathrm L_H}(A_j)$. Since $\\underline{\\mathrm L_H}$ is a Zariski sheaf, there is a unique $I\\subset A[\\mathbf x,x_n]$ such that $I$ is a $x_n$-lifting of $H$, $I\\otimes_{A}A_{a_i}=I_i$ and $I\\otimes_{A}A_{a_j}=I_j$. By Proposition \\ref{prop:geometric lifting} we conclude that $I$ belongs to $\\underline{\\mathrm{L}}_Y^{p(t)}(A)$ and is the unique ideal in $A[\\mathbf x,x_n]$ such that $I_i=I\\otimes_A A_{a_i}$ for every $i$. \n\\end{proof}\n\nRecall that a closed subscheme in $\\mathbb P^n_K$ is {\\em equidimensional} if all its components have the same dimension, in particular it has no embedded components. Thus, there exists an equidimensional lifting $W$ of a subscheme $Y$ only if $Y$ is equidimensional, i.e.~the ideal $I'\\subset K[\\mathbf x]$ defines an equidimensional scheme in $\\mathbb P^{n-1}_K$.\nWe say that a saturated ideal $I\\subset A[\\mathbf x,x_n]$ is {\\em equidimensional} if defines families of equidimensional subschemes. The next well-known result \nhighlights that base extension preserves the fibers on every $K$-point and, hence, their possible equidimensionality. \n\n\\begin{lemma}\\label{lemma:funtore equidimensionale}\nLet $W$ be a scheme over a $K$-algebra $A$. If $\\phi: A \\rightarrow B$ is a morphism of $K$-algebras, then the fibers of $W\\to \\textnormal{Spec}\\,(A)$ are isomorphic to the fibers of $W\\times_{\\textnormal{Spec}\\,(A)} \\textnormal{Spec}\\,(B)\\to \\textnormal{Spec}\\,(B)$ for every $K$-point.\n\\end{lemma}\n\n\\begin{proof} For the sake of completeness we give a proof of this statement.\nLet $\\phi: A\\to B$ be a morphism of $K$-algebras, $\\phi^{*}: \\textnormal{Spec}\\,(B) \\to \\textnormal{Spec}\\,(A)$ the corresponding morphism and $\\textnormal{Spec}\\,(K)\\to \\textnormal{Spec}\\,(B)$  the morphism associated to a $K$-point of $\\textnormal{Spec}\\,(B)$ (e.g.~\\cite[Chapter II, Exercise 2.7]{H}). Moreover, let $\\textnormal{Spec}\\,(K)\\to \\textnormal{Spec}\\,(A)$ be the morphism associated to the $K$-point of $\\textnormal{Spec}\\,(A)$ obtained by composition with $\\phi^{*}$ and $W\\times_{\\textnormal{Spec}\\,(A)} \\textnormal{Spec}\\,(K)$ the fiber on this $K$-point. Then, we obtain $(W\\times_{\\textnormal{Spec}\\,(A)} \\textnormal{Spec}\\,(B))\\times_{\\textnormal{Spec}\\,(B)} \\textnormal{Spec}\\,(K)\\simeq W\\times_{\\textnormal{Spec}\\,(A)} \\textnormal{Spec}\\,(K)$ due to the transitivity of base extension.\n\\end{proof}\n\n\\begin{definition}\\label{def:funtore lifting eq}\nLet $Y=\\textnormal{Proj}\\,(K[\\mathbf x]\/I')$ be an equidimensional closed subscheme of $\\mathbb P^{n-1}_K$ with Hilbert polynomial $p_Y(t)$ and $p(t)$ a Hilbert polynomial such that $\\Delta p(t)=p_Y(t)$. Thanks to Lemma \\ref{lemma:funtore equidimensionale} we define\n\\begin{itemize}\n\\item[(a)] The {\\em functor of equidimensional liftings of $Y$}, denoted by $\\underline{\\mathrm{L}}_{Y}^e: {\\text{Noeth-}K\\text{-Alg}}\\rightarrow {\\mathrm{Sets}}$,\nassociates to every Noetherian $K$-algebra $A$ the set \n$$\\underline{\\mathrm{L}}_{Y}^e(A):=\\{I \\subset A[\\mathbf x,x_n] : I \\text{ equidimensional lifting of } I' \\}$$\nand to every morphism of $K$-algebras $\\phi: A \\rightarrow B$ the map\n\\[\\begin{array}{rcl}\n\\underline{\\mathrm{L}}_{Y}^e(\\phi): \\underline{\\mathrm{L}}_{Y}^e(A)&\\rightarrow& \\underline{\\mathrm{L}}_{Y}^e(B)\\\\\n I&\\mapsto& I\\otimes_A B.\n\\end{array}\\]\n\n\\item[(b)] The {\\em functor of equidimensional liftings of $Y$ with Hilbert polynomial $p(t)$}, denoted by $\\underline{\\mathrm{L}}_Y^{p(t),e}:~{\\text{Noeth-}K\\text{-Alg}}\\rightarrow~{\\mathrm{Sets}}$, associates to every Noetherian $K$-algebra $A$ the set \n$$\\underline{\\mathrm{L}}_{Y}^{p(t),e}(A):=\\{I \\subset A[\\mathbf x,x_n]  : I \\text{ equidimensional lifting of } I' \\text{ with Hilbert polynomial } p(t)\\}$$\nand to every morphism of $K$-algebras $\\phi: A \\rightarrow B$ the map\n\\[\\begin{array}{rcl}\n\\underline{\\mathrm{L}}_{Y}^{p(t),e}(\\phi): \\underline{\\mathrm{L}}_{Y}^{p(t),e}(A)&\\rightarrow& \\underline{\\mathrm{L}}_{Y}^{p(t),e}(B)\\\\\n I&\\mapsto& I\\otimes_A B.\n\\end{array}\\]\n\\end{itemize}\n\\end{definition}\n\nBy definition, the functor $\\underline{\\mathrm{L}}_Y^{e}$ (resp.~$\\underline{\\mathrm{L}}_Y^{p(t),e}$) is a subfunctor of $\\underline{\\mathrm{L}}_Y$ (resp.~of $\\underline{\\mathrm{L}}_Y^{p(t)}$) and we have\n\\begin{equation}\\label{eq:co-product lifting equid}\n\\underline{\\mathrm{L}}_{Y}^e=\\coprod_{p(t)\\text{ admissible for liftings of $Y$ in } \\mathbb P_K^n} \\underline{\\mathrm{L}}_{Y}^{p(t),e}\n\\end{equation}\nsimilarly to formulas \\eqref{eq:co-product} and \\eqref{eq:co-product lifting} for $\\underline{\\mathrm{Hilb}^n}$ and $\\underline{\\mathrm{L}}_Y$.\n\n\n\n\\section{The functor $\\underline{\\mathrm{L}}_Y^{p(t),e}$ is representable}\n\nWe need some preliminary results. \n\n\\begin{proposition}\\label{prop:Gro2} \\cite[Proposition (2.3.4)(iii)]{Gro2}\nLet $S$ be a locally Noetherian $K$-scheme, $W$ be an element of $\\underline{\\mathrm{Hilb}}_{p(t)}^n(S)$ and $f: W \\rightarrow S$ the corresponding flat projection. For every irreducible closed subset $S'$ of $S$, every irreducible component $W'$ of $f^{-1}(S')$ is dominant on $S'$, i.e.~$f'=f_{\\vert W'} : W' \\rightarrow S'$ is dominant. \n\\end{proposition}\n\nIf $p_Y(t)$ is the Hilbert polynomial of $Y$, we can consider the Hilbert-flag scheme $\\mathcal Fl_{p_Y,p}$ (see \\citep{Kleppe}) and the projections $\\pi_1: \\mathcal Fl_{p_Y,p} \\longmapsto \\mathrm{Hilb}_{p_Y(t)}^{n-1}$ and  $\\pi_2: \\mathcal Fl_{p_Y,p} \\longmapsto \\mathrm{Hilb}_{p(t)}^n$. Thus, $\\pi_1^{-1}(Y)$ is the closed scheme consisting of the couples $(Y,W)$ where $W$ varies among all the closed subschemes of $\\mathbb P^n_K$ containing $Y$ and with Hilbert polynomial $p(t)$. We set $\\mathrm{Hilb}_{p(t),Y}^n:=\\pi_2(\\pi_1^{-1}(Y))$.\n\n\\begin{proposition}\\label{prop:flag}\n$\\mathrm{Hilb}_{p(t),Y}^n$ is a closed subscheme of $\\mathrm{Hilb}_{p(t)}^n$ which represents a closed subfunctor $\\underline{\\mathrm{Hilb}}_{p(t),Y}^n$ of $\\underline{\\mathrm{Hilb}}_{p(t)}^n$. \n\\end{proposition}\n\n\\begin{proof}\nFrom the definition it follows straightforwardly that $\\mathrm{Hilb}_{p(t),Y}^n:=\\pi_2(\\pi_1^{-1}(Y))\\simeq \\pi_1^{-1}(Y)$ is the closed subscheme of $\\mathrm{Hilb}_{p(t)}^n$\nconsisting of the points in $\\mathrm{Hilb}_{p(t)}^n$ corresponding to schemes containing $Y$. Thus, it represents the closed subfunctor of the Hilbert functor that associates to a scheme $S$ the set of subschemes $W$ of $\\mathbb P^n_K \\times S$ containing $Y\\times_{\\mathrm{Spec}(K)}~S$.\n\\end{proof}\n\n\\begin{theorem}\\label{th:schema che rappresenta}\nLet $Y$ be an equidimensional closed subscheme of $\\mathbb P^{n-1}_K$ with Hilbert polynomial $p_Y(t)$ and $p(t)$ a Hilbert polynomial such that $\\Delta p(t)=p_Y(t)$. Then, $\\underline{\\mathrm{L}}_Y^{p(t),e}$ is representable by a locally closed subscheme ${\\mathrm{L}}_Y^{p(t),e}$ of ${\\mathrm{Hilb}}_{p(t)}^n$.\n\\end{theorem}\n\n\\begin{proof} \nLet $S$ be a Noetherian $K$-scheme, $W$ an element of $\\underline{\\mathrm{Hilb}}_{p(t)}^n(S)$ and $f: W \\rightarrow S$ the corresponding flat projection. The fibers of $f$ in $W$ have degree equal to $\\deg(Y)$ and dimension equal to $\\dim(Y)+1$ because $\\Delta p(t)=p_Y(t)$. \n\nFor every irreducible closed subset $S'$ of $S$, let $W'$ be any irreducible component of $f^{-1}(S')$. By Proposition \\ref{prop:Gro2}, $W'$ is dominant on $S'$. Then, also $W'\\cap (\\mathcal H\\times S')$ is dominant on $S'$, because for every $s'\\in S'$ the fiber of $s'$ in $W'$ has dimension at least $1$ by construction, and hence $s'$ has a fiber in $W'\\cap (\\mathcal H\\times S')$ of dimension at least $0$. Indeed, the dimension of every fiber in $W'\\cap (\\mathcal H\\times S')$ is between $\\dim(Y)+1$ and $\\dim(Y)$.\n\nRecall that the dimension of the fibers of a dominant morphism is an upper semicontinuous function, namely the subset of $S'$ whose fibers in $W\\cap(\\mathcal H\\times S')$ have dimension less than or equal to $\\dim(Y)$ is open \\cite[Chapter I, section 8, Corollary 3]{Mumford}. Since this dimension cannot be strictly lower than $\\dim(Y)$ by the previous argument, all the fibers of the above open subset have dimension equal to $\\dim(Y)$. \n\nIn the above situation, if we assume that $W$ belongs to $\\underline{\\mathrm{Hilb}}_{p(t)}^n(U_e)$, where $U_e$ is the open subscheme of $\\mathrm{Hilb}_{p(t)}^n$ that parameterizes the families of equidimensional subschemes of $\\mathbb P^n_K$ with Hilbert polynomial $p(t)$  \\cite[Th\\'{e}or\\`{e}me (12.2.1)(iii)]{Gro3}, \nwe can also observe that the fibers in $W$ and $\\mathcal H \\times_{\\mathrm{Spec}(K)} S$ intersect properly, implying that the degree of the fibers in $W\\cap (\\mathcal H \\times_{\\mathrm{Spec}(K)} S)$ is less than or equal to $\\deg(Y)$ because the fibers and $\\mathcal H \\times_{\\mathrm{Spec}(K)} S$ are equidimensional (see \\cite[Corollary 18.5]{Harris92} in case of varieties). \n\nIf we also assume that $W$ belongs to $\\mathrm{Hilb}_{p(t),Y}^n$, so that $W$ contains $Y$, we obtain that the fibers in $W\\cap (\\mathcal H \\times_{\\mathrm{Spec}(K)} S)$ have degree equal to $\\deg(Y)$ because $Y\\times_{\\mathrm{Spec}(K)} S \\subseteq W\\cap (\\mathcal H \\times_{\\mathrm{Spec}(K)} S)$.\n\nWe can now conclude that there is an open subset in $\\mathrm{Hilb}_{p(t),Y}^n$ describing all subschemes $W$ such that $W\\cap (\\mathcal H \\times_{\\mathrm{Spec}(K)} S)=Y\\times_{\\mathrm{Spec}(K)} S$, namely $W$ is a lifting of $Y$. It is immediate that any equidimensional lifting of $Y$ belongs to this open subset of $\\mathrm{Hilb}_{p(t),Y}^n$ and, hence, locally closed subscheme of $\\mathrm{Hilb}_{p(t)}^n$ by Proposition \\ref{prop:flag}. \n\\end{proof}\n\n\\begin{remark} \nThe locally closed subscheme ${\\mathrm{L}}_Y^{p(t),e}$ of ${\\mathrm{Hilb}}_{p(t)}^n$ which has been introduced in the proof of Theorem \\ref{th:schema che rappresenta} completely describes the locally Cohen-Macaulay liftings of $Y$ when $Y$ is a zero-dimensional scheme. \n\\end{remark}\n\n\n\\section{The case of a quasi-stable initial ideal}\n\\label{sec:quasi-stable}\n\nThanks to Theorem \\ref{th:dove} we have that the liftings of a saturated homogeneous ideal $I'\\subset K[\\mathbf x]$ with $x_{n-1}$ generic belong to a Gr\\\"obner stratum over a monomial lifting $J$ of $J':=\\mathrm{in}(I')$. In this section we prove that if $J'$ is quasi-stable then $J$ is quasi-stable too (Theorem \\ref{th:lifting}). \n\nThe assumption that $J'$ is quasi-stable is not restrictive: indeed, this can be obtained by a change of coordinates on $I'$, and this change does not effect the scheme $Y=\\textnormal{Proj}\\,(K[\\mathbf x]\/I')$ from a geometric point of view.  Quasi-stability for initial ideals will allow us to use the techniques concerning marked bases over quasi-stable ideals developed in \\citep{BCR2} (see \\citep{ABRS} for the more general case of free modules).\n\n\\begin{lemma}\\label{lemma:ss tagliato}\nLet $J\\subseteq A[\\mathbf x,x_n]$ be a monomial ideal.\nIf $J_{\\geq s}$ is quasi-stable for some integer $s$, then $J$ is quasi-stable.\n\\end{lemma}\n\n\\begin{proof}\nIf $J_{\\geq s}$ is quasi-stable, it is enough to check the condition of Definition \\ref{def:stable} for every term $x^\\alpha \\in J$ with $\\vert\\alpha\\vert <s$. For every $x_i > \\min(x^\\alpha)$, take $x^\\alpha x_i^{s-\\vert\\alpha\\vert} \\in J_{\\geq s}$. Then, there is an integer $t$ such that $\\frac{x^\\alpha x_i^{s-\\vert\\alpha\\vert+t}}{\\min(x^\\alpha x_i^{s-\\vert\\alpha\\vert})}$ belongs to $J_{\\geq s}$, because $J_{\\geq s}$ is quasi-stable. We conclude by observing that $\\min(x^\\alpha)=\\min(x^\\alpha x_i^{s-\\vert\\alpha\\vert})$ by construction.\n\\end{proof}\n\n\\begin{theorem}\\label{th:lifting}\nIf $J'\\subseteq K[\\mathbf x]$ is a saturated quasi-stable  ideal then a monomial lifting $J\\subseteq A[\\mathbf x,x_n]$ of $J'$ is quasi-stable.\n\\end{theorem}\n\n\\begin{proof}\nConsider the ideal $L=\\frac{(J,x_n)}{(x_n)}\\cap K[\\mathbf x]$. Since quasi-stability is a property concerning the semigroup structure of the ideal generated by the minimal monomial basis of $J$, regardless of the coefficients of the polynomial ring, it is sufficient to prove that $L$ is quasi-stable.\n\nBy the hypothesis, we have $L^{\\textnormal{sat}}=J'$ and hence, if $s={\\textnormal{sat}}(L)$, then $L_{\\geq s}={J'}_{\\geq s}$ is quasi-stable. By Lemma \\ref{lemma:ss tagliato}(i), $L$ is quasi-stable too.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:lifting}\nLet $I'\\subseteq K[\\mathbf x]$ be a homogeneous saturated ideal and $I\\subseteq A[\\mathbf x,x_n]$ be a lifting of $I'$. If $\\mathrm{in}(I')$ is quasi-stable, then $\\mathrm{in}(I)$ is a quasi-stable lifting of $\\mathrm{in}(I')$. \n\\end{corollary}\n\n\\begin{proof}\nThis is a consequence of the fact that the smallest variable is generic for any quasi-stable ideal (see \\cite[Prop. 4.4(ii)]{Seiler2009II}), of Remark  \\ref{rem:generic and degrevlex}\\eqref{item:remgeneric} and of Theorems \\ref{th:lifting} and \\ref{th:dove}.\n\\end{proof}\n\n\\begin{example}\\label{ex:un primo calcolo} \nIn this example we apply Theorem \\ref{th:lifting}. Let $\\mathbf x:=\\{x_0,\\dots,x_3\\}$ and consider the saturated ideal $I'=(x_0^2,x_0x_1+x_1^2,x_0x_2)=(x_1^2,x_0) \\cap (x_2,x_1^3,x_0x_1+x_1^2,x_0^2) \\subseteq K[\\mathbf x]$. \nThe reduced Gr\\\"obner basis of $I'$ is $G'=\\{x_0x_2, x_0x_1 + x_1^2, x_0^2, x_1^2x_2, x_1^3\\}$, hence the initial ideal is the quasi-stable ideal $J':=\\mathrm{in}(I')=(x_0^2,x_0x_1,x_0x_2,$ $x_1^2x_2,x_1^3)$.\nThe Hilbert function of $\\frac{K[\\mathbf x]}{I'}$ is \n\\[\nh_{K[\\mathbf x]\/I'}(0)=1, h_{K[\\mathbf x]\/I'}(1)= 4  \\text{ and } h_{K[\\mathbf x]\/I'}(t)=2t+3 \\text{ for every }t\\geq 2, \n\\]\nhence $I'$ defines a curve $Y$ in $\\mathbb P^3_K$. A Hilbert polynomial $p(t)$ such that $\\Delta p(t)=2t+3$ must be of type $p(t)=t^2+4t+c$. \nAssume that $W\\subset \\mathbb P^4_K$ is a surface and is a lifting of $Y$ over a Noetherian $K$-algebra $A$. The Hilbert polynomial of $W$ is $p(t)=t^2+4t+c$ with $c\\geq 0$. Indeed, since $x_4$ is not a zero-divisor on $S\/I$, we have the short exact sequence\n$$ 0 \\rightarrow (A[\\mathbf x,x_4]\/I)_{t-1} \\xrightarrow{\\cdot x_4} (A[\\mathbf x,x_4]\/I)_t \\rightarrow (A[\\mathbf x,x_4]\/(I,x_4))_t \\rightarrow 0,$$\nthat gives $h_{A[\\mathbf x,x_4]\/(I,x_4)}(t) = \\Delta h_{A[\\mathbf x,x_4]\/I}(t)$, in particular $p_{A[\\mathbf x,x_4]\/{(I,x_4)}}(t)=\\Delta p_{A[\\mathbf x,x_4]\/{I}}(t)$ and $\\Delta h_W(t)\\geq h_Y(t)$ for every $t$. As a consequence, $h_W(t)\\geq \\sum_{i=0}^t h_Y(i)$, hence \n\\[\nh_{A[\\mathbf x,x_4]\/I}(0)= 1, h_{A[\\mathbf x,x_4]\/I}(1)= 5 \\text{ and } h_{A[\\mathbf x,x_4]\/I}(t)\\geq t^2+4t \\text{ for every }t\\geq 2, \n\\]\nand $c=0$ is the minimal possible value of the constant term $c$ for the Hilbert polynomial of $W$. \nWe now investigate the cases $c=0$ and $c=1$. In the following, we compute quasi-stable ideals by the algorithm described in \\citep{Bertone}. \n\nIf $c=0$, among $56$ possible quasi-stable saturated ideals there is a unique quasi-stable ideal $J:=J'\\cdot K[\\mathbf x,x_4]\\subset K[\\mathbf x,x_4]$ such that $\\left(\\frac{(J,x_4)}{(x_4)}\\right)^{{\\textnormal{sat}}} \\cap K[\\mathbf x]= J'$. Hence, in this case the liftings of $I'$ belong to the family $\\underline{\\mathrm{St}}_J(A)$ and exactly are the $x_4$-liftings of $I'$ because $J$ and $J'$ share the same generators, using Theorem \\ref{th:dove}.\n\nIf $c=1$,  among $176$ possible quasi-stable saturated ideals there are only the following $5$ quasi-stable ideals $J^{(i)}\\subset K[\\mathbf x,x_4]$ such that $\\left(\\frac{(J^{(i)},x_4)}{(x_4)}\\right)^{{\\textnormal{sat}}}\\cap K[\\mathbf x]=\\mathrm{in}(I')$:\n$$\\begin{array}{l}\nJ^{(1)}=(x_2x_0, x_1x_0, x_3x_0^2, x_2x_1^2, x_1^3, x_0^3),\\\\\nJ^{(2)}=(x_2x_0, x_0^2, x_3x_1x_0, x_2x_1^2, x_1^3, x_1^2x_0),\\\\\nJ^{(3)}=(x_1x_0, x_0^2, x_3x_2x_0, x_2^2x_0, x_2x_1^2, x_1^3) \n\\\\\nJ^{(4)}=(x_2x_0, x_1x_0, x_0^2, x_2x_1^2, x_3x_1^3, x_1^4),\\\\\nJ^{(5)}=(x_2x_0, x_1x_0, x_0^2, x_1^3, x_3x_2x_1^2, x_2^2x_1^2) \n\\end{array}\n$$\nHence, in this case the liftings of $I'$ belong to the union of Gr\\\"obner strata over the above quasi-stable ideals in the Hilbert scheme $\\mathrm{Hilb}_{t^2+4t+1}^4$. Now, we apply the construction described in the proof of Theorem \\ref{th:parametrizzazione} to these monomial ideals and obtain the following families of Gr\\\"obner bases for the ideals of type $H$, modulo the defining ideal of the Gr\\\"obner stratum over $J^{(k)}$:\n\n$J^{(1)}$: $G^{(1)}=\\{x_2x_0, x_0x_1+x_1^2, x_3x_0^2, x_2x_1^2, x_1^3, x_0^3 \\}$,\n\n$J^{(2)}$: $G^{(2)}(a)=\\{x_2x_0, \nx_0^2 + ax_0x_1 + ax_1^2, \nx_0x_1x_3+ x_1^2x_3,\nx_2x_1^2,\nx_1^3, \nx_1^2x_0\\}$ where $a\\in A$,\n\n$J^{(3)}$: $G^{(3)}(b,c)=\\{ x_0x_1+ x_1^2 + bx_0x_2,  \nx_0^2 + cx_0x_2, \nx_3x_2x_0,\nx_0x_2^2, \nx_2x_1^2,\nx_1^3\\}$ where $b,c\\in A$,\n\n$J^{(4)}$: $G^{(4)}=\\{x_2x_0, \nx_0x_1+x_1^2,\nx_0^2,\nx_2x_1^2,\nx_3x_1^3,\nx_1^4\\}$,\n\n$J^{(5)}$:  $G^{(5)}(d)=\\{x_0x_2, \nx_0x_1+x_1^2, \nx_0^2, \nx_1^3+dx_1^2x_2,  \nx_1^2x_2x_3,\nx_1^2x_2^2\\}$ where $d\\in A$.\n \n\\noindent In conclusion, for every $k=1,\\dots,5$, the liftings of $I'$ in $\\mathrm{St}_{J^{(k)}}$ are the $x_4$-liftings of the ideals generated by the corresponding Gr\\\"obner bases ${(G^{(k)}(a,b,c,d))}$, modulo the defining ideal of the Gr\\\"obner stratum over $J^{(k)}$.\n\\end{example}\n\n\\begin{example}\\label{ex:lCM} \nConsider a field $K$ with $\\mathrm{char}(K)=0$ and a $K$-algebra $A$. In the present example we test our methods in order to characterize the locally Cohen-Macaulay liftings in $\\mathbb P_A^3$ of the double point $Y\\subset \\mathbb P^2_K$ defined by the ideal $I'=(x_0,x_1^2)\\subset K[x_0,x_1,x_2]$, where $J'=\\mathrm{in}(I')=I'$ is quasi-stable. For every Hilbert polynomial $p(t)$, a lifting $W$ of $Y$ with Hilbert polynomial $p(t)$ has embedded or isolated points if and only if $W$ contains a lifting with Hilbert polynomial $p(t)-1$. \n\nFor $p(t)=2t+1$ we find a unique monomial quasi-stable lifting $J=(x_0,x_1^2)\\subset A[x_0,x_1,x_2,x_3]$ of $J'$.  Furthermore, $H=I'\\subset K[x_0,x_1,x_2]$ is the unique ideal such that $H^{\\textnormal{sat}}=I'$ and $\\mathrm{in}(H)=J$.\nHence, $\\underline{\\mathrm{L}}_Y^{2t+1}(A)$ consists of the $x_3$-liftings of the ideal $(x_0,x_1^2)$. These liftings are generated by the following Gr\\\"obner basis, which depends on the free parameters $\\alpha, \\beta, \\gamma, \\delta \\in A$:\n\\[\nG(\\alpha,\\beta,\\gamma,\\delta)=\\{x_{{0}}+\\alpha\\,x_{{3}},x_{{1}}^{2}+\\beta\\,x_{{1}}x_{{3}}+\n\\gamma\\,x_{{2}}x_{{3}}+\\delta\\,x_{{3}}^{2}\\}.\n\\]\n\nFor $p(t)=2t+2$, we find two monomial quasi-stable liftings $J^{(1)}=(x_0,x_1^3,x_1^2x_2)$ and $J^{(2)}=(x_0^2,x_0x_1,x_1^2,x_0x_2)$ of $J'$. By direct computation we see that $J^{(1)}$ (resp.~$J^{(2)}$) is the unique ideal in $K[x_0,x_1,x_2]$ such that its saturation is $I'$ and its initial ideal is $J^{(1)}$ (resp.~$J^{(2)}$). \nHence, $\\underline{\\mathrm{L}}_Y^{2t+2}(A)$ consists of the $x_3$-liftings of the ideals $J^{(1)}$ and $J^{(2)}$. We now study them in order to find the locally Cohen-Macaulay liftings of $Y$ with Hilbert polynomial $2t+2$.\n\nEvery $x_3$-lifting of $J^{(1)}$ \ndefines a plane curve with isolated or embedded points and hence is not locally Cohen-Macaulay. \n\nEvery $x_3$-lifting of $J^{(2)}$ is generated by polynomials of the following type\n\\begin{equation}\\label{eq:ex75groebner}\n\\begin{array}{c}\nx_{{0}}^{2}+c_{{1}}x_{{0}}x_{{3}}+ \\left( c_{{1}}c_{{8}}-c_{{8}}^{2} \\right) x\n_{{3}}^{2},\\quad\nx_{{0}}x_{{1}}+c_{{2}}c_{{8}}x_{{3}}^{2}+c_{{2}}x_{{0}}x_\n{{3}}+c_{{8}}x_{{1}}x_{{3}},\\\\\nx_{{1}}^{2}+c_{{3}}x_{{0}}x_{{3}}+c_{{4\n}}x_{{1}}x_{{3}}+c_{{5}}x_{{2}}x_{{3}}+ \\left( c_{{1}}c_{{3}}-c_{{2}\n}^{2}+c_{{2}}c_{{4}}-c_{{3}}c_{{8}}+c_{{5}}c_{{6}} \\right) x_{{3}}^{\n2},\\\\\nx_{{0}}x_{{2}}+c_{{6}}x_{{0}}x_{{3}}+c_{{7}}x_{{1}}x_{{3}}+c_{{8}}x_\n{{2}}x_{{3}}+ \\left(\\frac{c_{{4}}c_{{7}}}{2}+c_{{6}}c_{{8}} \\right) {x_{{3\n}}}^{2}\n\\end{array}\n\\end{equation}\nwith $c_1,\\dots,c_8\\in A$ such that $c_{{5}}c_{{7}}=c_{{3}}c_{{7}}=2\\,c_{{2}}c_{{7}}-c_{{4}}c_{{7}}=c_\n{{1}}c_{{7}}-2\\,c_{{7}}c_{{8}}=0$.\nThe assumptions on the $c_i$'s ensure that the polynomials in \\eqref{eq:ex75groebner} form a Gr\\\"obner basis.\n\nA $x_3$-lifting of $J^{(2)}$ is contained in the ideal generated by $G(\\alpha, \\beta, \\gamma, \\delta)$, with $\\alpha, \\beta, \\gamma, \\delta \\in A$, if and only if\n\\begin{equation}\\label{eq:ex75nonlCM}\n c_{{1}}=c_8=\\alpha,c_{{4}}=\\beta,c_{{5}}=\\gamma,c_{{2}}=c_{{3}}=c_{{6}}=c_{{7}}=0. \n \\end{equation}\nObserve that $\\delta=0$. Then,  the conditions \\eqref{eq:ex75nonlCM} characterize the liftings of the double point $Y$ which are not locally Cohen-Macaulay.\n\nSumming up, the locally Cohen-Macaulay liftings of $I'$ with Hilbert polynomial $2t+2$ are $x_3$-liftings of $J^{(2)}$ having Gr\\\"obner basis as in \\eqref{eq:ex75groebner} and such that the $c_i$'s do not satisfy equations~\\eqref{eq:ex75nonlCM}.\n\\end{example}\n\n\n\\section{Backgroud II: marked functor over a truncation ideal}\\label{sec:backgroundMarkedBases}\n\nThe results of Section \\ref{sec:quasi-stable} are preliminary for the proof of the representability of $\\underline{\\mathrm{L}}_Y^{p(t)}$, which is described in Section \\ref{sec:reprLYp} and uses the notion of marked functor. First, we need to recall the notion of Pommaret basis.\n\n\\begin{definition}\\citep{Seiler2009I}\\label{def:Pommaret}\nGiven a term $x^\\alpha$ and $x_j=min(x^\\alpha)$, the set $\\mathcal C_P(x^\\alpha):=\\{x^\\delta x^\\alpha \\ \\vert \\ \\delta_i=0 \\ \\forall i<j\\}$ is the \\emph{Pommaret cone} of $x^\\alpha$. Given a finite set $M$ of terms, its \\emph{Pommaret span} is $\\cup_{x^\\alpha\\in M} \\mathcal C_{\\mathcal P}(x^\\alpha)$. The finite set of terms $M$ is a \\emph{Pommaret basis $\\mathcal P((M))$} of the ideal $(M)$ if the Pommaret span of $M$ coincides with the set of terms in $(M)$ and the Pommaret cones of the terms in $M$ are pairwise disjoint. \n\\end{definition}\n\nA monomial ideal $J$ is quasi-stable if and only if it has a Pommaret basis $\\mathcal P(J)$ (see \\cite[Definition 4.3 and Proposition 4.4]{Seiler2009II}). A {\\em marked polynomial} is a polynomial $F$ together with a specified term of $\\mathrm{supp}(F)$ that will be called {\\em head term of $F$} and denoted by $\\mathrm{Ht}(F)$ \\citep{RStu}.\n\n\\begin{definition}\\label{def:cose marcate} \\cite[Definition 5.1]{CMR13}\nLet $J\\subset A[\\mathbf x,x_n]$ be a quasi-stable ideal.\n\nA {\\em $\\mathcal P(J)$-marked set} (or marked set over $\\mathcal P(J)$) $\\mathcal G$ is a set of homogeneous monic marked polynomials $F_\\alpha$ in $A[\\mathbf x,x_n]$ such that  the head terms $Ht(F_\\alpha)=x^\\alpha$ are pairwise different and form the Pommaret basis $\\mathcal P(J)$ of $J$, and $\\mathrm{supp}(F_\\alpha-x^\\alpha)\\subset \\mathcal N(J)$.\n\nA $\\mathcal P(J)$\\emph{-marked basis} (or marked basis over $\\mathcal P(J)$) $\\mathcal G$ is a $\\mathcal P(J)$-marked set such that $\\mathcal N(J)$ is a basis of $A[\\mathbf x,x_n]\/(\\mathcal G)$ as an $A$-module, i.e.~$A[\\mathbf x,x_n]=(\\mathcal G)\\oplus \\langle \\mathcal{N}\\, (J) \\rangle$ as an $A$-module.\n\\end{definition} \n\nLet $J\\subset A[\\mathbf x,x_n]$ be a saturated quasi-stable ideal and $m$ a non-negative integer. The ideal $J_{\\geq m}$ is called {\\em $m$-truncation} of $J$ and is quasi-stable, because $J$ is.\nReferring to \\citep{CMR13,LR2,BCR2}, we now recall the definition of marked functor over the $m$-truncation of a saturated quasi-stable ideal. \n\n\\begin{definition}\\label{def:markedscheme}\\citep{BCR2,LR2} \nThe {\\em marked functor} \\ $\\underline{\\mathrm{Mf}}_{\\mathcal P(J_{\\geq m})}:  \\underline{\\text{Noeth-}K\\text{-Alg}}\\rightarrow \\underline{\\mathrm{Sets}}$ \\ \nassociates to every Noetherian $K$-algebra $A$ the set\n$$\\underline{\\mathrm{Mf}}_{\\mathcal P(J_{\\geq m})} (A) :=\\{I\\text{ saturated ideal in }A[\\mathbf x,x_n]\\  : \\ A[\\mathbf x,x_n]=I_{\\geq m}\\oplus\\langle \\mathcal N(J_{\\geq m})\\rangle\\} =$$\n$$= \\{I\\text{ saturated ideal in }A[\\mathbf x,x_n]\\  : I_{\\geq m} \\text{ is generated by a } \\mathcal P(J_{\\geq m})\\text{-marked basis}\\}$$\nand to every $K$-algebra morphism  $\\phi : A\\rightarrow B$ the function  \n$${\\phi} \\colon \\underline{\\mathrm{Mf}}_{\\mathcal P(J_{\\geq m})}(A) \\rightarrow \\underline{\\mathrm{Mf}}_{\\mathcal P(J_{\\geq m})}(B) \\ \\hbox{ given by } \\  {\\phi} (I )= I\\otimes_A B.$$\n\\end{definition}\n\nThe marked functor $\\underline{\\mathrm{Mf}}_{\\mathcal P(J_{\\geq m})}$ is a {\\em representable subfunctor} of the Hilbert functor $\\underline{\\mathrm{Hilb}}^n_{p(t)}$, where $p(t)$ is the Hilbert polynomial of $A[\\mathbf x,x_n]\/J$, and we denote by $\\mathrm{Mf}_{\\mathcal P(J_{\\geq m})}$ its representing scheme \\cite[Theorem 6.6]{BCR2}.\n\n\\begin{remark} \nThe fact that $I$ belongs to $\\underline{\\mathrm{Mf}}_{\\mathcal P(J_{\\geq m})} (A)$ {\\em does not imply} that $I$ belongs to $\\underline{\\mathrm{Mf}}_{\\mathcal P(J)} (A)$ (e.g. \\cite[Example 3.8]{BCR2}). If $I$ belongs to $\\underline{\\mathrm{Mf}}_{\\mathcal P(J_{\\geq m})} (A)$, then the Hilbert function of $A[\\mathbf x,x_n]\/I_{\\geq m}$ is equal to the Hilbert function of $A[\\mathbf x,x_n]\/J_{\\geq m}$.\n\\end{remark}\n\n\\begin{remark} \nThe original definition of marked functor over a $m$-truncation of a quasi-stable ideal introduced in \\citep{LR2} stated:\n$$\\underline{\\mathrm{Mf}}_{\\mathcal P(J_{\\geq m})} (A) :=\\{M\\text{ homogeneous ideal in }A[\\mathbf x,x_n]\\  : \\ A[\\mathbf x,x_n]=M\\oplus\\langle \\mathcal N(J_{\\geq m})\\rangle\\}.$$\nOur presentation is equivalent to the original one because a homogeneous ideal $M\\subset A[\\mathbf x,x_n]$ such that $A[\\mathbf x,x_n]=M\\oplus\\langle \\mathcal N(J_{\\geq m})\\rangle$ is exactly of type $I_{\\geq m}$, where $I=M^{{\\textnormal{sat}}}$ (see \\cite[Corollary 3.7]{BCR2}).\n\\end{remark}\n\n\\section{The functor $\\underline{\\mathrm{L}}_Y^{p(t)}$ is representable}\\label{sec:reprLYp}\n\nWe can now constructively prove that the functor $\\underline{\\mathrm{L}}_Y^{p(t)}$ is representable. \n\nFrom now, $I'\\subset K[\\mathbf x]$ is a homogeneous saturated ideal with $J'=\\mathrm{in}(I')$ quasi-stable, $J\\subset A[\\mathbf x,x_n]$ is a monomial lifting of $J'$ and $Y$ is the projective subscheme defined by $I'$.\n\n\\begin{definition}\\label{def:sottofuntori aperti}\nWe denote by $\\underline{\\mathrm{L}}_{Y,J_{\\geq m}}^{p(t)}$ the subfunctor $\\underline{\\mathrm{L}}_Y^{p(t)}\\cap \\underline{\\mathrm{Mf}}_{\\mathcal P(J_{\\geq m})}:  \\underline{\\text{Noeth-}K\\text{-Alg}}\\rightarrow \\underline{\\mathrm{Sets}}$ \\ of $\\underline{\\mathrm{Hilb}}^n_{p(t)}$ associating to every Noetherian $K$-algebra $A$ the set\n$\\underline{\\mathrm{L}}_{Y,J_{\\geq m}}^{p(t)}(A) := \\underline{\\mathrm{L}}_Y^{p(t)}(A)~\\cap~\\underline{\\mathrm{Mf}}_{\\mathcal P(J_{\\geq m})}~(A)$ \nand to every $K$-algebra morphism  $\\phi : A\\rightarrow B$ the function  \n${\\phi} \\colon \\underline{\\mathrm{L}}_{Y,J_{\\geq m}}^{p(t)}(A) \\rightarrow \\underline{\\mathrm{L}}_{Y,J_{\\geq m}}^{p(t)}(B)$ given by ${\\phi}(I )= I\\otimes_A B$.\n\\end{definition}\n\n\\begin{lemma} \\label{lemma:sottofuntore aperto}\nLet $I'\\subset K[\\mathbf x]$ be a homogeneous saturated ideal with $J'=\\mathrm{in}(I')$ quasi-stable, $J\\subset A[\\mathbf x,x_n]$ a monomial lifting of $J'$ and $\\rho$ the satiety of $(J,x_n)\/(x_n)\\subset A[\\mathbf x]$. If $m\\geq \\rho$ then:\n\\begin{enumerate}[(i)]\n\\item A saturated ideal $I\\subset A[\\mathbf x,x_n]$ belongs to $\\underline{\\mathrm{L}}_{Y,J_{\\geq m-1}}^{p(t)}(A)$ if and only if $I$ is a lifting of $I'$ and $I_{\\geq m-1}$ has a $\\mathcal P(J_{\\geq m-1})$-marked basis.\n\\item\\label{lemma:sottofuntore aperto_ii} $\\underline{\\mathrm{L}}_{Y,J_{\\geq m-1}}^{p(t)}$ is an open subfunctor of $\\underline{\\mathrm{L}}_Y^{p(t)}$. \n\\item \\label{lemma:sottofuntore aperto_iii} $\\underline{\\mathrm{L}}_{Y,J_{\\geq m}}^{p(t)}$ is isomorphic to $\\underline{\\mathrm{L}}_{Y,J_{\\geq m-1}}^{p(t)}$\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nThe statements are direct consequences of Definition \\ref{def:sottofuntori aperti} and of properties of marked functors. In particular, item \\eqref{lemma:sottofuntore aperto_ii} is a consequence of the openness of  $\\underline{\\mathrm{Mf}}_{\\mathcal P(J_{\\geq m})}$ in $\\underline{\\mathrm{Hilb}}^n_{p(t)}$ (see \\citep{LR2} and \\cite[Proposition 6.13]{BCR2}); item \\eqref{lemma:sottofuntore aperto_iii} follows from the analogous property of marked functors (see \\cite[Theorem 5.7]{BCLR}, \\cite[Theorem 3.4]{LR2} if $J$ is strongly stable and \\cite[Section 6]{BCR2} in the more general case $J$ is quasi-stable).\n\\end{proof}\n\n\\begin{lemma}\\label{lemma:Pommaret}\nLet $J'\\subset K[\\mathbf x]$ be a saturated quasi-stable ideal, $J\\subset A[\\mathbf x,x_n]$ a monomial lifting of $J'$ and $\\rho$ the satiety of $(J,x_n)\/(x_n)\\subset A[\\mathbf x]$. If $m\\geq \\rho$ then: \n\\begin{itemize}\n\\item[(i)]$\\left(\\frac{(J,x_n)}{(x_n)}\\right)_{\\geq m}={J'}_{\\geq m}\\otimes A$, and \n\\item[(ii)] $\\mathcal P({J'}_{\\geq m})= \\{ x^\\alpha \\in \\mathcal P(J_{\\geq m}) \\ : \\  x^\\alpha \\hbox{ is not divisible by } x_n\\}$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nObserve that the terms of the Pommaret basis of $J$ are not divisible by $x_n$ \\cite[Remark 2.8]{BCR2}. Moreover, recall that in our setting we have $J'\\otimes A:=\\left(\\frac{(J,x_n)}{(x_n)}\\right)^{\\textnormal{sat}}$, where the ideal $\\frac{(J,x_n)}{(x_n)}=(\\mathcal P(J))\\cdot A[\\mathbf x]$ is quasi-stable and $m$-saturated because $m\\geq \\rho$. Thus, we obtain ${J'}_{\\geq m}\\otimes A=\\left(\\frac{(J,x_n)}{(x_n)}\\right)_{\\geq m}$ and then use the definition of Pommaret basis. \n\\end{proof}\n\n\\begin{lemma}\\label{lemma:sezioni ideali marcati}\nLet $J'\\subset K[\\mathbf x]$ be a saturated quasi-stable ideal, $J\\subset A[\\mathbf x,x_n]$ a monomial lifting of $J'$ and $\\rho$ the satiety of $(J,x_n)\/(x_n)\\subset A[\\mathbf x]$. For every $m\\geq \\rho$, if $I\\subset A[\\mathbf x,x_n]$ belongs to $\\underline{\\mathrm{Mf}}_{\\mathcal P(J_{\\geq m-1})}(A)$, then $\\left(\\frac{(I,x_n)}{(x_n)}\\right)_{\\geq m}$ belongs to $\\underline{\\mathrm{Mf}}_{\\mathcal P({J'}_{\\geq m})}(A)$.\n\\end{lemma}\n\n\\begin{proof}\nFor every $t\\geq m$ we have \n\\begin{equation}\\label{eq:uguaglianza Hilbert function}\nh_{A[\\mathbf x,x_n]\/(I,x_n)}(t)=h_{A[\\mathbf x,x_n]\/(J,x_n)}(t)=h_{K[\\mathbf x]\/J'}(t).\n\\end{equation}\nIndeed, $h_{A[\\mathbf x,x_n]\/I}(t-1)= h_{A[\\mathbf x,x_n]\/J}(t-1)$ because $I_{\\geq m-1}$ has a $\\mathcal P(J_{\\geq m-1})$-marked basis by hypothesis; for every $t\\geq 0$, we have $h_{A[\\mathbf x,x_n]\/(I,x_n)}(t)=\\Delta h_{A[\\mathbf x,x_n]\/I}(t)$ and $h_{A[\\mathbf x,x_n]\/(J,x_n)}(t)=\\Delta h_{A[\\mathbf x,x_n]\/J}(t)$, because $J$ and $I$ are saturated in $A[\\mathbf x,x_n]$ and $x_n$ is generic for $J$ and also for $I$, by \\cite[Theorem 3.5 and Corollary 3.7]{BCR2}; for every $t\\geq m$, we obtain $h_{A[\\mathbf x,x_n]\/(J,x_n)}(t)=h_{K[\\mathbf x]\/J'}(t)$, because $J$ is a lifting of $J'$ and then $\\left(\\frac{(J,x_n)}{(x_n)}\\right)_{\\geq m}={J'}_{\\geq m}\\otimes A$ by Lemma \\ref{lemma:Pommaret}.\n\nA $\\mathcal P(J_{\\geq m-1})$-marked set $\\mathcal G$ of $I_{\\geq m-1}$ is of the following type:\n\\begin{equation}\\label{eq:base marcata Im-1}\n\\mathcal G =\\left \\{f_\\alpha=x^\\alpha +  \\sum_{\\mathcal N(J)_{\\vert \\alpha\\vert} \\ni x^\\gamma} C_{\\alpha\\gamma} x^\\gamma \\ : \\\n\\mathrm{Ht}(f_\\alpha)=x^\\alpha\\in \\mathcal P(J_{\\geq m-1})\\right\\}\\subset K[\\mathcal C_{J_{\\geq m-1}}][\\mathbf x,x_n],\n\\end{equation}\nwhere $\\mathcal C_{J_{\\geq m-1}}$ is the set of the variables $C_{\\alpha\\gamma}$. \nLet $\\mathcal A_{J_{\\geq m-1}}\\subseteq K[\\mathcal C_{J_{\\geq m-1}}]$ be the defining ideal of $\\mathrm{Mf}_{\\mathcal P(J_{\\geq m-1})}$, that is the ideal exactly generated by all the relations which are satisfied by the variables $C_{\\alpha\\gamma}$ in order to obtain that $\\mathcal G$ is a $\\mathcal P(J_{\\geq m-1})$-marked basis. Indeed, the functor $\\underline{\\mathrm{Mf}}_{\\mathcal P(J_{\\geq m-1})}$ is represented by the scheme $\\textnormal{Spec}\\,\\Bigl({K[\\mathcal C_{J_{\\geq m-1}}]}\/{\\mathcal A_{J_{\\geq m-1}}}\\Bigr)$.\nWe now construct a $\\mathcal P({J'}_{\\geq m})$-marked basis for the ideal $\\left(\\frac{(I,x_n)}{(x_n)}\\right)_{\\geq m}$ by taking Lemma \\ref{lemma:Pommaret}  into account.  \n\nFor every $x^\\alpha\\in \\mathcal P(J_{\\geq m})_m$ that is not divisible by $x_n$, we consider its $J_{\\geq m-1}$-reduced form $\\mathrm{Nf}(x^\\alpha)$ modulo $I_{\\geq m-1}$  and the homogeneous polynomial $g_\\alpha:=x^\\alpha-\\mathrm{Nf}(x^\\alpha)\\in I_{\\geq m-1}$. Observe that $\\mathrm{Nf}(x^\\alpha)$ exists for properties of marked bases. Then, we take the following set that is a $\\mathcal P({J'}_{\\geq m})$-marked set by Lemma \\ref{lemma:Pommaret} and \\cite[Proposition 2.10]{BCR2}\n\\begin{equation}\\label{eq:base marcata Im}\n\\mathcal G':=\\{g_\\alpha|_{x_n=0} \\ : \\ x^\\alpha\\in \\mathcal P(J_{\\geq m})_m \\text{ and } x^\\alpha \\text{ not divisible by } x_n \\}\\cup\\{f_\\beta|_{x_n=0} \\ : \\ f_\\beta \\in \\mathcal G_{\\geq m+1}\\}.\n\\end{equation}\nThe polynomials of $\\mathcal G'$ belong to $(I,x_n)\/(x_n)$.\nBy construction, ${\\mathcal G'}_m$ is a linearly independent set of polynomials and so $h_{A[\\mathbf x,x_n]\/(J,x_n)}(m)=h_{A[\\mathbf x]\/J'}(m)=h_{A[\\mathbf x]\/(\\mathcal G')}(m)$. For every $t\\geq m+1$, by induction we suppose $\\left((I,x_n)\/(x_n)\\right)_{t-1}$ is equal to $({\\mathcal G'})_{t-1}$. We have $I_{t}=A[\\mathbf x, x_n]_1\\cdot I_{t-1}+\\langle \\mathcal G_{t}\\rangle$, hence\n\\[\n\\begin{array}{lll}\n\\left((I,x_n)\/(x_n)\\right)_{t}&=A[\\mathbf x]_1\\cdot \\left( (I,x_n)\/(x_n)\\right)_{t-1}+\\langle {\\mathcal G'}_{t}\\rangle & \\\\\n& =A[\\mathbf x]_1\\cdot (\\mathcal G')_{t-1}+\\langle {\\mathcal G'}_{t}\\rangle & =(\\mathcal G')_t.\n\\end{array}\n\\]\nThen, $\\mathcal G'$ is a $\\mathcal P({J'}_{\\geq m})$-marked set and generates $\\left(\\frac{(I,x_n)}{(x_n)}\\right)_{\\geq m}$. Moreover, by \\cite[Corollary 4.9]{BCR2} $\\mathcal G'$ is a $\\mathcal P({J'}_{\\geq m})$-marked basis because $h_{A[\\mathbf x,x_n]\/(I,x_n)}(t)=h_{K[\\mathbf x]\/J'}(t)$ for every $t\\geq m$, like observed in \\eqref{eq:uguaglianza Hilbert function}. Thus, $\\left(\\frac{(I,x_n)}{(x_n)}\\right)_{\\geq m}$ belongs to $\\underline{\\mathrm{Mf}}_{\\mathcal P({J'}_{\\geq m})}(A)$.\n\\end{proof}\n\n\\begin{remark}\\label{rem:insufficienza marked schemes}\nMarked schemes do not give us a characterization of liftings because ${I'}_{\\geq m}\\in \\underline{\\mathrm{Mf}}_{\\mathcal P({J'}_{\\geq m})}(K)$ does not imply $I'\\in \\underline{\\mathrm{Mf}}_{\\mathcal P(J')}(K)$ \\cite[Example 3.8]{BCR2} \\label{ex:BCR2}.\n\\end{remark}\n\nIn the proof of Lemma \\ref{lemma:sezioni ideali marcati} we have denoted by $\\mathcal A_{J_{\\geq m-1}}\\subseteq K[\\mathcal C_{J_{\\geq m-1}}]$ the defining ideal of $\\mathrm{Mf}_{\\mathcal P(J_{\\geq m-1})}$. In the same hypothesis of Lemma \\ref{lemma:sezioni ideali marcati}, we can also consider the $\\mathcal P({J'}_{\\geq m})$-marked set $\\mathcal G'$ of $\\left(\\frac{(I,x_n)}{(x_n)}\\right)_{\\geq m}$ which is given in formula \\eqref{eq:base marcata Im} and which is a marked basis under the conditions posed by the ideal $\\mathcal A_{J_{\\geq m-1}}$. \n\nLet $\\mathcal F$ be the set of the ${J'}_{\\geq m}$-normal forms modulo $I'$ of the polynomials in $\\mathcal G'$ (which can be computed by the reduced Gr\\\"obner basis of $I'$ with respect to degrevlex) and denote by $\\mathcal B_{{J'}_{\\geq m}}\\subseteq K[\\mathcal C_{J_{\\geq m-1}}]$ the ideal generated by the coefficients of the terms in the polynomials of $\\mathcal F$. Thus, $(\\mathcal G')=((I,x_n)\/(x_n))_{\\geq m}$ is contained in $I'$ under the conditions posed by the ideal $\\mathcal B_{{J'}_{\\geq m}}$. \n\nIn conclusion, the ideal $\\mathcal A_{J_{\\geq m-1}}+\\mathcal B_{{J'}_{\\geq m}}$ exactly gives the conditions that $\\mathcal G'$ is a marked basis and that $(\\mathcal G')=((I,x_n)\/(x_n))_{\\geq m}$ is contained in $I'$.\n\n\\begin{theorem}\\label{th:aperti}\nLet $J'\\subset K[\\mathbf x]$ be a saturated quasi-stable ideal, $J\\subset A[\\mathbf x,x_n]$ a monomial lifting of $J'$ and $\\rho$ the satiety of $(J,x_n)\/(x_n)\\subset A[\\mathbf x]$.  For every $m\\geq \\rho$, \nthe functor $\\underline{\\mathrm{L}}_{Y,J_{\\geq m-1}}^{p(t)}$ is representable by $\\textnormal{Spec}\\,\\Bigl({K[\\mathcal C_{J_{\\geq m-1}}]}\/{(\\mathcal A_{J_{\\geq m-1}}+\\mathcal B_{{J'}_{\\geq m}})}\\Bigr)$.\n\\end{theorem}\n\n\\begin{proof}\nBy Lemma \\ref{lemma:sottofuntore aperto}(i), a saturated ideal $I$ belongs to $\\underline{\\mathrm{L}}_{Y,J_{\\geq m-1}}^{p(t)}(A)$ if and only if it is a lifting of $I'$ and $I_{\\geq m-1}$ has a $\\mathcal P(J_{\\geq m-1})$-marked basis. \nBy Lemma \\ref{lemma:sezioni ideali marcati}, if $I\\subset A[\\mathbf x,x_n]$ is a saturated ideal such that $I_{\\geq m-1}$ has a $\\mathcal P(J_{\\geq m-1})$-marked basis $\\mathcal G$, we can assume that $\\mathcal G$ is of type  \\eqref{eq:base marcata Im-1} under the conditions posed by the ideal $\\mathcal A_{J_{\\geq m-1}}$.\nThen, the ideal $\\mathcal A_{J_{\\geq m-1}}+\\mathcal B_{{J'}_{\\geq m}}$ gives the conditions that the ideal $\\left(\\frac{(I,x_n)}{(x_n)}\\right)_{\\geq m}$ has a $\\mathcal P({J'}_{\\geq m})$-marked basis $\\mathcal G'$, like in \\eqref{eq:base marcata Im}, so that $(\\mathcal G')=((I,x_n)\/(x_n))_{\\geq m}$ is contained in $I'$. Hence, $\\left(\\frac{(I,x_n)}{(x_n)}\\right)_{\\geq m}$ is $m$-saturated, i.e.~$\\left(\\frac{(I,x_n)}{(x_n)}\\right)_{\\geq m}=\\left(\\left(\\frac{(I,x_n)}{(x_n)}\\right)^{\\textnormal{sat}}\\right)_{\\geq m}$, by \\cite[Corollary 3.7]{BCR2}. Now, we have $\\Bigl((I,x_n)\/(x_n)\\Bigr)^{{\\textnormal{sat}}}=I'$ by Hilbert polynomial arguments, hence $I$ is a lifting of $I'$ and we can conclude.\n\\end{proof}\n\nIn Algorithm \\ref{alg:computation in MS} we collect the main steps of the method applied in the proofs of Lemma \\ref{lemma:sezioni ideali marcati} and of Theorem \\ref{th:aperti} in order to construct the parameter schemes $\\mathrm{L}_{Y,J_{\\geq m-1}}^{p(t)}$.\n\n\\begin{algorithm}[!ht]\n\\caption{\\label{alg:computation in MS} Algorithm for computing the parameter schemes $\\mathrm{L}_{Y,J_{\\geq m-1}}^{p(t)}$ for the liftings of a saturated homogeneous ideal $I'\\subset K[\\mathbf x]$ over a Noetherian $K$-algebra $A$ in $\\mathrm{Hilb}_{p(t)}^n$.}\n\\begin{algorithmic}[1]\n\\STATE $\\textsc{LiftingMS}\\big(I',p(t)\\big)$\n\\REQUIRE $I'\\subset K[\\mathbf x]$ a saturated polynomial ideal with $\\mathrm{in}(I')$  quasi-stable.\n\\REQUIRE  $p(t)$ a Hilbert polynomial such that $\\Delta p(t)= p_Y(t)$, where $p_Y(t)$ is the Hilbert polynomial of the scheme $Y$ defined by $I'$.\n\\ENSURE  A set $\\mathfrak B$ containing parameter schemes for the liftings of $I'$ in $\\mathrm{L}_{Y,J_{\\geq m-1}}^{p(t)}$, for every quasi-stable lifting $J\\subset A[\\mathbf x,x_n]$ of $\\mathrm{in}(I')$ with Hilbert polynomial $p(t)$.\n\\STATE $\\mathcal L:=\\{ J \\subset A[\\mathbf x,x_n] \\ \\vert \\ J \\text{ quasi-stable lifting of } \\mathrm{in}(I') \\text{ with Hilbert polynomial } p(t)\\}$; \n\\STATE $\\mathfrak B=\\emptyset$;\n\\FOR{$J \\in \\mathcal L$}\n\\STATE $m:=\\varrho={\\textnormal{sat}}((J,x_n)\/(x_n))$;\n\\STATE let $\\mathcal A_{J_{\\geq m-1}}\\subseteq K[\\mathcal C_{J_{\\geq m}}]$ be the defining ideal of $\\mathrm{Mf}_{\\mathcal P(J_{\\geq m-1})}$;\n\\STATE let $\\mathcal G'$ be the set of formula \\eqref{eq:base marcata Im};\n\\STATE $\\mathcal B_{{J'}_{\\geq m}}:=(0)$;\n\\FOR{$g\\in \\mathcal G'$}\n\\STATE $\\mathcal B_{{J'}_{\\geq m}}:=\\mathcal B_{{J'}_{\\geq m}}+(\\mathrm{coeff}(\\overline{g}))$, where $\\overline{g}$ is the ${J'}_{\\geq m}$-normal form of $g$ modulo $I'$; \n\\ENDFOR\n\\STATE $\\mathfrak B:=\\mathfrak B\\cup \\{\\mathcal A_{J_{\\geq m-1}}+\\mathcal B_{{J'}_{\\geq m}}\\}$;\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{remark}\nUnlike\nthe ideal $\\mathfrak b_{J\\cap K[\\mathbf x]}$ in the proof of Theorem \\ref{th:parametrizzazione}, the ideal $\\mathcal B_{{J'}_{\\geq m}}\\subseteq K[\\mathcal C_{J_{\\geq m-1}}]$ \nis generated by polynomials which are not necessarily linear. In order to have linear generators, we could repeat the proof of Theorem \\ref{th:parametrizzazione} starting from the $\\mathcal P(J_{\\geq m})$-marked basis of $I_{\\geq m}$ instead of  the $\\mathcal P(J_{\\geq m-1})$-marked basis of $I_{\\geq m-1}$.\nHowever, the number of variables $\\mathcal C_{J_{\\geq m}}$ is much higher than the number of the variables $\\mathcal C_{J_{\\geq m-1}}$: indeed, the ideal $\\mathcal A_{J_{\\geq m-1}}$ can be obtained from $\\mathcal A_{J_{\\geq m}}$ by an elimination of variables (which is a time-consuming process). This elimination of variables applied on the linear generators of $\\mathcal B_{J'_{\\geq m}}\\subseteq K[\\mathcal C_{J_{\\geq m-1}}]$ gives generators of higher degree in a smaller number of variables.  If we use the process of the proof of Theorem \\ref{th:aperti}, the eliminable variables do not appear from the very beginning, allowing the embedding of $\\underline{\\mathrm{L}}_{Y,J_{\\geq m-1}}^{p(t)}$ in a rather small affine space, without an expensive elimination of variables (see \\cite[Section 5.1]{BCLR} for details).\n\\end{remark}\n\n\\begin{theorem}\\label{th:funtore di punti}\n$\\underline{\\mathrm{L}}_Y^{p(t)}$ is representable.\n\\end{theorem}\n\n\\begin{proof}\nWe check that $\\underline{\\mathrm{L}}_Y^{p(t)}$ is a Zariski sheaf and that there are rings $R_i$ and open subfunctors $\\alpha_i:h_{R_i}\\rightarrow \\underline{\\mathrm{L}}_Y^{p(t)}$ such that, for every field $\\mathbb K\\supseteq K$, $\\underline{\\mathrm{L}}_Y^{p(t)}(\\mathbb K)$ is the union of the images of $h_{R_i}(\\mathbb K)$ under $\\alpha_i$. Indeed, these two conditions imply that $\\underline{\\mathrm{L}}_Y^{p(t)}$ is isomorphic to the functor of points of a scheme, by \\cite[Theorem VI-14]{EH}.\n\nWe already observed that $\\underline{\\mathrm{L}}_Y^{p(t)}$ is a Zariski sheaf (Proposition \\ref{prop:Zariski sheaf}).\nFor the other condition, due to Theorems \\ref{th:aperti} and \\ref{th:lifting}, we can consider the finite set of the open subfunctors $\\underline{\\mathrm{L}}_{Y,J_{\\geq m-1}}^{p(t)}$ of the functor $\\underline{\\mathrm{L}}_Y^{p(t)}$, where $J$ varies among all the quasi-stable ideals in $K[\\mathbf x,x_n]$ that are liftings of $J'$ with $p(t)$ as Hilbert polynomial of $A[\\mathbf x,x_n]\/J$.\n\nBy Theorem \\ref{th:aperti}, for every of these ideals $J$, the open functor $\\underline{\\mathrm{L}}_{Y,J_{\\geq m-1}}^{p(t)}$ is the functor of points of the ring $R_J:=K[\\mathcal C_{J_{\\geq m}}]\/(\\mathcal A_{J_{\\geq m-1}}+\\mathcal B_{{J'}_{\\geq m}})$.\n\nAs recalled in Theorem \\ref{th:main features}(ii), for every non-negative integer $m$ and every $K$-algebra $A$ we have $\\underline{\\mathrm{St}}_{J}(A) \\ \\simeq \\ \\underline{\\mathrm{St}}_{{J}_{\\geq m}}(A)$, with the degrevelex order, and hence,\n\\begin{equation}\\label{eq:contenuti funtori}\n\\underline{\\mathrm{St}}_{J}(A) \\ \\simeq \\ \\underline{\\mathrm{St}}_{{J}_{\\geq m}}(A) \\ \\subseteq \\ \\underline{\\mathrm{Mf}}_{\\mathcal P({J}_{\\geq m})}(A).\n\\end{equation}\nThus, thanks to Theorem \\ref{th:dove}, for every $K$-algebra $A$ we obtain $\\underline{\\mathrm{L}}_Y^{p(t)}(A)=\\cup_J \\ \\underline{\\mathrm{L}}_{Y,J_{\\geq m-1}}^{p(t)}(A)$. This holds in particular for $A=\\mathbb K\\supseteq K$.\n\\end{proof}\n\n\\begin{remark}\\label{rem:stromme} \nConcerning the proof of Theorem \\ref{th:funtore di punti}, we can observe that the open subfunctors $\\underline{\\mathrm{L}}_{Y,J_{\\geq m-1}}^{p(t)}$ form an open covering of the functor $\\underline{\\mathrm{L}}_Y^{p(t)}$ (e.g.~\\cite[Definition 2.15]{Stromme}). Thus, \\cite[Proposition 2.16]{Stromme}  clarifies that a scheme defining a functor of points isomorphic to $\\underline{\\mathrm{L}}_Y^{p(t)}$ can be constructed by means of fiber products of the subfunctors $\\underline{\\mathrm{L}}_{Y,J_{\\geq m-1}}^{p(t)}$.\n\\end{remark}\n\nIn next example we show that there are liftings in a marked scheme over a quasi-stable ideal that do not belong to the Gr\\\"obner stratum over the same ideal.\n\n\\begin{example}\\label{ex:esempio bello}\nConsider the scheme $Y$ defined by the ideal $I'\\subset K[x_0,\\dots,x_3]=K[\\mathbf x ]$ of Example \\ref{ex:un primo calcolo} and the quasi-stable ideal $J^{(2)}=(x_2x_0, x_0^2, x_3x_1x_0, x_2x_1^2, x_1^3, x_1^2x_0)$. \nFollowing the proof of Lemma \\ref{lemma:sezioni ideali marcati} and the proof of Theorem \\ref{th:aperti}, we can construct the Noetherian $K$-algebra $K[\\mathcal C_{J^{(2)} _{\\geq m-1}}]\/(\\mathcal A_{J^{(2)}_{\\geq m-1}}+\\mathcal B_{{J'}_{\\geq m}})$ defining the functor $\\underline{\\mathrm{L}}_{Y,J^{(2)}_{\\geq m-1}}^{p(t)}$, with $p(t)=t^2+4t+1$, and $m=3$. \nThen, we observe that there are some liftings of $Y$ which belong to $\\underline{\\mathrm{L}}_{Y,{J^{(2)}}_{\\geq m-1}}^{p(t)}(K)$ but do not belong to $\\underline{\\mathrm{St}}_{{J^{(2)}}_{\\geq m-1}}(K)$. For instance, for every $e\\in K, e\\neq 0$, the ideal\n\\[\nI=(x_2x_0-ex_1^{2}-ex_1x_0,x_0^{2},x_3\nx_1^{2}+x_3x_1x_0,x_2x_1^{2},x_1^{3},\nx_1^{2}x_0,x_0x_1x_2)\n\\]\nis a lifting of $I'$, it has a marked basis over ${J^{(2)}}_{\\geq 2}$, hence it belongs to $\\underline{\\mathrm{L}}_{Y,{J^{(2)}}_{\\geq m-1}}^{p(t)}(K)$, but $\\mathrm{in}(I)=J^{(3)}$: this means that $I$ does not belong to $\\underline{\\mathrm{St}}_{{J^{(2)}}_{\\geq m-1}}(K)$. Moreover we observe that the marked basis of $I$ cannot be a Gr\\\"obner basis with respect to some term order, because  the term $x_1x_0$ is higher than $x_2x_0$ with respect to any term order.\nFor more information about the computations performed in this example, at the url http:\/\/wpage.unina.it\/cioffifr\/MaterialeCoCoALiftingGeometrico and the interested reader can find lists of generators for the ideals $\\mathcal A_{J^{(2)}_{\\geq m-1}}$ and $\\mathcal B_{{J'}_{\\geq m}}$ and a code written for CoCoA-4.7.5 \\citep{CoCoA} in order to construct a marked family.\n\\end{example}\n\n\n\\section*{Acknowledgements}\n\nThe investigations that gave rise to this paper began with a study shared with Margherita Roggero, to whom we are grateful for useful suggestions and stimulating discussions. We also thank the anonymous referees for useful comments and suggestions.\n\nThe first and second authors are members of GNSAGA (INdAM, Italy). The second author is partially supported by Prin 2015 (2015EYPTSB\\_011 - {\\em Geometry of Algebraic varieties}, Unit\\`{a} locale Tor Vergata, CUP  E82F16003030006).\n\n\n\\bibliographystyle{elsarticle-harv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{Sparse Sampling}\nIn training, we need to reduce the number of the point cloud for calculation of the WCL owing to limited GPU memory usage.\nWe uniformly sample the point clouds on the depth image, as shown in Fig.~\\ref{f:sample}.\nHere, $n_c$ and $n_r$ indicate vertical and horizontal grid point intervals, respectively.\n$m_c$ and $m_r$ indicate random offsets less than and equal to $n_c$ and $n_r$, respectively.\nWe cover the whole point clouds in training by randomly choosing $m_c$ and $m_r$ in each training iteration.\n\n\\begin{figure}[h]\n  \\begin{center}\n  \\hspace*{-5mm}\n      \\includegraphics[width=0.7\\hsize]{fig\/sampling.png}\n  \\end{center}\n      \\vspace*{-3mm}\n\t\\caption{\\small {\\bf Sparse sampling of point clouds.} Red dots are the position of sampled point clouds in the image coordinates.}\n  \\label{f:sample}\n  \\vspace*{-3mm}\n\\end{figure}\n\n\\subsection{Metrics of Depth Evaluation}\nIn quantitative analysis, we evaluate the estimated depth using seven metrics.\n``Abs Rel'', ``Sq Rel'', ``RMSE'', and ``RMSE log'' in Table~\\ref{tab:ev} are the means of the following values in the entire test data. \n\\begin{itemize}\n  \\item Abs Rel\\hspace{9mm}:\\hspace{2mm} $|D_{gt} - \\hat{D}|\/D_{gt}$\n  \\item Sq Rel\\hspace{11mm}:\\hspace{2mm} $(D_{gt} - \\hat{D})^2\/D_{gt}$\n  \\item RMSE\\hspace{11mm}:\\hspace{2mm} $\\sqrt{(D_{gt} - \\hat{D})^2}$\n  \\item RMSE log\\hspace{6mm}:\\hspace{2mm} $\\sqrt{(\\mbox{log}(D_{gt}) - \\mbox{log}(\\hat{D}))^2}$\n\\end{itemize}\nHere, $D_{gt}$ denotes the ground truth of the depth image and $\\hat{D}$ denotes estimated depth. The remaining three metrics are the ratios to satisfy with $\\delta < \\alpha$. $\\delta$ is calculated as follows:\n\\begin{eqnarray}\n    \\delta = \\mbox{max}(D_{gt}\/\\hat{D}, \\hat{D}\/D_{gt}).\n    \\label{eq:delta}\n\\end{eqnarray}\nAs with prior works, we have three $\\alpha$ as 1.25, 1.25$^2$ and 1.25$^3$.\n\n\\subsection{Ablation Study}\nTable~\\ref{tab:ab} shows the results of the ablation study in monodepth2~\\cite{godard2019digging} with our WCL; the upper part is for weighting factor $\\lambda_{w}$ and the lower part is for $n_c$ and $n_r$ of sparse sampling. At the top, $\\lambda_{w}$ is changed under fixed $n_c$ and $n_r$; at the bottom, $n_c$ and $n_r$ are changed under fixed $\\lambda_{w}$. From the top, it can be seen that increasing the weighting factor can improve the depth estimation performance but making it too large worsens the performance; in this case, the best result is obtained at $\\lambda_{w}=0.5$.\n\nIn the bottom part, better performance can be obtained at smaller $n_c$ and $n_r$ as geometric inconsistencies can be accurately penalized. However, as it is difficult to make $n_c$ and $n_r$ even smaller due to the GPU memory limitations, we decide $n_c=16$ and $n_r=4$ for the evaluation in Table~\\ref{tab:ev}.\n\\begin{table*}[h]\n  \\caption{{\\small {\\bf Ablation study on weighting value $\\lambda_w$(top) and sparse sampling parameters $n_c$ and $n_r$(bottom)} for monodepth2~\\cite{godard2019digging} + WCL. All metrics for evaluation are the  same as in Table~\\ref{tab:ev}. Input and output resolutions are 416$\\times$128.}}\n \n  \\begin{center}\n  \\resizebox{1.0\\columnwidth}{!}{\n  \\label{tab:ab}\n  \\begin{tabular}{|lccc|c|c|c|c||c|c|c|} \\hline\n    method & $\\lambda_w$ & $n_c$ & $n_r$ & Abs Rel & Sq Rel & RMSE & RMSE log & $\\delta < 1.25$ & $\\delta < 1.25^2$ & $\\delta < 1.25^3$ \\\\ \\hline\n    monodepth2~\\cite{godard2019digging} & $-$ & $-$ & $-$ & 0.128 & 1.087 & 5.171 & 0.204 & 0.855 & 0.953 & 0.978 \\\\\n    \\bf{+ WCL} & 0.1 & 16 & 4 & 0.127 & 0.995 & 5.039 & 0.204 & 0.853 & \\bf{0.954} & 0.979 \\\\ \n    & 0.3 & 16 & 4 & 0.125 & 0.965 & 5.019 & 0.202 & 0.856 & 0.953 & 0.979 \\\\  \n    & 0.5 & 16 & 4 & \\bf{0.123} & \\bf{0.920} & \\bf{4.990} & \\bf{0.201} & \\bf{0.858} & 0.953 & \\bf{0.980} \\\\\n    & 0.7 & 16 & 4 & 0.124 & 0.922 & 5.023 & \\bf{0.201} & 0.853 & \\bf{0.954} & \\bf{0.980} \\\\\n    & 1.0 & 16 & 4 & 0.128 & 0.993 & 5.158 & 0.205 & 0.847 & 0.950 & 0.979 \\\\ \n    & 2.0 & 16 & 4 & 0.132 & 1.045 & 5.201 & 0.207 & 0.845 & 0.949 & 0.978 \\\\ \\hdashline \n    \\bf{+ WCL} & 0.5 & 64 & 16 & 0.127 & 0.961 & 5.143 & 0.204 & 0.849 & 0.952 & 0.979 \\\\\n    & 0.5 & 32 & 16 & 0.127 & 0.976 & 5.052 & 0.203 & 0.852 & \\bf{0.953} & 0.979 \\\\ \n    & 0.5 & 64 & 8 & 0.126 & 0.957 & 5.081 & 0.203 & 0.853 & \\bf{0.953} & 0.979 \\\\       \n    & 0.5 & 32 & 8 & 0.126 & 0.933 & 5.039 & 0.203 & 0.853 & 0.952 & 0.979 \\\\\n    & 0.5 & 16 & 8 & 0.125 & 0.933 & 5.039 & 0.203 & 0.853 & 0.952 & 0.979 \\\\\n    & 0.5 & 32 & 4 & 0.125 & 0.938 & 5.006 & 0.202 & 0.854 & \\bf{0.953} & \\bf{0.980} \\\\      \n    & 0.5 & 16 & 4 & \\bf{0.123} & \\bf{0.920} & \\bf{4.990} & \\bf{0.201} & \\bf{0.858} & \\bf{0.953} & \\bf{0.980} \\\\ \\hline\n  \\end{tabular}\n  }\n\\end{center}\n \n\\end{table*}\n\\subsection{Additional Visualization of Estimated Depth Image}\nTo clarify the benefits of our WCL visually, we display the depth images for eight additional cases in Fig.~\\ref{f:pred_depth_1} and \\ref{f:pred_depth_2}.\nFig.~\\ref{f:pred_depth_1} and \\ref{f:pred_depth_2} show interpolated ground truth(GT) and estimated depth images by various methods, including four original works by \\cite{zhou2017unsupervised,mahjourian2018unsupervised,gordon2019depth,godard2019digging}, and these prior works with our WCL.\nOur WCL can sharpen the estimated depth image against the original methods and reduce any artifacts.\n\\begin{figure*}[t]\n  \\begin{center}\n \n      \\includegraphics[width=0.99\\hsize]{fig\/fig_depth_1.pdf}\n  \\end{center}\n     \n\t\\caption{\\small {\\bf Additional qualitative results of our proposed method (1).} The top row displays RGB images, the second row displays interpolated ground truth depth images, and the other rows display the estimated depth images, with and without our WCL. As the ground truth is sparse, we interpolate only for visualization. All estimated images are estimated from 416$\\times$128 RGB images.}\n  \\label{f:pred_depth_1}\n \n\\end{figure*}\n\n\\begin{figure*}[t]\n  \\begin{center}\n \n      \\includegraphics[width=0.99\\hsize]{fig\/fig_depth_2.pdf}\n  \\end{center}\n     \n\t\\caption{\\small {\\bf Additional Qualitative results of our proposed method (2).} The top row displays RGB images, the second row displays interpolated ground truth depth images, and the other rows display the estimated depth images, with and without our WCL. As the ground truth is sparse, we interpolate only for visualization.  All estimated images are estimated from 416$\\times$128 RGB images.}\n  \\label{f:pred_depth_2}\n \n\\end{figure*}\n\n\n\\section{Introduction}\nUnderstanding the three-dimensional (3D) structures of environments and objects is important for the navigation of autonomous vehicles and for robotic manipulation\\cite{thrun2002probabilistic,biswas2012depth}.\nIn recent years, depth estimation using RGB monocular images has been actively researched owing to the popularity of deep learning. Cameras can function as inexpensive and affordable sensors for autonomous vehicles and robots~\\cite{hirose2019deep,hirose2018gonet}.\nHence, they can be viable alternatives to expensive LIDARs. However, it is difficult to obtain a considerable number of depth image ground truths from LIDAR data, for application to machine learning techniques, and data collection is itself a challenge.\n\nSelf-supervised learning using videos captured by robots and vehicles enables the learning of depth and pose estimation networks, without ground truth depth images and poses. Here, pose estimation refers to the relative camera pose estimation between two consecutive images. The image reconstruction loss proposed by Zhou et al.~\\cite{zhou2017unsupervised} has significantly improved accuracy. Thereafter, various pluggable objectives, network structures, data augmentation methods, and masking methods for dynamic and occluded objects have been suggested to enable improvement~\\cite{zhou2017unsupervised,yang2018every,yang2018lego,mahjourian2018unsupervised,wang2018learning,casser2019depth,gordon2019depth,godard2019digging,fei2019geo,pillai2019superdepth}.\n\nIn this paper, we propose a novel pluggable objective to evaluate geometric inconsistencies with respect to these prior methods.\nAs the geometry of objects or environments does not depend on the camera pose, the estimated two sets of 3D point clouds in the same coordinates from different camera pose should be consistent.\n\nSeveral previous studies have proposed various objectives that attempt to penalize inconsistencies in 3D geometric constraints~\\cite{mahjourian2018unsupervised,gordon2019depth,fei2019geo,luo2020consistent}.\nHowever, their performance is limited owing to a geometrically inconsistent evaluation with an approximation and the dependence on other undifferentiable algorithms to obtain the coupling between point clouds~\\cite{mahjourian2018unsupervised,gordon2019depth,fei2019geo,luo2020consistent}.\nIn this work, we address these issues to improve the depth and pose estimation accuracy. Inspired by recent successful works on the Wasserstein distance for different tasks, such as generative modeling~\\cite{arjovsky2017wasserstein}, embedding~\\cite{huang2016supervised,courty2018learning}, and domain adaptation~\\cite{courty2017optimal}, we extend these applications to depth and pose estimation.\nThe proposed method employs the Wasserstein distance as a pluggable objective to measure the consistency between two point clouds and penalize it for more accurate estimation.\n\nIn contrast to the baselines, our method attempts to measure a geometric consistency from 3D point clouds, without any indirect process and bold approximation.\nIn addition, the mathematical formulation of our objective is smooth and symmetric, which is advantageous for an efficient and effective training process.\n\\begin{figure}[t]\n  \\begin{center}\n  \\hspace*{-5mm}\n      \\includegraphics[width=0.55\\hsize]{fig\/pull_fig_v5.png}\n  \\end{center}\n      \\vspace*{-3mm}\n\t\\caption{\\small {\\bf Evaluation of monocular depth estimation, with and without our proposed Wasserstein consistency loss (WCL) on the KITTI raw dataset, with an image size of 416$\\times$128.} Red markers are the results with the WCL. Gray markers are the baselines without the WCL. Our WCL can enhance the performance of monocular depth estimation. It should be noted that a smaller Sq Rel and RMSE log are better. Mathematical formulation of Sq Rel and RMSE log are explained in supplemental material.}\n  \\label{f:pull_fig}\n  \\vspace*{-3mm}\n\\end{figure}\nThe major contributions of this study are 1) the proposal of a novel objective, the Wasserstein consistency loss~(WCL), to evaluate consistency in 3D geometric constraints, and 2) comparative evaluation to demonstrate that the accuracy of several state-of-the-art approaches is improved by ``plugging in'' the WCL. \nTo the best of our knowledge, our work is the first application using the Wasserstein distance for depth and pose estimation.\nWe quantitatively and qualitatively demonstrate the benefits of our WCL using the KITTI dataset.\n\n\\section{Related Work}\n\\paragraph{Self-supervised monocular depth estimation}\nSelf-supervised depth estimation has recently become popularized ~\\cite{zhou2017unsupervised,vijayanarasimhan2017sfm,yang2018every,ummenhofer2017demon,yang2018lego,mahjourian2018unsupervised,wang2018learning,casser2019depth,gordon2019depth,godard2019digging,fei2019geo,pillai2019superdepth,guizilini2020robust,guizilini2019packnet,patil2020don,johnston2020self,zhang2020online,garg2016unsupervised,godard2017unsupervised}.\nZhou et al.~\\cite{zhou2017unsupervised} and Vijayanarasimhan et al.~\\cite{vijayanarasimhan2017sfm} demonstrated one of the first approaches for self-supervised monocular depth and pose estimation.\nTheir approach simultaneously trained the model to estimate depth and pose based on knowledge of structure from motion (SfM).\nGarg et al.~\\cite{garg2016unsupervised} and Godard et al.\\cite{godard2017unsupervised} proposed image reconstruction using SfM techniques between stereo images to estimate the depth, without pose estimation.\n\nSeveral works were subsequently presented to improve accuracy.\nCasser et al.~\\cite{casser2019depth} introduced a motion mask based on semantic segmentation results to remove the dynamic object effect.\nGodard et al.~\\cite{godard2019digging} proposed monodepth2, which had a modified image reconstruction loss, considering occlusion.\nCS Kumar et al.~\\cite{cs2018depthnet} applied a recurrent neural network to understand the environment geometry from a video.\nPillai et al.~\\cite{pillai2019superdepth} demonstrated that a higher resolution input image can enable a more accurate estimation.\nGuizilini et al.~\\cite{guizilini2020robust} proposed image reconstruction in different camera poses to update the neural network in semi-supervised learning efficiently.\nIn addition to these emergent techniques, the following section presents most works related to penalizing geometric inconsistencies between two point clouds estimated from different poses.\n\n\n\\paragraph{Penalization of Geometric Inconsistency}\nMahjourian et al.~\\cite{mahjourian2018unsupervised} proposed a 3D point cloud alignment loss (iterative closest point (ICP) loss) to penalize the inconsistencies between two point clouds.\nTheir approach employs an ICP to form the coupling between two point clouds estimated from the images captured at different poses.\nHowever, an ICP is not differentiable. Hence, the ICP loss separately computes the pose inconsistencies and the residual error between two point clouds, with approximation.\n\nGordon et al.~\\cite{gordon2019depth} proposed the depth and ego-motion estimation system in the wild by learning the intrinsic camera parameters.\nIn addition, they introduced a depth consistency loss to minimize the difference between two estimated depth images from different frames.\nThey shared a bi-linear sampler for the image reconstruction loss~\\cite{zhou2017unsupervised,jaderberg2015spatial} to determine the coupling and penalize any geometric inconsistency. As coupling is determined from the estimated depth itself, the positive effect of their depth consistency loss can be limited because the estimated depth will be indefinite.\n\nLuo et al.~\\cite{luo2020consistent} leveraged the optical flow to establish the coupling between point clouds and used these couplings for extracting 3D geometric constraints.\nAs the optical flow is estimated by a pre-trained neural network, the coupling performance largely depends on the domain of the dataset. It is reported that their approach would not work well on the KITTI dataset.\n\nFei et al.~\\cite{fei2019geo} introduced a semantically informed geometric loss to penalize deviations from a horizontal or vertical plane.\nThey leveraged the results of the semantic segmentation from a pre-trained network and an inertial measurement sensor to determine if the segmented areas belonged to a horizontal or vertical plane. Although the surface of some objects can be an exact horizontal or vertical plane, e.g., a wall, and road, their method cannot support most objects with complex shapes.\n\n\\section{Proposed Method}\n\\begin{figure*}[t]\n  \\begin{center}\n  \\hspace*{-5mm}\n      \\includegraphics[width=0.95\\hsize]{fig\/overview_v5.png}\n  \\end{center}\n      \\vspace*{-3mm}\n\t\\caption{\\small {\\bf Overview of our proposed approach.} We feed RGB images $I_A$ and $I_B$ into a neural network to estimate depth images $D_A$ and $D_B$, respectively. From $D_A$ and $D_B$, we obtain the point clouds $Q_A^A$ and $Q_B^B$, and $Q_A^B$, and $Q_B^A$, respectively, by using the intrinsic camera parameters and the estimated transformation matrices, $T_{A \\rightarrow B}$ and $T_{B \\rightarrow A}$, respectively. To penalize a geometric inconsistency, we propose the addition of the WCL $L_{wass}$ to the original cost functions, $L_{origin}$, of recent state-of-the-art approaches; this is done to train the neural networks.}\n  \\label{f:overview}\n \n\\end{figure*}\n\n\\begin{figure}[t]\n  \\begin{center}\n  \\hspace*{-3mm}\n    \\includegraphics[width=0.8\\hsize]{fig\/benefit7.png}\n  \\end{center}\n    \\vspace{-3mm}\n    \\caption{\\small {\\bf Overview of the baseline and our coupling.} The three red points in each image indicate the position of the estimated depth in the image space. The arrows between images at pose A and B indicate the coupling example. The number on the arrow is the assigned mass, which is delivered to the coupled points. The thickness of the arrow visually displays its assigned mass. Our coupling with the Wasserstein distance is a soft coupling with the preservation law, which can completely receive and deliver the same constant value (=$1\/m$=$1\/n$=1\/3). Our coupling is symmetric, which is different from the baselines.}\n  \\label{f:benefit}\n\\end{figure}\n\n\\subsection{Overview}\nFrom previous studies, it is known that the penalization of a geometric inconsistency can enhance depth and pose estimation accuracy ~\\cite{mahjourian2018unsupervised,gordon2019depth,fei2019geo,luo2020consistent}. \nInspired by these approaches, we propose a novel pluggable WCL, shown as $L_{wass}$ in Fig.~\\ref{f:overview}, for depth and pose estimation.\n$L_{wass}$ works by adding $L_{origin}$ from the target method to the original objective, as shown in Fig.~\\ref{f:overview}. In this paper, we focus on explaining $L_{wass}$, instead of the mathematical formulation of $L_{origin}$, which is the same as in prior works.\n\\paragraph{Preliminary} All prior works have commonly estimated the depth image as $D_A = f_{depth}(I_A)$, where $f_{depth}(\\cdot)$ is a neural network for depth estimation, and $I_A$ is the image at camera pose $A$.\nThe estimated depth image, $D_A$, can be projected on to a 3D point cloud $Q_A^{A}$.\n\\begin{eqnarray}\n    Q_A^A[i,j] = D_A[i,j] \\cdot K^{-1} [i,j,1]^T\n    \\label{eq:pc}\n\\end{eqnarray}\nHere, $i,j$ denotes the pixel position in the image coordinates, $K$ denotes an intrinsic camera parameter, and $Q_X^Y$ denotes a 3D point cloud at coordinate $X$ from $I_Y$.\nIt should be noted that $K$ is given as a constant matrix, except for the baseline~\\cite{gordon2019depth}.\nBy calculating \\eqref{eq:pc} for the entire image space, we obtain the point cloud $Q_A^A$ at coordinate $A$. \nBy applying the same process to $I_B$, we can further obtain $Q_B^B$.\nIn addition, $Q_A^A$ and $Q_B^B$ can be transformed into mutual coordinates as $Q_B^A$ and $Q_A^B$, respectively, by multiplying with the transformation matrix $T_{A \\rightarrow B}$ and $T_{B \\rightarrow A}$, respectively.\n$T_{A \\rightarrow B}$ and $T_{B \\rightarrow A}$ are estimated from $I_A$ and $I_B$, respectively, through another neural network.\n$(Q_A^A, Q_A^B)$ and $(Q_B^B, Q_B^A)$ are sets of estimated point clouds at coordinates A and B, respectively.\n\n\\paragraph{Overview of WCL} Similar to~\\cite{mahjourian2018unsupervised,gordon2019depth,luo2020consistent}, our WCL penalizes the inconsistency between two point clouds at the same coordinate. $L_{wass}$ is defined as follows\n\\begin{eqnarray}\n    L_{wass} = \\widetilde W^2(Q_A^A, Q_A^B) + \\widetilde W^2(Q_B^B, Q_B^A).\n    \\label{eq:Lwass}\n\\end{eqnarray}\nHere, $\\widetilde W^2(\\cdot)$ is the (squared) Wasserstein distance between two point clouds, as is subsequently described.\nWe add $\\lambda_w \\cdot L_{wass}$ to the original objective, $L_{origin}$, and update the neural networks for depth and pose estimation by minimizing the combined objective.\n$\\lambda_w$ is a weighting factor of $L_{wass}$ to balance $L_{origin}$.\nThe benefits of our WCL are,\n\\begin{enumerate}\n  \\item geometrically consistent penalization between 3D point clouds,\n \n  \\item smooth and symmetric objective,\n  \\item simple implementation.\n\\end{enumerate}\nIn the baselines~\\cite{mahjourian2018unsupervised,gordon2019depth,luo2020consistent}, the geometric consistency between two point clouds is calculated in two steps: 1) taking a coupling between two point clouds and 2) calculating the distance(e.g., Euclidean distance) between the aligned point clouds using the coupling. \n\nFig.~\\ref{f:benefit} shows a simple example of a coupling using three points ($m=n=3$)  on each image. \nHere, $m$ and $n$ are the size of two point clouds estimated from $I_A$ and $I_B$, respectively.\nIn the baseline coupling~\\cite{mahjourian2018unsupervised,gordon2019depth,luo2020consistent}, multiple geometrically different points on one image can correspond to one point on the other image and the coupling of $A \\rightarrow B$ and $B \\rightarrow A$ can be asymmetric, as shown in Fig.~\\ref{f:benefit}[a].\nThese behaviors are geometrically inconsistent.\nFurthermore, the baselines cause discontinuous coupling between each training iteration as updating a neural network for depth estimation changes the coupling, and the coupling itself is a hard coupling, that one point on $I_A$ can make a coupling only with one point on $I_B$ for $A \\rightarrow B$ coupling. The same is true for $B \\rightarrow A$ coupling.\nIt means that the whole assigned mass, $\\frac{1}{3} ( = \\frac{1}{m} = \\frac{1}{n})$, is delivered to exactly one point. Here, ``mass'' refers to the weight equally assigned to all points and is a term mainly used in optimal transport~\\cite{peyre2019computational}.\n\n\nOur proposed WCL can address these issues to improve the depth and pose estimation accuracy. \nIn contrast to the baselines, our coupling with the Wasserstein distance is a soft coupling in which the mass assigned is preserved, as shown in Fig.~\\ref{f:benefit}[b]. The sum of the delivered and received values are the same as the assigned mass $\\frac{1}{3}$.\nMoreover, our soft coupling by the Wasserstein distance is symmetric, as shown in Fig.~\\ref{f:benefit}[b].\nHence, our WCL can enforce avoidance of geometrically inconsistent penalization. \nIn addition, our WCL can simultaneously 1) take a soft coupling and 2) calculate the distance between two point clouds.\nThese processes of our WCL are totally differentiable without involving other external libraries (e.g., ICP) and results in a more stable learning compared to non-differentiable baselines~\\cite{mahjourian2018unsupervised,fei2019geo,luo2020consistent}.\nAs a result, this entire algorithm can be implemented only on a GPU. Furthermore, the code size of our WCL is relatively short, as shown later.\n\\subsection{WCL}\nWe introduce the \\emph{Wasserstein distance}~\\cite{peyre2019computational} and its approximation $\\widetilde W^2(\\cdot)$ used in $L_{wass}$, which provides a differentiable loss function that penalizes the geometric inconsistency between two point clouds.\n\n\\paragraph{Preliminary: Definition and Notation}\nIn this subsection, we express the point clouds as ${\\cal X}=\\{x_1,\\cdots,x_m\\}$ and ${\\cal Y}=\\{y_1,\\cdots,y_n\\}$, to simplify and generalize the mathematical formulation.\nHere, ${\\cal X}$ and ${\\cal Y}$ are a set of points in $\\mathbb{R}^3$.\n$\\langle \\vec{U}, \\vec{V}\\rangle$ denotes the inner product between two vectors (or matrices) of the same size. $\\onevec{n}$ is an $n$-dimensional vector, whose elements are all one.\n\n\\paragraph{Wasserstein Distance}\nThe \\emph{Wasserstein distance}\\footnote{Originally, the Wasserstein distance was defined for probabilistic distributions. Currently, point clouds can be regarded as mixtures of delta functions; hence, we can deal with point clouds using the Wasserstein distance.} is a metric between two point clouds.\nThis is defined as the sum of the squared Euclidean distances between two ``coupled'' points in ${\\cal X}$ and ${\\cal Y}$.\nAn example is shown in Fig.~\\ref{fig:wass}(a).\nSuppose two point clouds ${\\cal X}=\\{x_1, x_2, x_3\\}$ and ${\\cal Y}=\\{y_1, y_2, y_3\\}$, and consider a coupling $(x_1, y_3)$, $(x_2, y_1)$, $(x_3, y_2)$.\nThis coupling clearly minimizes the total distance between coupled points; no other coupling can reduce the total distance (e.g., the coupling in Fig.~\\ref{fig:wass}(c)).\nHowever, such an optimal coupling is unknown in advance; therefore, we need to determine the optimal coupling that minimizes the total distance (e.g., a non-optimal coupling in Fig.~\\ref{fig:wass}(c) does not provide the valid Wasserstein distance).\nEssentially, identifying such an optimal coupling corresponds to the following optimization problem~\\cite{peyre2019computational}:\n\\begin{align}\n    &\\min_{\\vec{P}}\\; \\langle \\vec{P}, \\vec{C}\\rangle\n    \\quad\n    \\text{subject to}\\; \\vec{P}\\in \\mathcal{U}_{m,n}.\\label{eq:wassdef}\n\\end{align}\n\nHere, $\\vec{C}$ is an $m$-by-$n$ Euclidean distance matrix whose $(i,j)$ element is the squared distance between $x_i$ and $y_j$ (i.e., $\\vec{C}_{ij}=\\|x_i-y_j\\|^2$), and $\\vec{P}$ is an $m$-by-$n$ coupling matrix whose $(i,j)$ element has amount of mass delivered from $x_i$ to $y_j$.\nMoreover, $\\mathcal{U}_{m,n}$ is a set of $m$-by-$n$ matrices that represent a valid coupling, formally defined by\n\\begin{align}\n    \\mathcal{U}_{m,n}=\\{\\vec{P}\\in\\mathbb{R}_{\\ge0}^{m\\times n}\\mid \\vec{P}\\onevec{n}=\\frac{\\onevec{m}}m, \\vec{P}^\\top\\onevec{m}=\\frac{\\onevec{n}}n\\}.\n    \\label{eq:wassconst}\n\\end{align}\nThis suggests that the mass assigned to each point in ${\\cal X}$ (i.e., $\\frac1m$) must be delivered to points in ${\\cal Y}$ \\emph{without overs and shorts}, and \\textit{vice versa}; this conservation law is entirely different to the existing geometric inconsistency losses.\nThe example in Fig.~\\ref{fig:wass}(b) represents the coupling matrix corresponding to Fig.~\\ref{fig:wass}(a).\nElements of $\\vec{P}$ corresponding to the coupled points (i.e., $(x_1, y_3)$, $(x_2, y_1)$, $(x_3, y_2)$) are filled with $\\frac13$, whereas the others are zero.\nThis $\\vec{P}$ satisfies the constraint \\eqref{eq:wassconst}; the sums of each row and column are $\\frac1m$ and $\\frac1n$, respectively (it should be noted that we have $m=n=3$ here).\n\nWith the optimal solution $\\vec{P}^*$, the Wasserstein distance is defined by $W^2({\\cal X},{\\cal Y})= \\langle\\vec{P}^*,\\vec{C}\\rangle$.\nIt is known that $W^2(\\cdot, \\cdot)$ defines a proper metric, i.e., the Wasserstein distance satisfies three axioms\\footnote{That is, $W({\\cal X},{\\cal Y})$ satisfies: (i) $W({\\cal X},{\\cal Y})=0\\Leftrightarrow {\\cal X}={\\cal Y}$ (identity of indiscernibles); (ii) $W({\\cal X},{\\cal Y})=W({\\cal Y},{\\cal X})$ (symmetry); (iii) $W({\\cal X},{\\cal Y})\\le W({\\cal X},{\\cal Z})+W({\\cal Z},{\\cal Y})$ (triangle inequality).} of metric~\\cite{peyre2019computational}; hence, we expect it to behave properly as a loss function.\nNote that $\\vec{P}$ provides soft coupling to allow weighted coupling with multiple points, although Fig.\\ref{fig:wass} shows the simplified case with only one-to-one coupling; we can define \\eqref{eq:wassdef}, even if $m\\ne n$, i.e., the size of ${\\cal X}$ and ${\\cal Y}$ are different.\n\n\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.6\\linewidth]{fig\/wass3.pdf}\n    \\caption{{\\bf Wasserstein distance.} (a) Optimal coupling; (b) Matrix $\\vec{P}$ corresponding to the coupling displayed in (a); (c) Non-optimal coupling; The blue numbers are the squared distances between two points. It should be noted that this figure is simplified; we can, in fact, treat soft coupling, which distributes the weight from one to many (see $\\mathcal{U}_{m,n}$). }\n    \\label{fig:wass}\n\\end{figure}\n\n\\paragraph{Computing WCL and its Gradient}\nWe train the neural networks by reducing the geometric inconsistency of two point clouds measured by the Wasserstein distance.\nThus, we need to compute $W^2({\\cal X},{\\cal Y})$ and its gradients with respect to ~${\\cal X}$ and ${\\cal Y}$.\n\\emph{Sinkhorn iteration} (Algorithm~\\ref{algo:sinkhorn}) allows us to compute $W^2({\\cal X},{\\cal Y})=\\langle \\vec{P}^*,\\vec{C}\\rangle$ very accurately, as well as its gradients.\nBenefits of the Sinkhorn iteration are two-fold:\n(1) \\emph{we can use GPUs} as it combines simple arithmetic operations;\n(2) \\emph{we can back-prop the iteration directly} through auto-gradient techniques equipped in most modern deep learning libraries (i.e., we do not need external libraries, unlike the ICP loss).\n\n\\begin{algorithm}[htbp]\n  \\caption{Computing WCL $\\widetilde W^2({\\cal X},{\\cal Y})$ by the Sinkhorn iteration. $\\varepsilon>0$ is a small constant.}\\label{algo:sinkhorn}\n  \\DontPrintSemicolon\n  \\KwData{Point clouds ${\\cal X}=\\{x_i\\}_{i=1}^m$, ${\\cal Y}=\\{y_j\\}_{j=1}^n$}\n  \\nl $\\vec{C}_{ij}=\\|x_i-y_j\\|^2 \\qquad 1\\le \\forall i\\le m, \\; 1\\le\\forall j\\le n$\\\n  \\nl $\\vec{G}\\leftarrow\\exp(-\\vec{C}\/\\varepsilon)$\\tcp*{\\footnotesize{Element-wise exp}}\n  \\nl $\\vec{v}\\leftarrow \\onevec{n}$\\tcp*{\\footnotesize{Initialize dual variable}}\n  \\nl \\While{Not converged}{\n      \\nl $\\vec{u}\\leftarrow \\frac1m\\onevec{m}\/\\vec{G}\\vec{v}$\\tcp*{\\footnotesize{Element-wise division}}\n      \\nl $\\vec{v}\\leftarrow \\frac1n\\onevec{n}\/\\vec{G}^\\top\\vec{u}$\n  }\n  \\nl \\textbf{{return}} \\, {$\\langle\\diag{\\vec{u}}\\vec{G}\\,\\diag{\\vec{v}},\\vec{C}\\rangle$ \\textbf{as} $\\widetilde W^2({\\cal X},{\\cal Y})$}\\;\n\\end{algorithm}\n\n\\paragraph{Remark}\nTo be exact, Algorithm~\\ref{algo:sinkhorn} is an approximation algorithm for \\eqref{eq:wassdef};\nit solves the optimization problem with an additional \\emph{entropy regularization} term $H(\\vec{P})$:\n\\begin{align}\n    &\n    \\min_{\\vec{P}} \\langle \\vec{P},\\vec{C}\\rangle-\\varepsilon H(\\vec{P}), \\quad \\text{subject to} \\; \\vec{P}\\in\\mathcal{U}_{m,n}\n    \\label{eq:regwass}\\\\\n    &\n    \\text{where}\\quad H(\\vec{P}):=-\\sum_{ij}\\vec{P}_{ij}(\\log\\vec{P}_{ij}-1).\\nonumber\n\\end{align}\nWith the optimal solution $\\vec{P}^\\dagger$ of this problem, we define the regularized Wasserstein distance by $\\widetilde W^2({\\cal X},{\\cal Y})=\\langle\\vec{P}^\\dagger,\\vec{C}\\rangle.$\nThis regularization term makes WCL smooth with respect to its inputs, resulting in stable training.\nAs can be seen, if $\\varepsilon\\to0$, \\eqref{eq:regwass} converges to the original optimization problem \\eqref{eq:wassdef};\nhence, a small $\\varepsilon>0$ gives a good approximation. However, such a small $\\varepsilon$ might cause numerical instability as a small $\\varepsilon$ reduces $\\vec{G}$ to an almost zero matrix at Line~2. \nTo improve numerical stability, we may use log-sum-exp at Lines 5 $-$ 6.\nFurthermore, we can compute Algorithm~\\ref{algo:sinkhorn} in parallel over batch dimension.\nAt Line~4, we can use any kind of stopping condition; here, we stop at 100 iterations.\nSee \\cite{cuturi2013sinkhorn,peyre2019computational} for details of the regularized Wasserstein distance.\n\n\\section{Experiment}\nWe apply our WCL to prior works and quantitatively and qualitatively evaluate them using the public dataset with comparison to the original method.\n\n\\subsection{Dataset}\nIn this study, we use the KITTI dataset~\\cite{Geiger2013IJRR}.\nAlthough it is known that a large input image size can lead to better performance~\\cite{pillai2019superdepth}, the main purpose of our experiment is to confirm any advantages against the target original method to be applied.\nHence, we use a 416$\\times$128 image size, similar to prior works~\\cite{zhou2017unsupervised,mahjourian2018unsupervised,gordon2019depth,godard2019digging}.\nHowever, we conduct an additional evaluation using an image size of 640$\\times$192, for monodepth2 only ~\\cite{godard2019digging} as their default image size is 640$\\times$192 in the public code. As with several prior works, we have two different datasets, split for the evaluation of depth and pose estimation. It should be noted that the models for depth and pose estimation are trained simultaneously on each dataset, but depth and pose are trained and evaluated on separate data.\n\\paragraph{Depth Estimation} We separate the KITTI raw dataset via the Eigen split~\\cite{eigen2014depth}. As a result, we have approximately 40,000 frames for training, 4,000 frames for validation, and 697 frames for the test.\nThe test data are chosen from 29 scenes of the KITTI raw dataset. \nAlthough the KITTI raw data include stereo images and LIDAR data, we use monocular images for both training and testing as the input data and use the LIDAR data as the ground truth only for testing.\n\\paragraph{Pose Estimation} Unlike the KITTI raw dataset, the KITTI odometry dataset has the ground truth of relative pose between consecutive images for testing.\nHowever, test data in the KITTI odometry dataset are partially included in the training data of the KITTI raw dataset.\nHence, we can not use the trained model on the KITTI raw dataset for the pose estimation evaluation on the KITTI odometry dataset.\nTherefore, we have both training and testing on the KITTI odometry dataset, as with the baseline methods~\\cite{zhou2017unsupervised,godard2019digging}.\nAlthough there are 22 sequences in the KITTI odometry dataset, we use 11 sequences, from 00 to 10. \nAfter training models on sequence 00 to 08, we test the model for pose estimation on sequence 09 and 10.\n\n\\subsection{Evaluation of Depth Estimation}\n\\paragraph{Training}\nIn the evaluation of depth estimation, we apply our WCL to four selected baselines~\\cite{zhou2017unsupervised,mahjourian2018unsupervised,gordon2019depth,godard2019digging}.\nFor \\cite{zhou2017unsupervised,godard2019digging}, we add $\\lambda_{w} \\cdot L_{wass}$ to their original cost function, $L_{origin}$, to train the model.\nBy contrast, as the original cost function of \\cite{mahjourian2018unsupervised,gordon2019depth} includes the ICP loss~\\cite{mahjourian2018unsupervised} and the depth consistency loss\\cite{gordon2019depth}, which also penalizes geometric inconsistencies, we remove them for the comparative evaluation with each objective.\nWe train all models by applying the same hyperparameters (e.g., mini-batch size, learning rate, data augmentation, and network structure) and training process (e.g., masking, training length, optimization, selection of input images $I_A$ and $I_B$ in Fig.~\\ref{f:overview}) as in the original code, except for the following hyperparameters about our WCL.\n\nWe determine the weighting values, $\\lambda_{w}$, for $L_{wass}$ as 7.0, 2.0, 3.0, and 0.5 in \\cite{zhou2017unsupervised}, \\cite{mahjourian2018unsupervised}, \\cite{gordon2019depth}, and \\cite{godard2019digging}, respectively.\n$\\lambda_w$ is a weighting factor that balances $L_{origin}$; therefore, there is no significance to the relative size of the values.\n$\\varepsilon$ in \\eqref{eq:regwass} is set as 0.001 to suppress the approximation and to stably calculate the WCL. \nIn addition, we uniformly sample the point clouds at a grid point on an image coordinate before feeding them into our WCL in \\eqref{eq:Lwass} owing to the limitation of GPU memory usage. \nAn explanation of the sampling method and an ablation study of the hyperparameters are shown in the supplementary materials.\n\n\n\\paragraph{Quantitative Analysis}\nTable~\\ref{tab:ev} displays the widely used seven metrics for the evaluation of depth estimation from monocular camera images. \nAs summarized in Table~\\ref{tab:ev}, the performance of all the baselines can be successfully improved on most metrics by adding our WCL.\nThe improvement for \\cite{zhou2017unsupervised} and \\cite{godard2019digging} is about 25$\\%$ and 15$\\%$ on the most advantageous metric, respectively. On the other hand, it is not quite as large for \\cite{mahjourian2018unsupervised} and \\cite{gordon2019depth} as they also penalize geometric inconsistencies using their own approach.\nHowever, our method still has explicit margins about 5$\\%$ on Sq Rel or RMSE against these baselines.\nFurthermore, monodepth2~\\cite{godard2019digging} + WCL can successfully obtain the best results, which are quite better than the other related works~\\cite{luo2020consistent,fei2019geo}\n\n\\begin{table*}[t]\n  \\caption{{\\small {\\bf Evaluation of depth estimation by self-supervised mono supervision on an Eigen split of the KITTI dataset.} We display seven metrics from the {\\bf estimated depth images less than 80 m}. ``+ WCL'' is the result obtained with our proposed method, in addition to the baseline method presented above. For the leftmost four metrics, smaller is better; for the rightmost three metrics, higher is better. $\\dagger$ and $\\ddagger$ indicate removing the ICP and depth consistency loss from the original cost function, respectively. The method of $\\ddagger$ is evaluated by the author. The $\\ast$ method uses the ground truth pose.}\n \n  \\begin{center}\n  \\resizebox{0.95\\columnwidth}{!}{\n  \\label{tab:ev}\n  \\begin{tabular}{|lc|c|c|c|c||c|c|c|} \\hline\n    method & image size & Abs Rel & Sq Rel & RMSE & RMSE log & $\\delta < 1.25$ & $\\delta < 1.25^2$ & $\\delta < 1.25^3$ \\\\ \\hline\n    Yang et al.~\\cite{yang2017unsupervised} & 416x128 & 0.182 & 1.481 & 6.501 & 0.267 & 0.725 & 0.906 & 0.963 \\\\\n    LEGO~\\cite{yang2018lego} & 416x128 & 0.162 & 1.352 & 6.276 & 0.252 & 0.783 & 0.921 & 0.969 \\\\\n    GeoNet~\\cite{yin2018geonet} & 416x128 & 0.155 & 1.296 & 5.857 & 0.233 & 0.793 & 0.931 & 0.973 \\\\\n    Fei et al.~\\cite{fei2019geo} & 416x128 & 0.142 & 1.124 & 5.611 & 0.223 & 0.813 & 0.938 & 0.975 \\\\\n    DDVO~\\cite{wang2018learning} & 416x128 & 0.151 & 1.257 & 5.583 & 0.228 & 0.810 & 0.936 & 0.974 \\\\\n    Yang et al.~\\cite{yang2018every} & 416x128 & 0.131 & 1.254 & 6.117 & 0.220 & 0.826 & 0.931 & 0.973 \\\\\n    Casser et al.~\\cite{casser2019depth} & 416x128 & 0.141 & 1.026 & 5.291 & 0.2153 & 0.8160 & 0.9452 & 0.9791 \\\\\n    Luo et al.~\\cite{luo2020consistent} $\\ast$ & 384x112 & 0.130 & 2.086 & 4.876 & 0.205 & 0.878 & 0.946 & 0.970 \\\\ \\hline\n   \n    Zhou et al.~\\cite{zhou2017unsupervised} & 416x128 & 0.208 & 1.768 & 6.856 & 0.283 & 0.678 & 0.885 & 0.957 \\\\\n   \n   \n    \\bf{+ WCL} & 416x128 & \\bf{0.171} & \\bf{1.316} & \\bf{6.080} & \\bf{0.255} & \\bf{0.755} & \\bf{0.915} & \\bf{0.966} \\\\ \\hdashline\n    vid2depth~\\cite{mahjourian2018unsupervised} & 416x128 & \\bf{0.163} & 1.240 & 6.220 & 0.250 & 0.762 & 0.916 & \\bf{0.968} \\\\\n    vid2depth~\\cite{mahjourian2018unsupervised} {\\scriptsize wo ICP loss}$\\dagger$ & 416x128 & 0.175 & 1.617 & 6.267 & 0.252 & 0.759 & 0.917 & 0.967 \\\\\n   \n    \\bf{+ WCL} & 416x128 & 0.165 & \\bf{1.226} & \\bf{5.892} & \\bf{0.246} & \\bf{0.767} & \\bf{0.918} & \\bf{0.968} \\\\ \\hdashline\n    Gordon et al.~\\cite{gordon2019depth} & 416x128 & 0.128 & 0.959 & 5.23 & 0.212 & \\bf{0.845} & \\bf{0.947} & 0.976 \\\\\n   \n   \n    Gordon et al.~\\cite{gordon2019depth} {\\scriptsize wo depth consis. loss}$\\ddagger$ & 416x128 & 0.129 & 0.945 & \\bf{5.211} & 0.214 & 0.839 & 0.944 & 0.976 \\\\    \n   \n    \\bf{+ WCL} & 416x128 & \\bf{0.125} & \\bf{0.915} & 5.231 & \\bf{0.210} & 0.844 & \\bf{0.947} & \\bf{0.977} \\\\ \\hdashline\n    monodepth2~\\cite{godard2019digging} & 416x128 & 0.128 & 1.087 & 5.171 & 0.204 & 0.855 & \\bf{0.953} & 0.978 \\\\\n   \n    \\bf{+ WCL} & 416x128 & \\bf{0.123} & \\bf{0.920} & \\bf{4.990} & \\bf{0.201} & \\bf{0.858} & \\bf{0.953} & \\bf{0.980} \\\\  \\hdashline\n    monodepth2~\\cite{godard2019digging} & 640x192 & 0.115 & 0.903 & 4.863 & 0.193 & \\bf{0.877} & \\bf{0.959} & \\bf{0.981} \\\\\n   \n    \\bf{+ WCL} & 640x192 & \\bf{0.114} & \\bf{0.813} & \\bf{4.705} & \\bf{0.191} & 0.874 & \\bf{0.959} & \\bf{0.981} \\\\ \\hline     \n  \\end{tabular}\n  }\n\\end{center}\n  \\vspace{-5mm}\n\\end{table*}\n\n\\paragraph{Qualitative Analysis}\nWe show the depth images of \\cite{zhou2017unsupervised,godard2019digging}, with and without our WCL, in Fig.~\\ref{f:pred_depth}.\nThe first row in Fig.~\\ref{f:pred_depth} shows the RGB image as the neural network input, the second and fourth rows are the depth images estimated by \\cite{zhou2017unsupervised} and \\cite{godard2019digging}, and the third and fifth rows are the depth images estimated by \\cite{zhou2017unsupervised} + WCL and \\cite{godard2019digging} + WCL, respectively.\nIn the estimated depth images, we display a white lined rectangle to highlight the advantage of our method.\nOur method can reduce the number of artifacts and sharpen estimation.\nThe depth images in additional cases estimated by various methods in Table~\\ref{tab:ev} are shown in the supplementary material.\n\n\\begin{figure*}[t]\n  \\begin{center}\n \n      \\includegraphics[width=0.99\\hsize]{fig\/fig_depth_corl.pdf}\n  \\end{center}\n     \n\t\\caption{\\small {\\bf Qualitative results of our proposed method.} The top row displays the input images for the trained neural network to estimate depth images and the other rows display the estimated depth images, with and without the WCL. The white rectangle box in the depth image highlights the advantages of the WCL.}\n  \\label{f:pred_depth}\n \n\\end{figure*}\n\\subsection{Evaluation of Pose Estimation}\nFor evaluation of pose estimation, we select two prior works~\\cite{zhou2017unsupervised,godard2019digging} to apply our WCL.\nWe could not evaluate \\cite{mahjourian2018unsupervised,gordon2019depth} for pose estimation, because the hyperparameters and codes for the pose estimation and evaluation are partially unclear from their released code.\nFor \\cite{zhou2017unsupervised,godard2019digging}, we train the models using the KITTI odometry dataset, with the same training conditions as the depth estimation.\n\nTable~\\ref{tab:ego} displays the absolute trajectory error (ATE) in meters for sequence 09 and 10 of the KITTI odometry dataset.\nWe show the mean and standard deviation of ATE over all overlapping 5 frame snippets. \nBy applying our WCL, the pose estimation accuracies of two prior studies~\\cite{zhou2017unsupervised,godard2019digging} are improved.\nAlthough the performance is slightly worse than ``ORB-SLAM (full)'', the original ORB-SLAM with loop closure using whole sequence images, the methods with our WCL outperforms ``ORB-SLAM (short)'', which takes 5 consecutive images like our method.\n\n\\begin{table}[t]\n  \\caption{{\\small {\\bf Evaluation of pose estimation on sequence 09 and 10 of the KITTI odometry dataset.} Results show the mean and standard deviation of the absolute trajectory error in meters. ``frame number'' is the number of input images used for pose estimation. The method with our WCL outperforms the original method.}}\n \n  \\begin{center}\n  \\resizebox{0.75\\columnwidth}{!}{\n  \\label{tab:ego}\n  \\begin{tabular}{|lc|c|c|c|} \\hline\n    method & image size & \\hspace{5mm} sequence 09 \\hspace{5mm} & \\hspace{5mm} sequence 10 \\hspace{5mm} & frame number \\\\ \\hline\n    ORB-SLAM (full) & full size & 0.014 $\\pm$ 0.008 & 0.012 $\\pm$ 0.011 & - \\\\\n    ORB-SLAM (short) & full size & 0.064 $\\pm$ 0.141 & 0.064 $\\pm$ 0.130 & - \\\\ \\hline\n   \n   \n    Zhou et al.~\\cite{zhou2017unsupervised} & 416x128  & 0.021 $\\pm$ 0.017 & 0.020 $\\pm$ 0.015 & 5 \\\\\n    \\bf{+ WCL} & 416x128  & \\bf{0.016} $\\pm$ \\bf{0.011} & \\bf{0.013} $\\pm$ \\bf{0.009} & 5 \\\\ \\hdashline\n    monodepth2~\\cite{godard2019digging} & 416x128  & 0.017 $\\pm$ \\bf{0.009} & 0.015 $\\pm$ \\bf{0.010} & 2\\\\\n    \\bf{+ WCL} & 416x128  & \\bf{0.016} $\\pm$ \\bf{0.009} & \\bf{0.014} $\\pm$ \\bf{0.010} & 2 \\\\  \\hdashline\n    monodepth2~\\cite{godard2019digging} & 640x192  & 0.017 $\\pm$ \\bf{0.008} & 0.015 $\\pm$ \\bf{0.010} & 2 \\\\\n    \\bf{+ WCL} & 640x192  & \\bf{0.016} $\\pm$ \\bf{0.008} & \\bf{0.014} $\\pm$ \\bf{0.010} & 2\\\\ \\hline     \n  \\end{tabular}\n  }\n\\end{center}\n  \\vspace{-0.5em}\n\\end{table}\n\n\\section{Conclusion}\nIn this paper, we proposed a novel WCL to penalize geometric inconsistencies, for depth and pose estimation.\nOur proposed approach employed the Wasserstein distance for measuring the consistency between two point clouds from different frames.\nOur WCL is a smooth and symmetric objective, which can suitably measure geometric consistency without using any other external and\/or non-differentiable libraries.\nTherefore, the neural network can be effectively and efficiently trained to obtain highly accurate depth estimation.\nIn the experiment, we applied our proposed WCL to several state-of-the-art baselines and confirmed the benefits of our method with healthy margins.\n\nThere are two remaining issues to be addressed in the future.\nFirst, the study of memory-saving methods for measuring geometric inconsistency.\nAs shown in the supplementary material, the performance of our WCL is restricted by the limitation on GPU memory.\nSecond, the occlusion when calculating Wasserstein distance.\nRelaxation of the coupling constraints and\/or masking for occluded point clouds will be required for more accurate estimation.\nOur WCL has potentials to adapt for this issue, because our WCL can be calculated for pairs of different number of point clouds as $m \\neq n$. \n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{intro}\n\nSeveral theorem provers\n(e.g.\\ Coq, Isabelle, PVS)\ninclude facilities to generate code in various programming languages\n(e.g.\\ C, Clean, Haskell, Lisp, Ocaml, Scala, Scheme, SML)\nfrom executable subsets of the provers' logical languages.\nThat way, code written in a prover's language,\npossibly verified to satisfy properties of interest,\ncan be run as code in a conventional programming language;\nverifying the correctness of\nthe generated code with respect to the prover's code\nis a separable problem, akin to compiler verification.\nWhen carrying out\nformal program synthesis by stepwise refinement\n\\cite{bmethod,zed,vdm,specware-www}\nin a prover\nto derive an implementation from a high-level specification,\na code generator can translate\nthe low-level (i.e.\\ fully refined) specification\nto the final program.\n\nACL2's tight integration with the underlying Lisp platform\nmay obviate the need for code generation,\nbecause executable ACL2 code can run efficiently as Lisp.\nAn APT program derivation \\cite{apt-www,apt-simplify,apt-isodata}\nmay end with an implementation in executable ACL2 that runs as Lisp.\nHowever, some applications require code in other languages,\nsuch as C for embedded systems or device drivers.\nTo synthesize this kind of code via an APT derivation,\na code generator for ACL2 can be used,\nsuch as ATJ \\cite{atj-deep,atj-shallow} \\citeman{JAVA____ATJ}{java::atj}\n(for Java).\n\nA typical code generator for a prover can be viewed as\na reification of a shallow embedding of\nthe prover's (executable) language in the programming language:\nconstructs of the prover's language\nare rendered as suitably equivalent constructs of the programming language.\nThe translation is centered on, and driven by, the prover's language,\nwhich is mimicked in the programming language.\nTherefore, unless the prover's language and the programming languages\nare sufficiently similar,\nthe generated code may not be very efficient (in time and space) and idiomatic;\nthis could be an issue for the aforementioned example applications.\n\nThis paper proposes an approach\nto turn the focus on the generated code,\nand to exert direct control over it,\nby flipping the direction of the embedding:\n(i) a shallow embedding of the programming language in the prover's language\ndefines representations of program constructs in the prover; and\n(ii) a code generator recognizes these representations\nand translates them to the represented code.\nThis is \\emph{code generation by inverse shallow embedding}:\nthe embedding is a (not necessarily reified) translation\nfrom the programming language to the prover's language,\nand the code generator is the (reified) inverse translation.\nIn contrast, \\emph{code generation by direct shallow embedding}\nis the more typical approach where\nthe code generator translates the prover's language to the programming language,\nthe translation being a shallow embedding of the former in the latter.\n\nThe program constructs shallowly embedded in the prover\nare oriented towards the programming language,\nand thus may not be idiomatic formulations in the prover's language.\nThis new code generation approach is designed\nfor program synthesis by stepwise refinement,\nwhere the final refinement steps turn idiomatic code in the prover's language\ninto provably equivalent shallowly embedded program code.\nThese final refinement steps, carried out under user guidance,\nafford fine-grained control on the exact form of the final program.\n\nThis new code generation approach is realized\nin ATC (\\textbf{A}CL2 \\textbf{T}o \\textbf{C}) \\citeman{C____ATC}{c::atc},\nthe C code generator for ACL2 described in this paper.\nATC is designed for use with the aforementioned APT:\nthe final steps of an APT derivation turn ACL2 code\ninto C code shallowly embedded in ACL2,\nwhich ATC turns into actual C code.\nThese final derivation steps take place within the ACL2 language,\nand their verification involves only the ACL2 language;\nthese steps may be carried out via proof-generating APT transformations,\nincluding ones tailored to ATC\nthat have been and are being developed at Kestrel.\nThese APT transformations are not discussed further in this paper,\nwhich concentrates on ATC.\n\nBesides the C code, ATC also generates ACL2 proofs\nof the correctness of the C code with respect to the ACL2 code.\nThe proofs are based on a formalization of the needed subset of C in ACL2.\nTogether with the ACL2-to-ACL2 proofs in an APT derivation,\nthe ACL2-to-C proof generated by ATC\nprovides an end-to-end correctness proof of\nthe synthesized C code that ends the derivation\nwith respect to the high-level specification that starts the derivation.\n\nATC supports a limited subset of C18 \\cite{c18},\nwhich includes\ninteger types,\noperations and conversions on them,\ninteger arrays with read and (destructive) write access,\nlocal variable declarations and assignments,\nloops of certain forms,\nconditional expressions and statements,\nand functions that may affect arrays.\nDespite its limitations, this set suffices for some interesting programs.\nATC has been and is being used in a growing number of applications at Kestrel,\nsupporting the viability of the approach.\n\n\\secref{embeddings} offers a perhaps original perspective on\nhow different kinds of language embedding correspond to\ndifferent code generation approaches.\n\\secref{shallow-codegen} describes\nthe shallow embedding of C in ACL2 and how ATC uses it to generate C code.\n\\secref{deep-proofgen} describes the formalization (i.e.\\ deep embedding)\nof C in ACL2\nand how ATC uses it to generate proofs.\n\\secref{future} discusses future work.\n\\secref{related} surveys related work.\nIn this paper, `fixtype' refers to a type\ndefined using the FTY library \\citeman{ACL2____FTY}{fty}.\n\n\n\\section{Perspective on Language Embedding and Code Generation}\n\\label{embeddings}\n\nAn \\emph{embedding} of a language $\\mathcal{L}$ in a language $\\mathcal{L}'$\nis a representation of $\\mathcal{L}$ in $\\mathcal{L}'$.\nIn a \\emph{deep} embedding,\nthe syntax of $\\mathcal{L}$ is represented\nvia $\\mathcal{L}'$ values that capture $\\mathcal{L}$ constructs,\nand the semantics of $\\mathcal{L}$ is represented\nvia $\\mathcal{L}'$ operations over those values;\nthe representation is explicit.\nIn a \\emph{shallow} embedding,\nthe syntax of $\\mathcal{L}$ is represented\nvia a translation of $\\mathcal{L}$ constructs to $\\mathcal{L}'$ constructs,\nand the semantics of $\\mathcal{L}$ is represented\nvia the $\\mathcal{L}'$ semantics of\nthe $\\mathcal{L}'$ counterparts of the $\\mathcal{L}$ constructs;\nthe representation is implicit.%\n\\footnote{Although representations are conceivable\nthat blur the line between deep and shallow embedding,\nthe distinction remains generally useful,\nand is commonly found in the literature\nand used in technical discourse.}\nAn embedding may apply to a subset of $\\mathcal{L}$.\nThe features of an embedding depend on the nature of $\\mathcal{L}$ and $\\mathcal{L}'$,\nin particular whether each one is a logical language or a programming language.\n\nGiven a logical language $\\mathcal{A}$ (e.g.\\ ACL2)\nand a programming language $\\mathcal{P}$ (e.g.\\ C or Java),\nthere are four possible kinds of embedding, based on direction and depth:\n(i) shallow embedding of $\\mathcal{A}$ in $\\mathcal{P}$;\n(ii) deep embedding of $\\mathcal{A}$ in $\\mathcal{P}$;\n(iii) shallow embedding of $\\mathcal{P}$ in $\\mathcal{A}$; and\n(iv) deep embedding of $\\mathcal{P}$ in $\\mathcal{A}$.\nEach kind corresponds to a different approach\nto generate $\\mathcal{P}$ code from $\\mathcal{A}$,\nparticularly for the first three kinds,\nwith the fourth kind being a little different.\nIn addition, the fourth kind plays a role in\nestablishing the correctness of the generated code for all four approaches.\n\n\n\\subsection{Direct Shallow Embedding}\n\\label{direct-shallow}\n\nA shallow embedding of $\\mathcal{A}$ in $\\mathcal{P}$\nis a translation of constructs in (an executable subset of) $\\mathcal{A}$\nto suitably equivalent constructs in $\\mathcal{P}$.\nThe translation can be used to generate $\\mathcal{P}$ from $\\mathcal{A}$.\nThe code generator reifies the embedding;\nembedding and code generation go in the same direction---%\nhence `direct'.\n\nThis is a typical code generation approach.\nThe constructs of $\\mathcal{A}$ are rendered in $\\mathcal{P}$:\nthe data types of $\\mathcal{A}$ are mapped to data types of $\\mathcal{P}$,\nthe operations of $\\mathcal{A}$ are mapped to operations of $\\mathcal{P}$,\nand so on.\nIf $\\mathcal{P}$ lacks suitable built-in counterparts\nfor certain $\\mathcal{A}$ constructs,\nsuch counterparts are defined in $\\mathcal{P}$ (e.g.\\ as libraries)\nand used as code generation targets.\n\nThis translation is centered on, and driven by, $\\mathcal{A}$,\nwhich is mimicked in $\\mathcal{P}$.\nUnless $\\mathcal{P}$ is sufficiently similar to $\\mathcal{A}$,\nthe generated code may look more like ``$\\mathcal{A}$ written in $\\mathcal{P}$''\nthan like idiomatic $\\mathcal{P}$;\nit may not be as efficient (in time and space)\nas the original $\\mathcal{A}$ code\nor as a handcrafted port to $\\mathcal{P}$.\nIn some applications,\nthe readability of the generated code may be unimportant,\nand its efficiency adequate;\nbut other applications, e.g.\\ for resource-constrained systems,\nmay have more stringent requirements.\n\nA realization of this code generation approach\nis ATJ in shallow embedding mode \\citeman{JAVA____ATJ}{java::atj},\nwhich generates Java code from ACL2.\nIt relies on AIJ \\citeman{JAVA____AIJ}{java::aij},\nwhich implements ACL2 data types and operations in Java.\nACL2 functions are translated to Java static methods,\norganized in Java classes corresponding to the ACL2 packages.\nACL2 \\code{let} bindings are translated to\nJava local variable declarations and assignments.\nAnd so on; see the references for more details.\n\n\n\\subsection{Direct Deep Embedding}\n\\label{direct-deep}\n\nA deep embedding of $\\mathcal{A}$ in $\\mathcal{P}$ is\nan interpreter of (an executable subset of) $\\mathcal{A}$ written in $\\mathcal{P}$.\nThis enables a simple code generation approach:\nthe code generator translates $\\mathcal{A}$ constructs\nto their deeply embedded counterparts in $\\mathcal{P}$,\nwhich the interpreter executes in $\\mathcal{P}$,\npossibly via a thin wrapper produced by the code generator.\nThe interpreter reifies the embedding;\nembedding and code generator go in the same direction---%\nhence `direct'.\n\nThis is an unconventional code generation approach,\nbut is a conceptually simple way to run $\\mathcal{A}$ in $\\mathcal{P}$.\nThe interpreter includes representations of\nthe abstract syntax of $\\mathcal{A}$,\nthe values and computation states of $\\mathcal{A}$,\nthe basic operations of $\\mathcal{A}$,\nand so on.\nThe translation carried out by the code generator is straightforward.\nThe $\\mathcal{P}$ code resulting from this code generation approach\nis even less idiomatic and efficient than discussed in \\secref{direct-shallow},\nbut it may be adequate for certain applications;\ndue to its simplicity, it is also fairly high-assurance\n(relevant if formal proofs are absent).\n\nA realization of this code generation approach\nis ATJ in deep embedding mode \\citeman{JAVA____ATJ}{java::atj},\nwhich generates Java code from ACL2.\nThe interpreter is AIJ \\citeman{JAVA____AIJ}{java::aij},\npart of which is the Java implementation of the ACL2 data types and operations\nmentioned in \\secref{direct-shallow}.\n\nThis code generation approach could become more interesting\nby accompanying it with partial evaluation.\nAccording to the first Futamura projection \\cite{parteval},\npartially evaluating an interpreter with respect to a program\namounts to compiling the program\nto the language that the interpreter is written in.\nThus, a partial evaluator for $\\mathcal{P}$ can be used to\npartially evaluate the $\\mathcal{A}$ interpreter (written in $\\mathcal{P}$)\nwith respect to (the $\\mathcal{P}$ representation of) the $\\mathcal{A}$ code.\nFor the aforementioned ATJ,\na partial evaluator for Java would be needed;\nthe partial evaluator may be written in any language, including ACL2.\n\n\n\\subsection{Inverse Shallow Embedding}\n\\label{inverse-shallow}\n\nA shallow embedding of $\\mathcal{P}$ in $\\mathcal{A}$\nis a translation of constructs in (a subset of) $\\mathcal{P}$\nto suitably equivalent constructs in $\\mathcal{A}$.\nThe inverse of this translation can be used\nto generate $\\mathcal{P}$ from $\\mathcal{A}$,\nrecognizing the $\\mathcal{P}$ constructs shallowly embedded in $\\mathcal{A}$\n(i.e.\\ recognizing the image of the embedding)\nand turning them into the actual $\\mathcal{P}$ constructs.\nThe code generator reifies the inverse of the embedding,\nwhile the embedding does not have to be reified;\nembedding and code generator go in opposite directions---%\nhence `inverse'.\n\nThis appears to be a novel code generation approach.\nIt is centered on, and driven by, $\\mathcal{P}$ rather than $\\mathcal{A}$:\nit enables users to specify the generated code exactly,\nmaking it as idiomatic and efficient as desired,\nand fit for the more demanding applications\nmentioned in \\secref{direct-shallow}.\n\nWhile the generated $\\mathcal{P}$ code can be idiomatic,\nits $\\mathcal{A}$ representation may look\nmore like ``$\\mathcal{P}$ written in $\\mathcal{A}$''\nthan like idiomatic $\\mathcal{A}$ code,\nand may be burdensome to write directly.\nThus, this code generation approach is designed\nfor program synthesis by stepwise refinement,\nwhere the final refinement steps turn idiomatic $\\mathcal{A}$\ninto the form required by the code generator,\ni.e.\\ into something in the image of the shallow embedding.\nThese refinement steps, carried out under user guidance,\nafford fine-grained control on the exact form of the final program.\nSince the code generator merely turns\n$\\mathcal{P}$ code shallowly embedded in $\\mathcal{A}$ into actual $\\mathcal{P}$ code,\nthe readability and the efficiency (in space and time) of the code\ndo not depend on the code generator:\nthey are the responsibility of the aforementioned final refinement steps;\nshifting this responsibility from the code generator\nto the stepwise refinement derivation\nis a deliberate and distinctive aspect of this code generation approach.\n\nA realization of this code generation approach is ATC, described in this paper,\nwhich generates C code from ACL2.\nATC is designed for use with APT,\nwhose transformations (some tailored to ATC) can be used\nto turn idiomatic ACL2 code into the form required by ATC.\n\n\n\\subsection{Inverse Deep Embedding}\n\\label{inverse-deep}\n\nA deep embedding of $\\mathcal{P}$ in $\\mathcal{A}$ is\na formalization of (a subset of) $\\mathcal{P}$ in $\\mathcal{A}$.\nThus, it is the basis for formal correctness proofs of\n$\\mathcal{P}$ code generated from $\\mathcal{A}$\nvia any of the three approaches described above,\nplaying a role alongside the three other kinds of embedding.\nThe formalization reifies the deep embedding.\n\nFurthermore, this kind of embedding can be used\nin a pop-refinement derivation \\cite{popref}, where:\n(i) the initial specification is\na predicate over $\\mathcal{P}$ programs\nthat characterizes the possible implementations; and\n(ii) the final specification is\na singleton predicate that selects one $\\mathcal{P}$ implementation\nin explicit syntactic form, from which the actual code is readily obtained.\nAll the specifications in the derivation, from initial to final,\nare written in $\\mathcal{A}$,\nbut their formulation requires a formalization of $\\mathcal{P}$,\ni.e.\\ the deep embedding.\nThere is no code generator as such,\nother than a simple obtainment of the implementation\nfrom the final specification predicate where it is deeply embedded.\nEmbedding and refinement to code go in opposite directions---%\nhence `inverse'.\n\nFor this code generation approach to be practical,\ntechniques must be developed to\nmake deeply embedded $\\mathcal{P}$ constructs ``emerge'' within $\\mathcal{A}$\nvia suitable refinement steps.\nAn idea worth pursuing is whether\nthe kind of code generator described in \\secref{inverse-shallow}\ncould be realized as one or more such refinement steps,\nto automatically turn shallowly embedded $\\mathcal{P}$ constructs\ninto deeply embedded ones.\n\nThe proofs generated by ATC are based on a formalization of C in ACL2,\nwhich is a deep embedding of C in ACL2.\nATC uses the code generation approach described in \\secref{inverse-shallow},\nnot the pop-refinement approach sketched above.\n\n\n\\section{C Shallow Embedding and Code Generation}\n\\label{shallow-codegen}\n\nATC generates C code from ACL2\naccording to the inverse shallow embedding approach\nexplained in \\secref{inverse-shallow}.\nATC relies on the definition of a shallow embedding of C in ACL2,\nwhose image ATC recognizes in ACL2 code and translates to actual C code.\n\n\n\\subsection{Shallow Embedding of C in ACL2}\n\\label{shallow}\n\nThe shallow embedding of C in ACL2 relied on by ATC consists of\n(i) an ACL2 model of C integers and arrays and\n(ii) a definition of how C expressions, statements, and functions\nare represented as ACL2 terms and functions.\nThe latter is embodied in the ATC user documentation,\nwhich spells out the representation in an almost formal way.\nThere is currently no implemented translation from C to ACL2.\n\n\n\\subsubsection{Integers}\n\nThe model includes all the C18 standard integer types except \\code{\\_Bool},\nnamely\n\\code{char}, \\code{short}, \\code{int}, \\code{long}, and \\code{long} \\code{long},\nboth \\code{signed} (the default) and \\code{unsigned}\n(but not the plain \\code{char} type);\n10 types in total.\nThe model includes\nunary integer operations\n(\\code{+}, \\code{-}, \\verb|~|, \\code{!}; 4 in total),\nbinary integer operations\n(\\code{+}, \\code{-}, \\code{*}, \\code{\/}, \\code{\\%},\n\\code{\\&}, \\code{|}, \\code{\\^}, \\code{<{}<}, \\code{>{}>},\n\\code{<}, \\code{>}, \\code{<=}, \\code{>=}, \\code{==}, \\code{!=};\n16 in total),%\n\\footnote{The non-strict operations \\code{\\&\\&} and \\code{||}\nare represented differently, as described later.}\nand integer type conversions (90 in total).\n\nThe exact format of the C integer types is implementation-dependent.\nThe model assumes two's complement without padding bits (as prevalent),\nand has ACL2 nullary functions for the sizes of the different types;\nthese nullary functions are referenced in other parts of the model\nto avoid hardwiring assumptions on sizes,\nwhich vary across popular C implementations.\nThese nullary functions are currently defined (to common values),\nbut they are planned to be made constrained instead\n(see \\secref{future}).\nThe C integer values are represented as ACL2 integers in appropriate ranges\n(defined via the nullary functions),\ntagged with an indication of their type.\nThe model has a fixtype for each C integer type.\n\nThe C integer operations and conversions are modeled as ACL2 functions,\nwhose guards capture the conditions under which\nthe results are well-defined in C18.\nFor example, \\code{int} addition is modeled as follows:\n\\begin{bcode}\n(defun add-sint-sint (x y) \\ccode{; addition of signed int and signed int}\n  (declare (xargs :guard (and (sintp x) (sintp y) (add-sint-sint-okp x y))))\n  (sint (+ (sint->get x) (sint->get y))))\n(defun add-sint-sint-okp (x y) \\ccode{; well-definedness condition}\n  (declare (xargs :guard (and (sintp x) (sintp y))))\n  (sint-integerp (+ (sint->get x) (sint->get y))))\n\\end{bcode}\nTwo \\code{int} values, recognized by \\code{sintp},\nare added by taking the underlying ACL2 integers via \\code{sint->get},\nadding them via ACL2's addition \\code{+},\nand turning the result into an \\code{int} value via \\code{sint},\nprovided that the result is representable as an \\code{int},\nwhose range is recognized by \\code{sint-integerp}.%\n\\footnote{In C18,\nsigned integer arithmetic operations are well-defined only if\nthe true result of the operation can be represented\nin the type of the result of the operation\n(which is determined from the types of the operands).\nC implementations may extend this well-definedness,\ne.g.\\ as two's complement wrap-around.}\n\nC unary and binary integer operations apply to any combination of operand types\nand have fairly complex rules about operand type conversions and result types;\nthe rules are not uniform across the different types.\nThe model has versions of the integer operations\nfor all possible combinations of operand types,\nwhose definitions capture those rules.\nThere are thousands of such functions,%\n\\footnote{(4 unary operations $\\times$ 10 types)\n$+$ (16 binary operations $\\times$ 100 type combinations)\n$=$ 1,640 functions.}\ngenerated via macros that exploit their available uniformity.\n\n\n\\subsubsection{Arrays}\n\nThe model includes monodimensional arrays of the above integer types,\nwith read and write access.\n\nA C array is modeled as a non-empty list of C integer values,\ntagged with an indication of its type.\nThe model has a fixtype for each C array integer type.\n\nRead and write access is modeled via ACL2 functions to read and write arrays.\nThe read functions return an array element from an array and an index;\nthe write functions return a new array\nfrom an old array, an index, and a value for the element.\nThese functions have guards requiring the index to be within the array bounds.\nSince an array may be indexed with any integer type in C,\nthere are versions of these functions\nfor all combinations of array and index types.\nThere are hundreds of such functions,%\n\\footnote{(2 read or write operations)\n$\\times$ (10 array types)\n$\\times$ (10 index types)\n$=$ 200 functions.}\ngenerated via macros that exploit their uniformity.\n\nArrays are not first-class entities in C,\nbut mere juxtapositions of their elements.\nAlthough the model just described\nappears to treat arrays as first-class entities,\ntheir use in the shallowly embedded C code\nis restricted in ways that make this modeling adequate;\nsee also \\secref{deep}.\n\n\n\\subsubsection{Expressions, Statements, and Functions}\n\nC functions are represented as ACL2 functions\nthat operate on (the ACL2 model of) C values.\nConsider the following C function:\n\\begin{bcode}\nint f(int x, int y, int z) \\{\n    return (x + y) * (z - 3);\n\\}\n\\end{bcode}\nThis is represented as follows in ACL2\n(where the ellipsis in the guard is explained later):\n\\begin{bcode}\n(defun |f| (|x| |y| |z|)\n  (declare (xargs :guard (and (sintp |x|) (sintp |y|) (sintp |z|) ...)))\n  (mul-sint-sint (add-sint-sint |x| |y|)\n                 (sub-sint-sint |z| (sint-dec-const 3))))\n\\end{bcode}\nThe reason for the vertical bars around the ACL2 symbols is that\na C identifier is represented as\nan ACL2 symbol whose \\code{symbol-name} is the identifier;\nif \\code{f} were used as the ACL2 function name instead of \\code{|f|},\nit would represent a C function named \\code{F}.\nThe correspondence between the body of \\code{|f|}\nand the return expression of \\code{f} is clear:\nthe ACL2 functions that model arithmetic operations\nhave been explained earlier;\nthe \\code{int} decimal (i.e.\\ in base 10) constant \\code{3}\nis represented via \\code{sint-dec-const}, which is part of the model.\nThe input and output types of \\code{f} are represented by\nthe corresponding guard conjuncts of \\code{|f|}\nand the fact that the body of \\code{|f|}\nreturns \\code{sintp} (via \\code{mul-sint-sint});\nsee also \\secref{codegen}.\n\nBesides C expressions as exemplified above,\nACL2 terms of certain forms represent C statements,\nwhich may contain blocks with local variable declarations.\nConsider the following C function:\n\\begin{bcode}\nunsigned int g(unsigned int x, unsigned int y) \\{\n    unsigned int z = 1U;\n    if (x < y) \\{ z = z + x; \\} else \\{ z = z + y; \\}\n    return 2U * z;\n\\}\n\\end{bcode}\nThis is represented as follows in ACL2:\n\\begin{bcode}\n(defun |g| (|x| |y|)\n  (declare (xargs :guard (and (uintp |x|) (uintp |y|))))\n  (let ((|z| (declar (uint-dec-const 1))))\n    (let ((|z| (if (boolean-from-sint (lt-uint-uint |x| |y|))\n                   (let ((|z| (assign (add-uint-uint |z| |x|))))\n                     |z|)\n                 (let ((|z| (assign (add-uint-uint |z| |y|))))\n                   |z|))))\n      (mul-uint-uint (uint-dec-const 2) |z|))))\n\\end{bcode}\nLocal variable declarations and assignments\nare both represented via \\code{let} bindings;\nthe two cases are distinguished by\nthe \\code{declar} and \\code{assign} wrappers,\nwhich are defined as identity functions.\nStatements, like the \\code{if} statement above,\nthat affect variables and are followed by more code\nare represented via \\code{let} bindings without any wrappers;\nboth branches of the \\code{if} term in ACL2\nmust end with (the latest values of) the affected variables.\nIf multiple variables were affected,\n\\code{mv-let} would be used to bind them to the \\code{if} term,\nwhose branches would have to end with an \\code{mv} of the variables.\nSince ACL2 is functional, variable updates are explicated.\nFunction \\code{boolean-from-sint} converts\nthe \\code{int} returned by the less-than operation to a boolean,\nas needed for \\code{if} tests in ACL2;\nthe conversion is not reflected in the C code,\nwhich uses scalars (including integers) in \\code{if} tests.\n\nC loops are represented as tail-recursive ACL2 functions.\nConsider the following modular factorial:\n\\begin{bcode}\nunsigned int h(unsigned int n) \\{\n    unsigned int r = 1U;\n    while (n != 0U) \\{ r = r * n; n = n - 1U; \\}\n    return r;\n\\}\n\\end{bcode}\nThis is represented as follows in ACL2:\n\\begin{bcode}\n(defun |h\\$loop| (|n| |r|) \\ccode{; representation of the loop of h}\n  (declare (xargs :guard (and (uintp |n|) (uintp |r|))))\n  (if (boolean-from-sint (ne-uint-uint |n| (uint-dec-const 0)))\n      (let* ((|r| (assign (mul-uint-uint |r| |n|)))\n             (|n| (assign (sub-uint-uint |n| (uint-dec-const 1)))))\n        (|h\\$loop| |n| |r|))\n    (mv |n| |r|)))\n(defun |h| (|n|) \\ccode{; representation of the function h}\n  (declare (xargs :guard (uintp |n|)))\n  (let ((|r| (declar (uint-dec-const 1))))\n    (mv-let (|n| |r|)\n      (|h\\$loop| |n| |r|)\n      (declare (ignore |n|)) \\ccode{; because n is not used after the loop}\n      |r|)))\n\\end{bcode}\nSince the loop function does not represent a C function,\nits name does not have to be a legal C identifier.\nIn \\code{|h|},\nthe two variables are \\code{mv-let}-bound to the loop function call,\nbecause the loop affects both.\nLoop functions may have a more elaborate structure than above,\nbut every control path must end\neither with a recursive call on the formal parameters\n(which must therefore be updated before the call)\nor with (the subset of) the formal parameters affected by the loop.\n\nC arrays are passed around as wholes in functional ACL2,\nbut they are passed around as pointers in C.\nThe ACL2 representation is correct so long as the arrays are treated in\na stobj-like \\citeman{ACL2____STOBJ}{stobj} single-threaded way,\nwhich is required in this shallowly embedded representation.\nConsider the following C function:\n\\begin{bcode}\nvoid i(unsigned char *a, int x, int y) \\{\n    a[x] = (unsigned char) 1;\n    a[y] = (unsigned char) 2;\n\\}\n\\end{bcode}\nThis is represented as follows in ACL2\n(where the ellipsis in the guard is explained later):\n\\begin{bcode}\n(defun |i| (|a| |x| |y|)\n  (declare (xargs :guard (and (uchar-arrayp |a|) (sintp |x|) (sintp |y|) ...)))\n  (let* ((|a| (uchar-array-write-sint |a| |x| (uchar-from-sint (sint-dec-const 1))))\n         (|a| (uchar-array-write-sint |a| |y| (uchar-from-sint (sint-dec-const 2)))))\n    |a|))\n\\end{bcode}\nArray writes are represented as \\code{let} bindings\nof the array variables to calls of the array write functions.\nThe C function returns nothing (i.e.\\ \\code{void}),\nbut since it affects the array,\nthe ACL2 function returns the updated array.\nIf multiple arrays were updated, they would all be returned via \\code{mv},\nwhile the C function would still return nothing.\nIf the C function returned a result besides affecting arrays,\nthe ACL2 function would return the result and the updated arrays via \\code{mv}.\nArrays may be updated in loops;\nin general, ACL2 loop functions return affected variables and arrays,\nwhile ACL2 non-loop functions return affected arrays and optionally a result.\n\nThe non-strict C operations \\code{\\&\\&} and \\code{||}\nare represented via ACL2's \\code{and} and \\code{or} macros,\nwhich are non-strict because they are defined in terms of \\code{if}.\nRepresenting \\code{\\&\\&} and \\code{||} via strict ACL2 functions\nwould require the evaluation of their second operand to be well-defined\n(i.e.\\ require its guards in ACL2 to be verified)\nregardless of the value of the first operand,\nwhich would be too restrictive,\npreventing many valid and useful C programs from being represented.\nFunctions like \\code{boolean-from-sint} shown earlier,\nas well as inverses like \\code{sint-from-boolean}\nthat are part of the shallow embedding,\nare used to convert between ACL2 booleans and C integers\nin the representation of C expressions involving \\code{\\&\\&} and \\code{||}.\n\nThe above is just an overview of the supported representations of C in ACL2.\nThe ATC reference documentation \\citeman{C____ATC}{c::atc}\nspells out the supported representations in full detail.\n\n\n\\subsection{C Code Generation}\n\\label{codegen}\n\nATC is invoked on a list of ACL2 functions\nthat represent C functions and C loops.\nIf an ACL2 function is not recursive, it represents a C function;\nif it is recursive, it represents a C loop.\nThe ACL2 functions must be listed in bottom-up order\n(i.e.\\ each function may call the preceding ones or itself,\nbut not the subsequent ones);\nthe C functions are generated in the same order,\ncurrently in a single \\code{.c} file.\nThe above requirements imply that\nrecursive C functions cannot be generated currently.\n\nATC requires all the ACL2 functions\nto be defined, in logic mode, and guard-verified.\nThis is critical for proof generation (see \\secref{proofgen}).\nThe fact that the loop functions are in logic mode means that\nthe loops must terminate under the guards,\nwhich may be added to the termination conditions via \\code{mbt},\nwhich ATC ignores as far as the representation of C code goes.\nThe fact that the functions are guard-verified means that\nall arrays are always accessed within their bounds\n(avoiding well-known problems in C),\nand that all arithmetic operations always have well-defined results\naccording to C18.\nThe ellipses in the guards of some of the examples in \\secref{shallow}\ninclude conditions needed for guard verification,\nnamely that the parameters of \\code{|f|} are in certain ranges\nand that the index parameters of \\code{|i|}\nare non-negative and less than the array's length.%\n\\footnote{No additional conditions are needed\nin the guards of \\code{|g|} and \\code{|h|} (and \\code{|h\\$loop|}),\nbecause unsigned arithmetic\nis always well-defined as wrap-around in C18.}\n\nATC checks that the names of the non-loop functions,\nand the names of the parameters of both loop and non-loop functions,\nare valid C ASCII identifiers.\nThe input types of the C functions are determined from the guards,\nwhich must include explicit conjuncts\nlike the ones in the examples in \\secref{shallow}.\nATC operates on the unnormalized bodies of the ACL2 functions\n\\citeman{ACL2____FUNCTION-DEFINEDNESS}{function-definedness}:\nit checks that they contain supported representations of C code,\ni.e.\\ that they are in the image of the shallow embedding.\nIn the process, ATC performs C-like type checking,\nto determine the types of \\code{let}-bound variables with \\code{declar},\nto determine the output types of the C functions,\nand to ensure that the generated C code is acceptable to C compilers;\nthe latter is not always ensured by guard verification,\nspecifically when there is dead code under the guards (by user mistake),\nwhich C compilers do not regard as dead.\n\nWhile this translation of ACL2 to C is conceptually relatively simple,\nits implementation is more complicated than anticipated.\nThere are a lot of detailed cases to consider and to check.\nGiving informative error messages when the checks fail takes some effort;\nthe current implementation may be improved in this respect.\nThe ATC code responsible for the checks and translation\nconsists of a few thousand lines (including documentation and blanks).\n\nATC generates C code\nvia an abstract syntax of (a sufficient subset of) C,\ndefined via algebraic fixtypes.\nThe C file is generated via a pretty-printer,\nwhich minimizes parentheses in expressions\nby considering the relative precedence of the C expression constructs.\n\nATC generates the C code, as well as the associated ACL2 proof events,\nvery quickly for all the examples tried so far\n(which are admittedly not very large).\nThis is expected, as ATC does a linear pass on the ACL2 code\nthat does not perform particularly intensive computations.\nHowever, the processing of the proof events by ACL2 is not always quick;\nsee \\secref{proofgen}.\n\n\n\\section{C Deep Embedding and Proof Generation}\n\\label{deep-proofgen}\n\nBesides C code,\nATC also generates ACL2 theorems asserting\nthe correctness of the C code with respect to the ACL2 code.\nThese assertions rely on a formalization in ACL2 of\nthe syntax and semantics of (a sufficient subset of) C,\ni.e.\\ a deep embedding of C in ACL2,\nwhich plays a role alongside the shallow embedding\n(see \\secref{inverse-deep}).\nThis formalization is more general than ATC, and is of independent interest.\n\n\n\\subsection{Deep Embedding of C in ACL2}\n\\label{deep}\n\nThe formalization of C in ACL2 consists of\n(i) an abstract syntax,\n(ii) a static semantics, and\n(iii) a dynamic semantics.\nBoth static and dynamic semantics are defined over the abstract syntax.\n\n\n\\subsubsection{Syntax}\n\nThe abstract syntax is currently the same one\nused for code generation (see \\secref{codegen}).\nIt captures the syntax of C after preprocessing.\n\nFuture versions of ATC may likely generate\nC code with at least some preprocessing directives,\nwhich would be convenient to incorporate\ninto the abstract syntax used for code generation;\nin this case, the abstract syntax would capture\nthe syntax of C before preprocessing.\nThus, at some point it may be appropriate for the C formalization\nto use its own separate abstract syntax,\nand in fact to have\none for the C syntax before preprocessing\nand one for the C syntax after preprocessing,\nto model more faithfully C18's translation phases.\n\nCurrently the formalization does not include the C concrete syntax.\nSince ATC's generated proofs currently apply to abstract syntax,\nthere is no immediate need to formalize concrete syntax.\n\n\n\\subsubsection{Static Semantics}\n\nThe static semantics of C\nconsists of decidable requirements\nthat must be satisfied by C code to be compiled and executed:\nevery referenced variable and function is in scope;\nevery operation is applied to operands of appropriate types;\nand so on.\nThese are described informally in the C18 standard.\n\nThese requirements are formalized via (executable) ACL2 code\nthat checks whether the C abstract syntax satisfies those requirements,\nanalogously to what C compilers do.\nThe checking code makes use of symbol tables,\ni.e.\\ data structures that capture which C symbols\n(e.g.\\ function and variable names) are in scope\nand what their types are.\nIf an abstract syntax entity violates any requirement,\nthe checking code returns an error result;\notherwise, it returns a non-error result\nthat may include inferred information about the checked abstract syntax entity.\nIn particular,\nthe successful checking of an expression yields\nthe (non-\\code{void}) C type of the expression,\nand the successful checking of a statement yields\na non-empty finite set of C types,\ncorresponding to the possible values that may be returned\n(including \\code{void} for code that completes execution\nwithout a \\code{return} or with one without expression);\nthe latter set is used to check that the body of a function\nalways returns something consistent with the function's return type.\n\n\n\\subsubsection{Dynamic Semantics}\n\nThe dynamic semantics of C\nis the execution behavior of C code\n(that satisfies the static semantic requirements),\ni.e.\\ how the execution of expressions, statements, etc.\\\nmanipulates values and memory.\nWhile this behavior is normally realized\nby compiling C code to machine code and running the latter,\nit can be described, as the C18 standard does,\nin terms of an abstract machine that directly executes C.\n\nThis abstract machine is formalized\nvia fixtypes that capture the machine states\nand via (executable) ACL2 code that manipulates machine states\naccording to the C expressions, statements, etc.\n\nThe model of machine states starts with the model of C integers and arrays\ndescribed in \\secref{shallow}, which is shared by deep and shallow embedding.\nC values are defined to consist of integers and pointers (for arrays);\na pointer consists of a type and an optional address (absent for null pointers),\nwhere an address is a natural number that is treated opaquely.%\n\\footnote{These addresses are just used to identify arrays.\nThey do not represent actual addresses in memory.\nThe formalization performs no arithmetic on them.\nOther kinds of entities, e.g.\\ strings, could be used instead.}\nImportantly, in this model the C values carry information about their types,\nwhich is used in the defensive checks described later.\nThe state of the variables in scope is\na stack (i.e.\\ list) of finite maps from identifiers to values:\nthe stack corresponds to the nested C scopes,\nand each finite map consists of the variables in the same scope.\nA frame consists of a function name and\na stack of variable scopes of the kind just described.\nA computation state consists of\na stack (i.e.\\ list) of frames, which captures the call stack,\nand a heap, which is a finite map\nfrom addresses (the ones used in non-null pointers)\nto arrays (the ones shared with the shallow embedding).\nThe current model of the heap is simple,\nwith arrays treated as wholes and accessed exclusively via their addresses.\n\nThe next component of the formalized dynamic semantics of C consists of\nACL2 functions that perform basic operations on computation states.\nThese include operations to:\npush and pop frames, and get the top frame;\npush and pop scopes (in the top frame);\ncreate variables (in the top scope in the top frame),\nand read and write variables\n(in some scope, looked up from innermost to outermost, in the top frame);\nread and write arrays (in the heap).\nThere are no operations to create or destroy arrays;\nthe read and write operations apply to externally created arrays.\n\nThe C functions in scope are captured via a function environment,\nwhich is a finite map from identifiers (function names)\nto information about the function (typed parameters, return type, and body).\nThe function environment for a C program is built\nby collecting all the C functions that form the program;\nit never changes during execution.\n\nThe C18 standard does not prescribe the order of expression evaluation;\nsince C expressions may have side effects,\ndifferent orders of evaluation may lead to different outcomes,\nwhich complicates formal modeling.\nThe formal model manages this complexity\nby partitioning expressions into pure ones (i.e.\\ free of side effects)\nand non-pure ones (i.e.\\ with possible side effects).\nWhile the pure ones may be freely nested\n(because their order of evaluation does not matter),\nthe non-pure ones may only appear in certain positions of the code\nthat force a unique order of evaluation:\nspecifically,\nassignments may only appear as expression statements.\nThese restrictions, which are not required in C18,\nare enforced in the subset of C covered by the formal model.\n\nThe last component of the formalized dynamic semantics of C consists of\nACL2 functions to execute expressions, statements, etc.\nThese functions are defensive,\ni.e.\\ they do not assume that\nthe code satisfies the static semantic requirements,\nand instead independently check those requirements on the dynamic data,\ni.e.\\ that a referenced variable is in the current frame,\nthat the values that an operation is applied to have appropriate types\n(recall the note above about values carrying type information),\nand so on;\nif any of these checks fails,\nthe execution functions return an error indication.%\n\\footnote{It remains to be formally proved that\nthe static semantics is sound with respect to the dynamic semantics,\nnamely that no such error indications are returned\nwhen executing code that satisfies the static requirements.\nThe proofs generated by ATC do not rely on this property,\nbut its proof would provide\na major validation of this formalization of C in ACL2.}\nThe ACL2 execution functions also check that\nthe results of integer operations and conversions are well-defined\nand that arrays are read and written within their bounds;\nif any of these checks fails,\nthe execution functions return an error indication.\nImportantly, this means that the dynamic semantics returns an error indication\nif the C code attempts an unsafe array access.\n\nThe ACL2 function to execute a pure expression\ntakes an expression and a computation state as inputs,\nand returns either a value or an error indication as output.\nExecuting a non-pure expression may involve executing a function call,\nwhich involves executing the statements in the function,\nand so on.\nThus, the ACL2 functions to execute\nnon-pure expressions, statements, functions, etc.\\\nare mutually recursive.\nBesides the syntactic entities they execute (statements etc.),\nthese functions take a computation state and a function environment as inputs\n(the latter is used to look up called functions),\nand they return a possibly updated computation state as output,\nalong with a result (e.g.\\ an optional value for a statement)\nor an error indication of the kind explained above.\n\nThese mutually recursive functions,\ntogether with the functions to execute pure expressions and lists thereof,\nformalize an (interpretive) big-step operational semantics:\neach function executes its syntactic construct to completion,\nby recursively executing the sub-constructs\nand combining their outcomes to yield the final outcome.%\n\\footnote{In contrast, a small-step operational semantics\nwould only execute part of a construct at each step.\nThis requires keeping track of\nwhich parts of a construct have been already executed\nand which parts must be executed next.\nThis is more complicated than just executing each construct to completion.\nAn example of small-step operational semantics,\nfor the ACL2 programming language (but still illustrating the point),\nis in the Community Books\n\\citeman{ACL2PL____ACL2-PROGRAMMING-LANGUAGE}\n        {acl2pl::acl2-programming-language}.}\nSince C code may not terminate in general,\nthe mutual recursion may not terminate,\nwhich makes it problematic to define in a theorem prover like ACL2.\nThis is solved by adding an artificial counter\nthat limits the depth of the mutual recursion:\nthe counter is decremented by 1 at each recursive call,\nand used as the measure of the mutual recursion,\nwhose termination proof is thus straightforward.\nThis does not sweep issues of (non-)termination under the rug:\nsee \\secref{proofgen}.\n\nThe ACL2 execution functions\nfor (pure and non-pure) expressions, statements, etc.\\\nuse the operations and conversions on integers discussed in \\secref{shallow},\nwhich are shared between deep and shallow embedding.\nThey also use the operations on computation states discussed above.\n\n\n\\subsection{C Proof Generation}\n\\label{proofgen}\n\nAlthough ATC's translation\nfrom the shallowly embedded C code in ACL2 to the actual C code\nis conceptually simple by design,\ngenerating ACL2 proofs of their equivalence\nis more laborious than anticipated.\nThese proofs consist of ACL2 events that ATC builds\nand then submits to ACL2 via \\code{make-event}.\n\n\n\\subsubsection{Code Constant}\n\nAs ATC computes the abstract syntax of the generated C code\nand pretty-prints it to a \\code{.c} file (see \\secref{codegen}),\nit also generates a \\code{defconst} event that defines a named constant\nfor the C code, which is a single translation unit in C18 terminology:%\n\\footnote{In the future, this may be generalized to\na collection of translation units that forms a generated program.\nEach translation unit is in a \\code{.c} or \\code{.h} file.}\n\\begin{bcode}\n(defconst *program* \\ccode{; default name, customizable by the user}\n    '(:transunit ...) \\ccode{; fixtype value of translation unit}\n\\end{bcode}\nThe other generated events refer to this constant\nto assert properties of the generated C code.\n\n\n\\subsubsection{Static Correctness}\n\nATC generates a \\code{defthm} event asserting that\nthe top-level checking function of the C static semantics\nsucceeds on the named constant described above:\n\\begin{bcode}\n(defthm *program*-well-formed\n    (equal (check-transunit *program*) \\ccode{; top-level static checking function}\n           :wellformed))\n\\end{bcode}\nSince this is a ground theorem, it is proved by execution:\nATC generates hints to prove it in the theory consisting of\nexactly the executable counterpart of that top-level checking function.\nThis proof is very quick for all the examples tried so far.\n\nThis establishes that the C code satisfies all the static semantic requirements,\nand can be therefore successfully compiled by C compilers.\nThis property is not always implied by the dynamic correctness theorems,\nin the presence of dead code under guards\nas briefly discussed in \\secref{codegen}.\n\n\n\\subsubsection{Dynamic Correctness}\n\nATC generates \\code{defthm} events asserting that, roughly speaking,\nexecuting each C function according to the dynamic semantics\nyields the same outcome as the representing ACL2 function.\nThe proofs are carried out via a symbolic execution of the C code\n(which is constant in the proofs, i.e.\\ \\code{*program*})\nthat turns the execution functions applied to the deeply embedded C code\ninto a form that can be matched with the shallowly embedded C code\nthat forms the ACL2 functions.\n\nThe general formulation is illustrated by\nthe theorem for function \\code{|f|} in \\secref{shallow}:\n\\begin{bcode}\n(defthm *program*-|f|-correct\n  (implies (and (compustatep compst)\n                (equal fenv (init-fun-env *program*))\n                (integerp limit)\n                (>= limit ...) \\ccode{; constant lower bound calculated by ATC}\n                (and (sintp |x|) (sintp |y|) (sintp |z|) ...)) \\ccode{; guard of |f|}\n           (equal (exec-fun (ident \"f\") (list |x| |y| |z|) compst fenv limit)\n                  (b* ((result (|f| |x| |y| |z|)))\n                    (mv result compst)))))\n\\end{bcode}\nIt says that executing the C function whose name is the identifier \\code{f}\non \\code{int} inputs \\code{|x|}, \\code{|y|}, \\code{|z|}\nsatisfying the guard of \\code{|f|},\non an arbitrary computation state \\code{compst},\nwith the function environment \\code{fenv} for the program,\nwith a sufficiently large recursion limit \\code{limit},\nyields the same result (value) as \\code{|f|} on the same inputs,\nand does not change the computation state.\nThe hypothesis on the limit ensures that\nexecution does not stop prematurely;\nthe lower bound, which is constant in this case,\nis calculated by ATC as it generates function \\code{f},\nbecause it knows how much recursion depth the execution of its body needs.\nATC generates hints (not shown) to prove this theorem,\nmainly consisting of a large theory\nfor symbolically executing the C code in ACL2,\nalong with a lemma instance for the guard theorem of \\code{|f|},\nwhich steers the symbolic execution away from returning error indications\ndue to non-well-defined arithmetic operations,\nsince they are all well-defined under the guard.\nThe fact that \\code{|f|} never returns error indications\nand that the execution result of \\code{f} is equal to that,\nmeans that execution never returns error indications:\nin general, this means that the C code generated by ATC\nalways has a well-defined behavior, including always accessing arrays safely.\nThis theorem also implicitly asserts that \\code{f} terminates:\nit is always possible to find a value of \\code{limit}\nthat satisfies the inequality hypothesis.\n\nTo prove the theorem above, ATC also generates a local theorem%\n\\footnote{Here `local' refers to the \\code{encapsulate}\nwith the ATC-generated events,\nonly a few of which are exported, free of hints.}\nsaying that \\code{|f|} returns an \\code{int};\nthis local theorem is referenced in the generated hints for the above theorem.\nAn analogous local theorem is generated by ATC for each ACL2 function,\nwith the applicable type(s).\n\nATC generates a correctness theorem for each C function.\nIf the C function affects arrays, the formulation is more complicated.\nThis is illustrated by the theorem for function \\code{|i|} in \\secref{shallow}:\n\\begin{bcode}\n(defthm *program*-|i|-correct\n  (b* ((|a| (read-array (pointer->address |a|-ptr) compst)))\n    (implies (and (compustatep compst)\n                  (equal fenv (init-fun-env *program*))\n                  (integerp limit)\n                  (>= limit ...) \\ccode{; constant lower bound calculated by ATC}\n                  (pointerp |a|-ptr)\n                  (not (pointer-nullp |a|-ptr))\n                  (equal (pointer->reftype |a|-ptr) (type-uchar))\n                  (and (uchar-arrayp |a|) ...)) \\ccode{; guard of |i|}\n    (equal (exec-fun (ident \"i\") (list |a|-ptr |x| |y|) compst fenv limit)\n           (b* ((|a|-new (|i| |a| |x| |y|)))\n             (mv nil (write-array (pointer->address |a|-ptr) |a|-new compst))))))\n\\end{bcode}\nSince C arrays are manipulated as wholes in ACL2 but via pointers in C,\na variable \\code{|a|-ptr} is introduced for the pointer to the array,\nwhile the array \\code{|a|} is the result of \\code{read-array}:\nthe call of \\code{exec-fun} takes \\code{|a|-ptr};\nthe call of \\code{|i|} takes \\code{|a|}.\nThe hypotheses on \\code{|a|-ptr} say that it is\na non-null pointer of the right type;\nthe guard hypothesis saying that \\code{|a|} is an array\nalso implicitly says that \\code{|a|-ptr} points to an existing array,\nbecause \\code{read-array} returns an error indication if that is not the case.\nThis hypothesis constrains the computation state to contain that array,\nwhich is thus no longer unconstrained\nas in the correctness theorem for \\code{|f|}.\nAlso unlike the correctness theorem for \\code{|f|},\nhere the computation state is updated by the execution of \\code{i}:\nthe new array, returned by \\code{|i|}, is \\code{|a|-new};\nthe new computation state is obtained by replacing\nthe array \\code{|a|} pointed by \\code{|a|-ptr} with \\code{|a|-new}.\nThe \\code{nil} that precedes the updated computation state\nrefers to the fact that \\code{i} returns no result.\nThe generated hints for this theorem are similar to\nthe ones for the correctness theorem for \\code{|f|}.\n\nATC also generates a correctness theorem for each C loop,\nrelating its execution to the ACL2 function that represents it.\nThis is illustrated by\nthe theorem for function \\code{|h\\$loop|} in \\secref{shallow}:\n\\begin{bcode}\n(defthm *program*-|h\\$loop|-correct\n  (b* ((|n| (read-var (ident \"n\") compst))\n       (|r| (read-var (ident \"r\") compst)))\n    (implies (and (compustatep compst)\n                  (not (equal (compustate-frames-number compst) 0))\n                  (equal fenv (init-fun-env *program*))\n                  (integerp limit)\n                  (>= limit ...) \\ccode{; non-constant lower bound calculated by ATC}\n                  (and (uintp |n|) (uintp |r|))) \\ccode{; guard of |h\\$loop|}\n             (equal (exec-stmt-while\n                     '... \\ccode{; test of the loop}\n                     '... \\ccode{; body of the loop}\n                     compst fenv limit)\n                    (b* (((mv |n|-new |r|-new) (|h\\$loop| |n| |r|)))\n                      (mv nil\n                          (write-var (ident \"n\")\n                                     |n|-new\n                                     (write-var (ident \"r\")\n                                                |r|-new\n                                                compst))))))))\n\\end{bcode}\nThe computation state is constrained to have at least one frame\n(i.e.\\ the number of frames is not 0)\nand two variables \\code{n} and \\code{r} in the top frame\n(i.e.\\ the variables accessed by the loop);\n\\code{|n|} and \\code{|r|} are bound to those variables' values.\nThe guard hypothesis saying that \\code{|n|} and \\code{|r|}\nare \\code{unsigned} \\code{int} values\nalso implicitly says that the variables exist,\nbecause \\code{read-var} returns an error indication if that is not the case.\nThe lower bound in the hypothesis on \\code{limit} is not constant here:\nit is a symbolic term that references the measure of \\code{|h\\$loop|},\nbecause the measure is related to the recursion limit needed to execute the loop;\nATC calculates this symbolic term as it generates the loop's code.\nThe \\code{exec-stmt-while} function is called on\nthe quoted test and body of the loop,\nmaking this theorem applicable as a rewrite rule\nin the proof of the correctness theorem for \\code{|h|}\n(more on this below).\nSince C loops affect variables (and possibly arrays) but do not return results,\nthe right side of the equality whose left side is the \\code{exec-stmt} call\nhas a form similar to the correctness theorem for function \\code{|i|} above,\nexcept that variables instead of arrays are updated in the computation state.\nThis theorem is proved by induction;\nthe generated hints make use of the termination theorem of \\code{|h\\$loop|},\nas well as of some local functions and theorems not discussed here.\n\nIf a C loop affects arrays besides variables,\nits correctness theorem combines characteristics of\nthe correctness theorems for \\code{|h\\$loop|} and for \\code{|i|}.\nThe theorem starts with \\code{b*} bindings for both variables and arrays.\nVariables for the pointers to the arrays are introduced.\nThe final computation state updates both variables and arrays,\nvia nested \\code{write-var} and \\code{write-array} calls.\n\nIf a C function or C loop affects multiple arrays,\nthe generated correctness theorem includes hypotheses saying that\nthe arrays are all at different addresses.\nSince the representing ACL2 function treats the arrays as wholes,\nupdating one leaves the others unchanged in the ACL2 representation.\nIn the C code, where the arrays are handled via pointers,\nupdates would not be independent if two pointers pointed to the same array.\nThe absence of aliasing is therefore\na critical hypothesis in the correctness theorems.\n\nThe correctness theorem for a C loop is used as a rewrite rule\nin the symbolic execution of\nthe correctness theorem for the C function that contains the loop.\nSimilarly, the correctness theorem for a C function is used as a rewrite rule\nin the symbolic execution of another C function that calls it.\nIn other words,\nthe generated correctness proofs build upon each other,\naccording to the call graph of the ACL2 functions\nthat represent the C functions and loops.\nFor this to work,\nthe theorems are formulated so that\nthe left sides of the rewrite rules match\nthe terms that arise during symbolic execution.\nIn particular, the recursion limit is a variable, \\code{limit},\nwhich the hypotheses constrain with a lower bound:\nthis ensures that any actual limit term\nthat arises during the symbolic expression is matched by the variable,\nand that the inequality hypothesis can be discharged.\nATC calculates the lower bound terms by considering\nnot only the code generated for the C function or loop,\nbut also the lower bound terms previously calculated\nfor all the C functions and loops that may be executed;\nthe more complex the call graph of the ACL2 functions,\nthe more complex the resulting lower bound terms.\n\nThe above is just an overview of how ATC generates dynamic correctness proofs.\nThere are several other complexities involved,\nsuch as canonical forms of the symbolic computation states,\nwhich are defined via slightly modified operations on computation states,\nand achieved via appropriate sets of rewrite rules.\nThe implementation of ATC includes documentation\nfor all the details of proof generation.\n\nSome correctness theorems are processed quickly by ACL2,\ne.g.\\ in less than 0.1 seconds.\nOther correctness theorems take several minutes.\nThe slow times seem mainly due to case splits that occur\nwhen dealing with code with several conditionals.\nThe granularity of the correctness proofs generated by ATC\nis at the function and loop level,\ni.e.\\ there is one theorem per C function or loop,\nproved by symbolically executing the function or loop as a whole,\nand matching that with the representing C functions.\nThis process may be slow even for relatively small C functions and loops,\nif they have enough conditionals.\nAn approach to overcome this problem is discussed in \\secref{future}.\n\nThe hypotheses in the correctness theorems generated by ATC\nmust be fulfilled by external C code that calls the generated C code,\nfor the correctness guarantees to hold.\nThis does not apply to the \\code{limit} hypotheses,\nwhich just serve to show that execution terminates,\nand which are easily satisfied with sufficiently large values of \\code{limit},\nwhich are not part of the data manipulated by the C code\n(they are an artifact of the model).\nThe satisfaction of the hypotheses about the types of the inputs\nmay be checked by C compilers.\nSatisfying the remaining hypotheses\nis the responsibility of the calling code;\nthese are the non-type portions of the guards\n(e.g.\\ that certain values are in certain ranges),\nand the fact that all the arrays are disjoint.\nEven though the formal model puts all the arrays in the heap,\nthe arrays passed to the C functions generated by ATC\ncould be allocated in the stack (by the callers) as well;\nthe heap of the formal model more generally represents\nexternally allocated memory.\n\n\n\\section{Future Work}\n\\label{future}\n\nAn obvious direction of future work is\nto add support for increasingly larger subsets of C.\nAlthough the current subset can represent some interesting programs,\nmany other interesting programs cannot be represented.\nWork is underway to add support for structures,\nwhich are commonly used in C.\nThe main challenge in supporting additional C features\nis perhaps the definition of their representations in the shallow embedding.\nGiven that ACL2 is a very different language from C,\ndefining such representations may require some thought.\nIn particular, representing aliasing may involve some complexity,\nbecause of the likely need to explicate some graph structure\nthat captures aliased data and that is passed through the ACL2 functions.\nHandling tree-shaped data without aliasing is comparatively simple,\nalong the lines of the current treatment of arrays,\nwhich may be generalized to includes structures,\npossibly mutually nested with arrays.\n\nBoth shallow and deep embedding currently comply to the C18 standard.\nHowever, some C implementations may extend the well-definedness of C constructs\n(e.g.\\ they may ensure that signed arithmetic is always well-defined,\nas two's complement wrap-around as in the x86 processor),\nand some applications may rely on this extended well-definedness.\nThe plan is to parameterize both shallow and deep embedding\nover certain implementation-defined characteristics,\ncaptured via ACL2 constrained functions;\ndifferent instantiations of these parameters\ncan be captured via hypotheses over these constrained functions,\nand these hypotheses can be used in the applications that require them.\nIn particular, the behavior of an arithmetic operation\nover operands of certain types\nwhose exact result is not representable in the result type\ncan be captured via a constrained function.\nFor C code strictly compliant to the C18 standard,\nthis constrained function can be hypothesized to return an error indication;\nfor C code tailored to an implementation based on x86 as above,\nthis constrained function can be hypothesized to return\nthe two's complement wrapped-around result.\nThe same approach is planned for the sizes of integer types,\nas alluded to in \\secref{shallow}.\n\nThe proof (in)efficiency issues discussed in \\secref{proofgen}\nmust be addressed soon.\nWhile there may be ways to mitigate the case splits\nby tweaking the symbolic execution,\na more promising approach is to generate finer-grained proofs,\nat the level of blocks, or even statements and expressions,\nas ATC generates these syntactic entities.\nThe finer-grained proofs will still use symbolic execution,\nbut in a more controlled and efficient way, avoiding case splits.\nIt should be possible to generate finer-grained proofs\nthat are processed in linear time over the size of the C code.\n\nATC could be extended with code generation modes\nbased on direct shallow embedding and direct deep embedding,\nanalogously to ATJ\n\\cite{atj-deep,atj-shallow} \\citeman{JAVA____ATJ}{java::atj}.\nThis may require the use of a garbage collector for C.\n\n\n\\section{Related Work}\n\\label{related}\n\nATC's shallow embedding of C in ACL2\nis similar to the shallow embedding of RAC (Restricted Algorithmic C) in ACL2\n\\cite[Chapter 15]{russinoff-book}.\nDespite their different purposes (code generation vs.\\ code verification),\nthey share the concern of representing C or RAC code in ACL2.\n\nATJ \\cite{atj-deep,atj-shallow} \\citeman{JAVA____ATJ}{java::atj}\nis a Java code generator for ACL2\nprimarily based on direct deep embedding and direct shallow embedding.\nIt also features partial support for inverse shallow embedding,\nwhich was in fact pioneered in ATJ,\nand then developed in full in ATC.\nWhile ATJ does not generate proofs yet,\nATC has been generating them from the outset.\nWhile ATC requires the ACL2 code translated to C\nto be in a very restricted form,\nATJ translates to Java a much larger subset of ACL2.\n\nSeveral other theorem provers include code generation facilities\n\\cite{coq-refman,isa-codegen,pvs-codegen}.\nThese are based on direct shallow embedding,\nwhich is quite different from ATC's inverse shallow embedding.\nFurthermore, the ACL2 language is quite different from those provers' languages,\nwhich are higher-order and strongly typed.\nThus, not many of the ideas from those provers' code generation facilities\nmay be relevant to ATC.\nThe C code generator for PVS \\cite{pvs-codegen}\nmay contain the most relevant ideas for ATC,\nbut likely more for a future code generation mode\nbased on direct shallow embedding.\n\nSeveral formalizations of C exist\n\\cite{c-asm,c-coq,c-hol,c-k,c-papaspyrou}.\nThese may contain ideas relevant to\nextending and improving the formalization of C in ACL2 described in this paper,\nwhich the proofs generated by ATC are based on.\n\n\n\\section*{Acknowledgements}\n\nThanks to\nRuben Gamboa,\nKarthik Nukala, and\nEric Smith\nfor using the C code generator\nand for providing valuable suggestions that led to improvements to the tool.\nThanks to Eric Smith and Karthik Nukala\nfor developing new APT transformations tailored to the C code generator.\nThanks to David Hardin for\nuseful discussions on C code generation and related topics.\n\n\n\\bibliographystyle{eptcs}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe theory of Sample Distribution  is an important branch of  Statistics and Probability which study the general problem of determining the distribution of functions of random vectors. It   provides a formal framework for  modelling, simulating and making statistical inference. To be more precise, let us fix   a probability measure space $\\left(\\Omega,\\Sigma,p\\right)$  and let $X$ be  a   (vector valued) random variable    i.e a $\\Sigma$-measurable function from $\\Omega$ to $\\mathbb{R}^k$, where $k\\in\\mathbb{N}$; $X$ is usually referred to as the data. Let $Y$ be any measurable function of the data $X$ i.e. $Y$ is a  random variable which satisfies $Y=\\phi(X)$ for some measurable function $\\phi:\\mathbb{R}^k\\to\\mathbb{R}^n$ with  $n\\in\\mathbb{N}$; $Y$ is usually called a statistic. \nThe problem which we address consists in finding the probability distribution of $Y$ knowing the distribution of $X$.\n\n\n\n\n Depending on the nature of the data, there are in general different approaches for finding the distribution of the statistic $Y$,  including the distribution function technique,  the moment-generating function technique and the change of variable technique (see e.g. \\citep[Section 4.1, page 122]{HoggCraig}). In the last case,  let us suppose for example that  the probability measure induced by $X$ is absolutely continuous with respect to the Lebesgue measure on $\\mathbb{R}^k$ and let  $f_X$ be its density function. Then if $k=n$ and $\\phi:\\mathbb{R}^k\\to\\mathbb{R}^k$ is a $C^1$-diffeomorphism, then the  change of variable $y=\\phi(x)$ yields, for every Borel subset $A\\subseteq\\mathbb{R}^k$,\n\\begin{align}\\label{int 2}\np\\left(\\phi(X)\\in A\\right)=p\\left(X\\in \\phi^{-1}(A)\\right)=\\int_{\\phi^{-1}(A)}f_X(x)\\,dx=\\int_{A}f_X(\\phi^{-1}(y))\\frac{1}{|\\mbox{det}J_\\phi(\\phi^{-1}(y))|}\\,dy.\n\\end{align}\nThe last equation implies that the probability measure induced by  $Y$ is absolutely continuous with respect to the Lebesgue measure on $\\mathbb{R}^k$ and   its density function $f_Y$ is  determined by the equation\n\\begin{align}\\label{int 1}\nf_Y(y)=f_X(\\phi^{-1}(y))\\frac{1}{|\\mbox{det}J_\\phi(\\phi^{-1}(y))|}, \\quad y\\in\\mathbb{R}^k.\n\\end{align}\nThis change of variables formula is widely used, for example,  in machine learning and is essential for some recent results in density estimation and\n\tgenerative modeling like normalizing flows (\\citep{Rezende}), NICE (\\citep{Dinh2}), or Real NVP (\\citep{Dinh1}). However  all uses of this formula in the machine learning literature that we are aware of are constrained by the  bijectivity and the differentiability of the map $\\phi$.\n\n\nIn this expository paper we extend formula \\eqref{int 1} to the more general case of statistics $Y=\\phi(X)$ defined by  locally  Lipschitz functions $\\phi:\\mathbb{R}^k\\to\\mathbb{R}^n$. The approach presented is mainly based upon the Coarea Formula proved by Federer in \\citep{Federer} which provides, in our setting,  an elegant tool to derive the  distribution of $Y$. This also allows  to get rid of the  differentiability and invertibility assumptions on $\\phi$ and to treat the case $k\\neq n$: this is  quite useful in many  problems of statistical inference and of machine learning (see e.g. \\citep{Civitkovic}).\n\nWhen $\\phi:\\mathbb{R}^k\\to \\mathbb{R}^n$ is a $C^1$-map  and  $m:=\\max\\{\\mbox{rank }(J\\phi(x)):x\\in\\mathbb{R}^k\\}$, our result states that the  probability  $p_Y$ induced by  $Y=\\phi(X)$ has a density function $f_Y$ with respect to the  Hausdorff measure $\\mathcal{H}^m$  on $\\phi(\\mathbb{R}^k)$\nwhich  satisfies  \n\\begin{align}\\label{eq intr}\n\tf_Y(y)=\n\t\\int_{\\phi^{-1}(y)}f_X(x)\\frac{1}{J_m\\phi(x)}\\,d\\mathcal{H}^{k-m}(x), &\\quad \\text{for $\\mathcal{H}^m$-a.e.}\\quad y\\in\\phi(\\mathbb{R}^k),\n\\end{align} \nwhere $J_m\\phi(x)$ is the $m$-dimensional Jacobian of $\\phi$ (see Definition \\ref{k-Jac}).  When  the Jacobian matrix $J\\phi$ has, at any point,  maximum rank we allow the map $\\phi$   to be only locally Lipschitz. We also consider the case of  $X$ having   probability  concentrated on  some $m$-dimensional sub-manifold $E\\subseteq\\mathbb{R}^k$. This case has  many applications in directional and axial statistics, morphometrics, medical\ndiagnostics, machine vision, image analysis and  molecular biology (see e.g \\citep{Bhattacharya} and references there in).\n\n\n\n We do not make any attempt to reach any novelty  but the evidence suggests that this result is not as universally known as it should be. Besides this we give also several examples.\n\n\n\n\\smallskip\nLet us briefly describe the content of the sections. In Section \\ref{preliminaries} we  introduce the main definitions and notation that we use throughout the paper.  Section \\ref{Section AC}    collects the main results we need  about  the theory of Hausdorff measures and the  Area and Coarea Formulas. In Section \\ref{Section Distribution} we  develop the method  proving Formula \\eqref{eq intr}:  when $J\\phi$ has maximum rank we also allow the map $\\phi$   to be locally Lipschitz. In Section \\ref{Section Generalization}  we gives  a further  generalization considering   the case of   random variables $X$ having   probability density functions $f_X$ with respect to  the Hausdorff measure $\\mathcal{H}^m_{\\vert E}$ concentrated on  some $m$-dimensional sub-manifold $E\\subseteq\\mathbb{R}^k$. \nFinally, in Section \\ref{Section apll}, we provide several examples which show how to apply the latter results in order to characterize the distribution of  algebra of random variables and how  to compute, in an easy way, the probability densities of some classic distributions including order statistics, degenerate normal distributions, Chi-squared and \"Student's t\" distributions. \n\n\n\\bigskip\n\\noindent\\textbf{Notation.} \nWe write $\\langle \\lambda,\\mu \\rangle=\\sum_{i}\\lambda_i\\mu_i$ to denote the inner product of $\\mathbb{R}^k$.\n When $f:\\mathbb{R}^k\\to\\mathbb{R}^n$ is a Lipschitz map we write $Jf$ to denote its  Jacobian matrix $\\left(\\frac{\\partial f_i}{\\partial x_j}\\right)_{i,j}$ which is defined a.e. on $\\mathbb{R}^k$. When $A=(a_{ij})\\in \\mathbb{R}^{n,k}$ is a real matrix we write $Ax$ to denote the linear operator \n$$\\phi:\\mathbb{R}^k\\to\\mathbb{R}^n,\\qquad x=(x_1,\\dots,x_k)\\mapsto x\\cdot A^t=(y_1,\\dots,y_n),\\quad y_i=\\sum_{j=1}^k a_{i,j}x_j.$$\n With this notation the Jacobian matrix $J(Ax)$ of $Ax$ satisfies $J(Ax)=A$. $I_k$ is the identity matrix of $\\mathbb{R}^{k,k}$. If  $\\left(\\Omega_1, \\Sigma_1\\right)$ and $\\left(\\Omega_2, \\Sigma_2\\right)$ are measurable spaces, a function $f :\\Omega_1\\to\\Omega_2$ is said to be $\\left(\\Sigma_1,\\Sigma_2\\right)$-measurable if $f^{-1}(B)\\in\\Sigma_1$ for all $B\\in\\Sigma_2$. \nUnless otherwise specified when $\\Omega_2=\\mathbb{R}^k$ we always choose $\\Sigma_2$ as the $\\sigma$-algebra $\\mathcal{B}(\\mathbb{R}^k)$ of all the Borel subsets of $\\mathbb{R}^k$ and in this case we simply say that $f$ is  $\\Sigma_1$-measurable. We finally write $\\mathcal L^k$ and $\\mathcal H^s$ to denote respectively the Lebesgue measure and the $s$-dimensional Hausdorff measure on $\\mathbb{R}^k$: under this notation we have in particular that  $\\mathcal L^k=\\mathcal H^k$ and that $\\mathcal H^0$ is the counting measure on $\\mathbb{R}^k$. \n\n\n\n\\bigskip\n\n\\noindent\\textbf{Acknowledgements.} The author thanks F. Durante  for several comments on a previous version of the manuscript.\n\n\\section{Preliminaries}\\label{preliminaries}\nIn this section we fix the main notation and collect the main  results we use concerning the  Probability theory. For a good survey on the topic  we refer  the reader, for example,  to \\citep[Chapter IV]{HalmosMeasure} and \\citep[Appendix A and B]{Schervish}.\n\n\nLet $\\mu,\\nu$ two (positive) measures defined on a  measurable space $\\left(\\Omega,\\Sigma\\right)$.   $\\nu$ is said to be \\emph{absolutely continuous} with respect to $\\mu$ and we write $\\nu\\ll\\mu$ if and only if $\\nu(B)=0$ for every $B\\in\\Sigma$ such that $\\mu(B)=0$.  $\\nu$ is said to have a  \\emph{density function} $f$ with respect to $\\mu$  if and only if  there exists a measurable positive function  $f:\\Omega\\to \\mathbb{R}^+$ such that \n\\begin{align*}\n\\nu(B)=\\int_B f\\,d\\mu,\\quad  \\text{for all}\\quad A\\in\\Sigma,\n\\end{align*}\n(note that $f$ is uniquely defined up to zero measure   sets). When $\\mu$ is $\\sigma$-finite, thanks to the Radon-Nikodym Theorem,  the latter two definitions coincide and  $\\frac{d\\nu}{d \\mu}:=f$ is called the \\emph{Radon-Nikodym derivative} of $\\nu$ with respect to $\\mu$ (see e.g. \\citep[Theorem 1.28]{AFPallara}).\n\n\nLet now $\\left(\\Omega,\\Sigma,p\\right)$ be a probability measure space i.e. a measure space with  $p(\\Omega)=1$ and let $k\\in\\mathbb{N}$. A $\\Sigma$-measurable function $X$ from $\\Omega$ to $\\mathbb{R}^k$ is called a   (vector) \\emph{random variable};  in statistical inference problems, $X$ is sometimes referred to as the given data. We write $p_X$ to denote the distribution of $X$ i.e. the measure induced by $X$ on $\\left(\\mathbb{R}^k, \\mathcal{B}(\\mathbb{R}^k)\\right)$ defined by\n\\begin{align*}\np_X(A)=p\\left(X\\in A\\right):=p(X^{-1}(A)),\\quad \\text{for all Borel set}\\quad A\\subseteq \\mathbb{R}^k.\n \\end{align*}\nWith a little abuse of terminology, $X$ is said to be an \\emph{absolutely continuous random variable} if and only if $p_X\\ll \\mathcal L^k$  i.e. $p_X$ is absolutely continuous with respect to the Lebesgue measure $\\mathcal L^k$. In this case the non-negative Radon-Nikodym derivative $f_X:=\\frac{dp_X}{d \\mathcal L^k}$ is called the    density  function of $X$ and it is defined through the relation\n\\begin{align*}\np_X(A)=\\int_{A}f_X(x)\\,dx,\\quad \\text{for all Borel set}\\quad A\\subseteq \\mathbb{R}^k.\n\\end{align*}\n$X$ is said to be a \\emph{discrete random variable} if and only if there exists   a countable subset $I=(a_i)_{i\\in N}$ of $\\mathbb{R}^k$ such that $p_X\\ll {\\mathcal{H}^0}_{\\vert I}$  i.e. $p_X$ is absolutely continuous with respect to the counting measure $\\mathcal{H}^0$ on $I$. In this case the density \n $f_X:=\\frac{dp_X}{d {\\mathcal{H}^0}_{\\vert I}}$ is also  called the probability mass function and it is defined through the relation\n\\begin{align*}\np_X(A)=\\sum_{i:a_i\\in A}f_X(a_i),\\quad \\text{for all subset}\\quad A\\subseteq \\mathbb{R}^k.\n\\end{align*}\nIn particular $p_X(a)=f_X(a)$, for all $a\\in A$.\n\n\n Let $X:\\Omega\\to\\mathbb{R}^k$ be a fixed random variable and let $n\\in\\mathbb{N}$; a random variable $Y:\\Omega\\to\\mathbb{R}^n$ is called a \\emph{statistic} (of the data $X$)  if it is a measurable function   of  $X$ i.e. $Y$  satisfies $Y=\\phi\\circ X$ for some $\\left(\\mathcal{B}(\\mathbb{R}^k), \\mathcal{B}(\\mathbb{R}^n)\\right)$ measurable function $\\phi:\\mathbb{R}^k\\to\\mathbb{R}^n$.\n\n\nFinally, let $X_1,\\dots, X_n$ be $n$ random variables where $X_i:\\Omega\\to\\mathbb{R}^k$, for $i=1,\\dots, n$. $X_1,\\dots, X_n$ are said to be  \\emph{independent} if  for every  Borel subset $A_1,\\dots A_n$ of $\\mathbb{R}^k$ and for every  $J\\subseteq \\{1,\\dots, n\\}$ one has \n\\begin{align*}\np\\left(\\bigcap_{i\\in J}X_i^{-1}(A_i)\\right)=\\prod_{i\\in J}p_{X_i}(A_i).\n\\end{align*}\nIn this case, if every $X_i$ is absolutely continuous with density function $f_i$, then $X$ is absolutely continuous and its density function satisfies $f_X(x_1,\\dots, x_n)=\\prod_{i=1,\\dots,n}f_i(x_i)$ for every $x_i\\in\\mathbb{R}^k$.\n\nIf, moreover, $X_1,\\dots, X_n$ are identically distributed i.e. $p_{X_i}=p_{X_j}:=q$ for every $i,j$, then  $X=\\left(X_1,\\dots, X_n\\right)$ is called a \\emph{random sample} from the distribution $q$; in this case, if every $X_i$ is absolutely continuous with density $f_i=f$,  then the density function of $X$ satisfies $f_X(x_1,\\dots, x_n)=\\prod_{i=1,\\dots,n}f(x_i)$.\n\n\\section{Area and Coarea Formulas}\\label{Section AC}\n\nIn this section we provide a brief introduction to the theory of Hausdorff measures and we collect the main results about   the  Area and Coarea Formulas proved by Federer in \\citep{Federer}.  For the related proofs, we refer the reader, for example,  to \\citep{AFPallara, Federer-book, GiaqModica} (and references therein).\n\nWe begin with the definition of the $s$-dimensional Hausdorff measure.\nLet $E\\subseteq\\mathbb{R}^n$, $\\epsilon>0$ and let   $\\left(B(x_i,r_i)\\right)_{i\\in\\mathbb{N}}$ be  a coverings of $E$ by a countable collections of balls $B(x_i,r_i)$  whose radii  satisfy $r_i\\leq \\epsilon$. For each $s\\geq 0$, let\n\\begin{align*}\n\\sigma_s(\\epsilon)= \\frac{\\pi^{s\/2}}{\\Gamma (1 + s\/2)}\\inf\\sum_{i\\in\\mathbb{N}} r_i^s,\n\\end{align*}\nwhere the infimum is taken over all such coverings. The monotonicity of $\\sigma_s$, with respect to $s$, implies that there exists the limit (finite or\ninfinite)\n\\begin{align*}\n\\mathcal{H}^s(E) := \\lim_{\\epsilon\\to 0^+}\\sigma_s(\\epsilon).\n\\end{align*}\nThis limit is called the $s$-dimensional Hausdorff measure of E. The Hausdorff measure $\\mathcal{H}^s$ satisfies Caratheodory's criterion therefore, the $\\sigma$-algebra of all the  $\\mathcal{H}^s$-measurable sets contains all the Borel subsets of $\\mathbb{R}^n$ (see e.g. \\citep[Proposition 2.49]{AFPallara}).\n\n\nIf $0<s\\leq n$ is a positive\ninteger and $E$ is  an $s$-dimensional smooth sub-manifold of $\\mathbb{R}^n$, then $\\mathcal{H}^s(E)$ is  the $s$-dimensional volume of $E$. In particular, $\\mathcal{H}^n(E) = \\mathcal L^n(E)$ for Lebesgue measurable subsets $E\\subseteq \\mathbb{R}^n$ (see, e.g., \\citep[Section 3.2]{Federer-book}).\n\n\n\nWhen $s=0$, $\\mathcal{H}^0$ coincides with the counting measure on $\\mathbb{R}^n$ which associates to any $E\\subseteq\\mathbb{R}^n$, its number of elements $|E|$ ($+\\infty$ in case the subset is infinite); in particular for every function $f:\\mathbb{R}^n\\to\\mathbb{R}$ one has \n\\begin{align*}\n\\int_E f(x)\\,d\\mathcal{H}^0(x)=\\sum_{x\\in E}f(x).\n\\end{align*}\n\n\nLet now  $\\phi:\\mathbb{R}^k\\to\\mathbb{R}^n$ be a locally Lipschitz map; we remark that \n  Rademacher's Theorem assures that $\\phi$ is a.e. differentiable  (see for example \\citep[Proposition 2.12 and 2.14]{AFPallara}).\n\n\n\n\n\\begin{defi}[$k$-dimensional Jacobian]\\label{k-Jac}\nLet $k,m,n\\in\\mathbb{N}$  and let $\\phi:\\mathbb{R}^k\\to\\mathbb{R}^n$ be a locally Lipschitz map. The $m$-dimensional Jacobian of $\\phi$ is defined by \n\\begin{align*}\n J_m \\phi(x):=\\sup\\left\\{\n \\frac{\\mathcal{H}^m\\Big(J\\phi(x)(P)\\Big)}{\\mathcal{H}^m(P)}\n\\;:\\; P\\;\\text{is a $m$-dimensional parallelepiped of $\\mathbb{R}^k$}\n \\right\\},\n \\end{align*}\n where  $J\\phi(x)(P)$ is the image of $P$ under the Jacobian  matrix $J\\phi(x)$ of $\\phi$ which exists for a.e. $x\\in\\mathbb{R}^k$. When $\\mbox{rank }(J\\phi(x))\\leq m$ then \n\\begin{align*}\n J_m \\phi(x)=\\sqrt{\\sum_{B}(\\mbox{det\\,}B)^2},\n \\end{align*}\n where the sum in the last equality  runs along all $m\\times m$ minors $B$ of  $J\\phi(x)$ (see e.g. \\citep[Section 3.6]{Morgan}).\n\\end{defi}\nNote that $J_m\\phi(x)=0$ if and only if $\\mbox{rank}(J\\phi(x))<m$ and that the Cauchy-Binet formula gives  in particular \n\\begin{align*}\nJ_n \\phi=\\sqrt{\\mbox{det}\\left(J\\phi\\cdot J\\phi^t\\right)}, \\qquad J_k \\phi=\\sqrt{\\mbox{det}\\left(J\\phi^t\\cdot J\\phi\\right)}.\n\\end{align*}\n\nIn the conventional situation $k=n=m$, the above definition gives back the classic Jacobian  $J_k \\phi=|\\mbox{det}J\\phi|$.\n\\bigskip\n\nThe next Theorem handles with the case $k\\leq n$ and it  can be seen as a generalization of the change of variables formula \\eqref{int 2} when the   invertibility and  $C^1$-regularity assumptions on $\\phi$ are dropped.  For its proof we refer the reader to \\citep[Theorem 3.2.11]{Federer}, \\citep[Theorem 2.80]{GiaqModica}.\n\n\\begin{teo}[Area formula]\\label{Area}\nLet $\\phi:\\mathbb{R}^k\\to\\mathbb{R}^n$ be a locally Lipschitz map with $k\\leq n$.\n\\begin{itemize}\n\\item[(i)]For any $\\mathcal{L}^k$-measurable set $E\\subseteq \\mathbb{R}^k$ the multiplicity function $y\\mapsto \\mathcal{H}^0\\left(E\\cap \\phi^{-1}(y)\\right)$ is $\\mathcal{H}^k$-measurable in $\\mathbb{R}^n$ and\n\\begin{align*}\n\\int_E J_k\\phi(x)\\,dx=\\int_{\\mathbb{R}^n}\\mathcal{H}^0\\left(E\\cap \\phi^{-1}(y)\\right)\\,d\\mathcal{H}^k(y).\n\\end{align*}\n\\item[(ii)] If $u$ is a positive measurable function,  or $u\\,J_k\\phi\\in L^1\\left(\\mathbb{R}^k \\right)$,   then\n\\begin{align*}\n\\int_{\\mathbb{R}^k}u(x) J_k\\phi(x)\\,dx=\\int_{\\mathbb{R}^n}\\int_{\\phi^{-1}(y)}u(x)\\,d\\mathcal{H}^0(x)\\;d\\mathcal{H}^k(y)=\\int_{\\mathbb{R}^n}\\sum_{x\\in \\phi^{-1}(y)}u(x)\\;d\\mathcal{H}^k(y).\n\\end{align*}\n\\end{itemize} \n\\end{teo}\n\n\nWhen $\\phi: E\\to \\mathbb{R}^n$ is injective, the last formula allows the computation of the area of the  Lipschitz parametrized $k$-dimensional manifold $\\phi(E)$ of $\\mathbb{R}^n$:\n\\begin{align}\\label{parametrized manifold}\n\\nonumber\\mathcal{H}^k(\\phi(E))&=\\int_E J_k\\phi\\,dx=\\int_E \\sqrt{\\mbox{det}(J\\phi^t\\cdot J\\phi)}\\,dx,\\\\[1ex]\n\\int_{\\phi(E)}g(y)\\,d\\mathcal{H}^k(y)&=\\int_E g(\\phi(x))J_k\\phi\\,dx=\\int_E g(\\phi(x))\\sqrt{\\mbox{det}(J\\phi^t\\cdot J\\phi)}\\,dx,\n\\end{align}\nwhere $g$ is any positive measurable function,  or $g\\in L^1\\left(\\phi(E),\\mathcal{H}^k \\right)$.\n Note that in the particular case of a Cartesian parametrization $\\phi(x) = (x,\\psi(x))$ one has  $J_k\\phi=\\sqrt{1+\\sum_B \\mbox{det}(B)^2}$\nwhere this times the sum   runs along all square minors $B$ of the Jacobian matrix $J\\psi$ of $\\psi$ (see \\citep[page 88]{AFPallara}). \n\n\n\\bigskip\n\n\nThe next Theorem treats, conversely, the case $k> n$ and it  can be seen as a generalization of the Fubini's theorem about the reduction of  integrals. For its proof we refer the reader to \\citep[Theorem 3.2.3]{Federer}, \\citep[Theorem 2.86]{GiaqModica}. \n\\begin{teo}[Coarea formula]\\label{Coarea}\nLet $\\phi:\\mathbb{R}^k\\to\\mathbb{R}^n$ be a locally Lipschitz map with $k> n$.\n\\begin{itemize}\n\\item[(i)]For any $\\mathcal{L}^k$-measurable set $E\\subseteq \\mathbb{R}^k$ the  function $y\\mapsto \\mathcal{H}^{k-n}\\left(E\\cap \\phi^{-1}(y)\\right)$ is $\\mathcal{L}^n$-measurable in $\\mathbb{R}^n$ and\n\\begin{align*}\n\\int_E J_n\\phi(x)\\,dx=\\int_{\\mathbb{R}^n}\\mathcal{H}^{k-n}\\left(E\\cap \\phi^{-1}(y)\\right)\\,dy.\n\\end{align*}\n\\item[(ii)] If $u$ is a positive measurable function, or $uJ_n\\phi\\in L^1\\left(\\mathbb{R}^k\\right)$, then \n\\begin{align*}\n\\int_{\\mathbb{R}^k}u(x) J_n\\phi(x)\\,dx=\\int_{\\mathbb{R}^n}\\int_{\\phi^{-1}(y)}u(x)\\,d\\mathcal{H}^{k-n}(x)\\;dy.\n\\end{align*}\n\\end{itemize} \n\\end{teo}\nWhen $\\phi$ is an orthogonal projection \n(e.g. $\\phi(x_1,\\cdots, x_k)=(x_{i_1},\\cdots, x_{i_n})$ where $\\{i_1,\\cdots, i_n\\}\\subseteq \\{1,\\cdots, k\\}$), then $J_n\\phi=1$,  the level sets of $\\phi$ are $(n-k)$-planes and the last formula corresponds to Fubini's theorem. \n\n\nApplying Theorem \\ref{Coarea} in the particular case $n=1$, then one has  $J_1\\phi(x)=|\\nabla \\phi(x)|$ and the formula in (ii) becomes\n\\begin{align}\\label{1d-Coarea}\n\\int_{\\mathbb{R}^k}u(x)\\, |\\nabla \\phi(x)|\\,dx=\\int_{\\mathbb{R}}\\int_{\\phi^{-1}(y)}u(x)\\,d\\mathcal{H}^{k-1}(x)\\;dy.\n\\end{align}\nIn the special case $\\phi(x)=|x|$, $J_n\\phi(x)=1$ for every $x\\neq 0$ and,  since the map sending $x\\mapsto rx$ changes $\\mathcal{H}^{k-1}$ by the factor $r^{k-1}$ (see e.g. \\citep[Proposition 2.49]{AFPallara}),   one has  \n\\begin{align*}\n\\int_{\\mathbb{R}^k}u(x)\\,dx=\\int_0^\\infty\\int_{|x|=r}u(x)\\,d\\mathcal{H}^{k-1}(x)\\;dr=\\int_0^\\infty r^{k-1}\\int_{|x|=1}u(x)\\,d\\mathcal{H}^{k-1}(x)\\;dr.\n\\end{align*}\n\n \n\n\\bigskip\n\nWe end the section by stating a generalization of  the  Coarea Formula to the case where the Lebesgue measure on the right hand side of the equations in Theorem \\ref{Coarea} is replaced by the Hausdorff measure $\\mathcal{H}^m$, where $m:=\\max\\{\\mbox{rank }(J\\phi(x)):x\\in\\mathbb{R}^k\\}$. For simplicity we suppose $f$ to be $C^1$-differentiable.\n\\begin{teo}\\label{Coarea gen}\n\tLet $k,n\\in\\mathbb{N}$ and  let  $\\phi:\\mathbb{R}^k\\to\\mathbb{R}^n$ be a $C^1$-map and let   $m:=\\max\\{\\mbox{rank }(J\\phi(x)):x\\in\\mathbb{R}^k\\}$ . The following properties hold.\n\t\\begin{itemize}\n\t\t\\item[(i)] For every  $\\mathcal{L}^k$-measurable set $E\\subseteq \\mathbb{R}^k$ the  function $y\\mapsto \\mathcal{H}^{k-m}\\left(E\\cap \\phi^{-1}(y)\\right)$ is $\\mathcal{H}^m$-measurable in $\\mathbb{R}^n$ and one has\n\t\t\\begin{align*}\n\t\t\t\\int_E J_m\\phi(x)\\,dx=\\int_{\\mathbb{R}^n}\\mathcal{H}^{k-m}\\left(E\\cap \\phi^{-1}(y)\\right)\\,d\\mathcal{H}^m(y).\n\t\t\\end{align*}\n\t\t\\item[(ii)] If $u$ is a positive measurable function, or $uJ_m\\phi\\in L^1\\left(E\\right)$, then \n\t\t\\begin{align*}\n\t\t\t\\int_{E}u(x) J_m\\phi(x)\\,dx=\\int_{\\mathbb{R}^n}\\int_{\\phi^{-1}(y)\\cap E}u(x)\\,d\\mathcal{H}^{k-m}(x)\\;d\\mathcal{H}^m( y).\n\t\t\\end{align*}\n\t\\end{itemize} \n\\end{teo}\n{\\sc{Proof.}} The proof is a consequence of  \\citep[Theorem 5.1, Theorem 5.2]{KristensenABridge}.\\qed\n\n\n\n\n\\bigskip\nThe next Remark clarifies some  positivity properties about   $k$-dimensional Jacobians. It  can be seen as a generalization of Sard's Theorem (see also \\citep[Lemma 2.73, Lemma 2.96, Remark 2.97]{AFPallara} and \\citep[Theorem 1.1]{KristensenABridge}).\n\n\n\n\\begin{os}\\label{oss J>0}\nLet  $\\phi:\\mathbb{R}^k\\to \\mathbb{R}^n$ be  a locally Lipschitz map and let us suppose, preliminarily, that  $J\\phi(x)$ has maximum rank for a.e. $x\\in\\mathbb{R}^k$.\n\\begin{itemize}\n\\item[(i)] if $k\\leq n$, then using (i) of  Theorem \\ref{Area}  with $E:=\\{x\\in\\mathbb{R}^k\\,:\\,J_k\\phi(x)=0\\}$ we get\n\\begin{align*}\n\\int_{\\mathbb{R}^n}\\mathcal{H}^0\\left(E\\cap \\phi^{-1}(y)\\right)\\,d\\mathcal{H}^k(y)=0.\n\\end{align*}\nThis implies $\\mathcal{H}^0\\left(E\\cap \\phi^{-1}(y)\\right)=0$ (i.e. $\\phi(E)\\cap \\{y\\}=\\emptyset $)  for  $\\mathcal{H}^k$-a.e. $y\\in\\mathbb{R}^n$. This yields  $\\mathcal{H}^k\\left(\\phi(E)\\right)$=0 and it  implies, in particular,  that $J_k\\phi>0$ on $\\phi^{-1}(y)$ for  $\\mathcal{H}^k$-a.e. $y\\in\\mathbb{R}^n$.\n\\item[(ii)] if $k\\geq n$, then using (i) of  Theorem \\ref{Coarea}  with $E:=\\{x\\in\\mathbb{R}^k\\,:\\,J_n\\phi(x)=0\\}$ we get\n\\begin{align*}\n\\int_{\\mathbb{R}^n}\\mathcal{H}^{k-n}\\left(E\\cap \\phi^{-1}(y)\\right)\\,dy=0.\n\\end{align*}\nThis yields $\\mathcal{H}^{k-n}\\left(E\\cap \\phi^{-1}(y)\\right)=0$   for  a.e. $y\\in\\mathbb{R}^n$ and it implies, in particular,  that $J_n\\phi>0$ $\\mathcal{H}^{k-n}$-a.e on $\\phi^{-1}(y)$ for   a.e. $y\\in\\mathbb{R}^n$.\n\\end{itemize}\n\\medskip\nLet us suppose, now, $\\phi:\\mathbb{R}^k\\to\\mathbb{R}^n$ to be  $C^1$ and let   $m:=\\max\\{\\mbox{rank }(J\\phi(x)):x\\in\\mathbb{R}^k\\}$.   Then  setting  $E:=\\{x\\in\\mathbb{R}^k\\,:\\,J_m\\phi(x)=0\\}$ and  using Theorem \\ref{Coarea gen} we get\n\\begin{align*}\n\\int_{\\mathbb{R}^n}\\mathcal{H}^{k-m}\\left(E\\cap \\phi^{-1}(y)\\right)\\,d\\mathcal{H}^m(y)=0.\n\\end{align*}\nThis implies that $J_m\\phi>0$ $\\mathcal{H}^{k-m}$-a.e on $\\phi^{-1}(y)$ for   $\\mathcal{H}^m$-a.e. $y\\in\\mathbb{R}^n$.\n\n\n\\end{os}\n\\section{Sample Distribution Theory}\\label{Section Distribution}\nLet $\\left(\\Omega,\\Sigma,p\\right)$ be a probability measure space, let $k\\in\\mathbb{N}$ and let $X:\\Omega\\to \\mathbb{R}^k$ be an absolutely  continuous random variable. Let $Y:=\\phi\\circ X$ be a statistic, where $\\phi:\\mathbb{R}^k\\to \\mathbb{R}^n$ is a measurable map and  $k\\in\\mathbb{N}$. In this section we prove that when  $\\phi$ is  locally Lipschitz then  \nthe probability measure $p_Y$ induced by $Y$ has a density function, with respect to some Hausdorff measure $\\mathcal{H}^m$ on $\\phi(\\mathbb{R}^k)\\subseteq\\mathbb{R}^n$, which can be computed explicitly in terms of an integral involving the  density function $f_X$ of  $X$.\n We recall preliminarily that the  Radon-Nikodym derivative of a measure  is uniquely defined up to   zero measure   sets: since, by  definition,  $p_Y$ is concentrated on $\\phi(\\mathbb{R}^k)$, in what follows we can always set  $f_Y(y)=0$ for  $y\\notin\\phi(\\mathbb{R}^k)$. \n\n\nWe start with the case $k\\leq n$.\n\n\n\\begin{teo}\\label{teo k<n}\nLet $\\left(\\Omega,\\Sigma,p\\right)$ be a probability measure space, let $k,n \\in\\mathbb{N}$ with $k\\leq n$ and let $X:\\Omega\\to \\mathbb{R}^k$ be an absolutely  continuous random variable with probability density function $f_X$. If $\\phi:\\mathbb{R}^k\\to \\mathbb{R}^n$ is a locally Lipschitz map  such that $\\mbox{rank }(J\\phi)=k$ a.e.,  then the probability measure induced by the  statistic $Y:=\\phi\\circ X$ is absolutely continuous with respect to the Hausdorff measure $\\mathcal{H}^k$ on $\\mathbb{R}^n$ i.e.  $p_{Y}\\ll \\mathcal{H}^k$. Its Radon-Nykodym derivative $\\frac{dp_y}{d\\mathcal \\mathcal{H}^k}$ is defined through the relation \n\\begin{align*}\np_Y(A)=p\\left(Y^{-1}(A)\\right)=\\int_{A}\\frac{dp_Y}{d\\mathcal \\mathcal{H}^k}(y)\\,d\\mathcal{H}^{k}(y),\\quad \\text{for all Borel  subset}\\quad A\\subseteq \\mathbb{R}^n.\n\\end{align*}\nIt satisfies\n\\begin{align*}\n\\frac{dp_Y}{d\\mathcal \\mathcal{H}^k}(y)&=\n\\int_{\\phi^{-1}(y)}f_X(x)\\frac{1}{J_k\\phi(x)}\\,d\\mathcal{H}^0(x)=\\sum_{\\phi(x)=y}f_X(x)\\frac{1}{J_k\\phi(x)}, &\\quad \\text{for $\\mathcal{H}^k$-a.e.}\\quad y\\in\\phi(R^k)\n\\end{align*}\nand it is $0$ otherwise.\n\n\nMoreover let us suppose that there exists a countable disjoint covering $\\bigcup_{i\\in\\mathbb{N}} E_i$ of a.e point  of $\\mathbb{R}^k$ (i.e. the set of points of $\\mathbb{R}^k$ which are not covered has $\\mathcal{L}^k$-measure zero)  such that on each measurable subset   $E_i$, the restriction map  $\\phi_i:=\\phi_{\\vert E_i}$ is a.e injective. Then \n\\begin{align*}\n\\frac{p_Y}{d\\mathcal \\mathcal{H}^k}(y)=\\sum_{i\\in \\mathbb{N}}f_X\\left(\\phi_i^{-1}(y)\\right)\\frac{1}{J_k\\phi\\left(\\phi_i^{-1}(y)\\right)},\\quad\\text{for $\\mathcal{H}^k$-a.e.}\\quad y\\in\\phi(R^k).\n\\end{align*}\n\\end{teo}\n{\\sc{Proof.}} Let $A\\subseteq \\mathbb{R}^n$ be a Borel set. Recalling Definition \\ref{k-Jac} and Remark \\ref{oss J>0},   $J_k\\phi>0$ on $\\phi^{-1}(y)$ for  $\\mathcal{H}^k$-a.e. $y\\in\\mathbb{R}^n$. Then using the Area Formula of Theorem \\ref{Area}  one has\n\\begin{align*}\np_Y(A)&=p\\left(Y^{-1}(A)\\right)=p\\left(X^{-1}\\left(\\phi^{-1}(A)\\right)\\right)=\\int_{\\phi^{-1}(A)}f_X(x)\\,dx\\\\[1ex]\n&=\\int_{\\mathbb{R}^n}\\int_{\\phi^{-1}(y)\\cap \\phi^{-1}(A)}f_X(x)\\frac{1}{J_k\\phi(x)}\\,d\\mathcal{H}^0(x)\\;d\\mathcal{H}^k(y)\\\\[1ex]\n&=\n\\int_{A}\\int_{\\phi^{-1}(y)}f_X(x)\\frac{1}{J_k\\phi(x)}\\,d\\mathcal{H}^0(x)\\;d\\mathcal{H}^k(y)=\\int_{A}\\sum_{x\\in \\phi^{-1}(y)}f_X(x)\\frac{1}{J_k\\phi(x)}\\;d\\mathcal{H}^k(y).\n\\end{align*}\nThis proved the first  required claim. The second assertion follows after observing that, under the given hypothesis, $\\phi^{-1}(y)=\\bigcup_i \\{\\phi_i^{-1}(y)\\}$ for every $y\\in \\phi(\\mathbb{R}^k)$.\n\\\\\\qed\n\\begin{os}\nWe note that if $k<n$, then $H^k$ is not $\\sigma$-finite on $\\mathbb{R}^n$, so we cannot directly use  the Radon-Nik\u00f3dym theorem in order to deduce from $p_{Y}\\ll \\mathcal{H}^k$ the existence of $\\frac{dp_Y}{d\\mathcal \\mathcal{H}^k}(y)$. Nevertheless in this case $p_Y$ is concentrated on $\\phi(\\mathbb{R}^k)$ which has $\\sigma$-finite $\\mathcal{H}^k$-measure and therefore the Radon-Nik\u00f3dym theorem applies using  $p_{Y}\\ll \\mathcal{H}^k_{\\vert \\phi(\\mathbb{R}^k)}$. \n\nIndeed if $\\mathbb{R}^k\\subseteq \\bigcup_{i\\in\\mathbb{N}} E_i$, where each $E_i$ is a Borel subset of $\\mathbb{R}^k$ such that \n$\\mathcal{L}^k\\left(E_i\\right)<\\infty$, then \n$\\phi\\left(\\mathbb{R}^k\\right)\\subseteq \\bigcup_{i\\in\\mathbb{N}} \\phi\\left(E_i\\right)$ \nand from \\citep[Proposition 2.49]{AFPallara} one has  $\\mathcal{H}^k\\left(\\phi\\left(E_i\\right)\\right)<\\mbox{Lip}(\\phi)^k \\mathcal{L}^k\\left(E_i\\right)<\\infty$.\n\\end{os}\nWhen  $k=n$, then recalling that $\\mathcal{H}^k=\\mathcal{L}^k$, the previous theorem implies that   $p_Y \\ll  \\mathcal{L}^k$.\n\\begin{cor}\\label{teo k=n}\nLet $\\left(\\Omega,\\Sigma,p\\right)$ be a probability measure space, let $k\\in\\mathbb{N}$ and let $X:\\Omega\\to \\mathbb{R}^k$ be an absolutely  continuous random variable with probability density function $f_X$. If $\\phi:\\mathbb{R}^k\\to \\mathbb{R}^k$ is a locally  Lipschitz map such that $\\mbox{rank }(J\\phi)=k$ a.e.,  then the statistic $Y:=\\phi\\circ X$ is an absolutely  continuous random variable  and its probability density function  $f_Y$  satisfies\n\\begin{align*}\nf_Y(y)=\\sum_{\\phi(x)=y}f_X(x)\\frac{1}{J_k\\phi(x)}, &\\quad \\text{for a.e.}\\quad y\\in\\phi(R^k)\n\\end{align*}\nand it is $0$ otherwise.  Moreover let us suppose that there exists a countable disjoint covering $\\bigcup_{i\\in\\mathbb{N}} E_i$ of a.e point  of $\\mathbb{R}^k$  such that on each measurable subset   $E_i$,  $\\phi_i:=\\phi_{\\vert E_i}$ is a.e injective. Then \n\\begin{align*}\nf_Y(y)=\\sum_{i\\in \\mathbb{N}}f_X\\left(\\phi_i^{-1}(y)\\right)\\frac{1}{J_k\\phi\\left(\\phi_i^{-1}(y)\\right)},\\quad \\text{for a.e.}\\quad y\\in\\phi(R^k).\n\\end{align*}\n\\end{cor}\n\\bigskip\n\nLet us now consider  the case $k>n$.\n\n\\begin{teo}\\label{teo k>n}\nLet $\\left(\\Omega,\\Sigma,p\\right)$ be a probability measure space, let $k,n \\in\\mathbb{N}$ with $k\\geq n$ and let $X:\\Omega\\to \\mathbb{R}^k$ be an absolutely  continuous random variable with probability density function $f_X$. If $\\phi:\\mathbb{R}^k\\to \\mathbb{R}^n$ is a locally Lipschitz map such that $\\mbox{rank }(J\\phi)=n$ a.e.,  then the statistic $Y:=\\phi\\circ X$ is an absolutely  continuous random variable (i.e.  $p_{Y}\\ll \\mathcal{L}^n$)  and its probability density function  $f_Y$  satisfies  \n\\begin{align*}\nf_Y(y)=\n\\int_{\\phi^{-1}(y)}f_X(x)\\frac{1}{J_n\\phi(x)}\\,d\\mathcal{H}^{k-n}(x), &\\quad \\text{for a.e.}\\quad y\\in\\phi(R^k)\n\\end{align*} \nand it is $0$ otherwise.\n\\end{teo}\n{\\sc{Proof.}} The case $k=n$ is the result of  the previous Corollary. Let us suppose $k>n$. Recalling Definition \\ref{k-Jac} and Remark \\ref{oss J>0},  $J_n\\phi>0$ $\\mathcal{H}^{k-n}$-a.e on $\\phi^{-1}(y)$ for   a.e. $y\\in\\mathbb{R}^n$. Let $A\\subseteq \\mathbb{R}^n$ be a Borel set. Then using the Coarea Formula of Theorem \\ref{Coarea}  one has\n\\begin{align*}\np_Y(A)&=p\\left(Y^{-1}(A)\\right)=p\\left(X^{-1}\\left(\\phi^{-1}(A)\\right)\\right)\\\\[1ex]\n&=\\int_{\\phi^{-1}(A)}f_X(x)\\,dx=\\int_{\\mathbb{R}^n}\\int_{\\phi^{-1}(y)\\cap \\phi^{-1}(A)}f_X(x)\\frac{1}{J_n\\phi(x)}\\,d\\mathcal{H}^{k-n}(x)\\;dy\\\\[1ex]\n&=\n\\int_{A}\\int_{\\phi^{-1}(y)}f_X(x)\\frac{1}{J_n\\phi(x)}\\,d\\mathcal{H}^{k-n}(x)\\;dy.\n\\end{align*}\nThis proved the required claim.\\\\\\qed\n\nFor  the reader's convenience we enlighten in the following corollary the particular case $n=1$ which is very useful in the applications and which follows from formula \\eqref{1d-Coarea}.   \n\n\\begin{cor}\\label{teo k>1}\nLet $\\left(\\Omega,\\Sigma,p\\right)$ be a probability measure space, let $k\\in\\mathbb{N}$  and let $X:\\Omega\\to \\mathbb{R}^k$ be an absolutely  continuous random variable with probability density function $f_X$. If $\\phi:\\mathbb{R}^k\\to \\mathbb{R}$ is a locally Lipschitz map such that $|\\nabla \\phi|>0$ a.e.,  then the statistic $Y:=\\phi\\circ X$ is an absolutely  continuous random variable   and has probability density function \n\\begin{align*}\nf_Y(y)=\n\\int_{\\phi^{-1}(y)}f_X(x)\\frac{1}{|\\nabla\\phi(x)|}\\,d\\mathcal{H}^{k-1}(x), &\\quad \\text{for a.e.}\\quad y\\in\\phi(R^k).\n\\end{align*} \n\\end{cor}\n\\medskip\n\\begin{os}The assumptions $p(\\Omega)=1$ was never used in the proof of the Theorems \\ref{teo k<n} and \\ref{teo k>n}. Indeed  analogous results hold  with $p_X$ replaced by an absolutely continuous measure on $\\mathbb{R}^k$. More precisely let $\\mu$ be  a measure defined on $\\left(\\mathbb{R}^k,\\mathcal{B}\\right)$  such that $\\mu \\ll \\mathcal L^k$ and let $\\frac{d\\mu}{d\\mathcal L^k}$ its Radon-Nykodym derivative. Let  $\\phi:\\mathbb{R}^k\\to\\mathbb{R}^n$ be a locally Lipschitz map whose Jacobian matrix $J\\phi$ has a.e. maximum rank.\n\\begin{itemize}\n\\item[(i)] If $k\\leq n$ then $\\mu \\phi^{-1} \\ll  \\mathcal{H}^k$ and for $\\mathcal{H}^k$-a.e. $y\\in \\phi(\\mathbb{R}^k)$one has\n\\begin{align*}\n\\frac{d\\mu \\phi^{-1}}{d\\mathcal \\mathcal{H}^k}(y)=\\int_{\\phi^{-1}(y)}\\frac{d\\mu}{d\\mathcal L^k}(x)\\frac{1}{J_k\\phi(x)}\\,d\\mathcal{H}^0(x)=\\sum_{\\phi(x)=y}\\frac{d\\mu}{d\\mathcal L^k}(x)\\frac{1}{J_k\\phi(x)}.\n\\end{align*}\n\\item[(ii)] If $k> n$ then $\\mu \\phi^{-1} \\ll  \\mathcal{L}^n$ and for a.e. $y\\in\\mathbb{R}^n$ one has\n\\begin{align*}\n\\frac{d\\mu \\phi^{-1}}{d\\mathcal{L}^n}(y)=\\int_{\\phi^{-1}(y)}\\frac{d\\mu}{d\\mathcal{L}^k}(x)\\frac{1}{J_n\\phi(x)}\\,d\\mathcal{H}^{k-n}(x).\n\\end{align*}\n\\end{itemize} \n\\end{os}\n\\bigskip\n\nWe end the section by applying Theorem \\ref{Coarea gen}  in order to  extend Theorem \\ref{teo k>n}  to the case of a  $C^1$-map $\\phi$ whose Jacobian could possibly have not maximum  rank.    In this case, setting $m:=\\max\\{\\mbox{rank }(J\\phi(x)):x\\in\\mathbb{R}^k\\}$, the induced probability $p_Y$  has a density function $f_Y$ with respect to the  Hausdorff measure $\\mathcal{H}^m$ on $\\phi(\\mathbb{R}^k)\\subseteq\\mathbb{R}^n$.\n\n\n\\begin{teo}\\label{teo Hausdorff}\nLet $\\left(\\Omega,\\Sigma,p\\right)$ be a probability measure space, let $k,n \\in\\mathbb{N}$ \n and let $X:\\Omega\\to \\mathbb{R}^k$ be an absolutely  continuous random variable with probability density function $f_X$. Let $\\phi:\\mathbb{R}^k\\to \\mathbb{R}^n$ be a $C^1$-map  and let $m:=\\max\\{\\mbox{rank }(J\\phi(x)):x\\in\\mathbb{R}^k\\}$. Then the induced probability measure $p_Y$ of the statistic $Y:=\\phi\\circ X$ has a density function $f_Y$ with respect to the  Hausdorff measure $\\mathcal{H}^m$ \n\n which  satisfies  \n\\begin{align*}\nf_Y(y)=\n\\int_{\\phi^{-1}(y)}f_X(x)\\frac{1}{J_m\\phi(x)}\\,d\\mathcal{H}^{k-m}(x), &\\quad \\text{for $\\mathcal{H}^m$-a.e.}\\quad y\\in\\phi(R^k)\n\\end{align*} \nand it is $0$ otherwise.\n\\end{teo}\n{\\sc{Proof.}} Recalling Definition \\ref{k-Jac} and Remark \\ref{oss J>0},  $J_m\\phi>0$ $\\mathcal{H}^{k-m}$-a.e on $\\phi^{-1}(y)$ for   $\\mathcal{H}^m$-a.e. $y\\in\\mathbb{R}^n$. Let $A\\subseteq \\mathbb{R}^n$ be a Borel set. Then using  Theorem \\ref{Coarea gen}  one has\n\\begin{align*}\np_Y(A)&=p\\left(Y^{-1}(A)\\right)=p\\left(X^{-1}\\left(\\phi^{-1}(A)\\right)\\right)\\\\[1ex]\n&=\\int_{\\phi^{-1}(A)}f_X(x)\\,dx=\\int_{\\mathbb{R}^n}\\int_{\\phi^{-1}(y)\\cap \\phi^{-1}(A)}f_X(x)\\frac{1}{J_m\\phi(x)}\\,d\\mathcal{H}^{k-m}(x)\\;d\\mathcal{H}^m(y)\\\\[1ex]\n&=\n\\int_{A}\\int_{\\phi^{-1}(y)}f_X(x)\\frac{1}{J_m\\phi(x)}\\,d\\mathcal{H}^{k-m}(x)\\;d\\mathcal{H}^m(y).\n\\end{align*}\nThis proves the required claim.\\\\\\qed\n\n\n\n\\section{Further generalizations}\\label{Section Generalization}\n\nIn this section we  briefly expose, for the interested reader,  a further  generalization of  Theorem \\ref{teo k>n} which covers  the case of  probabilities $p_X$  concentrated on some $m$-dimensional subset $E\\subseteq\\mathbb{R}^k$. This includes, for example, the cases of a random variable $X$ which is uniformly distributed over a generic subset $E$ and  of random variables taking values on $m$-dimensional manifolds.  Statistical analysis on   manifolds \n\thas many applications in directional and axial statistics, morphometrics, medical\n\tdiagnostics, machine vision and image analysis (see e.g \\citep{Bhattacharya} and references there in).  Amongst the many important applications, those arising, for example, from the analysis of  data on torus  play also a fundamental role in molecular biology in the study of the Protein Folding Problem. \n\nAlthough the following Theorem is valid for   countably $m$-rectifiable sets (see \\citep[Definition 5.4.1, Lemma 5.4.2]{Krantz-Parks}), we suppose for simplicity $E$ to be  an $m$-dimensional sub-manifold of $\\mathbb{R}^k$. \n\nWe state first the Area and Coarea formula relative to   sub-manifolds of $\\mathbb{R}^k$. If $J\\phi^E$ is the  tangential Jacobian matrix of $\\phi$ with respect to $E$ (see \\citep[Formula 11.1]{Maggi}), the $k$-tangential Jacobian $J_k^E\\phi$ is defined as in Definition \\ref{k-Jac}. For a rigorous introduction to tangential Jacobians  as well as for all the other details we refer the reader to \\citep[Chapter 3]{Federer}, \\citep[Section 5.3]{Krantz-Parks},  \\citep[Section 11.1]{Maggi}.\n\n\\begin{teo}\nLet $\\phi:\\mathbb{R}^k\\to\\mathbb{R}^n$ be a $C^1$-map  and let $E\\subseteq\\mathbb{R}^k$ an $m$-dimensional manifold. The following properties hold.\n\\begin{itemize}\n\\item[(i)]If $m\\leq n$ and $u$ is a positive measurable function, or $uJ_m^E\\phi\\in L^1\\left(E,\\mathcal{H}^m\\right)$, one has \n\\begin{align*}\n\\int_{E}u\\, J_m^E\\phi\\,d\\mathcal{H}^m(x)=\\int_{\\mathbb{R}^n}\\int_{E\\cap \\phi^{-1}(y)}u\\,d\\mathcal{H}^{0}(x)\\,d\\mathcal{H}^{m}(y).\n\\end{align*}\n\\item[(ii)]\nIf $m\\geq n$ and  $u$ is a positive measurable function, or $uJ_n^E\\phi\\in L^1\\left(E,\\mathcal{H}^m\\right)$,one has\n\\begin{align*}\n\\int_{E}u\\, J_n^E\\phi\\,d\\mathcal{H}^m(x)=\\int_{\\mathbb{R}^n}\\int_{E\\cap \\phi^{-1}(y)}u\\,d\\mathcal{H}^{m-n}(x)\\,dy.\n\\end{align*}\n\\end{itemize}\n\\end{teo}\n{\\sc{Proof.}} See \\citep[Theorem 2.91, 2.93]{Federer} and \\citep[Theorem 2.91, Theorem 2.93]{AFPallara}.\\qed\n\\bigskip\nThe same methods of proof used in Section \\ref{Section Distribution},  yield finally  the next result. Note that every  $m$-dimensional manifold $E\\subseteq\\mathbb{R}^k$ has $\\mathcal{H}^m$-$\\sigma$-finite measure.\n\\begin{teo}\nLet $\\left(\\Omega,\\Sigma,p\\right)$ be a probability measure space, let $k,n \\in\\mathbb{N}$ and let $E\\subseteq\\mathbb{R}^k$ an $m$-dimensional manifold. Let $X:\\Omega\\to E$ be a random variable having a  probability density function $f_X$ with respect to the  Hausdorff measure $\\mathcal{H}^m_{\\vert E}$ on $E$. Let  $\\phi:\\mathbb{R}^k\\to \\mathbb{R}^n$ be  a $C^1$-map whose tangential Jacobian  $J^E\\phi(x)$ has maximum rank at any point $x\\in E$.  The following properties hold.\n\\begin{itemize}\n\\item[(i)] If $m\\leq n$ then the probability measure $p_Y$ induced by the  statistic $Y:=\\phi\\circ X$ is absolutely continuous    with respect to the Hausdorff measure $\\mathcal{H}^m_{\\vert\\phi(E)}$ on $\\phi(E)$  and its density function $f_Y$  satisfies\n\\begin{align*}\nf_Y(y)&=\n\\int_{\\phi^{-1}(y)}f_X(x)\\frac{1}{J_m^E\\phi(x)}\\,d\\mathcal{H}^0(x)=\\sum_{\\phi(x)=y}f_X(x)\\frac{1}{J_m^E\\phi(x)}, &\\quad \\text{for $\\mathcal{H}^m$-a.e.}\\quad y\\in\\phi(E).\n\\end{align*}\n\\item[(ii)] If $m\\geq n$ then  the statistic $Y:=\\phi\\circ X$ is an absolutely  continuous random variable (i.e.  $p_{Y}\\ll \\mathcal{L}^n$)  and its probability density function  $f_Y$  satisfies  \n\\begin{align*}\nf_Y(y)=\n\\int_{\\phi^{-1}(y)}f_X(x)\\frac{1}{J_n^E\\phi(x)}\\,d\\mathcal{H}^{m-n}(x), &\\quad \\text{for a.e.}\\quad y\\in\\phi(E).\n\\end{align*} \n\\end{itemize}\n\\end{teo}\n\n\\section{Some applications}\\label{Section apll}\n\n\nIn this section we apply the results of the previous sections in order  to compute the density functions of some distributions in some   cases of relevant interest.\n\\subsection{First examples}\nIn these first examples we provide a density formula for random variables which are algebraic manipulations of absolutely continuous random variables. The first example, in particular, is used in Proposition \\ref{section chi squared} in order to find the density of the chi-squared distribution.\n\\begin{esem}[Square function]\\label{Square function}\nLet $\\left(\\Omega,\\Sigma,p\\right)$ be a probability measure space and  let $X:\\Omega\\to \\mathbb{R}$ be an absolutely  continuous random variable with probability density function $f_X$. We employ  Corollary \\ref{teo k=n} with  \\begin{align*}\n\\phi:\\mathbb{R}\\to [0,\\infty[,\\quad \\phi(t)=t^2,\\quad J_1(t)=2|t|.\n\\end{align*}\nThen  the statistic $Y:=X^2$ is an absolutely  continuous random variable  and its probability density function  $f_Y$  satisfies  \n\\begin{align*}\nf_Y(y)=\\frac{f_X(\\sqrt y)+f_X\\left(-\\sqrt y\\right)}{2\\sqrt y},\\quad \\text{for any}\\quad y>0.\n\\end{align*}\nMore generally let  $k\\in\\mathbb{N}$ and let  $X=(X_1,\\dots, X_k):\\Omega\\to \\mathbb{R}^k$ be an absolutely  continuous (vector valued) random variable with probability density function $f_X$. Then employing   Theorem \\ref{teo k>n} with \n\\begin{align*}\n\\phi:\\mathbb{R}^k\\to [0,\\infty[,\\quad \\phi(x)=\\|x\\|^2=x_1^2+\\cdots+x_k^2,\\quad J_1(x)=2\\|x\\|,\n\\end{align*}\n we get that the statistic $Y:=\\|X\\|^2=X_1^2+\\cdots+X_k^2$ is an absolutely  continuous random variable whose probability density function satisfies  \n\\begin{align*}\nf_Y(y)&=\\int_{\\|x\\|^2=y}\\frac{f_X(x)}{2\\|x\\|}\\,d\\mathcal{H}^{k-1}(x)=\\frac{1}{2\\sqrt y}\\int_{\\|x\\|=\\sqrt y}f_X(x)\\,d\\mathcal{H}^{k-1}(x),\\quad \\text{for any}\\quad y>0.\n\\end{align*}\n\\end{esem}\n\n\n\\begin{esem}[Affine transformations]\\label{Affine transformations}\nLet $\\left(\\Omega,\\Sigma,p\\right)$ be a probability measure space, let $k\\in\\mathbb{N}$ and  let $X:\\Omega\\to \\mathbb{R}^k$ be an absolutely  continuous random variable with probability density function $f_X$.  Let us consider the affine transformation \n\\begin{align*}\n\\phi:\\mathbb{R}^k\\to \\mathbb{R}^n,\\quad \\phi(x)=Ax+y_0,\n\\end{align*}\nwhere $A\\in\\mathbb{R}^{n\\times k}$,  $\\mbox{rank}(A)=m$ and $y_0\\in\\mathbb{R}^n$. Recalling Definition \\ref{k-Jac}, the $m$-dimensional Jacobian of $\\phi$ is given by\n\\begin{align*}\n\tJ_m\\phi(x)=\\sqrt{\\sum_{B}(\\mbox{det\\,}B)^2}=:A_m\n\\end{align*}\nwhere the sum   runs along all $m\\times m$ minors $B$ of  $A$.\n Then, using Theorem \\ref{teo Hausdorff},  the induced probability measure $p_Y$ of the statistic $Y:=AX+y_0$ has a density function $f_Y$ with respect to the  Hausdorff measure $\\mathcal{H}^m$ on the $m$-dimensional hyper-surface $\\phi\\left(\\mathbb{R}^k\\right)=\\{y=Ax+y_0:x\\in\\mathbb{R}^k\\}$ which  satisfies   \n\\begin{align*}\nf_Y(y)&=\n\\frac{1}{A_m}\\int_{Ax+y_0=y}f_X(x)\\,d\\mathcal{H}^{k-m}(x)\n\\\\[1ex]&=\\frac{1}{A_m}\\int_{\\mbox{Ker}(A)+x_y}f_X(x)\\,d\\mathcal{H}^{k-m}(x)\n, \\quad \\text{for}\\quad y\\in\\phi(R^k).\n\\end{align*} \nHere   for $y\\in\\phi(R^k)$,  $x_y\\in\\mathbb{R}^k$ is  any fixed solution of the equation $y=Ax_y+y_0$.\n\n\n When $m=n$, then $A_n=\\sqrt{\\mbox{det}(AA^T)}$ and the map $\\phi$ is surjective i.e. $\\phi(\\mathbb{R}^k)=\\mathbb{R}^n$. In this case theorem \\ref{teo k>n} implies that $p_y\\ll\\mathcal L ^n$ i.e.  $Y$ is an absolutely  continuous random variable. If moreover  $k=n$ and  $A\\in\\mathbb{R}^{k\\times k}$ is not-singular  then $A_k=|\\mbox{det } A|$ and in this case we have \n\\begin{align*}\nf_Y(y)=\n\\frac{1}{|\\mbox{det } A|}\\int_{Ax+y_0=y}f_X(x)\\,d\\mathcal{H}^{0}(x)=\\frac{f_X(A^{-1}(y-y_0))}{|\\mbox{det } A|}, &\\quad \\text{for}\\quad y\\in \\mathbb{R}^k.\n\\end{align*} \n\\end{esem}\n\n\n\n\n\\begin{esem}[Sum of variables and Sample mean]\nLet $\\left(\\Omega,\\Sigma,p\\right)$ be a probability measure space, let $k\\in\\mathbb{N}$ and  let $X=\\left(X_1,\\dots, X_k\\right):\\Omega\\to \\mathbb{R}^k$ be an absolutely  continuous random variable with probability density function $f_X$. We employ  Corollary \\ref{teo k>1} with  \n\\begin{align*}\n\\phi:\\mathbb{R}^k\\to  \\mathbb{R},\\quad  \\phi(t)=\\sum_{i=1}^k t_i,\\quad  J_1(t)=|\\nabla\\phi|=\\sqrt k.\n\\end{align*}\nThen  the statistic $Y:=\\sum_{i=1}^k X_i$ is an absolutely  continuous random variable  and its probability density function  $f_Y$  satisfies  \n\\begin{align*}\nf_Y(y)=\n\\int_{\\sum_{i=1}^k x_i=y}\\frac{f_X(x)}{\\sqrt k}\\,d\\mathcal{H}^{k-1}(x), &\\quad \\text{for a.e.}\\quad y\\in \\mathbb{R}.\n\\end{align*} \nLet us  set $x^{k-1}:=\\left(x_1,\\dots,x_{k-1}\\right)$ and let $\\psi(x^{k-1})=\\left(x^{k-1},\\,y-\\sum_{i=1}^{k-1}x_i\\right)$ be a parametrization of the hyperplane  $\\sum_{i=1}^k x_i=y$.   Using the area formula \\eqref{parametrized manifold}, the last integral becomes\n\\begin{align*}\nf_Y(y)=\n\\int_{\\mathbb{R}^{k-1}}f_X\\Big(x^{k-1},\\,y-\\sum_{i=1}^{k-1}x_i\\Big)\\,dx^{k-1}, &\\quad \\text{for a.e.}\\quad y\\in \\mathbb{R}.\n\\end{align*}\nIn the particular case $k=2$ and if $X_1$, $X_2$ are independent, the last formula gives the well known convolution form for the distribution of the random variable $X_1+X_2$:\n\\begin{align*}\nf_{X_1+X_2}(y)=\n\\int_{\\mathbb{R}}f_{X_1}(t)f_{X_2}(y-t)\\,dt, &\\quad \\text{for a.e.}\\quad y\\in \\mathbb{R},\n\\end{align*}\nwhere  $f_{X_1}, f_{X_2}$ are respectively the density function of the distribution generated by $X_1,X_2$.\n\n \nMoreover if  $X_1,\\dots,X_k$ are identically distributed and independent with common probability density function  $f:\\Omega\\to\\mathbb{R}$, then    (using also Example \\ref{Affine transformations} with $\\phi(x)=\\frac 1 kx$), the density function of  the sample mean $Z:=\\frac{1}{k}\\sum_{i=1}^k X_i$ is\n\\begin{align*}\nf_Z(y)= k\\, f_Y\\left(ky\\right)= k\n\\int_{\\sum\\limits_{i=1}^k x_i= k y}\\;\\prod_{i=1}^k f(x_i)\\,d\\mathcal{H}^{k-1}(x), &\\quad \\text{for a.e.}\\quad y\\in \\mathbb{R}.\n\\end{align*}\n\\end{esem}\n\n\\begin{esem}[Product and ratio of random variables]\\label{ratio}\nLet $\\left(\\Omega,\\Sigma,p\\right)$ be a probability measure space and  let $X:\\Omega\\to \\mathbb{R}^2,\\, X=\\left(X_1,X_2\\right)$ be an absolutely  continuous random variable with probability density function $f_X$. \n\\begin{itemize}\n\\item[(i)] Let us  employ  Corollary \\ref{teo k>1} with  \n\\begin{align*}\n\\phi:\\mathbb{R}^2\\to  \\mathbb{R},\\quad  \\phi(x_1,x_2)=x_1x_2,\\quad  J_1\\phi(x_1,x_2)=|\\nabla\\phi(x_1,x_2)|=\\sqrt{x_1^2+x_2^2}.\n\\end{align*}\nThen  the statistic $X_1X_2$ is an absolutely  continuous random variable  whose  probability density function    satisfies\n\\begin{align*}\nf_{X_1X_2}(y)=\n\\int_{x_1x_2=y}\\frac{f_X(x_1,x_2)}{\\sqrt {x_1^2+x_2^2}}\\,d\\mathcal{H}^{1}(x_1,x_2)=\\int_{\\mathbb{R}\\setminus\\{0\\}}f_X\\Big(t,\\frac y t\\Big)\\frac{1}{|t|}\\,dt, &\\quad \\text{for a.e.}\\quad y\\in \\mathbb{R},\n\\end{align*} \nwhere we parametrized  the hyperbole $x_1x_2=y$ by $\\psi(t)=\\left(t,\\frac y t\\right)$ and we used Formula \\eqref{parametrized manifold} to evaluate  the last integral.\n\n\\item[(ii)] Let us suppose $X_2\\neq 0$ a.e. and let us employ Corollary \\ref{teo k>1} with  \n\\begin{align*}\n\\phi:\\mathbb{R}^2\\to  \\mathbb{R},\\quad  \\phi(x_1,x_2)=\\frac{x_1}{x_2},\\quad  J_1\\phi(x_1,x_2)=|\\nabla\\phi(x_1,x_2)|=\\frac 1 {x_2^2}\\sqrt{x_1^2+x_2^2}.\n\\end{align*}\nThen  the statistic $\\frac{X_1}{X_2}$ is an absolutely  continuous random variable  whose  probability density function   satisfies\n\\begin{align*}\nf_{\\frac{X_1}{X_2}}(y)=\n\\int_{\\frac{x_1}{x_2}=y}f_X(x_1,x_2)\\frac{x_2^2}{\\sqrt {x_1^2+x_2^2}}\\,d\\mathcal{H}^{1}(x_1,x_2)=\\int_{\\mathbb{R}}f_X\\Big(ty, t\\Big)|t|\\,dt, &\\quad \\text{for a.e.}\\quad y\\in \\mathbb{R},\n\\end{align*} \nwhere we parametrized  the line $x_1=yx_2$ by $\\psi(t)=\\left(ty,t\\right)$ and we used  \\eqref{parametrized manifold} to evaluate  the last integral.\n\\item[(iii)]Let $X:\\Omega\\to \\mathbb{R}$ be an absolutely  continuous random variable such that $X\\neq 0$ a.e. and let  $f_X$ its probability density function. We employ  Corollary \\ref{teo k=n} with  \\begin{align*}\n\\phi:\\mathbb{R}\\setminus\\{0\\}\\to \\mathbb{R}\\setminus\\{0\\},\\quad \\phi(t)=\\frac 1 t,\\quad J_1(t)=|\\phi'(t)|=\\frac 1{t^2}.\n\\end{align*}\nThen  the statistic $\\frac 1 {X}$ is an absolutely  continuous random variable  whose probability density function satisfies    \n\\begin{align*}\nf_{\\frac 1 X}(y)=f_X\\left(\\frac 1 y\\right)\\frac{1}{y^2},\\quad \\text{for any}\\quad y\\neq 0.\n\\end{align*}\n\\end{itemize}\n\n\n\n\n\n\\end{esem}\n\\subsection{Order Statistics}\nLet  $S_k$ be  the set of all the permutations of the set $\\{1,\\dots,k\\}$. Let  $X=\\left(X_1,\\dots, X_k\\right):\\Omega\\to \\mathbb{R}^k$ be a  random variable and let us consider the map\n$$\\phi:\\mathbb{R}^k\\to\\mathbb{R}^k,\\quad x=(x_1,\\dots, x_k)\\mapsto (x_{(1)},\\dots, x_{(k)}),$$\nwhich associates to any vector $x$ its increasing rearrangement $(x_{(1)},\\dots, x_{(k)})$  i.e. $x_{(1)}\\leq \\dots\\leq x_{(k)}$.  The random variable $\\phi\\circ X:= \\left(X_{(1)},\\dots, X_{(k)}\\right)$ is the random vector of the so-called \\emph{Order Statistics} of $X$. In what follows, as an easy application of the results of the previous sections, we deduce  their  well known density functions. We start with the following Lemma which shows, in particular,  that $\\phi$ has unitary Jacobian. \n \n \\begin{lem}\n Let $n\\in\\mathbb{N}$ such that $n\\leq  k$, let  $I=\\{i_1,i_2,\\dots,i_n\\}\\subseteq\\{1,\\dots,k\\}$ a subset of indexes, where $|I|=n$ and $i_1<i_2<\\dots<i_n$. Let \n $$\\phi_I:\\mathbb{R}^k\\to\\mathbb{R}^n,\\quad x=(x_1,\\dots, x_k)\\mapsto (x_{(i_1)},\\dots, x_{(i_n)})$$\n where $(x_{(i_1)},\\dots, x_{(i_n)})$ is the vector obtained by extracting from the increasing rearrangement $(x_{(1)},\\dots, x_{(k)})$ the component corresponding to the indexes of $I$ (note that when $I=\\{1,\\dots,k\\}$ then  $\\phi_I=\\phi$).  Then the $n$-Jacobian $J_n\\phi_I$ of $\\phi_I$ satisfies $J_n\\phi_I=1$.\n \\end{lem}\n {\\sc{Proof.}} Let us suppose, without any loss of generality, $I=\\{1,2\\dots,n\\}$. For every fixed permutation $\\sigma\\in S_k$, let us consider the Borel subset $A_\\sigma\\subseteq\\mathbb{R}^k$ defined by\n \\begin{align*}\n A_\\sigma:=\\{x=(x_1,\\dots,x_k)\\in\\mathbb{R}^k\\;:\\; x_{\\sigma(1)}<x_{\\sigma(2)}<\\dots< x_{\\sigma(k)}\\}.\n \\end{align*}\nThe complementary set of $\\bigcup_{\\sigma\\in S_k} A_\\sigma$ is  the set \n$$F=\\{x=(x_1,\\dots,x_k)\\in\\mathbb{R}^k\\;:\\; \\exists\\, i\\neq j \\text{ s.t. } x_i=x_j \\}$$\nwhich satisfies  $\\mathcal{L}^k(F)=0$; therefore the collection of all  $A_\\sigma$ is a finite  disjoint covering  of a.e point  of $\\mathbb{R}^k$.\n\nLet us fix $\\sigma\\in S_k$; on $A_\\sigma$, $\\phi_I$ is injective and it  coincides with the permutation of indexes operator $T_\\sigma$ defined by $\\sigma$:\n\\begin{align*}\n\\phi_I(x):=T_\\sigma(x)=\\left(x_{\\sigma(1)},\\dots, x_{\\sigma(n)}\\right),\\quad \\forall x\\in A_\\sigma.\n\\end{align*}\nThe  Jacobian matrix of $T_\\sigma$ at any point $x\\in A_\\sigma$ is then  a  permutation  of the matrix $\\left(\\begin{array}{c}\n  I_n  \\\\ \n  \\hline\n  0 \n \\end{array}\\right)$.  Recalling Definition \\ref{k-Jac}, this implies in particular $J_n\\phi=1$ on $A_\\sigma$ which, by the arbitrariness of $\\sigma$,  proves the required claim.\n \\\\\\qed\n  Let now $X_1,\\dots,X_k$ be a sequence of  absolutely continuous, independent and identically distributed random variables. Let $f$ be the common  density function of each $X_i:\\Omega\\to\t\\mathbb{R}$ and let $F(y):=\\int_{-\\infty}^yf(t)\\,dt$ be the associated distribution function. In the following Proposition we compute  the density of the Order statistics of the vector  $X=\\left(X_1,\\dots,X_k\\right):\\Omega\\to \\mathbb{R}^k$. We remark that $X$ is an absolutely  continuous random variable having    density function given by $f_X(y)=\\prod_{i=1}^k f(y_i)$.\n \\begin{prop}\n\\begin{itemize}\n\\item[(i)] The density function of the    distribution of the vector $Y=\\left(X_{(1)},\\dots, X_{(k)}\\right)$ of the all order statistics satisfies\n \\begin{align*}\nf_Y(y)=k!\\prod_{i=1}^kf(y_i),\\quad \\forall y\\in \\mathbb{R}^k \\text{ s.t. } y_1<y_2<\\dots<y_k\n\\end{align*}\nand it is $0$ otherwise.\n\\item[(ii)] Let $i\\in\\mathbb{N}$ such that $i\\leq k$.  The density function of the   distribution of the order statistic $X_{(i)}$ satisfies\n\\begin{align*}\nf_{X_{(i)}}(y)=i {{k}\\choose{i}}\\,f(y)\\,F(y)^{i-1}\\,\\Big(1-F(y)\\Big)^{k-i}, \\quad \\forall y\\in\\mathbb{R}.\n\\end{align*}\n\\item[(iii)] Let $i,j\\in\\mathbb{N}$ such that $i<j\\leq k$.  The density function of the  distribution of the vector of the two order statistics $\\left(X_{(i)},X_{(j)}\\right)$ satisfies\n \\begin{align*}\nf_{\\left(X_{(i)},X_{(j)}\\right)}(y_1,y_2)=&\\frac{k!}{(i-1)!(j-i-1)!(k-j)!}f(y_1)f(y_2)\\\\[1ex]\n&\\times F(y_1)^{i-1}\\Big(F(y_2)-F(y_1)\\Big)^{j-i-1}\\Big(1-F(y_2)\\Big)^{k-j},\\quad \\forall y\\in \\mathbb{R}^2 \\text{ s.t. } y_1<y_2\n\\end{align*}\nand it is $0$ otherwise.\n\\end{itemize}\n \\end{prop}\n {\\sc{Proof.}} Using Corollary \\ref{teo k=n} and the previous Lemma, we get that  the random vector $Y=\\left(X_{(1)},\\dots, X_{(k)})\\right)$ of the Order Statistics of $X$    is an absolutely  continuous random variable   and its probability density function  $f_Y$  satisfies  \n\\begin{align*}\nf_Y(y)&=\\sum_{\\sigma\\in S_k}f_X\\left(T_\\sigma^{-1}(y)\\right)=\\sum_{\\sigma\\in S_k}f_X\\left(T_{\\sigma^{-1}}(y)\\right)=\\sum_{\\sigma\\in S_k}\\prod_{i=1}^kf(y_{\\sigma^{-1}(i)})\\\\\n&=\\sum_{\\sigma\\in S_k}\\prod_{j=1}^kf(y_j)=k!\\prod_{j=1}^kf(y_j),\\quad \\forall y\\in \\mathbb{R}^k \\text{ s.t. } y_1<y_2<\\dots<y_k.\n\\end{align*}\nand it is $0$ otherwise. This  proves (i).\n\n Let $i\\in\\mathbb{N}$ such that $i\\leq k$. Claim (ii)  can be proved  directly by applying, as in the previous step,   Theorem \\ref{teo k>n} and the previous Lemma or alternatively by integrating the joint density $f_Y$. Indeed if we write $\\hat{x_i}=\\left(x_1,\\dots x_{i-1}, x_{i+1},\\dots, x_k\\right)\\in\\mathbb{R}^{k-1}$ to denote the variable $x$ without the $x_i$ component and   if we set  $F=\\{\\hat{x_i}\\in\\mathbb{R}^{k-1} \\;:\\; x_1<x_2\\dots<x_{i-1}<y<x_{i+1}<\\dots<x_k\\}$ then we  obtain\n\\begin{align*}\nf_{X_{(i)}}(y)&=\\int_{\\{x\\in\\mathbb{R}^k:x_i=y\\}}f_Y\\left(x\\right)\\,d\\hat{x_i}= k!f(y)\\int_{F}\\prod_{j\\neq i}f(x_j)\\,d\\hat{x_i}\\\\\n&= k!f(y)\\int_{\\{x_1<\\dots<x_{i-1}<y\\}}\\prod_{j=1}^{i-1}f(x_j)\\,dx_1\\dots dx_{i-1}\\int_{\\{y<x_{i+1<}\\dots<x_{k}\\}}\\prod_{j=i+1}^{k}f(x_j)\\,dx_{i+1}\\dots dx_{k}.\n\\end{align*}\nSince the integrand of the first integral of the right hand side of the last equation is invariant under any  permutations of its variables then\n\\begin{align*}\n\\int_{\\{x_1<\\dots<x_{i-1}<y\\}}\\prod_{j=1}^{i-1}f(x_j)\\,dx_1\\dots dx_{i-1}=\\frac 1{(i-1)!}\\int_{\\{x_j<y,\\, \\forall j\\leq i-1\\}}\\prod_{j=1}^{i-1}f(x_j)\\,dx_1\\dots dx_{i-1}\\\\\n=\\frac 1{(i-1)!}\\prod_{j=1}^{i-1}\\int_{-\\infty}^y f(x_j)dx_j=\\frac 1{(i-1)!}F(y)^{i-1}.\n\\end{align*}\nAnalogously one has\n\\begin{align*}\n\\int_{\\{y<x_{i+1<}\\dots<x_{k}\\}}\\prod_{j=i+1}^{k}f(x_j)\\,dx_{i+1}\\dots d_{k}=\\frac 1{(k-i)!}\\left(1-F(y)\\right)^{k-i}.\n\\end{align*}\nThis gives \n\\begin{align*}\nf_{X_{(i)}}(y)=k!\\frac 1{(i-1)!(k-i)!}f(y)F(y)^{i-1}\\left(1-F(y)\\right)^{k-i}=i {{k}\\choose{i}}f(y)F(y)^{i-1}\\left(1-F(y)\\right)^{k-i}\n\\end{align*}\nwhich is the required claim.\n\nThe proof of (iii) follows similarly.\n\\\\\\qed\n\n\\begin{os}\n\t (i) The joint density function of three or more order statistics could be derived using similar  arguments.\n\t\n\t(ii) The same methods applies also when the random variables $X_1,\\dots,X_k$ are independent  but not identically distributed. For example let  $f_i(y)$ be the density function of  $X_i:\\Omega\\to\t\\mathbb{R}$.  Then in this case $f_X(y)=\\prod_{i=1}^kf_i(y_i)$ for every $y\\in\\mathbb{R}^k$. If $Y=\\left(X_{(1)},\\dots, X_{(k)})\\right)$ then  one obtains as before\n\t\\begin{align*}\n\t\tf_Y(y)=\\sum_{\\sigma\\in S_k}\\prod_{i=1}^kf_i(y_{\\sigma(i)}),\\quad \\forall y\\in \\mathbb{R}^k \\text{ s.t. } y_1<y_2<\\dots<y_k.\n\t\\end{align*}\n\\end{os}\n\n \n\\subsection{Normal distributions}\nLet $a\\in\\mathbb{R}$, $\\sigma> 0$ and let $X:\\Omega\\to \\mathbb{R}$ be a random variable. $X$ is said to have a \n\\emph{ Normal (or Gaussian) distribution}\n $p_X$, and we write $p_X\\sim  \\mathcal N\\left(a,\\sigma^2\\right)$, if  $p_X$ has density\n \\begin{align*}\n f_X(t)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\n \\exp\\left(-\\frac{|t-a|^2}{2\\sigma^2}\\right),\\quad t\\in\\mathbb{R}.\n \\end{align*}\nWhen $\\sigma=0$ we also write $p_X\\sim  \\mathcal N\\left(a,0\\right)$ with the understanding  that  $p_X$ is the dirac measure $\\delta_a$ at the point $a$. \nThe parameters $a$ and $\\sigma^2$ are called the mean and the variance of $X$, respectively.\n\n\nLet $k\\in\\mathbb{N}$,  $a\\in\\mathbb{R}^k$ and let $\\Sigma\\in\\mathbb{R}^{k,k}$ be a symmetric, and positive semi-definite matrix. A random variable $X=\\left(X_1,\\dots,X_k\\right):\\Omega\\to \\mathbb{R}^k$  is said to have a \\emph{(multivariate) normal distribution}  $\\mathcal{N}(a,\\Sigma)$  if \n\\begin{align*}\n\\langle\\lambda, X\\rangle \\sim \\mathcal{N}\\Big(\\langle\\lambda, a\\rangle, \\langle\\Sigma \\lambda,\\lambda \\rangle\\Big),\\quad \\forall \\lambda\\in\\mathbb{R}^k\n\\end{align*}\n(we write $\\langle \\lambda,\\mu \\rangle=\\sum_{i}\\lambda_i\\mu_i$ to denote the inner product of $\\mathbb{R}^k$).\nHere $a:=E\\left(X\\right)$ is the mean vector and $\\Sigma=\\left(\\sigma_{ij}\\right)_{i,j}:=\\mbox{Cov}(X)$ is the covariance matrix of $X$ i.e. $\\sigma_{i,j}=\\mbox{Cov}\\left(X_i,X_j\\right)$. The following very well known properties about Gaussian vectors are direct consequences of their definition  (see for example \\citep[Chapter 1]{bogachev}).\n\n\\begin{prop}\\label{Basic Normal}\nLet  $X=\\left(X_1,\\dots,X_k\\right):\\Omega\\to \\mathbb{R}^k$ be a random variable such that $X\\sim\\mathcal{N}\\left(a,\\Sigma\\right)$. \n\\begin{itemize}\n\\item[(i)] The mean vector $a$ and the Covariance matrix $\\Sigma$  uniquely characterized the Gaussian measure $p_X$.\n\\item[(ii)]  $X_i$, $X_j$ are independent if and only if $\\sigma_{ij}=\\mbox{Cov}(X_i,X_j)=0$. \n\\item[(iii)] For every matrix $A\\in\\mathbb{R}^{m,k}$ one has $AX\\sim\\mathcal{N}\\left(\\langle Aa\\rangle, A\\Sigma A^t\\right)$.\n\\item[(iv)]  When $\\Sigma$ is positive definite  we say that $p_X$ is \\emph{not-degenerate}: in this case  $X$ is  absolutely continuous and has density function \n\\begin{align*}\nf_X(x)=\\frac{1}{(2\\pi)^{\\frac k 2}\\mbox{det}(\\Sigma)^\\frac 1 2}\\exp{\\left(-\\frac{|\\Sigma^{-\\frac 1 2}(x-a)|^2}{2}\\right)}.\n\\end{align*}\n\\end{itemize}  \n\\end{prop}\n\n\\medskip\nIn the following Proposition we show  that, when the Covariance matrix $\\Sigma$ is degenerate,  $p_X$ has a density function  with respect to the  Hausdorff measure $\\mathcal{H}^m$ on  some hyperplane of $\\mathbb{R}^k$. In what follows we say that  that a matrix $P\\in \\mathbb{R}^{k,m}$, with  $m\\leq k$,  is  orthogonal if it has orthonormal columns; in this case $|Qy|=|y|$ for every $y\\in\\mathbb{R}^m$. \n\n\n\\begin{prop}\nLet $k\\in\\mathbb{N}$,  $a\\in\\mathbb{R}^k$ and let $\\Sigma\\in\\mathbb{R}^{k,k}$ be a positive semi-definite matrix with  $m=\\mbox{rank} (\\Sigma)\\geq 1$. Let  $X=\\left(X_1,\\dots,X_k\\right):\\Omega\\to \\mathbb{R}^k$ be a random variable. Then  $X\\sim\\mathcal{N}(a,\\Sigma)$ if and only if there exists an orthogonal matrix $P\\in \\mathbb{R}^{k,m}$ and $m$ independent random variables $Y_1,\\dots,Y_m$  which satisfies $Y_i\\sim \\mathcal{N}\\left(0,1\\right)$ for every $i=1,\\dots,m$ and  such that \n\\begin{align*}\nX=\\Sigma^{\\frac 1 2 }P\\,Y+a,\\quad Y=\\left(Y_1,\\dots,Y_m\\right).\n\\end{align*}\nMoreover the probability measure $p_X$ has a density function $f_X$ with respect to the  Hausdorff measure $\\mathcal{H}^m$ on  the hyperplane \n$$\\Sigma^{\\frac 12}P\\left(\\mathbb{R}^m\\right)+a=\\{x\\in\\mathbb{R}^k\\;:\\;x=\\Sigma^{\\frac 12}Py+a,\\;y\\in\\mathbb{R}^m\\}$$\n which  satisfies   \n\\begin{align*}\nf_X(x)=\\frac{1}{(2\\pi)^{\\frac m 2}\\Sigma^{\\frac 12}_m}\\exp{\\left(-\\frac{|y|^2}{2}\\right)}, \\quad x=\\Sigma^{\\frac 12}Py+a,\n\\end{align*}\nwhere  $\\Sigma^{\\frac 12}_m=\\prod_i{\\sqrt \\lambda_i}$ and  the product runs over all positive eigenvalues of $\\Sigma$ (counted with their multiplicities).\n\n\\end{prop}\n{\\sc{Proof.}} Let us prove the first claim and let  us suppose, preliminarily, that the Covariance matrix $\\Sigma$ is a diagonal matrix and, without any loss of generality, let us assume that its entries in the main diagonal are \n$$\\left(\\sigma_{11},\\dots,\\sigma_{mm}, 0,\\dots,0\\right),$$\n where  $\\sigma_{ii}>0$ for $i\\leq m$. Then from Proposition \\ref{Basic Normal}, $X_1,\\dots, X_m$ are independent and $X_i\\sim\\mathcal{N}(a_i,\\sigma_{ii})$; moreover $X_i=a_i$ a.e. for $i>m$. The required claim then immediately follows  setting $Y=\\left(Y_1,\\dots, Y_m\\right)$, with $Y_i=\\frac{X_i-a_i}{\\sqrt\\sigma_{ii}}$, and  $P=\\left(\\begin{array}{c}\n  I_m  \\\\ \n  \\hline\n  0 \n \\end{array}\\right)$, where $I_m$ is the identity matrix of $\\mathbb{R}^{m,m}$.\n\n\n\n In the general case let us diagonalize  the Covariance matrix $\\Sigma$: Let $Q\\in\\mathbb{R}^{k,k}$ be an  orthogonal matrix  such that $Q\\Sigma Q^t=D$, where $D$ is the diagonal matrix whose entries in the main diagonal are $\\left(\\lambda_1,\\dots,\\lambda_m, 0,\\dots,0\\right)$, where  the  $\\lambda_i>0$ are the positive eigenvalues of $\\Sigma$. From Proposition \\ref{Basic Normal} the vector $Z=QX$ satisfies $Z\\sim\\mathcal{N}\\left(Qa,D\\right)$; from the previous step there exists $Y=\\left(Y_1,\\dots,Y_m\\right)\\sim \\mathcal{N}\\left(0,I_m\\right)$ such that\n \\begin{align*}\n Z=D^{\\frac 1 2}\\left(\\begin{array}{c}\n  I_m  \\\\ \n  \\hline\n  0 \n \\end{array}\\right)Y+Qa.\n \\end{align*}\n Then since $Q\\Sigma^{\\frac 1 2 }Q^t=D^{\\frac 1 2 }$ we get \n \\begin{align*}\n X=Q^tZ=Q^tD^{\\frac 1 2}\\left(\\begin{array}{c}\n  I_m  \\\\ \n  \\hline\n  0 \n \\end{array}\\right)Y+a=\\Sigma^{\\frac 1 2} Q^t\\left(\\begin{array}{c}\n  I_m  \\\\ \n  \\hline\n  0 \n \\end{array}\\right)Y+a\n \\end{align*}\n and the claim follows with $P=Q^t\\left(\\begin{array}{c}\n  I_m  \\\\ \n  \\hline\n  0 \n \\end{array}\\right)$.\n\n\n\nFinally, to prove the second claim, let us apply the first step and  Example \\ref{Affine transformations} with $A=\\Sigma^{\\frac 1 2}P$ and $y_0=a$. Then we get that $X$ has a density function $f_X$ with respect to the  Hausdorff measure $\\mathcal{H}^m$ on  the hyperplane \n$$\\Sigma^{\\frac 12}P\\left(\\mathbb{R}^m\\right)+a=\\{x\\in\\mathbb{R}^k\\;:\\;x=\\Sigma^{\\frac 12}Py+a,\\;y\\in\\mathbb{R}^m\\}$$\n which  satisfies   \n\\begin{align*}\nf_X(x)&=\n\\frac{1}{(\\Sigma^{\\frac 12}P)_m}\\int_{\\Sigma^{\\frac 12}Py+a=x}f_Y(y)\\,d\\mathcal{H}^{0}(y), \\quad \\text{for}\\quad x\\in \\Sigma^{\\frac 12}P\\left(\\mathbb{R}^m\\right)+a.\n\\end{align*} \nSince $P$ has orthogonal columns then from Definition \\ref{k-Jac} we have $(\\Sigma^{\\frac 12}P)_m=\\prod{\\sqrt \\lambda_i}:=\\Sigma^{\\frac 12}_m$, where the product runs over all positive eigenvalues of $\\Sigma$. Moreover since $\\Sigma^{\\frac 12}P$ has maximum  rank, the equation  $x=\\Sigma^{\\frac 12}Py+a$ has a unique solution. Then\n\\begin{align*}\nf_X(x)=\\frac{1}{(2\\pi)^{\\frac m 2}\\Sigma^{\\frac 12}_m}\\exp{\\left(-\\frac{|y|^2}{2}\\right)}, \\quad x=\\Sigma^{\\frac 12}Py+a.\n\\end{align*}\n\\qed\n\n\\subsection{Chi-squared and Student's  distributions}\nLet $X:\\Omega\\to \\mathbb{R}^k$ be a Gaussian random vector whose  covariance matrix is the identity matrix $I_k$. If $X\\sim\\mathcal{N}\\left(0,I_k\\right)$ then the probability measure $p_{\\chi^2(k)}$ induced by  $|X|^2$ is called  \\emph{Chi-squared distribution with $k$-degrees of freedom} and we write $|X|^2\\sim \\chi^2(k)$.\n\n If $X$ is not-centred  i.e. $X\\sim\\mathcal{N}\\left(\\mu,I_k\\right)$ for some $\\mu\\in\\mathbb{R}^k\\setminus\\{0\\}$, then the measure $p_{\\chi^2(k,\\lambda)}$  induced by $|X|^2$ is called  \\emph{Non-central Chi-squared distribution with $k$-degrees of freedom and non-centrality parameter $\\lambda=|\\mu|^2>0$} and we write $|X|^2\\sim \\chi^2(k,\\lambda)$.\n\n\\medskip\nIn the next Proposition we derive the density function of $|X|^2$. In what follows we consider the gamma function $\\Gamma(r)=\\int_0^\\infty t^{r-1}e^{-r}\\,dr$, $r>0$ (see e.g. \\citep[page 255]{abramowitz+stegun}) and the modified Bessel function of the first kind $ I_{\\nu }$  defined for $y>0$ as   \n\\begin{align*}\nI_{\\nu }(y)=(y\/2)^{\\nu }\\sum _{j=0}^{\\infty }{\\frac {(y^{2}\/4)^{j}}{j!\\,\\Gamma (\\nu +j+1)}}=\\frac{(y\/2)^{\\nu }}{\\pi^{\\frac 1 2}\\Gamma\\left(\\nu+\\frac 1 2\\right)}\\int_0^\\pi e^{y\\cos\\theta}\\left(\\sin \\theta\\right)^{2\\nu}\\,d\\theta,\n\\end{align*}\n(see e.g. \\citep[Section 9.6 and Formula 9.6.20, page 376]{abramowitz+stegun}).\n\\begin{prop}[Chi-squared  Distribution]\\label{section chi squared}\nLet $X:\\Omega\\to \\mathbb{R}^k$ be a Gaussian random vector. If  $X\\sim\\mathcal{N}\\left(0,I_k\\right)$ then  the Chi-squared distribution $p_{\\chi^2(k)}$ induced by $|X|^2$ has density function\n\\begin{align*}\nf_{\\chi^2(k)}(y)=\\frac{1}{2^{\\frac k 2}\\Gamma\\left(\\frac k 2\\right)}y^{\\frac k 2-1}\\exp\\left(-\\frac y 2\\right),\\quad \\text{for any}\\quad y>0.\n\\end{align*}\nIf $X\\sim\\mathcal{N}\\left(\\mu,I_k\\right)$ for some $\\mu\\in\\mathbb{R}^k\\setminus\\{0\\}$ then, setting $\\lambda=|\\mu|^2>0$,  the Non-central Chi-squared distribution  $p_{\\chi^2(k,\\lambda)}$ induced by $|X|^2$ has density function\n\n\\begin{align*}\nf_{\\chi^2(k,\\lambda)}(y)&=\\frac{1}{2}\\exp\\left(-\\frac{y+\\lambda}2\\right)\\left(\\frac y\\lambda\\right)^{\\frac k 4-\\frac 1 2}I_{\\frac k 2 -1}\\left(\\sqrt{\\lambda y}\\right),\\quad \\text{for any}\\quad y>0.\n\\end{align*}\n\\end{prop}\n{\\sc{Proof.}} \nLet $X\\sim\\mathcal{N}\\left(0,I_k\\right)$; using Example \\ref{Square function} we have for any $y>0$\n\\begin{align*}\nf_{\\chi^2(k)}(y)&=\\frac{1}{2\\sqrt y}\\frac{1}{(2\\pi)^{\\frac k 2}}\\int_{|x|=\\sqrt y}\\exp\\left(-\\frac{|x|^2}2\\right)\\,d\\mathcal{H}^{k-1}(x)\n\\\\[1ex]&=\\frac{1}{2\\sqrt y}\\frac{1}{(2\\pi)^{\\frac k 2}}\\exp\\left(-\\frac{y}2\\right)\\mathcal{H}^{k-1}\\left(\\mathbb{S}^{k-1}\\right)y^{\\frac{k-1}2}=\\frac{1}{2^{\\frac k 2}\\Gamma\\left(\\frac k 2\\right)}y^{\\frac k 2-1}\\exp\\left(-\\frac y 2\\right)\n\\end{align*}\nwhich is the first claim. If  $X\\sim\\mathcal{N}\\left(\\mu,I_k\\right)$ for some $\\mu\\in\\mathbb{R}^k\\setminus\\{0\\}$, then using Example \\ref{Square function} again and the elementary  equality $|x-\\mu|^2=|x|^2+|\\mu|^2-2\\langle x,\\mu\\rangle$ we have for any $y>0$\n\\begin{align*}\nf_{\\chi^2(k)}(y)&=\\frac{1}{2\\sqrt y}\\int_{|x|=\\sqrt y}\\frac{1}{(2\\pi)^{\\frac k 2}}\\exp\\left(-\\frac{|x-\\mu|^2}2\\right)\\,d\\mathcal{H}^{k-1}(x)\\\\[1ex]\n&=\\frac{1}{2\\sqrt y (2\\pi)^{\\frac k 2}}\\exp\\left(-\\frac{y+\\lambda}2\\right)\\int_{|x|=\\sqrt y}\\exp\\langle x,\\mu\\rangle\\,d\\mathcal{H}^{k-1}(x)\n\\\\[1ex]&=\\frac{y^{\\frac {k}2-1}}{2(2\\pi)^{\\frac k 2}}\\exp\\left(-\\frac{y+\\lambda}2\\right)\\int_{|z|=1}\\exp\\langle \\sqrt y z,\\mu\\rangle\\,d\\mathcal{H}^{k-1}(z).\n\\end{align*}\nSince $\\mathcal{H}^{K-1}_{\\vert_{\\mathbb{S}^{k-1}}}$ is rotationally invariant,  up to an  orthogonal transformation of $\\mathbb{R}^k$ which maps $\\frac{\\mu}{|\\mu|}$ to $e_1=\\left(1,0,\\dots,0\\right)$, we can suppose $\\frac{\\mu}{|\\mu|}=e_1$. Using $k$-dimensional spherical coordinates to evaluate the last integral then   we have \n\\begin{align*}\n\\int_{|z|=1}\\exp\\langle \\sqrt y z,\\mu\\rangle\\,d\\mathcal{H}^{k-1}(z)&=\\int_{|z|=1}\\exp\\left(\\sqrt{y\\lambda}z_1\\right)\\,d\\mathcal{H}^{k-1}(z)\n\\\\[1ex]&=\\mathcal{H}^{K-2}(\\mathbb{S}^{K-2})\\int_0^\\pi \\exp\\left(\\sqrt{y\\lambda}\\cos\\theta\\right)(\\sin\\theta)^{k-2}\\,d\\theta\n\\\\[1ex]&=(2\\pi)^{\\frac k 2}\\left(\\lambda y\\right)^{-\\frac k 4+\\frac 1 2}I_{\\frac k 2 -1}\\left(\\sqrt{\\lambda y}\\right).\n\\end{align*}\nCombining the latter equalities gives the required last claim.\\qed\n\\bigskip\n\n\n Finally let  $X,Y:\\Omega\\to \\mathbb{R}$ be two independent random variables such that $X\\sim\\mathcal{N}\\left(0,1\\right)$ and  $Y\\sim\\chi^2(k)$. The probability measure $p_{T}$ induced by the random  variable $T=\\frac{X}{\\sqrt {Y\/k}}$ is called a \\emph{(Student's)  t-distribution with $k$-degrees of freedom}. In the next Proposition we use Corollary \\ref{teo k>1} and example  \\ref{ratio} in order to derive the density function of $p_{T}$.\n\n\n\\begin{prop}[Student's t-Distribution]\nLet $X,Y:\\Omega\\to \\mathbb{R}$ two independent random variables such that $X\\sim\\mathcal{N}\\left(0,1\\right)$ and  $Y\\sim\\chi^2(k)$.  The t-distribution $p_{T}$ induced by  $T=\\frac{X}{\\sqrt {Y\/k}}$ has density function\n\\begin{align*}\nf_T(y)=\\frac{\\Gamma\\left(\\frac{k+1}2\\right)}{\\sqrt{k\\pi}\\Gamma\\left(\\frac{k}2\\right)}\\left(1+\\frac{y^2}{k}\\right)^{-\\frac{k+1}2},\\quad \\text{for any}\\quad t\\in\\mathbb{R}.\n\\end{align*}\n\\end{prop}\n{\\sc{Proof.}}\nUsing Corollary \\ref{teo k>1} with $\\phi:\\mathbb{R}^+\\to\\mathbb{R}^+, \\phi(t)=\\sqrt {t\/k}$,  we have\n\\begin{align*}\nf_{\\sqrt {Y\/k}}(y)&=\\frac{2k^{\\frac k 2}}{2^{\\frac k 2}\\Gamma\\left(\\frac k 2\\right)}y^{ k -1}\\exp\\left(-\\frac {ky^2} 2\\right),\\qquad \\forall y>0.\n\\end{align*}\nThen applying  example  \\ref{ratio}  we get for $y\\in \\mathbb{R}$,\n\\begin{align*}\nf_{T}(y)=\\int_{0}^\\infty f_X(ty)f_{\\sqrt {Y\/k}}(t)t\\,dt\n=\\frac{2k^{\\frac k 2}}{2^{\\frac k 2}\\Gamma\\left(\\frac k 2\\right)\\sqrt{2\\pi}}\\int_{0}^\\infty \\exp\\left({-\\frac{t^2(y^2+k)}{2}}\\right)t^k\\,dt,\n\\end{align*} \nwhich with the substitution $s=t^2\\frac{y^2+k}{2}$ becomes\n\\begin{align*}\nf_{T}(y)=\\frac{1}{\\Gamma\\left(\\frac k 2\\right)\\sqrt{k\\pi}}\\left(1+\\frac{y^2}k\\right)^{-\\frac {k+1} 2 }\\int_{0}^\\infty e^{-s}s^{\\frac {k-1}2}\\,ds=\\frac{\\Gamma\\left(\\frac {k+1} 2\\right)}{\\Gamma\\left(\\frac k 2\\right)\\sqrt{k\\pi}}\\left(1+\\frac{y^2}k\\right)^{-\\frac {k+1} 2 }.\n\\end{align*}\n\\qed\n\n\n\\bibliographystyle{apalike}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nTexture-zeros, vanishing elements of fermion mass matrices, \nin quark and lepton mass matrix are successful to\npredict masses and mixing angles. However the origin of zero is not \nclear, which is a motivation of our model. As an approach for this problem \nwe consider the discrete flavor symmetry approach. This approach is studied by \nmany authors{\\cite{discrete symmetry}}. Flavor symmetry is expected to \nbe a clue to understand the masses and mixing angles of quarks and leptons because\nit reduces the number of free parameters in Yukawa couplings and some testable \npredictions of masses and mixing angles generally follow(see references in {\\cite{our model}}).\n\nA interesting point of our model is that dynamical realization of Texture-zeros in\nthe discrete flavor symmetry approach. In previous model, in order to derive \nTexture-zeros certain Yukawa couplings are forbidden by the discrete symmetry. \nIn our model however we consider that some of Higgs vacuum expectation values(VEVs) vanish \nby electroweak symmetry breaking(EWSB) in flavor basis, that is, we consider multi-Higgs system and \nTexture-zeros are derived by EWSB dynamically. \n\nIn our model we take $S_3$ symmetry, permutations of three objects, as the discrete symmetry. \nThe reasons why we adopt $S_3$ are that this symmetry is the smallest group of non \ncommutative discrete groups and $S_3$ has three \nirreducible representations, doublet $\\bf 2$, singlet ${\\bf 1_{S}}$, pseudo singlet \n${\\bf 1_{A}}$, so that it is easy to assign the flavor symmetry representations to three generations such as\n${\\bf 2} + {\\bf 1_{S}}$. In addition, we consider all the $S_{3}$ irreducible representations in this model.\n\n\\section{$S_3$ invariant mass matrix on supersymmetry}\n\nIn this section, the $S_3$ invariant mass matrices are presented{\\cite{mass matrix}}. \nWe consider supersymmetric theory and we suppose that two of three generations belong to $S_3$ doublets and the others are singlets. Using the following tensor product of $S_3$ doublet, $\\phi^{c} = \\sigma_{1} \\phi^{*}= \\sigma_{1}(\\phi^{*}_{1},\\phi^{*}_{2})^{T},\\ \\psi = (\\psi_{1},\\psi_{2})^{T}$,\n\\begin{equation} \n\t\\begin{array}{ccccccc}\n\t\t\\phi^{c} \\times \\psi &=& (\\phi_2 \\psi_2,\\phi_1 \\psi_1)^{T} & + & (\\phi_1 \\psi_2 - \\phi_2 \\psi_1) & + & (\\phi_1 \\psi_2 + \\phi_2 \\psi_1), \\\\\n\t\t    & & {\\bf 2} & & {\\bf 1_{A}} & & {\\bf 1_{S}} \n\t\\end{array}\n\\end{equation}\nthe $S_{3}$ invariant mass matrices are obtained as\n\\begin{equation}\n   M_{D}=\n      \\left(\n          \\begin{tabular}{cc|c}\n\t      $aH_{1}$ & $b H_{S} + c H_{A}$ & $d H_{2}$ \\\\\n\t      $b H_{S} - c H_{A}$ & $a H_{2}$ & $d H_{1}$ \\\\\n\t      \\hline\n\t      $e H_{2}$ & $e H_{1}$ & $ f H_{S}$\n\t        \\end{tabular}\n      \\right),\\qquad\n\t\tM_{R}=\n        \\left(\n\t    \\begin{tabular}{cc|c}\n\t\t    & $M_{1}$ &   \\\\\n\t      $M_{1}$ &   &  \\\\\n\t      \\hline\n\t             &    &  $M_{2}$\n\t    \\end{tabular}\n\t\\right),\n\\end{equation}\nwhere $a, b, \\cdots,f$ are independent Yukawa coupling constants, $M_1, M_{2}$ are majorana masses. Now we assign ${\\bf 1_{S}}$ to third generation temporarily. We reconfigure this assignment later. \n\n\\section{$S_3$ invariant Higgs scalar potential analysis}\nIn our model we consider the following eight Higgs bosons,\n\\begin{equation}\n\tH_{uS},H_{dS},H_{uA},H_{dA},H_{u1},H_{d1},H_{u2},H_{d2}.\n\\end{equation}\nOur purpose in this section is to discuss whether or not there are some vacuum patterns with no parameter relations in terms of vanishing-VEVs. \nTherefore we have to analyze eight equations at vacuum, which correspond to each Higgs field.\n\\begin{table}[t]\n\\begin{minipage}{0.5 \\textwidth}\n\t\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|} \\hline\n$v_{uS}$ & $v_{dS}$ & $v_{uA}$ & $v_{dA}$ & $v_{u1}$ & $v_{u2}$ \n& $v_{d1}$ & $v_{d2}$ \\\\ \\hline \\hline\n $0$ & $0$ & $0$ & $0$ & $0$ &     &     & $0$ \\\\ \\hline\n $0$ & $0$ & $0$ & $0$ &     & $0$ & $0$ &     \\\\ \\hline\n $0$ & $0$ & $0$ & $0$ &     &     &     &     \\\\ \\hline\n $0$ & $0$ &     &     & $0$ & $0$ & $0$ & $0$ \\\\ \\hline\n $0$ & $0$ &     &     & $0$ &     &     & $0$ \\\\ \\hline\n $0$ & $0$ &     &     &     & $0$ & $0$ &     \\\\ \\hline\n $0$ & $0$ &     &     &     &     &     &     \\\\ \\hline\n \\end{tabular}\n \\end{center}\n \\end{minipage}\n \\begin{minipage}{0.5 \\textwidth}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|} \\hline\n$v_{uS}$ & $v_{dS}$ & $v_{uA}$ & $v_{dA}$ & $v_{u1}$ & $v_{u2}$ \n& $v_{d1}$ & $v_{d2}$ \\\\ \\hline \\hline\n     &     & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\\\ \\hline\n     &     & $0$ & $0$ & $0$ &     &     & $0$ \\\\ \\hline\n     &     & $0$ & $0$ &     & $0$ & $0$ &     \\\\ \\hline\n     &     & $0$ & $0$ &     &     &     &     \\\\ \\hline\n     &     &     &     & $0$ & $0$ & $0$ & $0$ \\\\ \\hline\n     &     &     &     & $0$ &     &     & $0$ \\\\ \\hline\n     &     &     &     &     & $0$ & $0$ &     \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{minipage}\n\\caption{All possible minima of the scalar potential for $S_3$ singlet\nand doublet Higgs fields without tuning of Lagrangian parameters for\nelectroweak symmetry breaking. The blank entries denote\nnon-vanishing VEVs.}\n{\\label{summary}}\n\\end{table} \nIn general, these equations are the coupled equations through a common parameter which contains all the Higgs VEVs. However we can separate these equations into three parts for the singlet\n, the pseudo singlet\nand the doublet\nbecause vanishing-VEVs makes the equations trivial within each sector. \nTherefore analyzing the equations\n, possible 14 VEV patterns(Table {\\ref{summary}}) are obtained with no parameter relations in terms of vanishing-VEVs. \n\n\\section{Quark and lepton mass textures}\n\nIn previous section, we got 14 VEV patterns.\nNow let us analyze these patterns phenomenologically. At first we can \nobtain the most interesting pattern of 14 patterns. \nThis pattern is the following.\n\\begin{equation}\n\tv_{uS}=v_{dS}=v_{u1}=v_{d2}=0,\\qquad v_{uA},v_{dA},v_{u2},v_{d1}\\not= 0. {\\label{vev}}\n\\end{equation}\nThis pattern leads to the simplest texture(i.e.the maximal number of zero matrix elements) with non-trivial flavor mixing. Next we consider mass matrices obtained from this VEV pattern. We only took ${\\bf 2} + {\\bf 1_S}$ as the $S_3$ representations of three generation matter fields so that the $S_3$ charge assignments of matter fields has a complexity. For example we can assign ${\\bf 1_{S}}$ to any generation in general. As results of exhausting all the $S_3$ charge assignments for the quark sector and assuming $SU(5)$ grand unification{\\cite{su(5)GUT}}{\\cite{georgi jarlskog}} for the lepton sector, mass matrices and predictions of this model are derived as \n \n\\begin{equation}\n\tM_u =\n\t\t\\left(\n\t\t\t\\begin{array}{ccc}\n\t\t\t\t\t& b_u &\t\t\\\\\n\t\t\t\td_u &\t\t& f_u \\\\\n\t\t\t\t & - f_u & i_u\n\t\t\t\\end{array}\n\t\t\\right),\\qquad\n\tM_d =\n\t\t\\left(\n\t\t\t\\begin{array}{ccc}\n\t\t\t\t\t& b_d &\t\t\\\\\n\t\t\t\td_d &\te_d\t&  \\\\\n\t\t\t\t-e_d &  & i_d\n\t\t\t\\end{array}\n\t\t\\right), \\qquad\n\tM_e =\n\t\t\\left(\n\t\t\t\\begin{array}{ccc}\n\t\t\t\t\t& d_d &\t3e_d\t\\\\\n\t\t\t\tb_d &\t-3 e_d\t& \\\\\n\t\t\t\t &  & i_d\n\t\t\t\\end{array}\n\t\t\\right),\n\\end{equation}\n\\begin{equation}\n\tM_{\\nu} =\n\t\t\\left(\n\t\t\t\\begin{array}{ccc}\n\t\t\t\t& b_{\\nu} &\tc_{\\nu}\t\\\\\n\t\t\t\t-b_{\\nu} &\te_{\\nu}\t&  \\\\\n\t\t\t\tg_{\\nu} & & \n\t\t\t\\end{array}\n\t\t\\right),\\qquad\n\tM_R =\n\t\t\\left(\n\t\t\t\\begin{array}{ccc}\n\t\t\t\t\t& M_1 &\t\t\\\\\n\t\t\t\tM_1 &\t\t&  \\\\\n\t\t\t\t &  & M_2\n\t\t\t\\end{array}\n\t\t\\right),\n\\end{equation}\n\\begin{equation}\n\t\\left| V_{cb} \\right| = \\sqrt{\\frac{m_c}{m_t}},\\qquad \\left| V_{e3} \\right| \\ge 0.04{\\cite{PDG}}.\n\\end{equation}\nwhere blank entries denote zero and each parameter in $M_{u},M_{d},M_{e},M_{\\nu}$ such as $d_{u},d_{d},b_{\\nu}$ denote a product of a Yukawa coupling and a VEV, for example $d_{u} = d v_{u2}$.\n\\section{Higgs mass spectrum and $S_3$ soft breaking in B-term}\n\nThe $S_3$ potential has an enhanced global symmetry $SU(2) \\times U(1)^2$ and leads to massless Nambu-Goldstone bosons \nin the electroweak broken phases. It is therefore reasonable to softly break the flavor symmetry within the scalar potential.\nWe introduce the following supersymmetry-breaking soft terms which do not break phenomenological characters of the exact $S_3$ model.\n\\begin{equation}\n\tV_{\\not S_3} = b_{SD}H_{uS}H_{d2}+ b'_{SD}H_{u1}H_{dS}+ b_{AD}H_{uA}H_{d1}+ b'_{AD}H_{u2}H_{dA} + {\\rm h.c.}\n\\end{equation}\nThese soft terms have not only the same phenomenological characters as exact $S_3$ model but also a character which we can take the same VEV pattern as ({\\ref{vev}}) in previous section with no parameter relations.\n\n\\section{Tree level FCNC}\n\nSince there are multiple electroweak doublet Higgs bosons which couple to \nmatter fields, flavor-changing processes are mediated at classical \nlevel by these Higgs fields.\tWe can show that all but one have masses of the order of supersymmetry breaking parameters. Therefore the experimental observations of FCNC rare events would lead to a bound on the supersymmetry breaking scale.\nAmong various experimental constraints, we find the most important constraint comes from the neutral K meson mixing. For the heavy mass eigenstates, the tree-level $K_{L}-K_{S}$ mass difference $\\Delta m^{\\rm tree}_K$ is given by the matrix element of the effective Hamiltonian between K mesons{\\cite{FCNC}}.\n$\\Delta m^{\\rm tree}_K$ contains $M_{H}$, which is an average of the Higgs masses $1\/M^{2}_{H} = \\frac{1}{4}\\left( 1\/M^{2}_{H^{0}_{1}} + 1\/ M^{2}_{H^{0}_{2}} + 1\/M^{2}_{H^{0}_{3}}+1\/M^{2}_{H^{0}_{4}} \\right)$, and a free parameter $\\eta$, which contains the down type quark Yukawa couplings.\nIt is found that heavy Higgs masses are bounded from below so as to suppress the extra Higgs contribution compared with the standard model one, which bound is roughly given by\n\\begin{equation}\n  M_{H} \\ge\n\t    \\left\\{\n\t\t\t    \\begin{array}{cl}\n\t\t\t\t\t\t3.8 {\\rm TeV} &(\\eta = 0) \\\\\n\t\t\t\t\t\t1.4 {\\rm TeV} &(\\eta = 0.03) \\\\\n\t\t\t\t\t\\end{array}\n\t\t\t\t\\right.\n,\n\\end{equation}\nwhere we take $\\eta = 0,\\ 0.03$ as typical values.\n\n\\section{Summary}\n\nIn our model we have discussed the structure of Higgs potential and fermion mass \nmatrices in supersymmetric models with $S_3$ flavor symmetry and examined possible \nvanishing elements of quark and lepton mass matrices. As results of exhausting the \npatterns of flavor symmetry charges of matter fields, some predictions such that \nthe lepton mixing $V_{e3}$ is within the range which will be tested in near future \nexperiments are obtained. We have also discussed the physical mass spectrum of \nHiggs bosons and the tree level FCNC processes which is propagated by heavy Higgs fields. \nFrom the tree level FCNC process analysis, it is found that heavy Higgs masses, \nwhich is the order of soft supersymmetry breaking scale, are a few TeV.\n\t\n\n\\vspace*{12pt}\n\\noindent\n{\\bf Acknowledgement}\n\\vspace*{6pt}\n\n\\noindent\nThe Summer Institute 2006 is sponsored by the Asia \nPacific Center for Theoretical Physics and the BK 21 \nprogram of the Department of Physics, KAIST.\nWe would like to thank the organizers of Summer Institute 2006 and thank J. Kubo, T. Kobayashi and H. Nakano for helpful discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Stability of Dual Solution}\n\\label{appn:dual}\nWe first note that $D(\\alpha)$ is the dual of a convex function and has a unique minimizer $\\alpha^*$.\nThe function ${V(\\alpha) = D(\\alpha) - D(\\alpha^*)}$, hence, is a non-negative function and equals zero only at $\\alpha = \\alpha^*$.\nDifferentiating $V(\\alpha)$ with respect to time we get\n\\[\\dot{V}(\\alpha) = \\frac{\\partial V}{\\partial \\alpha}\\dot{\\alpha} = -\\gamma \\Big(\\sum_{i}{h_i} - B \\Big)^2 < 0.\\]\n\nTherefore, $V(\\cdot)$ is a Lyapunov function, and the system state will converge to optimum starting from any initial condition.\n\n\\section{Stability of Primal Solution}\n\\label{appn:primal}\nWe first note that since $W(\\mathbf{h})$ is a strictly concave function, it has a unique maximizer $\\mathbf{h}^*$.\nMoreover ${V(\\mathbf{h}) = W(\\mathbf{h}^*) - W(\\mathbf{h})}$ is a non-negative function and equals zero only at $\\mathbf{h} = \\mathbf{h}^*$.\nDifferentiating $V(\\cdot)$ with respect to time we obtain\n\\begin{align*}\n\\dot{V}(\\mathbf{h}) &= \\sum_{i}{\\frac{\\partial V}{\\partial h_i}\\dot{h_i}} \\\\\n&= -\\sum_{i}{\\left( U'_i(h_i) - C'(\\sum_{i}{h_i} - B) \\right) \\dot{h_i}}.\n\\end{align*}\n\nFor $\\dot{h_i}$ we have\n\\[\\dot{h_i} = \\frac{\\partial h_i}{\\partial t_i}\\dot{t_i}.\\]\nFor non-reset and reset TTL caches we have\n\\[\\frac{\\partial h_i}{\\partial t_i} = \\frac{\\lambda_i}{(1+\\lambda_i t_i)^2} \\qquad\\text{ and }\\qquad \\frac{\\partial h_i}{\\partial t_i} = \\lambda_i e^{-\\lambda_i t_i},\\]\nrespectively, and hence $\\partial h_i \/ \\partial t_i > 0$.\n\nFrom the controller for $t_i$ we have\n\\[t_i = k_i \\left( U'_i(h_i) - C'(\\sum_{i}{h_i} - B) \\right).\\]\n\nHence, we get\n\\[\\dot{V}(\\mathbf{h}) = -\\sum_{i}{k_i \\frac{\\partial h_i}{\\partial t_i} \\left( U'_i(h_i) - C'(\\sum_{i}{h_i} - B) \\right)^2} < 0.\\]\n\nTherefore, $V(\\cdot)$ is a Lyapunov function\\footnote{A description of Lyapunov functions and their applications can be found in~\\cite{srikant13}.}, and the system state will converge to $\\mathbf{h}^*$ starting from any initial condition.\n\n\\section{Stability of Primal-Dual Solution}\n\\label{appn:primal_dual}\nAs discussed in Section~\\ref{sec:opt}, the Lagrangian function for the optimization problem~\\eqref{eq:opt} is expressed as\n\\[\\mathcal{L}(\\mathbf{h}, \\alpha) = \\sum_{i}{U_i(h_i)} - \\alpha(\\sum_{i}{h_i} - B).\\]\nNote that $\\mathcal{L}(\\mathbf{h}, \\alpha)$ is concave in $\\mathbf{h}$ and convex in $\\alpha$, and hence first order condition for optimality of $\\mathbf{h}^*$ and $\\alpha^*$ implies\n\\begin{align*}\n\\mathcal{L}(\\mathbf{h}^*, \\alpha) &\\le \\mathcal{L}(\\mathbf{h}, \\alpha) + \\sum_{i}{\\frac{\\partial \\mathcal{L}}{\\partial h_i}(h_i^* - h_i)}, \\\\\n\\mathcal{L}(\\mathbf{h}, \\alpha^*) &\\ge \\mathcal{L}(\\mathbf{h}, \\alpha) + \\frac{\\partial \\mathcal{L}}{\\partial \\alpha}(\\alpha^* - \\alpha) .\n\\end{align*}\n\nAssume that the hit probability of a file can be expressed by $f(\\cdot)$ as a function of the corresponding timer value $t_i$, \\emph{i.e.}\\ ${h_i = f(t_i)}$. The temporal derivative of the hit probability can therefore be expressed as\n\\[\\dot{h_i} = f'(t_i) \\dot{t_i},\\]\nor equivalently\n\\[\\dot{h_i} = f'(f^{-1}(h_i)) \\dot{t_i},\\]\nwhere $f^{-1}(\\cdot)$ denotes the inverse of function $f(\\cdot)$. For notation brevity we define ${g(h_i) = f'(f^{-1}(h_i))}$. Note that as discussed in Appendix~\\ref{appn:primal}, $f(\\cdot)$ is an increasing function, and hence ${g(h_i)\\ge 0}$.\n\nIn the remaining, we show that $V(\\mathbf{h}, \\alpha)$ defined below is a Lyapunov function for the primal-dual algorithm:\n\\[V(\\mathbf{h}, \\alpha) = \\sum_{i}{\\int_{h_i^*}^{h_i}{\\frac{x - h_i^*}{k_i g(x)}\\mathrm{d} x}} + \\frac{1}{2\\gamma}(\\alpha - \\alpha^*)^2.\\]\nDifferentiating the above function with respect to time we obtain\n\\[\\dot{V}(\\mathbf{h}, \\alpha) = \\sum_{i}{\\frac{h_i - h_i^*}{k_i g(h_i)}\\dot{h_i}} + \\frac{\\alpha - \\alpha^*}{\\gamma}\\dot{\\alpha}.\\]\nBased on the controllers defined for $t_i$ and $\\alpha$ we have\n\\[\\dot{h_i} = g(h_i) \\dot{t_i} = k_i g(h_i) \\frac{\\partial \\mathcal{L}}{\\partial h_i},\\]\nand\n\\[\\dot{\\alpha} = -\\gamma\\frac{\\partial \\mathcal{L}}{\\partial \\alpha}.\\]\nReplacing for $\\dot{h_i}$ and $\\dot{\\alpha}$ in $\\dot{V}(\\mathbf{h}, \\alpha)$, we obtain\n\\begin{align*}\n\\dot{V}(\\mathbf{h}, \\alpha) &= \\sum_{i}{(h_i - h_i^*)\\frac{\\partial \\mathcal{L}}{\\partial h_i}} - (\\alpha - \\alpha^*)\\frac{\\partial \\mathcal{L}}{\\partial \\alpha} \\\\\n&\\le \\mathcal{L}(\\mathbf{h}, \\alpha) - \\mathcal{L}(\\mathbf{h}^*, \\alpha) + \\mathcal{L}(\\mathbf{h}, \\alpha^*) - \\mathcal{L}(\\mathbf{h}, \\alpha) \\\\\n&= \\Big(\\mathcal{L}(\\mathbf{h}^*, \\alpha^*) - \\mathcal{L}(\\mathbf{h}^*, \\alpha)\\Big) + \\Big(\\mathcal{L}(\\mathbf{h}, \\alpha^*) - \\mathcal{L}(\\mathbf{h}^*, \\alpha^*)\\Big) \\\\\n&\\le 0,\n\\end{align*}\nwhere the last inequality follows from \n\\[\\mathcal{L}(\\mathbf{h}, \\alpha^*)  \\le \\mathcal{L}(\\mathbf{h}^*, \\alpha^*)  \\le \\mathcal{L}(\\mathbf{h}^*, \\alpha),\\]\nfor any $\\mathbf{h}$ and $\\alpha$.\n\nMoreover, $V(\\mathbf{h}, \\alpha)$ is non-negative and equals zero only at $(\\mathbf{h}^*, \\alpha^*)$.\nTherefore, $V(\\mathbf{h}, \\alpha)$ is a Lyapunov function, and the system state will converge to optimum starting from any initial condition.\n\n\\end{appendices}\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this paper, we proposed the concept of utility-driven caching, and formulated it as an optimization problem with rigid and elastic cache storage size constraints. Utility-driven caching provides a general framework for defining caching policies with considerations of fairness among various groups of files, and implications on market economy for (cache) service providers and content publishers. This framework has the capability to model existing caching policies such as FIFO and LRU, as utility-driven caching policies.\n\nWe developed three decentralized algorithms that implement utility-driven caching policies in an online fashion and that can adapt to changes in file request rates over time. We prove that these algorithms are globally stable and converge to the optimal solution. Through simulations we illustrated the efficiency of these algorithms and the flexibility of our approach.\n\n\\section{Discussion}\n\\label{sec:discussion}\nIn this section, we explore the implications of utility-driven caching on monetizing the caching service and discuss some future research directions.\n\n\\subsection{Decomposition}\nThe formulation of the problem in Section~\\ref{sec:opt} assumes that the utility functions $U_i(\\cdot)$ are known to the system. In reality content providers might decide not to share their utility functions with the service provider. To handle this case, we decompose the optimization problem~\\eqref{eq:opt} into two simpler problems.\n\nSuppose that the cache storage is offered as a service and the service provider charges content providers at a constant rate $r$ for storage space. Hence, a content provider needs to pay an amount of $w_i = rh_i$ to receive hit probability $h_i$ for file $i$. The utility maximization problem for the content provider of file $i$ can then be written as\n\\begin{align}\n\\label{eq:opt_user}\n\\text{maximize} \\quad &U_i(w_i\/r) - w_i \\\\\n\\text{such that} \\quad &w_i \\ge 0 \\notag\n\\end{align}\n\nNow, assuming that the service provider knows the vector $\\mathbf{w}$, for a proportionally fair resource allocation, the hit probabilities should be\nset according to\n\\begin{align}\n\\label{eq:opt_network}\n\\text{maximize} \\quad &\\sum_{i=1}^{N}{w_i\\log{(h_i)}} \\\\\n\\text{such that} \\quad &\\sum_{i=1}^{N}{h_i} = B \\notag\n\\end{align}\n\nIt was shown in~\\cite{kelly97} that there always exist vectors $\\mathbf{w}$ and $\\mathbf{h}$, such that $\\mathbf{w}$ solves~\\eqref{eq:opt_user} and $\\mathbf{h}$ solves~\\eqref{eq:opt_network}; further, the vector $\\mathbf{h}$ is the unique solution to~\\eqref{eq:opt}.\n\n\\subsection{Cost and Utility Functions}\nIn Section~\\ref{sec:soft}, we defined a penalty function denoting the cost of using additional storage space. One might also define cost functions based on the consumed network bandwidth. This is especially interesting in modeling in-network caches with network links that are likely to be congested.\n\nOptimization problem~\\eqref{eq:opt} uses utility functions defined as functions of the hit probabilities. It is reasonable to define utility as a function of the hit \\emph{rate}. Whether this makes any changes to the problem, \\emph{e.g.}\\ in the notion of fairness, is a question that requires further investigation. One argument in support of utilities as functions of hit rates is that a service provider might prefer pricing based on request rate rather than cache occupancy. Moreover, in designing hierarchical caches a service provider's objective could be to minimize the internal bandwidth cost. This can be achieved by defining the utility functions as $U_i = -C_i(m_i)$ where $C_i(m_i)$ denotes the cost associated with miss rate $m_i$ for file $i$.\n\n\\subsection{Online Algorithms}\nIn Section~\\ref{sec:online}, we developed three online algorithms that can be used to implement utility-driven caching. Although these algorithms are proven to be stable and converge to the optimal solution, they have distinct features that can make one algorithm more effective in implementing a policy. For example, implementing the max-min fair policy based on the dual solution requires knowing\/estimating the file request rates, while it can be implemented using the modified primal-dual solution without such knowledge. Moreover, the convergence rate of these algorithms may differ for different policies. The choice of non-reset or reset TTL caches also has implications on the design and performance of these algorithms.\nThese are subjects that require further study.\n\n\\section{Online Algorithms}\n\\label{sec:online}\nIn Section~\\ref{sec:opt}, we formulated utility-driven caching as a convex optimization problem either with a fixed or an elastic cache size. However, it is not feasible to solve the optimization problem offline and then\nimplement the optimal strategy. Moreover, the system parameters can change over time. Therefore, we need algorithms\nthat can be used to implement the optimal strategy and adapt to changes in the system by collecting limited information.\nIn this section, we develop such algorithms.\n\n\\subsection{Dual Solution}\n\\label{sec:dual}\nThe utility-driven caching formulated in~\\eqref{eq:opt} is a convex optimization problem, and hence the optimal solution corresponds to solving the dual problem.\nThe Lagrange dual of the above problem is obtained by incorporating the constraints into the maximization by means of Lagrange multipliers\n\\begin{align*}\n\\text{minimize} \\quad &D(\\alpha, \\boldsymbol{\\nu}, \\boldsymbol{\\eta}) = \\max_{h_i}\\Bigg\\{ \\sum_{i=1}^{N}{U_i(h_i)} \\\\\n&\\qquad\\qquad\\qquad -\\alpha\\left[ \\sum_{i=1}^{N}{h_i} - B\\right] \\\\\n&\\qquad\\qquad\\qquad -\\sum_{i=1}^{N}{\\nu_i (h_i - 1)} + \\sum_{i=1}^{N}{\\eta_i h_i} \\Bigg\\}\\\\\n\\text{such that} \\quad &\\alpha \\ge 0, \\quad \\boldsymbol{\\nu}, \\boldsymbol{\\eta} \\ge \\mathbf{0}.\n\\end{align*}\nUsing timer based caching techniques for controlling the hit probabilities with $0 < t_i < \\infty$ ensures that $0 < h_i < 1$, and hence we have $\\nu_i = 0$ and $\\eta_i = 0$. \n\nHere, we consider an algorithm based on the dual solution to the utility maximization problem~\\eqref{eq:opt}. Recall that we can write the Lagrange dual of the utility maximization problem as\n\\[D(\\alpha) = \\max_{h_i}{\\left\\{ \\sum_{i=1}^{N}{U_i(h_i)}-\\alpha\\left[ \\sum_{i=1}^{N}{h_i} - B\\right] \\right\\}},\\]\nand the dual problem can be written as\n\\[\\min_{\\alpha \\ge 0}{D(\\alpha)}.\\]\n\nA natural decentralized approach to consider for minimizing $D(\\alpha)$ is to gradually move the decision variables towards the optimal point using the gradient descent algorithm.\nThe gradient can be easily computed as\n\\[\\frac{\\partial D(\\alpha)}{\\partial\\alpha} = -\\Big(\\sum_{i}{h_i} - B \\Big),\\]\nand since we are doing a gradient \\emph{descent}, $\\alpha$ should be updated according to the \\emph{negative} of the gradient as\n\\[\\alpha \\gets \\max{\\Big\\{0, \\alpha + \\gamma \\Big( \\sum_{i}{h_i} - B \\Big)\\Big\\}},\\]\nwhere $\\gamma > 0$ controls the step size at each iteration. Note that the KTT conditions require that $\\alpha \\ge 0$.\n\nBased on the discussion in Section~\\ref{sec:opt}, to satisfy the optimality condition we must have\n\\[U'_i(h_i) = \\alpha,\\]\nor equivalently\n\\[h_i = {U'_i}^{-1}(\\alpha).\\]\nThe hit probabilities are then controlled based on the timer parameters $t_i$ which can be set according to~\\eqref{eq:non_reset_t} and~\\eqref{eq:reset_t} for non-reset and reset TTL caches.\n\nConsidering the hit probabilities as indicators of files residing in the cache, the expression $\\sum_{i}{h_i}$ can be interpreted as the number of items currently in the cache, denoted here as $B_{curr}$. We can thus summarize the control algorithm for a reset TTL algorithm as\n\\begin{align}\n\\label{eq:dual_sol}\nt_i &= -\\frac{1}{\\lambda_i} \\log{\\Big(1 - {U'_i}^{-1}(\\alpha) \\Big)}, \\notag\\\\\n\\alpha &\\gets \\max{\\{0, \\alpha + \\gamma ( B_{curr} - B )\\}}.\n\\end{align}\nWe obtain an algorithm for a non-reset TTL cache by using the correct expression for $t_i$ in~\\eqref{eq:non_reset_t}.\n\nLet $\\alpha^*$ denote the optimal value for $\\alpha$. We show in Appendix~\\ref{appn:dual} that $D(\\alpha) - D(\\alpha^*)$ is a Lyapunov function and the above algorithm converges to the optimal solution.\n\n\\subsection{Primal Solution}\nWe now consider an algorithm based on the optimization problem in~\\eqref{eq:opt_soft} known as the \\emph{primal} formulation.\n\nLet $W(\\mathbf{h})$ denote the objective function in~\\eqref{eq:opt_soft} defined as\n\\[W(\\mathbf{h}) = \\sum_{i=1}^{N}{U_i(h_i)} - C(\\sum_{i=1}^{N}{h_i} - B).\\]\nA natural approach to obtain the maximum value for $W(\\mathbf{h})$ is to use the gradient ascent algorithm.\nThe basic idea behind the gradient ascent algorithm is to move the variables $h_i$ in the direction of the gradient\n\\[\\frac{\\partial W(\\mathbf{h})}{\\partial h_i} = U'_i(h_i) - C'(\\sum_{i=1}^{N}{h_i} - B).\\]\nSince the hit probabilities are controlled by the TTL timers, we move $h_i$ towards the optimal point by updating timers $t_i$.\nLet $\\dot{h_i}$ denote the derivative of the hit probability $h_i$ with respect to time. Similarly, define $\\dot{t_i}$ as the derivative of the timer parameter $t_i$\nwith respect to time. We have\n\\[\\dot{h_i} = \\frac{\\partial h_i}{\\partial t_i}\\dot{t_i}.\\]\nFrom~\\eqref{eq:hit_non_reset} and~\\eqref{eq:hit_reset}, it is easy to confirm that $\\partial h_i\/\\partial t_i > 0$ for non-reset and reset TTL caches.\nTherefore, moving $t_i$ in the direction of the gradient, also moves $h_i$s in that direction.\n\nBy gradient ascent, the timer parameters should be updated according to\n\\[t_i \\gets \\max{\\left\\{0, t_i + k_i\\Big[ U'_i(h_i) - C'(B_{curr} - B)  \\Big]\\right\\}},\\]\nwhere $k_i > 0$ is the step-size parameter, and $\\sum_{i=1}^{N}{h_i}$ has been replaced with $B_{curr}$ based on the same argument as in the dual solution.\n\nLet $\\mathbf{h}^*$ denote the optimal solution to~\\eqref{eq:opt_soft}. We show in Appendix~\\ref{appn:primal} that $W(\\mathbf{h}^*) - W(\\mathbf{h})$ is a Lyapunov function, and the above algorithm converges to the optimal solution.\n\n\\subsection{Primal-Dual Solution}\nHere, we consider a third algorithm that combines elements of the previous two algorithms.\nConsider the control algorithm\n\\begin{align*}\nt_i &\\gets \\max{\\{0,  t_i + k_i [ U'_i(h_i) - \\alpha]\\}}, \\\\\n\\alpha &\\gets \\max{\\{0, \\alpha + \\gamma (B_{current} - B)\\}}.\n\\end{align*}\nUsing Lyapunov techniques we show in Appendix~\\ref{appn:primal_dual} that the above algorithm converges to the optimal solution.\n\nNow, rather than updating the timer parameters according to the above rule explicitly based on the utility function, we can have update rules based on a cache hit or miss.\nConsider the following differential equation\n\\begin{equation}\n\\label{eq:t}\n\\dot{t_i} = \\delta_m(t_i, \\alpha)(1 - h_i)\\lambda_i - \\delta_h(t_i, \\alpha)h_i\\lambda_i,\n\\end{equation}\nwhere $\\delta_m(t_i, \\alpha)$ and $-\\delta_h(t_i, \\alpha)$ denote the change in $t_i$ upon a cache miss or hit for file $i$, respectively.\nMore specifically, the timer for file $i$ is increased by $\\delta_m(t_i, \\alpha)$ upon a cache miss, and decreased by $\\delta_h(t_i, \\alpha)$ on a cache hit.\n\nThe equilibrium for~\\eqref{eq:t} happens when $\\dot{t_i} = 0$, which solving for $h_i$ yields\n\\[h_i = \\frac{\\delta_m(t_i, \\alpha)}{\\delta_m(t_i, \\alpha) + \\delta_h(t_i, \\alpha)}.\\]\nComparing the above expression with $h_i = {U'_i}^{-1}(\\alpha)$ suggests that\n$\\delta_m(t_i, \\alpha)$ and $\\delta_h(t_i, \\alpha)$ can be set to achieve desired hit probabilities and caching policies.\n\nMoreover, the differential equation~\\eqref{eq:t} can be reorganized as\n\\[\\dot{t_i} = h_i \\lambda_i \\Big(\\delta_m(t_i, \\alpha)\/h_i - [\\delta_m(t_i, \\alpha) + \\delta_h(t_i, \\alpha)]\\Big),\\]\nand to move $t_i$ in the direction of the gradient $U'_i(h_i) - \\alpha$ a natural choice for the update functions can be\n\\[\\delta_m(t_i, \\alpha) = h_i U'_i(h_i), \\text{ and } \\delta_m(t_i, \\alpha) + \\delta_h(t_i, \\alpha) = \\alpha.\\]\n\nTo implement proportional fairness for example, these functions can be set as\n\\begin{equation}\n\\label{eq:prop_pd}\n\\delta_m(t_i, \\alpha) = \\lambda_i, \\text{ and } \\delta_h(t_i, \\alpha) = \\alpha - \\lambda_i.\n\\end{equation}\n\nFor the case of max-min fairness, recall from the discussion in Section~\\ref{sec:opt_identical} that a utility function that is content agnostic, \\emph{i.e.}\\ $U_i(h) = U(h)$, results in a max-min fair resource allocation. Without loss of generality we can have $U_i(h_i) = \\log{h_i}$. Thus, max-min fairness can be implemented by having\n\\[\\delta_m(t_i, \\alpha) = 1, \\text{ and } \\delta_h(t_i, \\alpha) = \\alpha - 1.\\]\nNote that with these functions, max-min fairness can be implemented without requiring knowledge about request arrival rates~$\\lambda_i$, while the previous approaches require such knowledge.\n\n\n\\subsection{Estimation of $\\lambda_i$}\n\\label{sec:estimate}\nComputing the timer parameter $t_i$ in the algorithms discussed in this section requires knowing the request arrival rates for most of the policies.\nEstimation techniques can be used to approximate the request rates in case such knowledge is not available at the (cache) service provider.\n\nLet $r_i$ denote the remaining TTL time for file $i$. Note that $r_i$ can be computed based on $t_i$ and a time-stamp for the last time file~$i$ was requested.\nLet $X_i$ denote the random variable corresponding to the inter-arrival times for the requests for file~$i$, and $\\bar{X_i}$ be its mean.\nWe can approximate the mean inter-arrival time as $\\hat{\\bar{X_i}} = t_i - r_i$. Note that $\\hat{\\bar{X_i}}$ defined in this way is a one-sample\nunbiased estimator of $\\bar{X_i}$. Therefore, $\\hat{\\bar{X_i}}$ is an unbiased estimator of $1\/\\lambda_i$. In the simulation section, we will use this estimator in computing the timer parameters for evaluating our algorithms.\n\n\n\\section{Utility Functions and Fairness}\n\\label{sec:fairness}\nUsing different utility functions in the optimization formulation~\\eqref{eq:opt} yields different timer values for the files.\nIn this sense, each utility function defines a notion of fairness in allocating storage resources to different files.\nIn this section, we study a number of utility functions that have important fairness properties associated with them.\n\\subsection{Identical Utilities}\n\\label{sec:opt_identical}\nAssume that all files have the same utility function, \\emph{i.e.}\\ $U_i(h_i) = U(h_i)$ for all $i$. Then, from~\\eqref{eq:c} we obtain\n\\[\\sum_{i=1}^{N}{{U'}^{-1}(\\alpha)} = N {U'}^{-1}(\\alpha) = B,\\]\nand hence\n\\[{U'}^{-1}(\\alpha) = B\/N.\\]\nUsing~\\eqref{eq:hu} for the hit probabilities we get\n\\[h_i = B\/N, \\quad \\forall{i}.\\]\n\nUsing a non-reset TTL policy, the timers should be set according to\n\\[t_i = \\frac{B}{\\lambda_i (N - B)},\\]\nwhile with a reset TTL policy, they must equal\n\\[t_i = -\\frac{1}{\\lambda_i}\\log{\\left(1-\\frac{B}{N}\\right)}.\\]\n\nThe above calculations show that identical utility functions yield identical hit probabilities for all files. Note that the hit probabilities computed above do not depend on the utility function.\n\n\\subsection{$\\boldsymbol{\\beta}$-Fair Utility Functions}\nHere, we consider the family of $\\beta$-fair (also known as \\emph{isoelastic}) utility functions given by\n\\[U_i(h_i) = \\left\\{ \\begin{array}{ll}\n w_i\\frac{h_i^{1-\\beta}}{1-\\beta} & \\beta \\ge 0, \\beta \\neq 1; \\\\\n & \\\\\n w_i \\log{h_i} & \\beta = 1,\n \\end{array} \\right. \\]\nwhere the coefficient $w_i \\ge 0$ denotes the weight for file $i$.\nThis family of utility functions unifies different notions of fairness in resource allocation~\\cite{srikant13}.\nIn the remainder of this section, we investigate some of the choices for $\\beta$ that lead to interesting special cases.\n\n\\subsubsection{$\\boldsymbol{\\beta = 0}$}\\hspace*{\\fill} \\\\\nWith $\\beta = 0$, we get $U_i(h_i) = w_i h_i$, and maximizing the sum of the utilities\ncorresponds to\n\\[\\max_{h_i}{\\sum_{i}{w_i h_i}}.\\]\n\nThe above utility function defined does not satisfy the requirements for a utility function mentioned in Section~\\ref{sec:model}, as it is not strictly concave.\nHowever, it is easy to see that the sum of the utilities is maximized when\n\\[h_i = 1, i=1,\\ldots, B  \\quad \\text{ and } \\quad h_i = 0, i=B+1,\\ldots, N,\\]\nwhere we assume that weights are sorted as ${w_1 \\ge \\ldots \\ge w_N}$.\nThese hit probabilities indicate that the optimal timer parameters are\n\\[t_i = \\infty, i=1,\\ldots, B  \\quad \\text{ and } \\quad t_i = 0, i=B+1,\\ldots, N.\\]\n\nNote that the policy obtained by implementing this utility function with $w_i = \\lambda_i$ corresponds to the Least-Frequently Used (LFU) caching policy,\nand maximizes the overall throughput.\n\n\\subsubsection{$\\boldsymbol{\\beta = 1}$}\\hspace*{\\fill} \\\\\nLetting $\\beta = 1$, we get $U_i(h_i) = w_i \\log{h_i}$,\nand hence maximizing the sum of the utilities corresponds to\n\\[\\max_{h_i}{\\sum_{i}{w_i \\log{h_i}}}.\\]\n\nIt is easy to see that ${U'_i}^{-1}(\\alpha) = w_i \/ \\alpha$, and hence using~\\eqref{eq:c} we obtain\n\\[\\sum_{i}{{U'_i}^{-1}(\\alpha)} = \\sum_{i}{w_i} \/ \\alpha = B,\\]\nwhich yields\n\\[\\alpha = \\sum_{i}{w_i} \/ B.\\]\nThe hit probability of file $i$ then equals\n\\[h_i = {U'_i}^{-1}(\\alpha) = \\frac{w_i}{\\sum_{j}{w_j}}B.\\]\n\nThis utility function implements a \\emph{proportionally fair} policy~\\cite{kelly98}.\nWith $w_i = \\lambda_i$, the hit probability of file $i$ is proportional to the request arrival rate $\\lambda_i$.\n\n\\subsubsection{$\\boldsymbol{\\beta = 2}$}\\hspace*{\\fill} \\\\\nWith $\\beta = 2$, we get $U_i(h_i) = -w_i\/h_i$, and maximizing the total utility corresponds to\n\\[\\max_{h_i}{\\sum_{i}{\\frac{-w_i}{h_i}}}.\\]\n\nIn this case, we get ${U'_i}^{-1}(\\alpha) = \\sqrt{w_i} \/ \\sqrt{\\alpha}$, therefore\n\\[\\sum_{i}{{U'_i}^{-1}(\\alpha)} = \\sum_{i}{\\sqrt{w_i}} \/ \\sqrt{\\alpha} = B,\\]\nand hence\n\\[\\alpha = \\Big(\\sum_{i}{\\sqrt{w_i}}\\Big)^2 \/ B^2.\\]\n\nThe hit probability of file $i$ then equals\n\\[h_i = {U'_i}^{-1}(\\alpha) = \\frac{\\sqrt{w_i}}{\\sqrt{\\alpha}} = \\frac{\\sqrt{w_i}}{\\sum_{j}{\\sqrt{w_j}}}B.\\]\n\nThe utility function defined above is known to yield minimum potential delay fairness. It was shown in~\\cite{kelly98} that the TCP congestion control protocol\nimplements such a utility function.\n\n\\subsubsection{$\\boldsymbol{\\beta \\rightarrow\\infty}$}\\hspace*{\\fill} \\\\\nWith $\\beta \\rightarrow\\infty$, maximizing the sum of the utilities corresponds to (see~\\cite{mo00} for proof)\n\\[\\max_{h_i}{\\min_{i}{h_i}}.\\]\n\nThis utility function does not comply with the rules mentioned in Section~\\ref{sec:model} for utility functions, as it is not strictly concave.\nHowever, it is easy to see that the above utility function yields\n\\[h_i = B\/N, \\quad \\forall{i}.\\]\n\nThe utility function defined here maximizes the minimum hit probability, and corresponds to the \\emph{max-min fairness}. Note that using identical utility functions for all files resulted in similar hit probabilities as this case. A brief summary of the utility functions discussed here is given in Table~\\ref{tbl:u}. \n\\begin{table*}[]\n\\centering\n\\caption{$\\beta$-fair utility functions family}\n\\begin{tabular}{ | c | c | c | c |}\n\\hline\n$\\beta$ & $\\max{\\sum_{i}{U_i(h_i)}}$ & $h_i$ & implication \\\\\n\\hline\n  0 & $\\max{\\sum{w_i h_i}}$ & $h_i = 1, i\\le B, h_i = 0, i \\ge B+1$ & maximizing overall throughput \\\\\n  1 & $\\max{\\sum{w_i \\log{h_i}}}$ & $h_i = w_i B \/ \\sum_{j}{w_j}$ & proportional fairness \\\\\n  2 & $\\max{-\\sum{w_i \/ h_i}}$ & $h_i = \\sqrt{w_i} B \/ \\sum_{j}{\\sqrt{w_j}}$ & minimize potential delay \\\\\n  $\\infty$ & $\\max{\\min{h_i}}$ & $h_i = B\/N$ & max-min fairness \\\\\n\\hline\n\\end{tabular}\n\\label{tbl:u}\n\\end{table*}\n\n\n\\section{Introduction}\nThe increase in data traffic over past years is predicted to continue more aggressively, with global Internet traffic in 2019 estimated to reach 64 times of its volume in 2005~\\cite{cisco14}.\nThe growth in data traffic is recognized to be due primarily to streaming of video on-demand content over cellular networks.\nHowever, traditional methods such as increasing the amount of spectrum or deploying more base stations are not sufficient to cope with this predicted traffic increase~\\cite{Andrews12, Golrezaei12}. Caching is recognized, in current and future Internet architecture proposals, as one of the most effective means to improve the performance of web applications.\nBy bringing the content closer to users, caches greatly reduce network bandwidth usage, server load, and perceived service delays~\\cite{borst10}.\n\n\n\nBecause of the trend for ubiquitous computing, creation of new content publishers and consumers, the Internet is becoming an increasingly heterogeneous environment where different content types have different quality of service requirements, depending on the content publisher\/consumer.\nSuch an increasing diversity in service expectations advocates the need for content delivery infrastructures with service differentiation among different applications and content classes.\nService differentiation not only induces important technical gains, but also provides significant economic benefits~\\cite{feldman02}.\nDespite a plethora of research on the design and implementation of \\emph{fair} and \\emph{efficient} algorithms for differentiated bandwidth sharing in communication networks, little work has focused on the provision of multiple levels of service in network and web caches.\nThe little available research has focused on designing controllers for partitioning cache space~\\cite{ko03, lu04}, biased replacement policies towards particular content classes~\\cite{kelly99}, or using multiple levels of caches~\\cite{feldman02}. These techniques either require additional controllers for fairness, or inefficiently use the cache storage.\n\nMoreover, traditional cache management policies such as LRU treat different contents in a strongly coupled manner that makes it difficult for (cache) service providers to implement differentiated services, and for content publishers to account for the valuation of their content delivered through content distribution networks.\nIn this paper, we propose a utility-driven caching framework, where each content has an associated utility and content is stored and managed in a cache so as to maximize the aggregate utility for all content. Utilities can be chosen to trade off user satisfaction and cost of storing the content in the cache.\nWe draw on analytical results for time-to-live (TTL) caches~\\cite{Nicaise14b}, to design caches with ties to utilities for individual (or classes of) contents.\nUtility functions also have implicit notions of fairness that dictate the time each content stays in cache.\nOur framework allows us to develop \\emph{online} algorithms for cache management, for which we prove achieve optimal performance. Our framework has implications for distributed pricing and control mechanisms and hence is well-suited for designing cache market economic models.\n\nOur main contributions in this paper can be summarized as follows:\n\\begin{itemize}\n\\item We formulate a utility-based optimization framework for maximizing aggregate content publisher utility subject to buffer capacity constraints at the service provider.\nWe show that existing caching policies, \\emph{e.g.}\\ LRU, LFU and FIFO, can be modeled as utility-driven caches within this framework.\n\\item By reverse engineering the LRU and FIFO caching policies as utility maximization problems, we show how the \\emph{characteristic time}~\\cite{Che01} defined for these caches relates to the Lagrange multiplier corresponding to the cache capacity constraint.\n\\item We develop online algorithms for managing cache content, and prove the convergence of these algorithms to the optimal solution using Lyapunov functions.\n\\item We show that our framework can be used in revenue based models where content publishers react to prices set by (cache) service providers without revealing their utility functions.\n\\item We perform simulations to show the efficiency of our online algorithms using different utility functions with different notions of fairness.\n\\end{itemize}\n\nThe remainder of the paper is organized as follows. We review related work in the next section. Section~\\ref{sec:model} explains the network model considered in this paper, and\nSection~\\ref{sec:opt} describes our approach in designing utility maximizing caches. In Section~\\ref{sec:fairness} we elaborate on fairness implications of utility functions, and in Section~\\ref{sec:reverse}, we derive the utility functions maximized by LRU and FIFO caches. In Section~\\ref{sec:online}, we develop online algorithms for implementing utility maximizing caches. We present simulation results in Section~\\ref{sec:simulation}, and discuss prospects and implications of the cache utility maximization framework in Section~\\ref{sec:discussion}. Finally, we conclude the paper in Section~\\ref{sec:conclusion}.\n\n\\section{Model}\n\\label{sec:model}\nConsider a set of $N$ files, and a cache of size $B$. We use the terms file and content interchangeably in this paper.\nLet $h_i$ denote the hit probability for content $i$.\nAssociated with each content, $i=1,\\ldots, N$, is a utility function $U_i:[0,1] \\rightarrow \\mathbb{R}$ that represents the ``satisfaction'' perceived by observing hit probability $h_i$.\n$U_i(\\cdot)$ is assumed to be increasing, continuously differentiable, and strictly concave.\nNote that a function with these properties is invertible. We will treat utility functions that do not satisfy these constraints as special cases.\n\n\\subsection{TTL Caches}\nIn a TTL cache, each content is associated with a timer~$t_i$.  Whenever a cache miss to content $i$ occurs, content $i$ is stored in the cache and its timer is set to $t_i$.\nTimers decrease at constant rate, and a content is evicted from cache when its timer reaches zero.\nWe can adjust the hit probability of a file by controlling the time a file is kept in cache.\n\nThere are two TTL cache designs:\n\\begin{itemize}\n\\item Non-reset TTL Cache: TTL is only set at cache misses, \\emph{i.e.}~TTL is not reset upon cache hits.\n\\item Reset TTL Cache: TTL is set each time the content is requested.\n\\end{itemize}\nPrevious work on the analysis of TTL caches~\\cite{Nicaise14} has shown that the hit probability of file $i$ for these two classes of non-reset and reset TTL caches can be expressed as \n\\begin{equation}\n\\label{eq:hit_non_reset}\nh_i = 1 - \\frac{1}{1 + \\lambda_i t_i},\n\\end{equation}\nand\n\\begin{equation}\n\\label{eq:hit_reset}\nh_i = 1 - e^{-\\lambda_i t_i},\n\\end{equation}\nrespectively, where requests for file $i$ arrive at the cache according to a Poisson process with rate $\\lambda_i$.\nNote that depending on the utility functions, different (classes of) files might have different or equal TTL values.\n\n\\section{Cache Utility Maximization}\n\\label{sec:opt}\nIn this section, we formulate cache management as a utility maximization problem.  We introduce two formulations, one where the buffer size introduces a hard constraint and a second where it introduces a soft constraint.\n\n\\subsection{Hard Constraint Formulation}\nWe are interested in designing a cache management policy that optimizes the sum of utilities over all files, more precisely,\n\\begin{align}\n\\label{eq:opt}\n\\text{maximize} \\quad &\\sum_{i=1}^{N}{U_i(h_i)} \\notag\\\\\n\\text{such that} \\quad &\\sum_{i=1}^{N}{h_i} = B \\\\\n& 0 \\le h_i \\le 1, \\quad i=1, 2, \\ldots, N. \\notag\n\\end{align}\nNote that the feasible solution set is convex and since the objective function is strictly concave and continuous, a unique maximizer, called the optimal solution, exists.  Also note that the buffer constraint is based on the {\\em expected} number of files not exceeding the buffer size and not the total number of files.\nTowards the end of this section, we show that the buffer space can be managed in a way such that the probability of \\emph{violating} the buffer size constraint vanishes as the number of files and cache size grow large.\n\nThis formulation does not enforce any special technique for managing the cache content, and any strategy that can easily adjust the hit probabilities can be employed. We use the TTL cache as our building block because it provides the means through setting timers to control the hit probabilities of different files in order to maximize the sum of utilities.\n\nUsing timer based caching techniques for controlling the hit probabilities with $0 < t_i < \\infty$ ensures that $0 < h_i < 1$, and hence, disregarding the possibility of $h_i = 0$ or $h_i = 1$, we can write the Lagrangian function as\n\\begin{align*}\n\\mathcal{L}(\\mathbf{h}, \\alpha) &= \\sum_{i=1}^{N}{U_i(h_i)}-\\alpha\\left[ \\sum_{i=1}^{N}{h_i} - B\\right] \\\\\n&= \\sum_{i=1}^{N}{\\Big[ U_i(h_i)-\\alpha h_i \\Big]} + \\alpha B,\n\\end{align*}\nwhere $\\alpha$ is the Lagrange multiplier.\n\nIn order to achieve the maximum in $\\mathcal{L}(\\mathbf{h}, \\alpha)$, the hit probabilities should satisfy\n\\begin{equation}\n\\label{eq:drv}\n\\frac{\\partial\\mathcal{L}}{\\partial h_i} = \\frac{\\mathrm{d} U_i}{\\mathrm{d} h_i} - \\alpha = 0.\n\\end{equation}\n\nLet $U'_i(\\cdot)$ denote the derivative of the the utility function $U_i(\\cdot)$, and define ${U'_i}^{-1}(\\cdot)$ as its inverse function.\nFrom~\\eqref{eq:drv} we get\n\\[U'_i(h_i) = \\alpha,\\]\nor equivalently\n\\begin{equation}\n\\label{eq:hu}\nh_i = {U'_i}^{-1}(\\alpha).\n\\end{equation}\nApplying the cache storage constraint we obtain\n\\begin{equation}\n\\label{eq:c}\n\\sum_{i}{h_i} = \\sum_{i}{{U'_i}^{-1}(\\alpha)} = B,\n\\end{equation}\nand $\\alpha$ can be computed by solving the fixed-point equation given above.\n\nAs mentioned before, we can implement utility maximizing caches using TTL based policies.\nUsing the expression for the hit probabilities of non-reset and reset TTL caches given in~\\eqref{eq:hit_non_reset} and~\\eqref{eq:hit_reset},\nwe can compute the timer parameters $t_i$, once $\\alpha$ is determined from~\\eqref{eq:c}.\nFor non-reset TTL caches we obtain\n\\begin{equation}\n\\label{eq:non_reset_t}\nt_i = -\\frac{1}{\\lambda_i}\\Big(1 - \\frac{1}{1 - {U'_i}^{-1}(\\alpha)}\\Big),\n\\end{equation}\nand for reset TTL caches we get\n\\begin{equation}\n\\label{eq:reset_t}\nt_i = -\\frac{1}{\\lambda_i}\\log{\\Big(1 - {U'_i}^{-1}(\\alpha)\\Big)}.\n\\end{equation}\n\n\\subsection{Soft Constraint Formulation}\n\\label{sec:soft}\nThe formulation in~\\eqref{eq:opt} assumes a hard constraint on cache capacity.\nIn some circumstances it may be appropriate for the (cache) service provider to increase the available cache storage at some cost to the file provider\nfor the additional resources\\footnote{One straightforward way of thinking this is to turn the cache memory disks on and off based on the demand.}.\nIn this case the cache capacity constraint can be replaced with a penalty function $C(\\cdot)$ denoting the cost for the extra cache storage.\nHere, $C(\\cdot)$ is assumed to be a convex and increasing function.\nWe can now write the utility and cost driven caching formulation as\n\\begin{align}\n\\label{eq:opt_soft}\n\\text{maximize} \\quad &\\sum_{i=1}^{N}{U_i(h_i)} - C(\\sum_{i=1}^{N}{h_i} - B) \\\\\n\\text{such that} \\quad &0 \\le h_i \\le 1, \\quad i=1,2,\\ldots, N. \\notag\n\\end{align}\n\nNote the optimality condition for the above optimization problem states that\n\\[U'_i(h_i) = C'(\\sum_{i=1}^{N}{h_i} - B).\\]\n\nTherefore, for the hit probabilities we obtain\n\\[h_i = {U'_i}^{-1}\\Big(C'(\\sum_{i=1}^{N}{h_i} - B)\\Big),\\]\nand the optimal value for the cache storage can be computed using the fixed-point equation\n\\begin{equation}\n\\label{eq:elastic_B}\nB^* = \\sum_{i=1}^{N}{{U'_i}^{-1}\\Big(C'(B^* - B)\\Big)}.\n\\end{equation}\n\n\\subsection{Buffer Constraint Violations}\n\\label{sec:violation}\nBefore we leave this section, we address an issue that arises in both formulations, namely how to deal with the fact that there may be more contents with unexpired timers than can be stored in the buffer.  This occurs in the formulation of (\\ref{eq:opt}) because the constraint is on the {\\em average} buffer occupancy and in (\\ref{eq:opt_soft}) because there is no constraint. Let us focus on the formulation in (\\ref{eq:opt}) first.  Our approach is to provide a buffer of size $B(1+\\epsilon )$ with $\\epsilon > 0$, where a portion $B$ is used to solve the optimization problem and the additional portion $\\epsilon B$ to handle buffer violations.  We will see that as the number of contents, $N$,  increases, we can get by growing $B$ in a sublinear manner, and allow $\\epsilon$ to shrink to zero, while ensuring that content will not be evicted from the cache before their timers expire with high probability.  Let  $X_i$ denote whether content $i$ is in the cache or not; $P(X_i =1) = h_i $.  Now Let $\\mathbb{E}\\bigl[\\sum_{i=1}^N X_i\\bigr] = \\sum_{i=1}^N h_i = B$. We write $B(N)$ as a function of $N$, and assume that $B(N) = \\omega (1)$. \n\\begin{theorem}\n\\label{thrm:violation}\nFor any $\\epsilon > 0$\n\\[\n \\mathbb{P}\\bigl(\\sum_{i=1}^N X_i \\ge B(N)(1+\\epsilon)\\bigr) \\le e^{-\\epsilon^2 B(N)\/3} .\n\\]\n\\end{theorem}\nThe proof follows from the application of a Chernoff bound.\n\nTheorem~\\ref{thrm:violation} states that we can size the buffer as $B(1+\\epsilon)$ while using a portion $B$ as the constraint in the optimization.  The remaining portion, $\\epsilon B$, is used to protect against buffer constraint violations. \nIt suffices for our purpose that ${\\epsilon^2 B(N) = \\omega (1)}$.  This allows us to select $B(N) = o(N)$ while at the same time selecting $\\epsilon = o(1)$. As an example, consider Zipf's law with $\\lambda_i = \\lambda\/i^s$, $\\lambda > 0$, $0 < s <1$,  $i=1,\\ldots, N$ under the assumption that $\\max{\\{t_i\\}} = t$ for some $t <\\infty$.  In this case, we can grow the buffer as $B(N) = O(N^{1-s})$ while \n$\\epsilon$  can shrink as $\\epsilon = 1\/N^{(1-s)\/3}$.  Analogous expressions can be derived for $s \\ge 1$.\n\nSimilar choices can be made for the soft constraint formulation.\n\n\n\n\n\n\n\n\\section{Related Work}\n\\subsection{Network Utility Maximization}\n\nUtility functions have been widely used in the modeling and control of computer networks, from stability analysis of queues to the study of fairness in network resource allocation; see~\\cite{srikant13, neely10} and references therein. Kelly~\\cite{kelly97} was the first to formulate the problem of rate allocation as one of achieving maximum\naggregate utility for users, and describe how network-wide optimal rate allocation can be achieved by having individual users control their transmission rates.\nThe work of Kelly~\\emph{et al.}~\\cite{kelly98} presents the first mathematical model and analysis of the behavior of congestion control algorithms for general topology networks.\nSince then, there has been extensive research in generalizing and applying Kelly's \\emph{Network Utility Maximization} framework to model and analyze various network protocols and architectures. This framework has been used to study problems such as network routing~\\cite{tassiulas92}, throughput maximization~\\cite{eryilmaz07}, dynamic power allocation~\\cite{neely03} and scheduling in energy harvesting networks~\\cite{huang13}, among many others. Ma and Towsley~\\cite{Ma15} have recently proposed using utility functions for the purpose of designing contracts that allow service providers to monetize caching.\n\n\n\\subsection{Time-To-Live Caches}\nTTL caches, in which content eviction occurs upon the expiration of a timer, have been employed\nsince the early days of the Internet with the Domain Name System (DNS) being an important application~\\cite{Jung03}. More recently, TTL caches have regained popularity, mostly due to admitting a general approach in the analysis of caches that can also be used to model replacement-based caching policies such as LRU. The connection between\nTTL caches and replacement-based (capacity-driven) policies was first established for the LRU policy by Che~\\emph{et al.}~\\cite{Che01} through the notion of cache \\emph{characteristic time}. The characteristic time was theoretically justified and extended to other caching policies such as FIFO and RANDOM~\\cite{Fricker12}. \nThis connection was further confirmed to hold for more general arrival models than Poisson processes~\\cite{Bianchi13}. Over the past few years, several exact and approximate analyses have been proposed for modeling single caches in isolation as well as cache networks using the TTL framework~\\mbox{\\cite{Nicaise14, Berger14}}.\n\nIn this paper, we use TTL timers as \\emph{tuning knobs} for individual (or classes of) files to control the utilities observed by the corresponding contents,\nand to implement \\emph{fair} usage of cache space among different (classes of) contents.\nWe develop our framework based on two types of TTL caches described in the next section.\n\n\n\n\\section{Reverse Engineering}\n\\label{sec:reverse}\nIn this section, we study the widely used replacement-based caching policies, FIFO and LRU, and show that their hit\/miss behaviors can be duplicated in our framework through an appropriate choice of utility functions.\n \nIt was shown in~\\cite{Nicaise14} that, with a proper choice of timer values, a TTL cache can generate the same statistical properties, \\emph{i.e.}~same hit\/miss probabilities, as FIFO and LRU caching policies. \nIn implementing these caches, non-reset and reset TTL caches are used for FIFO and LRU, respectively, with $t_i=T, i=1,\\ldots,N$ where $T$ denotes the \\emph{characteristic time}~\\cite{Che01} of these caches. For FIFO and LRU caches with Poisson arrivals the hit probabilities can be expressed as\n$h_i = 1 - 1\/(1+\\lambda_iT)$ and $h_i = 1 - e^{-\\lambda_i T}$, and $T$ is computed such that $\\sum_{i}{h_i} = B$.\nFor example for the LRU policy $T$ is the unique solution to the fixed-point equation\n\\[\\sum_{i=1}^{N}{\\left(1 - e^{-\\lambda_i T}\\right)} = B.\\]\n\n\nIn our framework, we see from~\\eqref{eq:hu} that the file hit probabilities depend on the Lagrange multiplier $\\alpha$ corresponding to the cache size constraint in~\\eqref{eq:opt}.\nThis suggests a connection between $T$ and $\\alpha$. Further note that the hit probabilities are increasing functions of $T$. On the other hand, since utility functions are concave and increasing, $h_i = {U'_i}^{-1}(\\alpha)$ is a decreasing\nfunction of $\\alpha$. Hence, we can denote $T$ as a decreasing function of $\\alpha$, \\emph{i.e.}~$T = f(\\alpha)$. \n\nDifferent choices of function $f(\\cdot)$ would result in different utility functions for FIFO and LRU policies. \nHowever, if we impose the functional dependence $U_i(h_i) = \\lambda_i U_0(h_i)$, then the equation $h_i = {U'_i}^{-1}(\\alpha)$ yields\n\\[h_i = {U'_0}^{-1}(\\alpha\/\\lambda_i).\\]\nFrom the expressions for the hit probabilities of the FIFO and LRU policies, we obtain $T = 1\/\\alpha$. In the remainder of the section, we use this to derive utility functions for the FIFO and LRU policies.\n\n\\subsection{FIFO}\nThe hit probability of file $i$ with request rate $\\lambda_i$ in a FIFO cache with characteristic time $T$ is\n\\[h_i = 1 - \\frac{1}{1 + \\lambda_i T}.\\]\nSubstituting this into~\\eqref{eq:hu} and letting $T = 1\/\\alpha$ yields\n\\[{U'_i}^{-1}(\\alpha) = 1 - \\frac{1}{1 + \\lambda_i \/ \\alpha}.\\]\nComputing the inverse of ${U'_i}^{-1}(\\cdot)$ yields\n\\[U'_i(h_i) = \\frac{\\lambda_i}{h_i} - \\lambda_i,\\]\nand integration of the two sides of the above equation yields the utility function for the FIFO cache \n\\[U_i(h_i) = \\lambda_i (\\log{h_i} - h_i).\\]\n\n\\subsection{LRU}\nTaking $h_i = 1 - e^{-\\lambda_i T}$ for the LRU policy and letting ${T = 1\/\\alpha}$ yields\n\\[{U'_i}^{-1}(\\alpha) = 1 - e^{-\\lambda_i\/\\alpha},\\]\nwhich yields\n\\[U'_i(h_i) = \\frac{-\\lambda_i}{\\log{(1-h_i)}}.\\]\nIntegration of the two sides of the above equation yields the utility function for the LRU caching policy\n\\[U_i(h_i) = \\lambda_i \\text{li}(1-h_i),\\]\nwhere $\\text{li}(\\cdot)$ is the logarithmic integral function\n\\[\\text{li}(x) = \\int_0^x{\\frac{\\mathrm{d} t}{\\ln{t}}}.\\]\n\nIt is easy to verify, using the approach explained in Section~\\ref{sec:opt}, that the utility functions computed\nabove indeed yield the correct expressions for the hit probabilities of the FIFO and LRU caches.\nWe believe these utility functions are unique if restricted to be multiplicative in\\footnote{We note that utility functions, defined in this context, are subject to affine transformations, \\emph{i.e.}~$aU+b$ yields the same hit probabilities as $U$ for any constant $a>0$ and $b$.} $\\lambda_i$.\n\n\\section{Simulations}\n\\label{sec:simulation}\nIn this section, we evaluate the efficiency of the online algorithms developed in this paper.\nDue to space restrictions, we limit our study to four caching policies: FIFO, LRU, proportionally fair, and max-min fair.\n\nPer our discussion in Section~\\ref{sec:reverse}, non-reset and reset TTL caches can be used with $t_i = T, i=1,\\ldots,N$ to implement caches with the same statistical properties as FIFO and LRU caches.\nHowever, previous approaches require precomputing the cache characteristic time $T$.\nBy using the online dual algorithm developed in Section~\\ref{sec:dual} we are able to implement these policies with no a priori information of $T$.\nWe do so by implementing non-reset and reset TTL caches, with the timer parameters for all files\nset as $t_i = 1\/\\alpha$, where $\\alpha$ denotes the dual variable and is updated according to~\\eqref{eq:dual_sol}.\n\nFor the proportionally fair policy, timer parameters are set to\n\\[t_i = \\frac{-1}{\\lambda_i}\\log{(1 - \\frac{\\lambda_i}{\\alpha})},\\]\nand for the max-min fair policy we set the timers as\n\\[t_i = \\frac{-1}{\\lambda_i}\\log{(1 - \\frac{1}{\\alpha})}.\\]\nWe implement the proportionally fair and max-min fair policies as reset TTL caches.\n\n\nIn the experiments to follow, we consider a cache with the expected number of files in the cache set to $B=1000$. Requests arrive for ${N = 10^4}$ files according to a Poisson process with aggregate rate one. File popularities follow a Zipf distribution with parameter ${s=0.8}$,~\\emph{i.e.}~${\\lambda_i = 1\/i^s}$. In computing the timer parameters we use estimated values for the file request rates as explained in Section~\\ref{sec:estimate}.\n\nFigure~\\ref{fig:dual} compares the hit probabilities achieved by our online dual algorithm with those computed numerically for the four policies explained above.\nIt is clear that the online algorithms yield the exact hit probabilities for the FIFO, LRU and max-min fair policies. For the proportionally fair policy however, the\nsimulated hit probabilities do not exactly match numerically computed values. This is due to an error in estimating $\\lambda_i, i=1,\\ldots, N$. Note that we use a simple estimator\nhere that is unbiased for $1\/\\lambda_i$ but biased for $\\lambda_i$. It is clear from the above equations that computing timer parameters for the max-min fair policy\nonly require estimates of $1\/\\lambda_i$ and hence the results are good. Proportionally fair policy on the other hand requires estimating $\\lambda_i$ as well,\nhence using a biased estimate of $\\lambda_i$ introduces some error.\n\nTo confirm the above reasoning, we also simulate the proportionally fair policy assuming perfect knowledge of the request rates.\nFigure~\\ref{fig:prop_exact} shows that in this case simulation results exactly match the numerical values.\n\n\\begin{figure}[h]\n\\centering\n \\begin{subfigure}[b]{0.50\\linewidth}\n  \t\\centering\\includegraphics[scale=0.20]{prop_dual_hit.eps}\n  \t\\caption{\\label{fig:prop_exact}}\n \\end{subfigure}%\n \\begin{subfigure}[b]{0.50\\linewidth}\n  \t\\centering\\includegraphics[scale=0.20]{prop_pd_hit_est.eps}\n  \t\\caption{\\label{fig:prop_pd}}\n \\end{subfigure}\n\\vspace{-0.25cm}\n \\caption{Proportionally fair policy implemented using the (a) dual algorithm with exact knowledge of $\\lambda_i$s, and (b) primal-dual algorithm with ${\\delta_m(t_i, \\alpha) = \\lambda_i}$ and ${\\delta_h(t_i, \\alpha) = \\alpha - \\lambda_i}$, with approximate $\\lambda_i$ values.}\n  \\label{fig:prop_fair}\n\\end{figure}\n\nWe can also use the primal-dual algorithm to implement the proportionally fair policy.\nHere, we implement this policy using the update rules in~\\eqref{eq:prop_pd}, and estimated values for the request rates.\nFigure~\\ref{fig:prop_pd} shows that, unlike the dual approach, the simulation results match the numerical values.\nThis example demonstrates how one algorithm may be more desirable than others in implementing a specific policy.\n\nThe algorithms explained in Section~\\ref{sec:online} are proven to be globally and asymptotically stable, and converge to the optimal solution.\nFigure~\\ref{fig:lru_dual_var} shows the convergence of the dual variable for the LRU policy.\nThe red line in this figure shows $1\/T=6.8\\times 10^{-4}$ where $T$ is the characteristic time of the LRU cache computed according to the discussion in Section~\\ref{sec:reverse}.\nAlso, Figure~\\ref{fig:lru_cache_size} shows how the number of contents in the cache is centered around the capacity $B$.\nThe probability density and complementary cumulative distribution function (CCDF) for the number of files in cache are shown in Figure~\\ref{fig:cs}.\nThe probability of violating the capacity $B$ by more than $10\\%$ is less than $2.5\\times 10^{-4}$. For larger systems, \\emph{i.e.}\\ for large $B$ and $N$, the probability of violating the \ntarget cache capacity becomes infinitesimally small; see the discussion in Section~\\ref{sec:violation}. This is what we also observe in our simulations.\nSimilar behavior in the convergence of the dual variable and cache size is observed in implementing the other policies as well.\n\n\\begin{figure}[h]\n\\centering\n \\begin{subfigure}[b]{0.5\\linewidth}\n  \t\\centering\\includegraphics[scale=0.21]{lru_dual_var.eps}\n  \t\\caption{\\label{fig:lru_dual_var}}\n \\end{subfigure}%\n \\begin{subfigure}[b]{0.5\\linewidth}\n  \t\\centering\\includegraphics[scale=0.21]{cs_conv.eps}\n  \t\\caption{\\label{fig:lru_cache_size}}\n \\end{subfigure}\n\\vspace{-0.25cm}\n \\caption{Convergence and stability of dual algorithm for the utility function representing LRU policy.}\n    \\label{fig:lru_dual}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n \\begin{subfigure}[b]{0.5\\linewidth}\n  \t\\includegraphics[scale=0.21]{cs_distr.eps}\n \\end{subfigure}%\n \\begin{subfigure}[b]{0.5\\linewidth}\n  \t\\includegraphics[scale=0.21]{cs_ccdf.eps}\n \\end{subfigure}\n\\vspace{-0.25cm}\n \\caption{Cache size distribution and CCDF from dual algorithm with the utility function representing LRU policy.}\n    \\label{fig:cs}\n\\end{figure}\n\n\n\\section{Final Remarks}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Appendix}\n\nThis appendix contains a description of the methodology used to solve for treatment strategies in chain graphs, the methodology used to train a Q-learner on the Karate Club network, and a brief description of community detection techniques that were used on the Karate Club network.\n\n\\subsection{Solving for treatment strategies in the chain graph via linear program}\n\\label{app:chain_lp}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.33\\columnwidth]{images\/chain_bases.png}\n    \\caption{Some risk profiles in a chain graph with 11 nodes for different preemptive treatment strategies. Treating node $i$ creates zero risk for individual $i$ and lowers the risk for neighbors as well.}\n    \\label{fig:chain_graph_bases}\n\\end{figure}\n\n\\subsubsection{Preemptive treatments}\nIn the preemptive setting, with only a single treatment, there are a limited number of possible interventions, corresponding to treating each of the members of the graph. \nEach of those interventions has a corresponding risk profile (Figure~\\ref{fig:chain_graph_bases}).\n\nTo find the strategy that minimizes average disease burden, it is sufficient to simply enumerate them and find the lowest.\n\nTo see if there is a distribution over strategies that gives a uniform risk profile, we try to find a convex combination of risk profiles that minimizes average risk and is subject to the constraints that all of the risks are equal. These are all linear constraints, so the problem is a linear program. We use scipy's {\\tt optimize.linprog} function as a solver.\n\n\\subsubsection{1-step precision treatments}\nWe also consider precision treatments after 1 step - i.e., waiting to see which patient is infected and responding with a treatment at that point.\n\nFor a chain graph with N members, we are now looking for N different reactive allocations, with each allocation being a convex combination of treatment risk profiles, similar to above, but with the initial patient already infected.\n\nThis problem has a similar form to above, but with N convex combination constraints instead of 1, and is also a linear program.\n\nWaiting for longer than one step before acting was not considered, though could be a reasonable strategy as well.\n\n\n\\subsection{Training a DQN network on the karate-club graph}\n\\label{app:training_dqn}\n\nA DQN agent was trained using dopamine's experiment runner.\nThe reward function at each step was the negative number of newly sick nodes.\n\nThe observation vector at each step was one-hot encoded health states. \n\nThe graph structure was not passed to the agent explicitly, but it was given the chance to learn by observing the simulations.\n\nHyperparameters of the agent were optimized using a black-box hyperparameter tuning system.\n\nThe range of settings explored were:\n\\begin{itemize}\n    \\item gamma: [0, 0.99]\n    \\item hidden layer size: [5, 500]\n    \\item learning rate: [$10^{-4}$, $10^0$]\n    \\item num iterations: [$10^2$, $10^5$]\n\\end{itemize}\nThe agent's stack size was set to 1; max steps per episode was 20, and the training steps per iteration was 500. All other arguments were left at default.\n\nThe best settings found by the hyperparameter tuner after 150 trials were:\n\\begin{itemize}\n    \\item gamma: 0.70\n    \\item hidden layer size: 106\n    \\item learning rate: 0.95\n    \\item num iterations: 2543\n\\end{itemize}\n\n\\subsection{Community detection in the karate-club graph}\n\\label{app:community_dection_karate}\nWe use networkx's implementation of the Girvan Newman algorithm for community detection and use the highest level in the hierarchy, creating two communities shown in Figure~\\ref{fig:karate_partition}. \n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\columnwidth]{karate_club_partition.png}\n    \\caption{Partitioning of the Karate club graph into 2 communities using the Girvan-Newman algorithm}\n    \\label{fig:karate_partition}\n\\end{figure}\n\\section{Background and Related Work}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images\/epidemic_progression.png}\n    \\caption{An illustration of the development of an epidemic on a barbell graph with a central individual.  Here, blue nodes are in the `Susceptible' state, red nodes are in the `Infected' state, and purple nodes are in the `Recovered' state.  `Recovered` is an abosrbing state. After t=4, there are not more transitions to take place.}\n    \\label{fig:barbell_evolution}\n\\end{figure}\n\nMathematical models for studying infectious disease spread have a long history~\\citep[e.g.,][]{longini1978optimization, hethcote1978immunization, patel2005finding, tennenbaum2008simple}, including explicit treatment of network structures~\\citep{newman2002spread, valente2012network, newman2018networks}, and evolving control policies~\\citep{sharomi2017optimal}. \n\n\\citet{keeling2012optimal} used analytical models to illustrate a tension between optimal and equitable distributions of vaccines in a simple case of two non-interacting (or barely interacting) populations. \\citet{yi2015fairness} study tradeoffs between fairness and effectiveness when considering different possible prioritization orderings of who receives vaccines in the case of limited resources. \\citet{salathe2010dynamics} discuss the importance of ``intercommunity-bridges'', individuals who have contact with multiple communities, in order to stem the spread of disease.\n\nEthical discussions of prioritizing individuals for treatment based on considerations like quality-adjusted life years, vulnerability, need, productivity, ability to treat others, and lottery have been discussed extensively~\\citep[e.g.,][]{emanuel2006should, persad2009principles, buccieri2013ethical, saunders2018equality}, but generally deal with the problem of a general order of prioritization rather than precision treatments and do not consider explicit social network structures.\n\nTo the best of our knowledge, the problem of optimally allocating vaccines within a social network in a step-by-step manner as the disease spreads has not been explicitly studied neither from an effectiveness nor fairness view. The turn-based precision version of the problem is closer to learning to play a strategy game like chess or go which has received a lot of attention in the reinforcement learning community~\\citep[e.g.,][]{atari, silver2017mastering}. \n\n\\subsection{Fairness in networks} \n\\label{sec:fairness_in_networks}\n\nMany recently proposed measurements of fairness focus either on group fairness~\\citep{hardt2016equality} or  individual-fairness~\\citep{dwork2012fairness}. Networks serve as an interesting setting because of they way that they can richly reproduce many societal structures.\n\n{\\bf Communities:} Social networks tend to organize in community structures~\\citep{girvan2002community, faust2005using}. Membership in these communities tends to be much more nuanced than strictly binary or categorical with individuals potentially belonging to many communities, possibly with differing degrees of identification~\\citep{palla2005uncovering} and with communities interacting at different scales or resolutions~\\citep{ronhovde2009multiresolution}. Using measures like overall burden to a community rather than probability of disease conditioned on being from a community, directly links fairness measures to group well-being. \n\n{\\bf Centrality:} Individuals in different positions in the graph may play different roles and be treated differently. Centrality of a node in a network refers to a measure of its influence on the network and can be characterized using a number of different measures~\\citep{lu2016vital}. Three important measures are {\\em degree centrality}: the number of immediate neighbors; {\\em betweenness-centrality}: the number of shortest paths that pass through this node; and {\\em eigenvector centrality}: uses the dominant eigenvector of the adjacency matrix.\n\nEven with the same set of actors, an individual's characterization in the network (both in terms of community and centrality) may be very different depending on how the edges in the network are defined. For example, in an workplace, the communities formed by who eats lunch together may be different from the communities of who attend meetings together. Thus centrality and community memberships of an individual should not be thought of as unique but depend crucially on the context being analyzed.\n\n\\section{Formalizing the precision contagion control problem}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\columnwidth]{images\/wait_and_see.png}\n    \\caption{Network structures to illustrate some of the complexity of the precision control task. {\\bf Case 1 - greedy suboptimal:} If one treatment is available per time-step, a greedy agent would prefer to treat node A whereas an agent that plans ahead would treat B then C. {\\bf Case 2 - wait and see:} If disease transfer occurs with with $\\tau = 0.5$ and only one treatment is available overall, an agent minimizing expected sick days should to wait to see if the disease transfers to each neighbor before deciding which side to treat.}\n    \\label{fig:greedy_suboptimal}\n\\end{figure}\nWe pose the precision contagion control problem in this section. In defining the problem, we build on work on discrete-time compartmental modeling of infectios disease \\cite{brauer2010discrete,hernandez2015discrete,liu2015effect,allen1994some}.\n\nA population of individuals, $V$, is connected by edges, $E$, forming a social network $N = (V, E)$. Individuals can be in one of three health states $\\{S, I , R\\}$ for susceptible, infected, recovered. The health of the network is $H \\in \\{S, I, R\\}^{|V|}$. Health evolves in discrete time steps. At time $t_0$ an initial set of nodes $V_0$ are infected. At every subsequent step step, disease spreads stochastically from infected to susceptible neighbors at every time step, with probability:\n\n\\begin{equation}\np_{S \\rightarrow I} = 1-(1-\\tau)^{\\#I(N)},\n\\end{equation}\nwhere $\\tau \\in [0, 1]$ is probability of transmission and $\\#I(N)$ is a count of infected neighbors.\n\nThe recovery process does not depend on neighborhood.\n\\begin{equation}\np_{I \\rightarrow R} = \\rho,\n\\end{equation}\n\nwhere $\\rho \\in [0, 1]$ is probability of recovery. \n\nAn agent allocates $N_t$ treatments at each step. Treatments effect a direct transition to the R state, and have no effect on individuals in states $\\{I, R\\}$ (i.e., treatments are {\\em preventative}). Figure~\\ref{fig:barbell_evolution} illustrates the disease being transmitted and individuals recovering in a simple network with no treatments bring allocated.\n\nThe precision contagion control problem consists of learning a treatment policy $\\pi(N, \\tau, \\rho, N_t):  H \\rightarrow (V \\cup \\{\\varnothing\\})^{N_t}$, i.e., finding the best next individuals to treat based on the current health state of the network. There is always an option to not use an available treatment which we denote by allocating that treatment a non-existent individual, $\\varnothing$.\n\nIn public health, uncertainties in the network structure as well as the logistical difficulties in implementing precision policies make it practically not worthwhile to optimize precision interventions in the way that we describe here. In this work, we consider fairness implications of this idealized precision setting that may become possible as health IT infrastructure makes it possible to target health interventions on the individual level. \n\nFinding optimal strategies for precision contagion control is not easy. The following properties of the problem, which we prove with example network structures, highlight some of the complexity of the task. \n\n{\\bf Greedy is not optimal:}  A greedy policy that aims to minimize expected total disease burden at every step in a greedy way can suffer from being sub-optimal (Figure~\\ref{fig:greedy_suboptimal}, Case 1).\n\n{\\bf Act now or wait?:} Strategies that wait to see how the disease progresses for some number of steps before allocating a treatment can be more effective than treating immediately (Figure~\\ref{fig:greedy_suboptimal}, Case 2). \n\nFrom a fairness perspective, one might be interested in finding treatment strategies that equalize outcomes in the population.\n\n{\\bf Uniform allocation may not produce uniform outcomes:} Some individuals are more or less at risk before intervention due to their position in the social network. Uniformly allocating treatment does not always create uniform outcomes. This will be explored in more detail in Section~\\ref{sec:chain}.\n\n\n\\section{Conclusion}\nThis is work in progress, but we believe that the precision contagion treatment problem, which we intend to contribute to the collection of simulated environments in the open-source {\\tt ML-fairness-gym}~\\citep{fairnessgym}, contains interesting questions of how to find fair allocation policies when individuals are highly intertwined. \nIt emphasizes the importance of {\\em fairness in  outcomes} rather than a narrower goal of fairness in e.g., classification accuracy (there is no classification or state estimation task in this problem -- just planning).\nIt also highlights how individuals with ``intermediate risk'' can end up being most at risk worst when policies are put in place to help the most at risk.\nDespite the idealized nature of the problem which assumes fully observed graph structures and highly precise treatment delivery, we hope that an increased understanding of the fairness implications of the underlying optimization problem with network effects can inform work in healthcare more broadly.\n\\section{Problem Statement}\nLet $K = \\{1, ..., N_p\\}$ be indices that represent individual members of a population with $N_p$ members and let $N_t$ be the number of treatments an agent can give out.  Let $A \\in P^{N_t}$ be an allocation of treatments to individuals in the population.  Finally, let $H = \\{S,I,R\\}^{N_p}$ be a health state, and let $\\tau$ be a schedule according to which treatments are given out -- perhaps only at $t=0$, or at timestep, or something else.\n\nThe agent's task is to find a probabilistic \\emph{allocation policy} $f: H \\rightarrow P(A)$ that maps health states to a distribution over allocations.  This function will be applied according to the schedule $\\tau$.  One way to go about this is to find the policy by solving an optimization problem that encodes a goal related to reducing infection.\n\nA few more definitions are required before we pose the optimization problems.  Let $x_i | t, f \\in \\{0, 1\\}$ be a random variable that indicates whether individual $i \\in K$ ever enters the \\texttt{Infected} state by time $t$ given allocation policy $f$.\n\nNow consider an agent whose only concern is to maximize utility -- that is, to minimize the number of individuals who get sick once the infectious disease has played out.  They can find this \\emph{maximum utility} policy by solving the following optimization problem:\n\\begin{equation}\n\\begin{aligned}\nf_{MU} = \\argmin_{f} \\quad & \\frac{1}{N_p} \\sum\\limits_{i=1}^{N_p} P(x_i = 1 | t=\\infty, f) \\\\\n\\end{aligned}\n\\end{equation}\n\nConsider now an agent who is concerned with a particular notion of fairness -- each member of the population should have the same probability of being infected.  An agent can find this \\emph{naively equitable} policy by adding a constraint to the problem above:\n\\begin{equation}\n\\begin{aligned}\nf_{NE} = \\argmin_{f} \\quad & \\frac{1}{N_p} \\sum\\limits_{i=1}^{N_p} P(x_i = 1 | t=\\infty, f) \\\\\n\\textrm{s.t.} \\quad & P(x_i = 1 | t=\\infty, f) = P(x_j = 1 | t=\\infty, f) \\: \\forall \\: i, j \\in K\n\\end{aligned}\n\\end{equation}\n\nWhy is the policy naive?  It could well be the case that there's a policy that \\emph{dominates} the naively equitable policy by offering an improvement for every individual over the naively equitable policy.  That is, an \\emph{undominated equitable} policy is a naively equitable policy for which:\n\\begin{equation}\n\\nexists f' \\; \\textrm{s.t.} \\; P(x_i = 1 | t=\\infty, f') < P(x_j = 1 | t=\\infty, f) \\: \\forall \\: i \\in K\n\\end{equation}\n\\section{Introduction}\n\nThe field of public health has long understood that different medical and health problems are not always easily decomposable into individual health measures, but can benefit from a networked analysis - viewing health as a population-level process.\nExamples of this type of analysis have long been applied to spread of infectious diseases, and more recently to other health factors like obesity~\\citep{christakis2007spread}, substance abuse~\\citep{rosenquist2010spread}, happiness~\\citep{fowler2008dynamic}, and even misinformation about health topics~\\citep{fernandez2015health}.\n\nIn network analyses, determining who benefits from an intervention is not as simple as observing who is directly affected by the intervention (e.g., who receives a vaccine), since neighbors and neighbors of neighbors are also affected indirectly.  With this in mind, we can still investigate how different health interventions differentially benefit individuals and communities within a larger population. These tradeoffs are important to study because they highlight how pursuing a coarse metric of success of an intervention averaged over an entire population (like e.g., minimizing total expected number of sick-days) can, under certain circumstances, leave parts of the population under-served.  \n\nWe study a stylized version of the public health task of epidemic control that highlights the networked setting. More specifically, we study the problem of optimally allocating vaccines within a social network in a step-by-step manner as the disease spreads.  We assume that the agents allocating vaccines can observe the disease states of individuals within the population and the underlying contact network structure.  We call this the {\\em precision disease contagion control} problem.\n\nThis work is relevant to fairness in machine learning for healthcare because it addresses class of optimization problems that appear repeatedly in the public health. We study this problem in simulation in order to illuminate core dynamics \\cite{epstein2008} -- with the rise of learned policies that attempt to optimize health outcomes on real data, the tensions at the root of the underlying optimization problem become increasingly relevant.\n\nWe propose this environment as a tool for researchers to explore the dynamics of disease under different contact and initial infection conditions, and to evaluate the relative performance of allocation approaches, learned or otherwise, that they may be interested in exploring.\n\nThis work is exploratory in nature.  We present several different contexts in which epidemics can spread and explore the efficacy and fairness of different treatment scenarios within those contexts.\n\n\\subsection{Contributions}\n\\begin{itemize}\n    \\item We pose a version of the precision disease contagion control problem and discuss heuristic and learned policies. \n    \\item We characterize some of the tradeoffs between efficiency and fairness in simple social network graphs.\n    \\item We open-source all of our code in an extensible manner to provide for reproduction and extensions of this work.\n\\end{itemize}\n\n\n\n\n\\section{Related Work}\nIn this section, we provide a brief overview of infectious disease modeling.\n\n\\subsection{Infectious Disease Epidemiology}\n\nInfectious disease epidemiology studies infectious disease at the population level.  Researchers in this field rely on simple models where human contact is associated with some probability of communicating disease.  Compartmental models, which are a central to the study, represent disease dynamics in populations by specifying a small number of mutually exclusive states that members of a population can occupy and some kind of mechanism by which disease propagates \\cite{hethcote2000mathematics}.\n\nIn this paper, we study the Susceptible-Exposed-Infected-Recovered (SEIR) subclass of compartmental models, where individuals exist in one of four mutually exclusive states: \\texttt{Susceptible}, \\texttt{Exposed}, \\texttt{Infected}, and \\texttt{Recovered}.  \\texttt{Susceptible} people are healthy but vulnerable to infection, \\texttt{Exposed} people are not yet sick but will become so after time passes, \\texttt{Infected} people are sick and spreading the disease, and \\texttt{Recovered} people were once sick but no longer are.  Note that not all stages need be present -- for example, the SI model only includes the \\texttt{Susceptible} and \\texttt{Infected} states.\n\nCompartmental models are not fully specified by the states that people can occupy.  Additional considerations include the representation of time, which is typically treated as continuous, but the discrete case is actively studied as well (such as in \\cite{brauer2010discrete,hernandez2015discrete,liu2015effect,allen1994some}).  \n\nIn addition, there is the question of how people come into contact.  Models typically assume `full mixing,' whereby all members of a population are equally likely to come into contact with all other members of a population.  However, there is a rich vein of work that assumes instead that contact proceeds along some explicitly-specified social network (for a detailed treatment, see \\cite{newman2002spread, newman2018networks}).\n\n\\subsection{Social welfare orderings and collective utility functions}\n\n\n\n\n\\section{Experiments}\n\n\nWe implement the simulated dynamics of this environment using the ML Fairness Gym~\\citep{fairnessgym}. Centrality is computed with the networkx package's centrality measures. The code for the environment, agents, and experiments will be published open-source and is intended to be contributed to the ML Fairness Gym collection of examples.\n\n\\subsection{Policies for contagion control}\nThere are many possible policies for contagion control. In these experiments we consider:\n\n{\\bf Random selection}: Treatments are uniformly at random allocated to individuals in the population. \n\n{\\bf Direct optimization:} Some networks are simple enough that policies can be computed by direct optimization of expected number of sick days.\n\n{\\bf Central-first heuristic}: Central-first heuristics treat more {\\em central} individuals before less-central individuals. Centrality is discussed in more detail in~\\ref{sec:fairness_in_networks}.\n\n{\\bf Deep Q-Network}: In cases where we cannot directly optimize over policies because the graph, we train a Deep Q-Network~\\citep{mnih2015human} with a single hidden layer using the Dopamine library~\\citep{castro18dopamine} for reinforcement learning. Details of the training can be found in Appendix~\\ref{app:training_dqn}.\n\n\n\\subsection{Barbell Graph}\n\nWe start by considering disease transmission in a barbell-like graph where a central, initially-infected node is connected to two cliques of different sizes through intermediaries.  Figure \\ref{fig:barbell_evolution} shows the natural progression of disease without intervention.\n\nThe barbell setting was previously considered by~\\citet{keeling2012optimal} as a clear example of conflict between minimizing disease in the total population, and treating communities equally.\n\nIf a single vaccine is available, the optimal policy is to treat on the side of the larger community to block the spread of disease. However, this policy is clearly unfair to the smaller community. If multiple vaccines are available and it is impossible to completely block the spread to one community, \\citet{keeling2012optimal} showed that the optimal allocation will change depending on the amount of vaccine available, but that for many values it would strongly favor one community or the other.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Chain graph}\n\\label{sec:chain}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\columnwidth]{images\/equalizing.png}\n    \\includegraphics[width=0.45\\columnwidth]{images\/chain.png}\n    \\caption{\n    (Left) A visualization of the ``Precision: Equalize'' strategy. Each row represents a possible initial infected patient and the agent's conditional distribution of who to treat in response.\n    (Right) Probability of infection of individuals in a chain graph using different treatment strategies. It is possible to ensure an equal probability of infection in the precision setting, but it does not have the lowest overall disease burden.}\n    \\label{fig:chain_treatments}\n\\end{figure}\n\nIn contrast to the barbell graph, the chain graph (where vertices are connected sequentially) has no clear group structure, but we can still consider likelihood of infection for individuals in different locations in the graph. For experiments we use a chain graph of size 11 with $\\tau=0.75, \\rho=1$ and a single treatment available.\n\nIf the initial infected patient is a uniformly chosen node in the network, the expected number of sick days with no intervention is higher for the central node and is lower at the two ends (Figure~\\ref{fig:chain_treatments} Right - No treatment). A random allocation strategy recapitulates this shape. \n\n{\\bf Preemptive treatment:} Treatment can be allocated preemptively, {\\em before} the first patient is infected. The preemptive strategy that optimizes average utility is to treat the middle node in the chain. Interestingly, with only one treatment available, there may not be a preemptive strategy (including randomized strategies) that equalizes the expected disease burden for all individuals. We verify this for our chain graph by posing the problem as a constrained linear program (Appendix~\\ref{app:chain_lp}) and determining that there is {\\em no feasible solution}. This is in contrast to equalized-opportunity classifiers in the classification setting~\\citep{hardt2016equality} which can always be constructed through randomization.\n\n{\\bf Precision treatment:} In the precision setting, we choose the individual to receive treatment {\\em conditional} on the observation of which individual becomes sick first. In this setting, the strategy that optimizes average utility is to treat the individual beside the sick patient, choosing the side with more people. Finding a precision policy that equalizes disease burden can be formulated as a constrained linear program (Appendix~\\ref{app:chain_lp}). Despite there being no preemptive treatment strategy that equalizes disease burden, a precision strategy {\\em does exist} (Figure~ \\ref{fig:chain_treatments} Left), though it has a higher average disease burden than the unconstrained policy. Figure \\ref{fig:chain_treatments} compares how the different treatment policies distribute disease burden over the population in the chain graph.  The random policy is dominated by the precision equalizing strategy, i.e., every individual in the network fares equally or better with the precision equalize strategy than the random strategy. \n\n\\subsection{Scale-free network}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images\/scale-free.png}\n    \\caption{Disease burden by centrality (using three different measures of centrality) for the members of the synthetic scale-free network~\\citep{holme2002growing}. Moderately central individuals fare the worst under centrality-based treatment.}\n    \\label{fig:general_graph}\n\\end{figure}\n\n\nMany real social networks have degree distributions that follow a power-law (or similar) distribution, meaning that a small number of nodes have very high connectivity.\n\n\n\n\nIn this setting, how will individuals with low centrality fare if high-centrality individuals are prioritized for treatment? They may be better off than they would have been under random allocation if centrality-based allocation methods control the epidemic more effectively; they may be worse off because they do not receive treatment themselves.\n\nWe simulate this scenario with a synthetic scale-free network~\\citep{holme2002growing} with 100 nodes and allowed an agent to allocate 1 treatment per turn. \nInfection spreads along contacts with $\\tau=0.25$ and individuals recover with $\\rho=0.01$.\n\nFigure \\ref{fig:general_graph} was generated by running 1000 simulations for 20 steps under three different centrality-based treatment allocation policies.  Note that moderately-central individuals have the highest epidemic size burden under centrality-based treatment because they do not enjoy the low risk that comes with being low centrality or the benefit offered to highly-central individuals by treatment.\n\n\n\\subsection{Karate club graph}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.95\\columnwidth]{images\/karate_club_centrality.png}\n    \n    \\caption{Disease burden by centrality (using three different measures of centrality) for each of the 34 members of the Karate club graph~\\citep{zachary1977information}. With natural progression (green circles), more central nodes tend to have a higher expected number of sick days. Central-first treating heuristics bring down everyone's expected disease burden, with the highest disease burden for individuals with intermediate centrality. The Deep Q-network does not succeed in finding strategies that beat heuristics.}\n    \\label{fig:karate_club_centrality}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.4\\columnwidth]{images\/karate_club_community.png}\n    \\caption{Violin plots showing disease burden in two communities in the karate club graph, for each of the treatment strategies. The distributions pictured are over 1000 simulation runs. Black lines indicate median runs of the simulation.}\n    \\label{fig:karate_club_community}\n\\end{figure}\n\nBeyond the synthetic graphs discussed above, we also consider the small scale Karate club graph \\citep{zachary1977information} which records the social interactions between 34 members of a karate club over three years, measuring the expected disease burden on individuals as a function of their centrality in the graph.\n\nSimulations were run 1000 times with parameters ($\\tau = \\rho = 0.5; N_{t} = 1$). Each simulation lasts 20 time-steps. Agents prioritized by centrality (experiments were repeated with the same set of random seeds for each of the 3 different measures of centrality) or using a deep Q-network~\\citep{atari}(Details in appendix~\\ref{app:training_dqn}). Ties between individuals with equal centrality were broken randomly. In each run, the initial infected patient was chosen uniformly. \n\nWith the natural disease progression, more central individuals tend to be {\\em more at risk}, but once central-first treatment policies are put in place, the worst-off individuals are those with intermediate centrality (everyone's risk reduces somewhat). \n\nLooking at expected total number of sick days in the entire population, the different central-first policies are all similar (No treatments: 58.01, Eigen-centrality: 25.56, Degree-centrality: 25.61; Betweenness-centrality: 25.17), but the individual burdens are different (Figure~\\ref{fig:karate_club_centrality}). The Deep Q-network (DQN) did not succeed in learning a policy that outperformed centrality heuristics, suggesting that a different network structure or training approach should be tried.\n\nThe Karate club graph naturally splits into two communities based on how members of the club sided in a particular conflict that led to the dissolution of the club (for more history on this graph, see~\\citet{zachary1977information}). Figure~\\ref{fig:karate_club_community} shows how the disease burden is spread over the two communities (details on the community detection procedure is in Appendix~\\ref{app:community_dection_karate}). On average, community 1 tends to fare a little bit better than community 2. This is even true with natural disease progression, suggesting that community 1 is somehow structured in a way that is less conducive to disease spread. \nThe DQN favors community 1 in an extreme way, but since it's performance is generally poor, its hard to draw strong conclusions from that.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n  Upon exhausting all of its available fuel, the iron core of a massive star ($\\gtrsim$8$M_\\odot$) collapses, often giving rise to an extremely powerful, bright explosion known as a core-collapse supernova (CCSN). Under very specific circumstances these CCSNe are also thought to be associated with another very powerful and luminous event known as a long-duration gamma-ray burst (LGRB). The circumstances that permit this extraordinary\npartnership are still not well understood, but it seems likely\nthat progenitors born in lower-metallicity environments (sub-solar) favour production of GRBs \\citep[see][]{2006AcA....56..333S,2006Natur.441..463F,2003A&A...406L..63F,2004MNRAS.352.1073T}.\n\n  The most popular theoretical model interprets the LGRB as\n  being caused by the core-collapse of a specific class of massive\n  star. This fundamental theory, coined as the `Collapsar' model, was\n  first conceived by \\citet{1993ApJ...405..273W} \\citep[also see][]{1999A&AS..138..499W}. The criteria for the creation of a LGRB\n  resulting from the collapse of a massive star are that a black-hole\n  must be formed upon core-collapse, either promptly or via fall-back\n  of ejected material, and that an accretion disk is allowed to form\n  around this black-hole which acts as a `central engine' causing the\n  release of collimated relativistic jets along the rotation axis of\n  the black-hole\n  \\citep{2006astro.ph.10276H,2008arXiv0801.4362Y,2008arXiv0803.3807T}. It\n  is these relativistic jets that form the LGRB and the associated\n  afterglow emission. These criteria are found to place \n  very specific\n  constraints on the type of progenitor star that can collapse to\n  produce a LGRB. The core of\n  the progenitor must be massive enough to produce a black-hole, \n  it must be rotating rapidly enough to form an accretion disk, and the star\n  needs to be stripped of\n  its hydrogen envelope, so that the resulting relativistic jets are\n  not inhibited from reaching the stellar surface. All of this points toward\n  rapidly rotating Wolf-Rayet stars as viable progenitors. \n   \\citep{2008arXiv0804.0014D,2006ApJ...637..914W,2006ARA&A..44..507W}\n\n Rapidly rotating O-stars which mix-up their core processed material\n to produce WR stars through chemically homogeneous evolution have been\nsuggested to \n  satisfy the requirements of the collapsar model \\citep{1987A&A...178..159M,2006ASPC..353...63Y,2006A&A...460..199Y,2006ApJ...637..914W}.\n This model prevents the depletion of angular\n momentum due to mass-loss and allows a massive star to become a WR star\n directly, avoiding the supergiant phase. \\citet{2006A&A...460..199Y} have estimated that for a\n rapidly rotating star to be the progenitor of an LGRB, prior to collapse\n it must have a helium core mass greater than 10M$_{\\odot}$, that is an\n initial mass greater than 25-30M$_{\\odot}$. \\citet{2006ApJ...637..914W}\n have modelled the WR mass-loss rates of \\citet{2005A&A...442..587V} and\n predict that the upper metallicity limit for forming LGRB may be as high\n as 0.3Z$_{\\odot}$. Apart from the single star progenitor model, a rapidly rotating massive star could lose its hydrogen envelope\n  and still retain a large fraction of its initial angular momentum if a close binary companion were to strip it of the envelope\n  rapidly enough for the progenitor to avoid spin down\n  \\citep{2007A&A...465L..29C,2007astro.ph..2652B,2007A&A...471L..29D}.\n\nWhatever the model for the progenitor of the LGRB, prior to collapse we require a very rapidly rotating, massive WR star. These constraints lead to the prediction that LGRBs favour a lower-metallicity environment and that CCSNe found in association with an LGRB will be classified as Type Ib or Type Ic, characterised by an absence of hydrogen in their spectra \n\\citep{2007ApJ...666.1024B,2003fthp.conf..171F,2003astro.ph..1006H}. To date many LGRBs have revealed `bumps' in their afterglow emission, thought to be evidence of associated SNe \\citep[see][]{1999Natur.401..453B, 2000ApJ...536..185G, 2003A&A...406L..33D,  2005ApJ...624..880L}. A few of these associated SNe have been adequately spectroscopically typed, all have been found to be broad-lined Type Ic SNe, helping to confirm the theory that LGRBs are the result of the collapse of hydrogen-stripped massive stars \\citep{1998Natur.395..670G,2003ApJ...591L..17S,2003Natur.423..847H,2004ApJ...609..952Z}. \\citet{2008AJ....135.1136M} have found that the SNe Type Ic-BL associated with an LGRB are found to inhabit a sample of host galaxies of significantly lower metallicity than the host galaxies of their Type Ic-BL counterparts with no associated observed LGRB. Interestingly, there has not been a SN Type Ic-BL without an associated observed LGRB that has been found to inhabit the same low-metallicity galaxy sample that the LGRB associated SNe inhabit.\n\n\\citet{2006Natur.441..463F} have found that LGRBs are preferentially\nfound in irregular galaxies which are \nsignificantly fainter, smaller and hence presumably of\nlower-metallicity than typical CCSNe hosts. \n\\citet{2005NewA...11..103S} make a similar point, stating that\nthe majority of LGRBs are found in extremely blue, sub-luminous\ngalaxies, similar to the Blue\nCompact Dwarf Galaxy population.\n\\citet{2006astro.ph..9208S} \nhas speculated that at primordial metallicities stars of up \nto 300M$_{\\odot}$ may form and if they have low mass-loss, \nthey may end their lives still retaining much of their original \nmass. \\citet{2002ApJ...567..532H} predict that those stars\nwithin the mass range 140M$_{\\odot}$-260M$_{\\odot}$ \nmay produce  pair-instability supernovae\n(PISNe), huge thermonuclear explosions with energies as high as\n$\\sim$10$^{53}$ ergs. Hence understanding SNe in low-metallicity \nenvironments locally could offer insights into high-z GRBs and\nearly Universe explosions. \nAn indication that these exotic PISN\nevents may not be beyond our present day reach is the recent SN 2006gy\nwhich has been reported to have a peak luminosity three times greater\nthan any other supernova ever recorded reaching a magnitude of -22\n\\citep{2006astro.ph.12617S}. Hypothesised as having had a Luminous\nBlue Variable as a progenitor with an initial mass of\n$\\sim$150M$_{\\odot}$ this has been suggested to be the first ever PISNe\ndetected \\citep[also\nsee][]{2007arXiv0708.1970L,2007ApJ...659L..13O,2007Natur.450..390W}.\nThe unusual SN 2006jc defines another class of peculiar low-metallicity\nevent, occurring spatially coincident with an outburst thought to be\nthat of an LBV just two years previously \\citep{2007Natur.447..829P}, \nwhich has also been postulated to be related to the PISN mechanism\n\\citep{2007Natur.450..390W}\n\nTo date there \nhave been many surveys specifically designed to search for SNe and\ndepending on the motivation of an individual survey generally one of\ntwo different survey strategies has been employed. The first of these\nstrategies is the \\textit{pointed survey strategy}; here we have a\nfixed catalogue of galaxies which are all individually\nobserved with a cadence of a few days and scanned for fresh SNe. These\nsurveys included the highly successful Lick Observatory Supernova\nSearch (LOSS) which uses the fully robotic 0.75m KAIT telescope to\nscan a catalogue of 7\\,500-14\\,000 nearby galaxies ($z \\lesssim 0.04$), \naiming to observe each\nindividual galaxy with a cadence of 3-5 days\n\\citep{2001ASPC..246..121F}. With\nthe advent of relatively inexpensive CCD technology, the ability of\namateur astronomers to detect nearby SNe using this pointed survey\nstrategy has become more than substantial and a large fraction of\nnearby SNe are now being discovered by this community\n\\citep{2006JAVSO..35...42G}.\nThe Southern inTermediate Redshift ESO\nSupernova Search (STRESS) produced a catalogue of $\\sim$43,000\n($0.05 < z < 0.6$) \ngalaxies within 21 fields observed with the ESO Wide-Field-Imager\nand searched for SNe in these hosts to determine \nrates and their dependency on host galaxy colour and \nredshift evolution \\citep{2008A&A...479...49B}.\n\nThe second strategy \nemployed by SN surveys is the \\textit{area survey strategy}; here an\narea of sky is surveyed repeatedly and image subtraction used to\nidentify transient events including SNe. The SN Legacy Survey uses\nthe CFHT MegaCam imager to image four one-square degree fields to\nsearch for SNe Type Ia with the motivation of improving the sampling\nof these SNe within the redshift range $0.2 < z < 1.0$\n\\citep{2005ASPC..339...60P, 2006A&A...447...31A}. The Equation of\nState SupErNova Cosmology Experiment (ESSENCE) uses the Mosaic Imager\non the Blanco 4m telescope to survey equatorial fields, that have\npreferably been observed with other wide-field surveys, to discover SN\nType Ia within the redshift range $0.15 < z < 0.74$\n\\citep{2007ApJ...666..674M}.  The Nearby SN Factory uses ultra-wide\nfield CCD mosaic images from the Near-Earth Asteriod Tracking (NEAT) and Palomar-Quest survey\nprograms to perform an area survey in a bid to find SN Type\nIa in the redshift range $0.03 < z < 0.08$\n\\citep{2002SPIE.4836...61A}. The SDSS-II Supernova Survey takes\nrepeated images of Stripe 82, a 300 square degree southern equatorial\nstrip of the sky, driven by the ambition of identifying and measuring\nthe lightcurves of SNe Type Ia in the intermediate redshift range\n$0.05 < z < 0.35$ \\citep{2008AJ....135..338F, 2008AJ....135..348S}. \nThe Texas Supernova Search search \n\\citep{2005AAS...20717102Q, 2007AAS...21110505Y}\nis somewhat unique in that it \nhas a small aperture telescope (0.45m) but very wide field (1.85 square\ndegrees) and is focussed on finding nearby SNe in wide area searches. \nThis survey discovered SN2006gy along with many other of the brightest\nknown SNe ever found and we shall discuss this further in \nSec. \\ref{sect:disc}.\n\n\nThe main driver of \nSN surveys that employ the pointed survey strategy is basically to\nfind as many SNe as possible regardless of type or characteristic.  To\nensure that these surveys are as efficient as possible at finding\nCCSNe, the galaxies catalogues used by the surveys generally consist\nonly of the most massive galaxies with the greatest star-formation\nrates. As a result these galaxy catalogues\ntend to be heavily biased towards galaxies with high metallicity. This\nover arching high-metallicity bias has more than likely placed us in\nthe situation where the vast majority of CCSNe that have occurred in\nlow-metallicity environments (such as low-luminosity, dwarf, irregular\ngalaxies) have remained undetected.  \\citet{2008ApJ...673..999P} have\nmatched the SAI supernova catalogue to the SDSS-DR4 catalogue of\nstar-forming galaxies with measured metallicities and it is clear that\nthe vast majority of the SNe considered have been detected in areas of\nrelatively high-metallicity; SN Type II occurring in host galaxies\nwith mean metallicity \\mbox{$12+{\\rm \\log(O\/H)}=8.94\\pm0.04$} and SN\nIb\/c at 9.06$\\pm$0.04.\n\nIt is inherently interesting then to search for SNe specifically in \nlow-metallicty environments, or at least without biasing the search\nto look at only high metallicity regimes. \nIn this paper we shall discuss several methods to \nsearch for low-metallicity CCSN\nevents. First we consider compiling a catalogue of nearby,\nlow-metallicity galaxies from pre-existing catalogues \n(taking SDSS DR5 as our primary survey source)\nand using either a single 2.0m telescope or a\nnetwork of 2.0m telescopes to perform a pointed survey of \nlow-metallicity galaxies in the hope of detecting a few CCSNe. \nSecondly we consider\nusing a future all-sky transient survey such as the Panoramic Survey\nTelescope and Rapid Response System (Pan-STARRS) to perform a\nvolume-limited survey for SNe and estimate how many may be found\nin low-metallicity galaxies. \nA third and final method we consider is to use a future all-sky\ntransient survey, limited only by\nthe limiting magnitude of the survey, to search for all CCSNe\nincluding those low-metallicity events.While this paper is \nprimarily aimed at determining numbers of low metallicity events\nthat could be found, the latter calculation gives an estimate of \nthe total number of CCSNe that are likely to be harvested from these\nupcoming surveys.\n\n\\section{Creating Galaxy Catalogues for the Various Survey Strategies}\\label{sec_a}\n\n  In order to produce catalogues of galaxies that we can use for the various survey strategies to be considered, we begin with the Sloan Digital Sky Survey (SDSS) \\citep{2007arXiv0707.3380A}. The SDSS Data\n  Release 5 (DR5) provides photometric\n  and spectroscopic data, within a wavelength range of 3500-9200$\\AA$, for\n  $\\sim$675\\,000 galaxies, over an area of 5\\,713 square degrees of the northern\n  hemisphere out to a redshift\n  of $z\\sim0.4$. In general we only want to detect relatively nearby CCSNe as we want to spectroscopically type and follow these SNe with relative ease, so to this end we introduce a distance limit of  $z=0.04$ to the SDSS spectroscopic catalogue. We have used\n  the SDSS DR5 website\\footnote{SDSS DR5 website:\n  \\it{http:\/\/www.sdss.org\/dr5}} to extract the out 44\\,041 galaxies within z\n  $=0.04$ along with data\n  including the petrosian magnitudes ({\\it\n  u, g, r, i} and  {\\it z}), the galactic extinctions in each filter determined following \\citet{1998ApJ...500..525S}, the spectroscopic\n  redshifts, the\n  {\\it r}-band fibre magnitudes and the line intensities of H$\\alpha$, H$\\beta$,\n  [OIII]$\\lambda5007$ and [NII]$\\lambda6584$ for each galaxy.\n  \n Of these 44\\,041 galaxies we extract out two separate samples of galaxies. The first of these samples is classified as the high signal-to-noise sample, containing galaxies with an SDSS defined line significance indicator `nSigma'  $>$ 7$\\sigma$ for all of the four spectral lines mentioned previously (as advised by DR5). The second sample is classified as the low signal-to-noise sample, containing galaxies that have not been included in the high signal-to-noise sample but exhibit `nSigma' $>$ 5$\\sigma$ in both H$\\alpha$ and H$\\beta$. The high signal-to-noise sample contains 20\\,632 galaxies and the low signal-to-noise sample contains 8\\,703 galaxies.\n  \n\\subsection{\\textbf{Removing AGN}}\\label{sec_aa}\n\n  Eventually we aim to determine a star-formation rate (SFR) for each\n  individual galaxy\n  in our sample in order to then determine their core-collapse SN rates\n  (CCSR).\n  As it is the young, massive, hot star population within each galaxy that\n  is the dominant source of hydrogen ionising radiation, it is possible to use\n  the H$\\alpha$ luminosity of these galaxies as a SFR indicator.\n  \n  A difficultly arises in that many galaxies within both the high signal-to-noise and the low signal-to-noise galaxy samples will host Active Galactic Nuclei\n  (AGN), which will also contribute to the galaxy's H$\\alpha$\n  luminosity. To remove these AGN contaminated galaxies\n  from the high signal-to-noise galaxy sample we use the following\n  diagnostic line provided by \\cite{2003MNRAS.346.1055K} to discriminate between\n  purely star-forming galaxies (SFGs) and galaxies that also host AGN:\n  \n\\begin{equation}\n  \\log\\left(\\frac{[\\rm{OIII}]\\lambda5007}{\\rm{H}\\beta}\\right)=\\frac{0.61}{\\log([\\rm{NII}]\\lambda6584\/\\rm{H}\\alpha)-0.05}+1.3\n  \\label{kauff_equ}\n\\end{equation}\n\n\\begin{figure}[t]\n  \\begin{center}\n  \\includegraphics[totalheight=0.2\\textheight]{BPT_dia.ps}\n  \\caption{\\textrm{[OIII]$\\lambda5007\/$H$\\beta$ vs\n  [NII]$\\lambda6584\/$H$\\alpha$ plot of 20\\,632 galaxies in our SDSS DR5 good signal-to-noise galaxy sample.\n  The diagnostic line from \\cite{2003MNRAS.346.1055K} discriminates between the purely star-forming galaxies in\n  our sample (those below the line) and those hosting AGN (above the line).}\\label{BPT_dia}}\n  \\end{center}\n\\end{figure}\n\n  \\noindent Fig. \\ref{BPT_dia} shows the SFGs found below this line and AGN\n  host galaxies above the line. We now have 18\\,350 high signal-to-noise SFGs. Concerning the low signal-to-noise galaxy sample , we do not have accurate enough spectral information to apply this diagnostic line to remove any unwanted AGN host galaxies. However, following the example of\n  \\citet{2004MNRAS.351.1151B} it is still possible to remove the AGN hosts from\n  the low signal-to-noise sample by requiring that [NII]$\\lambda6584$\/H$\\alpha > 0.6$ and that\n  nSigma $>$ 7$\\sigma$ in both lines. We now also have 6\\,000 low signal-to-noise SFGs. An overview of all galaxy sub-samples can be found\n  in Table \\ref{subsample_table}. The 17\\,409 unclassified galaxies are\n  predominantly early-type galaxies that show little signs of\n  recent star-formation, which is the reason that H$\\alpha$ has not been detected to\n  the significance levels that we have required. The lack of recent star-formation within these galaxies implies that they will\n  also be void of any future CCSN events and hence these galaxies are not of\n  interest to our survey.\n  \n  Having removed all of the AGN contaminated galaxies we now have two catalogues of SFGs which we shall refer to from now on as the high signal-to-noise SFG (HSFG) catalogue and the low signal-to-noise SFG (LSFG) catalogue.\n\n\\begin{table*}\n  \\begin{center}\n  \\caption{\\textrm{Hierarchy of galaxies extracted from the original SDSS\n    DR5 spectroscopic galaxy sample within z$=$0.04}\\label{subsample_table}}\n    \\begin{tabular}{c | c | c | c | c | c | c}\n      \\hline\\hline\n      \\multicolumn{7}{c}{\\bf All SDSS DR5 Galaxies [44\\,041]}\\\\\n      {\\bf Sample} & \\multicolumn{2}{|c|}{High signal-to-noise galaxies [20\\,632]} & \\multicolumn{2}{|c|}{Low signal-to-noise galaxies [8\\,703]}& \\multicolumn{2}{c}{Unclassified galaxies [17\\,409]}\\\\\n      {\\bf Sub-sample} & {SFGs [18\\,350]} & {AGN [2\\,282]} & {SFGs [6\\,000]} & {AGN [2\\,703]} & \\multicolumn{2}{c}{}\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n\\end{table*}\n\n  \n\\subsection{Measuring Oxygen Abundances}\\label{sec_b}\n\n\n  In order to select out low-metallicity galaxies from both the HSFG\n  and LSFG catalogues,\n  we define oxygen abundances using the empirical calibrations of\n  \\citet{2004MNRAS.348L..59P}. Within the range $8.12 \\lesssim 12+\\log(\\rm{O\/H}) < 9.05$ the\n  following empirical calibration is used:\n\n\\begin{center}  \n\\begin{equation}\n  12+\\log(\\rm{O\/H})=8.73-0.32\\log\\left(\\frac{[\\rm{OIII}]\\lambda5007\/\\rm{H}\\beta}{[\\rm{NII}]\\lambda6584\/\\rm{H}\\alpha}\\right),\n\\end{equation}\n\\end{center}\n\n  \\noindent and below $12+\\log(\\rm{O\/H})\\simeq 8.12$ the following calibration is used:\n\n\\begin{center}  \n  \\begin{equation}\n    12+\\log(\\rm{O\/H})=8.9+0.59\\log([\\rm{NII}]\\lambda6584\/\\rm{H}\\alpha).\n  \\end{equation}\n\\end{center}\n\n  The wavelengths of the emission lines used for the flux ratios in both of these calibrations\n  are separated by only a small amount and therefore their ratios are free of any extinction\n  effects. Of the sample of 18\\,350 HSFGs, \\citet{2006A&A...448..955I} have directly measured\n  metallicities for 209. They determined the\n  oxygen abundances by measuring the\n  [OIII]$\\lambda$4363\/[OIII]$\\lambda$5007 line ratio to calculate an\n  electron temperature $T_{e}$, and then derived the\n  abundances directly from the strengths of the [OII]$\\lambda$3727 (or\n  [OII]$\\lambda$7320, 7331 when [OII]$\\lambda$3727 was not available) and\n  [OIII]$\\lambda\\lambda$4959, 5007 emission lines. Comparing the\n  \\citeauthor{2006A&A...448..955I} $T_{e}$ measured abundances with our  \\citeauthor{2004MNRAS.348L..59P}\n  empirically calibrated abundances we find good agreement, apart\n  from four outlying galaxies (see Fig. \\ref{izo_fig}). These galaxies are\n  SDSS J124813.65-031958.2, SDSS J091731.22+415936.8, SDSS\n  J123139.98+035631.4 and SDSS J130240.78+010426.8.\n\n  \n  When viewed, these four\n  outliers seem to be dwarf galaxies that are in the same\n  line of sight as and possibly gravitationally bound\n  to much larger and presumably more metal rich galaxies.\n  Assuming that the contamination from these background galaxies has not been\n  adequately removed from the spectra of the dwarf galaxies by the SDSS reduction pipeline would explain\n  the discrepancy between the oxygen abundances measured by \\citeauthor{2006A&A...448..955I} and\n  our measurements. Viewing a random selection of the remaining galaxies from\n  the \\citeauthor{2006A&A...448..955I} sample reveal the blue compact dwarf galaxies expected from\n  their low-metallicity galaxy sample.\n  The fact that we may be over-estimating the oxygen abundances of a very\n  small fraction of galaxies should not concern us too much as we are trying to produce a\n  low-metallicity galaxy sample and not a high-metallicity sample that would then\n  possibly be contaminated by a few misplaced lower-metallicity galaxies. Choosing then\n  to ignore these four outlying galaxies we\n  find that our oxygen abundance measurements for the remaining\n  205 galaxies fall with an RMS scatter of 0.14 dex from the directly\n  measured abundances of the \\citeauthor{2006A&A...448..955I}\n  \n Recently \\cite{2008ApJ...673..999P} have taken a sample of 125 958 SFGs from SDSS DR4, with oxygen abundances derived in the same fashion as  \\cite{2004ApJ...613..898T} used for SFGs in DR2. The \\cite{2004ApJ...613..898T} method for deriving oxygen abundance estimates an individual galaxy's metallicty via a likelihood analysis which simultaneously fits multiple optical nebular emission lines to those predicted by the hybrid stellar-population plus photoionisation models of \\cite{2001MNRAS.323..887C}. A likelihood distribution of the metallicity is determined for each galaxy the median is taken as the best estimate of the galaxy metallicity. The \\cite{2004ApJ...613..898T} metallicities are essentially on the \\cite{2002ApJS..142...35K} abundance scale. Matching our catalogues of HSFGs and LSFGs against the sample of 125 958 SFGs of \\cite{2008ApJ...673..999P}, we find a common sample of 18 014 SFGs. The oxygen abundances that we measure with the \\citeauthor{2004MNRAS.348L..59P} method are typically $\\sim$0.2 dex below that of \\citeauthor{2004ApJ...613..898T}, in agreement with the findings of \\cite{2008AJ....135.1136M}. The cause of this discrepancy is debated but may either be due to certain parameters that produce temperature variations not being taken into consideration when deriving $T_{e}$ at higher-metallicity, which would lead to an under-estimation of the oxygen abundance measured on the \\citeauthor{2004MNRAS.348L..59P} scale which is calibrated with $T_{e}$ measured abundances \\citep{2005A&A...434..507S, 2006astro.ph..8410B}, or to an unknown problem with the photoionisation models used by  \\citeauthor{2004ApJ...613..898T} \\citep{2003ApJ...591..801K}. \n  \n\\subsection{SN Rate Indicator}\\label{sec_c}\n\n  Having measured the metallicity for each of the galaxies in our\n catalogues, we now wish to determine CCSRs. To do this we must first\n  determine SFRs for the galaxies and then determine the fraction of those stars\n  formed that will eventually end their lives as CCSNe. The\n  best indicator that we have for the SFR for each galaxy is its H$\\alpha$\n  luminosity. As alluded to previously, it is going to be the young, massive, hot stars\n  in purely star-forming galaxies that are the dominant source of\n  hydrogen ionising radiation, causing the galaxy's\n  H$\\alpha$ luminosity to be proportional to its recent SFR.\n  \n  \\citet{1998ARA&A..36..189K} has determined the following calibration\n  between a galaxy's SFR and its H$\\alpha$ luminosity:\n  \n\\begin{center}\n\\begin{equation}\n  {\\rm SFR_{H\\alpha}}({\\rm M}_{\\odot}{\\rm \\,\\,yr^{-1}})=\\frac{L_{{\\rm H}\\alpha}}{1.27\\times10^{34}({\\rm W})}\n  \\label{kennicutt_equ}\n\\end{equation}\n\\end{center}\n\n  \\noindent where the luminosity is measured in Watts.\n\n  Derived from their model fits, \\citet{2004MNRAS.351.1151B} also provide likelihood distributions for\n  the conversion factor between the H$\\alpha$ luminosity and the SFR for galaxies of\n  various mass ranges. They confirm that the Kennicutt calibration is a\n  very good {\\it typical} calibration, comparing well with\n  the median value for their sample. When considering the complete HSFG and LSFG catalogues it is acceptable to assume a median mass range for the galaxies and we employ\n  the Kennicutt calibration. However, when considering galaxies with\n  relatively poor metallicity we choose to use the most\n  probable conversion factor for the \\citeauthor{2004MNRAS.351.1151B} distribution with the lowest mass range (${\\rm \\log}M_{*}<8$) as this\n  distribution closest resembles the low-metallicity galaxies in our catalogues i.e. low-mass, blue,\n  dwarf, irregular galaxies.\n\n\\begin{center}  \n\\begin{equation}\n  {\\rm SFR_{H\\alpha}}({\\rm M}_{\\odot}{\\rm \\,\\,yr^{-1}})=\\frac{L_{{\\rm H}\\alpha}}{2.01\\times10^{34}({\\rm W})}\n\\end{equation}\n\\end{center}\n\n\\begin{figure*}[b]\n\\begin{center}  \n\\begin{equation}\n  {\\rm SFR_{H\\alpha}}({\\rm M}_{\\odot}{\\rm \\,\\,yr^{-1}})=10^{-0.4(r_{\\rm Petro} - r_{\\rm\n  fibre})}\\left[\\frac{S_{\\rm H\\alpha}\/S_{\\rm H\\beta}}{2.86}\\right]^{2.114}\\frac{L_{{\\rm H}\\alpha}}{1.27\\times10^{34}({\\rm W})}\n   \\label{AC_equ}\n\\end{equation}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}[b]\n\\begin{center}  \n\\begin{equation}\n  {\\rm SFR_{H\\alpha}}({\\rm M}_{\\odot}{\\rm \\,\\,yr^{-1}})=10^{-0.4(r_{\\rm Petro} - r_{\\rm\n  fibre})}\\left[\\frac{S_{\\rm H\\alpha}\/S_{\\rm H\\beta}}{2.86}\\right]^{2.114}\\frac{L_{{\\rm H}\\alpha}}{2.01\\times10^{34}({\\rm\n  W})}\n   \\label{AC_low_equ}\n\\end{equation}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[totalheight=0.2\\textheight]{izo_fig.ps}\n  \\caption{\\textrm{Comparing our \\citet{2004MNRAS.348L..59P} empirically calibrated oxygen abundances with those\n  directly measured by \\citet{2006A&A...448..955I} via measuring\n  the electron temperature reveals that our\n  measurements are reliable with an RMS scatter of 0.14 dex, the solid line depicting a one-to-one correspondence. Note the four outlying galaxies - see text for\n  details.}\\label{izo_fig}}\n  \\end{center}\n\\end{figure}\n\n  SDSS provides the H$\\alpha$ equivalent line width and also the continuum flux\n  at the wavelength of H$\\alpha$, the equivalent width times the continuum flux\n  giving the H$\\alpha$ flux which is then used to determine the H$\\alpha$\n  luminosity. A problem with the SDSS data is that the measured H$\\alpha$\n  flux for a galaxy is only the flux which falls within the 3''\n  fibre aperture of the SDSS multi-fibre spectrograph. Typically this is only a\n  fraction of the total galaxy flux, as the SDSS spectrograph fibre locks onto the\n  centre of the galaxy and any flux that falls outside of the 3'' fibre\n  aperture is lost. \\citet{2003ApJ...599..971H} have developed a very simple aperture-correction that can\n  be applied to the measured galaxy H$\\alpha$ luminosity to give an estimate of the total\n  galaxy  H$\\alpha$ luminosity. The aperture-correction takes account of the ratio between\n  the petrosian photometric $r$-band galaxy magnitude and the\n  synthetic $r$-band `fibre magnitude' determined from the galaxy spectrum. \n  \\citeauthor{2003ApJ...599..971H} also provide an extinction correction to be\n  used when determining the galaxy H$\\alpha$ SFR. The correction makes use of the Balmer decrement\n  and assumes the standard Galactic extinction law of\n  \\citet{1989ApJ...345..245C}. This gives\n  a final aperture and extinction corrected H$\\alpha$ luminosity SFR indicator\n  for our entire galaxy sample as given in Equation \\ref{AC_equ}, and for the low-metallicity galaxy sample as given in Equation \\ref{AC_low_equ}, where $S_{\\rm H\\alpha}$ and $S_{\\rm H\\beta}$ are the line \nfluxes corrected for stellar absorption according to \\citet{2003ApJ...599..971H}.\n   Having now determined a SFR for each of the galaxies in our\n  catalogues, we can compare these rates with their oxygen abundances\n  (Fig. \\ref{oxy_sfr}). It is\n  clear that in general the higher the galaxy SFR the higher the typical oxygen abundance. It can be reasoned that if the SFR of a galaxy is high, or has been high at any point in its history, there is an increased population of young, hot, massive stars which in turn leads to an increased rate of\n  CCSNe and therefore a greater rate of enrichment of the\n  ISM. The observed high-metallicity cutoff of the SDSS galaxies is probably due to a saturation suffered by [OIII] $\\lambda$4363 T$_{e}$ calibrated metallicities \\citep[see][]{2008arXiv0801.1849K}\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[totalheight=0.2\\textheight]{SFR_oxy.ps}\n  \\caption{\\textrm{H{$\\alpha$} determined star-formation rates using the low-mass range calibration of \\citet{2004MNRAS.351.1151B} of\n  18\\,350 galaxies\n  in our HSFG catalogue compared with their measured oxygen abundances. In general,\n  the greater the galaxy metallicity, the greater its star-formation rate - as\n  expected.}\n\\label{oxy_sfr}}\n  \\end{center}\n\\end{figure}\n\n  Given a SFR, it is relatively simple to convert to a CCSR by determining\n  the fraction of a stellar population that will eventually collapse to create\n  CCSNe. Following the method used by e.g. \\citet{2001MNRAS.324..325M}, we use an initial mass function (IMF) for a stellar\n  population to calculate the fraction of this population within the mass\n  range 8M$_{\\odot}<$ M $<$ 50M$_{\\odot}$, the same mass range for stars predicted to end\n  their lives as CCSNe. From this logic the CCSR is determined as:\n\n\\begin{center}  \n\\begin{equation} \n  {\\rm CCSR}=\\frac{\\int_{\\rm 8M_{\\odot}}^{\\rm 50M_{\\odot}}\\phi(m)dm}{\\int_{\\rm\n  0.1M_{\\odot}}^{\\rm 125M_{\\odot}}m\\phi(m)dm}\\times{\\rm SFR}\n  \\label{mattila_equ}\n\\end{equation}\n\\end{center}\n\n  where \\mbox{$\\phi(m)$} is the Salpeter IMF \\citep{1955ApJ...121..161S} with upper and\n  lower mass cut-offs of \\mbox{$0.1{\\rm M}_{\\odot}$} and \\mbox{$125{\\rm M}_{\\odot}$}. This conversion is\n  calculated to be:\n\n\\begin{center}  \n\\begin{equation} \n  {\\rm CCSR (SNe \\,\\,yr^{-1})}=0.007 \\times {\\rm SFR (M_{\\odot}\\,\\,yr^{-1})}\n  \\label{mattila_equ}\n\\end{equation}\n\\end{center}\n\n\\subsection{Additional nearby bright galaxies}\n\n  The target selection criteria for the SDSS DR5 spectroscopic sample\n  includes a\n  bright magnitude limit of \\mbox{$r\\sim$14.5} in order to avoid saturation and excessive\n  cross-talk in the spectrographs. As a result of this restriction many of\n  the nearby luminous galaxies have been omitted from the DR5 spectroscopic\n  sample. To account for these missing bright galaxies and \nconstruct a complete  galaxy catalogue we initially\n  select out the galaxies from the HyperLeda galaxy catalogue that match with\n  galaxies from the SDSS\n  DR5 {\\it photometric} survey with magnitudes $r<14.5$, assuming that the HyperLeda\n  catalogue contains all of the nearby luminous galaxies. From this catalogue\n  of matched galaxies we further select out those galaxies that are not found in the\n  SDSS DR5 {\\it spectroscopic} survey. We discover a total of 1\\,887 nearby\n  luminous galaxies included in the SDSS DR5 photometric survey that have\n  been omited from the spectroscopic survey.\n  \n  Of these 1\\,887 galaxies we wish to know the fraction that are star-forming\n  galaxies. HyperLeda provides a galaxy morphological classification for a\n  large number of the galaxies in the catalogue and we are able to remove all\n  galaxies with an early-type classification, as these galaxies\n  will generally have very low SFRs. But note that some early-type galaxies\n  have HII regions and some evidence of low-level star-formation\n  \\citep[e.g.][]{2005MNRAS.357.1337M}. The mean\n  $g-i$ colour of these removed early-type galaxies is 1.216, with a standard\n  deviation \\mbox{$\\sigma = 0.153$}. In an attempt to remove the remaining\n  non-starforming galaxies from this bright galaxy sample that do not have a morphological classification, we remove all\n  galaxies redward of one $\\sigma$ from the mean $g-i$ colour, i.e.\n  $g-i>1.063$. This results in 1\\,216 remaining star-forming galaxies. There is a possibility that this cut may also exclude a few starburst galaxies which have been heavily reddened due to the effects of extinction, but we are mainly concerned about low-metallicity galaxies, which are not greatly affected by extinction.\n  \n  As we have no spectral information for these galaxies from SDSS DR5, it is\n  necessary to use an alternative indicator to estimate SFR other than the\n  H$\\alpha$ luminosity. The $U$-band luminosity can be used as a suitable\n  indicator as developed by \\cite{2006ApJ...642..775M}:\n  \n\\begin{center}  \n\\begin{equation}\n  {\\rm SFR_{\\it U_{obs}}}({\\rm M}_{\\odot}{\\rm yr^{-1}})=(1.4\\pm1.1)\\times10^{-43}\\frac{L_{U_{obs}}}{\\rm ergs\\ s^{-1}},\n\\end{equation}\n\\end{center}\n\n  \\noindent where the SDSS $u$-band is transformed to \\mbox{$U_{\\rm vega}$} using the following\n  transformation from \\citet{2007AJ....133..734B}:\n  \n\\begin{center}  \n\\begin{equation}\n  U_{\\rm vega}=u-0.0140(u-g)+0.0556,\n\\end{equation}\n\\end{center}\n \n \\noindent and the distance moduli from HyperLeda are used to determine $L_{U_{obs}}$. The\n CCSR is then determined using Equation \\ref{mattila_equ}. Note\n that as \\citeauthor{2006ApJ...642..775M} have empirically calibrated this\n $U$-band SFR indicator from\n extinction-corrected H$\\alpha$ galaxy luminosities, there is no need to\n further correct this indicator for dust reddening. The 1\\,216 \n bright, nearby galaxies have a resulting $U$-band indicated CCSR of \\mbox{12.70\n CCSNe yr$^{-1}$}.\n\n\n  We now have three galaxy catalogues; a HSFG catalogue and a LSFG\n  catalogue, with measured metallcities and CCSRs, and a nearby \n  galaxy catalogue containing bright galaxies not found in the SDSS\n  spectroscopic galaxy catalogue. \n  We have estimates of SFRs and CCSRs for all these galaxies.\n  Taking our\n  catalogues of 18\\,350 HSFGs and 6\\,000 LSFGs and introducing various\n  upper limits on the individual galaxy metallicities and lower limits\n  on the CCSRs we are able to extract out separate galaxy\n  sub-samples. Table \\ref{SFG_rate_table} displays these sub-samples,\n  differing both in the number of galaxies they contain and their\n  estimated CCSR determined from the galaxy SFRs\n    derived from aperture and extinction-corrected H$\\alpha$ luminosities for the HSFG and LSFG catalogues, and from $U$-band luminosities for the \nnearby bright galaxy catalogue. For the samples of\n    galaxies with no metallicity constraint the SFRs have been derived using the\n    `typical' calibration of \\citet{1998ARA&A..36..189K} whereas the SFRs of the\n    sub-samples of galaxies with constraints on metallicity have been\n    derived using the lowest mass range calibration of \\citet{2004MNRAS.351.1151B}. These full catalogues or sub-samples of these catalogue can now be used to determine the feasibility of searching for low-metallicity CCSN events using various survey strategies. Figure \\ref{cum_dis} shows the cumulative\n  distribution of the combined CCSN rate from our HSFG and LSFG\n  catalogues, measured against oxygen abundance. It is clear that the\n  CCSR increases steeply with oxygen abundance of the galaxy sample.\n\n\n\\section{\\textbf{Strategy 1 : A Pointed survey of catalogued low-Metallicity galaxies}}\\label{sec_d}\n\n  \n  When determining the feasibility of using a pointed low-metallicity survey to search for CCSNe in a catalogue of low-metallicity galaxies we consider using the fully robotic 2.0m\n  Liverpool Telescope situated at the Observatorio del Roque de los Muchachos, La Palma. We shall also consider the using of a network of similar sized telescopes.\n  \n  There are three characteristics that we require of the galaxy catalogue that we shall use with this survey strategy; firstly, it must contain only a few hundred galaxies as too many galaxies in the catalogue would hinder our ability to observe each individual\n  galaxy frequently enough to ensure that we detect any SN that it may\n  host. Secondly, we require that the galaxies are of\n  sufficiently low metallicity in order to determine how the CCSNe that\n  they host differ from those hosted by their higher metallicity counterparts.\n  Finally, the galaxies must exhibit a suitably high enough CCSR to increase\n  the probability of detecting these CCSNe. The latter two requirements somewhat\n  contradict each other because metallicity\n  tends to scale with SFR in SFGs and therefore requiring a\n  low-metallicity\n  galaxy sample implies that we require galaxies with lower\n  SFRs, and hence {\\it lower} CCSRs. It is\n  therefore essential that we produce a galaxy catalogue that can act as\n  a compromise between these two conflicting requirements.\n    \n\\begin{table*}\n\\caption{\\textrm{The table shows the expected CCSN rates within the SDSS DR5\n    spectroscopic survey area (14\\% of the entire sky) out to a distance of \\mbox{$z\\sim0.04$}. }}\\label{SFG_rate_table}\n  \\begin{center}\n    \\begin{tabular}{c c c c c c c}\n      \\multicolumn{7}{c}{\\bf HSFG Catalogue}\\\\\n      \\hline\\hline\n      & \\multicolumn{6}{c}{\\bf INDIVIDUAL GALAXY SN RATE LIMITS}\\\\\n      & \\multicolumn{2}{c}{\\bf $>$ 0.0 SNe yr$^{-1}$} & \\multicolumn{2}{c}{\\bf $>$ 0.001 SNe yr$^{-1}$} & \\multicolumn{2}{c}{\\bf $>$ 0.01 SNe yr$^{-1}$}\\\\\n      \\hline\n      {\\bf 12+log(O\/H)} & {\\bf Galaxies} & {\\bf SNe yr$^{-1}$} & {\\bf Galaxies} & {\\bf SNe yr$^{-1}$} & {\\bf Galaxies} & {\\bf SNe yr$^{-1}$}\\\\\n      \\hline\n      {\\bf No Limit} & 18\\,350 & 115.6 & 13\\,974 & 113.4 & 2\\,557 & 73.6\\\\\n      {\\bf $<$ 8.4} & 8\\,019 & 13.2 & 3\\,650 & 11.2 & 120 & 2.3\\\\\n      {\\bf $<$ 8.3} & 4\\,290 & 6.9 & 1\\,830 & 5.9 & 73 & 1.3\\\\\n      {\\bf $<$ 8.2} & 1\\,713 & 3.1 & {\\bf 727} & {\\bf 2.8} & 42 & 0.8\\\\\n      {\\bf $<$ 8.1} & 537 & 1.0 & 209 & 0.9 & 16 & 0.3\\\\\n      \\hline\n      \\multicolumn{7}{c}{}\\\\\n      \\multicolumn{7}{c}{\\bf LSFG Catalogue}\\\\\n      \\hline\\hline\n      & \\multicolumn{6}{c}{\\bf INDIVIDUAL GALAXY SN RATE LIMITS}\\\\\n      & \\multicolumn{2}{c}{\\bf $>$ 0.0 SNe yr$^{-1}$} & \\multicolumn{2}{c}{\\bf $>$ 0.001 SNe yr$^{-1}$} & \\multicolumn{2}{c}{\\bf $>$ 0.01 SNe yr$^{-1}$}\\\\\n      \\hline\n      {\\bf 12+log(O\/H)} & {\\bf Galaxies} & {\\bf SNe yr$^{-1}$} & {\\bf Galaxies} & {\\bf SNe yr$^{-1}$} & {\\bf Galaxies} & {\\bf SNe yr$^{-1}$}\\\\\n      \\hline\n      {\\bf No Limit} & 6\\,000 & 34.1 & 3\\,691 & 33.2 & 901 & 22.9\\\\\n      {\\bf $<$ 8.4} & 1\\,757 & 0.8 & 116 & 0.3 & 6 & 0.1\\\\\n      {\\bf $<$ 8.3} & 1\\,025 & 0.3 & 50 & 0.1 & 0 & 0.0\\\\\n      {\\bf $<$ 8.2} & 401 & 0.1 & 18 & 0.0 & 0 & 0.0\\\\\n      {\\bf $<$ 8.1} & 116 & 0.0 & 4 & 0.0 & 0 & 0.0\\\\\n      \\hline \n      \\multicolumn{7}{c}{}\\\\\n      \\multicolumn{7}{c}{\\bf Nearby Bright Galaxy Catalogue}\\\\\n      \\hline\\hline\n       {\\bf } & {\\bf } & {\\bf Galaxies} & {\\bf } & {\\bf SNe yr$^{-1}$} & {\\bf } & {\\bf }\\\\\n      \\hline\n      {\\bf } &  & 1 216 &  & 12.70 &  & \\\\\n        \\hline \n\n    \\end{tabular}  \n  \\end{center}\n\\end{table*}\n  \n  \\begin{figure}\n  \\begin{center}\n  \\includegraphics[totalheight=0.3\\textheight]{cum-dist.eps}\n  \\caption{\\textrm{Cumulative distribution of the combined CCSN rate from our HSFG and LSFG catalogues, measured against oxygen abundance. It is clear that as oxygen abundance increases so to does the rate of CCSNe}}\\label{cum_dis}\n  \\end{center}\n\\end{figure}\n  \n  \n  \n  For the purpose of this survey strategy we decide that the galaxy catalogue that\n  optimally fits our requirements is the sub-sample of the HSFG cataglogue that\n  contain galaxies with oxygen abundances of less than 8.2\n  dex {\\it and} a CCSR greater than 1 CCSNe every 1\\,000 years. This low-metallicity catalogue contains\n  727 galaxies, a suitable number for a pointed survey, with an estimated\n  CCSR of \\mbox{2.8 CCSNe yr$^{-1}$}. It should be noted that this galaxy catalogue is extracted solely from the 5\\,713 square degrees of the sky that\n  SDSS DR5 spectroscopic survey covers, that is 14\\% of the entire sky.\n  It can be assumed that the rest of the sky contains similar density of\n  these low-metallicity galaxies. We shall return to this point later.\n  \n\\subsection{Monte-Carlo Simulations}\\label{sec_e}\n\n  Of the estimated \\mbox{2.8 CCSNe yr$^{-1}$} that this\n  low-metallicity galaxy catalogue will produce, we will only be able to detect a fraction\n  of these due to the practical limiting factors of a pointed survey. The\n  reason that a given SN would not be detected is simply due to its faintness at the point of\n  observation. The factors that will influence the\n  likelihood that\n  a SN will be detected when observed within our search are: whether\n  or not the galaxy that hosts the\n  CCSN is observable during the period of time when the CCSN\n  is\n  detectable, the type of CCSN (IIP, IIL, Ib, Ic or IIn) observed and its intrinsic\n  brightness, the distance to the host galaxy, the extinction towards the\n  CCSN, the exposure time and the age of the CCSN when observed (affected by the\n  cadence of the observations).\n  \n  By running a Monte-Carlo simulation, constrained by each of these parameters,\n  to randomly produce a sample of 100\\,000 possible SNe observable\n  within our search, we can infer the fraction of CCSNe that we should\n  actually detect.\n  \n\\subsection{Supernova Rates, Template Lightcurves and Distributions}\n\n  In order for the Monte-Carlo simulation (MCS) to accurately reproduce the\n  relative rates of the different types of CCSNe, we\n  use the observed rates compiled by \\citet{smartt_et_al}, given in Table\n  \\ref{smartt_rate}. These rates have been compiled within a time and volume-limited\n  sample, accounting for all SNe discovered within the eight year period between 1999 January 1\n  and 2006 December 2006 in galaxies with a recessional velocity \\mbox{$< 2\\,000\n  {\\rm km s}^{-1}$}. Apart from those SNe that\n  inhabit environments of heavy extinction or first observed late into their\n  evolution, it is expected that within this appointed distance limit \n  \\mbox{($\\mu = 32.3$)} all known types of SNe should have been bright enough to\n  have been detected, implying that these relative rates are\n  as free from any Malmquist bias as possible. CCSNe of type IIb have been merged with type Ib\/c,\n    and type IIn have been divided between those with a plateau phase in the\n    tail of their lightcurves, IIn\/P, and those with a linear phase, IIn\/L.\n  \n\\begin{table}\n \\caption{{\\textrm Relative CCSRs taken from\n    Smartt et al. (2007).}\\label{smartt_rate}}\n  \\begin{center}\n    \\begin{tabular}{ c c c c}\n      \\hline\\hline\n      {\\bf SN Type} & {\\bf Number} & {\\bf Relative Rate} & {\\bf Core-Collapse\n      Only}\\\\\n      \\hline\n      {\\bf Ia} & 25 & 24.8$\\%$ & -\\\\\n      {\\bf IIP} & 43 & 42.6$\\%$ & 56.6$\\%$\\\\\n      {\\bf IIL} & 2.5 & 2.4$\\%$ & 3.3$\\%$\\\\\n      {\\bf Ib\/c} & 28 & 27.7$\\%$ & 36.6$\\%$\\\\\n      {\\bf IIn\/P} & 1.25 & 1.2$\\%$ & 1.6$\\%$\\\\\n      {\\bf IIn\/L} & 1.25 & 1.2$\\%$ & 1.6$\\%$\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n\\end{table}\n  \n  We also supply template lightcurves of the various SNe for the MCS. For Type IIP we use\n  1999em as our template, taking the data from \\citet{2001ApJ...558..615H}. For Type IIL we use 1998S,\n  taking data for the rise to maximum light from \\citet{2000A&AS..144..219L} and\n  data for the tail from \\citet{2000MNRAS.318.1093F}. For Type Ib\/c we use\n  2002ap, taking data from \\citet{2003PASP..115.1220F}. For\n  SNe of Type IIn, we suggest that it is appropriate to divide the relative rate evenly\n  between Type IIn that exhibit a plateau phase in their\n  lightcurves and those that exhibit a linear phase, Type IIn\/P and Type\n  IIn\/L respectively. For the Type IIn\/P, we use\n  1994Y as the template taking data from \\citet{2001PASP..113.1349H}, allowing\n  1998S to provide the rise to maximum light. For Type IIn\/L we use 1999el as the\n  template taking data from \\citet{2002ApJ...573..144D}, again allowing 1998S\n  to provide the rise to maximum light (see Fig \\ref{templates} for comparison). \n  The following conversion from \\citet{2007AJ....133..734B} is used to transform the data for the\n  lightcurves to the Sloan $g$-band (note magnitudes are to be in AB system):\n\n\\begin{center}  \n\\begin{equation}\n  g=B-0.03517-0.3411(B-V)\n  \\label{blanton_equ}\n\\end{equation}\n\\end{center}  \n\n  The SN absolute magnitude distributions of \\citet{2002AJ....123..745R} are used in the MCS to provide weighted,\n  random distributions of peak magnitudes for the\n  SNe (Table \\ref{rich_table}). Equation \\ref{blanton_equ} is again\n  used to transform these distributions to the $g$-band, taking\n  a $(B-V)$ colour from the epoch of peak $g$-magnitude from our template\n  lightcurves.  $\\sigma$ is the range in the peak magnitude distribution and $g$-band magnitudes are given in the AB system. When choosing a filter to perform a SN search with the Liverpool telescope the $r$-band is superior to the $g$-band because it has a greater filter throughput, the CCD detector is more responsive in the $r$-band and also SNe are generally brighter in the $r$-band especially later in their evolution. However, the \\citeauthor{2002AJ....123..745R} distributions of SN peak magnitudes are given in the $B$-band which we can transform to the $g$-band but not to the $r$-band. It is for this reason that we simulate a SN survey in the $g$-band and then with these results we can then hypothesise the outcome of a search in the $r$-band.\n  \n\\begin{figure*}\n  \\begin{center}\n  \\includegraphics[totalheight=0.3\\textheight]{lightcurves.eps}\n  \\caption{\\textrm{Template lightcurves used within the Monte-Carlo\n  simulation. Lightcurves for SNe of type IIP are according to 1999em, type IIL to 1998S, type Ib\/c to 2002ap, type IIn\/P to 1994Y and\n  type IIn\/L to 1999el. Note that all magnitudes are in the AB system.}}\n  \\label{templates}\n  \\end{center}\n\\end{figure*}\n\n  \n\\begin{table}\n\\caption{\\textrm{Peak magnitude distributions from \\citet{2002AJ....123..745R}}\\label{rich_table}}\n  \\begin{center}\n     \\begin{tabular}{c c c c}\n     \\hline\\hline\n     {\\bf SN Type} & ${\\bf M_{B}}$ & ${\\bf M_{g}}$ & ${\\bf \\sigma}$\\\\\n     \\hline\n     {\\bf IIP} & -17.00 & -17.02 & 1.12\\\\\n     {\\bf IIL} & -18.03 & -18.00 & 0.90\\\\\n     {\\bf Ib\/c} & -18.04 &-18.22 & 1.39\\\\\n     {\\bf IIn\/P} & -19.15 & -19.24 & 0.92\\\\\n     {\\bf IIn\/L} & -19.15 & -19.24 & 0.92\\\\\n     \\hline\n     \\end{tabular}\n   \\end{center}\n\\end{table}\n\n\\subsection{Cadence, Distance and Extinction}\n\n  A major factor influencing the detection efficiency of any SN\n  search is the time between consecutive\n  observations of a galaxy hosting a SN. As this\n  time increases, so does the likelihood that we will not observe the\n  SN until it is well advanced along its lightcurve. This could result\n  in the magnitude dropping below our detection limit, hence failing to be detected in the\n  search. There are three parameters that affect the cadence of our\n  search and\n  that need to be included in the MCS. The first of these is the probability\n  that the SN will be in solar conjunction for a\n  fraction, if not all, of the time that it\n  remains detectable by our search. To account for this factor, we use\n  the celestial coordinates of the 727 galaxies within our low-metallicity galaxy catalogue and an almanac for the Liverpool Telescope to determine the fraction\n  of the year that each individual galaxy is observable, producing a\n  distribution with a mean fraction of 0.380. We then use this\n  distribution in the MCS to allocate a random amount of time to the\n  cadence of observation of each SN, to account for cases where a SN\n  would be unobservable and hence undetectable because its host galaxy is behind the sun.\n  \n  The second factor that affects the cadence is the\n  number of nights that cannot be used by the Liverpool Telescope to\n  perform our search. Reasons why the telescope could not be used on any\n  particular night include weather, technical\n  difficulties and scheduled maintenance nights. To\n  compensate for these nights in our MCS, we have taken nightly reports\n  from the Liverpool Telescope website\\footnote{LT website:\n  \\it{http:\/\/telescope.livjm.ac.uk\/}}, from 2005 August 1 to 2006 July 31,\n  and use these reports to determine a distribution of nights (58$\\%$ of nights in total) that\n  can be used for our search. This distribution of usable nights is included in our MCS when considering the cadence of observations.\n  \n  The final factor that will affect the cadence of observations, is the\n  number of galaxies that we aim to observe every night. This number is\n  influenced by the amount of telescope time dedicated to our\n  search each night; the\n  greater the amount of time on the telescope each night, the greater the number of galaxies we can observe per\n  night and the higher the cadence of observations. The number of galaxies\n  that can be observed each night is also affected by exposure time. The\n  benefit of increasing the exposure time is that we can search to a deeper magnitude limit, meaning\n  that fainter SNe are detectable and SNe are detectable for a greater period of time. It is essential therefore to run a number of MCSs in order to find\n  the optimal balance between the exposure time given to each galaxy and\n  the cadence of observations and hence enabling us to detect the greatest fraction of\n  CCSNe hosted by our low-metallicity galaxy catalogue.\n  \n  The final two parameters placed in our MCS are the host galaxy distances and\n  the extinction toward the SNe. The MCS will attribute to each SN a random host galaxy\n  distance, out to the distance limit of our search, which is \\mbox{z$=$0.04} (assuming the galaxies within\n  our search are spread homogeneously and isotropically throughout this\n  volume). Considering that the majority, if not all, of the galaxies\n  within our catalogue are low-luminosity, blue, dwarf galaxies with relatively\n  low metallicities, we assume that host galaxies will have produced very\n  little dust and will not contribute a great deal to any extinction toward\n  the CCSN.\n  To this end we simply attribute a typical Galactic extinction to each\n  SN \\mbox{(A$_{g}=0.3$)}.\n  \n  We note that a fraction of SNe remain undetected within spiral galaxies that are not observed face on \\citep[e.g. ][]{2003astro.ph.10859C}. This is due to the fact that the average extinction attributed to SNe in spiral galaxies with higher inclination angles is higher than that of face on spiral galaxies, making the SNe fainter and more difficult to detect. Implementation of a correction for this effect into the MCS, using a similar method to that of \\citet{2005MNRAS.362..671R}, could easily result in an under-estimation of the CCSRs observed in low-metallicity galaxies as these tend not be to grand-design spiral galaxies but rather dwarf, irregular galaxies where the effect of galaxy inclination would be negligible. We have therefore decided not to attempt to correct for this effect. This may make our\npredictions for the SN discovery rates in the full galaxy catalogues\n(with no metallicity limit) somewhat optimistic. However this may not lead to a significant over-estimate of events, as the SFRs that we calculate come from the observed galaxy H$\\alpha$ luminosities and in inclined spirals these will also be lower than in face on targets.\n  \n  \n\\subsection{Monte Carlo Simulation Results for a Single 2.0m Telescope}\\label{sec_f}\n\n  Having included each of these parameters in the MCS, we randomly\n  generate 100\\,000 SNe that would potentially be observable within\n  our search. The information gained about each SN produced by the simulation include\n  its type and its apparent magnitude at the point of observation. It is only if this magnitude is\n  above the limiting magnitude of our search can we register the SN as being\n  detected.\n  \n  By running the MCS multiple times for (a) the varying number of hours we aim to\n  observe with the LT every night and (b) the varying amount of exposure time\n  that we dedicate to each individual galaxy, we can determine the optimal\n  values of these parameters that shall enable us to detect the greatest\n  fraction of the CCSNe that we have predicted our low-metallicity galaxy\n  will produce. The results can be seen in Fig. \\ref{exp_time}.\n  \n\\begin{figure*}\n  \\begin{center}\n  \\includegraphics[totalheight=0.3\\textheight]{exp_time.eps}\n  \\caption{\\textrm{Percentage of CCSNe\n  detected from our MCSs for a differing number of hours observing per night\n  (labeled to the right of the plot). The percentage of CCSNe detected\n  increases as the nightly observing time increases, as this increases the\n  cadence of observations. As the individual exposure time per galaxies\n  increases, the limiting magnitude of the search deepens but the cadence of\n  observations decreases. There is therefore an optimal point at which\n  the exposure time and cadence of observation are balanced to detect the maximum\n  percentage of SNe.}}\n  \\label{exp_time}\n  \\end{center}\n\\end{figure*} \n\n  Studying the results of the MCSs we suggest that the\n  difference between the percentage of CCSNe detected while aiming to observe one\n  hour per night as opposed to two hours per night or more is not enough to\n  justify the extra observing time as the\n  fraction of CCSN detected only increases by a few percent. As the detection rates are double valued either side of the peak\n  detection rate, we suggest using a shorter exposure time in order to\n  increase cadence and thereby increase the probability of \n  discovering CCSNe earlier in their evolution as opposed to later. \n  \n   Aiming to observe one hour a night using 60 sec exposures allows each galaxy to be observed once every $\\sim$17 days while accessible with LT and enables 29.6\\% of CCSNe to be detected with\n  our search. When considering the catalogue of 727 low-metallicity galaxies which produce \\mbox{2.8 CCSNe yr$^{-1}$}, this equates to a detected CCSN rate of \\mbox{0.8 SNe yr$^{-1}$}\n  and requires a total amount of \\mbox{154hrs\n  yr$^{-1}$} telescope time. Remembering that the $r$-band is superior to the $g$-band for performing a SN search, we note that given a 60 sec exposure the AB limiting magnitudes for the Liverpool telescope are $g = 18.7$ mags and $r = 19.0$ mags. Given the average cadence of observations determined by our chosen survey parameters, a typical SN will be observed $\\sim$73 days post-explosion with $g-r=0.8$ mags. Given the difference in limiting magnitudes and the typical $g-r$ colour at the point of observation, we expect to detect SNe that are 1.1 magnitudes fainter with an $r$-band search than we would with a $g$-band search. Assuming that the template SN lightcurves and the distribution of SN peak magnitudes are similar in the $r$-band as compared to the $g$-band, we re-run the $g$-band MCS with a limiting magnitude 1.1 magnitudes fainter than previously used. From this simulation we predict that using the $r$-band will allow for $47.8\\%$ of CCSNe to be detected with our search. Again considering the catalogue of 727 low metallicity HSFGs which produce \\mbox{2.8 CCSNe yr$^{-1}$}, this equates to a detected CCSN rate of \\mbox{1.3 SNe yr$^{-1}$}.\n  \n  Assuming a typical Galactic extinction of A$_g=0.3$ (A$_r=0.22$) and an exposure\n  time of 60 seconds we estimate that the absolute\n  limiting magnitude ($25\\sigma$ significance) for this search at a distance of 20Mpc will be\n  $M_r=-12.7$, at 70Mpc it will be $M_r=-15.4$ and at 150Mpc it will be\n  $M_r=-17.1$. We plan to use the method of image matching and subtraction \\citep[see ][]{1998ApJ...503..325A} to detect SNe throughout this survey. Assuming original \nimages with equal depth, the process of image subtraction increases the\nnoise by roughly a factor of $\\sqrt{2}$. In addition, the detection of\nSNe inside their host galaxies will be always more difficult due to\nimage subtraction residuals caused by uncertainties in the alignment and\nmatching of the two images. Since we wish to avoid a large fraction of \nspurious detections, we require the relatively high significance level of \n25$\\sigma$ from the limiting magnitude of our search.\n\n  \n\\subsection{Surveying with a Network of 2.0m Telescopes}\\label{sec_f2}\n\n  As discussed previously SDSS DR5 covers only $\\sim$14\\% of the\n  entire sky. If we presently had the ability to compile a catalogue of low-metallicity\n  galaxies for the whole sky as complete as that for the area of the SDSS DR5\n  spectroscopic survey and a collection of seven 2.0m telescopes (or\n  three to four 2.0m telescopes with double the time allocation) we would expect to detect\n  roughly 7 times the CCSNe detected solely by the LT, i.e. $\\sim$9.3\n  CCSNe\/yr. We could consider using a network of six to eight 2.0m robotic telescopes similar to the RoboNet global network of 2.0m\n  telescopes consisting of the LT and the Faulkes Telescope North and the\n  Faulkes Telescope South to perform this kind of survey. Another advantage of using a network of telescopes in both the northern and southern hemispheres as opposed to a single telescope is that we gain a greater sky coverage and can therefore target a greater number of galaxies within our search.\n  \n  The two obstacles that we would encounter if we were to use this strategy to search for CCSNe are firstly the generous amount of telescope\n  time that we would require \\mbox{($\\sim$1\\,000 hrs yr$^{-1}$)} and secondly the present lack of\n  data required to compile an all sky galaxy catalogue. Compiling a catalogue\n  of all galaxies listed in the 2dF, 6dF, LEDA and SDSS DR5 within the redshift range\n  \\mbox{0$<$z$<$0.04} results in a catalogue containing a total of 103\\,549 individual galaxies, only a fraction of which will be star-forming. Apart from this we shall have to wait for PS1, the prototype\n  1.8m telescope of the Panoramic Survey Telescope and\n  Rapid Response System (Pan-STARRS), which shall cover an area of 3$\\pi$ of\n  the sky to a depth exceeding that of SDSS DR5, in order to compile a far more\n  complete all-sky low-metallicity galaxy catalogue.\n  \n  \n\\section{Strategy 2 : volume-limited searches with the Pan-STARRS all sky surveys} \\label{sec_g}\n\n  Having considered a dedicated pointed low-metallicity galaxy survey to search for CCSNe both with one and with a network of 2.0m telescopes, we\n  now turn our attention to an all-sky survey. Pan-STARRS is a\n  set of four 1.8m telescopes with wide field optics, each with a 7 square\n  degree field of view, which will be constructed by the University of Hawaii. The\n  prototype telescope PS1 has now achieved first light and\n  is set to go online during 2008. PS1 will have the capability of\n  surveying\n  the whole available sky in less than a week at a rate of 6\\,000 square degrees per\n  night, covering 30\\,000 square degrees of the sky per cycle\n  \\citep{PS1_DRM}.\n    Included in the list of tasks that the PS1 image reduction pipeline will\n  perform is the subtraction of every image from a reference image in order to\n  produce a database of residual transient objects that will include moving objects and static\n  variables such as SNe.\n\nThere are two different strategies that one would employ to search for\nSNe (or transients of any type), these are volume-limited searches and\nmagnitude-limited searches. We will consider both of these. A volume-limited search has the advantage that it can quantify true rates of \ntransients. In addition the radial limit can be chosen so that the \ntarget discoveries are bright enough to be followed in multi-wavelength\nstudies with complementary facilities (e.g. spectroscopic and photometric\nfollow-up with 2-8m telescopes). \n\n\n\\subsection{Monte Carlo Simulations}\\label{sec_h}\n\n   Of all the modes in which PS1 shall be run the 3$\\pi$ Steradian Survey\n   (covering 30,000 square degrees of the sky)\n   shall be the most effective when searching for nearby CCSNe. The survey aims to cover that whole sky 60 times in 3 years, that is 12 times\n   in each of the five filters {\\it g, r, i, z} and {\\it y}. This aim has already\n   taken historic weather patterns on Haleakala into consideration. The footprint of the survey\n   completely covers the entire footprint of the SDSS DR5 spectroscopic\n   survey and the AB limiting magnitude \\mbox{(25$\\sigma$)} in the $g$-band is stated\n   to be 21.5\n   mags for a single 60 second exposure \\citep{PS1_DRM}.\n   \n   Having an adapted MCS produce 100\\,000 potential SNe over the\n   redshift range of our galaxy samples \\mbox{($0<z<0.04$)} will again help us\n   to estimate the fraction of potentially observable CCSNe that we will\n   actually detect with PS1. We assume that we can take the\n   fraction of CCSNe detected in the $g$-band as the typical fraction that\n   would be detected in each of the five filters; indeed running the adapted MCS detailed in Section \\ref{sec_i} in an attempt to produce the detected CCSN rates of this nearby sample in all five filters, were the {\\it K}-corrections essentially behave as colour corrections at such low redshift, results in predicted detection efficiencies extremely close to those found if considering only the $g$-band. Concerning the cadence we have also taken the\n   PS1 survey strategy into consideration \\citep{PS1_DRM}, see\n   Figure \\ref{cad_fig}. \nA random extinction is assigned to each of the CCSNe\n   using a weighted distribution based on CCSRs taken from \\citet{2001MNRAS.324..325M} and\n   \\citet{1994ApJ...432...42S}. However for \n   low metallicity galaxies ($\\leq$8.2\\,dex) \n  only a typical Galactic extinction of \\mbox{A$_{g}=0.3$}\n   is assigned to the CCSNe, as at these metallicities we expect the galaxies\n   to be relatively dust free.\n   \n  We must also consider the fact that Pan-STARRS shall be covering a large fraction of the sky that is obscured by the Galactic Plane, a total of $\\sim$7\\,500 square degrees of the Pan-STARRS area. Any SN that is located with a line-of-sight through the Galactic Plane shall be extinguished above the typical Galactic extinction assumed. To allow for this we take the extinctions, as determined by \\citet{1998ApJ...500..525S}, of $\\sim$1\\,500 points positioned homogeneously across the expanse of the Galactic Plane visible with Pan-STARRS and determine a distribution of extinctions to be folded into the MCS for any SN falling behind the Galactic Plane. The cumulative distribution of these extinctions can be seen in Figure \\ref{ext_dist}.\n   \n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[totalheight=0.25\\textheight]{cummulative_ext.eps}\n  \\caption{\\textrm{Cumulative distribution of ~1\\,500 extinctions as measured through the fraction of the Galactic Plane which lies within the Pan-STARRS survey area.}}\n  \\label{ext_dist}\n  \\end{center}\n\\end{figure} \n   \n   \n   For the full galaxy catalogue (within $z \\leq 0.04$) \n  we can expect to detect \\mbox{66.9$\\%$} of all\n   CCSNe. When considering the lower metallicity samples we can expect to detect\n   74.2$\\%$ of CCSNe. The predicted detectable CCSRs for both the full galaxy catalogue\n   and for the lower-metallicity samples can be found in Table \\ref{PS1_rates}.\n   \n   These results are a vast improvement on the rates predicted for the\n   catalogued galaxy surveys with both a single 2.0m telescope and a network\n   of 2.0m telescopes. Considering that the Pan-STARRS survey area is\n   $\\sim$5.25 that of SDSS DR5 and given a complete galaxy catalogue of that\n   area, we can expect to detect a total of \\mbox{$\\sim$570 CCSNe yr$^{-1}$}, roughly 13 of which\n   will be from host galaxies with \\mbox{$12+{\\rm \\log(O\/H)}<8.2$}. \nWe would\nexpect that the PS1 survey itself will provide a photometric catalogue\nof at least the quality the depth of SDSS DR5 (most likely signficcantly\nbetter) over the whole sky and hence provide us with a means to identify\nSNe in nearby, faint blue hosts. \nOur estimate of \n   number of CCSNe expected to be detected in the low-metallicity galaxy sample can be considered as a lower\n   limit as there may be many more blue, compact, dwarf galaxies that have\n   not been detected and catalogued by SDSS DR5.\n The number of CCSNe expected to be detected\n   by PS1 in low-metallicity galaxies is hence about 50\\%\n times more than would be possible using a network of seven \n2.0m telescope in a dedicated pointed survey with 1000\\,hrs of dedicated\ntime. It would this seem much more efficient to use the PS1 survey\nto discover these are objects than invest in a dedicated pointed survey\nwith large numbers of 2.0m telescopes with standard CCD cameras\nand fields of view. \n   \n\\subsection{PS4}\n\n   Eventually three more 1.8m telescopes identical to PS1 will be added to\n   the Pan-STARRS network to create PS4. These four telescopes will be trained\n   on the same area of the sky simultaneously \n   to effectively perform a deeper all-sky\n   survey (with the same observing strategy as PS1), \ngiving a co-added 25$\\sigma$ limiting magnitude of $g$=22.2.\n   \n   At an \\textit{apparent} limiting magnitude of \\mbox{$m_{g}$=21.5}, PS1 has an \\textit{absolute} limiting\n   magnitude \\mbox{$M_{\\rm g}$=-14.7} at the boundary of our search ($z=0.04$). For\n   PS4 we would push the boundary of our search out to the point where the\n   absolute limiting magnitude is identical to that of our search with PS1 in\n   order to detect the same fraction of CCSNe as we have predicted for PS1.\n   This distance limit is calculated to be $z=0.056$.\n   Using the same \n   survey strategy as previously suggested for PS1 and introducing\n   the new limiting magnitude and distance limit, we predict that PS4 should discover a total of \\mbox{$\\sim$1\\,308 CCSNe yr$^{-1}$}, \\mbox{$\\sim$18.3 CCSNe yr$^{-1}$} being found in galaxies with \\mbox{$12+{\\rm \\log(O\/H)}<8.2$}. These results\n   can be found in Table \\ref{PS4_rates}.\n   \n   \\begin{table}\n\\caption{\\textrm{Estimated intrinsic and detectable CCSRs within the SDSS\n    DR5 survey area and $z<0.04$. Also shown are the rates extended to the PS1 survey area.}\\label{PS1_rates}}\n  \\begin{center}\n    \\begin{tabular}{c c c c c}\n      \\hline\\hline\n      & \\multicolumn{2}{c}{\\bf Total Sample} & \\multicolumn{2}{c}{\\bf Detected CCSNe yr$^{-1}$}\\\\\n      \\hline\n      {\\bf 12+log(O\/H)} & {\\bf Galaxies} & {\\bf CCSNe yr$^{-1}$} & {\\bf SDSS Area} & {\\bf PS1 Area}\\\\\n      \\hline\n      {\\bf No Limit} & 25\\,091 & 162.4 & 108.6 & 570.4\\\\\n      {\\bf $<$ 8.4} & 9\\,776 & 13.9 & 10.3 & 54.1\\\\\n      {\\bf $<$ 8.3} & 5\\,315 & 7.2 & 5.3 & 28.0\\\\\n      {\\bf $<$ 8.2} & 2\\,114 & 3.2 & 2.4 & 12.5\\\\\n      {\\bf $<$ 8.1} & 653 & 1.1 & 0.8 & 4.3\\\\\n      \\hline\n    \\end{tabular}  \n  \\end{center}\n\\end{table}\n\n\n\\begin{table}\n\\caption{\\textrm{Estimated intrinsic and detectable CCSRs within the SDSS\n    DR5 survey area and $z<0.056$. Also shown are the rates extended to the PS4 survey area.}\\label{PS4_rates}}\n  \\begin{center}\n    \\begin{tabular}{c c c c c}\n      \\hline\\hline\n      & \\multicolumn{2}{c}{\\bf Total Sample} & \\multicolumn{2}{c}{\\bf Detected CCSNe yr$^{-1}$}\\\\\n      \\hline\n      {\\bf 12+log(O\/H)} & {\\bf Galaxies} & {\\bf CCSNe yr$^{-1}$} & {\\bf SDSS Area} & {\\bf PS4 Area}\\\\\n      \\hline\n      {\\bf No Limit} & 42\\,335 & 372.4 & 249.1 & 1\\,308.0\\\\\n      {\\bf $<$ 8.4} & 12\\,104 & 24.6 & 18.3 & 95.8\\\\\n      {\\bf $<$ 8.3} & 6\\,053 & 11.6 & 8.6 & 45.2\\\\\n      {\\bf $<$ 8.2} & 2\\,309 & 4.7 & 3.5 & 18.3\\\\\n      {\\bf $<$ 8.1} & 680 & 1.2 & 0.9 & 4.8\\\\\n      \\hline\n    \\end{tabular}  \n  \\end{center}\n\\end{table}\n   \n    \n \\section{\\textbf{Strategy 3 : magnitude-limited searches\nwith future all-sky surveys}}\\label{sec_ii}\n \n   Thus far when considering the Pan-STARRS survey to search for CCSNe\n   we have limited ourselves to the known galaxy\n   population within $z \\lesssim 0.04$. If our aim is to detect\n   \\textit{all} of the CCSNe that we possibly can then we are by no\n   means utilizing the Pan-STARRS survey to its greatest\n   potential. The galaxy catalogue employed is\n   taken from SDSS DR5 and is therefore limited by a comparatively\n   shallow depth compared to the depth to which Pan-STARRS will be\n   able to detect CCSNe \\citep{PS1_DRM}.\n   \n   A far more powerful method that we could implement with\n   Pan-STARRS to detect CCSNe is to perform simply a magnitude-limited search\n   to detect CCSNe\n   over the whole sky and to as large distances as possible\n   following the example of surveys such as the\n   {\\it Nearby SN Factory} \\citep{2006NewAR..50..436C}. The inherent\n   difficulties that we will encounter with this kind of survey will\n   be trying to distinguish SNe from all other transient events and\n   variables such as classical novae, asteroids and AGN and further\n   distinguishing CCSNe from the thermonuclear (Type Ia) SNe.   \n   Assuming that it may be possible to overcome these\n    difficulties (and that we can probabilistically classify transient events)\n    we can\n   estimate how many CCSNe that we shall be able to detect with PS1, PS4 and\n   also compare these rates with the likes of the Large Synoptic Survey\n   Telescope (LSST) set to come online in 2016.\n   \n\\subsection{Adapted Monte-Carlo Simulations}\\label{sec_i}\n\n   In order to determine the total CCSRs that we are to expect from the all-sky\n   surveys PS1, PS4 and LSST, we must first ascertain to what redshift depth our SDSS\n   DR5 galaxy sample is relatively complete to. When targeting galaxies for the\n   spectroscopic survey SDSS DR5 has a lower magnitude limit of $r=17.77$.\n   This apparent magnitude limit equates to an absolute galaxy magnitude\n   limit of \\mbox{$M_{\\rm r}=-18$} being reached at a redshift of $z=0.033$. The\n   galaxy density distribution for SDSS DR5 can be seen in Figure\n   \\ref{sdss_galaxies}. There is a clear drop in the observed galaxy density at this\n   redshift and we choose to set this limit as the completeness limit of the\n   SDSS DR5 galaxy sample.\n   \n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[totalheight=0.2\\textheight]{sdss_distribution.ps}\n  \\caption{\\textrm{The galaxy distribution for the SDSS DR5 spectroscopic\n  survey galaxy sample. The solid curve represents the absolute magnitude\n  limit as a function of redshift given that the SDSS DR5 apparent magnitude\n  limit for the sample is $r=17.77$. An absolute magnitude limit of \\mbox{$M_{\\rm\n  r}=-18$} is reached at at redshift of 0.033 (dashed line). There is a clear\n  drop in the galaxy density at this redshift, indicated by the dashed line,\n  and we choose to use this limit as the completeness limit of the SDSS DR5\n  galaxy sample.}}\n  \\label{sdss_galaxies}\n  \\end{center}\n\\end{figure} \n  \n   With this completeness limit in mind we can now extrapolate the complete local\n   galaxy sample out to the observable redshift limits for the individual CCSN\n   surveys that we are considering. We assume that the galaxy\n   distribution throughout the observable universe is both homogeneous and\n   isotropic. To determine the observable redshift limits of each of the suggested\n   CCSN searches it is necessary  to model the CCSN detection distribution\n   for each of the surveys. This is made possible by using an adapted MCS to determine\n   the percentage of CCSNe that would be detected as a function of redshift.\n   The variant parameters included in the MCS that define one distribution from another are the\n   cadence and the limiting magnitudes of the survey. For PS1 and PS4 we\n   set the cadence to reflect the observing strategy for Pan-STARRS (Figure\n   \\ref{cad_fig}) and the 25$\\sigma$ limiting magnitudes outlined by\n   \\citet{PS1_DRM}, assuming that the limiting magnitude of PS4 is\n   achieved via co-adding four PS1 images (see Table \\ref{mag_table}).\n   \n\\begin{table}\n\\caption{\\textrm{The 25$\\sigma$ AB limiting magnitudes for PS1, PS4 and LSST}\\label{mag_table}}\n  \\begin{center}\n     \\begin{tabular}{c c c c}\n     \\hline\\hline\n     {\\bf FILTER} & {\\bf PS1} & {\\bf PS4} & {\\bf LSST}\\\\\n     \\hline\n     {\\it g} & 21.5 & 22.2 & 22.7\\\\\n     {\\it r} & 20.9 & 21.7 & 22.4\\\\\n     {\\it i} & 20.8 & 21.6 & 21.8\\\\\n     {\\it z} & 19.8 & 20.6 & 21.2\\\\\n     {\\it Y} & 18.4 & 19.1 & 19.9\\\\\n     \\hline\n     \\end{tabular}\n   \\end{center}   \n\\end{table}\n   \n   \n   The current LSST configuration sets the telescope at an 8.4m diameter with a 9.6\n   square degree field-of-view. It is planned that LSST will survey the whole\n   of the observable sky every 3 nights, eventually covering a total of 20\\,000 square\n   degrees. The primary focus of the survey is to detect transients and in\n   order to detect fast faint events the sky will be surveyed with\n   duplicate pairs of 15sec exposures. Using the LSST exposure calculator\n   this equates to the 25$\\sigma$ limiting magnitudes per single exposure as found in Table\n   \\ref{mag_table}.\n   \n   When considering a magnitude-limited survey it is important to note that we shall\n   detect CCSNe at relatively large redshifts compared to the volume-limited survey that we have previously discussed. With this increase in\n   survey depth we now encounter parameters in our search that we have\n   previously neglected due to fact that our searches have been relatively\n   nearby.These parameters that now have to be considered in our MCS are all\n   functions of redshift and hence could be neglected at low-redshift. They are the\n   star-formation history (SFH) of the universe, the redshifting\n   of light emitted by the CCSNe, the broadening of the CCSNe\n   lightcurves due to time-dilation and finally the retardation\n   of the CCSN rates again due to time-dilation.\n\n   {\\it Star-Formation History:\t} As star-formation was more prolific earlier\n   in the Universe's history than it is at present, star-formation is seen to\n   increase as redshift increases (at least to moderate redshifts of\n   $z\\sim2.5$). It is also important to note that approximately 50$\\%$ of all local star formation activity is obscured, while at a redshift of z$\\sim$1 this fraction could be as high as 80$\\%$ \\citep{2005A&A...440L..17T}. To incorporate the SFH of the universe into our simulations, but also taking into account that a greater fraction of this star-formation will be obscured by dust at higher redshifts, we include the far-ultraviolet (FUV) SFR evolution determined by \\citet{2005ApJ...619L..47S}, see Equation (\\ref{sfh_equ}). Compared with all other indicators of SFR, the FUV luminosity is the most inhibited by dust and therefore we would hope that by using this indicator we would only consider the star-formation that will give rise to CCSNe that are not so extinguished as to be invisible to our search.\n     \n  \n\\begin{equation}\n  \\dot{\\rho}_{*}(z)=(1+z)^{2.5\\pm0.7}.\n  \\label{sfh_equ}\n\\end{equation}\n\n\n   \n   {\\it K Corrections:\t} Comparing the bolometric fluxes of objects at various redshifts presents no\n   difficulty as we are comparing the total flux emitted by the objects at all\n   wavelengths. However, in practice we measure only a fraction of the total\n   flux of an object, redshifted to the observed wavelength and transmitted through a given\n   bandpass. This presents us with a challenge when we wish to simulate the\n   total number of CCSNe that an all-sky survey will detect, as we have so\n   far only considered their rest-frame (or emitted-frame) magnitudes but we\n   must now consider their redshifted-frame (or observed-frame)\n   magnitudes. To do\n   this we need to calculate a transformation known as the $K$ correction\n   \\citep{1968ApJ...154...21O,2002astro.ph.10394H}, which transform between\n   emitted-frame and observed-frame magnitudes:\n   \n\\begin{equation}\nm_R=M_Q+{\\rm DM}+K_{QR}\n\\label{kcor_four}\n\\end{equation}\n  \n \\begin{figure*}[!b]   \n\\begin{equation}\n  K_{QR}=-2.5{\\rm log}_{10}\\left[\\frac{1}{[1+z]}\\frac{\\displaystyle{\\int}\\lambda_{obs}F_{\\lambda}(\\lambda_{obs})R(\\lambda_{obs}){\\rm d}\\lambda_{obs}{\\displaystyle{\\int}\\lambda_{em}S^Q_{\\lambda}(\\lambda_{em})Q(\\lambda_{em}){\\rm d}\\lambda_{em}}}{\\displaystyle{\\int}\\lambda_{obs}S^R_{\\lambda}(\\lambda_{obs})R(\\lambda_{obs}){\\rm d}\\lambda_{obs}\\displaystyle{\\int}\\lambda_{em}F_{\\lambda}([1+z]\\lambda_{em})Q(\\lambda_{em}){\\rm d}\\lambda_{em}}\\right]\n \\label{kcor_nine}\n\\end{equation}\n\\end{figure*}\n\n\n\\noindent where $m_R$ is the apparent magnitude the the object as measured through observed-frame\nbandpass, $M_Q$ is the absolute magnitude of the object as measured through the emitted-frame\nbandpass, DM is the distance modulus of the object considered as\ndetermined from its luminosity\ndistance and $K_{QR}$ is the $K$ correction that transforms between these\nmagnitudes. The $K$ correction can be understood as the difference in\nmagnitude between two identical objects placed at the same distance; one traveling at a redshift of\n$z$ relative to the observer and measured through the observed-frame\nbandpass, the other at rest relative to the observer and measured through the\nemitted-frame bandpass.\n\n\nWe choose to use the $K$ correction as derived by \\citet{2002astro.ph.10394H} to account for the redshifting of the SN light in our MCS; see Equation \\ref{kcor_nine}, where \\mbox{$F_{\\lambda}(\\lambda)$} is the spectral density of flux of the\nobject considered taken from spectral data (measured in \\mbox{ergs cm$^{-2}$\ns$^{-1}$ \\AA$^{-1}$)}.\n$R(\\lambda)$ and $Q(\\lambda)$ are the observed-frame and emitted-frame filter\ntotal throughput functions, both\ndepending on the various parameters of the survey considered including the total\nthroughput of the atmosphere, the reflectivity of the telescope\nmirrors, the transmission of the corrector optics and filters and the quantum\nefficiency of the camera detector. $S^R_{\\lambda}(\\lambda)$ and\n$S^Q_{\\lambda}(\\lambda)$ are the spectral density of flux for the AB zero-magnitude standard source, which\nis a synthetic source equal to 3631 Jy (where \\mbox{1 Jy = 10$^{-26}$ W m$^{-2}$\nHz$^{-1}$)} for all frequencies \\citep{1983ApJ...266..713O}, as measured\nthrough the observed-frame and emitted-frame filters respectively.\n\n\nIt is essential to provide our MCS with\n$K$ corrections for each of the CCSNe simulated in order to determine\naccurate apparent magnitudes as measured through each of the five Pan-STARRS\nfilters. As the $K$ correction varies for each filter considered we must\nproduce five $K$ corrections for each CCSN; $K_{gg},K_{gr},K_{gi},K_{gz}$ and $K_{gy}$. Any\ngiven CCSN produced by the simulation is placed at a random redshift and is\nfirst observed at a random point in its evolution, therefore we require $K$ corrections\nover the entire redshift range of our surveys, for every possible epoch of\nobservation and for every CCSN type. It is then imperative that we have enough spectral information for each type of\nCCSN considered in order to cover the shear expanse of the parameters\ninvolved in calculating the $K$ corrections. We require spectral information\ncovering a large range of the evolution of each CCSN type including the rise\nto maximum light and through to over 300 days post-explosion in order to\naccommodate for the maximum time separating two consecutive observations in the same\nfilter according to the Pan-STARRS observation strategy implemented in the\nMCS (see Figure \\ref{cad_fig}). As the $K$ correction evolves with time we also require spectral data for a large\nnumber of epochs in order to achieve good time resolution over the evolution\nof each CCSN. Finally, a wide range in wavelength coverage is required in\norder to provide enough information to determine $K$ corrections for all five\nbandpasses whenever possible, remembering that the spectral data shall be\nredshifted according to the distance appointed to the CCSN by the MCS.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[totalheight=0.3\\textheight]{cir_cad.eps}\n\\caption{\\textrm{A schematic depicting the cadence of observation for any\ngiven point in the sky, taken from the Pan-STARRS observation strategy\n\\citep{PS1_DRM}. One Pan-STARRS year of 12 lunar months, approximately 352\ndays, is represented by a circle. With each observation epoch a double exposure is taken in order to\nidentify Near Earth Objects}\\label{cad_fig}}\n\\end{center}\n\\end{figure}\n\nFor CCSNe of Type IIP we have used the spectra of SN1999em\n\\citep{2001ApJ...558..615H, 2001ApJ...553..861L, 2000ApJ...545..444B}, for\nType IIL the spectra of SN1998S are used \\citep{2000MNRAS.318.1093F,\n2004MNRAS.352..457P, 2005ApJ...622..991F} and for Type Ib\/c the spectra of\nSN2002ap are used \\citep{2002MNRAS.332L..73G} supplemented with spectra from\nSN1990B \\citep[and Asiago Catalogue]{2001AJ....121.1648M} and SN1998bw\n\\citep{2001ApJ...555..900P} in order to attain the time coverage required. Due\nto the fact that Type IIn\/P and IIn\/L are relatively rare, there is not enough\nspectral data to cover both the time and wavelength coverage required to\ndetermine a full range of $K$ corrections for these CCSNe. To compensate for\nthis lack of data we allow the Type IIn\/P SN population to be represented\nby Type IIP SNe and the Type IIn\/L population to be represented by Type IIL\nSNe, adapting the Smartt et al. (2007) rates appropriately for the MCS.\n\nHaving calculated the thousands of possible $K$ corrections allowed by the\nrelevant parameters of the MCS, we then fit a third\norder polynomial to each set of $K$ correction data defined by CCSN type,\nfilter and redshift in order to provide a $K$ correction for all epochs\nrequired by the MCS (See Figure \\ref{kgi_fig})\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[totalheight=0.25\\textheight]{kgi_IIL_z0pt20.ps}\n  \\caption{\\textrm{A third order polynomial is fitted to each set of $K$\n  correction data, $K_{gg},K_{gr},K_{gi},K_{gz}$ and $K_{gy}$, for each CCSNe\n  type and each redshift position sampled over the entire redshift range of\n  the all-sky surveys considered. This example is of the $K_{gi}$ correction\n  for Type IIL SNe as sampled at a redshift of $z=0.2$}\\label{kgi_fig}.}\n  \\end{center}\n\\end{figure}\n\n\nThese polynomials are read by the MCS in order to determine a final accurate\napparent magnitude in each of the five filters, for each CCSN as measured at its specific redshift and\npoint in its evolution.\n\n   {\\it Time Dilation:\t} The broadening of the CCSN lightcurves due to time dilation at relatively\n   high redshift can be described simply as watching the CCSN\n   evolve in `slow motion' with respect to our frame of reference. This\n   enables us to sample the lightcurve of the CCSN with a greater frequency\n   within its frame of reference relative to our frame of reference. The\n   cadence hence goes as:\n   \n\\begin{center}  \n\\begin{equation}\n  \\rm{cadence}=\\rm{cadence}_{rest}\\times(1+z).\n\\end{equation}\n\\end{center}\n     \n   The same argument leads to a retardation of the detected CCSN rates,\n   giving:\n   \n\\begin{center}  \n\\begin{equation}\n  \\rm{CCSR}=\\rm{CCSR}_{rest}\/(1+z).\n\\end{equation}\n\\end{center}\n\n\n   \n   Figure \\ref{CCSN_distributions} shows the CCSN detection distributions for PS1, PS2\n   and LSST. Beyond the redshift limit where an absolute limiting magnitude\n   of $Mg=-20$ is attained the surveys will cease to detect all but the most\n   extreme of CCSNe were luminosity is concerned and hence it is at this\n   limit that we choose to mark the boundary of the observable universe when\n   considering CCSNe. These redshift limits are 0.37, 0.50 and 0.59 for PS1,\n   PS4 and LSST respectively.\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[totalheight=0.25\\textheight]{CCSN_25sig.eps}\n  \\caption{\\textrm{The solid lines show CCSN detection distributions for PS1,\n  PS4 and LSST, the dashed lines the absolute magnitude limits of the surveys\n  as a function of redshift. At a absolute limiting magnitude of $M_{\\rm\n  g}=-20$ the PS1, PS4 and LSST reach their respective limiting redshifts\n  depths of $z\\sim0.4$, $z\\sim0.5$ and $z\\sim0.6$.}}\n  \\label{CCSN_distributions}\n  \\end{center}\n\\end{figure} \n\n   Assuming that the galaxy density is homogeneous and isotropic\n   throughout the volumes contained by these redshifts, we extrapolate our\n   nearby local sample of galaxies and their predicted CCSRs to both the sky coverage areas and the respective redshifts for the\n   three surveys, taking into account the SFH of the universe, the\n   redshifting of light from distant CCSNe and the\n   time-dilation of both the CCSN lightcurves and the CCSRs. Using the CCSN detection distributions obtained from the\n   MCSs we then determine the detected CCSR for each of the surveys. The\n   results can be found in Table \\ref{CCSN_survey_rates}.\n   \n\n\\begin{table*}\n\\caption{\\textrm{Predicted CCSRs for PS1, PS4 and LSST}\\label{CCSN_survey_rates}}\n  \\begin{center}\n     \\begin{tabular}{c c c c c}\n     \\hline\\hline\n     & \\multicolumn{4}{c}{\\bf Detected Core-Collapse SN Rates (SNe yr$^{-1}$)}\\\\\n     \\hline\n     {\\bf 12+log(O\/H)} & {\\bf DR5 Sample ($z<0.033$)} & {\\bf PS1} & {\\bf PS4} & {\\bf LSST}\\\\\n     \\hline\n     {\\bf No Limit} & 103.5 & 24\\,095 & 68\\,620 & 160\\,249\\\\\n     {\\bf $<8.4$} & 9.8 & 3\\,046 & 8\\,343 & 19\\,627\\\\\n     {\\bf $<8.3$} & 5.3 & 1\\,640 & 4\\,491 & 10\\,566\\\\\n     {\\bf $<8.2$} & 2.3 & 786 & 2\\,153 & 5\\,064\\\\\n     {\\bf $<8.1$} & 0.9 & 278 & 762 & 1\\,794\\\\\n     \\hline\n     \\end{tabular}\n   \\end{center}\n\\end{table*}\n\n\\section{Discussion and comparison with other surveys}\n\\label{sect:disc}\n\nHaving considered three different survey strategies designed to search\nfor CCSNe, specifically in low-metallicity environments (a\npointed survey of catalogued low-metallicity galaxies \nusing both a single and a network of\n2.0m telescopes, a volume-limited survey using the Pan-STARRS all-sky\nsurveys and finally a magnitdue limited survey using future all-sky\nsurveys), it is now appropriate to compare these different\nstrategies, the numbers of CCSNe we expect to detect and the various\nlimitations of each strategy.\n\nUsing a single 2.0m telescope or a network of seven 2.0m telescopes to\nperform a pointed survey of low-metallicty galaxies\nwithin $z=0.04$ we expect to\ndetect approximately \\mbox{1.3 CCSNe yr$^{-1}$} and \\mbox{9.3 CCSNe\nyr$^{-1}$} respectively in host galaxies of very low-metallicity,\nexhibiting oxygen abundances \\mbox{$12+{\\rm \\log(O\/H)} < 8.2$}. When\nconsidering the relative numbers of similar events that we would hope\nto detect using a volume-limited\nsurvey with PS1 we estimate that we\nwould detect approximately \\mbox{12.5 CCSNe yr$^{-1}$} and, extending\nthe search to $z=0.056$, PS4 would detect \\mbox{18.3 CCSNe\nyr$^{-1}$}. A drawback of the pointed low-metallicity survey is the\nfact that $\\sim$1\\,000 hrs of telescope time would be required of the\nnetwork of seven 2.0m telescopes. Unlike the network of 2.0m\ntelescopes, the all-sky surveys have the additional\nadvantage of not\nbeing limited by the number of galaxies that they can feasibly observe\nand as a result shall be capable of observing CCSN events in all\nenvironments and not only those in the lowest-metallicity host\ngalaxies, helping us to reduce the possibility of missing any rare\nCCSN events of extreme interest and also gaining the capability of\ndetermining accurate relative rates of SNe within a large but limited\nvolume, expecting to detect a total of \\mbox{$\\sim$570 CCSNe\nyr$^{-1}$} with PS1 and \\mbox{$\\sim$1\\,300 CCSNe yr$^{-1}$} with PS4.\n\nTo estimate the amount of 8.0m telescope time that we would require to\nspectroscopically follow the $\\sim$13 CCSNe found with the PS1 volume-limited survey in low-metallicity environments \\mbox{($12+{\\rm\n\\log(O\/H)} < 8.2$)}, we consider a CCSN occurring at the average\ndistance of the survey, that is \\mbox{$\\sim$120 Mpc}. Depending on how\nearly we catch this SN, we hope to observe it at various epochs during\nits evolution and into its tail phase. Suggesting an average absolute\nmagnitude of -15 for these epochs equates as an observed apparent\nmagnitude of 20.7 at 120 Mpc, including a typical galactic extinction\nof 0.3 magnitudes. To determine the time required from a typical 8.0m\ntelescope for spectroscopic follow-up we use the VLT FORS2 exposure\ntime calculator, selecting a suitable grism and requiring a\nsignal-to-noise of 50 per spectral element for each spectrum. To reach\nthis signal-to-noise level we would require a $\\sim$3\\,000 sec on\nsource integration. If we were to spectroscopically follow the 13\nevents, requiring 5 epochs of observations per SN, we would then\nrequire $\\sim$70 hrs of 8.0m telescope time per year.\n\nAn obstacle that would hinder both the pointed survey of\nlow-metallicity galaxies and the volume-limited Pan-STARRS surveys is\nthe incompleteness of any galaxy catalogue that could presently be\nproduced.  In the former case one would obviously need an all-sky\ntarget list for a pointed survey as we have demonstrated that the SDSS\nfootprint does not have enough low-metallicity galaxies to make a\nsearch particularly fruitful (only 1.3 CCSNe\\,yr$^{-1}$ is likely to\nbe found). In the case of the wide area search of Pan-STARRS one would\nideally like to quantify all potential low-metallicity systems within\na local volume of the Universe so that new discoveries can be quickly\ncross-matched and the optical transients in these galaxies immediately\nselected for follow-up.  Hence it would be extremely useful to have an\nSDSS DR5 type catalogue over the entire sky.  As previously discussed\n(see \\ref{sec_f2}) it is currently possible to compile a catalogue of\n$\\sim$ $10^5$ galaxies within the sky area covered by Pan-STARRS and\nwithin the redshift range $z<0.04$.  This catalogue is far from\ncomplete, however when PS1 completes a full cycle of sky observations\nwe would have a deep photometric catalogue with which to search for\nCCSNe and classify the host galaxies.\n\nIt will be challenging to identify and select genuine CCSNe in the\nmagnitude-limited, all sky-surveys.  Without a defined volume limit it\nbecomes impossible to describe accurate relative rates of SNe and\nwithout the prior knowledge of the SN hosts it will be very difficult\nto segregate CCSNe from the large numbers of other transitory events,\nespecially more distant SNe Type Ia in apparently faint (but perhaps\nintrinsically bright) hosts. Most of the SNe are likely to be too\ndistant to have any catalogued host and one of the challenges is to\nclassify objects from the information gathered in the all-sky surveys\nthemselves e.g. lightcurve matching, identification and photometric\nredshift measurement of the host, estimated energy of the transient.\nIf it is possible to overcome these challenges the numbers of detected\nCCSNe expected from these unbiased, all-sky and magnitude-limited\nsurveys shall overwhelm those numbers from the other survey straegies\nfar above an order of magnitude (see Table \\ref{CCSN_survey_rates}).\n\nA pioneering survey in this area has been the Texas Supernova Search\n(TSS), which has now evolved into the ROTSE Supernova Verification\nProject (RSVP)\n\\citep{2005AAS...20717102Q, 2007AAS...21110505Y}.  The Robotic Optical\nTransient Search Experiment (ROTSE) presently consists of four 0.45m\nrobotic telescope located around the globe, each with a 1.85 square\ndegree field of view, dedicated to detecting and observing optical\ntransients, with an emphasis on GRBs. The TSS has used the ROTSE-IIIb\n(MacDonald Observatory Texas) to perform a\nwide-field search for nearby SNe. \nThe RSVP uses all four ROTSE-III telescopes\nto discover SNe and other transient variables, by imaging areas of\nthe sky that host nearby galaxy clusters. The typical cadence is 1-3\ndays with a typical limiting magnitude of $\\sim$18.5 mags. To date\nthe ROSTE-IIIb telescope has discovered 50 SNe in total at mean redshift\nof $z=0.049$; 34 Type Ia and 16 Type II (5 of which have been\nclassified as Ic and 5 as IIn). The most remarkable result of this\nsmall scale SN survey is that they have claim over the discovery of\nthe three most luminous SN ever detected; SN2006tf (-20.7 mag),\nSN2006gy (-22 mag) and SN2005ap (-22.7 mag). SN 2006tf and SN 2006gy\nwere both classified as a Type IIn \\citep{2007CBET..793....1Q,\n2006CBET..644....1Q} and possibly had Luminous Blue Variable (LBV)\nstar progenitors with explosions speculated to have been triggered by\npulsation pair instability \\citep{2008arXiv0804.0042S,\n2006astro.ph.12617S,2007Natur.450..390W}. SN 2005ap was classified as\na Type II \\citep{2005CBET..116....1Q} and exhibited a broad H$\\alpha$\nP-Cygni profile. The explosion of the progenitor, which still had its\nHydrogen envelope intact, was possibly powered by either a GRB type\nengine or a pair-instability eruption\n\\citep{2007ApJ...668L..99Q}. Although these type of super-luminous SNe\nmay be intrinsically rare, it is more likely that they have remained\nundiscovered in the past due to the fact that they have been missed or\nmisclassified as AGN in pointed surveys as they have occurred close to\nthe cores of bright galaxies. The image-subtraction techniques used by\narea surveys, such as TSS and RSVP, will ensure that these types of\ninteresting events will continue to be discovered. \nTaking a spectrum of the SN Type IIn 1998S\nat peak (Fassia et al. 2000) and redshifting it to estimate the\n$K$-corrections that SNe similar to 2006gy would require at various\nredshifts, we predict that we would detect these events to $z\\sim 0.6$\nfor PS1, $z\\sim 0.8$ for PS4 and $z\\sim 0.9$ for LSST. For PS4 and\nLSST these limits correspond to a search volume $\\sim 2.9$ and $\\sim\n4.4$ times greater than PS1.\n\n\\begin{figure*}[b]\n  \\begin{center}\n  \\includegraphics[totalheight=0.15\\textheight]{5-SDSS-NSF-GALAXIES.ps}\n  \\caption{\\textrm{Of the 12 SNe discovered by the Nearby Supernova Factory that were hosted by SFGs in the SDSS DR5 catalogue, 5 were hosted by galaxies with 12+log(O\/H) $<$ 8.4. The host shown are SDSS J140737.23+384047 (SN 2007fg), SDSS J160205.11+294338 (SN 2007fz), SDSS J135042.68+400208 (SNF20080324-010), SDSS J154404.32+275335 (SNF20080515-004) and SDSS J150031.40+552210 (SNF20080614-002).}\\label{NSF-gal}}\n  \\end{center}\n\\end{figure*}\n\n\nThe Nearby Supernova Factory (NSF) uses ultra-wide\nfield CCD mosaic images from the Near-Earth Asteriod Tracking (NEAT) and Palomar-Quest survey\nprograms with the aim of finding low-redshift SNe Ia\n\\citep{2002SPIE.4836...61A}. Within the volume-limited ($z<0.04$) search with PS1 we predict relative-rates of SNe II, SNe Ib\/c and SNe IIn as 57.8\\%, 38.2\\% and 4.0\\% and at mean redshifts of 0.0318, 0.0316 and 0.0321 respectively. In its lifetime the NSF had discovered and classified 85 CCSNe within $z<0.04$ with relative-rates of SNe II, SNe Ib\/c and SNe IIn of 67.1\\%, 20.0\\% and 12.9\\% at mean redshifts of 0.0283, 0.0292 and 0.0284 respectively. The reason that the relative-rates of CCSNe types predicted by our PS1 volume-limited search and those discovered by the NSF are somewhat discrepant may be due in part to the photometric screening of SN candidates that the NSF perform prior to spectroscopically classifying candidates. The NSF aim to find young SNe Ia that are nearby yet are within the Hubble-flow and therefore seem to classify SN candidates that are on the rise and that are close to the limiting-magnitude of their search (R $\\sim$ 20 mags), as can be seen from the relative-magnitude distributions of classified SNe in Table \\ref{NSF_PS1}.  However it could be argued that because of the high-metallicity bias of many current and historic SN searches, the relative-rates of nearby CCSNe compiled by  \\cite{smartt_et_al}, and used in our simulations, are also biased toward higher metallicity. A high-metallicity bias in these relative-rates would be seen as an over-estimation of the true SN Ib\/c to SN II ratio; as seen when compared to the relative-rates discovered by the NSF. Further evidence supporting this hypothesis results from cross-matching NSF discovered SNe with our catalogue of SDSS DR5 SFGs within $z<0.04$. A total of 12 CCSNe are matched with SDSS galaxies, 5 of which occurring in galaxies  with 12+log(O\/H)$<$8.4 (see Fig. \\ref{NSF-gal}). This high fraction of CCSNe discovered in low-metallicity environments may again be due to selection criteria of the NSF (possibly looking for SNe occurring in apparently faint hosts) but remains encouraging for our plans to search for low-metallicity CCSNe with future all-sky surveys.\n\n\n\\begin{table}\n\\caption{\\textrm{Comparison between the relative magnitudes of CCSNe we predict with a volume-limited ($z<0.04$) search with PS1 and those discovered within the same volume by the Nearby Supernova Factory.}\\label{NSF_PS1}}\n  \\begin{center}\n    \\begin{tabular}{c c c}\n      \\hline\\hline\n      {\\bf Magnitude Range} & {\\bf PS1 Predictions}&{\\bf Classified by NSF}\\\\\n      \\hline\n      $< 16$ & 3.5 \\% & 3.5 \\%\\\\\n     16 - 17 & 6.1 \\% & 5.9 \\%\\\\\n     17 - 18 & 12.6 \\% & 3.5 \\%\\\\\n      18 - 19 & 20.3 \\%& 47.1 \\%\\\\\n     19 - 20 & 25.4 \\% & 40.0 \\%\\\\\n     $>20$ & 32.2 \\% & 0.0 \\%\\\\\n    \\end{tabular}  \n  \\end{center}\n\\end{table}\n\nAnother future all-sky survey that has potential for SNe discoveries is \nGaia : the European Space Agency's `super-Hipparcos' satellite with\nthe objective of creating the most precise three-dimensional map of\nthe Galaxy \\citep{2005ASPC..338....3P}. The satellite shall have many\nother additional capabilities including the ability to detect nearby\nSNe (within a few hundred Mpc), and predicted for launch in December\n2011 it is a potential competitor of Pan-STARRS and\nLSST. \\cite{2003MNRAS.341..569B} have performed a feasibility study\nsimilar to this study for Gaia and have predicted that the satellite\nshall detect \\mbox{$\\sim$1\\,420 CCSNe yr$^{-1}$} using a magnitude-limited\nsurvey strategy. Capable of detecting objects brighter than 20th\nmagnitude, 1.5 magnitudes brighter than the $g$=21.5 limit for PS1,\nGaia has the ability to survey a volume of only an eighth of the depth\nthat PS1 shall survey. \\citet{2003MNRAS.341..569B} employed a galaxy\ncatalogue which more than likely neglected low-luminosity galaxies,\nwhich would result in the CCSN rate being under-predicted by a factor\nof up to $\\sim$2 fewer SNe. Hence if we scale the\n\\citeauthor{2003MNRAS.341..569B} numbers by $\\sim$16, the final\nnumbers should be comparable with our PS1 estimates, predicting\n\\mbox{$\\sim$22\\,720 CCSNe yr$^{-1}$}, which is very close to the rate\nthat we have predicted for PS1 using our MCSs, \\mbox{$\\sim$24\\,000\nCCSNe yr$^{-1}$} (see Table \\ref{CCSN_survey_rates}). However, we note\nthat these numbers are likely somewhat over optimistic due to the\ntreatment of host galaxy extinction in our MCS.\n\n\\section{Conclusions}\nHaving determined oxygen abundances, star-formation rates and CCSN rates for all\nspectroscopically typed star-forming galaxies in the Sloan Digital Sky\nSurvey Data Release 5 within $z=0.04$, we have used Monte-Carlo simulations to predict the fraction of\nthese CCSNe that we can expect to detect using different survey\nstrategies. Using a single 2m telescope (with a standard CCD camera) \nsearch we predict a detection rate of\n$\\sim$1.3 CCSNe yr$^{-1}$ in galaxies with metallicities below\n$12+\\log({\\rm O\/H})<8.2$ which are within a volume that will allow\ndetailed follow-up with 4m and 8m telescopes ($z=0.04$). With a\nnetwork of seven 2m telescopes we estimate $\\sim$9.3 CCSNe yr$^{-1}$ \ncould be found, although this would require more than \n1000\\,hrs of telescope time allocated to the network. \nWithin the same radial distance, a \nvolume-limited search with the future Pan-STARRS\nPS1 all-sky survey should uncover 12.5 CCSNe yr$^{-1}$  in low metallicity\ngalaxies. Over a period of a a few years this would allow a detailed comparison of their properties. We have also extended our \ncalculations to determine the total numbers of CCSNe that can potentially be \nfound in magnitude-limited surveys \nwith PS1 (24\\,000 yr$^{-1}$, within $z \\lesssim 0.6$), \nPS4 (69\\,000 yr$^{-1}$, within $z \\lesssim 0.8$ ) \nand LSST (160\\,000 yr$^{-1}$, within $z \\lesssim0.9 $) surveys.\n\nAll considered, a final strategy chosen to searching for CCSNe in low-metallicity environments shall realistically involve both a volume-limited and a magnitude-limited all-sky survey in order to include a volume-limited galaxy sample with which to accurately determine relative SN rates and have some prior knowledge of the host galaxy characteristics, but yet not exclude the potential of detecting rare CCSN events that would have otherwise been missed had we only considered the volume-limited survey strategy. \n\nWith the huge number of CCSNe predicted to be detected, these all-sky surveys are set to serve as a catalyst concerning our understanding of CCSNe; including their varying characteristics with metallicity, the relative rates of the various types of SNe and of extremely rare events similar to SN 2006jc, SN 2006gy and more than likely events the nature of which have not yet been observed. \n\n\n   \n\\begin{acknowledgements}\n\n    Funding for the Sloan Digital Sky Survey (SDSS) and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, and the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web site is http:\/\/www.sdss.org\/.\n\n    The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, The University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.\n    \n  This research has made use of the CfA Supernova Archive, which is funded in part by the National Science Foundation through grant AST 0606772.\n  \n  This work, conducted as part of the award \"Understanding the lives of\nmassive stars from birth to supernovae\" (S.J. Smartt) made under the\nEuropean Heads of Research Councils and European Science Foundation\nEURYI (European Young Investigator) Awards scheme, was supported by\nfunds from the Participating Organisations of EURYI and the EC Sixth\nFramework Programme. SJS and DRY thank the Leverhulme Trust and\nDEL for funding. SM acknowledges financial support from the Academy of Finland, project 8120503.\n\n\\end{acknowledgements}\n\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\begin{figure*}\n\\includegraphics{smearing-explanation.pdf}\n\\caption{The transition to the double fringes packet for a binary. \\rev{Gray} curves show the fringes of the individual components \\rev{with vertical dotted lines indicating the position of their centres}. The \\rev{black} curve is the detected fringe packet (sum). In comic strip order: (i) An unresolved binary superimposes two  fringe systems and achieve maximum fringe contrast; (ii) Contrast loss arises when the binary is resolved because individual fringe packets do not overlap exactly; (iii) The packet is elongated and loses its original shape when the binary is sufficiently separated, this is the transition to (iv) two separate fringe packets are clearly seen.  Cases (i), (ii) are standard in interferometry. Case (iv) is easily analysed, but helps to understand why smearing occurs: each fringe packet doesn't seem impacted, but the power in its fringes is diluted by the incoherent flux from the other one; the resulting visibility will be a constant, strictly smaller than one, independent \\rev{of} binary separation.  This paper focuses on case (iii) that has not been thoroughly studied in the optical. Some of the notations of the paper are also reported: $\\beta^{ij}$, the decentering of the fringe packet of an on-axis source due to instrumental effects; $\\xobj[ij]{o}$ the OPD position shift of the fringe packet for an off-axis source; $\\phiobj[ij]{o}$ the same as the latter expressed in terms of a phase shift.}\n\\label{fig:smearing-explanation}\n\\end{figure*}\n\nLong-baseline interferometry is an observational technique used from the optical \\citep{MIC20} to the radio domain \\citep{RYL46,PAW46} that allows to overcome the resolution limit of single-dish telescopes, as ultimately set by diffraction. To achieve such a goal an ideal interferometer measures the complex degree of coherence and relates this so-called complex visibility to the object intensity distribution through a Fourier transform \\citep{VCI34,ZER38}.  Practically speaking, interference fringes are formed and their contrast and shift will be used to retrieve partial or total information on the complex visibilities.\n\nThere are numerous sources of error and biases that have to be evaluated and as much as possible corrected in order to provide a proper estimation of the interferometric observables. Among them, \\emph{bandwidth smearing} occurs in finite bandwidth for \\rev{objects spanning an extended field of view}. The interferogram corresponding to a point-like source has a coherence length of the order of $R\\lambda$ where $R$ is the spectral resolution and $\\lambda$ the central wavelength.  For two points of an extended sourc\\rev{e} separated by a distance $\\theta$ along the projected baseline length $B$ corresponding to two telescopes of the array, individual fringe packets are shifted with \\rev{respect to} each other by an optical path difference $\\theta\/B$.  When the OPD shift $\\theta\/B$ becomes of the order of, or greater \\rev{than}, the fringe packet width, i.e. when $\\theta \\approx R\\lambda\/B$, the fringe packets of these points do not overlap correctly and bandwidth smearing of the interferogram occurs (see bottom left panel of Fig.~\\ref{fig:smearing-explanation}).  In other words, one can consider that the coherence length of an interferogram $R\\lambda$ corresponds to an angular extension on the sky $\\theta \\approx  R\\lambda\/B$: it is called the interferometric field of view.  Objects composed of multiple \\rev{incoherent} sources, either discrete or continuous, are affected by the smearing when their extent becomes of the order of the interferometric field of view.\n\nFigure \\ref{fig:smearing-explanation} shows an illustration of that effect applied to the case of a binary system. Each of the sources \\rev{contributes} with a fringe packet; the observed interferogram is their sum. The distance between the interferograms is proportional to the angular separation. We can distinguish four separation regimes: 1) the unresolved case; 2) the resolved case where the separation is a small fraction of the interferogram envelope; 3) the smeared regime where \\rev{the separation} is not anymore a small fraction and interferometric estimators are altered; 4) the ``double packet'' regime where two fringes packets are well separated.\n\nWhile this effect \\rev{has been} known for decades \\citep{THO73}, it cannot be remedied by calibration as other biases. This was analysed in a review by \\citet{BRI89} in the radio-astronomy context, in which the observer had no other choice but to define, somewhat heuristically, the best compromise between observing performance and limiting the bandwidth smearing. However, modern radio techniques, using \\textit{a posteriori} software recombination, can overcome the problem in many situations by using several phase centres, around which smearing does not occur. In the optical and the infrared, software recombination is not technically feasible and bandwidth smearing must be dealt with. \\citet{ZHA07} \\rev{recommend to limit} the field of view $\\theta$ to $1\/5$ of \\rev{the} theoretical value \\rev{of the interferometric field of view} i.e. $\\theta\\approx R\\lambda\/(5B)$ to remain in the resolved regime. For an interferometer working in the near-IR with 100\\,m baselines, this corresponds to 5--10 milliarcseconds of separation when a \\rev{wide-band filter ($\\lambda\/\\Delta\\lambda \\sim \\rev{5}$)} is used without spectral resolution.  The main leverage to increase the interferometric field of view is adapting the spectral resolution or the baseline length. However, it comes very often at a prohibitive sensitivity cost (spectral resolution) or a loss of spatial \\rev{resolution} (baseline length).\n\nIn this paper, we present the first analytic calculation of the bandwidth smearing effect on the two main optical interferometric observables, namely the squared visibility and closure phase. We restricted the calculation to temporally encoded interferograms, including the so-called Fourier mode \\rev{(a full scan of the fringe packet)} and the \\rev{temporal} ABCD \\rev{(a 4-point scan of a central fringe)}, which are among the most popular optical schemes. Fourier mode has been or is being used at COAST \\citep{COAST}, IOTA with IONIC \\citep{IONIC} and FLUOR \\citep{IOTAFLUOR}, CHARA with FLUOR \\citep{CHARAFLUOR} and CLIMB \\citep{CLIMB}, VLTI with VINCI \\citep{VINCI}, PIONIER \\citep{PIONIER2,PIONIER}, and MIDI \\citep{MIDI}. \\rev{Temporal} ABCD is the choice at PTI \\citep{PTI} and the Keck Interferometer \\citep{KI}.  It should be stressed that a similar line of reasoning can be used with very little adaptation to the 8-point time-encoded interferograms of NPOI \\citep{NPOI}, and, with more efforts, to spatially encoded interferograms such as in VLTI\/AMBER \\citep{AMBER} and static ABCD systems such as VLTI\/PRIMA \\citep{PRIMA}.\nThe derived formula\\newrev{e} can \\rev{be} applied to correct \\emph{any} squared visibility and closure phase analytic formula describing the object angular intensity distribution. We apply this corrective technique to the study of binary stellar systems. Indeed optical long baseline interferometry is a unique tool to study the astrometry of close binary systems with milli-arcsecond accuracy to provide direct means to measure accurate masses. Moreover very recently several attempts at searching for substellar companions \\citep{ABS11,ZHA11} are pushing the technique down to dynamical ranges where no adverse effects can be neglected. Since \\rev{most studies forgo} bandwidth smearing correction \\rev{without assessing the biases that may arise from such approximation}, we felt a proper treatment had become mandatory and would be useful in the future. For practical purposes we used the PIONIER instrument characteristics to provide an application of that work. PIONIER is currently being used at the Very Large Telescope Interferometer \\citep[VLTI,][]{VLTI} to combine four beams in the H band ($1.5$ to $1.8\\mu\\mathrm{m}$).\n\nSect.~\\ref{sec:hypnot} gives the analytic expression of the observables in the absence of atmospheric turbulence for an instrument working in fringe-scanning mode.  Section~\\ref{sec:bin} is an application of these formulas to a binary star, which allows us to analyse the bias that smearing produces on the interferometeric observables and the model-fitted parameters of the binary.  We also show there how simulated fringes of PIONIER are much better fitted with the smeared model than with the standard expression. Finally, section~\\ref{sec:atm} studies the impact of atmospheric turbulence on the observables, indicating that a moderate spectral resolution is enough to alleviate most of its effects.\n\n\\section{Modelling the bandwidth smearing: turbulence-free case}\n\\label{sec:hypnot}\n\\label{sec:ana}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{PIONIER-transmission.pdf}\n\\caption{Spectral transmission and fringe packet envelope for PIONIER, as measured on an internal calibration with the 3-channel spectral setting of the H band. The left column display\\newrev{s} the spectral transmission and instrumental phase for a telescope triplet, as contained in $\\textsflux[ij]{\\text{lamp}}(\\wavenum - \\wavenumzero)$. The right column shows the envelope and phase of the fringe packet, given by the Fourier transform of the latter. The slope of the instrumental phase translates into a fringe packet decentering, known as group delay.  \\rev{Phases are expressed in radians.}}\n\\label{fig:PT}\n\\end{figure*}\n\n\n\\begin{table}\n\\caption{Principal notations of this paper.}\n\\label{tab:notations}\n\\begin{tabular}{ll}\n  \\hline\\hline\n  \\multicolumn{2}{l}{Indexing}\\\\\n  $o$, $p$, $q$                      & Source number (index)\\\\\n  $i$, $j$, $k$                      & Station number (superscript)\\\\\n  \\hline\n  \\multicolumn{2}{l}{Space and spatial frequency variables}\\\\\n  $\\wavenum$, $\\wavenumzero$              & Wavenumber, central wavenumber\\\\\n  $\\xi = \\wavenum - \\wavenumzero$         & Reduced wavenumber\\\\\n  $\\opdvar$, $\\textopd[ij]$               & Optical path difference\\\\\n  \\textxobj[ij]{o}                        & Fringe packet position in a perfect instrument\\\\\n  $\\textphiobj[ij]{o} = 2\\pi\\wavenumzero\\textxobj[ij]{o}$\n  & Fringe packet phase in perfect instrument\\\\\n  $\\textphishift[ij]{}$                   & Instrumental group delay (see Fig.~\\ref{fig:smearing-explanation})\\\\\n                                          & $\\rightarrow$ Packet position is \\smash{$\\xobj[ij]{o} + \\phishift[ij]{}\/2\\pi\\wavenumzero$}\\\\\n  \\hline\n  \\multicolumn{2}{l}{Functions of wavenumber $\\wavenum$ or reduced wavenumber $\\xi$}\\\\\n  $\\sfluxstar{o}(\\wavenum)$               & Spectrum of a point source\\\\\n  $\\texttrans[i]{}(\\xi)$                  & Transmission through an arm\\\\\n  $\\loss[ij]{}(\\xi)$                      & Instrumental contrast\\\\\n  $\\insphi[ij]{}(\\xi)$                    & Instrumental differential phase\\\\\n  $\\textsflux[ij]{o}(\\xi)$                & The equivalent of $\\newrev{N_iN_j}$\\\\\n  \\hline\n  \\multicolumn{2}{l}{Functions of OPD $\\opdvar$ or phase $\\alpha = 2\\pi\\wavenumzero\\opdvar$}\\\\\n  $\\textphasor[ij]{}(\\opdvar)$            & Complex coherent flux\\\\\n  $\\textsmearing{}(\\alpha)$               & Complex smearing\\\\\n  $\\textenvband{}(\\alpha)$                & Smearing amplitude\\\\\n  $\\phiband{}(\\alpha)$                    & Smearing phase\\\\\n\\hline\n  \\multicolumn{2}{l}{Fluxes}\\\\\n  $\\textflux{o}$                          & Flux of a point source\\\\\n  $\\textflux{}$                           & Total flux\\\\\n  $\\textflux[ij]{op}$                     & Flux product equivalent\\\\\n\\hline\n  \\multicolumn{2}{l}{Other}\\\\\n  $\\dopd[ij]{}$                           & OPD scanning speed\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nIn order to introduce the basic concepts of the data processing for fringe-scanning mode instruments, we remind \\rev{the reader} here how observables are derived \\rev{in monochromatic light}.\n\nIgnoring the atmosphere and instrumental signatures, the interferogram of a binary on baseline $ij$ can be written as\n\\begin{equation}\n  N^{ij}(\\opdvar) = \\rev{N}_1 \\big[1 +  \\cos 2\\pi\\wavenum\\opdvar\\big] \n  + \\rev{N}_2\\big[ 1 + \\cos(2\\pi\\wavenum\\opdvar + \\phiobj[ij]{}) \\big]\\newrev{,}\n\\end{equation}\nwhere $\\rev{N}_1$ and $\\rev{N}_2$ are the fluxes of each component, $\\opdvar$ is the OPD \\rev{between the arms of the interferometer}, and $\\textphiobj[ij]{} = (2\\pi\\wavenum\\textbase[ij]\\cdot\\textpos{})$ is proportional to the binary separation $\\textpos{}$, the projected baseline $\\textbase[ij]$, and wavenumber $\\wavenum$.\n\nIt is convenient to use the coherent flux, a complex quantity representing the interferogram, from which the continuum $\\rev{N}_1 + \\rev{N}_2$ is removed and the negative frequencies are filtered out. In practice, one can take the Fourier transform of the interferogram, remove all frequencies but a small interval centred on the frequency of the fringes, and take the inverse Fourier transform. \\rev{The coherent flux can be written as}\n\\begin{equation}\n  \\phasor[ij]{}(\\opdvar) =  \\exp{2\\pi\\j\\wavenum\\opdvar} \n  \\big[ \\rev{N}_1 + \\rev{N}_2\\, \\exp{\\j\\phiobj[ij]{}} \\big].\n  \\label{eq:Mmono}\n\\end{equation}\n\nThe \\rev{square visibility amplitude} is obtained by dividing the power contained in the coherent flux by that in the continuum:  \n\\begin{equation}\n\\begin{split}\n  \\vsqPS[ij]{} &= \\frac{<|\\phasor[ij]{}(\\opdvar)|^2>_\\opdvar}{(\\rev{N}_1+\\rev{N}_2)^2}\\\\\n               &= 1 - \\frac{4 \\rev{N}_1 \\rev{N}_2}{(\\rev{N}_1+\\rev{N}_2)^2} \\sin^2 \\frac{\\phiobj[ij]{}}2,\n\\end{split}\n  \\label{eq:Vmono}\n\\end{equation}\n\\rev{where $<x>_\\opdvar$ means the average of variable $x$ over the OPD.} In practice, the power may be computed using the Fourier \\rev{transform} of the coherent flux, which is strictly equivalent (Parseval's identity).\n\nWhen a triplet of telescopes $ijk$ is used, the closure phase is used to obtain partial information on the phase because it is independent \\rev{of} atmospheric turbulence.  It is the argument of the bispectrum given by:\n\\begin{equation}\n\\begin{split}\n  \\bisp[ijk]{} &=\\ <\\phasor[ij]{}(\\opd[ij](t))\n                  \\phasor[jk]{}(\\opd[jk](t))\n                  \\phasor[ki]{}(\\opd[ki](t))>_\\opdvar\\\\\n             &= (N_1-N_2)^2 \n                  + 4\\rev{N}_1 \\rev{N}_2\n                         \\cos\\frac{\\phiobj[ij]{}}2 \n                         \\cos\\frac{\\phiobj[jk]{}}2 \n                         \\cos\\frac{\\phiobj[ki]{}}2 \n                    \\\\\n                    &\\quad -4\\j \\, \\rev{N}_1 \\rev{N}_2(\\rev{N}_1-\\rev{N}_2) \n                    \\sin\\frac{\\phiobj[ij]{}}2 \n                    \\sin\\frac{\\phiobj[jk]{}}2\n                    \\sin\\frac{\\phiobj[ki]{}}2 ,\n\\end{split}\n\\label{eq:Bmono}\n\\end{equation}\nwhere $\\textopd[ij]$, $\\textopd[jk]$, and $\\textopd[ki]$ are\nthe time-modulated OPDs on the three baselines, meeting the closure\nrelation  $\\textopd[ij] + \\textopd[jk] + \\textopd[ki] = 0$. (Eq.~\\ref{eq:Bmono} gives \\rev{a compact, generic expression for the bispectrum in the same way \\citet{LEB12} did for the specific case of high-contrast binaries.})\n\nThe goal of this section is to describe the coherent flux, squared visibility, and closure phase of time encoded interferograms processed by means of Fourier analysis, when observing a source of arbitrary geometry in finite bandwidth. In other words, we seek to generalise Eqs.~(\\ref{eq:Mmono}, \\ref{eq:Vmono},~\\& \\ref{eq:Bmono}) and provide ready-to-use formulas to fit object models to smeared data.  For the sake of simplicity we use a discrete formalism valid for a collection of point-like sources.   The results presented here are easily generalised to systems of resolved, compact sources (Appendix~\\ref{ap:syscomp}) and to any system with our summations over a finite number of point-like sources replaced by integrations on the plane of the sky.\n\n\nThe most \\rev{frequently used} notations and symbols used in this section are given in Table~\\ref{tab:notations}.  \n\n\n\\subsection{Interferogram}\n\\label{sec:interferogram}\n\nWe consider an interferometer with stations $i$, $j$, etc. separated  by a baseline $\\base[ij]$ operating in a spectral channel centred on wavenumber $\\wavenumzero$.  In the following developments we shall use $\\wavenum$, the wavenumber, and $\\xi = \\wavenum - \\wavenumzero$ as ``reduced'' wavenumber.  Without losing generality, we assume that we observe an object made of several point sources $o$, $p$, etc. with positions $\\textpos{o}$, $\\textpos{p}$, etc. \\rev{in} the plane of the sky and spectra $\\textsfluxstar{o}(\\wavenum)$, $\\textsfluxstar{p}(\\wavenum)$, etc.\n\nThe interferometer measures the complex coherent flux of the electromagnetic field by forming dispersed fringes on a detector.  In our case, fringes are obtained by a temporal modulation of the optical path difference (OPD) $\\opdvar$ around an ideal position $\\xobj[ij]{o}$. This position is related to the angular position of the source in the sky $\\pos{o}$ through the relation $\\xobj[ij]{o} = \\base[ij]\\ensuremath{\\!\\cdot\\!}\\pos{o}$. Each of the point sources contributes to a quasi-monochromatic interferogram per instrument spectral channel. Once the incoherent photometric contribution has been removed from the two telescopes and the negative frequencies have been filtered out in Fourier space, the  complex coherent flux of one source reads: \n\n\\begin{equation}\n  \\phasor[ij]{o}(\\xi,\\opdvar) = \n               2\\sflux[ij]{o} (\\xi)\n              \\exp{\n                2\\jpi(\\wavenumzero+\\xi)(\\xobj[ij]{o} + \\opdvar) \n              }\n  \\label{eq:phasormono}\n\\end{equation}\n\nwhere $\\sflux[ij]{o} (\\xi)$ is the \\rev{``instrumental'' coherent flux density} \\rev{primarily} due to the wavelength-dependent instrumental effects\\rev{, but also to some extent to the spectrum of the source.} We can define this coherent flux density as:\n\n\\begin{equation}\n\\sflux[ij]{o}(\\xi) = \\loss[ij]{}(\\xi)\\sqrt{\\trans[i]{}(\\xi)\\trans[j]{}(\\xi)}\n                    \\,\\exp{\\j \\insphi[ij]{}(\\xi)}\n                    \\,\\sfluxstar{o}(\\wavenumzero + \\xi)\n      \\label{eq:cohernorm}\n\\end{equation}\nwhere:\n\\begin{itemize}\n  \\item $\\loss[ij]{}(\\xi)$, is the instrumental visibility, or instrumental contrast loss, \\newrev{and} has different origins such as differential polarisation or wavefront aberrations; \n  \\item $\\insphi[ij]{}(\\xi)$, is the instrumental differential phase, \\newrev{and} arises from a difference of optical path lengths between the arms of the interferometer that depends on the wavelength. For example this can be the case when light travels through glass (e.g waveguides, dichroics) that do not have the same refraction index dependence as a function of wavelength.\n  \\item $\\trans[i]{}(\\xi)$, is the spectral transmission through an arm including detector efficiency.\n\\end{itemize}\nWe assume that these instrumental signatures do not depend on the \\newrev{OPD position in the interferogram}, which is a good approximation in fringe-scanning mode\\newrev{, since the OPD modulation is obtained through a few micrometres of air or vacuum, with negligible dispersion. In other words, we assume that the instrumental differential phase is a static term that is not impacted by the movement of the differential delay lines.} However, this is usually not true for spatially dispersed fringes \\citep[see][for a generic expression for the fringes]{TAT06}, so that our approach needs adaptation to instruments like AMBER \\citep{AMBER}.\n\n\nIt is now possible to describe the coherent flux for an arbitrary number of sources and across a wider spectral bandpass:\n\n\\begin{equation}\n  \\phasor[ij]{}(\\opdvar) = \n     \\intinf \\sum_o \\phasor[ij]{o}(\\xi, \\opdvar) \\idiff\\xi, \n  \\label{eq:phasorgen}\n\\end{equation}\n\nFor practical purposes we use the Fourier transform\n\\begin{equation}\n  \\IFT{f}(\\opdvar) =  \\intinf f(\\xi) \\exp{2i\\pi\\xi\\opdvar} \\idiff\\xi,\n\\end{equation}\nsubstitute Eq.~(\\ref{eq:phasormono}) into Eq.~(\\ref{eq:phasorgen}), and\nobtain \n\\begin{align}\n  \\phasor[ij]{} (\\opdvar) = \n         \\sum_o\n           2\\IFT{\n             \\sflux[ij]{o}\n            }(\\xobj[ij]{o} + \\opdvar)\n            \\,\\exp{2\\jpi\\wavenumzero\\opdvar + \\j\\phiobj[ij]{o}}.\n  \\label{eq:def:phasor}\n\\end{align}\nwhere $\\textphiobj[ij]{o} = 2\\pi\\wavenumzero\\textxobj[ij]{o}$. In the following, we will use the coherent flux expression (Eq.~\\ref{eq:def:phasor}) to compute the most \\rev{commonly used} interferometric observables i.e. the square visibility and the closure phase.  In practice, $\\textsflux[ij]{o}$ is not known a priori. However, it can be inferred from fringes obtained on an internal lamp.  The coherent flux of the lamp fringes yield $\\textsflux[ij]{\\text{lamp}}$ (see Eq.~\\ref{eq:def:phasor}).  \\rev{If both the spectrum of the source $\\textsflux[\\star]{o}$ and that of lamp $\\textsflux{\\text{lamp}}$ are known, $\\textsflux[ij]{o} = \\textsflux[ij]{\\text{lamp}} \\textstrans[ij]{\\text{int}} \\, (\\textsflux[\\star]{o}\/\\textsflux{\\text{lamp}})$ (see Eq.~\\ref{eq:cohernorm}) where $\\textstrans[ij]{\\text{int}}$ is the transmission of the interferometer before the calibration lamp.  The amplitude of the VLTI transmission is a smooth function of wavelength that can be considered constant.  Its phase results from dispersive elements in the optical path.  The optical elements of the VLTI before PIONIER are all in reflection and the most dispersive ones (the M9 dichroics) have been designed to display the least differential dispersion, so that the dispersion is dominated by the air in the non evacuated delay line. In the rest of this paper, we have considered near-zenithal observations for which the interferometric delay is small so that the air dispersion could be ignored as Appendix~\\ref{ap:gd} shows. While the presence of dispersion in non zenithal observations has a significant impact on the amount of smearing, it neither changes its order of magnitude nor the general conclusions of this paper. When the atmospheric dispersion must be tackled, it can be done either explicitly (Appendix~\\ref{ap:gd} explains how) or implicitly by letting the parameters of Sect.~\\ref{sec:isr} free in model fits, as \\citet{ZHA07} do for the spectral resolution.}\n\nAs an illustration, we show in the left panels of Fig.~\\ref{fig:PT} the spectral coherence transmission \\textsflux[ij]{\\text{lamp}} (amplitude and phase) that we measured on the internal source of PIONIER using three spectral channels across the H band on three baselines.  The right panels correspond to the coherent flux of the fringes \\textphasor[ij]{\\text{lamp}} (amplitude and phase). \n\n\n\\subsection{Instrumental spectral response}\n\\label{sec:isr}\nIn this paper, after providing generic formulas using Fourier formalism, we will also give closed form expressions for direct use.  To do so, we need an analytic description of the instrumental transmission ($\\texttrans[i]{}$) and differential phase ($\\textinsphi[ij]{}$).  PIONIER's \\rev{instrumental coherent flux density} is obtained on a calibration lamp (Fig.~\\ref{fig:PT}, left panels)\\newrev{. It} displays a near-quadratic behaviour of the differential phase and a spectral transmission intermediate between top-hat and Gaussian functions.\n\nWe therefore describe the instrumental differential phase as\n\\begin{equation}\n  \\insphi[ij]{} (\\xi) = \\insphi[ij]{}(0) + \\phishift[ij]{} (\\xi\/\\wavenumzero) + \\disp[ij]{} (\\xi\/\\wavenumzero)^2. \n  \\label{eq:hyp:insphi}\n\\end{equation}\n\nThe linear term $\\textphishift[ij]{}$ in the instrumental differential phase $\\textinsphi[ij]{}(\\xi)$ translates into a fringe packet shift of $\\textphishift[ij]{}\/2\\pi\\wavenumzero$ with respect to the nominal zero OPD (see Fig.~\\ref{fig:smearing-explanation}, bottom right panel).  It is called group delay.  In a single-spectral channel interferometer it is possible to zero it by means of fringe tracking. When several spectral channels are observed at the same time, it is no longer possible to do so in all channels simultaneously. \\rev{For instance, if a central \\rev{spectral} channel is centred at zero OPD, adjacent channels may be shifted with respect to it if there is a differential behaviour of the dispersive elements (such as waveguides, dichroics, or air whose refractive index depend on wavelength) in the beam paths before the recombiner. In the bottom panels of Fig.~\\ref{fig:PT} (baseline 1-3), the central \\rev{spectral} channel is approximately centred at zero OPD (the solid line on the right panel \\newrev{shows the envelope of the fringe packet, i.e. the amplitude of the coherent flux}) with a slope of the phase averaging to $\\approx 0$ (same line of the left panel). The adjacent channels feature some shift (dashed lines on the right panels) and non-zero phase slope (same lines on the left). Appendix~\\ref{ap:gd} gives a further description of the group delay and its correction through fringe-tracking.} \n\nThe quadratic term in the instrumental differential phase $\\disp[ij]{}$ has a less visible impact on the fringe packet.\n\nWe will give results both for Gaussian and top-hat transmissions of FWHM $\\dwavenum{}$:\n\\begin{align}\n  \\transG[i]{}(\\xi)  &= \\wideexp{-\\frac{4 \\log 2}{\\dwavenum{}^2} \\xi^2},\n    \\label{eq:hyp:bandpass}\\\\\n  \\transH[i]{}(\\xi)  &=\n    \\begin{cases}\n      1 \\quad &\\text{if $|\\xi| \\le \\dwavenum{}\/2$},\\\\\n      0 \\quad &\\text{otherwise}.\n    \\end{cases}\n\\end{align}\n\n\n \n\n\n\\subsection{Square visibility amplitude}\nThe square visibility amplitude is obtained from the coherent flux using:\n\\begin{equation}\n   \\vsqPS[ij]{} \n      = \\frac1{4\\normtot[ij]{}}\n       \\intinf \\phasor[ij]{}(\\opdvar)\n            \\!\\cdot\\! \\conj{\\phasor[ij]{}(\\opdvar)} \\idiff\\opdvar,\n   \\label{eq:def:vsqPS}\n\\end{equation}\nwhere \\textnormtot[ij]{} is a normalisation factor that relates to the total flux of the target ($\\propto \\textfluxtot{}^2$) and \\rev{$\\conj{x}$ stands for the complex conjugate of $x$}. In the first line of the previous equation, we substitute Eq.~(\\ref{eq:def:phasor}) and expand the product into a double sum to find\\rev{:}\n\\begin{equation}\n  \\vsqPS[ij]{}\n  = \\frac1{\\normtot[ij]{}} \\sum_{o,p} \n    \\exp{\\j(\\phiobj[ij]{o} - \\phiobj[ij]{p})}\n    \\intinf\n      \\IFT{\\sflux[ij]{o}}(\\xobj[ij]{o} + \\opdvar) \n      \\IFT{\\sflux[ij]{p}}(-\\xobj[ij]{p} - \\opdvar) \n      \\idiff\\opdvar.\n\\end{equation}\nUsing the change of variables $\\opdvar \\rightarrow u = \\opdvar + \\textxobj[ij]{o}$, a correlation of Fourier transforms is recognised and simplified into the Fourier transform of a product.  Thus,\n\\begin{equation}\n  \\vsqPS[ij]{} = \\frac1{\\normtot[ij]{}} \\sum_{o, p} \n      \\IFT{\\sflux[ij]{o}\\sflux[ji]{p}}(\\xobj[ij]{o} - \\xobj[ij]{p})\n                 \\exp{\\j(\\phiobj[ij]{o} - \\phiobj[ij]{p})}.\n\\end{equation}\n\nThe bandwidth smearing is contained in $\\IFT{\\sflux[ij]{o}\\sflux[ji]{p}}$.  It\ncan be made clearer by introducing the complex smearing\n\\begin{equation}\n  \\smearing[ij]{op}(\\alpha) = \\frac{\n     \\IFT{\\sflux[ij]{o}\\sflux[ji]{p}}(\\alpha \/ 2\\pi\\wavenumzero) \n            }{  \\IFT{\\sflux[ij]{o}\\sflux[ji]{p}}(0)},\n  \\label{eq:gen:V2smearing}\n\\end{equation}\n\\rev{where $\\alpha$ is an angular variable that is linked to the OPD by the relation $\\alpha = 2\\pi\\wavenumzero\\delta$.} It \\rev{is} convenient to use the amplitude and phase \\rev{of the smearing}: $\\textenvband[ij]{op} = |\\textsmearing[ij]{op}|$ is the contrast loss due to smearing and $\\textphiband[ij]{op} = \\arg \\textsmearing[ij]{op}$ is a phase shift induced by it.  We also define the flux product equivalent---the equivalent to $\\flux{o}\\flux{p}$ in the monochromatic case---as\n\\begin{equation}\n  \\norm[ij]{op} = \\intinf \\sflux[ij]{o}(\\xi)\\sflux[ji]{p}(\\xi)\\idiff\\xi.\n  \\label{eq:def:norm}\n\\end{equation}\nWith these definitions, we can rearrange the square visibility amplitude:\n\\begin{equation}\n\\begin{split}\n   \\vsqPS[ij]{} =\n   \\sum_o & \\frac{\\norm[ij]{oo}}{\\normtot[ij]{}} \n        + \\sum_{o < p}\n        \\Bigg[\\frac{2\\norm[ij]{op}}{\\normtot[ij]{}}\n        \\envband[ij]{op}(\\phiobj[ij]{o}-\\phiobj[ij]{p})\\\\\n        &\\times \\cos \\left(\\phiobj[ij]{o}-\\phiobj[ij]{p} \n        + \\phiband[ij]{op}(\\phiobj[ij]{o}-\\phiobj[ij]{p})\n           \\right) \\Bigg]. \n\\end{split}\n\\label{eq:gen:vsqPS}\n\\end{equation}\nThese results are independent of the instrumental phase $\\insphi[ij]{}$.  If $\\textenvband[ij]{op} = 1$ and $\\textphiband[ij]{op} = 0$ (no smearing) this formula is equivalent to the monochromatic case (Eq.~\\ref{eq:Vmono} in the case of a binary). In practice, model-fitting of square visibility amplitudes by multiple stellar systems uses Eqs.~(\\ref{eq:gen:V2smearing}, \\ref{eq:def:norm},~\\& \\ref{eq:gen:vsqPS}).  Knowledge of $\\textsflux[ij]{o}$, needed in Eqs.~(\\ref{eq:gen:V2smearing} \\& \\ref{eq:def:norm}), can be inferred from fringes obtained on a calibration lamp (or a calibrator) if the spectra of both lamp and source $o$ are known, as we discussed in Sect.~\\ref{sec:interferogram}. \n\n\\def\\ensuremath{V_\\text{ins}}{\\ensuremath{V_\\text{ins}}}\nWhen the different sources share the same spectrum, i.e. $\\sfluxstar{o}(\\xi) \\propto \\sfluxstar{p}(\\xi)$, we may express the visibility as a function of the individual fluxes \\textflux{o} and the total flux \\textfluxtot{}. In Eq.~\\ref{eq:gen:vsqPS}, we then use the flux products in lieu of the flux products equivalents, i.e. $\\textflux[ij]{op} = \\ensuremath{V_\\text{ins}}\\textflux{o}\\textflux{p}$ and $\\textflux[ij]{} = \\textflux{}^2$, where \n\\begin{equation}\n  \\ensuremath{V_\\text{ins}}^2 = \\intinf \\loss[ij]{}(\\xi)^2\\trans[i]{}(\\xi)\\trans[j]{}(\\xi)\n  \\sfluxstar{}(\\xi)^2 \\idiff\\xi\\, \\Big\/ \\intinf \\sfluxstar{}(\\xi)^2 \\idiff\\xi \n\\end{equation}\nis the ``instrumental'' square visibility amplitude.  Note that $\\ensuremath{V_\\text{ins}}$ also depends on the spectral profile. It only disappears in the calibration if the calibrator has the same spectral profile as the source.  \n\nIn the cases of the Gaussian and top hat transmissions with FWHM $\\dwavenum{}$ around central wavelength $\\wavenumzero$ and a constant contrast loss $\\loss[ij]{}$ in the spectral channel, the smearing is purely real\n($\\phiband[]{} = 0$) and\n\\begin{subequations}\n\\label{eq:easy:vsqPS}\n\\begin{align}\n  \\envbandH{}(\\alpha)     \n  &= \\sinc\\left(\\frac{\\alpha}{2\\resol{}}\\right),\n    \\label{eq:C:vsqPS}\n  \\\\\n  \\envbandG{}(\\alpha) \n  &= \\wideexp{\\left(\n     -\\frac{\\alpha^2}{32\\resol{}^2\\log 2} \n   \\right)},\n  \\label{eq:gauss:vsqPS}\n\\end{align}\n\\end{subequations}\nwhere $\\resol{} = \\wavenumzero \/ \\dwavenum{}$ is the spectral resolution. For small enough baselines, we have shown in Appendix~\\ref{ap:smallsmearing} that an exponential formula can be used by properly choosing the value of $\\resol{}$.  On real data, $\\resol{}$ will need to be set to a value that differs from the spectral resolution in order to account from the departure from Gaussian profile and the wavelength dependence of the contrast.   In practice, a model fit of smeared data may leave it as a free parameter. If high precision is needed, the asymmetry of the spectral band and the slope of $\\loss[ij]{}$ give a non zero $\\gamma$. Cubic developments for the smearing terms $\\textenvband[]{}$ and $\\textphiband[]{}$ are given in Appendix~\\ref{ap:smallsmearing}.\n\n\n\n \n\n\n\n\\subsection{Closure phase}\n\\label{sec:ana:clo}\nA triple correlation or its Fourier transform, the bispectrum, or an equivalent method, is generally used to determine the closure phase \\citep{LOH83,ROD86}. The determination of the closure phase in direct space uses the phase of the bispectrum, given by:\n\\begin{equation}\n\\bispDS[ijk]{} = \\intinf \n                    \\phasor[ij]{}(\\opd[ij](t)) \n                    \\phasor[jk]{}(\\opd[jk](t)) \n                    \\phasor[ki]{}(\\opd[\\rev{ki}](t)) \n                    \\idiff t,\n\\label{eq:bispDS:1}\n\\end{equation}\nwhere $t$ is time in the case of linear OPD variations. By substitution of Eq.~(\\ref{eq:phasorgen}) into Eq.~(\\ref{eq:bispDS:1}) and writing $\\textopd[ij](t) = \\textdopd[ij] t$\n\\begin{equation}\n  \\bispDS[ijk]{} =\n                  \\sum_{o,p,q} \n                    \\intinf\n                    \\phasor[ij]{o}(\\dopd[ij] t) \n                    \\phasor[jk]{p}(\\dopd[jk] t) \n                    \\phasor[ki]{q}(\\dopd[ki] t)\n                    \\idiff t.\n\\label{eq:def:bispDS}\n\\end{equation}\nIt follows from Eqs.~\\newrev{(\\ref{eq:def:phasor} \\& \\ref{eq:def:bispDS})} and closure relation $\\textdopd[ij] + \\textdopd[jk] + \\textdopd[ki] = 0$ that\n\\begin{equation}\n\\begin{split}\n  \\bispDS[ijk]{} &\\propto\n  \\sum_{o, p, q}\n    \\Bigg[\n      \\exp{i(\\phiobj[ij]{o} + \\phiobj[jk]{p} + \\phiobj[ki]{q})}\\\\\n      &\\times\n      \\intinf \n        \\IFTsflux[ij]{o}(\\xobj[ij]{o} + \\dopd[ij] t)\n        \\IFTsflux[jk]{p}(\\xobj[jk]{p} + \\dopd[jk] t)\n        \\IFTsflux[ki]{q}(\\xobj[ki]{q} + \\dopd[ki] t)\n        \\idiff t\n    \\Bigg].\n\\end{split}\n\\label{eq:int:bispDS}\n\\end{equation}\nUsing the change of variables $t \\rightarrow u = \\xobj[ij]{o}\/\\textopd[ij] + t$, a triple cross-correlation of Fourier transforms can be recognised and expressed as the two-dimensional Fourier transform \n\\begin{equation}\n  \\IFTtd{\\ f \\ }(\\opdvar_1, \\opdvar_2) \n    =  \\iintinf f(\\xi_1, \\xi_2) \n            \\exp{2\\j\\pi(\\xi_1\\opdvar_1 + \\xi_2\\opdvar_2)} \n             \\idiff\\xi_1\\idiff\\xi_2\n\\end{equation}\nof the triple product\n\\begin{equation}\n  \\striple[ijk]{opq}(\\xi_1, \\xi_2) = \n     \\sflux[ij]{o}(\\xi_1) \\sflux[jk]{p}(\\xi_2) \\sflux[ki]{q} \n     \\Big(\n     - \\frac{\\dopd[ij]}{\\dopd[ki]} \\xi_1 \n     - \\frac{\\dopd[jk]}{\\dopd[ki]} \\xi_2\n     \\Big).\n  \\label{eq:def:striple}\n\\end{equation}\nThe bispectrum therefore reads\n\\begin{equation}\n\\begin{split}\n  \\bispDS[ijk]{} \\propto \n  \\sum_{o, p, q} \\Bigg[\n    \\IFTstriple[ijk]{opq} \\Big(\n    \\phiobj[ij]{o} - \\frac{\\dopd[ij]}{\\dopd[ki]} \\phiobj[ki]{q},& \n        \\frac{\\dopd[jk]}{\\dopd[ki]} \\phiobj[ki]{q}\n        - \\phiobj[jk]{p}\n      \\Big)\\\\\n      &\\times \\exp{\\j\\left(\\phiobj[ij]{o} + \\phiobj[jk]{p} + \\phiobj[ki]{q}\\right)}\n    \\Bigg].\n\\end{split}\n\\end{equation}\nThe bandwidth smearing is contained in $\\IFTstriple[ijk]{opq}$.  In order to make it clearer we need to introduce several terms.  The triple flux product equivalent (corresponding to $\\flux{o}\\flux{p}\\flux{q}$ in the monochromatic case) is given by \n\\begin{equation}\n  \\triple[ijk]{opq}      = \\left| \\IFTstriple[ijk]{opq}(0, 0) \\right|,\n  \\label{eq:gen:triple}\n\\end{equation}\nthe ``instrumental'' closure phase by\n\\begin{equation}\n  \\insphi[ijk]{opq} = \\arg \\IFTstriple[ijk]{opq}(0, 0), \n  \\label{eq:gen:insphi}\n\\end{equation}\nand the smearing by\n\\begin{equation}\n  \\smearing[ijk]{opq}(\\phivar_1, \\phivar_2) = \n      \\IFTstriple[ijk]{opq}( \\phivar_1 \/ 2\\pi\\wavenumzero, \n                            -\\phivar_2 \/ 2\\pi\\wavenumzero)  \n      \\,\/\\,\\IFTstriple[ijk]{opq}(0, 0).\n  \\label{eq:gen:smearing}\n\\end{equation}\nThe  ``instrumental'' closure phase is a flux-weighted mean over the spectral channel and thus also depends on the spectrum of the source.  The triple flux product equivalent can be simplified to the triple flux product ($\\texttriple[ijq]{opq} \\propto \\textflux{o}\\textflux{p}\\textflux{q}$) when the sources have the same spectrum, i.e. $\\textsfluxstar{o}(\\xi) \\propto \\textsfluxstar{p}(\\xi)$.  Note that the instrumental closure phase cancels out in the calibration only if the sources $o$, $p$, $q$ and the calibrator all share the same spectrum.\n\nWith these notations, the bispectrum reads\n\\begin{equation}\n\\begin{split}\n  \\bispDS[ijk]{} \\propto \\sum_{o, p, q} \n    \\Bigg[\n      \\smearing[ijk]{opq}\n    \\Big(\n    \\phiobj[ij]{o} - &\\frac{\\dopd[ij]}{\\dopd[ki]} \\phiobj[ki]{q}, \n        \\frac{\\dopd[jk]}{\\dopd[ki]} \\phiobj[ki]{q}\n        - \\phiobj[jk]{p}\n      \\Big)\\\\\n      &\\times\\triple[ijk]{opq} \\exp{i\\left( \\phiobj[ij]{o} + \\phiobj[jk]{p} + \\phiobj[ki]{q}\n      + \\insphi[ijk]{opq}\n                 \\right)}\n    \\Bigg].\n\\end{split}\n\\label{eq:gen:bispDS}\n\\end{equation}\nIf $\\textsmearing[ijk]{opq} = 1$ (no smearing) and $\\insphi[ijk]{opq} = 0$ (no bandwidth-related differential phase), the formula is equivalent to the monochromatic case (Eq.~\\ref{eq:Bmono} for a binary). In practice,  Eqs.~(\\ref{eq:def:striple}, \\ref{eq:gen:triple}, \\ref{eq:gen:insphi}, \\ref{eq:gen:smearing}, \\& \\ref{eq:gen:bispDS}) allow us to to model fit multiple stellar systems to smeared interferometric data. The knowledge of $\\textsflux[ij]{o}$ needed in Eq.~(\\ref{eq:def:striple}) can be inferred from calibration fringes obtained on an internal lamp (or a calibrator) as discussed in Sect.~\\ref{sec:interferogram}.  \n\n\\rev{This modelling} can be further simplified using an analytic description of the bandpass.  In that case, Eqs.~(\\ref{eq:gen:bispDS}~\\& \\ref{eq:bispDS}) can be used for the model fit of closure phases. In our cases of top-hat and Gaussian transmission of FWHM \\dwavenum{}, with a linear instrumental phase, we reorder baselines so that $\\textdopd[ki]$ has the largest absolute value, and we can assume it negative without losing generality.  Then, the smearing can be simplified to\n\\begin{subequations}\n\\label{eq:bispDS}\n\\begin{align}\n  \\smearingH[ijk]{}(\\phivar_1, \\phivar_2) &\\propto \n    \\sinc\\left(\n \n    \\frac{\\phivar_1 + \\phishift[ijk]{}}{2\\resol{}}\n    \\right)\n    \\sinc\\left(\n      \\frac{\\phivar_2 + \\phishift[ijk]{}}{2\\resol{}}\n    \\right)\n\\label{eq:gate:bispDS}\n\\\\\n  \\smearingG[ijk]{}(\\phivar_1, \\phivar_2) &\\propto\n  \\exp{ \n    - \\frac{\n    \n         (\\phishift[ijk]{} + \\phivar_1)^2\n    \n       + (\\phishift[ijk]{} + \\phivar_2)^2 + \n          \\left(  \n            \\phishift[ijk]{}\n            - \\frac{\\dopd[jk]\\phivar_1 \n                  + \\dopd[ij]\\phivar_2} \n                   {\\dopd[ki]} \n          \\right)^2 \n      }{\n        16\\resol{}^2\\log 2 \n        \\left(1\n        + \\big(\\frac{\\dopd[ij]}{\\dopd[ki]}\\big)^2 \n        + \\big(\\frac{\\dopd[jk]}{\\dopd[ki]}\\big)^2\\right) \n      }\n  }.\n\\label{eq:gauss:bispDS}\n\\end{align}\n\\end{subequations}\nIn the equations above, the ``group delay closure'' is expressed as\n\\begin{equation}\n  \\phishift[ijk]{} = \n       \\frac{\\dopd[ki]\\phishift[ij]{} - \\dopd[ij] \\phishift[ki]{}}\n            {\\dopd[ki]}\n      .\n\\label{eq:bisp:gd}\n\\end{equation}\nThe group delay closure is the consequence of the incorrect centering of the three fringe packets on the three baselines of the telescope triplets.  Because of this de-centering, the centres of these packets are not scanned at the same time.  In order to yield a usable closure phase, there should still be an overlap in the time intervals when the high contrast part of the packets are scanned.  It means that the individual group delays \\textphishift[ij]{}, \\textphishift[jk]{}, and \\textphishift[ki]{}, and thus the group delay closure, should be of the order of a few times the spectral resolution or less ($\\textphishift[ijk]{} \\lesssim 2\\pi\\resol{}$).  Since this overlap in time depends on the relative scanning speeds along the baselines, the group delay closure depends on $\\dopd[ij]$, $\\dopd[jk]$, and $\\dopd[ki]$.  \n\nIn our analytic approach to the spectral transmission, the instrumental closure phase reduces to a constant term, independent of \\newrev{the} source\\newrev{s}\\begin{equation}\n  \\insphi[ijk]{} = \\insphi[ij]{}(0) + \\insphi[jk]{}(0) + \\insphi[ki]{}(0).\n\\end{equation}\n\nAppendix~\\ref{ap:disp} explains how to use the Gaussian formula if the the quadratic chromatic dispersion term $\\textdisp[ij]{}$ is non zero. \n\n\n\n\n\n\n\n\n\n\\section{Consequence on companion search}\n\\label{sec:bin}\n\n\\subsection{Bias on the interferometric observables}\n\\label{sec:bias:observables}\n\n\\begin{table}\n  \\caption{Test case used in \\rev{Figs.~\\ref{fig:ideal}~\\& \\ref{fig:phi:jitt}}. For the square visibility amplitude, the first baseline is used.  The spectral resolution is, by definition, the major source of smearing.  In addition, the visibility is slightly impacted by the spectral dispersion $\\disp[ij]{}$.  The closure phase is strongly impacted by the group delay closure $\\phishift[123]{}$ (indirectly by group delays and OPD modulation speeds) and moderately by the dispersion $\\disp[ij]{}$.}\n\\label{tab:testcase}\n\\begin{tabular}{ll}\n\\hline\\hline\nBinary flux ratio                & 0.6\\\\\nEffective bandpass               & Gaussian\\\\\nSpectral resolution              & lines: 7, 18, 42, contours: 3--100\\\\\nProjected telescope positions    & $(0, B, 0.4B)$\\\\\n\\textit{Corresponding baselines} & $(B, -0.6B, -0.4B)$\\\\\nOPD modulation along baselines   & $\\dopd[ij] = (\\dopd[12], -2\\dopd[12], \\dopd[12])$\\\\\nOPD bounds                       & $(\\pm 25\\lambda, \\mp 50\\lambda, \\pm 25\\lambda)$\\\\\nGroup delays                     & $\\phishift[ij]{} = (5, 0, -5)\\times2\\pi$\\\\\n\\textit{Corresponding group delay closure} \n                                 & $\\phishift[123]{} = 10\\times 2\\pi$\\\\\nSpectral dispersion              & $\\disp[ij]{} = 0$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{figure*}[p]\n\\subfigure[Square visibility amplitude]{\\includegraphics[width=\\linewidth]{ideal-visibility.pdf}\\label{fig:ideal:Vsq}}\n\\subfigure[Closure phase]{\\includegraphics[width=\\linewidth]{ideal-closure.pdf}\\label{fig:ideal:phi3}}\n\\caption{Square visibility amplitude (top) and closure phase (bottom) of a binary with flux ratio 0.6 (test case of Table~\\ref{tab:testcase}) observed with an interferometer with Gaussian bandpass under ideal atmospheric conditions and baselines $B$, $-0.6B$, $-0.4B$.  In both figures, \\emph{top panel:} interferometric observable as a function of binary separation (milliarcseconds at one micron wavelength for a 100\\,m baseline) for an infinite resolution and three spectral resolutions approximately matching those of PIONIER.  \\emph{bottom panel:} deviation of the smeared observable with respect to the infinite spectral resolution case, shown as contours in the separation-spectral resolution plane.  In the lowest panel, the behaviour change around spectral resolution $\\resol{} = 8$ is explained by the transition from the single spectral channel mode (group-delay free in ideal fringe tracking conditions, \\rev{since a single fringe packet can be centred around zero OPD, see Appendix~\\ref{ap:gd}}) to the multiple channel observation (where \\rev{the fringe packets of the different spectral channels are shifted with respect to each other and therefore cannot be simultaneously positioned at zero OPD by the fringe-tracker, see Appendix~\\ref{ap:gd}}).}\n\\label{fig:ideal}\n\\end{figure*}\n\n\nThe first impact of the smearing is a tapering of the peak-to-peak amplitude of the oscillation of the visibility with baseline, hour angle, or spectral\nchannel, due to the smearing amplitude $\\envband{}$.  The second \\newrev{impact} only concerns the closure phase in multi-channel observations\\rev{. I}t originates from the imperfect alignment of the fringe packets on baseline triplets, \nas measured by $\\phishift[ijk]{}$.  In order to make these influences clearer,\nwe give in Fig.~\\ref{fig:ideal} the interferometric observables of a binary with a high flux ratio 0.6, whose characteristics are given in Table~\\ref{tab:testcase}.\n\n\\paragraph{Square visibility amplitude.}\nFigure~\\ref{fig:ideal:Vsq}, top panel, shows the theoretical smearing of the visibility amplitude of a binary as a function of reduced separation $\\theta B\/\\lambda$ (in $\\mathrm{mas}\\cdot\\mathrm{hm}\\cdot\\mu\\text{m}\\smash{^{-1}}$) for three different spectral resolutions ($\\approx 7, 18, 42$) corresponding to the observing modes available on PIONIER at the VLTI. The lower panel of the figure displays the error on the square visibility occurring from not taking smearing into account, as a function of separation and spectral resolution.   The result is easily generalised to binaries of different flux ratios, as the relative error on the visibility $\\Delta|V^2| \/ |V^2|$ remains unchanged.\n\n\\paragraph{Closure phase.}\nFigure~\\ref{fig:ideal:phi3}, top panel, shows the theoretical closure phase of a binary for three different spectral resolutions ($\\approx 7, 18, 42$) corresponding to the observing modes available on PIONIER at the VLTI.  It can be seen at small separations  (5--10\\,$\\mathrm{mas}\\cdot\\mathrm{hm}\\cdot\\mu\\text{m}\\smash{^{-1}}$) that the intermediate spectral resolution ($\\approx 18$) shows more smearing than expected for these separations, in particular more than the broad-band $\\approx 7$ observing mode.  The reason lies in \\rev{the dispersive elements in the light beams of the interferometer and instrument that decentre fringe packets more in some spectral channels than in others, thus making it impossible to centre all fringes packets at the same time. (see the imperfect centering of some spectral channels of PIONIER in Fig.~\\ref{fig:PT} and a description of the group-delay tracking in Appendix~\\ref{ap:gd})}. This effect is not seen in the broad band, where \\rev{the single fringe packet of each baseline can be centred with a fringe tracker, thus eliminating the group-delay}.  This low-separation smearing approximately scales linearly with separation, as $f\\textphishift[ijk]{}\\theta\/\\resol{}\\smash{^2}$, where $f$ is the flux ratio of the binary, $\\theta$ the separation, and $\\textphishift[ijk]{}$ the group-delay closure (This can be obtained analytically by linearising Eq.~\\ref{eq:gauss:bispDS} and normalising by the bispectrum of a point-source calibrator.)  At larger separations ($\\gtrsim 10\\mathrm{mas}\\cdot\\mathrm{hm}\\cdot\\mu\\mathrm{m}\\smash{^{-1}}$ in Fig.~\\ref{fig:ideal:phi3}), the closure phase is impacted by a combination of the tapering of the oscillation of the visibility (a purely spectral resolution effect, as seen in the visibility in Fig.~\\ref{fig:ideal:Vsq}) and the instrumental phase, the impact is relatively complex, and we can only recommend to use Eq.~(\\ref{eq:gauss:bispDS}) to model it.  As an illustration, Fig.~\\ref{fig:closim} of Appendix~\\ref{ap:closim} compares the closure phase of the three spectral channels of PIONIER for a given configuration of the interferometer, and it is quite clear the the behaviour radically changes with channel and telescope triplet.\n\nThe lower panels displays the error on the closure phase occurring from not taking smearing into account, as a function of separation and spectral resolution. The figure shows a sharp discontinuity at resolution $\\resol{} = 8$ where the transition occurs from a single spectral channel (where the single fringe packet of each baseline is positioned at zero OPD by an ideal fringe-tracker) to spectrally dispersed fringes (with the fringe packets \\rev{of each baseline} that do not align well \\rev{because they are shifted with respect with each other by the instrumental phase}). Even for moderately resolved sources, percent precision requires a good enough spectral resolution ($\\resol{} \\gtrsim 40$ or more), adequate modelling of \\rev{bandwidth} smearing, or a good fringe-tracking on a single spectral channel at moderate spectral resolutions ($\\resol{} \\gtrsim 10$). \n\n\n\\subsection{Retrieving binary parameters}\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{binobs-uv.pdf}\n\\caption{$(u, v)$ coverage of a typical 100\\,m baseline 4T observation (K0-A1-G1-I1) at the VLTI for an object close to the meridian, with 3 observations over a few hours.}\n\\label{fig:uv}\n\\end{figure}\n\nWe assess here the bias on the binary parameters that smearing produces. In order to model the data as \\rev{realistically} as possible we build synthetic binary fringes corresponding to a \\rev{typical scenario}: near-zenith object observed in a sequence of three sets of fringes separated by one hour using a large telescope quadruplet at VLTI (see Fig.~\\ref{fig:uv} for $u$, $v$ coverage).  They are obtained from calibration fringes obtained by PIONIER on an internal calibration lamp, which can be considered as a point source observation for our purpose. Then, we feed \\rev{these} synthetic data to the PIONIER data reduction software and get visibility amplitudes and closure phases. They are calibrated using simulated fringes of a point-source calibrator.  They are then fit with a binary model to derive the parameters of the binary. In a first step, the model is that of an unsmeared binary (Eqs.~\\ref{eq:Vmono}~\\& \\ref{eq:Bmono}), then we use the smeared model of Sect.~\\ref{sec:ana} with Gaussian bandpass (Eq.~\\ref{eq:gauss:vsqPS}~\\& \\ref{eq:gauss:bispDS}). \\rev{Additional transmission effects of the VLTI from the telescope up to the internal calibration lamp, positioned after the delay lines, have been ignored: the near-zenith observations we consider here are dominated by PIONIER's instrumental effects (as we discuss in Sect.~\\ref{sec:interferogram}).  For non zenithal observations, where the interferometric delay in the delay lines is several tens of metres, the air dispersion in the delay lines becomes a factor of the same order of PIONIER's instrumental phase and can be modelled using Appendix~\\ref{ap:gd}.}\n\nIn our analysis, the separations in right ascension and declination are varied from $-30$ to 30\\,mas or approximately 10 times the angular resolution the interferometer and the magnitude differences from 0.1 to 3.3 (flux ratios from 0.05 and 0.95).  For each point triplet of parameters, the difference between the fitted values and the input gives us the bias on the binary position and magnitude difference.  The reduced chi square was determined assuming a 2\\% accuracy on visibilities and 20\\,mrad on closure phases typical of single-mode instrument performances on bright objects (like PIONIER).  Figure~\\ref{fig:binobs-bias} shows the \\rev{absolute values of the errors} and reduced chi square at each separation and position angle at the given magnitude difference of 0.55 (flux ratio of 0.6). In Figure~\\ref{fig:binobs-bias-2}, we \\rev{consider possible biases and give} the median value of the \\rev{error with} its confidence intervals for a given binary separation, considering all the position angles and flux ratios at that separation.\n\n\\begin{figure*}\n\\includegraphics{binobs-bias.pdf}\n\\caption{Quality of least-squares model fitting of binary parameters to smeared interferometric observables.  These observables are derived from PIONIER synthetic fringes in the 3-channel spectral resolution ($\\resol{} \\approx 20$) using the data reduction pipeline.  The contour plots give the \\newrev{absolute value of the error in the model fits} for each position of the secondary assuming a binary flux ratio of 0.6. \\text{Left:} the binary model assumes monochromatic light and absence of smearing. \\text{Right:} the binary model assumes a Gaussian bandpass and takes into account the smearing. \\text{Top:} \\rev{absolute value of the} error on the binary separation. \\text{Middle:} \\rev{absolute value of the} error on the magnitude difference. \\text{Bottom:} reduced chi squares assuming 2\\% error on square\\newrev{d} visibilities and 20~mrad on closure phases.}\n\\label{fig:binobs-bias}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=\\linewidth]{bias-smearing-allratios.pdf}\n\\caption{\\newrev{The solid lines give the median value of the errors on the fitted binary parameters} as a function of binary separation.  \\newrev{If non zero and systematically of one sign, the median indicates a bias. The grayed area are the} confidence intervals for the errors (dark gray 1-$\\sigma$, light gray 2-$\\sigma$). At a given separation, all binary orientations and flux ratios were considered. \\text{Left:} the binary model assumes monochromatic light and absence of smearing. \\text{Right:} the binary model assumes a Gaussian bandpass and takes into account the smearing. \\text{Top:} \\newrev{error} on the binary separation. \\text{Middle:} \\newrev{error} on the magnitude difference. \\text{Bottom:} reduced chi square.}\n\n\\label{fig:binobs-bias-2}\n\\end{figure*}\n\n\\paragraph{Smearing-free binary model.}  A binary model with the classical expression for the visibility amplitude and closure phase (Eqs.~\\ref{eq:Vmono}~\\& \\ref{eq:Bmono}) is fitted to synthetic PIONIER data with the three-channel spectral resolution.  \nThe left panel of Fig.~\\ref{fig:binobs-bias} displays from top to bottom the \\newrev{absolute value of the error} on the secondary's position, the \\newrev{absolute value of the error} on the magnitude difference, and the reduced chi square for errors of 2\\% and 20\\,mrad on individual measurements of square visibility amplitudes and closure phases respectively.   We checked that the results for other flux ratios are similar. The \\newrev{errors (with median value and confidence intervals)} for the parameters are given in Fig.~\\ref{fig:binobs-bias-2} (left panel) as a function of separation when the flux ratio is allowed to vary between detectable limits (0.05 to 0.95). \\newrev{The median value of the error indicates a bias, if it is non zero and consistently of one sign.} \n\nThe main impact of the smearing is a degradation of the goodness of fit at all separations, followed by errors on the flux ratio and separation at moderate separations, and a clear bias of both observables at larger separations. In our models, the secondary is dimmer than the input of the simulation \\newrev{more often than not} and the separation tends to be smaller \\newrev{more often than not}. \\newrev{(For instance, the confidence intervals on the errors of Fig.~\\ref{fig:binobs-bias-2} show that the error on the separation is approximately 5 times more likely to be negative than positive at a separation of 30\\,mas.)} The \\newrev{apparent dimming of the secondary} is easily explained by the tapering of the fringe contrast that occurs due to smearing. The \\newrev{bias on separation} is independent of smearing as we will see later on. \n\nEven at moderate separations (5--10\\,mas) the reduced chi square is around 3.  However, the errors on the flux ratio and positions become significant (50\\,$\\mu$as and 20\\,mmag) only at higher separations ($\\gtrsim 15$ mas), as Fig.~\\ref{fig:binobs-bias}.  At first sight, it seems to contradict the trend of Sect.~\\ref{sec:bias:observables}. In that section, we have found a significant smearing of the closure phase at small separations, as a result of the imperfect centering of fringe packets in an observation with multiple spectral channels.  We easily reconcile these findings by noting that, as an average over the spectral band, the group delay is zero, i.e. both ends of the bands have group delays of same magnitude but opposite signs; thus their respective impacts on the observables approximately cancel out in the fit. The deviation of the individual spectral channels from the average over the band still explains the larger chi square. (Fig.~\\ref{fig:closim} in Appendix~\\ref{ap:closim} shows how the closure phases are impacted differently for the three spectral channels of PIONIER in low resolution mode.) \n\n\\paragraph{Smeared binary model.}  We performed similar fits to synthetic smeared fringes of a binary \\rev{by} using the Gaussian formulas for the smearing (see Sect.~\\ref{sec:ana}).  The \\newrev{absolute values of the errors} on the position and flux ratio are given for a binary with a flux ratio of 0.6 in the right panel of Fig~\\ref{fig:binobs-bias}. The \\rev{errors} on the position and magnitude difference, and the quality of the fit are given in the right panel Fig.~\\ref{fig:binobs-bias-2} for a wide range of flux ratios.  \\newrev{In Fig.~\\ref{fig:binobs-bias-2}, the median value of the error indicates a bias if it is non zero and consistently of one sign.}\n\nTaking the smearing into account eliminates most of the errors and bias on the flux ratio. It also largely improves the quality of the fit, with a reduced chi square of 3 found at significant separations ($\\gtrsim 15$\\,mas) in \\rev{most cases}.  The errors on the separation are improved at all separations but \\rev{the bias remains at larger separations}. We \\rev{have found that the bias is related} to the uncertainty on the effective wavelength of the interferometer, which varies by $\\approx 0.1$\\% across baselines on PIONIER; this phenomenon is independent \\rev{of} our adequate modelling of the smearing. \\rev{It is difficult to calibrate in the first place, because a deviation of the pie\\newrev{z}o scan speed from its nominal value has exactly the same observable consequence. (We note that including a proper spectral calibration in the instrument would solve for this problem.)}  At 30 mas of separation, \\rev{a 0.1\\% bias translates into 30$\\,\\mu$as}, which is what we indeed find: \\rev{the solid lines in the top panels of Fig.~\\ref{fig:binobs-bias-2} show this bias both in the monochromatic model and the smeared one.} At specific binary parameters, seen as high \\rev{error} values islands on Fig.~\\ref{fig:binobs-bias}, \\rev{the discrepancy} originates from the difference between the smeared visibility and the Gaussian model: This happens close to smearing-induced phase jumps (see Fig.~\\ref{fig:closim} of Appendix~\\ref{ap:closim} for a comparison between Gaussian smearing and simulated values). High contrast binaries \\rev{do not feature these phase jumps} and are not impacted.  For precision work \\rev{of high to moderate flux ratio binaries, we strongly recommend to discard closure phases} close to predicted jumps. \n\n\n\n\n\n\n\n\n\n\n\\section{Modelling the atmosphere}\n\\label{sec:atm}\n\\label{sec:atm:temp}\n\nThe estimators of the interferometric observables have been chosen to be mostly immune to atmospheric biases in the typical interferometric regime of a moderately resolved source, \\rev{i.e. when bandwidth smearing can be ignored}. In this section, we investigate possible biases when \\rev{bandwidth smearing becomes significant}, as \\citet{ZHA07} did for IOTA's closure phase estimator.\n\nFor temporal scanning, it is possible to write the differential piston---the variable differential phase induced by the atmosphere---as a function of OPD since time and OPD are linked \\citep[see for instance][]{jitter}.  The jittered coherent flux can be expressed as a function of the ideal coherent flux\n\\begin{equation}\n  \\phasorjitt[ij]{}(\\opdvar) = \n     \\phasor[ij]{}(\\opdvar + \\piston[ij](\\opdvar))\n     \\wideexp{\\left[\n       -\\frac16 \\left(\n         \\pi\\wavenumzero \n          \\pderiv{\\piston[ij]}{\\opdvar}(\\opdvar)\n        \\right)^2\\right],\n     }\n  \\label{eq:coherjitt}\n\\end{equation}\n\\rev{where $\\textpiston[ij]$ is the atmospheric differential piston on baseline  $ij$.} The exponential term is the contrast loss due to piston variation during the integration, of the order of one millisecond for one OPD step of a temporal scan.  It bears the assumption that the spectral envelope of the fringes does not have features as sharp as the fringe frequency and that the integration during one OPD step is fast enough (of the order of \\rev{a} millisecond in practice) to allow for a linearisation of piston.  \n\n\\subsection{Orders of magnitude}\n\\label{sec:atm:om}\nAn analytic approach to the atmospheric turbulence can be taken, using the \nassumption that scanning is fast enough for the piston to vary linearly during\na sub-second scan, i.e. $\\textpist[ij]{} = \\textpist[ij]{0} + \n\\textpist[ij]{1} \\textopd[ij]$, where $\\textpist[ij]{0}$\nis the group-delay tracking error and $\\textpist[ij]{1}$ a rate of piston \nvariation during scan. $\\textpist[]{0}$ and $\\textpist[]{1}$ are random variables when statistics over a large number of scans are derived. Using this approach, the coherent flux is:\n\\begin{align}\n\\begin{split}\n  \\phasorjitt[ij]{} (\\opd[ij]) &= \n    \\sum_o\n    2 \\IFTsflux{o}(\\xobj[ij]{o} + (1 + \\pist[ij]{1})\\opd[ij] + \\pist[ij]{0})       \n    \\\\&\\qquad \\times \\exp{\n        i\\phiobj[ij]{o} \n     + 2i\\pi\\wavenumzero[(1+\\pist[ij]{1})\\opd[ij] + \\pist[ij]{0}]\n     - \\frac16 (\\pi\\wavenumzero\\pist[ij]{1})^2 \n    }.\n\\end{split}\n\\label{eq:atm:phasor}\n\\end{align}\nThis approach can be used to determine the orders of magnitude of the atmospheric effects.\n\n\\paragraph{Visibility.} \nThe piston variation term $1 + \\textpist[ij]{1}$ comes as a product of the OPD variable in Eq.~(\\ref{eq:atm:phasor}), so we recognise it as a scaling factor.  $\\textpist[ij]{0}$ is a mere shift of the central OPD and has no impact---the square visibility does not depend on centering.   Therefore, we can link the jittered visibility to the ideal case: \n\\begin{equation}\n  \\vsqPS[ij]{\\text{jit}} = \\frac{1}{1+\\pist[ij]{1}} \\vsqPS[ij]{\\text{ideal}}\n  \\wideexp{-\\frac13 (\\pi\\wavenumzero\\pist[ij]{1})^2}.\n\\end{equation}\nThe impact of atmospheric jitter is independent \\rev{of} the geometry of the source and, thus, smearing.  For all separations it can be calibrated out if science target and calibrators are observed with similar atmospheric conditions.\n\n\\paragraph{Closure phase.}  The group-delay tracking term $\\textpist[ij]{0}$ can be seen as a fringe shift that adds to the predicted fringe position $\\textphishift[ij]{} \\rightarrow \\textphishift[ij]{} + 2\\pi\\wavenumzero\\textpist[ij]{0}$ and the linear variation of the piston can be seen as a scanning velocity change $\\textdopd[ij] \\rightarrow \\textdopd[ij](1 + \\textpist[ij]{1})$.   With these substitutions, the formulas of Sect.~\\ref{sec:ana:clo} can be used directly to determine the jittered closure phase. As we have seen, the predominant impact of the bandwidth smearing on the closure phase is the fringe decentering $\\textphishift[ij]{}$, so we expect the group-delay tracking errors to be the main source of bias. \n\n\\subsection{Numerical modelling}\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{pdf\/jittered-interferogram.pdf}\n\\caption{One of the simulated temporal scans. The deformation of the envelope is correlated with the piston slope and the accordion-like features to variations of its slope.  \\textit{Top:} piston; \\textit{Bottom:} simulated fringes.}\n\\label{fig:interf:jitt}\n\\end{figure}\nIn the high frequency regime the pistons at the different stations can be considered as uncorrelated when the baselines are larger than the outer scale of turbulence $\\mathcal{L}_0$ \\citep{KEL07}.  With a median value $\\mathcal{L}_0 = 22$\\,m at Paranal \\citep{MAR00} baselines of the medium and large telescope quadruplets used with PIONIER normally fulfil the criterium.  At other sites, for smaller baselines, or under relatively uncommon atmospheric conditions at Paranal, the pistons can be correlated. This correlation decreases the amount of atmospheric jitter for given coherence time and seeing, which in turns tends to decrease the bias on the interferometric observables. Therefore, we model the random piston $\\piston[i](t)$ using its spectral density\n\\begin{equation}\n  \\DFT{\\piston[i]}(\\nu) = A\\nu^{-B} \\exp{\\j\\Phi^i(\\nu)},\n\\end{equation}\nwhere $A$ and $B$ are constants and $\\Phi^i(\\nu)$ is chosen randomly for each sampled temporal frequency $\\nu$.  For Kolmogorov turbulence, the fast scan ($\\ll 1$\\,s) regime has $B = 17\/6$ \\citep{CON95} but there is \\rev{experimental evidence \\citep{DIF03}} that the slope is not as steep at VLTI, \\rev{with simulations by \\citet{ABS06} explaining it in terms of the piston induced at high frequency by the adaptive optics (imperfect) correction \\citep[``bimorph piston'', see][]{VER01} and wavefront errors produced by the injection into single-mode waveguides \\citep[``coupled piston'', see][]{RUI01}. \\citet{LIN99} have also measured a deviation from the Kolmogorov behaviour at PTI.} We used $B = 2$, which experimentally reproduces well the accordion features of temporal scans obtained under below average atmospheric conditions (see Fig.~\\ref{fig:interf:jitt}). We normalise $A$ to match the group-delay tracking rms in the differential piston $\\piston[ij] = \\piston[j] - \\piston[i]$.\n\nBy substituting in Eq.~\\ref{eq:atm:phasor}, we perform a numerical integration of Eqs.~(\\ref{eq:def:vsqPS}~\\& \\ref{eq:def:bispDS}) and obtain the jittered\nvisibility amplitude and closure phase.\n\n\\begin{figure*}\n\\includegraphics{jittered-closure.pdf}\n\\caption{Bias on the closure phase resulting from atmospheric piston in temporal scans, assuming that the static smearing is correctly modelled. The x-axis shows the reduced binary separation in milliarcseconds-hectometres of baselines per micron of wavelength (below) or the OPD between binary components (above). \\textit{Top:} bias and statistical errors for three spectral resolutions corresponding to PIONIER at the VLTI. \\textit{Bottom panel:} bias in the spatial resolution-spectral resolution plane. The bias decreases quickly with spectral resolution.}\n\\label{fig:phi:jitt}\n\\end{figure*}\n\n\n\\subsection{Bias on the observables} \nAs we have seen in Sect.~\\ref{sec:atm:om} there is little bias of the atmosphere on the square visibility amplitude and we could confirm it numerically. However, the bias can be substantial on the closure phase. Figure~\\ref{fig:phi:jitt} displays in its top panel the bias on the closure phase of our test-case binary as a function of separation, for the three spectral resolutions $\\resol{} = 7$, 18, 42 corresponding to PIONIER's modes.  \\rev{For each separation, baseline, and spectral resolution considered in the simulation}, 100 random scans with a \\rev{remaining scatter of the fringe tracking of $6\\lambda$ (typical value by average conditions) have been generated.  The closure phase on the telescope triplet is the average closure phase of the scans.  To better identify the biases}, the closure phase of a \\rev{jitter-free observation} has been subtracted from the results. In the lower panel of the figure, the bias on the phase is given in the separation-spectral resolution plane.  As one can see, the impact of the atmosphere is very important at low resolution but quickly vanishes for $\\resol{} \\gtrsim 20$. For three spectral channels across a typical IR band, the error on the phase is at most a few degrees or less. \n\n\\section{Discussion \\& Conclusion}\n\n\\subsection{Impact of the instrument and visibility estimator}\n\nAs already discussed by \\citet{PER05}, the square visibility amplitude is impacted differently for different estimators that otherwise would be equivalent in the absence of smearing. Not only is the amount of smearing different but the behaviour can be changed.  Because it is a popular recombination method and it illustrates this argument, we have given the formulas for the smeared complex visibility of a time-modulated ABCD recombiner in Appendix~\\ref{ap:ABCD}. In Sect.~\\ref{sec:ana}, we have seen that the square visibility amplitude is not impacted by the fringe centering in full scans processed by Fourier analysis : in Eq.~(\\ref{eq:gen:V2smearing}), smearing is independent \\rev{of} absolute source position---only on source distances $\\textphiobj[ij]{o} - \\textphiobj[ij]{p}$---and group delay $\\textphishift[ij]{}$. Conversely, the ABCD visibility estimator shows explicit dependence on $\\textphiobj[ij]{o}$ and $\\textphishift[ij]{}$ (see for instance Eq.~\\ref{eq:ABCD:gauss:smearing}), and this propagates to the square visibility estimator.  \n\nAlso, we have clearly put in evidence that instrumental features such as the OPD modulation scheme \\rev{(ABCD or Fourier mode, stroke speeds on the different baselines)} or the chromatic dispersion have a strong impact on the closure phase. In particular, the smearing behaviour of the closure phase of PIONIER (Fig.~\\ref{fig:closim}) shows different trends on different triplets or different spectral channels: on one hand, different telescope triplets are impacted differently because of the different OPD modulations; on the other hand, different spectral channels of the same triplet behave in different manners, as a consequence of different chromatic signatures.  While the square visibility amplitude did not show a strong dependence on instrumental signature for full scans processed by Fourier analysis (Sect.~\\ref{sec:ana}), this is not necessarily the case.  For instance, a time-modulated ABCD method displays impact for both visibility and phase (see Eq.~\\ref{eq:ABCD:gauss:smearing} in Appendix~\\ref{ap:ABCD}).\n\n\\rev{We therefore stress} that each data reduction pipeline and each instrument require their own modelling of the smearing.  In this paper, we have provided a generic formalism which can be used as is for VLTI\/PIONIER and probably with little adaptation to other instruments that temporally scan most of the fringe packet.  \n\n\\subsection{When only part of the fringe packet is sensed}\n\nAlso, our developments make the implicit assumption that most of the flux of \\newrev{the} fringe packet is measured, i.e. that the OPD range is significantly larger than the FWHM of the fringe envelope. Actually, our developments still hold if the centres of the fringe packets originating from the different parts of the source are scanned but the extremities of the fringe packet are cropped, providing that the cropping is not too aggressive. \\rev{In the case of PIONIER, the partial cropping on some baselines does not prevent a good agreement between simulated fringed and our analytic development, as Fig.~\\ref{fig:closim} shows.}   \n\nHowever, it is clearly not the case in the ABCD method when a fringe-tracker locks the recombiner on the ``central'' fringe \\citep[e.g][]{SHA80}.  While the smearing can be derived theoretically for this method (see Appendix~\\ref{ap:ABCD}), \\rev{its magnitude will depend on the location of the fringe (i.e the OPD) onto which the fringe tracker locks. In the aforementionned Appendix it is shown that the visibility depends on the position of a source which in turns depends on the value of the group delay \\textphishift[ij]{} (see Eq.~\\ref{eq:ABCD:beta}). For relatively compact objects, the fringe tracker locks onto the brighter fringe or a local zero of the group delay and possible biases are calibrated out when observing an (almost) point-like calibrator under similar conditions.  When a source is smeared, the fringe tracker does not necessarily behave in the same manner on source and calibrator, since there is no longer an obvious location of a central fringe (e.g. in the extreme case of a double fringe packet, it may lock on either packet). Therefore,} it is quite likely that instruments sensing the central fringe of sources more resolved than a few beam widths \\rev{(i.e. a few times the resolution power of the interferometer) will lead to altered measurements}, unless \\rev{(a)} a high spectral resolution \\rev{is used ($\\resol{} \\gg \\textphishift[ij]{}$ in Eq.~\\ref{eq:ABCD:beta})} or \\rev{(b) the fringe tracking scheme can be modelled with enough detail to know on which part of a given smeared fringe packet it locks}.  In particular, instruments that target high accuracy astrometry with the ABCD method like GRAVITY \\citep{GRAVITY} and PRIMA \\citep{PRIMA} will require that both the tracking reference and the science target are not very resolved.\n\n\\subsection{Image reconstruction}\nOur approach clearly targets parametric analysis, by providing formulas to model fit interferometric data by systems of compact sources.  Image reconstruction however, usually relies on the Fourier relation between visibility and image, a relation which is broken in finite bandwidth.  Thus, image reconstruction is made difficult as \\cite{BRI89} already noted in radio interferometry.\n\n\n\\subsection{Dealing with bandwidth smearing in practice}\nThe angle of attack of radio astronomers to limit bandwidth smearing\n(see e.g \\citet{BRI89}), is to restrict its effects either by\nincreasing the spectral resolution to optimise the interferometric\nfield of view or centering the phase tracking delay optimally to\nreduce the radial spread. Optical interferometry users do not have\nnecessarily such a flexibility. One of the important differences\nbetween the wavelength regimes is that, in the optical, because the\narrays have \\rev{many fewer} telescopes, most of the users do not actually\nreconstruct images but rather model directly the interferometric\nobservables. This \\rev{has} been done to an extreme level of precision\nwhere visibilities are measured to a fraction of percent\n\\citep[e.g.][]{Absil:2008} and closure phases to a fraction of a degree\n\\citep[see e.g][]{Zhao:2011}. The particularly large impact of the\nsmearing, even for moderately resolved sources, undermine the idea\nthat the parameters for a large number of objects might be derived\neffortlessly using the traditional techniques.\n\nIt therefore appears reasonable to adopt a two step strategy to deal with\nbandwidth smearing first by \\emph{limiting the static instrumental smearing\nby design} and secondly by \\emph{operating the instrument under\nconditions that allow a proper modelling of the induced biases}.\n\n\\emph{Limiting the instrumental smearing.} We have seen that the ``group delay\nclosure'' is the major contributor to a static smearing effect in the closure\nphase \\rev{for instruments that operate in Fourier mode}; it depends on the\ngroup delays and the OPD modulation scheme. The scanning speed scheme can be\nchosen so as to minimise the average group delay closures. For the\n\\rev{temporal ABCD, visibility amplitudes and closures phases are directly\nimpacted by the group delay, and this mitigation can longer be used.  Since the\ngroup delay is mostly produced by a static chromatic dispersion in the instrument (waveguides, optical elements), an} integrated approach\nto differential dispersion and birefringence compensation can be attempted as\ndiscussed in \\citep{LAZ12}. Solutions exist that can provide guided or free\nspace optics instrument with dispersion compensation \\citep{Vergnole:2005}.\n\\rev{Correcting the air dispersion in the delay lines in real time may prove\nmore difficult to implement than static correction of the dispersion in the optical elements, so that evacuated delay lines are probably part of the solution for larger baseline lengths ($\\gg 100$\\,m) \\newrev{and at shorter wavelengths where the air dispersion is larger}.}\n\n\\emph{Modelling the biases.} We have shown that bandwidth smearing can be\nmodelled provided that, a moderate spectral resolution is used (the first\nobvious step) \\rev{and} the \\rev{estimators of the observables are properly\ncalculated}.  In very low spectral resolution or in full-band ($\\resol{} \\sim\n5$) observations atmospheric effects must also be decently constrained. For the\nlatter, initial studies \\citep[e.g.][]{LIN99,DIF03} have shown the correlation\nbetween atmospheric turbulence and low frequency statistics of piston but these\nare not necessarily well adapted to the sub second exposure\n\\citep[e.g.][]{ABS06}. Dedicated further characterisation of piston statistics\n\\rev{vs. monitored atmospheric properties} would be needed. In summary, the\nultimate tool to obtain a smeared source's \\rev{properties} will simulate the\ninstrumental visibility numerically taking the instrumental signatures, in\nparticular a dedicated spectral calibration, and the atmosphere into account.\n\n\\subsection{Concluding remarks}\n\n\\beginrevision\nOptical interferometry is increasingly used for precise measurements of high flux ratios and\/or separation. Application of this precision techniques range from the detection of hot dust components around debris-disc host stars or the search for direct detection of hot Jupiters to the accurate astrometry of binary systems in search of precise mass determination. \n\nWe have focused our work on a rarely studied effect that can alter significantly these astrophysical measurements, the so-called the bandwidth smearing. This bias-inducing phenomenon arises from the wavelength-dependence in the characteristics of the instrument, the atmosphere, and the source. We have modelled its impact by analysing its influence on the instrumental fringe contrast and determined how it alters the visibility amplitudes and closure phases. The magnitude of this effect will depend, for a given instrument, on the spectral resolution and the extension of the observed field of view and in some cases on the atmospheric piston.\n\nWe have demonstrated analytically how to calibrate for this degradation in the context of popular temporal fringe scanning instruments and applied this analysis to the specific case of binary systems by computing the error or biases induced on the separation vector and flux ratio.\n\nWe have further discussed ``real-life'' constraints such as the influence of the atmospheric piston, the use of different fringe encoding schemes or the imperfections of the fringe tracking quality.  We believe that the current analysis can be used with little effort to correct for potential bandwidth smearing biases in almost any astrophysical case.\n\\endrevision\n\n\\section*{Acknowledgements}\n\\rev{We would like to thank an anonymous referee and Chris Haniff who helped us to improve this paper.  This research has made use of NASA's Astrophysics Data System, the free softwares maxima, Yorick, and python. It has been supported by Comit\\'e Mixto ESO-Chile and Basal-CATA (PFB-06\/2007).}\n\n{\\footnotesize\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Methods}\n\\noindent\n$\\textbf{Crystal growth and magnetic characterizations.}$ $\\rm{(MnBi_2Te_4)(Bi_2Te_3)_{\\emph{n}}(\\emph{n} = 1, 2,...)}$ single crystals were grown by flux method \\cite{N33}. Mn powder, Bi lump and Te lump were weighed with the ratio of Mn: Bi: Te = 1: 8: 13 (MnTe: $\\rm{Bi_2Te_3}$ = 1: 4). The mixture was loaded into a corundum crucible sealed into a quartz tube. The tube was then placed into a furnace and heated to 1100 \u00b0C for 20 h to allow sufficient homogenization. After a rapid cooling to 600 \u00b0C at 5 \u00b0C\/h, the mixture was cooled slowly to 585 \u00b0C (581 \u00b0C) at 0.5 \u00b0C\/h for $\\rm{MnBi_4Te_7}$ ($\\rm{MnBi_6Te_{10}}$) and kept at this temperature for 2 days. Finally, the single crystals were obtained after centrifuging. The centimeter-scale plate-like $\\rm{MnBi_4Te_7}$ and $\\rm{MnBi_6Te_{10}}$ single crystals can be easily exfoliated. Magnetic measurements of $\\rm{MnBi_4Te_7}$ and $\\rm{MnBi_6Te_{10}}$ single crystals were measured by the vibrating sample magnetometer (VSM) option in a Quantum Design Physical Properties Measurement System (PPMS-9 T). The temperature-dependent magnetization measurements are described in detail in Supplementary Materials.\n\\bigskip\n\n\\noindent\n$\\textbf{Preparation of the ultra-thin samples.}$ The $\\rm{MnBi_4Te_7}$ and $\\rm{MnBi_6Te_{10}}$ flakes with different thicknesses were first mechanically exfoliated on a polydimethylsiloxane (PDMS) substrate by the Scotch tape method. The exfoliated samples on PDMS substrates were then dry-transferred onto 285 nm $\\rm{SiO_2}$\/Si substrates with evaporated gold films. Then, a layer of PMMA was spin-coated on the thin flakes for protection.\n\\bigskip\n\n\\noindent\n$\\textbf{AFM characterization.}$ The thickness of the ultra-thin samples wwas verified by the atomic force microscopy characterization using the Oxford Cypher S system in tapping mode. According to the height line profiles, the $\\rm{MnBi_4Te_7}$ and $\\rm{MnBi_6Te_{10}}$ were confirmed to possess an alternated lattice structure of BT (1 nm) + MBT (1.4 nm) and  BT (1 nm) + BT (1 nm) + MBT (1.4 nm). See more details in Supplementary Materials.\n\\bigskip\n\n\\noindent\n$\\textbf{RMCD measurements.}$ The RMCD measurements were performed based on the Attocube closed-cycle cryostat (attoDRY2100) down to 1.6 K and up to 9 T in the out-of-plane direction. The linearly polarized light of 633 nm HeNe laser was modulated between left and right circular polarization by a photoelastic modulator (PEM) and focused on the sample through a high numerical aperture (0.82) objective. The reflected light was detected by a photomultiplier tube (THORLABS PMT1001\/M). The magnetic reversal under external magnetic field was detected by the RMCD signal determined by the ratio of the a.c. component of PEM at 50.052 kHz and the a.c. component of chopper at 779 Hz (dealt by a two-channel lock-in amplifier Zurich HF2LI). The errors of ratio of FM and AFM components are determined by the instability of the data acquired during RMCD measurements.\n\\bigskip\n\n\\noindent\n$\\textbf{STEM characterization.}$ Samples for cross-sectional investigations were prepared by standard lift-out procedures using an FEI Helios NanoLab G3 CX focused ion beam system. To minimize sidewall damage and make the samples sufficiently thin to be electronically transparent, final milling was carried out at a voltage of 5 kV and a fine milling at 2 kV. Aberration-corrected STEM imaging was performed using a Nion HERMES-100 operating at an acceleration voltage of 60 kV and a probe forming semi-angle of 32 mrad. HAADF images were acquired using an annular detector with a collection semi-angle of 75-210 mrad. EELS measurements were performed using a collection semi-angle of 75 mrad, an energy dispersion of 0.3 eV per channel, and a probe current of $\\sim$20 pA. The Mn-$L$ (640 eV) and Te-$M$ (572 eV) absorption edges were integrated for elemental mapping after background subtraction. The original spectrum images were processed to reduce random noise using a principal component analysis (PCA) tool. HAADF image simulations were computed using the STEM\\_CELL software simulation package matching the microscope experimental settings described above and using a supercell with a thickness $\\sim$20 nm.\n\n\n\\section{\\label{sec:level1}DATA AVAILABILITY}\nThe data that support the findings of this study will be available at an open-access repository with a doi link, when accepted for publishing.\n\n\\section{\\label{sec:level3}ACKNOWLEDGEMENT}\nThis work was supported by the National Key R\\&D Program of China (Grants No. 2018YFA0306900, No. 2017YFA0206301, 2019YFA0308602, No. 2019YFA0308000, and 2018YFA0305800), the National Natural Science Foundation of China (Grants No. 62022089, No. 12074425, and No. 11874422), Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant No. XDB33000000), Beijing Natural Science Foundation (Grant No. JQ21018 and BJJWZYJH01201914430039), and the fundamental Research Funds for the Central Universities (E1E40209).\n\n\\section{Author contributions}\nY.Y., S.Y., and X.X. conceived the project, designed the experiments, analyzed the results and wrote the manuscript.  S.Y. and X.X. conducted the RMCD measurements. H.W. and T.X. grew the $\\rm{MnBi_4Te_7}$ and $\\rm{MnBi_6Te_{10}}$ bulk crystals. M.X., S.T., and H.L. grew the $\\rm{MnSb_2Te_4}$ bulk crystal. Y.H. prepared the few-layer samples. Y.P. and J.Y. performed the magnetic characterizations of the bulk crystals. R.G. performed the STEM characteristics under the supervision of W.Z. Y.Z. and Z.L. helped with results analysis. All authors discussed the results and contributed to the manuscript.\n\n\\section{ADDITIONAL INFORMATION}\nCompeting interests: The authors declare no competing financial interests.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIt is well known that Einstein's General Relativity (GR) is plagued by short-scale divergences, be it in the context of the curvature singularities inside black holes, or the big bang and related singularities in cosmology. In the context of point masses, the the gravitational field close to the mass becomes singular, and curvature invariants diverge. These singularities ``survive'' the Newtonian limit, where they resurface as unbounded tidal forces. Seemingly, general covariance is not the guiding principle to ameliorate the situation. On the other hand, these singularities are unphysical and have to be dealt with. What to do?\n\nOne may hope that these issues will be resolved once a consistent UV completion of GR (``quantum gravity'') is known. In the meantime, we can instead attempt to find modifications of GR that feature finite potentials, regarding these modifications as the effective field theory limit of some more fundamental concepts, hitherto unknown. Before jumping into technical details, let us consider some modifications of Newtonian gravity by considering the field of a $\\delta$-like mass distribution\n\\begin{align}\n\\label{eq:delta-density}\n\\rho = 4\\pi G m\\delta(r) \\, .\n\\end{align}\nThen, the Poisson equation gives the well-known Newtonian potential of a point mass,\n\\begin{align}\n\\bigtriangleup \\phi = \\rho \\, , \\quad \\phi(r) = -\\frac{Gm}{r} \\, .\n\\end{align}\nThis potential is singular, and it gives rise to infinite tidal forces. In this essay, we shall uphold Eq.~\\eqref{eq:delta-density}, that is, the notion of sharply concentrated densities. Then, a simple Pauli--Villars-type regularization scheme of the Poisson equation can ameliorate the situation:\n\\begin{align}\n\\label{eq:modified-poisson}\n\\bigtriangleup (1 + M^{-2}\\bigtriangleup)\\phi = \\rho \\, , \\quad \\phi(r) = -\\frac{Gm}{r} \\left( 1 - e^{-M r} \\right)\\, ,\n\\end{align}\nand $M$ is a large mass scale such that for $M \\rightarrow \\infty$ one recovers the original Poisson equation. For finite $M$, however, the potential is now finite at the origin, $\\phi(0) = -GMm$. Note, however, that it is not regular since its derivative does not vanish: $\\phi'(0) = GmM^2\/2$. If we think of the Newtonian limit of GR, this would imply that the corresponding spacetime has a conical singularity at the origin. There is another problem, however, which is generic to higher-derivative modifications:\n\nTypically, they bring along massive ghost degrees of freedom, which in turn lead to unstable vacua upon quantization. In the above example, the Green function is\n\\begin{align}\nD(r) = \\frac{1}{\\bigtriangleup(1 + M^{-2}\\bigtriangleup)} = \\frac{1}{\\bigtriangleup} - \\frac{M^2}{M^2 + \\bigtriangleup} \\, ,\n\\end{align}\nthe second term of which corresponding to a ghost of mass $M$. The particle spectrum of this theory, then, will not only feature a massless graviton of helicity 2, but also a massive ghost mode \\cite{VanNieuwenhuizen:1973fi,Stelle:1977ry}.\n\nA recent approach \\cite{Modesto:2011kw,Biswas:2011ar} that avoids the excitation of ghost modes at tree level \\cite{Shapiro:2015uxa} is aptly called \\emph{ghost-free gravity}. To see how this works, let us consider the following modification of the Poisson equation:\n\\begin{align}\ne^{f(\\bigtriangleup)} \\bigtriangleup \\phi = \\rho \\, ,\n\\end{align}\nwhere $f(\\bigtriangleup)$ is some polynomial of the Laplace operator defined as a formal power series (we are forced to introduce again at least one scale $M$ such that we can build the dimensionless combination $M^{-2}\\bigtriangleup$ that enters the power series). Then, the propagator is\n\\begin{align}\nD(r) = \\frac{1}{e^{f(\\bigtriangleup)} \\bigtriangleup} \\, , \n\\end{align}\nbut by construction there are no new poles since the exponential function is nowhere zero on the real axis. This can be made more rigorous in a mathematical sense by noticing that the exponential of a polynomial is a so-called \\emph{entire function} which does not have any poles at finite distance from the origin. Then, one can apply Picard's little theorem for entire functions to ensure that the denominator of the propagator never goes to zero (except at the graviton pole), which is a fundamental mathematical statement that surpasses simple plausibility arguments.\n\nAs it turns out (and as we shall see below), the gravitational potential for some generic choices of the function $f(\\bigtriangleup)$ is regular at the origin. However, as a theory with an infinite number of derivatives, this ghost-free gravity is necessarily non-local at some characteristic scale $\\ell \\equiv M^{-1}$. What is the nature of this non-locality? In this essay we shall treat it as classical, that is, $\\ell \\gg \\ell_\\text{Planck}$. This is consistent with regarding ghost-free gravity as an effective description of gravity, obtained by coarse-graining some underlying, more fundamental quantum description. In that sense, $\\ell > 0$ may be regarded as a reminder that ghost-free gravity may have a non-classical origin.\n\nLet us summarize: in both higher-derivative and infinite-derivative theories of gravity, one can attain a regular Newtonian potential that is (i) regular at the origin, and (ii) decays like $\\sim 1\/r$ for large distances. On the classical side there are various attempts to understand the non-linear regime of non-local gravity, whereas on the quantum side the perturbative structures are still to be fully understood.\n\nIn the remainder of this essay, we would like to venture in a somewhat orthogonal direction and study the gravitational field of point sources at mesoscopic distances away from the point source, that is, not directly at the origin, and also not very far away where the gravitational field approaches the standard Newtonian $1\/r$-behavior.\n\n\\section{A framework for linearized higher-derivative gravity}\nLet us now briefly sketch a more rigorous framework that we can employ to study the linearized gravitational field of matter distributions in both higher-derivative and infinite-derivative gravity on a flat Minkowski background. Writing the metric as $g{}_{\\mu\\nu} = \\eta{}_{\\mu\\nu} + h_{\\mu\\nu}$, the most general action quadratic in $h_{\\mu\\nu}$ in $D=d+1$ spacetime dimensions can be written as\n\\begin{align}\n\\begin{split}\nS &= \\frac{1}{2\\kappa} \\int \\mbox{d}^D x \\Big( \\frac12 h^{\\mu\\nu}\\,a(\\Box)\\Box\\,h_{\\mu\\nu}-h^{\\mu\\nu}\\,a(\\Box)\\partial_{\\mu}\\partial_{\\alpha}\\,h^{\\alpha}{}_{\\nu} +h^{\\mu\\nu}\\, c(\\Box)\\partial_{\\mu}\\partial_{\\nu} h \\\\\n&\\hspace{75pt}  - \\frac12 h\\,c(\\Box)\\Box h \n+ \\frac12 h^{\\mu\\nu}\\,\\frac{a(\\Box)-c(\\Box)}{\\Box}\\partial_{\\mu}\\partial_{\\nu}\\partial_{\\alpha}\\partial_{\\beta}\\,h^{\\alpha\\beta}\\Big) \\, ,\n\\end{split}\n\\end{align}\nwhere ``$\\Box$'' denotes the d'Alembert operator. The two functions $a(\\Box)$ and $c(\\Box)$ are non-local \\emph{form factors} that satisfy $a(0)=c(0)=1$ in order to reproduce linearized GR at large scales. The above action is indeed the most general one since the Bianchi identities relate the possible choices of functions $f(\\Box)$ to just two independent functions.\n\nFor the sake of simplicity, let us consider a simple case where $a(\\Box)=c(\\Box)$. Then, for a stress-energy tensor of a point mass, $T{}_{\\mu\\nu} = m \\delta^t_\\mu \\delta^t_\\nu \\delta^{(d)}(\\vec{r}\\,)$, and the metric ansatz\n\\begin{align}\n\\label{eq:metric-ansatz}\n\\mbox{d} s^2 = -\\left[1-2(d-2)\\phi\\right]\\mbox{d} t^2 + (1+2\\phi)\\mbox{d} \\vec{r}^{\\,2} \\, ,\n\\end{align}\nwhere $\\mbox{d} \\vec{r}^{\\,2} = \\mbox{d} x_1^2 + \\dots + \\mbox{d} x_d^2$ is the metric of flat space in Euclidean coordinates $x_i$ ($i=1,\\dots,d$), one obtains the field equations\n\\begin{align}\na(\\bigtriangleup)\\bigtriangleup \\phi = \\frac{\\kappa m}{d-1} \\delta^{(d)}(\\vec{r}\\,) \\, .\n\\end{align}\nThe Green function for this static problem takes the form\n\\begin{align}\n\\label{eq:green-function}\nD(r) = \\frac{1}{(2\\pi)^{\\frac d2} r^{d-2}} \\int\\limits_0^\\infty \\mbox{d} \\zeta \\frac{\\zeta^{\\frac{d-4}{2}}}{a(-\\zeta^2\/r^2)} J_{\\frac d2 - 1}(\\zeta) = \\frac{1}{2\\pi^2 r} \\int\\limits_0^\\infty \\mbox{d} \\zeta \\frac{\\sin \\zeta}{\\zeta} \\frac{1}{a(-\\zeta^2\/r^2)} \\, .\n\\end{align}\nwhere $J_n$ denotes the Bessel function of the first kind, and in the second equality we inserted $d=3$ (which we shall concern ourselves with in what follows). The gravitational potential is then given by\n\\begin{align}\n\\label{eq:potential-master}\n\\phi(r) = -\\frac{\\kappa m}{d-1} D(r) \\, ,\n\\end{align}\nand is easy to see that for $a=1$ one obtains the well-known result $\\phi(r) = -Gm\/r$ as obtained in the Newtonian limit of GR in four spacetime dimensions ($d=3$).\n\n\\section{Gravitational Friedel oscillations}\nGiven the general solution of the potential, Eq.~\\eqref{eq:potential-master}, we can now study it shape in various higher-derivative as well as infinite-derivative theories of gravity. The Green function \\eqref{eq:green-function} can either be evaluated analytically or numerically; for the sake of this essay, let us focus on analytical results that can easily be written down.\n\nTo that end we shall consider higher-derivative theories of the following class, call them $\\mathrm{HD_N}$,\n\\begin{align}\na(\\Box) = 1 + (-\\Box\/M^2)^N \\, , \\quad N \\in \\mathbb{N} \\, ,\n\\end{align}\nas well as a class of infinite-derivative ``ghost-free'' theories, call them $\\mathrm{GF_N}$:\n\\begin{align}\na(\\Box) = \\exp\\left[ (-\\Box\/M^2)^N \\right] \\, , \\quad N \\in \\mathbb{N} \\, .\n\\end{align}\nClearly these theories satisfy $a=1$ for $M \\rightarrow \\infty$, so for large scales they will reproduce linearized GR. The Green functions can be calculated analytically, and one obtains (see also \\cite{Frolov:2015usa})\n\\begin{align}\n\\begin{split}\n\\label{eq:green-functions}\n\\mathrm{GR}  : \\quad & D(r) = \\frac{1}{4\\pi r} \\, , \\\\\n\\mathrm{HD_1}: \\quad & D(r) = \\frac{1}{4\\pi r} \\left( 1 - e^{-M r} \\right) \\, , \\\\\n\\mathrm{HD_2}: \\quad & D(r) = \\frac{1}{4\\pi r} \\left[ 1 - e^{-Mr\/\\sqrt{2}} \\cos\\left( Mr\/\\sqrt{2} \\right) \\right] \\, , \\\\\n\\mathrm{HD_3}: \\quad & D(r) = \\frac{1}{4\\pi r} \\left[ 1 - \\tfrac13 e^{-Mr} - \\tfrac23 e^{-Mr\/2} \\cos\\left( \\sqrt{3}Mr\/2 \\right) \\right] \\, , \\\\\n\\mathrm{GF_1}: \\quad & D(r) = \\frac{\\text{erf}(M r\/2)}{4\\pi r} \\, , \\\\\n\\mathrm{GF_2}: \\quad & D(r) = \\frac{M}{6\\pi^2}\\Big[ 3 \\Gamma\\!\\left(\\tfrac54\\right) {}_1\\!F\\!{}_3\\left( \\tfrac14;~ \\tfrac12,\\tfrac34,\\tfrac54;~ y^2 \\right) -2y\\Gamma\\!\\left(\\tfrac34\\right) {}_1\\!F\\!{}_3\\left( \\tfrac34;~ \\tfrac54, \\tfrac32, \\tfrac74;~ y^2 \\right) \\Big] \\, , \\\\\n\\mathrm{GF_3}: \\quad & D(r) = \\frac{M}{\\pi}\\Big[ -\\frac{1}{\\Gamma\\left(-\\tfrac16\\right)}{}_{1\\!}F{}_{\\!5}\\left( \\tfrac16;~ \\tfrac13, \\tfrac 12, \\tfrac 23, \\tfrac56, \\tfrac76;\\,-z^3 \\right) - \\frac{z}{2\\sqrt{\\pi}} {}_{1\\!}F{}_{\\!5}\\left( \\tfrac12;~ \\tfrac23,\\tfrac56,\\tfrac76,\\tfrac43,\\tfrac32;\\,-z^3\\right) \\\\\n&\\hspace{62pt} + \\frac{3z^2}{10\\Gamma\\left(\\tfrac76\\right)} {}_{1\\!}F{}_{\\!5}\\left( \\tfrac56;~ \\tfrac76,\\tfrac43,\\tfrac32,\\tfrac53,\\tfrac{11}{6};\\,-z^3\\right) \\Big] \\, ,\n\\end{split}\n\\end{align}\nwhere ${}_{p\\!}F{}_{\\!q}(a_1,\\dots,a_p;\\,b_1,\\dots,b_q;\\,z)$ denotes the generalized hypergeometric function and we defined the dimensionless radial variables $y \\equiv M^2 r^2\/16$ and $z \\equiv M^2r^2\/36$. See Fig.~\\ref{fig:potentials} for a visualization of the Green functions: in both higher-derivative and infinite-derivative gravity they are finite at $r=0$, whereas for GR the Green function diverges at the origin.\n\n\\begin{figure}[!htb]\n\\centering\n\\subfloat[Higher derivative theories for $N=1,2,3$.]\n{\n    \\includegraphics[width=0.5\\textwidth]{potentials-hd-log-log.pdf}\n}\n\\subfloat[Infinite-derivative theories for $N=1,2,3$.]\n{\n    \\includegraphics[width=0.5\\textwidth]{potentials-gf-log-log.pdf}\n}\n\\caption{The Green functions of $\\mathrm{HD_N}$ and $\\mathrm{GF_N}$ theories visualized for $N=1,2,3$. Whereas $N=1$ approaches the $1\/r$ power law directly, there are oscillations in cases of $N=2,3$.}\n\\label{fig:potentials}\n\\end{figure}\n\nThis constitutes a major insight of these calculations in the literature, and, at the linear level, one can easily extend these studies to $p$-branes in higher-dimensional Minkowski space \\cite{Boos:2018bxf}. Observe, however, the particular shape of the Green functions a bit closer. There appears to be a substructure: whereas the $N=1$ Green functions decay like $1\/r$ for large values of the dimensionless radial distance $M r$, there exist noticeable oscillations in the potentials for the cases $N=2,3$ \\cite{Modesto:2011kw,Modesto:2016ofr,Edholm:2016hbt,Conroy:2017nkc,Boos:2018bxf}; for a visualization, see Fig.~\\ref{fig:potentials}.\n\nThese oscillations have direct consequences for the local energy density $\\rho$ perceived by a static observer whose 4-velocity is tangential to $\\ts{\\xi} = \\partial_t$. For the metric ansatz \\eqref{eq:metric-ansatz} one obtains\n\\begin{align}\n\\rho \\equiv G{}_{\\mu\\nu} \\xi{}^\\mu \\xi{}^\\nu = (1-d)\\bigtriangleup \\phi \\, ,\n\\end{align}\nwhere $G{}_{\\mu\\nu}$ denotes the linearized Einstein tensor. For $N=1$ theories, this quantity is positive definite, whereas for $N=2,3$ it undergoes oscillations around zero that decay in strength with distance $M r$. See a visualization of this behavior in Fig.~\\ref{fig:energy-densities}.\n\n\\begin{figure}[!htb]\n\\centering\n\\subfloat[Higher-derivative theories for $N=1,2,3$.]\n{\n    \\includegraphics[width=0.5\\textwidth]{energy-densities-hd-log.pdf}\n}\n\\subfloat[Infinite-derivative theories for $N=1,2,3$.]\n{\n    \\includegraphics[width=0.5\\textwidth]{energy-densities-gf-log.pdf}\n}\n\\caption{The absolute value of the local energy density $\\rho \\equiv G{}_{\\mu\\nu}\\xi{}^\\mu\\xi{}^\\nu$, $\\ts{\\xi} = \\partial_t$, undergoes oscillatory behavior in the cases $N=2,3$, whereas there are no oscillations in the case $N=1$ (both for higher-derivative and infinite-derivative theories). Close to the origin, $M r \\approx 0$, one has $\\rho > 0$; at the points of diverging slope, the energy density vanishes. Between these points, it switches its overall sign.}\n\\label{fig:energy-densities}\n\\end{figure}\n\nUsing these diagrams, we can extract some typical wavelengths: from Eq.~\\eqref{eq:green-functions} it is clear that the wavelengths of oscillation are constant in the cases $N=2,3$ in higher-derivative gravity due to the explicit appearance of trigonometric functions. For the infinite-derivative theories the behavior is more involved. The oscillations still scale with $M{}^{-1}$, but they decay with increasing distance $M r$. A rather qualitative fit gives\n\\begin{align}\n\\mathrm{GF_2} : \\quad M\\delta_2 \\sim 9.68 \\, (M r)^{-0.28} \\, , \\qquad\n\\mathrm{GF_3} : \\quad M\\delta_3 \\sim 8.28 \\, (M r)^{-0.16} \\, ,\n\\end{align}\nbut a closer inspection reveals that the precise wavelengths oscillate over and under these curves, see Fig.~\\ref{fig:wavelengths} for more details.\n\n\\begin{figure}[!htb]\n\\centering\n\\subfloat\n{\n    \\bgroup\n\t\\def1.2{1.2}\n\t\\footnotesize\n\t\\begin{tabular}{rccrcc}\n\t$M r$ & $M\\delta_2\/2$ & ~~ & $M r$ & $M\\delta_3\/2$ \\\\ \\hline\n\t 4.59 & 3.16 &&  4.65 & 3.23 \\\\\n\t 7.75 & 2.76 &&  7.88 & 3.02 \\\\\n\t10.51 & 2.53 && 10.90 & 2.86 \\\\\n\t13.04 & 2.38 && 13.76 & 2.75 \\\\\n\t15.42 & 2.26 && 16.51 & 2.66 \\\\\n\t17.68 & 2.17 && 19.17 & 2.58 \\\\\n\t19.85 & ---  && 21.75 & ---\n\t\\end{tabular}\n\t\\egroup\n}\n\\qquad\n\\subfloat{\\adjustbox{raise=-6pc}{\\includegraphics[width=0.5\\textwidth]{wavelength-fit.pdf}}}\n\\caption{We evaluated the zeroes of the energy density for $\\mathrm{GF_2}$ theory and $\\mathrm{GF_3}$ theory numerically, from which we can read off (half the) wavelength $M\\delta_N$. Contrary to the higher-derivative theories, in infinite-derivative theories the wavelengths are not constant, but decrease with increasing $M r$. To first approximation, this can be described by simple power laws.}\n\\label{fig:wavelengths}\n\\end{figure}\n\nIt seems that these oscillations occur irrespective of the precise modification method of GR. While they are absent for $N=1$, we should remark that the case $N=1$ is somewhat degenerate for both classes of theories considered: in the higher-derivative framework the potential is indeed finite at the origin, but it's first derivative is non-zero, leading to a conical singularity (as well as a diverging energy density, see Fig.~\\ref{fig:energy-densities}). As it turns out, the scalar case of $N=1$ infinite-derivative theory has time-dependent instabilities \\cite{Frolov:2016xhq}.\n\nIn other words: for all regular versions of both higher-derivative and infinite-derivative gravity, these oscillations do occur at distances where $M r \\sim \\mathcal{O}(1)$ before they decay roughly like a power law. Since these theories are classical, and hence $M \\ll m_\\text{Planck}$, the typical distance $r \\sim M^{-1} \\mathcal{O}(1)$ might be accessible to experiment at some point in time.\n\nOscillations of energy density, somewhat similar to the ones we described here, are well known in condensed matter physics where they are called \\emph{Friedel oscillations} \\cite{Friedel:1952,Friedel:1954,Friedel:1958}: upon insertion of a positively charged impurity into a cold metal the overall charge density around this impurity exhibits spatial  oscillations. This effect is usually calculated at 1-loop using the random phase approximation wherein the photon propagator picks up a fermion-loop as a correction term. In other words, the screening mechanism of an electric charge inside a cold metal is to be treated as a scattering problem \\cite{Altland:2006}.\n\nThere is also a physically intuitive explanation: in the Jellium model, electrons in a metal at low temperature fill up the Fermi sphere up to a maximum momentum of $k_\\ind{F}$ while the positive ions form a rigid background structure. Electrons close to the Fermi momentum $k_\\ind{F}$ are most prone to interact with the impurity, and since these electrons are non-local objects (scale of non-locality $\\sim k_\\ind{F}^{-1}$), they cannot compensate the positive charge exactly: they overcompensate the charge, and thereby induce a spatially oscillating charge distribution.\n\n\\section{Discussion and conclusions}\n\nIn the recent literature, there has been a lot of focus on (i) the classical non-linear behavior and (ii) the perturbative quantum structure of infinite-derivative gravity. In this essay, we pursued a somewhat different direction by focussing on the linearized theory at mesoscopic distances, $M r \\sim \\mathcal{O}(1)$, where both the gravitational potential as well we the local energy density exhibits fluctuations, the latter one assuming negative values in some regions. For values of $M \\gg m_\\text{Planck}$ these oscillations might become observable at some point in the future \\cite{Accioly:2016qeb,Accioly:2016etf,Perivolaropoulos:2016ucs}.\n\nIn an analogy to condensed matter physics and Friedel oscillations in cold metals, we think it may be appropriate to call the oscillations described in this essay \\emph{gravitational Friedel oscillations.} Since they do not only appear in higher-derivative theories (wherein they can be interpreted as spurious effects occurring at the Pauli--Villars regularization scale due to the presence of complex poles \\cite{Accioly:2016qeb,Giacchini:2016xns}) and they survive the ghost-free limit, we think that these oscillations are of some physical relevance.\n\nAt the present stage, the perturbative structure of infinite-derivative ``ghost-free'' gravity is not fully understood \\cite{Shapiro:2015uxa,Giacchini:2016xns,Biswas:2013kla,Talaganis:2016ovm} (see, however, the recent work \\cite{Calcagni:2018gke,Buoninfante:2018mre}), and it is also not clear whether ghost-ridden higher-derivative theories can be considered physically viable classical theories at distances close to the Pauli--Villars regularizations scale $M$. It would also be interesting to study the linearized gravitational field in other non-local modifications of gravity \\cite{Hehl:2009es} arising from non-local constitutive relations rather than from a quadratic tensor action with somewhat ad hoc non-local form factors.\n\nSince the oscillations occur in both higher-derivative and infinite-derivative classes of theories, quite independently from one another, we hope that the above observations may prove helpful in extracting observational criteria on the gravitational potential at mesoscopic distances.\n\n\\section*{Acknowledgements}\nThe author benefited from discussions with Valeri P.\\ Frolov, Hennadii Yerzhakov, and Andrei Zelnikov (all Edmonton), as well as Breno Loureiro Giacchini (Rio de Janeiro), and is moreover grateful for a Vanier Canada Graduate Scholarship administered by the Natural Sciences and Engineering Research Council of Canada as well as for the Golden Bell Jar Graduate Scholarship in Physics by the University of Alberta.\n\n\\begin{singlespace}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDifferent natural landforms such as valleys, forests, and other areas have a significant impact on the propagation of radio waves\\cite{b1}. The forest area is covered by dense trees, which makes the multipath fading and non-line-of-sight channel very complex, and differs from the effects of urban buildings\\cite{b7}. Some main factors affecting radio wave propagation, such as the distance between the transmitting and receiving antennas, the height of the antenna, and the type of ground objects, are reflected in the path loss formula as variable functions\\cite{b8}. However, in different geographical environments, topographic relief, vegetation height and density, climate, and other factors have various degrees of influence on propagation\\cite{b9}. Therefore, when these propagation models are applied in specific environments, the corresponding variable functions should be different, and it is necessary to identify a reasonable channel model. The accuracy of the channel model is very important for network deployment, because the inappropriate channel models will lead to significant reduction of network coverage\\cite{b10}. \n\nDifferent typical propagation models have different characteristics, and they are applicable to different environment. Typical propagation models include Okumura Hata model\\cite{b18}, cost-231 Hata model\\cite{b19}, SPM model\\cite{b20}, etc. These models are suitable for cities, suburbs and villages, but not for forest areas. In\\cite{b16}, the authors proposed the Erceg model which is based on extensive experimental data collected by AT\\&T Wireless Services across the United States in 95 existing macro cells at 1.9GHz. The terrains are classified in three categories. Category A is hilly terrain with moderate-to-heavy tree density, category B is hilly terrain with light tree density or flat terrain with moderate-to-heavy tree density and category C is mostly flat terrain with light tree density. Soon later, In\\cite{b17}, Stanford University proposed Stanford University Interim (SUI) channel model is a set of 6 channel models representing three terrain types and a variety of Doppler spreads, delay spread and line-of-sight\/non-line-of-site conditions that are typical of the continental US. The terrain type A, B, C are same as those defined earlier for Erceg model. However, these models were proposed earlier and are mainly applicable to the North American environment.\n\nOther scholars have proposed channel models for specific forest environment, in addition to the  typical channel models. In \\cite{b11}, the authors studied the oblique leaf path of roadside woodland, including three vegetation types, and obtained the attenuation loss results of oblique path at C-band yielding 0.9 dB overall improvement and up to 20 dB regional improvement in root mean square errors. In \\cite{b12}, the authors investigated the propagation behavior of 28-GHz millimeter wave in coniferous forests and model its basic transmission loss, and proposed novel fully automated site-specific models. The root-mean-square deviations between model predictions and simulation results are 11.3 dB for an ITU woodland model and 6.8 dB for a site-specific model published in this paper. In \\cite{b13}, the authors characterized the wireless channel for a short-range, temperate, medium density forest environment at the 5-GHz band. In \\cite{b14}, the authors presented measurement results and propose empirical models of ultra-wideband signal propagation in forest environments after recording more than 22000 measurements at 165 sites in four different forest environments in Virginia and Maryland. However, these works only focus on the single scenario.\n\nIn this paper, we carry out propagation measurement campaigns in two different types of forest areas, and  record the measured values of signal propagation loss in those areas. Owing to the large attenuation of the signal in the forest area, the propagation effect of the lower frequency band is better, and in order to study the communication status of the emergency frequency band, so we adopted the 605 MHz frequency band for measurement. Then we use three classical large-scale path loss models and forest excess attenuation models to characterize the measured data, providing a comprehensive model comparison. Through the analysis of the results, we develop a new forest-specific path loss model, which has better performance than representative existing models.\n\n\\section{Description of measurements}\n\n\n\nIn order to study the impact of different forest areas on signal propagation, we selected two different types of areas which are Jiaozi snow mountain and Pudu-river dry-hot valley where channel propagation measurement campaigns were conducted in March 2022.\n\n\\begin{figure}[htbp]\n\\centerline{\\includegraphics[width=8.5cm]{xueshan.pdf}}\n\\caption{Geographical environment of Jiaozi snow mountain}\n\\label{fig}\n\\end{figure}\n\nJiaozi snow mountain is located at the junction of Luquan County and Dongchuan District in Yunnan Province, China, with a maximum altitude of 4344.1 meters (m), a minimum altitude of 2300 m, and a relative height difference of more than 2000 m. Jiaozi snow mountain belongs to seasonal snow mountain, which is the lowest snow mountain in the northern hemisphere. There are 15000 mu of Abies lanceolata primary secondary forest and Rhododendron forest, which are typical dense forest scenes. Fig. 1 shows the geographical scenario of Jiaozi snow mountain.\n\nPudu-river dry-hot valley is located in the Pudu river area in Yunnan Province as well. There are seven vegetation types, 11 vegetation subtypes, 17 formation groups, and 28 formations in the reserve, including dry-hot valley hard leaf evergreen oak forest, semi-humid evergreen broad-leaved forest, mountaintop bryophyte dwarf forest, cold temperate shrub, and cold temperate meadow. Fig. 2 shows the geographical scenario of Pudu-river dry-hot valley.\n\n\n\\begin{figure}[htbp]\n\\centerline{\\includegraphics[width=8.5cm]{hegu.pdf}}\n\\caption{Geographical environment of Pudu-river dry-hot valley.}\n\\label{fig}\n\\end{figure}\n\n\n\\begin{figure}[htbp]\n\\centerline{\\includegraphics[width=8.5cm]{equip.pdf}}\n\\caption{Transmitter and receiver equipment. The backpack base station  on the left is transmitter and the handheld device on the right is receiver.}\n\\label{fig}\n\\end{figure}\n\n\n\n\\begin{table}[htbp]\n\\caption{MEASUREMENT INFORMATION}\n\\begin{center}\n\\begin{tabular}{|l|l|}\n\\hline\nBase station information                & Parameters or information  \\\\ \\hline\nTransmitting antenna height             & 1.5 m                       \\\\ \\hline\nCoordinate of Jiaozi snow mountain      & (102.848226, 26.0845327)   \\\\ \\hline\nCoordinate of Pudu-river dry-hot valley & (102.7342136, 26.02112129) \\\\ \\hline\nBase station transmission power         & 43 dBm                      \\\\ \\hline\nBase station antenna type               & Omnidirectional antennas   \\\\ \\hline\nTransmit antenna gain                   & 5 dBi                       \\\\ \\hline\nReceiving antenna gain                  & 0 dBi                       \\\\ \\hline\nTransmission band                       & 605 MHz                     \\\\ \\hline\nCell reference signal power                               & 15.2 dBm                    \\\\ \\hline                         \n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIn the propagation measurement campaigns, we used a backpack base station as the signal transmitter (Tx) in a fixed position, equipped with an omni-directional antenna, while a handheld device was used as the receiver (Rx), with an omni-directional receiving antenna inside. Tx and Rx devices are shown in Fig. 3. The transmit power of the base station is 43 dBm, the transmit antenna gain is 2.5 dBi, the receive antenna gain is 0 dBi, and the carrier frequency is 605 MHz. The relevant information of the measurements is provided in Table \\uppercase\\expandafter{\\romannumeral1}. We recorded the longitude and latitude of every transmission and reception position, adopted the continuous wave as the signal source to transmit the signal, and carried out the on-board test on the preset routes. \n\nWe collected and recorded the pilot signal received power at the on-board test cell phone. Since the test was aimed to obtain the path loss data in the actual network, the test data truly reflects the propagation of broadband signals in the local wireless environment. As there is no need to set up a base station by itself, this test scheme is simple and convenient. Fig. 4 illustrates the measurement trajectory, and different colors of trajectory indicate different reference signal received power where green represents the minimum and red represents the maximum.  The measurement data in Jiaozi snow mountain and Pudu-river dry-hot valley will be shown in Fig. 5.\n\n\n\n\n\\begin{figure}[t]\n\\centering \n\\subfigure[]{\n\\label{Fig.sub.1}\n\\includegraphics[width=8cm,height = 5cm]{xueshan_luxian.pdf}}\\\\\n\\subfigure[]{\n\\label{Fig.sub.2}\n\\includegraphics[width=8cm,height = 5cm]{hegu_luxian.pdf}}\n\\caption{Measurement trajectory in Jiaozi snow mountain and Pudu-river Dry-hot valley. The colors in the graphs indicate reference signal received power value, where green and red represent the lowest and highest power, respectively.}\n\\label{1}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\centering \n\\subfigure[]{\n\\label{Fig.sub.1}\n\\includegraphics[width=9cm]{tu4.eps}}\\\\\n\\subfigure[]{\n\\label{Fig.sub.2}\n\\includegraphics[width=9cm]{tu3.eps}}\n\\caption{Measurement data set in Jiaozi snow mountain and Pudu-river dry-hot valley.}\n\\label{1}\n\\end{figure}\n\n\n\n\n\n\\section{Experimental results}\n\nAfter obtaining the data set, we consider three path loss models as the baseline generic model which are the alpha-beta\u2013gamma (ABG) model\\cite{b2}\\cite{b6}, the close-in free-space reference distance (CI) model\\cite{b2}\\cite{b6}\\cite{b3}, and the free-space path loss (FSPL) model. The expressions of CI, ABG, and FSPL are as follows:\n\n\\begin{small}\n\\begin{equation}\n\\mathrm{P L}_{\\mathrm{CI} }=10 n \\log_{10}{\\frac{d}{d_{0} } }+20\\log_{10}{\\left ( \\frac{4\\pi\\times 10^{9}}{c}\\right )}+20 \\log_{10}{f} \n\\end{equation}\n\\end{small}\n\\begin{small}\n\\begin{equation}\n\\mathrm{P L}_{\\mathrm{ABG} }=10 \\alpha\\log_{10}{d} +\\beta +10 \\gamma\\log_{10}{f}  \n\\end{equation}\n\\end{small}\n\\begin{small}\n\\begin{equation}\n\\mathrm{P L}_{\\mathrm{FSPL} }=20\\log_{10}{(\\frac{4\\pi fd\\times 10^{9}}{c}  ) }\n\\end{equation}\n\\end{small}\n\n\\noindent where $n$ denotes the path loss exponent (PLE), $d_{0}$ is the close-in free-space reference distance and is set to 1 m\\cite{b2}, $d$ is the 3-D T-R separation distance in meters, $\\alpha$ and $\\gamma$ are coefficients showing the dependence of path loss on distance and frequency, respectively, $\\beta$ is an optimized offset value for path loss in decibels,  $f$ is the carrier frequency in GHz,  $c$ is the speed of light.\n\nNote that the CI model has a very similar form compared with the ABG model, but has fewer model parameters and more solid physical basis\\cite{b2}\\cite{b6}. Since additional attenuation is caused by the occlusion of vegetation in the forest area, we use the ITU horizontal forest model\\cite{b4} as the excess path loss model.The expressions of ITU horizontal forest model is as follows:\n\n\\begin{small}\n\\begin{equation}\n\\mathrm{P L}_{\\mathrm{ITU-H} }=A_m\\left[ 1-e^ {\\left( -d\\mu\/A_m \\right)} \\right]\n\\end{equation}\n\\end{small}\n\n\\noindent where $\\mu$ denotes the specific attenuation for very short vegetative paths (dB\/m) and $A_m$ denotes the maximum attenuation for one terminal within a specific type and depth of vegetation (dB). Next, we combine FSPL model and ITU horizontal forest model to fit the measured data\\cite{b5}. The expression of FSPL-H is given by:\n\n\\begin{small}\n\\begin{equation}\n\\mathrm{P L}_{\\mathrm{FSPL-H} }=20\\log_{10}{ (\\frac{4\\pi fd\\times 10^{9}}{c}  )} +A_m\\left[ 1-e^ {\\left( -d\\mu\/A_m \\right)} \\right]\n\\end{equation}\n\\end{small}\n\n\\begin{figure}[t]\n\\centering \n\\subfigure[]{\n\\label{Fig.sub.1}\n\\includegraphics[width=8.7cm]{tu1.eps}}\\\\\n\\subfigure[]{\n\\label{Fig.sub.2}\n\\includegraphics[width=8.5cm]{tu2.eps}}\n\\caption{Measured path loss data and fitting results of the two classical models, the ITU forest excess attenuation model, and the proposed BHF model.}\n\\label{1}\n\\end{figure}\n\n\nFor short-distance specific forest scenes, we build a simple but powerful scene-specific model by more carefully characterizing forest-specific propagation loss, which can simplify the expression and parameters compared with directly combining two types of models presented above. We name the model Beijing University of Posts and Telecommunications horizontal forest model (BHF). The expression of BHF is as follows:\n\n\\begin{small}\n\\begin{equation}\n\\mathrm{P L}_{\\mathrm{BHF}}=10\\alpha  \\log_{10}{d}+\\beta+\\zeta \\tanh (d \/20)+20 \\log_{10}{f} \n\\end{equation}\n\\end{small}\n\n\\noindent where $\\alpha$ is a coefficient showing the dependence of path loss on the conventional log-scaled distance, $\\beta$ is an optimized offset value for path loss in decibels, $\\zeta$ is a coefficient characterizing the path loss caused by vegetation attenuation.\n\n\n\\begin{table}[htbp]\n\\caption{Optimized Model Parameters In The Baseline And Proposed Path Loss Models}\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\nSite                   & Model  & \\begin{tabular}[c]{@{}l@{}}$n$(CI)\\\\$\\alpha$(ABG)\\\\$Am$(FSPL-H) \\\\ $\\alpha$(BHF)\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}$\\beta$(ABG)\\\\$\\mu$(FSPL-H) \\\\$\\beta$(BHF)\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}$\\gamma$(ABG)\\\\ $\\zeta$(BHF)\\end{tabular} \\\\ \\hline\nJiaozi snow            & CI     & 3.8                                                   & -                                                  & -                                             \\\\ \\cline{2-5} \nmountain               & ABG    & 2.9                                                   & 31.8                                              & 2.0                                           \\\\ \\cline{2-5} \n\\multicolumn{1}{|c|}{} & FSPL-H & 40.0                                                  & 1.2                                               & -                                              \\\\ \\cline{2-5} \n                       & BHF    & 1.6                                                   & -1305.2                                           & 1407.0                                        \\\\ \\hline\nPudu-river             & CI     & 4.0                                                   & -                                                  & -                                              \\\\ \\cline{2-5} \ndry-hot valley         & ABG    & 1.9                                                   & 57.7                                              & 2.0                                           \\\\ \\cline{2-5} \n                       & FSPL-H & 43.8                                                  & 4.6                                               & -                                              \\\\ \\cline{2-5} \n                       & BHF    & 0.8                                                   & 48.3                                              & 64.2                                          \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\nThe BHF model contains the negative exponential characteristic attenuation caused by vegetation, where the changes are made on the basis of ITU horizontal model, and the additional attenuation is close to the hyperbolic tangent function. Compared with the ITU horizontal model in (4), the function about the additional attenuation of vegetation changes more gently in the BHF model.  We fit the BHF model and three baseline models by using the least-square method to find the optimal model parameters. The optimized model parameters are shown in Table  \\uppercase\\expandafter{\\romannumeral2}. It can be seen from the Table \\uppercase\\expandafter{\\romannumeral2} that the coefficients of the CI and FSPL-H models are relatively stable, with little change across environments, due to the free-space reference term acting as an anchor point. In contrast, the coefficients of the ABG and BHF models are strongly influenced by the environment.\n\nFig. 6 shows the fitting results of the CI, ABG, FSPL-H, and BHF models. It can be observed from Fig. 6 that the CI model and ABG model are two straight lines. Because the relationship between the parameter term and the 3D T-R separation distance in the expression is multiplication, when other coefficients are given, the function between path loss and distance in the logarithmic scale is a linear function. Since the adaptation ability of straight lines is relatively limited, the fitting errors of CI and ABG models are large. The CI model has fewer parameter variables than the ABG model, rendering larger errors. The relationship between $d$ and parameters of the other two models are more complex, so the fitting effect is better. It can be seen that the BHF model has stronger capability of adapting with the trend of measured data in Fig. 6(b).\n\n\\begin{table}[htbp]\n\\caption{Overall Models Performance}\n\\begin{center}\n\\begin{tabular}{|l|lll|l|}\n\\hline\n-                           & \\multicolumn{3}{l|}{Traditional}                                & Proposed \\\\ \\hline\nModel                      & \\multicolumn{1}{l|}{CI}    & \\multicolumn{1}{l|}{ABG}  & FSPL-H & BHF          \\\\ \\hline\nJiaozi snow mountain       & \\multicolumn{1}{l|}{4.6} & \\multicolumn{1}{l|}{4.1} & 3.6   & 3.0         \\\\ \\hline\nPudu-river dry-hot valley  & \\multicolumn{1}{l|}{13.1}  & \\multicolumn{1}{l|}{9.7} & 9.3   & 8.3         \\\\ \\hline\nNumber of model parameters & \\multicolumn{1}{l|}{1}     & \\multicolumn{1}{l|}{2}    & 2      & 3            \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\nWe use root-mean-square error (RMSE) and the number of parameters to quantify the fitting effect of the models, which are given in Table \\uppercase\\expandafter{\\romannumeral3}. Note that for single frequencies, $\\gamma$ in the ABG model is set to 2, thus there are actually two model parameters in the ABG model.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nIt can be seen from Table \\uppercase\\expandafter{\\romannumeral3} that the fitting error for Jiaozi snow mountain is smaller than that for Pudu-river dry-hot valley in general,  because the different types and densities of vegetation and the geographical environment have an impact on the transmission of signals. The terrain of Pudu-river dry-hot valley is steeper than that of  Jiaozi snow mountain. Accoding to Table \\uppercase\\expandafter{\\romannumeral3},  the RMSEs of the BHF model are 3.0 dB and 8.3 dB for Jiaozi snow mountain and Pudu-river dry-hot valley, respectively. This shows that the BHF model proposed in this paper has the best fitting effect overall and is more suitable for the forest environment. \n\n \n\\section{Conclusion}\n\n In this paper, we have provided results from real-world measurement campaigns to assess channel characteristics for the forest environment. The signal measurement data near Jiaozi snow mountain and Pudu-river dry-hot valley are used to compare the attenuation of vegetation with comprehensive large-scale path loss models. Inspired by these results, we have developed a new site-specific path loss model. Compared with typical traditional models, the model proposed in this paper yields significantly smaller fitting errors with acceptable computational complexity, and is thus more suitable for the forest environment.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nA problem arising throughout both nuclear theory---from {\\it ab-initio} nuclear\ntheory \\cite{Ekstrom2019,Piarulli2016} to density functional theory (DFT)\n\\cite{Bender2003,schunck2019energy}---and supervised machine learning is the\nfitting of a model to data. Formally, given a computer model $m$ evaluated at\ninputs $\\bm{\\nu}_1, \\ldots, \\bm{\\nu}_{\\nd}$, one seeks a parameter vector $\\xb \\in \\R^{\\nx}$ so that\nthe outputs $m\\left(\\bm{\\nu}_1;\\xb\\right), \\ldots, m\\left(\\bm{\\nu}_{\\nd};\\xb\\right)$\nagree with data $\\db=[d_1, \\ldots, d_{\\nd}]$ within the assumed uncertainties. For example, the inputs $\\bm{\\nu}$\nmight characterize a particular  configuration of the atomic nucleus (defined by   the number of its\nproton and neutron constituents).  For such a case, the\ndata might include observables such as experimentally measured binding energies and  charge\nradii. Or, generally, the inputs could correspond to 64-pixel by 64-pixel images, and the data could represent labels such as ``cat'' or ``banana.'' \n\nAlthough fitting can result in many different formulations of optimization problems, the most common form in \nthe physical sciences \nfollows a $\\chi^2$-convention wherein independence of errors is assumed and one seeks to solve\n\\begin{equation}\n\\label{eq:function}\n\\min_{\\xb \\in \\R^{\\nx}} f(\\xb), \\qquad \\mbox{where } f(\\xb) = \\sum_{i=1}^{\\nd} \\left(\\frac{m\\left(\\bm{\\nu}_i;\\xb\\right) - d_i}{\\sigma_i}\\right)^2,\n\\end{equation}\nwith $\\sigma_1, \\ldots, \\sigma_{\\nd}>0$ often interpreted as experimental and\/or model error bars \\cite{DNR14}. \nMore general types of such objective functions (also called ``loss functions'' or ``penalty functions'') include those that take into account correlations, such as\n\\begin{equation*}\n\\label{eq:correlated}\nf(\\xb) = \\sum_{i=1}^{\\nd} \\sum_{j=1}^{\\nd} w_{i,j} \\left(m\\left(\\bm{\\nu}_i;\\xb\\right) - d_i\\right) \\left(m\\left(\\bm{\\nu}_j;\\xb\\right) - d_j\\right).\n\\end{equation*}\n\nIn this paper we address the squared-loss formulation in \\cref{eq:function},\nwhich we generalize as the finite sum of squares of nonlinear functions of the\nparameter vector $\\xb$; that is,\n\\begin{equation}\n\\label{eq:genfun}\nf(\\xb) = \\sum_{i=1}^{\\nd} F_i(\\xb)^2.\n\\end{equation}\nThroughout the text, we refer to these general functions, $F_i$, as \\emph{component functions}.\nSince objective functions of the form in \\cref{eq:genfun} are found throughout supervised learning, \nmany optimization methods used for training machine learning models are applicable here.\nIn contrast to\nstandard fitting problems that arise in nuclear theory, however, the number of data,\n$\\nd$, used when training machine learning models tends to be massive. For\nexample, as of August 2020, the open images dataset \n\\cite{openimages20}\ncontained nearly 60 million image labels.  \nWhen fitting nuclear models, \nthe value of $\\nd$ is typically many orders of magnitude smaller; this is the case in the study conducted in this paper. \n\nA natural question is thus whether the algorithms used to train machine learning models can benefit the physicist who has a computer model and desires to solve fitting problems. \nHere we investigate the strengths and limitations of different optimization algorithms for minimizing \\cref{eq:genfun} through a case study from nuclear theory. We focus on the ``derivative-free'' case where gradients \n$\\nablab f(\\xb), \\nablab F_1(\\xb),  \\ldots, \\nablab F_{\\nd}(\\xb)$ \nand higher-order derivatives \nare unavailable for use in the optimization. \nThis is often the setting when the computer models are composed of iterative components \\cite{more2014nd,WSS15} or when dependencies on legacy computer codes pose obstacles to algorithmic differentiation \\cite{Berz1996}, which is a key enabling technology for deep learning \\cite{2015arXiv150205767G}.\n\n\nIn \\cref{sec:algorithms} we summarize the set of optimization algorithms tested. We focus on methods for local optimization (i.e., those that do not seek to asymptotically cover the entire parameter space) since we are interested in assessing performance within a budget of function evaluations.\nSuch a budget limits the applicability of global optimization algorithms. \nOur case study, involving the fitting of the Fayans energy density functional (EDF) to\ndata across the nuclear chart, is described in \\cref{sec:fayans}. This problem was selected in part because it shares characteristics with many fitting problems.\nFor this problem, there are $\\nd=198$ data, $\\nx=13$ parameters to be optimized, and correlations among the errors $m\\left(\\bm{\\nu}_i;\\xb\\right) - d_i$ are evident.\nNumerical results are presented in \\cref{sec:numerical}, and we summarize both the\nconsistency and efficiency of the tested algorithms. Our performance measures\nemphasize how the efficiency of optimization methods, as measured in function evaluations, can change depending on one's ability to evaluate components $F_i(\\xb)$ concurrently. \n\nAlthough our focus is on optimization-based approaches for training, these could also be used in a larger framework of statistical calibration (e.g., as discussed in \\cite{HigdonJPG15}).\n \n\n\n\n\n\\section{Derivative-free optimization\/training algorithms}\n\\label{sec:algorithms}\n\nWe consider five algorithmic families of iterative methods for local, unconstrained derivative-free optimization. \nThe first two algorithms are deterministic, and the latter three are randomized\n(sometimes called ``stochastic''). For randomized algorithms, the \nsequence of points in parameter space at which the component functions will be evaluated is generated stochastically\/nondeterministically by the method. \n\nThe randomized algorithms considered in this study are designed to have the ability to vary the number of component functions evaluated in any one iteration. \nThroughout this paper, we refer to this number as the \\emph{batch size}, denoted $\\nb$.\nIn our experiments, as is typical in such batch-sampling-based algorithms, a\nbatch of size $\\nb$ is generated by sampling uniformly $\\nb$ many times without\nreplacement from the integers $\\{1, \\ldots, \\nd\\}$. Hence, the maximum batch size $\\nb=\\nd$ corresponds to evaluating all of the component functions.\n\nWe now briefly describe each of the algorithm types, along with their hyperparameters and our implementation. \nFor additional details on these and other derivative-free optimization methods, we refer the reader to \\cite{Conn2009a,LMW2019AN}.\n\n\n\\subsection{Deterministic algorithms}\n\nIn general, deterministic methods have the property that, given a starting point\nand hyperparameter values, the sequence of points in parameter space generated by the method will be\nthe same every time the optimization is repeated. The deterministic methods considered here also assume that all of the\n$\\nd$ component functions in \\cref{eq:genfun} are evaluated before the next\npoint in parameter space to be evaluated is determined. \nThat is, we address a batch-synchronous, rather than asynchronous, environment; the latter may be appropriate when the component function evaluation time varies significantly and\/or individual component functions depend on relatively few parameters \\cite{Recht2011hogwild}.\n\n\\subsubsection{Direct search algorithm.}\n\\label{sec:nelder}\n\nThe Nelder-Mead simplex algorithm \\cite{NelderMead} is a popular direct search algorithm for general derivative-free optimization \\cite{PhysRevLett.67.1334,pres07}. The version tested here is from the MATLAB routine \\texttt{fminsearch} based on \\cite{JCL98}. \n\nThe Nelder-Mead algorithm determines a new point for evaluation by performing operations on a simplex\ndefined by $\\nx+1$ previously evaluated affinely independent points\n\\cite{Conn2009a,Wright2012,AudetHare2017}. The particular choice of operations is dictated \nby the $f$ values associated with each of the simplex's vertices. \nSince the algorithm bases its decision on the complete evaluation of $f$, accepted points $\\xb_k$ monotonically decrease the objective: $f(\\xb_0) > f(\\xb_1) > \\ldots$. \nMultiple complete evaluations of $f$ may be required before an acceptable point is found, and each of these evaluations corresponds to $\\nd$ component function evaluations.\n\nThe sole hyperparameter in our Nelder-Mead implementation is the initial simplex size. This size can be interpreted as defining the size of the neighborhood within which Nelder-Mead begins its search. \n\n\\subsubsection{Model-based trust-region algorithm.}\n\\label{sec:pounders}\n\nPOUNDERS \\cite{SWCHAP14} is a deterministic method that exploits the structural form in \\cref{eq:genfun} by constructing a local surrogate model of each component function $F_i$. POUNDERS was used in the optimization of the UNEDF family of energy density functionals \\cite{UNEDF0,UNEDF1,UNEDF2} and chiral nucleon-nucleon interactions \\cite{Ekstrom13,EksJPG15}.  \n\nThe surrogate models in POUNDERS are updated with each new function evaluation, and the algorithm assumes that all $\\nd$ component functions are evaluated at each point. \nA new point to evaluate \nis obtained by locally minimizing an\naggregation of the component surrogate models.  Thus, unlike the Nelder-Mead\nmethod, POUNDERS requires and exploits full knowledge of the individual \ncomponent function values $F_1(\\xb_k), \\ldots, F_{\\nd}(\\xb_k)$. \nSimilar to Nelder-Mead, \nsince POUNDERS evaluates all component functions, accepted points monotonically decrease the objective, and multiple such evaluations of $\\nd$ components may be required before decrease in the function value is found.\n\nThe primary hyperparameter in POUNDERS is the radius used to define the initial neighborhood within which the surrogate models are constructed and optimized over.\n\n\n\\subsection{Derivative-free stochastic approximation}\n\\label{sec:sgd}\n\nIn supervised learning tasks of machine learning---a class of optimization problems containing \\cref{eq:genfun}---the\nworkhorse optimization method for obtaining (approximate) solutions to \\cref{eq:genfun} has been \nstochastic gradient methods \\cite{RobbinsMonro1951}. \nFor an excellent contemporary survey of stochastic gradient methods, see \\cite{BottCurtNoce16}. \nStochastic gradient methods as applied to \\cref{eq:genfun} resemble traditional gradient descent methods, \nthe basic iteration of which takes the form\n\\begin{equation}\\label{eq:gd_iteration}\n \\xb_{k+1}= \\xb_k-\\alpha_k\\nablab f(\\xb_k).\n\\end{equation}\nWhen $\\nd$ is large, however, it may become computationally prohibitive to\nevaluate, or even numerically estimate, the gradient of \\cref{eq:genfun}:\n\\begin{equation}\n\\label{eq:gradsum}\n\\nablab f(\\xb_k)=\\sum_{i=1}^{\\nd} \\nablab F_i^2(\\xb_k).\n\\end{equation}\n\nThus, stochastic gradient methods compute approximations to the gradient  \\cref{eq:gradsum} \nby including in the sum only a\nsampled batch of the component function indices $\\{1,\\dots,\\nd\\}$. \nIn its simplest form, a single $i(k)\\in\\{1,\\dots,\\nd\\}$ is chosen at random from a \ndiscrete (often uniform) distribution, and the basic iteration in \\cref{eq:gd_iteration} is replaced with\n\\begin{equation}\\label{eq:sgd_basic_iteration}\n \\xb_{k+1}= \\xb_k-\\alpha_k\\nd\\nablab F_{i(k)}^2(\\xb_k).\n\\end{equation}\nEffectively, the cost of performing  \\cref{eq:sgd_basic_iteration}\nis a factor of $\\nd$ times cheaper than the cost of performing \\cref{eq:gd_iteration}.\nThis represents significant computational savings in performing a single iteration when $\\nd$ is large,\nat the expense of using an inaccurate gradient approximation.\nMore generally, one can consider sampling a batch of component function indices\n$\\B_k\\subseteq\\{1,\\dots,\\nd\\}$ of size $\\nb(k)\\leq\\nd$,\nand replacing \\cref{eq:gd_iteration} with \n\\begin{equation}\\label{eq:sgd_batch_iteration}\n \\xb_{k+1} = \\xb_k-\\alpha_k\\displaystyle\\frac{\\nd}{\\nb(k)}\\displaystyle\\sum_{i\\in B_k}\\nablab F_{i}^2(\\xb_k).\n\\end{equation}\nThe rationale behind the random sampling approach in \\cref{eq:sgd_batch_iteration}\nis that the expected value (with\nrespect to the stochastic sampling of component function indices) of \n$\\displaystyle\\frac{\\nd}{\\nb(k)}\\displaystyle\\sum_{i\\in B_k}\\nablab F_{i}^2(\\xb_k)$\nis exactly\n\\cref{eq:gradsum}.  \nWe note that when $\\nb(k) < \\nd$, the step from $\\xb_k$ to $\\xb_{k+1}$ will be based on incomplete information; however, since the sampled batches will be independent from one iteration to the next, these methods probabilistically find a zero of the full gradient \\cref{eq:gradsum} when the step sizes decay fast enough. \n\nDating almost as far back as the earliest stochastic gradient methods \\cite{RobbinsMonro1951},\nderivative-free variants of the iterations in \\cref{eq:sgd_basic_iteration} and in \\cref{eq:sgd_batch_iteration} have been proposed \\cite{KieferWolfowitz}. \nAll these methods \nperform an analogous iteration \n\\begin{equation}\n\\label{eq:basic_update}\n\\xb_{k+1} = \\xb_k - \\alpha_k \\db_k, \\qquad k=0,1,2,\\ldots,\n\\end{equation}\nwith $\\db_k \\in \\R^{\\nx}$ serving as an estimate of a gradient quantity. \nThese algorithms differ in their selection of the step size $\\alpha_k>0$ and, most distinctively, the choice of direction $\\db_k$. \n\n\\subsubsection{Kiefer-Wolfowitz method.}\nIterations of type \\cref{eq:basic_update} are found in the Kiefer-Wolfowitz (KW) method \\cite{KieferWolfowitz}.\nThe KW method computes \\emph{finite differences of sampled functions} to\napproximate directional derivatives in each of the $\\nx$ coordinate directions. \nAlthough other variants exist  \\cite{LEcuyerYin1998,Kleinman1999}, \nin this paper we use the most common sampling described in the following.\n\nIn the $k$th iteration, we uniformly sample a \nbatch $B_k\\subseteq\\{1,\\dots,\\nd\\}$ of size $|B_k|=\\nb(k)$.  \nGiven a fixed finite-difference parameter $h$, \nforward differences are used to approximate the partial derivatives needed to\nestimate the gradients of the $\\nb(k)$ squared component functions associated\nwith the batch.  Specifically, we compute\n\\begin{equation}\n\\label{eq:kw-fs}\n\\db_k = \\frac{\\nd}{\\nb(k)} \\displaystyle  \\sum_{i\\in\\B_k} \n\\gb_i(\\xb_k;h), \n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq:FD}\n\\gb_i(\\xb_k;h) = \\frac{1}{h}\n\\left[\\begin{array}{c}\nF_i\\left(\\xb_k+h\\mathbf{e}_1\\right)^2-F_i\\left(\\xb_k\\right)^2 \\\\\n\\vdots \\\\\nF_i\\left(\\xb_k+h\\mathbf{e}_{\\nx}\\right)^2-F_i\\left(\\xb_k\\right)^2 \n\\end{array}\\right].\n\\end{equation}\nIn our experiments, we refer to the algorithm that uses \\cref{eq:kw-fs} as $\\db_k$ in \\cref{eq:basic_update}\nas ``KW.'' \n\nObserve that in KW, $\\nb(\\nx+1)$ component function evaluations are performed in a single iteration. \nAs with any method using \\cref{eq:basic_update}, both a sequence of step sizes $\\{\\alpha_k\\}$ and a sequence\nof batch sizes $\\{\\nb(k)\\}$ must be selected. \nAdditionally, the finite-difference parameter $h>0$ must be selected. \nFor the sake of simplicity in presentation,\nwe have chosen to keep $h$ fixed, with the immediate consequence that \n$\\db_k$ is a biased estimator of $\\nablab f(\\xb_k)$  for all $k$, even when $\\nb(k)=\\nd$. \n\n\\subsubsection{Bandit method.}\n\\label{sec:bandit}\nWe now consider members of a class of derivative-free methods that have become increasingly attractive in supervised learning over the past decade,\nthe so-called (two-point) bandit methods \\cite{Agarwal2010,Ghadimi2013,Duchi2015,Gasnikov2017,Shamir2017}. \nSimilar to KW, bandit methods can employ a batch\n$B_k\\subseteq\\{1,\\dots,\\nd\\}$ of component function indices in the computation\nof $\\db_k$, which is computed based on finite differences and employed in\niterations of the type \\cref{eq:basic_update}.\nIn each iteration of a bandit method, however, only \\emph{one} directional\nderivative is numerically approximated per element in the batch; in contrast, KW uses \n$\\nx$ partial derivatives per element.\nFor the basic iteration of what we refer to as the ``Bandit'' method in the following,\nthe direction vector $\\db_k$ becomes\n\\begin{equation}\n\\label{eq:bandit-fs}\n\\db_k = \\frac{\\nd}{\\nb(k)} \\sum_{i\\in\\B_k} \n\\left(\n\\frac{F_i\\left(\\xb_k+h\\mathbf{u}_k\\right)^2-F_i\\left(\\xb_k\\right)^2}{h} \n\\right)\\mathbf{u}_k,\n\\end{equation}\nwhere $\\mathbf{u}_k$ is a \\emph{randomized} direction.  In particular, \nwe sample $\\mathbf{u}_k$ uniformly from the surface of an $\\nx$-dimensional sphere centered at the origin and of unit radius.\nOnce again, the quantity $h$ in \\cref{eq:bandit-fs} denotes a fixed finite-difference parameter. \nIn the case where $\\nb(k)=\\nd$ for all $k$, the Bandit method is related to the\niteration used in the Gaussian smoothing method \\cite{Nesterov2015}.\nWe remark that even when $\\nb(k)=\\nd$, the Bandit method is still randomized because of the\nrandom directions $\\mathbf{u}_k$. \n\nWhereas a KW method involves $\\nb(\\nx+1)$ component function evaluations in a single iteration, \nthe Bandit method entails only $2\\nb$ component function evaluations in a single iteration. \nIn common with a KW method, however, the Bandit method requires a selection of\nthe finite-difference parameter $h$,\na sequence of step sizes $\\{\\alpha_k\\}$,\nand a sequence of batch sizes $\\{\\nb(k)\\}$. \n\n\n\\subsection{Adaptive sampling quasi-Newton method}\n\\label{sec:aqn}\n\nWe now consider adaptive sampling quasi-Newton (AdaQN) methods \n\\cite{BollapragadaICML18, RBSWshort19,RBSWlong20}, which iteratively construct a local quadratic surrogate model according to the sampled component functions and select\nsearch directions $\\db_k$ as an approximate minimizer of the quadratic surrogate\nmodel.\nThe quadratic surrogate model is updated at every iteration using the differences between current and previously evaluated forward-difference gradient approximations.  \nWhereas the KW and Bandit methods considered here use a prescribed sequence of batch sizes $\\{\\nb(k)\\}$, \nAdaQN adaptively increases the  \nbatch size. \nDifferent adaptive rules \\cite{RBSWshort19,RBSWlong20} will increase the batch sizes differently; and we consider one such rule, called the \\emph{norm test}, in this study. \n\nAdaQN computes a direction $\\db_k$ of the form similar to \\cref{eq:kw-fs}:\n\\begin{equation}\n\\label{eq:adaqn-fs}\n\\db_k = \\frac{\\nd}{\\nb(k)} \\Hb_k\\displaystyle  \\sum_{i\\in\\B_k} \n\\gb_i(\\xb_k;h),\n\\end{equation}\nwhere $\\gb_i$ is defined in \\cref{eq:FD} and $\\Hb_k$ is a quasi-Newton matrix defining the quadratic surrogate model, \nupdated such that $\\Hb_{k+1} \\mathbf{v}_k = \\xb_{k+1} - \\xb_k$, where \n\\begin{equation}\n\\label{eq:adqn-fs-hk}\n\\mathbf{v}_k =\n\\displaystyle  \\sum_{i\\in\\B_k} \\Big(\\gb_i(\\xb_{k+1};h) - \\gb_i(\\xb_k;h)\\Big).\n\\end{equation} \nUnlike KW, however, the step size $\\alpha_k$ in AdaQN is adaptively determined in each\niteration via a \\emph{stochastic backtracking line search}\n\\cite{BollapragadaICML18, RBSWshort19}, an automatic procedure that ensures\nsufficient decrease in a sampled function. This procedure \nrequires evaluating the currently sampled component functions \nalong the direction $\\db_k$, with the associated number of such evaluations,\n$l_k$, varying at each iteration;  typically, $l_k$ is less than $5$. Observe that in\nAdaQN, $2|B_k|(\\nx+1)$ component function evaluations are required to compute\n$\\db_k$ (and, for free, $\\mathbf{v}_k$) and $|B_k|l_k$ component function evaluations are required to compute $\\alpha_k$. \n\nThe primary hyperparameter in AdaQN is the initial batch size. All the other hyperparameters associated with AdaQN are set to their default values as specified in \\cite{RBSWlong20}. \n\n\n\n\\section{Case Study: Optimizing Fayans energy density functional}\n\\label{sec:fayans}\n\nA key pursuit in the understanding of atomic nuclei is a global (i.e., across the nuclear chart)  description of nuclei. For such a task, the microscopic tool of choice\nis nuclear DFT rooted in the\nmean-field approach \\cite{Bender2003}. An effective interaction in DFT is given by the EDF,\nwhose parameters are adjusted to experimental data.\nOver the past decade, increasingly refined EDFs  have been developed, with increasingly complex and computationally expensive computer models; see, for example, \\cite{UNEDF0,UNEDF1,UNEDF2}. Because of the expense of these computer models, their calibration has focused largely on point-\/optimization-based estimation. Bayesian approaches have been demonstrated with limited (e.g., 200 in \\cite{McDonnellPRL15}) model evaluations, with nontrivial failure (associated with convergence at insufficient fidelity levels) rates of roughly 9\\% of the designs present even in relatively narrow (in terms of posterior probability) regions of parameter space; see \\cite{HigdonJPG15}. \n\nWe focus on calibration of a Fayans EDF\n\\cite{Fayans1998}, for which computer models have recently been developed and\ndemonstrated to correct some systematic effects of other state-of-the-art\nfunctionals \\cite{Reinhard2017}. This functional form has recently sparked\nsignificant interest, especially in the context of charge radii measurements \\cite{Hammen2018,Miller2019,Gorges2019,deGroote2020}.  \n\n\n\\begin{table}[htb]\n\\caption{\\label{tab:observableClasses} Nine classes of physical\nobservables that constitute all observables included in the study\n\\cite{Reinhard2017}.}\n\\begin{indented}\n\\lineup\n\\item[]\\begin{tabular}{lcc}\n\\br\nClass & Symbol & Number of Observables\\cr\n\\mr\nBinding Energy                            & $E_B$                       & 63 \\cr\nCharge Radius                    & $r_{\\textrm{ch}}$          & 52 \\cr\nDiffraction Radius                        & $R_{\\textrm{diffr}}$        & 28 \\cr\nSurface Thickness                         & $\\sigma$                    & 26 \\cr\nNeutron Single-Level Energy               & $\\epsilon_{ls,n}$           & 5  \\cr\nProton Single-Level Energy                & $\\epsilon_{ls,p}$           & 5  \\cr\nDifferential Radii                           & $\\delta\\langle r^2 \\rangle$ & 3  \\cr\nNeutron Pairing Gap                       & $\\Delta E_n$                & 5  \\cr\nProton Pairing Gap                        & $\\Delta E_p$                & 11 \\cr\n\\mr\n& & $\\nd = 198$ \\cr\n\\br\n\\end{tabular}\n\\end{indented}\n\\end{table}\n\n\\subsection{Problem specification}\n\nThe computer model $m\\left(\\bm{\\nu};\\xb\\right)$  for the currently used Fayans EDF has $\\nx=13$ free model\nparameters and employs a pool of fit data for spherical nuclei that primarily comprises\nbulk properties of the nuclear ground state (energy, radii, surface thickness),\nthree-point binding energy differences to calibrate pairing strengths, and some\nisotopic differences of  root mean square  radii in calcium isotopes.  Specifically, the pool\nused for this study is that used to fit the new Fayans EDF\nFy($\\Delta r$) reported in \\cite{Reinhard2017} but with the even-odd staggering\nof binding energies replaced by the even-even data. \nThe total dataset\nconsists of $\\nd=198$ observables of different classes (see\nTable~\\ref{tab:observableClasses}) that are associated with 72 different spherical, ground-state,\neven-even nucleus configurations (with these configurations encapsulated by $\\bm{\\nu}$).  \nThe weights ($\\sigma_i$) associated with each observable in the pool are related to those in \\cite{Reinhard2017} and are detailed in the \nsupplemental material \\cite{suppl}. The data, weights, and model outputs together define the collection of component functions $F_1(\\xb), \\ldots, F_{198}(\\xb)$ used in \\cref{eq:genfun}.\n\nTo ensure that our optimizations solve the specified problem, we identified \ntransient platform errors,\ntransient software faults outside of our control, user error, and \nreproducible software faults in the Fayans model evaluation software\nas the classes of failures that can occur during an optimization run and that\nmust be understood and handled sensibly throughout the study.  \nWe developed scripts to scan over\nall optimization results and their associated metadata so that possible failures\ncould be flagged and manually inspected.  When a transient failure was positively\nidentified and  was determined to affect the data quality, the\nassociated optimization run could simply be rerun and the results manually verified as\nacceptable.  Since the failures associated with the Fayans model software, which are discussed further in\n\\cref{sec:FayansSW}, are reproducible, rerunning a failed optimization is not an option.  As\na result, schemes for handling this class of errors were developed and\nimplemented.  A detailed discussion of this handling is given in \\cref{sec:mods}.\n\n\\subsection{Fayans model evaluation software}\n\\label{sec:FayansSW}\nThe code that is used in our study to evaluate the Fayans \nEDF is derived from a code solving nonrelativistic nuclear \nHartree-Fock equations for spherically symmetric nuclei \n\\cite{Rei91aR}, which is under continuous development.  We have\nidentified two classes of reproducible software faults within\nthe Fayans model evaluation software.  The numerical methods used\ninternally by the code are iterative, and therefore the first class of\nfailures is the inability of the methods to satisfy \na specified stopping criterion within a given maximum\niteration budget. \nWhile a single computation that does not satisfy\nthis criterion would normally be deemed as a failed result, for this study and informed by the experience and knowledge of\nthe developer, we implemented a secondary stopping criterion.  This\ncriterion, which is a relaxation of the primary\ncriterion, is employed as follows.  If a computation has failed to\nachieve the primary stopping criterion within the budget but does\nachieve the secondary criterion within the budget, then the result is\nflagged as \\emph{marginally convergent}.  If, however, a computation\ndoes not satisfy either criterion within the budget, the associated\nmodel evaluation is flagged as \\emph{nonconvergent}.\n\nThe second class of failures contains those computations that could not\nbe run to completion because at runtime the code encountered a situation that\nwas incompatible with its computations.  Such failures, which are referred to as\n\\emph{runtime failures}, could arise because \nof exceptional conditions that cause internal algorithmic failures or because \nthe computation is being\nevaluated in a region of the parameter space for which the functional \nis unstable \\cite{Hellemans2013,Pastore2015}.\nWhen runtime\nfailures occur, the Fayans model code reports an error, execution is aborted,\nand the associated model evaluation result is flagged as failed.\nTo avoid such severe failures as much as possible, we have\nestablished empirical, ``reasonable'' bounds for the model parameters,\nwhere reasonable means that we want to avoid \ninstabilities as well as unphysical results (e.g., unbound\nmatter).  For details regarding the region of assumed stability of Fayans EDF  that is characterized by these bounds, see the supplemental material \\cite{suppl}.\n\nKnowledge of this region has not been programmed\ninto the optimization methods, and therefore any optimization\ncan evaluate the model at points outside the region of stability.  We expect that\nsome methods, such as the randomized methods, might have a greater propensity\nfor choosing points outside the region of stability and that the various methods might also differ in their ability to recover from such bad evaluations.  Our means for managing such potential difficulties is detailed next.\n\n\n\\subsection{Modifications to minimize and address possible failures}\n\\label{sec:mods}\n\nA necessary step in facilitating the automatic training of any simulation-based model is to ensure that error handling is adequately addressed. All of the methods of \\cref{sec:algorithms} use some form of the output  \n$\\left \\{F_i(\\xb) ; \\; i \\in B_k \\right\\}$\nat a queried point $\\xb$ to inform their internal decision-making.  Consequently, it is necessary to address what occurs if the evaluation $F_i(\\xb)$ fails for one or more components $i \\in B_k$. \nIn this paper, we seek to make minimal changes to the methods stated in\n\\cref{sec:algorithms} and instead modify the objective function to\naccount for the variety of situations that can be encountered as discussed in\n\\cref{sec:FayansSW}.  \nBefore detailing each of these modifications,\nwe stress that throughout this article, information about specific points in parameter\nspace is reported in the original unscaled space used by physicists.  However,\nthe optimization methods used in this study were implemented to work on a scaled version of the\nparameter space; hence, it is understood that the domain of the\nobjective function is the scaled space. Unless otherwise stated, the points in parameter\nspace discussed in the remainder of this section should be assumed to be with respect to\nthe scaled parameter space used for optimization.  For more information regarding the choice of \nscaling, we refer the reader to the supplemental material \\cite{suppl}.\n\n\\subsubsection{Projection.}\n\\label{sec:projection}\n\nThe tested optimization methods all were intended primarily for unconstrained optimization. We operate in such a setting here and do not provide any method with prior knowledge \nof valid combinations of parameters.\nWe observed that, depending on the quality of gradient estimators obtained by the randomized \nmethods, the directions $\\db_k$ could become so large as to generate steps into physically\nmeaningless or unstable regions of parameter space.  \nTo help such methods avoid divergence, we alter the objective function to include a projection \nonto an $\\ell_1$-ball centered around the\npoint $\\bar{\\xb}$.\nThe unscaled version of this point is\ngiven in the supplemental material \\cite{suppl}.\nBecause of the scaling, it is\nappropriate to use an isotropic $\\ell_1$-ball \nfor defining a reasonable region; that is, we compute the projection  \n\\begin{equation}\n\\label{eq:project}\n\\xb_{\\mathbf{P}} = \\displaystyle\\arg\\min_{\\yb\\in\\mathbf{P}} \\|\\yb-\\xb\\|_2, \n\\quad \\textrm{where } \n\\mathbf{P}= \\left\\{\\yb\\in \\R^{\\nx}: \\|\\yb-\\bar{\\xb}\\|_1\\leq 2\\right\\}.\n\\end{equation} \nOur choice of using the $\\ell_1$-norm to define $\\mathbf{P}$ is motivated by  \nour observation that failures are more likely to occur when many parameter components deviate significantly from $\\bar{\\xb}$.\n\nWe then pass the projected point $\\xb_{\\mathbf{P}}$ to the Fayans model simulation for\nevaluation. \nWe modify the\nobjective function \\cref{eq:genfun} by applying a multiplicative penalty to each residual $F_i(\\xb)$ based on the distance between $\\xb_{\\mathbf{P}}$ and $\\xb$; that is,\n\\begin{equation}\n\\label{eq:projection_modification}\n\\tilde{F_i}(\\xb) = F_i(\\xb_{\\mathbf{P}})\\left(1+ \\left\\|\\xb-\\xb_{\\mathbf{P}}\\right\\|_2^2\\right).\n\\end{equation}\nWe acknowledge that the replacement of each $F_i$ with $\\tilde{F_i}$\ncan introduce nonsmoothness \nat the boundary of $\\mathbf{P}$, even when we assume each $F_i$ is smooth in a neighborhood of the boundary of $\\mathbf{P}$. \n\n\\subsubsection{Observable convergence.}\n\\label{sec:convergence}\nTo account for marginally convergent and noncovergent results as well as\noccasional runtime failures, and\ninformed by the belief that convergent computations are more likely to\nindicate physically meaningful points in parameter space, we further modified\nthe observable data $\\tilde{F_i}(\\xb)$ in \\cref{eq:projection_modification} by\ncomputing\n\\begin{equation*}\n\\hspace{-55pt}\n\\label{eq:convergence_modification}\n\\hat{F_i}(\\xb) = \n\\left\\{ \n\\begin{array}{ll}\n\\tilde{F_i}(\\xb) &\\mbox{if the computation of }F_i\\left(\\xb_{\\mathbf{P}}\\right)\\mbox{ succeeded} \\\\\n(1 + \\lambda_m^2)\\tilde{F_i}(\\xb) & \\mbox{if the computation of }F_i\\left(\\xb_{\\mathbf{P}}\\right)\\mbox{ was marginally convergent} \\\\\n(1 + \\lambda_n^2)\\tilde{F_i}(\\xb) & \\mbox{if the computation of }F_i\\left(\\xb_{\\mathbf{P}}\\right)\\mbox{ was nonconvergent } \\\\\n(1 + \\sigmar^2)\\tilde{F_i}(\\xb) & \\mbox{if the computation of }F_i\\left(\\xb_{\\mathbf{P}}\\right)\\mbox{ had a runtime failure}, \n\\end{array}\n\\right.\n\\end{equation*}\nwhere $\\sigmar \\ge \\lambda_n \\ge \\lambda_m \\ge 0$ denote penalty parameters. In our study, we\nset $\\lambda_m=2, \\lambda_n=5$, and $\\sigmar=100$. With these considerations, our \nmodified objective function, seen by all of the optimization algorithms, is\n\\begin{equation}\n\\label{eq:modfun}\n\\hat{f}(\\xb) = \\sum_{i=1}^{\\nd} \\hat{F_i}(\\xb)^2.\n\\end{equation}\n\n\n\\subsubsection{Recovering from failed simulations.}\n\\label{sec:failures}\nIn our study, not even the use of the modified objective function\n$\\hat{f}(\\xb)$ can cover every possible failure case. \nWhen the Fayans simulation returned no output $\\hat{f}(\\xb)$ whatsoever---a situation that we refer to as a \\emph{hard failure}---none of the methods that we tested can continue. \nWe thus slightly modified the methods to handle hard failures. \nThe deterministic methods (POUNDERS and Nelder-Mead) were modified to terminate gracefully when a hard failure occurred,\nreturning the point in parameter space corresponding to the best-found objective value in the run up until the hard failure occurred. \nThe randomized methods were augmented with a simple backtracking procedure, like the one employed in AdaQN. \nAfter a direction $\\db$ was computed, if the function evaluation at the next suggested point $\\xb+\\db$ resulted in hard failure, then the direction $\\db$ was replaced by $0.1\\db$, and we reevaluated at $\\xb+\\db$. This process was repeated until the evaluation of $\\xb+\\db$ did not result in hard failure. \nAs we will see in the numerical results, the deterministic methods and AdaQN\nnever suggested a point that resulted in hard failure; but KW and Bandit did encounter hard failures, depending on the selection of hyperparameters. \n\n\n\\section{Numerical Results}\n\\label{sec:numerical}\n\nWe now study the performance of the algorithms from \\cref{sec:algorithms} on the function in \\cref{eq:modfun}.\nWe first tune the identified hyperparameters to obtain hyperparameter values to maximize the  performance of each algorithm. \nSince computational budgets may limit one's ability to perform comprehensive  hyperparameter tuning, the insensitivity to hyperparameter selection \n(as well as the variability overall) may be a key consideration in selecting an optimization algorithm. \nWe report on this sensitivity and perform a thorough study of each algorithm using the best hyperparameter values found.\n\nFor  this study, the results for all $\\nd=198$ component functions were\nstored at each evaluated point, even when an optimization method with $\\nb<198$ was not provided \n(or charged for) this full set of component function evaluations.\nStoring this information allowed us to\nreevaluate, during postprocessing, the true function (i.e., with all 198 component functions) \nfor every point queried.\n\nThe randomized algorithms of \\cref{sec:algorithms} require a forward-difference parameter $h>0$. \nIn our computations we use $h=5\\cdot 10^{-7}$.\nThis value was obtained by estimating the noise level in each $F_i^2$ following \\cite{more2011ecn}. These noise estimates were then used to determine $h$ following the procedure in \\cite{more2011edn}. Although variation was seen across different component functions $i$ and different directions in $\\R^{\\nx}$, the effect of this variation turned out to be mild, and hence we used a fixed difference parameter for all component functions. \n\n\n\n\\subsection{Tuning of hyperparameters}\n\\label{sec:tuning}\n\nFor our hyperparameter tuning procedure, we randomly selected 5 starting points from the\nsame $\\ell_1$-ball as in \\cref{eq:project}. \nWe ran each \nmethod with a budget of $700\\nd$ component function evaluations from each starting point. \nEach randomized method was run with three different seeds from each starting point, while deterministic methods were run once from each starting point. \n\nThe three main classes of hyperparameters, and the ways we chose to vary them, are defined below.\n\n\\subsubsection{Step-size hyperparameters.}\nEvery method that we tested, with the exception of AdaQN, requires some kind of (initial) step-size parameter. \nWhile POUNDERS and Nelder-Mead require a single radius parameter,\nthe stochastic approximation methods KW and Bandit require a \\emph{sequence} of step-size parameters \n$\\{\\alpha_k\\}$, as seen in \\cref{eq:gd_iteration}.\nFor all four methods, we chose to use a common set of step-size hyperparameters based on the set \n$J\\equiv \\{3,4,5,6,7\\}.$\n\nIn the case of POUNDERS, the hyperparameter value $\\alpha=2^{-j}, j\\in J$ sets the initial trust-region radius; \n and in the case of Nelder-Mead, the hyperparameter value $\\alpha=2^{-j}, j\\in J$ sets the initial simplex radius. \nFor the stochastic approximation methods, we opted to use a schedule of decaying step sizes\n$\\alpha_k = 2^{-j}\/(k+1), j\\in J$. \nEmploying such a harmonic sequence as the step-size schedule for stochastic approximation methods is in line with standard convergence theory for those methods. \nWe remark again that AdaQN employs adaptive step sizes and hence does not require a step-size hyperparameter. \n\n\\subsubsection{Batch-size hyperparameters.}\nEach of the stochastic methods\nrequires the specification of a batch-size parameter. Recall from \\cref{sec:sgd} that a batch is drawn uniformly from the $\\nd$ component functions and that such draws are independent from one draw to the next.\nIn each iteration, KW and Bandit methods require a batch size $\\nb$ of component function evaluations to compute a gradient estimator; recall \\cref{eq:gradsum}. Following standard practice, we chose to hold $\\nb$ constant for these methods. \nWhile AdaQN adaptively increases $\\nb$ during the course of an algorithm, it still requires an initial $\\nb$. \nFor all three  methods, we used a set of 4 common batch sizes\n$$\\nb\\in\\{11,33,99,198\\}.$$\nWe interpreted $\\nb$ as the constant batch size for KW and Bandit methods and as the initial batch size for AdaQN. \nObserve that all of our tested $\\nb$ divide $\\nd=198$, which is helpful for comparing the stochastic methods with the full-batch deterministic methods. \nMoreover, when $\\nb=\\nd$, \nKW and AdaQN are deterministic methods while Bandit is still a randomized method since it employs a random direction $\\mathbf{u}_k$ in each iteration.\n\n\n\n\n\n\\subsection{Performance metrics}\n\\label{sec:metrics}\nWe now discuss various measures of effort in order to compare the performance of the methods.\nWe label by $f_{s,*}$ the minimum function value evaluated over all runs \ninstantiated from the $s$th starting point $\\xb_{s,0}$, regardless of method, seed, and all relevant hyperparameters. \nWe say that a point\n$\\xb_{s,k}$ is\n\\emph{$\\tau$-optimal for starting point $\\xb_{s,0}$} provided \n\\begin{equation}\n \\label{eq:tau-optimality}\n \\displaystyle\\frac{f(\\xb_{s,k})-f_{s,*}}{f(\\xb_{s,0})-f_{s,*}}\\leq\\tau.\n\\end{equation}\nA point $\\xb_{s,k}$ satisfying \\cref{eq:tau-optimality} has achieved a fraction $\\tau$ of the best-known decrease from starting point $\\xb_{s,0}$.\n\nOur primary measure for any run is the best function value, $\\min_{k\\leq K} f(\\xb_{s,k})$, as a function of the number of points, $K$, evaluated during that run. We often report these results in terms of the number of component functions evaluated, that is, $\\sum_{k\\leq K} \\nb(k)$. This also allows us to track the number of component function evaluations needed to achieve $\\tau$-optimality, for a specified value of $\\tau\\in (0,1)$.\nNote that for some values of $\\tau$, not all runs may achieve $\\tau$-optimality; \nwhen a run fails to do so, we define the number of component function evaluations it required to attain $\\tau$-optimality as the budget of component function evaluations it was given.\n\n\n\\subsection{Results of hyperparameter tuning}\n\\label{sec:empirical}\nWe now show the results of hyperparameter tuning to search for ``best\" step sizes and\/or batch sizes, where appropriate. \nIn \\Cref{fig:pounders_nmead} we look at summary hyperparameter tuning results for POUNDERS and Nelder-Mead,\nand in \\Cref{fig:adaqn} we look at summary results for AdaQN. \nFor AdaQN, we \nchose  to tune only initial batch sizes.\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.92\\linewidth]{Fig1}\n\\caption{Tuning the step-size parameter for POUNDERS (left) and Nelder-Mead (right). \nThroughout such plots in this paper, the \nvertical axis shows the value of $\\hat{f}$, which is defined in \\cref{eq:modfun}, that is best among those seen within the specified number (horizontal axis) of evaluations. \nSolid lines denote median performance over all starting points (and stochastic replications), while the translucent bands denote the $25$th and $75$th percentiles of performance.\nThe number in parentheses in the legend denotes the \\emph{average} number of hard failures produced by the Fayans-model simulation during the run of the algorithm.\nThe solid black horizontal line denotes the value of $\\hat{f}(\\xb_1)$ in Table~\\ref{tab:BestResults}. \n\\label{fig:pounders_nmead}}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=.55\\linewidth]{Fig2}\n\\caption{Tuning the batch-size parameter for AdaQN. \\label{fig:adaqn}} \n\\end{figure}\n\nBased on the results for AdaQN in \\Cref{fig:adaqn}, \nwe chose to initialize $\\nb=11$, which finds the same quality of median solutions in terms of $\\hat{f}$ values as do other batch sizes toward the end of its budget\nbut identifies better solutions earlier on (in terms of the $25$th, $50$th, and $75$th percentiles). \n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=.7\\linewidth]{Fig3} \n\\caption{Median number of component function evaluations needed by POUNDERS and Nelder-Mead to attain $\\tau$-optimality, where $\\tau=0.01$. \n\\label{fig:pounders_nmead2}}\n\\end{figure}\n\n\n\nWe see in \\Cref{fig:pounders_nmead}  that the performance of POUNDERS is extremely robust to the selections of initial trust-region radius. \nFor Nelder-Mead, we observe  that its performance is not as independent of the simplex radius as POUNDERS is independent of its initial trust-region radius. This is summarized in \\Cref{fig:pounders_nmead2}, which follows the example set in \\cite{asi2019importance} and shows\nthe \\emph{median} amount of component function evaluations required by a method to attain $0.01$-optimality. \n\nBecause the POUNDERS performance was so similar for all initial trust-region radius values, we \nselected $\\alpha=2^{-4}$. \nFor Nelder-Mead, because the best final median function value occurred for $\\alpha=2^{-4}$, and because the median performance of $\\alpha=2^{-4}$ was nearly as fast as the median performance of $\\alpha=2^{-3}$ in finding $0.01$-optimal solutions, we selected $\\alpha=2^{-4}$. \n\nFor the remaining stochastic approximation methods, we first fix batch-size parameters and compare the median performance of step-size parameters. These results are shown, respectively, in \\Cref{fig:bandit-batchsizes} and \\Cref{fig:kw-batchsizes}.\nWe remark that in the case of $\\nb=11$, we did not test all the step-size parameters because of the extreme computational cost of running\nthese two methods with $\\nb=11$ in our computational setup. \nInstead, for $\\nb=11$ we tested the two step sizes that \nresulted in the fewest average hard failures from running only one seed per starting point.  \nIn \\Cref{fig:kw-batchsizes}, we see that for KW, \nthis selection of step sizes matches the selection of step sizes that performed best for $\\nb=33$.\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{Fig4.pdf}\n\n\\caption{Hyperparameter tuning results for Bandit; from left to right, and then top to bottom, are batch sizes 198, 99, 33, and 11.\n\\label{fig:bandit-batchsizes}}\n\\end{figure}\n\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{Fig5.pdf}\n\\caption{Hyperparameter tuning results for KW; from left to right, and then top to bottom, are batch sizes 198, 99, 33, and 11.\n\\label{fig:kw-batchsizes}}\n\\end{figure}\n\n\n\n\n\nIn \\Cref{fig:bandit-batchsizes}, we observe that for each fixed batch size, there is a step size that provides a clear best median performance. Unlike in the comparisons made for POUNDERS,  Nelder-Mead, and AdaQN, however,\nthe difference in the percentile performance between the Bandit methods and the average number of hard failures encountered across different runs should be taken into account. \nWith these three considerations in mind, for $\\nb=198$, we selected $\\alpha=2^{-3}$. \nFor $\\nb=99$, balancing the significantly lower number of hard failures encountered by $\\alpha=2^{-4}$ compared with $\\alpha=2^{-3}$, as well as the better $75$th percentile performance of $\\alpha=2^{-4}$, we selected $\\alpha=2^{-4}$. \nFor $\\nb=33$, because of the similar median final performance of $\\alpha=2^{-5}$ and $\\alpha=2^{-3}$, coupled with the better $75$th percentile performance of $\\alpha=2^{-5}$ and lower number of hard failures encountered by $\\alpha=2^{-5}$, we selected $\\alpha=2^{-5}$.\nFor $\\nb=11$, the choice of $\\alpha=2^{-5}$ was clear. \n\nIn \\Cref{fig:kw-batchsizes}, the choice for $\\nb=198$ and $\\nb=99$ was fairly easy to make, at $\\alpha=2^{-4}$ and $\\alpha=2^{-5}$, respectively. The choice for $\\nb=33$ was less clear; the median performance of $\\alpha=2^{-5}$ was not much worse than the median performance of $\\alpha=2^{-4}$; however, because the $75$th percentile performance of $\\alpha=2^{-5}$ was better than that of $\\alpha=2^{-4}$, and because $\\alpha=2^{-5}$ had fewer average hard failures, we selected $\\alpha=2^{-5}$. \nFor $\\nb=11$, we selected $\\alpha=2^{-5}$ because of its failure-free performance. \n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{Fig6.pdf}\n\\caption{Best step sizes of each batch size for Bandit (left) and KW (right). \\label{fig:best-stepsizes-compared}}\n\\end{figure}\n\nHaving downselected the step-size parameters per batch size, we now compare the methods across different batch sizes in \\Cref{fig:best-stepsizes-compared}.\nWe see that for KW, using the smallest tested batch size $\\nb=11$ with a step size of $\\alpha=2^{-5}$ is the best setting of hyperparameters. \nThe situation was less clear for Bandit methods. \nWhile the median performance within the budget of component function evaluations was best with batch size 11, this parameter combination exhibited \nmany hard failures; \nas a tradeoff between a smaller number of hard failures and \na reasonable median performance, we selected step size $\\alpha=2^{-4}$ and $\\nb=99$.  \n\n\\subsection{Comparing tuned methods on additional starting points}\nHaving performed the hyperparameter tuning in the preceding section to select appropriate hyperparameters for each of the five methods, we then ran the selected variant of each method on a larger set of problems.\nIn particular, we randomly generated twenty starting points (instead of five) from the\nsame $\\ell_1$-ball  as in \\cref{eq:project} and again\nran three seeds for each starting point for each of the randomized methods. \nThe budget for each method was extended to $1500\\nd(\\nx+1)$ component function evaluations, more than double the budget provided in the hyperparameter tuning runs.  These results are presented in \\Cref{fig:compare-all}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\linewidth]{Fig7.pdf}\n\\caption{Comparing the best variants of all methods. \n \\label{fig:compare-all}}\n\\end{figure}\n\nThe results as seen in \\Cref{fig:compare-all} are remarkable.\nEven in the full run, POUNDERS continues to exhibit an interesting phenomenon where after a fixed number of full-batch component function evaluations, the objective function found suddenly drops and exhibits very low variation in the $25$th to $75$th percentile band in decreasing to a final solution. This sort of robustness and consistency in performance is certainly desirable.\n\nIn terms of final median performance, AdaQN and Nelder-Mead find similar quality solutions as POUNDERS. \nOne could argue that the performance of Nelder-Mead, in terms of overall median and other percentile trajectories, is \ndominated by the performance of POUNDERS. \nThe performance of AdaQN is interesting in that it is not strictly dominated by the performance of POUNDERS. In fact, if one were  interested only in generating reasonable solutions (say $\\tau-$optimal solutions, where $\\tau\\approx 0.25$) in as few component function evaluations as possible, AdaQN is a better choice than POUNDERS. \nIf one were simultaneously interested in the robustness of the final solution, then AdaQN remains a strong choice. This is in contrast with KW, which also achieves gains fairly quickly but does not have the same final robustness as exhibited by AdaQN. For all but a few $\\tau$ values, the Bandit method is bested by KW, which may be  attributed partly to the nontrivial failure rate experienced by Bandit.\n\nOur comparisons thus far have measured computational expense in terms of the number of component function evaluations. Such metrics are fully justified in computing environments where a single component function can be evaluated at a time.\nFor sufficiently large parallel computing environments, resources are available to perform full batch (here $\\nb=\\nd=198$) evaluations simultaneously. We now examine the case between the extremes of \na single component function evaluated at a time and all $\\nd$ component functions evaluated at a time.\nMethods capable of subsampling (i.e., using batch sizes $\\nb<\\nd$) are potentially promising in such intermediate regimes.\n\nOur \\emph{resource utilization plots} illustrate these considerations. \nBy resource size we denote the number of component function evaluations \nthat can be simultaneously computed.\nGiven a resource size, we refer to the number of \\emph{rounds} as the iterations of such whole resources to evaluate the component function evaluations needed to \nachieve a performance metric (e.g., $\\tau$-optimality as in \\cref{eq:tau-optimality}). \nFor example, if the resource size is 11, then a method with $\\nb=198$ will use at least 198\/11=18 \nrounds to make an optimization step, while a method with $\\nb=11$ will potentially use only one such round. \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\linewidth]{Fig8}\n\\caption{Resource utilization plots with respect to $\\tau$-optimality (a: $\\tau=0.5$; b: $\\tau=0.1$) for the tuned methods shown in \\Cref{fig:compare-all}. The construction of these figures assumes that component functions can only be evaluated in parallel at a single point in parameter space at a time.  In addition, some choices of resource size result in poor performance when the size is incompatible with $\\nb$.  For example, evaluating 33 component functions with a resource size of 22 is charged two rounds (44 possible evaluations).  \nSimilarly, a method with nonadaptive batch size $\\nb$ will not be able to benefit, absent failures, from resource sizes larger than $\\nb$.  \n \\label{fig:rups}}\n\\end{figure}\n\n\\begin{table}\n\\caption{\\label{tab:initialevals}  Minimum number of initial component function evaluations to evaluate first (noninitialization) optimization step.}\n\\centering\n\\begin{tabular}[t]{lr}\n\\br\n  Method & $F_i$ evaluations\\\\\n\\mr\nAdaQN        & $(\\nx+2)\\nb(0)$  \\\\\nBandit       & $3 \\nb $ \\\\\nKW           & $(\\nx+2)\\nb$ \\\\\nNelder-Mead  & $(\\nx+2)\\nd$ \\\\\nPOUNDERS     & $(\\nx+2)\\nd$ \\\\\n\\br\n\\end{tabular}\n\\end{table}\n\nWe highlight computational environments where POUNDERS is not the most obvious choice by showing select values of $\\tau$ in the resource utilization plots in \\Cref{fig:rups}.  \nFor low demands on solution quality \n($\\tau=0.5$), Bandit methods are exceptionally good, identifying $0.5$-optimal solutions remarkably quickly. This is because Bandit requires very few evaluations to get started; \\Cref{tab:initialevals} shows that Bandit (with $\\nb=33$) will have evaluated its first step (to $\\xb_1$) after 99 component function evaluations. The left plot in \\Cref{fig:rups} shows that on over half the runs, this number (3 rounds at a resource level of 33) is sufficient for satisfying this coarse accuracy level. Nelder-Mead and POUNDERS show a similar behavior after their first step is performed, but this step requires 15 rounds at a resource level of 198 (90 rounds at a resource level of 33). AdaQN's smaller batch size allows it to outperform POUNDERS and Nelder-Mead at lower resource levels, but is insufficient for catching Bandit at the coarse accuracy $\\tau=0.5$. \n\n\nWhen we tighten the accuracy demands to $\\tau=0.1$, we see that the deterministic methods (i.e., those with a full batch $\\nb=198$) are again best, even for resource sizes as small as 11. This plot also shows that AdaQN's adaptive batch size allows it to remain competitive even at this tighter accuracy for resource sizes up to 99.\n\n\n\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nOur results show that the deterministic methods tested were insensitive to starting point in terms of finding a good objective function value with a limited number of $\\hat{f}$ evaluations. Furthermore, these methods were generally insensitive to hyperparameters, did not evaluate points that resulted in hard failures, and are attractive even if the expense of evaluating the Fayans model allowed for computing only a fraction (e.g., 11\/198=1\/18) of the component functions at a time. For problems where even smaller fractions are possible or when less accurate solutions are desired with even smaller computational budgets than those tested here, AdaQN appears especially promising. \nWe expect that such methods that can use smaller batch sampling will become more attractive as the number of fit data significantly increases (as in the case of traditional supervised learning applications).\n\nAs part of understanding the quality of results achieved in this study, we identified the best run, \nin terms of lowest $\\hat{f}$ value, for each of the 20 starting points.\nEleven of these 20 best results were found with POUNDERS, which also found the overall best result; seven by AdaQN; and two by Nelder-Mead.  \nAll 20 points of these best results are contained in $\\mathbf{P}$ and resulted in fully converged Fayans model evaluations.\n\\revised{The parameter values of two of these best points are presented in unscaled form in Table~\\ref{tab:BestResults}.}\n{The parameter values of two of these best points are presented in\nunscaled form in Table~\\ref{tab:BestResults}. To give an impression of typical\nparameters from previous fits, we also show the parameters for\nFy($\\Delta r$) \\cite{Reinhard2017} and Fy($\\Delta r$,HFB) \\cite{Miller2019,Reinhard2020}.} \n\n\n\n\\begin{table}\n\\caption{\\label{tab:BestResults}  Unscaled parameter values for two of the 20 best optimization results in the study.\n\\revised{and their associated objective function value.}\n{The parameters are given up\nto six digits, which suffices to reproduce the output values shown in \\Cref{tab:Chi2Comparison}.}\nThe point $\\xb_1$\nhad the lowest objective function value in the study and is chosen as the representative of\nthe group of the four best runs; the point $\\xb_{5}$ had the best result in\n\\revised{the second grouping of the remaining 16 best runs. For the definition of Fayans EDF parameters, see \\cite{suppl}.  $\\rho_\\mathrm{eq}$ is in fm$^{-3}$;  $E\/A, K, J, L$ are in MeV; other parameters are dimensionless. For comparison, parameter values of Fy($\\Delta r$) \\cite{Reinhard2017} and Fy($\\Delta r$,HFB) \\cite{Miller2019,Reinhard2020} EDFs are also given.}\n{the second grouping of the remaining 16 best runs. For the definition\nof Fayans EDF parameters, see \\cite{suppl}.  $\\rho_\\mathrm{eq}$ is in\nfm$^{-3}$;  $E\/A, K, J, L$ are in MeV; other parameters are\ndimensionless. As a guideline for typical model parameters, the values\nfor Fy($\\Delta r$) \\cite{Reinhard2017} and Fy($\\Delta r$,HFB)\n\\cite{Miller2019,Reinhard2020} EDFs are also given.} \n}\n\\begin{indented}\n\\lineup\n\\item[]\n\\begin{tabular}[t]{lll|ll}\n\\br\n  Parameter & $\\xb_1$ & $\\xb_{5}$   & \\multicolumn{1}{c}{Fy($\\Delta r$)}\n   & \\multicolumn{1}{c}{Fy($\\Delta r$,HFB)} \\\\\n\\mr\n$\\rho_\\mathrm{eq}$ \t\\cmmnt{RHO\\_NM}   & \\0\\00.165755   & \\0\\00.166182 & \\0\\00.160 & \\0\\00.164\\\\\n$E\/A$                     \\cmmnt{EOVERA}  & \\0\\-15.8715    & \\0\\-15.8780 & \\0\\-16.11 & \\0\\-15.86 \\\\\n$K$ \t                    \\cmmnt{COMPR} & 192.686        & 185.156   & 219 & 210.3\\\\\n$J$ \t                    \\cmmnt{ASYMM} & \\028.8018      & \\028.8467  & \\029 & \\028.1 \\\\\n$L$ \t                    \\cmmnt{DASYM} & \\035.6545      & \\031.5877  & \\030 & \\037.5\\\\\n${h_{2-}^\\mathrm{v}}$      \\cmmnt{H2VM}     & \\0\\07.08066    & \\0\\04.71124  & \\0\\01.2150 & \\022.8090 \\\\\n${a_+^\\mathrm{s}}$\t     \\cmmnt{ASP}  & \\0\\00.594920   & \\0\\00.620893 & \\0\\00.6047 & \\0\\00.56548 \\\\\n${h_{\\nabla}^\\mathrm{s}}$\t    \\cmmnt{HGRADP}& \\0\\00.510148   & \\0\\00.613192 & \\0\\00.6656 & \\0\\00.45795\\\\\n${\\kappa}$\t          \\cmmnt{C0NABJ}  & \\0\\00.192851   & \\0\\00.191370 & \\0\\00.18792 & \\0\\00.19833 \\\\\n${\\kappa'}$\t          \\cmmnt{C1NABJ}  & \\0\\00.0383998  & \\0\\00.0532395 & \\0\\0\\-0.0237 & \\0\\00.44008 \\\\\n${f_{\\mathrm{ex}}^\\xi}$\t  \\cmmnt{FXI}     & \\0\\0\\-3.70050  & \\0\\0\\-3.63760 & \\0\\0\\-4.4720 & \\0\\0\\-4.4556 \\\\\n${h_\\nabla^\\xi}$\t          \\cmmnt{HGRADXI} & \\0\\03.17494    & \\0\\03.48559 & \\0\\03.227 & \\0\\03.113 \\\\\n${h_{+}^\\xi}$\t          \\cmmnt{H1XI}    & \\0\\03.22592    & \\0\\03.13267 & \\0\\04.229 &  \\0\\04.2440\\\\[5pt]\n\\br\n\\end{tabular}\n\\end{indented}\n\\end{table}\n\n\\Cref{fig:MeanChi2PerClass} shows the outputs of these 20 points by observable class  (see Table~\\ref{tab:observableClasses}). For each observable class, by $\\chi^2$ we denote the contributions to $\\hat{f}(\\xb)$ from that observable class (and hence the sum over all observable classes is $\\hat{f}(\\xb)$). We normalized these $\\chi^2$ by the number of observables in the associated class to obtain the average $\\chi^2$ of each observable class, $\\overline{\\chi^2}$. \n\\Cref{fig:MeanChi2PerClass} suggests that the results can be partitioned into two groups.  This partitioning is  related not only to $\\overline{\\chi^2}$ but also to the values of $\\hat{f}$.  The four results labeled as Low $\\hat{f}$ corresponds to those results with $\\hat{f}$ less than 49; the 16 other results, labeled as High $\\hat{f}$, have a slightly higher $\\hat{f}$.  The results with lowest $\\hat{f}$ from each group are denoted by $\\xb_1$ and $\\xb_5$ in Table~\\ref{tab:BestResults}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.5\\linewidth]{Fig9.pdf}\n\\caption{Average $\\chi^2$ by observable class ($\\overline{\\chi^2}$) plotted for each of the 20 best\nresults obtained in the study.  In terms of this quantity, the 20 results clearly can be\npartitioned into two different groups.  The results in one such group are\ncolored blue and also correspond to the four results with the lowest $\\hat{f}$\nresults in the study.\\label{fig:MeanChi2PerClass}}\n\\end{figure}\n\nThe $\\chi^2$ and $\\overline{\\chi^2}$ values are given for the same two points in Table~\\ref{tab:Chi2Comparison}.  In general,  the low-$\\hat{f}$ cluster appears to fit radius-based observables better than does the high-$\\hat{f}$ cluster, but at the expense of the quality of fit to energy-based observables.  In particular, the fit to the isotopic differences of charge radii in Ca isotopes is better, but the fits to the two pairing gaps deteriorate.  \n\n\n\\begin{table}\n\\caption{\\label{tab:Chi2Comparison} Breakdown of the $\\chi^2$ and average $\\chi^2$ ($\\overline{\\chi^2}$) by \nobservable class (see\nTable~\\ref{tab:observableClasses}) for the two points in parameter space given\nin Table~\\ref{tab:BestResults}.  In bold are those values with potentially significant differences between the two groups of best results.} \n\\begin{indented}\n\\lineup\n\\item[]\n\\begin{tabular}[t]{l||ll|lll}\n\\br\n & \\multicolumn{2}{c|}{$\\xb_1$}       &             \\multicolumn{2}{c}{$\\xb_5$}  \\\\\nClass & $\\chi^2$ & $\\overline{\\chi^2}$ & $\\chi^2$ & $\\overline{\\chi^2}$\\\\\n\\mr\n$E_B$                       & \\09.64 &         0.153  & \\09.06 &         0.144  \\\\\n$R_{\\rm diffr}$             & \\09.49 &         0.339  & \\09.81 &         0.351  \\\\\n$r_{\\rm ch}$               &  16.41 &         0.316  &  17.95 &         0.345  \\\\\n$\\sigma$                    & \\02.48 &         0.095  & \\03.17 &         0.122  \\\\\n$\\epsilon_{ls,p}$           & \\00.78 &         0.156  & \\00.70 &         0.141  \\\\\n$\\epsilon_{ls,n}$           & \\03.64 &         0.728  & \\03.90 &         0.780  \\\\\n$\\delta\\langle r^2 \\rangle$ & \\00.25 & \\textbf{0.082} & \\01.31 & \\textbf{0.436} \\\\\n$\\Delta E_p$                & \\03.52 & \\textbf{0.320} & \\02.66 & \\textbf{0.242} \\\\\n$\\Delta E_n$                & \\02.12 & \\textbf{0.425} & \\01.12 & \\textbf{0.225} \\\\\n\\mr\n$\\hat{f}$\t            & 48.33  &                & 49.69  & \\\\\n\\br\n\\end{tabular}\n\\end{indented}\n\\end{table}\n\n\n\nThese results underscore the value of optimization methods being able to train physics models with few model evaluations. \nSuch efficiency allows one to perform several different optimizations (e.g., from different starting points, with different fit data) and thereby identify potentially different local minimizers. \nThe subsequent study of distinct local minima could be useful; the ability of a solution to model the desired physics often matters more than the final objective function value.\n\n\n\n\\section{Perspectives}\n\nIn this study, we addressed the calibration of the nuclear physics model Fayans EDF using the $\\chi^2$-minimization, which can be viewed as a supervised machine learning problem.  \nThe  model is somewhat computationally expensive and the  \nderivative information with respect to the model parameters is not available. To this end, we investigated the strengths and limitations of five algorithmic  families  of  iterative  methods  for  local,  unconstrained derivative-free optimization.  We considered two deterministic and three randomized methods. We analyzed hyperparameter tuning considerations and variability associated with the methods, and illustrated considerations for tuning in different computational settings. In total, nearly a half million CPU core hours were expended for this study, an indication of the infeasibility for doing thorough hyperparameter tuning and comparison for many nuclear physics model training problems.  \n\nFor the model considered, we conclude that  the  performance  of  POUNDERS, within a  budget  of  function  evaluations,  is  extremely  robust. The Fayans EDF optimization results obtained in this work are generally consistent with those of\nFy($\\Delta r$) \\cite{Reinhard2017} and Fy($\\Delta r$,HFB) \\cite{Miller2019,Reinhard2020} models, see  Table~\\ref{tab:BestResults}. In particular, the set $\\xb_1$, which performs very well on the $\\delta\\langle r^2 \\rangle$ class appears to be fairly close to Fy($\\Delta r$,HFB). The extension of the Fayans model to isovector pairing, suggested in  \\cite{Reinhard2017}, will be carried out in the following work, which will also contain  detailed discussion of resulting  quantified nuclear properties.\n\n\n\n\\section*{Acknowledgments}\nThe work at Argonne was \nsupported by the U.S.\\ Department of\nEnergy, Office of Science, Office of Advanced Scientific Computing\nResearch, applied mathematics and SciDAC programs under Contract No.\\\nDE-AC02-06CH11357 and by the NUCLEI SciDAC-4 collaboration.  This work was also supported by the U.S.\\ Department of Energy, Office of Science, Office of Nuclear Physics under award numbers DE-SC0013365 (Michigan State University) and DE-SC0018083 (NUCLEI SciDAC-4 collaboration).\nWe gratefully acknowledge the computing resources provided on Bebop, a \nhigh-performance computing cluster operated by the Laboratory Computing Resource \nCenter at Argonne National Laboratory.\n\n\n\n\\section*{References}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:Introduction}\nRapid advances in sensing and data acquisition technologies are increasingly resulting in individual data samples or signals structured by multiple \\textit{modes}. Examples include hyperspectral video (four modes; two spatial, one temporal, and one spectral), colored depth video (five modes; two spatial, one temporal, one spectral, and one depth), and four-dimensional tomography (four modes; three spatial and one temporal). Such data form multiway arrays and are called \\textit{tensor data}~\\cite{smilde2005multi,kolda2009tensor}.\n\nTypical feature extraction approaches that handle tensor data tend to collapse or vectorize the tensor into a long one-dimensional vector and apply existing processing methods for one-dimensional data. Such approaches ignore the structure and inter-mode correlations in tensor data. More recently, several works instead assume a structure on the tensor of interest through tensor decompositions such as the CANDECOMP\/PARAFAC (CP) decomposition~\\cite{harshman1970foundations}, Tucker decomposition~\\cite{tucker1963implications}, and PARATUCK decomposition~\\cite{kolda2009tensor} to obtain meaningful representations of tensor data. Because these decompositions involve fewer parameters, or degrees of freedom, in the model, inference algorithms that exploit such decompositions often perform better than those that assume the tensors to be unstructured. Moreover, algorithms utilizing tensor decompositions tend to be more efficient in terms of storage and computational costs: the cost of storing the decomposition can be substantially lower, and numerical methods can exploit the structure by solving simpler subproblems.\n\nIn this work, we focus on the problem of finding sparse representations of tensors that admit a Tucker decomposition. More specifically, we analyze the \\textit{dictionary learning} (DL) problem for tensor data. The traditional DL problem for vector-valued data involves constructing an overcomplete basis (dictionary) such that each data sample can be represented by only a few columns (atoms) of that basis~\\cite{aharon2006img}. To account for the Tucker structure of tensor data, we require that the dictionary underlying the vectorized versions of tensor data samples be \\textit{Kronecker structured} (KS). That is, it is comprised of \\textit{coordinate dictionaries} that independently transform various modes of the tensor data. Such dictionaries have successfully been used for tensor data representation in applications such as hyperspectral imaging, video acquisition, distributed sensing, magnetic resonance imaging, and the tensor completion problem (multidimensional inpainting)~\\cite{duarte2012kronecker,caiafa2013multidimensional}. To provide some insights into the usefulness of KS dictionaries for tensor data, consider the hypothetical problem of finding sparse representations of $1024 \\times 1024 \\times 32 $ hyperspectral images. Traditional DL methods require each image to be rearranged into a one-dimensional vector of length $2^{25}$ and then learn an unstructured dictionary that has a total of $(2^{25} p)$ unknown parameters, where $p \\geq 2^{25}$. In contrast, KS DL only requires learning three coordinate dictionaries of dimensions $1024 \\times p_1$, $1024 \\times p_2$, and $32 \\times p_3$, where $p_1,p_2\\geq 1024$, and $p_3 \\geq 32$. This gives rise to a total of $[1024 (p_1 + p_2) + 32p_3]$ unknown parameters in KS DL, which is significantly smaller than $2^{25} p$. While such ``parameter counting'' points to the usefulness of KS DL for tensor data, a fundamental question remains open in the literature: what are the theoretical limits on the learning of KS dictionaries underlying $K$th-order tensor data? To answer this question, we examine the KS-DL objective function and find sufficient conditions on the number of samples (or sample complexity) for successful local identification of \\textit{coordinate dictionaries} underlying the KS dictionary. To the best of our knowledge, this is the first work presenting such identification results for the KS-DL problem.\n\n\\subsection{Our Contributions}\\label{subsec:contr}\nWe derive sufficient conditions on the true coordinate dictionaries, coefficient and noise distributions, regularization parameter, and the number of data samples such that the KS-DL objective function has a local minimum within a small neighborhood of the true coordinate dictionaries with high probability. Specifically, suppose the observations are generated from a true dictionary $\\mb{D}^0 \\in \\mbb{R}^{m \\times p}$ consisting of the Kronecker product of $K$ coordinate dictionaries, $\\mb{D}_k^0 \\in \\mbb{R}^{m_k \\times p_k}, k \\in \\lr{1,\\dots,K}$, where $m = \\prod_{k=1}^Km_k$ and $p = \\prod_{k=1}^Kp_k$. Our results imply that $N = \\max_{k\\in[K]} \\Omega(m_kp_k^3\\varepsilon_k^{-2})$ samples are sufficient (with high probability) to recover the underlying coordinate dictionaries $\\mb{D}_k^0$ up to the given estimation errors $\\varepsilon_k, k \\in \\lr{1,\\dots,K}$.\n\n\\subsection{Relationship to Prior Work}\\label{subsec:prior}\n\nAmong existing works on structured DL that have focused exclusively on the Tucker model for tensor data, several have only empirically established the superiority of KS DL in various settings for 2nd and 3rd-order tensor data~\\cite{hawe2013separable,zubair2013tensor,caiafa2013multidimensional,roemer2014tensor,dantas2017learning,ghassemi2017stark}.\n\nIn the case of unstructured dictionaries, several works do provide analytical results for the dictionary identifiability problem~\\cite{aharon2006uniqueness,agarwal2013learning,agarwal2013exact,arora2013new, schnass2014identifiability,schnass2014local,gribonval2014sparse,jung2015minimax}. These results, which differ from each other in terms of the distance metric used, cannot be trivially extended for the KS-DL problem. In this work, we focus on the Frobenius norm as the distance metric. Gribonval et al.~\\cite{gribonval2014sparse} and Jung et al.~\\cite{jung2015minimax} also consider this metric, with the latter work providing minimax lower bounds for dictionary reconstruction error. In particular, Jung et al.~\\cite{jung2015minimax} show that the number of samples needed for reliable reconstruction (up to a prescribed mean squared error $\\varepsilon$) of an $m\\times p$ dictionary within its local neighborhood must be \\emph{at least} on the order of $N = \\Omega(mp^2\\varepsilon^{-2})$. Gribonval et al.~\\cite{gribonval2014sparse} derive a competing upper bound for the sample complexity of the DL problem and show that $N = \\Omega(mp^3\\varepsilon^{-2})$ samples are \\emph{sufficient} to guarantee (with high probability) the existence of a local minimum of the DL cost function within the $\\varepsilon$ neighborhood of the true dictionary. In our previous works, we have obtained lower bounds on the minimax risk of KS DL for 2nd-order~\\cite{shakeri2016minimax} and $K$th-order tensors~\\cite{shakeri2017sample,shakeri2016arxiv}, and have shown that the number of samples necessary for reconstruction of the true KS dictionary within its local neighborhood up to a given estimation error scales with the sum of the product of the dimensions of the coordinate dictionaries, i.e., $N = \\Omega(p\\sum_{k=1}^Km_kp_k\\varepsilon^{-2})$. Compared to this sample complexity lower bound, our upper bound is larger by a factor $\\max_{k} p_k^2$.\n\nIn terms of the analytical approach, although we follow the same general proof strategy as the vectorized case of Gribonval et\nal.~\\cite{gribonval2014sparse}, our extension poses several technical challenges. These include: ($i$) expanding the asymptotic objective function into a summation in which individual terms depend on coordinate dictionary recovery errors, ($ii$) translating identification conditions on the KS dictionary to conditions on its coordinate dictionaries, and ($iii$) connecting the asymptotic objective function to the empirical objective function using concentration of measure arguments; this uses the \\textit{coordinate-wise Lipschitz continuity} property of the KS-DL objective function with respect to the coordinate dictionaries. To address these challenges, we require additional assumption on the generative model. These include: ($i$) the true dictionary and the recovered dictionary belong to the class of KS dictionaries, and ($ii$) dictionary coefficient tensors follow the \\textit{separable sparsity} model that requires nonzero coefficients to be grouped in blocks~\\cite{caiafa2013computing,shakeri2016arxiv}.\n\n\\subsection{Notational Convention and Preliminaries} \\label{subsec:notation}\nUnderlined bold upper-case, bold upper-case and lower-case letters are used to denote tensors, matrices and vectors, respectively, while non-bold lower-case letters denote scalars. For a tensor $\\ul{\\X}$, its $(i_1,\\dots,i_K)$-th element is denoted as $\\underline{x}_{i_1\\dots i_K}$. The $i$-th element of vector $\\mathbf{v}$ is denoted by $v_i$ and the $ij$-th element of matrix $\\mb{X}$ is denoted as $x_{ij}$. The $k$-th column of $\\mb{X}$ is denoted by $\\mb{x}_k$ and $\\mb{X}_{\\mathcal{I}}$ denotes the matrix consisting of the columns of $\\mb{X}$ with indices $\\mathcal{I}$. We use $|\\mathcal{I}|$ for the cardinality of the set $\\mathcal{I}$. Sometimes we use matrices indexed by numbers, such as $\\mb{X}_1$, in which case a second index (e.g., $\\mathbf{x}_{1,k}$) is used to denote its columns. We use $\\mathop{\\mathrm{vec}}\\nolimits(\\mb{X})$ to denote the vectorized version of matrix $\\mb{X}$, which is a column vector obtained by stacking the columns of $\\mb{X}$ on top of one another. We use $\\diag{\\mb{X}}$ to denote the vector comprised of the diagonal elements of $\\mb{X}$ and $\\Diag{\\mb{v}}$ to denote the diagonal matrix, whose diagonal elements are comprised of elements of $\\mb{v}$. The elements of the sign vector of $\\mathbf{v}$, denoted as $\\mathop{\\mathrm{sign}}\\nolimits(\\mathbf{v})$, are equal to $\\mathop{\\mathrm{sign}}\\nolimits(v_i)= v_i\/|v_i|$, for $v_i \\neq 0$, and $\\mathop{\\mathrm{sign}}\\nolimits(v_i)=0$ for $v_i = 0$, where $i$ denotes the index of any element of $v$. We also use $\\sin(\\mb{v})$ to denote the vector with elements $\\sin(v_i)$ (used similarly for other trigonometric functions). Norms are given by subscripts, so $\\|\\mb{v}\\|_0$, $\\|\\mb{v}\\|_1$, and $\\|\\mb{v}\\|_2$ are the $\\ell_0$, $\\ell_1$, and $\\ell_2$ norms of $\\mathbf{v}$, while $\\|\\mb{X}\\|_2$ and $\\|\\mb{X}\\|_F$ are the spectral and Frobenius norms of $\\mb{X}$, respectively.\nWe use $[K]$ to denote $\\{1,2,\\dots,K\\}$ and $\\mb{X}_{1:K}$ to denote $\\{\\mb{X}_k\\}_{k=1}^K$.\n\nWe write $\\mb{X} \\otimes \\mb{Y}$ for the \\textit{Kronecker product} of two matrices $\\mb{X}\\in \\mbb{R}^{m\\times n}$ and $\\mb{Y}\\in \\mbb{R}^{p\\times q}$,\nwhere the result is an $mp \\times nq$ matrix and we have $\\|\\mb{X} \\otimes\\mb{Y} \\|_F = \\|\\mb{X}\\|_F\\| \\mb{Y}\\|_F$~\\cite{horn2012matrix}. We also use $\\bigotimes_{k \\in K} \\mb{X}_k \\triangleq \\mb{X}_1 \\otimes \\dots \\otimes \\mb{X}_K$ . We define $\\mb{H}_{\\mb{X}}\\triangleq (\\mb{X}^\\top \\mb{X})^{-1}$, $\\mb{X}^+ \\triangleq \\mb{H}_{\\mb{X}}\\mb{X}^\\top$, and $\\mb{P}_{\\mb{X}} \\triangleq \\mb{X} \\mb{X}^+$ for full rank matrix $\\mb{X}$. In the body, we sometimes also use $\\Delta f(\\mb{X};\\mb{Y}) \\triangleq f(\\mb{X}) - f(\\mb{Y})$.\n\nFor matrices $\\mb{X}_1$ and $\\mb{X}_2$ of appropriate dimensions, we define their distance to be $d(\\mb{X},\\mb{Y}) = \\|\\mb{X}-\\mb{Y}\\|_F$. For $\\mb{X}^0$ belonging to some set $\\mathcal{X}$, we define\n\t\\begin{align}\n\t\\mc{S}_{\\varepsilon}(\\mb{X}^0) \\triangleq \\lr{\\mb{X} \\in \\mathcal{X}: \\|\\mb{X} - \\mb{X}^0\\|_F = \\varepsilon}, \\nonumber \\\\\n\t\\mathcal{B}_{\\varepsilon}(\\mb{X}^0) \\triangleq \\lr{\\mb{X} \\in \\mathcal{X}: \\|\\mb{X} - \\mb{X}^0\\|_F < \\varepsilon}, \\nonumber \\\\\t\n\t\\bar{\\mathcal{B}}_{\\varepsilon}(\\mb{X}^0) \\triangleq \\lr{\\mb{X} \\in \\mathcal{X}: \\|\\mb{X} - \\mb{X}^0\\|_F \\leq \\varepsilon}.\n\t\\end{align}\nNote that while $\\mc{S}_{\\varepsilon}(\\mb{X}^0)$ represents the surface of a sphere, we use the term ``sphere\" for simplicity. We use the standard ``big-$\\mathcal{O}$'' (Knuth) notation for asymptotic scaling.\n\n\\subsubsection{Tensor Operations and Tucker Decomposition for Tensors}\nA tensor is a multidimensional array where the order of the tensor is defined as the number of dimensions in the array.\n\n\\textit{Tensor Unfolding: }A tensor $\\ul{\\X} \\in \\mbb{R}^{p_1 \\times p_2\\times \\dots \\times p_K}$ of order $K$ can be expressed as a matrix by reordering its elements to form a matrix. This reordering is called unfolding: the mode-$k$ unfolding matrix of a tensor is a $p_k \\times \\prod_{i \\ne k} p_i$ matrix, which we denote by $\\mb{X}_{(k)}$. Each column of $\\mb{X}_{(k)}$ consists of the vector formed by fixing all indices of $\\ul{\\X}$ except the one in the $k$th-order.\nThe $k$-rank of a tensor $\\ul{\\X}$ is defined by $\\mathop{\\mathrm{rank}}\\nolimits(\\mb{X}_{(k)})$; trivially, $\\mathop{\\mathrm{rank}}\\nolimits(\\mb{X}_{(k)}) \\leq p_k$.\n\n\\textit{Tensor Multiplication: } The mode-$k$ matrix product of the tensor $\\ul{\\X}$ and a matrix $\\mb{A} \\in \\mbb{R}^{m_k \\times p_k}$, denoted by $\\ul{\\X} \\times_k \\mb{A}$, is a tensor of size $p_1 \\times \\dots p_{k-1} \\times m_k \\times p_{k+1} \\dots \\times p_K$ whose elements are\n$\n\t(\\ul{\\X} \\times_k \\mb{A})_{i_1\\dots i_{k-1} j i_{k+1} \\dots i_K} = \\sum_{i_k=1}^{p_k} \\underline{x}_{i_1\\dots i_{k-1} i_k i_{k+1} \\dots i_K} a_{ji_k}.\n$\nThe mode-$k$ matrix product of $\\ul{\\X}$ and $\\mb{A}$ and the matrix multiplication of $\\mb{X}_{(k)}$ and $\\mb{A}$ are related~\\cite{kolda2009tensor}:\n\t\\begin{align}\n\t\\ul{\\Y} = \\ul{\\X} \\times_k \\mb{A} \\Leftrightarrow \\mb{Y}_{(k)} = \\mb{A} \\mb{X}_{(k)}.\n\t\\end{align}\n\t\n\\textit{Tucker Decomposition: } The Tucker decomposition decomposes a tensor into a \\textit{core tensor} multiplied by a matrix along each mode~\\cite{tucker1963implications,kolda2009tensor}. We take advantage of the Tucker model since we can relate the Tucker decomposition to the Kronecker representation of tensors~\\cite{caiafa2013computing}.\nFor a tensor $\\ul{\\Y} \\in \\mbb{R}^{m_1 \\times m_2 \\times \\dots \\times m_K}$ of order $K$, if $\\mathop{\\mathrm{rank}}\\nolimits(\\mb{Y}_{(k)})\\leq p_k$ holds for all $k \\in [K]$ then, according to the Tucker model, $\\ul{\\Y}$ can be decomposed into:\n\t\\begin{align} \\label{eq:UY_UX}\n\t\\ul{\\Y} = \\ul{\\X} \\times_1 \\mb{D}_1  \\times_2 \\mb{D}_2 \\times_ 3 \\dots \\times_K \\mb{D}_K,\n\t\\end{align}\nwhere $\\ul{\\X} \\in \\mbb{R}^{p_1 \\times p_2\\times \\dots \\times p_K}$ denotes the core tensor and $\\mb{D}_k \\in \\mbb{R}^{m_k \\times p_k}$ are factor matrices.\nThe following is implied by \\eqref{eq:UY_UX}~\\cite{kolda2009tensor}:\n\t\\begin{align*}\n\t\\mb{Y}_{(k)} = \\mb{D}_{k}\\mb{X}_{(k)}(\\mb{D}_{K} \\otimes \\dots \\otimes \\mb{D}_{k+1} \\otimes \\mb{D}_{k-1} \\otimes \\dots \\otimes \\mb{D}_1)^\\top.\n\t\\end{align*}\nSince the Kronecker product satisfies $\\mathop{\\mathrm{vec}}\\nolimits(\\mb{B}\\mb{X}\\mb{A}^\\top)=(\\mb{A} \\otimes \\mb{B})\\mathop{\\mathrm{vec}}\\nolimits(\\mb{X})$, \\eqref{eq:UY_UX} is equivalent to\n\t\\begin{align} \\label{eq:vecty_vectx}\n\t\\mathop{\\mathrm{vec}}\\nolimits(\\ul{\\Y}) = \\big( \\mb{D}_K \\otimes \\mb{D}_{K-1} \\otimes \\dots \\otimes \\mb{D}_1 \\big) \\mathop{\\mathrm{vec}}\\nolimits(\\ul{\\X}),\n\t\\end{align}\nwhere $\\mathop{\\mathrm{vec}}\\nolimits(\\ul{\\Y}) \\triangleq \\mathop{\\mathrm{vec}}\\nolimits(\\mb{Y}_{(1)})$ and $\\mathop{\\mathrm{vec}}\\nolimits(\\ul{\\X}) \\triangleq \\mathop{\\mathrm{vec}}\\nolimits(\\mb{X}_{(1)})$.\n\n\\subsubsection{Definitions for Matrices}\nWe use the following definitions for a matrix $\\mb{D}$ with unit-norm columns:\n$\\delta_s(\\mb{D})$ denotes the \\textit{restricted isometry property} ($\\mathsf{RIP}$) constant of order $s$ for $\\mb{D}$~\\cite{candes2008restricted}.\nWe define the \\textit{worst-case coherence} of $\\mb{D}$ as $\\mu_1(\\mb{D}) = \\max_{\\substack{i,j\\\\i \\neq j}} \\lra{\\mb{d}_i^\\top \\mb{d}_j}$.\nWe also define the \\textit{order-$s$ cumulative coherence} of $\\mb{D}$ as\n\t\\begin{align} \\label{eq:mu_1}\n\t\\mu_{s}(\\mb{D}) \\triangleq \\max_{|\\mc{J} |\\leq s} \\max_{j \\not\\in \\mc{J}}\n\t\t\\|\\mb{D}_{\\mc{J}}^\\top \\mb{d}_{j}\\|_1.\n\t\\end{align}\nNote that for $s=1$, the cumulative coherence is equivalent to the worst-case coherence and $\\mu_{s}(\\mb{D}) \\leq s \\mu_1(\\mb{D})$~\\cite{gribonval2014sparse}.\nFor $\\mb{D} = \\bigotimes_{k \\in [K]} \\mb{D}_k$, where $\\mb{D}_k$'s have unit-norm columns, $\\mu_1(\\mb{D}) = \\max_{k \\in [K]} \\mu_1(\\mb{D}_k)$~\\cite[Corollary 3.6]{jokar2009sparse} and it can be shown that\\footnote{The proof of \\eqref{eq:mu_s} is provided in Appendix C.}:\n\t\\begin{align}\\label{eq:mu_s}\n\t\\mu_s(\\mb{D}) &\\leq \\max_{k \\in [K]} \\mu_{s_k}(\\mb{D}_k)\n\t\t\\bigg( \\prod_{\\substack{i \\in [K], \\\\ i \\neq k}}  \\lrp{ 1+\\mu_{s_i-1}(\\mb{D}_i)} \\bigg).\n\t\\end{align}\t\n\nThe rest of the paper is organized as follows. We formulate the KS-DL problem in Section~\\ref{sec:model}. In Section~\\ref{sec:asymp}, we provide analysis for asymptotic recovery of coordinate dictionaries composing the KS dictionary and in Section~\\ref{sec:finite}, we present sample complexity results for identification of coordinate dictionaries that are based on the results of Section~\\ref{sec:asymp}. Finally, we conclude the paper in Section~\\ref{sec:discuss}. In order to keep the main exposition simple, proofs of the lemmas and propositions are relegated to appendices.\n\n\\section{System Model} \\label{sec:model}\nWe assume the observations are $K$th-order tensors $\\ul{\\Y} \\in \\mbb{R}^{m_1\\times m_2 \\times \\dots \\times m_K}$. Given generating \\textit{coordinate dictionaries} $\\mb{D}^0_k \\in \\mbb{R}^{m_k \\times p_k}$, \\textit{coefficient tensor} $\\ul{\\X} \\in \\mbb{R}^{p_1\\times p_2 \\times \\dots \\times p_K}$, and \\textit{noise tensor} $\\ul{\\N}$, we can write $\\mb{y} \\triangleq \\mathop{\\mathrm{vec}}\\nolimits(\\ul{\\Y})$ using \\eqref{eq:vecty_vectx} as\\footnote{We have reindexed $\\mb{D}_k$'s in \\eqref{eq:vecty_vectx} for ease of notation.}\n\t\\begin{align} \\label{eq:obs_model}\n\t\\mb{y} = \\bigg( \\bigotimes_{k \\in [K]} \\mb{D}_k^0 \\bigg) \\mb{x} + \\mb{w},\n\t\t\\quad \\|\\mb{x}\\|_0 \\leq s,\n\t\\end{align}\nwhere $\\mb{x}=\\mathop{\\mathrm{vec}}\\nolimits(\\ul{\\X}) \\in \\mbb{R}^{p}$ denotes the sparse generating coefficient vector, $\\mb{D}^0 = \\bigotimes \\mb{D}_k^0 \\in \\mbb{R}^{m\\times p}$ denotes the underlying KS dictionary, and $\\mb{w}=\\mathop{\\mathrm{vec}}\\nolimits(\\ul{\\N})  \\in \\mbb{R}^m$ denotes the underlying noise vector. Here, $\\mb{D}_k^0 \\in \\mathcal{D}_k = \\lr{ \\mb{D}_k \\in \\mbb{R}^{m_k\\times p_k}, \\|\\mb{d}_{k,j}\\|_2 = 1, \\forall j \\in [p_k]}$ for $k \\in [K]$, $p = \\prod_{k \\in [K]}p_k$ and $m = \\prod_{k \\in [K]}m_k$.\\footnote{Note that the $\\mathcal{D}_k$'s are compact sets on their respective oblique manifolds of matrices with unit-norm columns~\\cite{gribonval2014sparse}.} We use $\\bigotimes$ for $\\bigotimes_{k\\in[K]}$ in the following for simplicity of notation. We assume we are given $N$ noisy tensor observations, which are then stacked in a matrix $\\mb{Y} = [\\mb{y}_1,\\dots,\\mb{y}_N]$. To state the problem formally, we first make the following assumptions on distributions of $\\mb{x}$ and $\\mb{w}$ for each tensor observation.\n\n\\textit{Coefficient distribution:} We assume the coefficient tensor $\\ul{\\X}$ follows the random \\textit{``separable sparsity\"} model. That is, $\\mb{x}=\\mathop{\\mathrm{vec}}\\nolimits(\\ul{\\X})$ is sparse and the support of nonzero entries of $\\mb{x}$ is structured and random. Specifically, we sample $s_k$ elements uniformly at random from $[p_k]$, $k \\in [K]$. Then, the random support of $\\mb{x}$ is $\\lr{ \\mc{J} \\subseteq [p], |\\mc{J} |=s}$ and is associated with\n\\begin{align*}\n\t\\lr{\\mc{J}_1\\times \\mc{J}_2 \\times \\dots \\times \\mc{J}_K:\n\t\t\\mc{J}_k\\subseteq [p_k], |\\mc{J}_k|=s_k, k \\in [K]}\n\\end{align*}\nvia lexicographic indexing, where $ s=\\prod_{k \\in [K]} s_k$, and the support of $\\mb{x}_{1:N}$'s are assumed to be independent and identically distributed (i.i.d.). This model requires nonzero entries of the coefficient tensors to be grouped in blocks and the sparsity level associated with each coordinate dictionary to be small~\\cite{caiafa2013computing}.\\footnote{In contrast, for coefficients following the random non-separable sparsity model, the support of the nonzero entries of the coefficient vector are assumed uniformly distributed over $\\lr{\\mc{J} \\subseteq [p]: |\\mc{J}|=s}$.}\n\nWe now make the same assumptions for the distribution of $\\mb{x}$ as assumptions A and B in Gribonval et al.~\\cite{gribonval2014sparse}.\nThese include:\n($i$) $\\mbb{E}\\lr{\\mb{x}_\\mc{J} \\mb{x}_\\mc{J}^\\top |\\mc{J}} = \\mbb{E}\\lr{x^2}\\mb{I}_{s}$,\n($ii$) $\\mbb{E}\\lr{\\mb{x}_\\mc{J} \\boldsymbol{\\sigma}_\\mc{J}^\\top |\\mc{J}} = \\mbb{E} \\lr{|x|}\\mb{I}_{s}$, where $\\boldsymbol{\\sigma} = \\mathop{\\mathrm{sign}}\\nolimits(\\mb{x})$,\n($iii$) $\\mbb{E}\\lr{\\boldsymbol{\\sigma}_\\mc{J} \\boldsymbol{\\sigma}_\\mc{J}^\\top |\\mc{J}} = \\mb{I}_{s}$,\n($iv$) magnitude of $\\mb{x}$ is bounded, i.e., $\\|\\mb{x}\\|_2 \\leq M_x $ almost surely, and\n($v$) nonzero entries of $\\mb{x}$ have a minimum magnitude, i.e., $\\min_{j \\in \\mc{J}} |x_j| \\geq x_{\\mathrm{min}}$ almost surely.\nFinally, we define $\\kappa_x \\triangleq \\mbb{E}\\lr{|x|}\/\\sqrt{\\mbb{E} \\lr{x^2}}$ as a measure of the flatness of $\\mb{x}$ ($\\kappa_x \\leq 1$, with $\\kappa_x=1$ when all nonzero coefficients are equal~\\cite{gribonval2014sparse}).\n\n\\textit{Noise distribution:} We make following assumptions on the distribution of noise, which is assumed i.i.d. across data samples:\n($i$) $\\mbb{E}\\lr{\\mb{w} \\mb{w}^\\top} = \\mbb{E}\\lr{w^2} \\mb{I}_m $,\n($ii$) $\\mbb{E}\\lr{\\mb{w} \\mb{x}^\\top |\\mc{J}}=\\mbb{E}\\lr{\\mb{w} \\boldsymbol{\\sigma}^\\top |\\mc{J}}=\\mathbf{0}$, and\n($iii$) magnitude of $\\mb{w}$ is bounded, i.e., $\\|\\mb{w}\\|_2 \\leq M_w$ almost surely.\n\nOur goal in this paper is to recover the underlying coordinate dictionaries, $\\mb{D}^0_k$, from $N$ noisy realizations of tensor data.\nTo solve this problem, we take the empirical risk minimization approach and define\n\t\\begin{align}\n\t&f_\\mb{y} \\lrp{\\D_{1:K}} \\triangleq\n\t\t \\inf_{\\mb{x}' \\in \\mbb{R}^p } \\bigg\\{\n\t\t \\frac{1}{2} \\norm{\\mb{y} - \\lrp{\\bigotimes \\mb{D}_k } \\mb{x}'\n\t\t }_2^2\n\t\t +\\lambda\\|\\mb{x}'\\|_1 \\bigg\\},\n \t\t\\text{and} \\nonumber \\\\\n\t&F_\\mb{Y} \\lrp{\\D_{1:K}} \\triangleq\n\t\t \\frac{1}{N} \\sum_{n =1}^N f_{\\mb{y}_n}\\lrp{\\D_{1:K}} ,\n\t\\end{align}\nwhere $\\lambda$ is a regularization parameter. In theory, we can recover the coordinate dictionaries by solving the following regularized optimization program:\n\t\\begin{align}\n    \\min_{\\substack{\\mb{D}_k \\in \\mathcal{D}_k \\\\ k \\in [K]}}\n\t\tF_\\mb{Y} \\lrp{\\D_{1:K}}. \\label{eq:f_x}\n\t\\end{align}\nMore specifically, given desired errors $\\lr{\\varepsilon_k}_{k=1}^K$, we want a local minimum of~\\eqref{eq:f_x} to be attained by coordinate dictionaries $\\wh{\\D}_k \\in \\mathcal{B}_{\\varepsilon_k}(\\mb{D}^0_k), k \\in [K]$. That is, there exists a set $\\{\\wh{\\D}_k\\}_{k \\in [K]} \\subset \\lr{\\mb{D}_k \\in \\mathcal{B}_{\\varepsilon_k}(\\mb{D}^0_k)}_{k \\in [K]}$ such that $F_\\mb{Y}(\\wh{\\D}_{1:K}) \\leq F_\\mb{Y}(\\D_{1:K})$.\\footnote{We focus on the local recovery of coordinate dictionaries (i.e., $\\wh{\\D}_k \\in \\mathcal{B}_{\\varepsilon_k}(\\mb{D}^0_k)$) due to ambiguities in the general DL problem. This ambiguity is a result of the fact that dictionaries are invariant to permutation and sign flips of dictionary columns, resulting in equivalent classes of dictionaries. Some works in the literature on conventional overcome this issue by defining distance metrics that capture the distance between these equivalent classes~\\cite{agarwal2013exact,agarwal2013learning,arora2013new}.}\nTo address this problem, we first minimize the statistical risk:\n\t\\begin{align} \\label{eq:f_x_asym}\n\t&\\min_{\\substack{\\mb{D}_k \\in \\mathcal{D}_k \\\\ k \\in [K]}} f_\\mbb{P} \\lrp{\\D_{1:K}} \\triangleq\n\t\\min_{\\substack{\\mb{D}_k \\in \\mathcal{D}_k \\\\ k \\in [K]}}\n\t\t\\mbb{E}_\\mb{y} \\lr{ f_\\mb{y}\\lrp{\\D_{1:K}}}.\n\t\\end{align}\nThen, we connect $F_\\mb{Y} \\lrp{\\D_{1:K}}$ to $f_\\mbb{P} \\lrp{\\D_{1:K}}$ using concentration of measure arguments and obtain the number of samples sufficient for local recovery of the coordinate dictionaries. Such a result ensures that any KS-DL algorithm that is guaranteed to converge to a local minimum, and which is initialized close enough to the true KS dictionary, will converge to a solution close to the generating coordinate dictionaries (as opposed to the generating KS dictionary, which is guaranteed by analysis of the vector-valued setup~\\cite{gribonval2014sparse}).\n\n\\section{Asympototic Identifiability Results}\\label{sec:asymp}\nIn this section, we provide an identifiability result for the KS-DL objective function in \\eqref{eq:f_x_asym}. The implications of this theorem are discussed in Section~\\ref{sec:discuss}.\n\\begin{theorem}\\label{thm:asymp}\nSuppose the observations are generated according to \\eqref{eq:obs_model} and the dictionary coefficients follow the separable sparsity model of Section~\\ref{sec:model}. Further, assume the following conditions are satisfied:\n\t\\begin{align} \\label{eq:cond_k_i_p_i}\n\t&s_k \\leq \\frac{p_k}{8\\lrp{\\norm{\\mb{D}^0_k}_2+1}^2}, \\\\\\nonumber\n\t&\\max_{k \\in [K]} \\lr{\\mu_{s_k}(\\mb{D}^0_k)} \\leq \\frac{1}{4} ,  \\quad\n\t\\mu_s(\\mb{D}^0) <\\frac{1}{2},\n\t\\end{align}\nand\n\t\\begin{align} \\label{eq:cond_m_p}\n\t&\\frac{\\mbb{E}\\lr{x^2}}{M_x \\mbb{E}\\lr{|x|}} > \\frac{24\\sqrt{3}(4.5^{K\/2})K}{(1-2\\mu_s(\\mb{D}^0))} \\nonumber\\\\\n\t&\\qquad \\quad \\max_{k \\in [K]} \\lr{\n\t\t \\frac{s_k}{p_k}\n\t\t \\norm{ {\\mb{D}^0_k}^\\top\\mb{D}^0_k - \\mb{I}}_F\n\t\t \\lrp{ \\norm{\\mb{D}^0_k}_2+1}}.\n\t\\end{align}\nDefine\n\t\\begin{align} \\label{eq:cond_C_min_C_max}\n\t&C_{k,\\min} \\triangleq   8 (3^{\\frac{K+1}{2}})\\kappa_x^2\n\t\t\\lrp{\\frac{s_k}{p_k}}\n\t\t\\norm{{\\mb{D}_k^0}^\\top\\mb{D}_k^0 - \\mb{I}}_F  \\lrp{ \\norm{\\mb{D}^0_k}_2+1} ,\n\t\t \\nonumber \\\\\n\t&C_{\\max} \\triangleq \\frac{1}{3K(1.5)^{K\/2}} \\frac{\\mbb{E}\\lr{|x|}}{M_x} (1-2\\mu_s(\\mb{D}^0)).\n\t\\end{align}\nThen, the map $\\D_{1:K}  \\mapsto f_{\\mbb{P}}\\lrp{\\D_{1:K}}$ admits a local minimum $\\widehat{\\mb{D}}=\\bigotimes_{k \\in [K]} \\widehat{\\mb{D}}_k$ such that $\\widehat{\\mb{D}}_k \\in \\mathcal{B}_{\\varepsilon_k}(\\mb{D}^0_k)$, $k \\in [K]$, for any $\\varepsilon_k>0$ as long as\n\t\\begin{align} \\label{eq:cond_lambda}\n\t\\lambda \\leq \\frac{x_\\mathrm{min}}{8\\times 3^{(K-1)\/2}},\n\t\\end{align}\n\t\\begin{align} \\label{eq:cond_r_i}\n\t&\\frac{\\lambda C_{k,\\min}}{\\mbb{E}\\lr{|x|}} < \\varepsilon_k < \\frac{\\lambda C_{\\max}}{\\mbb{E}\\lr{|x|}}, \\ k \\in [K],\n\t\\end{align}\nand\n\t\\begin{align} \\label{eq:cond_noise}\n\t\\frac{M_w}{M_x}\n\t\t < 3(1.5)^{K\/2} \\bigg(\\frac{\\lambda K C_{\\max} }{\\mbb{E}\\lr{|x|}}\n\t\t - \\sum_{k \\in [K]} \\varepsilon_k\\bigg).\n\t\\end{align}\n\\end{theorem}\n\n\\subsection{Discussion}\n\nTheorem~\\ref{thm:asymp} captures how the existence of a local minimum for the statistical risk minimization problem depends on various properties of the coordinate dictionaries and demonstrates that there exists a local minimum of $f_{\\mbb{P}} \\lrp{\\D_{1:K}}$ that is in local neighborhoods of the coordinate dictionaries. This ensures asymptotic recovery of coordinate dictionaries within some local neighborhood of the true coordinate dictionaries, as opposed to KS dictionary recovery for vectorized observations~\\cite[Theorem 1]{gribonval2014sparse}.\n\nWe now explicitly compare conditions in Theorem~\\ref{thm:asymp} with the corresponding ones for vectorized observations~\\cite[Theorem 1]{gribonval2014sparse}.\nGiven that the coefficients are drawn from the separable sparsity model, the sparsity constraints for the coordinate dictionaries in \\eqref{eq:cond_k_i_p_i} translate into\n\t\\begin{align}\n\t\\frac{s}{p} = \\prod_{k \\in [K]} \\frac{ s_k}{ p_k}\n\t\t\\leq \\frac{1}{8^K \\prod_k \\lrp{\\norm{\\mb{D}^0_k}_2+1}^2} .\n\t\\end{align}\nTherefore, we have $\\dfrac{s}{p}= \\mathcal{O}\\lrp{ \\frac{1}{ \\prod_k \\norm{\\mb{D}^0_k}_2^2}}=\\mathcal{O}\\lrp{\\frac{1}{\\|\\mb{D}^0\\|_2^2}}$. Using the fact that $\\norm{\\mb{D}^0}_2 \\geq \\|\\mb{D}^0\\|_F\/ \\sqrt{m} = \\sqrt{p}\/\\sqrt{m}$, this translates into sparsity order\n$\ts = \\mathcal{O}\\lrp{ m}$. Next, the left hand side of the condition in \\eqref{eq:cond_m_p} is less than 1. Moreover, from properties of the Frobenius norm, it is easy to show that\n$\n\t\\norm{{\\mb{D}^0_k}^\\top\\mb{D}_k^0 - \\mb{I} }_F \\geq \\sqrt{p_k(p_k-m_k)\/m_k}.\n$\nThe fact that $\\norm{\\mb{D}_k^0}_2 \\geq \\sqrt{p_k}\/\\sqrt{m_k}$ and the assumption $\\mu_{s_k}(\\mb{D}_k^0)\\leq 1\/4$ imply that the right hand side of \\eqref{eq:cond_m_p} is lower bounded by $\\Omega\\lrp{ \\max_k s_k\\sqrt{(p_k-m_k)\/m_k^2}}$.\nTherefore, Theorem~\\ref{thm:asymp} applies to coordinate dictionaries with  dimensions $p_k \\leq m_k^2$ and subsequently, KS dictionaries with $p \\leq m^2$. Both the sparsity order and dictionary dimensions are in line with the scaling results for vectorized data~\\cite{gribonval2014sparse}.\n\n\\subsection{Proof Outline}\n\nFor given radii $0<\\varepsilon_k\\leq 2\\sqrt{p_k}, k \\in [K]$, the spheres $\\mc{S}_{\\varepsilon_k}(\\mb{D}^0_k)$ are non-empty. This follows from the construction of dictionary classes, $\\mc{D}_k$'s.\nMoreover, the mapping $\\D_{1:K} \\mapsto f_{\\mbb{P}}\\lrp{\\D_{1:K}}$ is continuous with respect to the Frobenius norm $\\|\\D_k-\\D'_k\\|_F$ on all $\\mb{D}_k,\\mb{D}'_k \\in \\mbb{R}^{m_k\\times p_k}, k \\in [K]$~\\cite{gribonval2015sample}. Hence, it is also continuous on compact constraint sets $\\mathcal{D}_k$'s.\nWe derive conditions on the coefficients, underlying coordinate dictionaries, $M_w$, regularization parameter, and $\\varepsilon_k$'s such that\n\t\\begin{align}  \\label{eq:f_p_r_def}\n\t\\Delta f_{\\mbb{P}}\\lrp{\\eps_{1:K}}\n\t\t\\triangleq \\inf_{\\mb{D}_k \\in \\mc{S}_{\\varepsilon_k}(\\mb{D}^0_k)}\n\t\t\\Delta f_{\\mbb{P}} \\lrp{ \\D_{1:K}; \\D^0_{1:K} } >0.\n\t\\end{align}\nThis along with the compactness of closed balls $\\bar{\\mathcal{B}}_{\\varepsilon_k}(\\mb{D}^0_k)$ and the continuity of the mapping $\\D_{1:K} \\mapsto f_{\\mbb{P}}\\lrp{\\D_{1:K}}$ imply the existence of a local minimum of $f_{\\mbb{P}}\\lrp{\\D_{1:K}}$ achieved by $\\wh{\\D}_{1:K}$ in open balls, $\\mathcal{B}_{\\varepsilon_k}(\\mb{D}_k^0)$'s, $k \\in [K]$.\n\n\nTo find conditions that ensure $\\Delta f_{\\mbb{P}}\\lrp{\\eps_{1:K}} > 0$, we take the following steps:\ngiven coefficients that follow the separable sparsity model, we can decompose any $\\mb{D}_\\mc{J}, |\\mc{J}|=s$, as\n\t\\begin{align} \\label{eq:Dj_D12}\n\t\\mb{D}_\\mc{J}  = \\bigotimes \\mb{D}_{k,\\mc{J}_k},\n\t\\end{align}\nwhere $\\ |\\mc{J}_k|=s_k$ for $k \\in[K]$.\\footnote{The separable sparsity distribution model implies sampling without replacement from columns of $\\mb{D}_k$.}\nGiven a generating $\\boldsymbol{\\sigma} = \\mathop{\\mathrm{sign}}\\nolimits(\\mb{x})$,\nwe obtain $\\widehat{\\mb{x}}$ by solving $f_\\mb{y}\\lrp{\\D_{1:K}}$ with respect to $\\mb{x}'$, conditioned on the fact that $\\mathop{\\mathrm{sign}}\\nolimits(\\widehat{\\mb{x}})=\\widehat{\\boldsymbol{\\sigma}}=\\boldsymbol{\\sigma}$. This eliminates the dependency of $f_\\mb{y} \\lrp{\\D_{1:K}}$ on $\\inf_{\\mb{x}'}$ by finding a closed-form expression for $f_\\mb{y}\\lrp{\\D_{1:K}}$ given $\\widehat{\\boldsymbol{\\sigma}}=\\boldsymbol{\\sigma}$, which we denote as $\\phi_\\mb{y}\\lrp{\\D_{1:K}|\\boldsymbol{\\sigma}}$. Defining\n\\begin{align}\n\\phi_{\\mbb{P}}\\lrp{\\D_{1:K}|\\boldsymbol{\\sigma}} \\triangleq \\mbb{E}\\lr{\\phi_\\mb{y}\\lrp{\\D_{1:K}|\\boldsymbol{\\sigma}}},\n\\end{align}\nwe expand $ \\Delta \\phi_{\\mbb{P}}\\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}}$  using \\eqref{eq:Dj_D12} and separate the terms that depend on each radius $\\varepsilon_k = \\|\\D_k-\\D^0_k\\|_F$ to obtain conditions for sparsity levels $s_k, k\\in [K]$, and coordinate dictionaries such that $\\Delta \\phi_{\\mbb{P}} \\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}} >0$.\nFinally, we derive conditions on $M_w$, coordinate dictionary coherences and $\\varepsilon_k$'s that ensure $\\widehat{\\boldsymbol{\\sigma}}=\\boldsymbol{\\sigma}$ and $\\Delta f_{\\mbb{P}} \\lrp{ \\D_{1:K};\\D^0_{1:K} } = \\Delta \\phi_{\\mbb{P}} \\lrp{\\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}}$.\n\n\n\\begin{remark}\nThe key assumption in the proof of Theorem~\\ref{thm:asymp} is expanding $\\mb{D}_{\\mc{J}}$ according to~\\eqref{eq:Dj_D12}. This is a consequence of the separable sparsity model for dictionary coefficients. For a detailed discussion on the differences between the separable sparsity model and the random sparsity model for tensors, we refer the readers to our earlier work~\\cite{shakeri2016minimax}.\n\\end{remark}\n\n\\begin{remark}\nAlthough some of the forthcoming lemmas needed of Theorem~\\ref{thm:asymp} impose conditions on $\\mb{D}_k$'s as well as true coordinate dictionaries $\\mb{D}^0_k$'s, we later translate these conditions exclusively in terms of $\\mb{D}_k^0$'s and $\\varepsilon_k$'s.\n\\end{remark}\n\n\nThe proof of Theorem~\\ref{thm:asymp} relies on the following propositions and lemmas. The proofs of these are provided in Appendix A.\n\n\\begin{proposition} \\label{prop:1}\nSuppose the following inequalities hold for $k \\in [K]$:\n\t\\begin{align} \\label{eq:cond_delt_s_prop}\n\ts_k \\leq \\frac{p_k}{8(\\|\\mb{D}_k^0\\|_2+1)^2}\\quad \\text{and} \\quad\n\t\\max_{k\\in [K]}&\\lr{\\delta_{s_k}(\\mb{D}_k^0)}\\leq \\frac{1}{4} .\n\t\\end{align}\nThen, for\n\t\\begin{align} \\label{eq:cond_lamb_1}\n\t\\bar{\\lambda} \\triangleq \\dfrac{\\lambda}{\\mbb{E}\\lr{|x|}}\\leq \\dfrac{1}{8\\times 3^{(K-1)\/2}},\n\t\\end{align}\nany collection of $\\lr{ \\varepsilon_k: \\varepsilon_k \\leq 0.15, k \\in [K]}$, and for all $\\mb{D}_k \\in \\mc{S}_{\\varepsilon_k}(\\mb{D}_k^0)$, we have :\n\t\\begin{align} \\label{eq:delt_p_LB}\n\t&\\Delta \\phi_{\\mbb{P}} \\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}}\n\t\t\\geq  \\frac{s\\mbb{E}\\{x^2\\}}{8} \\sum_{k \\in [K]}\n\t\t\\frac{\\varepsilon_k}{p_k}  \\lrp{\\varepsilon_k- \\varepsilon_{k,\\min}(\\bar{\\lambda})},\n\t\\end{align}\nwhere\n\t\\begin{align*}\n\t&\\varepsilon_{k,\\min} (\\bar{\\lambda})\n\t\\triangleq  \\frac{3^{(K-1)\/2}}{2} \\lrp{1.5^{\\frac{K-1}{2}}\n\t\t+2^{(K+1)}\\bar{\\lambda} } \\bar{\\lambda} C_{k,\\min}.\n\t\\end{align*}\nIn addition, if\n\t\\begin{align}  \\label{eq:cond_lamb_2}\n\t\\bar{\\lambda} \\leq \\frac{0.15}{\\max_{k \\in [K]} C_{k,\\min}},\n\t\\end{align}\nthen $\\varepsilon_{k,\\min}(\\bar{\\lambda})<0.15$.\nThus, $\\Delta \\phi_\\mbb{P}\\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}}  > 0 $ for all $\\varepsilon_k \\in (\\varepsilon_{k,\\min}(\\bar{\\lambda}),0.15], k \\in [K]$.\n\\end{proposition}\n\nThe proof of Proposition \\ref{prop:1} relies on the following lemmas as well as supporting lemmas from the analysis of vectorized data~\\cite[Lemmas~4,6,7,15,16]{gribonval2014sparse}.\n\n\n\\begin{lemma} \\label{def:P_H_PS}\nLet $\\mb{D} = \\bigotimes \\D_k $ where $\\delta_s(\\mb{D}_k)<1$ for $k \\in [K]$, and $\\mc{J}$ be a support set generated by the separable sparsity model. Then any $\\mb{D}_\\mc{J}, |\\mc{J}|=s$, can be decomposed as\n$\\mb{D}_\\mc{J}  = \\bigotimes \\mb{D}_{k,\\mc{J}_k}$,\nwhere $\\ |\\mc{J}_k|=s_k$ and $\\mathop{\\mathrm{rank}}\\nolimits (\\mb{D}_{k,\\mc{J}_k})=s_k$, for $k \\in[K]$. Also, the following relations hold for this model:\\footnote{The equations follow from basic properties of the Kronecker product~\\cite{horn2012matrix}.}\n\t\\begin{align} \\label{eq:P_H_Ps}\n\t\\bP_{\\D_{\\cJ}} = \\bigotimes \\bP_{\\D_{k,\\cJ_k}},   \\D_{\\cJ}^+ = \\bigotimes \\D_{k,\\cJ_k}^+,  \\bH_{\\D_{\\cJ}} = \\bigotimes \\bH_{\\D_{k,\\cJ_k}},\n\t\\end{align}\nwhere $\\mb{P}$ and $\\mb{H}$ are defined in Section~\\ref{subsec:notation}.\n\\end{lemma}\n\n\n\\begin{lemma} \\label{lem:Dtld}\nGiven $\\D_{1:K}$ and $\\D^0_{1:K}$, the difference\n\t\\begin{align} \\label{eq:otD_otDp}\n\t&\\bigotimes \\D_k  - \\bigotimes \\D^0_k  \\nonumber\\\\\n\t&\\qquad = \\sum _{k \\in [K]} \\widetilde{\\mb{D}}_{k,1} \\otimes \\dots \\otimes\n\t\t\\lrp{\\mb{D}_k - \\mb{D}^0_k}\t\\otimes \\dots \\otimes \\widetilde{\\mb{D}}_{k,K},\n\t\\end{align}\t\nwhere without loss of generality, each $\\widetilde{\\mb{D}}_{k,i}$ is equal to either $\\mb{D}^0_i$ or $\\mb{D}_i$, for $k \\in [K]$.\n\\end{lemma}\n\nWe drop the $k$ index from $\\wt{\\D}_{k,i}$ for ease of notation throughout the rest of the paper.\n\n\\begin{lemma} \\label{lemma:f_closedform}\nLet $\\boldsymbol{\\sigma} \\in \\{-1,0,1\\}^p$ be an arbitrary sign vector and $\\mc{J} = \\mc{J}(\\boldsymbol{\\sigma})$ be its support. Define\\footnote{The quantity $\\phi_\\mb{y}\\lrp{\\D_{1:K}|\\boldsymbol{\\sigma}}$ is not equal to $\\phi_\\mb{y}\\lrp{\\D_{1:K}}$ conditioned on $\\boldsymbol{\\sigma}$ and the expression is only used for notation.}\n\t\\begin{align} \\label{eq:delt_inf}\n\t\\phi_\\mb{y}\\lrp{\\D_{1:K}|\\boldsymbol{\\sigma}} \\triangleq \\inf_{\\substack{\\mb{x} \\in \\mbb{R}^p \\\\ \\mathop{\\mathrm{supp}}\\nolimits(\\mb{x}) \\subset \\mc{J}}}\n\t\t\\frac{1}{2} \\norm{\\mb{y} - \\lrp{\\bigotimes \\D_k } \\mb{x} }_2^2+ \\lambda {\\boldsymbol{\\sigma}}^\\top\\mb{x}.\n\t\\end{align}\nIf $\\mb{D}_{k,\\mc{J}_k}^\\top \\mb{D}_{k,\\mc{J}_k}$ is invertible for $k \\in [K]$, then $\\widehat{\\mb{x}}$ minimizes $\\phi_\\mb{y}\\lrp{\\D_{1:K}|\\boldsymbol{\\sigma}} $, where\n\t\\begin{align}  \\label{eq:xhat_closedform}\n\t\\widehat{\\mb{x}}_\\mc{J} = \\lrp{\\bigotimes \\D_{k,\\cJ_k}^+ } \\mb{y}  - \\lambda \\lrp{ \\bigotimes \\big( \\mb{D}_{k,\\mc{J}_k}^\\top\\mb{D}_{k,\\mc{J}_k} \\big)^{-1} }\\boldsymbol{\\sigma}_\\mc{J},\n\t\\end{align}\nand $\\widehat{\\mb{x}}_{\\mc{J}^c} = \\mathbf{0}$. Thus, $\\phi_\\mb{y}\\lrp{\\D_{1:K} |\\boldsymbol{\\sigma}}$ can be expressed in closed form as:\n\t\\begin{align} \\label{eq:delt_closedform}\n\t&\\phi_\\mb{y}\\lrp{\\D_{1:K}|\\boldsymbol{\\sigma}} = \\frac{1}{2}\\|\\mb{y}\\|_2^2\n\t\t- \\frac{1}{2} \\mb{y}^\\top\n\t\t\\lrp{ \\bigotimes \\bP_{\\D_{k,\\cJ_k}}}\\mb{y}  \\nonumber\\\\\n\t&\\ + \\lambda {\\boldsymbol{\\sigma}}_\\mc{J}^\\top\n\t\t\\lrp{ \\bigotimes \\D_{k,\\cJ_k}^+  } \\mb{y}\n\t\t-\\frac{\\lambda^2}{2}{\\boldsymbol{\\sigma}}_\\mc{J}^\\top \\lrp{ \\bigotimes \\bH_{\\D_{k,\\cJ_k}}} \\boldsymbol{\\sigma}_\\mc{J}.\n\t\\end{align}\n\\end{lemma}\n\n\\begin{lemma} \\label{lem:exp_phi}\nAssume $\\max\\lr{ \\delta_{s_k}(\\mb{D}_k^0),\\delta_{s_k}(\\mb{D}_k)}<1$ for $k \\in [K]$ and let $\\wt{\\D}_k$ be equal to either $\\mb{D}^0_k$ or $\\mb{D}_k$.\nFor\n\t\\begin{align}\n\t\\Delta \\phi_{\\mbb{P}} \\lrp{ \\D_{1:K};\\D^0_{1:K} \\big| \\boldsymbol{\\sigma}}\n\t\t\\triangleq \\phi_{\\mbb{P}} \\lrp{ \\D_{1:K}| \\boldsymbol{\\sigma}}\n\t\t- \\phi_{\\mbb{P}} \\lrp{ \\D^0_{1:K}| \\boldsymbol{\\sigma}},\n\t\\end{align}\nwe have\n\t\\begin{align}  \\label{eq:delt_ph_2}\n\t&\\Delta \\phi_{\\mbb{P}} \\lrp{ \\D_{1:K};\\D^0_{1:K} \\big| \\boldsymbol{\\sigma}} \\nonumber\\\\\n\t& =  \\frac{\\mbb{E}\\{x^2\\}}{2}\n\t\t\\sum_{k \\in [K]} \\mbb{E}_{\\mc{J}_1} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ {\\mb{D}^0_1}^\\top \\mb{P}_{\\widetilde{\\mb{D}}_{1,\\mc{J}_1}}\\mb{D}^0_1} }\\dots  \\nonumber\\\\\n\t&\\qquad \\qquad \\qquad \\qquad \\mbb{E}_{\\mc{J}_k} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{  {\\mb{D}^0_k}^\\top(\\mb{I}_{m_k} - \\bP_{\\D_{k,\\cJ_k}})\\mb{D}^0_k}} \\nonumber \\\\\n\t&\\qquad \\qquad \\qquad \\qquad \\dots\n\t\t\\mbb{E}_{\\mc{J}_K} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{{\\mb{D}^0_K}^\\top \\mb{P}_{\\widetilde{\\mb{D}}_{K,\\mc{J}_K}}\\mb{D}^0_K}}\n\t\t\\nonumber \\\\\n\t&- \\lambda \\mbb{E}\\{|x|\\}\n\t\t\\sum_{k \\in [K]} \\mbb{E}_{\\mc{J}_1} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ {\\widetilde{\\mb{D}}_{1,\\mc{J}_1}}^+\\mb{D}^0_1}} \\dots  \\nonumber\\\\\n\t&\\quad \\mbb{E}_{\\mc{J}_k} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{\\mb{I}_{s_k} - \\D_{k,\\cJ_k}^+ \\mb{D}^0_k}} \\dots\n\t\t\\mbb{E}_{\\mc{J}_K} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{  {\\widetilde{\\mb{D}}_{K,\\mc{J}_K}}^+ \\mb{D}^0_K}} \\nonumber \\\\\n\t&+  \\frac{\\lambda^2}{2}\n\t\t\\sum_{k \\in [K]} \\mbb{E}_{\\mc{J}_1} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ \\mb{H}_{\\widetilde{\\mb{D}}_1,\\mc{J}_1}} }\\dots  \\nonumber\\\\\n\t&\\quad \\mbb{E}_{\\mc{J}_k} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ \\mb{H}_{\\mb{D}^0_{k,\\mc{J}_k}}  - \\bH_{\\D_{k,\\cJ_k}} }\t} \\dots\n\t\t\\mbb{E}_{\\mc{J}_K} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ \\mb{H}_{\\widetilde{\\mb{D}}_{K,\\mc{J}_K}}}}.\n\t\\end{align}\n\\end{lemma}\n\n\\begin{lemma} \\label{lem:UB_mu_delt_RIP}\nFor any $\\mb{D}_k \\in \\mc{D}_k$ satisfying $\\mathsf{RIP}$ of order $s_k$, given $\\mc{J}_k \\subset [p_k]$ and $ |\\mc{J}_k|=s_k$, the following relations hold:\n\t\\begin{align}\n\t\\norm{\\mb{D}_{k,\\mc{J}_k}}_2 &= \\norm{{\\mb{D}_{k,\\mc{J}_k}}^\\top}_2\n\t\t\\leq \\sqrt{1+\\delta_{s_k}(\\mb{D}_k)}, \\label{eq:l2_delt} \\\\\n\t\\delta_{s_k}(\\mb{D}_k) &\\leq \\mu_{s_k-1}(\\mb{D}_k). \\label{eq:delt_mu}\n\t\\end{align}\n\\end{lemma}\n\n\n\\begin{lemma}[Lemma 4~\\cite{gribonval2014sparse}]\n\\label{lem:H_Dps}\nLet $\\mb{D}_k$'s be coordinate dictionaries such that $\\delta_{s_k}(\\mb{D}_k)<1$. Then for any $\\mc{J}_k \\subset p_k, |\\mc{J}_k|=s_k$, $\\bH_{\\D_{k,\\cJ_k}}$ exists and\n\t\\begin{align} \\label{eq:pso_cond}\n\t&\\norm{\\bH_{\\D_{k,\\cJ_k}}}_2 \\leq \\frac{1}{1-\\delta_{s_k}(\\mb{D}_k)}, \\quad\n\t\\norm{\\D_{k,\\cJ_k}^+}_2 \\leq \\frac{1}{\\sqrt{1-\\delta_{s_k}(\\mb{D}_k)}},\n\t\\end{align}\nand for any $\\mb{D}_k'$ such that $\\|\\D_k-\\D'_k\\|_F \\leq \\varepsilon_k < \\sqrt{1-\\delta_{s_k}(\\mb{D}_k)}$:\n\t\t\\begin{align} \\label{eq:cond_delt_r_i}\n\t\t&1-\\delta_{s_k}(\\mb{D}'_k) \\geq (\\sqrt{1-\\delta_{s_k}(\\mb{D}_k)} - \\varepsilon_k)^2 \\triangleq 1-\\delta_k.\n\t\t\\end{align}\n\\end{lemma}\n\n\n\\begin{lemma}[Lemma 6~\\cite{gribonval2014sparse}]\n\\label{lem:D_12_tet}\nGiven any $\\mb{D}_k^1,\\mb{D}_k^2 \\in \\mc{D}_k$, there exist $\\mathbf{V}_k \\in \\mbb{R}^{m_k \\times p_k}$ with $\\diag{{\\mb{D}^1_k}^\\top \\mathbf{V}_k}=\\mathbf{0}$ and $\\diag{\\mathbf{V}_k^\\top \\mathbf{V}_k}=\\mb{I}_{p_k}$ and a vector $\\boldsymbol{\\theta}_k \\triangleq \\boldsymbol{\\theta}_k(\\mb{D}_k^1,\\mb{D}_k^2) \\in [0,\\pi]^{p_k}$, such that\n\t\\begin{align}\n\t\\mb{D}_k^2 = \\mb{D}_k^1 \\mathbf{C}_k (\\boldsymbol{\\theta}_k) + \\mathbf{V}_k \\mathbf{S}_k(\\boldsymbol{\\theta}_k),\n\t\\end{align}\nwhere $\\mathbf{C}_k (\\boldsymbol{\\theta}_k) \\triangleq \\Diag{\\cos(\\boldsymbol{\\theta}_k) }$ and $\\mathbf{S}_k (\\boldsymbol{\\theta}_k)  \\triangleq \\Diag{\\sin(\\boldsymbol{\\theta}_k) }$. Moreover,\n\t\\begin{align} \\label{eq:tet_rk}\n\t&\\frac{2}{\\pi}\\theta_{k,j} \\leq \\|\\mb{d}^2_{k,j} - \\mb{d}^1_{k,j} \\|_2\n\t\t= 2\\sin \\lrp{\\frac{\\theta_{k,j}}{2}} \\leq\\theta_{k,j}, \\text{and} \\nonumber \\\\\n\t&\\frac{2}{\\pi} \\|\\boldsymbol{\\theta}_k\\|_2 \\leq \\|\\mb{D}_k^2 - \\mb{D}_k^1 \\|_F \\leq \\|\\boldsymbol{\\theta}_k\\|_2 ,\n\t\\end{align}\nwhere $j \\in [p_k]$.\nSimilarly, there exists $\\mathbf{V}_k'$ such that $\\mb{D}_k^1 = \\mb{D}_k^2 \\mathbf{C}_k (\\boldsymbol{\\theta}_k) + \\mathbf{V}'_k \\mathbf{S}_k(\\boldsymbol{\\theta}_k)$, where $\\diag{{\\mb{D}^2_k}^\\top \\mathbf{V}'_k}=\\mathbf{0}$.\n\\end{lemma}\n\n\n\\begin{lemma} \\label{lem:A_B_delt}\nFix $\\D_{1:K}$ and $\\D^0_{1:K}$, and suppose $\\lr{A_k},\\lr{B_k},\\lr{\\delta_k}$ satisfy the following:\n\t\\begin{align} \\label{eq:A_B_Delk}\n\t&A_k \\geq \\max\\lr{ \\|\\mb{D}_k^\\top \\mb{D}_k - \\mb{I}_{p_k}\\|_F,\\|{\\mb{D}_k^0}^\\top \\mb{D}_k^0 - \\mb{I}_{p_k}\\|_F } , \\nonumber \\\\\n\t&B_k \\geq \\max\\lr{ \\|\\mb{D}_k\\|_2, \\|\\mb{D}_k^0\\|_2 }, \\text{and}\\nonumber \\\\\n\t&\\delta_k \\geq \\max\\lr{ \\delta_{s_k}(\\mb{D}_k),\\delta_{s_k}(\\mb{D}_k^0) }.\n\t\\end{align}\nThen for all $ \\boldsymbol{\\theta}_k \\triangleq \\boldsymbol{\\theta}_k(\\mb{D}_k,\\mb{D}_k^0), k \\in [K]$, we have\n\t\\begin{align}  \\label{eq: delt_ph_3}\n\t&\\Delta\\phi_{\\mbb{P}}\\lrp{ \\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}}  \\nonumber\\\\\n\t&\\geq \\frac{s\\mbb{E}\\{x^2\\}}{2}\n\t\\sum_{k \\in [K]} \\frac{\\|\\boldsymbol{\\theta}_k\\|_2 }{p_k}\n\t\t\\bigg[\\|\\boldsymbol{\\theta}_k\\|_2\n\t\t\\bigg( 1 - \\frac{s_k}{p_k} \\frac{B^2_k}{1-\\delta_k}\n\t\t-\\bar{\\lambda} \\kappa_x^2\n\t\t\\delta_{-k}\\bigg)\n\t\t\\nonumber \\\\\n\t&\\qquad \\quad -\\bigg(\\delta_{-k}\n\t\t+2\\bar{\\lambda}\\prod_{i \\in [K]} \\frac{1}{1-\\delta_i}\\bigg)\n\t\t\\bar{\\lambda} \\kappa_x^2\n\t\t\\frac{s_k}{p_k} \\frac{2A_kB_k}{1-\\delta_k}\\bigg],\n\t\\end{align}\nwhere $\\bar{\\lambda} \\triangleq \\dfrac{\\lambda}{\\mbb{E}\\lr{|x|}} $ and $\\delta_{-k} \\triangleq \\prod_{\\substack{i \\in [K]\\\\ i \\neq k}}\n\t\t\\sqrt{\\dfrac{1+\\delta_i}{1-\\delta_i}} $.\n\\end{lemma}\n\n\nProposition~\\ref{prop:1} shows $\\Delta\\phi_{\\mbb{P}}\\lrp{ \\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}} > 0 $. However, given $\\widehat{\\mb{x}}$, the solution of $\\phi_\\mb{y}\\lrp{\\D_{1:K}|\\boldsymbol{\\sigma}}$, $\\widehat{\\boldsymbol{\\sigma}} = \\mathop{\\mathrm{sign}}\\nolimits\\lrp{\\widehat{\\mb{x}}}$ is not necessarily equal to the sign of the generating $\\boldsymbol{\\sigma}$. We derive conditions that ensure $\\widehat{\\mb{x}}$ is almost surely the unique minimizer of $f_\\mb{y}\\lrp{\\D_{1:K}}$ and $\\widehat{\\boldsymbol{\\sigma}}=\\boldsymbol{\\sigma}$.\nWe introduce the following proposition for this purpose.\n\n\\begin{proposition} \\label{prop:3}\nLet the generating coordinate dictionaries $\\{ \\mb{D}_k^0 \\in\\mc{D}_k\\}$ satisfy:\n\t\\begin{align} \\label{eq:delt_cond}\n\t\\mu_{s}(\\mb{D}^0) <  \\frac{1}{2} , \\quad  \\max_k\\{ \\delta_{s_k}(\\mb{D}_k^0)\\} <  \\frac{1}{4} .\n\t\\end{align}\nSuppose $\\bar{\\lambda} = \\dfrac{\\lambda}{\\mbb{E}\\lr{|x|}}\\leq \\dfrac{x_{\\min}}{2\\mbb{E}\\lr{|x|}}$ and\n\t\\begin{align} \\label{eq:prop_r_max}\n\t\\max_{k\\in [K]}\\{\\varepsilon_k\\} \\leq \\min\\lr{ \\bar{\\lambda}C_{\\max}, 0.15}.\n\t\\end{align}\nIf the following is satisfied:\n\t\\begin{align} \\label{eq:M_eps_M_al}\n\t\\frac{M_w}{M_x}\n\t\t < 3(1.5)^{K\/2} \\bigg( \\bar{\\lambda} K C_{\\max}  - \\sum_{k \\in [K]} \\varepsilon_k\\bigg),\n\t\\end{align}\nthen for any $\\D_{1:K}$ such that $\\mb{D}_k \\in \\mc{S}_{\\varepsilon_k}(\\mb{D}^0_k)$, for $k \\in [K]$, $\\widehat{\\mb{x}}$ that is defined in~\\eqref{eq:xhat_closedform} is almost surely the minimizer of the map $\\mb{x}' \\mapsto \\frac{1}{2}\\norm{\\mb{y}- \\lrp{\\bigotimes \\D_k } \\mb{x}'}_2^2 +\\lambda \\|\\mb{x}'\\|_1 $ and \t\n$\\Delta\\phi_{\\mbb{P}}\\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}} = \\Delta f_{\\mbb{P}} \\lrp{\\D_{1:K};\\D^0_{1:K}}$.\n\\end{proposition}\n\n\\begin{remark}\nNote that $\\mu_s(\\mb{D}^0) < \\frac{1}{2}$ in \\eqref{eq:delt_cond} can be satisfied by ensuring that the right hand side of~\\eqref{eq:mu_s} is less than $\\frac{1}{2}$. One way this can be ensured is by enforcing strict conditions on coordinate dictionaries; for instance, $\\mu_{s_k}(\\mb{D}^0_k)\\leq \\frac{1}{2^K}$.\n\\end{remark}\n\nThe proof of Proposition~\\ref{prop:3} relies on the following lemmas and~\\cite[Lemmas 10--13]{gribonval2014sparse}.\n\n\\begin{lemma}[Lemma 13~\\cite{gribonval2014sparse}]\n\\label{lem:a_hat_min_cond}\nAssume $\\mu_s(\\mb{D}) <\\dfrac{1}{2}$. If\n\t\\begin{align} \\label{eq:lem8_cod}\n\t\\min_{j \\in \\mc{J}} \\left| x_j \\right| \\geq 2\\lambda, \\\n\t\t\\text{and} \\\n\t\t\\norm{\\mb{y} - \\mb{D} \\mb{x}}_2 < \\lambda (1-2\\mu_s(\\mb{D}))\n\t\\end{align}\nhold for generating $\\mb{x}$, then $\\widehat{\\mb{x}}$ defined in \\eqref{eq:xhat_closedform} is the unique solution of $\\min_{\\mb{x}'} \\frac{1}{2}\\norm{\\mb{y} - \\lrp{\\bigotimes \\D_k } \\mb{x}' }_2  +\\lambda\\|\\mb{x}'\\|_1$.\n\\end{lemma}\n\n\n\\begin{lemma} \\label{lem:mu_mu0_rel}\nFor any $\\mb{D}^0=\\bigotimes \\D^0_k $ and $\\mb{D} = \\bigotimes \\D_k $  such that\n$\\mb{D}_k \\in \\bar{\\mathcal{B}}_{\\varepsilon_k}(\\mb{D}^0_k)$, for $k \\in [K]$, suppose the following inequalities are satisfied:\n\t\\begin{align} \\label{eq:UB_mu_delt}\n\t\\max_{k \\in [K]} \\{\\delta_{s_k}(\\mb{D}_k^0)\\} \\leq \\frac{1}{4},\n\t\t\\quad \\text{and}\n\t\t\\quad\t\\max_{k \\in [K]} \\varepsilon_k \\leq 0.15.\n\t\\end{align}\nThen, we have\n\t\\begin{align} \\label{eq:mu_mu0}\n\t\\mu_s(\\mb{D}) \\leq  \\mu_s(\\mb{D}^0) + 2(1.5)^{K\/2}\\sqrt{s} \\bigg(\\sum_{k \\in [K]} \\varepsilon_k\\bigg).\n\t\\end{align}\t\n\\end{lemma}\n\n\n\\begin{IEEEproof}[Proof of Theorem~\\ref{thm:asymp}]\nTo prove this theorem, we use Proposition~\\ref{prop:1} to show that $\\Delta \\phi_{\\mbb{P}}\\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}} > 0$, and then use Proposition~\\ref{prop:3} to show that $\\Delta \\phi_{\\mbb{P}}\\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}} =\\Delta f_{\\mbb{P}}\\lrp{\\D_{1:K};\\D^0_{1:K}}$.\nThe assumptions in \\eqref{eq:cond_k_i_p_i} ensure that the conditions in \\eqref{eq:cond_delt_s_prop} and \\eqref{eq:delt_cond} are satisfied for Proposition~\\ref{prop:1} and Proposition~\\ref{prop:3}, respectively. Assumptions \\eqref{eq:cond_m_p} and \\eqref{eq:cond_lambda} ensure that the conditions in \\eqref{eq:cond_lamb_1} and \\eqref{eq:cond_lamb_2}  are satisfied for Proposition~\\ref{prop:1},  $\\bar{\\lambda}\\leq \\dfrac{x_{\\min}}{2\\mbb{E}\\lr{|x|}}$ holds for Proposition~\\ref{prop:3}, and $\\max_{k \\in [K]}\\{C_{k,\\mathrm{min}}\\}<C_{\\max}$. Hence, according to Proposition~\\ref{prop:1},  $\\Delta \\phi_{\\mbb{P}}\\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}} > 0 $ for all $\\varepsilon_k \\in (\\bar{\\lambda}C_{k,\\min},0.15], k \\in [K]$. Finally, using the assumption in \\eqref{eq:cond_noise} implies $\\Delta \\phi_{\\mbb{P}}\\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}} =\\Delta f_{\\mbb{P}}\\lrp{\\D_{1:K};\\D^0_{1:K}}$ for all $\\varepsilon_k \\leq \\bar{\\lambda}C_{\\max}, k \\in [K]$. Furthermore, the assumption in \\eqref{eq:cond_lambda} implies $C_{\\max}\\bar{\\lambda} \\leq 0.15$. Consequently, for any $\\lr{ \\varepsilon_k>0,k \\in [K]}$ satisfying the conditions in \\eqref{eq:cond_r_i}, $\\D_{1:K} \\rightarrow f_{\\mbb{P}}\\lrp{\\D_{1:K}}$ admits a local minimum $\\widehat{\\mb{D}}= \\bigotimes \\widehat{\\mb{D}}_k$ such that $\\widehat{\\mb{D}}_k \\in \\mathcal{B}_{\\varepsilon_k}(\\mb{D}^0_k), k \\in [K]$.\n\\end{IEEEproof}\n\n\n\n\\section{Finite Sample Identifiability Results} \\label{sec:finite}\n\nWe now focus on leveraging Theorem~\\ref{thm:asymp} and solving~\\eqref{eq:f_x} to derive finite-sample bounds for KS dictionary identifiability. Compared to Gribonval et al.~\\cite{gribonval2014sparse}, who use Lipschitz continuity of the objective function with respect to the larger KS dictionary, our analysis is based on  ``coordinate-wise Lipschitz continuity\" with respect to the coordinate dictionaries.\n\n\\begin{theorem}\\label{thm:finite_n}\nSuppose the observations are generated according to \\eqref{eq:obs_model} and the dictionary coefficients follow the separable sparsity model of Section~\\ref{sec:model} such that \\eqref{eq:cond_k_i_p_i} to \\eqref{eq:cond_noise} are satisfied. Next, fix any $\\xi \\in (0,\\infty)$. Then, for any number of observations satisfying\n\t\\begin{align} \\label{eq:smplCmp}\n\tN = \\max_{k \\in [K]}\n\t\t&\\Omega \\bigg(\n\t\t\\frac{p_k^2 (\t\\xi+m_kp_k) } {(\\varepsilon_k - \\varepsilon_{k,\\min}(\\bar{\\lambda}))^2}\n\t\t\\nonumber\\\\\n\t&\\qquad \\bigg( \\frac{2^{K}(1 + \\bar{\\lambda}^2) M_x^2}\n\t\t{s^2\\mbb{E}\\{x^2\\}^2}\n\t\t+ \\bigg(\\frac{M_w}{s\\mbb{E}\\{x^2\\}} \\bigg)^2 \\bigg) \\bigg),\n\t\\end{align}\nwith probability at least $1-e^{-\t\\xi}$, $\\D_{1:K} \\mapsto F_\\mb{Y}\\lrp{\\D_{1:K}}$ admits a local minimum $\\widehat{\\mb{D}}=\\bigotimes \\widehat{\\mb{D}}_k$ such that $\\widehat{\\mb{D}}_k \\in \\mathcal{B}_{\\varepsilon_k}(\\mb{D}^0_k)$, for $k \\in [K]$.\n\\end{theorem}\n\n\\subsection{Discussion}\nLet us make some remarks about implications of Theorem~\\ref{thm:finite_n}. First, sample complexity has an inverse relationship with signal to noise ratio ($\\mathop{\\mathrm{SNR}}\\nolimits$),\\footnote{Sufficient conditioning on $N$ implies $\\mathcal{O}$-scaling for sample complexity.} which we define as\n\t\\begin{align}\n\t\\mathop{\\mathrm{SNR}}\\nolimits \\triangleq \\frac{\\mbb{E}\\{\\|\\mb{x}\\|_2^2\\}}{\\mbb{E}\\{\\|\\mb{w}\\|^2_2\\}} = \\frac{s\\mbb{E}\\{x^2\\}}{m\\mbb{E}\\{w^2\\}}.\n\t\\end{align}\nLooking at the terms on the right hand side of~\\eqref{eq:smplCmp} in Theorem~\\ref{thm:finite_n}, $M_x\/(s\\mbb{E}\\lr{x^2})$ is related to the deviation of $\\|\\mb{x}\\|_2$ from its mean, $\\mbb{E}\\lr{\\|\\mb{x}\\|_2}$, and depends on the coefficient distribution, while $M_w\/(s\\mbb{E}\\lr{x^2})$ is related to $1\/\\mathop{\\mathrm{SNR}}\\nolimits$ and depends on the noise and coefficient distributions.\n\nSecond, we notice dependency of sample complexity on the recovery error of coordinate dictionaries. We can interpret $\\varepsilon_k$ as the recovery error for $\\mb{D}^0_k$. Then, the sample complexity scaling in \\eqref{eq:smplCmp} is proportional to $\\max_k \\varepsilon_k^{-2}$.\nWe note that the sample complexity results obtained in~\\cite{gribonval2014sparse} that are independent of $\\varepsilon \\triangleq \\norm{\\mb{D}-\\mb{D}^0}_F$ only hold for the noiseless setting and the dependency on $\\varepsilon^{-2}$ is inevitable for noisy observations~\\cite{gribonval2014sparse}. Furthermore, given the condition on the range of $\\varepsilon_k$'s in \\eqref{eq:cond_r_i}, $\\varepsilon_k$'s cannot be arbitrarily small, and will not cause $N$ to grow arbitrarily large.\n\nThird, we observe a linear dependence between the sample complexity scaling in \\eqref{eq:smplCmp} and coordinate dictionaries' dimensions, i.e., $\\max_k \\mathcal{O}(m_k p_k^3)$. Comparing this to the $\\mathcal{O}(mp^3)=\\mathcal{O}\\lrp{\\prod_k m_kp_k^3}$ scaling in the unstructured DL problem~\\cite{gribonval2014sparse}, the sample complexity in the KS-DL problem scales with the dimensions of the largest coordinate dictionary, as opposed to the dimensions of the larger KS dictionary.\n\n\\begin{table}\n\\caption{\\small Comparison of upper and lower bounds on the sample complexity of dictionary learning for vectorized DL and KS DL.}\n\\label{table:1}\n\\centering\n\\begin{tabular}{l|C{1.7cm}|c| N} \\cline{2-3}\n & Vectorized DL & KS DL & \\\\ [20pt] \\hline\n\t\\multicolumn{1}{|c|} {Minimax Lower Bound}\n\t&$\\dfrac{mp^2}{\\varepsilon^2}$~\\cite{jung2015minimax}\n\t&$\\dfrac{p\\sum_{k } m_k p_k}{\\varepsilon^2 }$~\\cite{shakeri2016arxiv}\n\t& \\\\ [20pt]\\hline\n\t\\multicolumn{1}{|c|} {Achievability Bound} & $\\dfrac{mp^3}{\\varepsilon^2}$~\\cite{gribonval2014sparse}\n\t& $\\max\\limits_k \\dfrac{m_kp_k^3}{\\varepsilon_k^2} $ & \\\\ [20pt]\\hline\n\\end{tabular}\n\\end{table}\n\nWe also compare this sample complexity upper bound scaling to the sample complexity lower bound scaling in our previous work~\\cite[Corollary 1]{shakeri2016minimax}, where we obtained $N = \\Omega\\lrp{p\\sum_k m_kp_k\\varepsilon^{-2}\/K}$ as a \\emph{necessary condition for recovery of KS dictionaries}.\\footnote{We have the following relation between $\\varepsilon$ and $\\varepsilon_k$'s:\n\t\\begin{align*}\n\t\\varepsilon \t\\leq\t\\sum_{k \\in [K]} \\bigg(\n\t\t\\prod_{\\substack{i \\in [K]\\\\i \\neq k}} \\norm{\\wt{\\D}_k}_F \\bigg)\n\t\t\\norm{\\mb{D}_k-\\mb{D}^0_k}_F\n\t\t\\leq \\sqrt{p} \\sum_{k \\in [K]} \\varepsilon_k.\n\t\\end{align*}\nAssuming all $\\varepsilon_k$'s are equal, this then implies $\\varepsilon_k^2 \\geq \\varepsilon^2\/(K^2 p)$.}\nIn terms of overall error $\\varepsilon$, our result translates into $N = \\max_k \\Omega\\lr{ 2^K K^2p(m_kp_k^3)\\varepsilon^{-2}}$ as a \\emph{sufficient} condition for recovery of coordinate dictionaries. The lower bound depended on the average dimension of the coordinate dictionaries, $\\sum_k m_kp_k\/K$, whereas we observe here a dependence on the dimensions of the coordinate dictionaries in terms of the maximum dimension, $\\max_k m_kp_k$. We also observe an increase of order $\\max_k p_k^2$ in the sample complexity upper bound scaling. This gap suggests that tighter bounds can be obtained for lower and\/or upper bounds. A summary of these results is provided in Table~\\ref{table:1} for a fixed $K$.\n\n\\subsection{Proof Outline}\n\nWe follow a similar approach used in~\\cite[Theorem 2]{gribonval2014sparse} for vectorized data.\nWe show that, with high probability,\n\t\\begin{align}\n\t\\Delta F_\\mb{Y} (\\eps_{1:K}) \\triangleq \\inf_{\\mb{D}_k \\in \\mc{S}_{\\varepsilon_k}(\\mb{D}_k^0)} \\Delta F_\\mb{Y} \\lrp{ \\D_{1:K};\\D^0_{1:K}}\n\t\\end{align}\nconverges uniformly to its expectation,\n\t\\begin{align}\n\t\\Delta f_{\\mbb{P}}(\\eps_{1:K})\\triangleq \\inf_{\\mb{D}_k \\in \\mc{S}_{\\varepsilon_k}(\\mb{D}^0_k)}\n\t\t\\Delta f_{\\mbb{P}} \\lrp{ \\D_{1:K}; \\D^0_{1:K} }.\n\t\\end{align}\nIn other words, with high probability,\t\n\t\\begin{align}\n\t\\lra{\\Delta F_\\mb{Y} (\\eps_{1:K}) - \\Delta f_{\\mbb{P}}(\\eps_{1:K}) } \\leq \\eta_N,\n\t\\end{align}\nwhere $\\eta_N$ is a parameter that depends on the probability and other parameters in the problem.\nThis implies $\\Delta F_\\mb{Y} (\\eps_{1:K}) \\geq \\Delta f_{\\mbb{P}}(\\eps_{1:K}) - 2\\eta_N $.\nIn Theorem~\\ref{thm:asymp}, we obtained conditions that ensure $\\Delta f_{\\mbb{P}}(\\eps_{1:K})> 0$. Thus, if $2\\eta_N < \\Delta f_{\\mbb{P}}(\\eps_{1:K})$ is satisfied, this implies $\\Delta F_\\mb{Y} (\\eps_{1:K})> 0$, and we can use arguments similar to the proof of Theorem~\\ref{thm:asymp} to show that\n$\\D_{1:K}  \\mapsto F_\\mb{Y}\\lrp{\\D_{1:K}}$ admits a local minimum $\\widehat{\\mb{D}}=\\bigotimes \\widehat{\\mb{D}}_k$, such that $\\widehat{\\mb{D}}_k \\in \\mathbf{B}_{\\varepsilon_k}(\\mb{D}^0_k)$, for $k \\in [K]$.\n\nIn Theorem~\\ref{thm:asymp}, we showed that under certain conditions, $f_{\\mbb{P}}(\\D_{1:K};\\D^0_{1:K}) = \\Delta \\phi_\\mbb{P}\\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}}$.\nTo find $\\eta_N$, we uniformly bound deviations of $\\D_{1:K} \\mapsto \\Delta \\phi_\\mb{y}\\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}}$ from its expectation on $\\lr{ \\mc{S}_{\\varepsilon_k}(\\mb{D}^0_k)}_{k=1}^K$.\nOur analysis is based on the \\textit{coordinate-wise Lipschitz continuity} property of $\\Delta \\phi_\\mb{y}\\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}}$ with respect to coordinate dictionaries. Then, to ensure $ 2\\eta_N < \\Delta \\phi_\\mbb{P}\\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}}$, we show that $2\\eta_N$ is less than the right-hand side of~\\eqref{eq:delt_p_LB} and obtain conditions on the sufficient number of samples based on each coordinate dictionary dimension and recovery error.\n\nThe proof of Theorem~\\ref{thm:finite_n} relies on the following definition and lemmas. The proofs of these are provided in Appendix B.\n\n\\begin{definition}[Coordinate-wise Lipschitz continuity]\nA function $f: \\mc{D}_1 \\times \\dots \\times \\mc{D}_K \\rightarrow \\mbb{R}$ is coordinate-wise Lipschitz continuous with constants $(L_1,\\dots,L_K)$ if there exist real constants $\\lr{L_k \\geq 0}_{k=1}^K$, such that for $\\lr{\\mb{D}_k,\\mb{D}'_k \\in \\mc{D}_k}_{k=1}^K$:\n\t\\begin{align}\n\t\\lra{ f\\lrp{\\D_{1:K}} - f\\lrp{\\D'_{1:K}} }\n\t\t\\leq \\sum_{k \\in [K]} L_k \\norm{\\mb{D}_k - \\mb{D}'_k}_F.\n\t\\end{align}\n\\end{definition}\n\n\\begin{lemma}[Rademacher averages~\\cite{gribonval2014sparse}]\\label{lem:rad}\nConsider $\\mathcal{F}$ to be a set of measurable functions on measurable set $\\mc{X}$ and $N$ i.i.d. random variables $X_1,\\dots,X_N \\in \\mc{X}$. Fix any $\\xi \\in (0,\\infty)$. Assuming all functions are bounded by $B$, i.e., $|f(X)|\\leq B$, almost surely, with probability at least $1-e^{-\\xi}$:\n\t\\begin{align} \\label{eq:rad_gau}\n\t& \\sup_{f \\in \\mathcal{F}} \\bigg( \\frac{1}{N}\n\t\t\\sum_{n \\in [N] } f \\lrp{X_n}\n\t\t- \\mbb{E}_{X} \\lr{ f \\lrp{X}} \\bigg)  \\nonumber \\\\\n\t&\\quad\n\t\\leq 2\\sqrt{\\frac{\\pi}{2}} \\mbb{E}_{X,\\beta_{1:N}}\n\t\t\\bigg\\{ \\sup_{f \\in \\mathcal{F}}\n\t\t\\bigg( \\frac{1}{N}\n\t\t\\sum_{n \\in [N] } \\beta_n f \\lrp{ X_n} \\bigg) \\bigg\\}\n\t\t+ B\\sqrt{\\frac{2\\xi}{N}},\n\t\\end{align}\nwhere $\\beta_{1:N}$'s are independent standard Gaussian random variables.\n\\end{lemma}\n\n\n\\begin{lemma} \\label{lemma:delt_m_T_dev}\nLet $\\mathcal{H}$ be a set of real-valued functions on $\\mb{D}_k \\in \\overline{\\mathcal{B}}_{\\varepsilon_k}(\\mb{D}^0_k), k \\in [K]$, that are bounded by $B$ almost everywhere and are coordinate-wise Lipschitz continuous with constants $(L_1,\\dots,L_K)$ .\nLet $h_1,h_2,\\dots,h_N$ be independent realizations from $\\mathcal{H}$ with uniform Haar measure on $\\mathcal{H}$. Then, fixing $\\xi \\in (0,\\infty)$, we have with probability greater than $1-e^{-\\xi}$ that:\n\t\\begin{align} \\label{eq:dev}\n\t&\\sup_{\\substack{ \\mb{D}_k \\in \\overline{\\mathcal{B}}_{\\varepsilon_k}(\\mb{D}^0_k) \\\\ k \\in [K]}}\n\t\t \\bigg| \\frac{1}{N} \\sum_{n \\in [N]}\n\t\t h_n(\\D_{1:K}) - \\mbb{E} \\lr{ h(\\D_{1:K})} \\bigg|   \\nonumber\\\\\n\t&\\qquad \\quad \\leq 4\\sqrt{\\frac{\\pi}{2N}} \\bigg(\\sum_{k \\in [K]} L_k\\varepsilon_k \\sqrt{Km_kp_k} \\bigg)\n\t\t+  B \\sqrt{\\frac{2\\xi}{N}}.\n\t\\end{align}\n\\end{lemma}\n\n\\begin{lemma}[Lemma 5~\\cite{gribonval2014sparse}]\n\\label{lem:H_Ps}\nFor any $\\delta_k<1$, $\\mb{D}_k,\\mb{D}_k'$ such that $\\max(\\delta_{s_k}(\\mb{D}_k),\\delta_{s_k}(\\mb{D}'_k))\\leq \\delta_k$, and $\\mc{J}_k \\subset p_k, |\\mc{J}_k|=s_k$, we have\n\t\\begin{align} \\label{eq:PH_PHp}\n\t&\\|\\mb{I} - \\D_{k,\\cJ_k}^+ \\mb{D}'_{k,\\mc{J}_k}\\|_2 \\leq (1-\\delta_k)^{-1\/2} \\|\\D_k-\\D'_k\\|_F, \\nonumber \\\\\n\t&\\|\\bH_{\\D_{k,\\cJ_k}}-\\bH_{\\D'_{k,\\cJ_k}} \\|_2 \\leq 2(1-\\delta_k)^{-3\/2} \\|\\D_k-\\D'_k\\|_F, \\nonumber \\\\\n\t&\\|\\D_{k,\\cJ_k}^+ - {\\D'}_{k,\\cJ_k}^+ \\|_2\\leq 2(1-\\delta_k)^{-1} \\|\\D_k-\\D'_k\\|_F,\\text{and}\\nonumber \\\\\n\t&\\|\\bP_{\\D_{k,\\cJ_k}} -\\bP_{\\D'_{k,\\cJ_k}} \\|_2 \\leq 2(1-\\delta_k)^{-1\/2} \\|\\D_k-\\D'_k\\|_F.\n\t\\end{align}\n\\end{lemma}\n\n\\begin{lemma} \\label{lem:phi_m_T1_lip}\nConsider $\\mb{D}^0_k \\in \\mc{D}_k$ and $\\varepsilon_k$'s such that $\\varepsilon_k < \\sqrt{1-\\delta_{s_k}(\\mb{D}_k^0)}$, for $k \\in [K]$ and define\n $\\sqrt{1-\\delta_k} \\triangleq \\sqrt{1-\\delta_{s_k}(\\mb{D}_k^0)} - \\varepsilon_k>0$. The function $\\Delta \\phi_\\mb{y}\\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}}$ is almost surely coordinate-wise Lipschitz continuous on $\\lr{ \\mathcal{B}_{\\varepsilon_k}(\\mb{D}^0_k)}_{k=1}^K$ with Lipschitz constants\n \t\\begin{align} \\label{eq:lipsch_const_h}\n \tL_k \\triangleq (1-\\delta_k)^{-1\/2}\n\t\t \\bigg(& M_x \\bigg( \\prod_{k \\in [K]} \\sqrt{1+\\delta_{s_k}(\\mb{D}^0_k)}\\bigg)\n\t\t+M_w \\nonumber\\\\\n\t&+ \\lambda\\sqrt{s}\n\t\t \\prod_{k \\in [K] } (1-\\delta_k)^{-1\/2}  \\bigg)^2 ,\n \t\\end{align}\nand $\\lra{\\Delta \\phi_\\mb{y}\\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}}}$ is almost surely bounded on $\\lr{ \\mathcal{B}_{\\varepsilon_k}(\\mb{D}^0_k)}_{k=1}^K$ by $\\sum_{k \\in [K]} L_k\\varepsilon_k$.\n\\end{lemma}\n\n\n\n\\begin{IEEEproof}[Proof of Theorem 2]\nFrom Lemmas~\\ref{lemma:delt_m_T_dev} and \\ref{lem:phi_m_T1_lip}, we have that with probability at least $1-e^{-\\xi}$:\n\t\\begin{align} \\label{eq:delt_finite_UB}\n\t& \\sup_{\\substack{\\mb{D}_k \\in \\overline{\\mathcal{B}}_{\\varepsilon_k}(\\mb{D}^0_k)\\\\ k \\in [K]}}\n\t\t\\big| \\Delta \\phi_\\mb{y}\\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}} - \\Delta \\phi_\\mbb{P} \\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}} \\big|  \\nonumber\\\\\n\t&\\qquad \\quad \\quad \\leq \\sqrt{\\frac{2}{N}}\\sum_{k \\in [K]} L_k\\varepsilon_k \\lrp{ 2\\sqrt{\\pi m_kp_k}\n\t\t+ \\sqrt{\\xi}},\n\t\\end{align}\nwhere $L_k$ is defined in \\eqref{eq:lipsch_const_h}.\nFrom \\eqref{eq:delt_finite_UB}, we obtain $ \\Delta \\phi_\\mb{y}\\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}} >\\Delta \\phi_\\mbb{P} \\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}} - 2\\eta_N$ where $\\eta_N = \\sqrt{\\frac{2}{N}}\\sum_{k \\in [K]} L_k\\varepsilon_k \\lrp{ 2\\sqrt{\\pi m_kp_k}\n\t\t+ \\sqrt{\\xi}}$. In Theorem~\\ref{thm:asymp}, we derived conditions that ensure $\\Delta f_\\mb{y} (\\D_{1:K};\\D^0_{1:K}) =  \\Delta \\phi_\\mb{y}\\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}} $ and $\\Delta f_{\\mbb{P}}(\\D_{1:K};\\D^0_{1:K})=\\Delta \\phi_\\mbb{P} \\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}}$. Therefore, given that the conditions in Theorem~\\ref{thm:asymp} are satisfied, $\\Delta F_\\mb{Y} (\\eps_{1:K}) > \\Delta f_{\\mbb{P}}(\\eps_{1:K}) - 2\\eta_N $, and the existence of a local minimum of $F_\\mb{Y}(\\D_{1:K})$ within radii $\\varepsilon_k$ around $\\mb{D}_k^0$, $k \\in [K]$, is guaranteed with probability at least $1-e^{-\\xi}$ as soon as $2\\eta_N < \\Delta f_\\mbb{P} (\\eps_{1:K}) $. According to \\eqref{eq:delt_p_LB}, $\t\\Delta \\phi_\\mbb{P} \\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}}\n\\geq \\dfrac{s\\mbb{E}\\{x^2\\}}{8} \\sum_{k \\in [K]} \\dfrac{\\varepsilon_k}{p_k}\n\\lrp{ \\varepsilon_k -  \\varepsilon_{k,\\min}(\\bar{\\lambda})}$; therefore, it is sufficient to have for all $k \\in [K]$:\n\t\\begin{align*}\n\t\\sqrt{\\frac{8}{N}} L_k\\varepsilon_k \\lrp{ 2\\sqrt{\\pi m_kp_k}\n\t\t+ \\sqrt{\\xi}}\n\t    < \\frac{s\\mbb{E}\\{x^2\\}\\varepsilon_k\\lrp{\\varepsilon_k - \\varepsilon_{k,\\min}(\\bar{\\lambda})} }{8p_k},\n\t\\end{align*}\nwhich translates into $N \\geq \\max_{k \\in [K]} N_k$, where\n\t\\begin{align} \\label{eq:N_k}\n\t&N_k= \\lrp{ 2\\sqrt{\\pi m_kp_k} + \\sqrt{\\xi}}^2\n\t\\lrp{ \\frac{2^{4.5}L_k p_k }{s\\mbb{E} \\{x^2\\} (\\varepsilon_k - \\varepsilon_{k,\\min}(\\bar{\\lambda}))}}^2 .\n\t\\end{align}\t\t\nFurthermore, we can upper bound $L_k$ by\n\t\\begin{align} \\label{eq:Rk_UB}\n\tL_k &\\numrel{\\leq}{r_Rk} \\sqrt{2}\\bigg(1.25^{K\/2} M_x + M_w + 2^{K\/2} \\lambda \\sqrt{s} \\bigg)^2 \\nonumber \\\\\n\t&\\numrel{\\leq}{r_lmb_Mx} \\sqrt{2}c_1 \\bigg(\\big(1.25^{K} +  2^{K} \\bar{\\lambda}^2 \\big) M_x^2 + M_w^2 \\bigg),\n\t\\end{align}\nwhere $c_1$ is some positive constant, \\eqref{r_Rk} follows from the fact that given the assumption in~\\eqref{eq:cond_delt_s_prop}, assumptions in Lemma~\\ref{lem:phi_m_T1_lip} are satisfied with $\\sqrt{1-\\delta_k}\\geq \\sqrt{1\/2}$ for any $\\varepsilon_k\\leq 0.15$, and \\eqref{r_lmb_Mx} follows from the following inequality:\n\t\\begin{align*}\n\t\\lambda\n\t\t= \\bar{\\lambda} \\mbb{E}\\lr{ |x| }\n\t\t= \\dfrac{1}{s}\\bar{\\lambda} \\mbb{E}\\lr{ \\norm{\\mb{x}}_1 }\n\t\t\\leq \\dfrac{1}{\\sqrt{s}} \\bar{\\lambda} \\mbb{E}\\lr{ \\norm{\\mb{x}}_2 }\n\t\t\\leq\\dfrac{1}{\\sqrt{s}} \\bar{\\lambda} M_x.\n\t\\end{align*}\nSubstituting \\eqref{eq:Rk_UB} in \\eqref{eq:N_k} and using $\\lrp{ \\sqrt{\\xi} +  2\\sqrt{\\pi m_kp_k} }^2 \\leq c_2 (\\xi + m_kp_k)$ for some positive constant $c_2$, we get\n\t\\begin{align*} %\n\t&N_k =\n\t\t\\Omega \\bigg(\n\t\tp_k^2 (m_kp_k+\\xi)\n\t\t\\bigg( \\frac{2^{K}(1 + \\bar{\\lambda}^2) M_x^2 + M_w^2}\n\t\t{s^2\\mbb{E}\\{x^2\\}^2(\\varepsilon_k - \\varepsilon_{k,\\min}(\\bar{\\lambda}))^2} \\bigg) \\bigg)\n\t\t\\nonumber \\\\\n\t&=\n\t\t\\Omega \\bigg(\n\t\t\\frac{p_k^2 (m_kp_k+\\xi) } {(\\varepsilon_k - \\varepsilon_{k,\\min}(\\bar{\\lambda}))^2}\n\t\t\\bigg( \\frac{2^{K}(1 + \\bar{\\lambda}^2) M_x^2}\n\t\t{s^2\\mbb{E}\\{x^2\\}^2}\n\t\t+ \\frac{M_w^2}{s^2\\mbb{E}\\{x^2\\}^2} \\bigg) \\bigg).\n\t\\end{align*}\nand $N \\geq \\max_{k \\in [K]} N_k $.\n\\end{IEEEproof}\n\n\\begin{remark}\nTo bound deviations of $\\Delta \\phi_\\mb{y}\\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}}$ from its mean,\nwe can also use the bound provided in~\\cite[Theorem 1]{gribonval2015sample} that prove uniform convergence results using covering number arguments for various classes of dictionaries. In this case, we get $\\eta_N \\leq c\\sqrt{\\dfrac{\\lrp{\\sum_k m_kp_k + \\xi}\\log N}{N}}$ for some constant $c$, where an extra $\\sqrt{\\log N}$ term appears compared to \\eqref{eq:dev}. Therefore, Lemma~\\ref{lemma:delt_m_T_dev} provides a tighter upper bound.\n\\end{remark}\n\n\n\n\n\\section{Conclusion} \\label{sec:discuss}\nIn this paper, we focused on local recovery of coordinate dictionaries comprising a Kronecker-structured dictionary used to represent $K$th-order tensor data. We derived a sample complexity upper bound for coordinate dictionary identification up to specified errors by expanding the objective function with respect to individual coordinate dictionaries and using the coordinate-wise Lipschitz continuity property of the objective function. This analysis is local in the sense that it only guarantees existence of a local minimum of the KS-DL objective function within some neighborhood of true coordinate dictionaries. Global analysis of the KS-DL problem is left for future work.\nOur results hold for dictionary coefficients generated according to the separable sparsity model. This model has some limitations compared to the random sparsity model and we leave the analysis for the random sparsity model for future work also. Another future direction of possible interest includes providing practical KS-DL algorithms that achieve the sample complexity scaling of Theorem~\\ref{thm:finite_n}.\n\n\n\n\\section*{Appendix A}\n\\begin{IEEEproof}[Proof of Lemma~\\ref{lem:Dtld}]\nTo prove the existence of such a formation for any $K\\geq 2$, we use induction.\nFor $K=2$, we have\n\t\\begin{align} \\label{eq:k2}\n\t\\lrp{\\mb{D}_1\\otimes \\mb{D}_2} &- \\lrp{\\mb{D}_1^0\\otimes \\mb{D}_2^0} \\nonumber\\\\\n\t&= \\lrp{\\mb{D}_1 - \\mb{D}_1^0} \\otimes \\mb{D}^0_2 + \\mb{D}_1 \\otimes \\lrp{\\mb{D}_2 - \\mb{D}_2^0} \\nonumber \\\\\n\t&= \\lrp{\\mb{D}_1 - \\mb{D}_1^0} \\otimes \\mb{D}_2 + \\mb{D}^0_1 \\otimes \\lrp{\\mb{D}_2 - \\mb{D}_2^0} .\n\t\\end{align}\nFor $K$ such that $K>2$, we assume the following holds:\n\t\\begin{align} \\label{eq:kK}\n\t&\\bigotimes_{k \\in [K]} \\mb{D}_k - \\bigotimes_{k \\in [K]} \\mb{D}_k^0 \\nonumber\\\\\n\t&\\qquad = \\sum _{k \\in [K]} \\widetilde{\\mb{D}}_{k,1} \\otimes \\dots \\otimes\n\t\t\\lrp{\\mb{D}_k - \\mb{D}^0_k}\t\\otimes \\dots \\otimes \\widetilde{\\mb{D}}_{k,K}.\n\t\\end{align}\nThen, for $K+1$, we have:\n\t\\begin{align} \\label{eq:kKp1}\n\t&\\bigotimes_{k \\in [K+1]} \\mb{D}_k - \\bigotimes_{k \\in [K+1]} \\mb{D}_k^0 \\nonumber\\\\\n\t&= \\bigg( \\bigotimes_{k \\in [K]} \\mb{D}_k \\bigg) \\otimes \\mb{D}_{K+1}\n\t\t-  \\bigg( \\bigotimes_{k \\in [K]} \\mb{D}_k^0 \\bigg) \\otimes \\mb{D}_{K+1}^0 \\nonumber\\\\\n\t&\\numrel{=}{r_k2} \\bigg( \\bigotimes_{k \\in [K]} \\mb{D}_k -\\bigotimes_{k \\in [K]} \\mb{D}_k^0 \\bigg)\n\t\t\\otimes \\mb{D}_{K+1}^0  \\nonumber\\\\\n\t&+ \\bigg( \\bigotimes_{k \\in [K]} \\mb{D}_k \\bigg) \\lrp{\\mb{D}_{K+1} - \\mb{D}_{K+1}^0} \\nonumber\\\\\n\t&\\numrel{=}{r_kK} \\bigg( \\sum _{k \\in [K]} \\widetilde{\\mb{D}}_{k,1} \\otimes \\dots \\otimes  \\lrp{\\mb{D}_k - \\mb{D}^0_k}\t\\otimes \\dots \\otimes \\widetilde{\\mb{D}}_{k,K} \\bigg)\n\t\t \\nonumber \\\\\n\t&\\qquad \\qquad \\otimes \\mb{D}_{K+1}^0 + \\bigg( \\bigotimes_{k \\in [K]} \\mb{D}_k \\bigg) \\lrp{\\mb{D}_{K+1} - \\mb{D}_{K+1}^0} \\nonumber\\\\\n\t&\\numrel{=}{r_allcases}\n\t \t\\sum _{k \\in [K+1]} \\widetilde{\\mb{D}}_{k,1} \\otimes \\dots \\otimes  \\lrp{\\mb{D}_k - \\mb{D}^0_k}\t\\otimes \\dots \\otimes \\wt{\\D}_{k,K+1} ,\n\t\\end{align}\nwhere \\eqref{r_k2} follows from \\eqref{eq:k2}, \\eqref{r_kK} follows from \\eqref{eq:kK} and \\eqref{r_allcases} follows from replacing $\\mb{D}^0_{K+1}$ with $\\wt{\\D}_{k,K+1}$ in the first $K$ terms of the summation and $\\mb{D}_k$'s with $\\wt{\\D}_{K+1,k}$, for $k \\in[K]$, in the $(K+1)$th term of the summation.\n\\end{IEEEproof}\n\n\\begin{IEEEproof}[Proof of Lemma~\\ref{lemma:f_closedform}]\nUsing the same definition as Gribonval et al.~\\cite[Definition 1]{gribonval2014sparse}, taking the derivative of $\\phi_\\mb{y}\\lrp{\\D_{1:K}|\\boldsymbol{\\sigma}}$ with respect to $\\mb{x}$ and setting it to zero, we get the expression in~\\eqref{eq:xhat_closedform} for $\\widehat{\\mb{x}}$. Substituting $\\widehat{\\mb{x}}$ in~\\eqref{eq:delt_inf}, we get\n \t\\begin{align*}\n\t&\\phi_\\mb{y}\\lrp{\\D_{1:K}|\\boldsymbol{\\sigma}} = \\frac{1}{2}\n\t\t\\bigg[\n\t\t\\|\\mb{y}\\|_2^2 - \\lrp{ \\lrp{\\bigotimes \\mb{D}_{k,\\mc{J}_k}^\\top} \\mb{y} -\\lambda \\boldsymbol{\\sigma}_\\mc{J} }^\\top \\nonumber\\\\\n\t&\\qquad \\lrp{ \\bigotimes (\\mb{D}_{k,\\mc{J}_k}^\\top \\mb{D}_{k,\\mc{J}_k})^{-1} }\n\t\t\\lrp{ \\lrp{\\bigotimes \\mb{D}_{k,\\mc{J}_k}^\\top} \\mb{y} -\\lambda \\boldsymbol{\\sigma}_\\mc{J} }\n\t\t\\bigg]\\nonumber\\\\\n\t&\\qquad \\qquad \\quad  \\numrel{=}{r_dP} \\frac{1}{2}\\|\\mb{y}\\|_2^2\n\t\t- \\frac{1}{2} \\mb{y}^\\top \\lrp{\\bigotimes \\bP_{\\D_{k,\\cJ_k}}} \\mb{y} \\nonumber\\\\\n\t&\\qquad + \\lambda \\boldsymbol{\\sigma}_\\mc{J}^\\top \\lrp{\\bigotimes \\D_{k,\\cJ_k}^+}\\mb{y}\n\t\t-\\frac{\\lambda^2}{2}\\boldsymbol{\\sigma}_\\mc{J}^\\top \\lrp{ \\bigotimes \\bH_{\\D_{k,\\cJ_k}}}  \\boldsymbol{\\sigma}_\\mc{J},\n\t\\end{align*}\nwhere \\eqref{r_dP} follows from \\eqref{eq:P_H_Ps}.\n\\end{IEEEproof}\n\n\n\\begin{IEEEproof}[Proof of Lemma~\\ref{lem:exp_phi}]\nWe use the expression for $\\phi_\\mb{y}\\lrp{\\D_{1:K}|\\boldsymbol{\\sigma}}$ from~\\eqref{eq:delt_closedform}. For any $\\mb{D}=\\bigotimes \\mb{D}_k,\\mb{D}'=\\bigotimes \\mb{D}'_k$, $\\mb{D}_k,\\mb{D}_k' \\in \\mc{D}_k$, we have\n\t\\begin{align} \\label{eq:delt_ph_1}\n\t&\\Delta \\phi_\\mb{y}\\lrp{ \\D_{1:K};\\D'_{1:K} | \\boldsymbol{\\sigma}}\n\t= \\phi_\\mb{y}\\lrp{\\D_{1:K}|\\boldsymbol{\\sigma}} - \\phi_\\mb{y}\\lrp{\\D'_{1:K}|\\boldsymbol{\\sigma}} \\nonumber \\\\\n\t&\\quad= \t\\frac{1}{2} \\mb{y}^\\top  \\lrp{ \\bigotimes \\bP_{\\D'_{k,\\cJ_k}} - \\bigotimes \\bP_{\\D_{k,\\cJ_k}} } \\mb{y} \\nonumber\\\\\n\t&\\qquad- \\lambda \\boldsymbol{\\sigma}_\\mc{J}^\\top \\lrp{ \\bigotimes {\\D'}_{k,\\cJ_k}^+ - \\bigotimes \\D_{k,\\cJ_k}^+ } \\mb{y}  \\nonumber \\\\\n\t&\\qquad + \\frac{\\lambda^2}{2}\\boldsymbol{\\sigma}_\\mc{J}^\\top \\lrp{ \\bigotimes \\bH_{\\D'_{k,\\cJ_k}} - \\bigotimes \\bH_{\\D_{k,\\cJ_k}} } \\boldsymbol{\\sigma}_\\mc{J}.\n\t\\end{align}\nWe substitute $\\mb{y} = \\lrp{\\bigotimes \\mb{D}^0_k} \\mb{x} +\\mb{w} = \\lrp{\\bigotimes \\mb{D}^0_{k,\\mc{J}_k} }\\mb{x}_{\\mc{J}} +\\mb{w}$\nand break up the sum in \\eqref{eq:delt_ph_1} into 6 terms:\n\t\\begin{align} \\label{eq:delt_t}\n\t\\Delta \\phi_\\mb{y}\\lrp{ \\D_{1:K};\\D'_{1:K} | \\boldsymbol{\\sigma}} = \\sum_{i\\in[6]} \\Delta \\phi_i \\lrp{ \\D_{1:K};\\D'_{1:K} | \\boldsymbol{\\sigma}} ,\n\t\\end{align}\t\nwhere\n\t\\begin{align} \\label{eq:delt_phi}\n\t& \\Delta \\phi_1 \\lrp{ \\D_{1:K};\\D'_{1:K} | \\boldsymbol{\\sigma}}\n\t\t=  \\frac{1}{2} {\\mb{x}}^\\top \\lrp{\\bigotimes \\mb{D}^0_k }^\\top  \\nonumber\\\\\n\t&\\qquad\t\\lrp{ \\bigotimes \\bP_{\\D'_{k,\\cJ_k}} - \\bigotimes \\bP_{\\D_{k,\\cJ_k}} }\n\t\t\\lrp{\\bigotimes \\mb{D}^0_k } \\mb{x} \\nonumber\\\\\n\t&\\numrel{=}{r_Dtild}  \\frac{1}{2} {\\mb{x}}^\\top  \\lrp{\\bigotimes \\mb{D}^0_k }^\\top\n\t\t\\bigg( \\sum_{k \\in [K]} \\mb{P}_{\\widetilde{\\mb{D}}_{1,\\mc{J}_1}} \\otimes \\dots \\otimes\n\t\t\\nonumber\\\\\t\t\n\t&\\qquad \\lrp{\\bP_{\\D'_{k,\\cJ_k}} - \\bP_{\\D_{k,\\cJ_k}}}\t\\otimes \\dots \\otimes \\mb{P}_{\\widetilde{\\mb{D}}_{K,\\mc{J}_K}}\\bigg)\n\t\t\\lrp{\\bigotimes \\mb{D}^0_k } \\mb{x} \\nonumber \\\\\n\t&= \\frac{1}{2} {\\mb{x}}^\\top\n\t\t\\bigg( \\sum_{k \\in [K]} \\lrp{{\\mb{D}^0_1}^\\top \\mb{P}_{\\widetilde{\\mb{D}}_{1,\\mc{J}_1}}\\mb{D}^0_1} \\otimes \\dots \\otimes  \\nonumber\\\\\n\t&\\qquad \\lrp{{\\mb{D}^0_k}^\\top(\\bP_{\\D'_{k,\\cJ_k}} - \\bP_{\\D_{k,\\cJ_k}})\\mb{D}^0_k}\t\\otimes \\dots \\otimes \\nonumber \\\\\n\t&\\qquad\n\t\t\\lrp{{\\mb{D}^0_K}^\\top \\mb{P}_{\\widetilde{\\mb{D}}_{K,\\mc{J}_K}}\\mb{D}^0_K} \\bigg)\n\t\t\\mb{x} , \\nonumber \\\\\n\t&\\Delta \\phi_2 \\lrp{ \\D_{1:K};\\D'_{1:K} | \\boldsymbol{\\sigma}}\n\t%\n\t%\n\t%\n\t%\n\t= \\mb{w}^\\top\n\t\t\\bigg( \\sum_{k \\in [K]} \\lrp{\\mb{P}_{\\widetilde{\\mb{D}}_{1,\\mc{J}_1}}\\mb{D}^0_1}\n\t\t\\otimes \\dots \\otimes \\nonumber\\\\\n\t&\\qquad\n\t\t\\lrp{(\\bP_{\\D'_{k,\\cJ_k}} - \\bP_{\\D_{k,\\cJ_k}}) \\mb{D}^0_k}\t\t\t\n\t\t\\otimes \\dots \\otimes\n\t\t\\lrp{ \\mb{P}_{\\widetilde{\\mb{D}}_{K,\\mc{J}_K}} \\mb{D}^0_K}\\bigg)\n\t\t\\mb{x}, \\nonumber \\\\\n\t&\\Delta \\phi_3 \\lrp{ \\D_{1:K};\\D'_{1:K} | \\boldsymbol{\\sigma}}\n\t%\n\t%\n\t%\n\t\t= \\frac{1}{2} \\mb{w}^\\top\n\t\t\\bigg( \\sum_{k \\in [K]} \\mb{P}_{\\widetilde{\\mb{D}}_{1,\\mc{J}_1}} \\otimes \\dots \\otimes \\nonumber\\\\\n\t&\\qquad \\lrp{\\bP_{\\D'_{k,\\cJ_k}} - \\bP_{\\D_{k,\\cJ_k}}}\n\t\t\\otimes \\dots \\otimes  \\mb{P}_{\\widetilde{\\mb{D}}_{K,\\mc{J}_K}} \\bigg)\n\t\t\\mb{w}, \\nonumber \\\\\n\t&\\Delta \\phi_4 \\lrp{ \\D_{1:K};\\D'_{1:K} | \\boldsymbol{\\sigma}}\n\t%\n\t%\n\t%\n\t\t= - \\lambda \\boldsymbol{\\sigma}_\\mc{J}^\\top\n\t\t\\bigg( \\sum_{k \\in [K]} \\lrp{ {\\widetilde{\\mb{D}}_{1,\\mc{J}_1}}^+\\mb{D}^0_1} \\otimes \\dots \\otimes \t\\nonumber\\\\\n\t&\\qquad\n\t\t\\lrp{({\\D'}_{k,\\cJ_k}^+ - \\D_{k,\\cJ_k}^+ ) \\mb{D}^0_k} \\otimes \\dots \\otimes\n\t\t\\lrp{  {\\widetilde{\\mb{D}}_{K,\\mc{J}_K}}^+ \\mb{D}^0_K} \\bigg)\n\t\t\\mb{x}, \\nonumber\\\\\n\t&\\Delta \\phi_5 \\lrp{ \\D_{1:K};\\D'_{1:K} | \\boldsymbol{\\sigma}}\n\t%\n\t%\n\t\t= - \\lambda \\boldsymbol{\\sigma}_\\mc{J}^\\top\n\t\t\\bigg( \\sum_{k \\in [K]} {\\widetilde{\\mb{D}}_{1,\\mc{J}_1}}^+ \\otimes \\dots \\otimes \\nonumber\\\\\n\t&\\qquad \\lrp{{\\D'}_{k,\\cJ_k}^+ - \\D_{k,\\cJ_k}^+ }\t\\otimes \\dots \\otimes\n\t\t {\\widetilde{\\mb{D}}_{K,\\mc{J}_K}}^+ \\bigg) \\mb{w}, \\text{and}\\nonumber \\\\\n\t&\\Delta \\phi_6 \\lrp{ \\D_{1:K};\\D'_{1:K} | \\boldsymbol{\\sigma}}\n\t%\n\t%\n\t%\n\t\t= \\frac{\\lambda^2}{2}\\boldsymbol{\\sigma}_\\mc{J}^\\top\n\t \t\\bigg( \\sum_{k \\in [K]} \\mb{H}_{\\widetilde{\\mb{D}}_{1,\\mc{J}_1}} \\otimes \\dots \\otimes \\nonumber\\\\\n\t &\\qquad \\lrp{\\bH_{\\D'_{k,\\cJ_k}} - \\bH_{\\D_{k,\\cJ_k}} }\n\t \t\\otimes \\dots \\otimes\n\t\t\\mb{H}_{\\widetilde{\\mb{D}}_{K,\\mc{J}_K}} \\bigg) \\boldsymbol{\\sigma}_\\mc{J},\n\t\\end{align}\nwhere \\eqref{r_Dtild} follows from Lemma~\\ref{lem:Dtld} and analysis for derivation of $\\lr{\\Delta \\phi_i\\lrp{ \\D_{1:K};\\D'_{1:K} | \\boldsymbol{\\sigma}} }_{i=2}^6 $ are omitted due to space constraints.\nNow, we set $\\mb{D}' = \\mb{D}^0$ and take the expectation of $\\Delta \\phi_\\mb{y}\\lrp{ \\D_{1:K};\\{\\mb{D}^0_k\\} | \\boldsymbol{\\sigma}}$ with respect to $\\mb{x}$ and $\\mb{w}$. Since the coefficient and noise vectors are uncorrelated,\n\t\\begin{align*}\n\t\\mbb{E} \\lr{ \\Delta \\phi_2 \\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}}}=\\mbb{E} \\lr{\\Delta \\phi_5 \\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}} } = 0.\n\t\\end{align*}\nWe can restate the other terms as:\n\t\\begin{align} \\label{eq:delt_tr}\n\t &\\Delta \\phi_1 \\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}}  \\nonumber\\\\\n\t &\\numrel{=}{r_Imk} \\frac{1}{2}\\mathop{\\mathrm{Tr}}\\nolimits \\bigg[\\mb{x}_\\mc{J}  {\\mb{x}}^\\top_\\mc{J}\n\t\t \\sum_{k \\in [K]} \\lrp{{\\mb{D}^0_1}^\\top \\mb{P}_{\\widetilde{\\mb{D}}_{1,\\mc{J}_1}}\\mb{D}^0_1} \\otimes \\dots \\otimes  \\nonumber\\\\\n\t&\\lrp{{\\mb{D}^0_k}^\\top(\\mb{I}_{m_k} - \\bP_{\\D_{k,\\cJ_k}})\\mb{D}^0_k}\t\n\t\t\\otimes \\dots \\otimes\n\t\t\\lrp{{\\mb{D}^0_K}^\\top \\mb{P}_{\\widetilde{\\mb{D}}_{K,\\mc{J}_K}}\\mb{D}^0_K}\n\t\t\\bigg] ,\\nonumber \\\\\n\t& \\Delta \\phi_3 \\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}}  \\nonumber\\\\\n\t&= \\frac{1}{2} \\mathop{\\mathrm{Tr}}\\nolimits \\bigg[ \\mb{w} \\mb{w}^\\top\n\t\t\\bigg( \\sum_{k \\in [K]} \\mb{P}_{\\widetilde{\\mb{D}}_{1,\\mc{J}_1}} \\otimes \\dots \\otimes\n\t\t\\lrp{\\bP_{\\D^0_{k,\\cJ_k}} -\\bP_{\\D_{k,\\cJ_k}}} \\nonumber\\\\\n\t&\\qquad \\otimes \\dots \\otimes \\mb{P}_{\\widetilde{\\mb{D}}_{K,\\mc{J}_K}}\\bigg) \\bigg],\n\t\t\\nonumber\\\\\n\t &\\Delta \\phi_4 \\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}}  \\nonumber\\\\\n\t &\\numrel{=}{r_Isk} - \\lambda \\mathop{\\mathrm{Tr}}\\nolimits \\bigg[  \\mb{x}_\\mc{J} \\boldsymbol{\\sigma}_\\mc{J}^\\top\n\t\t\\bigg( \\sum_{k \\in [K]} \\lrp{ {\\widetilde{\\mb{D}}_{1,\\mc{J}_1}}^+\\mb{D}^0_1} \\otimes \\dots \\otimes  \\nonumber\\\\\n\t&\\qquad \\lrp{\\mb{I}_{s_k} - \\D_{k,\\cJ_k}^+ \\mb{D}^0_k}\t\\otimes \\dots \\otimes\n\t\t\\lrp{  {\\widetilde{\\mb{D}}_{K,\\mc{J}_K}}^+ \\mb{D}^0_K}\\bigg)  \\bigg] ,\\text{and}\\nonumber \\\\\n\t& \\Delta \\phi_6 \\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}}  \\nonumber\\\\\n\t&= \\frac{\\lambda^2}{2} \\mathop{\\mathrm{Tr}}\\nolimits \\bigg[ \\boldsymbol{\\sigma}_\\mc{J} \\boldsymbol{\\sigma}_\\mc{J}^\\top\n\t \t\\bigg( \\sum_{k \\in [K]} \\mb{H}_{\\widetilde{\\mb{D}}_{1,\\mc{J}_1}} \\otimes \\dots \\otimes  \\nonumber\\\\\n\t&\\qquad \\lrp{\\mb{H}_{\\mb{D}^0_{k,\\mc{J}_k}}  - \\bH_{\\D_{k,\\cJ_k}} }\t\\otimes \\dots \\otimes\n\t\t\\mb{H}_{\\widetilde{\\mb{D}}_{K,\\mc{J}_K}} \\bigg) \\bigg],\n\t\\end{align}\nwhere \\eqref{r_Imk} and \\eqref{r_Isk} follow from the facts that $\\mb{P}_{\\mb{D}^0_{k,\\mc{J}_k}}\\mb{D}_k^0 =\\mb{D}_k^0$ and ${\\mb{D}^0}^+_{k,\\mc{J}_k} \\mb{D}^0_k = \\mb{I}_{s_k}$, respectively. Taking the expectation of the terms in~\\eqref{eq:delt_tr}, we get\n\t\\begin{align} \\label{eq:jam_T}\n\t& \\mbb{E} \\lr{ \\Delta \\phi_1 \\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}} } \\nonumber \\\\\n\t&\\numrel{=}{r_tr_eq}  \\frac{\\mbb{E}\\{x^2\\}}{2}\\mbb{E}_\\mc{J}\n\t\t\\bigg\\{ \\sum_{k \\in [K]} \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ {\\mb{D}^0_1}^\\top \\mb{P}_{\\widetilde{\\mb{D}}_{1,\\mc{J}_1}}\\mb{D}^0_1} \\dots \\nonumber\\\\\n\t&\\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ {\\mb{D}^0_k}^\\top(\\mb{I}_{m_k} - \\bP_{\\D_{k,\\cJ_k}})\\mb{D}^0_k} \\dots\n\t\t\\mathop{\\mathrm{Tr}}\\nolimits \\lrb{{\\mb{D}^0_K}^\\top \\mb{P}_{\\widetilde{\\mb{D}}_{K,\\mc{J}_K}}\\mb{D}^0_K}\\bigg\\}\n\t\t\\nonumber \\\\\n\t&=  \\frac{\\mbb{E}\\{x^2\\}}{2}\n\t\t\\sum_{k \\in [K]} \\mbb{E}_{\\mc{J}_1} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ {\\mb{D}^0_1}^\\top \\mb{P}_{\\widetilde{\\mb{D}}_{1,\\mc{J}_1}}\\mb{D}^0_1} }  \\dots\\nonumber\\\\\n\t&\\qquad \\mbb{E}_{\\mc{J}_k} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{  {\\mb{D}^0_k}^\\top(\\mb{I}_{m_k} - \\bP_{\\D_{k,\\cJ_k}})\\mb{D}^0_k}} \\dots \\nonumber \\\\\n\t& \\qquad\n\t\t\\mbb{E}_{\\mc{J}_K} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{{\\mb{D}^0_K}^\\top \\mb{P}_{\\widetilde{\\mb{D}}_{K,\\mc{J}_K}}\\mb{D}^0_K}}, \\nonumber \\\\\n\t& \\mbb{E} \\{ \\Delta \\phi_3 \\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}} \\}  \\nonumber \\\\\n\t&= \\frac{\\mbb{E}\\{w^2\\}}{2}\n\t\t\\mbb{E}_\\mc{J}\n\t\t\\bigg\\{ \\mathop{\\mathrm{Tr}}\\nolimits \\bigg[\\sum_{k \\in [K]} \\mb{P}_{\\widetilde{\\mb{D}}_{1,\\mc{J}_1}} \\otimes \\dots \\otimes  \\nonumber\\\\\n\t&\\qquad \\lrp{\\bP_{\\D^0_{k,\\cJ_k}} -\\bP_{\\D_{k,\\cJ_k}}} \\otimes \\dots \\otimes \\mb{P}_{\\widetilde{\\mb{D}}_{K,\\mc{J}_K}} \\bigg] \t\n\t\t\\bigg\\}\n\t\t\\nonumber \\\\\n\t&= \\frac{\\mbb{E}\\{w^2\\}}{2}\n\t\t\\mbb{E}_\\mc{J}\n\t\t\\bigg\\{ \\sum_{k \\in [K]}  \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ \\mb{P}_{\\widetilde{\\mb{D}}_{1,J_1}} } \\dots\n\t\t \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ \\bP_{\\D^0_{k,\\cJ_k}} -\\bP_{\\D_{k,\\cJ_k}}} \\nonumber\\\\\n\t&\\qquad \\dots  \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{  \\mb{P}_{\\widetilde{\\mb{D}}_{K,\\mc{J}_K}} } \\bigg\\} \\nonumber\\\\\n\t&\\numrel{=}{r_proj_tr} 0, \\nonumber \\\\\n\t& \\mbb{E} \\lr{ \\Delta \\phi_4 \\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}} } \\nonumber \\\\\n\t& =- \\lambda \\mbb{E}\\{|x|\\}\n\t\t\\sum_{k \\in [K]} \\mbb{E}_{\\mc{J}_1} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ {\\widetilde{\\mb{D}}_{1,\\mc{J}_1}}^+\\mb{D}^0_1}} \\dots \\nonumber\\\\\n\t&\\quad \\mbb{E}_{\\mc{J}_k} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{\\mb{I}_{s_k} - \\D_{k,\\cJ_k}^+ \\mb{D}^0_k}} \\dots\n\t\t\\mbb{E}_{\\mc{J}_K} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{  {\\widetilde{\\mb{D}}_{K,\\mc{J}_K}}^+ \\mb{D}^0_K}},\n\t \\nonumber \\\\\n\t& \\mbb{E} \\lr{ \\Delta \\phi_6 \\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}} } \\nonumber \\\\\n\t&= \\frac{\\lambda^2}{2}\n\t\t\\sum_{k \\in [K]} \\mbb{E}_{\\mc{J}_1} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ \\mb{H}_{\\widetilde{\\mb{D}}_{1,\\mc{J}_1}}} }\\dots  \\nonumber\\\\\n\t&\\quad \\mbb{E}_{\\mc{J}_k} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ \\mb{H}_{\\mb{D}^0_{k,\\mc{J}_k}}  - \\bH_{\\D_{k,\\cJ_k}} }\t} \\dots\n\t\t\\mbb{E}_{\\mc{J}_K} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ \\mb{H}_{\\widetilde{\\mb{D}}_{K,\\mc{J}_K}}}} .\n\t\\end{align}\t\nwhere \\eqref{r_tr_eq} follows from the relation $\\mathop{\\mathrm{Tr}}\\nolimits(\\mb{A}\\otimes \\mb{B})=\\mathop{\\mathrm{Tr}}\\nolimits[\\mb{A}]\\mathop{\\mathrm{Tr}}\\nolimits[\\mb{B}]$~\\cite{horn2012matrix} and \\eqref{r_proj_tr} follows from the fact that $\\bP_{\\D_{k,\\cJ_k}}$'s are orthogonal projections onto subspaces of dimension $s_k$ and $ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{\\bP_{\\D^0_{k,\\cJ_k}} -\\bP_{\\D_{k,\\cJ_k}}}=s_k - s_k=0$.\nAdding the terms in \\eqref{eq:jam_T}, we obtain the expression in \\eqref{eq:delt_ph_2}.\n\\end{IEEEproof}\n\n\\begin{IEEEproof} [Proof of Lemma~\\ref{lem:UB_mu_delt_RIP}]\nEquation \\eqref{eq:l2_delt} follows from the definition of $\\mathsf{RIP}$ and \\eqref{eq:delt_mu} follows from Gerschgorin's disk theorem~\\cite{HornJohnson,horn2012matrix,golub2012matrix}.\n\\end{IEEEproof}\n\n\n\\begin{IEEEproof} [Proof of Lemma~\\ref{lem:A_B_delt}]\nTo lower bound $\\Delta \\phi_\\mbb{P}\\lrp{ \\D_{1:K}; \\D^0_{1:K} | \\boldsymbol{\\sigma}}$, we bound each term in~\\eqref{eq:delt_ph_2} separately. For the first term $\\mbb{E}\\lr{ \\Delta \\phi_1 \\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}} }$, we have\n\t\\begin{align} \\label{eq:P_Dt_LB}\n\t&\\mbb{E}_{\\mc{J}_k} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ {\\mb{D}^0_k}^\\top \\mb{P}_{\\wt{\\D}_{k,\\mc{J}_k}}\\mb{D}^0_k} }\t\n\t%\n\t%\n\t\t=  \\mbb{E}_{\\mc{J}_k} \\lr{ \\norm{\\mb{P}_{\\wt{\\D}_{k,\\mc{J}_k}}\\mb{D}^0_{k,\\mc{J}_k}}_F^2} .\n\t\\end{align}\nIf $\\wt{\\D}_k = \\mb{D}_k^0$, then\n\t\\begin{align}\n\t\\mbb{E}_{\\mc{J}_k} \\lr{ \\norm{ \\bP_{\\D^0_{k,\\cJ_k}} \\mb{D}^0_{k,\\mc{J}_k}}_F^2} &\n\t\t%\n\t\t\\numrel{=}{r_Esp} \\frac{s_k}{p_k} \\norm{\\mb{D}^0_k}_F^2 = s_k,\n\t\\end{align}\nwhere~\\eqref{r_Esp} follows from~\\cite[Lemma 15]{gribonval2014sparse}. If $\\wt{\\D}_k = \\mb{D}_k$, then \t\n\t\\begin{align*}\n\t& \\mbb{E}_{\\mc{J}_k} \\lr{ \\norm{\\mb{P}_{\\mb{D}_{k,\\mc{J}_k}}\\mb{D}^0_{k,\\mc{J}_k}}_F^2}\n\t\\numrel{=}{r_DC} \\mbb{E}_{\\mc{J}_k} \\lr{ \\norm{ [\\mb{D}_k \\mb{C}_k^{-1}]_{\\mc{J}_k}}_F^2 }  \\nonumber \\\\\n\t& \\numrel{=}{r_exp} \\frac{s_k}{p_k} \\norm{\\mb{D}_k\\mb{C}_k^{-1}}_F^2\n\t\t\\numrel{=}{r_d_j_1} \\frac{s_k}{p_k}\n\t\t\\sum_{j=1}^{p_k} \\frac{1}{\\cos^2(\\theta_{(k,j)})}\n\t\t\\numrel{\\geq}{r_cos_ineq} \\frac{s_k}{p_k} p_k = s_k,\n\t\\end{align*}\nwhere \\eqref{r_DC} is a direct consequence of Lemma~\\ref{lem:D_12_tet}; we can write $\\mb{D}^0_k = \\mb{D}_k\\mb{C}_k^{-1} - \\mathbf{V}_k\\mb{T}_k$ where $\\mb{C}_k = \\Diag{\\cos(\\boldsymbol{\\theta}_{k})}$, $\\mb{T}_k = \\Diag{\\tan(\\boldsymbol{\\theta}_k)}$ and $\\theta_{k,j}$ denotes the angle between $\\mb{d}_{k,j}$ and $\\mb{d}^0_{k,j}$. Hence $\\bP_{\\D_{k,\\cJ_k}} \\mb{D}^0_{k,\\mc{J}_k} =  [\\mb{D}_k \\mb{C}_k^{-1}]_{\\mc{J}_k}$. Moreover,~\\eqref{r_exp} follows from~\\cite[Lemma 15]{gribonval2014sparse}, \\eqref{r_d_j_1} follows from the fact that $\\|\\mb{d}_{k,j}\\|_2=1$, and \\eqref{r_cos_ineq} follows from the fact that $\\cos (\\theta_{k,j})<1$. Similarly, we have\n\t\\begin{align}\n\t& \\mbb{E}_{\\mc{J}_k} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ {\\mb{D}^0_k}^\\top(\\mb{I}_{m_k} - \\bP_{\\D_{k,\\cJ_k}})\\mb{D}^0_k} } \\nonumber\\\\\n\t&\\qquad = \\mbb{E}_{\\mc{J}_k} \\lr{ \\norm{(\\mb{I}_{m_k} - \\bP_{\\D_{k,\\cJ_k}})\\mb{D}^0_{k,\\mc{J}_k}}_F^2 }  \\nonumber \\\\\n\t&\\qquad \\numrel{\\geq}{r_diff_p_LB} \\frac{s_k}{p_k} \\|\\boldsymbol{\\theta}_k\\|_2^2 \\lrp{ 1 - \\frac{s_k}{p_k} \\frac{B^2_k}{1-\\delta_k} },\n\t\\end{align}\nwhere~\\eqref{r_diff_p_LB} follows from similar arguments as in Gribonval et al.~\\cite[Equation (72)]{gribonval2014sparse}.\nPutting it all together, we have\n\t\\begin{align} \\label{eq:delt1_LB}\n\t\\mbb{E} &\\lr{ \\Delta \\phi_1 \\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}} } \\nonumber\\\\\n\t&\\geq \\frac{\\mbb{E}\\{x^2\\}}{2}\n\t\t\\sum_{k \\in [K]} \\bigg(\\prod_{\\substack{i \\in [K]\\\\ i \\neq k}} s_i\\bigg) \\frac{s_k}{p_k} \\|\\boldsymbol{\\theta}_k\\|_2^2\n\t\t\\lrp{ 1 - \\frac{s_k}{p_k} \\frac{B^2_k}{1-\\delta_k} } \\nonumber \\\\\n\t&= \\frac{s \\mbb{E}\\{x^2\\}}{2}\n\t\t\\sum_{k \\in [K]} \\frac{\\|\\boldsymbol{\\theta}_k\\|_2^2 }{p_k}\n\t\t\\lrp{ 1 - \\frac{s_k}{p_k} \\frac{B^2_k}{1-\\delta_k} }.\n\t\\end{align}\n\nNext, to lower bound $\\mbb{E}\\lr{ \\Delta \\phi_4 \\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}} }$, we upper bound $\\lra{\\mbb{E}\\lr{ \\Delta \\phi_4 \\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}} }}$. If $\\wt{\\D}_k = \\mb{D}_k^0$, we have\n\t\\begin{align}\n\t\\mbb{E}_{\\mc{J}_k} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ {\\mb{D}^0}_{k,\\mc{J}_k}^+\\mb{D}^0_{k,\\mc{J}_k}}}\n\t\t= \\mbb{E}_{\\mc{J}_k} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ \\mb{I}_{s_k}}} = s_k,\n\t\\end{align}\notherwise, if $\\wt{\\D}_k = \\mb{D}_k$, we get\n\t\\begin{align}\n\t&\\left| \\mbb{E}_{\\mc{J}_k} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ {\\mb{D}_{k,\\mc{J}_k}}^+\\mb{D}^0_k}} \\right| \\nonumber\\\\\n\t&\\qquad \\numrel{\\leq}{r_tr_spn} s_k \\mbb{E}_{\\mc{J}_k} \\lr{ \\norm{ {\\mb{D}}_{k,\\mc{J}_k}^+\\mb{D}^0_{k,\\mc{J}_k}}_2}\n\t\t\\nonumber \\\\\n\t&\\qquad  \\leq  s_k \\mbb{E}_{\\mc{J}_k} \\lr{ \\| {\\mb{D}}_{k,\\mc{J}_k}^+\\|_2\\|\\mb{D}^0_{k,\\mc{J}_k} \\|_2}\n\t\t\\nonumber \\\\\n\t&\\qquad \\numrel{\\leq}{r_5} s_k \\lrp{ \\frac{1}{\\sqrt{1-\\delta_{s_k}(\\mb{D}_k)}} }\n\t\t\\lrp{ \\sqrt{1+\\delta_{s_k}(\\mb{D}^0_k)}}\n\t\t\\nonumber \\\\\n\t&\\qquad  \\numrel{\\leq}{r_delts} s_k \\sqrt{\\frac{1+\\delta_k}{1-\\delta_k}},\n\t\\end{align}\nwhere \\eqref{r_tr_spn} follows from the fact that for a square matrix $\\mb{A} \\in \\mbb{R}^{q\\times q}$, $\\mathop{\\mathrm{Tr}}\\nolimits \\lrb{\\mb{A}} \\leq q\\|\\mb{A}\\|_2$, \\eqref{r_5} follows from \\eqref{eq:l2_delt} and \\eqref{eq:pso_cond} and \\eqref{r_delts} follows from \\eqref{eq:A_B_Delk}. Similar to~\\cite[Equation (73)]{gribonval2014sparse}, we also have\n\t\\begin{align}\n\t&\\left| \\mbb{E}_{\\mc{J}_k} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\big[\\mb{I}_{s_k} - \\D_{k,\\cJ_k}^+ \\mb{D}^0_k\\big]} \\right|\n\t\\nonumber\\\\\n\t&\\qquad \\leq \\frac{s_k}{p_k} \\frac{\\|\\boldsymbol{\\theta}_k\\|_2^2}{2}\n\t\t+ \\frac{s_k^2}{p_k^2} \\frac{A_kB_k}{1-\\delta_k}\\|\\boldsymbol{\\theta}_k\\|_2.\n\t\\end{align}\nThus, defining $\\delta_{-k} \\triangleq \\prod_{\\substack{i \\in [K]\\\\ i \\neq k}} \\sqrt{\\dfrac{1+\\delta_i}{1-\\delta_i}}$, we get\n\t\\begin{align}  \\label{eq:delt4_LB}\n\t &\\mbb{E} \\lr{ \\Delta \\phi_4 \\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}} } \\nonumber\\\\\n\t&\\quad \\geq - \\lambda \\mbb{E}\\{|x|\\}\n\t\t\\sum_{k \\in [K]}\\delta_{-k}\\bigg( \\prod_{\\substack{i \\in [K]\\\\ i \\neq k}}s_i\n\t\t \\bigg) \\nonumber\\\\\n\t&\\quad \\qquad \\lrp{\\frac{s_k}{p_k} \\frac{\\|\\boldsymbol{\\theta}_k\\|_2^2}{2}\n\t\t+ \\frac{s_k^2}{p_k^2} \\frac{A_kB_k}{1-\\delta_k}\\|\\boldsymbol{\\theta}_k\\|_2}\n\t\t \\nonumber \\\\\n\t&\\quad  = - \\lambda s \\mbb{E}\\{|x|\\}  \\sum_{k \\in [K]} \\frac{\\delta_{-k}}{p_k}\n\t\t\\lrp{ \\frac{\\|\\boldsymbol{\\theta}_k\\|_2^2}{2}\n\t\t+ \\frac{s_k}{p_k} \\frac{A_kB_k}{1-\\delta_k}\\|\\boldsymbol{\\theta}_k\\|_2}.\n\t\\end{align}\n\t\nTo lower bound $\\mbb{E}\\lr{ \\Delta \\phi_6 \\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}} }$, we upper bound $\\lra{\\mbb{E}\\lr{ \\Delta \\phi_6 \\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}} }}$. For any $\\wt{\\D}_k$, we have\n\t\\begin{align}\n\t\\lra{ \\mbb{E}_{\\mc{J}_k} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ \\mb{H}_{\\wt{\\D}_{k,\\mc{J}_k}} }} }\n\t\t&\\leq \\mbb{E}_{\\mc{J}_k} \\lr{ s_k \\norm{ \\mb{H}_{\\wt{\\D}_{k,\\mc{J}_k}} }_2} \t\n\t\t\\numrel{\\leq}{r_H_UB} \\frac{s_k}{1-\\delta_k},\n\t\\end{align}\nwhere \\eqref{r_H_UB} follows from \\eqref{eq:pso_cond} and \\eqref{eq:A_B_Delk}. Similar to Gribonval et al.~\\cite[Equation (74)]{gribonval2014sparse}, we also have\n\t\\begin{align*}\n\t\\left| \\mbb{E}_{\\mc{J}_k} \\lr{ \\mathop{\\mathrm{Tr}}\\nolimits \\lrb{ \\mb{H}_{\\mb{D}^0_{k,\\mc{J}_k}} - \\bH_{\\D_{k,\\cJ_k}} }} \\right|\n\t\\leq \\frac{s_k^2}{p_k^2} \\frac{4A_kB_k}{(1-\\delta_k)^2}\\|\\boldsymbol{\\theta}_k\\|_2.\n\t\\end{align*}\nThus, we get\n\t\\begin{align}  \\label{eq:delt6_LB}\n\t& \\mbb{E} \\{ \\Delta \\phi_6 \\lrp{ \\D_{1:K};\\D^0_{1:K} | \\boldsymbol{\\sigma}} \\} \\nonumber\\\\\n\t&\\quad \\geq -  \\frac{\\lambda^2}{2}\n\t\t\\sum_{k \\in [K]} \\bigg( \\prod_{\\substack{i \\in [K]\\\\ i \\neq k}}\n\t\t\\frac{s_i}{1-\\delta_i}\\bigg)\n\t\t\\lrp{ \\frac{s_k^2}{p_k^2} \\frac{4A_kB_k}{(1-\\delta_k)^2}\\|\\boldsymbol{\\theta}_k\\|_2}\n\t\t\\nonumber \\\\\n\t&\\quad = - \\frac{\\lambda^2s}{2}\n\t\t\\sum_{k \\in [K]} \\frac{1}{p_k}\n\t\t\\bigg( \\prod_{i \\in [K]} \\frac{1}{1-\\delta_i}\\bigg)\n\t\t\\lrp{ \\frac{s_k}{p_k} \\frac{4A_kB_k}{1-\\delta_k}\\|\\boldsymbol{\\theta}_k\\|_2}.\n\t\\end{align}\nAdding~\\eqref{eq:delt1_LB}, \\eqref{eq:delt4_LB}, and \\eqref{eq:delt6_LB}, we get \\eqref{eq: delt_ph_3}.\n\\end{IEEEproof}\n\n\\begin{IEEEproof} [Proof of Proposition \\ref{prop:1}]\nTo show that $\\Delta\\phi_{\\mbb{P}}\\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}} > 0$, we use Lemma~\\ref{lem:A_B_delt} and prove that the right hand side of \\eqref{eq: delt_ph_3} is positive under certain conditions. First, we ensure the conditions in \\eqref{eq:cond_delt_r_i} and \\eqref{eq:A_B_Delk} hold for Lemma~\\ref{lem:H_Dps} and Lemma~\\ref{lem:A_B_delt}, respectively. We set $\\delta_k = \\dfrac{1}{2}$, $\\delta_{s_k}(\\mb{D}_k) = \\dfrac{1}{2}$ and $\\delta_{s_k}(\\mb{D}^0_k) = \\dfrac{1}{4}$, for $k \\in [K]$. For $\\varepsilon_k\\leq 0.15 $, this ensures:\n\t\t\\begin{align}\n\t\t&\\sqrt{1-\\delta_{s_k}(\\mb{D}_k)}\n\t\t\t\\geq \\sqrt{1-\\delta_{s_k}(\\mb{D}^0_k)} - \\varepsilon_k, \\text{ and}\n\t\t\t%\n\t\t\t\\nonumber \\\\\n\t\t&\\max\\lr{ \\delta_{s_k}(\\mb{D}^0_k), \\delta_{s_k}(\\mb{D}_k)}  \\leq \\delta_k,\n\t\t\\end{align}\nand implies $\\delta_k<1$ (condition for Lemmas~\\ref{lem:exp_phi} and \\ref{lem:H_Ps}).\nNext, we find conditions that guarantee:\n\t\\begin{align} \\label{eq:tet_2_p1}\n\t\\frac{s_k}{p_k} \\frac{B^2_k}{1-\\delta_k}\n\t\t+\\bar{\\lambda} \\kappa_x^2\n\t\t\\delta_{-k}\n\t\t\\numrel{=}{r_deltmk} \\frac{2B^2_k s_k}{p_k}\n\t\t+\\bar{\\lambda} \\kappa_x^2\n\t\t\\lrp{3}^{(K-1)\/2}\n\t\t\\leq \\frac{1}{2},\n\t\\end{align}\nwhere \\eqref{r_deltmk} follows from replacing $\\delta_k$ with $\\dfrac{1}{2}$.\nIf we take $\\dfrac{s_k}{p_k} \\leq \\dfrac{1}{8B_k^2}$ and $\\bar{\\lambda}\\leq \\dfrac{1}{8\\times 3^{(K-1)\/2}}$, given the fact that $\\kappa_x^2\\leq 1$, \\eqref{eq:tet_2_p1} is satisfied.\\footnote{These numbers are chosen for a simplified proof and can be modified.}\nConsequently, we can restate~\\eqref{eq: delt_ph_3} as\n\t\\begin{align} \\label{eq:eb_1}\n\t&\\Delta\\phi_{\\mbb{P}}\\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}}\n\t\t\\geq \\frac{s \\mbb{E}\\{x^2\\}}{4}\n\t\t\\sum_{k \\in [K]} \\frac{\\|\\boldsymbol{\\theta}_k\\|_2 }{p_k}\n\t\t\\bigg[\\|\\boldsymbol{\\theta}_k\\|_2 \\nonumber\\\\\n\t&\\qquad -8 \\lrp{3^{(K-1)\/2}\n\t\t+2^{(K+1)}\\bar{\\lambda} }\n\t\t\\bar{\\lambda} \\kappa_x^2\n\t\t\\frac{s_k}{p_k} A_kB_k\\bigg].\n\t\\end{align}\nFrom~\\cite[Proof of Proposition 2]{gribonval2014sparse}, we use the following relations:\n\t\\begin{align} \\label{eq:A0_A}\n\tB_k\\leq B^0_k  +\\varepsilon_k \\leq B^0_k +1, \\quad\n\tA_k \\leq A^0_k + 2B_k\\varepsilon_k, \\quad k\\in [K],\n\t\\end{align}\nwhere $A^0_k \\triangleq \\norm{{\\mb{D}_k^0}^\\top \\mb{D}_k^0 - \\mb{I}_{p_k}}_F$ and $B^0_k \\triangleq  \\norm{\\mb{D}_k^0}_2$ and \\eqref{eq:A0_A} follows from matrix norm inequalities~\\cite{gribonval2014sparse}.\nDefining $\\gamma_k \\triangleq 16 \\bigg(3^{(K-1)\/2}\n\t\t+2^{(K+1)}\\bar{\\lambda} \\bigg)   \\bar{\\lambda}\\kappa_x^2\n\t\t\\dfrac{B_k^2 s_k}{p_k}$ for $k \\in [K]$ and using $\\kappa_x^2 \\leq 1$, we have\n\t\\begin{align} \\label{eq:gam}\n\t\\gamma_k &\\leq\n\t\t2\\bigg(3^{(K-1)\/2}\n\t\t+\\frac{2^{(K+1)}}{8\\times 3^{(K-1)\/2}} \\bigg)\n\t\t\\bigg(\\frac{1}{8\\times 3^{(K-1)\/2}} \\bigg) \\nonumber\\\\\n\t\t&\\leq 2\\lrp{\\frac{1}{8}+\\frac{4}{64}}\n\t\t\\leq\\frac{1}{2}.\n\t\\end{align}\nThen, for $\\mb{D}_k \\in \\mc{S}_{\\varepsilon_k}(\\mb{D}_k^0)$, $k \\in [K]$, we get\n\t\\begin{align}\n\t&\\Delta\\phi_{\\mbb{P}}\\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}}  \\nonumber\\\\\n\t&\\qquad \\numrel{\\geq}{r_tet_rk} \\frac{s\\mbb{E}\\{x^2\\}}{4}\n\t\t\\sum_{k \\in [K]} \\frac{\\varepsilon_k}{p_k}\n\t\t\\lrp{ \\varepsilon_k - \\frac{\\gamma_k}{2} \\frac{A_k}{B_k}\t}\n\t\t\\nonumber \\\\\n\t&\\qquad\t\\numrel{\\geq}{r_A_B} \\frac{s\\mbb{E}\\{x^2\\}}{4}\n\t\t\\sum_{k \\in [K]} \\frac{\\varepsilon_k}{p_k}\n\t\t\\lrp{ \\varepsilon_k - \\frac{\\gamma_k}{2} \\frac{A_k^0+2B_k\\varepsilon_k}{B_k}\t}\n\t\t\\nonumber \\\\\n\t&\\qquad\t\\geq \\frac{s\\mbb{E}\\{x^2\\}}{4}\n\t\t\\sum_{k \\in [K]} \\frac{\\varepsilon_k}{p_k}\n\t\t\\lrp{ \\varepsilon_k(1-\\gamma_k) - \\frac{\\gamma_k}{2} \\frac{A_k^0}{B_k}\t}\n\t\t\\nonumber \\\\\n\t&\\qquad\t\\numrel{\\geq}{r_gam} \\frac{s\\mbb{E}\\{x^2\\}}{8}\n\t\t\\sum_{k \\in [K]} \\frac{\\varepsilon_k}{p_k}\n\t\t\\lrp{ \\varepsilon_k - \\gamma_k \\frac{A_k^0}{B_k}\t},\n\t\\end{align}\nwhere \\eqref{r_tet_rk} follows from \\eqref{eq:eb_1}, \\eqref{r_A_B} follows from \\eqref{eq:A0_A}, and \\eqref{r_gam} follows from \\eqref{eq:gam}.\nHence, we can write\n\t\\begin{align} \\label{eq:Delt_LB_rmin}\n\t\\Delta\\phi_{\\mbb{P}}\\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}} \\geq& \\frac{s\\mbb{E}\\{x^2\\}}{8}\n\t\t\\sum_{k \\in [K]} \\frac{\\varepsilon_k}{p_k}\n\t\t\\lrp{ \\varepsilon_k -  \\varepsilon_{k,\\min}(\\bar{\\lambda})},\n\t\\end{align}\nwhere we define\n\t\\begin{align}\n\t& \\varepsilon_{k,\\min} (\\bar{\\lambda})\n\t\\triangleq  \\gamma_k \\frac{A_k^0}{B_k} \\nonumber\\\\\n\t&= 16 \\lrp{3^{(K-1)\/2}\n\t\t+2^{(K+1)}\\bar{\\lambda} }\n\t\t\\bar{\\lambda}\\kappa_x^2\n\t\t\\frac{s_k}{p_k}  A_k^0 B_k \\nonumber \\\\\n\t&=\\frac{2}{3^{(K+1)\/2}} \\bigg(3^{(K-1)\/2}\n\t\t+2^{(K+1)}\\bar{\\lambda} \\bigg)  \\bar{\\lambda} C_{k,\\min},\n\t\\end{align}\nand $C_{k,\\min}$ is defined in \\eqref{eq:cond_C_min_C_max}.\nThe lower bound in \\eqref{eq:Delt_LB_rmin} holds for any $\\varepsilon_k \\leq 0.15$ and $\\mb{D}_k \\in \\mc{S}_{\\varepsilon_k}(\\mb{D}_k^0)$, $k \\in [K]$.\nFinally, since $3^{(K-1)\/2} +2^{(K+1)}\\bar{\\lambda} \\leq 0.5\\times 3^{(K+1)\/2}$,\nthe assumption\n$\\bar{\\lambda} \\leq 0.15\/(\\max_{k \\in [K]} C_{k,\\min})$\nimplies that  $\\varepsilon_{k,\\min}(\\bar{\\lambda}) \\leq 0.15$ for $k\\in[K]$.\nTherefore, $\\Delta\\phi_{\\mbb{P}}\\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}} > 0 $ for all $\\varepsilon_k \\in (\\varepsilon_{k,\\min}(\\bar{\\lambda}),0.15]$, $k \\in [K]$.\n\\end{IEEEproof}\n\n\n\\begin{IEEEproof}[Proof of Lemma~\\ref{lem:mu_mu0_rel}]\nConsidering $j \\not \\in \\mc{J}$, associated with $\\lrp{j_1,\\dots,j_k} \\not \\in \\lrp{\\mc{J}_1\\times \\dots \\times \\mc{J}_K}$, we have\n\t\\begin{align}\n\t&\\|\\mb{D}_\\mc{J}^\\top \\mb{d}_j\\|_1  \\nonumber\\\\\n\t&\\numrel{\\leq}{r_tr_ineq} \\|{\\mb{D}_\\mc{J}^0}^\\top \\mb{d}^0_j\\|_1 + \\|{\\mb{D}_\\mc{J}^0}^\\top (\\mb{d}_j-\\mb{d}^0_j)\\|_1\n\t\t+ \\|(\\mb{D}_\\mc{J}-\\mb{D}_\\mc{J}^0)^\\top \\mb{d}_j\\|_1\\nonumber \\\\\n\t&\\leq \\mu_s(\\mb{D}^0)\n\t\t+ \\sqrt{s} \\lrb{ \\|{\\mb{D}_\\mc{J}^0}^\\top( \\mb{d}_{j}-\\mb{d}_{j}^0) \\|_2\n\t\t+ \\|(\\mb{D}_\\mc{J}-\\mb{D}^0_\\mc{J})^\\top\\mb{d}_j\\|_2} \\nonumber \\\\\n\t&\\leq \\mu_s(\\mb{D}^0) + \\sqrt{s}\n\t\t\\bigg[ \\norm{ \\bigotimes {\\mb{D}^0_{k,\\mc{J}_k}}^\\top}_2\n\t\t\\norm{ \\bigotimes \\lrp{ \\mb{d}_{k,j_k} - \\mb{d}^0_{k,j_k} }}_2 \\nonumber\\\\\n\t&\\qquad + \\norm{\\bigotimes \\mb{D}_{k,\\mc{J}_k} -\\bigotimes \\mb{D}^0_{k,\\mc{J}_k}}_2\n\t\t \\norm{\\mb{d}_j}_2 \\bigg] \\nonumber \\\\\n\t&\\numrel{\\leq}{r_d0_mu} \\mu_s(\\mb{D}^0) + \\sqrt{s}\n\t\t\\bigg[ \\bigg( \\prod_{k \\in [K]} \\sqrt{1+\\delta_{s_k}(\\mb{D}^0_k)} \\bigg)\n\t\t\\nonumber\\\\\n\t&\\qquad \\bigg( \\sum_{k \\in [K]} \\norm{ \\widetilde{\\mb{d}}_{1,j_1}}_2\\dots\n\t\t\\norm{\\mb{d}_{k,j_k} - \\mb{d}^0_{k,j_k} }_2 \\dots \\\n\t\t\\norm{ \\widetilde{\\mb{d}}_{k,j_K}}_2 \\bigg)  \\nonumber \\\\\n\t &\\quad\n\t \t+ \\sum_{k \\in [K]} \\norm{ \\wt{\\D}_{1,\\mc{J}_1}}_2\\dots\n\t\t\\norm{\\mb{D}_{k,\\mc{J}_k} - \\mb{D}^0_{k,\\mc{J}_k} }_2 \\dots \\\n\t\t\\norm{ \\wt{\\D}_{k,\\mc{J}_k}}_2 \\bigg] \\nonumber \\\\\n\t&\\numrel{\\leq}{r_RIP_d} \\mu_s(\\mb{D}^0) + \\sqrt{s}\n\t\t\\bigg[ \\bigg( \\prod_{k \\in [K]} \\sqrt{1+\\delta_{s_k}(\\mb{D}^0_k)} \\bigg)\n\t\t\\bigg( \\sum_{k \\in [K]} \\varepsilon_k \\bigg) \\nonumber\\\\\n\t&\\qquad + \\sum_{k \\in [K]} \\bigg( \\prod_{\\substack{i \\in [K] \\\\ i \\neq k}}\n\t\t\\norm{\\wt{\\D}_{i,\\mc{J}_i}}_2 \\bigg) \\varepsilon_k\n\t\t \\bigg] \\nonumber \\\\\n\t%\n\t%\n\t%\n\t%\n\t%\n\t&\\numrel{\\leq}{r_asump} \\mu_s(\\mb{D}^0) + 2(1.5)^{K\/2}\\sqrt{s}  \\bigg( \\sum_{k \\in [K]} \\varepsilon_k \\bigg),\n\t\\end{align}\t\nwhere \\eqref{r_tr_ineq} follows from the triangle inequality, \\eqref{r_d0_mu} follows from \\eqref{eq:otD_otDp}, \\eqref{r_RIP_d} follows from \\eqref{eq:delt_mu}, and, \\eqref{r_asump} follows from substituting the upper bound value from \\eqref{eq:UB_mu_delt} for $\\delta_{s_k}(\\mb{D}_k^0) $. For  $\\wt{\\D}_i = \\mb{D}_i^0$, $\\norm{\\mb{D}_{i,\\mc{J}_i}^0}_2 \\leq \\sqrt{1 + \\delta_{s_i}(\\mb{D}^0_i)}  \\leq \\sqrt{\\frac{5}{4}} < 1.5$ and for $\\wt{\\D}_i = \\mb{D}_i$, according to \\eqref{eq:A0_A}, we have $\\norm{\\mb{D}_{i,\\mc{J}_i}}_2 \\leq \\norm{\\mb{D}_{i,\\mc{J}_i}^0}_2 + \\varepsilon_i \\leq \\sqrt{\\frac{5}{4}}+0.15 < 1.5$.\n\\end{IEEEproof}\n\n\\begin{IEEEproof}[Proof of Proposition~\\ref{prop:3}]\nWe follow a similar approach to Gribonval et al.~\\cite{gribonval2014sparse}. We show that the conditions in~\\eqref{eq:lem8_cod} hold for Lemma~\\ref{lem:a_hat_min_cond}. We have\n\t\\begin{align}\n\t& \\norm{\\mb{y} - \\lrp{ \\bigotimes \\mb{D}_k} \\mb{x} }_2 \\nonumber\\\\\n\t&\\leq \\norm{ \\lrp{ \\bigotimes \\mb{D}^0_{k,\\mc{J}_k} - \\bigotimes \\mb{D}_{k,\\mc{J}_k} }\n\t\t\\mb{x}_\\mc{J}} _2+\\|\\mb{w}\\|_2 \\nonumber \\\\\n\t&\\leq M_x\n\t\t\\sum _{k \\in [K]} \\big\\| \\wt{\\D}_{1,\\mc{J}_1} \\otimes \\dots \\otimes\n\t\t\\lrp{\\mb{D}^0_{k,\\mc{J}_k} - \\mb{D}_{k,\\mc{J}_k} }\t\\otimes \\dots \\otimes \\nonumber\\\\\n\t&\\qquad  \\wt{\\D}_{K,\\mc{J}_K}\\big\\|_2  +M_w \\nonumber \\\\\n\t&\\leq M_x\n\t\t\\sum _{k \\in [K]} \\norm{ \\wt{\\D}_{1,\\mc{J}_1}}_2 \\dots\n\t\t\\norm{\\mb{D}^0_{k,\\mc{J}_k} - \\mb{D}_{k,\\mc{J}_k} }_2 \\dots \\norm{ \\wt{\\D}_{K,\\mc{J}_K}}_2  \\nonumber\\\\\n\t&\\qquad +M_w\n\t\t\\nonumber \\\\\n\t&\\leq M_x\n\t\t\\sum _{k \\in [K]}\n\t\t\\bigg( \\prod_{\\substack{i \\in [K] \\\\ i \\neq k}}\n\t\t\\norm{ \\wt{\\D}_{i,\\mc{J}_i }}_2 \\bigg) \\varepsilon_k\n\t\t +M_w \\nonumber \\\\\n\t&\\numrel{\\leq}{r_delt_leq} (1.5)^{(K-1)\/2} M_x  \\sum_{k \\in [K]} \\varepsilon_k  + M_w,\n\t\\end{align}\nwhere \\eqref{r_delt_leq} follows from \\eqref{eq:delt_cond} and the fact that for  $\\wt{\\D}_i = \\mb{D}_i^0$, $\\norm{\\mb{D}_{i,\\mc{J}_i}^0}_2 \\leq \\sqrt{1 + \\delta_{s_i}(\\mb{D}^0_i)}  \\leq \\sqrt{\\frac{5}{4}} < 1.5$ and for $\\wt{\\D}_i = \\mb{D}_i$, according to \\eqref{eq:A0_A}, we have $\\norm{\\mb{D}_{i,\\mc{J}_i}}_2 \\leq \\norm{\\mb{D}_{i,\\mc{J}_i}^0}_2 + \\varepsilon_i \\leq \\sqrt{\\frac{5}{4}}+0.15 < 1.5$.\nHence, we get\n\t\\begin{align} \\label{eq:lmb_m_err}\n\t&\\lambda (1-2\\mu_s(\\mb{D}))- \\norm{\\mb{y} - \\lrp{ \\bigotimes \\mb{D}_k} \\mb{x}}_2 \\nonumber \\\\\n\t&\\geq \\lambda (1-2\\mu_s(\\mb{D})) -  (1.5)^{(K-1)\/2} M_x  \\sum_{k \\in [K]} \\varepsilon_k  -M_w\n\t\t\\nonumber \\\\\n\t&\\numrel{\\geq}{r_mu_mu0} \\lambda (1-2\\mu_s(\\mb{D}^0))\n\t\t- (1.5)^{K\/2} \\lrp{ 4\\lambda \\sqrt{s} + (1.5)^{-1\/2}M_x} \\nonumber\\\\\n\t&\\qquad \\sum_{k \\in [K]} \\varepsilon_k\n\t\t-M_w \\nonumber \\\\\n\t&\\numrel{\\geq}{r_lam_m_a}  \\lambda (1-2\\mu_s(\\mb{D}^0)) - 3(1.5)^{K\/2} M_x \\sum_{k \\in [K]} \\varepsilon_k -M_w\n\t\\nonumber \\\\\n\t&=  3(1.5)^{K\/2}  M_x \\bigg( K \\bar{\\lambda} C_{\\max} -  \\sum_{k \\in [K]} \\varepsilon_k  \\bigg) -M_w,\n\t\\end{align}\nwhere \\eqref{r_mu_mu0} follows from \\eqref{eq:mu_mu0} and \\eqref{r_lam_m_a} follows from \\eqref{eq:lem8_cod} ($2\\lambda \\sqrt{s} \\leq x_{\\min}\\sqrt{s}\\leq M_x$) and~\\eqref{eq:mu_mu0}.\nIf $\\varepsilon_k < C_{\\max} \\bar{\\lambda} $, $k \\in [K]$, the assumption on the noise level in \\eqref{eq:M_eps_M_al} implies that the right-hand side of \\eqref{eq:lmb_m_err} is greater than zero and $\\lambda (1-2\\mu_s(\\mb{D}))>  \\norm{\\mb{y} - \\lrp{ \\bigotimes \\mb{D}_k} \\mb{x}}_2 $. Thus, according to Lemma~\\ref{lem:a_hat_min_cond}, $\\widehat{\\mb{x}}$ is almost surely the unique solution of $\\min_{\\mb{x} } \\frac{1}{2}\\norm{\\mb{y} - \\lrp{ \\bigotimes \\mb{D}_k} \\mb{x}' }_2  +\\lambda\\|\\mb{x}'\\|_1$ and $\\Delta\\phi_{\\mbb{P}}\\lrp{\\D_{1:K},\\D^0_{1:K}|\\boldsymbol{\\sigma}} = \\Delta f_{\\mbb{P}} \\lrp{\\D_{1:K},\\D^0_{1:K}}$.\n\\end{IEEEproof}\n\n\n\n\\section*{appendix B}\n\n\n\\begin{IEEEproof} [Proof of Lemma~\\ref{lemma:delt_m_T_dev}]\nAccording to Lemma~\\ref{lem:rad}, we have to upper bound\n$\n\t\\mbb{E} \\lr{ \\sup_{\\mb{D}_k \\in \\overline{\\mathcal{B}}_{\\varepsilon_k}(\\mb{D}^0_k), k \\in [K]}\\lra{ \\frac{1}{N}\n\\sum_{n \\in [N] } \\beta_n h_n(\\D_{1:K}) }}\n$. Conditioned on the draw of functions $h_1,\\dots,h_N$, consider the Gaussian processes\n$\n\tA_{\\D_{1:K}} = \\frac{1}{N} \\sum_{n \\in [N]} \\beta_n h_n(\\D_{1:K})\n$ and\n$\n\tC_{\\D_{1:K}} = \\sqrt{\\frac{K}{N}}\n\t\t\\sum_{k \\in [K]} \\bigg( L_k \\sum_{i \\in [m_k]} \\sum_{j \\in [p_k]}\n\t\t\\zeta_{ij}^k  (\\mb{D}_k-\\mb{D}_k^0)_{ij} \\bigg)\n$,\nwhere $\\lr{\\beta_n}_{n=1}^N$'s and $\\lr{\\zeta_{ij}^k},k\\in [K],i\\in [m_k],j\\in [p_k]$'s are independent standard Gaussian vectors.\nWe have\n\t\\begin{align}\n\t&\\mbb{E} \\lr{\\lra{A_{\\D_{1:K}} - A_{\\D'_{1:K}}}^2} \\nonumber\\\\\n\t&\\qquad= \\frac{1}{N^2} \\bigg|\n\t\t\\sum_{n \\in [N]} h_n(\\D_{1:K})- h_n(\\D'_{1:K}) \\bigg|^2 \\nonumber \\\\\n\t&\\qquad \\numrel{\\leq}{r_lpsh_h} \\frac{1}{N}\n\t\t\\bigg(\\sum_{k \\in [K]} L_k\\|\\D_k-\\D'_k\\|_F \\bigg)^2\\nonumber \\\\\n\t&\\qquad \\numrel{\\leq}{r_cs} \\frac{K}{N}\\sum_{k \\in [K]} L_k^2\\|\\D_k-\\D'_k\\|_F^2\\nonumber \\\\\n\t&\\qquad = \\mbb{E}\\lr{\\lra{C_{\\D_{1:K}} - C_{\\D'_{1:K}}}^2},\n\t\\end{align}\nwhere \\eqref{r_lpsh_h} follows from coordinate-wise Lipschitz continuity of $h$ and \\eqref{r_cs} follows from Cauchy-Schwartz inequality. Hence, using Slepian's Lemma~\\cite{massart2007concentration}, we get\n\t\\begin{align}\n\t\\mbb{E} \\bigg\\{ \\sup_{\\substack{\\mb{D}_k \\in \\overline{\\mathcal{B}}_{\\varepsilon_k}(\\mb{D}^0_k)  \\\\ k \\in [K]}}\n\t\tA_{\\D_{1:K}} \\bigg\\}\n\t&\\leq \\mbb{E} \\bigg\\{ \\sup_{\\substack{ \\mb{D}_k \\in \\overline{\\mathcal{B}}_{\\varepsilon_k}(\\mb{D}^0_k)  \\\\ k \\in [K]}}\n\t\tC_{\\D_{1:K}} \\bigg\\} \\nonumber \\\\\n\t& = \\sqrt{\\frac{K}{N}} \\bigg(\\sum_{k \\in [K]}  L_k\\varepsilon_k \\mbb{E} \\big\\{\\|\\boldsymbol{\\zeta}^k\\|_F \\big\\}\\bigg)\\nonumber \\\\\n\t& =\\sqrt{\\frac{K}{N}} \\bigg(\\sum_{k \\in [K]} L_k\\varepsilon_k \\sqrt{m_kp_k}\\bigg).\n\t\\end{align}\nThus, we obtain\n$\n\t\\mbb{E} \\lr{\\sup_{\\substack{ \\mb{D}_k \\in \\overline{\\mathcal{B}}_{\\varepsilon_k}(\\mb{D}^0_k)  \\\\ k \\in [K]}}\n\t\t\\lra{\\frac{1}{N} \\sum_{n \\in [N]} \\beta_nh_n(\\D_{1:K}) } }\n\t\\\\ \\leq 2\\sqrt{\\frac{K}{N}}\\lrp{\\sum_{k \\in [K]} L_k\\varepsilon_k \\sqrt{m_kp_k}}.\n$\n\\end{IEEEproof}\t\n\n\n\n\\begin{IEEEproof}[Proof of Lemma~\\ref{lem:phi_m_T1_lip}]\nWe expand $\\Delta \\phi_\\mb{y} \\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}}$ according to \\eqref{eq:delt_t} and bound each term of the sum separately. Looking at the first term, we get\n\t\\begin{align}\n\t& \\lra{ \\Delta \\phi_1 \\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}} }\n\t\t\\numrel{=}{r_exp_phi1} \\bigg| \\frac{1}{2}\\mb{x}^\\top\n\t\t{\\mb{D}^0 }^\\top\n\t\t\\bigg( \\sum_{k \\in [K]} \\mb{P}_{\\widetilde{\\mb{D}}_{1,\\mc{J}_1}}\n\t\t\\otimes \\dots \\otimes  \\nonumber\\\\\n\t&\\qquad\n\t\t\\lrp{\\bP_{\\D'_{k,\\cJ_k}} - \\bP_{\\D_{k,\\cJ_k}}}\t\\otimes \\dots \\otimes \\mb{P}_{\\widetilde{\\mb{D}}_{K,\\mc{J}_K}}\\bigg)\n\t\t\\mb{D}^0\n\t\t\\mb{x} \\bigg| \\nonumber \\\\\n\t&\\numrel{\\leq}{r_D0_D0k} \\frac{1}{2} \\norm{\\mb{x}}_2^2\n\t\t\\bigg(\\prod_{k \\in [K]} \\norm{ \\mb{D}^0_{k,\\mc{J}_k} }_2^2 \\bigg)\n\t\t\\bigg( \\sum_{k \\in [K]}\n\t\t\\norm{\\bP_{\\D^0_{k,\\cJ_k}} - \\bP_{\\D_{k,\\cJ_k}}}_2 \\nonumber\\\\\n\t&\\qquad \\bigg( \\prod_{\\substack{i\\in [K] \\\\ i \\neq k}}\n\t\t\\norm{\\mb{P}_{\\wt{\\D}_{i,\\mc{J}_i}}}_2  \\bigg)\\bigg)\n\t\t\\nonumber \\\\\n\t&\\numrel{\\leq}{r_p_rip1} M_x^2\n\t\t\\bigg( \\prod_{k \\in [K]}\\big(1+\\delta_{s_k}(\\mb{D}^0_k)\\big) \\bigg)\n\t\t\\nonumber\\\\\n\t&\\qquad \\bigg( \\sum_{k \\in [K]}(1 - \\delta_k)^{-1\/2}\\|\\D_k-\\D^0_k\\|_F \\bigg) ,\n\t\\end{align}\nwhere \\eqref{r_exp_phi1} follows from \\eqref{eq:delt_phi}, \\eqref{r_D0_D0k} follows from the fact that $\\norm{\\mb{D}^0_\\mc{J}}_2 = \\prod_{k \\in [K]} \\norm{ \\mb{D}^0_{k,\\mc{J}_k} }_2$, and \\eqref{r_p_rip1} follows from  the definition of $\\mathsf{RIP}$, equation \\eqref{eq:PH_PHp}, and $\\big\\|\\mb{P}_{\\wt{\\D}_{i,\\mc{J}_i}}\\big\\|_2=1$. Following a similar approach and expanding the rest of the terms, we get\n\t\\begin{align*}\n\t&\\lra{ \\Delta \\phi_2 \\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}} } \\nonumber\\\\\n\t&\\leq \\norm{\\mb{w}}_2 \\norm{\\mb{x}}_2\n\t\t\\bigg( \\prod_{k \\in [K]} \\norm{ \\mb{D}^0_{k,\\mc{J}_k} }_2^2 \\bigg)\n\t\t\\nonumber\\\\\n\t&\\qquad \\bigg( \\sum_{k \\in [K]} \\norm{\\bP_{\\D^0_{k,\\cJ_k}} - \\bP_{\\D_{k,\\cJ_k}} }_2\n\t\\bigg( \\prod_{\\substack{i\\in [K] \\\\ i \\neq k}} \\norm{\\mb{P}_{\\wt{\\D}_{i,\\mc{J}_i}}}_2  \\bigg)\\bigg)\n\t\\nonumber \\\\\n\t&\\numrel{\\leq}{r_p_rip} 2M_w M_x\n\t\t\\bigg( \\prod_{k \\in [K]}\\big(1+\\delta_{s_k}(\\mb{D}^0_k)\\big)^{1\/2} \\bigg)\n\t\t\\nonumber\\\\\n\t&\\qquad \\bigg( \\sum_{k \\in [K]} (1 - \\delta_k)^{-1\/2}\\|\\D_k-\\D^0_k\\|_F \\bigg), \\nonumber \\\\\n\t&\\lra{ \\Delta \\phi_3 \\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}} }\n\t\t\\leq  \\frac{1}{2} \\norm{\\mb{w}}_2^2 \\nonumber\\\\\n\t& \\qquad \\bigg( \\sum_{k \\in [K]}\n\t\t\\norm{\\bP_{\\D^0_{k,\\cJ_k}} - \\bP_{\\D_{k,\\cJ_k}}}_2\t\n\t\t\\bigg( \\prod_{\\substack{i\\in [K] \\\\ i \\neq k}} \\norm{\\mb{P}_{\\wt{\\D}_{i,\\mc{J}_i}}}_2  \\bigg)\n\t\t\\bigg) \\nonumber \\\\\n\t&\\leq M_w^2 \\bigg( \\sum_{k \\in [K]} (1 - \\delta_k)^{-1\/2}\\|\\D_k-\\D^0_k\\|_F \\bigg),\\nonumber \\\\\n\t&\\lra{ \\Delta \\phi_4 \\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}} }\n\t\t= \\lambda \\norm{\\boldsymbol{\\sigma}_\\mc{J}}_2 \\norm{\\mb{x}}_2\n\t\t\\bigg( \\prod_{k \\in [K]} \\norm{\\mb{D}^0_{\\mc{J}_k}}_2 \\bigg)\n\t\t\\nonumber\\\\\n\t&\\qquad \\bigg( \\sum_{k \\in [K]}\n\t\t\\norm{{\\D^0}_{k,\\cJ_k}^+ - \\D_{k,\\cJ_k}^+}_2\n\t\t\\bigg( \\prod_{\\substack{i\\in [K] \\\\ i \\neq k}}  \\norm{ {\\wt{\\D}_{i,\\mc{J}_i}}^+}_2 \\bigg)\n\t\t \\bigg) \\nonumber \\\\\n\t&\\numrel{\\leq}{r_ps} 2\\lambda \\sqrt{s} M_x\n\t\t \\bigg( \\prod_{k \\in [K]}\\big(1+\\delta_{s_k}(\\mb{D}^0_k)\\big)^{1\/2} \\bigg) \\nonumber\\\\\n\t&\\qquad \\bigg( \\sum_{k \\in [K]} (1-\\delta_k)^{-1}\n\t\t \\bigg( \\prod_{\\substack{i \\in [K] \\\\ i \\neq k}}\n\t\t (1-\\delta_i)^{-1\/2} \\bigg) \\|\\D_k-\\D^0_k\\|_F \\bigg) ,\\nonumber \\\\\n\t& \\lra{ \\Delta \\phi_5 \\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}} }\n\t= \\lambda \\norm{\\boldsymbol{\\sigma}_\\mc{J}}_2 \\norm{\\mb{w}}_2 \\nonumber\\\\\n\t&\\qquad \\bigg( \\sum_{k \\in [K]}\n\t\t\\norm{{\\D^0}_{k,\\cJ_k}^+ - \\D_{k,\\cJ_k}^+ }_2\t\n\t\t\\bigg( \\prod_{\\substack{i\\in [K] \\\\ i \\neq k}}  \\norm{ {\\wt{\\D}_{i,\\mc{J}_i}}^+}_2 \\bigg)\n\t\t\\bigg)\t\\nonumber \\\\\n\t&\\leq 2\\lambda \\sqrt{s}M_w \\nonumber\\\\\n\t&\\qquad \\bigg( \\sum_{k \\in [K]} (1-\\delta_k)^{-1}\n\t\t \\bigg( \\prod_{\\substack{i \\in [K] \\\\ i \\neq k}}\n\t\t (1-\\delta_i)^{-1\/2} \\bigg)  \\|\\D_k-\\D^0_k\\|_F \\bigg) ,\\nonumber \\\\\n\t&\\lra{ \\Delta \\phi_6 \\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}} }\n\t\t= \\frac{\\lambda^2}{2} \\norm{\\boldsymbol{\\sigma}_\\mc{J}}_2^2  \\nonumber \\\\\n\t&\\qquad\n\t \t\\bigg( \\sum_{k \\in [K]}\n\t\t\\norm{\\bH_{\\D^0_{k,\\cJ_k}} - \\bH_{\\D_{k,\\cJ_k}} }_2\t\n\t\t\\bigg( \\prod_{\\substack{i\\in [K] \\\\ i \\neq k}}  \\norm{\\mb{H}_{\\wt{\\D}_{i,\\mc{J}_i}} }_2 \\bigg)\\bigg)\t\n\t\t\\nonumber \\\\\n\t&\\numrel{\\leq}{r_h} \\lambda^2 s\n\t\t\\bigg( \\sum_{k \\in [K]} (1-\\delta_k)^{-\\frac{3}{2}}\n\t\t \\bigg( \\prod_{\\substack{i \\in [K] \\\\ i \\neq k}}\n\t\t (1-\\delta_i)^{-1} \\bigg) \\|\\D_k-\\D^0_k\\|_F\\bigg),\n\t\\end{align*}\nwhere \\eqref{r_ps} and \\eqref{r_h} follow from \\eqref{eq:pso_cond} and \\eqref{eq:PH_PHp}. Adding all the terms together, we get\n\t\\begin{align}\n\t&\\lra{\\Delta \\phi_\\mb{y}\\lrp{\\D_{1:K};\\D^0_{1:K}|\\boldsymbol{\\sigma}}}  \\leq\n\t\t\\sum_{k \\in [K]} L_k \\|\\D_k-\\D^0_k\\|_F.\n\t\\end{align}\nwhere $L_k$ is defined in \\eqref{eq:lipsch_const_h}.\n\\end{IEEEproof}\t\n\\section*{appendix C}\n\\begin{IEEEproof}[Proof of the coherence relation for KS dictionaries]\nTo prove \\eqref{eq:mu_s}, we define the set $\\mathcal{A} = \\lr{\\forall j_k \\in \\mc{J}_k, (j_1,\\dots,j_K) \\not\\in (\\mc{J}_1,\\dots,\\mc{J}_K)}$. We have\n\t\\begin{align}\n\t\\mu_s(\\mb{D})&= \\max_{|\\mc{J}|\\leq s} \\max_{j \\not\\in \\mc{J}}\\|\\mb{D}_\\mc{J}^\\top \\mb{d}_j\\|_1 \\nonumber \\\\\n\t&= \\max_{\\substack{|\\mc{J}_k|\\leq s_k\\\\ k \\in [K]}}\n\t\t\\max_{\\mathcal{A}}\n\t\t\\norm{\\lrp{\\bigotimes \\mb{D}_{k,\\mc{J}_k}^\\top}\\lrp{\\bigotimes \\mb{d}_{k,j_k}}}_1\\nonumber \\\\\n\t&= \\max_{\\substack{|\\mc{J}_k|\\leq s_k\\\\ k \\in [K]}}\n\t\t\\max_{\\mathcal{A}}\n\t\t\\norm{\\bigotimes \\mb{D}_{k,\\mc{J}_k}^\\top\\mb{d}_{k,j_k}}_1 \\nonumber\\\\\n\t&=  \\max_{\\substack{|\\mc{J}_k|\\leq s_k\\\\ k \\in [K]}}\n\t\t\\max_{\\mathcal{A}}\n\t\t\\prod_{k \\in [K]} \\norm{\\mb{D}_{k,\\mc{J}_k}^\\top\\mb{d}_{k,j_k}}_1  \\nonumber\\\\\n\t&\\leq \\max_{k \\in [K]} \\mu_{s_k}(\\mb{D}_k)\n\t\t\\bigg( \\prod_{\\substack{i \\in [K], \\\\ i \\neq k}}  \\lrp{ 1+\\mu_{s_i-1}(\\mb{D}_i)} \\bigg).\n\t\\end{align}\t\n\n\\end{IEEEproof}\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSolar sytem progenitors, from the deeply embedded sources to\nthe weak line T Tauri Stars (WTTSs), are sources of high\nenergy radiation (X-ray to UV); the total energy\nradiated in this range goes from $\\sim 0.02$~L$_{\\odot}$\nmeasured in the very young\nsources through their X-ray radiation to the 0.2 L$_{\\odot}$ radiated in the\nUV during the T Tauri phase (or Phase~TT) (Preibish 2004, G\\'omez de Castro 2008).\nLater on, in the Weak line T Tauri Phase the energy released drop to\n$\\sim 10^{-3}$~L$_{\\odot}$ radiated in X-ray. Being TTSs\nintrinsically cool stars (with $\\log T_{\\rm eff}\\sim$ 6500-3600~K),\nsurrounded by cool accretion disks radiating at infrared\nwavelengths, the source\nof this energy must be searched in the release of magnetic energy.\n\nIt is well known that the mediation of magnetic fields in the\naccretion process is able to heat up the plasmas since a fraction of the\ngravitational energy lost during accretion is invested in field\namplification and dynamo action thus, radiative loses are pushed\ntowards the high energy range. Unfortunately, in the early phases\n(ages $< 0.1$Myr) most of this radiation is reabsorbed by the\ndense circumstellar environment (Av$>3$) and only the hard X-ray\nradiation is able to escape from the system providing direct\ninformation on the evolution of the accretion process. After 1\nMyr, extinction drops enough to make the engine accessible to UV\nwavelengths; current technologies allow to carry out high\nresolution spectroscopy in the UV range which is also extremely\nrich in spectral tracers so a single echelle spectra can provide\ninformation on molecular, atomic and ionized gas (from singly\nionized gas to very high ionization species such as  Fe~XII,\nFe~XVIII or  Fe~XXI). Thus, UV spectroscopy is an extremely\nefficient tool to study solar system progenitors from 1 Myr on with the\ncurrent technology and these application will be discussed in detail\nbelow.  However, one can foresee a future when\nmicroarcseconds UV imaging  will be available and\nstudies similar to those being run on the Sun, will be feasible\nfrom 1 Myr old Suns all the way down into the main sequence while\nthe young planetary disk settles down and life begins to grow.\nThis review deals with this; with a description of our\ncurrent understanding of the evolution from the T Tauri phase to\nthe modern Sun and with a non-technologically biased ambitious\nview of what we could learn from challenging new UV observatories.\nThis review has been written after the end of the 1st. conference\nof the network for UV astronomy held in El Escorial in May 2007\nwhere some challenging projects for new space observatories were\npresented; you should find references to some of them in this text.\n\n\\section{Physics to be understood I: the gravito-magnetic engine}\n\nDuring the phase TT, stars count on an energy source\nwhich is not available during the main sequence evolution: gravitational\nenergy from the infalling material. This extra energy is released either\nthrough shocks on the stellar surface or through the gravito-magnetic\ninteraction between the star and the disk.\n\nShocks release the kinetic energy of the infalling material into\nheating at the impact point. If matter infall occurs along the\nfield lines, all the gravitational energy is damped into heating\nand the gas may reach temperatures as high as $\\sim 10^6$K. The\ndominant output radiation is produced by the photoionized preshock\ninfalling gas radiating mainly in the UV range (G\\'omez de Castro\n\\& Lamzin 1999; Gullbring et al. 2000). As the density of the\ninfalling gas column is high ($n_e \\simeq 10^9-10^{12}$~cm$^{-3}$)\nthe thickness of the radiating column is expected to be negligible\ncompared with the stellar radius thus, accretion shocks are\nobserved as {\\it hot spots} on the stellar surface. As such, they\nare expected to produce a rotationally modulated signal that has\nbeen detected in monitoring campaigns of some stars both in optical \n(see i.e. Petrov et al. 2001;\nBouvier et al. 2003) and UV (G\\'omez de Castro \\& Fern\\'andez\n1996; G\\'omez de Castro \\& Franqueira 1997a). An important result \nof these campaigns is that only $\\sim\n50$\\% of the UV continuum excess is rotationally modulated. Thus,\na significant fraction of the UV excess is not produced by the\naccretion shocks even in sources where rotational modulation\nhas been detected. However, this excess decreases as the star\napproaches the main sequence (see Fig.~1).\n\n\\begin{figure}[]\n\\centerline{\\includegraphics[width=18pc]{gomezdecastro_fig1.ps}}\n\\caption{The (UV-V, V) colour -- magnitude diagram for the\nT Tauri stars observed with the IUE satellite in the Taurus region.\nThe crosses represent cool TTSs (spectral types later than\n$\\sim $ K3) and the open circles warm TTSs (spectral types\nearlier than $\\sim$ K3). The location of the main sequence is\nmarked by the spectral types. The stars closer to the main sequence\nare the WTTSs (from G\\'omez de Castro 1997).}\n\\end{figure}\n\n\nIn fact, the major source of high energy radiation is the dissipation\nof the magnetic and mechanical energy produced by the gravito-magnetic\nengine. A simple analogy can be made with a self-regulated hydraulic\nturbine: the potential energy of the gas falling from the disk into the\nstellar magnetic field drives to the generation of electric currents\ndue to the Lorentz force that, in turn, create new field components\nthrough dynamo action. There are however, a great number of uncertainties\nin the way the system self-regulates and also on the dependence of the\nengine details on initial conditions such as the effective gravity of\nthe star, the role of stellar radiation and magnetic field on the engine\nperformance and the role of the ionizing radiation produced by the engine on\nthe evolution of the mass storage, the disk.\n\nThe Sun, itself, provides important clues to understand the\nphysics of the gravito-magnetic engine and its evolution. At the\nbase of the Sun convective layer, the tachocline marks the\nlocation of the shear layer between the rigid body rotation of the\nradiative core and the differentially rotating convective\nenvelope. The tachocline is significantly prolate; it is centered\nat 0.69R$_{\\odot}$ at the equator and 0.72R$_{\\odot}$ at latitude\n60$^o$ (Basu \\& Antia, 2003). The tachocline thickness is $\\sim\n0.04$R$_{\\odot}$. The angular velocity profile based on\nhelioseismic inversions shows that at a latitude of about 35$^o$,\nthe radial gradient changes sign becoming negative for latitudes\n$> 35^o$. This latitude marks the limit of the two latitude belts\nwhere the overwhelming majority of sunspots occur. There are also\nsome indications of the meridional flow moving equatorwards below\nthis latitude and polewards above it (see Miesch 2005). The Solar\nwind is (magnetic)\nlatitude dependent during solar minimum;\nabove $\\sim 35^o$ is fast (1000~km\/s) and\nthin, below is slower (300~km\/s) and denser (see Fig.2 from\nUlysses data).  The current paradigm\nfor how solar dynamo operates includes: (1) field amplication in a\nturbulent downflow ($\\alpha$ effect) that is pumped downward by\nconvection and accumulate in the overshoot region and the\ntachocline; (2) field amplification and organization into toroidal\nflux tubes and sheets by differential rotation in the tachocline;\n(3) magnetic instabilities (buoyancy) drives the field to the surface and\n(4) the Coriolis force acting on the rising structures depends on the\nlatitude producing a latitude-dependent emergence of bipolar magnetic\nstructures.\n\n\n\\begin{figure}[]\n\\centerline{\\includegraphics[width=18pc]{gomezdecastro_fig2.eps}}\n\\caption{{\\it ``Solar wind observations collected by the Ulysses spacecraft\nduring two separate polar orbits of the Sun, six years apart,\nat nearly opposite times in the solar cycle. Near solar minimum\n(left) activity is focused at low altitudes, high-speed solar wind\nprevails, and magnetic fields are dipolar. Near solar maximum (right),\nthe solar winds are slower and more chaotic, with fluctuating magnetic\nfields.''} (From NASA Solar Probe Web (solarprobe.gsfc.nasa.gov),\ncourtesy of Southwest Research Institute and the Ulysses\/SWOOPS team)\n}\n\\end{figure}\n\n\nGoing backwards in time, during the phase TT,\nthe problem becomes complicated by the\npresence an additional ``tachocline'' or differentially rotating\nregion attached to the convective layer. This {\\it external\ntachocline} connects the star with the accretion disk which\nrotates significantly faster than the stellar surface; rotation\nperiods during the TT phase are about 7-8 days ($\\Omega _* =\n0.8-0.9$ day$^{-1}$) while the Keplerian frequency is:\n$$\n\\Omega _k = 11.1  {\\rm day}^{-1} \\lgroup \\frac {M}{M_{odot}} \\rgroup ^{1\/2}\n\\lgroup \\frac {r}{3 R_{odot}} \\rgroup ^{-3}\n$$\nKeplerian disk corotation radius is at,\n$$\nr_{\\rm co} = (7.2 - 6.9)R_{\\odot}  \\lgroup \\frac {M}{M_{odot}} \\rgroup ^{1\/2}\n$$\nTo avoid this large shear, the magnetosphere will grow to balance the\ntoroidal component of the flux with the angular momentum of the infalling\nmatter (Ghosh \\& Lamb 1979) thus,\n$$\n\\frac{B_p B_t}{4 \\pi} 4\\pi r^2 \\Delta r \\simeq \\dot M r V_k\n$$\nwhere $B_p$ and $B_t$ are the poloidal and toroidal components of the\nfield respectively, $r$ is the magnetosphere radius, $\\Delta r$ is\nthe thickness of the shear layer, $\\dot M$ is the accretion rate and\n$V_k$ is the Keplerian velocity at the magnosphere radius. For typical\nT~Tauri stars parameters:\n$$\nr_{\\rm mag}= 4.4 R_{\\odot} \\gamma ^{2\/7} \\lgroup \\frac {B_*}{1kG}\n\\rgroup^{4\/7} \\lgroup \\frac {\\dot M}{10^{-8} M_{\\odot} {\\rm\nyr}^{-1}} \\rgroup^{-2\/7} \\lgroup \\frac {M_*}{M_{\\odot}}\n\\rgroup^{-1\/7}\n$$\nwhere $\\gamma ^{2\/7}$ is a factor about unity ($\\gamma =\n(B_t\/B_p)(\\Delta r \/r)$, see Lamb, 1989). Notice that the main\nuncertainties in the physics, namely the ratio between the\ntoroidal and the poloidal components and the relative thickness of\nthe {\\it ``external tachocline''} are enclosed in this factor.\nAs in the Solar\ninterior, the shear region is fed by turbulent, magnetized\nmaterial though this comes from the accretion disk instead of\nthe convective layer. The turbulent disk dynamo is fed by  the\nmagneto-rotational instability in the acretion disk. Shear\namplifies the field producing a strong toroidal component; an\nexternal dynamo sets in. This toroidal field and\nthe associated magnetic pressure push the field lines outwards\nfrom the disk rotation axis, inflating and opening  them in a {\\it\nbutterfly-like pattern} reminiscent of the helmet streamers in\nthe solar corona, so producing a current layer between the\nstellar and the disk dominated regions as displayed in Fig~3.\nMagnetic field dissipation in the current layer produces high\nenergy radiation and particles. The magnetic link between the star\nand the disk is broken and reestablished continuously by magnetic\nreconnection. The opening angle of the current layer, as well as\nits extent, depends on the stellar and disk fields, the accretion\nrate and the ratio between the inner disk radius and the stellar\nrotation frequencies. Hot, pressure driven outflows are produced\nfrom the region closer to the rotation axis while cool\ncentrifugally driven flows are produced by the disk; plasmoids are\nejected from the current layer generating a third outflowing\ncomponent.\n\n\\begin{figure}[]\n\\centerline{\\includegraphics[width=18pc]{gomezdecastro_fig3.eps}}\n\\caption{The interaction between the stellar magnetic field and the\ndisk twists the stellar field lines due to the differential rotation.\nThe toroidal magnetic field generated out of the poloidal flux and\nthe associated pressure tends to push the field lines outwards,\ninflating them, and eventually braking the magnetic link between\nthe star and the disk (boundary between regions I and II).\nThree basic regions can be defined:\nRegion I dominated by the stellar wind, Region II dominated by\nthe disk wind and Region III dominated by stellar magnetospheric\nphenomena. The dashed line traces the boundaries between this\nthree regions. The continuous lines indicate the topology\nof the field and the shadowed areas represent regions where\nmagnetic reconnection events are likely to occur, producing\nhigh energy radiation and particles (from G\\'omez de Castro 2004).}\n\\end{figure}\n\nDisk-star interaction has been investigated by  means of numerical\nsimulations since the early works by Goodson et al (1997) till the\nlast results ({\\it i.e.} von Rekowski \\& Brandenburg 2006). They\nshow that the fundamental  mechanism for disk winds formation is\nrobust; numerical simulations with different parameters (disk\/star\nfields) and initial conditions produce disk winds. Stellar winds\nare much more sensitive to the physical conditions and specially\nto the stellar field; compare the results of simulations assuming\nthat the stellar field is a magnetic dipole (von Rekowski \\&\nBrandenburg 2004) with those of simulations where the stellar\nfield is prescribed through the action of the stellar dynamo (von\nRekowski \\& Brandenburg, 2006). In fact, the characteristics of\nthe accretion flow  and the winds (dominant driver, temperature,\nterminal velocity, density, variability) depend on the physical\nproperties of the system such as the degree of magnetization of\nthe disk, the characteristics of the disk dynamo and  the stellar\nfield.\n\nThe bulk of the energy produced in this engine is released at UV\nand X-ray wavelenghts as in the Sun atmosphere. In the very early\nepochs, when extinction is high ($A_V \\geq 3$), only the X-ray\nradiation from the engine is detected. Later on, about 1 Myr,\nextinction drops and the engine can be studied in the UV. Only in\nthe UV, the various components of the engine can be defined and\nstudied as well as their evolution, starting during  the phase~TT\nall the way down into the main sequence.\n\n\\section{Physics to be understood II: the\nimpact of the engine on disk evolution}\n\nThough often neglected, the impact of the engine in the inner disk\nevolution is enormous. On the one hand, the engine adds a significant\npoloidal component in the inner disk thus favouring gas motions\nperpendicular to the disk as shown by the numerical simulations,\non the other hand, the engine is a source of highly energetic\nradiation where part of the dissipation is produced at heights of\nsome few stellar radii above the disk in the inflating current\nlayer; this high latitude illumination favours energy absorption\nby the disk. Both together act to increase the disk scale height and\nthe absorption of the radiation henceforth producing the\nrarification of the disk\natmosphere and favoring disk evaporation close to the star.\nTo achieve the evaporation of a standard optically thin accretion disk,\nthe sound speed should be comparable to the keplerian velocity thus,\n$$\nT = \\frac{G M_*\/r_{\\rm mag}}{\\mu m_H \/ \\gamma \\kappa}\n= 3.13 \\times 10^7 K \\lgroup \\frac {M_*}{M_{\\odot}}\n\\rgroup \\lgroup \\frac {r_{\\rm mag}}{4.4 R_{\\odot}}\n\\rgroup ^{-1}\n$$\nwhere $\\mu$ is the mean molecular weight, $\\gamma $ is the\npolytropic index and $\\kappa $ is Boltzmann constant.\nHowever, this value relaxes in the presence of a poloidal\nfield as the expected in the disk-magnetosphere interface so,\n$$\nT = 3.13 \\times 10^7 K \\frac {\\beta}{1+\\beta}\n\\lgroup \\frac {M_*}{M_{\\odot}}\\rgroup\n\\lgroup \\frac {R_{\\rm mag}}{4.4 R_{\\odot}}\n\\rgroup ^{-1}\n$$\nwhere $\\beta$ is the rate between magnetic and thermal pressures.\nThus, for highly magnetized environments, $T$ may drop\narbitrarily. For thin accretion disks\\footnote{See Frank et al 2002\nfor the standard prescription of the thin disk density and temperature\nas a function the accretion rate, radius and stellar mass.} penetrated\nby the stellar dipolar field, $B_*$,\n\n\\begin{eqnarray}\n\\beta &=&  \\frac {\\gamma \\kappa T\/ \\mu M_H}{B^2 \/4 \\pi \\rho} \\\\\n&=&\n4.77  \\lgroup \\frac {M_*}{M_{\\odot}}\\rgroup ^{7\/8}\n\\lgroup \\frac {\\dot M}{10^{-8}M_{\\odot}{\\rm yr}^{-1}}\\rgroup ^{17\/20} \\\\\n&\\times & \\lgroup \\frac {r}{4.4 R_{\\odot}}\\rgroup ^{-21\/8}\n\\lgroup \\frac {B_*}{1kG}\\rgroup ^{-2} \\\\\n\\end{eqnarray}\n\\noindent\nwhere $r$ is the disk radius, $B_*$ the stellar magnetic field and\n$\\dot M$ the accretion rate. Note that\n$\\beta $ drops to 0.02 for accretion rates of $10^{-9}$M$_{\\odot}$yr$^{-1}$.\n\n Another important phenomenon to be considered is that the disk\nis not unlocked from the engine so disk material should be\nsubjected to the propagation of the Alfv\\'en waves, shear waves\nand global alfv\\'en oscillations driven from the interface.\nIn summary, we might expect the inner rim of the disk to be hot\nwith temperatures of about $10^4$K well above the\ntemperature of dust sublimation.\n\nThe role of far-UV radiation fields and high energy particles in\nthe disk chemical equilibrium is now beginning to be understood.\nBergin et al. (2003) showed how strong Ly$\\alpha$ emission may\ncontribute to the observed enhancement of CN\/HCN in the disk. The\npenetration of UV photons coming from the engine in a dusty disk\ncould produce an important change in the chemical composition of\nthe gas allowing the growth of large organic molecules. In this\ncontext, UV photons photodissociating organic molecules at\n$\\lambda > 1500$~\\AA\\ could play a key role in the chemistry of\nthe inner regions of the disk, while those photodissociating H$_2$\nand CO will control the chemistry of the external layers of the\ndisk directly exposed to the radiation from the central engine.\n\n\n\n\n\\section{Lessons learned from UV (spectroscopic) observations}\n\nThe first observations of pre-main sequence (PMS) stars in the UV\nwere carried out with the International Ultraviolet Explorer (IUE)\n(1979-1997). The observations showed that pre-main sequence stars\nhave UV fluxes exceeding those of main sequence stars by a factor\nof about 50. In fact, the UV excess decreases as the stars\napproach the main sequence as shown in Fig~1.\n\nUV radiation provides direct information on the interaction\nbetween the star and the disk. This includes all the various\ncomponents mentioned above: the shear layer, i.e.the {\\it external\ntachocline}, the\nwind, the enhanced magnetospheres, mass ejections from\nreconnecting loops, shocks between the various wind components\n(among themselves and also with the disk material) and as well as\nthe inner regions of the disk. There is a recent review on UV\nobservations of pre-main sequence stars and young planetary disks\n(G\\'omez de Castro et al 2006) where a detailed accounting of the\nwork carried out since the IUE times is summarized. Thus, I should\nconcentrate on the main lessons learned from IUE and HST\nobservations\\footnote{Some observations have also been obtained\nwith FUSE but its small effective area has allowed to observed\nonly the brightest of the TTSs and some Vega-like disks.} that are:\n\n\\subsection{About the accretion flow}\n\nThe actual measurements of infalling gas in the UV are scarce. The\nare hints of accretion in the large extent of the red wings of the\nmain UV resonance lines (Herczeg et al 2005) or through the\ndetection of redshifted absorption components on the profiles\nof the most prominent emission lines.\nHowever, the only target for which there is clear spectroscopic\nevidence of accretion shocks (in the UV) is RY~Tau (see Fig.~4).\nTwo observations of the same star obtained in 1993 and 2001 show that\nthere is a variable redshifted component. From this single\nobservation three important properties are learnt:\n\\begin{enumerate}\n\\item As described above, UV radiation from accretion shocks\nis predicted to be produced in the preshock region on scale\nheights significantly smaller than the stellar radius. Thus, it is\nexpected that only matter falling onto the visible hemisphere can\nbe detected at UV wavelengths. The fact that the variable flux\ncomponent is redwards shifted supports these theoretical\nexpectations. Moreover, the broadening shows that infalling matter\nshould cover a significant fraction of the hemisphere to account\nfor the broad distribution of projected velocities in the\ninfalling gas.\n\n\\item The red wing extends to\nvelocities of 250~km\/s which corresponds to the free-fall velocity\\footnote{\nRY Tau mass is 1.63~M$_{\\odot}$ and radius 2.4~R$_{\\odot}$ according to\nHartigan et al 1995} from 1.7 R$_*$ which is much smaller than\nthe fiducial values derived for the inner disk radius.\n\n\\item The UV excess is not only produced by accretion;\nalso the wind contributes to it. Thus accretion rates derived from\nthe UV excess, assuming that it is caused just by magnetospheric\ninfall, are overestimated.\n\n\\end{enumerate}\n\n\nIt also adds to our understanding of {\\it magnetospheric\naccretion}. Magnetospheric accretion was originally proposed to\nexplain the large broadening of the TTSs H$\\alpha $ lines\n(Muzerolle et al 1998) later detected also in the UV\nresonance lines of CIV, CIII (Ardila et al 2002, Herczeg et al 2005).\nTypical line widths are about\n200-300 km\/s that exceed by far what expected from the rotation\nvelocities of the TTSs ($10-20$ km\/s) even if the corotating\nmagnetosphere is postulated to extend to some 4-5 stellar radii.\nInfall adds a radial velocity component to rotation of the\nradiating gas. As free fall velocity is:\n$$\nv_ {ff} \\simeq 315 {\\rm km s} ^{-1} \\lgroup \\frac {M_*}{M_{\\odot}}\n\\rgroup ^{1\/2} \\lgroup \\frac {R_*}{R_{\\odot}} \\rgroup ^{-1\/2}\n$$\nThe observed\nprofiles broadenings can be reproduced without difficulty. The\nobserved broadening of H$\\alpha$ or Mg~II lines do not vary\nsignificantly requiring that, at least, the spatial average of the\naccretion flow is rather stable.  Strong variations in the\naccretion flow, like the reported from RY~Tau UV observations, should\nalso show in the large scale magnetosphere tracers (i.e.\nH$\\alpha$ or Mg~II profiles).\n\n\\begin{figure}[]\n\\centerline{\\includegraphics[width=18pc]{gomezdecastro_fig4.eps}}\n\\caption{ SiIII] and CIII] UV lines observed in RY~Tau (from\nG\\'omez de Castro \\& Verdugo 2007); RSDP processed data are\nplotted with a thin line and the 3-pixels average profile with a\nthick line. The rest wavelength of the lines and the velocity of\nthe unresolved jet at $\\simeq -80$ km\/s (from G\\'omez de Castro \\&\nVerdugo 2001 and Hamann 1994) are marked with dashed lines. {\\it\nLeft panel}: Observations obtained in Dec. 31st, 1993 with the\nGHRS. {\\it Right panel}: Observations obtained in March 27th,\n2001. Both lines show an excess of flux in the red wing compared\nwith the 1993 observations; this excess is shaded in the figure.}\n\\end{figure}\n\n\n\n\\subsection{About the wind}\n\nThere is another possibility to broaden the line profiles: adding\na radial velocity component associated with the outflow. The\npresence of magnetic fields and the relevance of centrifugal\nlaunching drives to the formulation of the velocity field in the\nwind by means of three components: axial component (along the\nrotation axis), radial expansion from the axis and the azimutal\ntoroidal component (rotation around the axis); in figure 5 there\nis a representation of the three components for a warm centrifugal\nwind model (from G\\'omez de Castro \\& Ferro-Font\\'an 2005). A\nrapid radial expansion close to the star (to guarantee that the\nwind density and temperature are about the observed\nn$_e=10^{10}$cm$^{-3}$ and T$_e = 20-30 \\times 10^{3}$K values)\ncould produce similar effects in the profiles than those predicted\nby magnetospheric infall. Several attempts have been made to\nreproduce the wind profiles with cold winds from accretion disks\n(Ferro-Font\\'an \\& G\\'omez de Castro 2003), warm disk winds\n(G\\'omez de Castro \\& Ferro-Font\\'an, 2005), coronal winds from\naccretion disks (Ferreira \\& Casse, 2004) and winds driven\nby the star-disk interaction (G\\'omez de Castro \\& von Rekoswki, 2008).\n\n\\begin{figure}[]\n           \\includegraphics[width=18pc]{gomezdecastro_fig5.eps}\n\\caption{The basic kinematics of MHD centrifugal winds is outlined\nfrom the simple semiempirical model of centrifugally driven MHD\nwinds with thermal pressure of G\\'omez de Castro \\& Ferro-Font\\'an\n(2005). {\\it Left:} Velocity field in the flow.\n$V_r$ (solid) and $V_z$ (dashed) are the components of the\nvelocity along the $r$\nand $z$ axes, respectively. The toroidal component of the\nvelocity, $V_{t}$ is scaled with respect to the keplerian\nvelocity of the disk at the radius from which the wind is ejected.\n {\\it Right:} Line profiles generated by the wind in ring\nof gas  at different heights along the\nz-axis. }\n\\end{figure}\n\nCold disk winds fail to reproduce the high temperatures observed.\nThe original wind temperature is as low as the disk one and\nheating has to be extracted from photoionization by the central\nsource. However, this radiation is able to heat the gas only to\nmild temperatures of about 10$^4$K. Warm disk winds produce\nprofiles that are too narrow to reproduce all the observations;\nthis is because the vertical thermal pressure push forces the\ngrowth of the radial velocity component to heights were the plasma\nis already too cool. Finally, winds driven from the star-disk\ninteraction also produce narrower profiles than the observed in\nsome sources as shown in Fig.6. According their UV forbidden lines\nprofiles, TTSs can be classified in two groups: stars with\nbroadenings with full width half maximum about 150~km~s$^{-1}$\nthat can be adjusted with the current models and stars with\nextremely broad profiles ($> 250$km~s$^{-1}$). The source of the\nvery large broadenings have to be seek in other structures such as\nion belts or plasma rings as the resolved in RW~Aur (see Sect.\n4.3).\n\n\\begin{figure}[]\n           \\includegraphics[width=18pc]{gomezdecastro_fig6.eps}\n\\caption{{\\it Left:} Si~III] profiles of the TTSs observed\nwith HST; notice the very different line broadenings.\n{\\it Right:} Predicted Si~III] profiles for winds generated\nby means of the interaction between a stellar magnetosphere\nwith stellar field 1kG and an accretion disk undergoing\n$\\alpha ^2 \\Omega$-dynamo effect (e.g. Krause \\& Raedler, 1980)\nfrom G\\'omez de Castro \\& von Rekowski (2008).\n}\n\\end{figure}\n\nFinally, UV observations have clearly proved that:\n\\begin{enumerate}\n\\item  {\\it Warm winds are latitude dependent on scales\ncomparable to the stellar radius}. As an example, the Mg~II\nresonance doublet has been observed in a broad sample of 17 TTSs\n(adding IUE and HST samples); this is the largest sample of TTSs\nobserved in a single UV spectral line.  These  lines can be\ngenerically described as broad, asymmetric emission lines with\ntypical full widths at 10 \\% intensity of few hundreds km\/s. The\nbroad blueward shifted absorption component characteristic of\nmass-loss was detected in few sources (Penston \\& Lago, 1983,\nImhoff \\& Appenzeller, 1989) but not in all of them; the degree of\nabsorption (the asymmetry of the line) varies from not absorption\nto full absorption of the bluewards shifted emission (see G\\'omez\nde Castro, 1997).\n\\item {\\it The collimated flow, the jet, radiates in the UV as\nwell as the bow-shock and the Herbig-Haro objects}. Basically\nall data have been obtained with the IUE satellite (see\nG\\'omez de Castro \\& Robles 2001 for a compilation).\nRecent observations obtained with  Hopkins Ultraviolet Telescope\n(HUT) have shown that it is still unclear how line\nradiation is excited, at least, in HH2; in particular\nO~VI is not detected as was expected in high\nexcitation Herbig-Haro objects where line radiation is\npredicted to be produced in strong radiative shocks\nwhere the shock kinetic energy is damped into heating\n(Raymond et al 1997).\n\n\\end{enumerate}\n\n\\subsection{About the inner disk and ion belts}\n\nStrong continuum FUV emission (1300--1700 \\AA ) has\nbeen detected recently from some stars with bright molecular disks\nincluding GM~Aur, DM~Tau, and LkCa~15, together with inner disk\ngaps of few AUs (Bergin et al. 2004). This emission is likely due\nto energetic photoelectrons mixed into the molecular layer that\nlikely indicates the existence of a very hot component in the\ninner disk.\n\nHigh-resolution HST\/STIS spectra have revealed, for the first\ntime, the rich UV molecular emission in CTTSs. H$_2$ fluorescence\nemission has now been studied in detail in the nearest CTTS,\nTW~Hya, and the richness of the spectrum is overwhelming: Herczeg\net al. (2002) detected 146 Lyman-band H$_2$ lines. The observed\nemission is likely produced in the inner accretion disk, as are\nthe infrared CO and H$_2$O lines.   From these UV data, Herczeg et\nal. (2004) estimated that the warm disk surface has a column\ndensity of $N_{H_2} = 3.2 \\times 10^{18}$~cm$^{-2}$, temperature\nof $T=2500$~K, and filling factor of H$_2$ as seen from the source\nof the Ly$\\alpha$ emission of 0.25$\\pm$0.08. The observed spectrum\nshows that some ground electronic state H$_2$ levels with\nexcitation energies as large as 3.8 eV are pumped by Ly$\\alpha$.\nThese highly excited levels may be formed by dissociative\nrecombination of H$^+_3$, which in turn may be formed by reactions\ninvolving X-rays and UV photons from the star. Also DF~Tau and\nV836~Tau H$_2$ emission seems to arise from the disk (Herczeg et\nal 2006).\n\nIn addition to this molecular component, there is increasing\nevidence of the existence of ion belts\/rings around some TTSs. An\nion belt has been detected around the TTS, RW~Aur (G\\'omez de\nCastro \\& Verdugo, 2003). A corotation radius of 4.4~$R_*$ is\nderived and a $\\log T_e (K) \\simeq 4.7$ and $\\log n_e (cm^{-3}) =\n11.6$ are estimated. This was the first detection of such an\nstructure around a classical TTS. In addition, there are\nindications of a similar structure around AB~Dor, a weak line TTS\n(see Fig.~7). The structure is resolved, as in RW~Aur, because\nthere is an inner hole that allows separating the stellar\/wind\ncontribution from the belt. However in a 5.7 hours time lapse the\ndouble peaked profile is lost, and  the inner part of the profile\nis filled in again with emission (G\\'omez de Castro 2002) .\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=17pc]{gomezdecastro_fig7.eps}}\n\\caption[]{SiIII] profiles of AB~Dor obtained with the HST\/GHRS\n(see G\\'omez de Castro 2002 for more details). In the bottom\npanel, the profile at phase ($\\phi$) 0.329 is overplotted\n(dashed line) on the profiles at $\\phi = 0.794$ (continuous line),\nfor comparison.}\n\\end{figure}\n\n\n\n\\subsection{About the interaction between the disk and the wind}\n\nAB~Dor, a very bright nearby 30~Myr old star, is the only young\nstar that has been well monitored in the UV for flares. Nine events were\ndetected during 10.63~hours of monitoring with HST\/GHRS!.\nThe C~IV and Si~IV UV line profiles produced by most of the events\nare narrow and redshifted, indicating hot gas falling onto the\nstar during the flare. However, the strongest event produced a\nvery broad profile with narrow absorption slightly blueshifted.\nThis profile lasted a few kiloseconds and thus the broad wings are\nmost likely tracing the front shock of a CIR (G\\'omez de Castro\n2002). In the solar system, there are three very different types\nof  ``flares'', which are sudden increases of the high energy\nradiation and particles flux: magnetic flares (magnetic\nreconnection events), corotating interaction regions or CIRs\n(shock fronts formed by the interaction between the slow and the\nfast component of the solar wind), and coronal mass ejections.\nThis classification also applies to TTSs and their circumstellar\nenvironments. High-resolution UV spectroscopic monitoring is\nrequired to disentangle the possible mechanisms for flares in\nproto-stellar systems and to study their impact in young planetary disks\nevolution as well as on planetary atmospheres embryos.\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=18pc]{gomezdecastro_fig8.eps}}\n\\caption[]{The C~IV 1548~\\AA\\ profile of AB~Dor during a normal\nstellar flare (left) and a transient feature probably associated\nwith a CIR (right). Both events lasted several kiloseconds. The\nleft profile is typical of three events that occured during the\nshort monitoring time, while the profile on the right was observed\nonly once. Note the presence of a narrow absorption and the very\nbroad line wings in the right panel profile (see G\\'omez de Castro\n2002 for more details). }\n\\end{figure}\n\n\n\n\n\\section{Summary: the key observables}\n\nIn brief, the radiation produced by the accretion engine\n(including magnetospheres, outflows, accretion, inner disk\nand shock between winds and young planetary disks) is produced\nin the UV. To separate the various contributions is necessary\neither very high spatial resolution or moderate time resolution.\n\nCurrent surveys show that although there are some few nearby TTSs\nand WTTSs sparsely distributed around the Sun (AB~Dor at 14.9~pc\nor TW~Hya at 56~pc) the nearest star forming complexes are\nconcentrated in a {\\it Star Formation Belt} (SFB) at  140~pc\naround the Sun which includes Taurus, Auriga-Perseus, Ophiuchus,\nLupus and Chamaleon molecular clouds and several thousands of\npre-main sequence stars forming in various environments (clustered\nas in Ophichus, sparse as in Taurus). Resolving spatial scales of\na tenth of the solar radius in the SFB would allow to study the\nconnection between the star and the outflow in full detail. For\nthis purposes spatial resolutions of 3.3 micro arcseconds are\nrequired; thus, for a fiducial wavelength of 1500\\AA\\ the aperture\nmust be about 10~km. Such a long baseline interferometry should be\ncarried out in the space and the Stellar Imager project (Carpenter\net al 2008) represents a first atempt to such an ambitious\nproject.\n\nHowever, the requirements are not so strong to resolve the inner\ndisk structure, the disk wind and the plasmoids ejection from the\ncurrent layer between the stellar and the disk wind. In such a\ncase, spatial resolutions of 1-0.5 milliarcseconds (mas) would be\nenough to map the SFB sources thus requiring apertures of 20~m and\neffective areas about 10$^4$ times those of HST\/STIS; research in\nnew coatings and detectors in the UV field (Kappelmann \\&\nBarnstedt, 2007) as well as a clever optical design may account\nfor a factor of ten but, still, a larger, $\\sim 30$m aperture will\nbe required to get SNR$\\simeq 10$ in the C~IV line in reasonable\nexposure times (few hours). There is an ongoing project that\nsatisfies these requirements, the Fresnel Interferometer (see\nKoechlin et al 2008) and some projections on the expected\nperformance of the interferometer on the mapping of the engine are\nplotted in Fig.~9.\n\nBoth space interferometer projects are under study by their\nnational space agencies and, in case they succeed, they will\nbe available about 2030. Is there anything else to be done\n{\\it in the meantime}?. The answer is definitely positive, time\nmapping will allow us to resolve the structures since the\nvariability time scales are not the same for all the\nphenomena and they do not produce the same inprint in the\nspectra (neither in temperatures, densities or velocities).\nSome examples of the power of this technique have\nalready been shown in this contribution.\n\nHigh resolution spectroscopy (R$\\sim 50,000$) is enough to\ndiscriminate among the various components thus, scaling with\nthe fluxes of the weakest H$_2$ lines (from Heczeg et al\n2002) detected with the HST\/STIS, a factor of 10 increase\nof the effective area with respect to HST\/STIS is required\nto reach most of the sources\nin the SFB. The COS instrument in HST will provide\na factor of 10 sensitivity increase with respect to STIS\nin the 1150-1700\\AA\\ range because of its optimized optical\ndesign (see Froening et al 2008), unfortunately\nthe orbital constrains of HST do not favour monitoring\nprograms. High orbit missions alike the WSO-UV (see Shustov et al\n2008) are better suited for this purpose.\n\nAn additional factor of 10\nwould be required to obtain SNR$\\sim 10$ in exposure times\nof few minutes; this short time scales are necessary to\nmap variations in flare time scales as shown in\nFig.~8 for the pre-main sequence stars in the SFB.\nUnfortunately, spectroscopic monitorings of the flaring activity in the\nSFB will have to wait for future missions, with collecting\nareas about 8-10~m, preferably located at the L2.\n\n\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=26pc]{gomezdecastro_fig9.eps}}\n\\caption[]{Theoretical prediction of the Si~III] emissivity from\nnumerical simulations of star-disk interaction (G\\'omez de Castro\n\\& Rekowski, 2008). The stellar magnetosphere is assumed to be\ndipolar with a field strength at the surface of 1kG. The\nmagnetosphere interacts with the disk which is under a moderate\n$\\alpha$-dynamo effect\\footnote{The magnetic field in the disk is\nassumed to be generated by a standard $\\alpha ^2 \\Omega $ dynamo\n(e.g. Krauser \\& Raedler 1980) where $\\alpha$ is the mean-field\n$\\alpha$ effect and $\\Omega$ the angular velocity of the plasma.\n$\\alpha$ scaling is: $\\alpha = -0.1 \\frac{z}{z_0} \\frac {\\chi\n_{\\rm disk} (r, z)}{1+ V_A^2\/C_s^2}$ where $\\chi _{\\rm disk}$ is\nthe disk profile (see von Rekowski et al 2003), $z_0$ is the disk\nhalf-thickness, $V_A$ is the local Alfv\\'en velocity and $C_s$ is\nthe local sound speed.} . The inner disk wind is\nmagnetocentrifugally accelerated ({\\it top panel}).\n\nThe convolution of the theoretical prediction with the point\nspread function of the 30m Fresnel Interferometer\nFII has been carried out by Laurent Koechlin\nand Truswin Raksasataya.\nThe convolution is shown for three inclinations\n(0$^o$, 45$^o$ and 90$^o$) and three distances to the Earth\n(15~pc, 40~pc and 140~pc). Notice that the inner ring is resolved\neven for 140~pc.\n }\n\\end{figure*}\n\n\n\n\n\n\n\\acknowledgments\n\nThis work has been supported by the Ministry of Education of Spain\nthrough grant AYA2007-67726 and the Comunidad Aut\\'onoma de Madrid\nthrough grant CAM-S-0505\/ESP\/0237\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusion}\n\\label{sec:conclusion}\n\\vspace{-0.4em}\nWe hypothesised that explicitly modeling the internal structure of complex labels for morphological tagging improves the overall tagging accuracy over the baseline with monolithic tags.\nTo test this hypothesis, we experimented with three approaches to model composite morphological tags in a neural sequence tagging framework.\nExperimental results on 49 languages demonstrated the advantage of modeling morphological labels as sequences of category values, whereas the superiority of this model is especially pronounced on smaller datasets.\nFurthermore, we showed that, in contrast to baselines, our models are capable of predicting labels that were not seen during training.\n\n\\section{Analysis and Discussion}\n\\label{sec:discussion}\n\n\n\\paragraph{OOV label accuracy}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.95\\columnwidth]{fig\/oov_labels_seq_crop}\n\\caption{OOV label accuracies of the \\textsc{Seq} model.}\n\\label{fig:oov_labels_seq}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\columnwidth]{fig\/category_errors_crop}\n\\caption{Average error rates of distinct morphological categories for \\textsc{Seq} and \\textsc{Mc} models.}\n\\label{fig:cat_errors}\n\\end{figure}\n\n\nOur models are able to predict labels that were not seen in the training data. Figure~\\ref{fig:oov_labels_seq} presents the accuracy of test tokens with OOV labels obtained with our best performing \\textsc{Seq} model plotted against the number of OOV label types. The datasets with zero accuracy are omitted. The main observation is that although the OOV label accuracy is zero for some languages, it is above zero on ca. half of the datasets---a result that would be impossible with \\textsc{MarMoT} or \\textsc{Mc} baselines.\n\n\n\n\n\n\n\\paragraph{Error Analysis}\nFigure~\\ref{fig:cat_errors} shows the largest error rates for distinct morphological categories for both \\textsc{Seq} and \\textsc{Mc} models averaged over all languages. \nWe observe that the error patterns are similar for both models but the error rates of the \\textsc{Seq} model are consistently lower as expected. \n\n\\iffalse\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\columnwidth]{fig\/feature_confusion_crop}\n\\caption{Most common category-value confusion patterns.}\n\\label{fig:feature_confusion}\n\\end{figure}\n\nTo better understand the nature of these errors, we looked at the confusion patterns of category-value pairs. Ten most common confusion patterns over all languages are shown on Figure~\\ref{fig:feature_confusion}. For each confusion pattern, we plot both its average occurrence count together with the average proportion with respect to the first category-value pair in the pattern. We only show confusion patterns for which the average proportion is at least $0.7$. \nThe most common source of errors is predicting a value for a category that has not been annotated in the test set, e.g. predicting a \\emph{case} value for a noun which does not have an annotated \\emph{case} attribute. Since the error patterns are similar for both models, we conclude that both \\textsc{Seq} and \\textsc{Mc} models learn roughly the same classification function but due to the flexibility in composing the label, the \\textsc{Seq} model makes less errors. \n\\fi\n\n\\paragraph{Stability Analysis}\nTo assess the stability of our predictions, we picked five languages from different families and with different corpus size, and performed five independent train\/test runs for each language.\nTable~\\ref{tbl:stability} summarises the results of these experiments and demonstrates a reasonably small variance for all languages. \nFor all languages, except for Finnish, the worst accuracy of the \\textsc{Seq} model was better than the best accuracy of the \\textsc{Mc} model, confirming our results that in those languages, the \\textsc{Seq} model is consistently better than the \\textsc{Mc} baseline.\n\n\\begin{table}[t]\n\\centering\n\\small\n\\begin{tabular}{lcc}\n\\toprule\nDataset         & \\textsc{Seq}     & \\textsc{Mc} \\\\\n\\midrule\nFinnish     & 93.24 $\\pm$ 0.12 &  93.20 $\\pm$ 0.07 \\\\\nGerman      & 88.45 $\\pm$ 0.21 &  87.74 $\\pm$ 0.17 \\\\\nHungarian   & 84.51 $\\pm$ 0.54 &  80.68 $\\pm$ 0.48 \\\\\nRussian     & 91.08 $\\pm$ 0.18 &  90.13 $\\pm$ 0.15 \\\\\nTurkish     & 90.29 $\\pm$ 0.24 &  89.16 $\\pm$ 0.27 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Mean accuracy with standard deviation over five independent runs for \\textsc{Seq} and \\textsc{Mc} models.}\n\\label{tbl:stability}\n\\vspace{-1em}\n\\end{table}\n\n\\paragraph{Hyperparameter Tuning}\nIt is possible that the hyperparameters tuned on Finnish are not optimal for other languages and thus, tuning hyperparameters for each language individually would lead to different conclusions than currently drawn.\nTo shed some light on this issue, we tuned hyperparameters for the \\textsc{Seq} and \\textsc{Mc} models on the same subset of five languages.\nWe first independently optimised the dropout rates on word embeddings, encoder's LSTM inputs and outputs, as well as the number of LSTM layers.\nWe then performed a grid search to find the optimal initial learning rate, the learning rate decay factor and the decay step.\nValue ranges for the tuned parameters are given in Table~\\ref{tbl:tuning_grid}.\n\n\\begin{table}[h]\n\\centering\n\\small\n\\begin{tabular}{lc}\n\\toprule\nParameter         & Values \\\\\n\\midrule\nWord embedding dropout      & $\\{0, 0.1, \\dots, 0.5\\}$ \\\\\nLSTM input dropout          & $\\{0, 0.1, \\dots, 0.5\\}$ \\\\\nLSTM input dropout          & $\\{0, 0.1, \\dots, 0.5\\}$ \\\\\nNumber of LSTM layers       & $\\{1, 2\\}$ \\\\\n\\midrule\nInitial learning rate       & $\\{0.01, 0.1, 1, 2\\}$ \\\\\nLearning rate decay factor  & $\\{0.97, 0.98, 0.99, 1\\}$ \\\\\nDecay step                  & $\\{1250, 2500, 5000\\}$ \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{The grid values for hyperparameter tuning.}\n\\label{tbl:tuning_grid}\n\\vspace{-1em}\n\\end{table}\n\n\nTable~\\ref{tbl:tuning} reports accuracies for the tuned models compared to the mean accuracies reported in Table~\\ref{tbl:stability}.\nAs expected, both tuned models demonstrate superior performance on all languages, except for German with the \\textsc{Seq} model.\nHyperparameter tuning has a greater overall effect on the \\textsc{Mc} model, which suggests that it is more sensitive to the choice of parameters than the \\textsc{Seq} model.\nStill, the tuned \\textsc{Seq} model performs better or at least as good as the \\textsc{Mc} model on all languages.\n\n\\begin{table}[t]\n\\centering\n\\small\n\\begin{tabular}{lcc|cc}\n\\toprule\nDataset     & \\textsc{Seq} & Gain  & \\textsc{Mc} & Gain \\\\ \n\\midrule\nFinnish     & 93.44 & $+0.20$ & 93.43 & $+0.23$ \\\\\nGerman      & 88.35 & $-0.10$ & 88.14 & $+0.40$ \\\\\nHungarian   & 85.56 & $+1.05$ & 82.29 & $+1.61$ \\\\\nRussian     & 91.44 & $+0.36$ & 90.74 & $+0.61$ \\\\\nTurkish     & 90.56 & $+0.27$ & 89.32 & $+0.16$ \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Accuracies of the tuned \\textsc{Seq} and \\textsc{Mc} models compared to the mean accuracies in Table~\\ref{tbl:stability}.}\n\\label{tbl:tuning}\n\\end{table}\n\n\\paragraph{Comparison with Previous Work}\nSince UD datasets have been in rapid development and different UD versions do not match, direct comparison of our results to previously published results is difficult.\nStill, we show the results taken from \\citet{heigold2017}, which were obtained on UDv1.3, to provide a very rough comparison.\nIn addition, we compare our \\textsc{Seq} model with a neural tagger presented by \\citet{Dozat2017}, which is similar to our \\textsc{Mc} model, but employs a more sophisticated encoder.\nWe train this model on UDv2.1 on the same set of languages used by \\citet{heigold2017}.\n\nTable~\\ref{tbl:result_comparison} reports evaluation results for the three models.\nThe \\textsc{Seq} model and Dozat's tagger demonstrate comparable performance.\nThis suggests that the \\textsc{Seq} model can be further improved by adopting a more advanced encoder from \\citet{Dozat2017}.\n\n\n\\begin{table}\n\\centering\n\\small\n\\begin{tabular}{lcc|c}\n\\toprule\nDataset & \\textsc{Seq} & Dozat & Heigold \\\\\n\\midrule\nArabic & \\textbf{93.84} & 92.85 & 93.78 \\\\\nBulgarian & 97.04 & \\textbf{97.25} & 95.14 \\\\\nCzech & \\bf 95.39 & 95.22 & 96.32 \\\\\nEnglish & 94.80 & \\textbf{94.81} & 93.32 \\\\\nEstonian & 93.30 & \\bf 93.90 & 94.25 \\\\\nFinnish & 93.41 & \\textbf{93.73} & 93.52 \\\\\nFrench & \\textbf{96.39} & 95.90 & 94.91 \\\\\nHindi & 91.75 & \\textbf{92.36}  & 90.84 \\\\\nHungarian & \\textbf{84.12} & 82.84 & 77.59 \\\\\nRomanian & 97.16 & \\textbf{97.20} & 94.12 \\\\\nRussian-SynTagRus & \\textbf{96.67} & 96.20 & 96.45 \\\\\nTurkish & \\textbf{90.70} & 90.22 & 89.12 \\\\\n\\midrule\nAverage & \\bf 93.71 & 93.54 & 92.45 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Accuracies for the \\textsc{SEQ} model, \\citet{Dozat2017} and \\citet{heigold2017}.}\n\\label{tbl:result_comparison}\n\\vspace{-1em}\n\\end{table}\n\n\n\\iffalse\n\\paragraph{Runtime}\nWe perform runtime analysis on Nvidia Tesla P100 graphics processor.\nIn the training phase, the \\textsc{Seq} model processes 43 sentences per second on average.\nAlthough it is ca 4.5 times slower than the \\textsc{Mc} model\\footnote{Note that the batch size is 5 for \\textsc{Seq} and 20 for \\textsc{Mc} model.}, the overall training time is still reasonable, ranging from few hours for smaller corpora, such as Turkish, to several days for the largest corpora, such as Czech.\nThe inference speed is similar for both models, with the \\textsc{Seq} model running at a rate of 460 sentences per second on average with a batch size of 50.\n\n\n               \n\n\n\\fi\n\\section{Experimental Setup}\n\\label{sec:experiments}\n\n\\begin{table*}\n\\centering\n\\ssmall\n\\renewcommand{\\arraystretch}{0.98}\n\\tabcolsep=0.11cm\n\\input{tbl_corpus_stats}\n\\caption{\\small Descriptive statistics for all UDv2.1 datasets. For training sets we report the number of word tokens and types, the average (Avg) and maximum (Max) tags per word type, the proportion of word types for which pre-trained embeddings were available (\\% Emb) and the size of the morphological tagset (\\# Tags). For the test sets, we also give the total number of tokens and types, the proportion of OOV words (\\% OOV) and the number of OOV tag tokens and types.}\n\\label{tbl:corpus_stats}\n\\end{table*}\n\nThis section details the experimental setup. We describe the data, then we introduce the baseline models and finally we report the hyperparameters of the models.\n\n\\subsection{Data}\nWe run experiments on the Universal Dependencies version 2.1~\\citep{nivre2017}.\nWe excluded corpora that did not include train\/dev\/test split, word form information\\footnote{French-FTB and Arabic-NYUAD}, or morphological features\\footnote{Japanese}. Additionally, we excluded corpora for which pre-trained word embeddings were not available.\\footnote{Ancient Greek and Coptic}\nThe resulting dataset contains 69 corpora covering 49 different languages.\nTagsets were constructed by concatenating the POS and morphological annotations of the treebanks.\nTable~\\ref{tbl:corpus_stats} gives corpus statistics. We present type and token counts for both training and test sets. For training set, we also show the average and maximum number of tags per word type and the size of the morphological tagset. \nFor the test set, we report the proportion of out-of-vocabulary (OOV) words as well as the number of OOV tag tokens and types.\n\nIn the encoder, we use fastText word embeddings \\citep{bojanowski2017} pre-trained on Wikipedia.\\footnote{\\ssmall \\url{https:\/\/github.com\/facebookresearch\/fastText}}\nAlthough these embeddings are uncased, our model still captures case information by means of character-level embeddings.\nIn Table~\\ref{tbl:corpus_stats}, we also report for each language the proportion of word types for which the pre-trained embeddings are available.\n\n\n\\subsection{Baseline Models}\n\nWe use two models as baseline: the CRF-based \\textsc{MarMoT} \\citep{mueller2013} and the regular neural multiclass classifier.\n\n\\paragraph{MarMoT (\\textsc{MMT})}\n\\textsc{MarMoT}\\footnote{\\url{http:\/\/cistern.cis.lmu.de\/marmot\/}} is a CRF-based morphological tagger which has been shown to achieve competitive performance across several languages \\citep{mueller2013}.\n\\textsc{MarMoT} approximates the CRF objective using a pruning strategy which enables training higher-order models and handling large tagsets.\nIn particular, the tagger first predicts the POS part of the label and based on that, constrains the set of possible morphological labels.\nFollowing the results of \\citet{mueller2013}, we train second-order models. We tuned the regularization type and weight on German development set and based on that, we use L2 regularization with weight 0.01 in all our experiments.\n\n\\paragraph{Neural Multiclass classifier (\\textsc{Mc})}\nAs the second baseline, we employ the standard multiclass classifier used by both \\citet{heigold2017} and \\citet{yu2017}.\nThe proposed model consists of an LSTM-based encoder, identical to the one described above in section~\\ref{sec:encoder}, and a softmax classifier over the full tagset.\nThe tagset sizes for each corpora are shown in Table~\\ref{tbl:corpus_stats}.\nDuring preliminary experiments, we also added CRF layer on top of softmax, but as this made the decoding process considerably slower without any visible improvement in accuracy, we did not adopt CRF decoding here.\nThe multiclass model is shown in Figure~\\ref{fig:neural_models} (d).\n\nThe inherent limitation of both baseline models is their inability to predict tags that are not present in the training corpus. Although the number of such tags in our data set is not large, it is nevertheless non-zero for most languages.\n\n\n\\subsection{Training and Parametrisation}\\label{sbsec:training}\nSince tuning model hyperparameters for each of the 69 datasets individually is computationally demanding, \nwe optimise  parameters on Finnish---a morphologically complex language with a reasonable dataset size---and apply the resulting values to other languages.\nWe first tuned the character embedding size and character-LSTM hidden layer size of the encoder on the \\textsc{Seq} model and reused the obtained values with all other models.\nWe tuned the batch size, the learning rate and the decay factor for the \\textsc{Seq} and \\textsc{Mc} models separately since these models are architecturally quite different.\nFor the \\textsc{McMl} and \\textsc{HMcMl} models we reuse the values obtained for the \\textsc{Mc} model.\nThe remaining hyperparameter values are fixed.\nTable~\\ref{tbl:parameters} lists the hyperparameters for all models.\n\nWe train all neural models using stochastic gradient descent for up to 400 epochs and stop early if there has been no improvement on development set within 50 epochs.\nFor all models except \\textsc{Seq}, we decay the learning rate by a factor of 0.98 after every 2500 batch updates.\nWe initialise biases with zeros and parameter matrices using Xavier uniform initialiser~\\citep{Glorot2010}.\n\nWords in training sets with no pre-trained embeddings are initialised with random embeddings.\nAt test time, words with no pre-trained embedding are assigned a special UNK-embedding.\nWe train the UNK-embedding by randomly substituting the singletons in a batch with the UNK-embedding with a probability of 0.5.\n\n\\begin{table}[t]\n\\centering\n\\footnotesize\n\\tabcolsep=0.11cm\n\\begin{tabular}{lrr}\n\\toprule\n & \\textsc{Seq} & \\textsc{Other NN} \\\\\n\\midrule\n\\textbf{Encoder} & &   \\\\\nWord embedding size & 300 & 300 \\\\\nCharacter embedding size & 100 & 100 \\\\ \nCharacter LSTM hidden layer size & 150 & 150  \\\\\nWord embedding dropout & 0.5 & 0.5 \\\\\nLSTM layers & 1 & 1 \\\\\nLSTM hidden state size & 400 & 400 \\\\\nLSTM input dropout & 0.5 & 0.5 \\\\\nLSTM state dropout & 0.3 & 0.3 \\\\\nLSTM output dropout & 0.5 & 0.5 \\\\\n\\midrule\n\\textbf{Decoder} & &   \\\\\nLSTM hidden state size & 800 & 800 \\\\\nTag embedding size & 150 & -- \\\\\n\\midrule\n\\textbf{Training} & & \\\\\nInitial learning rate & 1.0  & 1.0 \\\\\nBatch size & 5 & 20 \\\\\nMaximum epochs & 400 & 400 \\\\\nLearning rate decay factor & -- & 0.98 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Hyperparameters for neural models.}\n\\label{tbl:parameters}\n\\end{table}\n\n\n\n\n\\section{Introduction}\n\\label{sec:introduction}\n\n\n\nThe common approach to morphological tagging combines the set of word's morphological features into a single monolithic tag and then, similar to POS tagging, employs multiclass sequence classification models such as CRFs \\citep{mueller2013} or recurrent neural networks \\citep{labeau2015,heigold2017}.\nThis approach, however, has a number of limitations.\nFirstly, it ignores the intrinsic compositional structure of the labels and treats two labels that differ only in the value of a single morphological category as completely independent; compare for instance labels \\textsc{[POS=noun,Case=Nom,Num=Sg]} and \\textsc{[POS=noun,Case=Nom,Num=Pl]} that only differ in the value of the \\textsc{Num} category.\nSecondly, it introduces a data sparsity issue as the less frequent labels can have only few occurrences in the training data.\nThirdly, it excludes the ability to predict labels not present in the training set which can be an issue for languages such as Turkish where the number of morphological tags is theoretically unlimited \\citep{yuret2006}.\n\nTo address these problems we propose to treat morphological tags as composite labels and explicitly model their internal structure. \nWe hypothesise that by doing that, we are able to alleviate the sparsity problems, especially for languages with very large tagsets such as Turkish, Czech or Finnish, and at the same time also improve the accuracy over a baseline using monolithic labels.\nWe explore three different neural architectures to model the compositionality of morphological labels.\nIn the first  architecture, we model all morphological categories (including POS tag) as independent multiclass classifiers conditioned on the same contextual word representation.\nThe second architecture organises these multiclass classifiers into a hierarchy---the POS tag is predicted first and the values of morphological categories are predicted conditioned on the value of the predicted POS.\nThe third architecture models the label as a sequence of morphological category-value pairs.\nAll our models share the same neural encoder architecture based on bidirectional LSTMs to construct contextual representations for words \\citep{lample2016}.\n\nWe evaluate all our models on 49 UD version 2.1 languages.\nExperimental results show that our sequential model outperforms other neural counterparts establishing state-of-the-art results in morphological tagging for most languages.\nWe also confirm that all neural models perform significantly better than a competitive CRF baseline.\nIn short, our contributions can be summarised as follows:\n\\begin{enumerate}[label=\\arabic*),topsep=0em,noitemsep]\n\\item We propose to model the compositional internal structure of complex morphological labels for morphological tagging in a neural sequence tagging framework;\n\\item We explore several neural architectures for modeling the composite morphological labels;\n\\item We find that tag representation based on the sequence learning model achieves state-of-the art performance on many languages.\n\\item We present state-of-the-art morphological tagging results on 49 languages on the UDv2.1 corpora.\n\\end{enumerate}\n\n\n\\section*{Acknowledgments}\nThis work was supported by the Estonian Research Council (grants no. 2056, 1226 and IUT34-4).\n\n\n\n\\section{Neural Models}\n\\label{sec:models}\n\n\\begin{figure*}[]\n\\centering\n\\includegraphics[width=0.85\\textwidth]{fig\/neural_models_crop.pdf}\n\\caption{Neural architectures for modeling complex morphological labels: a) Multiclass Multilabel model (\\textsc{McMl}), b) Hierarchical Multiclass Multilabel model (\\textsc{HMcMl}), c) Sequence model (\\textsc{Seq}) and d) Multiclass baseline model (\\textsc{Mc}). \nCorrect labels are shown with a green border, incorrect labels have a red dotted border.}\n\\label{fig:neural_models}\n\\end{figure*}\n\nWe explore three different neural architectures for modeling morphological labels: multiclass multi\\-label model that predicts each category value separately, hierarchical multiclass multilabel model where the values of morphological features depend on the value of the POS, and a sequence model that generates morphological labels as sequences of feature-value pairs.\n\n\n\\subsection{Notation}\nGiven a sentence $w_1, \\dots, w_n$ consisting of $n$ words, we want to predict the sequence $t_1, \\dots, t_n$ of morphological labels for that sentence.\nEach label $t_i = \\{f_{i0}, f_{i1}, \\dots, f_{im}\\}$ consists of a POS tag ($f_{i0} \\equiv \\textsc{POS}$) and a sequence of $m$ category values. \nFor each word $w_i$, the encoder computes a contextual vector $h_i$, which captures information about the word and its left and right context.\n\n\\subsection{Decoder Models}\n\n\\paragraph{Multiclass Multilabel model (\\textsc{McMl})}\nThis model formulates the morphological tagging as a multiclass multilabel classification problem. \nFor each morphological category, a separate multiclass classifier is trained to predict the value of that category (Figure~\\ref{fig:neural_models} (a)).\nBecause not all categories are always present \nfor each POS (e.g., a noun does not have a \\textit{tense} category), we extend the morphological label of each word by adding all features that are missing from the annotated label and assign them a special value that marks the category as ``off''.\nFormally, the model can be described as:\n\\begin{equation}\np(t|h)_{\\textsc{McMl}} = \\prod_{j=0} ^ M p(f_j|h),\n\\end{equation}\nwhere $M$ is the total number of morphological categories (such as case, number, tense, etc.)\nobserved in the training corpus.\nThe probability of each feature value is computed with a softmax function:\n\\begin{equation*}\np(f_j|h)_{\\textsc{McMl}} = \\text{softmax}(W_j h + b_j),\n\\end{equation*}\nwhere $W_j$ and $b_j$ are the parameter matrix and bias vector for the $j$th morphological feature ($ j=0, \\dots, M$).\nThe final morphological label for a word is obtained by concatenating predictions for individual categories while filtering out off-valued categories.\n\n\\paragraph{Hierarchical Multiclass Multilabel model (\\textsc{HMcMl})}\nThis is a hierarchical version of the \\textsc{McMl} architecture that models the values of morphological categories as directly dependent on the POS tag (Figure~\\ref{fig:neural_models} (b)):\n\\begin{equation}\np(t|h)_{\\textsc{HMcMl}} = p(\\textsc{pos}|h)\\prod_{j=1} ^ M p(f_j|\\textsc{pos},h)\n\\end{equation}\nThe probability of the POS is computed from the context vector $h$ using the respective parameters:\n\\begin{equation*}\np(\\textsc{pos}|h) = \\text{softmax}(W_{\\textsc{pos}} h + b_{\\textsc{pos}})\n\\end{equation*}\nThe POS-dependent context vector $l$ is obtained by concatenating the context vector $h$ with the unnormalised log probabilities of the POS:\n\\begin{equation*}\nl = [h; W_{\\textsc{pos}} h + b_{\\textsc{pos}}]\n\\end{equation*}\nThe probabilities of the morphological features are computed using the POS-dependent context vector:\n\\begin{equation*}\np(f_j|\\textsc{pos}, h) = \\text{softmax}(W_j l + b_j) \\hspace{2mm} j=1, \\dots, M\n\\end{equation*}\n\n\\paragraph{Sequence model (\\textsc{Seq})}\nThe \\textsc{Seq} model predicts complex morphological labels as sequences of category values. This approach is inspired from neural sequence-to-sequence models commonly used for machine translation \\citep{Cho2014a,Sutskever2014}.\nFor each word in a sentence, the \ndecoder uses  a unidirectional LSTM network (Figure~\\ref{fig:neural_models} (c)) to generate a sequence of morphological category-value pairs based on the context vector $h$  \nand the previous predictions.\nThe probability of a morphological label $t$ is under this model:\n\\begin{equation}\np(t|h)_{\\textsc{Seq}} = \\prod_{j=0} ^ m p(f_j|f_0, \\dots, f_{j-1}, h)\n\\end{equation}\n\nDecoding starts by passing the start-of-sequence symbol  as input.\nAt each time step, the decoder computes the label context vector $g_j$ based on the previously predicted category value, previous label context vector and the word's context vector.\n\\begin{equation*}\ng_j = \\text{LSTM}([f_{j-1}; h], g_{j-1})\n\\end{equation*}\nThe probability of each morphological feature-value pair is then computed with a softmax.\n\\begin{equation*}\np(f_j|g_j)_{\\textsc{Seq}} = \\text{softmax}(W_{\\textsc{seq}} g_j + b_{\\textsc{seq}})\n\\end{equation*}\nAt training time, we feed correct labels as inputs while at inference time, we greedily emit the best prediction from the set of all possible feature-value pairs.\nThe decoding terminates once the end-of-sequence symbol is produced.\n\n\n\\subsection{Encoder}\n\\label{sec:encoder}\nWe adopt a standard sequence tagging encoder architecture for all our models.\nIt consists of a  bidirectional LSTM network that maps words in a sentence into context vectors using character and word-level embeddings.\nCharacter-level word embeddings are constructed with a bidirectional LSTM network and they capture useful information about words' morphology and shape.\nWord level embeddings are initialised with pre-trained embeddings and fine-tuned during training.\nThe character and word-level embeddings are concatenated and passed as inputs to the bidirectional LSTM encoder.\nThe resulting hidden states $h_i$ capture contextual information for each word in a sentence.\nSimilar encoder architectures have been applied recently with notable success to morphological tagging \\citep{heigold2017,yu2017} as well as several other sequence tagging tasks \\citep{lample2016,Chiu2016,ling2015}.\n\n\\section{Related Work}\n\\label{sec:related}\n\n\nMost previous work on modeling the internal structure of complex morphological labels has occurred in the context of morphological disambiguation---a task where the goal is to select the correct analysis from a limited set of candidates provided by a morphological analyser. The most common strategy to cope with a large number of complex labels has been to predict all morphological features of a word using several independent classifiers whose predictions are later combined using some scoring mechanism \\citep{hajic1998,hajic2000,smith2005,yuret2006,zalmout2017,kirov2017}. \n\\citet{inoue2017} combined these classifiers into a multitask neural model sharing the same encoder, and predicted both POS tag and morphological category values given the same contextual representation computed by a bidirectional LSTM. \nThey showed that the multitask learning setting outperforms the combination of several independent classifiers on tagging Arabic.\nIn this paper, we experiment with the same architecture, termed as multiclass multilabel model, on many languages.\nAdditionally, we extend this approach and explore a hierarchical architecture where morphological features directly depend on the POS tag. \n\n\n\nAnother previously adopted approach involves modeling complex morphological labels as sequences of morphological feature values \\citep{hakkani2000,schmid2008}. In neural networks, this idea can be implemented with recurrent sequence modeling. Indeed, one of our proposed models generates morphological tags with an LSTM network. Similar idea has been applied for the morphological reinflection task \\citep{kann2016,faruqui2016} where the sequential model is used to generate the spellings of inflected forms given the lemma and the morphological label of the desired form. In morphological tagging, however, we generate the morphological labels themselves.\n\nAnother direction of research on modeling the structure of complex morphological labels involves structured prediction models~\\citep{mueller2013,mueller2015,malaviya2018,lee2011}.\n\\citet{lee2011} introduced a factor graph model that jointly infers morphological features and syntactic structures.\n\\citet{mueller2013} proposed a higher-order CRF model which handles large morphological tagsets by decomposing the full label into POS tag and morphology part.\n\\citet{malaviya2018} proposed a factorial CRF to model pairwise dependencies between individual features within  morphological labels and also between labels over time steps for cross-lingual transfer.\nRecently, neural morphological taggers have been compared to the CRF-based approach \\citep{heigold2017,yu2017}.\nWhile \\citet{heigold2017} found that their neural model with bidirectional LSTM encoder surpasses the CRF baseline, the results of \\citet{yu2017} are mixed with the convolutional encoder being slightly better or on par with the CRF but the LSTM encoder being worse than the CRF baseline. \n\nMost previous work on neural POS and morphological tagging has shared the general idea of using bidirectional LSTM for computing contextual features for words \\citep{ling2015,huang2015,labeau2015,ma2016,heigold2017}.\nThe focus of the previous work has been mostly on modeling the inputs by exploring different character-level representations for words \\citep{heigold2016,santos2014,ma2016,inoue2017,ling2015,rei2016}.\nWe adopt the general encoder architecture from these works, constructing word representations from characters and using another bidirectional LSTM to encode the context vectors. \nIn contrast to these previous works, our focus is on modeling the compositional structure of the complex morphological labels. \n\nThe morphologically annotated Universal Dependencies (UD) corpora~\\citep{nivre2017} offer a great opportunity for experimenting on many languages. \nSome previous work have reported results on several UD languages \\citep{yu2017,heigold2017}. \nMorphological tagging results on many UD languages have been also reported for parsing systems that predict POS and morphological tags as preprocessing \\citep{andor2016,straka2016,straka2017}.\nSince UD treebanks have been in constant development, these results have been obtained on different UD versions and thus are not necessarily directly comparable. We conduct experiments on all UDv2.1 languages and we aim to provide a baseline for future work in neural morphological tagging\n\n\\section{Results}\n\\label{sec:results}\n\n\\begin{table*}\n\\centering\n\\ssmall\n\\renewcommand{\\arraystretch}{0.91}\n\\tabcolsep=0.095cm\n\\input{tbl_results}\n\\caption{\\small Morphological tagging accuracies on UDv2.1 test sets for MarMot (\\textsc{MMT})  and \\textsc{Mc} baselines as well as for \\textsc{McMl}, \\textsc{HMcMl} and \\textsc{Seq} compositional models. The left section shows the full \\textsc{Pos+Morph} tag results, the middle section gives accuracies for OOV words only, the right-most section shows the POS tagging accuracy. The best result in each section for each language is in bold. The languages are color-coded according to the training set size, lighter color denotes larger training set: cyan (<20K), violet (20K-50K), magenta (50K-100K), pink (>100K).}\n\\label{tbl:results}\n\\end{table*}\n\nTable~\\ref{tbl:results} presents the experimental results. We report tagging accuracy for all word tokens and also for OOV tokens only. \nA full morphological tag is considered correct if both its POS and all morphological features are correctly predicted. \n\nFirst of all, we can confirm the results of \\citet{heigold2017} that the performance of neural morphological tagging indeed exceeds the results of a CRF-based model. In fact, all our neural models perform significantly better than \\textsc{MarMoT} ($p<0.001$).\\footnote{As indicated by Wilcoxon signed-rank test.}\n\n\nThe best neural model on average is the \\textsc{Seq} model, which is significantly better from both the \\textsc{Mc} baseline as well as the other two compositional models, whereby the improvement is especially well-pronounced on smaller datasets. We do not observe any significant differences between \\textsc{McMl} and \\textsc{HMcMl} models neither on all words nor OOV evaluation setting.\n\nWe also present POS tagging results in the right-most section of Table~\\ref{tbl:results}. Here again, all neural models are better than CRF which is in line with the results presented by \\citet{plank2016}. For POS tags, the \\textsc{HMcMl} is the best on average. It is also significantly better than the neural \\textsc{Mc} baseline, however, the differences with the \\textsc{McMl} and \\textsc{Seq} models are insignificant.\n\nIn addition to full-tag accuracies, we assess the performance on individual features.\nTable~\\ref{tbl:features} reports macro-averaged F1-cores for the \\textsc{Seq} and the \\textsc{Mc} models on universal features.\nResults indicate that the \\textsc{Seq} model systematically outperforms the \\textsc{Mc} model on most features.\n\n\n\\begin{table}[t]\n\\centering\n\\small\n\\renewcommand{\\arraystretch}{0.91}\n\\tabcolsep=0.095cm\n\\begin{tabular}{lrrr|lrrr}\n\\toprule\nFeature & \\textsc{Seq} & \\textsc{Mc} & \\# & Feature & \\textsc{Seq} & \\textsc{Mc} & \\# \\\\\n\\midrule\nPOS & \\textbf{91.03} & 90.20 & 69 & NumType & \\textbf{89.68} & 87.82 & 54 \\\\\nNumber & \\textbf{94.02} & 93.05 & 63 & Polarity & \\textbf{93.83} & 92.86 & 54 \\\\\nVerbForm & \\textbf{91.29} & 89.86 & 61 & Degree & \\textbf{87.44} & 84.12 & 48 \\\\\nPerson & \\textbf{89.02} & 87.52 & 60 & Poss & \\textbf{94.52} & 93.60 & 44 \\\\\nTense & \\textbf{92.96} & 91.31 & 59 & Voice & \\textbf{88.40} & 82.85 & 42 \\\\\nPronType & \\textbf{89.83} & 88.81 & 58 & Definite & \\textbf{95.26} & 94.10 & 37 \\\\\nMood & \\textbf{87.34} & 85.40 & 58 & Aspect & \\textbf{89.76} & 87.71 & 29 \\\\\nGender & \\textbf{89.31} & 87.78 & 55 & Animacy & \\textbf{86.22} & 83.73 & 19 \\\\\nCase & \\textbf{88.90} & 87.04 & 55 & Polite & 75.76 & \\textbf{80.48} & 10 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Performance of \\textsc{Seq} and \\textsc{Mc} models on individual features reported as macro-averaged F1-scores. \n}\n\\label{tbl:features}\n\\end{table}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThere is a long and rich history of modeling the response of visual cortex neurons to stimuli extending back to the work of Hubel and Wiesel on simple and complex cells \\cite{hubel1962receptive}. \nIn recent years, artificial neural networks (ANNs) have achieved state-of-the-art performance predicting neural responses to natural stimuli \\cite{Antolik2016,Yamins2016,Klindt2017,Cadena2019,Batty2016,Sinz2018,Yamins2014,Vintch2015,Ecker2018,Walker2019}. \nThese models are accurate enough that the stimuli that maximally excite a neuron can be computed \\textit{in silico}, and when tested \\textit{in vivo} indeed drive neurons effectively \\cite{Walker2019,Bashivan2019}.\nHowever, these approaches place the computational burden of optimizing network parameters \\textit{after} extensive data from a neuron has been collected, which prohibits their use in real-time closed-loop experiments.\nTo avoid this optimization step, we wanted a model that can predict the response of a novel neuron to any stimulus, conditioned on a set of $K$ observed stimulus-response pairs -- essentially performing $K$-Shot prediction on neural responses.\n\n\\citet{Garnelo2018a} aptly describe how Neural Processes (NPs) can help solve this problem:\n\\emph{``Meta-learning models share the fundamental motivations of NPs as they shift workload from training time to test time. NPs can therefore be described as meta-learning algorithms for few-shot function regression''}. \nNPs achieve this by embedding input and output measurements into a latent space that maps to a space of functions, essentially learning the distribution over functions and a method to infer the posterior over functions given limited samples \\cite{Garnelo2018a,Garnelo2018,Kim2019}. \n\nA significant advance in modeling visual responses with ANNs was using convolutional neural networks with a factorized readout between the tuning function's location and properties  \\cite{Klindt2017, Sinz2018}.\nWe found that NPs struggle to learn the space of tuning functions from stimulus-response samples without such a factorized representation. \nThus, we developed a Factorized Neural Process (FNP), which is composed of stacking multiple NPs.\nA key insight for this was that by passing the latent variable computed by early layers to deeper layers, in addition to the observations, we could obtain a factorized latent space while retaining the representational power and efficiency of NPs. \nWe used a two-layer FNP applied to visual responses, where the first NP produces a latent variable for the tuning function's location that the second NP uses to infer the tuning function's properties.\nWe found that a FNP trained on simulated data generalizes to new neurons, successfully inferring the tuning function's location and properties and predicting the responses to unseen stimuli. An FNP trained on neural responses from the mouse primary visual cortex made predictions with comparable accuracy to state-of-the-art approaches, and made these predictions almost \\emph{100 times faster}.\n\nIn short, our contributions in this work include: \\circled{1}  We reformulate the problem of predicting the response of neurons to visual stimuli as a K-shot regression problem, removing the time consuming step of optimizing network parameters for each newly acquired neuron. \\circled{2} We develop a Factorized Neural Process that embeds the observed stimuli-response pairs into a latent space representing the tuning function that is partitioned into location and tuning function properties. \\circled{3} We train this Factorized Neural Process for Neural Processes end-to-end on simulated data and show it approaches the ground truth predictions as the size of the observation set increases. \\circled{4} We found that this approach performs comparably to state-of-the-art predictive models on responses from mouse visual cortex neurons while improving estimation speed by multiple orders of magnitude. The code is available at \\url{https:\/\/github.com\/peabody124\/fnp_neurips2020}.\n\n\\section{Neural Processes for Neural Processes}\n\nThe core steps that allow a NP to efficiently meta-learn a $K$-shot regression model are (1) encoding each element from a set of observed input-output observations into a representation space, (2) aggregating the elements in that representation space (typically by taking the mean) to produce a sufficient statistic of the observed set, (3) a conditional decoder that maps the aggregated representation to a function used for prediction of new observations, and (4) training this over many different sets of observations from different sample functions, i.e. meta-learning the distribution over tuning functions \\cite{Garnelo2018}. Our approach is largely based on \\citet{Garnelo2018a}, which expanded on \\citet{Garnelo2018} by introducing a stochastic variable used by the conditional decoder. NPs were further extended to include attention in \\citet{Kim2019}, which we do not use in this work. \n\nFirst, we describe the data generation process we seek to model: Let $\\mathcal F : \\mathcal X \\rightarrow \\mathcal Y$ be the space of all tuning functions that map from images to neural responses. An individual neuron corresponds to a sample function, $f \\in \\mathcal F$, from which we get $K$ observations $O_K=\\{(\\boldsymbol x_i, y_i)\\}_{i=0}^{i<K}$ where $y_i \\sim f(\\boldsymbol x_i)$,  $\\boldsymbol x \\in \\mathcal X$ and $y \\in \\mathcal Y$. The model should maximize the likelihood of a new input-output observation, called the target: $p_\\theta(y_t | \\boldsymbol x_t, O_K)$ with $y_t \\sim f(\\boldsymbol x_t)$, i.e. $K$-shot prediction.  \n\nFollowing the formulation of stochastic NPs \\cite{Garnelo2018a}, this predictions is split into \\emph{encoding} the set of observations into a posterior distribution over a latent space, $p_\\theta(\\boldsymbol z|O_K)$ with $\\boldsymbol z \\in \\mathbb R^D$, and \\emph{conditionally decoding} it with a predictive distribution conditioned on the latent variable and target input, $p_\\theta(y_t|\\boldsymbol x_t, \\boldsymbol z)$:\n\\begin{equation}\np_\\theta(y_t | \\boldsymbol x_t, O_K) =\\int p_\\theta(y_t|\\boldsymbol x_t, \\boldsymbol z)\\, p_\\theta(\\boldsymbol z|O_K) \\, \\mathrm d \\boldsymbol z\n\\label{eq:np_latent}\n\\end{equation}\nThe encoder computes the distribution over $\\boldsymbol z$ using a learned embedding of individual observations that are aggregated into a sufficient statistic: $s_K=\\frac{1}{K}\\sum_{i<K} h_\\theta(\\mathbf x_i, y_i)$, with $h_\\theta: \\mathcal X \\times \\mathcal Y \\rightarrow \\mathbb R^{D_2}$ implemented as a neural network. The dimensionality of this statistic, $D_2$, does not need to match the dimensionality of the latent variable, $D$, but it is used to parameterize a distribution for the latent variable, such as Gaussian: $p_\\theta(\\boldsymbol z | O_K) \\sim \\mathcal N(\\mu_\\theta(s_K), \\sigma_\\theta(s_K))$. The decoder function is also implemented as a distribution that is parameterized by the output of a neural network, in our case determining the mean rate of a Poisson distribution $\\lambda_\\theta : \\mathcal X \\times \\mathbb R^D \\rightarrow \\mathbb R$ and $p_\\theta(y_t|\\boldsymbol x_t, \\boldsymbol z) \\sim \\mathtt {Poisson}[\\lambda_\\theta (\\boldsymbol x_t, \\boldsymbol z)]$.\n\nOur formulation differs from \\cite{Garnelo2018, Kim2019} in a few details. First, they used a variational approximation to train the encoder. Because of our approach to efficient K-Shot training, described below in Section~\\ref{section:k_shot_training}, that involves predicting single samples we can directly optimize Eq.~\\ref{eq:np_latent} with samples of $\\boldsymbol z$. Second, and more critically, for NPs applied to image completion, individual observations are pixels with $\\mathcal X \\in \\mathbb R^2$ and $h_\\theta$ was implemented with an MLP \\cite{Garnelo2018,Garnelo2018a, Kim2019}. To apply this to direct regression of neural responses, $h_\\theta$ must embed entire stimulus-response associations where the domain of $\\mathcal X$ are images $\\mathbb R^{\\mathrm{width} \\times \\mathrm{height}}$.\n\nWe attempted to apply a NP to predicting visual neuron response by including a convolutional neural network (CNN) in $h_\\theta$ to extract the relative stimulus features and optimizing Eq.~\\ref{eq:np_latent}. \nHowever, we could not achieve good predictive performance despite trying numerous architecture alterations. \nThis motivated us to develop the Factorized Neural Process.\n\n\\subsection{Factorized Neural Process}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.8\\linewidth]{high_level_factorized.pdf}\n  \\caption{Overview of K-shot neural prediction using a Factorized Neural Process.}\n  \\label{fig:factorized_latent_process}\n\\end{figure}\n\nPrior work using neural networks to model visual responses has demonstrated the importance of identifying the location of a receptive field before trying to model the exact properties. For example, early work used a two stage approach which first computed a spike-triggered average to localize the receptive fields and then stimuli were cropped and centered on this location to build predictive models \\cite{Batty2016}. In Section~\\ref{section:tuning_position} we make a link between a spike-triggered average and our position encoder. \nWhen using CNNs to predict neural responses, \\citet{Klindt2017} showed it is helpful to factorize the tuning function into a location and the tuning properties within that location. We used a similar approach to factor our latent space into visual location and response properties.\n\nAggregation in NPs should be invariant to permutation of the order of elements in the set, because changing the order inputs are presented to the function should not change the representation of that function (at least under the generative model of our data stated above). Aggregation using the mean, such as used in the NPs above, is one aggregation operation with this property. \\citet{Zaheer2017} demonstrated that it applies to a class of deeper network architectures, which they termed Deep Sets. These assure permutation invariance when composing multiple operations, provided each operation (layer) transforms and passes individual elements from a set along with the result of an invariant aggregation operation (such as mean). This insight encouraged us to develop a FNP by composing multiple NPs. \n\nIn a FNP, the encoder, $h_\\theta^l$, for each layer $l$ receives both observed samples and the latent variable samples from lower layers, $z^l$ (which is an empty set for the first layer):\n\\begin{equation}\n    s_K^l :=  \\frac{1}{K}\\sum_{i<K} h^l_\\theta(\\boldsymbol x_i, y_i, \\left\\{ \\boldsymbol z_K^j \\right\\}^{j<l}), \\qquad\n    \\boldsymbol z_K^l \\sim p_\\theta^l \\left(\\boldsymbol z_K^l | s_K^l\\right)\n    \\label{eq:s_k_l}\n\\end{equation}\nThe decoder receives all the latent variables, which can be marginalized out to give the  predictive model for a $L$ layer FNP:\n\\begin{equation}\n    p_\\theta(y_t | \\boldsymbol x_t, O_K) = \\int _{\\boldsymbol z} p_\\theta \\left( y_t | \\boldsymbol x_t , \\left\\{ \\boldsymbol  z_K^j \\right\\} \\right) \\prod_{l=0}^{l<L} p_\\theta^l \\left(\\boldsymbol z^l| s_K^l\\right) \\mathrm{d} \\boldsymbol z_K^0 ... \\mathrm{d} \\boldsymbol z_K^{L-1}\n    \\label{eq:fnp}\n\\end{equation}\n\n\\subsection{FNP Model for Visual Responses}\n\nFollowing the motivation above, we use a two-layer FNP to partition the latent space into the tuning function position, $\\boldsymbol z_K^p$, and the tuning function properties within that location, $\\boldsymbol z_K^w$ (Figure \\ref{fig:factorized_latent_process}). The factorized probability (following Eq.~\\ref{eq:fnp}) is thus:\n\\begin{equation}\np_\\theta(y_t | \\boldsymbol x_t, O_K) = \\int _{\\boldsymbol z_K^p, \\boldsymbol z_K^w} p_\\theta\\left(y_t | \\boldsymbol x_t , \\boldsymbol  z_K^w, \\boldsymbol z_K^p\\right) p_\\theta\\left(\\boldsymbol z_K^w | s^w_K\\right) \\, p_\\theta \\left( \\boldsymbol z_K^p | s^p_K \\right)  \\mathrm{d} \\boldsymbol z_K^p \\mathrm{d} \\boldsymbol z_K^w \\\\\n\\label{eq:vision_fnp}\n\\end{equation}\nThe latent space acquires this factorization from the inductive bias in the architecture of the conditional decoder (Section~\\ref{section:response_predictor}), which uses a similar factorization between tuning position and properties as \\citet{Klindt2017}. With this conditional decoder, we empirically found certain architectures helped the encoders learn an appropriate embedding for these latent variables, which we describe in Sections \\ref{section:tuning_position} and \\ref{section:properties_encoding}. All components use a common Group Convolutional Neural Network to compute the relevant features space from the stimuli, which we will describe next. \n\n\\subsection{Group Convolutional Network}\n\nOne of the most striking features of V1 neurons are their sensitivity to stimulus orientation \\cite{hubel1962receptive}, so CNN-based models expend a great deal of capacity representing the same feature at different orientations. \\citet{Cohen2016, Cohen2016a} developed  Group Convolutional Neural Networks (G-CNNs), which are rotationally equivariant network that reuse parameters across orientations, similarly to how CNNs reuse parameters across space. G-CNNs have improved performance on a number of image classifications tasks \\cite{Weiler2017,Bekkers2018,Lafarge2020} including the modeling of visual cortex neurons \\cite{Ecker2019,Ustyuzhaninov2019}. They have also been combined with a DenseNet \\cite{Huang2017} architecture when modeling histology images \\cite{Veeling2018}. Our stimuli were grayscale images, $\\mathcal X \\in \\mathbb R ^{H_{\\boldsymbol x} \\times W_{\\boldsymbol x} }$ that were transformed to a reduced spatial dimension with multiple feature maps: $g_\\theta:\\mathbb R^{H_{\\boldsymbol x} \\times W_{\\boldsymbol x} } \\rightarrow \\mathbb R^{H \\times W \\times C} $.\n\n\nOur G-CNN Dense Blocks followed the bottleneck DenseBlock architecture of \\cite{Huang2017} (other than using group convolutions) of 1) Batch Normalization \\cite{Ioffe2015}, 2) non-linearity, 3)  $1\\!\\times\\! 1$ group convolution, 3) Batch Normalization, 4) non-linearity, 5) spatial group convolution, 6) concatenating this output to the inputs. Specific architectural parameters are described in the experiments. Following \\citet{Bekkers2018} and \\citet{Veeling2018}, images were passed through an initial layer that lifted them into a group representation, followed by the G-CNN Dense Blocks, and a subsequent projection from the group representation (i.e. a tensor with an additional dimension representing orientation) down to a representation over image space by flattening the group and channel dimensions together. An additional $1\\!\\times\\! 1$ convolution \\cite{Szegedy2015,Lin2013} reduce this to the final channel depth. We found using the G-CNN DenseNet architecture resulted in faster and more reliable training than a standard CNN, although we did not perform exhaustive searches over architectures. \n\n\\subsection{Tuning function position encoder}\n\\label{section:tuning_position}\n\nThe first layer is designed to produce a distribution for a latent variable centered at the location of the tuning function, $\\boldsymbol z_K^p$. The latent has two dimensions corresponding to horizontal and vertical position.\nTo compute this, we found it was important to preserve spatial structure from $g_\\theta(\\boldsymbol x)$ when performing the set aggregation, which we did by averaging the embedded observations into an image analogous to a trainable, non-linear, spike-triggered average:\n\\begin{equation*}\n    s_K^p =  \\frac{1}{K} \\sum_{i<K} h^p_\\theta \\left( \\left[ g_\\theta(\\boldsymbol  x_i) ,  y_i g_\\theta(\\boldsymbol  x_i) , y_i  \\right] \\right), \\qquad s_K^p \\in \\mathbb R^{H \\times W}\n\\end{equation*}\nWhere $[\\cdot,\\cdot]$ concatenates on the channel dimension (and tiles non-spatial values, such as $y_i$, along the spatial dimension) and  $h^p_\\theta$ is a series of $1\\!\\times\\! 1$ convolutions that outputs a single channel; the first two $1\\!\\times\\! 1$ retain the same channel depth as $g_\\theta$ followed by an ELU activation \\cite{Clevert2015}. We interpret $\\mbox{softmax}(s_K^p)$ as a spatial distribution over the receptive field location of a given neuron. In order to sample a single spatial readout point, we separately computed the marginal means $\\boldsymbol\\mu(s_K^p)$ and standard deviations $\\boldsymbol\\sigma(s_K^p)$ of $\\mbox{softmax}(s_K^p)$, and used them to parameterize a truncated normal distribution $\\mathcal N_{\\mid}$, cut at the same height and width as the output of the G-CNN ($H \\times W$): \n\\begin{equation*}\n    \\boldsymbol  z_K^p \\sim \\mathcal N_{\\mid}\\left(\\boldsymbol \\mu(s_K^p), \\boldsymbol \\sigma(s_K^p) \\right) , \\qquad \\boldsymbol z_K^p, \\boldsymbol \\mu(s_K^p), \\boldsymbol \\sigma(s_K^p) \\in \\mathbb R^2.\n\\end{equation*}\nWe also experimented with learning an MLP to transform $s_K^p \\rightarrow (\\boldsymbol \\mu, \\boldsymbol \\sigma)$. While this worked, it was much slower to train than the Gaussian approximation of $\\mbox{softmax}(s_K^p)$ we used.\n\n\\subsection{Tuning function properties encoder}\n\\label{section:properties_encoding}\nThe second layer of the FNP computes the tuning function's properties at the location determined by the first layer:\n\\begin{equation}\n    s_K^w  = \\frac{1}{K} \\sum_{i<K} h^w_\\theta \\left( \\left [\\mathcal T \\left( g_\\theta(\\boldsymbol  x_i\\right), \\boldsymbol z_K^p) , y_i \\right] \\right),\n    \\label{eq:tuning_position}\n\\end{equation}\nwhere $\\mathcal T: \\mathbb{R}^{H \\times W \\times C} \\times \\mathbb R^2 \\rightarrow \\mathbb{R}^C$ is a differentiable operator that performs interpolation on the image features at a particular location to preserve differentiability with respect to the spatial location. We use a spatial transformer layer for that \\citep{jaderberg2016spatial}.\nEssentially, \\eqref{eq:tuning_position} selects the features computed by the G-CNN at a given location, concatenates them with the response, and passes them through a two-layer MLP, $h_\\theta^w$.\n\nWe split $s_K^w=(\\boldsymbol \\mu_K^w, \\boldsymbol \\Sigma_K^w)$ into a mean and covariance (more specifically the vectorized lower triangular component of the covariance) and use it to parameterize the distribution over the $D$-dimension latent variable for tuning function's properties within that location:\n\\begin{equation*}\n    \\boldsymbol z_K^w \\sim \\mathcal N(\\boldsymbol \\mu_K^w, \\boldsymbol \\Sigma_K^w) , \\qquad \\boldsymbol z_K^w, \\boldsymbol \\mu_K \\in \\mathbb R^D.\n\\end{equation*}\n\n\\subsection{Conditional Decoder}\n\\label{section:response_predictor}\nThe latent variables for tuning function position and tuning function properties, $\\boldsymbol z_K^p$, $\\boldsymbol z_K^w$, summarize the observed set of stimulus-response pairs, $O_K$. They define a $K$-shot regression tuning function that we use them to predict the response to new target stimuli, $\\boldsymbol x_t$.\n\\begin{equation*}\n    \\lambda(\\boldsymbol x_t, \\boldsymbol z_K^p, \\boldsymbol z_K^w) := \\mathtt{ELU}\\left(\\left[\\mathcal T \\left( g_\\theta\\left(\\boldsymbol  x_t\\right), \\boldsymbol z_K^p\\right), 1\\right] \\cdot u_\\theta\\left(\\boldsymbol z_K^w\\right) \\right) + 1\n\\end{equation*}\nSimilar to prior work \\cite{Klindt2017, Ecker2019, Walker2019,Sinz2018,Cadena2019}, our response predictor uses a linear weighting of the features from the convolutional network selected at the neuron's tuning function's location. Because we sometimes used a different dimensional representation for tuning function's properties than the feature map, we used an additional two-layer MLP, $u_\\theta$ to match the dimensions. We also found this additional transformation of the latent improved performance, although did not experiment with this extensively. The predictive response probabilities for $y_t$ are modeled as a Poisson distribution with mean rate $\\lambda$:\n\\begin{equation}\np_\\theta(y_t|\\boldsymbol x_t, \\boldsymbol z_K^p, \\boldsymbol z_K^w) \\sim \\mathtt {Poisson}[\\lambda(\\boldsymbol x_t, \\boldsymbol z_K^p, \\boldsymbol z_K^w)]\n\\label{eq:conditional_decoder}\n\\end{equation}\n\n\\subsection{Efficient K-Shot Training}\n\\label{section:k_shot_training}\nEach training sample for this model consists of a set of K stimuli and responses with an additional target pair and it must be trained over many sets. If implemented na\\\"ively and sampling a range of set sizes, training is computationally prohibitive as it requires passing thousands of images through the G-CNN per sample. Additionally, in this case the loss is determined by the prediction of a single sample, thus the gradient is fairly noisy. To greatly accelerate training we used two techniques.\n\nThe first technique was including the responses of multiple neurons to the same stimuli in a minibatch. This allowed the computation of $g_\\theta(\\boldsymbol x)$ to be reused across the neurons, at the expense of GPU memory. This memory is proportional to the number of stimuli $\\times$ the number of neurons, because Eq.~\\ref{eq:tuning_position} is computed in parallel for all neurons and must be stored during each step to allow backpropagation through the G-CNN. \n\nThe second technique was for a set of $T$ trials, we computed the $K$-shot prediction for every set size up to $T-1$ by predicting the response on the next trial. Because aggregation involved computing the mean outputs along the set dimension, this could be efficiently implemented with a cumulative sum that excluded the current element divided by the set size. This allowed us to test $T-1$ predictions rather than 1 for each training set with almost no increase in computation. Thus the per-neuron loss function we minimized was:\n\\begin{equation*}\n    \\mathcal L(\\theta) = -\\sum_{k=0}^{T-1} \\log p(y_{k+1} | \\boldsymbol x_{k+1}, O_k)\n\\end{equation*}\nWith the likelihood approximated by the FNP described in Eq. \\ref{eq:vision_fnp}.  We do not compute the full marginalization in Eq.~\\ref{eq:vision_fnp} but estimate it using Monte Carlo samples from $\\boldsymbol z_k^w$ and $\\boldsymbol z_k^p$ drawn for each value of $k$. The distribution parameters are optimized using the reparameterization trick \\cite{Kingma2014}.\n\nIn prior work with NPs (that had much less computationally expensive inputs), the full $T$ trials were randomly partitioned into the observation and target set to learn generalization over set size \\cite{Garnelo2018, Garnelo2018a, Kim2019}. We leave it to future work to determine if this can be efficiently done with our model, for example generating multiple splits for the encoded values in a single batch, and if it improves training speed or final performance.\n\n\n\n\\section{Experiments with Simulated Neural Responses}\n\nWe first validated that using our approach, a trained FNP could infer the ground truth tuning function from simulated simple and complex visual neurons. Experiments using different architectures and hyperparameters were managed using DataJoint \\cite{Yatsenko2015}. Please see the Appendix for the architecture details used in the presented results.\n\n\\subsection{Simulated responses}\n\\label{section:simulated_responses}\nSimulated neurons had a linear component to their receptive field that was generated using a Gabor kernel, $k_\\phi$, with parameters $\\phi$ including $x$ and $y$ location, orientation, frequency, width, phase offset and scale. Cells could either be simple or complex. Responses for simple cells were generated with the inner product between the receptive field kernel and the stimuli, followed by a ReLU non-linearity. Complex cells used an energy-based model which involved a second kernel, $k_{\\phi+}$, with the same parameters as the first but with the phase offset by $\\pi\/2$. Responses were then sampled from a Poisson distribution for a given cell. \n\\begin{equation*}\n    \\lambda_\\phi(\\boldsymbol x) = \n    \\begin{cases}\n    \\mathtt{ReLU} \\left( \\boldsymbol x \\cdot k_\\phi \\right)  & \\text{, if $\\phi$ simple} \\\\\n    \\sqrt{ \\left( \\boldsymbol x \\cdot k_\\phi \\right)^2 + \\left( \\boldsymbol x \\cdot k_{\\phi+} \\right)^2} & \\text{, if $\\phi$ complex}.\n    \\end{cases},\n    \\qquad r_i \\ \\sim \\mathtt{Poisson}[\\lambda_\\phi(\\boldsymbol x_i)]\n\\end{equation*}\n\nTo simulate the finite data available in real experiments, we used a fixed set of 5000 neuron parameters and 10,000 stimuli in the training data and a different set in the validation data. Stimuli were sampled from ImageNet downsampled to $16 \\times 16$ or $32 \\times 32$ \\cite{Russakovsky2015,Chrabaszcz2017}. Each sample used a random stimulus order with a fixed number of trials.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.5\\linewidth]{accuracy.pdf}\n  \\caption{Predictive accuracy versus ground truth. Legend indicates the maximum observation set used during training. See \\nameref{section:predictive_accuracy} for the definition of $\\Delta LL$.}\n  \\label{fig:accuracy}\n\\end{figure}\n\n\\subsection{Predictive Accuracy versus observation set size}\n\n\\label{section:predictive_accuracy}\nTo quantify the predictive performance of this model for different numbers of observations, we computed the difference between the negative log-likelihood of the response under the $K$-shot predictive distribution to the ground truth model. This was averaged across many neurons:\n\\begin{equation*}\n    \\Delta LL_k = \\left< -\\log p_\\theta\\left(y_{k+1} | \\boldsymbol x_{k+1}, O_k\\right) + \\log p\\left(y_{k+1} | \\lambda_\\phi\\left(\\boldsymbol x_{k+1}\\right)\\right) \\right>\n\\end{equation*}\n\nWe found that as as the size of the observation set increased, the predictive accuracy improved up to several hundred observations and then began to saturate. We also found that increasing the maximum set size used during training had a slight benefit in the asymptotic performance when increasing from 512 to 1024 trials, but with little benefit beyond this (Fig. \\ref{fig:accuracy}). These were averaged over three different seeds, with each fit producing similar performance. \n\nDespite trying a number of architecture variations, we could not get the asymptotic performance to quite reach ground truth. However, it performed well, with a $\\Delta LL$ of $0.4$ corresponding to a correlation coefficient between the ground truth mean response, $\\lambda_\\phi\\left(\\boldsymbol x\\right)$, and the model prediction mean, $\\lambda_{K=1024}(\\boldsymbol x)$, of $0.8$. \n\n\\subsection{Latent variables accurately capture  the tuning function}\n\\label{section:reconstruction}\n\n\\begin{figure}[b]\n  \\centering\n  \\includegraphics[width=0.8\\linewidth]{rf_accuracy.pdf}\n  \\caption{a) the correlation between the location latent variable, $z_k^p$, and the ground truth for increasing observations. b) Reconstruction of receptive fields (RF). Each row corresponds to a different cell with the bottom half being complex cells. The first column shows the ground truth kernels, the second column is the RF reconstructed by the gradient method, and the remaining block shows the maximally exciting images computed using increasing numbers of observations. Ground truth kernels of complex cells use pseudocolor to reflect the two phases in the energy model and any reconstruction of this energy model with the same orientation and location is equally valid, regardless of the phase.}\n  \\label{fig:rf_accuracy}\n\\end{figure}\n\nWe confirmed the information about the tuning function was correctly factorized by computing the correlation coefficient between the latent variable for location, $\\boldsymbol z_k^p$, and the ground truth location of the kernel, $k_\\phi$. We found that with only 64 observations there was a correlation of $0.8$, and it reached nearly $1.0$ with 256 observations (Fig.~\\ref{fig:rf_accuracy}a). \n\nWe then asked if the latent variables $(\\boldsymbol z_K^p, \\boldsymbol  z_K^w)$ from 1024 observations were sufficient to reconstruct the receptive field. First, we computed receptive fields as the gradient of the tuning function conditioned on the latent variables: \n\\begin{equation*}\n    RF_\\nabla^K = \\nabla_{\\boldsymbol x} \\left([\\mathcal T \\left( g_\\theta(\\boldsymbol x), \\boldsymbol z_K^p \\right), 1] \\cdot u_\\theta(\\boldsymbol z_K^w) \\right)\n\\end{equation*}\nFor simple cells the gradient showed a good correspondence to the kernel used to generate the responses, $k_\\phi$ (Fig. \\ref{fig:rf_accuracy}b). For complex cells it was in the correct location, but did not show the same structure as the kernel. This is expected as complex cells are not well described by a single kernel. We then computed the maximally exciting images (MEIs) for a neuron similarly to \\citet{Walker2019} by maximizing the predicted response, conditioned on the latent variables sampled after an increasing number of observations:\n\\begin{equation*}\n    MEI_K = \\argmax_{\\boldsymbol x} \\left([\\mathcal T \\left( g_\\theta(\\boldsymbol x), \\boldsymbol z_K^p \\right), 1] \\cdot u_\\theta(\\boldsymbol z_K^w) \\right) - \\kappa \\Vert \\boldsymbol x \\Vert\n\\end{equation*}\nWith $\\kappa=0.01$ to regularize the images.  As desired, MEIs computed with more observations converged towards the ground truth kernels, with complex cells having an anticipated random phase offset.\n\n\\subsection{Latent dimensionality and stimulus complexity}\n\\label{section:tuning_complexity}\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\linewidth]{latent.pdf}\n  \\caption{\\textbf{Left}: The difference between the predictive log likelihood and ground truth as the dimension of the tuning function properties latent variable increases. Each line reflects an increasingly complex RF space. \\textbf{Right}: The correlation between the ground truth mean firing rate and the predicted mean firing rate for the same data.}\n  \\label{fig:latent_dimension}\n\\end{figure}\n\nWe also studied how important the tuning function property's latent dimension $D$, with $\\boldsymbol z_K^w \\in \\mathbb R^D$, was to the predictive performance by increasing it from 2 to 64 (all experiments above used 64). We did this with different complexities of the simulated receptive fields by reducing the number of parameters in $\\phi$ that were randomized. In all experiments the orientation and location of the tuning function was randomized ($\\phi \\in \\mathbb R^3$). We increased the tuning function dimensions by then additionally randomizing (in order): frequency, width, phase offset, simple only versus simple and complex cells, and scale. Because this analysis involved refitting many models, we performed it with $16\\times 16$ stimuli. We found the performance improved with greater model capacity (tuning function properties latent dimension) and this impact was much more pronounced for more complex (higher dimensional) tuning functions (Fig.~\\ref{fig:latent_dimension}). Randomizing the phase offset produced the greatest reduction in predictive accuracy, although performance still remained quite good with high correlations between the model predictions and ground truth. Encouragingly, including complex cells did not produce a significant change in performance.\n\n\\section{Experiments with real neural responses}\n\nWe next tested our approach on real visual responses recorded with the same experimental paradigm as in \\citet{Walker2019}, and found it had a comparable predictive performance to optimization-based approaches. The data consists of pairs of neural population responses and grayscale visual stimuli sampled and cropped from ImageNet, isotropically downsampled to $64\\times 36$\\,px, with a resolution of $0.53$\\,ppd (pixels per degree of visual angle). The neural responses were recorded from layer L2\/3 of the primary visual cortex (area V1) of the mouse, using a wide field two photon microscope. A single scan contained the responses of approximately 5000--9000 neurons to up to 6000 images. \n\nWe trained an FNP on 57,533 mouse V1 neurons collected across 19 different scans and tested it on 1000 neurons from a hold-out scan (i.e. never seen during training). \nDuring testing, we assigned the latent variables assigned to their mean values: $\\boldsymbol z_K^p:=\\boldsymbol \\mu(s_K^p)$ and $\\boldsymbol z_K^w:=\\boldsymbol \\mu_K^w$, and used these in Eq.~\\ref{eq:conditional_decoder}. \nWe measured the $K$-shot predictive accuracy for each neuron as the correlation between the predicted mean from the conditional decoder, $\\lambda(\\boldsymbol x_t, \\boldsymbol z_K^p, \\boldsymbol z_K^w)$, and the real responses, $y_t$, for the remaining trials. \nIn agreement with synthetic data, the predictive accuracy improves rapidly with the first several hundred trials and continues to improve with additional observations (Fig.~\\ref{fig:real_predictions}). \nWe compared the performance of our FNP to an optimization based approach similar to \\citet{Klindt2017}, adapted for mouse V1, which we reference as Per Neuron Optimization (PNO). We measured the predictive performance of PNO similarly to FNP, on the same 1000 neurons with the readout optimized with $K$ trials and used to predict the response to the remaining stimuli. \nExcitingly, FNP performs well and with 1k images is almost as accurate as PNO (which is optimized for those individual cells), and even \\emph{outperforms} it for smaller numbers of observations (Fig.~\\ref{fig:real_predictions}). This likely arises because the FNP learns the prior distribution over tuning functions, which has a greater influence with less data. Please see the Appendix for details of both FNP and PNO fitting and testing.\n\nThese experiments also demonstrated the speed improvements for inferring the tuning function of a newly recorded neurons that FNP was designed for. While fitting the FNP to the training data took 6 days using two V100 GPUs, computing the latent variables for one thousand neurons with $K=1000$ took only 250~ms on a 1080Ti. This is in comparison to PNO which takes from 20~s to compute the readout using a pretrained CNN (Supplementary Table~\\ref{table:times}). Thus an FNP is two orders of magnitude faster, enabling real-time inference of tuning functions within the time of a single stimulus presentation.\n\n\\begin{figure}\n  \\begin{center}\n\n  \\includegraphics[width=0.5\\linewidth]{figures\/optimization_v_prediction.pdf}\n\n  \\end{center}\n  \\caption{Performance of a FNP for $K$-shot prediction for new neurons compared a traditional approach with per-neuron optimization (PNO) for $K$ up to 1000 trials}\n  \\label{fig:real_predictions}\n\\end{figure}\n\n\\section{Discussion}\n\nUsing a Factorized Neural Process, we are able to learn a distribution over tuning functions that can be conditioned on a set of observed stimulus-response pairs and predict the response to novel stimuli. We first focused on simulated data from simple and complex cells where we could compare the inferred tuning functions to the ground truth. Importantly, the model performed equally well when including complex cells, which is not possible for classical techniques like spike-triggered average that similarly accumulate sufficient statistics. The fact that the asymptotic log likelihood for predictions did not reach the ground truth also indicates there is room to increase the model capacity, although the correlation between the ground truth and model predictions exceeded $0.8$. \nFollowing prior work \\cite{Klindt2017, Ecker2019, Walker2019,Sinz2018}, we restricted ourselves to a decoder that was a factorized linear readout on output of $g_\\theta$, but learning a more powerful decoder could also improve the capacity. \nWe then tested our approach on data from the mouse primary visual cortex in response to natural images. We found the trained FNP predicted the responses to test data with comparable accuracy as a model specifically optimized for those neurons, and even exceeded the performance when conditioned on less than 500 trials. Additionally, the FNP made these predictions orders of magnitudes more quickly than an optimization-based approach, thus opening the door to real-time, closed-loop inference of tuning functions updated after every stimulus presentation.\n \nThis work was motivated by real-time experimentation, but during an experiment the best way to know how a neuron responds to a stimulus is to measure it. The real need is using the observations to rapidly generate stimuli to test a hypothesis. We envision combining a FNP for rapid inference with a generator network that takes the latent representations as input and is trained \\textit{in silico} prior to experiments to generate stimuli to illicit a maximal responses or reduce the uncertainty in the latent representations. We believe this general approach of training a more powerful model prior to experiments that is capable of rapid, real-time inference will be a powerful tool for the neuroscience community, and that this approach using FNPs will facilitate it.\n\n\\section*{Broader Impact}\n\nWe hope this approach will be useful to the Neuroscience community and that Factorized Neural Processes may have even broader applications for modeling functions. The ability to perform real-time, closed-loop experiments and to performances inferences with less data may reduce the amount of time to record from animals or the number of experimental sessions. \nWe do not believe this methodology or the demonstrated application will disadvantage anyone.\n\n\\begin{ack}\nRJC thanks the Research Accelerator Program of the Shirley Ryan AbilityLab for support during residency.\nFHS is supported by the Carl-Zeiss-Stiftung and acknowledges the support of the DFG Cluster of Excellence \"Machine Learning \u2013 New Perspectives for Science\", EXC 2064\/1, project number 390727645.\nSupported by the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior\/Interior Business Center (DoI\/IBC) contract number D16PC00003. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Disclaimer: The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IARPA, DoI\/IBC, or the U.S. Government. \\end{ack}\n\n\\small\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Large parton densities in the nuclear wave function}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=5cm,angle=-90]{dhj-evol.eps}\n\\hspace{0.5cm}\n\\includegraphics[width=5cm,angle=-90]{dhj-pred.eps}\n\\caption{Forward particle production in d+Au collisions at RHIC. The left plot shows the importance of including both the large-$x$ DGLAP evolution of the dilute deuteron and the\nsmall-$x$ CGC evolution of the dense nucleus. The right plots shows the excellent description of the spectra shapes, and the K factors needed to obtain the normalization.}\n\\end{center}\n\\end{figure}\n\nWhen probing small distances inside a hadron or nucleus with a hard process, one resolves their partonic constituents. Increasing the energy of the scattering process at a fixed momentum transfer allows to probe lower-energy partons, with smaller energy fraction $x.$ As the parton densities in the hadronic\/nuclear wave function grow with decreasing $x,$ they eventually become so large that a non-linear (yet weakly-coupled) regime is reached, called saturation, where partons do not interact with the probe independently anymore, but rather behave coherently. \n\nThe Color Glass Condensate (CGC) is an effective theory of QCD \\cite{cgcrev} which aims at describing this part of the wave function. Rather than using a standard Fock-state decomposition, it is more efficient to describe it with collective degrees of freedom, more adapted to account for the collective behavior of the small-$x$ gluons. The CGC approach uses classical color fields: \n\\begin{equation}\n|h\\rangle=|qqq\\rangle+|qqqg\\rangle+\\dots+|qqqg\\dots ggg\\rangle+\\dots\\quad\n\\Rightarrow\\quad|h\\rangle=\\int D\\rho\\ \\Phi_{x_A}[\\rho]\\ |\\rho\\rangle\n\\label{cgc}\\ .\\end{equation}\nThe long-lived, large-$x$ partons are represented by a strong color source\n$\\rho\\!\\sim\\!1\/g_S$ which is static during the lifetime of the short-lived small-$x$ gluons, whose dynamics is described by the color field $A^\\mu\\!\\sim\\!1\/g_S.$ The arbitrary separation between the field and the source is denoted $x_A.$\n\nWhen probing the CGC with a dilute object carrying a weak color charge, the color field $A^\\mu$ is directly obtained from $\\rho$ via classical Yang-Mills equations:\n\\begin{equation}\n[D_\\mu,F^{\\mu\\nu}]=\\delta^{-\\nu}\\rho\\ ,\n\\end{equation}\nand it can be used to characterize the CGC wave function $\\Phi_{x_A}[A^-]$\n(in the $A^+\\!=\\!0$ gauge). This wave function is a fundamental object of this picture, it is mainly a non-perturbative quantity, but the $x_A$ evolution can be computed perturbatively. Requiring that observables are independent of the choice of $x_A,$ a functional renormalization group equation can be derived. In the leading-logarithmic approximation which resums powers of $\\alpha_S\\ln(1\/x_A),$ the JIMWLK equation describes the evolution of $|\\Phi_{x_A}|^2$ with $x_A.$\n\nThe information contained in the wave function, on gluon number and gluon correlations, can be expressed in terms of n-point correlators, probed in scattering processes. These correlators consist of Wilson lines averaged with the CGC wave function, and resum powers of $g_S A^-$ (therefore both multiple scatterings and non-linear QCD evolution are taken into account). For instance in the case of single inclusive gluon production in pA collisions, the CGC is described by its (all-twist) unintegrated gluon distribution, obtained from the 2-point function \\cite{mygprod}. More exclusive observables involve more complicated correlators. \n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=6.8cm]{dau.eps}\n\\hspace{0.5cm}\n\\includegraphics[width=6.8cm]{ppb.eps}\n\\caption{Two-particle production at forward rapidities in pA collisions. The $\\Delta\\phi$ spectrum is displayed at RHIC (left) and LHC (right) energies. When decreasing $p_{T_2}$ at fixed $y_2,$ the correlation in azimuthal angle is suppressed. At the LHC, smaller values of $x_A$ are probed, and the azimuthal angle correlation is more suppressed as indicated by the vertical axis; the peak is also less pronounced.}\n\\end{center}\n\\end{figure}\n\nForward particle production in pA collisions allows to investigate the non linear QCD dynamics of high-energy nuclei with a probe well understood in QCD. Indeed, while such processes are probing small-momentum partons in the nuclear wave function, only high-momentum partons of the proton contribute to the scattering ($\\sqrt{s} x_p\\!=\\!k e^y$ and $\\sqrt{s} x_A\\!=\\!k e^{-y}$ with $k$ and $y$ denoting transverse momentum and rapidity), and that involves standard parton distribution functions. In two-particle production, contrary to single particle production, the CGC cannot be described only by its unintegrated gluon distribution, the so-called $k_T$-factorization framework is not applicable.\n\nIt was not obvious that the CGC picture (\\ref{cgc}), which requires small values of $x_A,$ would be relevant at present energies. One of the most acclaimed successes came in the context of d+Au collisions at RHIC: the prediction that the yield of high-$p_T$ particles at forward rapidities in d+Au collisions is suppressed compared to A pp collisions, and should decrease when increasing the rapidity, was confirmed\n\\cite{jyrev}. In Fig.1 the $dAu\\!\\to\\!hX$ $p_T$ spectra computed in the CGC approach \\cite{dhj} is compared to RHIC data, and the description of the slope is impressive. The need of K factors to describe the normalization could be expected since this is a leading-order based calculation. Improving the calculation with the next-leading evolution has yet to be done.\n\nThe focus should now shift towards more exclusive observables like two-particle production $pA\\!\\to\\!h_1h_2X.$ In particular the correlations in azimuthal angle between the produced hadrons should be suppressed compared to pp collisions. Predictions for the process $pA\\to h_1h_2X$ are shown in Fig.2, for RHIC and the LHC \\cite{mytpc}. $k_1,$ $k_2$ and $y_1,$ $y_2$ are the transverse momenta and rapidities of the final state hadrons, and the azimuthal angle spectra are displayed. It is obtained that the perturbative back-to-back peak of the azimuthal angle distribution is reduced by initial state saturation effects. As the momenta decrease, the angular distribution broadens.\n\n\\section{Particle production in the Glasma}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=6.5cm]{mpi.eps}\n\\hspace{0.5cm}\n\\includegraphics[width=7.5cm]{horizon.eps}\n\\caption{Left: typical leading-order diagram for particle production in the Glasma, multiple partonic interactions are crucial when low values of $x$ are being probed in the nuclear wave functions. Right: the space-time location of events that may correlate two particles is the intersection of their past light-cones. Correlations between particles widely separated in rapidity are due to early-time dynamics.}\n\\end{center}\n\\end{figure}\n\nThe Glasma is the result of the collision of two CGCs, and it provides a weak-coupling description of the early stages after a high-energy heavy-ion collision. Each nuclear wave function is characterized by a strong color charge, and the field describing the dynamics of the small-x gluons is the solution of\n\\begin{equation}\n[D_\\mu,F^{\\mu\\nu}]=\\delta^{+\\nu}\\rho_1+\\delta^{-\\nu}\\rho_2\\ .\n\\end{equation}\nThe field after the collision is non-trivial \\cite{glasma}: it has a strong component ($A^\\mu\\sim1\/g_s$), a component which is particle like ($A^\\mu\\sim1$), and components of any strength in between. To understand how this pre-equilibrium system thermalizes, one needs to understand how the Glasma field decays into particles. Right after the collision, the strong field component contains all modes.\nThen, as the field decays, modes with $p_T>1\/\\tau$ are not part of the strong component anymore, and for those a particle description becomes more appropriate. After a time of order $1\/Q_s,$ this picture breaks down, and it has been a formidable challenge to determine weather a fast thermalization can be achieved within this framework.\n\nA problem which can be more easily addressed is particle production. The difficult task is to express the cross-section in terms of the Glasma field, taking into account multiple partonic interactions, as pictured in Fig.3 (left). Because of the flux-tube structure of its color field $A^\\mu,$ the Glasma is a natural candidate to explain the ridge-shaped two-particle correlations observed at RHIC, as well as three-particle correlations \\cite{ridge}. The ridge is collimated in azimuthal angle because of the radial flow which happens at a later stage, but since the ridge is several units long in rapidity, it is due to early time dynamics: this is explained in Fig.3 (right) which shows the space-time picture of the collision. In the forward light-cone, lines of constant proper time $\\tau=\\sqrt{x^+x^-}$ are hyperbolae and lines of constant rapidity $\\eta=\\frac12\\log(x^+\/x^-)$ are straight lines from the origin. For two final-state particles separated by the rapidity $\\Delta\\eta,$ causality imposes that they can be correlated only by events which happened at\n\\begin{equation}\n\\tau<\\tau_{f.o.}\\ e^{-\\Delta\\eta\/2}\\ ,\n\\end{equation}\nwhere the freeze-out time $\\tau_{f.o.}$ denote the time of last interaction. While the features of the ridge are qualitatively explained by the Glasma, a quantitative description is needed.\n\n\\section{Energy loss of high-$p_T$ partons in the QCD plasma}\n\nHard probes are believed to be understood well enough to provide clean measurements of the properties of the QGP formed in heavy-ion collisions. A large amount of work has been devoted to understand what happens to a quark (of high energy $E,$ mass $M$ and Lorentz factor  $\\gamma=E\/M$) as it propagates through a thermalized plasma\n\\cite{jqrev}. Multiple scatterings are a main ingredient of the perturbative QCD (pQCD) description of how a quark losses energy, until it thermalizes or exits the medium (see Fig.4).\n\nAt lowest order with respect to $\\alpha_s,$ quantum fluctuations in a quark wave function consist of a single\ngluon, whose energy we denote $\\omega$ and transverse momentum $k_\\perp.$ The virtuality of that\nfluctuation is measured by the coherence time, or lifetime, of the gluon $t_c=\\omega\/k_\\perp^2.$\nShort-lived fluctuations are highly virtual while longer-lived fluctuations are more easily put on shell when\nthey interact. The probability of the fluctuation is $\\alpha_sN_c,$ up to a kinematic factor which for heavy\nquarks suppresses fluctuations with $\\omega>\\gamma k_\\perp.$ This means that when gluons are put\non-shell, they are not radiated in a forward cone around a heavy quark. This suppression of the available\nphase space for radiation, the {\\it dead-cone} effect, implies less energy loss for heavier quarks.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=4.5cm]{cartoon.eps}\n\\hspace{1cm}\n\\includegraphics[width=6.5cm]{raa.eps}\n\\caption{Left: production of high-energy partons in a hard process, which then lose energy propagating through the plasma. Some quantum fluctuations in their wave function are put on shell while interacting with the medium and become emitted radiation.\nRight: the resulting particle production in AA collisions is suppressed ($R_{AA}<1$) compared to independent nucleon-nucleon collisions. The suppression is large for light hadrons, and similar for heavy mesons (those data are displayed in the figure), which is difficult to accommodate in a weakly-coupled QCD description.}\n\\end{center}\n\\end{figure}\n\nIn pQCD, medium-induced gluon radiation is due to multiple scatterings of the virtual gluons.\nIf, while undergoing multiple scattering, the virtual gluons pick up enough transverse momentum to be put on shell,\nthey become emitted radiation. The accumulated transverse momentum squared picked up by a gluon of coherence\ntime $t_c$ is\n\\begin{equation}\np_\\perp^2=\\mu^2 \\frac{t_c}{l}\\equiv\\hat{q}\\ t_c\n\\end{equation}\nwhere $\\mu^2$ is the average transverse momentum squared picked up in each scattering, and\n$l$ is the mean free path. These medium properties are involved through the ratio\n$\\hat{q}=\\mu^2\/l.$\n\nSince only the fluctuations which pick up enough transverse momentum are freed ($k_\\perp<p_\\perp$),\nthe limiting value can be obtained by equating $k_\\perp^2$ with\n$p_\\perp^2=\\hat{q}\\omega\/k_\\perp^2:$\n\\begin{equation}\nk_\\perp<(\\hat{q}\\omega)^{1\/4}\\equiv Q_s(\\omega)\\ .\n\\end{equation}\nThe picture is that highly virtual fluctuations with $k_\\perp>Q_s$ do not have time to pick up enough $p_\\perp$ to be freed, while the longer-lived ones with $k_\\perp<Q_s$ do. That transverse momentum $Q_s$ which controls which gluons are freed and which are not is called the saturation scale. With heavy quarks, one sees that due to the dead cone effect, the maximum energy a radiated gluon can have is $\\omega=\\gamma k_\\perp=\\gamma Q_s$ (and its coherence time is $t_c=\\gamma\/Q_s)$. This allows to\nestimate the heavy-quark energy loss:\n\\begin{equation}\n-\\frac{dE}{dt}\\propto\\alpha_sN_c\\frac{\\gamma Q_s}{\\gamma\/Q_s}=\\alpha_s N_c Q_s^2\\ .\n\\label{eloss}\\end{equation}\nThe saturation momentum in this formula is the one that corresponds to the fluctuation which dominates the energy loss: $Q_s=(\\hat{q}\\gamma)^{1\/3}.$\n\nFor a plasma of extend $L\\!<\\!t_c=\\gamma^{2\/3}\/\\hat{q}^{1\/3},$ formula (\\ref{eloss}) still holds but with $Q_s^2=\\hat{q}L.$ These are the basic ingredients of more involved phenomenological calculations, but after comparisons with data, it has remained unclear if this perturbative approach can describe the suppression of high$-p_\\perp$ particles. For instance, at RHIC temperatures, the value $\\hat{q}\\sim 1-3\\ \\mbox{GeV}^2\/\\mbox{fm}$ is more natural than the $5-10\\ \\mbox{GeV}^2\/\\mbox{fm}$ needed to describe the data on light hadron production. If one accepts to adjust $\\hat{q}$ to this large value, then the $D$ and $B$ mesons are naturally predicted to be less suppressed than light hadrons, which is not the case (see Fig.4).\n\nWhile the present pQCD calculations should still be improved, and may be shown to work in the future, this motivated to think about strongly-coupled plasmas. The tools to address the strong-coupling dynamics in QCD are quite limited, however for the $N=4$ Super-Yang-Mills (SYM) theory, the AdS\/CFT correspondence is a powerful approach used in many studies. The findings for the strongly-coupled SYM plasma may provide insight for gauge theories in general, and some aspects may even be universal. One interesting result is that the total energy loss of hard probes goes as $\\Delta E\\propto L^3$ at strong coupling \\cite{us}, instead of the $L^2$ law at weak coupling.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nMagnetic fields are observed on various spatial scales of the\nuniverse: they are detected in planets and stars \\citep{DL09,R12},\nin the interstellar medium \\citep{C12}, and on galactic scales \\citep{B16}.\nAdditionally, observational lower limits on intergalactic\nmagnetic fields have been reported\n\\citep{NV10,DCRFCL11}.\nContrary to the high magnetic field strengths observed on scales below\nthose of galaxy clusters, which can be explained by dynamo amplification\n\\citep[see, e.g.,][]{BS05},\nintergalactic magnetic fields, if confirmed, are most likely of primordial origin.\nBecause of their often large energy densities, magnetic fields can play\nan important role in various astrophysical objects, a prominent\nexample being the $\\alpha\\Omega$ dynamo in solar-like stars that explains\nstellar activity\n\\citep[see, e.g.,][]{P55,P79,M78,KR80,ZRS83,C14}.\n\nWhile there is no doubt about the significant role of magnetic fields in the\ndynamics of the\npresent-day universe, their origin and evolution over cosmic times remain a\nmystery \\citep{R87,GR01,W02,KZ08}.\nNumerous scenarios for the generation of primordial magnetic fields have been\nsuggested\nin the literature. The proposals span from inflation-produced magnetic fields\n\\citep{TW88} to field generation during cosmological phase transitions\n\\citep{SOJ97}.\nEven though strong magnetic fields\ncould be generated shortly after the Big Bang, their strength subsequently\ndecreases in cosmic expansion unless they undergo further amplification.\nBe this as it may, the presence of primordial magnetic fields can affect the\nphysics of the early universe.\nFor example, it has been shown that primordial fields could have\nsignificant effects on the matter power spectrum by suppressing the formation\nof small-scale structures \\citep{KEtAl13,PEtAl15}.\nThis, in turn, could influence cosmological structure formation.\n\nThe theoretical framework for studying the evolution of magnetic fields is\nmagnetohydrodynamics (MHD).\nIn classical plasma physics,\nthe system of equations includes the induction equation,\nwhich is derived from the Maxwell equations and Ohm's law and describes\nthe evolution of magnetic fields,\nthe continuity equation for the fluid density,\nand the Navier-Stokes equation governing the evolution of the\nvelocity field.\n\nAt high energies, for example, in the quark-gluon plasma of the early universe,\nhowever, an additional quantity needs to be taken into account, namely\nthe chiral chemical potential.\nThis quantity is related to an asymmetry between the number densities of\nleft-handed fermions (spin antiparallel to the momentum) and right-handed\nfermions (spin parallel to the momentum). This leads to an additional \ncontribution to the electric current along the magnetic field, known as the\n{\\em chiral magnetic effect} (CME).\nThis phenomenon was discovered by \\cite{Vilenkin:80a} and\nwas later carefully investigated using different\ntheoretical approaches in a number of studies \\citep{RW85,Tsokos:85,\nJS97,AlekseevEtAl1998,Frohlich:2000en,Frohlich:2002fg,\nFukushima:08,Son:2009tf}.\n\nThe CME causes a small-scale dynamo instability \\citep{JS97}, which has also\nbeen revealed from a kinetic description of chiral plasmas \\citep{AY13}.\nThe evolution equation for a nonuniform\nchiral chemical potential has been derived in \\citet{BFR12,BFR15} who used it\nto study the inverse magnetic cascade that results in an\nincrease of the characteristic scale of the magnetic field.\n\\citet{BFR12} have shown that the chiral asymmetry can survive down to energies\nof the order of $10\\,{\\rm MeV}$, due to coupling to an effective axion field.\nThese studies triggered various investigations related to\nchiral MHD turbulence \\citep{Pavlovic:2016gac,Y16}\nand its role in the early universe \\citep{TVV12,DvornikovSemikoz2017},\nas well as in neutron stars \\citep{DvornikovSemikoz2015,SiglLeite2016}.\n\nRecently, a systematic theoretical analysis of the system of \\emph{chiral MHD}\nequations, including the back-reaction of the magnetic field on the chiral\nchemical potential, and the coupling to the plasma velocity field has been\nperformed in \\cite{REtAl17}, referred to here as Paper~I.\nThe main findings of Paper~I include a modification of MHD waves by the\nCME and different kinds of laminar and turbulent dynamos.\nBesides the well-studied laminar chiral dynamo caused by\nthe CME, a chiral--shear dynamo in the presence of a shearing velocity\nwas discussed there.\nIn addition, a mean-field theory of chiral MHD in the presence\nof small-scale nonhelical turbulence was developed in Paper~I,\nwhere a new chiral $\\alpha_\\mu$ effect\nnot related to a kinetic helicity has been found.\nThis effect results from an interaction of chiral magnetic fluctuations\nwith fluctuations of the electric current caused by the tangling magnetic\nfluctuations.\n\nIn the present paper, we report on\nnumerical simulations that confirm and further substantiate the chiral\nlaminar and turbulent dynamos found in Paper~I.\nTo this end, we have implemented the chiral MHD equations in the\n{\\sc Pencil Code}\\footnote{http:\/\/pencil-code.nordita.org\/},\na high-order code suitable for compressible MHD turbulence.\nDifferent situations are considered, from laminar dynamos\nto chiral magnetically driven turbulence\nand large-scale dynamos in externally forced turbulence.\nWith our direct numerical simulations (DNS), we are able to study the dynamical\nevolution of a plasma that includes chiral effects in a large domain of the\nparameter space.\nGiven that the detailed properties of relativistic astrophysical plasmas,\nin particular the initial chiral asymmetry and chiral feedback mechanisms,\nare not well understood at present, a broad analysis of various\nscenarios is essential.\nThe findings from DNS can then be used to explore the possible evolution of \nastrophysical plasmas under different assumptions. \nThese applications should not be regarded as realistic descriptions of \nhigh-energy plasmas; they aim to find out under what conditions the\nCME plays a significant role in the evolution of a\nplasma of relativistic charged fermions (electrons) and to test the\nimportance of chirality flips changing the handedness of the fermions.\nWe are not pretending that the regimes accessible to our simulations\nare truly realistic in the context of the physics of the early universe or in\nneutron stars.\n\nThe outline of the present paper is as follows.\nIn Section~\\ref{sec:simulations} we review the governing equations and\nthe numerical setup,\nand we discuss the physics related to the two main nonlinear effects\nin chiral MHD, which lead to different scenarios of the magnetic\nfield evolution.\nIn Section~\\ref{sec:lamdyn} we present numerical results on laminar chiral\ndynamos.\nIn Section~\\ref{sec_turbdynamo1} we show how magnetic fields, amplified by\nthe CME, produce turbulence (chiral magnetically driven turbulence).\nWe discuss how this gives rise to the chiral $\\alpha_\\mu$ effect.\nWe also study this effect in Section~\\ref{sec_turbdynamo} for a system\nwhere external forcing is employed to produce turbulence.\nAfter a discussion of chiral MHD in astrophysical and cosmological processes in\nSection~\\ref{sec_astro}, we draw conclusions in Section~\\ref{sec_concl}.\n\n\n\n\\section{Chiral MHD in numerical simulations}\n\\label{sec:simulations}\n\n\\subsection{Equations of chiral MHD}\n\\label{sec:eqs-chiral MHD}\n\nThe system of chiral MHD equations\nincludes the induction equation for the magnetic field $\\bm{B}$,\nthe Navier-Stokes equation for the velocity field $\\bm{U}$ of the plasma,\nthe continuity equation for the plasma density $\\rho$,\nand an evolution equation for the normalized chiral chemical potential\n$\\mu$:\n\\begin{eqnarray}\n\\frac{\\partial \\bm{B}}{\\partial t} &=&\n\\bm{\\nabla}   \\times   \\left[{\\bm{U}}  \\times   {\\bm{B}}\n- \\eta \\, \\left(\\bm{\\nabla}   \\times   {\\bm{B}}\n- \\mu {\\bm{B}} \\right) \\right] ,\n\\label{ind-DNS}\\\\\n\\rho{D \\bm{U} \\over D t}&=& (\\bm{\\nabla}   \\times   {\\bm{B}})  \\times   \\bm{B}\n-\\bm{\\nabla}  p + \\bm{\\nabla}  {\\bm \\cdot} (2\\nu \\rho \\mbox{\\boldmath ${\\sf S}$} {})\n+\\rho \\bm{f} ,\n\\label{UU-DNS}\\\\\n\\frac{D \\rho}{D t} &=& - \\rho \\, \\bm{\\nabla}  \\cdot \\bm{U} ,\n\\label{rho-DNS}\\\\\n\\frac{D \\mu}{D t} &=& D_5 \\, \\Delta \\mu\n+ \\lambda \\, \\eta \\, \\left[{\\bm{B}} {\\bm \\cdot} (\\bm{\\nabla}   \\times   {\\bm{B}})\n- \\mu {\\bm{B}}^2\\right]\n -\\Gamma_{\\rm\\!f}\\mu,\n\\label{mu-DNS}\n\\end{eqnarray}\nwhere $\\bm{B}$ is normalized such that the magnetic energy\ndensity is $\\bm{B}^2\/2$ (without the $4\\pi$ factor), and\n$D\/D t = \\partial\/\\partial t + \\bm{U} \\cdot \\bm{\\nabla}$ is the\nadvective derivative.\nFurther, $\\eta$ is the microscopic magnetic diffusivity,\n$p$ is the fluid pressure,\n${\\sf S}_{ij}={\\textstyle{1\\over2}}(U_{i,j}+U_{j,i})-{\\textstyle{1\\over3}}\\delta_{ij} {\\bm \\nabla}\n{\\bm \\cdot} \\bm{U}$\nare the components of the trace-free strain tensor $\\mbox{\\boldmath ${\\sf S}$} {}$ (commas denote\npartial spatial derivatives),\n$\\nu$ is the kinematic viscosity,\nand $\\bm{f}$ is the turbulent forcing function.\n\nEquation~(\\ref{mu-DNS}) describes the evolution of the chiral\nchemical potential $\\mu_5 \\equiv\\mu_L - \\mu_R$, with $\\mu_L$ and $\\mu_R$ being\nthe chemical potentials of left- and right-handed chiral fermions, which is normalized such that\n$\\mu = (4 \\ensuremath{\\alpha_{\\rm em}} \/\\hbar c) \\mu_5$ has the dimension of an inverse length.\nHere $D_5$ is the diffusion constant of the chiral chemical potential $\\mu$, and\nthe parameter $\\lambda$, referred to in Paper~I as\nthe \\textit{chiral feedback parameter},\ncharacterizes the strength of the\ncoupling of the electromagnetic field to $\\mu$.\nThe expression of the feedback term in Equation~(\\ref{mu-DNS}) was\nderived in Paper~I and is valid for the limit of small magnetic\ndiffusivities.\nFor hot plasmas, when $k_{\\rm B} T \\gg \\max(|\\mu_L|,|\\mu_R|)$,\nthe parameter $\\lambda$ is given\nby\\footnote{The definition of $\\lambda$ in the case of a degenerate Fermi\ngas will be given in Section~\\ref{sec:PNS}.}\n\\begin{eqnarray}\n  \\lambda=3 \\hbar c \\left({8 \\ensuremath{\\alpha_{\\rm em}} \\over k_{\\rm B} T} \\right)^2,\n\\label{eq_lambda}\n\\end{eqnarray}\nwhere $\\ensuremath{\\alpha_{\\rm em}} \\approx 1\/137$ is the fine structure constant,\n$T$ is the temperature, $k_{\\rm B}$ is the Boltzmann constant,\n$c$ is the speed of light, and $\\hbar$ is the reduced Planck constant.\nWe note that\n$\\lambda^{-1}$ has the dimension of energy per unit length.\nThe last term in Equation~(\\ref{mu-DNS}), proportional to $\\Gamma_{\\rm\\!f} \\mu$,\ncharacterizes chirality flipping processes due to finite fermion masses.\nThis term is included in a phenomenological way.\nThe detailed\ndependence of $\\Gamma_{\\rm\\!f}$ on the plasma parameters in realistic systems\nis still not fully understood.\nIn most of the runs,\nthe chirality flipping effect is neglected because we concentrate in this paper\non the high-temperature regime, where the other terms in\nEquation~(\\ref{mu-DNS}) dominate.\nHowever, we study its effect on the nonlinear evolution of\n$\\mu$ in Section~\\ref{sec_flip}.\n\nWe stress that the effects related to the chiral anomaly cannot be\nseparated from the rest of the equations. This is one of the\nessential features of the chiral MHD equations that we are studying.\nThe equations interconnect the chiral chemical potential to the\nelectromagnetic field.\nHowever, the chiral anomaly couples the electromagnetic field\nnot directly to the chiral chemical potential but to\nthe chiral charge density, a conjugate variable in the sense of statistical\nmechanics. \nThe parameter $\\lambda$ is nothing but a susceptibility, that is,\na (inverse) proportionality coefficient\nquantifying the response of the\naxial charge to a change in the chiral chemical potential; see Paper~I.\n\nThe system of equations~(\\ref{ind-DNS})--(\\ref{mu-DNS}) and their range of\nvalidity have been discussed in detail in Paper~I.\nBelow we present a short summary of the assumptions made in deriving these\nequations.\nWe focus our attention on an isothermal plasma, $T=\\mathrm{const}$.\nThe equilibration rate of the temperature gradients is related to the shortest\ntimescales of the plasma (of the order of the plasma frequency or below) and is\nmuch shorter than the time-scales that we consider in the present study.\nFor an isothermal equation of state, the pressure $p$ is related to\nthe density $\\rho$ via $p=c_{\\rm s}^2\\rho$, where $c_{\\rm s}$ is the sound speed.\nWe apply a one-fluid MHD model that\nfollows from a two-fluid plasma model\n\\citep{Artsimovich-Sagdeev,Melrose-QuantumPlasmadynamicsMagnetized,Biskamp:97}.\nThis implies that we do not consider here kinetic effects and effects related to\nthe two-fluid plasma model.\nWe note that the MHD formalism is valid for scales above the mean free path that can be \napproximated as \\citep{ArnoldEtAl2000}\n\\begin{equation}\n  \\ell_\\mathrm{mfp} \\approx \\frac{1}{(4 \\pi \\alpha_\\mathrm{em})^2 \\ln{((4 \\pi \\alpha_\\mathrm{em})^{-1\/2})}} \\frac{\\hbar c}{k_{\\rm B} T}.\n\\label{eq_mfp}\n\\end{equation}\nFurther, we study the nonrelativistic bulk motion of a highly relativistic plasma.\nThe latter leads to a term in the Maxwell equations that destabilizes\nthe nonmagnetic equilibrium and causes an exponential growth of\nthe magnetic field.\nSuch plasmas arise in the description of certain astrophysical systems,\nwhere, for example,\na nonrelativistic plasma interacts with cosmic rays consisting of\nrelativistic particles with small number density;\nsee, for example, \\cite{Schlickeiser-book}.\nWe study the case of small magnetic diffusivity\ntypical of many astrophysical systems with large magnetic Reynolds numbers, so\nwe neglect terms of the order of $\\sim O(\\eta^2)$ in the electric field; see Paper I.\n\n\nA key difference in the induction equations of chiral\nand classical MHD is the last term\n$\\propto \\bm{\\nabla} \\times (\\eta \\, \\mu \\, {\\bm{B}})$ in Equation~(\\ref{ind-DNS}).\nThis is reminiscent of mean-field dynamo theory, where a mean magnetic field $\\overline{\\mbox{\\boldmath $B$}}{}}{$ is\namplified by an $\\alpha$ effect due to a term $\\propto \\bm{\\nabla} \\times (\\alpha \\, {\\overline{\\mbox{\\boldmath $B$}}{}}{})$\nin the mean-field induction equation, which results in an $\\alpha^2$ dynamo.\nIn analogy with mean-field dynamo theory, we use the name\n\\textit{$v_\\mu^2$ dynamo}, introduced in Paper~I, where\n\\begin{equation}\n  v_\\mu\\equiv\\eta\\mu_0\n  \\label{eq:16}\n\\end{equation}\nplays the role of $\\alpha$ (see Equation~(\\ref{ind-DNS})),\nand $\\mu_0$ is the initial value of the normalized chiral chemical potential.\nThese different notions are motivated by the fact that the $v_\\mu$ effect\nis not related to any turbulence effects; that is, it is not determined by\nthe mean electromotive force, but originates from the\nCME; see Paper~I for details.\nWe will discuss the differences between chiral and classical MHD in more\ndetail in Section~\\ref{sec:difference}.\n\nThe system of Equations~(\\ref{ind-DNS})--(\\ref{mu-DNS}) implies a conservation law:\n\\begin{equation}\n\\frac{\\partial }{\\partial t} \\left({\\lambda \\over 2}  {\\bm A} {\\bm \\cdot} \\bm{B}\n+ \\mu \\right) + \\bm{\\nabla}  {\\bm \\cdot} \\bm{F}_{\\rm tot} = 0,\n\\label{CL}\n\\end{equation}\nwhere\n\\begin{equation}\n\\bm{F}_{\\rm tot} =\n{\\lambda \\over 2} \\left({\\bm \\bm{E}} \\times   {\\bm A} + \\bm{B} \\, \\Phi\\right)\n-  D_5 \\bm{\\nabla}  \\mu\n\\end{equation}\nis the flux of total chirality and\n$\\bm{B} = {\\bm \\nabla} {\\bm \\times} {\\bm A}$,\nwhere ${\\bm A}$ is the vector potential, ${\\bm \\bm{E}}=\n- c^{-1} \\, \\{ {\\bm \\bm{U}} {\\bm \\times} {\\bm{B}} + \\eta \\, \\big(\\mu {\\bm{B}}  -\n{\\bm \\nabla} {\\bm \\times} {\\bm{B}} \\big) \\}$ is the electric field, $\\Phi$ is the\nelectrostatic potential, $\\lambda$ is assumed to be constant,\nand the chiral flipping term, $-\\Gamma_{\\rm\\!f}\\mu$,\nin Equation~(\\ref{mu-DNS}) is assumed to be negligibly small.\nThis implies that the total chirality is a conserved quantity:\n\\begin{equation}\n  \\label{cons_law}\n\\frac\\lambda 2 \\langle \\bm{A} \\cdot \\bm{B}\\rangle + \\langle \\mu \\rangle = \\mu_0 = \\mathrm{const},\n\\end{equation}\nwhere $\\langle \\mu \\rangle$\nis the value of the chemical potential and\n$ \\langle \\bm{A} \\cdot \\bm{B}\\rangle \\equiv V^{-1}\\int \\bm{A} \\cdot \\bm{B} \\, dV$\nis the mean magnetic helicity density over the volume $V$.\n\n\\subsection{Chiral MHD equations in dimensionless form}\n\nWe study the system of chiral MHD equations~(\\ref{ind-DNS})--(\\ref{mu-DNS})\nin numerical simulations to analyze various laminar and turbulent dynamos,\nas well as the production of turbulence by the CME.\nIt is, therefore, useful to rewrite this system of equations in\ndimensionless form, where velocity is measured in units of the sound speed\n$c_\\mathrm{s}$, length is measured in units of $\\ell_\\mu \\equiv \\mu_0^{-1}$, so time\nis measured in units of $\\ell_\\mu\/c_\\mathrm{s}$,\nthe magnetic field is measured in units of $\\sqrt{\\overline{\\rho}} \\, c_\\mathrm{s}$,\nfluid density is measured in units of $\\overline{\\rho}$\nand the chiral chemical potential is measured in units of $\\ell_\\mu^{-1}$,\nwhere $\\overline{\\rho}$ is the volume-averaged density.\nThus, we introduce the following dimensionless functions, indicated by a\ntilde:\n${\\bm \\bm{B}}=\\sqrt{\\overline{\\rho}} \\, c_\\mathrm{s}\\tilde{\\bm \\bm{B}}$,\n${\\bm \\bm{U}}=c_\\mathrm{s}\\tilde{\\bm \\bm{U}}$,\n$\\mu=\\ell_\\mu^{-1} \\tilde\\mu$\nand $\\rho= \\overline{\\rho} \\, \\tilde\\rho$.\nThe chiral MHD equations in dimensionless form are given by\n\\begin{eqnarray}\n\\frac{\\partial \\tilde\\bm{B}}{\\partial \\tilde t}  &=&   \\tilde{\\bm \\nabla}\n{\\bm \\times} \\biggl[\\tilde{\\bm \\bm{U}} {\\bm \\times} \\tilde{\\bm{B}}  + {\\rm Ma}_\\mu \\,\n\\Big(\\tilde\\mu \\tilde{\\bm{B}}  - \\tilde{\\bm \\nabla} {\\bm \\times} {\\tilde\\bm{B}} \\Big)\n\\biggr],\n\\label{ind-NS}\\\\\n\\tilde \\rho {D \\tilde \\bm{U} \\over D \\tilde t}&=& (\\tilde \\bm{\\nabla} \\times \\tilde{\\bm{B}})  \\times\n\\tilde \\bm{B}\n-\\tilde \\bm{\\nabla} \\tilde \\rho + {\\rm Re}_\\mu^{-1} \\tilde\\bm{\\nabla} {\\bm \\cdot} (2\\nu \\tilde\n\\rho \\mbox{\\boldmath ${\\sf S}$} {})\n+ \\tilde \\rho \\bm{f} ,\n\\nonumber\\\\\n\\label{UU-NS}\\\\\n\\frac{D \\tilde\\rho}{D \\tilde t} &=& - \\tilde\\rho \\, \\tilde\\bm{\\nabla}  \\cdot \\tilde\\bm{U} ,\n\\label{rho-NS}\\\\\n\\frac{D \\tilde\\mu}{D \\tilde t}  &=& D_\\mu \\, \\tilde\\Delta \\tilde\\mu\n+ \\Lambda_\\mu \\, \\Big[{\\tilde\\bm{B}} {\\bm \\cdot} (\\tilde\\bm{\\nabla} {\\bm \\times}\n{\\tilde\\bm{B}})\n- \\tilde\\mu {\\tilde\\bm{B}}^2 \\Big]\n- \\tilde \\Gamma_\\mathrm{f} \\tilde \\mu ,\n\\label{mu-NS}\n\\end{eqnarray}\nwhere we introduce the following dimensionless parameters:\n\\begin{itemize}\n\\item{Chiral Mach number:\n\\begin{eqnarray}\n{\\rm Ma}_\\mu = \\frac{\\eta\\mu_0}{c_\\mathrm{s}} \\equiv \\frac{v_{\\mu}}\n{c_\\mathrm{s}},\n\\label{Ma_mu_def}\n\\end{eqnarray}\n}\n\\item{Magnetic Prandtl number:\n\\begin{eqnarray}\n{\\rm Pr}_{_{\\rm M}} = \\frac{\\nu}{\\eta},\n\\end{eqnarray}\n}\n\\item{Chiral Prandtl number:\n\\begin{eqnarray}\n{\\rm Pr}_\\mu = \\frac{\\nu}{D_5} ,\n\\end{eqnarray}\n}\n\\item{Chiral nonlinearity parameter:\n\\begin{eqnarray}\n  \\lambda_\\mu = \\lambda \\eta^2 \\overline{\\rho} ,\n\\label{eq_lambamu}\n\\end{eqnarray}\n}\n\\item{Chiral flipping parameter:\n\\begin{eqnarray}\n  \\tilde\\Gamma_\\mathrm{f} = \\frac{\\Gamma_\\mathrm{f}}{\\mu_0 c_\\mathrm{s}} .\n\\end{eqnarray}\n}\n\\end{itemize}\nThen, $D_\\mu={\\rm Ma}_\\mu \\, {\\rm Pr}_{_{\\rm M}} \/ {\\rm Pr}_\\mu$,\n$\\, \\Lambda_\\mu =\\lambda_\\mu\/{\\rm Ma}_\\mu$\nand ${\\rm Re}_\\mu = \\left({\\rm Ma}_\\mu \\, {\\rm Pr}_{_{\\rm M}}\\right)^{-1}$.\n\n\n\\subsection{Physics of different regimes of magnetic field evolution}\n\nThere are two key nonlinear effects that determine the dynamics\nof the magnetic field in chiral MHD.\nThe first nonlinear effect is determined by\nthe conservation law~(\\ref{CL}) for the total chirality,\nwhich follows from the induction equation\nand the equation for the chiral magnetic potential.\nThe second nonlinear effect is determined by the\nLorentz force in the Navier-Stokes equation.\n\nIf the evolution of the magnetic field starts from a very small\nforce-free magnetic field, the\nsecond nonlinear effect, due to the\nLorentz force, vanishes if we assume that\nthe magnetic field remains force-free.\nThe magnetic field is generated by the chiral\nmagnetic dynamo instability\nwith a maximum growth rate $\\gamma^\\mathrm{max}_\\mu=v_\\mu^2\/4 \\eta$\nattained at the wavenumber $k_\\mu=\\mu_0\/2$ \\citep{JS97}.\n\nSince the total chirality is conserved, the increase of\nthe magnetic field in the nonlinear regime results in a decrease of the\nchiral chemical potential, so that the characteristic scale\nat which the growth rate is maximum increases in time.\nThis nonlinear effect has been interpreted in terms of\nan inverse magnetic cascade \\citep{BFR12}.\nThe maximum saturated level of the magnetic field can be\nestimated from the conservation law~(\\ref{CL}):\n$B_{\\rm sat} \\sim (\\mu_0 k_{\\rm M}\/\\lambda)^{1\/2} < \\mu_0\/\\lambda^{1\/2}$.\nHere, $\\mu_{\\rm sat} \\ll \\mu_0$ is the chiral chemical potential at saturation with\nthe characteristic wavenumber $k_{\\rm M} < \\mu_0$, corresponding\nto the maximum of the magnetic energy.\n\nHowever, the growing force-free magnetic field cannot stay\nforce-free in the nonlinear regime of the magnetic field evolution.\nIf the Lorentz force does not vanish, it generates small-scale velocity\nfluctuations.\nThis nonlinear stage begins when the nonlinear term ${\\bm{U}}\\times{\\bm{B}}$ in\nEquation~(\\ref{ind-DNS}) is of the order of the dynamo generating\nterm $v_\\mu {\\bm{B}}$, that is, when the velocity reaches the level\nof $U \\sim v_\\mu$.\nThe effect described here results in the production of\nchiral magnetically driven turbulence,\nwith the level of turbulent kinetic energy being determined\nby the balance of the nonlinear term, $(\\bm{U} \\cdot \\bm{\\nabla})\\bm{U}$, in\nEquation~(\\ref{UU-DNS}) and the Lorentz force, $(\\bm{\\nabla} \\times {\\bm{B}}) \\times \\bm{B}$,\nso that the turbulent velocity can reach the Alfv\\'en speed\n$v_{\\rm A}=({\\bm{B}^2}\/\\overline{\\rho})^{1\/2}$.\n\nThe chiral magnetically driven turbulence causes\ncomplicated dynamics:\nit produces the mean electromotive force that includes\nthe turbulent magnetic diffusion\nand the chiral $\\alpha_\\mu$ effect that\ngenerates large-scale magnetic fields; see Paper~I.\nThe resulting large-scale magnetic fields are concentrated\nat the wavenumber $k_\\alpha = 2 k_\\mu (\\ln {\\rm Re}_{_\\mathrm{M}})\/(3{\\rm Re}_{_\\mathrm{M}})$,\nfor ${\\rm Re}_{_\\mathrm{M}} \\gg 1$; see Paper~I.\nThe saturated value of the large-scale\nmagnetic field controlled by the conservation law~(\\ref{CL}), is\n$B_{\\rm sat} \\sim (\\mu_0 k_\\alpha\/\\lambda)^{1\/2}$.\nHere, ${\\rm Re}_{_\\mathrm{M}}$ is the magnetic Reynolds number based on the integral\nscale of turbulence and the turbulent velocity at this scale.\n\nDepending on the chiral nonlinearity parameter $\\lambda_\\mu$\n(see Equation~\\ref{eq_lambamu}), there\nare either two or three stages of magnetic field evolution.\nIn particular, when $\\lambda_\\mu$ is very small,\nthere is sufficient time to produce turbulence and excite the large-scale dynamo,\nso that the magnetic field evolution includes three stages:\n\\begin{asparaenum}[\\it (1)]\n\\item{\nthe small-scale chiral dynamo instability,}\n\\item{\nthe production of chiral magnetically driven MHD turbulence\nand the excitation of a large-scale dynamo instability, and}\n\\item{\nthe saturation of magnetic helicity and magnetic field growth\ncontrolled by the conservation law~(\\ref{CL}).}\n\\end{asparaenum}\n\nIf $\\lambda_\\mu$ is not very small, such that the saturated value\nof the magnetic field is not large, there is not enough time to excite\nthe large-scale dynamo instability. In this case,\nthe magnetic field dynamics includes two stages:\n\\begin{asparaenum}[\\it (1)]\n\\item{\nthe chiral dynamo instability, and}\n\\item{\nthe saturation of magnetic helicity and magnetic field growth\ncontrolled by the conservation law~(\\ref{CL}) for the total chirality.}\n\\end{asparaenum}\n\n\\subsection{Characteristic scales of chiral magnetically driven turbulence}\n\\label{sec_chiScales}\n\nIn the nonlinear regime, once turbulence is fully developed,\nsmall-scale magnetic fields can be excited over a broad range of wavenumbers\nup to the diffusion cutoff wavenumber.\nUsing dimensional arguments and numerical simulations,\n\\cite{BSRKBFRK17} found that, for chiral magnetically driven turbulence,\nthe magnetic energy spectrum $E_{\\rm M}(k,t)$ obeys\n\\begin{equation}\n   E_{\\rm M}(k,t)=C_\\mu\\,\\overline{\\rho}\\mu_0^3\\eta^2k^{-2},\n\\label{Cmu}\n\\end{equation}\nwhere $C_\\mu\\approx16$ is a chiral magnetic Kolmogorov-type constant.\nHere, $E_{\\rm M}(k,t)$ is normalized such that\n$\\mathcal{E}_{\\rm M} = \\int E_{\\rm M}(k)~\\mathrm{d}k=\\bra{\\bm{B}^2}\/2$\nis the mean magnetic energy density.\nIt was also confirmed numerically in \\cite{BSRKBFRK17}\nthat the magnetic energy spectrum $E_{\\rm M}(k)$ is bound from above\nby $C_\\lambda \\mu_0\/\\lambda$,\nwhere $C_\\lambda\\approx1$ is another empirical constant.\nThis yields a critical minimum wavenumber,\n\\begin{equation}\n   k_\\lambda=\\sqrt{\\overline{\\rho}\\lambda \\frac{C_\\mu}{C_\\lambda}}\\,\\mu_0\\eta,\n\\label{klambda}\n\\end{equation}\nbelow which the spectrum will no longer be proportional to $k^{-2}$.\n\nThe spectrum extends to larger wavenumbers up to a diffusive cutoff wavenumber\n$k_{\\rm diff}$.\nThe diffusion scale for magnetically produced turbulence\nis determined by the condition $\\mbox{\\rm Lu}(k_{\\rm diff})=1$,\nwhere $\\mbox{\\rm Lu}(k)=v_{\\rm A}(k)\/\\eta k$ is the scale-dependent Lundquist number,\n$v_{\\rm A}(k)=(\\bra{\\bm{B}^2}_k\/\\overline{\\rho})^{1\/2}$\nis the scale-dependent Alfv\\'en speed,\nand $\\bra{\\bm{B}^2}_k=2\\int_{k_\\lambda}^{k} E_{\\rm M}(k)~\\mathrm{d}k$.\nTo determine the Alfv\\'en speed, $v_{\\rm A}(k)$, we integrate Equation~(\\ref{Cmu}) over $k$\nand obtain\n\\begin{equation}\nv_{\\rm A}(k)=\\eta \\mu_0\\left({2\\,C_\\mu\\,\\mu_0\\over k_\\lambda}\\right)^{1\/2}\n\\left(1 - {k_\\lambda \\over k}\\right)^{1\/2}.\n\\end{equation}\nThe conditions $\\mbox{\\rm Lu}(k_{\\rm diff})=1$ and $k_{\\rm diff} \\gg k_\\lambda$ yield\n\\begin{equation}\n   k_{\\rm diff}=\\sqrt{2}\\left(\\frac{C_\\mu C_\\lambda}{\\lambda_\\mu}\\right)^{1\/4}\\mu_0\n              \\approx2.8\\,\\lambda_\\mu^{-1\/4}\\,\\mu_0.\n\\label{eq_kdiffchi}\n\\end{equation}\nNumerical simulations reported in \\cite{BSRKBFRK17} have been performed for\n$0.75 \\leq k_{\\rm diff}\/\\mu_0 \\leq 75$.\nIn the present DNS, we use values in the range from 4.5 to 503.\n\n\n\\subsection{Differences between chiral and standard MHD}\n\\label{sec:difference}\n\nThe system of equations~(\\ref{ind-DNS})--(\\ref{mu-DNS}) describing chiral\nMHD exhibits the following key differences from standard MHD:\n\\begin{itemize}\n \\item{\n     The presence of the term\n     ${\\bm \\nabla} \\times (\\eta \\, \\mu \\, {\\bm{B}})$\n     in Equation~(\\ref{ind-DNS}) causes a chiral dynamo\n     instability and results in chiral magnetically driven turbulence.}\n \\item{\n      Because of the finite value of $\\lambda$, the presence of a helical magnetic\n      field affects the evolution of $\\mu$; see Equation~(\\ref{mu-DNS}).}\n \\item{\n      For $\\Gamma_\\mathrm{f}=0$, the total chirality,\n      $\\int ({\\textstyle{1\\over2}} \\lambda {\\bm A} {\\bm \\cdot} \\bm{B} + \\mu) \\, dV$,\n      is strictly conserved, and not just in the limit $\\eta\\to 0$.\n      This conservation law determines the level of the saturated magnetic field.}\n  \\item{\n      The excitation of a large-scale magnetic field is caused by\n      (i)  the combined action of the chiral dynamo instability\n           and the inverse magnetic cascade due to the conservation\n           of total chirality, as well as by\n      (ii) the chiral $\\alpha_\\mu$ effect resulting in\n           chiral magnetically driven turbulence.\n           This effect is not related to kinetic helicity\n           and becomes dominant at large fluid and magnetic\n           Reynolds numbers; see Paper~I.}\n\\end{itemize}\n\n\nThe chiral term in Equation~(\\ref{ind-DNS}) and the evolution\nof $\\mu$ governed by Equation~(\\ref{mu-DNS}) are responsible for\ndifferent behaviors in chiral and standard MHD.\nIn particular, in standard MHD,\nthe following phenomena and a conservation law are established:\n\\begin{itemize}\n \\item{\n     The magnetic helicity $\\int {\\bm A} {\\bm \\cdot} \\bm{B} \\, dV$\n     is only conserved in the limit of $\\eta \\to 0$.}\n  \\item{\n      Turbulence does not have an intrinsic source.\n      Instead, it can be produced externally by a stirring force,\n      or due to large-scale shear at large fluid Reynolds numbers,\n      the Bell instability in the presence of a cosmic-ray current,\n      \\citep{RNBE2012,BL14},\n      the magnetorotational instability \\citep{HGB95,BNST95},\n      or just an initial irregular magnetic field \\citep{BKT15}.\n      }\n  \\item{\n      A large-scale magnetic field can be generated by:\n      (i) helical turbulence with nonzero mean kinetic helicity\n           that is produced either by external helical forcing or by\n           rotating, density-stratified, or inhomogeneous turbulence\n           (so-called mean-field $\\alpha^2$ dynamo);\n      (ii) helical turbulence with large-scale shear, which\n           results in an additional mechanism of large-scale dynamo action\n           referred to as an $\\alpha \\Omega$ or $\\alpha^2 \\Omega$ dynamo\n           \\citep{M78,P79,KR80,ZRS83};\n     (iii) nonhelical turbulence with large-scale shear, which causes\n           a large-scale shear dynamo \\citep{vishniac1997,RK03,RK04,sridhar2010,sridhar2014};\n     and\n     (iv) in different nonhelical deterministic flows due to negative\n     effective magnetic diffusivity \\citep[in Roberts flow IV, see][]{Devlen+13},\n     or time delay of an effective pumping velocity of the magnetic field\n     associated with the off-diagonal components of the $\\alpha$\n     tensor that are either antisymmetric (known as the $\\gamma$\n     effect) in Roberts flow III or symmetric in Roberts flow II;\n     see \\cite{Rheinhardt+14}.\n     All effects in items (i)--(iv) can work in chiral MHD as well.\n     However, which one of these effects is dominant depends on the flow\n     properties and the\n     governing parameters.\n}\n  \\end{itemize}\n\n\n\\subsection{DNS with the {\\sc Pencil Code}}\n\\label{sec:Penci-Code}\n\nWe solve Equations~(\\ref{ind-NS})--(\\ref{mu-NS})\nnumerically using the {\\sc Pencil Code}.\nThis code uses sixth-order explicit finite differences in space\nand a third-order accurate time-stepping method \\citep{BD02,Bra03}.\nThe boundary conditions are periodic in all three directions.\nAll simulations presented in Sections~\\ref{sec:lamdyn} and~\\ref{sec_turbdynamo1}\nare performed without external forcing of turbulence.\nIn Section~\\ref{sec_turbdynamo} we apply a turbulent forcing function\n$\\bm{f}$ in the Navier-Stokes equation, which consists of random plane\ntransverse white-in-time, unpolarized waves.\nIn the following, when we discuss numerical simulations,\nall quantities are considered as dimensionless quantities, and we drop the\n``tildes'' in Equations~(\\ref{ind-NS})--(\\ref{mu-NS}) from now on.\nThe wavenumber $k_1=2\\pi\/L$ is based on the size of the box $L=2\\pi$.\nIn all runs, we set $k_1=1$, $c_\\mathrm{s}=1$ and the mean fluid density $\\overline{\\rho}=1$.\n\n\n\n\\begin{table}\n\\centering\n\\caption{\nOverview of Runs for the Laminar $v_\\mu^2$ Dynamos\n(Reference Run in Bold)}\n     \\begin{tabular}{l|lllll}\n      \\hline\n      \\hline\n      \\\\\n\tsimulation \t& ${\\rm Pr}_{_{\\rm M}}$        & $\\lambda_\\mu$  & $\\dfrac{{\\rm Ma}_\\mu}{10^{-3}}$ & $\\dfrac{k_\\lambda}{10^{-4}\\mu_0}$ & $\\dfrac{k_\\mathrm{diff}}{\\mu_0}$\t\\\\\t                        \t\t\\\\\n      \\hline\n      \tLa2-1B  \t& $1.0$\t\t& $1\\times10^{-8}$\t& $2$\t& $4.0$ & $283$ \\\\\n \tLa2-2B  \t& $0.5$\t\t& $4\\times10^{-8}$\t& $4$\t& $8.0$ & $200$ \\\\\n     \tLa2-3B  \t& $0.2$\t\t& $2.5\\times10^{-7}$\t& $10$\t& $20$ & $126$ \\\\\n       \tLa2-4B  \t& $2.0$\t\t& $2.5\\times10^{-9}$\t& $1$\t& $2.0$ & $400$ \\\\\n      \tLa2-5B  \t& $1.0$\t\t& $1\\times10^{-9}$\t& $1.5$\t& $1.3$ & $503$ \\\\\n        La2-5G  \t& $1.0$\t\t& $1\\times10^{-8}$\t& $1.5$\t& $4.0$ & $283$ \\\\\n        La2-6G  \t& $1.0$\t\t& $1\\times10^{-5}$\t& $2$\t& $130$ & $50$  \\\\\n     \tLa2-7B   \t& $1.0$\t\t& $1\\times10^{-9}$\t& $3$\t& $4.0$ & $283$ \\\\\n        La2-7G   \t& $1.0$\t\t& $1\\times10^{-9}$\t& $3$\t& $4.0$ & $283$ \\\\\n    \tLa2-8B   \t& $1.0$\t\t& $1\\times10^{-9}$\t& $5$\t& $4.0$ & $283$ \\\\\n        La2-8G   \t& $1.0$\t\t& $1\\times10^{-9}$\t& $5$\t& $4.0$ & $283$ \\\\\n    \tLa2-9B   \t& $1.0$\t\t& $1\\times10^{-9}$\t& $10$\t& $4.0$ & $283$ \\\\\n        La2-9G   \t& $1.0$\t\t& $1\\times10^{-9}$\t& $10$\t& $4.0$ & $283$ \\\\\n     \tLa2-10B  \t& $1.0$\t\t& $1\\times10^{-5}$\t& $20$\t& $130$ & $50$ \\\\\n      \tLa2-10Bkmax  \t& $1.0$\t\t& $1\\times10^{-5}$\t& $20$\t& $130$ & $50$ \\\\\n     \tLa2-10G  \t& $1.0$\t\t& $1\\times10^{-5}$\t& $20$\t& $4.0$ & $283$ \\\\\n    \tLa2-11B   \t& $1.0$\t\t& $1\\times10^{-9}$\t& $50$\t& $1.3$ & $503$ \\\\\n    \tLa2-11G   \t& $1.0$\t\t& $1\\times10^{-8}$\t& $50$\t& $4.0$ & $283$ \\\\\n      \tLa2-12B  \t& $1.0$\t\t& $1\\times10^{-9}$\t& $2$\t& $1.3$ & $503$ \\\\\n      \tLa2-13B  \t& $1.0$\t\t& $1\\times10^{-7}$\t& $2$\t& $13$ & $159$ \\\\\n      \tLa2-14B  \t& $1.0$\t\t& $3\\times10^{-9}$\t& $2$\t& $2.2$ & $382$ \\\\\n      \t\\textbf{La2-15B}  & $\\mathbf{1.0}$& $\\mathbf{1\\times10^{-5}}$& $\\mathbf{2}$ & $\\mathbf{130}$ & $\\mathbf{50}$\\\\\n      \tLa2-16B  \t& $1.0$\t\t& $3\\times10^{-8}$\t& $2$\t& $6.9$ & $215$ \\\\\n      \\hline\n      \\hline\n    \\end{tabular}\n  \\label{table_simulations_vmu2}\n\\end{table}\n\\medskip\n\n\n\\section{Laminar chiral dynamos}\n\\label{sec:lamdyn}\n\n\nIn this section, we study numerically laminar chiral dynamos\nin the absence of any turbulence (externally or chiral magnetically driven).\n\n\\subsection{Numerical setup}\n\nParameters and initial conditions for all\nlaminar dynamo simulations are listed in\nTables~\\ref{table_simulations_vmu2} and~\\ref{table_simulations_vmushear}.\nAll of these simulations are two-dimensional and have a resolution of $256^2$.\nRuns with names ending with `B' are with the initial conditions\nfor the magnetic field in the form of a Beltrami magnetic field:\n$B(t=0)=10^{-4}$(0, sin$\\,x$, cos$\\,x$),\nwhile runs with names ending with `G' are initiated with Gaussian noise.\nThe initial conditions for the velocity field for the laminar $v_\\mu^2$ dynamo\nare $U(t=0)=(0,0,0)$, and for the laminar chiral--shear dynamos\n(the $v_\\mu^2$--shear or $v_\\mu$--shear dynamos) are\n$U(t=0)=(0, S_0~\\mathrm{cos}\\,x, 0)$, with\nthe dimensionless shear rate $S_0$ given for all runs.\n\nWe set the chiral Prandtl number $\\Prmu=1$ in all runs.\nIn many runs the magnetic Prandtl number ${\\rm Pr}_{_{\\rm M}}=1$ (except in several\n\nruns for the laminar $v_\\mu^2$ dynamo, see Table~\\ref{table_simulations_vmu2}).\nThe reference runs for the laminar $v_\\mu^2$ dynamo (La2-15B)\nand the chiral--shear dynamos (LaU-4G) are shown in bold in\nTables~\\ref{table_simulations_vmu2} and~\\ref{table_simulations_vmushear}.\nThe results of numerical simulations are compared with\ntheoretical predictions.\n\n\n\\subsection{Laminar $v_\\mu^2$ dynamo}\n\\label{sec_laminara2}\n\nWe start with the situation without an imposed fluid flow,\nwhere the chiral laminar $v_\\mu^2$ dynamo can be excited.\n\n\\subsubsection{Theoretical aspects}\n\nIn this section, we outline the theoretical predictions\nfor a laminar chiral dynamo; for details see Paper~I.\nTo determine the chiral dynamo growth rate, we seek a solution of\nthe linearized Equation~(\\ref{ind-DNS})\nfor small perturbations of the following form:\n$\\bm{B}(t,x,z)=B_y(t,x,z) {\\bm e}_y + \\bm{\\nabla}   \\times  [A(t,x,z) {\\bm e}_y]$,\nwhere ${\\bm e}_y$ is the unit vector in the $y$ direction.\n\nWe consider the equilibrium configuration:\n$\\mu=\\mu_0={\\rm const}$ and ${\\bm{U}}_0=0$.\nThe functions $B_y(t,x,z)$ and $A(t,x,z)$ are determined by the equations\n\\begin{eqnarray}\n \\frac{\\partial A(t,x,z)}{\\partial t} &=& v_\\mu \\, B_y + \\eta \\Delta A,\n\\label{A-eq} \\\\\n\\frac{\\partial B_y(t,x,z)}{\\partial t}&=& -v_\\mu \\, \\Delta A + \\eta \\Delta B_y ,\n\\label{By-eq}\n\\end{eqnarray}\nwhere $v_\\mu=\\eta \\, \\mu_0$,  $\\Delta=\\nabla_x^2 + \\nabla_z^2$,\nand the remaining components of the magnetic field are given by\n$B_x=-\\nabla_z A$ and $B_z=\\nabla_x A$.\nWe seek a solution to Equations~(\\ref{A-eq}) and~(\\ref{By-eq}) of the form\n$A, B_y \\propto \\exp[\\gamma t + i (k_x x + k_z z)]$.\nThe growth rate of the dynamo instability is given by\n\\begin{eqnarray}\n\\gamma = |v_\\mu \\, k| - \\eta k^2 ,\n\\label{gamma}\n\\end{eqnarray}\nwhere $k^2=k_x^2 + k_z^2$.\nThe dynamo instability is excited (i.e., $\\gamma> 0)$ for $k < |\\mu_0|$.\nThe maximum growth rate of the dynamo instability,\n\\begin{eqnarray}\n\\gamma^{\\rm max}_\\mu = \\frac{v_\\mu^2}{4 \\eta},\n\\label{gamma-max}\n\\end{eqnarray}\nis attained at\n\\begin{equation}\nk_\\mu =\\frac12|\\mu_0|.\n\\label{eq_kmax}\n\\end{equation}\n\n\n\\subsubsection{Time evolution}\n\n\\begin{figure}[t]\n\\begin{center}\n   \\includegraphics[width=\\columnwidth]{La2_15B__ts}\n\\end{center}\n\\caption[]{\\textbf{Laminar $v_\\mu^2$ dynamo:}\ntime evolution of $B_\\mathrm{rms}$ (solid black line),\n$\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle$ (dashed gray line),\n$\\mu_\\mathrm{rms}$ (multiplied by\n$2\/\\lambda$, dotted blue line), and\n$\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle +2\\mu_\\mathrm{rms}\/\\lambda$\n(dash-dotted red line) for reference run La2-15B\n(see table \\ref{table_simulations_vmu2}).\n}\n\\label{fig__La2_15B__ts}\n\\end{figure}\n\nIn Figure~\\ref{fig__La2_15B__ts} we show the time evolution\nof the rms magnetic field $B_\\mathrm{rms}$,\nthe magnetic helicity $\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle$,\nthe chemical potential $\\mu_\\mathrm{rms}$ (multiplied by a factor of\n$2\/\\lambda$), and $\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle\n+2\\mu_\\mathrm{rms}\/\\lambda$ for reference run La2-15B.\nIn simulations, the time is measured in units of diffusion time\n$t_\\eta=(\\eta k_1^2)^{-1}$.\nThe initial conditions for the magnetic field are chosen in the form of\na Beltrami field on $k=k_1=1$.\n\nThe magnetic field is amplified\nexponentially over more than four orders\nof magnitude until it saturates after roughly eight diffusive times.\nWithin the same time, the magnetic helicity\n$\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle$ increases\nover more than eight orders of magnitude.\nSince the sum of magnetic helicity and $2 \\mu\/\\lambda$ is conserved,\nthe chemical potential $\\mu$ decreases, in a nonlinear era\nof evolution, from the initial value $\\mu_0=2$ to $\\mu=1$,\nresulting in a saturation of the laminar $v_\\mu^2$ dynamo.\n\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{Gamma_etamu}\n\\end{center}\n\\caption[]{\\textbf{Laminar $v_\\mu^2$ dynamo:}\ngrowth rates as a function of ${\\rm Ma}_\\mu$,\nfor simulations with $\\mu_0=2$.\nThe black line is the theoretical prediction for the maximum growth rate\n$\\gamma^\\mathrm{max}_\\mu$\n(see Equation~\\ref{gamma-max}) that is attained at\n$k_\\mu=\\mu_0\/2=1$\n(see Equation~\\ref{eq_kmax}).\nThe runs with Gaussian initial fields, shown as red diamonds,\nlie on the theoretically predicted\n$\\gamma^\\mathrm{max}_\\mu$.\nThe dotted line corresponds to the theoretical prediction for the growth rate\n$\\gamma(k=1)$ at the scale of the box.\nThe runs with an initial magnetic Beltrami field on $k=1$,\nshown as blue diamonds, lie on the theoretically predicted dotted\ncurve $\\gamma(k=1)$.\n}\n\\label{fig__Gamma_eta}\n\\end{figure}\n\n\\subsubsection{Dynamo growth rate}\n\nIn Figure~\\ref{fig__Gamma_eta}, we show the growth rate of the magnetic field\nas a function of the chiral Mach number, ${\\rm Ma}_\\mu$.\nThe black solid line\nin this figure shows the theoretical prediction for the maximum\ngrowth rate $\\gamma^\\mathrm{max}_\\mu$ that is attained at $k_\\mu=\\mu_0\/2=1$;\nsee Equations~(\\ref{gamma-max}) and (\\ref{eq_kmax}).\nWhen the initial magnetic field is distributed over all spatial scales,\nlike in the case of initial magnetic Gaussian noise, in which\nthere is a nonvanishing magnetic field at\n$k_\\mu$; that is inside the computational domain,\nthe initial magnetic field is excited with the maximum growth rate\nas observed in the simulations.\nConsequently, the runs with Gaussian initial fields shown as red diamonds in\nFigure~\\ref{fig__Gamma_eta}, lie on the theoretical curve\n$\\gamma^\\mathrm{max}_\\mu$.\nThe dotted line in Figure~\\ref{fig__Gamma_eta} corresponds to the theoretical\nprediction for the growth rate $\\gamma$ at the scale of the box $(k=1)$.\nThe excitation of the magnetic field from an initial Beltrami field on $k=1$\noccurs with growth rates in agreement with the theoretical dotted curve;\nsee blue diamonds in Figure~\\ref{fig__Gamma_eta}.\n\n\n\\subsubsection{Dependence on initial conditions}\n\n\\begin{figure}[t]\\begin{center}\n\\includegraphics[width=\\columnwidth]{La2_B_t}\n\\end{center}\n\\caption[]{\\textbf{Laminar $v_\\mu^2$ dynamo:}\nTime evolution of $B_\\mathrm{rms}$ for two different initial conditions.\nThe black line is for the dynamo instability started from an initial\nBeltrami field at $k=1$ (run La2-10B), while the blue line\nis for an initial Beltrami field with\n$k=10$ (run La2-10Bkmax).\nFits in different regimes are indicated by thin lines.\nBoth runs are for the initial value $\\mu_0=20$, so that $k_\\mu=10$;\nand $\\gamma^\\mathrm{max}_\\mu=0.1$ (see~Equation~\\ref{gamma-max}).\n}\n\\label{fig__La2_B_t}\n\\end{figure}\n\nThe initial conditions for the magnetic field are important mostly at early times.\nIf the magnetic field is initially concentrated on the box scale, we expect to\nobserve a growth rate $\\gamma(k=1)$ as given by Equation~(\\ref{gamma}).\nAt later times, the spectrum of the magnetic field can, however, be changed,\ndue to mode coupling, and be amplified with a larger growth rate.\nThis behavior is observed in Figure~\\ref{fig__La2_B_t}, where\nan initial Beltrami field with $k=10$ is excited with maximum growth rate,\nsince $\\mu_0=20$.\nIn Figure~\\ref{fig__La2_B_t} we also consider another situation\nwhere the dynamo is started from an initial Beltrami field with $k=1$ (La2-10B).\nIn this case, the dynamo starts with a growth rate $\\gamma=0.019$,\nwhich is consistent with the theoretical prediction for $\\gamma(k=1)$.\nLater, after approximately $0.4\\,t_\\eta$, the dynamo growth\nrate increases up to the value $\\gamma=0.07$, which is close to\nthe maximum growth rate $\\gamma^\\mathrm{max}_\\mu=0.1$.\n\n\n\\subsubsection{Saturation}\n\nThe parameter $\\lambda$ in the evolution Equation~(\\ref{mu-DNS}),\nor the corresponding dimensionless parameter $\\lambda_\\mu$\nin Equation~(\\ref{mu-NS}), for the chiral chemical potential determines\nthe nonlinear saturation of the chiral dynamo.\nWe determine the saturation value of the\nmagnetic field $B_\\mathrm{sat}$ numerically for different values of $\\lambda_\\mu$;\nsee Figure~\\ref{fig_Bsat_lambda}.\nWe find that the saturation value of the magnetic field\nincreases with decreasing $\\lambda_\\mu$.\nThis can be expected from the conservation law (\\ref{CL}).\nIf the initial magnetic energy is very small,\nwe find from Equation~(\\ref{CL}) the following estimate\nfor the saturated magnetic field during laminar chiral dynamo action:\n\\begin{eqnarray}\n   B_\\mathrm{sat} \\sim \\left[\\frac{\\mu_0(\\mu_0 -\\mu_\\mathrm{sat})}{\\lambda}\\right]^{1\/2},\n\\label{eq_Bsat}\n\\end{eqnarray}\nwhere $\\mu_\\mathrm{sat}$ is the chiral chemical potential at saturation, and\nwe use the estimate $A$ by $2 B \/\\mu_0$.\nInspection of Figure~\\ref{fig_Bsat_lambda} demonstrates\na good agreement between\ntheoretical (solid line) and numerical results (blue diamonds).\n\n\\begin{figure}[t]\\begin{center}\n\\includegraphics[width=\\columnwidth]{Bsat_lambda}\n\\end{center}\n\\caption{{\\bf Laminar $v_\\mu^2$ dynamo:}\nthe saturation magnetic field strength for simulations\nwith different $\\lambda_\\mu$.\nDetails for the different runs, given by labeled blue diamonds, can be found in\nTable~\\ref{table_simulations_vmu2}.}\n\\label{fig_Bsat_lambda}\n\\end{figure}\n\n\n\\subsubsection{Effect of a nonvanishing flipping rate}\n\\label{sec_flip}\n\nIn this section, we consider the influence of a nonvanishing chiral\nflipping rate on the $v_\\mu^2$ dynamo.\nA large flipping rate $\\Gamma_\\mathrm{f}$ decreases the chiral chemical\npotential $\\mu$; see Equation~(\\ref{mu-DNS}). It can stop\nthe growth of the magnetic field caused by the chiral dynamo instability.\n\nQuantitatively, the influence of the flipping term can be estimated by\ncomparing the last two terms of Equation~(\\ref{mu-DNS}).\nThe ratio of these terms is\n\\begin{eqnarray}\n   f_\\mu \\equiv \\frac{\\Gamma_\\mathrm{f}\\mu_0}{\\lambda\\eta\\mu_0 B_\\mathrm{sat}^2}\n   = \\frac{\\Gamma_\\mathrm{f}}{\\eta\\mu_0^2},\n\\label{eq_fmu}\n\\end{eqnarray}\nwhere we have used Equation~(\\ref{eq_Bsat}) with $\\mu_\\mathrm{sat} \\ll \\mu_0$\nfor the saturation value of the magnetic field strength.\nIn Figure~\\ref{fig_ts_flip} we present the time evolution of\n$B_\\mathrm{rms}$ and $\\mu_\\mathrm{rms}$\nfor different values of $f_\\mu$.\nThe reference run La2-15B, with zero flipping rate ($f_\\mu=0$),\nhas been repeated with a finite flipping term.\nAs a result, the magnetic field grows more\nslowly in the nonlinear era, due to the flipping effect, and it\ndecreases the saturation level of the magnetic field; see Figure~\\ref{fig_ts_flip}.\nFor larger values of $f_\\mu$, the chiral chemical potential\n$\\mu$ decreases quickly, leading to strong quenching of the $v_\\mu^2$ dynamo;\nsee the blue lines in Figure~\\ref{fig_ts_flip}.\n\n\\begin{figure}[t]\\begin{center}\n\\includegraphics[width=\\columnwidth]{ts_flip.ps}\n\\end{center}\n\\caption{{\\bf Laminar $v_\\mu^2$ dynamo:}\ntime evolution of the chiral chemical potential $\\mu_{\\rm rms}$ (black lines) and\nthe magnetic field $B_{\\rm rms}$ (blue lines)\nfor $f_\\mu=0$ (solid), $f_\\mu = 0.0025$ (dashed), and $f_\\mu = 0.01$ (dotted).\n}\n\\label{fig_ts_flip}\n\\end{figure}\n\n\n\\subsection{Laminar chiral--shear dynamos}\n\\label{sec_laminarashear}\n\n\nIn this section, we consider laminar chiral dynamos\nin the presence of an imposed shearing velocity.\nSuch a nonuniform velocity profile can be created\nin different astrophysical flows.\n\n\\subsubsection{Theoretical aspects}\n\nWe start by outlining the theoretical predictions\nfor laminar chiral dynamos in the presence\nof an imposed shearing velocity; for details see Paper~I.\nWe consider the equilibrium configuration specified by the shear velocity\n${\\bm{U}}_{\\rm eq}=(0,S\\, x,0)$,\nand $\\mu=\\mu_0=$ const.\nThis implies that the fluid has nonzero vorticity\n${\\bm W} = (0,0,S)$ similar to differential (nonuniform) rotation.\nThe functions $B_y(t,x,z)$ and $A(t,x,z)$ are determined by\n\\begin{eqnarray}\n&&\\frac{\\partial A(t,x,z)}{\\partial t} = v_\\mu \\,  B_y + \\eta \\Delta A,\n\\label{A1-eq}\\\\\n&&\\frac{\\partial B_y(t,x,z)}{\\partial t} = - S\\nabla_z A - v_\\mu \\,  \\Delta A\n+ \\eta \\Delta B_y .\n\\label{By1-eq}\n\\end{eqnarray}\nWe look for a solution to Equations~(\\ref{A1-eq})\nand~(\\ref{By1-eq}) of the form $A, B_y \\propto \\exp[\\gamma t + i (k_x x + k_z z-\n\\omega t)]$.\nThe growth rate of the dynamo instability and the frequency of the dynamo waves\nare given by\n\\begin{eqnarray}\n \\gamma = {|v_\\mu \\, k| \\over \\sqrt{2}} \\,\n\\left\\{1 + \\left[1 + \\left({S k_z \\over v_\\mu \\, k^2}\\right)^2 \\right]^{1\\over 2}\n\\right\\}^{1\\over 2} - \\eta k^2\n\\label{eq_gamma_aS}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n \\omega=  {\\rm sgn} \\left(\\mu_0  k_z\\right) \\, {S k_z \\over \\sqrt{2} k}\n\\,\n\\left\\{1 + \\left[1 + \\left({S k_z \\over v_\\mu \\, k^2}\\right)^2\\right]^{1\\over 2}\n\\right\\}^{-{1\\over 2}} ,\n\\nonumber\\\\\n\\label{omega10}\n\\end{eqnarray}\nrespectively.\nThis solution describes a laminar $v_\\mu^2$--shear dynamo\nfor arbitrary values of the shear rate $S$.\n\n\nNext, we consider a situation where the shear term on the\nright side of Equation~(\\ref{By1-eq}) dominates,\nthat is, where $|S \\nabla_z A| \\gg |v_\\mu \\, \\Delta A|$.\nThe growth rate of the dynamo instability\nand the frequency of the dynamo waves are then given by\n\\begin{eqnarray}\n&& \\gamma = \\left({ |v_\\mu  \\, S \\, k_z| \\over 2}\\right)^{1\/2} - \\eta k^2 ,\n\\label{gamma1}\\\\\n&& \\omega= {\\rm sgn} \\left(\\mu_0  k_z\\right) \\, \\left({|v_\\mu  \\, S \\,\nk_z| \\over 2}\\right)^{1\/2} .\n\\label{omega}\n\\end{eqnarray}\nThe dynamo is excited for $k < |v_\\mu \\, S \\, k_z \/2\\eta^2|^{1\/4}$.\nThe maximum growth rate of the dynamo instability\nand the frequency $\\omega=\\omega (k=k_z^\\mu)$\nof the dynamo waves are attained at\n\\begin{equation}\n  k_z^\\mu ={1 \\over 4} \\left({2|S \\, v_\\mu| \\over \\eta^2} \\right)^{1\/3},\n\\label{kz-max}\n\\end{equation}\nand are given by\n\\begin{eqnarray}\n&& \\gamma^{\\rm max}_\\mu\n  = {3 \\over 8} \\left({S^2 \\, v_\\mu^2 \\over 2\\eta}\\right)^{1\/3}\n  - \\eta k_x^2,\n\\label{gam-max}\\\\\n&& \\omega(k=k_z^\\mu)\n  = {{\\rm sgn} \\left(v_\\mu \\, k_z\\right) \\over 2\\eta}  \\,\n  \\left({S^2 \\, v_\\mu^2 \\over 2 \\eta}\\right)^{1\/3} .\n\\label{omega-max}\n\\end{eqnarray}\nThis solution describes the laminar $v_\\mu$--shear dynamo.\n\n\n\\subsubsection{Simulations of the laminar $v_\\mu$--shear dynamo}\n\n\n\\begin{table}\n\\centering\n\\caption{\nOverview of Runs for the Chiral--Shear Dynamos\n(Reference Run in Bold)}\n     \\begin{tabular}{l|llllll}\n      \\hline\n      \\hline\n      \\\\\n\tsimulation \t& $\\lambda_\\mu$  \t& $\\dfrac{{\\rm Ma}_\\mu}{10^{-3}}$ & $u_S$\t& $\\dfrac{k_\\lambda}{10^{-4}\\mu_0}$ & $\\dfrac{k_\\mathrm{diff}}{\\mu_0}$\t\\\\\\\\\t                        \t\t\\\\\n      \\hline\n   \tLaU-1B\t   \t& $1\\times10^{-9}$\t& $2.0$\t& 0.01\t& 1.3 & 503 \\\\\n   \tLaU-1G\t   \t& $1\\times10^{-9}$\t& $2.0$\t& 0.01\t& 1.3 & 503 \\\\\n   \tLaU-2B\t   \t& $1\\times10^{-9}$\t& $2.0$\t& 0.02\t& 1.3 & 503 \\\\\n   \tLaU-2G\t   \t& $1\\times10^{-9}$\t& $2.0$\t& 0.02\t& 1.3 & 503 \\\\\n   \tLaU-3B\t   \t& $1\\times10^{-9}$\t& $2.0$\t& 0.05\t& 1.3 & 503 \\\\\n   \tLaU-3G\t   \t& $1\\times10^{-9}$\t& $2.0$\t& 0.05\t& 1.3 & 503 \\\\\n   \tLaU-4B\t   \t& $1\\times10^{-9}$\t& $2.0$\t& 0.10\t& 1.3 & 503 \\\\\n   \t\\textbf{LaU-4G}   & $\\mathbf{1\\times10^{-5}}$\t& $\\mathbf{2.0}$ & $\\mathbf{0.10}$ & $\\mathbf{126}$ &  $\\mathbf{50}$ \\\\\n   \tLaU-5B\t   \t& $1\\times10^{-9}$\t& $2.0$\t& 0.20 \t& 1.3 & 503 \\\\\n   \tLaU-5G\t   \t& $1\\times10^{-9}$\t& $2.0$\t& 0.20  & 1.3 & 503 \\\\\n   \tLaU-6B\t   \t& $1\\times10^{-9}$\t& $2.0$ & 0.50\t& 1.3 & 503 \\\\\n   \tLaU-6G\t   \t& $1\\times10^{-9}$\t& $2.0$ & 0.50\t& 1.3 & 503 \\\\\n   \tLaU-7G\t   \t& $1\\times10^{-8}$\t& $10$  & 0.01\t& 4.0 & 283 \\\\\n   \tLaU-8G\t   \t& $1\\times10^{-8}$\t& $10$  & 0.05\t& 4.0 & 283 \\\\\n   \tLaU-9G\t   \t& $1\\times10^{-8}$\t& $10$  & 0.10\t& 4.0 & 283 \\\\\n   \tLaU-10G\t\t& $1\\times10^{-8}$\t& $10$\t& 0.50\t& 4.0 & 283 \\\\\n      \\hline\n      \\hline\n    \\end{tabular}\n  \\label{table_simulations_vmushear}\n\\end{table}\n\n\n\\begin{figure}[t]\n\\begin{center}\n   \\includegraphics[width=\\columnwidth]{LaU_9G__ts}\n\\end{center}\n\\caption[]{\\textbf{Laminar $v_\\mu$--shear dynamo:}\ntime evolution of the magnetic field\n$B_\\mathrm{rms}$, the velocity $u_\\mathrm{rms}$, the magnetic helicity\n$\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle$,\nthe chemical potential $\\mu_\\mathrm{rms}$ (multiplied by a factor\nof $2\/\\lambda$), and $\\langle \\mathbf{A} \\cdot \\mathbf{B}\n\\rangle +2\\mu_\\mathrm{rms}\/\\lambda$\n(run LaU-4G).}\n\\label{fig_AlphaShear_t}\n\\end{figure}\n\nSince our simulations have periodic boundary conditions, we model shear\nvelocities as $U_S=(0, u_S \\cos x, 0)$.\nThe mean shear velocity $\\overline{u}_S$ over half the box is\n$\\overline{u}_S = (2\/\\pi) u_S$.\nIn Figure~\\ref{fig_AlphaShear_t} we show the time evolution of the magnetic field\n(which starts to be excited from a Gaussian initial field),\nthe velocity $u_\\mathrm{rms}$, the magnetic helicity\n$\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle$,\nthe chemical potential $\\mu_\\mathrm{rms}$ (multiplied by a factor\nof $2\/\\lambda$), and $\\langle \\mathbf{A} \\cdot \\mathbf{B}\n\\rangle +2\\mu_\\mathrm{rms}\/\\lambda$ for run LaU-4G.\nThe growth rate for the chiral--shear dynamo (the $v_\\mu^2$--shear dynamo)\nis larger than that for the laminar chiral dynamo (the $v_\\mu^2$--dynamo).\nAfter a time of roughly $0.03~t_\\eta$, the system enters a nonlinear\nphase, in which the velocity field is affected by the magnetic field,\nbut the magnetic field can still increase slowly.\nSaturation of the dynamo occurs after approximately $0.1~t_\\eta$.\n\nFor Gaussian initial fields, we have observed a short delay in the\ngrowth of the magnetic field.\nIn both cases, the dynamo growth rate increases with increasing shear.\nAs for the chiral $v_\\mu^2$ dynamo, we observe perfect\nconservation of the quantity $\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle +\n2\\mu_\\mathrm{rms}\/\\lambda$ in the simulations of the laminar $v_\\mu$--shear\ndynamo.\n\n\\begin{figure}[t]\n  \\begin{center}\n    \\includegraphics[width=\\columnwidth]{Beltrami_GammaOmega_Us}\n\\end{center}\n\\caption[]{\\textbf{Laminar $v_\\mu$--shear dynamo:}\ngrowth rate (top panel) and dynamo frequency (bottom panel) as a function\nof the mean shear $\\overline{u}_S$ for the Beltrami initial field\n(runs LaU-$n$B with $n=1$--$6$; see Table~\\ref{table_simulations_vmushear}).}\n\\label{Gamma_Us_Beltrami}\n\\end{figure}\n\nIn Figure~\\ref{Gamma_Us_Beltrami} we show\nthe theoretical dependence of the growth rate $\\gamma$ and the dynamo frequency\n$\\omega$ on the shear velocity $\\overline{u}_S$\nfor Beltrami initial conditions at different wavenumbers;\nsee Equations~(\\ref{gamma1}) and~(\\ref{gam-max}).\nThe dynamo growth rate is estimated from an exponential fit.\nThe result of the fit depends slightly on the fitting regime, leading\nto an error of the order of 10\\%.\nThe dynamo frequency is determined afterward by dividing the magnetic field\nstrength by $\\mathrm{exp}(\\gamma t)$ and fitting a sine function.\nDue to the small amplitude and a limited number of periods of dynamo waves,\nthe result is sensitive to the fit regime considered.\nHence we assume a conservative error of 50\\% for the dynamo frequency.\nThe blue diamonds correspond to the numerical results.\nWithin the error bars, the theoretical and numerical results are in agreement.\n\n\\subsubsection{Simulations of the laminar $v_\\mu^2$--shear dynamo}\n\n\\begin{figure}[t]\n\\begin{center}\n   \\includegraphics[width=\\columnwidth]{a2shear_Gamma_Us}\n\\end{center}\n\\caption[]{\\textbf{Laminar $v_\\mu^2$--shear dynamo:}\ngrowth rate $\\gamma$ as a function of\nmean shear $\\overline{u}_S$.\nFor comparison, we plot the maximum growth rate of $v_\\mu^2$ dynamo\n(\\ref{gamma-max}) and of the $v_\\mu$--shear dynamo (\\ref{gam-max}).\nThe solid black line is the theoretically predicted maximum growth rate\n(see Equation~\\ref{eq_gamma_aS}).\n}\\label{fig_Gamma_Us_aShear}\n\\end{figure}\n\n\nThe growth rate of chiral--shear dynamos versus mean shear in\nthe range between $u_S=0.01$\nand $0.5$ is shown in Figure~\\ref{fig_Gamma_Us_aShear}.\nWe choose a large initial value of the chemical potential, i.e.\\\n$\\mu_0=10$, to ensure that $k_\\mathrm{max}$ is\ninside the box for all values of $\\overline{u}_S$.\nWe overplot the growth rates found from the simulations with the maximum growth\nrate given by Equation (\\ref{eq_gamma_aS}).\nIn addition, we show the theoretical predictions for the limiting cases\nof the $v_\\mu^2$ and\n$v_\\mu$--shear dynamos; see Equations~(\\ref{gamma-max}) and (\\ref{gam-max}).\nInspection of Figure~\\ref{fig_Gamma_Us_aShear} shows that the results\nobtained from the simulations agree with theoretical predictions.\n\n\n\n\n\\section{Chiral magnetically driven turbulence}\n\\label{sec_turbdynamo1}\n\n\nIn this section we show that the CME\ncan drive turbulence via the Lorentz force in the Navier-Stokes equation.\nWhen the magnetic field increases exponentially, due to the\nsmall-scale chiral magnetic dynamo with growth rate $\\gamma$,\nthe Lorentz force,\n$({\\bm \\nabla} {\\bm \\times} {\\bm{B}}) {\\bm \\times} \\bm{B}$,\nincreases at the rate $2\\gamma$.\nThe laminar dynamo occurs only up to the first nonlinear phase,\nwhen the Lorentz force starts to produce turbulence\n(referred to as chiral magnetically driven turbulence).\nWe will also demonstrate here that, during the second nonlinear phase,\na large-scale dynamo is excited by the chiral $\\alpha_\\mu$ effect\narising in chiral magnetically driven turbulence.\nThe chiral $\\alpha_\\mu$ effect was studied using different\nanalytical approaches in Paper~I.\nThis effect is caused by an interaction of the CME\nand fluctuations of the small-scale\ncurrent produced by tangling magnetic fluctuations.\nThese fluctuations are generated by tangling of the large-scale\nmagnetic field through sheared velocity fluctuations.\nOnce the large-scale magnetic field becomes strong enough,\nthe chiral chemical potential decreases, resulting in the saturation\nof the large-scale dynamo instability.\n\nThis situation is similar to that of\ndriving small-scale turbulence via the Bell instability\nin a system with an external cosmic-ray current\n\\citep{B04,BL14}, and the generation of a\nlarge-scale magnetic field by the Bell turbulence; see \\citet{RNBE2012}\nfor details.\n\n\\subsection{Mean-field theory for large-scale dynamos}\n\\label{sec_meanfieldMHD}\n\nIn this section, we outline the theoretical predictions\nfor large-scale dynamos based on mean-field theory;\nsee Paper~I for details.\nThe mean induction equation is given by\n\\begin{eqnarray}\n\\frac{\\partial \\overline{\\mbox{\\boldmath $B$}}{}}{}{\\partial t} &=&\n\\bm{\\nabla}   \\times   \\left[\\overline{\\bm{U}}  \\times   \\overline{\\mbox{\\boldmath $B$}}{}}{\n+ (\\overline{v}_\\mu + \\alpha_\\mu) \\overline{\\mbox{\\boldmath $B$}}{}}{\n- (\\eta+ \\, \\eta_{_{T}})\\bm{\\nabla}   \\times   \\overline{\\mbox{\\boldmath $B$}}{}}{\\right], \\nonumber\\\\\n\\label{ind4-eq}\n\\end{eqnarray}\nwhere $\\overline{v}_\\mu = \\eta \\overline{\\mu}_{0}$, and we consider the following equilibrium\nstate: $\\overline{\\mu}_{\\rm eq}=\\overline{\\mu}_{0}={\\rm const}$ and ${\\bm \\overline{\\bm{U}}}_{\\rm eq}=0$.\nThis mean-field equation contains additional terms that are\nrelated to the chiral $\\alpha_\\mu$ effect and the turbulent magnetic diffusivity\n$\\eta_{_{T}}$.\nIn the mean-field equation, the chiral $v_\\mu$ effect is replaced\nby the mean chiral $\\overline{v}_\\mu$ effect.\nNote, however, that at large fluid and magnetic Reynolds numbers, the $\\alpha_\\mu$ effect\ndominates the $\\overline{v}_\\mu$ effect.\n\nTo study the large-scale dynamo,\nwe seek a solution to Equation~(\\ref{ind4-eq}), for small perturbations in\nthe form\n$\\overline{\\mbox{\\boldmath $B$}}{}}{(t,x,z)=\\overline{B}_y(t,x,z) {\\bm e}_y + \\bm{\\nabla}   \\times\n[\\overline{A}(t,x,z) {\\bm e}_y]$,\nwhere ${\\bm e}_y$ is the unit vector directed along the $y$ axis.\nThe functions $\\overline{B}_y(t,x,z)$ and $\\overline{A}(t,x,z)$ are determined by\n\\begin{multline}\n  \\frac{\\partial \\overline{A}(t,x,z)}{\\partial t}\n  =(\\overline{v}_\\mu + \\alpha_\\mu)\\, \\overline{B}_y\n + (\\eta+ \\, \\eta_{_{T}}) \\, \\Delta \\overline{A},\n \\label{me-A-eq}\n\\end{multline}\n\\begin{multline}\n  \\frac{\\partial \\overline{B}_y(t,x,z)}{\\partial t}\n  =-(\\overline{v}_\\mu + \\alpha_\\mu) \\, \\Delta \\overline{A}\n + (\\eta+ \\, \\eta_{_{T}}) \\, \\Delta \\overline{B}_y ,\n\\label{me-By-eq}\n\\end{multline}\nwhere $\\Delta=\\nabla_x^2 + \\nabla_z^2$, and the other components of the magnetic\nfield are $\\overline{B}_x=-\\nabla_z \\overline{A}$ and $\\overline{B}_z=\\nabla_x \\overline{A}$.\n\nWe look for a solution of the mean-field equations~(\\ref{me-A-eq})\nand~(\\ref{me-By-eq}) in the form\n\\begin{eqnarray}\n  \\overline{A}, \\overline{B}_y \\propto \\exp[\\gamma t + i (k_x x + k_z z)],\n\\end{eqnarray}\nwhere the growth rate of the large-scale dynamo instability is given by\n\\begin{eqnarray}\n\\gamma = |(\\overline{v}_\\mu + \\alpha_\\mu)\\, k| - (\\eta+ \\, \\eta_{_{T}}) \\, k^2,\n\\label{gamma_turb}\n\\end{eqnarray}\nwith $k^2=k_x^2 + k_z^2$.\nThe maximum growth rate of the large-scale dynamo instability, attained at\nthe wavenumber\n\\begin{equation}\n   k \\equiv k_\\alpha\n   ={|\\overline{v}_\\mu + \\alpha_\\mu|\n   \\over 2(\\eta+ \\, \\eta_{_{T}})},\n\\label{kmax_turb}\n\\end{equation}\nis given by\n\\begin{eqnarray}\n\\gamma^{\\rm max}_\\alpha\n= {(\\overline{v}_\\mu + \\alpha_\\mu)^2\\over 4 (\\eta+ \\, \\eta_{_{T}})}\n= {(\\overline{v}_\\mu + \\alpha_\\mu)^2\\over 4 \\eta \\, (1 + \\, {\\rm Re}_{_\\mathrm{M}}\/3)}.\n\\label{gammamax_turb}\n\\end{eqnarray}\nFor small magnetic Reynolds numbers,\n${\\rm Re}_{_\\mathrm{M}}=u_0 \\ell_0\/\\eta = 3 \\eta_{_{T}}\/\\eta$, this equation\nyields the correct result for the laminar $v_\\mu^2$ dynamo;\nsee Equation~(\\ref{gamma-max}).\n\nAs was shown in Paper~I, the CME\nin the presence of turbulence gives rise to the chiral $\\alpha_\\mu$ effect.\nThe expression for $\\alpha_\\mu$\nfound for large Reynolds numbers and a weak\nmean magnetic field is\n\\begin{eqnarray}\n  \\alpha_\\mu = - {2 \\over 3} \\overline{v}_\\mu \\ln {\\rm Re}_{_\\mathrm{M}}.\n\\label{alphamu}\n\\end{eqnarray}\nSince the $\\alpha_\\mu$ effect in homogeneous turbulence\nis always negative, while the $\\overline{v}_\\mu$ effect is positive,\nthe chiral $\\alpha_\\mu$ effect decreases the $\\overline{v}_\\mu$ effect.\nBoth effects compensate each other at ${\\rm Re}_{_\\mathrm{M}}=4.5$ (see Paper~I).\nHowever, for large fluid and magnetic Reynolds numbers, $\\overline{v}_\\mu \\ll\n|\\alpha_\\mu|$, and we can neglect $\\overline{v}_\\mu$ in these equations.\nThis regime corresponds to the large-scale $\\alpha_\\mu^2$ dynamo.\n\n\n\\subsection{DNS of chiral magnetically driven turbulence}\n\nWe have performed a higher resolution $(576^3)$ three-dimensional\nnumerical simulation to study chiral magnetically driven turbulence.\nThe chiral Mach number of this simulation is ${\\rm Ma}_\\mu=2\\times10^{-3}$,\nthe chiral nonlinearity parameter is $\\lambda_\\mu=2\\times10^{-7}$, and the\nmagnetic and the chiral Prandtl numbers are unity.\nThe velocity field is initially zero, and the magnetic field is Gaussian noise,\nwith $B=10^{-6}$.\n\n\\begin{figure}[t]\n\\centering\n  \\subfigure{\\includegraphics[width=\\columnwidth]{LamTurb_ts_mu20}}\n\\caption{\n{\\bf Chiral magnetically driven turbulence.}\nTime evolution for different quantities.\n}\n  \\label{fig_LTts}\n\\end{figure}\n\nThe time evolution of $B_\\mathrm{rms}$, $u_\\mathrm{rms}$,\n$\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle$,\n$\\mu_\\mathrm{rms}$ (multiplied by\n$2\/\\lambda$), and $\\langle \\mathbf{A} \\cdot \\mathbf{B}\n\\rangle +2\\mu_\\mathrm{rms}\/\\lambda$\nof chiral magnetically driven turbulence\nis shown in the top panel of Figure~\\ref{fig_LTts}.\nFour phases can be distinguished:\n\\begin{asparaenum}[\\it (1)]\n\\item{The kinematic phase of small-scale chiral dynamo instability\nresulting in exponential growth of small-scale magnetic field\ndue to the CME.\nThis phase ends approximately at $t=0.05 t_\\eta$.\n}\n\\item{The first nonlinear phase resulting in production\nof chiral magnetically driven turbulence.\nIn this phase, $u_\\mathrm{rms}$ grows from very weak noise\nover seven orders of magnitude up to nearly the equipartition value\nbetween turbulent kinetic and magnetic energies,\ndue to the Lorentz force in the Navier-Stokes equation.\n}\n\\item{The second nonlinear phase resulting in large-scale dynamos.\nIn particular, the evolution of $B_\\mathrm{rms}$ for $t > 0.12 t_\\eta$\nis affected by the velocity field.\nDuring this phase, the velocity stays approximately constant,\nwhile the magnetic field continues to increase\nat a reduced growth rate in comparison with that of the\nsmall-scale chiral dynamo instability.\nIn this phase, we also observe the formation of inverse energy transfer\nwith a $k^{-2}$ magnetic energy spectrum that was previously found\nand comprehensively analyzed by \\cite{BSRKBFRK17} in DNS of chiral MHD\nwith different parameters.\n}\n\\item{The third nonlinear phase resulting in saturation of the large-scale\ndynamos, which ends at $\\approx 0.45 t_\\eta$ when the\nlarge-scale magnetic field reaches the maximum value.\nThe conserved quantity $\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle\n+2\\mu_\\mathrm{rms}\/\\lambda$ stays constant over all four phases.\nSaturation is caused by the $\\lambda$ term in the evolution equation of the chiral\nchemical potential, which leads to a decrease of $\\mu$ from its initial value to 1.\n}\n\\end{asparaenum}\n\nThe middle panel of Figure~\\ref{fig_LTts} shows the measured growth rate of\n$B_\\mathrm{rms}$ as a function of time.\nIn the kinematic phase, $\\gamma$ agrees\nwith the theoretical prediction for the\nlaminar chiral dynamo instability; see Equation~(\\ref{gamma-max}),\nwhich is indicated by the dashed red horizontal line in the middle panel of\nFigure~\\ref{fig_LTts}.\nDuring this phase, the growth rate\nof the velocity field, given by the dotted gray line in Figure~\\ref{fig_LTts}, is\nlarger by roughly a factor of two than that of the magnetic field.\nThis is expected when turbulence is driven via the Lorentz force, which is\nquadratic in the magnetic field.\n\nOnce the kinetic energy is of the same order\nas the magnetic energy, the growth rate of\nthe magnetic field decreases abruptly by a factor of more than five.\nThis is expected in the presence of turbulence, because\nthe energy dissipation of the magnetic field is increased by turbulence due to\nturbulent magnetic diffusion.\nAdditionally, however, a positive contribution to the\ngrowth rate comes from the chiral $\\alpha_\\mu$ effect\nthat causes large-scale dynamo instability.\n\nThe time evolution of the ratio of the mean magnetic field to the total\nfield, $\\overline{B}\/B_\\mathrm{rms}$, is presented in the bottom panel\nof Figure~\\ref{fig_LTts}.\nThe mean magnetic field grows faster than the rms of the total magnetic\nfield in the time interval between 0.14 and 0.2 $t_\\eta$.\nDuring this time, the large-scale (mean-field) dynamo operates,\nso magnetic energy is transferred to larger spatial scales.\nWe now determine, directly from DNS,\nthe growth rate of the large-scale dynamo using\nEquation~(\\ref{gamma_turb}).\nTo this end, we determine the Reynolds number and the strength of the $\\alpha_\\mu$\neffect using the data from our DNS.\nWhereas the rms velocity is a direct output of the simulation, the turbulent\nforcing scale can be found from analysis of the energy spectra.\nThe theoretical value based on these estimates at the time $0.2\\,t_\\eta$ is\nindicated as the solid red horizontal line in the middle\npanel of Figure~\\ref{fig_LTts}.\n\n\\begin{figure}[t]\n\\centering\n  \\subfigure{\\includegraphics[width=\\columnwidth]{LamTurb_spec_mu20}}\n\\caption{\n{\\bf Chiral magnetically driven turbulence.}\nMagnetic (blue lines) and kinetic (black lines) energy\nspectra are calculated at equal time differences,\nand the very last spectra are shown as solid lines.}\n  \\label{fig_LTspec}\n\\end{figure}\n\n\nThe evolution of kinetic and magnetic energy spectra is\nshown in Figure~\\ref{fig_LTspec}.\nWe use equal time steps between the different spectra, covering the whole\nsimulation time.\nThe magnetic energy, indicated by blue\nlines, increases initially at $k=\\mu_0\/2=10$, which agrees with the theoretical\nprediction for the chiral laminar dynamo.\nThe magnetic field drives a turbulent spectrum of the kinetic energy, as can\nclearly be seen in\nFigure~\\ref{fig_LTspec} (indicated by black lines in Figure~\\ref{fig_LTspec}).\nThe final spectral slope of the kinetic energy is roughly $-5\/3$.\nThe magnetic field continues to grow at small wavenumbers,\nproducing a peak at $k=1$ in the final stage of the time evolution.\n\n\\begin{figure}[t]\n\\centering\n   \\subfigure{\\includegraphics[width=\\columnwidth]{LamTurb_kmax_t_mu20}}\n\\caption{\n{\\bf Chiral magnetically driven turbulence.}\nThe black solid line shows the inverse correlation length, $k_{\\rm M}$,\nof the magnetic energy, defined by Equation (\\ref{eq_kcorr}), as a\nfunction of time $t$.\nUsing this wavenumber and the rms velocity, the fluid and magnetic\nReynolds numbers are estimated (see Equation~\\ref{eq_Rmkcorr}), that is\nshown by the dashed blue line.}\n  \\label{fig_LTkmax}\n\\end{figure}\n\n\nWe determine the correlation length of the magnetic field from\nthe magnetic energy spectrum via\n\\begin{equation}\n\\xi_{\\rm M}(t)\\equiv\n  k_{\\rm M}^{-1}(t) = \\frac{1}{\\mathcal{E}_{\\rm M}(t)} \\int k^{-1}\n  E_{\\rm M}(k,t)~\\mathrm{d}k .\n\\label{eq_kcorr}\n\\end{equation}\nThe wavenumber $k_{\\rm M}$ so defined coincides (up to a numerical factor of order unity) with\nthe so-called tracking solution, $\\Delta \\mu_{\\rm tr}$ in \\citet{BFR12}.\nThere it was demonstrated that, in the course of evolution, the chiral\nchemical potential follows $k_{\\rm M}(t)$.\nAnd, indeed, the evolution of $k_{\\rm M}$, shown in Figure~\\ref{fig_LTkmax},\nstarts at around 10 (the value of $\\mu_0\/2$ in this simulation) and then decreases to\n$k_{\\rm M} = k_1$ (corresponding to the simulation box size) at $t\\approx0.18~t_\\eta$.\nInterestingly, the chemical potential is affected by\nmagnetic helicity only at much later times, as can be seen in\nFigure~\\ref{fig_LTkmax}.\nBased on the wavenumber, $k_{\\rm M}$, we estimate\nthe Reynolds numbers as\n\\begin{equation}\n  {\\rm Re}_{_\\mathrm{M}} = \\mathrm{Re} = \\frac{u_\\mathrm{rms}}{\\nu k_{\\rm M}} .\n\\label{eq_Rmkcorr}\n\\end{equation}\nFigure~\\ref{fig_LTkmax} shows that the Reynolds number increases exponentially,\nmostly due to the fast increase of $u_\\mathrm{rms}$, and saturates\nlater at ${\\rm Re}_{_\\mathrm{M}}\\approx 10^2$.\nSimilarly, the turbulent diffusivity can be estimated as\n\\begin{equation}\n  \\eta_{_{\\rm T}} = \\frac{u_\\mathrm{rms}}{3~k_{\\rm M}} .\n\\label{eq_etaTkcorr}\n\\end{equation}\nDuring the operation of the mean-field large-scale dynamo, we find\nthat $\\eta_{_{\\rm T}}\\approx2.4\\times10^{-3}$,\nwhich is about 24 times larger than the molecular diffusivity $\\eta$.\nUsing these estimates, we determine\nthe chiral magnetic $\\alpha_\\mu$ effect from Equation~(\\ref{alphamu}).\nThe large-scale dynamo growth rate (\\ref{gamma_turb}) is\nshown as the solid red horizontal line in the middle panel of Figure~\\ref{fig_LTts}\nand is in agreement with the DNS results shown as the black solid line.\n\n\n\\begin{figure}[t]\n\\centering\n  \\subfigure{\\includegraphics[width=\\columnwidth]{LamTurb_ts4_mu20}}\n  \\subfigure{\\includegraphics[width=\\columnwidth]{LamTurb_gammak_mu20}}\n\\caption{\n{\\bf Chiral magnetically driven turbulence.}\nThe evolution of the magnetic energy $E_{\\rm M}$ on different wavenumbers $k$\n(top panel).\nThe growth rate as a function of $k$ in different time intervals as given in the\nplot legend. The black line corresponds to a fit, while the theoretical\nexpectations are given as a red\nline.}\n  \\label{fig_LT_gammak}\n\\end{figure}\n\nFurther analysis of the evolution of the magnetic field at different wavenumbers\nis presented in Figure~\\ref{fig_LT_gammak}.\nIn the top panel, we display the magnetic energy at various\nwavenumbers as a function of time.\nIn the kinematic phase, for $t<0.1~t_\\eta$, the fastest amplification\noccurs at $k=10$, as can also be seen in the energy spectra.\nAt wavenumbers $k<k_\\mu$ there is an initial phase of magnetic dissipation,\nfollowed by an exponential increase of the field.\nThe rapid transition between the two phases, which occurs at\n$t=0.05\\,t_\\eta$ for $k=30$ in our example, may lead to the impression of an\ninterpolation between long time steps.\nIn reality, however, the range $t=0$--$0.1\\,t_\\eta$ is\nresolved by approximately 500 time steps.\nAt $t\\approx0.18\\,t_\\eta$, the magnetic field grows only at $k=1$.\nThis confirms the idea that a large-scale (mean-field) dynamo operates.\nIn the next two panels, we compare the observed growth rates as a function\nof wavenumber at different time intervals.\nThe middle panel of Figure~\\ref{fig_LT_gammak}\nshows the growth rate in the laminar phase, where we find\ngood agreement with the theoretical predictions below $k\\approx20$.\nThe resulting value for the growth rate\ndepends on the accuracy of the fitting, and\na typical error of 10\\% is shown by\na gray uncertainty band in the middle panel of Figure~\\ref{fig_LT_gammak}.\nAlso, the observed growth rate of the mean-field dynamo, which we find from\nfitting growth rates in the time interval $0.17$--$0.24\\,t_\\eta$, is comparable to\nthe prediction from mean-field theory, using our estimates for the Reynolds\nnumber (\\ref{eq_Rmkcorr}) and the turbulent diffusivity (\\ref{eq_etaTkcorr}).\nAs the mean-field dynamo phase is followed by the nonlinear phase, the growth\nrate is more sensitive to the fitting regime.\nHence we indicate a 30\\% uncertainty band in this phase.\nThe time intervals for the two different fitting regimes are indicated by gray\narrows in the top panel.\n\n\\subsection{The effect of a strong initial magnetic field}\n\nThe effect of changing the chiral nonlinearity parameter\n$\\lambda_\\mu$ is explored in \\cite{BSRKBFRK17},\nwho considered values between $2\\times10^{-6}$ and $200$.\nUsing dimensional analysis and simulations, they showed that the extension\nof the inertial range of the turbulence\nis approximately $\\lambda_\\mu^{-1\/2}\/4$.\nThe ratio $\\mu\/k_\\lambda$ is approximately $660$ in our reference\nrun for chiral magnetically driven turbulence, which was presented in the last section.\n\n\\cite{BSRKBFRK17} found that $E_{\\rm M}(k,t)$ is bound from above\nby the value of $\\mu\/\\lambda$.\nIt is interesting to note that this also applies when the initial\nmagnetic field strength exceeds this limit.\nTo demonstrate this, we now present a simulation with an\ninitial magnetic energy spectrum $\\propto k^4$ for $k\/k_1<60$\nand exponential decrease for larger $k$ with $v_{\\rm A}\/c_{\\rm s}=0.089$.\nWe use $\\mu_0=40$, $\\eta=5\\times10^{-5}$,\n$\\overline{\\rho}\\lambda=8\\times10^8$,\n${\\rm Ma}_\\mu=0.0014$ and $\\lambda_\\mu=2$.\nThe result is shown in \\Fig{pspec_select_288_3D_kf60_mu040_lambda1e9d_B3em3}.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{spec_highB2}\n\\end{center}\\caption[]{\n{\\bf Chiral magnetically driven turbulence.}\nEvolution of the magnetic (blue lines) and kinetic (black lines) energy spectrum\nfor a run with large initial\nmagnetic field on a small spatial scale.\nThe initial spectra are shown as thick solid lines; later spectra have\nequal time intervals up to 0.025 $t_\\eta$ (shown in thick dashed lines).\nAbove $t = 0.025~t_\\eta$, the time intervals increase by a factor of two,\nuntil the final spectra are reached, presented here as thin solid lines.\nThe horizontal dashed gray line shows $1\/\\lambda_\\mu$, the upper limit\npredicted by the chiral conservation law, and the vertical gray line shows the\nscale where\nthe growth rate of the small-scale chiral instability\nreaches its maximum.\n}\n\\label{pspec_select_288_3D_kf60_mu040_lambda1e9d_B3em3}\n\\end{figure}\n\nAt early times, $E_{\\rm M}\/(\\eta^2 \\mu_0)$ overshoots the value of\n$(\\mu\/\\lambda)\/(\\eta^2 \\mu_0)=1\/\\lambda_\\mu$,\nbut after a short time it follows this limit almost precisely.\nThis shows that the bound on $E_{\\rm M}(k,t)$ is obeyed even when\nthe initial field strength exceeds this value.\n\n\n\\subsection{Stages of chiral magnetically driven turbulence}\n\\label{sec:stages}\n\nThis DNS demonstrates that the\nmagnetic field evolution proceeds in the following\ndistinct stages:\n\\begin{asparaenum}[\\it (1)]\n\\item{Small-scale chiral dynamo instability.}\n\\item{First nonlinear stage when the Lorentz force drives\nsmall-scale turbulence.}\n\\item{Formation of inverse energy transfer\nwith a $k^{-2}$ magnetic energy spectrum;\nsee \\cite{BSRKBFRK17} for details.}\n\\item{Generation of large-scale magnetic fields\nby chiral magnetically driven turbulence.}\n\\item{Decrease of the chemical potential and saturation\nof the large-scale chiral dynamo.}\n\\end{asparaenum}\nAlthough the magnetic field cannot grow any further, the spectrum\ncontinues to move to smaller wavenumbers in a shape-invariant fashion\n\\citep[see][]{BK17}.\nThis implies that the magnetic integral scale\n$\\xi_{\\rm M}\\equiv k_{\\rm M}^{-1}$ continues to grow and\nthe magnetic energy continues to decrease proportional to $t^{-2\/3}$\nwith $\\langle \\bm{B}^2 \\rangle \\xi_{\\rm M} \\approx {\\rm const}$.\n\n\n\n\n\n\\section{DNS of large-scale dynamos in forced, nonhelical, and homogeneous\nturbulence}\n\\label{sec_turbdynamo}\n\nIn this section, we study the evolution of the magnetic field in the presence\nof forced, nonhelical, and homogeneous turbulence in order\nto control the turbulence parameters in the chiral MHD simulations.\nChiral dynamos in forced turbulence can be described by the mean-field chiral\nMHD equations. The theoretical results related to\nthe mean-field chiral dynamos obtained in Paper~I\nhave been outlined in Section~\\ref{sec_meanfieldMHD}.\n\n\\subsection{DNS setup for externally forced turbulence}\n\nTo study chiral large-scale dynamos, we perform three-dimensional DNS\nwith externally forced turbulence and a spatial resolution of $200^3$.\nIn run Ta2-10, the resolution is $280^3$ (see Table~\\ref{table_simulations_forced}).\nTurbulence is driven via the forcing term $\\bm{f}(\\mathbf{x},t)$ in\nEquation~(\\ref{UU-DNS}).\nThe forcing function is nonhelical and localized around the wavenumber\n$k_\\mathrm{f}$; see \\cite{HBD2004} for details.\nFor the runs presented in the following,\nwe choose $k_\\mathrm{f}= 4$ and 10.\nThese values are small enough for the fluid and magnetic Reynolds numbers,\n${\\rm Re} = u_\\mathrm{rms}\/(\\nu k_\\mathrm{f})$ and\n${\\rm Re}_{_\\mathrm{M}} = u_\\mathrm{rms}\/(\\eta k_\\mathrm{f})$, respectively, to be sufficiently\nlarge for turbulence to develop.\nAt the same time, $k_\\mathrm{f}$ is large enough for a clear separation between the\nbox scale and the forcing scale, allowing to study\nof mean-field (large-scale) dynamos.\nIn the numerical simulations, we vary ${\\rm Ma}_\\mu$, $\\lambda_\\mu$, and ${\\rm Re}$\n(see Table~\\ref{table_simulations_forced}).\n\nFor comparison with the results from mean-field theory, the simulations need to\nfulfill the following criteria:\\\\\n\\begin{itemize}\n\\item{\nTo capture the maximum amplification inside the numerical domain with\n$k_\\mathrm{box}=2\\pi\/L_\\mathrm{box}=1$, the condition $k_\\mathrm{max}>1$ needs\nto be fulfilled.\nAs shown in Equation (\\ref{kmax_turb}), $k_\\mathrm{max}$ is proportional\nto $\\eta\/\\eta_{_{\\rm T}}$, which is inversely proportional to the magnetic\nReynolds number ${\\rm Re}_{_\\mathrm{M}}$.\nAs a result, the chemical potential needs to be sufficiently large for\n$k_\\mathrm{max}>1$.}\n\\item{\nDue to nonlocal effects, the turbulent diffusivity $\\eta_{_{\\rm T}}$\nis generally scale-dependent and\ndecreases above $k_\\mathrm{f}$ \\citep{BRS08}.\nFor comparison with mean-field theory, the chiral dynamo\ninstability has to occur on scales $k < k_\\mathrm{f}$, where\n$\\eta_{_{\\rm T}} \\approx u_\\mathrm{rms}\/(3k_\\mathrm{f})$.\nNote, however, that\nthe presence of a mean kinetic helicity in the system\ncaused by the CME (see Paper~I)\ncan increase the turbulent diffusivity $\\eta_{_{\\rm T}}$\nfor moderate magnetic Reynolds numbers by up to 50\\% \\citep{BSR17}.}\n\\item{\nTo simplify the system, we\navoid classical small-scale dynamo action, which\noccurs at magnetic Reynolds numbers larger than\n${\\rm Re}_{_{\\mathrm{M,crit}}}\\approx50$.}\n\\end{itemize}\n\n\n\n\\begin{table}[t]\n\\caption{\nOverview of Runs With Externally Forced Turbulence\n(Reference Run in Bold)}\n\\centering\n     \\begin{tabular}{l|lllllll}\n      \\hline\n      \\hline\n      \\\\\n\t~ \t        & $\\mu_0$ \t& $\\dfrac{{\\rm Ma}_\\mu}{10^{-3}}$ \t& $\\dfrac{\\lambda_\\mu}{10^{-6}}$  \t& $\\dfrac{k_\\lambda}{10^{-3} \\mu_0}$ & $\\dfrac{k_\\mathrm{diff}}{\\mu_0}$ & $k_\\mathrm{f}$ \t& ${\\rm Re}_{_\\mathrm{M}}$\t\t\\\\\t\n\t~ \t        & \t\t& ~\t\t \t& ~\t\t  \t& ~\t\t& ~\t\t& ~\t \t& (early $\\rightarrow$ late)\t\t\\\\\t                \\hline\n      \tTa2-1  \t& $20$\t\t& $8$\t& $16$ \t& 160\t\t  \t& 4.5\t& $10$ \t\t& $24 \\rightarrow 19$   \\\\\n      \tTa2-2  \t& $20$\t\t& $4$\t& $4.0$ & 80\t\t  \t& 63\t& $10$ \t\t& $36 \\rightarrow 28$ \t\\\\\n      \tTa2-3  \t& $20$\t\t& $8$\t& $16$ \t& 160\t\t  \t& 45\t& $10$ \t\t& $16 \\rightarrow   14$ \t\\\\\n      \tTa2-4  \t& $20$\t\t& $4$\t& $4.0$ & 80\t\t  \t& 63\t& $10$ \t\t& $4 \\rightarrow  13$ \t\\\\\n      \t\\textbf{Ta2-5} & $\\mathbf{20}$\t& $\\mathbf{8}$ & $\\mathbf{160}$ & $\\mathbf{51}$ \t& $\\mathbf{80}$\t& $\\mathbf{10}$ \t& $\\mathbf{24} \\rightarrow\\mathbf{18}$ \t\\\\\n      \tTa2-6  \t& $20$\t\t& $8$\t& $1.6$ & 51\t\t  \t& 80\t& $10$ \t\t& $16 \\rightarrow 14$ \t\\\\\n      \tTa2-7  \t& $30$\t\t& $12$\t& $32$ \t& 230\t\t  \t& 38\t& $4$ \t\t& $42 \\rightarrow  58$ \t\\\\\n      \tTa2-8  \t& $30$\t\t& $9$\t& $18$ \t& 160\t\t  \t& 43\t& $4$ \t\t& $58 \\rightarrow  65$ \t\\\\\n      \tTa2-9  \t& $30$\t\t& $9$\t& $13.5$ & 150\t\t  \t& 47\t& $4$ \t\t& $82 \\rightarrow  74$ \t\\\\\n      \tTa2-10  & $40$\t\t& $8$\t& $16$ \t& 160\t\t  \t& 45\t& $4$ \t\t& $119 \\rightarrow 107$ \\\\\n      \\hline\n      \\hline\n    \\end{tabular}\n  \\label{table_simulations_forced}\n\\end{table}\n\n\n\\subsection{DNS of chiral dynamos in forced turbulence}\n\\label{sec:forced-chiral-dynamos}\n\n\\begin{figure}[t]\n\\centering\n  \\includegraphics[width=\\columnwidth]{ts__TurbRef}\n\\caption{\n{\\bf Externally forced turbulence.}\nTime evolution of the magnetic field, the velocity field, and\nthe chemical potential, as well as the mean value of the magnetic\nhelicity (top panel).\nThe middle panel shows the growth rate of $B_\\mathrm{rms}$ as a function of\ntime (solid black line).\nThe red lines are theoretical expectations in different dynamo phases.\nIn the bottom panel, the ratio of the mean magnetic field to the total field\n$B_\\mathrm{rms}$ is presented.}\n\\label{fig_alphamu_ts}\n\\end{figure}\n\nThe time evolution of different quantities in our reference run is presented in\nFigure~\\ref{fig_alphamu_ts}.\nThe magnetic field first increases first exponentially,\nwith a growth rate $\\gamma \\approx 60~t_\\eta^{-1}$, which is\nabout a factor of 1.6 lower than that expected for the laminar $v_\\mu^2$ dynamo;\nsee the middle panel of Figure~\\ref{fig_alphamu_ts}.\nThis difference seems to be caused by the presence of random forcing;\nsee discussion below.\nAt approximately 0.2 $t_\\eta$, the growth rate decreases to a value\nof $\\gamma\\approx 15~t_\\eta^{-1}$ consistent with that of the mean-field\nchiral $\\alpha_\\mu^2$ dynamo, before saturation occurs at $0.4\\,t_\\eta$.\nThe evolution of $B_\\mathrm{rms}$ is comparable qualitatively\nin chiral magnetically produced turbulence; see Figure~\\ref{fig_LTts}.\nAn additional difference from the latter\nis the value of $u_\\mathrm{rms} \\approx 0.1$ for\nexternally forced turbulence, which is controlled by the intensity\nof the forcing function.\nAn indication of the presence of a mean-field dynamo is the evolution of\n$\\overline{B}\/B_\\mathrm{rms}$ in the bottom panel of\nFigure~\\ref{fig_alphamu_ts}, which reaches a value of unity at $0.3\\,t_\\eta$.\n\nThe energy spectra presented in Figure~\\ref{fig_alphamu_spec} support the\nlarge-scale dynamo scenario.\nFirst, the magnetic energy increases at all scales,\nand, at later times, the maximum of the magnetic energy\nis shifted to smaller wavenumbers, finally producing a\npeak at $k=1$, i.e., the smallest possible wavenumber in our periodic domain.\n\nA detailed analysis of the growth of magnetic energy is presented in\nFigure~\\ref{fig_alphamu_specana}.\nIn the first phase, the growth rate of the magnetic field is\nindependent of the wavenumber $k$ (see top panel), due to a coupling\nbetween different modes.\nThe growth rate measured in this phase is less than that\nin the laminar case (see middle panel), due to a scale-dependent\nturbulent diffusion caused by the random forcing.\n\nWithin the time interval $(0.22$--$0.28)\\,t_\\eta$, only the magnetic field at\n$k=1$ increases.\nThis is clearly seen in the bottom panel of Figure~\\ref{fig_alphamu_specana},\nwhere we show the evolution of the magnetic energy at different wavenumbers $k$.\nThe growth rate of the mean-field dynamo, which is determined\nat $k=1$, agrees with the result from\nmean-field theory, given by Equation~(\\ref{gammamax_turbRm}).\nThere is a small dependence of the resulting mean-field growth rate on the\nexact fitting regime.\nIf the phase of the\nmean-field dynamo is very short, changing the fitting range can affect the\nresult by a factor up to 30 \\%.\nWe use the latter value as an estimate of the uncertainty in the growth rate, and,\nin addition, indicate an error of 20 \\% in determining the Reynolds number,\nwhich is caused by the temporal variations of $u_\\mathrm{rms}$.\n\n\\begin{figure}[t]\n\\centering\n   \\subfigure{\\includegraphics[width=\\columnwidth]{spec__TurbRef}}\n\\caption{\n{\\bf Externally forced turbulence.}\nEvolution of kinetic (black lines) and magnetic energy spectra (blue lines)\nfor the reference run Ta2-5.\nThe ratio $\\mu_0\/\\lambda$ is indicated by the horizontal dashed line.}\n\\label{fig_alphamu_spec}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n   \\subfigure{\\includegraphics[width=\\columnwidth]{Emag_t__TurbRef}}\n   \\subfigure{\\includegraphics[width=\\columnwidth]{gamma_k__TurbRef}}\n\\caption{\n{\\bf Externally forced turbulence.}\nTime evolution of the magnetic energy at different wavenumbers $k$\n(top panel).\nThe remaining panels show the growth rates as a function of scale in\ndifferent fit intervals.}\n\\label{fig_alphamu_specana}\n\\end{figure}\n\n\n\\subsection{Dependence on the magnetic Reynolds number}\n\nBased on the mean-field theory developed in Paper~I, we expect\nthe following.\nUsing the expression for the $\\alpha_\\mu$ effect\ngiven by\nEquation (\\ref{alphamu}), the maximum growth rate (\\ref{gammamax_turb})\nfor the mean-field dynamo can be rewritten as a function of the magnetic\nReynolds number:\n\\begin{eqnarray}\n\\gamma_{\\rm max} ({\\rm Re}_{_\\mathrm{M}}) = \\frac{\\overline{v}_\\mu^2 (1 - 2\/3~\\ln{\\rm Re}_{_\\mathrm{M}} )^2}{4 \\eta \\,\n(1 + \\, {\\rm Re}_{_\\mathrm{M}}\/3)} ,\n\\label{gammamax_turbRm}\n\\end{eqnarray}\nwhere the ratio $\\eta_{_{\\rm T}}\/\\eta = {\\rm Re}_{_\\mathrm{M}}\/3$.\n\nWe perform DNS with different Reynolds numbers to\ntest the scaling of $\\gamma_{\\rm max}({\\rm Re}_{_\\mathrm{M}})$ given by\nEquation~(\\ref{gammamax_turbRm}).\nThe parameters of the runs with externally forced\nturbulence are summarized in Table \\ref{table_simulations_forced}.\nWe vary $\\nu \\, (=\\eta$), the forcing wavenumber $k_\\mathrm{f}$, as well as the\namplitude of the forcing,\nto determine the function $\\gamma_{\\rm max}({\\rm Re}_{_\\mathrm{M}})$.\nIn the initial phase, $u_\\mathrm{rms}$ is constant in time.\nOnce large-scale turbulent dynamo action occurs,\nthere are additional minor\nvariations in $u_\\mathrm{rms}$, because the system is already\nin the nonlinear phase.\nThe nonlinear terms in the Navier-Stokes equation lead to a modification of\nthe velocity field at small spatial scales, which affects the value of\n$u_\\mathrm{rms}$ and results in\nthe small difference between\nthe initial and final values of the Reynolds numbers\n(see Table \\ref{table_simulations_forced}).\n\nAccording to Equation~(\\ref{kmax_turb}),\nthe wavenumber associated with the maximum growth rate\nof the large-scale turbulent\ndynamo instability decreases with increasing ${\\rm Re}_{_\\mathrm{M}}$.\nIn order to keep this mode inside the computational domain and hence to\ncompare the measured growth rate with the maximum one given by\nEquation~(\\ref{gammamax_turbRm}),\nwe vary the value of $\\mu_0$ in our simulations.\nThe variation of $\\mu_0$ and the additional variation of $\\eta$ for scanning\nthrough the ${\\rm Re}_{_\\mathrm{M}}$ parameter space, implies that ${\\rm Ma}_\\mu$ changes\ncorrespondingly.\n\nThe values of the nonlinear parameter $\\lambda$ should be within a certain range.\nIndeed, the saturation value of the magnetic field, given by\nEquation~(\\ref{eq_Bsat}), is proportional to $\\lambda^{-1\/2}$.\nIn order for the Alfv\\'en velocity not to exceed the sound speed,\nwhich would result in a very small time step in DNS,\n$\\lambda$ should not be below a certain value.\nOn the other hand, $\\lambda$ should not be too large,\nas in this case the dynamo would saturate quickly, and there is only a very short\ntime interval of the large-scale dynamo.\nIn this case, determining the growth rate of the mean-field dynamo,\nand hence comparing with the mean-field theory, is difficult.\n\n\\begin{figure}[t]\n\\centering\n  \\includegraphics[width=\\columnwidth]{gamma_Rm}\n\\caption{\n{\\bf Externally forced turbulence and chiral magnetically driven turbulence.}\nThe normalized growth rate $\\gamma~\\eta\/v_\\mu^2$ of the magnetic\nfield as a function of the magnetic Reynolds number ${\\rm Re}_{_\\mathrm{M}}$.\nThe gray data points show the\ngrowth rate in the initial, purely kinematic phase of the simulations.\nThe blue data points show the measured growth rate of the magnetic field\non $k=1$, when the large-scale dynamo occurs.\nThe diamond-shaped data points represent simulations of forced turbulence, while\nthe dot-shaped data points refer to the case of\nchiral magnetically driven turbulence.\nThe growth rate observed in the initial laminar phase\nfor the case of chiral magnetically driven turbulence\nis shown at ${\\rm Re}_{_\\mathrm{M}}=2$, with the left arrow indicating that the actual ${\\rm Re}_{_\\mathrm{M}}$ is much\nlower and out of the plot range at this time; see Figure~\\ref{fig_LTkmax}.}\n  \\label{fig_gamma_Re}\n\\end{figure}\n\nIn Figure~\\ref{fig_gamma_Re} we show the normalized growth\nrate $\\gamma~\\eta\/v_\\mu^2$ of the magnetic\nfield as a function of the magnetic Reynolds number ${\\rm Re}_{_\\mathrm{M}}$.\nThe gray data points show the growth rate in the initial,\npurely kinematic phase of the simulations.\nThe blue data points show the measured growth rate of the magnetic field\non $k=1$, when the large-scale dynamo occurs.\nFor comparison of the results with externally forced turbulence\n(indicated as diamond-shaped data points), we show\nin Figure~\\ref{fig_gamma_Re} also the results obtained for the\ndynamo in chiral magnetically driven turbulence, which are indicated as dots.\n\nIn DNS with externally forced turbulence, we see in all cases a\nreduced growth rate due to mode coupling.\nContrary to the case with externally forced turbulence,\nin DNS with the chiral magnetically driven turbulence,\nwe do initially observe the purely laminar dynamo\nwith the growth rate given by Equation~(\\ref{gamma-max}),\nbecause there is no mode coupling in the initial phase\nof the magnetic field evolution in this case.\nOn the other hand, the measured growth rates of the mean-field dynamo\nin both cases agree (within the error bars) with the growth rates\nobtained from the mean-field theory.\n\n\n\n\n\n\\section{Chiral MHD dynamos in astrophysical relativistic plasmas}\n\\label{sec_astro}\n\n\nIn this section, the results for the\nnonlinear evolution of the chiral chemical potential,\nthe magnetic field, and the turbulent state of the plasma found in this paper\nare applied to astrophysical relativistic plasmas.\nWe begin by discussing the role of chiral dynamos in the early universe and\nidentify conditions under which the CME affects the generation and evolution of\ncosmic magnetic fields.\nFinally, in Section~\\ref{sec:PNS}, we examine the importance of the CME\nin proto-neutron stars (PNEs).\n\n\n\\subsection{Early Universe}\n\\label{sec:early-universe}\n\nIn spite of many possible mechanisms that can produce magnetic fields in the\nearly universe\n\\citep[see, e.g.,][for reviews]{W02,WEtAl12,DN13,Giovannini:2003yn,S16},\nunderstanding the origin of cosmic magnetic fields remains an open problem.\nTheir generation is often associated with nonequilibrium events in the universe\n(e.g., inflation or phase transitions).\nA period of particular interest is the electroweak (EW) epoch, characterized\nby temperatures of $\\unit[10^{15}]{K}$ ($k_{\\rm B} T \\sim \\unit[100]{GeV}$).\nSeveral important events take place around this time: the electroweak\nsymmetry gets broken, photons appear while intermediate vector bosons\nbecome massive, and the asymmetry between matter and antimatter appears\nin the electroweak baryogenesis scenario \\citep{KRS85};\nsee, for example, the review by \\citet{MRM12}.\nMagnetic fields of appreciable strength can be generated as a consequence of\nthese events \\citep{V91,Olesen:1992np,Enqvist:1993np,Enqvist:1994dq,\nVachaspati:1994ng,GGV95, 1996PhLB..380..253D,BBM96,V01,Semikoz:2010zua}.\nTheir typical correlation length $\\xi_{\\rm M}^{(\\rm ew)} \\sim (\\ensuremath{\\alpha_{\\rm em}} T)^{-1}$\ncorresponds to only a few centimeters today -- much less than the observed correlation\nscales of magnetic fields in galaxies or galaxy clusters.\nTherefore, in the absence of mechanisms that can increase\nthe comoving scale of the magnetic field beyond\n$\\xi_{\\rm M}^{(\\rm ew)}$, such fields were deemed to be irrelevant\nto the problem of cosmic magnetic fields \\citep[for discussion,\nsee, e.g.,][]{Durrer:2003ja,Caprini:2009pr,Saveliev:2012ea,KTBN13}.\n\nThe situation may change if (i) the magnetic fields are helical and\n(ii) the plasma is turbulent.\nIn this case, an inverse transfer of magnetic energy may develop,\nwhich leads to a shift of the typical scale of the magnetic field to\nprogressively larger scales \\citep{BEO96, CHB01, BJ04, KTBN13}.\nThe origin of such turbulence has been unknown.\nAn often considered paradigm is that a random magnetic field, generated at small\nscales, produces turbulent motions via the Lorentz force.\nHowever, continuous energy input is required.\nIf this is not the case, the magnetic field decays:\n$\\langle \\bm{B}^2 \\rangle \\sim t^{-2\/3}$ as the correlation scale grows\n\\citep{BM99PhRvL, KTBN13}, so that $\\langle \\bm{B}^2 \\rangle \\xi_{\\rm M} = {\\rm const}$.\n\nIn the present work, we demonstrated that the presence of a finite chiral\ncharge in the plasma at the EW epoch is sufficient to satisfy the above\nrequirements (i) and (ii).\nAs a result,\n\\begin{asparaenum}[\\it (1)]\n\\item helical magnetic fields are excited,\n\\item turbulence with large ${\\rm Re}_{_\\mathrm{M}}$ is produced, and\n\\item the comoving correlation scale increases.\n\\end{asparaenum}\nWe discuss each of these phases in detail below.\n\n\n\\subsubsection{Generation and evolution of cosmic magnetic fields in the\npresence of a chiral chemical potential}\n\\label{subsec_cosmologybounds}\n\nAlthough it is not possible to perform numerical simulations with\nparameters matching those of the early universe, the results of the\npresent paper allow us to make qualitative predictions about the fate\nof cosmological magnetic fields generated at the EW epoch in the\npresence of a chiral chemical potential.\n\nAll of the main stages of the magnetic field evolution,\nsummarized in Section~\\ref{sec:stages}, can occur in the early universe\n(a sketch of the main phases is provided in \\Fig{fig:phases}).\n\n\\begin{figure}[!t]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{sketch_phases_2}\n  \\caption{\n  \\textbf{Chiral MHD dynamos in the early universe.}\nSketch of the different phases of the chiral mean-field dynamo.\nFrom left to right:\nsmall-scale chiral dynamo (phase 1), large-scale turbulent dynamo (phase 2), and\nsaturation (phase 3).\nAfter saturation of the dynamo, the magnetic magnetic field dissipates.\nThe upper horizontal dotted line shows the initial value of $\\mu$ and the lower\none the ``saturation limit,'' given by Eq.~\\protect\\eqref{eq:2}.}\n  \\label{fig:phases}\n\\end{figure}\n\nPhase 1.\nAt this initial stage, the small-scale chiral dynamo instability develops at\nscales around $\\xi_\\mu$, where\n\\begin{equation}\n  \\xi_\\mu \\equiv \\frac{2}{|\\mu_0|} ,\n  \\label{eq_mu-1}\n\\end{equation}\nand\n\\begin{equation}\n  \\label{eq:1}\n  \\mu_0 \\approx 4\\ensuremath{\\alpha_{\\rm em}}\\frac{\\mu_5}{\\hbar c}\\approx\n\\unit[1.5\\times10^{14}]{cm^{-1}}\\frac{\\mu_5}{\\unit[100]{GeV}}.\n\\end{equation}\nThe chemical potential $\\mu_5$ can be approximated by the thermal energy\n$k_{\\rm B} T$ for order-of-magnitude estimates.\nIn what follows, we provide numerical estimates for $\\mu_5 = 100\\,{\\rm GeV}$ which\ncorresponds to the typical thermal energy of relativistic particles\nat the EW epoch.\nThe characteristic energy at the quantum chromodynamics phase transition\nis $\\approx 100\\,{\\rm MeV}$ where the quark--gluon plasma turns into hadrons.\nWe stress, however, that the MHD formalism is only valid if the scales\nconsidered are larger than the mean free path given by Equation~(\\ref{eq_mfp}).\nComparing the chiral instability scale $k_\\mu^{-1}$ with $\\ell_\\mathrm{mfp}$ \nresults in the condition \n$\\mu_5 \\ll k_{\\rm B} T\\, 4\\pi^2 \\ensuremath{\\alpha_{\\rm em}} \\ln{((4\\pi \\ensuremath{\\alpha_{\\rm em}})^{-1\/2})}$. \nStrictly speaking, modeling a system that does not fulfill this condition \nrequires full kinetic theory as described, for example, in \\citet{CPWW13} or\nin \\citet{AY13}.\n\nThe growth rate of an initially weak magnetic field in the linear\nstage of the chiral dynamo instability is given by \\Eq{gamma-max}:\n\\begin{equation}\n\\label{eq:9}\n  \\gamma^{\\rm max}_\\mu = \\frac{\\mu_0^2\\eta}4 \\approx 2.4\\times 10^{19}T_{100}^{-1}\\,{\\rm s}^{-1}.\n\\end{equation}\nFor the value of the magnetic diffusivity $\\eta = c^2\/(4 \\pi \\sigma)$\nin the early universe, we adopted the conductivity $\\sigma$ from\nEquation~(1.11) of \\cite{ArnoldEtAl2000}.\nNumerically,\n\\begin{equation}\n\\eta(T)={7.3\\times 10^{-4}}\\,{\\hbar c^2\\overk_{\\rm B} T}\\approx\n{4.3\\times10^{-9}}T_{100}^{-1}\\,{\\rm cm}^2\\,{\\rm s}^{-1} ,\n\\label{eq_rDiffvA}\n\\end{equation}\nwhere $T_{100}=1.2\\times10^{15}\\,{\\rm K}$ (so that $k_{\\rm B} T_{100} =\\unit[100]{GeV}$).\nAs a result, the number of $e$-foldings over one Hubble time $t_H$ is\n$$\\gamma^{\\rm max}_\\mu t_H \\gg 1,$$ where\n\\begin{equation}\n  \\label{eq:tH}\n  t_H = H^{-1}(T) \\approx 4.8\\times10^{-11}\\,g_{100}^{-1\/2}T_{100}^{-2}\\,\\,{\\rm s}\n\\end{equation}\n(here $g_*$ is the number of relativistic degrees of freedom and $g_{100}=g_*\/100$).\nWe should stress that this picture has been known before and was described\nin many previous works \\citep{JS97,Frohlich:2000en,Frohlich:2002fg,BFR12}.\n\nWe note that a nonzero chiral flipping rate\n$\\Gamma_\\mathrm{f}$ has been discussed in the literature\n\\citep{CDEO92,BFR12,DS15,BFR15,SiglLeite2016}.\nIn Section~\\ref{sec_flip}, we have found\nin numerical simulations that the flipping term affects the evolution\nof the magnetic field only\nfor large values of $f_\\mu$, when the flipping term is of the order of or\nlarger than the $\\lambda_\\mu$ term in Equation~(\\ref{mu-NS});\nsee also Equation~(\\ref{eq_fmu}) and Figure~\\ref{fig_ts_flip}.\nWhen adopting the estimate in \\citet{BSRKBFRK17} of $f_\\mu\\approx 1.6\\times10^{-7}$,\nchirality flipping is not likely to play a significant role for\nthe laminar $v_\\mu^2$ dynamo in the early universe at very high temperatures\nof the order of $100 \\,{\\rm GeV}$.\nHowever, $\\Gamma_\\mathrm{f}$ depends on the ratio $m_e c^2\/(k_{\\rm B} T)$ and\nthus suppresses all chiral effects once the universe has cooled down to\n$k_{\\rm B} T \\approx m_e c^2$ \\citep{BFR12}.\nAt this point, we stress again that the true value of $\\mu_0$ is unknown\nand has here been set to the thermal energy in Equation~(\\ref{eq:1}).\nIf it turns out that the initial value of the chiral chemical potential\nis much smaller than the thermal energy, $f_\\mu$ becomes larger, and\nthe flipping rate can play a more important role already during the initial\nphases of the chiral instability in the early universe.\nThis scenario is not considered in the following discussion.\n\nIn the regime of the laminar $v_\\mu^2$ dynamo, one could reach\n$\\mathcal{O}(10^9)$ $e$-folds over the Hubble time $t_{\\rm H}$; see lower panel\nof Figure~\\ref{fig:chiral_turbulence}.\nHowever, as shown in this work, already after a few hundred $e$-foldings,\nthe magnetic field starts to excite turbulence via the Lorentz force.\nThis happens once the magnetic field is no longer force-free.\nOnce the flow velocities reach the level $v_\\mu = \\mu_0\\eta$, nonlinear\nterms are no longer small, small-scale turbulence is produced, and\nthe next phase begins.\n\n\nPhase 2.\nThe subsequent evolution of the magnetic field depends on the strength\nof the chiral magnetically excited turbulence.\nThis has been shown in the mean-field analysis of \\citet{REtAl17} and\nis confirmed by the present work; see, for example, Figure~\\ref{fig_gamma_Re}.\nThe growth rate and instability scale depend on the magnetic Reynolds\nnumber; see Equations~(\\ref{gamma_turb})--(\\ref{gammamax_turb}).\nThe maximum growth rate for ${\\rm Re}_{_\\mathrm{M}} \\gg 1$ is given by\n\\begin{equation}\n  \\label{eq:10}\n \\gamma^{\\rm max}_{\\alpha} = \\gamma_\\mu^{\\rm max} \\frac{4}3 \\frac{(\\ln {\\rm Re}_{_\\mathrm{M}})^2}{{\\rm Re}_{_\\mathrm{M}}},\n\\end{equation}\nwhere $\\gamma_\\mu^{\\rm max}$ is given by \\Eq{eq:9}.\nFor the early universe, it is impossible to determine the exact value of the\nmagnetic Reynolds number from the numerical simulations, but one\nexpects ${\\rm Re}_{_\\mathrm{M}} \\gg 1$ and we show in \\Fig{fig:chiral_turbulence}\nthat, in a wide range of magnetic Reynolds numbers,\n$1 \\ll {\\rm Re}_{_\\mathrm{M}} \\ll 6\\times 10^{12}$, the number of $e$-foldings during one\nHubble time is much larger than $1$.\nThe turbulence efficiently excites magnetic fields at scales much larger than\n$\\xi_\\mu$ (Figure~\\ref{fig:chiral_turbulence}, top panel).\n\n\\begin{figure}[!t]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{TurbDyn_EU}\n  \\caption[Turbulence-driven instability in the early universe]\n  {\\textbf{Chiral MHD dynamos in the early universe. }\n    The ratios between $\\xi_\\alpha$ of the turbulence-driven dynamo (Eq.~(\\ref{kmax_turb}))\n    and scale $\\xi_\\mu$ (Equation~\\protect\\eqref{eq_mu-1}), as well as the ratio\n    between $\\xi_\\mu$ and the Hubble radius at different temperatures.\n    In the top panel, furthermore, the ratio $\\xi_\\mu\/\\xi_\\lambda$ is presented.\n    Maximum growth rates over the Hubble time for laminar ($\\gamma_\\mu^{\\rm max}$)\n    and turbulent ($\\gamma_\\alpha^{\\rm max}$) regimes are shown in the bottom panel.\n  }\\label{fig:chiral_turbulence}\n\\end{figure}\n\nUsing dimensional analysis and DNS, \\citet{BSRKBFRK17} demonstrated\nthat the resulting spectrum of the magnetic fields behaves as\n$E_{\\rm M}\\propto k^{-2}$ between $k_\\mu$ and $k_\\lambda$, given by\nEquation~(\\ref{klambda}).\nThe wavenumber $k_\\lambda$ depends on the nonlinearity parameter $\\lambda$,\ndefined by Equation~(\\ref{eq_lambda}), which, in the early universe, is given by\n\\begin{equation}\n  \\lambda=3 \\hbar c\\,\\left({8\\ensuremath{\\alpha_{\\rm em}}\\overk_{\\rm B} T}\\right)^2\\approx1.3\\times10^{-17}\\,T_{100}^{-2}\\,\n\\,{\\rm cm}\\,{\\rm erg}^{-1}.\n  \\label{eq_lambda_1}\n\\end{equation}\nWe note that this expression is, strictly speaking, only valid when\n$k_{\\rm B} T \\gg \\max(|\\mu_L|,|\\mu_R|)$ and modifications might be expected \noutside of this regime.\nFurther, the mean density of the plasma\n\\begin{equation}\n   \\overline{\\rho}=\\frac{\\pi^2}{30}\\,g_*\\frac{(k_{\\rm B} T)^4}{\\hbar^3c^5}\n      \\approx  7.6\\times10^{26}g_{100}T_{100}^4\\,{\\rm g}\\,{\\rm cm}^{-3}.\n\\end{equation}\nThe ratio $\\xi_\\lambda\/\\xi_\\mu = k_\\mu\/k_\\lambda$ is presented in the top\npanel of \\Fig{fig:chiral_turbulence}, but we note that the exact numerical\ncoefficient in the condition $k_\\mu\/k_\\lambda \\gg 1$ might depend on ${\\rm Re}_{_\\mathrm{M}}$.\n\nPhase 3.\nThe stage of large-scale turbulent dynamo action ends with the\n\\emph{saturation phase} (see Section~\\ref{sec:stages} and\n\\Fig{fig:phases}).\nAt this stage, the total chiral charge (determined by the initial\nconditions) gets transferred to magnetic helicity.\nAs shown in \\citet{BFR12} (see also~\\citet{JS97} for earlier work, as well\nas \\cite{TVV12} and \\cite{Hirono:2015rla} for more discussion), and confirmed by\nnumerical simulations in \\citet{BSRKBFRK17} and in the present work,\nthe chiral chemical potential $\\mu$ follows $k_{\\rm M}$ at this stage\nand thus decreases with time.\nTherefore, most of the chiral charge will be transferred with time into magnetic helicity,\n\\begin{equation}\n  \\langle \\bm{A} \\cdot \\bm{B}\\rangle \\simeq \\xi_{\\rm M} \\langle \\bm{B}^2 \\rangle \\to \\frac{2\\mu_0}\\lambda ,\n  \\label{eq:2}\n\\end{equation}\nswitching off the CME (the end of Phase 3 in \\Fig{fig:phases}).\n\n\\subsubsection{Chiral MHD and cosmic magnetic fields}\n\\label{sec:end_chiral_MHD}\n\nMagnetic fields produced by chiral dynamos are fully helical.\nOnce the CME has become negligible, the subsequent phase of decaying\nhelical turbulence begins and the\nmagnetic energy decreases, while the magnetic correlation length increases\nin such a way that the magnetic helicity~\\eqref{eq:2} is conserved\nfor very small magnetic diffusivity \\citep{BM99PhRvL, KTBN13}.\n\nBased on \\Eq{eq:2}, one can estimate the magnetic helicity \\emph{today}; see also \\cite{BSRKBFRK17}.\nTaking as an estimate for the chiral chemical potential $\\mu_5 \\sim k_{\\rm B} T$ (this means that the density of the chiral charge is of the order of the\nnumber density of photons), one finds\n\\begin{multline}\n  \\label{eq:Br17:3}\n  \\bra{\\bm{B}^2}\\xi_{\\rm M} \\simeq\n  \\frac{\\hbar c}{4\\ensuremath{\\alpha_{\\rm em}}} \\frac{g_{0}}{g_\\ast} n_\\gamma^{(0)}\n  \\simeq 6\\times 10^{-38}\\,{\\rm G}^2\\,{\\rm Mpc} .\n\\end{multline}\nHere, the present number density of photons is $n_\\gamma^{(0)} = 411\\,{\\rm cm}^{-3}$,\nand the ratio $g_{0}\/g_\\ast\\approx3.36\/106.75$ of the effective\nrelativistic degrees of freedom today and at the EW epoch appears,\nbecause the photon number density dilutes as $T^3$ while the magnetic helicity\ndilutes as $a^{-3}$.\nWe recall that, to arrive at the numerical value in $\\,{\\rm G}^2\\,{\\rm Mpc}$\ngiven in Equation~(\\ref{eq:Br17:3}), an additional $4\\pi$ factor\nwas applied to convert to Gaussian units.\n\nUnder the assumption that the spectrum of the cosmic magnetic field is sharply\npeaked at some scale\n$\\xi_0$ (as is the case in all of the simulations presented here),\nthe lower bounds on magnetic fields, inferred\nfrom the nonobservation of GeV cascades from TeV sources\n\\citep{NV10,Tavecchio:10,Dolag:10} can be directly translated into\na bound on magnetic helicity today. The observational bound scales\nas $|\\bm{B}| \\propto \\xi_0^{-1\/2}$ for $\\xi_0 < 1\\,{\\rm Mpc}$ \\citep{NV10} and\ntherefore $\\bra{\\bm{B}^2}\\xi_0 = {\\rm const} > 8\\times 10^{-38}\\,{\\rm G}^2\\,{\\rm Mpc}$.\nThe numerical value is obtained using the most conservative bound\n$|\\bm{B}| \\ge 10^{-18}\\,{\\rm G}$ at $1\\,{\\rm Mpc}$ (\\citealt{DCRFCL11},\nsee also~\\citealt{DN13}).\nThese observational constraints for intergalactic magnetic fields are\ncompared to the magnetic field produced in chiral MHD for different values\nof the initial chiral chemical potential in Figure~\\ref{fig_B_xi__comov}.\n\nThe limit given by Equation~(\\ref{eq:Br17:3}) is quite general.\nIt does not rely on chiral MHD or the CME, but simply\nreinterprets the bounds of \\cite{NV10}, \\cite{Tavecchio:10}, \\cite{Dolag:10},\nand \\cite{DCRFCL11} as bounds on magnetic helicity.\nGiven such an interpretation, we conclude that \\emph{if cosmic magnetic fields\nare helical and have a cosmological origin, then at some moment in the history\nof the universe the density of chiral charge was much larger\nthan $n_\\gamma(T)$}.\nThis chiral charge can be, for example, in the form of magnetic helicity or of\nchiral asymmetry of fermions, or both.\nTo generate such a charge density, some new physics beyond the Standard Model of\nelementary particles is required.\nBelow we list several possible mechanisms that can generate large initial\nchiral charge density:\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{B_xi__comov}\n  \\caption{\n    {\\bf Chiral MHD dynamos in the early universe.}\nThe magnetic field strength resulting from a chiral dynamo as a function of correlation length in comoving\nunits and comparison with observational constraints.\nThe differently colored lines show the chiral magnetically produced magnetic field\nstrength in the range between the injection length $\\mu^{-1}$ and the\nsaturation length $k_\\lambda^{-1}$; see Equations~(\\ref{eq_mu-1}) and\n(\\ref{klambda}), respectively.\nThe colors indicate different values of the chiral chemical potential:\nRed refers to the value of $\\mu_0$ given in Equation~(\\ref{eq:1}), blue to\n$10^{-2}\\mu_0$, and purple to $10^{2}\\mu_0$.\nThe dashed gray line is an upper limit on the intergalactic magnetic field from\nZeeman splitting.\nSolid gray lines,\nrefer to the lower limits reported by \\citet{NV10} (``NV10'') and \\citet{DCRFCL11} (``D+11''), respectively.\nThe vertical dotted gray lines show the\nhorizon at $k_{\\rm B} T = 100\\,{\\rm GeV}$ and $100\\,{\\rm MeV}$ correspondingly.\nThe thin colored arrows refer to the nonlinear evolution of magnetic fields in\nan inverse cascade in helical turbulence up to the final value as given in\n\\citet{BJ04}\n(line ``BJ04'').\n}\n  \\label{fig_B_xi__comov}\n\\end{figure}\n\n\\begin{asparaenum}[\\it (1)]\n\\item The upper bound in \\Eq{eq:Br17:3} assumes that only one fermion of the\nStandard Model developed a chiral asymmetry $\\sim n_\\gamma$.\n  Many fermionic species are present in the plasma at the electroweak epoch.\n  They all can have a left--right asymmetric population of comparable size,\n  increasing the total chirality by a factor $\\mathcal{O}(10)$, which\n  makes the estimate~(\\ref{eq:Br17:3}) consistent with the lower bound\n  from \\citet{DCRFCL11}.\n  One should check, of course, whether for more massive fermions the\n  chirality flipping rate is much slower than the dynamo growth rate\n  determined by Equation~(\\ref{eq:9}).\n\\item The estimate~(\\ref{eq:Br17:3}) assumed that left--right asymmetry was\ncreated via thermal processes. Of course, new physics at the EW epoch can result in\nnonthermal production of chiral asymmetry (e.g.\\ via decays of some long-lived\nparticles), thus leading to $n_5 \\gg n_\\gamma$ and so increasing the limit\n(\\ref{eq:Br17:3}).\n\\item\n  The left--right asymmetry may be produced as a consequence of the decay\n  of helical \\emph{hypermagnetic} fields prior to the EW epoch.\n  Such a scenario, relating hypermagnetic helicity to the\n  chiral asymmetry, has been discussed previously, such as in\n  \\cite{Giovannini:1997eg} and \\cite{Semikoz:2012ka}.\n  A conservation law similar to that of~(\\ref{cons_law}) exists also for\n  hypermagnetic fields, and the decay of the latter may cause asymmetric\n  populations of left and right states.\n\\item\nIn our analysis, we have not taken into account the chiral vortical effect\n\\citep{Vilenkin:79}.\nFor nonvanishing chemical potential, it leads to an additional current\nalong the direction of vorticity \\citep[see, e.g.,][]{TVV12}.\n\\end{asparaenum}\n\nFrom the point of view of chiral MHD, the value of $\\mu_0$ (to which\nthis bound is proportional) is just an initial condition and therefore\ncan take arbitrary values.\nOnce an initial condition with a large value of $\\mu_0$ has been\ngenerated, the subsequent evolution (as described above) does not require\nany new physics.\n\nMoreover, the coupled evolution of magnetic helicity and chiral chemical\npotential is \\emph{unavoidable} in the relativistic plasma and should be an\nintegral part of relativistic MHD (as was discussed in Paper~I).\n\n\n\n\n\\subsection{PNSs and the CME}\n\\label{sec:PNS}\n\n\nIn this section, we explore whether the CME and chiral dynamos can play a\nrole in the development of strong magnetic fields in neutron stars.\nA PNS is a stage of stellar evolution after\nthe supernova core collapse and before the cold and dense neutron star is\nformed \\citep[see, e.g.,][]{PRPLM99}.\nPNSs are characterized by high temperatures\n(typically $k_{\\rm B} T \\sim \\unit[\\mathcal{O}(10)]{MeV}\\gg m_e c^2$),\nlarge lepton number density (electrons Fermi energy $\\mu_e \\sim$ a few\nhundreds of MeV), the presence of turbulent flows in the interior, and quickly\nchanging environments.\nOnce the formation of a neutron star is completed, its magnetic\nfield can be extremely large.\nNeutron stars that exceed the quantum electrodynamic limit\n$B_\\mathrm{QED}\\equiv m_e^2c^3\/(e \\hbar)\\approx4.4\\times10^{13}~\\,{\\rm G}$ are known as\n``magnetars'' \\citep[see, e.g.,][for recent reviews]{MPM15,TZ15,KB17}.\nThe origin of such strong magnetic fields remains unknown, although\nmany explanations have been proposed; see, for example, \\citet{DT92}, \n\\cite{AWML03}, and \\cite{FW06}.\n\nThe role of the CME in the physics of (proto)neutron stars and\ntheir contribution to the generation of strong magnetic fields have been\ndiscussed in a number of works\n\\citep{Charbonneau:2009ax,Ohnishi:2014uea,Dvornikov:2015lea,DS15,\nGKR15,Dvornikov:2016cmz,SiglLeite2016,Yamamoto:2015gzz}.\n\n\n\\subsubsection{Chiral MHD in PNSs}\n\\label{sec:chiral-mhd-PNS}\n\nDuring the formation of a PNS, electrons and protons are converted into neutrons,\nleaving behind left-handed neutrinos.\nThis is known as the Urca process\n\\citep[$e + p \\to n + \\nu_e$;][]{Haensel:95}.\nIf the chirality-flipping timescale, determined by the electron's\nmass, is longer than the instability scale, the net chiral\nasymmetry in the PNS can lead to the generation\nof magnetic fields. This scenario has been discussed previously\n\\citep{Ohnishi:2014uea,SiglLeite2016,GKR15}.\nThe chiral turbulent dynamos discussed in this work can be relevant for\nthe physics of PNSs and can affect our conclusions\nabout the importance of the CME.\nHowever, to make a detailed quantitative analysis, a number of factors\nshould be taken into account:\n\\begin{asparaenum}[\\it (1)]\n\\item The rate of the Urca process is strongly temperature dependent\n  \\citep{LPPH91,Haensel:95}.\n  The temperatures inside PNSs are only known with large uncertainties, and\n  the cooling occurs on a scale of seconds \\citep[see, e.g.,][]{PRPLM99},\n  making estimates of the Urca rates uncertain by orders of magnitude.\n\\item The chirality flipping rate that aims to restore the depleted population of\n  left-chiral electrons is also expected to be temperature dependent\n  \\citep[see, e.g.,][]{GKR15,SiglLeite2016}.\n\\item The neutrinos produced via the Urca process are trapped in the\n  interior of a PNS and can release the chiral asymmetry back into the plasma\n  via the $n + \\nu_e \\to e + p$ process.\n  Therefore, only when the star becomes transparent to neutrinos\n  (as the temperature drops to a few MeV) does the creation of chiral asymmetry\n  can become significant.\n\\end{asparaenum}\n\n\nModeling the details of PNS cooling and neutrino propagation is\nbeyond the scope of this paper.\nBelow we perform the estimates that demonstrate that chiral MHD can\nsignificantly change the picture of the evolution of a PNS.\n\n\n\\subsubsection{Estimates of the relevant parameters}\n\nAn upper limit of the chiral chemical potential can be estimated by\nassuming that\n$n_\\mathrm{L}=0$ and $n_\\mathrm{R}=n_e$ (all left-chiral electrons have been\nconverted into neutrinos, and the rate of chirality flipping is much slower than\nother relevant processes).\nThis leads to the estimate $\\mu_5 \\simeq \\mu_e$ and correspondingly\n\\begin{equation}\n  \\mu_{\\rm max} = 4\\ensuremath{\\alpha_{\\rm em}} \\frac{\\mu_e}{\\hbar c} \\approx \\unit[4\\times 10^{11} ]{cm}^{-1}\\left(\\frac{\\mu_e}{\\unit[250]{MeV}}\\right),\n  \\label{eq_muupper_NS}\n\\end{equation}\nwhere we have used a typical value of the electron's Fermi energy $\\mu_e$\n\\citep{PRPLM99}.\nFor an ultrarelativistic \\emph{degenerate} electron gas (i.e., when\n$\\mu_e \\gg k_{\\rm B} T \\gg m_e c^2$), the relation between the number density\nof electrons, $n_e$, and their Fermi energy, $\\mu_e$, is\n\\begin{equation}\n  \\label{eq:5}\n  \\mu_e = \\hbar c(2\\pi^2 n_e)^{1\/3}\\approx 250\\,{\\rm MeV} \\left(\\frac{n_e}{\\unit[10^{38}]{cm^{-3}}}\\right)^{1\/3}.\n\\end{equation}\nThe interior of neutron stars is a conducting medium whose\nconductivity is estimated to be \\citep{BPP69,Kelly1973}:\n\\begin{eqnarray}\n  \\sigma(T) &=& \\sqrt 3\\left(\\frac4\\pi\\right)^{3\/2}\\frac{\\hbar^4 c^2}{e\\,m_p^{3\/2}}\\frac{n_e^{3\/2}}{k_{\\rm B}^2 T^2}\\\\\n  &\\approx& 1\\times  10^{27}\\left(\\frac{1~\\mathrm{MeV}}{k_{\\rm B} T}\\right)^{2}\\left(\\frac{n_e}{\\unit[10^{38}]{cm^{-3}}}\\right)^{3\/2}\\,{\\rm s}^{-1}\n\\label{eq_sigmaNS}\n\\end{eqnarray}\n(there is actually a difference in the numerical coefficient $\\mathcal{O}(1)$\nbetween the results of \\citet{BPP69} and \\citet{Kelly1973}).\nUsing Equation~(\\ref{eq_sigmaNS}), we find the magnetic diffusion coefficient to be\n\\begin{equation}\n  \\label{eq:3}\n  \\eta(T)\n  \\approx 7\\times 10^{-8}\\,{\\rm cm}^2\\,{\\rm s}^{-1} \\left(\\frac{k_{\\rm B} T}{1~\\mathrm{MeV}}\\right)^{2}\\left(\\frac{\\unit[10^{38}]{cm^{-3}}}{n_e}\\right)^{3\/2}.\n\\end{equation}\nTherefore, we can determine the\nthe maximum growth rate of the small-scale\nchiral instability (\\ref{gamma-max}) as\n\\begin{equation}\n  \\label{eq:4}\n  \\gamma^{\\rm max}_\\mu = \\frac{\\mu_{\\rm max}^2\\eta}4 \\approx 2\\times 10^{15}\\,{\\rm s}^{-1} \\left(\\frac{\\mu_e}{\\unit[250]{MeV}}\\right)^2 \\left(\\frac{k_{\\rm B} T}{1~\\mathrm{MeV}}\\right)^{2}.\n\\end{equation}\nWe see that over a characteristic time $\\tau_\\mathrm{cool}\\sim 1 \\,{\\rm s}$\n(the typical cooling time), the magnetic field would increase by many\n$e$-foldings.\nIn fact, using a flipping rate of $\\Gamma_\\mathrm{f}=10^{14} \\,{\\rm s}$, as\nsuggested in \\citet{GKR15} for $\\mu_\\mathrm{e}= 100 \\,{\\rm MeV}$ and $k_{\\rm B} T=30 \\,{\\rm MeV}$,\nwe find that $f_\\mu$ ranges from\n$\\approx 9\\times 10^{-3}$ down to $\\approx 9\\times 10^{-7}$ for the range\nbetween $k_{\\rm B} T = 1 \\,{\\rm MeV}$ and $k_{\\rm B} T = 100 \\,{\\rm MeV}$.\nHence the evolution of the chemical potential\nand the chiral dynamo\nis weakly affected by flipping reactions.\n\nAs in Section~\\ref{subsec_cosmologybounds}, the phase of the small-scale\ninstability ends when turbulence is excited.\nIt should be stressed, however, that unlike the early universe, the\ninteriors of PNSs are expected to be turbulent with high ${\\rm Re}_{_\\mathrm{M}}$ even in\nthe absence of chiral effects\n(with ${\\rm Re}_{_\\mathrm{M}}$ as large as $10^{17}$); see \\cite{TD93}.\nTherefore, the system may find itself in the forced turbulence regime of Section~\\ref{sec:forced-chiral-dynamos}.\nFigure~\\ref{fig:turbulence_PNS} shows that in a wide range of magnetic\nReynolds numbers, one can have many $e$-foldings over a typical timescale of\nthe PNS and that the scale of the magnetic field can reach macroscopic size.\n\n\\begin{figure}[!t]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{scales_NS}\n  \\caption[Laminar and turbulent scales in PNS]{\n \n  \\textbf{Chiral MHD dynamos in PNSs.}\nLaminar and turbulent growth rate multiplied by the\ncooling timescale (top panel) and the characteristic scales of chiral MHD\nnormalized by the typical radius of the PNS $r_{\\rm NS} \\sim 10\\,{\\rm km}$\n(bottom panel).\nThe estimates are presented as a function of ${\\rm Re}_{_\\mathrm{M}}$.\nThe initial value of the chiral chemical potential is assumed at the\nlevel~(\\protect\\ref{eq_muupper_NS}) and the we use $\\mu_e =\\unit[250]{MeV}$.\nSince the conductivity is temperature dependent, the ratios including $\\eta$ are\npresented for both $k_{\\rm B} T = 1~\\mathrm{MeV}$ and $k_{\\rm B} T = 10~\\mathrm{MeV}$.\n  }\n  \\label{fig:turbulence_PNS}\n\\end{figure}\n\n\\subsubsection{Estimate of magnetic field strengths}\n\nA dedicated analysis, taking into account temperature and density evolution of\nthe PNS as well as its turbulent regimes, is needed to make detailed predictions.\nHere we will make the estimates of the strength of the magnetic field,\nsimilar to Section~\\ref{sec:early-universe} above.\nTo this end, we use the conservation law~(\\ref{cons_law}), assuming\n$\\mu_0 = \\mu_{\\rm max}$.\nIn the PNS case, the plasma is degenerate, and therefore the relation between\n$n_5$ and $\\mu_5$ is given by\n\\begin{equation}\n  n_5 = \\frac{\\mu_5}{3\\pi^2}(3\\mu_e^2 + \\pi^2 T^2)\n  \\label{eq:7}\n\\end{equation}\n(in the limit $\\mu_5 \\ll T$).\n{\nAs a result, the chiral feedback parameter\n$\\lambda$ is\n\\begin{equation}\n  \\label{lambda_PNS}\n  \\lambda_{\\rm PNS} = \\frac{\\hbar c\\pi^2}{2}\\left(\\frac{8\\ensuremath{\\alpha_{\\rm em}}}{\\mu_e}\\right)^2,\n\\end{equation}\nwhich determines the wavenumber $k_\\lambda$; see Equation~(\\ref{klambda}).\nThe corresponding length scale $\\xi_\\lambda = k_\\lambda^{-1}$ is presented in\nthe top panel of Figure~\\ref{fig:turbulence_PNS}, where we assume a\nmean density of the PNS of $\\overline{\\rho}_{\\rm PNS} = 2.8\\times10^{14}~\\mathrm{g}\\mathrm{cm}^{-3}$.\n\nUsing \\Eqs{eq_muupper_NS}{lambda_PNS}, we find\n\\begin{eqnarray}\n  \\label{eq:8}\n  (B^2\\xi)_{\\max}\n   &=&\\frac{4\\pi\\mu_{\\rm max}}{\\lambda_{\\rm PNS}} = \\frac{\\mu_e^3}{2(\\hbar c)^2\\ensuremath{\\alpha_{\\rm em}}}\\\\\n  &\\approx& 1.4\\times 10^{24}\\,{\\rm G}^2\\,{\\rm cm}\\left(\\frac{\\mu_e}{\\unit[250]{MeV}}\\right)^3.\n\\end{eqnarray}\nAssuming for the maximum correlation scale $\\xi_{\\rm PNS} \\sim 1\\,{\\rm cm}$\n(see \\Fig{fig:turbulence_PNS}), we find that\nmagnetic field strength is of the order of\n\\begin{equation}\n  \\label{eq:6}\n  B_{\\rm max} \\approx 1.2\\times 10^{12}\\,{\\rm G} \\left(\\frac{\\mu_e}{\\unit[250]{MeV}}\\right)^{3\/2}\n  \\left(\\frac {1\\,{\\rm cm}}{\\xi_{\\rm M}}\\right)^{1\/2} .\n\\end{equation}\nNotice that the estimate~(\\ref{eq:8}) is independent of $T$\n(but depends strongly on the assumed value of $\\mu_e$).\n\n\nOur estimates have demonstrated that the chiral MHD could be capable of generating\nstrong small-scale magnetic fields. \\emph{Therefore, chiral effects should be\nincluded in the modeling of evolution of PNSs.}}\n\n\n\n\n\n\\section{Conclusions}\n\\label{sec_concl}\n\nIn this work, we have presented results from numerical simulations of chiral MHD\nthat include the temporal and spatial evolution of magnetic fields,\nplasma motions, and the chiral chemical potential.\nThe latter, characterizing the asymmetry between left- and right-handed fermions,\ngives rise to the CME, which\nresults in the excitation of a small-scale chiral dynamo instability.\n\nOur numerical simulations are performed for the system of\nchiral MHD equations~(\\ref{ind-DNS})--(\\ref{mu-DNS}) that was derived in\nPaper~I.\nThis system of equations is valid for plasmas with high electric conductivity,\nthat is, in the limit of high and moderately high Reynolds numbers.\nChiral flipping reactions are neglected in most of the simulations.\nIn the majority of the runs, the initial conditions are a very weak magnetic seed\nfield and a high chiral chemical potential.\nBoth initially force-free systems and systems with external forcing of\nturbulence are considered.\nWith our numerical simulations, we confirm\nvarious theoretical predictions of the chiral laminar\nand turbulent large-scale dynamos discussed in Paper~I.\n\nOur findings from DNS can be summarized as follows:\n\\begin{itemize}\n\\item{\nThe evolution of magnetic fields studied here in DNS agrees with\nthe predictions made in Paper~I for all types of laminar dynamos.\nIn particular, the scalings of\nthe maximum growth rate of the chiral dynamo instability\n$\\gamma_\\mu\\propto v_\\mu^2$ for the\n$v_\\mu^2$ dynamo (see Figure~\\ref{fig__Gamma_eta}) and\n$\\gamma_\\mu\\propto (S v_\\mu)^{2\/3}$\nfor the $v_\\mu$--shear dynamo (see Figure~\\ref{Gamma_Us_Beltrami}) have been\nconfirmed.\nAdditionally, the transitional regime of an $v_\\mu^2$--shear dynamo, where the\ncontributions from the $v_\\mu^2$- and shear terms\nare comparable, agrees with theoretical predictions, as can be seen in\nFigure~\\ref{fig_Gamma_Us_aShear}.\nIn our DNS, the scale-dependent amplification of the magnetic field\nin the laminar chiral dynamo is observed in the energy spectra; see, for example,\nFigure~\\ref{fig_LTspec} where the maximum\ngrowth rate of the $v_\\mu^2$ dynamo instability is attained\nat wavenumber $k_\\mu=\\mu_0\/2$.\n}\n\\item{\nThe conservation law~(\\ref{CL}) for total chirality\nimplies a maximum magnetic field strength\nof the order of $B_\\mathrm{sat}\\approx(\\mu_0 \\xi_\\mathrm{M}\/\\lambda)^{1\/2}$.\nThis dependence of $B_\\mathrm{sat}$ on the chiral nonlinearity parameter\n$\\lambda$ has been confirmed numerically and is presented in\nFigure~\\ref{fig_Bsat_lambda}.\n}\n\\item{\nThe CME can drive turbulence efficiently via the Lorentz force,\nwhich has been demonstrated in our numerical simulations\nthrough the measured growth rate of the turbulent velocity, which is\nlarger by approximately a factor of two than that of the magnetic field;\nsee, for example, the middle panel of Figure~\\ref{fig_LTts}.\n}\n\\item{\nIn the presence of small-scale turbulence, the large-scale dynamo\noperates due to the chiral $\\alpha_\\mu$ effect,\nwhich is not related to the kinetic helicity;\nsee Equation~(\\ref{alphamu}).\nIn the limit of large magnetic Reynolds numbers,\nthe maximum growth rate of the large-scale dynamo instability\nis reduced by a factor of\n$(4\/3)(\\ln{\\rm Re}_{_\\mathrm{M}})^2\/{\\rm Re}_{_\\mathrm{M}}$ as\ncompared to the laminar case; see Equation~(\\ref{gammamax_turbRm}).\nThe dynamo growth rate is close to this prediction\nof mean-field chiral MHD for both\nchiral magnetically produced turbulence and\nfor externally driven turbulence; see\nsee Figure~\\ref{fig_gamma_Re}.\n}\n\\item{\nUsing DNS, we found a new scenario of the magnetic field evolution\nconsisting of three phases\n(see also the schematic overview in Figure~\\ref{fig:phases}):\n\\begin{asparaenum}[\\it (1)]\n\\item small-scale chiral dynamo instability;\n\\item production of small-scale turbulence, inverse transfer of magnetic energy,\nand generation of a large-scale magnetic field by the chiral $\\alpha_\\mu$ effect;\n\\item\nsaturation of the large-scale chiral dynamo\nby a decrease of the CME\ncontrolled by the conservation law for\nthe total chirality:\n$\\lambda \\, \\bra{{\\bm A} {\\bm \\cdot} \\bm{B}}\/2 + \\bra{\\mu} = \\mu_0$.\n\\end{asparaenum}\nThe previously discussed scenario of magnetic field\nevolution caused by the CME \\citep{BFR12}\ndid not include the second phase.}\n\\end{itemize}\n\nWhile the results summarized above have been obtained\nin simulations of well-resolved periodic domains,\nastrophysical parameters are beyond the regime accessible to DNS.\nHence we can only estimate the effects of the chiral anomaly in relativistic\nastrophysical plasmas, like in the early universe or in neutron stars.\nThe main conclusions from the astrophysical applications are the following:\n\\begin{itemize}\n\\item{The chiral MHD scenario found in DNS may help to explain the origin of\nthe magnetic field observed in the interstellar\nmedium. The chiral dynamo instability produces helical magnetic\nfields initially at\nsmall spatial scales and simultaneously drives turbulence,\nwhich generates a magnetic field on large scales.\nAfter the chiral chemical potential has been transformed into magnetic\nhelicity during the dynamo saturation phase,\nthe magnetic field cascades to\nlarger spatial scales according to the phenomenology of decaying MHD\nturbulence.\nWe have estimated the values of $\\mu_0$ and $\\lambda$ for the early universe.\nThese parameters determine the time and spatial scales associated with\nthe chiral dynamo instability\n(see Figure~\\ref{fig:chiral_turbulence}) and the maximum magnetic\nhelicity (see Equation~\\ref{eq:Br17:3}).\nOur estimates for magnetic fields produced by chiral dynamos in the\nearly universe are consistent with the observational lower limits\nfound by \\citet{DCRFCL11} (see Figure~\\ref{fig_B_xi__comov})\nif we assume that the initial chiral chemical potential is of the order of\nthe thermal energy density.\n}\n\\item{In PNSs, chiral dynamos\noperating in the first tens of seconds after the supernova explosion\ncan produce magnetic fields of approximately $10^{12}\\,{\\rm G}$ at a magnetic\ncorrelation length of $1\\,{\\rm cm}$; see Equation~(\\ref{eq:6}).\nHowever, we stress that many questions remain open, especially regarding the\ngeneration of a chiral asymmetry and the role of the chiral flipping term\nin PNEs.\n}\n\\end{itemize}\n\nFinally, we stress again that the parameters and the initial conditions,\nincluding the initial chiral asymmetry, are unknown in the astrophysical \nsystems discussed in this paper. \nHence, the purpose of our applications should be classified as a study\nof the conditions under which the CME plays a significant \nrole in the evolution of a plasma of relativistic charged fermions. \nWith the regimes accessible to our simulations not being truly realistic in the \ncontext of the physics of the early universe and \nin neutron stars, our applications have a rather exploratory nature. \nIn this sense, our results from DNS can be used to answer\nthe question in which area of plasma physics -- the physics of the early\nuniverse, the physics of neutron stars, or the physics of heavy ion\ncollisions---the CME is important and can modify the\nevolution of magnetic fields.\n\n\n\\begin{acknowledgements}\nWe thank the anonymous referee for constructive criticism that\nimproved our manuscript.\nWe acknowledge support from Nordita, which is funded by the\nNordic Council of Ministers, the Swedish Research Council, and the two host\nuniversities, the Royal Institute of Technology (KTH) and\nStockholm University.\nThis project has received funding from the\nEuropean Union's Horizon 2020 research and\ninnovation program under the Marie Sk{\\l}odowska-Curie grant\nNo.\\ 665667.\nWe also acknowledge the University of Colorado's support through\nthe George Ellery Hale visiting faculty appointment.\nSupport through the NSF Astrophysics and Astronomy Grant Program (grant 1615100),\nthe Research Council of Norway (FRINATEK grant 231444),\nand the European Research Council (grant number 694896) are\ngratefully acknowledged. Simulations presented in this work have been performed\nwith computing resources\nprovided by the Swedish National Allocations Committee at the Center for\nParallel Computers at the Royal Institute of Technology in Stockholm.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nThe celebrated laws of black hole thermodynamics~\\cite{Bek72,Bek74,Haw74,Haw75} ascribe physical properties to the event horizon of a black hole. However, the event horizon is defined globally, as the boundary of the past of future infinity. Thus, the location of the thermodynamic object depends on the future history of the spacetime. For example, an observer in a perfectly flat spacetime region might already be inside a black hole, if a null shell is collapsing outside their past light-cone. By causality, Hawking radiation and the first and second law of black hole thermodynamics should have no manifestation for such an observer. Conversely, once a black hole has formed, its thermodynamic properties should be observable at finite distance, regardless of whether the collapsed region already coincides with the true event horizon, or is headed for substantial growth in the distant future.\n\nHere we consider the problem of finding a geometric object that is locally defined, and which obeys a classical law analogous to one of the laws of thermodynamics. We will focus on the second law, whose manifestation in classical General Relativity is the statement that the area of certain surfaces cannot decrease. For the cross-sections of an event horizon this was proven by Hawking in 1971~\\cite{Haw71}, but as noted above the event horizon is not locally defined.\n\nWe will formulate and prove a new area theorem. It is obeyed by what we shall call a {\\em future (or past) holographic screen}, $H$. $H$ is a hypersurface foliated by marginally (anti-)trapped surfaces, which are called {\\em leaves}. This definition is local, unlike that of an event horizon. It requires knowledge only of an infinitesimal neighborhood of each leaf.\n\\begin{figure*}[ht]\n\\subfigure[]{\n\\includegraphics[width=0.45 \\textwidth]{1aPRD.pdf}\n}\n\\subfigure[]{\n \\includegraphics[width=0.45 \\textwidth]{1bPRD.pdf}\n}\n\\caption{Penrose diagrams showing examples of holographic screens. The green diagonal lines show a null slicing of the spacetime; green dots mark the maximal area sphere on each slice. These surfaces combine to form a holographic screen (blue lines); we prove that their area increases monotonically in a uniform direction on the screen (blue triangles). (a) A black hole is formed by collapse of a star (inner shaded region); later another massive shell collapses onto the black hole (outer shaded region). At all other times an arbitrarily small amount of matter accretes (white regions); this suffices to satisfy our generic conditions. The black hole interior contains a future holographic screen that begins at the singularity and asymptotes to the event horizon. It is timelike in the dense regions and spacelike in the dilute regions.  (b) In a closed universe filled with dust, marginally antitrapped spheres form a past holographic screen in the expanding region; its area increases towards the future. Marginally trapped spheres form a future holographic screen in the collapsing region; its area increases towards the past. The equator of the three-sphere at the turnaround time (black circle) belongs to neither the past nor the future screen; it is extremal in the sense of Ref.~\\cite{HubRan07}.}\n\\label{fig-examplesPRD}\n\\end{figure*} \nA future holographic screen exists (nonuniquely) in generic spacetimes that have a future event horizon. It is disjoint from the event horizon but it may asymptote to it; see Fig.~\\ref{fig-examplesPRD}a. Past holographic screens exist in expanding universes such as ours, regardless of whether they have a past event horizon. Because $H$ is not defined in terms of distant regions, past and future holographic screens can exist in spacetimes with no distant boundary at all, such as a recollapsing closed universe; see Fig.~\\ref{fig-examplesPRD}b. Our area law applies to all future and past holographic screens.\n\n\\noindent{\\bf Relation to previous work} The notion of future or past holographic screen has roots in two distinct bodies of research, which had not been connected until now. It can be regarded as a refinement of the notion of ``preferred holographic screen hypersurface''~\\cite{CEB2}, which need not have monotonic area. Alternatively, it can be viewed as a generalization of the notion of ``dynamical horizon'', which obeys a straightforward area law but is not known to exist in many realistic solutions. We will now discuss these two connections for context and attribution; see also~\\cite{BouEng15a}. We stress, however, that our theorem and proof are self-contained. They rely only on classical General Relativity, and not, for example, on any conjecture about semiclassical or quantum gravity.\n\nFirst, let us discuss the relation to the holographic principle. (See~\\cite{Tho93,Sus95,FisSus98} for earlier work and Ref.~\\cite{RMP} for a review.) To an arbitrary codimension 2 spatial surface $B$, one can associate a light-sheet~\\cite{CEB1}: a null hypersurface orthogonal to $B$ with everywhere nonpositive expansion (i.e., locally nonincreasing area), in the direction away from $B$. The covariant entropy bound (Bousso bound)~\\cite{CEB1} is the conjecture that the entropy of the matter on the light-sheet cannot exceed the area of $B$, in Planck units. The conjecture has broad support; it has been proven in certain limiting regimes~\\cite{FlaMar99,BouFla03,StrTho03,BouCas14a,BouCas14b}.\n\nThere are four null directions orthogonal to any surface. In each direction, the orthogonal null congruence generates a null hypersurface with boundary $B$. The expansion in opposing directions, such as future-outward and past-inward, differs only by a sign. In typical settings, therefore, there will be two directions with initially negative expansion, each of which gives rise to a light-sheet. For example, a sphere in Minkowski space admits light-sheets in the future and past inward directions, but not in the outward directions. A large enough sphere near the big bang is anti-trapped: it admits light-sheets in the past inward and outward directions. Spheres near the singularity of a black hole are trapped: the light-sheets point in the future inward and outward directions.\n\nHowever, it is possible to find surfaces that are {\\em marginal}: they have vanishing expansion in one opposing pair of null directions. Hence they admit a pair of light-sheets whose union forms an entire null slice of the spacetime~\\cite{CEB1}. In fact, in strongly gravitating regions one can readily construct a continuous family of marginal surfaces, which foliate a hypersurface called ``preferred holographic screen hypersurface''. The opposing pairs of light-sheets attached to each leaf foliate the spacetime. The Bousso bound is particularly powerful when applied to these light-sheets. It constrains the entropy of the entire spacetime, slice by slice, in terms of the area of the leaves. All quantum information in the spacetime can be stored on the leaves, at no more than about one qubit per Planck area. In this sense the world is a hologram.\n\nFor event horizons, a classical area theorem~\\cite{Haw71} preceded the interpretation of area as physical entropy~\\cite{Bek72}. For holographic screens, the present work belatedly supplies a classical area law for an object whose relevance to geometric entropy had long been conjectured~\\cite{CEB2}. What took so long?\n\nIn fact, the notion of ``preferred holographic screen hypersurface'' lacked a key refinement, without which our theorem would not hold: the distinction between past and future holographic screens. The leaves of a ``preferred holographic screen hypersurface'' are {\\em marginal}, that is, one orthogonal null congruence has vanishing expansion. However, they were not required to be either marginally trapped, or marginally anti-trapped. That is, no definite sign was imposed on the expansion of the second, independent orthogonal null congruence. Fig.~\\ref{fig-examplesPRD}b shows a spacetime in which a ``preferred holographic screen hypersurface'' fails to obey an area law. Once we distinguish between marginally trapped and anti-trapped surfaces, however, we recognize that there are in fact two disconnected objects: a past and a future holographic screen. Each obeys an area law, as our proof guarantees, but in different directions of evolution. This is analogous to the distinction between past and future event horizons. From this perspective, it is not surprising that ``preferred holographic screen hypersurfaces'' fail to satisfy an area law without the refinement we introduce here.\n\nThis brings us to the second body of research to which the present work owes debt. Previous attempts to find a quasi-local alternative to the event horizon culminated in the elegant notions of a future outer trapping horizon (FOTH)~\\cite{Hay93,Hay97,HayMuk98} or dynamical horizon~\\cite{AshKri02,AshKri03,AshGal} (see~\\cite{AshKri04,Booth05} for reviews). In a generic, classical setting their definitions are equivalent: a dynamical horizon is a spacelike hypersurface foliated by marginally trapped surfaces. \n\n``Preferred holographic screen hypersurface'' was a weaker notion than future holographic screen; ``dynamical horizon'' is a stronger notion. It adds not only the crucial refinement from marginal to marginally trapped, but also the requirement that the hypersurface be spacelike. This immediately implies that the area increases in the outward direction~\\cite{Hay93,AshKri02}. (Note the brevity of the proof of Theorem~\\ref{thm-area} below, which alone would imply an area law without the need for any of the previous theorems, if a spacelike assumption is imposed.) \n\nHowever, our present work shows that the spacelike requirement is not needed for an area theorem. This is important, because the spacelike requirement is forbiddingly restrictive~\\cite{BoothBrits}: no dynamical horizons are known to exist in simple, realistic systems such as a collapsing star or an expanding universe dominated by matter, radiation, and\/or vacuum energy. \n\nThus, the notion of a dynamical horizon (or of a FOTH) appears to be inapplicable in a large class of realistic regions in which gravity dominates the dynamics. We are not aware of a proof of nonexistence. But we show here that an area theorem holds for the more general notion of future holographic screen, whose existence is obvious and whose construction is straightforward in the same settings. Thus we see little reason for retaining the additional restriction to hypersurfaces of spacelike signature, at least in the context of the second law.\n\nIn the early literature on FOTHs\/dynamical horizons, future holographic screens were already defined and discussed, under the name ``marginally trapped tube''~\\cite{AshKri04}.\\footnote{The definition of ``trapping horizon''~\\cite{Hay93} excludes the junctions between inner and outer trapping horizons and thus precludes the consideration of such objects as a single hypersurface.} Ultimately, two separate area laws were proven, one for the spacelike and one for the timelike portions of the future holographic screens. These follow readily from the definitions.  The first, for FOTHs\/dynamical horizons, was mentioned above. The second states that the area decreases toward the future along any single timelike portion (known as ``future inner trapping horizons''~\\cite{Hay93} or ``timelike membranes''~\\cite{AshKri04}). \n\nIn these pioneering works, no unified area law was proposed for ``marginally trapped tubes''\/future holographic screens. Perhaps this is because it was natural to think of their timelike portions as future directed and thus area-decreasing. Moreover, the close relation to ``preferred holographic screen hypersurfaces''~\\cite{CEB2} was not recognized, so the area of leaves lacked a natural interpretation in terms of entropy.\\footnote{It is crucial that the entropy associated with the area of leaves on a future holographic screen $H$ is taken to reside on the light-sheets of the leaves, as we assert, and not on $H$ itself.  The latter choice---called a ``covariant bound'' in Refs.~\\cite{He:2007qd, He:2007dh, He:2007xd, He:2008em} but related to~\\cite{BakRey99} and distinct from~\\cite{CEB1}---is excluded by a counterexample~\\cite{KalLin99} and would not lead to a valid Generalized Second Law.} And finally, it is not immediately obvious that an area law can hold once timelike and spacelike portions are considered together. Indeed, the central difficulty in the proof we present here is our demonstration that such portions can only meet in ways that uphold area monotonicity for the entire future holographic screen under continuous flow. A key element of our proof builds on relatively recent work~\\cite{Wal10QST}.\n\nThere is an intriguing shift of perspective in a brief remark in later work by Booth {\\em et al.}~\\cite{BoothBrits}. After explicitly finding a ``marginally trapped tube'' (i.e., what we call a future holographic screen) in a number of spherically symmetric collapse solutions, the authors point out that it could be considered as a single object, rather than a collection of dynamical horizon\/``timelike membrane'' pairs. They note that with this viewpoint the area increases monotonically in the examples considered. Our present work proves that this behavior is indeed general.\n\nAnalogues of a first law of thermodynamics have been formulated for dynamical horizons and trapping horizons. We expect that this can be extended to future holographic screens. However, here we shall focus on the second law and its classical manifestation as an area theorem.\n\n\\noindent{\\bf Outline \\ }  In Sec.~\\ref{sec-screens}, we give a precise definition of future and past holographic screens, and we establish notation and nomenclature. We also describe a crucial mathematical structure derived from the foliation of $H$ by marginally (anti-)trapped leaves $\\sigma(r)$: there exists a vector field $h^a$ tangent to $H$ and normal to its leaves, which can be written as a linear combination of the orthogonal null vector fields $k^a$ and $l^a$. Its integral curves are called fibers of $H$. \n\nIt is relatively easy to see that the area of leaves is monotonic if $h^a l_a$ has definite sign, i.e., if $H$ evolves towards the past or exterior of each leaf. The difficulty lies in showing that it does so everywhere.\n\nOur proof is lengthy and involves nontrivial intermediate results. Given an arbitrary two-surface $\\sigma$ that splits a Cauchy surface into complementary spatial regions, we show in Sec.~\\ref{sec-csss} that a null hypersurface $N(\\sigma)\\supset \\sigma$  partitions the entire spacetime into two complementary spacetime regions: $K^+(\\sigma)$, the future-and-interior of $\\sigma$; and $K^-(\\sigma)$, the past-and-exterior of $\\sigma$.\n\nIn Sec.~\\ref{sec-ssmono}, we consider a hypersurface foliated by Cauchy-splitting surfaces $\\sigma(r)$. We prove that $K^+(r)$ grows monotonically under inclusion, if the surfaces $\\sigma(r)$ evolve towards their own past-and-exterior. This puts on a rigorous footing the equivalence (implicit in the constructions of~\\cite{CEB2}) between foliations of $H$ and null foliations of spacetime regions. The proofs in Sec.~\\ref{sec-monosplit} do not use all of the properties of $H$; in particular they do not use the marginally trapped property of its leaves. Thus our results up to this point apply to more general classes of hypersurfaces.\n\nIn Sec.~\\ref{sec-arealaw}, we do use the assumption that the leaves of $H$ are marginally trapped, and we combine it with the monotonicity of $K^+(r)$ that we established for past-and-exterior evolution. This allows us to show that the evolution of leaves $\\sigma(r)$ on a future holographic screen $H$ must be {\\em everywhere} to the past or exterior (assuming the null energy condition and certain generic conditions). This is the core of our proof. We then demonstrate that such evolution implies that the area $A(r)$ of $\\sigma(r)$ increases strictly monotonically with $r$. \n\nWe close Sec.~\\ref{sec-arealaw} with a theorem establishing the uniqueness of the foliation of a given holographic screen. The holographic screens themselves are highly nonunique. For example, one can associate a past (future) holographic screen with any observer, by finding the maximal area surfaces on the past (future) light-cones of each point on the observer's worldline.\n\n\n\n\n\\section{Holographic Screens}\n\\label{sec-screens}\n\nWe assume throughout this paper that the spacetime is globally hyperbolic (with an appropriate generalization for asymptotically AdS geometries~\\cite{Wal10QST,EngWal14}). We assume the null curvature condition (NCC): $R_{ab}k^{a}k^{b}\\geq 0$ where $k^{a}$ is any null vector. In a spacetime with matter satisfying Einstein's equations this is equivalent to the null energy condition: $T_{ab}k^{a}k^{b}\\geq 0$.\n\n\\begin{defn}\nA {\\em future holographic screen}~\\cite{CEB2} (or {\\em marginally trapped tube}~\\cite{AshGal, AshKri04}) $H$ is a smooth\nhypersurface admitting a foliation by marginally trapped surfaces called {\\em leaves}.\n\nA {\\em past holographic screen} is defined similarly but in terms of marginally anti-trapped surfaces. Without loss of generality, we will consider future holographic screens in general discussions and proofs.\n\nBy {\\em foliation} we mean that every point $p\\in H$ lies on exactly one leaf.  A {\\em marginally trapped surface} is a codimension 2 compact spatial surface $\\sigma$ whose two future-directed orthogonal null geodesic congruences satisfy\n\\begin{eqnarray}\n\\theta_{k} &=& 0 \\label{marginal}~,\\\\\n\\theta_{l} &<&  0 \\label{trapped}~.\n\\end{eqnarray} \nThe opposite inequality defines ``marginally anti-trapped'', and thus, past holographic screens. Here $\\theta_{k}= \\hat{\\nabla}_a k^a$ and $\\theta_{l} = \\hat{\\nabla}_a l^a$ are the null expansions~\\cite{Wald} (where $\\hat{\\nabla}_{a}$ is computed with respect to the induced metric on $\\sigma$), and $k^{a}$ and $l^{a}$ are the two future directed null vector fields orthogonal to $\\sigma$.\n\nWe will refer to the $k^a$ direction as {\\em outward} and to the $l^a$ direction as {\\em inward}. For screens in asymptotically flat or AdS spacetimes, these notions agree with the intuitive ones. Furthermore, in such spacetimes any marginally trapped surface, and hence any holographic screen, lies behind an event horizon. However, holographic screens may exist in cosmological spacetimes where an independent notion of outward, such as conformal infinity, need not exist (e.g., a closed FRW universe).  In this case the definition of $H$ requires only that there exist some continuous assignment of $k^a$ and $l^a$ on $H$ such that all leaves are marginally trapped. See Fig.~\\ref{fig-examplesPRD} for examples of holographic screens.\n\\end{defn}\n\n\\begin{defn}\nThe defining foliation of $H$ into leaves $\\sigma$ determines a $(D-2)$-parameter family of leaf-orthogonal curves $\\gamma$, such that every point $p\\in H$ lies on exactly one curve that is orthogonal to $\\sigma(p)$. We will refer to this set of curves as the {\\em fibration of $H$}, and to any element as a {\\em fiber} of $H$.\n\\end{defn}\n\n\\begin{conv} \\label{conv-rh}\nThus it is possible to choose a (non-unique) {\\em evolution parameter} $r$ along the screen $H$ such that $r$ is constant on any leaf and increases monotonically along the fibers $\\gamma$. We will label leaves by this parameter: $\\sigma(r)$. \n\nThe tangent vectors to the fibers define a vector field $h^a$ on $H$. For any choice of evolution parameter the normalization of this vector field can be fixed by requiring that the function $r$ increases at unit rate along $h^a$: $h(r)=h^a\\, (dr)_a=1$. (Since $H$ can change signature, unit normalization of $h^a$ would be possible only piecewise, and hence would not be compatible with the desired smoothness of $h^a$.)\n\\end{conv}\n\n\\begin{rem}\nSince fibers are orthogonal to leaves, a tangent vector field $h^a$ can be written as a (unique) linear combination of the two null vector fields orthogonal to each leaf:\n\\begin{equation}\nh^{a} = \\alpha l^{a} + \\beta k^{a}\n\\label{eq-hlk}\n\\end{equation}\nMoreover, the foliation structure guarantees that $h^a$ vanishes nowhere: it is impossible to have $\\alpha=\\beta=0$ anywhere on $H$. (These remarks hold independently of the requirement that each leaf be marginally trapped.) \\label{rem-h}\n\\end{rem}\n\\begin{figure}[ht]\n\n\\includegraphics[width=2in]{AlphaBetaDiagram.pdf}\n\\caption{The null vectors $l^{a}$ and $k^{a}$ orthogonal to a leaf $\\sigma$ of the foliation of $H$ at some point. The evolution of $H$ is characterized by vector $h^a$ normal to the leaves and tangent to $H$. Depending on the quadrant $h^a$ points to, $H$ evolves locally to the future, exterior, past, or interior (clockwise from top).}\n\\label{fig-st}\n\n\\end{figure}\n\n\\begin{conv} \\label{conv-stnotation}\nAs shown in Fig.~\\ref{fig-st}, $h^a$ is spacelike and outward-directed if $\\alpha<0, \\beta>0$; timelike and past-directed if $\\alpha<0, \\beta<0$; spacelike and inward-directed if $\\alpha>0, \\beta<0$; and finally, timelike and future-directed if $\\alpha>0, \\beta>0$. We denote such regions, in this order (and somewhat redundantly): $S_{-+},T_{--},S_{+-},T_{++}$. \n\\end{conv}\n\\begin{rem} \\label{rem-result}\nOur key technical result below will be to demonstrate that $\\alpha$ cannot change sign on $H$. Thus on a given screen $H$, either only the first two, or only the second two possibilities are realized. (The latter case can be reduced to the former by taking $r\\to -r$.)\n\\end{rem}\n\\begin{rem} \\label{rem-borders}\nBecause $\\alpha$ and $\\beta$ cannot simultaneously vanish, $S_{+-}$ and $S_{--}$ regions cannot share a boundary or be separated by a null region; they must be separated by a timelike region. Similarly $T_{++}$ and $T_{--}$ regions must be separated by a spacelike region.\n\\end{rem}\n\nBelow we will consider only holographic screens that satisfy additional technical assumptions:\n\\begin{defn} \\label{def-technical}\nA holographic screen $H$ is {\\em regular} if\n\\begin{enumerate}[(a)]\n\\item \\label{def-technical1} the {\\em first generic condition} is met, that $R_{ab} k^a k^b + \\varsigma_{ab}\\varsigma^{ab}>0$ everywhere on $H$, where $\\varsigma_{ab}$ is the shear of the null congruence in the $k^a$-direction;\n\\item \\label{def-technical2} the {\\em second generic condition} is met: let $H_+$, $H_-$, $H_0$ be the set of points in $H$ with, respectively, $\\alpha>0$, $\\alpha<0$, and $\\alpha=0$. Then $H_0= \\dot H_- = \\dot H_+$. \n\\item \\label{def-technical3} every inextendible portion $H_i\\subset H$ with definite sign of $\\alpha$ either contains a complete leaf, or is entirely timelike.\n\\item \\label{def-technical4} every leaf $\\sigma$ splits a Cauchy surface $\\Sigma$ into two disjoint portions $\\Sigma^\\pm$.\n\\end{enumerate} \n\\end{defn}\nAnalogous assumptions have been used in the more restricted context of dynamical horizons. The first generic condition is identical to the regularity condition of~\\cite{AshGal}. Together with the null curvature condition, $R_{ab}k^a k^b\\geq 0$, it ensures that the expansion of the $k^a$-congruence becomes negative away from each leaf. The second generic condition excludes the degenerate case where $\\alpha$ vanishes along $H$ without changing sign. Either condition excludes the existence of an open neighborhood in $H$ with $\\alpha=0$. Both are aptly called ``generic'' since they can fail only in situations of infinitely fine-tuned geometric symmetry and matter distributions.\nThe third assumption is substantially weaker than the definition of a dynamical horizon, since we do not require global spacelike signature of $H$. The fourth assumption will play a role analogous to the assumption of achronality of the dynamical horizon. It holds in typical spacetimes of interest (including settings with nontrivial spatial topology, such as $S^1\\times S^2$, as long as the holographic screen is sufficiently localized on the sphere). We leave the question of relaxing some or all of these assumptions to future work.\n\n\\begin{rem} \\label{rem-s0exists}\nAssumption~\\ref{def-technical}.c and Remark~\\ref{rem-borders} imply that $H$ contains at least one complete leaf with definite sign of $\\alpha$.\n\\end{rem}\n\\begin{conv}\n\\label{conv-orient}\nLet $\\sigma(0)\\subset H$ be an arbitrary leaf with definite sign of $\\alpha$. We will take the parameter $r$ to be oriented so that $\\alpha< 0$ on $\\sigma(0)$, and we take $r=0$ on $\\sigma(0)$. By convention \\ref{conv-rh} this also determines the global orientation of the vector field $h^a$. For past holographic screens, it is convenient to choose the opposite convention, $\\alpha>0$ on $\\sigma(0)$.\n\\end{conv}\n\n\n\\section{Leaves Induce a Monotonic Spacetime Splitting}\n\\label{sec-monosplit}\n\nIn this section, we will use only a subset of the defining properties of a holographic screen. In Sec.~\\ref{sec-csss}, we examine the implications of Assumption~\\ref{def-technical}.d, that each leaf split a Cauchy surface. We show that a null surface orthogonal to such a leaf splits the entire spacetime into two disconnected regions  $K^\\pm(\\sigma)$.\n\nIn Sec.~\\ref{sec-ssmono}, we use the foliation property of the holographic screen. (However, nowhere in this section do we use the condition that each leaf be marginally trapped, or Assumptions~\\ref{def-technical}.a-c.) We show that in portions of $H$ where $\\alpha$ is of constant sign, the sets $K^\\pm(\\sigma(r))$ satisfy inclusion relations that are monotonic in the evolution parameter $r$. \n\nTogether these results imply that an $\\alpha<0$ foliation of any hypersurface $H$ into Cauchy-splitting surfaces $\\sigma$ induces a null foliation of the spacetime, such that each null hypersurface $N(\\sigma)$ splits the entire spacetime into disconnected regions $K^\\pm(\\sigma)$. \n\nIn the following section, we will add the marginally trapped condition and the remaining technical assumptions, to show that on a future holographic screen, $\\alpha$ {\\em must}\\\/ have constant sign.\n\n\\subsection{From Cauchy Splitting to Spacetime Splitting}\n\\label{sec-csss}\n\nBy Assumption~\\ref{def-technical}.d, every leaf $\\sigma$ splits a Cauchy surface $\\Sigma$ into two disconnected portions $\\Sigma^+$ and $\\Sigma^-$:\n\\begin{equation}\n\\Sigma = \\Sigma^+ \\cup \\sigma \\cup \\Sigma^-~,~~\\sigma = \\dot\\Sigma^\\pm~.\n\\end{equation}\nWe take $\\Sigma^\\pm$ to be open in the induced topology on $\\Sigma$, so that $\\Sigma^\\pm\\cap\\sigma=\\varnothing$.\nWe consider the following sets shown in Fig.~\\ref{fig-sets}a:\n\\begin{itemize} \n\\item $I^+(\\Sigma^+)$, the chronological future of $\\Sigma^+$: this is the set of points that lie on a timelike future-directed curve starting at $\\Sigma^+$. (Note that this set does not include $\\Sigma^+$.)\n\\item $D^-(\\Sigma^+)$, the past domain of dependence of $\\Sigma^+$: this is the set of points $p$ such that every future-directed causal curve through $p$ must intersect $\\Sigma^+$. (This set does include $\\Sigma^+$.)\n\\item Similarly, we consider $I^-(\\Sigma^-)$ and $D^+(\\Sigma^-)$.\n\\end{itemize} \n\n\\begin{figure*}[ht]\n\n\\subfigure[ ]{\n\n\\includegraphics[height=0.4 \\textwidth]{SigmaDecompositionv2.pdf}\n\\label{fig-sets-a}}\n\\qquad\n\\subfigure[]{\n\\includegraphics[height=0.4\\textwidth]{Ksv2.pdf}\n\\label{fig-sets-b}}\n\\caption{(a) Each leaf $\\sigma$ splits a Cauchy surface. This defines a partition of the entire spacetime into four regions, given by the past or future domains of dependence and the chronological future or past of the two partial Cauchy surfaces. (b) The pairwise unions $K^\\pm$ depend only on $\\sigma$, not on the choice of Cauchy surface. They can be thought as past and future in a null foliation defined by the lightsheets $N$.}\n\\label{fig-sets}\n\\end{figure*}\n\n\n\\begin{defn}\\label{def-sets}\nFrom the Cauchy-splitting property of $\\sigma$, it follows\\footnote{The proofs of the following statements are straightforward and use only well-known properties of $I^\\pm$ and $D^\\pm$.} that the four sets defined above have no mutual overlap. However they share null boundaries:\n\\begin{align} \nN^+(\\sigma) & \\equiv & \\dot I^+(\\Sigma^+) -\\Sigma^+ = \\dot D^+(\\Sigma^-) - I^-(D^+(\\Sigma^-)) \\\\\nN^-(\\sigma) & \\equiv & \\dot I^-(\\Sigma^-) - \\Sigma^- = \\dot D^-(\\Sigma^+) - I^+(D^-(\\Sigma^+))\n\\end{align} \nNote that $N^+(\\sigma)\\cap N^-(\\sigma)=\\sigma$. We define\n\\begin{eqnarray} \nK^+(\\sigma) & \\equiv & I^+(\\Sigma^+) \\cup D^-(\\Sigma^+)-N^+(\\sigma)~; \\\\\nK^-(\\sigma) & \\equiv & D^+(\\Sigma^-) \\cup I^-(\\Sigma^-)-N^-(\\sigma)~; \\\\\nN(\\sigma) &\\equiv & N^+(\\sigma) \\cup N^-(\\sigma)\n\\end{eqnarray}\nThus\n\\begin{equation}\nN(\\sigma) = \\dot K^+(\\sigma) = \\dot K^-(\\sigma)~;\n\\end{equation}\nand the sets $N$, $K^+$, and $K^-$ provide a partition of the spacetime (Fig.~\\ref{fig-sets}b). \n\\end{defn}\n\n\\begin{lem}\\label{lem-secondN}\nThere exists an independent characterization of $N^+$, $N^-$, and thus of $N$:\n$N^+(\\sigma)$ is generated by the future-directed null geodesic congruence orthogonal to $\\sigma$ in the $\\Sigma^-$ direction up to intersections: $p\\in N^+(\\sigma)$ if and only if no conjugate point or nonlocal intersection with any other geodesic in the congruence lies between $\\sigma$ and $p$.\n\\end{lem}\nThis follows from a significantly strengthened version of Theorem 9.3.11 in Ref.~\\cite{Wald}, a proof of which will appear elsewhere. Similarly $N^-$ is generated by the past-directed $\\sigma$-orthogonal null congruence towards $\\Sigma^+$. (Hence if $\\sigma$ is marginally trapped then $N^\\pm$ both are light-sheets of $\\sigma$~\\cite{CEB1}.)\n\n\\begin{cor}\\label{cor-onlysigma}\nLemma~\\ref{lem-secondN} implies that $N$ depends only on $\\sigma$, not on the Cauchy surface $\\Sigma$.  Moreover, the sets $K^+$ and $K^-$ are then uniquely fixed by the fact that $N$ splits the spacetime: $K^+$ is the largest connected set that contains $I^+(N)$ but not $N$.\n\\end{cor}\nThus our use of $\\sigma$ (as opposed to $\\Sigma^+$ and\/or $\\Sigma^-$) as the argument of the sets $K^\\pm$, $N^\\pm$ is appropriate. \n\n\\subsection{Monotonicity of the Spacetime Splitting}\n\\label{sec-ssmono}\n\nUntil now, we have only used the Cauchy-splitting property of $\\sigma$. We will now consider a family of such leaves, $\\sigma(r)$, that foliate a hypersurface ${\\cal H}$. (We use this notation instead of $H$, in order to emphasize that ${\\cal H}$ need not satisfy the additional assumptions defining a future holographic screen.) A tangent vector field $h^a$ can be defined as described in Remark~\\ref{rem-h}, with decomposition $h^a = \\alpha l^a + \\beta k^a$ into the null vectors orthogonal to each leaf. We take $\\Sigma^+$ to be the side towards which the vector $l^a$ points. (This convention anticipates Sec.~\\ref{sec-arealaw}. In the current section, $k^a$ and $l^a$ need not be distinguished by conditions on the corresponding expansions.) To simplify notation, we denote $K^+(\\sigma(r))$ as $K^+(r)$, etc.\n\n\\begin{thm}\\label{thm-kmono}\nConsider a foliated hypersurface ${\\cal H}$ with tangent vector field $h^a$ defined as above. Suppose that $\\alpha<0$ on all leaves $\\sigma(r)$ in some open interval, $r\\in I$. Then\n\\begin{equation}\n\\bar K^+(r_1)\\subset K^+(r_2)~,\n\\label{eq-kmono}\n\\end{equation}\nor equivalently $K^-(r_1)\\supset \\bar K^-(r_2)$, for all $r_1, r_2 \\in I$ with $r_1<r_2$. That is, the sets $K^\\pm(r)$ are monotonic in $r$ under inclusion; and the monotonicity is strict in the strong sense that the entire boundary  $N(r_1)$ of the open set $K^+(r_1)$ is contained in the open set $K^+(r_2)$.\\footnote{It is not difficult to strengthen this theorem by proving the converse. However this requires using assumption \\ref{def-technical}.b which is used nowhere else in this Section. Moreover, the converse is not needed in this paper.}\n\\end{thm}\n\\begin{proof}\nWe will first prove the inclusion monotonicity of $K^\\pm$ under an infinitesimal evolution step $r\\to r+\\delta r$. The assumption that $\\alpha<0$ implies that ${\\cal H}$ locally evolves towards the past or exterior of its leaves: for sufficiently small $\\delta r<0$,\n\\begin{equation}\n\\sigma(r+\\delta r)\\subset K^-(r)~.\n\\label{eq-rdr}\n\\end{equation}\nSince $K^-(r)\\cap K^+(r)=\\varnothing$, it follows that ${\\cal H}$ cannot locally evolve into the future or interior of any of its leaves:\n\\begin{equation}\n\\sigma(r+\\delta r)\\cap K^+(r)=\\varnothing~.\n\\end{equation}\nLet $\\delta {\\cal H}$ be the small portion of ${\\cal H}$ between $r$ and $r+\\delta r$; the above results imply that\n\\begin{equation}\n\\delta {\\cal H} \\subset K^-(r)~,~~~\\delta {\\cal H} \\cap K^+(r)=\\varnothing~.\n\\end{equation}\n\nBy Corollary~\\ref{cor-onlysigma}, we may choose the sets $\\Sigma^+(r)$ to suit our convenience. \nIt is instructive to consider first the special case where we can find a Cauchy surface such that $\\Sigma^+(r+\\delta r) = X$, where\n\\begin{equation}\nX\\equiv \\bar\\Sigma^+(r)\\cup \\delta {\\cal H}~,\n\\label{eq-x}\n\\end{equation}\nand we recall that an overbar denotes the closure of a set.  This is the case when $\\beta>0$ between $\\sigma(r)$ and $\\sigma(r+\\delta r)$, i.e., if $\\delta {\\cal H}$ is spacelike. Both the future of a set, and the past domain of dependence of a set cannot become smaller when the set is enlarged; hence,\n\\begin{eqnarray} \nI^+(X) & \\supset & I^+(\\Sigma^+(r))~,\\nonumber \\\\\nD^-(X) & \\supset & D^-(\\Sigma^+(r))~,\n\\label{eq-idsuper}\n\\end{eqnarray} \nand so the infinitesimal version of Eq.~(\\ref{eq-kmono}) follows trivially from the definition of $K^+$.\n\nNow consider the general case, with no restriction on the sign of $\\beta$. Thus, $\\delta {\\cal H}$ may be spacelike, timelike ($\\beta<0$), or null ($\\beta=0$); indeed, it may be spacelike at some portion of $\\sigma(r)$ and timelike at another. One can still define the submanifold $X$ as the extension of $\\Sigma^+(r)$ by $\\delta {\\cal H}$, as in Eq.~(\\ref{eq-x}); see Fig.~\\ref{fig-deltah}. Again, this extension cannot decrease the future of the set, nor its past domain of dependence,\\footnote{The future of a set is defined for arbitrary sets. The domain of dependence is usually defined only for certain sets, for example for closed achronal sets in Ref.~\\cite{Wald}. Here we extend the usual definition to the more general set $X$: $p\\in D^-(X)$ iff every future-inextendible causal curve through $p$ intersects $X$. This is useful for our purposes; however, we caution that certain theorems involving $D^\\pm$ need not hold with this broader definition.} as described in Eq.~(\\ref{eq-idsuper}).\n\nHowever, $X$ need not be achronal and hence, it need not lie on any Cauchy surface. In this case, we consider a new Cauchy surface that contains $\\sigma(r+\\delta r)$. Because $\\alpha<0$, this surface can be chosen so that $\\Sigma^+(r+\\delta r)$ is nowhere to the future of $X$; see Fig.~\\ref{fig-deltah}. Since $X$ and $\\Sigma^+(r+\\delta r)$ share the same boundary $\\sigma(r+\\delta r)$, $\\alpha>0$ then implies that $X$ is entirely in the future of $\\Sigma^+(r+\\delta r)$: \n\\begin{equation}\nX\\subset I^+(\\Sigma^+(r+\\delta r))\n\\label{eq-xfut}\n\\end{equation}\nMoreover, the set $X$ together with $\\bar\\Sigma^+(r+\\delta r)$ forms a ``box'' that bounds an open spacetime region $Y$, such that \n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=3.0in]{NestedK2.pdf}\n\\caption{Proof that $K^+(r)$ grows monotonically under inclusion, for any foliation $\\sigma(r)$ of a hypersurface $\\cal H$ with $\\alpha<0$. See the main text for details and definitions.}\n\\label{fig-deltah}\n\\end{center}\n\\end{figure}\n\\begin{equation}\nY\\subset  I^+(\\Sigma^+(r+\\delta r))~.\n\\label{eq-ysubi}\n\\end{equation}\nAll future-directed timelike curves that pass through $\\Sigma^+(r+\\delta r)$ enter $Y$ and then can exit $Y$ only through $X$. Hence $D^-(X) \\subset Y \\cup D^-(\\Sigma^+(r+\\delta r))$. Since $\\alpha<0$, for all points outside of $Y \\cup D^-(\\Sigma^+(r+\\delta r))$ there exist future-directed timelike curves that evade $X$. Hence equality holds:\n\\begin{equation}\nD^-(X) = Y \\cup D^-(\\Sigma^+(r+\\delta r))~.\n\\label{eq-dmx}\n\\end{equation}\nTo obtain the infinitesimal inclusion relation, \n\\begin{equation}\nK^+(r+\\delta r)\\supset K^+(r)~,\n\\label{eq-minimono}\n\\end{equation}\nby Eq.~(\\ref{eq-idsuper}) it suffices to show that $K^+(r+\\delta r)\\supset I^+(X)\\cup D^-(X)$. Indeed if $p\\in I^+(X)$ by Eq.~(\\ref{eq-xfut}) $p\\in I^+(\\Sigma^+(r+\\delta r))\\subset K^+(r+\\delta r)$. And if $p\\in D^-(X)$ then by Eqs.~(\\ref{eq-dmx}) and (\\ref{eq-ysubi}) we again have $p\\in  K^+(r+\\delta r)$.\n\nTo obtain the stricter relation\n\\begin{equation}\nK^+(r+\\delta r)\\supset \\bar K^+(r)~,\n\\label{eq-minimonos}\n\\end{equation}\nwe note that $\\sigma(r)\\subset X$; hence by Eq.~(\\ref{eq-xfut}), for every point $p\\in\\sigma(r)$ there exists a timelike curve from $\\Sigma^+(r+\\delta r)$ to $p$. This curve can be continued along the null generator of $N^+(r)$ starting at $p$ to a point $q\\in N^+(r)$, and then slightly deformed into a timelike curve connecting $p$ to $q$. By Lemma~\\ref{lem-secondN}, every point in $N^+(r)$ lies on a generator starting at $\\sigma(r)$. Hence, $N^+(r)\\subset K^+(r+\\delta r)$. A similar argument yields $N^-(r)\\subset K^+(r+\\delta r)$. Since $N(r)=N^+(r)\\cup N^-(r)$ and $\\bar K^+(r) = K^+(r)\\cup N(r)$, Eq.~(\\ref{eq-minimonos}) follows. \n\nTo extend Eq.~(\\ref{eq-minimonos}) to Eq.~(\\ref{eq-kmono}), one may iterate the above infinitesimal construction. The only way this could fail is if the iteration gets stuck because the steps $\\delta r$ have to be taken ever smaller to keep Eq.~(\\ref{eq-rdr}) satisfied. Suppose therefore that the iteration can only reach an open set $(r,r_*)$ but no leaves in the set $(r_*,r_2)$. But this contradicts the assumption that $\\alpha<0$ at $r_*$.\n\\end{proof}\n\n\\section{Area Law}\n\\label{sec-arealaw}\n\nIn this section, we prove our main result: that the area of the holographic screen is monotonic. The most difficult part of this task is proving that $\\alpha$ cannot change sign on $H$, Theorem~\\ref{thm-alpha}).\nWe then prove our Area Theorem~\\ref{thm-area}.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=0.35 \\textwidth]{lemma3D.pdf}\n\\caption{An example illustrating Lemma~\\ref{monotonicity}: in Minkowski space, the spatial sphere $\\chi$ is tangent to the null plane $N$ at $p$ and lies outside the past of $N$ near $p$. It is easy to see that this implies that $\\chi$ is a cross-section of a future light-cone that shares one null generator with $N$.  In this example it is obvious that $\\chi$  expands faster than $N$ at $p$, as claimed in Lemma~\\ref{monotonicity}.}\n\\label{fig-lemma}\n\\end{figure}\nWe begin by stating a useful Lemma.\n\\begin{lem}  \\label{monotonicity}\nLet $N$ be a null hypersurface and let $\\chi$ be a spacelike surface tangent to $N$ at a point $p$. That is, we assume that one of the two future-directed null vectors orthogonal to $\\chi$, $\\kappa^a$, is also orthogonal to $N$ at $p$. We may normalize the (null) normal vector field to $N$ so that it coincides with $\\kappa^a$ at $p$.  Let $\\theta^{(\\chi)}$ be the null expansion of the congruence orthogonal to $\\chi$ in the $\\kappa^a$ direction, and let $\\theta^{(N)}$ be the null expansion of the generators of $N$. Then:\n\\begin{itemize} \n\\item If there exists an open neighborhood $O(p)\\cap \\chi$ that lies entirely outside the past of $N$,\\footnote{I.e., there exists no past directed causal curve from any point on $N$ to any point in $O(p)\\cap \\chi$.} then $\\theta^{(\\chi)}\\geq \\theta^{(N)}$ at $p$.\n\\item If there exists an open neighborhood $O(p)\\cap \\chi$ that lies entirely outside the future of $N$, then $\\theta^{(\\chi)}\\leq \\theta^{(N)}$ at $p$.\n\\end{itemize} \n\\end{lem}\n\\begin{proof}\nSee Lemma A in Ref.~\\cite{Wal10QST}. Our Lemma is stronger but the proof is the same; so instead of reproducing it here, we offer Fig.~\\ref{fig-lemma} to illustrate the result geometrically. It generalizes to null hypersurfaces an obvious relation in Riemannian space, between the extrinsic curvature scalars of two codimension 1 surfaces that are tangent at a point in a Riemannian space but do not cross near that point.\n\\end{proof}\n\n\n\n\\begin{thm} Let $H$ be a regular future holographic screen with leaf-orthogonal tangent vector field $h^{a}=\\alpha l^{a} + \\beta k^{a}$, whose orientation is chosen so that $\\alpha<0$ at the leaf $\\sigma(0)$. Then $\\alpha< 0$ everywhere on $H$.   \\label{thm-alpha}\n\\end{thm}\n\n\\begin{proof}\nBy contradiction: suppose that $H$ contains a point with $\\alpha\\geq 0$. It immediately follows that the subset $H_+\\subset H$ of points with $\\alpha> 0$ is nonempty, since by Assumption~\\ref{def-technical}.b, $\\alpha=0$ can occur only as a transition between $\\alpha<0$ and $\\alpha>0$ regions. Let $\\sigma(0)$ be the complete leaf that exists by Remark~\\ref{rem-s0exists} and has $r=0$, $\\alpha<0$, by Convention~\\ref{conv-orient}. By continuity of $\\alpha$, there exists an open neighborhood of $\\sigma(0)$ where $\\alpha<0$. \n\n\\begin{figure}[t]\n\\includegraphics[height=0.3 \\textwidth]{Forbidden.pdf}\n\\caption{The four types of spacelike-timelike transitions on a future holographic screen that would violate the monotonicity of the area, and which our proof in Sec.~\\ref{sec-arealaw} will exclude. Near $\\sigma(0)$, the area increases in the direction of the arrow. On the far side of the ``bend'' the area would decrease, in the same direction. There are other types of spacelike-timelike transitions which preserve area monotonicity under uniform flow; these do arise generically (see Fig.~\\ref{fig-examplesPRD}a).}\n\\label{fig-sphsym}\n\\end{figure}\nWe first consider the case where $H_+$ has a component in the $r>0$ part of $H$ (cases 1 and 2 in Fig.~\\ref{fig-sphsym}). Let $\\sigma(1)$ be the ``last slice'' on which $\\alpha\\leq 0$, i.e., we use our freedom to rescale $r$ to set\n\\begin{equation}\n1=\\inf\\{r:r>0,\\sigma(r)\\cap H_+ \\neq \\varnothing\\}\n\\end{equation}\nBy the second generic condition~\\ref{def-technical}.b, $\\alpha<0$ for all leaves $\\sigma(r)$ with $0<r<1$; hence by Theorem~\\ref{thm-kmono} we have $K^-(0)\\supset \\bar K^-(1)$. Since the former set is open and the latter is closed, there exists an open neighborhood of $\\bar K^-(1)$ that is contained in $K^-(0)$. Thus for sufficiently small $\\epsilon$ we have\n\\begin{equation}\nK^-(0)\\supset K^-(1+\\epsilon)~.\n\\label{eq-nocon}\n\\end{equation}\n\nBy continuity of $\\alpha$, $\\sigma(1)$ must contain at least one point with $\\alpha=0$. Let $p$ denote this point; or, if there is more than one such point, let $p$ denote a connected component of the set of points with $\\alpha=0$ on $\\sigma(1)$. Since there is no point with $\\alpha=\\beta=0$, there exists an open neighborhood $O(p)\\subset H$ in which $\\beta$ has definite sign.  (Note that we do not assume that $\\beta$ is of fixed sign for $0<r<1$.) \n\\begin{figure*}[htb]\n\\centering\n\\subfigure[ ]{\n\\centering\n\\includegraphics[width=0.35 \\textwidth]{Case1Left.pdf}}\n\\qquad\n\\subfigure[]{\n\\centering \\includegraphics[width=0.35\\textwidth]{Case1Right.pdf}\n}\n\\caption{A case 1 transition ($S_{-+}\\to T_{++}$) is impossible. The proof crucially involves the intersection $\\phi$ of a light-sheet $N$ originating just behind the assumed transition, with the region prior to the transition. For further details see the main text. (a) Spacetime diagram with two spatial directions suppressed. (b) Diagram of the holographic screen $H$ only, with only one spatial direction suppressed. Vertical lines correspond to leaves; top and bottom edge should be identified. Portions of different signature are indicated by shading and labels. In general, the transition boundary (thick red line) will not coincide with a leaf (thin black vertical lines).}\n\\label{fig-Case1LR}\n\\end{figure*}\n\n\n\\vskip .6cm\n\\noindent {\\bf Case 1 \\ } We now specialize further to the case where $\\beta>0$ in $O(p)$, so that the assumed sign change from $\\alpha<0$ to $\\alpha>0$ corresponds to a transition of $h^a$ from spacelike-outward ($S_{-+}$) to timelike-future-directed ($T_{++}$). The following construction is illustrated in Fig. \\ref{fig-Case1LR}.\n\nLet $\\sigma^+(1+\\epsilon)$ be the set of points with $\\alpha>0$ on the leaf $\\sigma(1+\\epsilon)$. If there is more than one connected component, we choose $\\sigma^+(1+\\epsilon)$ to be the component at least one of whose fibers intersects $p$. By choosing $\\epsilon$ sufficiently small, we can ensure that $\\sigma^+(1+\\epsilon) \\subset O(p)$. Let $\\Gamma$ be the set of fibers that pass through $\\sigma^+(1+\\epsilon)$. \n\nBecause $\\alpha>0$, all fibers in $\\Gamma$ enter $K^-(1+\\epsilon)$ as we trace them back to smaller values of $r$. But $\\sigma(0)$ is entirely outside of this set: by definition, $\\sigma(0)\\cap K^-(0)=\\varnothing$, so Eq.~(\\ref{eq-nocon}) implies $\\sigma(0)\\cap K^-(1+\\epsilon)=\\varnothing$. Hence, all fibers in $\\Gamma$ also intersect $N(1+\\epsilon)$, at some positive value of $r< 1+\\epsilon$. Because $\\beta>0$ in $O(p)$, this intersection will be with $N^-(1+\\epsilon)$. By smoothness and the second generic assumption, the intersection will consist of one point per fiber. (Otherwise a fiber would coincide with a null generator of $N^{-}(1+\\epsilon)$ in a closed interval.) \nThe set of all such intersection points, one for each fiber in $\\Gamma$, defines a surface $\\phi$, and the fibers define a continuous, one-to-one map $\\sigma^+(1+\\epsilon)$ to $\\phi$. Similarly, the closures of both sets, $\\bar\\sigma^+(1+\\epsilon)$ and $\\bar\\phi$ are related by such a map. Note that these two sets share the same boundary at $r=1+\\epsilon$.\n\nLet $R$ be the minimum value of $r$ on the intersection: $R\\equiv\\inf\\{r(q): q\\in \\bar\\phi\\}$. Since $\\bar\\sigma^+(1+\\epsilon)$ is a closed subset of a compact set, it is compact; and by the fiber map, $\\bar\\phi$ is also compact. Therefore $R$ is attained on one or more points in $\\bar\\phi$. Let $Q$ be such a point. Since $R<1$ but $\\dot\\phi\\subset \\sigma(1+\\epsilon)$, $Q\\notin\\dot\\phi$, and hence $Q$ represents a local minimum of $r$. Hence the leaf $\\sigma(R)$ is tangent to the null hypersurface $N^{-}(1+\\epsilon)$ at $Q$.\n\nSince $Q$ achieves a global minimum of $r$ on $\\bar\\phi$, $\\sigma(R)$ lies nowhere in the past of $N^{-}(1+\\epsilon)$ in a sufficiently small open neighborhood of $Q$. For suppose there existed no such neighborhood. Then fibers arbitrarily close to the one containing $Q$ (and hence connected to $\\sigma^+(1+\\epsilon)$ would still be inside $K^-(1+\\epsilon)$ at $R$. Hence we could find a value $r<R$ on $\\phi$ by following such a fiber to smaller values of $r$ until it leaves $K^{-}(1+\\epsilon)$. But this would contradict our construction of $Q$ as a point that attains the minimum value of $r$ on $\\phi$.\n\nThus, Lemma~\\ref{monotonicity} implies that $\\theta_k^{\\sigma(R)}\\geq\\theta_k^{N^{-}(1+\\epsilon)}$ at $Q$. By the first generic assumption, the latter expansion is strictly positive, so we learn that $\\theta_k^{\\sigma(R)}>0$ at $Q$.  But this contradicts the defining property of holographic screens, that all leaves are marginally trapped ($\\theta_k^{\\sigma(r)}=0$ for all $r$). \n\n\\vskip .6cm\n\\noindent {\\bf Case 2 \\ }  Next we consider the case where $\\beta<0$ in the neighborhood of the assumed transition from $\\alpha<0$ to $\\alpha>0$ that begins at $r=1$ (see Fig.~\\ref{fig-sphsym}). This corresponds to the appearance of a spacelike-inward-directed region within a timelike-past-directed region: $T_{--}\\to S_{+-}$.\n\nWe note that the direct analogue of the above proof by contradiction fails: tracing back the generators from $\\sigma^+(1+\\epsilon)$ to $\\sigma(0)$, one finds that they pass through $N^+(1+\\epsilon)$, rather than $N^-(1+\\epsilon)$. But $N^+$ has negative expansion by the first generic condition, whereas $N^-$ had positive expansion. There is no compensating sign change elsewhere in the argument; in particular, the tangent leaf $\\sigma(R)$ with vanishing expansion again lies nowhere in the past of $N^+$ in a neighborhood of the tangent point $Q$. Thus no contradiction arises with Lemma~\\ref{monotonicity}.\n\nInstead, we show that every case 2 transition implies the existence of a case 1 transition at a {\\em different}\\\/ point on $H$, under the reverse flow $r\\to c-r$. Since we have already shown that case 1 transitions are impossible, this implies that case 2 transitions also cannot occur. \n\nLet us first illustrate this argument in the simple case where the transition occurs entirely on a single leaf: $\\alpha<0$ for $0\\leq r<1$, $\\alpha=0$ at $r=1$, and $\\alpha>0$ for $1<r\\leq 2$. Under a reversal of the flow, $r\\to 2-r$, $\\alpha$ and $\\beta$ both change sign. With this flow direction, the latter region now contains a leaf $\\sigma(0)$ on which $\\alpha<0$, and thus the starting point of our case 1 proof.  The putative sign change of $\\alpha$ corresponds to a case 1 transition $S_{-+}\\to T_{++}$. The case 1 proof by contradiction rules out this transition.\n\n\\begin{figure*}[htb]\n\\centering\n\\subfigure[ ]{\n\\centering\n\\includegraphics[height=0.25 \\textwidth]{Case2A.pdf}}\n\\qquad\n\\subfigure[]{\n\\centering \\includegraphics[height=0.25\\textwidth]{Case2B.pdf}\n}\n\\caption{A case 2 transition ($T_{--}\\to S_{+-}$) is impossible. By assumption, the $\\alpha>0$ region contains a complete leaf $\\sigma(2+\\epsilon)$. In the text we show that the complete-leaf region begins at some leaf $\\sigma(2)$ where a $T_{--}\\to S_{+-}$ boundary comes to an end: either the original one (a), or a different one containing a $T_{--}$ region with no complete leaf (b). The endpoint (green dot) becomes the starting point of a case 1 transition ($S_{-+}\\to T_{++}$) under reversal of the flow direction; but this case has already been ruled out.}\n\\label{fig-case2proof}\n\\end{figure*}\n\n\nIn general, the case 2 transition need not occur on a single leaf, so we shall assume for contradiction only that $\\alpha$ first becomes positive at some point on or subset of $\\sigma(1)$, as in the case 1 proof, and that $\\beta<0$ in a neighborhood of this set. Let $\\tilde H_+$ denote the connected region with $\\alpha>0$ that begins at this transition. Since the transition is $T_{--}\\to S_{+-}$, $\\tilde H_+$ contains some spacelike points; and hence by Def.~\\ref{def-technical}.c, $\\tilde H_+$ contains a complete leaf with $\\alpha>0$. We use our freedom to rescale $r$ to set\n\\begin{equation}\n2=\\inf\\{r:r>0,\\sigma(r)\\subset \\tilde H_+\\}\n\\label{eq-leaf2}\n\\end{equation}\nBy the second generic assumption, Def.~\\ref{def-technical}.b, this choice implies the existence of an open interval $(2,2+\\epsilon)$ such that every leaf in this interval is a complete leaf with $\\alpha>0$. Let us call this intermediate result (*); see Fig.~\\ref{fig-case2proof} which also illustrates the remaining arguments.\n\nWe now consider the boundary $B$ that separates the $\\alpha<0$ from the $\\alpha>0$ region, i.e., the connected set of points with $\\alpha=0$ that begins at $r=1$. Because $\\alpha$ and $\\beta$ cannot simultaneously vanish, we have $\\beta<0$ in an open neighborhood of all of $B$. Thus, $B$ separates a $T_{--}$ region at smaller $r$ from a $S_{+-}$ region at larger $r$. We note that $B$ must intersect every fiber, or else $H_+$ would not contain a complete leaf. Moreover, $B$ must end at some $r_*\\leq 2$, or else there would be points with $\\alpha<0$ in the interval $(2,2+\\epsilon)$, in contradiction with (*). \n\nIf $r_*=2$ then under the reverse flow starting from the complete leaf at $r=2+\\epsilon$ there is a case 1 transition at $r=2$ from $S_{-+}$ to $T_{++}$, and we are done. This is shown in Fig.~\\ref{fig-case2proof}a.  \n\nThe only remaining possibility is that $B$ ends at some $r_*\\in (1,2)$; this is shown in Fig.~\\ref{fig-case2proof}b. Then every leaf with $r\\in (r_*,2)$ must contain points with $\\alpha<0$, or else there would be a complete leaf with $\\alpha>0$ at some $r<2$, in contradiction with Eq.~(\\ref{eq-leaf2}). Therefore each leaf with $r\\in (r_*,2)$ must intersect one or more $\\alpha<0$ regions $\\tilde H_-^{(i)}$ that are disconnected from the $T_{--}$ region bounded by $B$. None of these regions $\\tilde H_-^{(i)}$ can contain a complete $\\alpha<0$ leaf, because this would imply that $\\tilde H_+$ does not contain a complete $\\alpha>0$ leaf. From Def.~\\ref{def-technical}.c it follows that each region $\\tilde H_-^{(i)}$ is everywhere timelike, i.e., of type $T_{--}$. But this implies that a $T_{--}$ region ends at $r=2$ where $\\alpha$ becomes positive. Moreover, the $S_{+-}$ region in which the $T_{--}$ region ends has complete leaves in some open interval $(2,2+\\epsilon)$ by our result (*). Thus we find again that under the reverse flow starting from the complete leaf at $r=2+\\epsilon$ there is a case 1 transition at $r=2$ from $S_{-+}$ to $T_{++}$.\n\nWe have thus established that a case 2 transition at $r=1$ implies a case 1 transition at the same or a larger value of $r$, after reversal of the direction of flow. Since case 1 transitions are impossible, we conclude that case 2 transitions are also impossible.\n\\begin{figure*}[ht]\n\n\\includegraphics[width= 0.85\\textwidth]{Panelv3.pdf}\n\\caption{(a) Case 3 is ruled out analogously to case 1, by contradiction.  (b) Case 4 is analogous to case 2: the transition is impossible because it would imply a case 3 transition elsewhere on $H$, under reversal of the flow direction.}\n\\label{fig-cases}\n\\end{figure*}\n\n\n\\vskip .6cm\n\\noindent {\\bf Cases 3 and 4} \nOur consideration of cases 1 and 2 has ruled out the possibility of points with $\\alpha>0$ at any $r>0$. (Recall that $r=0$ corresponds to a complete leaf with $\\alpha<0$.) We must now also rule out the possibility that $\\alpha$ might be positive in the region $r<0$; this corresponds to cases 3 and 4 in Fig.~\\ref{fig-sphsym}. Again, assume for contradiction that such a transition occurs, and focus on the transition nearest to $r=0$. We may rescale $r$ so that this transition ends at $r=-1$. That is, $\\alpha<0$ for all $r\\in (-1,0)$, but all leaves in some interval $(-(1+\\epsilon),-1)$ contain points with $\\alpha>0$. Again, a further case distinction arises depending on the sign of $\\beta$ at this transition.\n \nThe proof of case 3 (Fig.~\\ref{fig-cases}a), where $\\beta>0$ at the transition, proceeds exactly analogous to that of case 1. Fibers that connect the offending region to $r=0$ must cross the null hypersurface $N^+(-(1+\\epsilon))$, implying the existence of a leaf $\\sigma(R)$, $-1<R<0$ that is tangent to $N^+(-(1+\\epsilon))$ and nowhere to the future of $N^+(-(1+\\epsilon))$. But $N^+$ contracts at this tangent point whereas $\\sigma(R)$ has vanishing expansion, in contradiction with the second part of Lemma~\\ref{monotonicity}.\n\nThe proof of case 4 (Fig.~\\ref{fig-cases}b) proceeds analogous to that of case 2, by showing that a case 4 transition at $r=-1$ implies the existence of a transition at some $r\\leq -1$ that is recognized as a case 3 transition after reversal of the flow direction, and hence ruled out.\n\n\\end{proof}\n\n\\begin{figure*}[ht]\n\\includegraphics[width=6.5in]{ZigZagBoth.pdf}\n\\caption{Proof that $\\alpha<0$ implies an area law, by a zig-zag construction. For any pair of infinitesimally nearby leaves, consider the intersection $\\hat\\sigma$ of the light-sheet pair $N(r)$ tangent to the marginal direction $k^a$, with the lightsheet $L^+(r+\\delta r)$ generated by $l^a$. To first order in $\\delta r$, the area is constant as we leave $H$ from $\\sigma(r)$ along $N(r)$ to $\\hat \\sigma$, and the area increases as we follow $L^+$ from $\\hat\\sigma$ back onto $H$. This construction assumes only that $\\alpha<0$, so that evolution is in the $-l^a$ direction; it does not require $H$ to have uniform signature near $\\sigma(r)$.  (a) Timelike case, with one spatial dimension suppressed and relevant 2-surfaces labeled. (b) Spacelike case, with all spatial dimensions suppressed. In this plot, we choose to label the relevant null surfaces; the two-surfaces are labeled by their area not by their name.}\n\\label{fig-zigzag}\n\\end{figure*}\nWe now state and prove the area law.\n\\begin{thm} \\label{thm-area}\nThe area of the leaves of any regular future holographic screen $H$ increases strictly monotonically:\n\\begin{equation}\n\\frac{dA}{dr}>0~.\n\\label{eq-area}\n\\end{equation}\n\\end{thm}\n\\begin{proof} \nBy Theorem~\\ref{thm-alpha}, $\\alpha<0$ everywhere on $H$. In regions where $\\beta$ is of definite sign, the result would then follow from the analysis of Hayward~\\cite{Hay93} (using a 2+2 lightlike formalism) or that of Ashtekar and Krishnan~\\cite{AshKri02} who used a standard 3+1 decomposition. It should be straightforward to generalize their proofs to the case where $\\beta$ may not have definite sign on some or all leaves. However, since this would necessitate the introduction of additional formalism, we will give here a simple, geometrically intuitive proof. Our construction is shown in Fig.~\\ref{fig-zigzag}. \n\nConsider two infinitesimally nearby leaves at $r$ and $r+dr$, $dr>0$. Construct the null hypersurface $N(r)$ in a neighborhood of $\\sigma(r)$. Also, construct the null hypersurface $L^+(r+dr)$ generated by the future-directed null geodesics with tangent vector $l^a$, in a neighborhood of $\\sigma(r+dr)$. By Theorem~\\ref{thm-alpha}, for sufficiently small $dr$ these null hypersurfaces intersect on a two-dimensional surface $\\hat\\sigma(r,r+dr)$, such that every generator of each congruence lies on a unique point in $\\hat\\sigma(r,r+dr)$. \n\nNote that in regions where $H$ is spacelike, $\\beta>0$, the intersection will lie in $N^+(r)$; if $H$ is timelike, $\\beta<0$, the intersection will lie in $N^-(r)$; but this makes no difference to the remainder of the argument. Crucially, Theorem~\\ref{thm-alpha} guarantees that the intersection always lies in $L^+(r+dr)$, and never on $L^-(r+dr)$, the null hypersurface generated by the past-directed null geodesics with tangent vector $-l^a$.\nWe now exploit the defining property of $H$, that each leaf is marginally trapped ($\\theta^{\\sigma(r)}_k=0$). This implies\n\\begin{eqnarray} \nA[\\hat\\sigma]-A[\\sigma(r)] & = & O(dr^2)~;\\\\\nA[\\sigma(r+dr)]-A[\\hat\\sigma] & = & O(dr)>0~.\n\\end{eqnarray} \nHence, the area increases linearly in $dr$ between any two nearby leaves $\\sigma(r)$, $\\sigma(r+dr)$. This implies that the area increases strictly monotonically with $r$.\n\n\\end{proof}\n\n\\begin{cor}\\label{cor-quant}\nThe above construction implies, more specifically, that the area of leaves increases at the rate\n\\begin{equation}\n\\frac{dA}{dr} = \\int_{\\sigma(r)} \\sqrt{h^{\\sigma(r)}}~ \\alpha \\theta^{\\sigma(r)}_l~.\n\\label{eq-specarea}\n\\end{equation}\nwhere $h_{ab}^{\\sigma(r)}$ is the induced metric on the leaf $\\sigma(r)$ and $h^{\\sigma(r)}$ is its determinant. Note that the integrand is positive definite since $\\alpha<0$ and all leaves are marginally trapped; in this sense the area theorem is local. However, the theorem applies to complete leaves only, not to arbitrary deformations of leaves.\n\\end{cor}\n\n\\begin{cor}\\label{cor-past}\nFor past holographic screens, we recall the contrasting convention that $\\alpha>0$ on $\\sigma(0)$. The above arguments then establish that $\\alpha>0$ everywhere on $H$. Eqs.~(\\ref{eq-area}) and (\\ref{eq-specarea}) hold as an area theorem. \n\\end{cor}\n\n\\begin{rem}\\label{rem-arrow}\nWe note that the area increases in the outside or future direction along a past holographic screen. With an interpretation of area as entropy, the holographic screens of an expanding universe thus have a standard arrow of time. \n\\end{rem}\n\\begin{rem}\\label{rem-backarrow}\nBy contrast, the area increases in the outside or {\\em past} direction along a future holographic screen. Thus,  {\\em the arrow of time runs backwards on the holographic screens inside black holes, and near a big crunch}. Perhaps this intriguing result is related to the difficulty of reconciling unitary quantum mechanics with the equivalence principle~\\cite{Haw76,AMPS,Bou12c,AMPSS,Bou13,MarPol13,Bou13a,Bou13b}.\n\\end{rem}\n\nWe close with a final theorem that establishes the uniqueness of the foliation of $H$:\n\\begin{thm}\nLet $H$ be a regular future holographic screen with foliation $\\{\\sigma(r)\\}$. Every marginally trapped surface $s\\subset H$ is one of the leaves $\\sigma(r)$. \n\\end{thm}\n\\begin{proof}\nBy contradiction: suppose that $s$ is marginally trapped and distinct from any $\\sigma(r)$.  Thus $s$ intersects the original foliation in a nontrivial closed interval $[r_1,r_2]$ and is tangent to $\\sigma(r_1)$ and $\\sigma(r_2)$. The $\\theta=0$ null vector field orthogonal to $s$ must coincide with $k^a$ at the tangent point with $\\sigma(r_2)$. Since $r_1<r_2$, Theorems~\\ref{thm-alpha} and \\ref{thm-kmono} imply that $N(r_2)$ does not everywhere coincide with the null hypersurface orthogonal to $s$ with tangent vector $k^a$ at $\\sigma(2)$. Lemma B in Ref.~\\cite{Wal10QST} then implies that $\\theta_{k^{(s)}}^{(s)}\\neq 0$ somewhere on $s$, in contradiction with the assumption that $s$ is marginally trapped.\n\\end{proof}\n\n\\vskip .3cm\n\\indent {\\bf Acknowledgments} \nIt is a pleasure to thank S.~Fischetti, D.~Harlow, G.~Horowitz, W.~Kelly, S.~Leichen\\-auer, D.~Marolf, R.~Wald, and A.~Wall for discussions and correspondence. NE thanks the Berkeley Center for Theoretical Physics and the UC Berkeley Physics Department for their hospitality. The work of RB is supported in part by the Berkeley Center for Theoretical Physics, by the National Science Foundation (award numbers 1214644 and 1316783), by fqxi grant RFP3-1323, and by the US Department of Energy under Contract DE-AC02-05CH11231. The work of NE is supported in part by the US NSF Graduate Research Fellowship under Grant No. DGE-1144085 and by NSF Grant No. PHY12-05500.\n\n\\bibliographystyle{utcaps}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nElectricity networks have become pivotal in our societies attempt to \ndecrease the emission of greenhouse gasses. \nFlexible electricity networks must constantly mediate the tension between the demand-agnostic production of \nelectricity from renewable sources\nand the supplier-agnostic consumption we have been accustomed to.\nFlexible electricity networks continuously coordinate and optimize operations through ICT systems. \nAn overlay data grid conveys information about the state of the electricity grid, \nas well as the status of demand and production of electricity in households and industry. \nData is thus the basis for decisions that affect electricity costs and availability \nof assets on the electricity grid. It is crucial that these decisions are formed and monitored according to a well-defined governance model. No such data governance model exists today.\nThey are the topic of this paper.\n\nWe focus on the role of \"Virtual Power Plants\". Briefly for now, a virtual power plant, or VPP,\norganizes the flows of data between many asset owners (individuals or companies) and an aggregator that takes decisions about tariffs and asset availability.\nSince VPPs are a proposed way to integrate renewable energy sources and decrease the reliance on fossil fuels, they, simply by their function, inherit an aura of progressiveness, \nsustainability and collaboration. \nBut the powerful position they hold, collecting sensitive personal data about individuals and remotely controlling assets in their households, opens many possibilities for exploitation. These must be counteracted by a strong governance model.\n\nIn this paper, we explore the issues around governance of the data that flows from individuals to aggregator within a VPP. We introduce the frame of \nunionization as a way to address these issues, and discuss how unions can be incorporated into data governance models to increase transparency, accountability and trust.\n\nIn Section II, we detail the role of virtual power plants in flexible electricity networks. Section III defines the problem in terms of data sources and data flows within the system. In Section IV, we introduce the frame of unionization as a way to address these problems, and Section V describes the principles for data governance derived from this frame. Section VI presents selected work that we find relevant for this issue, and Section VII summarizes the findings of the paper.\n\n\\section{Virtual Power Plants}\n\nIn this section we introduce the concept of Virtual Power Plants. We describe the motivation for flexibility in electrical networks, the inherently asymmetrical structure of VPPs, and the issues surrounding personal data collection.\n\n\\subsection{The Need for Flexibility}\nMany energy-intensive sectors, including transportation and heating, are increasingly moving towards electrification. \nBut the green promise of electrification can only be kept if the electrons flowing in our grids are from low-emission renewable energy sources. \nAnd while renewable generation capacity is growing, this is in itself not sufficient. \n\nRenewable energy sources --- such as wind, solar and tidal --- are \\emph{non-dispatchable}, meaning that we cannot control their production. \nWe can turn generators off, but we cannot generate electricity from these sources at will. The conditions of our environment must allow it. \nSometimes, the wind is not blowing, the sun is not shining and the tidal currents are still. But our consumption of electricity continues regardless.\nThis creates large gaps in time between production and load, which is an issue as there is currently no viable solutions for large-scale energy storage.\\footnote{\n  Pumped-storage hydroelectricity is the primary large-scale energy storage solution, but it needs specific natural features (such as reservoirs) to be applicable.\n} \nThe electrical grid operators constantly balance the production and the consumption of energy in order to maintain a stable grid. \nSo even if we generate enough renewable energy to match our demand on average, the need to balance consumption and non-dispatchable generation on a minute-to-minute basis is an on-going problem.\n\nA proposed way to balance a grid with a large amount of non-dispatchable generation is to incorporate \\emph{demand-side flexibility}~\\cite{mohlerChapter23Energy2017}. \nThe load on the grid is constrained due to physical limits of the infrastructure, so flexibility, or supply-following load, eases this constraint by managing the electrical demand \n\u2013 making demand dispatchable to balance the uneven renewable generation and its storage. \nTemporary energy storage and other types of dynamic demand response are crucial to managing renewable energy flexibly on the grid. \n\nBut in order to achieve this flexibility, the future electricity networks, as envisioned by especially \\cite{tanejaFlexibleLoadsFuture2013a} and \\cite{katzInformationcentricEnergyInfrastructure2011}, will \nneed to be digitally networked electrical grids that continuously coordinate and optimize operation through ICT systems (see also \\cite{bondyFunctionalReferenceArchitecture2015, heArchitectureLocalEnergy2008,\namanPredictionModelsDynamic2015,sianoDemandResponseSmart2014}). This data infrastructure is a layer on top of the electrical grid, transporting information about the state of both industrial and domestic demand and \ngeneration. It organizes flows of data concerning home and individual energy use. These flows of data can be used to make decisions on how to balance the grid or on how to store electricity: Data is Electricity!\n\nThe data infrastructure associated with the electricity system relies on personal data. For example, home usage data in a 1-hour resolution can disclose whether \nyou are home \\cite{liisbergHiddenMarkovModels2016}, and sub-second resolution can reveal which movies you are watching \\cite{grevelerMultimediaContentIdentification2012}. Data governance is essential \nto manage the impact of such data infrastructures on society.\n\nA proposed component of this flexibly-managed smart grid, is what has popularly been coined \\emph{Virtual Power Plants}\\footnote{\n  First proposed in \\cite{veseyVirtualUtility1997} as \"virtual utilities\"\n} (VPPs). A VPP provides flexibility by aggregating a host of smaller \\emph{distributed energy resources} (DERs) --- such as solar panels, domestic batteries, electric vehicles or electric heating systems --- using their combined capacity to provide services normally reserved for industrial scale power plants.\n\n\\subsection{VPP Asymmetry}\n\nThe idea of VPPs is an emerging concept that has yet to see complete consensus on its definition. In \\cite{adu-kankamCollaborativeVirtualPower2018} a literature review is carried out which identified a convergence towards the following definition:\n\\emph{\"A VPP is a virtual entity involving multiple stakeholders and\ncomprising decentralized multi-site heterogeneous technologies, formed\nby aggregating dispatchable and non dispatchable distributed energy\nsources and energy storage systems, including electric vehicles and\ncontrollable loads. It is supported by information and communication\ntechnology to form the equivalent of a single virtual power plant with\ncapacity to manage and coordinate its operations, ensuring power and\ninformation flow among its stakeholders in order to minimize generation\ncosts, maximize profits, and enhance participation in demand\nresponse programs as well as trade within the electricity market.\"}\n\\cite[p.~281]{adu-kankamCollaborativeVirtualPower2018}\n\n\\begin{figure*}[tb]\n  \\begin{center}\n    \\includegraphics[width=0.8\\textwidth]{figures\/VPP_structure.pdf}\n    \\caption{Overall structure of a VPP. The aggregator controls a portfolio of assets --- such as heat pumps, domestic batteries or EVs --- and participates in the energy market or provides flexibility services to power systems operators.}\n    \\label{fig:vpp-structure}\n  \\end{center}\n\\end{figure*}\n\nThe subset of VPPs of interest for this paper, are those in which the DERs, or \\emph{assets}, are owned by individuals outside the organization that owns the VPP. The VPP as a concept includes all of the stakeholders, but we are particularly interested in relationship between the \\emph{asset-owners} and the \\emph{aggregator}. An asset-owner is a stakeholder that owns an individual asset in the portfolio of the aggregator. An example is a homeowner with a heat pump, electric car or domestic battery that can be remotely controlled. The aggregator\\footnote{\n  The term \"aggregator\" is sometimes used as a synonym for VPP, which is again a sign that the dust has not completely settled around the terminology in this field.\n} is the stakeholder controlling the ICT system that collects data and controls the assets. The relationship between asset, aggregator and VPP is shown in \\autoref{fig:vpp-structure}. \n\n\\subsection{VPPs and Personal Data}\n\nThe ICT system at the core of the VPP is dependent on several data flows in order to function. The most interesting and complex from a governance perspective is the flow from asset-owner to aggregator. This data flow contains electricity consumption information from the home (and potentially electric car), which is highly personal data. Behavior patterns can be inferred from electricity consumption, and the data acts as the basis for decisions made about the control of the asset, which influences both power cost and availability for the asset-owner. Because of the nature and purpose of this data, the data governance model employed within the VPP is crucial for the development of trust between asset-owners and aggregator.\n\nIn a demonstration project conducted as part of the Danish \\textsc{EcoGrid 2.0}-project \\cite{heinrichEcoGridLargescaleField2020}, an aggregator was implemented, controlling heat pumps in 300 private households over the course of 3 years \\cite{mullerLargescaleDemonstrationPrecise2019}. This aggregator remotely sent throttling commands to the heat pumps, decreasing load on the grid at certain times. The aggregator collected individual electricity consumption measurements every 5 minutes, and used it as input to its model along with weather service data. In the context of \\textsc{EcoGrid 2.0} the homeowners were found to have a high degree of trust in the project, attributed in part to the technicians from the local community-owned utility, who installed and serviced the connected heat pumps \\cite{pallesenArticulationWorkMiddle2018}.\n\nThe mass use of internet-connected services and devices that collect and store data of our use and behaviors have led to a continuous and pervasive monitoring of our lived lives. This data collection is generally not approved by the users, who find the data collection \"creepy\" \\cite{shklovskiLeakinessCreepinessApp2014}. Still, despite regular public outcries, people continue to use the services, handing over their data. This has typically been described as an expression of the so-called \"privacy paradox\" \\cite{norbergPrivacyParadoxPersonal2007,barnesPrivacyParadoxSocial2006}, which describes the discrepancy between intentions and behaviors when it comes to personal information disclosure. \n\nRecent research suggest that this \"paradox\" represents wide-spread \"digital resignation\" \\cite{draperCorporateCultivationDigital2019a}: People desire to control their personal information, but feel unable to do so, rendering them passive. \nA study on data leakage in mobile apps conducted in \\cite{shklovskiLeakinessCreepinessApp2014} indicated that people were opposed to this practice of data collection, but could see no alternative except to stop using the smartphone as a smartphone altogether.\nThe organizations collecting data benefit from this resignation, cultivating it through obfuscation measures, such as complicated privacy policies and transparency initiatives that allows partial access but not control \\cite{draperCorporateCultivationDigital2019a}.  \n\nWhat differentiates the case of VPPs from data collection through mobile apps is in part the novelty of the domain. Continuos monitoring of electrical consumption is a new type of surveillance in a primary private sphere, the home, and unlike the phones and smart speakers we bring into the home ourselves, this monitoring becomes part of the infrastructure that makes the house. Finally, the aggregator has direct control of the home, controlling electricity availability and cost. The aggregator is not just sensing, but also actuating.\n\n\\section{Problem Definition}\n\nA VPP can be viewed as a system that collects data from various sources and combines these in order to make decisions about the behavior of its assets and, by consequence, the lives of the asset-owners. In order to define the problem of data governance between the asset-owners and aggregator, we must first define the different types of data sources and flows within the system. From these we identify the relationships that these flows represent, and the problems to address.\n\n\\subsection{Data Sources}\n\nFor an aggregator to function within a VPP, the minimum data needed is the current status of the assets. In addition to this, the aggregator needs the privileges to remotely control the assets, which is tightly coupled to data access. Status data from a single asset, such as battery charge level or heat pump temperature, becomes richer when collected over time, and can be used to model the behavior of residents. These models can estimate when assets are idle or homes are vacant, which is highly relevant information for a VPP. But the richness of this data also makes it valuable for many purposes outside the scope of a VPP, such as profiling with the purpose of targeting or manipulation.\n\nData from assets alone does not allow for much meaningful decision-making. The crucial property of VPPs is the ability to optimize for outside parameters. For commercial aggregators, this data might come from electricity markets, detailing the market prices of electricity now or in the future, allowing the aggregator to optimize for price and profit by buying power in bulk when it is cheap (e.g \\cite{vayaOptimalBiddingStrategy2015, bessaOptimizedBiddingEV2012}). Or it might come from DSOs, detailing bids for flexibility services in certain parts of the grid. For VPPs where profit is not the only goal, other relevant data sources might be the current generation of renewable energy, if the goal is to increase utilization of renewable energy, or the weather, if the goal is to stock up on energy in preparation for cold or extreme weather.\n\n\n\\subsection{Data Flows}\n\nThe data created at each source flows to the aggregator to be used in its decision engine, deciding where to increase or decrease electricity consumption. We can distinguish between two discrete types of data flows: The \\emph{core flows}, data which flows from the assets to the aggregator, and the \\emph{supplementary flows}, which is data from external sources used in the optimization algorithm of the aggregator (see \\autoref{fig:data-flows-1}).\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=\\columnwidth]{figures\/data-flows-1.pdf}\n    \\caption{Data flows to the aggregator. The upper part represents the assets in the aggregators portfolio, while the lower part represents the data sources on which the aggregator bases its decisions.}\n    \\label{fig:data-flows-1}\n  \\end{center}\n\\end{figure}\n\n\\subsubsection{Core Flows}\nThe core flows are the basis of the capabilities of the VPP, carrying the current status, and thereby potential, of the portfolio. Each of the core flows represent the relationship between an organization (the aggregator) and an individual (the asset-owner). The core flows are numerous, a one-to-many relationship, and each of the core flows are in practice interchangeable. The aggregator simply needs enough capacity, it is less important exactly which assets that capacity is comprised of. If a core flow is discontinued or unavailable, the VPP can either continue with slightly decreased capacity or replace the flow with a similar asset.\n\n\\subsubsection{Supplementary Flows}\nThe supplementary flows are the basis of the direction of the VPP, carrying the current outside parameters that are used when optimizing for system goals. The supplementary data flows have the characteristic in common that they are between two organizations. They are one-to-one relationships that are not interchangeable. Each supplementary flow contains data to be used in the optimization loop of the aggregators decision engine. If a supplementary data flow is discontinued or unavailable, the VPP might not be able to make decisions, rendering it idle.\n\nBoth the two categories of data flows, the core and supplementary flows, are necessary for the VPP to function. But due to the interchangeability, the individual core flow is of lower importance than the individual supplementary flow. \n\n\\subsection{Vulnerability and Conflicting Goals} \\label{seq:conflicting-goals}\n\nThe nature of these data flows creates an asymmetrical power relation between the aggregator and the asset-owner. The aggregator has control over the electricity cost and availability for the asset-owners and access to valuable and personal data about them. This makes issues around data governance --- how the data is used in the decision engine of the aggregator, and to what degree the personal data is analyzed and shared --- important to the asset-owners. Simultaneously, the asset-owner is in a poor position to make demands or bargain with the aggregator, as the single data flow the asset-owner represents is not critical to the function of the VPP. Each individual asset-owner has very limited leverage in such a negotiation.\n\nThis becomes an issue when the aggregator and asset-owners have conflicting goals. The workings of a system such as a VPP are a manifestation of its \\emph{function} or \\emph{purpose} \\cite{meadowsThinkingSystemsPrimer2008}, to borrow terms from systems theory, which can be seen as the high-level direction of the system. This direction is shaped by the structural properties of the system, the context it is placed in, and the \\emph{goals} of the stakeholders. Goals in this context represents properties of the \\emph{desired direction} of the system, and are unique for each stakeholder. Examples of goals for a VPP include maximizing profits, lowering prices for end-consumers, ensuring grid stability or transitioning to renewable energy. The goals of a stakeholder are often not explicitly stated and their internal compatibilities are complex. Goals $A$ and $B$ can be said to be \"conflicting\" if adjusting the direction of the system to address $A$ negatively impacts $B$.    \n\nFor example, the aggregator presumably has a goal of maximizing profits, and the asset-owners have a goal of preserving the integrity of their personal sphere. Given the issue of selling data about the behavior of asset-owners to third parties, these two goals are in conflict. However, the decision about whether to sell the data or not, is placed with the aggregator. \n\n\nIf an asset-owner wishes to alter the direction of the VPP, they are in a precarious position to do so. This is due to the asymmetry of political power and leverage, and the consequence of such a conflict: If the asset-owner stops the cooperation with an aggregator, the asset-owner loses all of the perks of the service, while the aggregator only loses a fraction of its capacity.\n\nIt is this governance question, of how the asset-owners can influence the way their data is being utilized in a system where their leverage is limited, that is addressed in this paper.\n\n\n\\section{The Unionization Frame}\n\nIn this section we present the problem in the frame of \\emph{unionization}, inspired by the consensus-seeking trade unions of Northern Europe. From this context we derive principles for data governance in the context of VPPs.\n\n\\subsection{Managing Power Asymmetries Through Unions}\n\nThe core data flows of a VPP can be seen as analogous to the relations of employment in classical capitalism, where the production is contingent on the pooling of a significant amount of labor. This labor is secured through individual contracts (or simply day-wages, meaning no contracts at all), which leaves the worker in an asymmetrical power relation to the employer similar to the relationship described in \\autoref{seq:conflicting-goals}.\n\nThe primary ways of addressing the asymmetry between worker and employer has been government regulations and the forming of \\emph{trade unions}. Trade unions mediate the flow of labor from workers to employer, and through this mediation gain leverage when negotiating wages and conditions. This is what is known as \\emph{collective bargaining}. Should the employer refuse to meet the demands of the trade union representing the workers, they can invoke a \\emph{strike}, removing the employers access to labor and thereby halting production. \n\nIn a more general sense, unions can be seen as \\emph{aggregators of power}. Whether in relation to labour, consumer rights, student conditions, or a host of other domains, the fundamental concept remains: By organizing, the interchangeability and replaceability of the individual relation is countered by collective action.\n\nThe main concepts from traditional trade unions can be used within a VPP as a way to obtain leverage and negotiating power in the relationship between aggregator and asset-owner. We propose the creation of a new institutional actor - the \\emph{data union}, which individual asset owners can join. The data union has clearly articulated and defined goals and mediates on behalf of the asset owners, controlling all (or a significant amount of) the core data flows. This allows for collective bargaining, by using the possibility of halting or limiting the core flow (a \"strike\") as leverage.\nThe structure of unionized data flows is shown in \\autoref{fig:data-flows-2}.\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=\\columnwidth]{figures\/data-flows-2.pdf}\n    \\caption{Unionized data flows to the aggregator. The core flows, the data from the assets, flows though a node of union-controlled infrastructure, allowing for leverage in negotiations with the aggregators}\n    \\label{fig:data-flows-2}\n  \\end{center}\n\\end{figure}\n\n\nThis mediation can be implemented in various ways, offering the union different degrees of power. The union could act like a traditional trade union, bargaining contract conditions on behalf on its members and, in case of a conflict, encouraging members to stop the data flow to the aggregator by turning of the connectivity of their collective assets. Since this union is interacting with an ICT system, there is also the possibility for much more direct and dynamic involvement of the union in the data flows. This could either be done by granting the union control over the assets in a way that trumps the aggregator, allowing them to dynamically start and stop the core flow. Alternatively the data flows could be rerouted to go through union-controlled infrastructure before being forwarded to the aggregator, creating a single point of control between the union and the aggregator.\n\nIf the union chooses to physically route the data through union-controlled infrastructure, this allows for a wider range of possibilities for control of data usage. One such possibility is anonymization through continuous de-identification. An aggregator needs to know which resources are available, but it does not strictly need to know which asset belongs to which asset-owner. By rehashing the identifiers of asset-owners at set intervals in time, the union can limit the aggregators ability to reason about the individual asset-owners behavioral patterns, while still allowing them access to all resource information. Controlling infrastructure also allows the union to construct a valuable dataset from which more general patterns and information can be deduced, something that has otherwise only been the privilege of the aggregator.\n\n\n\\subsection{Organizing Communities}\n\nTraditionally, trade unions have been organized around fairly coherent occupational groups, whose members had similar conditions, experiences and demands as a base for organization. However, the diversification and specialization of the workforce have coincided with a great decline in union membership and power through the 1980s and 90s \\cite{visserTradeUnionDecline2007, brysonIntroductionCausesConsequences2011, bellTradeUnionDecline1998, waddoupsTradeUnionDecline2005}. \n\nOne way to address these issues has been to organize unions around \\emph{\"place-based communities\"} other than the work\\emph{place}, which can be seen by unions advocating for change on a social and political level that goes beyond worker self-interest \\cite{holgateCommunityOrganisingUK2015}. As described in \\cite{chaseEarlyTradeUnionism2017}, early trade unions were anchored in the local community, and \\cite{levesqueLocalGlobalActivating2002} argues that a return to local community unions is \"at the heart of union renewal strategies\".\n\nThis harmonizes with the idea of a VPP data union. In contrast to many other asymmetrical relations of information, which can be highly globalized and detached from the physical spaces occupied by its users, VPPs are necessarily concentrated within a common locality, determined by the topology of the electrical grid. A VPP must have a significant amount of capacity in an area to offer services to the DSOs and compete on the local energy market. This means that the asset-owners have in common a relationship to that area, and the localized community could provide an arena to organize in.\n\nClearly, the hard borders of the grid topology will not reflect the fluid and dynamic borders of local communities, but the common relationship and frame of reference of the asset-owners, that the locality provides, can serve as the basis for the construction of a new community centered around electricity- and data-based issues. The data union would be a key part of this community.\n\n\\subsection{Enabling Productive Conflicts}\n\nConflicts between stakeholders can be seen as a steering mechanism for a system, altering its direction according to the goals of those involved. But in order for this mechanism to work, these conflicts have to be productive. And to enable productive conflicts, each side must acknowledge the power of the other. This creates an even foundation for the negotiations surrounding the conflict, taking into account the goals and interests of both sides. \n\n\nBy unionizing, the asset-owners can force the aggregator to acknowledge their power, leveraging their ability for collective action. This can be done by halting or manipulating the flows of data or electricity to and from the assets. And while these collective actions are not without costs for the asset-owners, the mere ability to disrupt the functionality of the aggregator increases leverage.\n\n\n\\section{Governance Principles}\n\nWe have presented the issue of power asymmetry and exploitation within a VPP in the frame of unionization, and now precede to describe which principles for data governance that can be derived from this frame.\n\nTo describe data governance models in general terms, we use the framework proposed by Khatri \\& Brown, \\cite{khatriDesigningDataGovernance2010}\\footnote{\n  Adapted from the IT governance model presented in \\cite{weillITGovernanceHow2004}\n} partitioning the issues of governance into five decision domains: \\emph{data principles}, \\emph{data quality}, \\emph{metadata}, \\emph{data access} and \\emph{data lifecycle}. These domains influence each other, with the data principles informing the decisions and models in the remaining four domains. This relationship is visualized in \\autoref{fig:domains}.\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=\\columnwidth]{figures\/decision-domains.pdf}\n    \\caption{The five decision domains of data governance as defined in \\cite{khatriDesigningDataGovernance2010}}\n    \\label{fig:domains}\n  \\end{center}\n\\end{figure}\n\n\\subsection{Unionization of Data Subjects}\n\nThe basis for the following data governance principles, is the constitution of a data union that mediates the goals of the asset-owners.\n\nBy creating a union the asset-owners can engage in collective bargaining with the aggregator, negotiating the terms of the service that collects and processes information about their lives. The union replaces the one-to-many aggregator-to-asset-owner relationship with a one-to-one aggregator-to-union relationship. This gives the aggregator a single point of contact with the unionized asset-owners, and constitutes a single point of leverage, allowing asset-owners to make demands of the aggregator.\n\n\n\\subsection{Data Principles and Representation}\n\nThe decision domain of \\emph{data principles} is concerned with \"clarifying the role of data as an asset\", its uses and purposes, and the communication of its uses. In other words, this is where the direction of all other decisions surrounding data governance is made. In the case of VPPs this includes whether selling or sharing data with third parties is part of the business model, whether modeling the behavior of individual assets should be allowed, which supplementary data sources the decision engine should be based on, and so on. \n\nThese principles should define a clear business owner of data assets, and a governing body should be constituted to oversee the approvement and enforcement of data usage in the context of the defined purpose and direction. One example of such a governing body is an \"Enterprise Data Committee\". A committee of this sort would be an obvious place for union representation, allowing the interests and values of the asset-owners to be heard when discussing and reviewing data usage. It would provide an organizational frame where the union can impact the direction of the VPP by leveraging its control of the core data flow.\n\n\n\\subsection{Disruptive Measures for Asserting Power}\n\nThe data governance model should include concrete methods that allow for the asset-owners to disrupt the functioning of the VPP, if the current data usage does not live up to their demands. This could be done by limiting or halting the core flow of data, the VPPs basis for operation, thereby putting pressure on the aggregator to respond to their demands.\n\nThese measures are intended as a last resort, as none of the stakeholders are interested in a purely idle VPP. But its existence is important to establish the fact that the goals and demands of the asset-owners should be taken into account, in order to avoid damaging consequences for the aggregator.\n\nThis is in line with the Danish model for collective bargaining, where measures are taken to avoid disruptive conflicts such as a strike. This tradition is based on extensive negotiations towards a reconciliation that is acceptable to all parties. Only if the parties are unable to resolve the conflict, with government-assisted mediation, a strike is allowed. Still, these negotiations are always based on the fact that a strike is possible.\n\n\n\\subsection{Accountability Needs Transparency}\n\nKhatri \\& Brown describe the \\emph{\"locus of accountability of decision making\"} in a decision domain to denote those held accountable if the handling of data does not live up to the data governance model. But before we can hold anybody accountable, we need to be able to assert that something has gone wrong. \n\nThe aggregator and asset-owners might have conflicting goals, but the aggregator is responsible for the majority of the data collection, storage, processing, and usage. This means that the aggregator does not necessarily have an interest in divulging information about actions that diverges from the data principles. Therefore the asset-owners must implement measures to ensure a high degree of transparency, enabling them to audit the actions of the aggregator. \n\nFrom this we can see that transparency and auditing procedures around the implementation of the use of the data asset is crucial when designing a data governance model for VPPs.\n\n\n\\section{Related Work}\n\nWe present a couple of selected related works that offer other perspectives on similar or related issues of data and governance. We present the notion of \"data commons\", derived from commons theory, and \"\\textsc{Shift}\", a future design project that addresses the asymmetrical relations of information within an existing trade union. \n\n\\subsection{Data Commons}\n\nDrawing on commons theory from the field of economic governance (e.g. \\cite{ostromGoverningCommonsEvolution1990}), the frame of \"data commons\" seeks to describe the accumulated data of users as a \\emph{common pool resource} (CPR). \n\nThe main issue addressed in commons theory is how to ensure sustainable governance of a CPR that is \\emph{rivalrous}, meaning that its use by one precludes its use by another, and where the locally optimal strategies for the individual agents leads to a suboptimal strategy when combined. A key example is fisheries, with the CPR being the stock of fish in the sea: Each fisher benefits more, in the short term, by catching as much as possible, but the combination of each fisher pursuing this strategy results in overfishing and eventual depletion and erosion of the CPR.\\footnote{\n  While often called the \"tragedy of the commons\", it is worth noting that more recent studies show how, in practice, it is not a preordained tragedy, but much richer, more ambiguous, what \\cite{councilDramaCommons2002} describe as a \"drama of the commons\".\n}\n\nData is unlike the traditional environmental CPRs, as it does not directly erode with overuse. Data can be copied and copied without loss. However, data and knowledge can be seen as rivalrous if value is not strictly measured as monetary worth, but instead described along three dimensions: \\emph{economic}, \\emph{sociological} and \\emph{identity} (as described in \\cite{aragonWhereCommonsMeet2011}). These dimensions become intuitively clearer if we think of sustainability as \\emph{responsibility}: Irresponsible management is not confined to actions that diminish economic value, but also actions that affect sociological structures in a negative way or is at odds with the identities of the stakeholders. The primary solution to CPR issues is the creation of a locally anchored \\emph{stewardship} role that oversees the usage of the resource.\n\nA significant amount of work has been done applying commons theory to the non-depletable resources of \\emph{knowledge} and \\emph{information} (e.g. \\cite{hessUnderstandingKnowledgeCommons2007, hessIdeasArtifactsFacilities2003}). \nIn regard to data governance and commons theory, most research has been into the field of health care data (e.g. \\cite{taylorWhatResponsibleSustainable2019,purtovaHealthDataCommon2017,evansBarbariansGateConsumerDriven2016}) and \"smart cities\" (e.g. \\cite{beckwithDataFlowSmart2019}).\n\n\\iffalse\nThe point of divergence for VPPs and the \"data commons\" frame is the context in which the data is created, leading to different notions of ownership. In the case of both smart cities and health data, the data is created by citizens\/users interacting with infrastructure that is owned by governments or corporations, leading to multiple parties claiming ownership of the data. This ambiguous notion of ownership makes the data more common and less private, leading to a data commons. The data created by the asset-owners in a VPP on the other hand, is created by interacting with self-owned infrastructure, resulting in a more clear role of owner, and more possibilities for control.\n\nOne can think of a union as a certain type of stewardship organization, overseeing the use of a resource that its members can claim ownership of and control over. The union can choose to allow certain versions of its data to become a data common, allowing open access to all, but the critical point is that this is essentially the unions decision to make.\n\\fi\n\n\n\\iffalse\n\\subsection{POTs - Protective Optimization Technologies}\n\nLooking beyond stakeholders that are a direct part of the system, a self-optimizing system, such as an aggregator, can also cause harm to individuals and organizations who inhabit the application domain of the system. As presented in \\cite{kulynychPOTsProtectiveOptimization2020}, these \\emph{negative externalities}, a term borrowed from economics, are not \n\\fi\n\n\n\\subsection{\\textsc{Shift} - A Platform Union}\n\nIn this paper, we have been looking at the unionization of data creation. The \\textsc{Shift} project \\cite{slavnovaSHIFT2017} is a future design project that takes the reverse approach. Based around OPR, the union of russian truck drivers, the \\textsc{Shift} project proposes a strategic roadmap for the union, aiming to secure the living conditions of truckers as the development of automated solutions (\"self-driving trucks\") threatens to render them without employment.\n\nTo combat this threat, \\textsc{Shift} plans to equip the trucks of their members with sensors (camera, GPS, accelerometer) and collect data from these in order to create a dataset that would be of great value when developing these automated solutions. The goal of this is to either use the dataset as leverage when bargaining with employers, or to simply circumvent the employers completely and use the dataset to develop union-owned automation, retrofitting the trucks of their members with autopilot systems, turning the union into a logistics company.\n\nWhile speculative, this project has interesting parallels to the problem discussed in this paper, the primary being that the users own the assets, in this case the trucks. This gives them freedom to create, terminate, and redirect data flows to and from those assets. The strategic roadmap also includes the consolidation of the union through a digital platform, allowing for a sense of community despite the extremely distributed nature of truck driving. A similar solution might also be beneficiary in a VPP context for asset-owners to discuss internal issues. \n\n\n\\section{Conclusion}\n\nIn this article we have presented the frame of unionization, inspired by trade unions, as a way to describe the inherent problem of asymmetric relations of information in VPPs and its consequences for data governance models. We model the flows of data within a VPP as the \\emph{core data flows} from assets to the aggregator, and the \\emph{supplementary data flows} from external sources. The core flows are necessary for the VPP to function, while the supplementary flows provide context based on which the aggregator makes decisions. \n\nFor the aggregator, the individual assets are expendable and interchangeable, as long as the capacity is sufficient, much like a classical employer-worker relationship. By forming a union, the individual asset-owners are able to leverage their combined control over the assets to influence the data flow, processing and storage within the VPP.\n\nFrom this frame we have derived a set of principles for data governance models within VPPs: \n\n\\begin{LaTeXdescription}\n\n  \\item [Collective bargaining] \n  The asset-owners should be able to bargain collectively about the conditions and purposes of the data flows. This includes which supplementary data flows to include and how to utilize them.\n  \n  \\item [Representation] \n  The asset-owners should be represented in a central organizational governing body, which is in charge of defining and overseeing the data principles.\n\n  \\item [Disruptive measures] \n  The asset-owners should be able to limit or halt core data flow, allowing them to assert power and symmetry in the relation with the aggregator.\n\n  \\item [Accountability] \n  Transparency measures should be put in place to ensure the asset-owners ability to audit the data usage performed by the aggregator, in order to detect misuse and assign accountability\n\n\\end{LaTeXdescription}\n\n\nFinally we have presented the issue in relation to data-oriented trade unions and the idea of \"data commons\", which both have similar traits to the data flows within VPPs.\n\n\n\\printbibliography\n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nTheory and experiments have been exploring possible  violations of the Lorentz symmetry \nfor decades. On the other hand,  Lorentz symmetry has been very successful in \nthe Standard Model of Particle Physics and in General Relativity. Low energy Lorentz \nviolation possibly originates from high energy physics, such as quantum gravity and \nstring theory. A well-known example is that non-commutative geometry in string theory due to \na non-zero tensor $B_{\\mu \\nu}$ on a D-brane leads to Lorentz \nsymmetry violation~\\cite{Seiberg:1999vs}.  Recently, a new Lorentz symmetry violating \ntheory of gravity, Ho\\v{r}ava-Lifshitz Gravity~\\cite{Horava:2009uw}, has been developed.\nIn this theory, gravity is power-counting renormalizable and unitarity is maintained. More\ngenerally, it is possible to detect the hint of high energy physics from low energy \neffective field theory since the high energy phyiscs leads to a modification of the \ndispersion relations of different particles~\\cite{Smolin:2003rk}. Motivated by this, \nthere have been many searches for signs of Lorentz symmetry violating phenomena in \nterrestrial, astrophysical and cosmological settings\n~\\cite{AmelinoCamelia:1997gz,Bear:2000cd,Gleiser:2001rm,Sudarsky:2002ue,Jacobson:2002ye}, and the constraints on the magnitude of such phenomena are becoming better \nand better. \n\nThe fact that different particles have different limiting speeds will open up the possibility of \nCerenkov radiation for kinetic reasons, which is forbidden if Lorentz symmetry is kept. \nThe most stringent constraint is from the highest energy cosmic rays. If Lorentz symmetry \nis broken, then there is no reason to keep the limiting speed the same in the dispersion \nrelation of different particles. That the high energy cosmic rays which are possibly composed \nof hadrons, travelling astrophysical distances and times with  energy of about \n$3 \\times 10^{11} GeV$ gives the following constraint on the limiting speeds of protons $c_p$\nand photons $c_{\\gamma}$ ~\\cite{Coleman:1998ti}:\n\\begin{equation} \\label{problem}\n\\frac{c_p-c_\\gamma}{c_\\gamma} < 10^{-23} \\, .\n\\end{equation}\nEven if the two limiting speeds are the same at tree level, if there is no specific symmetry it is \nhard to keep them the same at all orders without fine-tuning.  Moreover, dimension three \noperators which break Lorentz invariance are also highly constrained~\\cite{Colladay:1996iz}.\nThe requirement is that their coupling constants should be much less than the Lorentz violation scale from simple dimensional counting.\n\n\nHere we introduce supersymmetry as the candidate to solve the naturalness problem \nmentioned above for theories with Lorentz symmetry violation. Supersymmetry is an \nextended symmetry of spactime. In general, the algebra of supersymmetry is\nbuilt on the Poincare algebra. But here we are considering theories with Lorentz \nsymmetry violation and thus the initial symmetry of space-time is not the Poincare \nsymmetry. In the context of Ho\\v{r}ava-Lifshitz gravity the initial symmetry is\n$SO(3)$, the group of spatial rotations. Here, we will construct a new supersymmetry \nalgebra starting from $SO(3)$ rather than the Lorentz group $SO(3,1)$. \n\nIn a supersymmetric theory the bosons and fermions which are in the same multiplet\nwill by symmetry have the same dispersion relation.  This will help us solve the naturalness\nproblem (\\ref{problem}). Therefore, if all the particle are in the same multiplet, such as in \n$N=8$ supergravity or $N=4$ super-Yang-Mills theory, the naturalness problem \n(\\ref{problem}) should be completely absent. If not all particles are in the same multiplet, \nfor example in the Minimal Supersymmetric Standard Model (MSSM), a gauge symmetry \nin combination with supersymmetry can keep the limiting speed of all  particles the same. \n\nIn \\cite{GrootNibbelink:2004za,Bolokhov:2005cj,Berger:2001rm}, the authors studied Lorentz violation in supersymmetric models in which the superalgebra itself did not violate the Lorentz \nsymmetry, which is motivated from the low energy effective field theory point of view. \nAt high energy, the superalgebra needs to be changed because the spacetime symmetry\nexplicitly breaks Lorentz invariance. To study this problem, we need to\nwork in the context of a specific theory with Lorentz violation at high energy, which \nis the reason that we here use Ho\\v{r}ava-Lifshitz theory to construct supersymmetry. \nThis theory is complete in the ultraviolet (UV), and in the infrared (IR) the Lorentz symmetry \nis emergent. The supersymmetry constructed here can be extended to other Lorentz \nviolating theories.  \n\nWe shall start with the spacetime symmetry and scaling symmetry of the Lorentz violating \nLifshitz theory in Sec.~\\ref{symmetry}. We derive the supersymmetry generators and their\ncommutators in Sec.~\\ref{lagrangian} from the free field Lagrangian. Then we will use\nsuperspace language to construct the susy langrangian and interactions in \nSec.~\\ref{superspace}. So far, the discussion does not contain gauge fields. By \nintroducing a gauge symmetry in Sec.~\\ref{gauge}, we can ensure that different particles in\ndifferent multiplets also have the same limiting speed, which solve the naturalness problem. \nIn Sec.~\\ref{break}, we will discuss the relationship between susy breaking and Lorentz \nviolation, and we will give the conclusion in the  last section.\n\n\n\\section{$SO(3)$ $\\times$ Scaling Symmetry}\n\\label{symmetry}\n\nLifshitz theory explicitly breaks Lorentz invariance, and the remaining\nsymmetry is  $SO(3) \\times$ Scaling Symmetry, spatial rotations\nand the anisotropic scaling between space and time introduced in\n\\cite{Horava:2009uw}. With these symmetries, there \nis no reason that the limiting velocities of all species be the same.\nDemanding agreement with the cosmic ray experiments would require\na fine-tuning of the parameters in the model to unbelievable accuracy. \nIn order to avoid having to do this fine-tuning, we try to extend the $SO(3)$ \nsymmetry by adding supersymmetry.\n\nThe Lorentz symmetry $ISO(3,1)$ algebra is composed of commutators of the translation generator $P_\\mu$ and the spactime rotation generator $M_{\\mu \\nu}$,\n\\begin{equation}\n[P_\\mu,P_\\nu]=0\n\\end{equation}\n\\begin{equation}\n[M_{\\mu\\nu} , P_\\lambda]=i \\left(   \\eta_{\\nu \\lambda} P_\\mu -\\eta_{\\mu \\lambda} P_{\\nu}   \\right)\n\\end{equation}\n\\begin{equation}\n[M_{\\mu\\nu}, M_{\\lambda \\sigma}]=\n i\\left( \\eta_{\\nu\\lambda}M_{\\mu \\sigma} +\\eta_{\\mu \\sigma}M_{\\nu\\lambda} -\n \\eta_{\\mu \\lambda}M_{\\nu \\sigma}-\\eta_{\\nu \\sigma}M_{\\mu \\lambda}\\right) \\ .\n\\end{equation}\nTo introduce supersymmetry, we add two-component Weyl spinor generator \n$Q_\\alpha$ and $\\bar{Q}_{\\dot\\alpha}$ to compose \nthe extended algebra,\n\\begin{equation}\n[M_{\\mu \\nu}, Q_{\\alpha}]= -i\\left(\\sigma_{\\mu \\nu}\\right)_\\alpha^{\\ \\beta} Q_\\beta\n\\end{equation} \n\\begin{equation}\n[M_{\\mu \\nu}, \\bar{Q}^{\\dot\\alpha}]= -i\\left(\\bar\\sigma_{\\mu \\nu}\\right)^{\\dot\\alpha}_{\\ \\dot\\beta} \\bar Q^{\\dot\\beta}\n\\end{equation} \n\\begin{equation}\n[P_\\mu,Q_{\\alpha}]=[P_\\mu,\\bar{Q}_{\\dot\\alpha}]=0\n\\end{equation}\n\\begin{equation}\n\\left\\{ Q_{\\alpha}, Q_{\\beta}  \\right\\} =\\left\\{ \\bar{Q}_{\\dot\\alpha},\\bar{Q}_{\\dot\\beta}  \\right\\}=0\n\\end{equation}\n\\begin{equation}\n\\left\\{ Q_{\\alpha},\\bar{Q}_{\\dot\\beta} \\right\\}=2 \\sigma^\\mu_{\\alpha \\dot \\beta} P_{\\mu} \\ . \\label{susyalgebra}\n\\end{equation}\n\nIn our case we do not have boost invariance. Hence, we set the boost generator \n\\begin{equation} \nM_{0i}=0  \\ ,\n\\end{equation} \nwhich reduces the spacetime symmetry  to $SO(3)$. The symmetry group $SO(3)$ is \nequivalent to $SU(2)$, while  $SO(3,1)$ is equivalent to $SU(2) \\times SU(2)$. \nTo obtain a supersymmetric generator algebra in the case of  SO(3) the two spinors can \nbe identified as\n\\begin{equation}\nQ_{\\alpha}^{*} = \\bar Q_{\\dot \\alpha} \\ .\n\\end{equation}\n\nThe other symmetry that we mentioned in Lifshitz theory is scaling symmetry, which posits \nthat the Lagrangian is invariant under the following anisotropic scaling of space and time: \n\\begin{equation}\n\\textbf{x} \\rightarrow e^{-\\Omega} \\textbf{x} \\ , \\ \\ \\ \\ \\ \\ t \\rightarrow  e^{- z \\Omega} t \\ ,\n\\end{equation}\nwhere z is taken to be an integer chosen as $z=3$ to obtain a theory of gravity in 4 spacetime dimension.\nIf we consider the momentum conjugate of the spacetime coordinate, the relationship between energy and momentum is as follows,\n\\begin{equation}\n(\\Delta p)^{z} \\sim \\Delta E \\ ,\n\\end{equation}\nwhere $p$ is the spatial momentum. This relationship will be reflected in the dispersion relation.\n\nIn the following section we will discuss how the scaling \nsymmetry impacts on the supersymmetry from which we can derive the right form of the spinor generator and of the \nrenormalizable supersymmetric Lagrangian. It will modify the susy algebra (\\ref{susyalgebra}). \n\n\\section{Supersymmetric Lagrangian}\n\\label{lagrangian}\n\nIn this section, we will from the most basic boson and fermion Lagrangian derive the \nexplicit form of the spinor generator. \nThe simplest free boson Lagrangian with two scalar fields is \n\\begin{equation}\n\\mathcal{L}_s= \\partial_0 \\phi^* \\partial_0 \\phi - \\partial_i\\partial^2\\phi^*  \\partial_i\\partial^2\\phi \\ .\n\\end{equation} \nAs a first step, we neglected other kinetic terms which are super-renormalizable in the \n$z=3$ Lifshitz theory. These additional terms are important when discussing  \nlow-energy phenomenology. However, just from the simplified case taken above, \nwe can clearly see the physics. Given the above bosonic Lagrangian, the\nLagrangian for the two Weyl fermions which are the supersymmetric partners\nof the bosons is as follows\n\\begin{equation}\n\\mathcal{L}_f= i \\psi^{\\dagger} \\sigma^0 \\partial_0 \\psi +i\\psi^{\\dagger} \\bar\\sigma^i \\partial_i \\partial^2 \\psi \\ ,\n\\end{equation}\nwhere $\\sigma^0$ is the identity matrix and $\\bar\\sigma^i$ is conjugate of $\\sigma^i$ \n \nThe susy transformation should change $\\phi$ to $\\psi$,\n\\begin{equation}\n\\delta \\phi = \\epsilon \\psi \n\\end{equation}\n\\begin{equation}\n\\delta \\phi^*= \\epsilon^{\\dagger} \\psi^\\dagger \\ ,\n\\end{equation}\nwhile the transformation from fermions to bosons can be derived from \nthe requirement that the symmetry is obeyed by the \naction. The result is\n\\begin{equation}\n\\delta \\psi =-i \\epsilon^\\dagger \\partial_0 \\phi -i \\sigma^i  \\epsilon^\\dagger \\partial_i \\partial^2 \\phi\n\\end{equation} \n\\begin{equation}\n\\delta \\psi^\\dagger =i \\epsilon \\partial_0 \\phi^* +i \\epsilon\\sigma^i   \\partial_i \\partial^2 \\phi^* \\ .\n\\end{equation} \nThe whole action is invariant under these transformation up to some total derivatives. \nThe commutator of the supersymmetric spinor generators \nis derived from the commutator $\\delta_1\\delta_2 -\\delta_2\\delta_1$.\n\\begin{equation}\n\\left\\{ Q_{\\alpha},\\bar{Q}_{\\dot\\beta} \\right\\}= 2 \\sigma^0 P_0 +2\\sigma^i P^2 P_i  \\ ,\n\\end{equation}\nwhere $P^2 = P^i P_i$.\n\nIf supersymmetry is not broken, it is easy to see that the speed of light of the two \nsuperpartners are the same, because the supersymmetric transformation \nfixes the coefficients in front of $p^3$ for fermions and of $p^6$ for bosons. \n\n\n\\section{Superspace and Superderivatives}\n\\label{superspace}\n\nIn this section, we will derive the superspace formulation of the modified supersymmetry proposed in the previous section. Because the dimension of the superfield is zero in the $z=3$ Lifshitz theory, the D-term and F-term in the theory are different from the case of regular supersymmetry. In the end, we will give the interaction Lagrangian which includes some new interactions between bosons and fermions.\n\nConstructing supersymmetry can proceed by two methods, one is by direct construction \nas above, and the other is from the superspace formalism making use of general superfields.\nIn the following we illustrate this superfield method,  a method which makes it easy to see why \nthe ``speed of light\" is the same for different species, and which also makes it easy to \ndeal with interactions. Below, $\\theta$ denotes the superspace variable. \n\nA superfield transforms as follows under the action of the extended spacetime symmetry,\n\\begin{equation}\n S(x^\\mu, \\theta, \\theta^\\dagger ) \\rightarrow \\exp (i(\\xi Q + \\xi^\\dagger Q^{\\dagger} -a^{\\mu}P_\\mu)) S(x^\\mu, \\theta, \\theta^\\dagger ) \n\\end{equation}\nwhere the generators are \n\\begin{equation}\n P_\\mu=i\\partial_\\mu\n\\end{equation}\n\n\\begin{equation}\n i Q_\\alpha = \\frac{\\partial}{\\partial \\theta^\\alpha} - i\\hat{\\sigma}^\\mu_{\\alpha \\beta} \\theta^{\\dagger \\beta} \\partial_\\mu\n\\end{equation}\n\n\\begin{equation}\n i Q^\\dagger_\\beta = \\frac{\\partial}{\\partial \\theta^{\\dagger\\alpha}} - i\\theta^{ \\alpha}  \\hat\\sigma^\\mu_{\\alpha \\beta}  \\partial_\\mu \\ .\n\\end{equation}\n$\\hat\\sigma$ is a newly defined sigma matrix in the Lifshitz theory,\n\\begin{equation}\n \\hat\\sigma^0 = \\sigma^0 \n\\end{equation}\n\\begin{equation}\n \\hat\\sigma^i = \\sigma^i f(P) \\ .\n\\end{equation}\nThe information about the scaling symmetry of Lifshitz theory is hiding in the redefined sigma matrix $\\hat\\sigma^\\mu$. After the redefinition, the susy algebra and superfield are written in the same form as the regular supersymmetry.\n$f(P)$ is a function of the magnitude of the spatial momentum. In the $z=2$ case, the \nspinor generator algebra is simpler, $\\{Q_\\alpha, \\bar {Q}_\\beta\\}= 2 P^2$. In Sec.~\\ref{lagrangian}, we considered the marginal kinetic operator, $f(p)=P^2$. \nIncluding some super-renormalizable kinetic operators, $f(P)=1+P^2$, and in the low energy limit, the supersymmetry algebra \nreturns to the algebra with Lorentz symmetry. \n\n\nFrom the spinor generator algebra,\n\\begin{equation}\n \\left\\{ Q_{\\alpha},{Q}^\\dagger_{\\beta} \\right\\}= 2 \\hat\\sigma^\\mu P_\\mu \\ ,\n\\end{equation}\nwe can derive  the fermionic derivatives $D_\\alpha$ and  $D^\\dagger_\\beta$, which anticommute \nwith the supersymmetric generator, and are written as follows\n\\begin{equation}\n D_\\alpha= \\frac{\\partial}{\\partial \\theta^\\alpha} +i\\hat{\\sigma}^\\mu_{\\alpha \\beta} \\theta^{\\dagger \\beta} \\partial_\\mu\n\\end{equation}\n\\begin{equation}\n D^\\dagger_\\beta = -\\frac{\\partial}{\\partial \\theta^{\\dagger\\alpha}} - i\\theta^{ \\alpha}  \\hat\\sigma^\\mu_{\\alpha \\beta}  \\partial_\\mu \\ .\n\\end{equation}\n\nApplying the superderivatives as a constraint, the chiral superfield $\\Phi$ can be defined, which\nsatisfies\n\\begin{equation}\n D^\\dagger_\\beta \\Phi=0 \\ .\n\\end{equation}\nAlmost all the formulas are the same as in the case of regular susy if we use the \nredefined $\\hat\\sigma$. In particular, the superfield can be expanded in superspace as\n\\begin{eqnarray}\n \\Phi(x^\\mu,\\theta,\\theta^\\dagger) &=& \\phi +\\sqrt{2} \\theta \\psi +\\theta \\theta F + i \\theta \\hat\\sigma^\\mu \\theta^\\dagger \\partial_\\mu \\phi + \\frac{i}{\\sqrt{2}}\n\\theta \\theta \\theta^\\dagger \\hat{\\bar \\sigma}^\\mu\\partial_\\mu \\psi   \\nonumber\\\\\n&& -\\frac{1}{4} (\\partial_0\\partial^0 + \\partial_i\\partial^i P^4) \\phi \\theta\\theta \\theta^\\dagger \\theta^\\dagger \\ ,\n\\end{eqnarray}\nwhere\n\\begin{equation}\n \\partial_0\\partial^0 + \\partial_i\\partial^i P^4 \\equiv \\hat \\partial_\\mu \\hat \\partial^\\mu\n\\end{equation}\nand where $F$ is the auxiliary field.\n\n\nThe dimension of all the operators in the superfield can be derived by power counting,\n\\begin{center}\n\\begin{tabular*}{0.5\\textwidth}{@{\\extracolsep{\\fill}} c|cccc}\n  &  $\\phi$ and $\\Phi$    &   $\\psi$  &  $\\theta$   & $F$  \\\\\n\t\\hline\ndim  & $\\frac{3-z}{2}$ & $\\frac{3}{2}$ &   $-\\frac{z}{2}$& $z$  \\\\\n\\end{tabular*}\n\\end{center}\nThe kinetic term comes from the D term of $\\Phi^\\dagger \\Phi$\n\\footnote{Please note the different partial derivative operators for fermions and bosons \nin the formula, one of which has a hat, while the other one does not have a hat.  }\n\\begin{eqnarray}\n \\left[ \\Phi^{\\dagger}_i \\Phi_j \\right]_D &=& \\left[ \\Phi^{\\dagger}_i \\Phi_j \\right]_{coeff. of \\theta^\\dagger \\theta^\\dagger \\theta \\theta}\\nonumber\\\\\n &=& F_i^\\dagger F_j +\\frac{1}{2} \\hat\\partial_\\mu \\phi^\\dagger_i \\hat\\partial^\\mu\\phi_j - \\frac{1}{4}  \\phi^\\dagger_i \\hat\\partial_\\mu \\hat\\partial^\\mu\\phi_j\n- \\frac{1}{4} \\hat\\partial_\\mu \\hat\\partial^\\mu \\phi^\\dagger_i \\phi_j \\nonumber\\\\\n&&+ \\frac{i}{2} \\psi_i^\\dagger \\hat{\\bar \\sigma}^\\mu \\partial_\\mu \\psi_j +\\frac{i}{2} \\partial_\\mu\\psi_i^\\dagger \\hat{\\bar \\sigma}^\\mu  \\psi_j  \\ .\n\\end{eqnarray}\n\nThere are new terms in the Lagrangian of the form,\n\\begin{equation}\n\\mathcal{L}_{NEW}=   [(\\Phi^\\dagger \\Phi)^n ]_D \\ ,\n\\end{equation} \nwhere $n$ is an integer, and in the limit of low energies, the smaller $n$ terms \ndominate, since the scale of Lorentz violation suppresses these terms by a factor \n$(\\frac{\\phi}{M})^n$. And there are other possible terms $[\\Phi^\\dagger \\Phi \\Phi^n]_D + h.c.$ in the lagrangian which are forbidden by $U(1)$ or other symmetries.\n\n\nThe superpotential is constructed by holomorphic function of superfield,\n\\begin{equation}\nW(\\Phi)=\\sum_{i=1}^{\\infty} g_n \\Phi^n\n\\end{equation}\nand yields the interaction Lagrangian,\n\\begin{equation}\n\\mathcal{L}_{int}=\\int \\d^4 x \\left[ W \\right]_F \\ .\n\\end{equation}\n\\begin{equation}\n\\mathcal{L}_{int} = - \\left(\\frac{\\partial W(\\phi)}{\\partial \\phi_i}\\right) \\cdot  F - \\frac{1}{2} \\frac{\\partial W(\\phi)}\n{\\partial \\phi_i \\phi_j} \\psi_i \\psi_j + h.c. \\, .\n\\end{equation}\nThe first term includes $\\left(\\frac{\\partial W(\\phi)}{\\partial \\phi_i}\\right)^2$, since \nthere is a $F^2$ term from the kinetical D-term. However, in Lifshitz theory, we introduce \nmany new D-terms which contain other terms depending on $F$, for example, the term \n$[\\lambda(\\Phi^\\dagger \\Phi)^2]_D$. There are no four fermion terms from the F-term due to the fact that $\\theta^4=0$, \nbut the newly introduced D term will give a four-fermion interaction. We write down the terms having F and four fermion interaction,\n\\begin{equation}\n\\left[ \\lambda (\\Phi^\\dagger \\Phi)^2 \\right]_D  \\supset \\lambda (2 \\phi^\\dagger \\phi F^\\dagger F  - 2 F^\\dagger \\psi \\psi \\phi ^\\dagger-2 F \\bar \\psi \\bar \\psi \\phi + \\lambda \\bar \\psi \\bar \\psi \\psi \\psi)\n\\end{equation}\nAfter some calculation, we obtain the interaction terms written without the auxiliary field $F$,\n\\begin{equation}\n\\mathcal{L}_{int}=\\frac{ - \\left( \\frac{\\partial W}{ \\partial \\phi} - 2\\lambda \\phi \\bar \\psi \\bar \\psi \\right) \\left(\\frac{\\partial W^\\dagger}{ \\partial \\phi^\\dagger} - 2\\lambda \\phi^\\dagger  \\psi  \\psi   \\right)}{1+2\\lambda \\phi^\\dagger \\phi} + \\lambda \\bar \\psi \\bar \\psi \\psi \\psi - (\\frac{1}{2} \\frac{\\partial W(\\phi)}\n{\\partial \\phi_i \\phi_j} \\psi_i \\psi_j+ h.c)\n\\end{equation}\nThis potential could be from usual scalar field vacuum expectation value, and also \npossibly from fermion condensation. Also, from the interaction Lagrangian, there \nare new phenomena which follow from the low energy effective field theory and which depend \non the form of the superpotential. \n\n\n\n\\section{Gauge field and Gaugino}\n\\label{gauge}\n\nBosons and fermions in the same supermultiplet have the same limiting speed because of\nsupersymmetry. It is essential to see whether we can arrange for particles in different \nmultiplets to have identical limiting speed by introducing a gauge symmetry and relating\nparticles in different supermultiplets by gauge transformations. If two supermultiplets \nare totally decoupled, then there is no reason that keep the coefficients of the kinetic \nterms the same. \n\nFirst of all, consider the kinetic term of a chiral superfield which is derived from \n$\\mathcal{L}_{kin}= \\left[ \\Phi^\\dagger \\Phi \\right]_D$. The supersymmetry \ntransformation $Q$ fixes the form of $D_\\alpha$, and then the form of the chiral \nsupermultiplets is the same across all multiplets, and therefore the limiting speed of the two chiral supermultiplets are the same. We will see the coefficients of \nvector supermultiplets are also the same as that of the chiral supermultiplet since \nthe gauge symmetry relates the two types of supermultiplets. \n  \nA general vector supermultiplet can be written as \n\\begin{eqnarray}\n V(x,\\theta,\\theta^\\dagger) &=& C +i \\theta \\chi -i \\theta^\\dagger \\chi^\\dagger +\\frac{1}{2} i \\theta \\theta \\left[ M + i N  \\right] -\\frac{1}{2} i \\theta^\\dagger \\theta^\\dagger [M-iN] \\nonumber\\\\\n &&\\theta \\tilde \\sigma_1^\\mu \\theta^\\dagger V_\\mu +i\\theta \\theta \\theta^\\dagger\\left[ \\lambda^\\dagger +\\frac{i}{2} \\tilde{\\bar\\sigma}_2^\\mu \\partial_\\mu \n  \\chi \\right] -i \\theta^\\dagger\\theta^\\dagger \\theta \\left[ \\lambda + \\frac{i}{2} \\tilde{\\sigma}_2^\\mu \\partial_\\mu \n  \\chi \\right] \\nonumber\\\\\n&& +\\frac{1}{2}\\theta\\theta \\theta^\\dagger\\theta^\\dagger \\left[ D-\\frac{1}{2} \\hat\\partial_\\mu \\hat\\partial^\\mu C \\right] \\ ,\n\\end{eqnarray}\nwhere $\\tilde{\\sigma}_{1 or 2}$ is some newly defined sigma matrix  which only obeys the $SO(3)$ \nspatial symmetry. And $C$, $M$, $N$, $D$ are scalar fields, $\\chi$, $\\lambda$ are Weyl spinor, and $V_\\mu$ is a vector field. \nA gauge transformation leads to the following change in the form of a superfield: \n\\begin{equation}\n V(x,\\theta,\\theta^\\dagger) \\rightarrow  V(x,\\theta,\\theta^\\dagger) + i(\\Phi(x,\\theta, \\theta^\\dagger)-\\Phi^\\dagger(x,\\theta,\\theta^\\dagger) ) \\ ,\n\\end{equation}\nwhere \n\\begin{eqnarray}\n \\Phi(x^\\mu,\\theta,\\theta^\\dagger)-\\Phi^\\dagger(x,\\theta,\\theta^\\dagger)  &=& (\\phi-\\phi^\\dagger) +\\sqrt{2} (\\theta \\psi- \\theta^\\dagger \\bar \\psi) +\\theta \\theta( F -F^\\dagger)+ i  \\theta \\hat\\sigma^\\mu \\theta^\\dagger \\partial_\\mu (\\phi-\\phi^\\dagger)  \\nonumber\\\\&& + \\frac{i}{\\sqrt{2}}\n\\theta \\theta \\theta^\\dagger \\hat {\\bar \\sigma}^\\mu \\partial_\\mu \\psi -\\frac{i}{\\sqrt{2}}\n\\theta^\\dagger \\theta^\\dagger \\theta \\hat { \\sigma}^\\mu \\partial_\\mu \\bar \\psi \n -\\frac{1}{4} \\theta\\theta \\theta^\\dagger \\theta^\\dagger \\hat \\partial_\\mu \\hat \\partial^\\mu (\\phi-\\phi^\\dagger) \n\\end{eqnarray}\nThe way to construct a gauge symmetry is to introduce a chiral superfield, by which \nthe sigma term is fixed, $\\tilde{\\sigma} =\\hat{\\sigma}$. \nTherefore, if the gauge field is coupled to the chiral supermultiplet, the coefficients \nof the kinetic terms are the same for all particles. \n\n\n\n\nThe component $V_\\mu$ in the vector supermultiplet is a regular gauge field, and the field strength is derived from the field strength superfield $W_\\alpha = D^{\\dagger2}D_\\alpha V$ ,\n\\begin{equation}\nW_\\alpha (y,\\theta) =i \\lambda_\\alpha - \\left[  \\delta_\\alpha^\\beta D(y) +\\frac{1}{2}i(\\hat \\sigma^\\mu \\hat {\\bar\\sigma}^\\nu)_\\alpha^\\beta V_{\\mu\\nu} (y) \\right]\\theta_\\beta\n+ \\theta^2 \\hat\\sigma^\\mu_{\\alpha\\beta} \\partial \\lambda^{\\dagger \\beta}\n\\end{equation}\nwhere $V_{\\mu\\nu}$ is the field strength\n\\begin{equation} \nV_{\\mu\\nu} = \\partial_{\\mu} V_\\nu - \\partial_\\nu V_\\mu \\ .\n\\end{equation}\nThe Lagrangian of the Super-Yang-Mills field is the F-term of $W_\\alpha W^\\alpha$,\n\\begin{eqnarray}\n  \\mathcal L &=& \\frac{1}{4} \\left[ W_\\alpha W^{\\alpha} \\right]_F +h.c. \\nonumber\\\\\n  &=& -\\frac{1}{4}\\hat V ^{\\mu\\nu}\\hat V_{\\mu\\nu} +i\\lambda \\hat \\sigma^\\mu \\partial_\\mu \\lambda^\\dagger -\\frac{1}{4} \\hat V ^{\\mu\\nu} \\left( * \\hat V_{\\mu\\nu}\\right)  \\ ,\n\\end{eqnarray}\nwhere the redefined field strength is \n\\begin{equation} \n\\hat V_{\\mu\\nu} = \\hat \\partial_{\\mu} V_\\nu - \\hat \\partial_\\nu V_\\mu \\ ,\n\\end{equation}\nand all the derivative terms in the chiral field change from being normal partial \nderivatives to gauge covariant derivatives, but note that $\\hat \\sigma^{\\mu}$ does not change. \nThe covariant kinetic term of chiral superfield is as follows, \n\\begin{equation}\n\\int d^4\\theta \\Phi^\\dagger \\e {gV}\\Phi \\ .\n\\end{equation}\n\nThe limiting speed of the gauge field is same as that of the gaugino, and also \nthe same as that of a chiral field. \n\nNote that our construction of the gauge theory is different from the suggestion of \n~\\cite{Chen:2009ka}. In order to make the Lagrangian gauge invariant, it follows\nfrom the gauge transformation of a derivative term of a matter field that we need \nintroduce the gauge field with the infinitesimal transformation rule,\n\\begin{equation}\n\\delta A_\\mu = \\hat \\partial_\\mu  \\epsilon + i f \\epsilon A_\\mu \\ ,\n\\end{equation}\nwhere $f$ is the structure constant. In the Abelian gauge field case, $f=0$.\n\n\\section{SUSY Breaking}\n\\label{break}\n\nThe $N=0$ Lagrangian contains terms such as $\\partial_i^2 \\phi \\phi^4$, which is \nrenormalizable in the $z=3$ case, so the loop corrections will make the speed of light \nrun with the scale. In the supersymmetric Lifshitz theory, there is the same interaction \nterm containing derivatives in this way. This will lead to a quantum correction of \nthe speed of light, but the supersymmetry will cancel the loop effect between bosons and \nfermions. Hence, this renders the theory free of the need of fine-tuning the speed of \nlight of different particles. \n\nIf SUSY is broken, there is a soft supersymmetry breaking term introduced which make \nthe speed of light of particles different. One way to keep the difference small or absent is \nto break supersymmetry at a low energy scale (Lorentz symmetry is emergent which is equivalent to $z=1$ Lifshitz theory), and then the correction to the speed of light is  \nnot a soft term any more, and cannot yield different corrections to fermions and bosons \nfrom the Hidden Sector. \nOn the other hand, if supersymmetry breaks around or above the scale of Lorentz symmetry\nviolation, the speed of light becomes related to a soft term, which makes it suffer \nfrom the fine-tuning problem to obtain agreement with the results from the cosmic ray \nexperiments. Therefore, we obtain the constraint that the susy breaking scale \nshould be smaller than the Lorentz violation scale.\n\n\n\\section{Discussion}\n\\label{dis}\n\nIn this paper, we have constructed an $N=1$ supersymmetric theory with scalar fields, \ngauge fields and fermion fields from a UV complete theory, the $z=3$ Lifshitz theory. \nThe different particles have the same kinetic Lagrangian and hence the same limiting \nspeed, which can solve the naturalness problem in theories with Lorentz symmetry violation. \n\nOur starting point is a supersymmetry algebra which is different from what is presented in previous papers \non supersymmetric Lorentz violating theory~\\cite{GrootNibbelink:2004za,Belich:2003fa,Berger:2001rm}. \nHowever, in the low energy limit, the supersymmetry algebra will be the same. Concerning \nthe dimension three Lorentz violating operators, they do not arise in the Lifshitz theory, \nand thus this is also true in the supersymmetric case.\n\nSince supersymmetry will make the loop calculations much different from the \n$N=0$ field theory case, it is important to calculate the renormalization group and \nfind  the fixed point of the Lifshitz theory in the supersymmetric framework. \nIn the original paper on Ho\\v{r}ava-Lifshitz theory~\\cite{Horava:2009uw}, \na power-counting renormalizable theory of gravity, the author pointed that in the UV, and in \nthe $z=3$ case, there is a free-field fixed point. In \\cite{Dhar:2009dx,Dhar:2009am}, \nthe authors studied the $z=3$ Lifshitz theory with fermions and gauge fields and \ncalculated its renormalization group. And in \\cite{Iengo:2009ix}, the authors studied the one-loop renormalization of scalar field in Lifshitz theory.\nBy introducing supersymmetry, there should \nbe some changes in the renormalization group equations and in the conclusions\nconcerning fixed points.\n\n\n\\section*{Acknowledgements}\nWe would like to thank \nRobert Brandenberger, Alessandro Cerioni, Keshav Dasgupta, Maxim Pospelov,Oriol Pujolas\nfor helpful discussions. \n\n\n\n\n\n\n\n\n\n\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe $L_{\\omega_1\\omega}$-formulas are infinitary formulas in which the disjunctions and conjunctions are over countable sets, and the strings of quantifiers are finite.  We consider $L_{\\omega_1\\omega}$-formulas with only finitely many free variables.  There is no prenex normal form for $L_{\\omega_1\\omega}$-formulas.  We cannot, in general, bring the quantifiers to the front.  However, we can bring the negations inside, and this gives a kind of normal form.  Formulas in this normal form are classified as $\\Sigma_\\alpha$ or $\\Pi_\\alpha$ for countable ordinals $\\alpha$.  \n\n\\begin{enumerate}\n\n\\item  A formula $\\varphi(\\bar{x})$ is $\\Sigma_0$ and $\\Pi_0$ if it is finitary quantifier-free.\n\n\\item  Let $\\alpha > 0$.  \n\n\\begin{enumerate}\n\n\\item  $\\varphi(\\bar{x})$ is $\\Sigma_\\alpha$ if it is a countable disjunction of formulas $(\\exists\\bar{u})\\psi(\\bar{x},\\bar{u})$, where $\\psi$ is $\\Pi_\\beta$ for some $\\beta < \\alpha$.\n\n\\item  $\\varphi(\\bar{x})$ is $\\Pi_\\alpha$ if it is a countable conjunction of formulas $(\\forall\\bar{u})\\psi(\\bar{x},\\bar{u})$, where $\\psi$ is $\\Sigma_\\beta$ for some $\\beta < \\alpha$. \n\n\\end{enumerate}\n\n\\end{enumerate}\n\nWe use special notation for some further classes of formulas.\n\n\\begin{enumerate}\n\n\\item  A formula is $\\Sigma_{<\\alpha}$ (resp. $\\Pi_{<\\alpha}$) if it is $\\Sigma_\\beta$ (resp. $\\Pi_\\beta$) for some $\\beta <\\alpha$.\n\n\\item  A formula is $d$-$\\Sigma_\\alpha$ if it is the conjunction of a formula that is $\\Sigma_\\alpha$ and one that is $\\Pi_\\alpha$.\n\n\\end{enumerate}\n\n\\noindent\n\\textbf{Negations}.  For a $\\Sigma_\\alpha$ (resp. $\\Pi_\\alpha$) formula $\\varphi$, in normal form, we write $neg(\\varphi)$ for the natural $\\Pi_\\alpha$ (resp. $\\Sigma_\\alpha$) formula in normal form that is logically equivalent to the negation of $\\varphi$.   \n\n\\bigskip\n\\noindent\n\\textbf{Computable infinitary formulas}.  Roughly speaking, the \\emph{computable infinitary formulas} are formulas of $L_{\\omega_1\\omega}$ in which the infinite disjunctions and conjunctions are over c.e.\\ sets.  For more about computable infinitary formulas, see \\cite{AK}.  We classify the computable infinitary formulas as computable $\\Sigma_\\alpha$ or computable $\\Pi_\\alpha$ for computable ordinals $\\alpha$. We may refer to computable $\\Sigma_{<\\alpha}$ formulas, or computable $d$-$\\Sigma_\\alpha$ formulas.\n\n\\bigskip\n\nScott \\cite{Scott} proved the following.\n\n\\begin{thm} [Scott Isomorphism Theorem]\n\nLet $\\mathcal{A}$ be a countable structure for a countable language $L$.  Then there is a sentence of $L_{\\omega_1\\omega}$ whose countable models are just the isomorphic copies of $\\mathcal{A}$.  \n\n\\end{thm}\n\nA sentence with the property above is called a \\emph{Scott sentence} for $\\mathcal{A}$.  The complexity of an optimal Scott sentence for a structure $\\mathcal{A}$ measures the internal complexity of $\\mathcal{A}$.   \n\n\\bigskip\n\nFor a countable language $L$, let $C$ be a countably infinite set of new constants.  Identifying the constants with natural numbers, we suppose that $C = \\omega$.  Let $Mod(L)$ be the class of $L$-structures with universe $\\omega$, and for a computable infinitary sentence $\\psi$ in the language $L \\cup C$, let $Mod(\\psi)$ be the class of $L$-structures with universe $\\omega$ that satisfy $\\psi$.  There is a natural topology on $Mod(L)$, generated by basic open (actually clopen) neighborhoods of the form $Mod(\\varphi)$, for $\\varphi$ a finitary quantifier-free sentence in the language $L\\cup C$.  We define the Borel hierarchy of classes $K\\subseteq Mod(L)$ as follows.\n\n\\begin{enumerate}\n\n\\item  $K$ is $\\mathbf{\\Sigma_0}$ and $\\mathbf{\\Pi_0}$ if it is a basic clopen neighborhood.\n\n\\item  For $0 < \\alpha < \\omega_1$, \n\n\\begin{enumerate}\n\n\\item  $K$ is $\\mathbf{\\Sigma_\\alpha}$ if it is a countable union of sets each of which is $\\mathbf{\\Pi_\\beta}$ for some $\\beta < \\alpha$.\n\n\\item  $K$ is $\\mathbf{\\Pi_\\alpha}$ if it is a countable intersection of sets each of which is $\\mathbf{\\Sigma_\\beta}$ for some $\\beta<\\alpha$.\n\n\\item $K$ is $\\mathbf{d}$-$\\mathbf{\\Sigma_{\\alpha}}$ if it is a difference of $\\Sigma_{\\alpha}$ sets.\n\n\\end{enumerate}\n\n\\end{enumerate}\n\nVaught \\cite{Vaught} proved the following.\n\n\\begin{thm} [Vaught]\n\\label{Vaught}\n\nFor a set $K\\subseteq Mod(L)$, closed under isomorphism, $K$ is $\\mathbf{\\Pi_\\alpha}$ in the Borel hierarchy iff there is a \n$\\Pi_\\alpha$ sentence $\\varphi$ of $L_{\\omega_1\\omega}$ such that $K = Mod(\\varphi)$.  \n\n\\end{thm}\nIt is easy to see, as a corollary of Vaught's Theorem, that the same holds for $\\Sigma_{\\alpha}$ and $d$-$\\Sigma_{\\alpha}$ sets and sentences.\n\n\nIf $L$ is a computable language, then we have also the \\emph{effective} Borel hierarchy.  Let $K\\subseteq Mod(L)$.  \n\n\\begin{enumerate}\n\n\\item  $K$ is effective $\\Sigma_0$ and effective $\\Pi_0$ if it is a basic clopen neighborhood.\n\n\\item  For $0 < \\alpha < \\omega_1^{CK}$, \n\n\\begin{enumerate}\n\n\\item  $K$ is effective $\\Sigma_\\alpha$ if it is a c.e.\\ union of sets each of which is effective $\\Pi_\\beta$ for some $\\beta < \\alpha$,\n\n\\item  $K$ is effective $\\Pi_\\alpha$ if it is a c.e.\\ intersection of sets each of which is effective $\\Sigma_\\beta$ for some $\\beta < \\alpha$.\n\n\\end{enumerate}\n\n\\end{enumerate}\n\nWe may effectively identify elements of $Mod(L)$ with elements of $2^\\omega$, and then the effective Borel sets are exactly the hyperarithmetical sets of functions in $2^\\omega$.  Vanden Boom \\cite{VB} proved the effective analogue of Vaught's Theorem.   \n\n\\begin{thm} [Vanden Boom]\n\\label{Vanden Boom}\n\nFor a set $K\\subseteq Mod(L)$, closed under isomorphism, $K$ is $\\Pi_\\alpha$ in the effective Borel hierarchy iff there is a computable $\\Pi_\\alpha$ sentence $\\varphi$ of $L_{\\omega_1\\omega}$ such that $K = Mod(\\varphi)$.  \n\n\\end{thm}\n\nMontalb\\'{a}n \\cite{Montalban} proved that for a countable ordinal $\\alpha\\geq 1$, a countable structure $\\mathcal{A}$ has a $\\Pi_{\\alpha+1}$ Scott sentence iff the orbits of all tuples are defined by $\\Sigma_\\alpha$ formulas.  The implication $\\Leftarrow$ is as in the proof of Scott's Theorem.  For the implication $\\Rightarrow$, Montalb\\'{a}n's proof was clever.  We shall use the ideas from his proof to obtain further results.      \n\n\\bigskip\n\nIn Section 2, we show that for a countable ordinal $\\alpha\\geq 2$, if $\\mathcal{A}$ has a $\\Pi_\\alpha$ Scott sentence, then the orbits of all tuples in $\\mathcal{A}$ are defined by $\\Sigma_{<\\alpha}$ formulas.  (Montalb\\'{a}n's Theorem gives this in the case where $\\alpha$ is a successor ordinal.)\nFor limit $\\alpha$, the implication $\\Leftarrow$ fails.  There are familiar structures $\\mathcal{A}$ such that the orbits of all tuples are defined by $\\Sigma_{<\\omega}$ formulas but there is no $\\Pi_\\omega$ Scott sentence.  \nNext, we give an effective version of Montalb\\'{a}n's theorem, saying that for a computable ordinal $\\alpha\\geq 2$, if $\\mathcal{A}$ has a computable $\\Pi_\\alpha$ Scott sentence, then the orbits of all tuples are defined by computable $\\Sigma_{<\\alpha}$ formulas.  Even for successor ordinals $\\alpha$, the implication $\\Leftarrow$ fails.    \nWe construct an example of a computable structure such that the orbits of all tuples are defined by finitary quantifier-free formulas, but there is no computable $\\Pi_2$ Scott sentence.\n    \nIn Section 3, we consider further results that can be proved using ideas from Section 2.  A.\\ Miller showed that if $\\mathcal{A}$ has a $\\Pi_\\alpha$ Scott sentence and a $\\Sigma_\\alpha$ Scott sentence, then it has a Scott sentence that is $d$-$\\Sigma_{<\\alpha}$.  The proof was based on a result of D.\\ Miller \\cite{DMiller} on separators for disjoint sets axiomatized by $\\Pi_\\alpha$ sentences.  \nOur Theorem \\ref{effectiveAMiller} is an effective version of the result of A.~Miller, saying that if $\\mathcal{A}$ has a computable $\\Pi_\\alpha$ Scott sentence and a computable $\\Sigma_\\alpha$ Scott sentence, then it has a Scott sentence that is computable $d$-$\\Sigma_{<\\alpha}$. In \\cite{DMiller}, D.\\ Miller gave an effective version of his result, which, unfortunately, was not sufficient to prove Theorem \\ref{effectiveAMiller}.  We give a direct proof of Theorem \\ref{effectiveAMiller}.  Then, in Theorem \\ref{effectiveDMiller}, we give an effective version of the result of D.\\ Miller that would have served to prove Theorem \\ref{effectiveAMiller}.\n\nIn Section 4, we consider finitely generated groups.  For such a group, there is always a $\\Sigma_3$ Scott sentence, and if the group is computable, then there is a computable $\\Sigma_3$ Scott sentence (see \\cite{KS}).  Often, however, there is a simpler Scott sentence.  In fact, the second author had conjectured that every finitely generated group has a $d$-$\\Sigma_2$ Scott sentence, and every computable finitely generated group has a computable $d$-$\\Sigma_2$ Scott sentence.  Recently, Harrison-Trainor and Ho \\cite{HH} gave an example disproving both conjectures.  We show that if $G$ is a finitely generated group, then $G$ has a $d$-$\\Sigma_2$ Scott sentence iff there is a generating tuple whose orbit is defined by a $\\Pi_1$ formula, and if $G$ is a computable finitely generated group, then $G$ has a computable $d$-$\\Sigma_2$ Scott sentence iff there is a generating tuple whose orbit is defined by a computable $\\Pi_1$-formula. \n\nRecall that the definition above of $Mod(L)$, all $L$-structures have universe $\\omega$.  Throughout this paper, all structures given are assumed to be countably infinite with universe $\\omega$, and all structures we build are guaranteed by our techniques to have universe $\\omega$.\n\n\\section{Varying Montalb\\'{a}n's Theorem}\n\nMontalb\\'{a}n \\cite{Montalban} proved the following.\n\n\\begin{thm} [Montalb\\'{a}n]\n\\label{Montalban}\n\nSuppose $\\alpha\\geq 1$ is a countable ordinal, and let $\\mathcal{A}$ be a countable structure for a countable language $L$.  Then $\\mathcal{A}$ has a $\\Pi_{\\alpha+1}$ Scott sentence iff the automorphism orbit of each tuple is defined by a $\\Sigma_\\alpha$ formula.\n\n\\end{thm}\n\nFor the implication $\\Leftarrow$, the proof is as for Scott's Isomorphism Theorem.  \n\n\\begin{proof} [Proof of $\\Leftarrow$]\n\nFor each $\\bar{a}$, let $\\varphi_{\\bar{a}}(\\bar{x})$ be a $\\Sigma_{\\alpha}$ formula that defines the orbit of $\\bar{a}$.  For each $\\bar{a}$, we determine a sentence $\\rho_{\\bar{a}}$ as follows.   \n\n\\begin{itemize}\n\n\\item  $\\rho_\\emptyset$: $\\bigwedge_b (\\exists x)\\varphi_b(x)\\ \\&\\ (\\forall x)\\bigvee_b\\varphi_b(x)$\n\n\\item  $\\rho_{\\bar{a}}$:  $(\\forall\\bar{u})[\\varphi_{\\bar{a}}(\\bar{u})\\rightarrow\n(\\bigwedge_b (\\exists x)\\varphi_{\\bar{a},b}(\\bar{u},x)\\ \\&\\ (\\forall x)\\bigvee_b\\varphi_{\\bar{a},b}(x))]$\n\\end{itemize}\nOur Scott sentence is the conjunction of the sentences $\\rho_{\\bar{a}}$.  It is not difficult to see that this is $\\Pi_{\\alpha+1}$.\n\\end{proof}\n\nThe implication $\\Rightarrow$ in Montalb\\'{a}n's result also holds for limit ordinals, with no change in the proof.  Here is the statement.      \n\n\\begin{thm}\n\\label{MontalbanVariant}\n\nLet $\\mathcal{A}$ be a countable structure for a countable language $L$.  Let $\\alpha\\geq 2$.  If $\\mathcal{A}$ has a $\\Pi_\\alpha$ Scott sentence, then the orbit of each tuple is defined by a $\\Sigma_{<\\alpha}$ formula.\n\n\\end{thm} \n\nIn our account of Montalb\\'{a}n's proof, we use a ``consistency property.''  This is a family of sets of sentences arising in a kind of Henkin construction, developed by Makkai for producing models of $L_{\\omega_1\\omega}$ sentences.  See \\cite{Keisler} for a discussion of consistency properties.  The definition that we give below is not standard, but it suits our needs.      \n\n\\begin{defn}\n\nLet $L$ be a countable language, and let $C$ be a countably infinite set of new constants.  A \\emph{consistency property} is a non-empty set $\\mathcal{C}$ of finite sets $S$ of sentences, each obtained by substituting constants from $C$ for the free variables in an $L_{\\omega_1\\omega}$ formula in normal form, such that the following conditions hold:\n\n\\begin{enumerate}\n\n\\item  for $S\\in\\mathcal{C}$, if $\\varphi\\in S$, where $\\varphi = \\bigwedge_i (\\forall\\bar{u}_i)\\varphi_i(\\bar{u}_i)$, then for each $i$ and each appropriate tuple of constants $\\bar{c}$, there exists $S'\\supseteq S$ in $\\mathcal{C}$ with $\\varphi_i(\\bar{c})\\in S'$,\n\n\\item  for $S\\in\\mathcal{C}$,\nif $\\varphi\\in S$, where $\\varphi = \\bigvee_i (\\exists\\bar{u}_i)\\varphi_i(\\bar{u}_i)$, then for some $i$ and $\\bar{c}$, there exists $S'\\supseteq S$ in $\\mathcal{C}$ with $\\varphi_i(\\bar{c})\\in S'$, \n\n\\item  for $S\\in\\mathcal{C}$, for each finitary quantifier-free $L$-formula $\\varphi(\\bar{x})$ and appropriate tuple $\\bar{c}$, there exists $S'\\supseteq S$ in $\\mathcal{C}$ with $\\pm\\varphi(\\bar{c})\\in S'$,\n\n\\item  for $S\\in\\mathcal{C}$, if $F$ is an $n$-place function symbol, and $c_1,\\ldots,c_n\\in C$, there is a constant $d\\in C$ such that for some $S'\\supseteq S$ in $\\mathcal{C}$, the sentence $F(c_1,\\ldots,c_n) = d$ is in $S'$, \n\n\\item  for $S\\in\\mathcal{C}$, and distinct $c,c'\\in C$, the sentence $c = c'$ is not in $S$, \n\n\\item  for $S\\in\\mathcal{C}$, the set of finitary quantifier-free sentences in $S$ is consistent.\n\n\\end{enumerate}\n\n\\end{defn}\n\n\\begin{lem}\n\nIf $\\mathcal{C}$ is a consistency property, then we can form a countable chain $(S_n)_{n\\in\\omega}$ of elements of $\\mathcal{C}$ such that the set $\\{S_n:n\\in\\omega\\}$ is also a consistency property.  \n\n\\end{lem}\n\n\\begin{proof} [Proof sketch]\n\nThis is clear just from the fact that the sets $S$ in $\\mathcal{C}$ are finite, and there are only finitely many clauses (in the definition of consistency property) asking for extensions.\n\\end{proof}\n\n\\begin{prop}\n\nLet $(S_n)_{n\\in\\omega}$ be a countable chain such that $\\{S_n:n\\in\\omega\\}$ is a consistency property.  \nThen the set of atomic sentences and negations of atomic sentences in $\\cup_n S_n$ is the atomic diagram of a well defined structure $\\mathcal{B}$, with universe equal to $C$. Moreover, all sentences $\\varphi\\in \\cup_n S_n$ are true of the appropriate elements in $\\mathcal{B}$.\n\n\\end{prop}\n\n\\begin{proof}\n\nFor an $n$-placed relation symbol $R$, we let $R^\\mathcal{B}$ be the set of $(c_1,\\ldots,c_n)$ such that $R(c_1,\\ldots,c_n)\\in \\cup_n S_n$.  Conditions (3) and (6) guarantee that $R^\\mathcal{B}$ is a well-defined relation on $C^n$.  Suppose $F$ is an $n$-placed function symbol in $L$. Then $F^\\mathcal{B}(c_1,\\ldots,c_n) = d$ if the sentence $F(c_1,\\ldots,c_n) = d$ is in some $S_n$.  Conditions (4), (5), and (6) guarantee that $F^\\mathcal{B}$ is well-defined. For $c,d\\in C$, the sentence $c = d$ is in $\\cup_n S_n$ iff the constants $c$ and $d$ are actually the same.  An easy induction on terms $\\tau(\\bar{x})$ shows that $\\tau^\\mathcal{B}(\\bar{c}) = d$ iff the sentence $\\tau(\\bar{c}) = d$ is in $\\cup_n S_n$.  Then an easy induction on finitary quantifier-free formulas $\\varphi$ shows that $\\mathcal{B}\\models\\varphi(\\bar{c})$ iff the sentence $\\varphi(\\bar{c})$ is in $\\cup_n S_n$.\nFinally, an easy induction on sentences shows that if $\\varphi\\in\\cup_n S_n$, then $\\mathcal{B}\\models\\varphi$.    \n\\end{proof}       \n\nWe want a consistency property $\\mathcal{C}$ that produces models of the $\\Pi_\\alpha$ Scott sentence \n$\\varphi = \\bigwedge_i(\\forall\\bar{u}_i)\\varphi_i(\\bar{u}_i)$, where $\\varphi_i(\\bar{u}_i) = \\bigvee_j (\\exists\\bar{v}_{i,j})\\psi_{i,j}(\\bar{u}_i,\\bar{v}_{i,j})$.  We also want to control the complexity of the sentences that appear in $S\\in\\mathcal{C}$.  Instead of putting $\\varphi$ into various sets $S\\in \\mathcal{C}$, we add to the six conditions in the definition of consistency property a seventh condition guaranteeing that $\\varphi$ is witnessed.     \n\\begin{enumerate}\n\\setcounter{enumi}{6}\n\n\\item  for $S\\in\\mathcal{C}$, for each $i$ and each appropriate $\\bar{c}$, there exist $j$, and an appropriate $\\bar{d}$ such that for some $S'\\supseteq S$ in $\\mathcal{C}$, $\\psi_{i,j}(\\bar{c},\\bar{d})\\in S'$.  \n\n\\end{enumerate}\n\nIf $\\mathcal{C}$ is a consistency property satisfying Conditions (1)--(7), then there is a chain $(S_n)_{n\\in\\omega}$ of elements of $\\mathcal{C}$ such that $\\{S_n:n\\in\\omega\\}$ also satisfies Conditions (1)--(7).  For any such chain $(S_n)_{n\\in\\omega}$ the resulting structure is a model of $\\varphi$.  Our consistency property $\\mathcal{C}$ will be the set of finite sets $S$ of sentences in the language $L\\cup C$, each $\\Sigma_\\beta$ or $\\Pi_\\beta$ for some $\\beta$ such that $\\beta+1 <\\alpha$ (and each, recall, obtained by substituting constants from $C$ for the free variables in an $L_{\\omega_1\\omega}$ formula in normal form), where some interpretation of the constants from $C$ appearing in the sentences of $S$, mapping distinct constants to distinct elements of $\\mathcal{A}$, makes all of these sentences true.  (Since the set of sentences is finite, there are only finitely many constants from $C$ to assign to elements of $\\mathcal{A}$.)  It is easy to see that this satisfies Conditions (1)--(7).  \n\n\\bigskip\n\\noindent\n\\textbf{Claim}:  For each tuple $\\bar{a}$ of distinct elements, our $\\mathcal{C}$ fails to satisfy the following further condition.       \n\n\\begin{enumerate}\n\\setcounter{enumi}{7}\n\n\\item  for each $S\\in\\mathcal{C}$, and for each $\\bar{c}$, a tuple of distinct constants of the same length as $\\bar{a}$, there is some $\\Pi_{<\\alpha}$ formula $\\psi(\\bar{x}) = \\bigwedge_i (\\forall\\bar{u}_i)\\psi_i(\\bar{x},\\bar{u}_i)$, true of $\\bar{a}$, and some $S'\\supseteq S$ in $\\mathcal{C}$ such that for some $i$ and some $\\bar{d}$, $neg(\\psi_i(\\bar{c},\\bar{d}))\\in S'$.  (Note that for some $\\beta$ such that $\\beta+1 <\\alpha$, $\\psi_i(\\bar{c},\\bar{d})$ is $\\Sigma_\\beta$, and $neg(\\psi_i(\\bar{c},\\bar{d}))$ is $\\Pi_\\beta$.)  \n\n\\end{enumerate}\n\n\\begin{proof} [Proof of Claim]\n\nIf our $\\mathcal{C}$ satisfied Conditions (1)--(8),  then, we would have countable chains $(S_n)_{n\\in\\omega}$ yielding models of $\\varphi$ with no tuple satisfying all of the $\\Pi_{<\\alpha}$ formulas true of $\\bar{a}$.  Since $\\varphi$ is a Scott sentence for $\\mathcal{A}$, this is impossible.  \n\\end{proof}\n\nBy the Claim, there must be some set of sentences $S\\in\\mathcal{C}$ and some tuple of distinct constants $\\bar{c}$ from $C$, of the same length as $\\bar{a}$, that witness the failure of Condition (8).  Let $\\bar{c}'$ be the tuple of all constants from $C$, other than $\\bar{c}$, appearing in $S$, and let $\\chi(\\bar{c},\\bar{c}')$ be the finite conjunction of the sentences in $S$ and sentences expresssing that the elements of $\\bar{c}, \\bar{c}'$ are pairwise distinct.  Then $\\mathcal{A}$ satisfies the $\\Sigma_{<\\alpha}$ sentence $(\\exists\\bar{x})\\chi(\\bar{a},\\bar{x})$; and for all $\\Pi_{<\\alpha}$ formulas $\\psi$ true of $\\bar{a}$, $\\mathcal{A}$ satisfies the $\\Pi_{<\\alpha}$ sentences logically equivalent to $(\\forall \\bar{u})[(\\exists\\bar{x})\\chi(\\bar{u},\\bar{x})\\rightarrow\\psi(\\bar{u})]$.  Thus, we have a $\\Sigma_{<\\alpha}$ formula $(\\exists\\bar{x})\\chi(\\bar{u},\\bar{x})$ that generates the complete $\\Pi_{<\\alpha}$ type of $\\bar{a}$.  \nFor each $\\bar{a}$, let $\\varphi_{\\bar{a}}(\\bar{u})$ be a $\\Sigma_{<\\alpha}$ formula that generates the complete $\\Pi_{<\\alpha}$ type of $\\bar{a}$.  We claim that these formulas define the orbits.  To show this, it is enough to prove the following lemma.   \n\n\\begin{lem}\n\\label{orbits}\n\nThe family $\\mathcal{F}$ of finite functions taking $\\bar{a}$ to a tuple $\\bar{b}$ satisfying $\\varphi_{\\bar{a}}$ has the back-and-forth property.\n\n\\end{lem}\n\n\\begin{proof} [Proof of Lemma] \n\nWe first show that for any $\\bar{a}$ and $\\bar{b}$, if $\\bar{b}$ satisfies $\\varphi_{\\bar{a}}$, then $\\bar{a}$ satisfies $\\varphi_{\\bar{b}}$.  To see this, note that $neg(\\varphi_{\\bar{b}}(\\bar{x}))$ is $\\Pi_{<\\alpha}$.  If this were true of $\\bar{a}$, then it would be true of $\\bar{b}$, a contradiction.  Suppose $\\bar{b}$ satisfies $\\varphi_{\\bar{a}}(\\bar{x})$.  For any $d$, there exists $c$ such that $\\bar{a},c$ satisfies $\\varphi_{\\bar{b},d}(\\bar{x},y)$.  To see this, note that $(\\forall y) neg(\\varphi_{\\bar{b},d}(\\bar{x},y))$ is \n$\\Pi_{<\\alpha}$, so if it were true of $\\bar{a}$, then it would also be true of $\\bar{b}$, a contradiction.  If $\\bar{a},c$ satisfies $\\varphi_{\\bar{b},d}(\\bar{x},y)$, then $\\bar{b},d$ satisfies $\\varphi_{\\bar{a},c}(\\bar{x},y)$, so we can go back.  \nNow, suppose that $\\bar{b}$ satisfies $\\varphi_{\\bar{a}}(\\bar{x})$, and take $c$.  Since $\\bar{a}$ satisfies $\\varphi_{\\bar{b}}(\\bar{x})$, the argument above says that there exists $d$ such that $\\bar{b},d$ satisfies $\\varphi_{\\bar{a},c}(\\bar{x},y)$, and then $\\bar{a},c$ satisfies $\\varphi_{\\bar{b},d}(\\bar{x},y)$.  Therefore, we can go forth.  Hence, for each $\\bar{a}$, $\\varphi_{\\bar{a}}(\\bar{x})$ is a $\\Sigma_{<\\alpha}$ formula that defines the orbit of $\\bar{a}$.  This completes the proof of Theorem \\ref{MontalbanVariant}.\n\\end{proof}\n\nBelow, we give a pair of examples.  \n\n\\bigskip\n\\noindent\n\\textbf{Example 1}. Let $\\mathcal{A}$ be an ordering of type $\\omega^\\omega$.  Then the orbits of all tuples in $\\mathcal{A}$ are defined by $\\Sigma_{<\\omega}$ formulas (in fact, the natural defining formulas are computable $\\Sigma_{<\\omega}$).  However, $\\mathcal{A}$ has no $\\Pi_\\omega$ Scott sentence.\n\n\\begin{proof}\n\nWe use the following familiar results (see \\cite{AK}).\n\n\\bigskip\n\\noindent\n\\textbf{Facts}:\\  \n\n\\begin{enumerate}\n\n\\item  $\\omega^\\omega\\leq_\\omega\\omega^{\\omega+1}$,  \n\n\\item  for each $\\beta < \\omega^\\omega$, there are computable $\\Sigma_{<\\omega}$ formulas $\\lambda(x)$ and $\\mu(x,y)$ such that $\\lambda(x)$ holds iff the interval to the left of $x$ has order type $\\beta$ and $\\mu(x,y)$ holds iff the interval between $x$ and $y$ has order type $\\beta$.  \n\n\\end{enumerate}\nIt follows from Fact 1 and a well-known result of Karp (see \\cite{Keisler} or \\cite{KK}) that every $\\Pi_\\omega$ sentence true of $\\omega^\\omega$ is true of $\\omega^{\\omega+1}$.  Therefore, $\\omega^\\omega$ has no $\\Pi_\\omega$ Scott sentence.  Take a tuple $\\bar{a} = (a_1,\\ldots,a_n)$.  Ordinals are rigid, so to define the orbit of $\\bar{a}$, we define the tuple itself.  Say that the interval to the left of $a_i$ has type $\\beta_i$.  Applying Fact 2, we get a computable $\\Sigma_{<\\omega}$ formula $\\lambda_i(x_i)$ saying that the interval to the left of $x_i$ has type $\\beta_i$.  The conjunction of the formulas $\\lambda_i(x_i)$ defines the tuple $\\bar{a}$.       \n\\end{proof}\n\n\\noindent\n\\textbf{Example 2}.  Let $\\mathcal{A}$ be an expansion of the ordering of type $\\omega^\\omega$ with a unary predicate $U_0$ for the interval $[0,\\omega)$ and unary predicates $U_n$ for the interval $[\\omega^n,\\omega^{n+1})$, for $n\\geq 1$.  Again, we have computable $\\Sigma_{<\\omega}$ formulas defining the orbits of all tuples.  We have a computable $\\Pi_\\omega$ Scott sentence.  This is the conjunction of a computable $\\Pi_2$ sentence saying $(\\forall x)\\bigvee_n U_n(x)$, a finitary $\\Pi_2$ sentence saying that $<$ is a linear ordering of the universe, with all elements of $U_n$ before all elements of $U_{n+1}$, and computable $\\Sigma_{<\\omega}$ sentences saying what is the order type of $U_n$.       \n\n\\bigskip\n\nWe turn to the effective version of Theorem \\ref{MontalbanVariant}.  \n\n\\begin{thm}\n\\label{effectiveMontalban}\n\nLet $\\mathcal{A}$ be a structure for a computable language $L$ (the structure need not have a computable copy).  Suppose $\\alpha\\geq 2$.  If $\\mathcal{A}$ has a computable $\\Pi_{\\alpha}$ Scott sentence, then the orbit of each tuple is defined by a computable $\\Sigma_{<\\alpha}$ formula.    \n\n\\end{thm}\n\n\\begin{proof}\n\nThe proof essentially the same as that for Theorem \\ref{MontalbanVariant}.  The corresponding rules for a consistency property involve computable infinitary formulas, so the conjunctions and disjunctions are over c.e. sets of indices.  Our particular consistency property $\\mathcal{C}$ will consist of the finite sets of computable $\\Pi_\\beta$ and computable $\\Sigma_\\beta$ sentences, for $\\beta+1 < \\alpha$, such that some interpretation of the constants, mapping distinct constants to distinct elements of $\\mathcal{A}$, makes all of the sentences true. These technical changes necessitate no significant change in the argument that constructs, for each tuple $\\bar{a}$, a computable \n$\\Sigma_{<\\alpha}$ formula true of $\\bar{a}$ that implies all computable $\\Pi_{<\\alpha}$ formulas true of $\\bar{a}$.    \n\nFor each $\\bar{a}$, let $\\varphi_{\\bar{a}}(\\bar{x})$ be a $\\Sigma_{<\\alpha}$ formula that implies all computable $\\Pi_{<\\alpha}$-formulas true of $\\bar{a}$.  To show that for each $\\bar{a}$, the orbit is defined by $\\varphi_{\\bar{a}}$, the following analogue of Lemma \\ref{orbits} suffices; the proof is exactly the same as above.\n\n\\begin{lem}\n\\label{orbits.effective}\n\nThe family $\\mathcal{F}$ of finite functions taking a tuple $\\bar{a}$ to a tuple $\\bar{b}$ satisfying $\\varphi_{\\bar{a}}$ has the back-and-forth property.\n\n\\end{lem}\n\\end{proof}\n\nWe do not have the effective version for the other implication even in the case where $\\alpha$ is a computable successor ordinal.     \n\n\\begin{prop}\n\nThere is a computable structure $\\mathcal{A}$ such that the orbits of all tuples are defined by computable $\\Sigma_1$ (even finitary quantifier-free) formulas, but there is no computable $\\Pi_2$ Scott sentence. \n\n\\end{prop}\n\n\\begin{proof}\n\nThe proof owes much to that of Badaev \\cite{B}, showing that there is a computable enumeration of a ``discrete'' set of functions that is not ``effectively discrete''.  We start with a computable subtree $T$ of $2^{<\\omega}$ with the following features.\n\n\\begin{enumerate}\n\n\\item  There are no terminal nodes.\n\n\\item  There is just one non-isolated path $p$, where this is non-computable.\n\n\\end{enumerate}\n\nWe may construct $T$ such that for all $\\sigma\\in T$, $\\sigma 0\\in T$, and the only non-isolated path has the form $0^{s_0} 1^{k_0}0^{s_1}1^{k_1}\\ldots$, where $(k_n)_{n\\in\\omega}$ is a list of the elements of the halting set, in increasing order, and $s_n$ is the number of steps in the halting computation of $\\varphi_{k_n}(k_n)$.\nAt stage $0$, we put $\\emptyset$ into $T$.  Suppose that we have determined $T_s$ at stage $s$, where $T_s$ is the set of nodes in $T$ of length at most $s$.  At stage $s+1$, we add $\\sigma 0$ for all $\\sigma$ of length $s$.  In addition, we consider $\\varphi_{k,s+1}(k)$ for all $k\\leq s+1$.  For the computations that halt, we arrange the $k$'s to form $k_0 < k_1 < \\ldots < k_r$, and determine the appropriate halting times $s_0,\\ldots,s_r$.  We put into $T_{s+1}$ the appropriate initial segment of the sequence $0^{s_0}1^{k_0}\\cdots 0^{s_r}1^{k_r}$.  Note that if we are inserting a new $k_i$, it is because $s_i = s+1$, and we already had the appropriate initial segment in $T_s$.  At stage $s+1$, the nodes just described are the only ones that we add to $T_{s+1}$.  Therefore, the tree $T = \\bigcup_{s \\in \\omega} T_s$ is computable.    \n\nWe turn the tree $T$ into a class of structures.  The language $\\mathcal{L}$ consists of unary predicates $U_n$ for $n\\in\\omega$. In each $\\mathcal{L}$-structure $\\mathcal{A}$, we have infinitely many elements $a$ representing each isolated path $f$ in $T$, in that if $f(n) = 0$, then $\\mathcal{A}\\models \\neg{U_na}$ and if $f(n) = 1$, then $\\mathcal{A}\\models U_na$.  \n\nWe can give a computable set of axioms for the elementary first order theory of these structures.  For $\\sigma\\in T$, we have a finitary quantifier-free formula $\\sigma(x)$ that is the conjunction of $U_nx$ for $\\sigma(n) = 1$ and $\\neg{U_nx}$ for $\\sigma(n) = 0$.  For each $n$, we have a finitary quantifier-free formula $T_n(x)$ that is the disjunction over $\\sigma\\in T\\cap 2^n$ of the formulas $\\sigma(x)$.  Consider the axioms $(\\forall x) T_n(x)$ for all $n$ and $(\\exists ^{\\geq n} x)\\sigma(x)$ for $\\sigma\\in T$.  Let $T^*$ be the theory generated by these axioms.  \n\n\nThe countable models of $T^*$ all have infinitely many elements representing each isolated path.  In addition, they may have one or more elements representing the non-isolated path.  To see that the axioms generate a complete theory, we note that any finitary sentence mentions only finitely many $U_m$, say for $m < n$.  The reducts of the various countable models of $T^*$ to this smaller language are all isomorphic.        \n\nConsider the model of $T^*$ with no elements representing the non-isolated path.  Clearly, this model has a computable copy $\\mathcal{A}$ with universe $A = \\omega$.  To see this, computably partition the set $A$ into infinitely many infinite, computable sets.  For each $\\sigma \\in T$, consider the infinite path composed of $\\sigma$ followed by all $0$'s; assign all of the elements $a$ from one of the infinite sets in the partition to this path.  Note that if $\\sigma_i$ isolates the path represented by $a_i$, then the conjunction of the formulas giving the equalities on the $a_i$ and the formulas $\\sigma_i$ generates the complete elementary first order type of $\\bar{a}$.  Moreover, since the language has only unary predicates, this formula actually defines the orbit of $\\bar{a}$.  Similarly, for any model $\\mathcal{D}$ of $T^*$, $\\mathcal{A}$ is isomorphic to an elementary substructure of $\\mathcal{D}$, and any mapping that sends each element $a \\in \\mathcal{A}$ to an element $d \\in \\mathcal{D}$ representing the same isolated path is an elementary embedding.  \n\n\nLet $\\mathcal{B}$ be the model of $T^*$ with just one  element $b$ representing the non-isolated path.  We write $\\mathcal{A}$ for the substructure of $\\mathcal{B}$ isomorphic to the structure $\\mathcal{A}$ above.  We want to show that any computable $\\Pi_2$ sentence true of $\\mathcal{A}$ is true of $\\mathcal{B}$.  It is enough to show that any computable $\\Sigma_2$ sentence true of $\\mathcal{B}$ is true of $\\mathcal{A}$.  Take a computable $\\Sigma_2$ sentence $\\varphi = \\bigvee (\\exists\\bar{u}_i)\\varphi_i(\\bar{u}_i)$ true of $\\mathcal{B}$.  Say $\\mathcal{B}\\models\\varphi_i(b,\\bar{a})$, where $\\varphi_i(x,\\bar{v})$ is computable $\\Pi_1$, $\\bar{u}_i = x,\\bar{v}$.  Let $\\delta(\\bar{v})$ be a finitary quantifier-free formula generating the type of $\\bar{a}$.  Let $p(x,\\bar{v})$ be the c.e.\\ set of finitary universal conjuncts of $\\varphi_i(\\bar{u}_i)$.  Let $\\Gamma$ consist of the computable set of axioms for $T^*$, plus $\\delta(\\bar{c})$, plus \n$p(d,\\bar{c})$.  \n\nLet $f$ be the non-isolated path through $T$.  We cannot have $\\Gamma\\vdash\\sigma(d)$, for all finite $\\sigma\\subseteq f$, since then $f$ would be computable.  So, for some $\\sigma\\subseteq f$, $\\Gamma\\cup\\{\\neg{\\sigma(d)}\\}$ has a model $\\mathcal{C}$, where the elements of $\\bar{c}$ and $d$ necessarily all represent isolated paths.  Now, by what we noted above, the reduct of $\\mathcal{C}$ to the language $\\mathcal{L}$ has an elementary substructure isomorphic to $\\mathcal{A}$, and we may suppose that $\\bar{c}$ and $d$ are in this elementary substructure, and that $\\bar{a}$ is mapped to $\\bar{c}$.  Therefore, for the element $a'$ mapped to $d$, $\\mathcal{A}\\models\\varphi_i(a',\\bar{a})$, as required.    \\end{proof} \n\nThe next result gives conditions sufficient to guarantee that a computable structure has a computable $\\Pi_{\\alpha+1}$ Scott sentence. \n\n\\begin{prop}\n\nLet $\\alpha\\geq 1$ be a computable ordinal.  Suppose $\\mathcal{A}$ is a computable structure, and there is a $\\Sigma_\\alpha$ Scott family $\\Phi$ consisting of computable $\\Sigma_\\alpha$ formulas, with no parameters.  Then $\\mathcal{A}$ has a computable $\\Pi_{\\alpha+1}$ Scott sentence.    \n\n\\end{prop}\n\n\\begin{proof}\n\nWe can prove the following.  \n\n\\noindent\n\\textbf{Claim}:  There is a computable function taking each tuple $\\bar{c}$ to a computable $\\Sigma_\\alpha$ formula $\\varphi_{\\bar{c}}(\\bar{x})$ that defines the orbit of $\\bar{c}$.\n\n\\begin{proof} [Proof of Claim]\n\nLet $a$ be a notation for $\\alpha$ (see \\cite{AK} for a technical exposition of Kleene's system of notations for computable ordinals).  We may suppose that all elements of $\\Phi$ have indices of the form $(\\Sigma,a,e)$.  Let $R$ be the $\\Sigma_\\alpha$ relation consisting of pairs $(\\bar{c},e)$ such that the formula $\\psi_{\\bar{c},e}$ with index $(\\Sigma,a,e)$ is an element of $\\Phi$ that is true of $\\bar{c}$.  We can construct a computable sequence of computable $\\Sigma_\\alpha$ formulas $\\tau_{\\bar{c},e}$, defined for all $\\bar{c}$ and all $e$, built up out of $\\top$ and $\\bot$, such that $\\tau_{\\bar{c},e}$ is logically equivalent to $\\top$ if $(\\bar{c},e)\\in R$ and to $\\bot$ otherwise.\nWe let $\\varphi_{\\bar{c}}(\\bar{x})$ be the disjunction over all $e$, of the formulas $\\tau_{\\bar{c},e}\\ \\&\\ \\psi_{\\bar{c},e}(\\bar{x})$.     \n\\end{proof}   \n\nUsing the formulas $\\varphi_{\\bar{c}}(\\bar{x})$ from the Claim, we can build a computable $\\Pi_{\\alpha+1}$ Scott sentence as Scott did.  We take the conjunction of the computable $\\Sigma_\\alpha$ sentence $\\varphi_\\emptyset$ and the computable $\\Pi_{\\alpha+1}$ sentences $\\rho_{\\bar{a}}$ saying  \n\\[(\\forall\\bar{x})[\\varphi_{\\bar{a}}(\\bar{x})\\rightarrow ((\\forall y)\\bigvee_b\\varphi_{\\bar{a},b}(\\bar{x},y)\\ \\&\\ \\bigwedge_b(\\exists y)\\varphi_{\\bar{a},b}(\\bar{x},y))]\\ .\\]  \\end{proof}                            \n\nRecall that in an example above, we showed that for a computable ordering of type $\\omega^\\omega$, the orbits of all tuples are defined by computable $\\Sigma_{<\\omega}$ formulas, but there is no $\\Pi_\\omega$ Scott sentence.   \n\n\n\\section{Varying the results of A.\\ Miller and D.\\ Miller} \n\nA.\\ Miller \\cite{AMiller} proved that for a countable ordinal $\\alpha\\geq 2$, if $\\mathcal{A}$ has a $\\Sigma_\\alpha$ Scott sentence and a $\\Pi_\\alpha$ Scott sentence, then it has one that is $d$-$\\Sigma_{<\\alpha}$.  We say a little about A.\\ Miller's proof.  The case where $\\alpha$ is a limit ordinal is trivial; in fact, if $\\mathcal{A}$ has a $\\Sigma_\\alpha$ Scott sentence $\\varphi$, then there is a $\\Sigma_{<\\alpha}$ Scott sentence.  The sentence $\\varphi$ is a countable disjunction of formulas $\\varphi_i$, each of which is $\\Sigma_{<\\alpha}$.  One of the disjuncts $\\varphi_i$ is true in $\\mathcal{A}$, and this is a $\\Sigma_{<\\alpha}$ Scott sentence.  \n\nThe interesting case is as follows.      \n\n\\begin{thm} [A.\\ Miller]\n\nFor a countable ordinal $\\alpha\\geq 1$, if $\\mathcal{A}$ has a Scott sentence that is $\\Pi_{\\alpha+1}$ and one that is $\\Sigma_{\\alpha+1}$, then it has one that is $d$-$\\Sigma_\\alpha$.\n\n\\end{thm}\n\n\\begin{proof} [Sketch of proof]\n\nA.\\ Miller used a result of D.\\ Miller \\cite{DMiller} saying that for disjoint sets $A,B\\subseteq Mod(L)$ both axiomatized by $\\Pi_{\\alpha+1}$ sentences, there is a separator (i.e., a set containing $A$ and disjoint from $B$) that is a countable union of sets axiomatized by $d$-$\\Sigma_\\alpha$ sentences.  Suppose that $\\mathcal{A}$ has Scott sentences $\\varphi$ and $\\psi$, where $\\varphi$ is $\\Pi_{\\alpha+1}$ and $\\psi$ is $\\Sigma_{\\alpha+1}$.  Applying the result of D.\\ Miller to the disjoint sets $A = Mod(\\varphi)$ and $B = Mod(neg(\\psi))$, we get a separator which is $Mod(\\gamma)$ for some sentence $\\gamma$ which is a countable disjunction of $d$-$\\Sigma_{\\alpha}$ sentences. Since $Mod(neg(\\psi))$ is the complement of $Mod(\\varphi)$ in $Mod(L)$, $Mod(\\gamma) = Mod(\\varphi)$; so $\\gamma$ is a Scott sentence for $\\mathcal{A}$. Thus, $\\mathcal{A}$ satisfies one of the disjuncts of $\\gamma$, and this is also a Scott sentence for $\\mathcal{A}$.\n\\end{proof}\n\nOur goal is to prove the following.\n\n\\begin{thm}\n\\label{effectiveAMiller}\n\nFor a computable ordinal $\\alpha\\geq 2$, if $\\mathcal{A}$ has a Scott sentence that is computable $\\Pi_\\alpha$ and one that is computable $\\Sigma_\\alpha$, then there is one that is computable $d$-$\\Sigma_{<\\alpha}$.   \n\n\\end{thm}\n\nAgain, the case where $\\alpha$ is a limit ordinal is trivial. We want to prove that if $\\mathcal{A}$ has one Scott sentence that is computable $\\Pi_{\\alpha+1}$ and another that is computable $\\Sigma_{\\alpha+1}$, then there is one that is computable $d$-$\\Sigma_\\alpha$.\n\nD.\\ Miller gave an effective version of his separation theorem, saying that if $A$ and $B$ are disjoint subsets of $Mod(L)$, axiomatized by $\\Pi_{\\alpha+1}$ sentences in the admissible fragment $L_{\\omega_1^{CK}}$, then there is a separator that is a disjoint union of sets axiomatized by $d$-$\\Sigma_\\alpha$ formulas in $L_{\\omega_1^{CK}}$.  This is not good enough for our purposes.  We give a direct proof of the following.\n\n\\begin{lem} [Main Lemma]  \n\nLet $\\alpha\\geq 1$ be a computable ordinal.  If $\\mathcal{A}$ has a computable $\\Sigma_{\\alpha+1}$ Scott sentence $\\varphi$ and a computable $\\Pi_{\\alpha+1}$ Scott sentence $\\psi$, then there is a computable $d$-$\\Sigma_\\alpha$ Scott sentence.\n\n\\end{lem}\n\n\\begin{proof}\n\nThe sentence $\\varphi$ has the form $\\bigvee_{i\\in W} (\\exists\\bar{u}_i)\\varphi_i(\\bar{u}_i)$, where each $\\varphi_i$ is computable $\\Pi_\\alpha$, and $W$ is a c.e. set.  For some $i$ and some $\\bar{a}$, we have $\\mathcal{A}\\models\\varphi_i(\\bar{a})$.  By Theorem \\ref{effectiveMontalban}, the orbit of $\\bar{a}$ is defined by a computable $\\Sigma_\\alpha$ formula $\\gamma(\\bar{u})$.  Note that $(\\exists\\bar{u})\\gamma(\\bar{u})$ is computable $\\Sigma_\\alpha$, and $(\\forall\\bar{u})(\\gamma(\\bar{u})\\rightarrow\\varphi_i(\\bar{u}))$ is logically equivalent to a computable $\\Pi_\\alpha$ sentence.  The conjunction of these is a Scott sentence \nfor~$\\mathcal{A}$.  \n\\end{proof}\n\nBelow, we give an effective version of D.\\ Miller's result, which would suffice to prove Theorem \\ref{effectiveAMiller}.    \n\\begin{thm}\n\\label{effectiveDMiller}\n\nLet $L$ be a computable language.  For a computable ordinal $\\alpha\\geq 2$, suppose $A$ and $B$ are disjoint subsets of $Mod(L)$ axiomatized by computable $\\Pi_{\\alpha}$ sentences.  Then then there is a separator that is the union of a countable family of sets each of which is axiomatized by a computable $d$-$\\Sigma_{< \\alpha}$ sentence.\n\n\\end{thm}\n       \n\\begin{proof}\n\nLet $A = Mod(\\varphi)$, and let $B = Mod(\\psi)$, where $\\varphi$ and $\\psi$ are computable $\\Pi_\\alpha$ sentences.  Say that\n$\\varphi = \\bigwedge_{i \\in W} (\\forall\\bar{u}_i)\\varphi_i(\\bar{u}_i)$, where $W$ is a c.e. set, and each $\\varphi_i(\\bar{u}_i)$ has the form \n$\\bigvee_{j \\in W_{e_i}}(\\exists\\bar{v}_{i,j}) \\xi_{i,j}(\\bar{u}_i,\\bar{v}_{i,j})$, where each $\\xi_{i,j}$ is $\\Pi_{\\beta}$ for some $\\beta$ such that $\\beta+1 < \\alpha$, and each $W_{e_i}$ is a c.e. set whose index $e_i$ (according to some canonical indexing of the c.e. sets) is determined computably from the index $i$ of $\\varphi_i$.  \n\nLet $C$ be an infinite computable set of new Henkin constants, corresponding to the natural numbers.  For each $\\mathcal{A}\\in A$, let $\\mathcal{C}_{\\mathcal{A}}$ be the consistency property consisting of the finite sets $S$ of sentences in the language $L\\cup C$, each computable $\\Sigma_\\beta$ or $\\Pi_\\beta$ for some $\\beta$ such that $\\beta+1 <\\alpha$ (and each, recall, obtained by substituting constants from $C$ for the free variables in a computable $L_{\\omega_1\\omega}$ formula in normal form), where some interpretation of the constants from $C$ appearing in the sentences of $S$, mapping distinct constants to distinct elements of $\\mathcal{A}$, makes all of these sentences true.  Recall the special computable $\\Pi_\\alpha$ sentence $\\varphi$ and its sub-formulas, $\\varphi_i$ and $\\xi_{i,j}$.  The set $\\mathcal{C}_{\\mathcal{A}}$ is a consistency property $\\mathcal{C}$ satisfying the following condition:   \n\n\\begin{itemize}[label=$\\star$] \n\\item  For each $S\\in\\mathcal{C}$, for each $i \\in W$ and each appropriate $\\bar{c}$, there exist $j \\in W_{e_i}$ and an appropriate $\\bar{d}$ such that for some $S'\\supseteq S$ in $\\mathcal{C}$, $\\xi_{i,j}(\\bar{c},\\bar{d})\\in S'$.\n\\end{itemize}\n\nFor any consistency property $\\mathcal{C}$ satisfying $\\star$, there is a chain $(S_n)_{n\\in\\omega}$ of elements of $\\mathcal{C}$ such that $\\{S_n:n\\in\\omega\\}$ is also a consistency property satisfying $\\star$.  For any such chain, the resulting structure is a model of $\\varphi$.    \n\nWe now consider the other special computable $\\Pi_\\alpha$ sentence $\\psi$.  Say that $\\psi = \\bigwedge_{i \\in W'}(\\forall\\bar{u}_i)\\psi_i(\\bar{u}_i)$, where $W'$ is a c.e. set, and for each $i$, $\\psi_i(\\bar{u}_i) = \\bigvee_{j \\in W_{e_i}}(\\exists\\bar{v}_j)\\theta_{i,j}(\\bar{u}_i,\\bar{v}_j)$, where $\\theta_{i,j}$ is $\\Pi_\\beta$ for some $\\beta$ such that $\\beta+1 < \\alpha$, and each $W_{e_i}$ is a c.e. set whose index $e_i$ (according to some canonical indexing of the c.e. sets) is determined computably by the index $i$ of $\\varphi_i$.  Since $A = Mod(\\varphi)$ and $B = Mod(\\psi)$ are disjoint subsets of $Mod(L)$, for any $\\mathcal{A}$ satisfying $\\varphi$, $\\mathcal{C}_{\\mathcal{A}}$ cannot satisfy the following added condition, which would witness the truth of $\\psi$:\n\n\\begin{itemize}[label=$\\star \\star$] \n\n\\item  For each $S\\in\\mathcal{C}$, for all $i \\in W'$ and all $\\bar{c}$ appropriate for $\\bar{u}_i$, there exists $S'\\supseteq S$ such that for some $j \\in W_{e_i}$ and $\\bar{d}$, we have $\\theta_{i,j}(\\bar{c},\\bar{d})\\in S'$.  \n\n\\end{itemize}\n\nIt follows that there must exist $S\\in\\mathcal{C}_{\\mathcal{A}}$, and some $i$ and $\\bar{c}$ appropriate for $\\bar{u}_i$, such that for all $j$ in the c.e.\\ set $W_{e_i}$ and all $\\bar{d}$, $S\\cup\\{\\theta_{i,j}(\\bar{c},\\bar{d})\\}$ is not satisfied by any assignment in $\\mathcal{A}$.  Let $\\bar{c}'$ be the tuple of constants from $C$, other than $\\bar{c}$, that appear in $S$, and let $\\chi(\\bar{c},\\bar{c}')$ be the conjunction of $S$ and sentences expressing that the tuples $\\bar{c}, \\bar{c}'$ are disjoint and the elements of $\\bar{c}'$ are distinct. \n\nNow, although $\\bar{c}$ is appropriate for the tuple of variables $\\bar{u}_i$, it might be that the tuple $\\bar{c}$ assigns the same constant to multiple variables; i.e, the tuple $\\bar{c}$ could list the same constant multiple times. Therefore, to define $\\chi(\\bar{u}_i, \\bar{x})$ unambiguously, for a given constant in $\\bar{c}$, substitute the variable from $\\bar{u}_i$ with least index to which this element of $\\bar{c}$ was assigned.  Finally, let $\\rho(\\bar{u}_i, \\bar{x})$ be the conjunction of $\\chi(\\bar{u}_i, \\bar{x})$ and formulas that express the pairwise equality of any elements of $\\bar{u}_i$ to which the same constant in $\\bar{c}$ was assigned, and the pairwise inequality of any elements of $\\bar{u}_i$ to which different constants in $\\bar{c}$ were assigned.  Then \n\\[\\mathcal{A}\\models (\\forall\\bar{u}_i)[((\\exists\\bar{x})\\rho(\\bar{u}_i,\\bar{x})) \\rightarrow (\\bigwedge_{j \\in W_{e_i}}(\\forall\\bar{v}_{i,j})(neg(\\theta_{i,j}(\\bar{u}_i,\\bar{v}_{i,j}))))]\\] .\n\n\n\\noindent Note that $(\\exists\\bar{u}_i \\bar{x})\\rho(\\bar{u}_i,\\bar{x})$ is computable $\\Sigma_{<\\alpha}$, and \\[(\\forall\\bar{u}_i)[((\\exists\\bar{x})\\rho(\\bar{u}_i,\\bar{x})) \\rightarrow (\\bigwedge_{j \\in W_{e_i}}(\\forall\\bar{v}_{i,j})(neg(\\theta_{i,j}(\\bar{u}_i,\\bar{v}_{i,j}))))]\\] is logically equivalent to a computable $\\Pi_{<\\alpha}$ sentence.  Both sentences are true in $\\mathcal{A}$.  They cannot both be true in any model of $\\psi$, for then there would be a tuple satisfying $neg(\\psi_i(\\bar{u}_i))$.  The conjunction gives a computable $d$-$\\Sigma_{<\\alpha}$ sentence that is true in $\\mathcal{A}$ and not true in any model of $\\psi$.  Let $M_{\\mathcal{A}}$ be the class of models for this sentence. As our separator, we take the union of the sets $M_\\mathcal{A}$.  While there may be uncountably many models $\\mathcal{A}$ of $\\varphi$, there are only countably many pairs of computable infinitary sentences. Hence, our separator is the union of a countable family of sets $S_{\\mathcal{A}}$.  \n\\end{proof} \n\n\\section{Finitely generated groups}       \n\nKnight and Saraph \\cite{KS} observed that every computable finitely generated group has a computable $\\Sigma_3$ Scott sentence.  However, for many kinds of computable finitely generated groups, there is a computable $d$-$\\Sigma_2$ Scott sentence.  In particular, this is so for finitely generated free groups \\cite{Free}, finitely generated Abelian groups, the infinite dihedral group of rank $2$  \\cite{KS}, further variants of the dihedral group \\cite{Raz}, polycyclic groups, and certain groups of interest in geometric group theory (lamplighter and Baumslag-Solitar groups) \\cite{Ho}.  \nBased on the known examples, Ho and Knight had conjectured that every computable finitely generated group has a computable $d$-$\\Sigma_2$ Scott sentence.  Knight also conjectured that every finitely generated group (not necessarily computable) has a $d$-$\\Sigma_2$ Scott sentence (not necessarily computable $d$-$\\Sigma_2$).  \nRecently, Harrison-Trainor and Ho \\cite{HH} gave an example of a computable finitely generated group that does not have a $d$-$\\Sigma_2$ Scott sentence, thereby disproving both conjectures.     \n\nHere we give necessary and sufficient conditions for a finitely generated group to have a $d$-$\\Sigma_2$ Scott sentence.  We also give necessary and sufficient conditions for a computable finitely generated group to have a computable $d$-$\\Sigma_2$ Scott sentence.  We show that for a finitely generated group, there is a $d$-$\\Sigma_2$ Scott sentence iff for some generating tuple, the orbit is defined by a $\\Pi_1$ formula iff for each generating tuple, the orbit is defined by a $\\Pi_1$ formula.  For a computable finitely generated group, there is a computable $d$-$\\Sigma_2$ Scott sentence iff for some generating tuple, the orbit is defined by a computable $\\Pi_1$ formula iff for each generating tuple, the orbit is defined by a computable $\\Pi_1$ formula.      \n\\subsection{Finitely generated groups with a $d$-$\\Sigma_2$ Scott sentence}\n\nIn \\cite{KS}, it is observed that a computable finitely generated group has a computable $\\Sigma_3$ Scott sentence.  \nThroughout the rest of this section, we use the notation $\\langle\\bar{x}\\rangle \\cong \\langle\\bar{y}\\rangle$ to represent the computable $\\Pi_1$ formula that says $\\bar{x}$ and $\\bar{y}$ satisfy the exact same relators and non-relators.  Note that if $\\bar{a}$ is a fixed tuple of a group $G$, then the set of relators and non-relators satisfied by $\\bar{a}$ is a set computable in $G$.  The notation $\\langle\\bar{x}\\rangle \\cong \\langle\\bar{a}\\rangle$ represents the conjunction of the formulas of the forms $\\bar{w}(\\bar{x}) = 1$ and $\\bar{w}(\\bar{x})\\not= 1$ that are true of $\\bar{a}$ in $\\mathcal{A}$.  This formula should not be thought of as including $\\bar{a}$ as parameters; instead, it represents a $\\Pi_1$ formula in~$\\bar{x}$.\n\n\\begin{prop}\n\nEvery finitely generated group has a $\\Sigma_3$ Scott sentence.  \n\n\\end{prop}\n\n\\begin{proof}\n\nLet $G$ be a group with generating tuple $\\bar{a}$.  As in \\cite{KS}, we get a Scott sentence saying that $(\\exists \\bar{x})[\\langle\\bar{x}\\rangle\\cong \\langle\\bar{a}\\rangle\\ \\&\\ (\\forall y)\\bigvee_w w(\\bar{x}) = y]$.\n\\end{proof}\n\nWe can prove the following.       \n\n\\begin{thm}\n\\label{characterization}\n\nFor a finitely generated group $G$, the following are equivalent:\n\n\\begin{enumerate}\n\n\\item  $G$ has a $\\Pi_3$ Scott sentence,\n\n\\item  $G$ has a $d$-$\\Sigma_2$ Scott sentence\n\n\\item  for some generating tuple, the orbit is defined by a $\\Pi_1$ formula\n\n\\item  for each generating tuple, the orbit is defined by a $\\Pi_1$ formula. \n\n\\end{enumerate} \n\n\\end{thm}\n\n\\begin{proof}\n\nClearly, $(2)\\Rightarrow (1)$.  Using the result of A.\\ Miller, together with the fact that $G$ has a $\\Sigma_3$ Scott sentence, we get $(1)\\Rightarrow (2)$.  To complete the proof, we will show that $(1)\\Rightarrow (4)\\Rightarrow (3)\\Rightarrow (1)$.  For $(1)\\Rightarrow (4)$, suppose $G$ has a $\\Pi_3$ Scott sentence, and let $\\bar{a}$ be a generating tuple.  By the result of Montalb\\'{a}n, the orbit of $\\bar{a}$ is defined by a computable $\\Sigma_2$ formula \n$\\varphi(\\bar{x}) = \\bigvee_i(\\exists\\bar{u}_i)\\varphi_i(\\bar{x},\\bar{u}_i)$, where \n$\\varphi_i$ is $\\Pi_1$.  Take $i$ and $\\bar{b}$ such that $G\\models\\varphi_i(\\bar{a},\\bar{b})$.  For some tuple of words $\\bar{w}$, $G\\models\\bar{w}(\\bar{a}) = \\bar{b}$.  Then the orbit of $\\bar{a}$ is defined by the $\\Pi_1$ formula \n$\\varphi_i(\\bar{x},\\bar{w}(\\bar{x}))$.  Clearly, $(4)\\Rightarrow (3)$.  To show that $(3)\\Rightarrow (1)$, suppose $\\bar{a}$ is a generating tuple with orbit defined by a $\\Pi_1$ formula $\\psi(\\bar{u})$.  We show that for all tuples $\\bar{b}$, the orbit is defined by a $\\Sigma_2$ formula.  Suppose \n$G\\models\\bar{b} = \\bar{w}(\\bar{a})$.  Then the orbit of $\\bar{b}$ is defined by the \n$\\Sigma_2$-formula\n$\\varphi(\\bar{x}) = (\\exists\\bar{u})(\\psi(\\bar{u})\\ \\&\\ \\bar{x} = \\bar{w}(\\bar{u}))$.  Then by the result of Montalb\\'{a}n, $G$ has a $\\Pi_3$ Scott sentence.      \n\\end{proof}\n\nIn this proof of Theorem \\ref{characterization} above, we used Miller's result to show that if $G$ has a $\\Pi_3$ Scott sentence, then it has a $d$-$\\Sigma_2$ Scott sentence.  There is an alternative proof, using the following result of Ho \\cite{Ho}.  \n\n\\begin{lem} [Generating Set Lemma]\n\nLet $G$ be a computable finitely generated group, and suppose $\\varphi(\\bar{x})$ is a computable $\\Sigma_2$ formula, satisfied in $G$, such that all tuples satisfying $\\varphi(\\bar{x})$ generate $G$.  Then $G$ has a computable $d$-$\\Sigma_2$ Scott sentence.  \n\n\\end{lem} \n\n\\begin{proof}\n\nLet $G = \\langle \\bar{a} \\rangle$, where $\\bar{a}$ satisfies the formula $\\varphi(\\bar{x})$.  \n\nHo's Scott sentence is the conjunction of the following:\n\\begin{enumerate}\n\n\\item  the computable $\\Pi_2$ sentence saying $(\\forall\\bar{x})[\\varphi(\\bar{x})\\rightarrow (\\forall y)\\bigvee_{w} w(\\bar{x}) = y]$,\n\n\\item the computable $\\Sigma_2$ sentence saying $(\\exists \\bar{x})[\\varphi(\\bar{x})\\ \\&\\ \\langle\\bar{x}\\rangle\\cong \\langle \\bar{a} \\rangle]$.\n\\end{enumerate}\n\\end{proof} \n\nWe automatically have the following non-effective analogue of Ho's result.\n\n\\begin{lem}\n\nSuppose $G$ is a finitely generated group, and there is a $\\Sigma_2$ formula $\\varphi(\\bar{x})$, satisfied in $G$, and such that all tuples satisfying $\\varphi(\\bar{x})$ generate $G$.  Then $G$ has a $d$-$\\Sigma_2$ Scott sentence.\n\n\\end{lem}  \n\n\\begin{proof} [Alternative proof of Theorem \\ref{characterization}]\n\nIf there is a $\\Pi_3$ Scott sentence, then by the result of Montalb\\'{a}n, there is a $\\Sigma_2$ formula defining the orbit of a generating tuple.  Then by the analogue of the result of Ho, there is a $d$-$\\Sigma_2$ Scott sentence.  \n\\end{proof} \n\n\\subsection{Computable finitely generated groups with a computable $d$-$\\Sigma_2$ Scott sentence} \n\nIn \\cite{KS}, it is observed that every computable group has a computable $\\Sigma_3$ Scott sentence.  We can prove the following.   \n\n\\begin{thm}\n\\label{characterization.effective}\n\nFor a computable finitely generated group $G$, the following are equivalent:\n\n\\begin{enumerate}\n\n\\item  there is a computable $\\Pi_3$ Scott sentence,\n\n\\item  there is a computable $d$-$\\Sigma_2$ Scott sentence,\n\n\\item  for some generating tuple, the orbit is defined by a computable $\\Pi_1$ formula,\n\n\\item  for each generating tuple, the orbit is defined by a computable $\\Pi_1$ formula.  \n\n\\end{enumerate}\n\n\\end{thm}\n\n\\begin{proof}\n\nWe show that $(2)\\Rightarrow (1)\\Rightarrow (4)\\Rightarrow (3)\\Rightarrow (2)$.\nFor $(2)\\Rightarrow (1)$, we note that a computable $d$-$\\Sigma_2$ Scott sentence may be regarded as a computable $\\Pi_3$ Scott sentence.  \nFor $(1)\\Rightarrow (4)$, if $G$ has a computable $\\Pi_3$ Scott sentence, and \n$\\bar{a}$ is a generating tuple, then by Theorem \\ref{effectiveMontalban}, the orbit of $\\bar{a}$ is defined by a computable $\\Sigma_2$ formula $\\varphi(\\bar{x}) = \\bigvee_{i \\in W}(\\exists\\bar{u}_i)\\varphi_i(\\bar{x},\\bar{u}_i)$, where each $\\varphi_i$ is computable $\\Pi_1$, and $W$ is some c.e. set.  For some $i \\in W$ and some $\\bar{b}$, $G\\models\\varphi_i(\\bar{a},\\bar{b})$.  For some tuple of words $\\bar{w}$, $G\\models\\bar{w}(\\bar{a}) = \\bar{b}$.\nThen the orbit of $\\bar{a}$ is defined by the computable $\\Pi_1$ formula \n$\\varphi_i(\\bar{x},\\bar{w}(\\bar{x}))$.\nClearly, $(4)\\Rightarrow (3)$.\nFor $(3)\\Rightarrow (2)$, let $\\bar{a}$ be a generating tuple whose orbit is defined by a computable $\\Pi_1$ formula.  We may regard this as a computable $\\Sigma_2$ formula.  By the result of Ho, $G$ has a computable $d$-$\\Sigma_2$ Scott sentence.  \n\\end{proof}\n\n\\subsection{Example of Harrison-Trainor and Ho}\n\nHo and Harrison-Trainor \\cite{HH} gave the definition below. They considered not just finitely generated groups, but more general finitely generated structures.  We consider only groups.\n\n\\begin{defn} [Harrison-Trainor-Ho]\n\nA finitely generated group $G$ is \\emph{self-reflective} if there is a generating tuple $\\bar{a}$ and a tuple $\\bar{b}$ generating a proper subgroup $H$, such that\n\\[(G,\\bar{a})\\cong (H,\\bar{b})\\] \nand every existential formula true of $\\bar{b}$ in $G$ is true of $\\bar{b}$ in $H$.  \n\n\\end{defn} \n\n\\begin{prop}\n\nLet $G$ be a finitely generated group.  Then $G$ is self-reflective iff there is a generating tuple $\\bar{a}$ whose orbit is not defined by a $\\Pi_1$ formula.\n\n\\end{prop} \n\n\\begin{proof}\n\n$\\Rightarrow$  Suppose that $G$ is self-reflective, witnessed by $\\bar{a}$ and $\\bar{b}$, where $\\bar{a}$ generates $G$, $\\bar{b}$ generates a proper subgroup $H$, and all existential formulas true of $\\bar{b}$ in $G$ are also true of $\\bar{b}$ in $H$.  Since $(G,\\bar{a})\\cong (H,\\bar{b})$, these existential formulas are also true of $\\bar{a}$ in $G$. Therefore, the universal formulas true of $\\bar{a}$ in $G$ are also true of $\\bar{b}$ in $G$.  This means that any $\\Pi_1$ formula true of $\\bar{a}$ is also true of $\\bar{b}$. Then the orbit of $\\bar{a}$ is not defined by a $\\Pi_1$ formula.\n\n$\\Leftarrow$  Suppose that $G$ is generated by a tuple $\\bar{a}$ whose orbit is not defined by a $\\Pi_1$ formula.  Let $\\varphi(\\bar{x})$ be the $\\Pi_1$ formula obtained as the conjunction of the universal formulas true of $\\bar{a}$.  There must be some $\\bar{b}$ satisfying $\\varphi(\\bar{x})$ and not generating $G$.  Let $H$ be the subgroup generated by $\\bar{b}$.  Then $(G,\\bar{a})\\cong (H,\\bar{b})$, and all existential formulas true of $\\bar{b}$ in $G$ are true of $\\bar{a}$ in $G$ and hence true of $\\bar{b}$ in $H$.  This means that $G$ is self-reflective.     \n\\end{proof} \n\nHarrison-Trainor and Ho constructed a computable finitely generated group $G$ that is self-reflective.  They also showed that for any such group, the index set is $m$-complete $\\Sigma^0_3$.  In this way, they arrived at the fact that their group $G$ has no computable $d$-$\\Sigma_2$ Scott sentence.  Relativizing, they got the fact that for any set $X$, the set of $X$-computable indices for copies of $G$ is $m$-complete $\\Sigma^0_3$ relative to $X$, so there is no $X$-computable $d$-$\\Sigma_2$ Scott sentence.  It follows that the group $G$ has no $d$-$\\Sigma_2$ Scott sentence.    \n\\section{Problems}\n\n\\begin{enumerate}\n\n\\item  Is there a finitely presented group that is self-reflective?  \n\n\\item  Is there a precise sense in which most finitely generated groups are not self-reflective?  Can we say in terms of limiting density that the typical finitely generated group has a $d$-$\\Sigma_2$ Scott sentence?   \n\n\\item  Is there a computable finitely generated group with a $d$-$\\Sigma_2$ Scott sentence but no computable $d$-$\\Sigma_2$ Scott sentence?  \n\n\\item  Give necessary and sufficient conditions for a computable structure $\\mathcal{A}$ to have a computable \n$\\Pi_{\\alpha+1}$ Scott sentence.  \n\n\\end{enumerate}  \n     \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Appendix A: Low-temperature setup}\n Our low-temperature setup consists of an adiabatic nuclear demagnetization stage, incorporated into a commercial dilution fridge.  The nuclear stage was made from high-purity copper, and vacuum annealed after machining.  The copper stage is attached to the microkelvin plate, where the experimental cell and melting curve thermometer are mounted, via a superconducting heat switch made of high-purity indium wires.  The sample cell housing the liquid $^3$He and Helmholtz resonators is designed after Ref.~\\cite{Pollanen12}, with sintered copper powder heat exchangers \\cite{Krusius78} of total surface area 84 m$^2$. With $^4$He impurities of 1\\% in our $^3$He, this results in a surface coverage of $\\approx1\/4$ of a monolayer, consistent with diffuse boundary scattering \\cite{Freeman90} and the results of our Ginzburg-Landau calculations in the main text.\n\n\\section*{Appendix B: Device geometry}\n\nThe Helmholtz resonator geometry is depicted in Figure SM 1. The basin diameter, $\\diameter$, channel width, $w$, and channel length (after bonding), $l$, are common to all three devices.  The devices differ by their etch depth, and hence helium confinement (see Table \\ref{tab:DeviceMeasurements}).\n\n\\begin{figure*}[b]\n\\centering\n\\begin{tikzpicture}\n\n\\begin{scope}[]\n\\draw (0,0) rectangle (6,9);\n\\draw (3,4.5) circle (2.1);\n\\draw[fill=white,white] (2.4,0) rectangle (3.6,9);\n\\draw (2.4,0) rectangle (3.6,9);\n\\draw[fill=white,white] (3,4.5) circle (2.082);\n\\draw[fill=gray,gray] (2.5,4.5) rectangle (3.5,9);\n\\draw[fill=gray,gray] (3,4.5) circle (2.0);\n\\draw[dashed] (-1,1.5) -- (6,1.5);\n\\draw[dashed] (-1,2.4) -- (3,2.4);\n\\draw[<->] (-0.5,1.5) -- (-0.5,2.4);\n\\node[left] at (-0.5,1.95) {$l$=1.38 mm};\n\n\\draw[<->] (0.9,4.5) -- (5.1,4.5);\n\\node[below,white] at (3,4.5) {$\\diameter$ = 7 mm};\n\n\\draw[dashed] (2.4,9) -- (2.4,9.6);\n\\draw[dashed] (3.6,9) -- (3.6,9.6);\n\\draw[<->] (2.4,9.5) -- (3.6,9.5);\n\\node[above] at (3,9.6) {$w$ = 1.6 mm};\n\\end{scope}\n\n\\begin{scope}[shift={(7,0)}]\n\\draw (0,0) rectangle (6,9);\n\n\\draw (3,4.5) circle (2.1);\n\\draw[fill=white,white] (2.4,0) rectangle (3.6,9);\n\\draw (2.4,0) rectangle (3.6,9);\n\\draw[fill=white,white] (3,4.5) circle (2.082);\n\\draw[fill=gray,gray] (2.5,0) rectangle (3.5,4.5);\n\\draw[fill=gray,gray] (3,4.5) circle (2.0);\n\n\\draw[dashed] (0,7.5) -- (6,7.5) node[right]{dice line};\n\\draw [<-] (5.15,4.5) -- (7,4.5) node[right]{basin};\n\\draw [<-] (4.7,4) -- (7,4) node[right]{Al electrode};\n\n\\draw [<-] (3.6,1) -- (7,1) node[right]{channel};\n\\end{scope}\n\\end{tikzpicture}\n\\label{fig:ChipGeometry}\n\\caption{Diagram of a Helmholtz resonator top and bottom chip. The dice line indicates where the chip has been diced to expose the electrode for wiring.}\n\\end{figure*}\n\n\\begin{table*}[]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n\\begin{tabular}[c]{@{}c@{}}Device \\\\ Number\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Basin\\\\ Diameter\\\\ (mm)\\end{tabular} &\n\\begin{tabular}[c]{@{}c@{}}Electrode\\\\ Diameter\\\\ (mm)\\end{tabular} &\n\\begin{tabular}[c]{@{}c@{}}Channel\\\\ Width\\\\ (mm)\\end{tabular} &\n\\begin{tabular}[c]{@{}c@{}}Electrode\\\\ Width\\\\ (mm)\\end{tabular} &\n\\begin{tabular}[c]{@{}c@{}}Channel\\\\ Length\\\\ (mm)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Electrode\\\\ Height\\\\ (nm)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Etch\\\\ Depth\\\\ (nm)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Basin\\\\ Confinement\\\\ (nm)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Channel\\\\ Confinement\\\\ (nm)\\end{tabular} \\\\ \\hline\n1  & 7.00 & 6.80 & 1.60 & 1.40 & 1.38 & 59 & 348 & 577 & 636  \\\\\\hline\n2  & 7.00 & 6.90 & 1.60 & 1.50 & 1.38 & 61 & 433 & 745 & 805  \\\\\\hline\n3  & 7.00 & 6.80 & 1.60 & 1.40 & 1.38 & 60 & 564 & 1007 & 1067\\\\\\hline\n\\end{tabular}\n\\caption{Summary of device geometries.}\n\\label{tab:DeviceMeasurements}\n\\end{table*}\n\nIt should be noted that the overlap of the two electrodes inside of the basin means that the confinement is $\\sim 60$ nm smaller in the basin than in the channel.  While this would result in different behavior for the fluid in the basin, it is irrelevant to our measurement as our experiment is insensitive to the fluid inside the basin.  This is because there is essentially no superflow inside the basin. In Fig.~SM~2, we show a finite element simulation of a Helmholtz resonator velocity profile.  The Helmholtz mode is dominated by the motion of fluid in the channels.  As our measurement probes the superfluid fraction via superflow, it is also dominated by the superfluid in the channels.  Furthermore, since the geometric factor connecting the resonant frequency to the superfluid fraction depends on the area of the Helmholtz resonator in which superflow occurs, any signal due to the small channel-basin interface can be safely discounted. Similarly, the small gap between the electrode and channel wall can be ignored due to it making up a small fraction of the total area.\n\n\\begin{figure*}[h]\n    \\includegraphics[width=.7\\textwidth]{SI3.png}\n    \\caption{Finite element model of normalized fluid velocity for a Helmholtz resonance.  The Helmholtz resonance is dominated by motion of the fluid in the channel, hence the resonance frequency in our experiment is dominated by the superflow in the channel.  }\n \\label{fig:flow}\n \\centering\n\\end{figure*}\n\n\\section*{Appendix C: Thermal Effects}\nA detailed model of thermal effects in our Helmholtz resonator geometry has been developed in Ref.~\\citenum{Souris17}.  This model concerns the thermal resistance between the helium in the basin and the surrounding helium via the helium in the channel, as well as the thermal resistance at the boundary between the helium and solid surfaces (called the Kapitza resistance \\cite{Kapitza41}) and the thermal resistance through the quartz.  Heat flux through the chip meets boundary resistance at each of the material interfaces: $^3$He to aluminum, aluminum to quartz (which is negligibly small), and quartz to $^3$He. Additionally there is some thermal resistance while propagating through each material.\n\nTo estimate the Kapitza resistance, Bekarevich and Khalatnikov have developed a model for thermal resistance between $^3$He and a solid surface that is applicable for temperatures below 0.2 K where energy flux is transmitted primarily through collective excitations of zero sound \\cite{Bekarevich61}. The model predicts the resistance over an area $A_{\\text{tot}}$ to be\n\\begin{equation}\n    A_{\\text{tot}}R_KT^3 = \\frac{5h^3}{8\\pi^5k_B^4} \\left(\\frac{\\rho}{\\rho_{\\text{He}}} \\right) \\frac{c^3}{v_F} \\left(\\frac{10^{-7}}{aF+b\\Phi}\\right),\n    \\label{eq:KapitzaModel}\n\\end{equation}\nwhere $\\rho$ and $c$ are the density and transverse speed of sound in the solid (i.e. aluminum or quartz), $v_F$ is the Fermi velocity of $^3$He, $a$ and $b$ are constants describing the diffusivity of quasiparticle scattering, and $F$ and $\\Phi$ are material constants which depend on the ratio of the transverse and longitudinal speeds of sound in the solid. Following the analysis of copper by Anderson et al.~\\cite{Anderson64}, we take $F=1.6$, $\\Phi=1$, $a=0.38$, and $b=0.05$. The difference in these values for aluminum and quartz is expected to be relatively small. \n\nWe consider the resulting Kaptiza resistance at 20 bar, where we measure the largest gap between the two observed phase transitions. At this pressure $\\rho_{\\text{He}} = 108.7$ kg\/m$^3$ and $v_F = 38.7$ m\/s. The material parameters for aluminum are taken to be $\\rho_{\\text{Al}} = 2730$ kg\/m$^3$ and $c_{\\text{Al}} = 3240$ m\/s. For quartz, they are $\\rho_{\\text{Q}} = 2650$ kg\/m$^3$ and $c_{\\text{Q}} = 2210$ m\/s. The total area of one side of the device including the basin and two channels is $4.29\\times 10^{-5}$ m$^2$. At $1.6$ mK, approximately where the PDW to B-phase transition occurs, the thermal boundary resistance for aluminum and quartz respectively are thus\n\\begin{equation}\n    R_{K,\\text{Al}} = 3.12 \\times 10^{5}~\\textrm{K\/W}\n\\end{equation}\nand\n\\begin{equation}\n    R_{K,\\text{Q}} = 9.62 \\times 10^{4}~\\textrm{K\/W}.\n\\end{equation}\nEach of these boundary resistances is added in series with the thermal resistances of aluminum, $R_{\\text{Al}}$, and quartz, $R_{\\text{Q}}$, to get the chip resistance. Given that the electrode thickness is four orders of magnitude smaller than the quartz thickness it can safely be neglected. Taking into account the two chip surfaces and two superfluid channels, which are all added in parallel, we find the total thermal resistance to be\n\\begin{equation}\n    R_{\\text{tot}} = \\left( \\frac{2}{R_{K,\\text{Al}}+R_{\\text{Q}}+R_{K,\\text{Q}} } +  \\frac{2}{R_{\\text{He}}} \\right)^{-1}.\n\\end{equation}\nThe thermal resistance across $0.5$ mm of quartz is found to be $R_{\\text{Q}} = 758$ K\/W by extrapolating data from \\cite{Vollhardt13} to $1.6$ mK. For superfluid $^3$He-B at 20 bar the thermal resistance is obtained from \\cite{Hall86}. The result is found to be $R_{\\text{He}}= 2.07\\times 10^{4}$ K\/W. Adding these resistances according to Eq.~4 gives\n\\begin{equation}\n    R_{\\text{tot}} = 9.85 \\times 10^3~\\textrm{K\/W}.\n\\end{equation}\nTherefore, the power required to produce a thermal gradient equal to the temperature gap between transitions of $0.2$ mK would be \n\\begin{equation}\n    \\dot{Q} = \\frac{\\Delta T}{R_{\\text{tot}}} = 20.3~\\textrm{nW}.\n\\end{equation}\nThis number rules out the possibility of heating effects contributing to our signal for the following reasons.  First, this number would have to apply to each device, since they each have identical electrodes, and would imply that the heat leak from the measurement alone would have to be larger than 60.9 nW.  This is roughly 200 times larger than expected heat leak into the entire nuclear demagnetization stage \\cite{Haard01}. Furthermore, prior experimental evidence \\cite{Harrison79} suggests that Eq.~\\ref{eq:KapitzaModel} typically over estimates the thermal resistance, which means the actual required power would be even higher than predicted. This calculation should therefore be taken as a lower limit on the power required to produce the observed temperature gap.\n\nSecond, such heating could not explain the different widths of the PDW state shown in the main text.  Since the electrodes in each device are identical, one expects each device to have identical heating and hence produce identical thermal gradients, inconsistent with the varying widths of the PDW phase observed in Fig.~4 of the main text.  Third, in our geometry heating effects would result in a thermal gradient across the helium channel, which is the only helium that contributes to the Helmholtz signal, and not two distinct regions of temperature.  Hence from such a heating model, one would expect a single broad phase transition, and not two sharp phase transitions.  Fourth, one cannot reconcile this level of heating from Joule heating as our electrodes are superconducting inside the basin and along the channel.  The only Ohmic losses in our electrical circuit would exist where the measurement leads are silver epoxied to the electrodes, but this contact is made in the surrounding bath of helium and cannot produce heating inside the helium channel of the Helmholtz resonator.  Fifth and finally, we note that the temperatures at which the observed phase transitions occur do not depend on the measurement drive, further ruling out any thermal effects from the measurement.\n\nIn light of these thermal calculations and experimental observations, we conclude that the stable phase transitions observed in our devices are genuine and do not result from any untoward thermal effects. \n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Appendix}\n\\subsection{Experimental Setup}\\label{supp:setup}\n\\paragraph{Details of Datasets}\nImages in MNIST and Omniglot are of size $28 \\times 28$, whereas ones in CIFAR-10 are $32 \\times 32$. \nWe center crop images in CelebA with $148 \\times 148$ bounding boxes and resize them into $64 \\times 64$ following~\\citep{DBLP:conf\/icml\/LarsenSLW16,dinh2016density}.\nOmniglot has $50$ different alphabets and $2089$ characters. \nEach alphabet has $41$ characters on average and each character belongs to only one alphabet. \nThere are $11$ images per character on average. \nThe appearance of different alphabets vary a lot while the appearance of different characters within the same alphabet are similar. \nThe original number of training images and test images are $24345$ and $8070$ respectively. \n\n\\paragraph{Details of Baselines}\n\nWe use the open-source implementation of \nDRAW\\footnote{\\url{https:\/\/github.com\/czm0\/draw_pytorch}},\nVQ-DRAW\\footnote{\\url{https:\/\/github.com\/unixpickle\/vq-draw\/tree\/official-release}}, \nAIR\\footnote{\\url{https:\/\/github.com\/addtt\/attend-infer-repeat-pytorch}},\nWGAN\\footnote{\\url{https:\/\/github.com\/martinarjovsky\/WassersteinGAN}}, and PixelCNN++\\footnote{\\url{https:\/\/github.com\/pclucas14\/pixel-cnn-pp}}. \n\nFor VAEs, we use an architecture with the number of parameters comparable to our model. \nWe follow the hyperparameters used in the original paper if they are given. \n\nFor DRAW, we use the same set of hyperparameters for both MNIST and Omniglot, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, the number of glimpses is 64, the hidden size of LSTM is 256, the latent size is 100, the read-size is $2 \\times 2$, and the write-size is $5 \\times 5$. \nFor experiments on CIFAR-10 and CelebA, we use the number of glimpses as 64, the hidden size of LSTM as 400, the latent size as 200, the read-size as $ 5 \\times 5$, and the write-size as $5 \\times 5$.\n\n\nExperiments on VQ-DRAW follow the default setting as the released code. \nFor AIR, we set the maximum number of steps as $3$, following the setting in \\citep{eslami2016attend}.\nSince AIR is originally tested on Multi-MNIST, where the number of digits, sizes of digits, and locations of the digits vary. \nTo adapt AIR to datasets we use, we increase the present-probability in their prior as well as the box size and reduce the variance of the location in the prior. \nWe try several groups of hyperparameters (present-probability in $\\{0.01,0.6,0.8\\}$, object size in $\\{8,12,20\\}$, std of object location in $\\{1.0, 0.1, 0.001\\}$, and object scale in $\\{0.5,1,1.2,2\\}$) for the prior.\nThe best hyperparameters we found are present-probability $= 0.8$, object size $= 20$, std of object location $=0.001$ for all datasets. \nThe object scale is set to $0.7$ for CelebA and $1.2$ for the rest.\nWe also tune the weight of the KL term in their loss function.\nSpecifically, we try the KL weight equal to $1$ or $2$ for the ``what-to-draw\" latent variable in their model.\nWe found $2$ is better and fix it for the rest of experiments. \n\nWe show generated images from AIR on MNIST in Fig. \\ref{fig:vis_appendix_AIR_MNIST} to give more intuition on why AIR has a low FID score. \nFrom Fig. \\ref{fig:vis_appendix_AIR_MNIST_a}, we can see that the reconstruction quality of AIR is reasonable.\nIn terms of generation, as illustrated in Fig. \\ref{fig:vis_appendix_AIR_MNIST_b}, AIR fails to generate realistic images which is consistent with numbers reported in Table \\ref{table:main_res:NLL}, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, AIR has worse NLLs on both MNIST and Omniglot datasets compared to other models. . \nOne reason is that the prior of AIR is independent across time steps and is still far from being as good as the posterior (\\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot, comparing Fig. \\ref{fig:vis_appendix_AIR_MNIST_a} with Fig. \\ref{fig:vis_appendix_AIR_MNIST_b}). \nFor example, the model chooses a location to write at each time step without depending on where it wrote previously. \nTherefore, during sampling, the box location tends to jump in the canvas in an arbitrary manner. \nIn our experiments, we set the variance of the object location in the prior to be small, but parts (boxes) are still placed on the canvas somewhat randomly. \nThis severely degrades the sample quality since parts are not well aligned.\n\n\\begin{figure}\n\\centering \n\\begin{subfigure}[t]{0.48\\textwidth}\n    \\includegraphics[width=\\textwidth,trim={0 7.45cm 0 0},clip,valign=m]{fig\/vis_sample\/air\/mnist\/recont_step_266000.png}\n    \\caption{Reconstruction of AIR on MNIST. Each pair of images consist of the original MNIST data on the left and the reconstructed image from AIR on the right. The red boxes indicate the predicted location at each time step. We can see that AIR is able to reconstruct the original MNIST data. The model keeps refining the digits at the same location across different time steps. \n   \n    } \n    \\label{fig:vis_appendix_AIR_MNIST_a}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.48\\textwidth}\n    \\includegraphics[width=\\textwidth,trim={0 6.45cm 0cm 0},clip,valign=m]{fig\/vis_sample\/air\/mnist\/sample_step_248000.png} \n    \\caption{Sampled images from AIR. The red boxes indicate the generated locations at each time step. The locations across time steps vary in a somewhat arbitrary manner. Moreover, the appearance of some generated digits per step are too dim to be visible. }\n    \\label{fig:vis_appendix_AIR_MNIST_b}\n\\end{subfigure}\n\\caption{Sampled and reconstructed images from AIR.}\n\\label{fig:vis_appendix_AIR_MNIST}\n\\end{figure}\n\n\\paragraph{Details of Our Architecture} \n\nFor our prior model, we use a eight layer Vision Transformer \\citep{dosovitskiy2020image} with hidden size $64$ where each time step is one node and the sequence length is equal to the maximum time step. \nWe use dropout with probability $0.1$ for Transformer which gives better training losses compared to the prior without dropout. \nGiven a canvas at $t$, we first have a shallow CNN to extract the feature. The shallow CNN consists of two layers with hidden size 16 and a ReLU unit after the first convolutional layer. \nThe canvas feature is then added to the positional embedding to serve as input feature to Transformer. \nThe prior model can attend to the feature of the canvas at any time step $< t$ and then predict the ``what-to-draw\" ${\\mathbf{z}}_{\\text{id}}$, ``where-to-draw\" ${\\mathbf{z}}_{\\text{loc}}$, and ``whether-to-draw\" ${\\mathbf{z}}_{\\text{is}}$. \nThe positional embedding follows \\citep{vaswani2017attention} and encodes the generation step index. \n\nFor experiments on Omniglot and MNIST datasets, our encoder consists of four convolutional layers, while the decoder has two convolutional layers with stride $2$, two convolutional layers with stride $1$ and two transposed convolutional layers. \nAll convolutional layers are followed by Batch Normalization and a ReLU activation function. \nThere are three independent MLP heads to predict the approximated posteriors at $T$ time steps. \nIn particular, we uniformly divide the 2D feature map into $T$ parts and feed the $t$-th feature to get the corresponding ${\\mathbf{z}}_{\\text{id}}$, ${\\mathbf{z}}_{\\text{loc}}$, and ${\\mathbf{z}}_{\\text{is}}$ at step $t$. \nWe set the canvas size equal to the image size for MNIST, Omniglot, and CIFAR-10 while use a downsampled ($32 \\times 32$) canvas for CelebA. \n\n\\paragraph{Details of Training} \nWe first train our prior model for $200$ epochs with batch size $64$ and learning rate $1e^{-4}$. \nAfter pre-training, we choose the prior model with the lowest validation loss and fix it during the training of the full model. \nThe full model is trained with batch size $150$ and learning rate $1e^{-3}$. \nWe use the Adam optimizer for training the model.  \nThe patch size and the number of parts in the part bank are $5 \\times 5$ and $50$ for MNIST and Omniglot, and $4 \\times 4$ and $200$ for CIFAR-10 and CelebA.\nThe maximum number of steps for MNIST and Omniglot is $36$ and the maximum number of steps for CIFAR-10 and CelebA is $64$.\n\n\\paragraph{Optimization}\nOur latent space contains discrete distribution, which makes the loss difficult to optimize. We use gumbel-softmax~\\citep{maddison2016concrete, jang2016categorical} for gradient estimation.\n\nWe can expand the KL term in our objective as follow: \n\\begin{align} \\small  \\nonumber\n    & D_{\\mathrm{KL}}( q({\\mathbf{z}} \\vert {\\mathbf{x}}) \\Vert p({\\mathbf{z}}) ) \\\\ \\nonumber\n    & = \\mathbb{E}_{q({\\mathbf{z}} \\vert {\\mathbf{x}})} \\left[ \\log \\frac{q({\\mathbf{z}} \\vert {\\mathbf{x}})}{p({\\mathbf{z}})} \\right] \\\\   \\nonumber\n    & = \\mathbb{E}_{q({\\mathbf{z}} \\vert {\\mathbf{x}})} \\Big[ \\sum_{t=1}^T \n    \\log {\n    q_\\phi({\\mathbf{z}}_{\\text{id}}^{t} \\vert {\\mathbf{x}}) q_\\phi({\\mathbf{z}}_{\\text{loc}}^{t} \\vert {\\mathbf{x}}) q_\\phi({\\mathbf{z}}_{\\text{is}}^{t} \\vert {\\mathbf{x}})\n    } \\\\ & - \\log {\n    p_{\\theta}({\\mathbf{z}}_{\\text{id}}^{t} \\vert {\\mathbf{c}}^{<t}, {\\mathbf{z}}_{\\text{loc}}^{<t}) \n    p_{\\theta}({\\mathbf{z}}_{\\text{loc}}^{t} \\vert {\\mathbf{c}}^{<t}, {\\mathbf{z}}_{\\text{loc}}^{<t}) \n    p_{\\theta}({\\mathbf{z}}_{\\text{is}}^{t} \\vert {\\mathbf{c}}^{<t}, {\\mathbf{z}}_{\\text{loc}}^{<t}) } \\Big], \n\\end{align}\nwhere we can apply the Monte Carlo estimators. \n\n\\subsection{More Results on Generation}\nIn Fig \\ref{fig:vis_sec61}, we show more visualization of sampled images from different models on all datasets. \nFrom the figure, we can see that our method clearly outperforms VAE and VQ-DRAW and is comparable to WGAN on all datasets (even better than WGAN on Omniglot).\nWe also compare our model against state-of-the-art likelihood based generative models in Table \\ref{table:main_res:NLL}.\nWe again approximate the likelihood of VAE-family of models using importance sampling with $50$ samples.\nFrom the table, we can see that ours is comparable to other structured image models but worse than other generic models including the vanilla VAE.\nThis is likely caused by two facts. \nFirst, the construction and the dimension of the latent space between ours and other VAEs are very different which would affect the numerical values of ELBO.\nSecond, likelihood (or ELBO) in general is not a faithful metric that well captures the visual quality of images. \n\n\n\\renewcommand{\\arraystretch}{1.5\n\\begin{figure*}\n    \\centering\n    \\settowidth\\rotheadsize{Omniglot}     \n    \\begin{tabular}{@{\\hspace{0mm}}c@{\\hspace{0.5mm}}c@{\\hspace{1mm}}c@{\\hspace{1mm}}c@{\\hspace{1mm}}c@{\\hspace{1mm}}c@{\\hspace{1mm}}c@{\\hspace{1mm}}c@{\\hspace{1mm}}c}\n        \n            \\rothead{\\centering MNIST} &\n        \\includegraphics[width=0.24\\textwidth,trim={0 9.5cm 0 0},clip,valign=m]{fig\/vis_sample\/mnist_ours\/sample_192_step_-1.png} & \n        \\includegraphics[width=0.24\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/vqdraw\/mnist\/10k_sample_E0_step_18.png} & \n       \n        \\includegraphics[width=0.24\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/wgan\/mnist\/vis.png} & \n        \\includegraphics[width=0.24\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/vae\/mnist\/vis.png} & \n       \n       \n            \\\\ \\addlinespace[1mm]\n            \\rothead{\\centering Omniglot} &\n        \\includegraphics[width=0.24\\textwidth,trim={0 9.5cm 0 0},clip,valign=m]{fig\/vis_sample\/omniglot_ours\/sample_799_step_3.png} & \n        \\includegraphics[width=0.24\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/vqdraw\/omniglot\/vis.png} & \n       \n        \\includegraphics[width=0.24\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/wgan\/omniglot\/fake_samples_50000.png} & \n        \\includegraphics[width=0.24\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/vae\/omniglot\/vis.png} & \n       \n       \n            \\\\ \\addlinespace[1mm]\n            \\rothead{\\centering CIFAR} &\n        \\includegraphics[width=0.24\\textwidth,trim={0 10.8cm 0 0},clip,valign=m]{fig\/vis_sample\/cifar_ours\/sample_150_step_-1.png} & \n        \\includegraphics[width=0.24\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/vqdraw\/cifar\/vis.png} & \n       \n        \\includegraphics[width=0.24\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/wgan\/cifar\/vis.png} & \n        \\includegraphics[width=0.24\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/vae\/cifar\/vis.png} & \n       \n       \n            \\\\ \\addlinespace[1mm]\n            \\rothead{\\centering CelebA} &\n        \\includegraphics[width=0.24\\textwidth,trim={0 21cm 0 0},clip,valign=m]{fig\/vis_sample\/celeba_ours\/sample_step_129.png} &\n        \\includegraphics[width=0.24\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/vqdraw\/celeba\/vis.png} & \n       \n        \\includegraphics[width=0.24\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/wgan\/celeba\/vis.png} & \n        \\includegraphics[width=0.24\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/vae\/celeba\/vis.png} & \n       \n       \n       \n       \n       \n       \n        \\\\ \n        {\\footnotesize  } \n        & {\\footnotesize (a) Ours} \n        & {\\footnotesize (b) VQ-DRAW} \n       \n        & {\\footnotesize (c) WGAN}\n        & {\\footnotesize (d) VAE}\n       \n       \n        \\\\\n    \\end{tabular}\n   \n    \\caption{Visualization of generated images from different models on different dataset.}\n   \n    \\label{fig:vis_sec61}\n\\end{figure*}\n\n\n\\begin{table}\n\\centering\n\\resizebox{0.49\\textwidth}{!}{\n    \\begin{tabular}{lcccc}\n        \\toprule\n        {\\bf Method} & {\\bf MNIST} & {\\bf Omniglot}  & {\\bf CIFAR-10} & {\\bf CelebA} \\\\\n                     & 28$\\times$28 & 28 $\\times$ 28 & 32$\\times$32 & 64$\\times$64 \\\\\n\n\n        \\midrule \n        VAE       & $\\leq$ 87.00 & $\\leq$ 107.90 & $\\leq$ 5.70 & $\\leq$ 5.76 \\\\\n        NVAE & $\\leq$ 79.60 & $\\leq$ 92.80 & $\\leq$ \\textbf{2.91} & $\\leq$  $\\textbf{2.06}$ \\\\\n        BIVA & $\\leq$ \\textbf{78.41} & $\\leq$ \\textbf{91.34} & $\\leq$ 3.08 & $\\leq$ 2.48   \\\\\n        Real NVP  & 80.34 & 99.60 & 3.49 & 3.07 \\\\\n\n        \\midrule \n        PixelCNN++   &  79.92       & \\textbf{90.82}      &  \\textbf{3.10}       & \\textbf{2.26} \\\\\n        AIR          & $\\leq$ 128.40  & $\\leq$ 116.08 & * & * \\\\ \n        DRAW         & $\\leq$ \\textbf{66.07}  & $\\leq$ 96.54 & * & *  \\\\ \n        Ours         & $\\leq$ 98.92 & $\\leq$ 129.73 & $\\leq$ 5.48 & $\\leq$ 5.89 \\\\\n\n        \\bottomrule\n    \\end{tabular}\n}\n\\caption{Comparison against the state-of-the-art likelihood-based generative models. The performance is measured in bits\/dimension (bpd) for all the datasets but MNIST and Omniglot in which negative log-likelihood in nats is reported (the lower the better in all cases). * entries are incomparable since they are using continues likelihood for data. } \n\\label{table:main_res:NLL}\n\\end{table}\n\n\n\\subsection{More Results on Low-Data Learning}\n\nIn Fig \\ref{fig:vis_sec62}, we show more visualization of sampled images from our model under different percentages of training data. \nAs one can see from the figure, the visual quality of our samples improves as more training data is included.\nEven with $0.1\\%$ training data, our model could still generate plausible faces on CelebA dataset.\n\n\n\n\\subsection{Ablation Study}\nTo figure out a reasonable range of values fro the patch size $K$ and the size of part bank $B$, we measure the visual quality of the output of our heuristic parsing algorithm using the peak signal-to-noise ratio (PSNR).\nIn particular, we paste all the selected parts returned by our heuristic parsing on a blank canvas and compute the PSNR between the original image with the created canvas. \nPSNR is computed as PSNR=$20 \\log_{10}(\\text{MAX}_I) - 10 \\log_{10}(\\text{MSE})$, where $\\text{MAX}_I$ is the maximum value of the pixel intensity and $\\text{MSE}$ is the mean squared error between the pasted canvas and the original image.\nFor the threshold $\\epsilon$ used in our heuristic parsing algorithm, we set its value as $0.01$ based on the best PSNR.\n\n\n\n\\renewcommand{\\arraystretch}{1.5\n\\begin{figure*}\n    \\centering\n    \\settowidth\\rotheadsize{{\\scriptsize Alphabet-Split-Omniglot}}\n   \n   \n   \n   \n    \\begin{tabular}{@{\\hspace{0mm}}c@{\\hspace{0.5mm}}c@{\\hspace{0.5mm}}c@{\\hspace{0.5mm}}c@{\\hspace{1mm}}c@{\\hspace{1mm}}c@{\\hspace{1mm}}c\n        \n            \\rothead{\\centering {\\scriptsize Sample-Split-MNIST}} &\n        \\includegraphics[width=0.32\\textwidth,trim={0 4.1cm 0 0},clip,valign=m]{fig\/vis_low\/vloc\/mnist\/001\/vis.png} &\n        \\includegraphics[width=0.32\\textwidth,trim={0 4.1cm 0 0},clip,valign=m]{fig\/vis_low\/vloc\/mnist\/01\/vis.png} &\n        \\includegraphics[width=0.32\\textwidth,trim={0 4.1cm 0 0},clip,valign=m]{fig\/vis_low\/vloc\/mnist\/03\/vis.png} \n       \n      \n      \n       \n       \n       \n          \\\\   [-0mm]\n          {\\footnotesize  } \n        & {\\footnotesize (a) 0.1\\% } \n        & {\\footnotesize (b) 1\\%} \n        & {\\footnotesize (c) 3\\%}\n       \n       \n       \n       \n            \\\\  [-0mm]\n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n           \n            \\rothead{\\centering {\\scriptsize Alphabet-Split-Omniglot}} &\n        \\includegraphics[width=0.32\\textwidth,trim={0 4.1cm 0 0},clip,valign=m]{fig\/vis_low\/vloc\/aomni\/03\/vis.png} &\n        \\includegraphics[width=0.32\\textwidth,trim={0 4.1cm 0 0},clip,valign=m]{fig\/vis_low\/vloc\/aomni\/05\/vis.png} &\n        \\includegraphics[width=0.32\\textwidth,trim={0 4.1cm 0 0},clip,valign=m]{fig\/vis_low\/vloc\/aomni\/10\/vis.png} &\n       \n       \n       \n       \n            \\\\ \n          [-0mm]\n          {\\footnotesize  } \n        & {\\footnotesize (a) 3\\%}\n        & {\\footnotesize (b) 5\\%}\n        & {\\footnotesize (c) 10\\%}\n       \n       \n       \n       \n        \\\\ [-0mm]\n            \\rothead{\\centering {\\scriptsize Sample-Split-CIFAR}} &\n        \\includegraphics[width=0.32\\textwidth,trim={0 4.65cm 0 0},clip,valign=m]{fig\/vis_low\/vloc\/cifar\/001\/vis.png} &\n        \\includegraphics[width=0.32\\textwidth,trim={0 4.65cm 0 0},clip,valign=m]{fig\/vis_low\/vloc\/cifar\/01\/vis.png} &\n        \\includegraphics[width=0.32\\textwidth,trim={0 4.65cm 0 0},clip,valign=m]{fig\/vis_low\/vloc\/cifar\/03\/vis.png} &\n       \n       \n       \n       \n       \n          \\\\    [-0mm]\n          {\\footnotesize  } \n        & {\\footnotesize (a) 0.1\\% } \n        & {\\footnotesize (b) 1\\%} \n        & {\\footnotesize (c) 3\\%}\n       \n       \n       \n       \n            \\\\ [-0mm]\n            \\rothead{\\centering {\\scriptsize Sample-Split-CelebA}} &\n        \\includegraphics[width=0.32\\textwidth,trim={0 9.3cm 0 0},clip,valign=m]{fig\/vis_low\/vloc\/celeba\/001\/vis.png} &\n        \\includegraphics[width=0.32\\textwidth,trim={0 9.3cm 0 0},clip,valign=m]{fig\/vis_low\/vloc\/celeba\/01\/vis.png} &\n        \\includegraphics[width=0.32\\textwidth,trim={0 9.3cm 0 0},clip,valign=m]{fig\/vis_low\/vloc\/celeba\/03\/vis.png} &\n       \n       \n       \n       \n       \n          \\\\   [-0mm]\n          {\\footnotesize  } \n        & {\\footnotesize (a) 0.1\\% } \n        & {\\footnotesize (b) 1\\%} \n        & {\\footnotesize (c) 3\\%}\n       \n       \n       \n       \n        \\\\ [-0mm]\n    \\end{tabular}\n   \n    \\caption{Visualization of generated images from our model on all datasets under different percentages of training data.}\n   \n    \\label{fig:vis_sec62}\n\\end{figure*}\n\n\\section{Conclusion}\\label{sect:conclusion}\n\nIn this paper, we propose a structured latent variable model which generates images in a part-by-part fashion.\nWe construct a non-parametric distribution over the latent variables that describe the appearance of image parts.\nWe then propose a Transformer based auto-regressive prior to capture sequential dependencies among parts.\nAt last, we develop a heuristic parsing algorithm to effectively pre-train the prior.\nExperiments on several challenging datasets show that our model significantly outperforms previous structured image models and is on par with state-of-the-art generic generative models.\nFuture work include exploring more expressive encoder, multi-scale generation, and other structured priors beyond auto-regressive models.\n\n\\section{Experiments}\\label{sect:exp}\n\nWe test our model on four datasets, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, MNIST \\citep{lecun1998gradient}, Omniglot \\citep{lake2015human}, CIFAR-10 \\citep{Krizhevsky2009a}, and CelebA \\citep{liu2015faceattributes}, from three different perspectives, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, image generation under full training data, image generation under low-data (\\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, less data than the original training set), and the latent space editing. \nMore details on datasets can be found in Appendix.\n\n\n\\paragraph{Baselines} \nWe compare our method with two classes of models, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, 1) structured image models including DRAW \\citep{gregor2015draw}, AIR \\citep{eslami2016attend}, PixelCNN++ \\citep{Salimans2017PixeCNN}, and VQ-DRAW \\citep{nichol2020vq}; 2) generic generative models including VAE, 2sVAE \\citep{DBLP:conf\/iclr\/DaiW19}, NVAE \\citep{vahdat2020nvae}, WGAN \\citep{arjovsky2017wasserstein}, snGAN \\citep{DBLP:journals\/corr\/abs-1802-05957} and WGAN-GP \\citep{gulrajani2017improved}. \nFor all baselines, we use the publicly released code. \nWe control the number of parameters of all methods roughly the same. \nMore details on hyperparameters can be found in Appendix.\n\n\n\\paragraph{Evaluation Metric} For all experiments, we compute the FID score \\citep{Heusel2017GANsTB} and the negative log likelihood (NLL) (if applicable). \nWe draw 10K samples from each model and compute the FID score w.r.t\\onedot} \\def\\dof{d.o.f\\onedot 10K images in the test set. \nFor evaluating NLL of VAEs, we use $50$ importance weighted samples to get a tighter bound as in \\citep{DBLP:journals\/corr\/BurdaGS15}.\nWe train all models for $500$ epochs and test the checkpoint with the best validation FID score.\n\n\n\\begin{table}\n\\vspace{-0.5cm}\n\\centering\n\\resizebox{\\linewidth}{!}{\n    \\begin{tabular}{lcccc}\n        \\toprule\n        {\\bf Method} & {\\bf MNIST} & {\\bf Omniglot}  & {\\bf CIFAR-10} & {\\bf CelebA} \\\\ \n        & 28$\\times$28 & 28 $\\times$ 28 & 32$\\times$32 & 64$\\times$64 \\\\  \n        \\midrule\n        VAE   & 16.13           & 31.97 & 106.7 & 70 \\\\ \n        2sVAE* & 12.6            &  - & 72.9 & 44.4 \\\\ \n        NVAE  & \\textbf{7.93}   & \\textbf{6.84} & 55.97 & \\textbf{14.74}  \\\\ \n        snGAN & -               & - & \\textbf{14.2} & -   \\\\ \n        WGAN  & 10.28           & 13.99 & 54.82 & 40.29 \\\\ \n        WGAN GP* & -            & - & 42.18 & 30.3 \\\\ \n\n        \\midrule\n        PixelCNN++ & 11.38 & 9.18 & 68.00 & 72.46 \\\\ \n        AIR      & 482.69 & 373.93 & 673.93 & 399.41 \\\\\n        DRAW     & 27.07 & 47.22 & 162.00 & 157.00 \\\\ \n        VQ-DRAW  & 19.64 & 56.03 & 80.19 & 41.87 \\\\ \n        Ours & \\textbf{2.48} & \\textbf{5.53} & \\textbf{62.72} & \\textbf{41.10} \\\\ \n\n        \\bottomrule\n    \\end{tabular}\n}\n\\caption{Comparison of sample qualities (lower FID score is better). * entries are from 2sVAE paper. - means unavailable.} \n\\label{table:main_res}\n\\end{table}\n\n\n       \n       \n       \n       \n       \n       \n       \n       \n       \n\n\\vspace{-0.1cm}\n\\subsection{Image Generation}\\label{subsect:image_generation}\n\nWe first compare our model with all competitors on image generation under the full training data.\nAs shown in Table \\ref{table:main_res}, our model significantly outperforms other structured image models in terms of the FID score. \nWe also report NLLs of all models (if applicable) in Appendix. \nOur model is comparable to the state-of-the-art generic generative models like NVAE and WGAN-GP.\nMoreover, our model is easier to train compared to other structured image models.\nFor example, while training AIR, we found it requires careful tuning to avoid a diverging loss. \nNote that AIR does not perform well on these datasets, which is likely due to their independent priors (\\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot, latent locations drawn from independent Gaussians) over different time steps. \nTherefore, parts drawn at different steps tend to be randomly placed on the canvas, leading to worse FID scores.  \nWe provide more visualization of our generated canvases and images in Fig. \\ref{fig:vis_sec41}.\nIt is clear that our canvases do capture the rough objects\/scenes in the images.\nWe show more visual comparisons with other models in Fig.~\\ref{fig:vis_sec61} and the Appendix.\n\n\\begin{table}[]\n  \n    \\centering\n    \\resizebox{\\linewidth}{!}{\n    \\begin{tabular}{lll|ccccc}\n    \\toprule    \n    K & M & $\\lambda$      & PSNR   & FID   & NLL  & BCE    & KLD   \\\\    \n    \\midrule\n    \\midrule\n    5 & 50 & 50          & \\textbf{17.71}  & \\textbf{5.53}  & 129.73    & \\textbf{89.95}  & 54.80 \\\\ \n    6 & 50 & 50          & 16.38  & 12.86 & \\textbf{118.37} & 108.81 & 25.62 \\\\\n    8 & 50 & 50          & 15.60  & 22.00 & 132.98 & 130.39 & \\textbf{21.34} \\\\    \n    \\midrule\n    \\midrule\n    5 & 50 & 50          & 17.71  & \\textbf{5.53}  & 129.73    & 89.95  & 54.80 \\\\ \n    5 & 10 & 50          & 15.77  & 15.02 & 127.22 & 115.53 & \\textbf{27.37} \\\\\n    5 & 200 & 50         & \\textbf{19.50}  & 6.08  & \\textbf{115.53} & \\textbf{89.94}  & 38.41 \\\\\n    \\midrule\n    \\midrule\n    5 & 50 & 50           & 17.71  & \\textbf{5.53}  & 129.73    & \\textbf{89.95}  & 54.80 \\\\ \n    5 & 50 & 0            & 17.71  & 35.27 & 110.64 & 112.07 & \\textbf{25.67} \\\\\n    5 & 50 & 1            & 17.71  & 5.58  & \\textbf{107.72} & 92.27  & 34.40 \\\\\n    5 & 50 & 100          & 17.71  & 5.74  & 134.54 & 92.23  & 57.96 \\\\\n    \\bottomrule\n    \\end{tabular}\n    }\n\\caption{Ablation study of patch size $K$ (top), part bank size $M$ (middle), and regularization $\\lambda$ (bottom) on Omniglot.}\n\\vspace{-0.4cm}\n\\label{tab:ablation}\n\\end{table} \n\n\n\\begin{figure*}[t]\n    \\centering\n     \\begin{subfigure}[b]{0.33\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{fig\/exp_few\/dmnist_fid.pdf}\n         \\vspace{-0.6cm}\n         \\caption{~}\n         \\label{fig:dmnist_fid}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.33\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{fig\/exp_few\/omni_fid.pdf}\n         \\vspace{-0.6cm}\n         \\caption{~}\n         \\label{fig:omni_fid}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.33\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{fig\/exp_few\/aomni_fid.pdf}\n         \\vspace{-0.6cm}\n         \\caption{~}\n         \\label{fig:aomni_fid}\n     \\end{subfigure}\\\\     \n     \\begin{subfigure}[b]{0.33\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{fig\/exp_few\/mnist_fid.pdf}\n         \\vspace{-0.6cm}\n         \\caption{~}\n         \\label{fig:mnist_fid}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.33\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{fig\/exp_few\/cifar_fid.pdf}\n         \\vspace{-0.6cm}\n         \\caption{~}\n         \\label{fig:cifar_fid}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.33\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{fig\/exp_few\/celeba_fid.pdf}\n         \\vspace{-0.6cm}\n         \\caption{~}\n         \\label{fig:celeba_fid}\n     \\end{subfigure}     \n   \n   \n   \n   \n   \n   \n   \n   \n\n   \n   \n    \\vspace{-0.3cm}\n    \\caption{FID score vs. percentage of training data for low-data regimes. Different splits are indicated in the subcaptions. Our method consistently outperforms other structured image models. All experiments are run with three random seeds. } \n    \\vspace{-0.4cm}\n    \\label{fig:low_data}\n\\end{figure*}\n\n\\vspace{-0.1cm}\n\\subsection{Image Generation with Low-Data}\\label{subsect:exp_low_data}\n\n\nIn this part, we investigate our model and other baselines under low-data regimes.\nThe reasoning is similar to \\citep{rezende2016one,lake2015human}, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, if our model could generate high-fidelity images with less training data, it is likely that our model better exploits the compositionality within the data.\nUnder each low-data setting, our model constructs the part bank only using the given limited training data, which still generalizes well to unseen images empirically. \nWe group the experiments based on how we split the original dataset to create the low-data regime, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, split by digits in MNIST, split by characters in Omniglot, split by alphabets in Omniglot, and split by samples in Omniglot and CelebA.  \nWe design such splits since only MNIST has the information of digits and only Omniglot has the information of alphabets and characters.\nAll results are shown in Fig \\ref{fig:low_data}. \n\n\\paragraph{Split by Digits}\nWe split the MNIST dataset based on digits, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, class labels.\nIn particular, there are $10$ digits in total each of which occupies $10\\%$ of the training data.\nWe train all models with different proportions of training data, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot, $10\\%, 20\\%, \\cdots, 80\\%$ which contain digits $\\{0\\}, \\{0,1\\}, \\cdots, \\{0,1,2,\\cdots, 7\\}$ respectively.\nWe use the original hold-out test set to evaluate different methods which contains all $10$ digits. \nAs shown in Fig. \\ref{fig:dmnist_fid}, we can see that our model performs similarly with other models in the extreme cases (\\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, $10\\%$ and $20\\%$) and outperforms others with other percentages.\nNote that the curves of DRAW and VQ-DRAW increase a bit after certain proportions which are likely due to their instable training.\n\n\n\n\\paragraph{Split by Characters}\nIn this setting, we split Omniglot (containing $2089$ characters in total and $11$ images per character on average) based on characters.\nWe randomly sample certain proportions (shown in the X-axis of Fig. \\ref{fig:omni_fid}) of characters to form the new training set and validate on the rest. \nWe report the final test performance on the original hold-out test set.\nSince the test set contains different characters, the task is more difficult than a plain split by samples.\nFrom Fig. \\ref{fig:omni_fid}, we can see that our method consistently outperforms others under all proportions of training data.\n\n\\paragraph{Split by Alphabets}\nIn this setting, we split Omniglot (containing $50$ alphabets in total) based on alphabets. \nWe randomly sample certain proportions (shown in the X-axis of Fig. \\ref{fig:aomni_fid}) of alphabets to form the new training set and validate on the rest. \nPerformances are reported on the hold-out test set.\nSince the test set contains different alphabets and different alphabets in Omniglot have drastically different appearances, the task is extremely challenging than other splits.\nOur method achieves the best FID scores on all proportions except $3\\%$ as shown in Fig. \\ref{fig:aomni_fid}.\nUnder the $3\\%$ case, we only use $1$ alphabet for training and constructing the part bank.\nThe slight inferior performance of our model is likely caused by that the part bank may not be expressive enough.\n\n\n\\begin{figure*}[t]\n    \\begin{center}\n        \\includegraphics[width=0.98\\linewidth]{fig\/sec4_3.pdf} \n    \\end{center} \n    \\vspace{-0.5cm}\n    \\caption{Latent space editing. Given images A and B, we encode them to obtain the latent canvases. Then we compose a new canvas by placing a portion of canvas B on top of canvas A. Finally, we decode an image using the composed canvas.} \\label{fig:vis_inter}\n    \\vspace{-0.2cm}\n\\end{figure*} \n\n\n\\renewcommand{\\arraystretch}{1.5\n\\begin{figure*}\n    \\centering\n    \\settowidth\\rotheadsize{Sampled Canvases}\n    \\begin{tabular}{@{\\hspace{0mm}}c@{\\hspace{0.5mm}}c@{\\hspace{1mm}}c@{\\hspace{1mm}}c@{\\hspace{1mm}}c}\n        \\rothead{\\centering Sampled Images} &\n        \\includegraphics[width=0.23\\textwidth,trim={0 9.5cm 0 0},clip,valign=m]{fig\/vis_sample\/mnist_ours\/sample_192_step_-1.png} & \n        \\includegraphics[width=0.23\\textwidth,trim={0 9.5cm 0 0},clip,valign=m]{fig\/vis_sample\/omniglot_ours\/sample_799_step_3.png} & \n        \\includegraphics[width=0.23\\textwidth,trim={0 10.8cm 0 0},clip,valign=m]{fig\/vis_sample\/cifar_ours\/sample_150_step_-1.png} & \n        \\includegraphics[width=0.23\\textwidth,trim={0 21cm 0 0},clip,valign=m]{fig\/vis_sample\/celeba_ours\/sample_step_129.png} \n        \\\\ \\addlinespace[1mm]\n        \\rothead{\\centering Sampled Canvases} &\n        \\includegraphics[width=0.23\\textwidth,trim={0 0 0 9.5cm},clip,valign=m]{fig\/vis_sample\/mnist_ours\/sample_192_step_-1.png}&\n        \\includegraphics[width=0.23\\textwidth,trim={0 0 0 9.5cm },clip,valign=m]{fig\/vis_sample\/omniglot_ours\/sample_799_step_3.png}&\n        \\includegraphics[width=0.23\\textwidth,trim={0  0 0 10.8cm},clip,valign=m]{fig\/vis_sample\/cifar_ours\/sample_150_step_-1.png}&\n        \\includegraphics[width=0.23\\textwidth,trim={0  0 0 21cm},clip,valign=m]{fig\/vis_sample\/celeba_ours\/sample_step_129.png} \\\\         \n        {\\footnotesize  } &{\\footnotesize (a) MNIST} &{\\footnotesize (b) Omniglot}&{\\footnotesize (c) CIFAR-10}& {\\footnotesize (d) CelebA } \\\\\n    \\end{tabular}\n    \\vspace{-0.3cm}\n    \\caption{Visualization of sampled canvases (${\\mathbf{z}} \\sim p({\\mathbf{z}})$) and images (${\\mathbf{x}} \\sim p({\\mathbf{x}} \\vert {\\mathbf{z}})$) generated by our model. Same positions within the top and bottom rows correspond to one $({\\mathbf{x}}, {\\mathbf{z}})$ pair.}\n    \\vspace{-0.4cm}\n    \\label{fig:vis_sec41}\n\\end{figure*}\n\n\n\\paragraph{Split by Samples}\nIn this setting, we randomly sample certain proportions (shown in the X-axis of bottom subfigures in Fig. \\ref{fig:low_data}) of samples to form the new training data set. \nWe test this setting on MNIST, CIFAR-10, and CelebA.\nIt is clear that our method outperforms others under most of the proportions on MNIST.\nOn CIFAR-10 and CelebA, ours is superior to others except for WGAN under almost all proportions.\nThe gap between ours and WGAN may be attributed to the different learning principles, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, maximum likelihood vs. adversarial training.\n\n\\vspace{-0.2cm}\n\\subsection{Interpretability}\\label{sect:exp:in}\n\nIn this section, we demonstrate the advantage of our interpretable latent space via interactively editing\/composing the latent canvas.\nIn particular, given two images A and B, we first obtain their latent canvases via the encoder.\nThen we manually compose a new canvas by placing a portion of the canvas B on top of the canvas A.\nAt last, we generate a new image by feeding this composed canvas to the decoder.\nSince each part in the canvas corresponds to a local region in the sampled image, this replacement would only bring local changes, thus ensuring the spatial compositionality. \nNote that this can not be achieved by common VAEs since even a slight modification to one dimension of the latent space would typically cause global changes w.r.t\\onedot} \\def\\dof{d.o.f\\onedot appearance, color, \\emph{etc}\\onedot} \\def{\\bm{s}}{\\emph{vs}\\onedot.\nAs shown in Fig \\ref{fig:vis_inter}, we can synthesize new images by placing certain semantically meaningful parts (\\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot, eyeglasses, hair, beard, face) from B to A.\n\\renewcommand{\\arraystretch}{1.5\n\\begin{figure*}[t]\n    \\centering\n    \\settowidth\\rotheadsize{Omniglot}     \n   \n   \n   \n   \n    \\begin{tabular}{@{\\hspace{0mm}}c@{\\hspace{0.5mm}}c@{\\hspace{1mm}}c@{\\hspace{1mm}}c@{\\hspace{1mm}}c@{\\hspace{1mm}}c@{\\hspace{1mm}}c@{\\hspace{1mm}}c@{\\hspace{1mm}}c}\n            \\rothead{\\centering Omniglot} &\n        \\includegraphics[width=0.23\\textwidth,trim={0 9.5cm 0 0},clip,valign=m]{fig\/vis_sample\/omniglot_ours\/sample_799_step_3.png} & \n        \\includegraphics[width=0.23\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/vqdraw\/omniglot\/vis.png} & \n       \n        \\includegraphics[width=0.23\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/wgan\/omniglot\/fake_samples_50000.png} & \n        \\includegraphics[width=0.23\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/vae\/omniglot\/vis.png} & \n       \n       \n            \\\\ \\addlinespace[1mm]\n            \\rothead{\\centering CIFAR} &\n        \\includegraphics[width=0.23\\textwidth,trim={0 10.8cm 0 0},clip,valign=m]{fig\/vis_sample\/cifar_ours\/sample_150_step_-1.png} & \n        \\includegraphics[width=0.23\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/vqdraw\/cifar\/vis.png} & \n       \n        \\includegraphics[width=0.23\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/wgan\/cifar\/vis.png} & \n        \\includegraphics[width=0.23\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/vae\/cifar\/vis.png} & \n       \n       \n            \\\\ \\addlinespace[1mm]\n            \\rothead{\\centering CelebA} &\n        \\includegraphics[width=0.23\\textwidth,trim={0 21cm 0 0},clip,valign=m]{fig\/vis_sample\/celeba_ours\/sample_step_129.png} &\n        \\includegraphics[width=0.23\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/vqdraw\/celeba\/vis.png} & \n       \n        \\includegraphics[width=0.23\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/wgan\/celeba\/vis.png} & \n        \\includegraphics[width=0.23\\textwidth,trim={0 0 0 0},clip,valign=m]{fig\/vis_sample\/vae\/celeba\/vis.png} & \n       \n       \n       \n       \n       \n       \n        \\\\ \n        {\\footnotesize  } \n        & {\\footnotesize (a) Ours} \n        & {\\footnotesize (b) VQ-DRAW} \n       \n        & {\\footnotesize (c) WGAN}\n        & {\\footnotesize (d) VAE}\n       \n       \n        \\\\\n    \\end{tabular}\n   \n    \\caption{Visualization of generated images from different models on different dataset.}\n   \n    \\label{fig:vis_sec61}\n\\end{figure*}\n\n\n\\vspace{-0.2cm}\n\\subsection{Ablation Study}\nWe show the ablation study of important hyperparameters on Omniglot, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, the patch size $K$, the size of the part bank $M$, and the regularization weight $\\lambda$. \nWe use the peak signal-to-noise ratio (PSNR) to measure the quality of our heuristic parsing algorithm (see more details in Appendix).\nWe also report the FID score, NLL, binary cross-entropy, and the KL divergence.\nResults are listed in Table \\ref{tab:ablation}. \nFrom the table, we find that the quality of our heuristic parsing improves as the part bank size $M$ increases and the patch size $K$ decreases. \nThere are some inherent trade-offs on choosing the values of $M$ and $K$.\nIntuitively, a large $M$ would make the learning harder since we need to deal with a higher dimensional categorical distribution, whereas a small $M$ would have inferior performance due to the lack of expressiveness.\nA large $K$ is hard to learn since it requires a large $M$ to ensure the expressiveness, while a small $K$ requires more generation steps.\nWe hence identify a range of $K$ and $M$ based on these trade-offs and then choose the values with the best FID scores.\nMoreover, adding regularization weight $\\lambda > 0$ improves the performance compared to disabling it ($\\lambda = 0$).\n$\\lambda = 50$ empirically gives the best result.\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n \n\n\n\n\n\n\\section{Introduction}\\label{sect:intro}\n\nHumans understand and create images in a structural way.\nGiven a scene, one can quickly recognize different objects and their functional relationships.\nWhile drawing a picture, humans naturally draw one part at a time and iteratively improve the drawing.\nHowever, many existing generative models like variational autoencoders (VAEs) \\citep{kingma2013auto} and generative adversarial networks (GANs) \\citep{goodfellow2014generative} generate all pixels of an image simultaneously, which is unnatural and hardly interpretable.\nFor example, the latent space of VAEs is a high-dimensional vector space which is hard to visualize and meaningfully manipulate.\nTherefore, building more human-like and interpretable models for image understanding and generation has been an important quest \\citep{lake2015human}.\nNote that good generative models could often facilitate both image understanding and generation (\\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot, a good encoder in VAEs could recognize specific patterns in the image, whereas a good decoder could generate images with high quality).\n\nIn this paper, we propose a structured probabilistic model for image generation, which follows the line of Deep Recurrent Attentive Writer (DRAW) \\citep{gregor2015draw} and Attend-Infer-Repeat (AIR) networks \\citep{eslami2016attend} that mimic how humans draw in a part-by-part fashion.\nIn particular, we sequentially decide whether-to-draw, generate parts (what-to-draw) and their locations (where-to-draw), and draw them on a latent canvas from which the final image is decoded.\nOur main contributions are as follows.\n\nFirst, instead of using Gaussian latent variables as in DRAW and AIR which are too limited to capture the large variation of the appearance of the individual image parts, we resort to a non-parametric categorical distribution, which is more expressive, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot, capturing multi-modalities in the appearance distribution.\nWe build a dictionary of exemplar parts by performing k-medoids clustering on random local patches collected from images.\nThis categorical distribution is then defined over the discrete choice of the exemplar parts.\nBy doing so, we convert the hard what-to-draw problem to an easier which-to-choose problem.\n\nSecond, instead of using recurrent neural networks (RNNs) to construct the encoder, prior, and decoder like what DRAW and AIR do, we simplify the overall model by using a Transformer-based prior and plain convolutional neural networks (CNNs) based encoder and decoder. \nTransformer excels at capturing dependencies among parts and is easier to train compared to RNNs since it does not carry hidden states over generation steps.\nThis design choice significantly eases the learning and makes our model more scalable.\n\nAt last, we propose a heuristic algorithm to parse the latent variables given an image so that we can effectively pre-train our prior network.\nMoreover, at each generation step, we allow the model to decide whether-to-draw by sampling from a learned Bernoulli distribution.\nIf yes, then we draw the generated part in the generated location of the canvas. \nOtherwise, we skip this step and leave the canvas unchanged.\nTherefore, although fixing the total number of generation steps, the effective number of steps varies from image to image, thus being more flexible.\n\nWe extensively benchmark our model on four datasets, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, MNIST, Omniglot, CIFAR-10, and CelebA.\nIn terms of the sample quality, our model is significantly superior to other structured generative models like DRAW \\citep{gregor2015draw} and AIR \\citep{eslami2016attend} and is comparable to state-of-the-art generic generative models like WGAN \\citep{arjovsky2017wasserstein}.\nFurthermore, under small portions of training data, our model consistently outperforms other structured generative models and is comparable to the best generic generative model. \nThe strong performances under the low-data regime support that our model better exploits the compositionality in the data, since otherwise it would require much more training samples to reach satisfying performances.\nWe also demonstrate the interpretability of our model by manipulating sampled images in a compositional way through latent space editing.\n\nThe rest of the paper is organized as follows. \nIn Section \\ref{sect:relate_work}, we discuss the related work.\nThen we introduce the details of our model in Section \\ref{sect:model}.\nSection \\ref{sect:exp} provides experimental results.\nSection \\ref{sect:conclusion} concludes the paper.\n\n\n\n\\subsubsection*{References}}\n\\usepackage{mathtools}\n\\usepackage{booktabs}\n\\usepackage{tikz}\n\n\n\\input{math_commands.tex}\n\\usepackage{microtype}\n\\usepackage{graphicx}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{mathrsfs}\n\n\\usepackage{xspace}\n\\usepackage{xcolor}\n\\usepackage{caption}\n\\usepackage{subcaption}\n\\usepackage{rotating}\n\\usepackage{makecell}\n\\usepackage[export]{adjustbox}\n\n\\usepackage{algpseudocode} \n\\usepackage{algorithm} \n\n\\definecolor{ourspecialtextcolor}{rgb}{0.528, 0.471, 0.701}\n\\hypersetup{\n    colorlinks = true,\n    linkbordercolor = {white},\n\tlinkcolor = black, \n\tcitecolor=ourspecialtextcolor, \n\turlcolor=blue\n}\n\n\\makeatletter\n\\DeclareRobustCommand\\onedot{\\futurelet\\@let@token\\@onedot}\n\\def\\@onedot{\\ifx\\@let@token.\\else.\\null\\fi\\xspace}\n\\def\\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot{\\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot}\n\\def\\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot{\\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot}\n\\def\\emph{c.f}\\onedot} \\def\\Cf{\\emph{C.f}\\onedot{\\emph{c.f}\\onedot} \\def\\Cf{\\emph{C.f}\\onedot}\n\\def\\emph{etc}\\onedot} \\def{\\bm{s}}{\\emph{vs}\\onedot{\\emph{etc}\\onedot} \\def{\\bm{s}}{\\emph{vs}\\onedot}\n\\defw.r.t\\onedot} \\def\\dof{d.o.f\\onedot{w.r.t\\onedot} \\def\\dof{d.o.f\\onedot}\n\\def\\emph{a.k.a}\\onedot{\\emph{a.k.a}\\onedot}\n\\def\\emph{et al}\\onedot{\\emph{et al}\\onedot}\n\\makeatother\n\n\\newcommand{\\renjie}[1]{{\\color{orange}{\\bf\\sf [Renjie: #1]}}}\n\\newcommand{\\rich}[1]{{\\color{blue}{\\bf\\sf [Rich: #1]}}}\n\\newcommand{\\raquel}[1]{{\\color{red}{\\bf\\sf [Raquel: #1]}}}\n\\newcommand{\\sanja}[1]{{\\color{magenta}{\\bf\\sf [Sanja: #1]}}}\n\\newcommand{\\xiaohui}[1]{{\\color{violet}{\\bf\\sf [Xiaohui: #1]}}}\n\\newcommand{NP-DRAW}{NP-DRAW}\n\\newcommand{NP-DRAW-fix}{NP-DRAW-fix}\n\n\n\\newcommand{\\swap}[3][-]{#3#1#2}\n\\makeatletter\n\\newcommand\\email[2][]%\n   {\\newaffiltrue\\let\\AB@blk@and\\AB@pand\n      \\if\\relax#1\\relax\\def\\AB@note{\\AB@thenote}\\else\\def\\AB@note{\\relax}%\n        \\setcounter{Maxaffil}{0}\\fi\n      \\begingroup\n        \\let\\protect\\@unexpandable@protect\n        \\def\\thanks{\\protect\\thanks}\\def\\footnote{\\protect\\footnote}%\n        \\@temptokena=\\expandafter{\\AB@authors}%\n        {\\def\\\\{\\protect\\\\\\protect\\Affilfont}\\xdef\\AB@temp{#2}}%\n         \\xdef\\AB@authors{\\the\\@temptokena\\AB@las\\AB@au@str\n         \\protect\\\\[\\affilsep]\\protect\\Affilfont\\AB@temp}%\n         \\gdef\\AB@las{}\\gdef\\AB@au@str{}%\n        {\\def\\\\{, \\ignorespaces}\\xdef\\AB@temp{#2}}%\n        \\@temptokena=\\expandafter{\\AB@affillist}%\n        \\xdef\\AB@affillist{\\the\\@temptokena \\AB@affilsep\n          \\AB@affilnote{}\\protect\\Affilfont\\AB@temp}%\n      \\endgroup\n       \\let\\AB@affilsep\\AB@affilsepx\n}\n\n\\makeatother\n\n\n\\title{NP-DRAW: A Non-Parametric Structured Latent Variable Model \\\\ for Image Generation}\n\n\\author[1,2]{{Xiaohui Zeng}}\n\\author[1,2,4]{Raquel Urtasun}\n\\author[1,2,4]{Richard Zemel}\n\\author[1,2,3]{Sanja Fidler}\n\\author[1,2]{Renjie Liao}\n\\affil[1]{University of Toronto,  ${}^2$ Vector Institute,  ${}^{3}$ NVIDIA, ${}^{4}$ Canadian Institute for Advanced Research } \n\\email{\\tt  \\{xiaohui, urtasun, zemel, fidler,  rjliao\\}@cs.toronto.edu}\n\n\\begin{document}\n\\maketitle\n\n\\begin{abstract}\nIn this paper, we present a non-parametric structured latent variable model for image generation, called NP-DRAW, which sequentially draws on a latent canvas in a part-by-part fashion and then decodes the image from the canvas.\nOur key contributions are as follows. \n1) We propose a non-parametric prior distribution over the appearance of image parts so that the latent variable ``what-to-draw'' per step becomes a categorical random variable.\nThis improves the expressiveness and greatly eases the learning compared to Gaussians used in the literature.\n2) We model the sequential dependency structure of parts via a Transformer, which is more powerful and easier to train compared to RNNs used in the literature.\n3) We propose an effective heuristic parsing algorithm to pre-train the prior.\nExperiments on MNIST, Omniglot, CIFAR-10, and CelebA show that our method significantly outperforms previous structured image models like DRAW and AIR and is competitive to other generic generative models.\nMoreover, we show that our model's inherent compositionality and interpretability bring significant benefits in the low-data learning regime and latent space editing. Code is available at  \n\n\\url{https:\/\/github.com\/ZENGXH\/NPDRAW}. \n\\end{abstract}\n\n\n\\input{intro.tex}\n\\input{related.tex}\n\\input{method.tex}\n\\input{exp.tex}\n\\input{conclusion.tex}\n\n\n\\begin{acknowledgements} \nThis work was supported by NSERC and DARPA's XAI program. SF acknowledges the Canada CIFAR AI Chair award at the Vector Institute. RL was supported by RBC Fellowship. We would like to thank Yang Song, Yujia Li, David Duvenaud, and anonymous reviewers for valuable feedback.\n\\end{acknowledgements}\n\n\n\n\\section{Model}\\label{sect:model}\n\n\\begin{figure*}[h]\n    \\begin{center} \n    \\includegraphics[width=\\textwidth]{fig\/model_fig.pdf}\n    \\end{center}    \n    \\vspace{-0.5cm}\n    \\caption{The generative process of our model. At each time step, a Transformer-based auto-regressive prior generates latent variables including what-to-draw ${\\mathbf{z}}_{\\text{id}}$ (choosing a part from the part bank), where-to-draw ${\\mathbf{z}}_{\\text{loc}}$ (choosing one of the discretized locations), and whether-to-draw ${\\mathbf{z}}_{\\text{is}}$ (whether skipping the current step). The sampled image is generated from the final canvas via a decoder.}\n    \\vspace{-0.4cm}\n    \\label{fig:model}\n\\end{figure*}\n\nIn this section, we introduce our model NP-DRAW, which follows the general framework of VAEs \\citep{kingma2013auto}.\nOur generative model $p_{\\theta}({\\mathbf{x}} \\vert {\\mathbf{z}}) p_{\\theta}({\\mathbf{z}})$, parameterized by $\\theta$, first generates some latent variables ${\\mathbf{z}}$ from the prior $p_{\\theta}({\\mathbf{z}})$ and then generates an image ${\\mathbf{x}}$ conditioned on ${\\mathbf{z}}$ via the decoder $p_{\\theta}({\\mathbf{x}} \\vert {\\mathbf{z}})$.\nGiven an image ${\\mathbf{x}}$, the inference of latent variables ${\\mathbf{z}}$ is implemented by an amortized encoder network $q_{\\phi}({\\mathbf{z}} \\vert {\\mathbf{x}})$ parameterized by $\\phi$.\nIdeally, ${\\mathbf{z}}$ should describe attributes of the image, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot, appearances and structural relationships of objects and parts.  \nMany existing latent variable models, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot, VAEs, use a single vector ${\\mathbf{z}}$, which hardly captures such structural information. \n\n\n\\paragraph{Part-based Image Representations} \n\nInspired by part-based models in image and object modeling \\citep{hinton1979some,fischler1973representation,felzenszwalb2005pictorial,zhu2007stochastic}, we introduce multiple groups of latent variables ${\\mathbf{z}} = \\{{\\mathbf{z}}^{1}, \\dots, {\\mathbf{z}}^{T}\\}$ where each group ${\\mathbf{z}}^{t}$ corresponds to the drawing of a part of the image and $T$ is the total number of generation steps.\nHere we use parts to refer to local spatial regions of images. \nThey could be semantically meaningful parts of objects like wheels of cars or some recurring regions of the scene.\nIn particular, at the $t$-th generation step, the group ${\\mathbf{z}}^{t}$ describes an image part in terms of its location ${\\mathbf{z}}_{\\text{loc}}^{t}$, its appearance ${\\mathbf{z}}_{\\text{id}}^{t}$, and whether we draw it ${\\mathbf{z}}_{\\text{is}}^{t}$ on the latent canvas ${\\mathbf{c}}^{t}$, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, ${\\mathbf{z}}^{t} = [{\\mathbf{z}}_{\\text{loc}}^{t}, {\\mathbf{z}}_{\\text{id}}^{t}, {\\mathbf{z}}_{\\text{is}}^{t}]$.\nAlthough we fixed $T$ as a hyperparameter following DRAW, we allow the model to learn to skip a step (\\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, ${\\mathbf{z}}_{\\text{is}}^{t} = 0$) during the generation.\nTherefore, the effective number of steps could vary from image to image.\nThis modeling choice is not restrictive if one pre-specifies a reasonably large $T$.\nAlso, we found that such a sequence of Bernoulli random variables $\\{{\\mathbf{z}}_{\\text{is}}^{t}\\}$ is easier to learn compared to the geometric distribution used by AIR.\nFor example, AIR tends to learn very small values of $T$ (\\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot, $T=2$ or $3$) in practice.\nWe will explain how to construct the probability distributions of these latent variables in Section \\ref{sect:prior}.\n\n\n\\paragraph{Overall Generative Process}\nThe overview of our generative process is shown in Fig. \\ref{fig:model}.\nThe sampling from the prior $p_{\\theta}({\\mathbf{z}}^{1}, \\dots, {\\mathbf{z}}^{T})$ is like drawing one part at a step on a blank canvas for $T$ steps (with the possibility to skip some generation steps), which should produce the rough objects\/scenes of the image.\nThe conditional sampling from the decoder $p_{\\theta}({\\mathbf{x}} \\vert {\\mathbf{z}}^{1}, \\dots, {\\mathbf{z}}^{T})$ then generates the image by completing the fine details on the canvas.\nIn the following, we will explain the design choices of the individual components in detail.\n\n\n\\subsection{Non-parametric Structured Prior}\\label{sect:prior}\n\nAs discussed above, our prior model $p_{\\theta}({\\mathbf{z}}^{1}, \\dots, {\\mathbf{z}}^{T})$ aims at sequentially drawing the main objects\/scenes of an image one part at a time for $T$ steps in total.\nApparently, parts within an image have dependencies.\nFor example, some background parts may have strong correlation in terms of their appearance, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot, two parts cropped from an image of sea may highly likely be blue.\nSome foreground parts may often appear together like wheels and windows of a car, sometimes even under certain relative poses.\nTo capture such dependencies, we employ a auto-regressive model in the latent space,\n\\begin{align}\\label{eq:auto-regressive-prior}\n    p_{\\theta}({\\mathbf{z}}^{1}, \\dots, {\\mathbf{z}}^{T}) & = \\prod_{t=1}^{T} p_{\\theta}({\\mathbf{z}}^{t} \\vert {\\mathbf{z}}^{<t}) \n\\end{align}\nwhere $p_{\\theta}({\\mathbf{z}}^{t} \\vert {\\mathbf{z}}^{<t})$ represents the distribution of ${\\mathbf{z}}^{t}$ conditioned on $\\{{\\mathbf{z}}^{1}, \\cdots, {\\mathbf{z}}^{t-1}\\}$.\nRecall that ${\\mathbf{z}}^{t} = [{\\mathbf{z}}_{\\text{loc}}^{t}, {\\mathbf{z}}_{\\text{id}}^{t}, {\\mathbf{z}}_{\\text{is}}^{t}]$.\nWe decompose the conditional distribution in Eq. (\\ref{eq:auto-regressive-prior}) as below,\n\\begin{align}\\label{eq:conditional}\n    p_{\\theta}({\\mathbf{z}}^{t} \\vert {\\mathbf{z}}^{<t}) = & \\underbrace{p_{\\theta}({\\mathbf{z}}_{\\text{id}}^{t} \\vert {\\mathbf{c}}^{<t}, {\\mathbf{z}}_{\\text{loc}}^{<t})}_{\\text{Appearance}} \\times \\underbrace{p_{\\theta}({\\mathbf{z}}_{\\text{loc}}^{t} \\vert {\\mathbf{c}}^{<t}, {\\mathbf{z}}_{\\text{loc}}^{<t})}_{\\text{Location}} \\nonumber \\\\ \n    & \\times \\underbrace{p_{\\theta}({\\mathbf{z}}_{\\text{is}}^{t} \\vert {\\mathbf{c}}^{<t}, {\\mathbf{z}}_{\\text{loc}}^{<t})}_{\\text{Skip}},\n\\end{align}\nwhere ${\\mathbf{c}}^{<t}$ is the collection of past canvases.\nWe introduce the canvas in the conditioning to better explain the model.\nThe canvas at time step $t$ is a deterministic function of all latent variables generated so far, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, ${\\mathbf{c}}^{t} = g({\\mathbf{z}}_{\\text{loc}}^{t}, {\\mathbf{z}}_{\\text{id}}^{t}, {\\mathbf{z}}_{\\text{is}}^{t}, {\\mathbf{c}}^{t-1})$.\nWe will explain the canvas update function $g$ in Section \\ref{subsect:canvas_update}.\n${\\mathbf{c}}^{0}$ is initialized to be empty.\nThe above three probability terms correspond to modeling the appearance of the part, the location of the part, and whether to skip the current generation step respectively. \n\n\n\n\\paragraph{Transformer Backbone}\nTo model the dependencies of all previous canvases and locations which are of variable lengths, we leverage Transformer \\citep{vaswani2017attention} as the backbone.\nWe use the same architecture as Vision Transformer (ViT) \\citep{dosovitskiy2020image}.\nSpecifically, at time step $t$, we feed a length-$(t-1)$ sequence of 2-channel images (each image corresponds to a time step in the past) to Transformer. \nThe first channel of an image contains the latent canvas at that time step whereas the second one is a binary mask which encodes the location of the part (pixels belong to the part are set to $1$ otherwise $0$) generated at that time step.\nThe canvas contains the appearance information of all parts drawn so far.\nThe binary masks help the model aware of the locations of the recently generated parts which often provide strong clues about where to draw the next part.\nWe found conditioning on the previously generated locations empirically improves the performance.\nOur design choice facilitates more efficient learning compared to RNNs used by DRAW and AIR, since Transformer enjoys the parallel training and is easier to optimize. \nThe process is illustrated in Fig. \\ref{fig:transformer}.\nIn the following, we explain how to construct the probabilities from the embeddings of Transformer in detail.\n\n\n\\subsubsection{What-to-Draw: Non-parametric Appearance Modeling for Image Parts}\\label{subsect:what}\n\nTo model the appearance of individual parts, a straightforward idea is to model $p_{\\theta}({\\mathbf{z}}_{\\text{id}}^{t} \\vert {\\mathbf{c}}^{<t}, {\\mathbf{z}}_{\\text{loc}}^{<t})$ using a parametric family of distributions, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot, Gaussians as used in DRAW and AIR.\nHowever, such a parametric form has the limited expressiveness and hardly captures the large variation of the appearance in natural images.\nWe instead take a non-parametric approach by first constructing a bank of exemplar parts through clustering and then representing ${\\mathbf{z}}_{\\text{id}}^{t}$ as a categorical random variable over the choice of exemplars.\n\nIn particular, we first collect raw patches with size $K \\times K$ from the training dataset through random sampling.\nThen we vectorize the patches and perform K-medoids clustering to obtain the bank of exemplar parts which are the $M$ cluster centroids $B = \\{\\tilde{{\\mathbf{x}}}_1, \\cdots, \\tilde{{\\mathbf{x}}}_M\\}$.\nTherefore, ${\\mathbf{z}}_{\\text{id}}^{t}$ takes value from the possible indices of parts, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, $1$ to $M$.\n\nTo construct the distribution $p_{\\theta}({\\mathbf{z}}_{\\text{id}}^{t} \\vert {\\mathbf{c}}^{<t}, {\\mathbf{z}}_{\\text{loc}}^{<t})$, we first pool the embeddings of each 2-channel image returned by a projection layer and add to the positional embeddings which encode the information of time steps.\nWe then feed the feature to the transformer encoder which uses the self-attention to improve the embedding.\nThe categorical distribution ${\\mathbf{z}}_{\\text{id}}^{t}$ is output by a MLP head (a size-$64$ hidden layer and a ReLU activation function~\\citep{nair2010rectified}) which takes the embedding as input. \n\n\n\n\\begin{figure}[t]\n    \\begin{center} \n    \\includegraphics[width=\\linewidth]{fig\/transformer.pdf}\n    \\end{center}    \n    \\vspace{-0.5cm}\n    \\caption{The Transformer-based conditional probability in Eq. (\\ref{eq:conditional}). The input is a sequence of 2-channel images containing the canvas and the location map.}\n    \\vspace{-0.4cm}\n    \\label{fig:transformer}\n\\end{figure}\n\n\n\\subsubsection{Where-to-Draw: Location Modeling}\\label{subsect:where}\n\nPrevious methods model the distribution of locations of parts via Gaussians which requires truncating the out-of-image ones.\nFor simplicity, we model the locations of parts by discretizing the images into a 2D grid so that a part can only center on a grid.\nTherefore, ${\\mathbf{z}}_{\\text{loc}}^{t}$ is a categorical random variable taking values from the possible indices of the 2D grid.\nWe discretize the image so that each grid is of size $K \\times K$ (\\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, same size as a part), thus disallowing parts to have overlap and greatly simplifying the pipeline.\nIn practice, we found learning with such a categorical distribution is easier compared to Gaussians.\nWe use the same embedding returned by the transformer encoder and predict the categorical distribution using a separate MLP head, which has one size-$64$ hidden layer and a ReLU activation function~\\citep{nair2010rectified}.\n \n\\subsubsection{Whether-to-Draw and Canvas Update}\\label{subsect:canvas_update}\n\nWhile generating both ``what-to-draw'' ${\\mathbf{z}}_{\\text{id}}^{t}$ and ``where-to-draw'' ${\\mathbf{z}}_{\\text{loc}}^{t}$ at time step $t$, we allow the model to choose whether to draw it on the canvas by sampling a per-step Bernoulli random variable ${\\mathbf{z}}_{\\text{is}}^{t}$.\nThe distribution is again constructed based on a separate MLP head that takes the embedding from the transformer encoder as input.\nThis mechanism allows the model to have varying numbers of generation steps from image to image.\nWe update the canvas as below,\n\\begin{align}\\label{eq:canvas_update}    \n    {\\mathbf{c}}^{t} & = g ( {\\mathbf{z}}_{\\text{loc}}^{t}, {\\mathbf{z}}_{\\text{id}}^{t}, {\\mathbf{z}}_{\\text{is}}^{t}, {\\mathbf{c}}^{t-1}) \\nonumber \\\\ \n    & = \\left({\\mathbf{c}}^{t-1} \\right)^{1 - {\\mathbf{z}}_{\\text{is}}^{t}} \\left( \\max \\left({\\mathbf{c}}^{t-1}, \\text{Draw}(\\tilde{{\\mathbf{x}}}_{{\\mathbf{z}}_{\\text{id}}^{t}}, {\\mathbf{z}}_{\\text{loc}}^{t}) \\right) \\right)^{{\\mathbf{z}}_{\\text{is}}^{t}}.\n\\end{align}\nIf ${\\mathbf{z}}_{\\text{is}}^{t} = 1$, the ``Draw'' function generates a mask (same size as the canvas) where the region specified by ``where-to-draw'' ${\\mathbf{z}}_{\\text{loc}}^{t}$ is filled by values from an exemplar part corresponding to ``what-to-draw'' ${\\mathbf{z}}_{\\text{id}}^{t}$.\nThe values outside this designated region are zero.\nThen we obtain the new canvas via a element-wise maximum between the previous canvas and the generated mask.\nIf ${\\mathbf{z}}_{\\text{is}}^{t} = 0$, then we keep the canvas unchanged.\nThe whole update process is deterministic.\n\n\n\\begin{algorithm*}[t]\n\\caption{Heuristic Parsing of Latent Variables}\n\\label{alg:parsing} \n\\begin{algorithmic}[1]\n    \\State \\textbf{Input}: image ${\\mathbf{x}}$, patch bank $B = \\{\\tilde{{\\mathbf{x}}}_1, \\cdots, \\tilde{{\\mathbf{x}}}_M\\}$ containing $K \\times K$ exemplar patches, threshold $\\epsilon = 0.01$\n    \\State Discretize image (padding if needed) into grids so that each grid is $K \\times K$ (assuming there are $T$ such grids)    \n    \\For {$t=1,2,\\ldots,T$}   \\Comment{Parsing latent variables at $t$-th step}\n    \\State ${\\mathbf{z}}_\\text{loc}^t$ = t\n    \\State $\\text{loc}^t$  = Center location of the $t$-th grid \n    \\State ${\\mathbf{x}}_c$ = CropImg(${\\mathbf{x}}$, $\\text{loc}^t$) \\Comment {Crop a $K \\times K$ patch centered at $\\text{loc}^{t}$}\n    \\State ${\\mathbf{z}}_\\text{id}^t$ = NearestNeighbor$(B, {\\mathbf{x}}_c)$ \\Comment{Find the index of the nearest patch to ${\\mathbf{x}}_c$ from $B$}\n    \\State Cost = $\\Vert {\\mathbf{x}}_c - \\tilde{{\\mathbf{x}}}_{{\\mathbf{z}}_\\text{id}^t} \\Vert - \\Vert {\\mathbf{x}}_c - \\mathbf{0} \\Vert$ \\Comment{Cost of pasting this patch vs. an empty patch}\n    \\State ${\\mathbf{z}}_{\\text{is}}^{t} = 1$ \\textbf{if} Cost $\\le \\epsilon$ \\textbf{else} $0$\n    \\EndFor\n    \\State \\textbf{Output}: ${\\mathbf{z}} = \\left( ({\\mathbf{z}}_{\\text{loc}}^{1}, {\\mathbf{z}}_{\\text{id}}^{1}, {\\mathbf{z}}_{\\text{is}}^{1}), \\dots, ({\\mathbf{z}}_{\\text{loc}}^{T}, {\\mathbf{z}}_{\\text{id}}^{T}, {\\mathbf{z}}_{\\text{is}}^{T}) \\right)$\n\\end{algorithmic}\n\\end{algorithm*}\n\n\n\n\n\\subsubsection{Heuristic Parsing for Pre-Training Prior}\\label{sect:pretrain-prior}\n\nLearning with a structured prior in VAEs is not easy as demonstrated by vector quantized VAEs (VQ-VAEs) \\citep{oord2017neural}. \nTo improve the learning, we also adopt the pre-training strategy as what VQ-VAEs do.\nBefore we explain details of the heuristic parsing, let us denote the general probabilistic parsing of latent variables ${\\mathbf{z}}$ given an image ${\\mathbf{x}}$ as $p_h({\\mathbf{z}} \\vert {\\mathbf{x}})$. \nTo pre-train the prior, we ideally need to,\n\\begin{align}\\label{eq:pretrain_prior_ideal}\n    \\min_{\\theta} \\quad D_{\\mathrm{KL}}( \\int p_h({\\mathbf{z}} \\vert {\\mathbf{x}}) p_{\\text{data}}({\\mathbf{x}}) \\mathrm{d} {\\mathbf{x}} \\Vert p_{\\theta}({\\mathbf{z}})).\n\\end{align}\nHowever, this is impossible since we do not know the data distribution $p_{\\text{data}}({\\mathbf{x}})$.\nIn practice, we observe samples from $p_{\\text{data}}({\\mathbf{x}})$ and then parse ${\\mathbf{x}}$ to get samples of latent variables ${\\mathbf{z}}$.\nTherefore, the sampled version of the objective becomes,\n\\begin{align}\\label{eq:pretrain_prior}\n    \\max_{\\theta} \\quad \\sum_{{\\mathbf{x}} \\sim p_{\\text{data}}({\\mathbf{x}})} \\sum_{{\\mathbf{z}} \\sim p_h({\\mathbf{z}} \\vert {\\mathbf{x}})} \\log p_{\\theta}({\\mathbf{z}}).\n\\end{align}\nNote that if the parsing is deterministic, then the objective becomes the maximum log likelihood of the observed pairs $({\\mathbf{x}}, {\\mathbf{z}})$.\nIn our case, we propose an effective heuristic parsing algorithm to deterministically infer the latent variables ${\\mathbf{z}}$ given an image ${\\mathbf{x}}$ as shown in Alg. \\ref{alg:parsing}.\nIn other words, $p_h({\\mathbf{z}} \\vert {\\mathbf{x}}) = \\delta({\\mathbf{z}} = \\text{Parsing}({\\mathbf{x}}))$ is a delta distribution.\nThe idea is as follows.\nWe first discretize the observed image into grids.\nThen for each grid, we search the nearest patch in the part bank and check whether it is closer to the observed patch than an empty patch.\nIf so, we use its index as the parsed value for the latent variable of the appearance.\nOtherwise, we choose to skip this grid. \nUsing Alg. \\ref{alg:parsing}, we can collect $({\\mathbf{x}}, {\\mathbf{z}})$ pairs and pre-train the prior using Eq. (\\ref{eq:pretrain_prior}).\n\n\\subsection{Encoder \\& Decoder}\n\nSimilar to VAEs, we approximate the posterior by learning an encoder parameterized by $\\phi$.\nFor simplicity, we assume the following conditionally independent encoder,\n\\begin{equation}\n     q_\\phi({\\mathbf{z}} \\vert {\\mathbf{x}}) = \\prod_{t=1}^{T} q_\\phi({\\mathbf{z}}_{\\text{id}}^{t} \\vert {\\mathbf{x}}) q_\\phi({\\mathbf{z}}_{\\text{loc}}^{t} \\vert {\\mathbf{x}}) q_\\phi({\\mathbf{z}}_{\\text{is}}^{t} \\vert {\\mathbf{x}}).\n\\end{equation}\nWe implement the encoder as a CNN which takes the image ${\\mathbf{x}}$ as input.\nDistributions of latent variables are constructed using separate MLP heads which take the flatten feature map of the CNN as input.\nOur encoder CNN consists of $5$ convolutional layers with hidden size $128$, followed by Batch Normalization~\\citep{ioffe2015batch} and a ReLU activation function~\\citep{nair2010rectified}.\nThe first $3$ convolutional layers use kernel size $3$, stride size $2$, and padding size $1$. \nThe last two convolutional layers use stride $1$. \nNote that one could further improve the encoder by using a Transformer-based auto-regressive posterior which captures the dependencies across time steps.\nWe leave it for future exploration.\n\nTo decode an image, suppose we finish the sampling from the prior and get a canvas ${\\mathbf{c}}^{T}$ at the last time step $T$. \nThe canvas should already roughly form the objects\/scenes of the image.\nThen we feed the canvas to a CNN decoder to generate the final image with refined details.\nOur decoder CNN consists of $2$ convolutional layers with stride $2$, and two residual blocks, followed by two transposed convolutional layers. \nAll layers have hidden size $128$ followed by Batch Normalization and ReLU activations. \nIn particular, we model the output image as,\n\\begin{align}\n    p_{\\theta}({\\mathbf{x}} \\vert {\\mathbf{z}}) = \\text{OutputDistribution}({\\mathbf{x}}; \\text{CNN}({\\mathbf{c}}^{T})),\n\\end{align}\nwhere we choose the output distribution for pixel intensities as Bernoulli for MNIST and Omniglot datasets and as the discretized logistic mixture for CIFAR10 and CelebA following \\citep{Salimans2017PixeCNN}.\n\n\\subsection{Loss Functions}\n\nThe standard objective for training a VAE model is to maximize the evidence lower bound (ELBO): \n\\begin{equation}\n    \\mathcal{L}_{\\text{ELBO}} = \\mathbb{E}_{q({\\mathbf{z}} \\vert {\\mathbf{x}})} [\\log p({\\mathbf{x}} \\vert {\\mathbf{z}})] - D_{\\mathrm{KL}}( q({\\mathbf{z}} \\vert {\\mathbf{x}}) \\Vert p({\\mathbf{z}}) )\n\\end{equation}\n\nHowever, we found that if we train the model with ELBO alone, the behavior of the learned posterior deviates from our expectation, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, the canvas should roughly form the objects\/scenes of the image. \nTherefore, we add an extra regularization term defined as \n\\begin{equation}\n    \\mathcal{L}_{\\text{reg}} = - D_{\\mathrm{KL}}( p_h({\\mathbf{z}} \\vert {\\mathbf{x}}) \\Vert q({\\mathbf{z}} \\vert {\\mathbf{x}}) ).\n\\end{equation}\nThis term could guide the posterior to behave like our heuristic parser.\nTherefore, we train our model by maximizing the following full objective\n\\begin{equation}\n    \\mathcal{L} = \\mathcal{L}_{\\text{ELBO}} + \\lambda \\mathcal{L}_{\\text{reg}},\n\\end{equation}\nwhere $\\lambda$ is the weight of the regularization term. \nWe set $\\lambda$ as $50$ for experiments on MNIST and Omniglot, and $500$ for experiments on CIFAR-10 and CelebA. \nThe value of $\\lambda$ is chosen so that the scaled regularization loss is in the same order of magnitude with ELBO. \nDuring training, we fix the prior for simplicity. \nExploring learning the prior together with the encoder and the decoder is left to future work.\n\n\\iffalse\n\n\\begin{table}[h]\n    \\centering\n    \\begin{tabular}{c|c|c|c} \n        $s_p$ \/ k & \n            5      & 10    & 50     \\\\ \\toprule\n        1  & -     & 39.0  & -      \\\\ \n        3  & 20.9  & 23.8  & 26.4   \\\\\n        5  & 17.3  & 19.4  & 23.4   \\\\ \n        10 & 13.3  & 14.8  & 17.6   \\\\ \n    \\end{tabular}\n    \\caption{Maximum PSNR value, we can see that the patch size weights more than the number of cluster to achieve a good reconstruction performance. Smaller patches gives better reconstruction, but has harder pattern for the full model to capture (reason?).}\n    \\label{tab:pb}\n\\end{table}\n\n\\begin{figure}[b]\n    \\centering\n   \n   \n   \n    \\caption{the reconstruction example. The bottom row shows the reconstruction with k=5 and patch size 5, (PlaceHolder) }\n    \\label{fig:pb}\n\\end{figure}\n    \n\\begin{figure}[b]\n    \\begin{center}\n       \n       \n        \\fbox{\\rule[-.5cm]{0cm}{4cm} placeholder \\rule[-.5cm]{4cm}{0cm}}\n    \\end{center} \n    \\label{fig:patch_bank:collect}\n    \\caption{Different ways to collect the patches}\n\\end{figure}\n\n\nGiven the original image $I$ and the reconstructed image $I'$, we can measure the Peak signal-to-noise ratio (PSNR).\n\nTo reconstruct the image $I$ with the patch bank, we need to decide 1) the number of patches we need to represent the canvas 2) which patches from the bank is needed.\n\nThe latent canvas is created by pasting the selected patches from our prepared patch bank, ${\\tens{E}}$ with size $k\\times s_p \\times s_p $, onto a canvas. Here $k$, $s_p$ is the size of the patch bank and the patch. We first at most $N$ patches from the patch bank and the corresponding locations. We denote their index as $z_{id}=\\{z^1_{id},z^2_{id}, ..., z^N_{id}\\}$, the location as $z_{loc}=\\{z^1_{loc},..,z^N_{loc}\\}$, where\n\\begin{equation}\n    z^i_{id} \\sim \\text{Cat}(\\pi^i), i \\in \\{1,2,..,N\\}\n\\end{equation}\n\\begin{equation}\n    z^i_{loc} \\sim \\mathcal{N}(\\mu^i_{loc}, \\sigma^i_{loc}), i \\in \\{1,2,..,N\\}\n\\end{equation}\n\nOur prior model will be \n\\begin{equation}\n    p_\\theta(z) = \\prod_{i=1}^{N} p_\\theta(z^i | z^{<i}),\n\\end{equation}\nwhere $z=(z_{id}, z_{loc})$.  \n\n\\textbf{Greedy search for hand-crafted GT}. We train the prior model with the hand-crafted GT, denoted as $p_h(z|x)$. There are different setting of our models: 1) fixed location: the model only predict the patch selection at fixed location. 2) vary location: the model will predict the patch selection, and patch location and a binary variable indicating the existence of the patch or not. \n\nTo get the GT for 1), given image $I$, for each possible location $(x, y)$, we find the closet patches ${\\tens{E}}[k] \\in \\mathbb{R}^{s_p \\times s_p}$ base on the distance $$ || I(x-\\frac{s_p}{2}:x+\\frac{s_p}{2}, y-\\frac{s_p}{2}:y+\\frac{s_p}{2}) - {\\tens{E}}[k] ||_2$$ and paste the nearest patch onto canvas. For the overlapped region, we take the maximum value over different patches. We evaluate the PSNR score over different size of patches and the number of clusters, as shown in Fig~\\ref{fig:pb} and Table~\\ref{tab:pb}. For different dataset, we choose the $s_p$ and $k$ which gives a good trade off between efficiency and being enough expressive.\n\n\\fi\n\\subsection{Other Evaluation Tasks in Related Work}\n\n\\subsubsection{Unsupervised Classification}\nThe SCAE paper perform unsupervised classification (also called unsupervised class discovery in images) on three dataset: MNIST, SVHN, CIFAR10. It compared to mutual information based method. Two classifier based on the SCAE model is proposed: \n\\begin{enumerate}\n    \\item LIN-MATCH: clustering into 10 clusters (kmeans) + bipartite graph matching to find the permutation of cluster indices that minimizes the classification error\u2014this is standard practice in unsupervised classification,\n    \\item LIN-PRED: train a linear classifier without changing weight of the main model \n\\end{enumerate}\n\n\\subsubsection{Cluttered MNIST Classification To Test the Attention Mechanism}\nUsed in the DRAW paper.\n\n\\subsubsection{Counting}\nUsed in AIR paper, count the Multi-MNIST dataset. AIR paper also test the generalization ability by training AIR model on images with K digits and testing it on non-K digits. \n\n\\subsubsection{Representational Power}\nUsed in AIR paper, we split\nthis data into training and test and consider two\ntasks: (a) predicting the sum of the two digits, and (b) determining if the\ndigits appear in an ascending order.\n\n\\subsubsection{Others}\n1) AIR paper also works with 3D scene. 2) VQVAE test the generalization on audio, and video. \n\n\\section{More Details for Related Works}\n\\subsection{Part Capsule Auto-encoder (PCAE)} \\label{sec:pcae} \n\\textbf{More details} For the $m^{th}$ capsule, the encoder predict $r_m \\in \\mathbb{R}^6$ ( rotation, translation, scale, shear) and presence $d_m \\in [0, 1]$, the decoder predict the template $T_m=T_m^{alpha}, T_m^c$. For MNIST, they set the size of template is $11\\times11$ and number of capsules  $M=24$. The image likelihood $ p(x) $ can be written as \n\\begin{equation}\n    p(x) = \\prod_{i, j} \\sum_{m}^M \\underbrace{(d_m \\hat{T}_{m, i, j}^{alpha})}_{\\text{mixing prob}} \\mathcal{N}(y_{i, j}| \\hat{T}_{m,i,j}^c; \\sigma_y^2),\n\\end{equation}\nwhere $\\hat{T}_m = \\text{TransformImage}(T_m, r_m)$. The model is optimized by MLE. The key is the design of using $r_m$ to transform $T_m$, which forces the model to learn the structure. The paper reports unsupervised classification performance.  \n\n\\subsection{Human Concept Learning} \\label{sec:hcl} \nThe generation process can be viewed in Fig~\\ref{fig:bpl}. In their cases, the primitives set is extracted from the Omniglot dataset. They first break every strokes in the dataset into sub-strokes (55,000), according to the time pause threshold, and then perform GMM to create the primitives set (size 1250). Given the breaking procedure, most of the distribution can be directly estimated on the training dataset. The paper reports performance on one-shot learning. \n\nThere are two main stages for generation, \n\\begin{itemize}\n    \\item generate character type $\\varphi = \\{\\kappa, S, R\\}$, where $\\kappa$ is number of stroke $S=\\{S_1, ..., S_\\kappa\\}$ and their spatial relation $R=\\{R_1, ..., R_\\kappa\\}$, its joint distribution is written at the bottom of Fig~\\ref{fig:bpl}. For each stroke $S_i=\\{s_{i}^1, ... , s_{i}^{n_i}\\}$, need to generate its template trajectory (a sequence of sub-strokes $\\{z_i^j\\}$: type $z_{i}^j$ and scale). $P(\\varphi) = p(\\kappa) \\prod_i^\\kappa P(S_i)P(R_i|S_{<i})$, $R\\in$ \\{independent, start, end, along\\}. \n    \\item generate the raw image given the character $\\varphi$, where some random transformation will be performed to create variant images. $P(I|\\varphi, \\theta_{trans})$\n\\end{itemize}\n\n\\textit{Empirical distribution directly:} number of strokes $p(\\kappa)$, number of sub-parts $p(n_i | \\kappa)$, transition of the primitive type $z_{i}^j \\in \\{1,..,1250\\}$, $z_{i}^j \\sim p(z_{i}^j | z_{i}^{j-1})$. \n\n\\textit{Estimated by fitting the motor program:} 1) the probabilities of different relationship type $\\theta_R$, and 2) the token-level (image-level) variance for shape, scale and attachment variables. \n\n\\begin{itemize}\n\\item [+] can be used for one-shot learning: the fitted program is able to generate variant of the same object \n\\item [-] The produce is too hand-crafted. \n\\end{itemize}\n\n\\paragraph{Task setup}\nThree tasks are proposed to evaluate one-shot generalization~\\cite{lake2015human,DBLP:conf\/icml\/RezendeMDGW16}, from easy to hard: \n\\begin{enumerate}\n    \\item unconditional (free) generation\n    \\item generation of novel variations of a given exemplar\n    \\item generation of representative samples from a novel alphabet\n\\end{enumerate}\n\n\n\n\\section{Related Work}\\label{sect:relate_work}\n\nGenerative models for images have been studied for several decades \\citep{dayan1995helmholtz,zhu1997minimax,freeman2000learning,crandall2005spatial,jin2006context}.\nIn the context of deep learning, generative models for images have been extensively studied from many aspects or principles, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot, \ndeep belief nets \\citep{hinton2006fast}, \ndeep Boltzmann machines \\citep{salakhutdinov2009deep},\nvariational auto-encoders (VAEs) \\citep{kingma2013auto,rezende2014stochastic,vahdat2020nvae,razavi2019generating}, \ngenerative adversarial networks (GANs) \\citep{goodfellow2014generative,arjovsky2017wasserstein,gulrajani2017improved,brock2018large}, \ngenerative moment matching networks \\citep{li2015generative,dziugaite2015training},\nnormalizing flows \\citep{rezende2015variational,dinh2016density,kingma2018glow,papamakarios2019normalizing}, \nenergy based models (EBMs) \\citep{du2019implicit,song2019generative}, \nand deep diffusion models \\citep{sohl2015deep,ho2020denoising}.\nOur work also relates to examplar VAE~\\citep{norouzi2020exemplar} which use a non-parametric prior with exemplars sampled from the training set. \nHowever, most of the above generative models generate all pixel values of an image at once, which contrasts with how humans draw pictures.\n\nIn this paper, we focus on a class of deep generative models that sequentially generate an image in a part-by-part fashion.\nOne of the first such models is the Deep Recurrent Attention Writer (DRAW) \\citep{gregor2015draw}.\nIt is a sequential variational auto-encoding framework with the encoder and the decoder parameterized by two recurrent neural networks (RNNs) respectively.\nDuring inference, the encoder uses the spatial attention mechanism to decide where-to-read and then constructs the approximated posterior.\nDuring generation, the decoder also leverages spatial attention to decide where-to-write and creates what-to-write.\nConvolutional DRAW \\citep{gregor2016towards} further improves DRAW by leveraging convolutions to construct the encoder\/decoder and learning the prior.\nAttend-Infer-Repeat (AIR) networks \\citep{eslami2016attend} enriches this model class by additionally learning to choose the appropriate number of inference\/generation steps.\nAIR was later extended to a sequential version (SeqAIR) \\citep{kosiorek2018sequential}, which models videos and can handle dynamically changing objects.\nMore recently, VQ-DRAW \\citep{nichol2020vq} introduces vector quantization in the latent space and decides what-to-draw by choosing one of the proposals output by a network instead of learning a Gaussian.\nArguably, PixelCNN \\citep{van2016conditional} can be viewed as an extreme case of this model class that generates one pixel at a time conditioned on previously generated ones without considering a latent space.\nThere are also stroke based generative models like SPIRAL~\\citep{ganin2018synthesizing}, Cose~\\citep{aksan2020cose}, and SketchEmbedNet~\\citep{wang2020sketchembednet}. SPIRAL generates images through a sequence of strokes while Cose and SketchEmbedNet focus on generating sketch images.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nIn recent years, privacy technologies have flourished and have received more attention. With the growth of the Internet, more and more users have sought out the ability to privately and securely communicate. These technologies are designed to provide anonymity and privacy for their users including whistleblowers and others who need to be able to communicate anonymously. Along with these new technologies came a rise in cybercrime. Anonymity is something that cybercriminals always desire and these technologies provide an easy way to achieve it. This led to an explosion in untraceable ransomware, Tor botnets, and other crimes that cannot easily be tracked back to an attacker. With the rise of Bitcoin, payments can also be made anonymously and cannot be tracked back to the attacker. The privacy infrastructure was designed to help users, but also led to this steep rise in cybercrime and a steep increase in the difficulty of catching these cyber criminals. This paper looks at how this privacy infrastructure is subverted for attacks and how serious of a threat this poses to the Internet as a whole. Section II will outline a few privacy technologies, Section III will outline the research analyzed within this paper, and Section IV will present conclusions drawn from this research.\n\n\\section{PRIVACY TECHNOLOGIES}\nAs usage of the Internet has exploded, the need for security and privacy when using the Internet has also increased dramatically. To meet these rising needs, many new privacy technologies have been created from secure protocols, like TLS\/SSL, to Tor, a network that allows for anonymous Internet usage, to Bitcoin which allows anonymous payments to be sent over the Internet. These technologies were all created to help provide security and privacy over the Internet in order to allow for services to be provided online. Whistleblowers and individuals living under totalitarian regimes can share information, individuals can anonymously browse the Internet, and individuals can check their bank accounts and make purchases online. The ideals that these technologies aimed to provide were positive, but were also very useful for cyber criminals, leading to many crimes and attacks that subvert these privacy technologies for nefarious means.\n\n\\subsection{SSL\/TLS}\nSSL\/TLS is a technology that was designed to allow for secure communication across the Internet. It is a protocol that was built on top of TCP that involves a session between a client and a server and then connections related with that session. It allows for one or two way authentication, confidentiality, and message integrity, all of which make the communication more secure and less vulnerable to attacks. SSL\/TLS has been widely deployed in order to provide these features to users, especially with HTTPS, which allows for secure browsing of the Internet and the ability to securely make payments over the Internet.\n\n\\subsection{Tor}\nTor is a network that aims to provide anonymous Internet browsing and communication. It is built on the idea of Onion routing, which involves messages passing through multiple servers in such a way that each node only knows the information about the previous and next nodes. In this infrastructure, the first node will only know about the user and second node, not the destination, that the last node will only know about the destination and the previous node, not the sender, and that intermediate nodes only know about the previous and next nodes. This is done by encrypting messages with a key for each node so that as the messages maneuver through the network, each layer of encryption is peeled off. Tor also allows decoupling of the IP address and the address used to visit the website so that the server hosting the site cannot easily be tracked down. \n\nTor does not provide end to end security, as it only encrypts the information within the network, so if end to end security is needed, it is provided by the end server. There have been a few recorded vulnerabilities in the Tor network, but it is widely considered to be very secure. The foundation of the Tor network led to the ability to browse the Internet and communicate anonymously, with the latter being very important to government and military entities, but, along with this, came an increase in sophisticated cybercrime, like ransomware and secure botnets, and the ability to create websites that offer illicit drugs, murder for hire, and many other illegal activities.\n\n\\subsection{Bitcoin}\nBitcoin is the first successful cryptocurrency. It is a form of currency that is completely electronic and is not backed by some other means like a government. By design, it allows for Bitcoins to be transferred anonymously but also securely. It is based on a chain of transactions, which is used to ensure that nobody is lying about a transaction and allows for Bitcoins to be transferred. Bitcoins are also only tied to an address, called a wallet, which is not tied directly to an individual's identity. Since the currency is not connected to an identity, a transaction cannot be tracked back to see who it came from. The ability to make anonymous payments over the Internet can be very useful, but can also be used in many crimes. One major crime that became popular with the rise of Bitcoin is ransomware, which is malware that will infect a device, encrypt important files, and then require payment in order to get the key that will unlock the files. With the invent of Bitcoin, attackers can require that payment is made in the form of Bitcoins, making the payment untraceable. Bitcoin also lead to an increase in other crimes, like purchasing illegal drugs and even hiring hitmen, since the payment cannot be traced.\n\n\\section{RESEARCH ON SUBVERTING PRIVACY TECHNOLOGIES}\n\n\\subsection{Botnet over Tor: The Illusion of Hiding [1]}\n\nThis research was performed by Casenove and Miraglia from VrijeUniversiteit in Amsterdam, The Netherlands. It analyzes how botnets try to guarantee anonymity of the Command and Control node, addresses the problems with Tor-based botnets, and shows how full anonymity is never achieved.\n\nThe researchers found that as botnets began to make use of the Tor network, they typically used it to hide the connection, as best as possible, from infected devices to the command and control node. This was successful in its intended purpose of hiding the communication, but the act of using the Tor network often led to abnormal traffic that revealed the fact that a device was infected with a botnet. This was ultimately counterproductive to the ultimate purpose of the botnet; while the control servers were more difficult to find and take down, it was more obvious that the infected devices were infected, which makes them easier to fix. The researchers then discuss the fact that botnets that make use of Tor are still susceptible to many attacks that affect non-Tor based botnets. Crawling the address space to determine how many hosts are infected is still feasible and so is using traffic analysis at the exit nodes to determine the address of a command and control node. Using Tor in this manner has made it more difficult to track down the command server, but it does not solve all issues relating to botnets, but as these botnets evolve and make better use of Tor, they will be increasing difficult to undermine.\n\nThis research demonstrates that botnets that make use of privacy technologies, like Tor, are already here and are becoming more and more sophisticated. The bots analyzed in this research were only making minimal use of Tor and were not attempting to hide the fact that Tor was running on the infected devices, yet they were still successful when only using privacy technologies minimally. As these botnets become more sophisticated, they will become more difficult to track and take down. They will be better able to hide their communications and hide themselves on the infected devices. While botnets that make use of Tor are relatively new, they will continue to improve and use more privacy features, highlighting the threat that comes from subverting privacy technologies and the need to get ahead of these types of attacks.\n\n\\subsection{OnionBots: Subverting Privacy Infrastructure for Cyber Attacks [2]}\n\nThis research was performed by Sanatinia and Noubir from Northeastern University in Boston, MA. The goal of this research was to introduce a method by which a botnet could operate using the Tor infrastructure and then propose a method for neutralizing bots that are using the Tor network.\n\nThe proposed method for creating a botnet that uses the Tor network relies on a self-healing, low degree, and low diameter graph of bots. New nodes would join the botnet by contacting a list of current bots and then generating a connection between itself and that bot. This would be done in such a way that the bots would have a small number of neighbors, but the shortest path to any other bot would also be small. These nodes would also maintain the network by healing when a node leaves the graph. Since these nodes are using the Tor network, there is a decoupling between the IP address of a node and its .onion address. This allows the nodes to continually change their .onion address while still allowing neighbors to be able to communicate to it and allowing the Command and Control node to control it. The research shows that this botnet would be able to maintain a low degree and diameter while also not becoming partitioned if many nodes drop out of the network. It would also be difficult to track and take down since it is built on the Tor network and the devices can frequently change their .onion addresses in order to make finding the infected devices more difficult.\n\nThe research then proposed a method for defeating these so-called OnionBots. The technique is called Sybil Onion Attack Protocol - SOAP. The basis of the technique is that a node controlled by a defender that is part of the botnet is able to get a neighboring node to drop all of its other links and connect only to nodes the defender set up. The defender does this by creating new nodes that are under its control (replicas) that then connect to the target node. Part of the design of the botnet states that a node should connect to the neighboring nodes with the lowest degree and drop its neighbors with the highest degree in order to do so. The defender can exploit this by having the new nodes connect to the target node and report having very few neighbors. This will cause the target node to drop its neighbors and connect to the defender controlled nodes. Once the targeted node is only connected to the defender owned nodes, it is then disconnected from the botnet and is neutralized.\n\nThe goal of this research was to show that it is possible to create a botnet that operates using the Tor network for privacy and anonymity and show that there are ways to counteract this type of botnet. The threat posed by botnets operating over the Tor network is severe. Botnets have already been used in many crimes and attacks across the Internet, including the DDoS attack launched in October 2016 that targeted Dyn and prevented access to numerous websites. Botnets are already difficult to track down and stop, but botnets that make use of the Tor network would be extremely difficult to take down and could degrade service within the Tor network as a whole, leading to issues for its intended users. As the previous research showed, Tor based botnets already exist, and while they are not very sophisticated yet, this research shows that this class of botnets can continue to improve and become more successful. Ultimately, the research argues that these issues should be dealt with preemptively, including making changes to these privacy technologies if needed. Subverting these privacy technologies is a significant threat to any user of the Internet and a preemptive strategy is the best strategy that can be adopted at this point as it would protect against these technologies being used for crimes and can still provide the services they provide to users who are using them for non-criminal means. If the threat of subverting Tor for botnet use is not dealt with preemptively and quickly, it could become an enormous issue for the entire Internet and lead to a significant increase in untrackable crimes.\n\n\\subsection{Preparing for Malware that Uses Covert Communication Channels: The Case of Tor-based Android Malware [3]}\n\nThis research was performed by Kioupakis and Serrelis from AMC Metropolitan College in Amaroussio, Greece. It proposes a methodology for creating Android based malware that uses the Tor network to extract and steal information from the device and then proposes a method to mitigate this type of malware. \n\nThe researchers first discuss the fact that there are already two known cases of Android based malware that made use of the Tor network in some capacity, although it was typically used to retrieve information from the Command and Control bot. After analyzing these previous botnets, they proposed a design, including requirements and module implementations, for an Android based malware that makes use of the Tor network to steal information from the device. This malware targets the most popular version of Andoird, Jelly Bean, and would not be caught by anti-malware implementations that currently exist. It would be able to sit on Android devices, undetected, and siphon information that is then transmitted back to a central server, possibly compromising personal and confidential information that is contained in or sent from the device.\n\nAfter describing in detail how the malware is built, they propose an anti-malware system that can help to detect and stop malware that makes use of the Tor network. This anti-malware system would sniff packets off of the network and would shut down an interface if it is sending Tor traffic and it is not in a list of devices approved to use the Tor network. Current malware mitigations would not be able to detect the malware they designed, so new techniques were required that could detect and stop this type of malware. They propose that this type of anti-malware be developed preemptively in order to detect and stop future malware variants that will make use of Tor.\n\nThe research demonstrates that Tor can be used for many purposes within the broad scope of cybercrime and the security community needs to begin to take note and get ahead of these uses. If malware is able to run on Android devices undetected and anonymously send data back to a Command and Control server, then this is a very large threat to an enormous number of users. Android has one of the largest user bases in the world, so any malware that affects these devices could be extremely effective and lucrative. A reactive approach to this issue could lead to the loss of personal information for millions of users before this malware is identified and stopped. Tor has many useful purposes, but the protections it provides to cyber criminals must be minimized in order to protect Tor and the Internet's future use, even if it means making modifications to the Tor network or the devices that use it. Malware that uses privacy technologies is already here and will continue to become more stealthy and secure, demonstrating the need to attack the issue now before it becomes even more prevalent.\n\n\\subsection{Ransomware: Emergence of the Cyber-Extortion Menace [4]}\n\nThis research was performed by Hampton and Baig from the Security Research Institute at Edith Cowan University in Perth, Australia. The goal of this research was to show the trends that have occurred in the history of ransomware and make obvious the idea that security professionals need to get ahead of these issues instead of constantly being behind and reacting to the issue.\n\nThe researchers analyzed twenty-nine variants of ransomware that belonged to nine different families. For each variant, they looked for twenty-two different features that these variants could possess to see what features have been kept and also see what recent trends are emerging. They found that cryptographically secure encryption was not used until around 2013 and in recent years, there has been an increase in the use of Tor and cryptocurrencies like Bitcoin. The researchers suggest that the limited data makes it difficult to make predictions, but history has shown that future generations learn from the success and failure of past generations, and as of now, using Tor and other privacy technologies seems to be successful. Their analysis suggests that future variants of ransomware will make more use of privacy technologies in order to protect the cybercriminals and make it that much more difficult to track them down.\n\nThe researchers suggest, similar to the research described above, that security researchers need to get ahead of the trends instead of reacting as they occur. This is not possible in every situation, but if researchers can look towards the future and try to predict what might occur, it can be mitigated before it becomes a larger issue. They also describe the fact that vulnerability analysis has become a systematic and well vetted process with numerous databases detailing almost every imaginable vulnerability, yet malware, specifically ransomware, analysis has not had the same prominence and success. Ransomware is currently a massive business bringing in millions of dollars per year for cybercriminals and it affects individuals, businesses, and the government alike. The researchers ultimately advocate for creating a formal approach for documenting and analyzing malware, especially ransomware, so that security professionals can prevent more malware variants from springing up, which has been happening rapidly in the past few years. Once this process is in place, security researchers can begin to counteract these strains of malware, including reducing the ability of these strains to use privacy technologies for nefarious means.\n\n\\section{Conclusions}\n\nAs all of this research has shown, the possibility of subverting privacy technologies for nefarious means is very real. These attacks are not just a future possibility; they are here now and appear to be extremely successful. These technologies, like Tor and Bitcoin, were designed to provide users with a sense of privacy and anonymity when using the Internet. Tor was designed to help people communicate in an anonymous and untraceable way, which is great for military use of for whistleblowers who are afraid of persecution. Bitcoin was developed to allow people to make monetary transactions that were not tied to their identity and could not be traced back to them. These technologies were all designed and implemented with the intent to help people, but they are also being used to commit crimes and even help terrorists.\n\nAddressing these issues is a real concern that must be met with proactive measures. Far too many times, especially with security, we are reactive to situations. Many protocols and, in fact, the basis of the Internet was designed without security in mind and it was only added when bad things began to happen. There was a swift reaction to these issues and those problems were fixed. In order to stop the viral spread of cybercrime, made much easier with these technologies, there needs to be a proactive plan. As the research from Hampton and Baig stated, we need to create a system of standardization and understanding when it comes to malware and cybercrime research. We also need to continue to push for more research like the research presented here. This research looked at future possibilities and explored what may be coming in order to make suggestions of how it can be stopped now. If we, as a community, can look ahead and understand how issues may arise in the future, we can fix them now and stop the current rise of cybercrime, and also prevent new variants from popping up in the future. \n\nThe privacy technologies discussed in this paper were designed for good reason and provide many useful services, but in order for them to be truly successful and usable, there need to be mitigations in place that prevent cybercriminals from exploiting the protections provided. The research above discusses future threats that undermine privacy technologies, like Tor and Bitcoin, but the threat of subverting privacy technologies for criminal means is here now and must be dealt with in a proactive manner in order to stop these crimes and prevent them from appearing in the future to the greatest extent possible.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction.}\nIn recent years there has been much interest in the concept\nof quantum groups and quantum algebras \\cite{Drin} \\cite{Jimb}\n\\cite{FRT} \\cite{Wor87C}. Originating from investigations on the quantum\ninverse scattering methods and the Yang-Baxter equations, the quantum\ngroups have found various applications in theoretical physics such as\nintegrable fields theories, statistical models and conformal field theories\nin two-dimensions (see for example \\cite{Dobn} \\cite{CFZ} \\cite{LPS}).\n\nAnother major field starting in the seventies and still attracting a lot of\nattention among physicists and mathematicians is the field of super-symmetry.\nThe extension of the activity on quantum groups to this field was started\nwith the paper of Manin \\cite{Man89}, where the multi-parametric quantum\ndeformation of the supergroup $GL(m\/n)$ was introduced. Since then intensive\ninvestigations concerning both structures and representations of the quantum\nsupergroups were completed (see for example \\cite{dota} \\cite{Nky98} and\nreferences therein).\n\nIn Ref \\cite{Man89}, Manin extended the notion of quantum space \\cite{Man88}\nto that of quantum super-space, called also quantum super-plane, of which\nthe defining quadratic relations remain invariant under linear\ntransformations. These endomorphisms constitute the quantum supergroup.\n From a rigorous mathematical point of view, the quantum superplane appears\nin this approach as a comodule over the corresponding quantum supergroup.\nThe quantum (super)space has been then envisioned by many as a paradigm for\nthe general program of quantum deformed physics. The most hopped for\napplications includes a possible role in a future quantized theory\nof gravity \\cite{Maj}.\nIn this direction many efforts have been accomplished in order to\ndevelop its differential structure \\cite{KU92}. But all these constructions\nare made in the sense of \\cite{WZ90} and not, as far as we know, in the\nsense of Woronowicz because of the lack of the Hopf structure of the quantum\nsuper-plane.\n\nMore recently, one of us \\cite{tahri} has introduced the quantum plane as a quantum group according to the\nFaddeev-Reshetikhin-Takhtajan approach \\cite{FRT}.\nIn the present paper we extend the\nabove investigation to the $1+1$-dimensional quantum super-plane. In this new\napproach the quantum super-plane appears as a quantum sub-supergroup\nof $SL_q(1\/1)$. We give its super-matrix element, its\ncorresponding R-matrix and its Hopf structure. The latter appears\nto be cocommutative. We then realize its dual Hopf superalgebra starting\nfrom postulated initial pairings. We also construct\na right invariant differential\ncalculus on it and we deduce its corresponding quantum\nLie superalgebra which is as commutation superalgebra,\nup to a redefinition of the generators, the\nclassical $N=1$ supersymmetric algebra, but its Hopf structure\nis a non-cocommutative $q$-deformed one. This quantum Lie superalgebra is\nisomorphic to the one we obtained by the above duality.\n\nThe paper is organized as follows. In section 2, we recall the Manin\napproach. In section 3, the $1+1$-dimensional quantum superplane $K_q^{1\/1}$\nis made in the FRT approach, we also discuss some important properties\nof the quantum supermatrix considered to be a group element of $K_q^{1\/1}$.\nIn section 4, we find the dual Hopf superalgebra. In section 5,\nwe construct a right invariant differential calculus.\nIn section 6, we derive the quantum Lie superalgebra, for which we give\nthe Hopf structure in section 7. Finally, we give some concluding remarks\nin section 8.\n\n\\section{Review of the Manin approach}\nThe simplest example illustrating this approach is the one parameter quantum\ndeformation $GL_q(1\/1)$ of the supergroup $GL(1\/1)$ of $2\\times 2$\nsupermatrices with two bosonic and two fermionic matrix elements.\nThe construction of this quantum supergroup is well studied in\n\\cite{Man89} \\cite{SSV90} \\cite{CFFS}. It is generated by the\nelements of a quantum supermatrix:\n\\beq\\label{supmatr}\nM~=~\\bl(\\bra{lcr}\n&a&\\be\\\\\n&\\ga &d\n\\era\\br)\n\\eeq\nwhich obey the following supercommutation relations:\n\\beq\\label{glq}\n\\bra{lcr}\n&a\\be=q\\be a,\\hfill &d\\be=q\\be d\\hfill\\\\\n&a\\ga=q\\ga a,\\hfill &d\\ga=q\\ga d\\hfill\\\\\n&\\be\\ga +\\ga\\be=0,\\hfill &\\be^2=\\ga^2=0\\\\\n&ad-da=-\\lambda\\be\\ga, &\\\\\n\\era\n\\eeq\nwhere $\\lambda=q-q^{-1}$ and the generators $a$ and $d$ are even\nwith parity $\\p a=\\p d=0$\nand $\\be$, $\\ga$ are odd with parity $\\p\\be=\\p\\ga=1$(called Grassmanian).\nThe supermatrix $M$ defines the basic representation of $GL_q(1\/1)$,\ni.e., $M$ is considered as a group element. Thus, in this sense, the quantum\nsupergroup $GL_q(1\/1)$ appears as a deformation of the algebra of\npolynomials function on $GL(1\/1)$, the later is recovered when one\ntakes $q\\to 1$.\n\nThe relations (\\ref{glq}) may be succinctly expressed\nin terms of the graded RTT equation as\n\\beq\\label{rtt}\nRM_1M_2=M_2M_1R,\n\\eeq\nwhere\n\\beq\\label{tensor}\n\\bra{ll}\n&(M_1)^{ij}_{kl}=(M\\ot 1)^{ij}_{kl}=(-1)^{(\\p k(\\p j+ \\p l))}M^i_k\\de^j_l\\\\\n&(M_2)^{ij}_{kl}=(1\\ot M)^{ij}_{kl}=(-1)^{(\\p i(\\p k +\\p l))}M^j_l\\de^i_k\n\\era\n\\eeq\nand the matrix $R$ is given by:\n\\beq\\label{rmatr}\nR~=~\n\\pmatrix{&q&0&0&0\\cr\n&0&1&0&0\\cr\n&0&\\lambda&1&0\\cr\n&0&0&0&q^{-1}\\cr}\n\\eeq\nwhich is a solution of the graded quantum\nYang-Baxter equation:\n\\beq\nR_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}\n\\eeq\narising from the associativity requirement of $GL_q(1\/1)$.\n\nIn this formulation one can define the quantum superdeterminant:\n\\beq\\bra{lr}\n{\\cal D}=Sdet(M)&=ad^{-1}-\\be d^{-1}\\ga d^{-1}\\\\\n&=d^{-1}a-d^{-1}\\be\\ga d^{-1}\\\\\n\\era\n\\eeq\nprovided $d^{-1}$ exists, and it can be checked that it is central, i.e.,\nit commutes with all elements of $M$. Therefore, by imposing\nthe relation ${\\cal D}=1$, we may define a quantum deformation\n$SL_q(1\/1)$ by analogy with the classical restriction to the special\nlinear supergroup. The inverse supermatrix can be now defined.\nIt is given by\n\\beq \\label{inverse}\nM^{-1}=\n\\pmatrix{\n&a^{-1}+a^{-1}\\be d^{-1}\\ga a^{-1} & -a^{-1}\\be d^{-1}\\cr\n&{-d^{-1}\\ga a^{-1}}&d^{-1}+d^{-1}\\ga a^{-1}\\be d^{-1}\\cr\n}.\n\\eeq\nSo the Hopf structure of $GL_q(1\/1)$ is given by the following:\n\\beq\\label{hopfalg}\n\\De(M)=M\\dot{\\ot }M=\n\\pmatrix{\n& a\\ot a+\\be\\ot\\ga &a\\ot\\be+\\be\\ot d\\cr\n&\\ga\\ot a +d\\ot\\ga&\\ga\\ot \\be+d\\ot d\\cr\n}\n\\eeq\n\\beq\\label{counit}\n\\vep(M)=\n\\pmatrix{\n&1&0\\cr\n&0&1\\cr\n}\n\\eeq\n\\beq\nS(M)=M^{-1}.\n\\eeq\n\nNow let us recall some statements related to the supermatrix $M$\nconsidered as group element (or a point) of $GL_q(1\/1)$.\n\\nt If one takes two copies $M$ and $M'$ such that their elements pairwise supercommute, then the products $MM'$ and $M'M$ are group elements\nof $GL_{q}(1\/1)$. Furthermore, the inverse $M^{-1}$ is\nactually a group element of $GL_{q^{-1}}(1\/1)$ rather than $GL_q(1\/1)$,\nand, in general, the n-th power of $M$,\n$n\\in {\\ZZ}$, is a group element of $GL_{q^n}(1\/1)$.\n\nManin has defined in correspondence with the quantum\nsupergroup what is called the quantum superplane or `quantum superspace'\nand the dual of it. This quantum superplane or, rather, the polynomial\nfunction ring on it, denoted by $K_q^{1\/1}$, is generated\nby two coordinates $x$ bosonic with parity $\\p x=0$ and $\\te$\nfermionic with parity $\\p\\te=1$ such that:\n\\beq\\label{supplane}\\bra{llrr}\n&x\\te~=~q\\te x&\\qquad\\qquad(q\\neq 0,1)\\\\\n&\\te^2~=~0.&\\\\\n\\era\n\\eeq\nThe dual, denoted by ${}^*K_q^{1\/1}$, is generated by coordinates\n$\\xi$ with parity $1$ and $y$ with parity $0$ and the following\nsupercommutation relations:\n\\beq\\label{dqsp}\\bra{llrr}\n&\\xi y~=~q^{-1}y \\xi&\\qquad\\qquad(q\\neq 0,1)\\\\\n&\\xi^2~=~0.&\\\\\n\\era\n\\eeq\nThe quantum supergroup $GL_q(1\/1)$ appears as\nendomorphisms of the quantum superplanes\n$K_q^{1\/1}$ and ${}^*K_q^{1\/1}$ under which the quadratic relations\n(\\ref{supplane}) and (\\ref{dqsp}) remain invariant. So Manin proved\nthe equivalence between the defining quantum supergroup supercommutation\nrelations (\\ref{glq}) and the quadratic relations (\\ref{supplane}) and\n(\\ref{dqsp}).\nIn a more rigorous mathematical point of view, the quantum superplanes\n$K_q^{1\/1}$ and ${}^*K_q^{1\/1}$ are viewed as comodules over $GL_q(1\/1)$\nvia the following (left) coactions\n\\beq\\label{lcoac}\n\\de(X)=M\\dot{\\ot }X=\n\\pmatrix{\n& a\\ot x+\\be\\ot\\te \\cr\n&\\ga\\ot x +d\\ot\\te \\cr\n}\n\\eeq\n\\beq\\label{lcoad}\n\\de(\\Te)=M\\dot{\\ot }\\Te=\n\\pmatrix{\n& a\\ot \\xi+\\be\\ot y \\cr\n&\\ga\\ot \\xi +d\\ot y \\cr\n}\n\\eeq\nwhere $X$ and $\\Te$ are column vectors of components $x, \\te$ and\n$\\xi, y$, respectively.\n\n\\section{Quantum superplane in new approach.}\nLet us consider a particular copy of $GL_q(1\/1)$ where\n$a=d=x$ and $\\be=\\ga=\\te$\nthat is:\n\\beq\\label{matra}\nA~=~\\pmatrix{\n&x &\\te\\cr&\\te&x\\cr},\n\\eeq\nin this case the commutation relations (\\ref{glq}) reduce\nto the quadratic relations (\\ref{supplane}) defining the quantum superplane\n$K^{1\/1}_q$. Note that the superdeterminant of $A$ is equal to $1$,\nso it may be viewed also as a particular copy of $SL_q(1\/1)$.\nIts quantum inverse supermatrix is given by\n\\beq\\label{inva}\nA^{-1}=\n\\pmatrix{\n&x^{-1}&-x^{-1}\\te x^{-1}\\cr\n&-x^{-1}\\te x^{-1}& x^{-1}\\cr}.\n\\eeq\nThis leads to our following main result.\\\\\\\\\n\n\\nt{{\\bf Thoerem.}\\\\{\\it\nThe quantum Manin superplane $K_q^{1\/1}$ generated by the coordinates\n$x$ with parity $0$ and $\\te$ with parity $1$ and extended by\nthe inverse $x^{-1}$ satisfying the quadratic relations\n$x\\te=q\\te x$ and $\\te^2=0$ is a quantum supergroup generated by\nelements of the supermatrix of type $A$ modulo the graded $RTT$\nequation (\\ref{rtt}) where the matrix $R$ is given by (\\ref{rmatr}).\nIts Hopf structure is given explicitly by:\n\\beq\\label{hopfsup}\\bra{lllll}\n&\\De(x)=x\\ot x+\\te \\ot\\te,\\\\\n&\\De(\\te)=\\te\\ot x +x\\ot\\te,\\\\\n&\\vep(x)=1,\\\\\n&\\vep(\\te)=0,\\\\\n&S(x)=x^{-1}\\\\\n&S(\\te)=-x^{-1}\\te x^{-1}.\n\\era\\eeq\n}\nWe see that it is a cocommutative Hopf superalgebra.\n\nThere are curious properties applicable to this quantum supermatrix\n considered as a group element of $K_q^{1\/1}$. They assert that:\\\\\na) If $A$ and $A'$ are two copies of $K_q^{1\/1}$ such that their\nentries pairwise supercommute, then the products\n$AA'$ and $A'A$ are group elements of $K_q^{1\/1}$.\\\\\nb) the n-th power of $A$ is a group element of $K_{q^{n}}^{1\/1}$ given by:\n\\beq{\\label{pmata}}\nA^n=\\pmatrix{\n&x^n&[n]_q\\te x^{n-1}\\cr\n&[n]_q\\te x^{n-1}&x^n\\cr},\n\\eeq\nwhere $[n]_q=(1-q^n)\/(1-q)$. Note that if we take $n=-1$ we find the\ninverse (\\ref{inva}) (since $[-1]_q=-q^{-1}$). Furthermore, if there\nexists an integer $n$ such that $q^n=1$, then the n-th power of $A$\n(\\ref{pmata}) reduces to\n\\beq{\\label{rpmat}}\nA^n=\\pmatrix{\n&x^n&0\\cr\n&0&x^n\\cr},\n\\eeq\nwhich gives a nilpotent supermatrix when $x^n=0$ or unit supermatrix\nwhen $x^n=1$.\n\nSo in this sense, the quantum superplane $K_q^{1\/1}$ can be considered as\na quantum subsupergroup of $SL_q(1\/1)$.\n\n\\section{The dual of $K_q^{1\/1}$}\nWe will apply here the method introduced by Sudbery in \\cite{Sud}, where\nhe obtained ${{\\cU}_q}(sl(2))\\ot {\\cU}(u(2))$ as the algebra of tangent vectors\nat the identity, and generalized to more complicated algebras by Dobrev\n\\cite{Dob1}. The application of this method to the multiparameter\nsupergroup $GL_{u{\\bf q}}(m\/n)$ is given in \\cite{dota}. It consists of\nusing the duality between (super)bialgebras or Hopf (super)algebras to\nobtain the unknown dual of a known (super)algebra.\n\nTwo bialgebras $\\cU$ and $\\cA$ are said to be in duality \\cite{Abe}\nif there exists a doubly nondegenerate bilinear form\n\\beq{\\label{dubi}}\n\\lag~,~\\rad :~{\\cU\\times\\cA\\longrightarrow\\CC} \\quad  (u,a)\\lpt\\lag~u,a~\\rad\n\\quad {u\\in \\cU},~a\\in\\cA\n\\eeq\nsuch that, for $u,~v\\in {\\cU}$, $a,~b\\in\\cA$\n\\beq\\label{prdu}\n\\bra{ll}\n&\\lag~u,ab~\\rad =\\lag~\\de_{\\cU}(u),a\\ot b~\\rad ,\\\\\n&\\lag~uv,a~\\rad =\\lag~u\\ot v,\\de_{\\cA}(a)~\\rad ,\\\\\n&\\lag~u,1_{\\cA}~\\rad =\\vep_{\\cU}(u),\\quad\n\\lag~1_{\\cU},a~\\rad =\\vep_{\\cA}(a).\\\\\n\\era\n\\eeq\nFor dual Hopf algebras one adds\n\\beq{\\label{dhal}}\n\\lag~\\ga_{\\cU}(u),a~\\rad~=~\\lag~u,\\ga_{\\cA}(a)~\\rad.\n\\eeq\nTo extend this to superalgebras or Hopf superalgebras, we take into account\nthat the tensor product is also graded, and, if (using Sweedler's notation)\n$\\de_{\\cU}=\\sum u_{(1)}\\ot u_{(2)},~\\de_{\\cA}=\\sum a_{(1)}\\ot a_{(2)}$, then\n\\beq\\label{grdu}\n\\bra{ll}\n&\\lag~u,ab~\\rad =\\sum (-1)^{\\widehat{u_{(2)}}\\widehat{a}}\n\\lag~u_{(1)},a~\\rad\\lag~u_{(2)},b~\\rad ,\\\\\n&\\lag~uv,a~\\rad =\\sum (-1)^{\\widehat{v}\\widehat{a_{(1)}}}\n\\lag~u,a_{(1)}~\\rad\\lag~v,a_{(2)}~\\rad ~.\\\\\n\\era\n\\eeq\nIt is enough to define the pairing (\\ref{dubi}) between the generating\nelements of the two algebras. The pairing may then be extended to all\nelements of $\\cU$ and $\\cA$ by using relations (\\ref{prdu}).\n\nTo find the unknown superalgebra, it is enough to give the pairing\nbetween the generating elements of the unknown superalgebra with an arbitrary\nelement of the PBW basis of the known superalgebra. So, we first need\nto fix a PBW basis of $K_q^{1\/1}$. This basis consists of monomials\n\\beq{\\label{pbwb}}\nh=x^m\\te^n\n\\eeq\nwhere $m\\in{\\ZZ}$ and $n\\in\\{0,1\\}$. Let us then denote the dual\nsuperalgebra by ${\\cU}_q(k(1\/1))$ and its generating elements by\n$\\chi$ and $\\phi$. Following \\cite{Dob1} we shall postulate the\npairing $\\lag Z,h\\rad$ $Z=\\chi,~\\phi$, $h$ from (\\ref{pbwb}), as\nwe use the classical tangent vector at the identity :\n\\beq\\label{tvai}\n\\bra{ll}\n&\\lag~\\chi,h~\\rad =m\\de_{n,0},\\\\\n&\\lag~\\phi,h~\\rad =\\de_{n,1}.\\\\\n\\era\n\\eeq\nWe also note that, from the duality properties, it follows\n\\beq{\\label{duco}}\n\\lag~1_{\\cU},h~\\rad =\\vep(h)=\\de_{n,0}.\n\\eeq\nTo obtain the commutation relations between the generators $\\chi$ and\n$\\phi$, we first need to evaluate the action of their bilinear\nproduct on the elements of $K_q^{1\/1}$. Using the defining relations\nwe obtain\n\\beq\\label{bpro}\n\\bra{ll}\n&\\lag~\\chi\\phi,h~\\rad =(m+1)\\de_{n,0},\\\\\n&\\lag~\\phi\\chi,h~\\rad =(m+1)\\de_{n,1},\\\\\n&\\lag~(\\phi)^2,h~\\rad =-{\\frac{1-q^{-2m}}{1-q^{-2}}}\\de_{n,0}.\\\\\n\\era\n\\eeq\nThus we have the commutation relations :\n\\beq\\label{cxph}\n\\bra{ll}\n&\\chi\\phi~-~\\phi\\chi ~=~0,\\\\\n&(\\phi)^2~=~-\\frac{1-q^{-2\\chi}}{1-q^{-2}}.\\\\\n\\era\n\\eeq\nThe Hopf structure of this superalgebra may be deduced by using the\nduality. We start with the coproduct, i.e., we shall use the relations\n\\beq{\\label{dcop}}\n\\lag~Z,h~\\rad =\\lag~\\de_{\\cU}(Z),h_1\\ot h_2~\\rad\n\\eeq\nfor every splitting $h=h_1h_2$. Thus we get\n\\beq{\\label{coda}}\n\\bra{ll}\n&\\de_{\\cU}(\\chi)~=~\\chi\\ot 1~+~1\\ot\\chi,\\\\\n&\\de_{\\cU}(\\phi)~=~\\phi\\ot q^{-\\chi}~+~1\\ot\\phi~.\\\\\n\\era\n\\eeq\nthe counit relations in ${\\cU}_q(k(1\/1))$ are given by\n\\beq{\\label{cocu}}\n\\vep_{\\cU}(Z)~=~0,\\quad Z=\\chi,~\\phi\n\\eeq\nwhich follows easily from\n\\beq{\\label{duep}}\n\\lag~Z,1_{\\cA}~\\rad ~=~0~=~\\vep_{\\cU}(Z).\n\\eeq\nFinally, the antipode map of ${\\cU}_q(k(1\/1))$ follows from\n\\beq{\\label{duan}}\n\\lag~\\ga_{\\cU}(Z),h~\\rad ~=~\\lag~Z,\\ga_{\\cA}(h)~\\rad.\n\\eeq\nIt is given by\n\\beq{\\label{anda}}\n\\bra{ll}\n&\\ga_{\\cU}(\\chi)~=~-\\chi,\\\\\n&\\ga_{\\cU}(\\phi)~=~-q^{\\chi}\\phi~.\\\\\n\\era\n\\eeq\n\nNow let us make the following redefinition:\n\\beq{\\label{redp}}\n\\phi'~=~iq^{(\\chi -1)\/2}\\phi,\\quad i^2=-1\n\\eeq\nthen we get the following quantum deformed enveloping superalgebra\n(dropping the primes)\n\\beq\\label{ncxp}\n\\bra{ll}\n&\\chi\\phi~-~\\phi\\chi ~=~0,\\\\\n&(\\phi)^2~=~[\\chi]\\\\\n\\era\n\\eeq\nwhere $[x]=(q^{x}-q^{-x})\/(q-q^{-1})$. Its Hopf structure is now given by:\n\\beq{\\label{nhst}}\n\\bra{ll}\n&\\de_{\\cU}(\\chi)~=~\\chi\\ot 1~+~1\\ot\\chi,\\\\\n&\\de_{\\cU}(\\phi)~=~\\phi\\ot q^{-\\chi\/2}~+~q^{\\chi\/2}\\ot\\phi~.\\\\\n&\\vep_{\\cU}(\\chi)~=~\\vep_{\\cU}(\\phi)~=~0,\\\\\n&\\ga_{\\cU}(\\chi)~=~-\\chi,\\quad \\ga_{\\cU}(\\phi)~=~-\\phi~.\\\\\n\n\\era\n\\eeq\n\n\\section{Differential calculus on  $K_q^{1\/1}$}\nThe differential calculus on quantum group was initiated by\nWoronowicz \\cite{Wor87} \\cite{Wor89}.\nHere we follow the same techniques used in \\cite{SWZ91}\nto construct the right invariant calculus on the two-parameter\ndeformation of $GL(2)$ and extended in \\cite{SVZ90} and\n\\cite{BT92} to the one and two-parameter deformation of $GL(1\/1)$,\nrespectively.\n\nWe introduce at first the exterior differential $d$ which satisfies the\nnilpotency and the graded Leibniz rule,\n\\beq\\label{leibr}\n\\bra{ll}\n&d^2=0,\\\\\n&d(fg)=(df)g~+~(-1)^{\\p f}f(dg),\\\\\n\\era\n\\eeq\nwhere $f$, $g$ are functions of the variables $x$, $\\te$ and ${\\p f}$\nis the parity of $f$. In analogy with the classical differential geometry,\none can construct from $A$ and the one-forms $dA$, via the matrix valued one-forms $\\Om$ , the right-invariant Cartan-Maurer forms as\n\\beq\\label{cmf}\n\\Om=(dA)A^{-1}=\n\\Bl(\n\\bra{llrr}\n&\\om&v\\\\\n&v&\\om\n\\era\n\\Br).\n\\eeq\nWriting this as $dA=\\Om A$, one can express the basic one-forms\n$dx$ and $d\\te$ in terms of Cartan-Maurer forms as\n\\beq\\label{bof}\n\\bra{llrr}\n&dx=\\om x +v\\te\\\\\n&d\\te=v x+\\om\\te.\\\\\n\\era\n\\eeq\nThe exterior differential $d$ can then be written as\n\\beq\\label{exod}\n\\bra{llrr}\nd~&=~dx\\pa_x +d\\te\\pa_{\\te}\\\\\n&=~\\om D_x+vD_{\\te}.\\\\\n\\era\n\\eeq\n\nUsing the nilpotency of the differential\n$d$ and (\\ref{bof}) it follows that:\n\\beq\\label{dcmf}\\bra{l}\nd\\om=\\om^2-v^2\\\\\ndv=\\om v -v\\om \\\\\n\\era\n\\eeq\nwhich we call the Cartan-Maurer equations.\n\nNow we want to calculate the $q$-deformed supercommutation relations between\nthe basic variables $x$, $\\te$ and the Cartan-Maurer forms. Using a similar\nrestriction as in \\cite{SVZ90} due to the right-invariance of the\nCartan-Maurer forms, we suggest\n\\beq\\label{crvof}\\bra{llllr}\n&xv=qvx,&\\te v=qv\\te,&\\\\\n&x\\om=F\\om x,&\\te\\om=-F\\om\\te,\n\\era\n\\eeq\nwhere $F$ is an arbitrary complex number. Then the action of $d$\non the quadratic relations (\\ref{supplane}) do not lead to\nany inconsistency, while the consistence of $d$ with (\\ref{crvof})\nrequires the following:\n\\beq\\label{module1}\\bra{llllr}\n&\\om v= v\\om& & \\\\\n&\\om^2= kv^2,& &k={\\frac{q^2-F}{F(F+1)}}.\n\\era\n\\eeq\nUsing this, the Cartan-Maurer equations (\\ref{dcmf}) reduce to\n\\beq\\label{rdcmf}\\bra{l}\nd\\om=(k-1)v^2\\\\\ndv=0\\\\\n\\era\n\\eeq\n\n\\section{Quantum Lie superalgebra }\n\nNow let us return to the expression of $d$ in (\\ref{exod}), the\nelements $D_x$ and $D_{\\te}$ generate the quantum Lie superalgebra\nassociated to the quantum superplane $K_q^{1\/1}$. To calculate the\nsupercommutation relations between these generators we use the nilpotency\nof $d$,\n\n\\beq\\label{niof}\\bra{ll}\n0=d^2(f) &=~d(\\om D_xf +vD_{\\te}f)\\cr\n&=~(d\\om)D_xf+(dv)D_{\\te}f-\\om(\\om D_x+vD_{\\te})D_xf+\\cr\n&+~v(\\om D_x+vD_{\\te})D_{\\te}f,\\cr\n\\era\n\\eeq\nwhere $f$ is an arbitrary function of the variables $x$, $\\te$.\nThen, using (\\ref{module1}) and (\\ref{rdcmf}), it follows that\n\\beq\\label{algebre1}\n\\bra{ll}\n&D_xD_{\\te} -D_{\\te} D_x=0\\\\\n&D_{\\te}^2=kD_x^2+(1-k)D_x.\n\\era\n\\eeq\nWe will denote this superalgebra by $k_{q,F}(1\/1)$. Note that\nif we make the following change:\n\\beq\\label{ndx}\nT_x=kD_x^2 +(1-k)D_x,\n\\eeq\nthen we get\n\\beq\\label{algebre2}\n\\bra{ll}\n&T_xD_{\\te} -D_{\\te} T_x=0,\\\\\n&D_{\\te}^2=T_x,\n\\era\n\\eeq\nwhich are the commutation relation of the `classical' $N=1$\nsupersymmetric algebra. So the\nquantum deformation does not affect the Lie superalgebra structure.\n\nNow it is possible to calculate the supercommutation relations between\nthe generators of $k_{q,F}(1\/1)$ and the basic coordinates.\nTo this end, we use the graded Leibniz rule by comparing the coefficients\nof the Cartan-Maurer forms when acting by $d$ on $xf$ or $\\te f$.\nThen we obtain\n\\beq\\label{module2}\\bra{llrr}\n&D_xx=x+FxD_x,&D_x\\te =\\te+F\\te D_x\\\\\n&D_{\\te} x=\\te +qxD_{\\te},&D_{\\te}\\te=x-q\\te D_{\\te}.\n\\era\n\\eeq\nOn the other hand, if we multiply the second relation in (\\ref{algebre1})\nfrom the right by $x$ or $\\te$ and then pull it to the left using the\ncommutation relations (\\ref{module2}), then we get\n\\beq\\label{ccod}\nD_{\\te}^2-\\frac{kF^2}{q^2}D_x^2-\\frac{F(1+k)}{q^2}D_x=0\n\\eeq\nfrom which it follows that $F$ must be equal to\n$q^2$ or $q$. We will see in the next section that these\ntwo solutions are equivalent.\n\n\\section{Hopf structure of the superalgebra $k_{q,F}(1\/1)$}\nNow we shall use the graded Leibniz rule for arbitrary\nfunctions $f$ and $g$ of coordinates $x$, $\\te$ to derive the Hopf structure\nof $k_{q,F}(1\/1)$. An arbitrary function is\nin fact a linear combination of monomials in the basis (\\ref{pbwb}).\n\nWe first need to calculate the commutation relations\nbetween the Cartan-Maurer one-forms and functions. From (\\ref{crvof})\nwe have\n\\beq\\label{ccmf}\n\\bra{ll}\n&f\\om=(-1)^{\\p f}wF^Nf\\\\\n&fv=vq^Nf.\n\\era\n\\eeq\nwhere $N$ is an operator number acting on monomial as\n\\beq\\label{opnu}\nN(x^m\\te^n)=(m+n)x^m\\te^n,\n\\eeq\nand may be related to the operator $D_x$. To see this we calculate\nthe action of $D_x$ on functions which may be achieved by calculating its\ncommutation relation with $h$. Indeed we have\n\\beq\\label{crwh}\nD_{x}h=\\frac{1-F^{m+n}}{1-F}h~+~F^nhD_x\n\\eeq\nfrom which we get\n\\beq\\label{rdxn}\nD_x=\\frac{1-F^N}{1-F}\n\\eeq\n\nTherefore, using the graded Leibniz rule applied to the product of functions\n$f$ and $g$, and the expression of $d$ in terms of $D_x$ and $D_{\\te}$\n(\\ref{exod}) with the help of (\\ref{ccmf}), this leads to the following:\n\\beq\\label{dofg}\\bra{ll}\n&D_x(fg)=D_x(f)g+F^NfD_x(g),\\\\\n&D_{\\te}(fg)=D_{\\te}(f)g+(-1)^{\\p f}q^NfD_{\\te}(g),\\\\\n\\era\n\\eeq\nfrom which we get the coproduct:\n\\beq\\label{copk}\n\\bra{ll}\n&\\de(D_x)=D_x\\ot 1+F^N\\ot D_x\\\\\n&\\de(D_{\\te})=D_{\\te}\\ot 1+q^N\\ot D_{\\te}\n\\era\n\\eeq\nwhere use was made of the graded tensor product: $(a\\ot b)(f\\ot g)=\n(-1)^{{\\p b}{\\p f}}a(f)\\ot b(g)$, $a, b\\in  k_{q,F}(1\/1)$\nand $f, g\\in K_q^{1\/1}$. The counit and antipode relations in\n$k_{q,F}(1\/1)$ may be calculated by using the basic axioms\nof Hopf superalgebras \\cite{Abe} \\cite{Man89}:\n\\beq{\\label{axha}}\n\\bra{ll}\n(\\vep_k\\ot id)\\de_k(u)~=~u,\\\\\nm(id\\ot\\ga_k)\\de_k(u)~=~\\vep_k(u),\\\\\n\\era\n\\eeq\nwhere $u\\in k_{q,F}(1\/1)$ and $m$ is the multiplication.\nSo we obtain\n\\beq{\\label{canr}}\n\\bra{ll}\n&\\vep_k(D_x)~=~\\vep_k(D_{\\te})~=~0,\\\\\n&\\ga_k(D_x)~=~-F^{-N}D_x,\\quad \\ga_k(D_{\\te})~=~-q^{-N}D_{\\te}.\\\\\n\\era\n\\eeq\nThus, we have a $q$-deformed non-cocommutative Hopf\nstructure depending on the parameter $q$. Furthermore,\nwe note that the two forms of (\\ref{copk}), and (\\ref{canr})\naccording to the two solutions of $F=q^2$ or $q$ are consistent\nwith the redefinition (\\ref{ndx}). Therefore, we may view the two solutions\nof $F$ as corresponding to changes of basis for the quantum enveloping\nsuperalgebra and are equivalent as such. So, it is convenient to choose\nthe solution $F=q^2$ which is the simplest one. We can then transform\nthis superalgebra to the form obtained in Section 4 (c.f., (\\ref{ncxp})\n(\\ref{nhst})) by making the following changes:\n\\beq\\label{comp}\n\\bra{ll}\n&D_x~\\equiv~{\\frac{1-q^{2\\chi}}{1-q^2}}~,\\\\\n&\\\\\n&D_{\\te}~\\equiv~q^{(\\chi-1)\/2}\\phi\\\\\n\\era\n\\eeq\nwhich are consistent with the commutation relations and the Hopf structure.\n\n\\section{Concluding remarks }\nWe have shown that the 1+1-dimensional quantum superplane, which is\nwell known in the literature as comodule over its corresponding\nquantum supergroup, may be incorporated in the Faddeev-Reshetikhin-Takhtajan\napproach to quantum (super)groups. We have given its supermatrix element,\nits corresponding R-matrix and its Hopf structure which\nis a cocommutative one. We then derived its dual Hopf superalgebra starting\nfrom a postulated initial pairing. We also constructed a right\ndifferential calculus on it and we deduced the corresponding quantum\nLie superalgebra. As a commutation superalgebra, the latter coincides with\nthe classical N=1 supersymmetric algebra, but its Hopf structure is a\nnon-cocommutative $q$-deformed one. We also prove that there is an\nisomorphism between this quantum Lie superalgebra and the one we obtained\nusing postulated duality.\nTo finish let us stress that the concept of $q$-deformation of Lie\n(super)group and (super)space is physically consistent when it is\nperformed in the context of Hopf (super)algebra. So we expect that\nthis new approach to the quantum superplane and the differential\ncalculus introduced on it here will be a starting point to construct\nmore consistent $q$-deformed physical models.\n\n{\\vskip 8mm\n{\\bf Acknowledgments.}~~~\nOne of the authors MEF is grateful for hospitality at the Abdus Salam ICTP\n, where part of this work was achieved, to Prof. R. Floreanini, and to Prof. S. Randjbar-Daemi for reading this paper.  }\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec_intro}\n\nConsider a set $M$ of random straight lines in $\\mathbb{R}^{d}$ and their intersections $M \\cap B$ with an object $B$~\\cite{Santalo, Solomon}. These intersections define an ensemble of chords $c$ through $B$: if the measure of random lines is normalized (so that it is a probability measure), then this set of chords can be seen as a random variable, whose associated probability density is usually named chord length distribution (if the object is non-convex, a line may generate several chords, and in this case each chord contributes to the chord length distribution). Furthermore, if the random straight lines obey a uniform density (in a sense that will be made rigorous in Sec.~\\ref{sec_length}), then the mean chord length $\\langle c \\rangle$ is simply related to the volume $V$ and to the surface $\\Sigma$ of the body $B$ by~\\cite{Santalo}\n\\begin{equation}\n\\langle c \\rangle = \\eta_d \\frac{V}{\\Sigma},\n\\end{equation}\nwhere $\\eta_d$ is a constant depending only on the dimension $d$ ($\\eta_2 = \\pi$ and $\\eta_3 = 4$). This fundamental result goes under the name of Cauchy's formula. The mean chord length $\\langle c \\rangle$ is a rather intuitive quantity describing the `mean size' of an object and plays an important role in stereology~\\cite{Underwood} and image analysis~\\cite{Serra}, since probing structures with random lines allows characterizing a given material. The mean chord length has also become a standard tool in transport theory and nuclear reactor analysis, since the pioneering work of Dirac and co-workers~\\cite{Dirac}, especially in the context of external irradiation~\\cite{Case}. Moreover, Cauchy's formula has practical applications in various fields such as acoustics~\\cite{Kingman}, ecology~\\cite{McIntyre}, and is key to the study of random media~\\cite{Torquato}. In 1981, Bardsley and Dubi~\\cite{Bardsley} based on the linear Boltzmann equation showed that Cauchy's formula applies more generally to the stochastic paths of Pearson random walks~\\cite{Pearson} with exponentially distributed flight lengths: for such Markovian processes, the average length $\\langle l \\rangle$ travelled in a bounded body $B$ is again (and surprisingly) given by $\\langle l \\rangle = \\eta_d V\/\\Sigma$, when the walkers start uniformly and isotropically on the surface of $B$. This means in particular that the travelled length for exponential Pearson walks is somewhat universal, in that $\\langle l \\rangle$ is expressed only in terms of the volume and the surface of the traversed medium, independent of the specific details of the geometry and of the walk (such as the mean free path, for instance). This strong result unfortunately went almost unnoticed, until it was independently rediscovered (by following a different approach) by Blanco and Fournier in 2003~\\cite{Blanco}, in the context of theoretical ecology (the study of animal displacements through confined geometries). Since then, Cauchy's original statement has attracted intense research efforts, mostly aimed at extending the range of applicability of the formula. The quantity $\\langle l \\rangle$ was first related to the average length travelled by walkers starting uniformly within the body $B$~\\cite{MazzoloEuro}, and successively to the average number of collisions performed in $B$~\\cite{Mazzolo05}. B\\'enichou and co-workers, based on a backward Chapman-Kolmogorov equation, extended the formula to the higher moments of the travelled length (in relation with the problem of residence times of stochastic motions in bounded domains), explored the effects of mixed boundary conditions (Dirichlet and Neumann) at the surface $\\Sigma$, and showed that remarkable ergodic-like properties apply to the average travelled length~\\cite{Benichou}. Recently, these results were further generalized by resorting to the Feynman-Kac path-integral approach, which allows explicit formulas to be derived for the travelled lengths and the number of collisions of exponential Pearson walks in the presence of both absorption and branching (this latter representing, e.g., fission events in reactor physics, or the birth of new individuals in the spreading of bacteria as well as of pathogens in epidemics)~\\cite{Zoia3}.\n\nSuch findings show that Cauchy's formula and exponential Pearson random walks are intimately connected. However, it is well-known that many real-world stochastic transport phenomena do not display exponentially distributed jump lengths (see, e.g., the discussion in~\\cite{Zoia2, Zoia6, Zoia4}), and one might naturally wonder whether similar universal properties still hold for these processes. Actually, some heuristic findings suggest that Cauchy's formula applies to arbitrary (isotropic) random walks whenever the random walk enters the domain $B$ with a length distribution compatible with equilibrium~\\cite{Benichou, BlancoPRL}. In particular, this was formally proven (by resorting to integral geometry arguments) for the quite restrictive class of walks having a single collision in $B$ before leaving the domain~\\cite{Mazzolo09}.\n\nThe purpose of this paper is to show under which conditions Cauchy's formula is valid for arbitrary Pearson walks. We begin our analysis by deriving a key result for the average number of collisions in $B$, within the framework of transport theory. Then, by building upon this result and by using the standard tools of geometric probabilities, we proceed further to derive a Cauchy's formula for the average lengths of the random paths. We conclude by considering an extension of these results to the case of purely absorbing random walks.\n\n\\section{A Cauchy's formula for the average number of collisions}\n\\label{sec_N}\n\nIn a series of recent papers~\\cite{Zoia1, Zoia5, Zoia3}, it has been shown that the travelled length and collision statistics of exponential Pearson walks in bounded domains can be straightforwardly determined by resorting to the Feynman-Kac path-integral formalism. In particular, this approach allows deriving generalized Cauchy's formulas for the moments of any order of lengths and collisions, and accounts also for the possibility of birth-death events leading to branching paths. However, the key argument underlying the applicability of the path-integral approach (at least for the travelled length, in the form discussed in the references above) is the Markovian nature of exponential Pearson walks, and this property is unfortunately not preserved when considering general Pearson walks with arbitrarily distributed jumps. While it is in principle possible to extend the path-integral approach to non-Markovian continuous-time processes (see, e.g.,~\\cite{Turgeman} for the case of anomalous diffusion with power-law memory kernels), we will not pursue this idea here. In this paper, we will consider instead a simpler strategy, and resort to standard transport theory. At the same time, we initially restrict our attention to diffusive walks (in other words, we neglect birth-death events).\n\nConsider the random walk of a particle from a point-source ${\\cal S}= \\delta({\\mathbf r}-{\\mathbf r_0}) \\delta(\\boldsymbol{\\omega} -\\boldsymbol{\\omega}_0 )$ located at ${\\mathbf r}_0$ on the surface of $B$. The particle is emitted in the inward direction ${\\boldsymbol{\\omega}}_0$, and moves at constant speed. The stochastic trajectory is composed of a series of flights of random length, interrupted by collisions (whereupon the particle is scattered, i.e., randomly changes its direction $\\boldsymbol{\\omega}$), and terminates when the particle crosses the surface of $B$ for the first time. This formally corresponds to imposing Dirichlet (or `vacuum') boundary conditions on the surface of $B$: a particle crossing the boundary is considered to be lost and can not re-enter the domain. In order to fully characterize collisions, we introduce the density $C({\\boldsymbol{\\omega}}|{\\boldsymbol{\\omega}'},{\\mathbf r})$, namely, the conditional probability density of changing direction from ${\\boldsymbol{\\omega}'}$ to ${\\boldsymbol{\\omega}}$, given a scattering event at ${\\mathbf r}$. The displacement law for the first jump (which characterizes the particle entrance in $B$, from the starting point ${\\mathbf r_0}$ and in direction ${\\boldsymbol{\\omega}}_0$) is denoted $H({\\mathbf r}|{\\mathbf r_0},{\\boldsymbol{\\omega}_0})$. All other jump lengths following a collision at a point ${\\mathbf r'}$ in $B$ (if any) obey the density $T({\\mathbf r}|{\\mathbf r'},{\\boldsymbol{\\omega}'})$, where ${\\boldsymbol{\\omega}'}$ is the current direction of the walker. We enumerate successive collisions by a discrete index $g$.\n\nIt follows that the quantity\n\\begin{equation}\nK({\\mathbf r},{\\boldsymbol{\\omega}}|{\\mathbf r'},{\\boldsymbol{\\omega}'}) = T({\\mathbf r}|{\\mathbf r'},{\\boldsymbol{\\omega}})C({\\boldsymbol{\\omega}}|{\\boldsymbol{\\omega}'},{\\mathbf r'})\n\\end{equation}\nrepresents the density of particles entering a collision at the $(g+1)$-th generation with coordinates $\\left\\lbrace {\\mathbf r},{\\boldsymbol{\\omega}} \\right\\rbrace$, having entered a collision at the $g$-th generation with coordinates $\\left\\lbrace {\\mathbf r'},{\\boldsymbol{\\omega}'} \\right\\rbrace$. We furthermore introduce the incident particle density $\\Psi_g({\\mathbf r},{\\boldsymbol{\\omega}}|{\\mathbf r_0},{\\boldsymbol{\\omega}_0})$, which is defined such that $N_{A} = \\int_A d{\\mathbf r} \\int_{\\Omega_d} d{\\boldsymbol{\\omega}} \\Psi_g({\\mathbf r},{\\boldsymbol{\\omega}}|{\\mathbf r_0},{\\boldsymbol{\\omega}_0})$ is the average number of collisions in the region $A$ at the $g$-th generation. Then, the stationary incident particle density will be given by\n\\begin{equation}\n\\Psi({\\mathbf r},{\\boldsymbol{\\omega}}|{\\mathbf r_0},{\\boldsymbol{\\omega}_0})=\\lim_{N \\to \\infty} \\sum_{g=1}^N \\Psi_g({\\mathbf r},{\\boldsymbol{\\omega}}|{\\mathbf r_0},{\\boldsymbol{\\omega}_0}),\n\\end{equation}\nprovided that such limit exists~\\cite{Spanier, Lux}. As customary, the source contribution is not taken into account in the definition of $\\Psi({\\mathbf r},{\\boldsymbol{\\omega}}|{\\mathbf r_0},{\\boldsymbol{\\omega}_0})$~\\cite{Spanier}. It can be shown that the stationary incident particle density satisfies the linear integral transport equation~\\cite{Lux, Zoia5}\n\\begin{equation}\n\\Psi({\\mathbf r},{\\boldsymbol{\\omega}}|{\\mathbf r_0},{\\boldsymbol{\\omega}_0})= \\int d{\\mathbf r}' \\int d\\boldsymbol{\\omega}' K({\\mathbf r},{\\boldsymbol{\\omega}}|{\\mathbf r'},{\\boldsymbol{\\omega}'})  \\Psi({\\mathbf r}', \\boldsymbol{\\omega}'|{\\mathbf r_0},{\\boldsymbol{\\omega}_0})+ H({\\mathbf r}|{\\mathbf r}_0,{\\boldsymbol{\\omega}_0}) \\delta(\\boldsymbol{\\omega} -\\boldsymbol{\\omega}_0 ) ,\n\\label{integral_transport_equations_one}\n\\end{equation}\nwith Dirichlet boundary conditions $\\Psi({\\mathbf r},\\boldsymbol{\\omega}|{\\mathbf r_0},{\\boldsymbol{\\omega}_0}) = 0$ for $\\boldsymbol{\\omega}_0$ directed outwards. Generally speaking, closed-form solutions to Eq.~\\eqref{integral_transport_equations_one} can hardly be found for arbitrary geometries~\\cite{Wasow}. However, our aim is now to average Eq.~\\eqref{integral_transport_equations_one} over an appropriate source condition at the boundary. Having in mind the case of straight paths and of exponential Pearson walks, we choose here an isotropic and uniform particle distribution on the surface of $B$ (a $\\mu-$ randomness in the language of stochastic geometry, which is (up to a constant) the unique measure invariant under the group of motion~\\cite{Santalo}). This choice allows the case of straight paths to be recovered when the density $H$ is such that the first jumps are much larger than the domain size. In this case, the probability measure at the surface reads~\\cite{Benichou, Zoia3}\n\\begin{equation}\n\\eta_d \\frac{d\\Sigma}{\\Sigma} \\frac{d\\boldsymbol{\\omega}_0}{\\Omega_d} (\\boldsymbol{\\omega}_0 \\cdot \\bf n),\n\\end{equation}\nwhere $\\Omega_d=2\\pi^{d\/2}\/\\Gamma(d\/2)$ is the surface of the unit sphere in dimension $d$ and\n\\begin{equation}\n\\eta_d = \\sqrt{\\pi}(d-1)\\frac{\\Gamma \\left(\\frac{d-1}{2} \\right)}{\\Gamma \\left( \\frac{d}{2} \\right)}\n\\end{equation}\nis a dimension-dependent normalization constant, equal to twice the inverse of the average height of the $d$-dimensional unit shell, $\\boldsymbol{n}$ being the unit vector normal to the surface and pointing inwards (remark that in~\\cite{Benichou} the normal was taken to be pointing outwards). We can then properly define the average of $\\Psi({\\mathbf r},{\\boldsymbol{\\omega}}|{\\mathbf r_0},{\\boldsymbol{\\omega}_0})$ over the surface of $B$ by taking\n\\begin{equation}\n\\langle \\Psi \\rangle_\\Sigma({\\mathbf r},{\\boldsymbol{\\omega}}) = \\eta_d \\int \\frac{d\\Sigma }{\\Sigma} \\int \\frac{d\\boldsymbol{\\omega}_0}{\\Omega_d} (\\boldsymbol{\\omega}_0   \\cdot \\mathbf{n}) \\Psi({\\mathbf r},{\\boldsymbol{\\omega}}|{\\mathbf r_0},{\\boldsymbol{\\omega}_0}).\n\\label{eq.mean_surf}\n\\end{equation}\nIn the following, we will furthermore assume that scattering is isotropic, which implies\n\\begin{equation}\nC({\\boldsymbol{\\omega}}|{\\boldsymbol{\\omega}'},{\\mathbf r'}) =  \\frac{1}{\\Omega_d}.\n\\end{equation}\nUnder this hypothesis, taking the surface average of Eq.~\\eqref{integral_transport_equations_one} leads to\n\\begin{align}\n\\eta_d \\int \\frac{d\\Sigma }{\\Sigma} \\int \\frac{d\\boldsymbol{\\omega}_0}{\\Omega_d} (\\boldsymbol{\\omega}_0 \\cdot \\mathbf{n}) \\Psi({\\mathbf r},{\\boldsymbol{\\omega}}|{\\mathbf r_0},{\\boldsymbol{\\omega}_0}) & =\\int d{\\mathbf r}' \\, T({\\mathbf r}|{\\mathbf r'},{\\boldsymbol{\\omega}}) \\int \\frac{d\\boldsymbol{\\omega}'}{\\Omega_d}\\left[ \\eta_d \\int \\frac{d\\Sigma }{\\Sigma} \\int \\frac{d\\boldsymbol{\\omega}_0}{\\Omega_d} (\\boldsymbol{\\omega}_0   \\cdot \\mathbf{n}) \\Psi({\\mathbf r'},{\\boldsymbol{\\omega}'}|{\\mathbf r_0},{\\boldsymbol{\\omega}_0})   \\right] \\notag \\\\\n   & + \\eta_d \\int \\frac{d\\Sigma }{\\Sigma} \\int \\frac{d\\boldsymbol{\\omega}_0}{\\Omega_d} (\\boldsymbol{\\omega}_0 \\cdot \\mathbf{n})\n         H({\\mathbf r}|{\\mathbf r}_0,{\\boldsymbol{\\omega}_0}) \\delta(\\boldsymbol{\\omega} -\\boldsymbol{\\omega}_0 ).\n\\end{align}\nThen, by making use of the definition of the surface average given in Eq.~\\eqref{eq.mean_surf}, we can rewrite\n\\begin{equation}\n\\langle \\Psi \\rangle_\\Sigma({\\mathbf r},{\\boldsymbol{\\omega}}) = \\int d{\\mathbf r}' \\int \\frac{d\\boldsymbol{\\omega}'}{\\Omega_d} \\, T({\\mathbf r}|{\\mathbf r'},{\\boldsymbol{\\omega}}) \\langle \\Psi \\rangle_\\Sigma({\\mathbf r'},{\\boldsymbol{\\omega}'}) + \\frac{\\eta_d} {\\Omega_d \\Sigma}  \\int d\\Sigma (\\boldsymbol{\\omega} \\cdot \\mathbf{n}) H({\\mathbf r}|{\\mathbf r}_0,{\\boldsymbol{\\omega}}).\n\\label{eq.psi.average}\n\\end{equation}\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=3.in,height=2.2in]{fig1.eps}\n\\setlength{\\abovecaptionskip}{15pt} \n\\caption{Two-dimensional projection in the frame $(0,{\\boldsymbol{\\omega}},\\mathbf e_2)$.}\n\\label{fig1}\n\\end{figure}\n\nWe now ask whether there exists a first-jump distribution $H$ such that the resulting incident collision density $\\langle \\Psi \\rangle_\\Sigma({\\mathbf r},{\\boldsymbol{\\omega}})$ in Eq.~\\eqref{eq.psi.average} is uniform in $B$, i.e., $\\langle \\Psi \\rangle_\\Sigma({\\mathbf r},{\\boldsymbol{\\omega}}) = \\langle \\Psi \\rangle_\\Sigma$, which intuitively corresponds to an equilibrium condition for the particles (see, e.g., the discussion in~\\cite{BlancoPRL}). If this is the case, we obtain\n\\begin{equation}\n\\langle \\Psi \\rangle_\\Sigma - \\langle \\Psi \\rangle_\\Sigma\\int d{\\mathbf r}' \\, T({\\mathbf r}|{\\mathbf r}',{\\boldsymbol{\\omega}}) = \\frac{\\eta_d} {\\Omega_d \\Sigma} \\int d\\Sigma (\\boldsymbol{\\omega} \\cdot \\mathbf{n}) H({\\mathbf r}|{\\mathbf r}_0,{\\boldsymbol{\\omega}}).\n\\label{eq.psi.average.aux}\n\\end{equation}\nBy applying the divergence theorem at the r.~h.~s.~of Eq.~\\eqref{eq.psi.average.aux} we get then\n\\begin{equation}\n\\frac{\\Omega_d \\Sigma}{\\eta_d} \\langle \\Psi \\rangle_\\Sigma \\left[1- \\int d{\\mathbf r}' \\, T({\\mathbf r}|{\\mathbf r}',{\\boldsymbol{\\omega}}) \\right] = - \\int d{\\mathbf r}_0 \\nabla \\cdot \\left[ \\boldsymbol{\\omega} H({\\mathbf r}|{\\mathbf r}_0,{\\boldsymbol{\\omega}}) \\right].\n\\label{eq.psi.average.aux2}\n\\end{equation}\nObserve that the integrals in Eq.~\\eqref{eq.psi.average.aux} are over the volume of $B$, whereas the jump densities $ T({\\mathbf r}|{\\mathbf r'},{\\boldsymbol{\\omega}})$ and $H({\\mathbf r}|{\\mathbf r}_0,{\\boldsymbol{\\omega}})$ are non-zero only when $\\mathbf r'-\\mathbf r$ and $\\mathbf r - \\mathbf r_0$, respectively, are aligned with the direction $\\boldsymbol{\\omega}$ (this is illustrated in Fig.~\\ref{fig1}). We can therefore make use of this property to perform the integration on an appropriate basis.\n\nLet us define $(D)$ the line passing by $\\mathbf r$ in direction $\\boldsymbol{\\omega}$, and denote $\\mathbf 0$ the intersection of $(D)$ with the surface in the direction opposite to $\\boldsymbol{\\omega}$ (see Fig.~\\ref{fig1}). We can then work in the ortho-normal frame $(\\mathbf{0},\\boldsymbol{\\omega}, \\mathbf e_2, \\ldots, \\mathbf e_d)$, where $\\mathbf e_2, \\ldots, \\mathbf e_d$ are a set of $d-1$ vectors orthogonal to $\\boldsymbol{\\omega}$. In this orthonormal basis, $\\mathbf r =  r \\boldsymbol{\\omega}$, with $r>0$, and the densities $H$ and $T$ take the form\n\\begin{align}\n\\label{T_delta}\nH({\\mathbf r}|{\\mathbf r}_0,{\\boldsymbol{\\omega}}) &= h(r-{r_0}_1) \\delta({r_0}_2) \\ldots \\delta({r_0}_d)\\\\\nT({\\mathbf r}|{\\mathbf r}',{\\boldsymbol{\\omega}}) &= t(r-r_1') \\delta(r_2') \\ldots \\delta(r_d') ,\\notag\n\\end{align}\nwhere $h(x)$ is the probability density of the first jump length and $t(x)$ that of the following jump lengths. In the same basis, we have\n\\begin{equation}\n\\nabla \\cdot \\left[ \\boldsymbol{\\omega} H({\\mathbf r}|{\\mathbf r}_0,{\\boldsymbol{\\omega}}) \\right] = \\partial_{{r_0}_1} h(r-{r_0}_1) \\delta({r_0}_2) \\ldots \\delta({r_0}_d) = - h'(r-{r_0}_1)\\delta({r_0}_2) \\ldots \\delta({r_0}_d).\n\\label{H_delta}\n\\end{equation}\nReplacing Eqs.~\\eqref{T_delta} and~\\eqref{H_delta} into Eq.~\\eqref{eq.psi.average.aux2} and performing the integrations over delta distributions, we obtain\n\\begin{equation}\n\\frac{\\Omega_d \\Sigma}{\\eta_d} \\langle \\Psi \\rangle_\\Sigma \\left[1- \\int_0^c dr_1' \\, t(r-r_1') \\right] = \\int_0^c d{r_0}_1 h'(r-{r_0}_1),\n\\label{eq.p.and.t}\n\\end{equation}\nwhere $c$ is the length of the chord resulting from the intersection of $(D)$ with $B$. Since both functions $t(x)$ and $h(x)$ must vanish for $x<0$ (jump lengths being positive), we immediately get\n\\begin{equation}\n\\frac{\\Omega_d \\Sigma}{\\eta_d} \\langle \\Psi \\rangle_\\Sigma \\left[1- \\int_0^r dx \\, t(x) \\right] = h(r).\n\\end{equation}\nFurthermore, $h(r)$ is a probability density, and normalization yields\n\\begin{equation}\n\\langle \\Psi \\rangle_\\Sigma = \\frac{\\eta_d}{\\Omega_d \\Sigma \\lambda},\n\\label{eq.mean.psi}\n\\end{equation}\nwhere $\\lambda = \\int_0^{\\infty} dx \\, t(x) x$ is the average jump length of an unconstrained walker. Hence, we have the sought condition on $H$, namely,\n\\begin{equation}\nh(r) = \\frac{1}{\\lambda} \\int_r^{\\infty} dx \\, t(x).\n\\label{eq.h}\n\\end{equation}\nObserve that we should require $\\lambda$ to be finite in order for $h(r)$ to be properly normalized. For instance, if $t(x)$ is a L\\'evy-stable density, with $t(x)=x^{-1-\\alpha}$ for $x \\to \\infty$~\\cite{levy}, $\\lambda$ is not finite for $0 < \\alpha \\leq 1$ and Eq.~\\eqref{eq.h} does not apply.  \n\nEquation~\\eqref{eq.h} can be seen a condition relating the first-jump density $H$ to the jump density $T$ in $B$: if this condition is met, then the stationary collision density in $B$ is uniform, i.e., particles are at equilibrium. Moreover, since the mean number of collisions in $B$ is given by the integral over the phase space\n\\begin{equation}\n\\langle N \\rangle_\\Sigma  = \\int_V  d{\\mathbf r} \\int_{\\Omega_d} d\\boldsymbol{\\omega} \\langle \\Psi \\rangle_\\Sigma,\n\\label{phase_space}\n\\end{equation}\nreplacing Eq.~\\eqref{eq.mean.psi} into Eq.~\\eqref{phase_space} leads to\n\\begin{equation}\n\\langle N \\rangle_\\Sigma = \\frac{\\eta_d}{\\lambda} \\frac{V}{\\Sigma},\n\\label{eq.mean.N}\n\\end{equation}\nwhich is a generalization to arbitrary Pearson random walks of the Cauchy's formula derived in~\\cite{Mazzolo05} for the average number of collisions of exponential Pearson walks. \n\nThe relevance of the assumption of isotropic scattering can be understood by considering, e.g., random walks having a constant jump length $a$ (i.e., $t(x)=\\delta(x-a)$), very small as compared to the domain size. Such walks enter the domain with a jump length that is at most equal to $a$ (see Eq.~\\eqref{eq.h}). In this case, the scattering distribution can not be arbitrary: for instance, a systematically back-scattering angular distribution would force particles to immediately leave from the boundaries, so that the domain would not be entirely probed. Then, a reasonable choice is to assume an isotropic distribution. For the particular case of exponentially distributed Pearson walks, the isotropy condition can be actually relaxed by demanding detailed balance to be satisfied at each collision~\\cite{Bardsley}.\n\nWe might further wonder whether there exists a process for which $h(x)=t(x)$, i.e., the law of the first jump coincides with that of the others. By imposing then $t(r)= (1\/\\lambda) \\int_r^{\\infty} dx \\, t(x)$, it immediately follows that this requirement is met by exponential Pearson random walks with density $t(x)=h(x)=(1\/\\lambda)\\exp(-x\/\\lambda)$. This is intuitively due to the Markovian (memoryless) nature of such walks (see, e.g.,~\\cite{Blanco, MazzoloEuro, Benichou}).\n\n\\section{Average travelled length}\n\\label{sec_length}\n\nWe will now address the issue of determining the average travelled length in $B$. Assume that the same hypotheses as above hold concerning the stochastic process and the distribution of the starting points. Let $P_n$ be the probability that a trajectory entering the domain has exactly $n$ collisions inside $B$ ($P_0$ is thus the probability that the trajectory is a chord, which precisely happens if the stochastic path has no collisions in $B$). Let us furthermore define $\\langle\\,L_n\\,\\rangle_\\Sigma$ the mean length travelled by paths constrained to have performed a number $n$ of collisions in $B$. Since trajectories having exactly $n$ collisions form a complete set of disjoint events, the mean length $\\langle\\,L\\,\\rangle_\\Sigma$ of trajectories inside the domain will be given by\n\\begin{equation}\n\\label{Lmean_start}\n \\langle\\,L\\,\\rangle_\\Sigma=\\sum_{n=0}^{\\infty} P_n \\langle\\,L_n\\,\\rangle_\\Sigma.\n\\end{equation}\nAs the walk is constrained to start from the surface and is terminated at the first exit from the surface, the quantity $\\langle\\,L\\,\\rangle_\\Sigma$ is directly related to the so-called first-return times to the boundary (the velocity of the particle being assumed constant)~\\cite{Blanco}. As shown in Fig.~\\ref{fig2}, stochastic trajectories consist of a series of segments which can enter from the surface, lie entirely within $B$, or leave from the surface. In order to fully characterize these distinct configurations, we introduce the following notation:\n\\begin{itemize}\n\\renewcommand{\\labelitemi}{\\textbullet}\n\\item $\\tilde{c}$, a random chord (the path undergoes no collisions inside the domain), with associated density $p_{\\tilde{c}}$.\n\\item $\\tilde{r}_{in}$, the first jump length inside $B$ (the starting point being uniformly distributed on the surface), with associated density $p_{in}$.\n\\item $\\tilde{r}_{out}$, the last jump length (the final point lies outside the domain), with associated density $p_{out}$.\n\\item $\\tilde{s}$, a segment entirely contained the domain, with associated density $p_{\\tilde{s}}$.\n\\end{itemize}\nThe tilde is used here to recall that all these quantities are random variables depending on the realizations of $h(x)$ (for segments starting from the surface) or $t(x)$ (for segments starting inside $B$).\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=5.8in,height=2.in]{fig2.eps}\n\\setlength{\\abovecaptionskip}{15pt}  \n\\caption{A schematic representation of trajectories having $0,1,2$ collisions inside $B$ and their associated lengths.}\n\\label{fig2}\n\\end{figure}\n\nWith these notations, we can identify\n\\begin{align} \n   & \\langle\\,L_0\\,\\rangle_\\Sigma = \\langle\\,\\tilde{c}\\,\\rangle                   \\notag\\\\\n   & \\langle\\,L_n\\,\\rangle_\\Sigma = \\langle\\,\\tilde{r}_{in}\\,\\rangle + (n-1)\\langle\\,\\tilde{s}\\,\\rangle + \\langle\\,\\tilde{r}_{out}\\,\\rangle  \\mathrm{~~for~~} n \\geq 1  \\notag.\n\\end{align}\nReplacing these quantities in Eq.~\\eqref{Lmean_start} leads then to\n\\begin{equation}\n\t\\langle\\,L\\,\\rangle_\\Sigma = \\langle\\,\\tilde{c}\\,\\rangle P_0 + (\\langle\\,\\tilde{r}_{in}\\,\\rangle+\\langle\\,\\tilde{r}_{out}\\,\\rangle)\\sum_{n=1}^{\\infty} P_n + \\langle\\,\\tilde{s}\\,\\rangle \\sum_{n=1}^{\\infty} (n-1)P_n.\n\\end{equation}\nFrom normalization, $\\sum_{n=0}^{\\infty} P_n=1$, and observe that by definition $\\langle\\,N\\,\\rangle_\\Sigma=\\sum_{n=1}^{\\infty} n P_n$. It follows that\n\\begin{equation}\n\\label{L_start}\n \\langle\\,L\\,\\rangle_\\Sigma =  \\langle\\,\\tilde{c}\\,\\rangle P_0 + (\\langle\\,\\tilde{r}_{in}\\,\\rangle+\\langle\\,\\tilde{r}_{out}\\,\\rangle )(1- P_0) + \\langle\\,\\tilde{s}\\,\\rangle \\left[\\langle\\,N\\,\\rangle_\\Sigma -(1-P_0)\\right]  .\n\\end{equation}\nWe have now to explicitly compute each term appearing in this equation. The term $\\langle\\,N\\,\\rangle_\\Sigma=\\langle\\,c\\,\\rangle\/\\lambda$, where $\\langle\\,c\\,\\rangle=\\eta_d V\/S$, is known from the results derived in the previous Section.\n\n\\subsection{The term $P_0$}\n\\label{sec_P_0}\n\nWe begin by considering the term $P_0$, which is the probability that the final point of the first jump actually falls outside $B$, i.e., that the length $r$ of the first jump is larger than the supported chord. (The chord length distribution is defined as  $\\mathrm {F(c) = Prob\\{c(M) \\leq c:M \\cap B \\not= \\varnothing \\} }$, measured with the uniform density $\\mathrm{M}$ of random lines in the sense of the theory of geometric probability (the $\\mu-$ randomness)~\\cite{Santalo,Solomon}, and $f(c)= dF(c)\/dc$ is  the corresponding density function. This assumptions is sometimes denoted IUR-chords (Isotropic Uniform Random chords), since this randomness stems from the body being exposed to a uniform, isotropic field of straight infinite\nlines~\\cite{Kellerer}). From this definition we have\n\\begin{equation}\n \\displaystyle P_0 = \\mathrm{Prob[} c \\le r \\mathrm{]} = \\int_0^{\\infty} dr \\, h(r) \\int_0^{r} dc \\, f(c) =  \\int_0^{\\infty} dc \\, f(c) \\int_{c}^{\\infty} dr \\, h(r) .\n\\end{equation}\nFrom Eq.~\\eqref{eq.h} we can rewrite\n\\begin{align}\n\\int_{c}^{\\infty}  dr \\, h(r) & = \\frac{1}{\\lambda} \\int_{c}^{\\infty} dr \\int_r^{\\infty} dx \\, t(x) =  \\frac{1}{\\lambda} \\int_c^{\\infty} dx \\, t(x) \\int_{c}^{x} dr =  \\frac{1}{\\lambda} \\int_c^{\\infty} dx \\, t(x) (x-c) \\notag\\\\\n\t\t\t\t\t\t\t& = 1 -\\frac{c}{\\lambda} + \\frac{1}{\\lambda} \\int_0^{c} dx \\, t(x) (c-x).\n\\label{eq_g(r)}\n\\end{align}\nThen, replacing this expression in the definition of $P_0$ yields (after having permuted the order of integration)\n\\begin{equation}\n  \\label{P_0_final}\n\tP_0 = \\frac{1}{\\lambda} \\int_0^{\\infty} dx \\, t(x) \\int_0^{x} dc \\, f(c)(x-c).\n\\end{equation}\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=1.8in,height=1.4in]{fig3.eps}\n\\setlength{\\abovecaptionskip}{15pt} \n\\caption{First jump lengths inside $B$ are smaller than supported chords.}\n\\label{fig3}\n\\end{figure}\n\n\\subsection{Proof that $p_{in} = p_{out}$.}\n\nThe random path begins with a jump from the surface whose length (independent of whether the jump falls or not inside $B$) obeys the density $h(x)$. Thus, the probability density $p_{in}(x)$ of the first jump inside $B$ is proportional to $h(x) \\times \\mathrm{Prob[jump ~of ~length~} x \\in B]$. This means that the chord supported by the first jump is larger than the jump length (see Fig.~\\ref{fig3}) and is given by $\\int_x^{\\infty} dc \\, f(c)$. As a consequence, the density of the first jump inside $B$ will be given by\n\\begin{equation}\n p_{in}(x) = \\frac{\\displaystyle h(x) \\int_x^{\\infty} dc \\, f(c)} {\\displaystyle \\int_0^{\\infty} dx \\, h(x) \\int_x^{\\infty} dc \\, f(c)}.\n\\end{equation}\nWe introduce now the ray distribution function: a ray of length $r$ is defined by the distance of a point inside $B$ to the frontier $\\partial B$ of $B$. Let $G(r) = \\mathrm{Pr\\{|P_1 P_2| \\leq r: P_1\\in B, P_2 \\in \\partial B \\} }$ be the partition function of the rays. Then, $g(r) = dG(r)\/dr$ is the corresponding density function~\\cite{Mazzolo04}. The ray distribution is related to the chord length distribution by~\\cite{Dixmier}\n\\begin{equation}\n \\displaystyle g(r) = \\frac{1}{\\langle\\,c\\,\\rangle} \\int_r^{\\infty} dc \\, f(c).\n\\label{def_g(r)}\n\\end{equation}\nThe density of the first jump inside $B$ can therefore be written in the more compact form\n\\begin{equation}\n p_{in}(x) = \\frac{\\displaystyle h(x) g(x)} {\\displaystyle \\int_0^{\\infty} dx \\, h(x) g(x)}.\n\\end{equation}\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=2.in,height=1.7in]{fig4.eps}\n\\setlength{\\abovecaptionskip}{15pt} \n\\caption{First jumps outside $B$ generate last jumps inside $B$ (formally, a ray).}\n\\label{fig4}\n\\end{figure}\n\nThe probability density $p_{out}(r)$ of the last jump (this definition excludes the case of chords) can be obtained by following a similar strategy. Actually, $p_{out}(r)$ is proportional to $g(r) \\times \\mathrm{Prob[jump ~of ~length~} x \\not\\in B]$, which means that the jump length is larger than the supported ray and is thus given by $\\int_r^{\\infty} dx \\, t(x)$. The function $g(r)$ stems from the collision density being uniform inside the domain, as shown above. By recollecting all these arguments, we are led to \n\\begin{equation}\n p_{out}(r) = \\frac{\\displaystyle g(r) \\int_r^{\\infty} dx \\, t(x)} {\\displaystyle \\int_0^{\\infty} dr g(r) \\int_r^{\\infty} dx \\, t(x)}.\n\\end{equation}\nDividing both numerator and denominator by $\\lambda$, we recognize the density $h(r) = 1\/\\lambda \\int_r^{\\infty} dx \\, t(x)$ in Eq.~\\eqref{eq.h}, and we thus obtain\n\\begin{equation}\n p_{out}(r) = \\frac{\\displaystyle h(r) g(r)} {\\displaystyle \\int_0^{\\infty} dr \\, h(r) g(r)}.\n\\end{equation}\nIt is then apparent that $p_{in}$ and $p_{out}$ are identically distributed. This strong property stems from $\\langle \\Psi \\rangle_\\Sigma$ being uniform, which is a consequence of the particular choice of $h(r)$. We therefore drop the subscript \\textit{in} and \\textit{out}, and name both variables $\\tilde{r}$. The mean value $\\langle\\,\\tilde{r}\\,\\rangle$ is given by\n\\begin{equation}\n\\displaystyle \\langle\\,\\tilde{r}\\,\\rangle = \\frac{\\displaystyle  \\int_0^{\\infty} dr \\, h(r) g(r)r} {\\displaystyle\\int_0^{\\infty} dr \\, h(r) g(r)}.\n\\label{mean_r_start}\n\\end{equation}\nReplacing Eq.~\\eqref{def_g(r)} in the denominator of Eq.~\\eqref{mean_r_start} gives \n\\begin{align}\n \\displaystyle  \\int_0^{\\infty} dr \\, h(r) \\, g(r)  & = \\displaystyle \\int_0^{\\infty} dr \\, h(r) \\frac{1}{\\langle\\,c\\,\\rangle} \\left[1-\\int_0^r dc \\, f(c) \\right]\\notag \\\\\n\t\t& = \\frac{1}{\\langle\\,c\\,\\rangle} \\left[1-\\int_0^{\\infty} dr \\, h(r) \\int_0^r dc \\, f(c) \\right]\\notag \\\\\n\t\t& = \\frac{1-P_0}{\\langle\\,c\\,\\rangle} ,\n\\end{align}\nand thus we get\n\\begin{equation}\n\\label{mean_r_final}\n \\displaystyle \\langle\\,\\tilde{r}\\,\\rangle = \\displaystyle \\frac{\\langle\\,c\\,\\rangle}{1-P_0}  \\int_0^{\\infty} dr \\, h(r) g(r)r.\n\\end{equation}\n\n\\subsection{The term $\\langle\\,\\tilde{c}\\,\\rangle$}\n\nThe conditional probability density of sampling a chord at the first jump is proportional to the probability density of chords $f(c)$ (unconstrained chords of length $c$), times the probability that the length of the first jump is larger than the chord, which is $\\int_c^{\\infty} dr \\, h(r)$. The quantity $p_{\\tilde{c}}(c)$ is thus given by\n\\begin{equation}\n \\displaystyle p_{\\tilde{c}}(c) = \\frac{\\displaystyle f(c) \\int_c^{\\infty} dr \\, h(r)} {\\displaystyle\\int_0^{\\infty} dc \\, f(c) \\int_c^{\\infty} dr \\, h(r)},\n\\end{equation}\nfrom which follows\n\\begin{equation}\n \\displaystyle \\langle\\,\\tilde{c}\\,\\rangle = \\frac{\\displaystyle \\int_0^{\\infty} dc \\, f(c)  \\, c \\int_c^{\\infty} dr \\, h(r)} {\\displaystyle\\int_0^{\\infty} dc \\, f(c) \\int_c^{\\infty} dr \\, h(r)}.\n\\label{c_mean_start}\n\\end{equation}\nBy observing that the denominator in Eq.~\\eqref{c_mean_start} is actually $P_0$, $\\langle\\,\\tilde{c}\\,\\rangle$ is therefore expressed as\n\\begin{equation}\n \\displaystyle \\langle\\,\\tilde{c}\\,\\rangle = \\frac{1}{P_0} \\displaystyle \\int_0^{\\infty} dc \\, f(c) c \\int_c^{\\infty} dr \\, h(r) = \\frac{1}{P_0} \\displaystyle \\int_0^{\\infty} dr \\, h(r)  \\int_0^{r} dc \\, f(c) c .\n \\label{mean_c_final}\n\\end{equation}\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=2in,height=1.4in]{fig5.eps}\n\\setlength{\\abovecaptionskip}{15pt} \n\\caption{Jumps entirely contained in $B$ must be smaller than the supported ray.}\n\\label{fig5}\n\\end{figure}\n\n\\subsection{The term $\\langle\\,\\tilde{s}\\,\\rangle$}\n\nThe conditional probability density of having a jump of length $r$ entirely inside $B$ is proportional to the density $t(x)$, times the probability that this jump has a length smaller than the supported ray (see Fig.~\\ref{fig5}). This last probability being equal to $\\int_{x}^{\\infty} dr \\,g(r) $, we thus get\n\\begin{equation}\n p_{\\tilde{s}}(x)= \\frac{\\displaystyle  t(x)  \\int_x^{\\infty} dr \\, g(r)} {\\displaystyle \\int_0^{\\infty} dx \\, t(x)  \\int_x^{\\infty} dr \\, g(r)}.\n\\end{equation}\nHence,\n\\begin{equation}\n \\displaystyle \\langle\\,\\tilde{s}\\,\\rangle = \\frac{\\displaystyle \\int_0^{\\infty} dx \\, t(x) x \\int_x^{\\infty} dr \\, g(r)} {\\displaystyle \\int_0^{\\infty} dx \\, t(x)  \\int_x^{\\infty} dr \\, g(r)} .\n \\label{mean_s_start}\n\\end{equation}\nThe integral over $g(r)$ can be easily expressed in terms of the chords density $f(c)$, namely,\n\\begin{equation}\n\\label{int_g_r}\n \\displaystyle \\int_x^{\\infty} dr \\, g(r) = \\int_x^{\\infty} dr \\, \\frac{1}{\\langle\\,c\\,\\rangle} \\int_r^{\\infty} dc \\, f(c) = \\frac{1}{\\langle\\,c\\,\\rangle} \\left[ \\langle\\,c\\,\\rangle - x + \\int_0^x dc \\, f(c)(x-c) \\right].\n\\end{equation}\nReplacing this identity in the denominator of Eq.~\\eqref{mean_s_start} and thanks to Eq.~\\eqref{P_0_final}, we get\n\\begin{equation}\n \\displaystyle \\int_0^{\\infty} dx \\, t(x)  \\int_x^{\\infty} dr \\, g(r) = 1-\\frac{\\lambda}{\\langle\\,c\\,\\rangle}(1-P_0).\n \\label{denom_mean_s}\n\\end{equation}\nThe numerator of Eq.~\\eqref{mean_s_start} can be rearranged as follows:\n\\begin{align}\n\t\\displaystyle \\int_0^{\\infty} dx \\, t(x) \\int_0^{x} du \\int_x^{\\infty} dr \\, g(r) & = \\lambda - \\int_0^{\\infty} dx \\, t(x) \\int_0^{x} du \\int_0^{x} dr \\, g(r) \\notag \\\\\n  & = \\lambda - \\int_0^{\\infty} du \\int_u^{\\infty} dx \\, t(x) \\left[\\int_0^{\\infty} dr \\, g(r) - \\int_x^{\\infty} dr \\, g(r) \\right]\\notag \\\\\n  & = \\lambda - \\lambda \\int_0^{\\infty} dr \\, h(r) \\int_0^{\\infty} du \\, g(u) - \\lambda \\int_0^{\\infty} dr \\int_r^{\\infty} du \\, g(u) \\int_r^{u} dx \\, t(x). \n\\end{align}\nThen, using Eq.~\\eqref{eq.h} and the identity $\\int_0^{r} du \\, g(u) = r g(r) + \\frac{1}{\\langle\\,c\\,\\rangle} \\int_0^{r} dc \\, f(c) c$ yields for the numerator\n\\begin{equation}\n\\label{num_mean_s}\n   \\displaystyle \\int_0^{\\infty} dx \\, t(x) \\, x \\int_x^{\\infty} dr \\, g(r) = \\lambda \\left[1- 2 \\int_0^{\\infty} dr \\, g(r) h(r) r - \\frac{1}{\\langle\\,c\\,\\rangle} \\int_0^{\\infty} dr \\, h(r) \\int_0^{r} dc \\, f(c) c\\right]. \n\\end{equation}\nReplacing the expression of both numerator and denominator in Eq.~\\eqref{mean_s_start} leads to\n\\begin{equation}\n\\label{mean_s_final}\n \\displaystyle \\langle\\,\\tilde{s}\\,\\rangle = \\frac{\\lambda}{1-\\frac{\\lambda}{\\langle\\,c\\,\\rangle}(1-P_0)} \\left[1- 2 \\int_0^{\\infty} dr \\, g(r) h(r) r - \\frac{1}{\\langle\\,c\\,\\rangle} \\int_0^{\\infty} dr \\, h(r) \\int_0^{r} dc \\, f(c) c\\right] .\n\\end{equation}\nReplacing finally Eqs.~\\eqref{P_0_final},~\\eqref{mean_r_final},~\\eqref{mean_c_final} and~\\eqref{mean_s_final} in Eq.~\\eqref{L_start} gives\n\\begin{align} \n \\langle\\,L\\,\\rangle_\\Sigma & = \\displaystyle \\langle\\,\\tilde{c}\\,\\rangle P_0 + 2 \\displaystyle \\langle\\,\\tilde{r}\\,\\rangle(1- P_0) + \\displaystyle \\langle\\,\\tilde{s}\\,\\rangle \\left[\\displaystyle \\langle\\,N\\,\\rangle -(1-P_0)\\right] \\notag\\\\\n       & = \\frac{1}{P_0} \\displaystyle \\int_0^{\\infty} dr \\, h(r)  \\int_0^{r} dc \\, f(c) c  \\, \\scriptstyle{\\times} \\displaystyle \\, P_0  \\notag \\\\\n       & ~~+ 2 \\scriptstyle{\\times} \\displaystyle \\frac{\\langle\\,c\\,\\rangle}{1-P_0}  \\int_0^{\\infty} dr \\, h(r) g(r) r \\scriptstyle{\\times} \\displaystyle (1- P_0) \\notag \\\\\n       & ~~+ \\frac{\\lambda}{1-\\frac{\\lambda}{\\langle\\,c\\,\\rangle}(1-P_0)} \\left[1- 2 \\int_0^{\\infty} dr \\, g(r) h(r) r - \\frac{1}{\\langle\\,c\\,\\rangle} \\int_0^{\\infty} dr \\, h(r) \\int_0^{r} dc \\, f(c) \\, c\\right] \\scriptstyle{\\times} \\displaystyle \\left[\\frac{\\langle\\,c\\,\\rangle}{\\lambda}-(1-P_0)\\right]\\notag \\\\\n       & = \\langle\\,c\\,\\rangle.\n\\end{align}\n\nHence follows the central result of this paper: the average length travelled in $B$ by isotropic Pearson random walks with arbitrarily distributed jump lengths satisfies the Cauchy's formula\n\\begin{equation}\n \\langle\\,L\\,\\rangle_\\Sigma = \\eta_d \\frac{V}{\\Sigma},\n\\label{equa_cauchy_L_final} \n\\end{equation}\nprovided that the first jump obeys $h(r)=(1\/\\lambda) \\int_r^{\\infty} dx \\, t(x)$ and that the walkers enter $B$ uniformly and isotropically. Correspondingly, as shown above, the average number of collisions in $B$ satisfies\n\\begin{equation}\n \\langle\\,N \\,\\rangle_\\Sigma = \\frac{\\eta_d}{\\lambda} \\frac{V}{\\Sigma} =\\frac{\\langle\\,L\\,\\rangle_\\Sigma}{\\lambda}.\n\\label{equa_cauchy_N_final} \n\\end{equation}\n\n\\section{Absorbing domains}\n\nWe conclude our analysis by considering the possibility that the random walks are uniformly absorbed in $B$, which is the case, e.g., for neutrons traversing capturing media. We will focus on purely absorbing domains (the particle trajectories terminate at the first collision), where the probabilistic tools developed in the previous Sections can be straightforwardly applied. In particular, we can show that for this configuration the average number $\\langle\\,N\\,\\rangle_\\Sigma$ of collisions for trajectories starting on the surface is related to the average number $\\langle\\,N\\,\\rangle_{V}$ of collisions for trajectories starting uniformly in the volume. Analogously, a similar relation exists for the travelled lengths.\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=4.in,height=1.8in]{fig6.eps}\n\\setlength{\\abovecaptionskip}{15pt} \n\\caption{Only jumps falling outside $B$ at the first step contribute to $P_0$ (left) and to $P_{out}$ (right).}\n\\label{fig6}\n\\end{figure}\n\nLet us begin from the average number of collisions. We have already introduced $P_0$, namely the probability for a jump starting on the surface to land outside $B$. In the same way, we define $P_{out}$ as the probability for a jump starting in $B$ to land outside $B$. The quantities $\\langle\\,N\\,\\rangle_\\Sigma$ and $\\langle\\,N\\,\\rangle_{V}$ are then respectively given by\n\\begin{align}\n\\label{mean_N_start}\n\t\\langle\\,N\\,\\rangle_{\\Sigma} & = 1 - P_0 \\notag \\\\\n    \\langle\\,N\\,\\rangle_{V}      & =   1 -P_{out}. \n\\end{align}\nThe explicit expression for $P_0$ is provided by Eq.~\\eqref{P_0_final}, whereas the quantity $P_{out}$ can be determined as follows. Suppose that a walker is isotropically and uniformly emitted in $B$, with jump length density $t(x)$: then, $P_{out}$ corresponds to the probability that the jump length $x$ is larger than the ray of length $r$ (whose density is $g(r)$) in the direction of the jump (see Fig.~\\eqref{fig6}). Then, it follows\n\\begin{equation}\n \\displaystyle P_{out} = \\int_0^{\\infty} dx \\, t(x) \\int_0^{x} dr \\, g(r). \n\\label{P_out_start}\n\\end{equation}\nFrom Eq.~\\eqref{int_g_r}, we immediately get\n\\begin{equation}\n \\displaystyle \\int_0^{x} dr \\, g(r) = \\frac{1}{\\langle\\,c\\,\\rangle} \\left[ x - \\int_0^x dc \\, f(c)(x-c) \\right].\n\\end{equation}\nReplacing this expression in Eq.~\\eqref{P_out_start} yields\n\\begin{align}\n\tP_{out} & = \\frac{1}{\\langle\\,c\\,\\rangle} \\bigg[ \\lambda - \\int_0^{\\infty} dx \\, t(x) \\int_0^x dc \\, f(c)(x-c) \\bigg]       \\notag \\\\\n            & =   \\frac{\\lambda(1 -P_0)}{\\langle\\,c\\,\\rangle}.\n\\label{P_out_final}\n\\end{align}\nBy resorting to Eq.~\\eqref{mean_N_start}, we finally obtain\n\\begin{equation}\n\t\\langle\\,N\\,\\rangle_{\\Sigma} = \\frac{\\langle\\,c\\,\\rangle}{\\lambda} \\Big[ 1 - \\langle\\,N\\,\\rangle_{V} \\Big].\n\\label{mean_N_final}\n\\end{equation}\n\nWe consider next the travelled lengths. We define $\\langle\\,L\\,\\rangle_{V}$ as the average travelled length when the walkers start uniformly and isotropically in $B$. For purely absorbing media, since trajectories end at the first collision, we have\n\\begin{align} \n\\label{mean_L_start}\n\t\\langle\\,L\\,\\rangle_{\\Sigma} & = \\langle\\,\\tilde{c}\\,\\rangle P_0 + \\langle\\,\\tilde{r}\\,\\rangle (1 - P_0) \\notag \\\\\n     \\langle\\,L\\,\\rangle_{V}      & = \\langle\\,\\tilde{r}\\,\\rangle P_{out} + \\langle\\,\\tilde{s}\\,\n\\rangle (1- P_{out}),\n\\end{align}\nwhere $\\langle\\,\\tilde{c}\\,\\rangle,\\langle\\,\\tilde{r}\\,\\rangle$ and $\\langle\\,\\tilde{s}\\,\\rangle$ are respectively defined by Eqs.~\\eqref{mean_c_final},~\\eqref{mean_r_final}, and~\\eqref{mean_s_final}. Replacing the explicit expressions for these quantities and those for the probabilities $P_0$ and $P_{out}$ in Eq.~\\eqref{mean_L_start} yields\n\\begin{equation}\n \\langle\\,L\\,\\rangle_{\\Sigma}  = \\int_0^{\\infty} dr \\, h(r) \\int_0^{r} dc \\, f(c) c + \\langle\\,c\\,\\rangle  \\int_0^{\\infty} dr \\, g(r) h(r) r,\n\\label{mean_L_sigma_final}\n\\end{equation}\nand\n\\begin{equation}\n \\langle\\,L\\,\\rangle_{V}  = \\lambda \\left[1- \\int_0^{\\infty} dr \\, g(r) h(r) r - \\frac{1}{\\langle\\,c\\,\\rangle} \\int_0^{\\infty} dr \\, h(r) \\int_0^{r} dc \\, f(c) \\right].\n \\label{mean_L_V_final}\n\\end{equation}\nThese equations can be finally combined, and we thus obtain\n\\begin{equation}\n \\langle\\,L\\,\\rangle_{\\Sigma} = \\langle\\,c\\,\\rangle \\Big[ 1 - \\frac{1}{\\lambda} \\langle\\,L\\,\\rangle_{V} \\Big].\n\\label{mean_L_V_S}\n\\end{equation}\nObserve that these formulas, which have been here established for general Pearson walks in purely absorbing media, are surprisingly identical to those previously derived for exponential Pearson walks in the presence of absorption~\\cite{Zoia3}. In sharp contrast with the case of diffusive random walks, Eq.~\\eqref{mean_L_V_S} depends on the details of the underlying process and of the geometry (because of the $\\langle\\,L\\,\\rangle_{V}\/\\lambda$ term) and has therefore lost its universal character.\n\n\\section{Conclusions}\n\nIn this paper we have examined under which conditions Cauchy's formulas hold true for diffusive Pearson walks with arbitrarily distributed jump lengths. If the walkers enter the domain $B$ uniformly and isotropically, and the first jump length obeys $h(r) = 1\/\\lambda \\int_r^{\\infty} dx \\, t(x)$, where $t(x)$ is the probability density of the jump lengths in $B$ and $\\lambda = \\int_0^{\\infty} dx \\, x t(x)$ is the average jump size, we have shown that the mean length $\\langle L \\rangle_{\\Sigma} $ travelled by the particles from the surface to the first exit from $B$ (which is proportional to the mean first-return time of the walk) satisfies the Cauchy's formula and is thus universal, in that it does not depend on the features of the random walk. In particular, $\\langle L \\rangle_{\\Sigma} $ is expressed in terms of the ratio between the volume and the surface of $B$ alone, up to a dimension-dependent constant (in three dimensions, we have for instance $\\langle L \\rangle_{\\Sigma} =4 V\/\\Sigma$). A similar result has been shown to hold also for the average number of collisions performed by the walkers in $B$, which reads $\\langle N \\rangle_{\\Sigma}  = \\langle L \\rangle_{\\Sigma} \/\\lambda$. \n\nThese findings apply to a broad class of diffusive random walks, and as such extend the range of applicability of Cauchy's formulas, which have been so far established for Pearson walks with exponentially distributed jump lengths. Moreover, the analysis of the case of purely absorbing media seems to suggest that the generalized Cauchy's formulas of~\\cite{Zoia3}, derived for exponential Pearson walks, may actually hold true for arbitrary Pearson walks (under the same hypothesis concerning the first jump length density). Investigations are ongoing and will be the subject of further research.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nFor the last decade, motion deblurring has been an active research topic in computer vision.\nMotion blur is produced by relative motion between camera and scene during the exposure where blur kernel, {\\it i.e.} point spread function (PSF), is spatially non-uniform.\nIn blind non-uniform deblurring problem, pixel-wise blur kernels and corresponding sharp image are estimated simultaneously.\n\nEarly works on motion deblurring~\\cite{chan1998total,cho2009fast,fergus2006removing,shan2008high,xu2010two} focus on removing spatially uniform blur in the image.\nHowever, the assumption of uniform motion blur is often broken in real world due to nonhomogeneous scene depth and rolling motion of camera.\nRecently, a number of methods~\\cite{cho2007removing,gupta2010single,hu2014joint,hu2012fast,ji2012two,kim2014segmentation,sun2015learning,whyte2012non,zheng2013forward} have been proposed for non-uniform deblurring.\nHowever, they still can not completely handle non-uniform blur caused by scene depth variation.\nThe main challenge lies in the difficulty of estimating the scene depth with only single observation, which is highly ill-posed.\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\vspace*{0.15in}\n\t\t\\subfloat[]{\\includegraphics[width=0.25\\textwidth]{figure\/figure1_Blurred.pdf}}\\hspace*{-0.01in}\n\t\t\\subfloat[]{\\includegraphics[width=0.25\\textwidth]{figure\/figure1_Latent.pdf}}\\hspace*{-0.01in}\n\t\t\\subfloat[]{\\includegraphics[width=0.25\\textwidth]{figure\/figure1_Depth.pdf}}\\hspace*{-0.01in}\n\t\t\\subfloat[]{\\includegraphics[width=0.25\\textwidth]{figure\/figure1_Motion.pdf}}\t\n\t\\end{center}\n\t\\vspace*{-0.05in}\n\t\\caption{The proposed algorithm jointly estimates latent image, depth map, and camera motion from a single light field. (a) Center-view of blurred light field sub-aperture image. (b) Deblurred image of (a). (c) Estimated depth map. (d) Camera motion path and orientation (6-DOF).}\n\t\\label{fig:comp0}\n\n\\end{figure}\nA light field camera ameliorates the ill-posedness of single-shot deblurring problem of the conventional camera.\n4D light field is equivalent to multi-view images with narrow baseline, {\\it i.e.} sub-aperture images, taken with an identical exposure~\\cite{ng2005light}.\nConsequently, motion deblurring using light field can be leveraged by its multi-dimensional nature of captured information.\nFirst, strong depth cue is obtained by employing multi-view stereo matching between sub-aperture images.\nIn addition, different blurs in the sub-aperture images can help the optimization converge more fast and precise.\n\nIn this paper, we propose an efficient algorithm to jointly estimate latent image, sharp depth map, and 6-DOF camera motion from a blurred single 4D light field as shown in Figure~\\ref{fig:comp0}.\nIn the proposed light field blur model, latent sub-aperture images are formulated by 3D warping of the center-view sharp image using the depth map and the 6-DOF camera motion.\nThen, motion blur is modeled as the integral of latent sub-aperture images during the shutter open.\nNote that the proposed center-view parameterization reduces light field deblurring problem in lower dimension comparable to a single image deblurring.\nThe joint optimization is performed in an alternating manner, in which the deblurred image, depth map, and camera motion are refined during iteration.\nThe overview of the proposed algorithm is shown in Figure~\\ref{fig:comp1}.\nIn overall, the contribution of this paper is summarized as follows.\n\\begin{itemize}\n  \\item[$\\bullet$] We propose a joint method which simultaneously solves deblurring, depth estimation, and camera motion estimation problems from a single light field.\n  \\item[$\\bullet$] Unlike the previous state-of-the-art algorithm, the proposed method handles blind light field motion deblurring under 6-DOF camera motion.\n  \\item[$\\bullet$] Practical and extensible blur formulation that can be extended to any multi-view camera system.\n\\end{itemize}\n\\section{Related Works}\n\n\\textbf{Conventional Single Image Deblurring.}~One way to effectively remove the spatially-variant motion in a conventional single image is to first find the motion density function (MDF) and then generate the pixel-wise kernel from this function~\\cite{gupta2010single,hu2014joint,hu2012fast}.\nGupta~et al.~\\cite{gupta2010single} modeled the camera motion in discrete 3D motion space comprising $x$, $y$ translation and in-plane rotation.\nThey performed deblurring by iteratively optimizing the MDF and the latent image that best describe the blurred image.\nSimilar model was used by Hu and Yang~\\cite{hu2012fast} in which MDF was modeled with 3D rotations.\nThese methods of using MDF well parameterize the spatially-variant blur kernel into low dimensions.\nHowever, modeling the motion blur using MDF only in depth varying images is difficult, because the motion blur is determined by both camera motion and scene depth.\nIn~\\cite{hu2014joint}, the image was segmented by the matting algorithm, and the MDF and representative depth values \u200b\u200bof each region were found through the expectation-maximization algorithm.\n\\begin{figure*}[t]\n\t\\vspace*{0.15in}\n\t\\centering\n\t\\scalebox{1.1}{\n\t\t\\includegraphics[width=0.90\\textwidth]{figure\/figure2_compress.pdf}\n\t}\n\t\\vspace*{-0.05in}\n\t\\caption{Overview of the proposed algorithm. The proposed algorithm jointly estimates the latent image, depth map, and camera motion from a single light field.\n\t\\label{fig:comp1}\n\\end{figure*}\n\nA few methods~\\cite{kim2014segmentation,sun2015learning} estimated linear blur kernels locally, and they showed acceptable results for the arbitrary scene depth.\nKim and Lee~\\cite{kim2014segmentation} jointly estimated the spatially varying motion flow and the latent image.\nSun~et al.~\\cite{sun2015learning} adopted a learning method based on convolutional neural network~(CNN) and assumed that the motion was locally linear.\nHowever, the locally linear blur assumption does not hold in large motion.\n\\vspace*{0.3cm}\n\n\\noindent\\textbf{Video and Multi-View Deblurring.}~Xu and Jia~\\cite{xu2012depth} decomposed the image region according to the depth map obtained from a stereo camera and recombined them after independent deblurring.\nRecently, several methods~\\cite{cho2012video,hyun2015generalized,park2017joint,sellent2016stereo,wulff2014modeling} have addressed the motion blur problem in video sequences.\nVideo deblurring shows good performance, because it exploits optical flow as a strong guide for motion estimation.\n\\vspace*{0.3cm}\n\n\\noindent\\textbf{Light Field Deblurring.}~Light field with two plane parameterization is equivalent to  multi-view images with narrow baseline.\nIt contains rich geometric information of rays in a single-shot image.\nThese multi-view images are called sub-aperture images and individual sub-aperture images show slightly different blur pattern due to the viewpoint variation.\nIn last a few years, several approaches~\\cite{chandramouli2014motion,dansereau2016motion,jin2015bilayer,snoswell2014lf,srinivasan2017light} have been proposed to perform motion deblurring on the light field.\nChandramouli~et~al.~\\cite{chandramouli2014motion} addressed the motion blur problem in the light field for the first time.\nThey assumed constant depth and uniform motion to alleviate the complexity of the imaging model.\nConstant depth means that the light field has little information about 3D scene structure, which depletes the advantages of light field.\nJin~et al.~\\cite{jin2015bilayer} quantized the depth map into two layers and removed the motion blur in each layer.\nTheir method assumed that the camera motion is in-plane translation and utilized depth value as a scale factor of translational motion.\nAlthough their model handles non-uniform blur kernel related to the depth map, a more general depth variation and camera motion should be considered for application to real-world scenes.\nDansereau~et al.~\\cite{dansereau2016motion} applied the Richardson-Lucy deblurring algorithm to the light field with non-blind 6-DOF motion blur.\nAlthough their method dealt with 6-DOF motion blur, it was assumed that the ground truth camera motion was known.\nUnlike~\\cite{dansereau2016motion}, in this paper, we address the problem of blind deblurring which is a more highly ill-posed problem.\nSrinivasan~et al.~\\cite{srinivasan2017light} solved the light field deblurring under 3D camera motion path and showed visually pleasing result.\nHowever, their methods do not consider 3D orientation change of the camera.\n\nIn contrast to the previous works of light field deblurring, the proposed method completely handles 6-DOF motion blur and unconstrained scene depth variation.\n\n\n\n\\section{Motion Blur Formulation in Light Field}\n\\label{sec3}\n\nA pixel in a 4D light field has four coordinates, {\\em i.e.} $(x,y)$ for spatial and $(u,v)$ for angular coordinates.\nA light field can be interpreted as a set of $u\\times v$ multi-view images with narrow baseline, which are often called sub-aperture images~\\cite{ng2006digital}.\nThroughout this paper, a sub-aperture image is represented as $I(\\mathbf{x},{\\mathbf{u}})$ where $\\mathbf{x}=(x,y)$ and $\\mathbf{u}=(u,v)$.\nFor each sub-aperture image, the blurred image $B(\\mathbf{x},{\\mathbf{u}})$ is the average of the sharp images ${I}_{t}(\\mathbf{x},{\\mathbf{u}})$ during the shutter open over $[t_0 , t_1]$ as follows:\n\\begin{align}\nB(\\mathbf{x},{\\mathbf{u}}) = \\int_{t_0}^{t_1} {I}_{t}(\\mathbf{x},{\\mathbf{u}}) dt.\n\\label{eq:blurEq}\n\\end{align}\n\nFollowing the blur model of~\\cite{park2017joint,sellent2016stereo}, we approximate all the blurred sub-aperture images by projecting a single latent image with 3D rigid motion.\nWe choose the center-view ($\\mathbf{c}$) of sub-aperture images and the middle of the shutter time ($t_r$) as the reference angular position and the time stamp of the latent image.\nWith above notations, the pixel correspondence from each sub-aperture image to the latent image $I_{t_r}(\\mathbf{x},\\mathbf{c})$ is expressed as follows:\n\\begin{align}\n{I}_{t}(\\mathbf{x},{\\mathbf{u}})={I}_{t_r}(w_{t}(\\mathbf{x},\\mathbf{u}),{\\mathbf{c}}),\n\\end{align}\nwhere\n\\begin{align}\nw_{t}(\\mathbf{x},\\mathbf{u})=\\Pi_{\\mathbf{c}}(\\mathrm{P}^{\\mathbf{c}}_{t_r}(\\mathrm{P}^{\\mathbf{u}}_{t})^{\\scriptscriptstyle -1}\\Pi^{\\scriptscriptstyle -1}_{\\mathbf{u}}(\\mathbf{x},D_{t}(\\mathbf{x},\\mathbf{u}))).\n\\label{eq:warping}\n\\end{align}\n$w_{t}(\\mathbf{x},\\mathbf{u})$ computes the warped pixel position from $\\mathbf{u}$ to $\\mathbf{c}$, and from $t$ to $t_r$.\n$\\Pi_{\\mathbf{c}}$, $\\Pi^{\\scriptscriptstyle -1}_{\\mathbf{u}}$ are the projection and back-projection function between the image coordinate and the 3D homogeneous coordinate using the camera intrinsic parameters.\nMatrices $\\mathrm{P}^{\\mathbf{c}}_{t_r}$ and $\\mathrm{P}^{\\mathbf{u}}_{t}\\in SE(3)$ denote the 6-DOF camera pose at the corresponding angular position and the time stamp. $D_{t}(\\mathbf{x},\\mathbf{u})$ is the depth map at the time stamp $t$.\n\nIn the proposed model, the blur operator $\\Psi(\\cdot)$ is defined by approximating the integral in (\\ref{eq:blurEq}) as a finite sum as follows:\n\\begin{align}\nB(\\mathbf{x},\\mathbf{u})\\approx (\\Psi\\circ I)(\\mathbf{x},\\mathbf{u}),\n\\end{align}\nwhere\n\\begin{align}\n(\\Psi\\circ I)(\\mathbf{x},\\mathbf{u}) = \\frac{1}{M}\\sum^{M-1}_{m=0}I_{t_r}(w_{t_m}(\\mathbf{x},\\mathbf{u}), \\mathbf{c}).\n\\label{eq:blurOp}\n\\end{align}\nIn (\\ref{eq:blurOp}), $t_m$ is $m_{th}$ uniformly sampled time stamp during the interval $[t_0,t_1]$.\n\nOur goal is to formulate $(\\Psi\\circ I)(\\mathbf{x},\\mathbf{u})$ with only center-view variables, {\\em i.e.} $I_{t_r}(\\mathbf{x},\\mathbf{c})$, $D_{t_r}(\\mathbf{x},\\mathbf{c})$, and $\\mathrm{P}^{\\mathbf{c}}_{t_0}$.\n$\\mathrm{P}^{\\mathbf{u}}_{t_m}$ and $D_{t_m}(\\mathbf{x},\\mathbf{u})$ are variables related to $\\mathbf{u}$ in the warping function (\\ref{eq:blurOp}).\nTherefore, we parameterize $\\mathrm{P}^{\\mathbf{u}}_{t_m}$ and $D_{t_m}(\\mathbf{x},\\mathbf{u})$ by employing center-view variables.\nBecause the relative camera pose  $\\mathrm{P}^{\\scriptscriptstyle \\mathbf{c\\rightarrow u}}$ is fixed over time, $\\mathrm{P}^{\\mathbf{u}}_{t_m}$ is expressed by $\\mathrm{P}^{\\mathbf{c}}_{t_0}$ and $\\mathrm{P}^{\\mathbf{c}}_{t_1}$ as follows:\n\\begin{align}\n\\mathrm{P}^{\\mathbf{u}}_{t_m}=\\mathrm{P}^{\\scriptscriptstyle \\mathbf{c\\rightarrow u}}\\mathrm{P}^{\\mathbf{c}}_{t_m},\n\\end{align}\n\\begin{align}\n\\mathrm{P}^{\\mathbf{c}}_{t_m}=\\exp(\\frac{m}{M}\\log(\\mathrm{P}^{\\mathbf{c}}_{t_1}{(\\mathrm{P}^{\\mathbf{c}}_{t_0})}^{\\scriptscriptstyle -1}))\\mathrm{P}^{\\mathbf{c}}_{t_0},\n\\label{eq:motionPath}\n\\end{align}\nwhere $\\exp$ and $\\log$ denote the exponential and logarithmic maps between Lie group $SE(3)$ and Lie algebra $\\mathfrak{se}(3)$ space~\\cite{blanco2010tutorial}.\nTo minimize the viewpoint shift of the latent image, we assume $\\mathrm{P}^{\\mathbf{c}}_{t_1}=(\\mathrm{P}^{\\mathbf{c}}_{t_0})^{\\scriptscriptstyle -1}$ which makes $\\mathrm{P}^{\\mathbf{c}}_{t_m}$ an identity matrix when $t_m=t_r$.\nNote that we use the camera path model used in~\\cite{park2017joint,sellent2016stereo}.\nHowever, the B\\'{e}zier camera path model used in~\\cite{srinivasan2017light} can be directly applied to (\\ref{eq:motionPath}) as well.\n$D_{t_m}(\\mathbf{x},\\mathbf{u})$ is also represented by $D_{t_r}(\\mathbf{x},\\mathbf{c})$ by forward warping and interpolation.\n\nIn order to estimate all blur variables in the proposed light field blur model, we need to recover the latent variables, {\\it i.e.} $I_{t_r}(\\mathbf{x},\\mathbf{c})$, $D_{t_r}(\\mathbf{x},\\mathbf{c})$, and $\\mathrm{P}^{\\mathbf{c}}_{t_0}$.\nWe model an energy function as follows:\n\\begin{align}\n\\begin{split}\nE & =  \\sum_{\\mathbf{u}}\\sum_{\\mathbf{x}}\\lambda_u\\|(\\Psi\\circ I)(\\mathbf{x},\\mathbf{u})-B(\\mathbf{x},{\\mathbf{u}})\\|_1\\\\\n& + \\lambda_L\\sum_{\\mathbf{x}}\\|\\nabla I_{t_r}(\\mathbf{x},\\mathbf{c})\\|_2+\\lambda_D\\sum_{\\mathbf{x}}\\|\\nabla D_{t_r}(\\mathbf{x},\\mathbf{c})\\|_2.\n\\label{eq:energy}\n\\end{split}\n\\end{align}\nThe data term imposes the brightness consistency between the input blurred light field and the restored light field.\nNotice that the L1-norm is employed in our approach as in~\\cite{kim2014segmentation}, where it effectively removes the ringing artifact around object boundary and provides more robust deblurring results on large depth change.\nThe last two terms are the total variation~(TV) regularizers~\\cite{beck2009fast} for the latent image and the depth map, respectively.\n\nIn our energy model, $D_{t_r}(\\mathbf{x},\\mathbf{c})$ and $\\mathrm{P}^{\\mathbf{c}}_{t_0}$ are implicitly included in the warping function (\\ref{eq:blurOp}).\nThe pixel-wise depth $D_{t_r}(\\mathbf{x},\\mathbf{c})$ determines the scale of the motion at each pixel.\nAt the boundary of an object where depth changes abruptly, there is a large difference of the blur kernel size between the near and farther objects.\nIf the optimization is performed without considering this, the blur will not be removed well at the boundary of the object.\n\nSimultaneously optimizing the three variables is complicated because the warping function (\\ref{eq:blurOp}) has severe nonlinearity.\nTherefore, our strategy is to optimize three latent variables in an alternating manner.\nWe minimize one variable while the others are fixed. The optimization (\\ref{eq:energy}) is carried out in turn for the three variables.\nThe L1 optimization is approximated using iterative reweighted least square (IRLS)~\\cite{scales1988robust}.\nThe optimization procedure converges in small number of iterations~$(<10)$.\n\nAn example of the iterative optimization is illustrated in Figure~\\ref{fig3} which shows the benefit of the iterative joint estimation of sharp depth map and latent image.\nThe initial depth map from the blurred light field is blurry as shown in Figure~\\ref{fig3}(c).\nHowever, both depth maps and latent images get sharper as the iteration continues as shown in Figure~\\ref{fig3}(d).\n\\begin{figure*}[t]\n\t\\vspace*{0.15in}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[width=0.265\\textwidth]{figure\/figure3_1.pdf}}\\hspace*{-0.05in}\n\t\\subfloat[]{\\includegraphics[width=0.25\\textwidth]{figure\/figure3_2.pdf}}\\hspace*{-0.05in}\n\t\\subfloat[]{\\includegraphics[width=0.265\\textwidth]{figure\/figure3_3.pdf}}\\hspace*{-0.05in}\n\t\\subfloat[]{\\includegraphics[width=0.25\\textwidth]{figure\/figure3_4.pdf}\n\n\t\\vspace*{-0.05in}\n\\caption{Example of the iterative joint estimation. The proposed method converges in small number of iteration.\n(a)$\\sim$(b) Input blurred image and deblurring results by iteration.\n(c)$\\sim$(d) Initial blurred depth map and depth estimation results by iteration.\n}\n\t\\label{fig:comp2}\n\t\\label{fig3}\t\n\\end{figure*}\n\n\\section{Joint Estimation of Latent Image, Camera Motion, and Depth Map}\n\\label{sec4}\n\n\\subsection{Update of the Latent Image}\n\\label{sec:4.1}\n\nThe proposed algorithm first updates the latent image $I_{t_r}(\\mathbf{x},\\mathbf{c})$.\nIn our data term, the blur operator (\\ref{eq:blurOp}) is simplified as the linear matrix multiplication, if $D_{t_r}(\\mathbf{x},\\mathbf{c})$ and $\\mathrm{P}^{\\mathbf{c}}_{t_0}$ remain fixed.\nUpdating the latent image is equivalent to minimizing (\\ref{eq:energy}) as follows:\n\\begin{align}\n\\min_{I^{\\mathbf{c}}_{t}}\\sum_{\\mathbf{u}}\\|K^{\\mathbf{u}}I^{\\mathbf{c}}_{t_r}-B^{\\mathbf{u}} \\|_1 + \\lambda_L\\|\\nabla I^{\\mathbf{c}}_{t_r}\\|_2.\n\\label{eq:imageUpdate}\n\\end{align}\n$I^{\\mathbf{c}}_{t_r}$, $B^{\\mathbf{u}}\\in\\mathbb{R}^n$ are vectorized images and ${K}^{\\mathbf{u}}\\in\\mathbb{R}^{n\\times n}$ is the blur operator in square matrix form, where $n$ is the number of pixels in the center-view sub-aperture image.\nTV regularization serves as a prior to the latent image with clear boundary while eliminating the ringing artifacts.\n\n\\subsection{Update of the Camera Pose and Depth Map}\n\\label{sec:4.2}\n\nSince (\\ref{eq:blurOp}) is a non-linear function of $D_{t_r}(\\mathbf{x},\\mathbf{c})$ and $\\mathrm{P}^{\\mathbf{c}}_{t_0}$, it is necessary to approximate it in a linear form for efficient computation.\nIn our approach, the blur operation (\\ref{eq:blurOp}) is approximated as a first-order expansion.\nLet $D_{0}(\\mathbf{x},\\mathbf{c})$ and $\\mathrm{P}^{\\mathbf{c}}_{0}$ denote the initial variables, then (\\ref{eq:blurOp}) is approximated as follow:\n\\begin{align}\n\\begin{split}\n&(\\Psi\\circ I)(\\mathbf{x},\\mathbf{u})\\\\\n&= B_0(\\mathbf{x},\\mathbf{u})+ \\textstyle \\frac{\\partial B_0}{\\partial \\mathbf{f}}(\\frac{\\partial\\mathbf{f}}{\\partial D_{t_r}(\\mathbf{x},\\mathbf{c})}\\Delta D_{t_r}(\\mathbf{x},\\mathbf{c})+\\frac{\\partial\\mathbf{f}}{\\partial \\varepsilon_{t_0}}\\varepsilon_{t_0}),\n\\end{split}\n\\end{align}\nwhere\n\\begin{align}\nB_0(\\mathbf{x},\\mathbf{u})=(\\Psi\\circ I)(\\mathbf{x},\\mathbf{u})\\vert_{D_{t_r}(\\mathbf{x},\\mathbf{c})=D_{0}(\\mathbf{x},\\mathbf{c}),\\mathrm{P}^{\\mathbf{c}}_{t_0}=\\mathrm{P}^{\\mathbf{c}}_{0}},\n\\end{align}\nNote that $\\mathbf{f}$ is motion flow generated by warping function, and $\\varepsilon_{t_0}$ denotes six-dimensional vector on $\\mathfrak{se}(3)$.\nThe partial derivatives related to $D_{t_r}(\\mathbf{x},\\mathbf{c})$ and $\\varepsilon_{t_0}$ are given in~\\cite{blanco2010tutorial}.\n\nOnce it is approximated using $\\Delta D_{t_r}(\\mathbf{x},\\mathbf{c})$ and $\\varepsilon_{t_0}$, (\\ref{eq:energy}) can be optimized using IRLS.\nThe resulting $\\Delta D_{t_r}(\\mathbf{x},\\mathbf{c})$ and $\\varepsilon_{t_0}$ are incremental values for the current $D_{t_r}(\\mathbf{x},\\mathbf{c})$ and $\\mathrm{P}^{\\mathbf{c}}_{t_0}$, respectively.\nThey are updated as follows:\n\\begin{align}\n\\begin{split}\n&D_{t_r}(\\mathbf{x},\\mathbf{c})=D_{t_r}(\\mathbf{x},\\mathbf{c})+\\Delta D_{t_r}(\\mathbf{x},\\mathbf{c}),\\\\\n&\\mathrm{P}^{\\mathbf{c}}_{t_0} = \\exp (\\varepsilon_{t_0})\\mathrm{P}^{\\mathbf{c}}_{t_0},\n\\end{split}\n\\end{align}\nwhere $\\mathrm{P}^{\\mathbf{c}}_{t_0}$ is updated through the exponential mapping of the motion vector $\\varepsilon_{t_0}$.\n\nFigure~\\ref{fig3} shows the initial latent variables and final outputs.\nAfter joint estimation, both the latent image and the depth map become clean and sharp.\n\nThe proposed blur formulation and joint estimation approach are not limited to the light field but can also be applied to images obtained from a stereo camera or general multi-view camera system. The only property of the light field we use is that sub-aperture images are equivalent to the images obtained from multi-view camera array.\nNote that the proposed method is not limited to a simple motion path model (moving smoothly in $\\mathfrak{se}(3)$ space).\nMore complex parametric curves, such as the B\\'{e}zier curve used in the prior work~\\cite{srinivasan2017light}, can be directly applied only if they are differentiable.\\\\\n\n\\begin{figure}[t]\n\t\\vspace*{0.15in}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[width=0.248\\columnwidth]{figure\/figure4_1.pdf}}\\hspace*{-0.01in}\n\t\\subfloat[]{\\includegraphics[width=0.248\\columnwidth]{figure\/figure4_2.pdf}}\\hspace*{-0.01in}\n\t\\subfloat[]{\\includegraphics[width=0.248\\columnwidth]{figure\/figure4_3.pdf}}\\hspace*{-0.01in}\n\t\\subfloat[]{\\includegraphics[width=0.248\\columnwidth]{figure\/figure4_4.pdf}}\\hspace*{-0.01in\n\t\\vspace*{-0.05in}\n\\caption{\nExample of camera motion initialization on a synthetic light field.\n(a) Blurred input light field.\n(b) Ground truth motion flow.\n(c) Sun~et al.~\\cite{sun2015learning} (EPE = 3.05),\n(d) Proposed initial motion (EPE = 0.95).\nIn (b) and (d), the linear blur kernels are approximated only using the end points of camera motion for the visualization.}\n\t\\label{fig4}\t\n\n\\end{figure}\n\\subsection{Initialization}\n\\label{sec:initialization}\n\nSince deblurring is a highly ill-posed problem and the optimization is done in a greedy and iterative fashion, it is important to start with good initial values.\nFirst, we initialize the depth map using the input sub-aperture images of the light field.\nIt is assumed that the camera is not moving and (\\ref{eq:energy}) is minimized to obtain the initial $D_{t_r}(\\mathbf{x},\\mathbf{c})$.\nMinimizing (\\ref{eq:energy}) becomes a simple multi-view stereo matching problem.\nFigure~\\ref{fig:comp2}(c) shows the initial depth map which exhibits  fattened object boundary.\n\nCamera motion $\\mathrm{P}^{\\mathbf{c}}_{t_0}$ is initialized from the local linear blur kernels and initial scene depth.\nWe first estimate the local linear blur kernel of $B(\\mathbf{x},{\\mathbf{c}})$ using~\\cite{sun2015learning}.\nThen, we fit the pixel coordinates moved by the linear kernel and the re-projected coordinates by the warping function as follows:\n\\begin{align}\n\\min_{\\mathrm{P}^{\\mathbf{c}}_{t_0}}\\sum^N_{i=1}\\|w_{t_0}(\\mathbf{x}_i,\\mathbf{c})-l(\\mathbf{x}_i)\\|^2_2\n\\label{eq:fitting}\n\\end{align}\nwhere $\\mathbf{x}_i$ is the sampled pixel position and $l(\\mathbf{x}_i)$ is the point that $\\mathbf{x}_i$ is moved by the end point of the linear kernel.\n$\\mathrm{P}^{\\mathbf{c}}_{t_0}$ is obtained by fitting $\\mathbf{x}_i$ moved by $w_{t_0}(\\cdot,\\mathbf{c})$ and $l(\\cdot)$.\n$\\mathrm{P}^{\\mathbf{c}}_{t_0}$ is the only variable of $w_{t_0}(\\cdot,\\mathbf{c})$ since the scene depth is fixed to the initial depth map.\nIn our implementation, RANSAC is used to find the camera motion that best describes the pixel-wise linear kernels.\n$N$ is the number of random samples, which is fixed to 4.\n\nFigure~\\ref{fig4} shows an example of camera motion initialization.\nIt is shown that \\cite{sun2015learning} underestimates the size of the motion (upper blue rectangle) and produces noisy motion where the texture is insufficient (lower blue rectangle).\n\n\n\t\n\\section{Experimental Results}\n\nThe proposed algorithm is implemented using Matlab on an Intel i7 7770K @ 4.2GHz with 16GB RAM and is evaluated for both synthetic and real light fields.\nSynthetic light field is generated using \\textit{Blender}~\\cite{blender} for qualitative as well as quantitative evaluation.\nIt includes 6~types of camera motion for 3~different scenes in which each light field has $7\\times7$ angular structure of 480$\\times$360 sub-aperture images.\nSynthetic blur is simulated by moving the camera array over a sequence of frames $(\\geq40)$ and then by averaging the individual frames.\nOn the other hand, real light field data is captured using Lytro Illum camera which generates $7\\times7$ angular structure of 552$\\times$383 sub-aperture images.\nWe generate the sub-aperture images from light field using the toolbox~\\cite{bok2014geometric} which provides the relative camera poses between sub-aperture images.\nLight fields are blurred by moving camera quickly under arbitrary motion, while the scene remains static.\nIn our implementation, we fixed most of the parameters except $\\lambda_D$ such that $ \\lambda_u=15, \\lambda_c = 1, \\lambda_L=5$.\n$\\lambda_D$ is set to a larger value for a real light field ($\\lambda_D=400$) than for synthetic data ($\\lambda_D=20$).\n\nFor quantitative evaluation of deblurring, we use both peak signal to noise ratio~(PSNR) and structural similarity~(SSIM).\nNote that PSNR and SSIM are measured by the maximum (best) ones among individual PSNR and SSIM values computed between the deblurred image and the ground truth images (along the motion path) as adopted in~\\cite{kohler2012recording}.\nFor comparison with light field depth estimation methods, we use the relative mean absolute error~(L1-rel) defined as\n\\begin{equation}\n\\text{L1-rel}(D,\\hat{D})=\\frac{1}{n}\\sum_{i}\\frac{|D_{i}-\\hat{D}_{i}|}{\\hat{D}_{i}},\n\\end{equation}\nwhich computes the relative error of the estimated depth $\\hat{D}$ to the ground truth depth $D$.\nThe accuracy of camera motion estimation is measured by the average end point error (EPE) to the end point of ground truth blur kernels.\nIn our evaluation, we compute the EPE by generating an end point of blur kernel using the estimated camera motion and ground truth depth.\nWe compare the performance of the proposed algorithm to linear blur kernel methods that directly computes the EPE between the ground truth and their pixel-wise blur kernel.\n\n\\begin{figure*}[ht]\n\t\\begin{center}\t\t\n\t\t\\vspace*{0.15in}\n\t\t\\includegraphics[width=0.24\\textwidth]{figure\/figure567_ryan_1_Blurred.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.24\\textwidth]{figure\/figure567_ryan_1_THKim.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.24\\textwidth]{figure\/figure567_ryan_1_JianSun.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.24\\textwidth]{figure\/figure567_ryan_1_Ours_nocurve.pdf\n\t\t\\vspace*{0.01in}\n\t\t\n\t\t\\includegraphics[width=0.24\\textwidth]{figure\/figure567_cocacola_5_Blurred.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.24\\textwidth]{figure\/figure567_cocacola_5_THKim.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.24\\textwidth]{figure\/figure567_cocacola_5_JianSun.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.24\\textwidth]{figure\/figure567_cocacola_5_Ours_nocurve.pdf\n\t\t\\vspace*{-0.12in}\n\t\n\t\t\\subfloat[]{\\includegraphics[width=0.24\\textwidth]{figure\/figure567_mvg_5_Blurred.pdf}}\\hspace*{0.01in}\n\t\t\\subfloat[]{\\includegraphics[width=0.24\\textwidth]{figure\/figure567_mvg_5_THKim.pdf}}\\hspace*{0.01in}\n\t\t\\subfloat[]{\\includegraphics[width=0.24\\textwidth]{figure\/figure567_mvg_5_JianSun.pdf}}\\hspace*{0.01in}\n\t\t\\subfloat[]{\\includegraphics[width=0.24\\textwidth]{figure\/figure567_mvg_5_Ours_nocurve.pdf}\n\t\t\n\t\\end{center}\t\n\t\\vspace*{-0.23in}\n\t\\caption{Deblurring result for real light field dataset with comparison to local linear blur kernel deblurring methods. (a) Blurred input image. (b) Result of Kim and Lee~\\cite{kim2014segmentation}. (c) Sun~et al.~\\cite{sun2015learning}. (d) Proposed algorithm.}\n\n\t\\label{qual:deblur_real1}\n\\end{figure*}\n\\begin{figure*}[!t]\n\t\\begin{center}\t\n\t\t\\includegraphics[width=0.24\\textwidth]{figure\/figure567_doll_rot_large_Blurred.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.24\\textwidth]{figure\/figure567_doll_rot_large_ZheHu.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.24\\textwidth]{figure\/figure567_doll_rot_large_Srinivasan_nocurve.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.24\\textwidth]{figure\/figure567_doll_rot_large_Ours_nocurve.pdf\n\t\t\\vspace*{0.01in}\n\t\t\n\t\t\\includegraphics[width=0.24\\textwidth]{figure\/figure567_friends_1_Blurred.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.24\\textwidth]{figure\/figure567_friends_1_ZheHu.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.24\\textwidth]{figure\/figure567_friends_1_Srinivasan_nocurve.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.24\\textwidth]{figure\/figure567_friends_1_Ours_nocurve.pdf\n\t\t\\vspace*{-0.12in}\n\t\t\n\t\t\\subfloat[]{\\includegraphics[width=0.24\\textwidth]{figure\/figure567_playground_1_Blurred.pdf}}\\hspace*{0.01in}\n\t\t\\subfloat[]{\\includegraphics[width=0.24\\textwidth]{figure\/figure567_playground_1_ZheHu.pdf}}\\hspace*{0.01in}\n\t\t\\subfloat[]{\\includegraphics[width=0.24\\textwidth]{figure\/figure567_playground_1_Srinivasan_nocurve.pdf}}\\hspace*{0.01in}\n\t\t\\subfloat[]{\\includegraphics[width=0.24\\textwidth]{figure\/figure567_playground_1_Ours_nocurve.pdf}\n\t\\end{center}\n\t\\vspace*{-0.23in}\n\t\\caption{Deblurring result for real light field dataset with comparison to global camera motion estimation methods. (a) Blurred input image. (b)  Result of Hu~et al.~\\cite{hu2014joint}. (c) Srinivasan et al.~\\cite{srinivasan2017light}. (d) Proposed algorithm.}\n\t\\label{qual:deblur_real2}\n\n\\end{figure*}\n\n\\subsection{Light Field Deblurring}\n\n\\textbf{Real Data.}~Figure~\\ref{qual:deblur_real1} and Figure~ \\ref{qual:deblur_real2} show the light field deblurring results for blurred real light field with spatially varying blur kernels.\nIn Figure~\\ref{qual:deblur_real1}, the result is compared with the existing motion deblurring methods~\\cite{kim2014segmentation,sun2015learning} which utilize motion flow estimation.\nIt is shown that the proposed algorithm reconstructs sharper latent image better than others.\nNote that \\cite{kim2014segmentation,sun2015learning} show satisfactory performance only for small blur kernels.\n\nFigure~\\ref{qual:deblur_real2} shows the comparison results with the deblurring method based on the global camera motion model~\\cite{hu2014joint,srinivasan2017light}.\nIn comparison with~\\cite{srinivasan2017light}, we deblur only cropped regions shown in the yellow boxes of Figure~\\ref{qual:deblur_real2}(c) due to GPU memory overflow ($>$12GB) for larger spatial resolution.\n\n\\cite{hu2014joint} assumes the scene depth is piecewisely planar. Therefore, it cannot be generalized to arbitrary scene, yielding unsatisfactory deblurring result.\n\\cite{srinivasan2017light} estimates the reasonably correct camera motion of the blurred light field while their output is less deblurred.\nNote that \\cite{srinivasan2017light} can not handle the rotational camera motion which produces completely different blur kernels from translational motion.\nOn the other hand, the proposed algorithm fully utilizes the 6-DOF camera motion and the scene depth, yielding outperforming results for the arbitrary scene.\n\nThe light field deblurring experiments with real data show that the proposed algorithm works robustly even for the hand-shake motion which does not match the proposed motion path model.\nThe proposed algorithm showed superior deblurring performance for both natural indoor and outdoor scenes, which confirms the robustness of the proposed algorithm to noise and depth level.\n\\begin{figure*}[t]\n\t\\begin{center}\t\n\t\t\\vspace*{0.15in}\n\t\t\\includegraphics[width=0.2\\textwidth]{figure\/figure567_StaticScene_Blurred.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.2\\textwidth]{figure\/figure567_StaticScene_ZheHu.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.2\\textwidth]{figure\/figure567_StaticScene_THKim.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.2\\textwidth]{figure\/figure567_StaticScene_JianSun.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.2\\textwidth]{figure\/figure567_StaticScene_Ours.pdf}\\hspace*{0.01in}\n\t\t\\vspace*{0.03in}\t\t\n\\\\\t\t\\includegraphics[width=0.2\\textwidth]{figure\/figure567_Fruits_small_rotation_Blurred.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.2\\textwidth]{figure\/figure567_Fruits_small_rotation_ZheHu.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.2\\textwidth]{figure\/figure567_Fruits_small_rotation_THKim.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.2\\textwidth]{figure\/figure567_Fruits_small_rotation_JianSun.pdf}\\hspace*{0.01in}\n\t\t\\includegraphics[width=0.2\\textwidth]{figure\/figure567_Fruits_small_rotation_Ours.pdf}\\hspace*{0.01in}\n\t\t\\vspace*{-0.11in}\t\t\n\\\\\t\t\\subfloat[]{\\includegraphics[width=0.2\\textwidth]{figure\/figure567_Baseball_large_forward_Blurred.pdf}}\\hspace*{0.01in}\n\t\t\\subfloat[]{\\includegraphics[width=0.2\\textwidth]{figure\/figure567_Baseball_large_forward_ZheHu.pdf}}\\hspace*{0.01in}\n\t\t\\subfloat[]{\\includegraphics[width=0.2\\textwidth]{figure\/figure567_Baseball_large_forward_THKim.pdf}}\\hspace*{0.01in}\n\t\t\\subfloat[]{\\includegraphics[width=0.2\\textwidth]{figure\/figure567_Baseball_large_forward_JianSun.pdf}}\\hspace*{0.01in}\n\t\t\\subfloat[]{\\includegraphics[width=0.2\\textwidth]{figure\/figure567_Baseball_large_forward_Ours.pdf}}\\hspace*{0.01in}\n\t\t\\vspace*{-0.13in}\n\\\\\n\t\\end{center}\n\t\\vspace*{-0.05in}\n\t\\caption{Deblurring result for synthetic light field. (a) Blurred input light field. (b) Result of Hu~et al.~\\cite{hu2014joint}. (c)  Kim and Lee~\\cite{kim2014segmentation}. (d)  Sun~et al.~\\cite{sun2015learning}. (e) Proposed algorithm.}\n\t\\label{qual:deblur_synthetic}\n\t\\vspace*{-0.05in}\n\\end{figure*}\n\n\\begin{table*}[t]\n\t\\caption{Quantitative evaluation of deblurring on synthetic light field dataset (in PSNR and SSIM).}\n\t\\centering\n\\renewcommand{\\arraystretch}{1.1}{\n\t\\scalebox{0.8}{\n\t\t\\begin{tabular}{ccccccccccccc}\n\t\t\t\\toprule\n\t\t\t& \\multicolumn{2}{c}{Forward} &  \\multicolumn{2}{c}{Rotation} & \\multicolumn{2}{c}{Translation} & \\multicolumn{2}{c}{Forward+Rot.} & \\multicolumn{2}{c}{Forward+Tran.} & \\multicolumn{2}{c}{Rot.+Tran.}\\\\\n\t\t\t\\multicolumn{1}{c}{Methods}& PSNR & SSIM & PSNR & SSIM & PSNR & SSIM & PSNR & SSIM & PSNR & SSIM & PSNR & SSIM \\\\\n\t\t\t\\cmidrule(r){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-5}\\cmidrule(lr){6-7}\\cmidrule(lr){8-9}\\cmidrule(lr){10-11}\\cmidrule(l){12-13}\n\t\t\t\\multicolumn{1}{c}{Input}& 20.72 & 0.740 & 20.37 & 0.731 & 21.82 & 0.758 & 19.84 & 0.723 & 20.30 & 0.731 & 19.79 & 0.728 \\\\\n\t\t\t\\multicolumn{1}{c}{Hu~et al.~\\cite{hu2014joint}}& 20.00 & 0.716 & 19.42 & 0.704 & 21.42 & 0.745 & 19.18 & 0.701 & 19.43 & 0.699 & 19.24 & 0.711 \\\\\n\t\t\t\\multicolumn{1}{c}{Kim and Lee~\\cite{kim2014segmentation}}& 20.06 & 0.721 & 19.78 & 0.714 & 21.42 & 0.749 & 19.32 & 0.706 & 19.65 & 0.708 & 19.34 & 0.714 \\\\\n\t\t\t\\multicolumn{1}{c}{Sun~et al.~\\cite{sun2015learning}}& 27.69 & 0.896 & 27.68 & 0.881 & 25.41 & 0.856 & 27.40 & 0.874 & 27.23 & 0.899 & 26.42 & 0.868 \\\\\n\t\t\t\\multicolumn{1}{c}{Proposed Method}& \\bf{29.24} & \\bf{0.915} & \\bf{29.14} & \\bf{0.913} & \\bf{26.99} & \\bf{0.876} & \\bf{28.92} & \\bf{0.905} & \\bf{28.91} & \\bf{0.922} & \\bf{27.85} & \\bf{0.893} \\\\\n\t\t\t\\midrule\n\t\t\t\\multicolumn{1}{c}{Input~(cropped)}& 21.01 & 0.758 & 21.19 & 0.746 & 19.39 & 0.698 & 21.73 & 0.758 & 21.67 & 0.782 & 20.50 & 0.745 \\\\\n\t\t\t\\multicolumn{1}{c}{Srinivasan~et al.~\\cite{srinivasan2017light}}& 17.15 & 0.730 & 19.02 & 0.652 & 16.28 & 0.620 & 19.17 & 0.660 & 16.20 & 0.726 & 16.38 & 0.626 \\\\\n\t\t\t\\multicolumn{1}{c}{Proposed Method}& \\bf{27.15} & \\bf{0.871} & \\bf{27.32} & \\bf{0.870} & \\bf{25.30} & \\bf{0.836} & \\bf{28.83} & \\bf{0.904} & \\bf{28.01} & \\bf{0.901} & \\bf{25.88} & \\bf{0.867} \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t}\n}\n\t\\label{quan:deblur}\n\t\\end{table*}\n\\noindent\n\n\\textbf{Synthetic Data.}~The performance of the proposed algorithm is evaluated using synthetic light field dataset, as shown in Figure~\\ref{qual:deblur_synthetic} and Table~\\ref{quan:deblur}.\nThe synthetic data consists of forward, rotation, in-plane translation motion and their combinations.\nIn Figure~\\ref{qual:deblur_synthetic}, we visualize and compare the deblurring performance with existing motion flow methods~\\cite{kim2014segmentation,sun2015learning} and a camera motion method~\\cite{hu2014joint}.\nIn all examples, the proposed algorithm produces sharper deblurred images than others as shown clearly in the cropped boxes.\n\nTable~\\ref{quan:deblur} shows the quantitative comparison of deblurring performance by measuring PSNR and SSIM to the ground truth.\nIt shows that the proposed joint estimation algorithm significantly outperforms the others.\nSun~et al.~\\cite{sun2015learning} achieves comparable performance to the proposed algorithm in which CNN is trained with MSE loss.\nOther algorithms achieve minor improvement from the input image because the assumed blur models are simple and inconsistent with the ground truth blur.\n\nFor the comparison with \\cite{srinivasan2017light}, we crop the each light field to $200\\times200$ because of the GPU memory overflow. Note that we use the original setting of \\cite{srinivasan2017light}.\n\\cite{srinivasan2017light} shows lower performance than the input blurred light field due to the spatial viewpoint shift as in the output of \\cite{srinivasan2017light}.\nSince the original point exists at the end point of the camera motion path in \\cite{srinivasan2017light}, the viewpoint shift occurs when the estimated 3D motion is large.\nIt is observed that this is an additional cause to decrease PSNR and SSIM when the estimated 3D motion is different from the ground truth.\nThe proposed algorithm estimates the latent image with ignorable viewpoint shift because the origin is located in the middle of the camera motion path.\n\n\\subsection{Light Field Depth Estimation}\n\n\n\nTo show the performance of light field depth estimation, we compare the proposed method with several state-of-the-art  methods~\\cite{chen2014light,jeon2015accurate,tao2015depth,wang2015occlusion,williem2016robust}.\nFor comparison, all blurred sub-aperture images are independently deblurred using~\\cite{sun2015learning} before running their own depth estimation algorithms.\n\nFigure~\\ref{qual:depth} shows the visual comparison of estimated depth map generated by different methods, which confirms that the proposed algorithm produces significantly better depth map in terms of accuracy and completeness.\nSince independent deblurring of all sub-aperture images does not consider correlation between them, conventional correspondence and defocus cue do not produce reliable matching, yielding noisy depth map.\nOnly the proposed joint estimation algorithm results in sharp and unfattened object boundary, and produces the closest result to the ground truth.\n\nQuantitative performance comparison of depth map estimation is shown in Table~\\ref{quan:depth}.\nFor three synthetic scenes with three different motion for each scene, the average L1-rel error of the estimated depth map is computed and compared.\nThe comparison clearly shows that the proposed method produces the lowest error in all types of camera motion.\nNote that the second best result is achieved by Chen et al.~\\cite{chen2014light}, which is relatively robust in the presence of motion blur because bilateral edge preserving filtering is employed for cost computation.\n\n\\begin{figure*}[t]\n\t\\begin{center}\n\t\t\\vspace*{0.15in}\n\t\t\\subfloat[]{\\includegraphics[width=0.245\\textwidth]{figure\/figure8_1.pdf}}\\hspace*{0.01in}\n\t\t\\subfloat[]{\\includegraphics[width=0.245\\textwidth]{figure\/figure8_2.pdf}}\\hspace*{0.01in}\n\t\t\\subfloat[]{\\includegraphics[width=0.245\\textwidth]{figure\/figure8_3.pdf}}\\hspace*{0.01in}\n\t\t\\subfloat[]{\\includegraphics[width=0.245\\textwidth]{figure\/figure8_4.pdf}}\\hspace*{0.01in}\\\\\n\t\t\\vspace*{-0.125in}\t\t\t\n\t\t\\subfloat[]{\\includegraphics[width=0.245\\textwidth]{figure\/figure8_5.pdf}}\\hspace*{0.01in}\n\t\t\\subfloat[]{\\includegraphics[width=0.245\\textwidth]{figure\/figure8_6.pdf}}\\hspace*{0.01in}\n\t\t\\subfloat[]{\\includegraphics[width=0.245\\textwidth]{figure\/figure8_7.pdf}}\\hspace*{0.01in}\n\t\t\\subfloat[]{\\includegraphics[width=0.245\\textwidth]{figure\/figure8_8.pdf}}\\hspace*{0.01in}\n\t\\end{center}\t\n\t\\vspace*{-0.2in}\n\t\\caption{Depth estimation results on blurred light field. (a) Blurred center sub-aperture image. (b) Ground truth depth. (c) Result of Jeon et al.~\\cite{jeon2015accurate}. (d) Williem and Park~\\cite{williem2016robust}. (e) Tao et al.~\\cite{tao2015depth}. (f) Wang et al.~\\cite{wang2015occlusion}. (g) Chen et al.~\\cite{chen2014light}. (h) Proposed algorithm.}\n\t\\label{qual:depth}\n\\end{figure*}\n\n\\begin{table}[!t]\n\t\\caption{Comparison of depth estimation (in average L1-rel error).}\n\t\\centering\n\\renewcommand{\\arraystretch}{1.1}{\n\t\\scalebox{1.1}{\n\t\t\\begin{tabular}{ccccc}\n\t\t\t\\toprule\n\t\t\tMethods & Forward & Rotation & Trans. & Overall\\\\\n\t\t\t\\cmidrule(r){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-4}\\cmidrule(l){5-5}\n\t\t\tChen et al.~\\cite{chen2014light} & 0.0251 & 0.0326 & 0.0331 & 0.0303 \\\\\n\t\t\tTao et al.~\\cite{tao2015depth} & 0.0251 & 0.0359 & 0.0371 & 0.0327 \\\\\n\t\t\tWang et al.~\\cite{wang2015occlusion} & 0.0312 & 0.0377 & 0.0400 & 0.0363 \\\\\n\t\t\tJeon et al.~\\cite{jeon2015accurate} & 0.0835 & 0.0916 & 0.0921 & 0.0891 \\\\\n\t\t\tWilliem and Park~\\cite{williem2016robust} & 0.0615 & 0.0895 & 0.0966 & 0.0825 \\\\\n\t\t\tProposed Method & \\bf{0.0198} & \\bf{0.0150} & \\bf{0.0243} & \\bf{0.0197} \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t}\n}\n\t\\vspace*{-0.2in}\n\t\\label{quan:depth}\n\\end{table}\n\nThe depth estimation experiment demonstrates that consistent deblurring between sub-aperture images is essential.\nThe cues used in conventional depth estimation are seriously hampered in the independently deblurred image.\nHowever, the proposed joint estimation algorithm results in robust and accurate depth estimation, providing the justification of solving deblurring and depth estimation in a joint manner.\n\n\n\\subsection{Camera Motion Estimation}\nTable~\\ref{quan:motion} shows the EPE of the estimated motion on synthetic light field dataset.\nCompared with other methods~\\cite{kim2014segmentation,sun2015learning}, the proposed method improves the accuracy of the estimated motion significantly.\nIn particular, a large gain is obtained in the rotational motion, which indicates that the rotational motion cannot be modeled accurately as a linear blur kernel used in~\\cite{kim2014segmentation,sun2015learning}.\n\nFigure~\\ref{fig:motion} shows the motion estimation results compared to the ground truth motion.\nSince the camera orientation changes while the camera is moving, the 6-DOF camera motion can not be recovered properly by~\\cite{srinivasan2017light}.\nAs shown in Figure~\\ref{fig:motion}(b) and Figure~\\ref{fig:motion}(c), the deblurring results are similar to the input, because the motion can not converge to the ground truth.\nIn contrast, the proposed algorithm converges to the ground truth 6-DOF motion and also produces the sharp deblurring result.\n\\begin{figure}[t]\n\\begin{center}\n\t\\subfloat[]{\\includegraphics[width=0.24\\columnwidth]{figure\/figure9_input.pdf}}\\hspace*{0.05in\n\t\\subfloat[]{\\includegraphics[width=0.24\\columnwidth]{figure\/figure9_quad.pdf}}\\hspace*{0.03in}\t\t\t\n\t\\subfloat[]{\\includegraphics[width=0.24\\columnwidth]{figure\/figure9_cubic.pdf}}\\hspace*{0.03in}\n\t\\subfloat[]{\\includegraphics[width=0.24\\columnwidth]{figure\/figure9_ours.pdf}}\\hspace*{0.03in}\n\t\n\t\\vspace*{-0.05in}\n\\end{center}\n\\caption{Deblurring and camera motion estimation result for synthetic light field with comparison to \\cite{srinivasan2017light}.\n\t(a) Input light field and ground truth camera motion.\n\t(b) Result of Srinivasan et al.~\\cite{srinivasan2017light} (quadratic).\n\t(b) Srinivasan et al.~\\cite{srinivasan2017light} (cubic).\n\t(d) Proposed algorithm.\n}\n\\vspace*{-0.1in}\n\\label{fig:motion}\n\\end{figure}\n\\begin{table}[t]\n\\caption{Comparison of motion estimation (in EPE).}\n\\centering\n\\renewcommand{\\arraystretch}{1.1}{\n\n\t\\scalebox{1.1}{\n\t\t\\begin{tabular}{cccc}\n\t\t\t\\toprule\n\t\t\tMethods & Forward & Rotation & Translation \\\\\n\t\t\t\\cmidrule(r){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(l){4-4}\n\t\t\tKim and Lee~\\cite{kim2014segmentation} & 2.153 & 3.317 & 1.989 \\\\\n\t\t\tSun et al.~\\cite{sun2015learning} & 1.492 & 2.557 & 1.810 \\\\\n\t\t\tProposed Method & \\bf{0.325} & \\bf{0.171} & \\bf{0.590}\\\\\n\t\t\t\\bottomrule\n\t\\end{tabular}}}\n\t\\label{quan:motion}\n\t\\vspace*{-0.05in}\n\\end{table}\n\n\\section{Conclusion}\nIn this paper, we presented the novel light field deblurring algorithm that estimated latent image, sharp depth map, and camera motion jointly.\nFirstly, we modeled all the blurred sub-aperture images by center-view latent image using 3D warping function.\nThen, we developed the algorithm to initialize the 6-DOF camera motion from the local linear blur kernel and scene depth.\nIn the iterative joint optimization, the nonlinear energy minimization was solved efficiently using IRLS.\nThe evaluation on both synthetic and real light field data showed that the proposed model and algorithm worked well with general camera motion and scene depth variation.\n\n\\bibliographystyle{splncs}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:intro}\nThe Standard Model of particle physics has proven successful in describing the elementary particles and their interactions \\cite{pdg}. However, it still falls short in explaining many of the basic features of the nucleon, features that to this day remain the objects of intensive research: spin \\cite{pspin,pspin2}, size \\cite{pradius1,pradius2,pradius3}, intrinsic structure \\cite{pstructure,pstructure2} and abundance, \\textit{i.e.} the excess of nucleons compared to antinucleons in the universe \\cite{pasym}.\n\nOne of the nucleons, the proton, is the most stable composite system we know. In order to study its properties, we therefore need to distort or break it by scattering something on it, for example an electron, or by adding some energy and thereby excite it. A third option is to replace one or several of the building blocks \\cite{granados}. The latter is the main concept of \\textit{hyperon physics}: one or several light $u$ or $d$ quarks in the nucleon is replaced by strange ones.\\footnote{In principle, one can also replace it with a charm or a bottom quark, but the scope of this paper is strange hyperons.} The mass of the strange quark is $\\approx$ 95 MeV, which is $\\geq$ 20 times larger than the light $d$ and $u$ quark masses. The strange quark in a hyperon is therefore expected to behave a bit differently than the light quarks, for example it will be less relativistic. Furthermore, a larger part of the mass of a hyperon comes from the quarks compared to the nucleon. However, the mass of the strange quark is much smaller than the mass of the hyperon itself, in contrast to the more than ten times heavier charm quark. Hence, strange hyperons are sufficiently similar to nucleons for comparisons to be valid, for example assuming approximate SU(3) flavour symmetry.\n\nBy being unstable, hyperons reveal more of their features than protons. In particular, the weak, parity violating and thereby \\textit{self-analysing} decay of many ground-state hyperons make their spin properties experimentally accessible. This makes hyperons a powerful diagnostic tool that can shed light on various physics problems, \\textit{e.g.} non-perturbative production dynamics, internal structure and fundamental symmetries.\n\nIn this paper, we outline the assets of hyperon physics to be exploited by the future PANDA (antiProton ANnihilation at DArmstadt) experiment with an antiproton beam at FAIR (Facility for Antiproton and Ion Research) in Darmstadt, Germany. We describe in detail a comprehensive simulation study that demonstrates the feasibility of the planned hyperon physics programme, and discuss the impact and long-term perspectives.\n\n\\section{The PANDA experiment}\n\\label{sec:panda}\n\n\\begin{figure*}[t]\n\\centering\n\\begin{tikzpicture}\n\\node[anchor=south west, inner sep=0] (image) {\\includegraphics[width=\\textwidth]{PANDA_full-set-up.png}};\n\t\\begin{scope}[x={(image.south east)},y={(image.north west)}]\n        \n        \\begin{scope}\n       \n            \\node[align=center, red] at (0.25,0.95) {TS};\n            \\node[align=center, red] at (0.65,0.7) {FS};\n           \n           \n           \n           \n           \n        \\end{scope}\n\t\\end{scope}\n\\end{tikzpicture}\n\\caption{(Colour online) Overview of the full PANDA setup. The antiproton beam will go from left to right, whereas the target jets\/pellets from top to bottom. The left part of the detector surrounds the interaction point and is the Target Spectrometer (TS), whereas the right part is the Forward Spectrometer (FS).}\n\\label{fig:panda}\n\\end{figure*} \n\nThe PANDA experiment, which is currently under construction at FAIR \\cite{fair} in Darmstadt, Germany, offers a broad programme for studies of the strong interaction and fundamental symmetries \\cite{panda}. The unique combination of an antiproton beam in the intermediate energy range and a nearly 4$\\pi$ detector with vertex- and tracking devices, multiple particle identification (PID) detectors and calorimeters, give excellent conditions for a new generation of hyperon physics experiments.\n\nThe High Energy Storage Ring (HESR) will deliver an antiproton beam with momenta ranging from 1.5 GeV\/$c$ up to 15 GeV\/$c$ \\cite{hesr}. In the start-up phases, referred to as \\textit{Phase One} and \\textit{Phase Two}, the HESR will be able to accumulate up to $10^{10}$ antiprotons in 1000~s. In the final \\textit{Phase Three}, the luminosity will be ramped up by the Recuperated Experimental Storage Ring (RESR), allowing up to $10^{11}$ antiprotons to be injected and stored in the HESR. The HESR will offer stochastic cooling resulting in a beam momentum spread of better than $5\\cdot 10^{-5}$. The antiproton beam will impinge on a hydrogen cluster jet or pellet target, which during Phase One will result in an average luminosity of $\\approx 10^{31}$ cm$^2$s$^{-1}$~\\cite{target}. At low energies, the luminosity will be about a factor of two lower. During Phase Three, the design luminosity of $\\approx 2\\cdot10^{32}$ cm$^2$s$^{-1}$ will be achieved. \n\nThe PANDA detector, shown in Fig. \\ref{fig:panda} and described in detail in Ref.~\\cite{pandadet}, is divided into two parts: the target spectrometer (TS) and the forward spectrometer (FS). The TS covers polar angles of $> 10^{\\mathrm{o}}$ in the horizontal direction and $> 5^{\\mathrm{o}}$ in the vertical direction, whereas the FS covers polar angles $<10^{\\mathrm{o}}$. The TS provides timing and vertexing by the silicon micro vertex detector (MVD). The MVD is also used for tracking together with the gas-filled straw tube trackers (STT). The polar angle range of the latter is $22^{\\mathrm{o}} < \\theta < 140^{\\mathrm{o}}$. In order to bridge the acceptance between the STT and the FS, the gas electron multiplier detectors (GEM) is designed to track particles emitted below 22$^\\circ$. Time-of-flight detectors (TOF), made of scintillating tiles, offer excellent time resolution. By providing the reaction time $t_0$, it improves the resolution of the track parameters, and increases the particle identification capabilities. Detection of internally reflected Cherenkov light (DIRC) offer independent PID and an electromagnetic calorimeter (EMC) with lead-tungstate (PbWO$_4$) crystals will measure energies between 10 MeV and 7 GeV. The laminated yoke of the solenoid magnet, outside the barrel EMC, is interleaved with sensitive layers to act as a range system for the detection and identification of muons. Measurement of the charge and momenta are possible thanks to the bending of particle trajectories by a solenoid magnet providing a field of up to 2.0 Tesla. \n\nThe FS will consist of six straw tube stations for tracking, a dipole magnet, a ring imaging Cherenkov counter (RICH) detector for PID as well as a TOF for timing and PID. The energies of the forward going, electromagnetically interacting particles, will be measured by a Shashlyk electromagnetic calorimeter. A muon range system, using sensors interleaved with absorber layers, is placed at the end of the FS.\n\nThe luminosity will be determined by using elastic antiproton-proton scattering as the reference channel. The differential cross section of this process can be calculated with extremely high precision at small angles, where the Coulomb component dominates~\\cite{luminosity}. At polar angles within 3-8~mrad, the scattered antiproton will be measured by a luminosity detector consisting of four layers of thin monolithic active pixels sensors made of silicon~\\cite{luminosity}. \n\nPANDA will feature, as one of the first experiments, a time-based data acquisition system (DAQ) without hard-ware triggers. Data will instead be read out as a continuous stream using an entirely software-based selection scheme. This change of paradigm is driven by the large foreseen reaction rates, resulting in huge amounts of data to be stored.\n\n\nThe feasibility studies presented in this work are performed within the common simulation and analysis framework PandaROOT \\cite{PANDAROOT}. It comprises the complete simulation chain, including Monte Carlo event generation, particle propagation and detector response, hardware digitization, reconstruction and calibration, and data analysis. PandaROOT is derived from the FairROOT framework \\cite{FAIRROOT} which in turn is based on ROOT \\cite{ROOT}. \n\n\\section{Hyperon production with antiproton probes}\n\\label{sec:hyperonphys}\nThe focus of this paper is $\\Lambda$ and $\\Xi^-$ hyperon production in the $\\bar{p}p \\to \\bar{Y}Y$ reaction, where $Y$ refers to the octet hyperons $\\Lambda$, $\\Xi^-$. Understanding the production and decay of these hyperons is crucial in order to correctly interpret experimental analyses of heavier hyperons. The study of excited multi-strange hyperons constitutes an important part of the PANDA physics programme and is described in more detail in Ref. \\cite{jennypthesis}. However, octet hyperons are interesting in their own right. The $\\Lambda$ and $\\Xi^-$ hyperons considered in this work, predominantly decay into charged final state particles which makes them straight-forward to measure experimentally.\nIn the following, we will discuss how the self-analysing decays can shed light on various aspects of fundamental physics and the advantages of antiproton probes in hyperon studies.\n\n\\subsection{Weak two-body decays}\n\\label{sec:features}\nAll ground-state hyperons except the $\\Sigma^0$ decay weakly through a process that has a parity violating component. This means that the direction of the decay products depends on the spin direction of the mother hyperon. In Fig. \\ref{fig:hypdecay}, the two-body decay of a spin 1\/2 hyperon $Y$ into a spin 1\/2 baryon $B$ and a pseudoscalar meson $M$, is illustrated. The angular distribution of $B$ in the rest system of $Y$ is given by \\cite{pdg,bigibook}\n\\begin{equation}\nW(\\cos\\theta^B)=\\frac{1}{4\\pi}(1+\\alpha P_y^{Y}(\\cos\\theta_{Y}) \\cos\\theta^B),\n\\label{eq:decay}\n\\end{equation}\n\n\\noindent where $P^Y_y(\\cos\\theta_{Y})$ is the polarisation with respect to some reference axis $\\hat y$. $P^Y_y$ carries information about the production process and therefore depends on the collision energy and the scattering angle. The decay asymmetry parameter $\\alpha$ is the real part of the product between the parity violating and the parity conserving decay amplitudes, $T_s$ and $T_p$ \\cite{leeyang}. \nEq. (\\ref{eq:decay}) demonstrates how the experimentally measurable decay angular distribution is related to quantities with physical meaning, \\textit{i.e.} $P^Y_y$ and $\\alpha$. This feature makes hyperons a powerful diagnostic tool. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.35\\textwidth]{AngDist3BW.png}\n\\end{center}\n\\caption{The $Y \\to BM$ decay, with the spin direction of $Y$ along the $y$-axis.}\n\\label{fig:hypdecay}\n\\end{figure} \n\\subsection{Scientific case}\n\\label{sec:scientific}\n\nAntihyperon-hyperon pair production in antiproton-proton annihilation, $\\bar{p}p \\to \\bar{Y}Y$, provides excellent conditions for hyperon studies, since\n\\begin{itemize}\n    \\item Antihyperons and hyperons, also with double- and triple strangeness, can be produced in two-body processes at low energies, where the number of partial waves is small. This makes the production process parameterizable in a close to model-independent way.\n    \\item Antihyperons and hyperons can be studied simultaneously, under symmetric conditions.\n    \\item The production cross sections for single- and double-strange hyperons are known to be large \\cite{tord} which results in large count rates also for modest luminosities.\n    \\end{itemize}\n\nThe scale of strangeness production is governed by the mass of the strange quark, $m_s \\approx 95$ MeV\/$c^2$. This is far below the scale where perturbative QCD breaks down ($\\approx$ 1 GeV) but close to the QCD cut-off scale ($\\Lambda_{QCD}$). Therefore, the relevant degrees of freedom in processes involving strange quarks are unclear: quarks and gluons, or hadrons?  \n\nSingle-strange hyperon production in $\\bar{p}p \\to \\bar{Y}Y$ has been modeled using quark-gluon degrees of freedom \\cite{quarkgluon}, meson exchange \\cite{kaonexchange} and a combination of the two \\cite{quarkgluonhadron}. The production of double-strange hyperons requires interactions at shorter distances, since it either implies annihilation of two quark-antiquark pairs \\cite{XiQG,kaidalov}, or exchange of two kaons \\cite{XiMEX}. Spin observables, accessible for hyperons through their self-analysing decays, are particularly powerful in differentiating between models, since they are sensitive to the production mechanism. Spin observables can also give information about possible polarized strangeness content in nucleons \\cite{alberg} and final state interactions \\cite{haidenbauerFSI}. It is important to have a solid understanding of the latter also when interpreting results from antihyperon-hyperon pair production with other probes. One example is the $e^+e^- \\to Y\\bar{Y}$ process, from which the time-like electromagnetic form factors are determined. In Ref. \\cite{haidenbauerFSI}, the $\\Lambda$ complex form factors are predicted based on potential models fitted to PS185 data \\cite{PS185} on spin observables in the $\\bar{p}p \\to \\bar{\\Lambda}\\Lambda$ reaction \\cite{haidenbauerPot}. It was found that the form factors are sensitive to the $\\bar{\\Lambda}\\Lambda$ final state interaction and that spin observables are necessary in order to discriminate between them \\cite{haidenbauerFSI}. This has been done in a recent measurement of $\\Lambda$ form factors by the BESIII collaboration \\cite{bes3prl}.\n\nHyperon decays can provide one piece to the puzzle of nucleon abundance, more commonly referred to as the \\textit{matter-antimatter asymmetry} puzzle. According to the present paradigm, equal amounts of matter and antimatter should have been produced in the Big Bang. Unless the initial matter-antimatter imbalance was fine-tuned, a dynamical enrichment of matter with respect to antimatter must have occurred, \\textit{i.e.} Baryogenesis. However, this is only possible if i) processes exist that violate baryon number conservation ii) processes exist that violate C and CP symmetry, and iii) the aforementioned processes occured outside thermal equilibrium \\cite{pasym}. With hyperons, criterion ii) can be tested. CP symmetry means that hyperons and antihyperons have the same decay patterns, but with reversed spatial coordinates. For two-body hyperon decays, it means that the decay asymmetry parameters, \\textit{e.g.} $\\alpha$ in Eq. (\\ref{eq:decay}), have exactly the same value but with opposite sign compared to the corresponding antihyperon parameter, \\textit{i.e.} $\\alpha = -\\bar{\\alpha}$. The large production rates and the symmetric hyperon and antihyperon conditions make the $\\bar{p}p \\to \\bar{Y}Y$ a suitable reaction for searching for CP violation. Hyperon-antiproton studies have been carried out recently with BESIII in $e^+e^- \\to Y\\bar{Y}$, a reaction that is similar to $\\bar{p}p \\to \\bar{Y}Y$ in the sense that it is a two-body reaction that is symmetric in particle-antiparticle observables. These studies show that the precision can be greatly improved by several orders of magnitudes if the production process can be pinned down \\cite{goran,goranand,bes3nature}. Hence, a proper understanding of the $\\bar{p}p \\to \\bar{Y}Y$ reaction mechanism constitutes a crucial milestone in future large-scale CP studies with PANDA at FAIR.\n\n\\subsection{State of the art}\n\\label{sec:stateoftheart}\n\n\\subsubsection{Hyperon production in $\\bar{p}p$ annihilations}\n\nThe large amount of high-quality data on single-strange hyperons \\cite{tord,PS185,PS185164} produced in antiproton-proton annihilation, partly with a polarised target, led to important insights. For instance, it was found that the $\\bar{\\Lambda}\\Lambda$ pair is produced almost exclusively in a spin triplet state. From this, conclusions about the $\\Lambda$ quark structure can be drawn: the spin of the $\\Lambda$ is carried by the strange quark, while the light $u$ and $d$ quarks form a spin-0 \\textit{di-quark}. Theoretical investigations based on the aforementioned quark-gluon approach \\cite{quarkgluon}, kaon exchange \\cite{kaonexchange} and a combined approach \\cite{quarkgluonhadron}, reproduced this finding. However, no model so far describes the complete spin structure of the reaction. The models extensions into the double-strange sector \\cite{XiQG,XiMEX} have not been tested due to the lack of data -- only a few bubble-chamber events exist for $\\Xi^-$ and $\\Xi^0$ from $\\bar{p}p$ annihilations \\cite{Musgrave1965}. In Ref. \\cite{XiMEX}, $\\bar{\\Xi}^+$ emitted in the forward-direction in the center of mass frame are predicted to be in a triplet state, while backward-going $\\bar{\\Xi}^+$ are in a singlet state, in contrast to the $\\Lambda$ case that is in a spin-triplet state irrespective of the angle \\cite{PS185}. With future data from PANDA, this prediction can be tested. The hope is also that new spin structure data of $\\bar{p}p \\rightarrow \\bar{Y}Y$ reactions will trigger the activity of the theory community and lead to a deeper understanding of strange reaction dynamics.  \n\n\\subsubsection{CP symmetry in hyperon decays}\n\nThe existence of CP violation for spinless mesons is experimentally well-established in the strange and bottom sector \\cite{pdg} and recently also in the charm sector \\cite{LHCbcharm}. It is also incorporated in the Standard Model, through the Cabibbo-Kobayashi-Maskawa mechanism \\cite{cabibbo,kobayashi}. However, Standard Model deviation from CP symmetry would result in a matter-antimatter asymmetry of eight orders of magnitude smaller than the observed one \\cite{werner}. Hence, this problem is intimately connected to the search for physics beyond the Standard Model. The spin-carrying baryons could give new insights into CP violation, since spin behaves differently from momentum under parity flip. However, the only indication of CP violation in a baryon decay, observed very recently by the LHCb collaboration \\cite{LHCb}, was not confirmed in a later study with larger precision by the same experiment \\cite{LHCb2}. Two-body decays of strange hyperons provide a cleaner search-ground, but require large data samples. The most precise CP test in the strange sector so far is provided by the HyperCP collaboration. The proton angular distributions from the $\\Xi^- \\to \\Lambda \\pi^-, \\Lambda \\to p\\pi^-$ chain was studied, along with the corresponding antiproton distributions from the $\\bar{\\Xi}^+$ decay chain. The result was found to be consistent with CP symmetry with a precision of $10^{-4}$ \\cite{hyperCP}. The most precise test for the $\\Lambda$ hyperon was obtained recently by the BESIII collaboration \\cite{bes3nature}. They analysed $\\bar{\\Lambda}\\Lambda$ pair production from $J\/\\Psi$ using a multi-dimensional method. The good precision for a relatively modest sample size ($\\approx$ 420 000 $\\bar{\\Lambda}\\Lambda$ events) demonstrates the merits of exclusive measurements of polarised and entangled hyperon-antihyperon pairs, produced in two-body reactions. The most remarkable finding was however that the decay asymmetry parameter $\\alpha_{\\Lambda}$ was found to be $0.750 \\pm 0.009 \\pm 0.004$, \\textit{i.e.} 17\\% larger than the PDG world average of $0.642$ at the time \\cite{pdg}. This average was calculated from measurements made in the 1960s and 1970s, based on the proton polarimeter technique \\cite{cronin}. In the 2019 update of the PDG, the old measurements are discarded and instead, the BESIII value is established as the recommended one. In a re-analysis of CLAS data, the $\\alpha_{\\Lambda}$ was calculated to be $0.721 \\pm 0.006 \\pm 0.005$. This is between the old average and the new BESIII value, though much closer the to the latter \\cite{claslambda}. More high-precision measurements from independent experiments will be valuable not only to establish the correct decay asymmetry, but also to understand the difference between old and new measurements.\n\n \n\\section{Formalism}\n\\label{sec:formalism}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\n\\includegraphics[width=0.95\\linewidth]{Hyperons_coordpbarpBW.png}\n\n\\caption{The reference system of the $\\bar{p}p \\to \\bar{Y}Y$ reaction.}\n\\label{fig:refsys}\n\\end{center}\n\\vspace{-11pt}\n\\end{figure}\n\\noindent Consider an antiproton beam impinging on a hydrogen target, producing a $\\bar{Y}Y$ pair. Then the rest systems of the  outgoing hyperons can be defined as in Fig. \\ref{fig:refsys}: the $\\hat{y}_Y$ and $\\hat{y}_{\\bar{Y}}$ axes as the normal of the production plane, spanned by the incoming antiproton beam and the outgoing antihyperon in the centre of mass system of the reaction. The $\\hat{z}_Y$ ($\\hat{z}_{\\bar{Y}}$) is defined along the direction of the outgoing hyperon (antihyperon) and the $\\hat{x}_Y$ ($\\hat{x}_{\\bar{Y}}$) is obtained by the cross product of the $y$ and $z$ direction:  \n\\begin{equation}\n\t\\hat{z}_Y = \\frac{\\vec{p}_{Y}}{|\\vec{p}_{Y}|}, \\hat{y}_Y = \\frac{\\vec{p}_{beam} \\times \\vec{p}_{Y}}{|\\vec{p}_{beam} \\times \\vec{p}_{Y}|}, \\hat{x}_Y = \\hat{y}_Y \\times \\hat{z}_Y,\n\\end{equation}\nwhere $\\vec{p}_Y$ is the momentum vector of the outgoing hyperon and $\\vec{p}_{beam}$ is the momentum of the initial beam.\n\nInterference between complex production amplitudes has a polarising effect on the outgoing hyperon and antihyperon, even if the initial state is unpolarised. In PANDA, the beam and target will be unpolarised. Since the $\\bar{p}p \\to \\bar{Y}Y$ reaction is a strong, parity-conserving process, the polarisation of the outgoing hyperon and antihyperon can only be non-zero in the direction along the normal of the production plane, \\textit{i.e.} $\\hat{y}_Y$($\\hat{y}_{\\bar{Y}})$ in Fig. \\ref{fig:refsys}. In the case of a spin 1\/2 hyperon $Y$ (antihyperon $\\bar{Y}$) decaying into a spin 1\/2 baryon $B$ (antibaryon $\\bar{B}$) and a meson $M$ (antimeson $\\bar{M}$), the angular distribution of the decay baryon and antibaryon can be parameterised as:\n\\begin{align}\n\\label{eq:ppbar}\n    &I(\\theta_Y,\\theta^B,\\theta^{\\bar{B}}) = N[1 + \\alpha\\sum_i P^Y_{i}(\\theta_Y)\\cos\\theta_{i}^{B} \\nonumber\\\\ &+ \\bar{\\alpha}\\sum_j P^{\\bar{Y}}_{j}(\\theta_Y)\\cos\\theta_{j}^{\\bar{B}} + \\alpha\\bar{\\alpha} \\sum_{ij} C^{Y\\bar{Y}}_{ij}(\\theta_Y)\\cos\\theta_{i}^{B}\\cos\\theta_{j}^{\\bar{B}}]\n  \n \\end{align}\nwhere $i,j = x,y,z$ and the opening angle $\\cos\\theta_{i}^B$ $(\\cos\\theta_{j}^{\\bar{B}})$ is taken between the direction of the final state baryon $B$ (antibaryon $\\bar{B}$) and the axis $i$ $(j)$ in the rest system of the hyperon (antihyperon). The $P^Y_i(\\theta_Y)$ denote the vector polarisation and the $C^{\\bar{Y}Y}_{ij}(\\theta_Y)$ the spin correlation of the antihyperon and hyperon with respect to the axes $i,j = x,y,z$. The $\\theta_Y$ angle is defined in the reaction CMS system. With the unpolarised beam and target foreseen with PANDA and the reference system defined in Fig. \\ref{fig:refsys}, most spin variables must be zero due to parity conservation. The only non-zero spin variables are $P^Y_y$, $P^{\\bar{Y}}_y$, $C^{Y\\bar{Y}}_{xz}$, $C^{Y\\bar{Y}}_{zx}$, $C^{Y\\bar{Y}}_{xx}$, $C^{Y\\bar{Y}}_{yy}$ and $C^{Y\\bar{Y}}_{zz}$ \\cite{erikthesis,paschke}. Of these, only five are independent since $P^Y_y = P^{\\bar{Y}}_y$ and $C^{Y\\bar{Y}}_{xz} = C^{Y\\bar{Y}}_{zx}$.\n\nThe angular distribution can also be expressed with a matrix formulation. Then, one first defines the 4D vectors\n\\begin{align}\n\t&k_{\\bar{B}}=(1,\\cos\\theta_{x}^{\\bar{B}},\\cos\\theta_{y}^{\\bar{B}},\\cos\\theta_{z}^{\\bar{B}}) \\\\\n\t&k_B=(1,\\cos\\theta_{x}^{B},\\cos\\theta_{y}^{B},\\cos\\theta_{z}^{B}).\n\\end{align}\n\n\\noindent In addition, a matrix with spin observables and decay parameters can be defined in the following way\n\n\\begin{equation}\n\tD_{\\mu\\nu}=\\begin{pmatrix}\n\t\t1 & \\alpha P^Y_{x} & \\alpha P^Y_{y} & \\alpha P^Y_{z} \\\\\n\t\t\\bar{\\alpha} P^{\\bar{Y}}_{x} & \\bar{\\alpha}\\alpha C^{\\bar{Y}Y}_{xx} & \\bar{\\alpha}\\alpha C^{\\bar{Y}Y}_{xy} & \\bar{\\alpha}\\alpha C^{\\bar{Y}Y}_{xz} \\\\\n\t\t\\bar{\\alpha} P^{\\bar{Y}}_{y} & \\bar{\\alpha}\\alpha C^{\\bar{Y}Y}_{yx} & \\bar{\\alpha}\\alpha C^{\\bar{Y}Y}_{yy} & \\bar{\\alpha}\\alpha C^{\\bar{Y}Y}_{yz} \\\\\n\t\t\\bar{\\alpha} P^{\\bar{Y}}_{z} & \\bar{\\alpha}\\alpha C^{\\bar{Y}Y}_{zx} & \\bar{\\alpha}\\alpha C^{\\bar{Y}Y}_{zy} & \\bar{\\alpha}\\alpha C^{\\bar{Y}Y}_{zz}\n\t\\end{pmatrix},\n\\end{equation}\n\n\\noindent where $\\mu= 0,1,2,3$ or $0, x, y, z$ for the antihyperon and $\\nu= 0,1,2,3$ or $0, x, y, z$ for the hyperon. Since parity is conserved in strong interactions, the spin observables matrix in the $\\bar{p}p \\to \\bar{Y}Y$ reduces to \n\n\\begin{align}\n\t&D_{\\mu\\nu}=\\begin{pmatrix}\n\t\t1 & 0 & D_{02} & 0 \\\\\n\t\t0 & D_{11} & 0 & D_{13} \\\\\n\t\tD_{20} & 0 & D_{22} & 0 \\\\\n\t\t0 & D_{31} & 0 & D_{33}\n\t\\end{pmatrix}\\nonumber\\\\\n\t&=\\begin{pmatrix}\n\t\t1 & 0 & \\alpha P^Y_{y} & 0 \\\\\n\t\t0 & \\bar{\\alpha}\\alpha C^{\\bar{Y}Y}_{xx} & 0 & \\bar{\\alpha}\\alpha C^{\\bar{Y}Y}_{xz} \\\\\n\t\t\\bar{\\alpha} P^{\\bar{Y}}_{y} & 0 & \\bar{\\alpha}\\alpha C^{\\bar{Y}Y}_{yy} & 0 \\\\\n\t\t0 & \\bar{\\alpha}\\alpha C^{\\bar{Y}Y}_{zx} & 0 & \\bar{\\alpha}\\alpha C^{\\bar{Y}Y}_{zz}\n\t\\end{pmatrix}.\n\\end{align}\n\n\\noindent Then the angular distribution, expressed in matrix form, becomes\n\n\\begin{equation}\n\tI(\\theta_{\\bar{B}},\\phi_{\\bar{B}},\\theta_B,\\phi_B)=\\frac{1}{16\\pi^2}k_{\\bar{B}}D_{\\mu\\nu}k_B^T.\\label{eq:pdfspincorrmatrix}\n\\end{equation}\n\nFrom the spin correlations, one can calculate the \\textit{singlet fraction}:\n\n\\begin{equation}\n\tF_S = \\frac{1}{4}(1+C^{\\bar{Y}Y}_{xx}-C^{\\bar{Y}Y}_{yy}+C^{\\bar{Y}Y}_{zz}).\n\t\\label{eq:spinfrac}\n\\end{equation}\n\n\\noindent In its original form, derived in Ref.\\cite{durand}, it equals the expectation value of the product of the Pauli matrices \\textbf{$\\sigma_{\\bar{Y}}$}$\\cdot$\\textbf{$\\sigma_{Y}$} which is a number between -3 and 1. In Eq. \\ref{eq:spinfrac}, it has been rewritten to stay between 0 and 1. If $F_S = 0$, all $\\bar{Y}Y$ states are produced in a spin triplet state whereas $F_S = 1$ means they are all in a singlet state. If the spins are completely uncorrelated, the singlet fraction equals 0.25. \n\n\n\n\n\n\\section{Simulations of hyperon production in PANDA}\n\\label{sec:simulations}\n\nIn order to estimate the expected hyperon reconstruction efficiency with PANDA, and to quantify its sensitivity to spin observables, a comprehensive simulation study of two key channels has been performed. We have simulated the reactions\n\n\\begin{itemize}\n    \\item $\\bar{p}p \\rightarrow \\bar{\\Lambda}\\Lambda, \\bar{\\Lambda} \\to \\bar{p}\\pi^+, \\Lambda \\to p \\pi^-$ at $p_{beam} = 1.64$ GeV\/$c$;\n    \\item $\\bar{p}p \\rightarrow \\bar{\\Xi}^+\\Xi^-, \\bar{\\Xi}^+ \\to \\bar{\\Lambda}\\pi^+, \\bar{\\Lambda} \\to \\bar{p}\\pi^+, \\Xi^- \\to \\Lambda \\pi^-, \\\\\\Lambda \\to p \\pi^-$ at $p_{beam} = 4.6$ GeV\/$c$ and $p_{beam} = 7.0$ GeV\/$c$.\n\\end{itemize}\n\n\\noindent The channels have been chosen since the most prominent decay channel in each case leaves only charged particles in the final state. Even though PANDA will be capable of measuring both neutral and charged final states, charged final states are more straight-forward and can be reconstructed with better resolution. Hence, channels with charged final state particles serve as a first benchmark in the overall PANDA hyperon performance check-list. The beam momentum $p_{beam} = 1.64$ GeV\/$c$ for the $\\bar{p}p \\rightarrow \\bar{\\Lambda}\\Lambda$ was chosen since it coincides with a large data set collected by the PS185 experiment \\cite{PS185164}. The PS185 measurement of the cross section, the angular distribution and the spin observables can be compared with new data from one of the first foreseen data taking periods with PANDA. This allows for a systematic comparison between PANDA and a completely independent previous experiment, hence providing important guidance for all future hyperon studies with PANDA.\n\nNeither differential cross sections nor spin observables of the $\\bar{p}p \\rightarrow \\bar{\\Xi}^+\\Xi^-$ reaction have been studied before, and the goal of PANDA is therefore to contribute with completely new insights. For the double-strange $\\Xi^-$, the chosen beam momenta coincide with the hyperon spectroscopy campaign (4.6 GeV\/$c$, see Ref. \\cite{jennynstar}) and the $X(3872)$ line-shape campaign (7 GeV\/$c$, see Ref. \\cite{xscan}). \n\nSince hyperons have relatively long life-time ($10^{-10}$ s), they travel a measurable distance before decaying. This makes the track reconstruction a challenging task \\cite{walter,michael,jenny} since most standard algorithms assume that all tracks originate in the beam-target interaction point. The simulation study presented here is focused on Phase One of PANDA. A realistic PandaROOT implementation of the Phase One conditions was used \\cite{phaseone}. Some simplifications were made due to limitations in the current version of the simulation software:\n\\begin{itemize}\n    \\item The general track reconstruction algorithms that can handle tracks originating far from the interaction point, are still under development and have not yet been deployed as a part of the standard PandaROOT package. Therefore, an ideal pattern recognition algorithm has been used, combined with some additional criteria on the number of hits per track in order to mimic realistic conditions.\n    \n    \\item The particle identification method is not yet stabilised and therefore, ideal PID matching was used. It was shown in Ref. \\cite{erikthesis} that the event selection of non-strange final state particles (\\textit{i.e.} decay products of $\\Lambda$ and $\\Xi^-$) can be performed without PID thanks to the distinct topology of hyperon events. Ideal PID however considerably reduces the run-time due to combinatorics, and was therefore used in the reconstruction. \n    \n\\end{itemize}  \n\\noindent In order to mimic the conditions of real pattern recognition, each track in the target spectrometer was required to contain either 4 hits in the MVD, or in total 6 hits in the MVD + STT + GEM. Tracks in the forward spectrometer are required to contain at least 6 hits in the FTS. \\footnote{The minimum number of hits to fit a circle is three, but additional hits are needed in order to verify that the hits come from a real track and to resolve ambiguities.}\n\\subsection{Signal sample}\nIn total, $10^6$ events were generated for $\\bar{\\Lambda}\\Lambda$ and $\\bar{\\Xi}^+\\Xi^-$ \\cite{walter} using the EvtGen generator \\cite{evtgen}.\nThe $\\bar{\\Lambda}\\Lambda$ sample was weighted using a parameterization of data from PS185, that revealed a strongly forward-peaking $\\bar{\\Lambda}$ distribution in the CMS system of the reaction \\cite{PS185,PS185164}. The $\\bar{\\Xi}^+\\Xi^-$ final state has never been studied and was therefore generated both with an isotropic angular distribution and with a forward-peaking distribution, using a parameterisation from $\\bar{p}p \\to \\bar{\\Sigma}^0\\Lambda$ production in Ref. \\cite{sigma6}. In this way, we can estimate the sensitivity of the reconstruction efficiency to the underlying angular distribution. This is particularly important in a fixed-target, two-spectrometer experiment like PANDA.\n\nIn hyperon-antihyperon pair production in $\\bar{p}p$ annihilations, the $\\theta_Y$ dependence of the spin observables is not straight-forward to parameterize in contrast to the $e^+e^-$ case \\cite{goran}, since more than two production amplitudes can contribute. However, the spin observables must satify some constraints: i) they need to stay within the interval $[-1,1]$ and ii) they need to go to zero at extreme angles, \\textit{i.e.} $\\theta_Y = 0^{\\mathrm{o}}$ and $\\theta_Y = 180^{\\mathrm{o}}$. The latter is because at these angles, the incoming beam is either parallel or anti-parallel to the outgoing antihyperon. Their cross product, giving the direction of the normal of the production plane, is thus not defined.\n\nThe data in this study were weighted according to \n\n\\begin{equation}\nP^Y_y(\\theta_Y) = \\sin2\\theta_Y  \n\\end{equation}\n\nand \n\n\\begin{equation}\nC^{\\bar{Y}Y}_{ij}(\\theta_Y) = \\sin\\theta_Y. \n\\end{equation}\n\n\\noindent since they satisfy the constraints and since this gives a polarisation with a shape that resembles real data \\cite{PS185,PS185164}. \n\n\\subsection{Background samples}\n\n\\subsubsection{Background to $\\bar{p}p \\to \\bar{\\Lambda}\\Lambda$}\n\nGeneric hadronic background is denoted $\\bar{p}p\\to X$, where $X$ refers to \\textit{any} allowed final state. Such processes are simulated using the Dual Parton Model (DPM) generator \\cite{pbarx}, based on a phenomenological model that incorporates Regge theory, topological expansions of QCD, and concepts from the parton model. From this, energy dependencies are obtained of hadron-hadron cross sections with large number of particles with small transverse momenta with respect to the collision axis. Since the strong coupling is large for such processes, perturbation theory is not applicable. Instead, a topological expansion is employed, where the number of colours $N_c$ or flavours $N_f$ is the expansion parameter.\n\nThe total cross section of all $\\bar{p}p\\to X$ processes is around three orders of magnitude larger than that of the $\\bar{p}p\\to\\bar{\\Lambda}\\Lambda$. The expected ratio of produced generic background and signal events can be estimated from simulations using:\n\\begin{equation}\n\t\\frac{N_{X}}{N_{\\mathrm{signal}}} = \\frac{\\sigma(\\bar{p}p \\to X)}{\\sigma (\\bar{p}p \\to \\bar{\\Lambda}\\Lambda)\\mathrm{BR}(\\Lambda \\to p \\pi)^2},\n\\end{equation}\nwhere $\\sigma (\\bar{p}p \\to \\bar{\\Lambda}\\Lambda) = 64.1 \\pm 0.4 \\pm 1.6$ $\\mu$b is the production cross section \\cite{PS185164_1}, $\\mathrm{BR}(\\Lambda \\to p \\pi) = 63.9 \\pm 0.5 \\%$ is the branching ratio \\cite{pdg} and $\\sigma(\\bar{p}p \\to X) = 96 \\pm 3$ mb \\cite{CERNHERA}\n \n \n In order to estimate the expected background contamination, one should ideally produce a realistic amount of background events with respect to the signal. This would however require $3.6\\times 10^3$ DPM events per signal event which in turn implies more than $10^9$ DPM events. Since this would take an unreasonably long time to simulate, a smaller background sample has been generated and then weighted to give the expected signal-to-background ratio.\n\nAmong the numerous channels included in the generic background, the non-resonant  $\\bar{p}p \\to \\bar{p}p\\pi^+\\pi^-$ process is particularly important. This is because it has the same final state particles as the process of interest $i.e.$ $\\bar{p}p \\to \\bar{\\Lambda}\\Lambda\\to \\bar{p}p\\pi^+\\pi^-$ and a cross section that is of the same order of magnitude as the signal process \\cite{CERNHERA,EASTMAN197329,LYS1973610}. Though included in the DPM generator, its cross section has not been tuned to real data. Therefore, this reaction has been simulated separately. The number of simulated events, the cross sections and the weights when calculating signal-to-background ratios, are given in Table \\ref{tab:samplesize1}\n\n\\begin{table}[ht]\n\t\t\\centering\n\t\t\\begin{tabular}{c|c|c|c}\n\t\tChannel & $\\bar{\\Lambda}\\Lambda$ & $\\bar{p}p\\pi^+\\pi^-$ & DPM \\\\ \\hline\n\t\tSample & $9.75\\cdot10^5$ & $9.74\\cdot10^5$ & $9.07\\cdot10^6$\\\\\n\t\tCross section [$\\mu$b] & 64.1 \\cite{PS185164_1} & 15.4 & 96 000\\\\\n\t\tWeight & 1.00 & 0.590 & 395 \\\\\t\n\t\t\\hline\n\t\t\\end{tabular}\n\t\t\\caption{Sample sizes, cross sections and weights for the simulation study at 1.64 GeV\/$c$. The non-resonant cross section has been calculated from the average of Refs. \\cite{CERNHERA,EASTMAN197329,LYS1973610}}\n\t\t\\label{tab:samplesize1}\n\t\\end{table}\n\n\\subsubsection{Background to $\\bar{p}p \\to \\bar{\\Xi}^+\\Xi^-$}\n\nAlso in this case, generic $\\bar{p}p\\to X$ processes are studied with the DPM generator to understand the background. \n\nThe expected production ratio of generic background and signal is given by\n\n\\begin{equation}\n\t\\frac{N_{X}}{N_{\\mathrm{signal}}} = \\frac{\\sigma(\\bar{p}p \\to X)}{\\sigma (\\bar{p}p \\to \\bar{\\Xi}^+\\Xi^-)\\mathrm{BR}(\\Xi\\to\\Lambda\\pi)^2 \\mathrm{BR}(\\Lambda\\to p\\pi)^2}. \\label{xsecratio}\n\\end{equation}\n\nThe cross sections $\\sigma(\\bar{p}p \\to X)$ at the beam momenta 7.3 GeV\/$c$ (the tabulated value closest to 7.0 GeV\/$c$) and 4.6 GeV\/$c$ are $58.3 \\pm 1.3$ mb at $p_{\\mathrm{beam}} = 7.3$ GeV\/c \\cite{dpm7} and $68.8 \\pm 0.8$ mb \\cite{dpm46}, respectively. From Eq. (\\ref{xsecratio}), we see that for each simulated signal event, at least $4.76\\cdot10^5$ DPM events must be simulated to obtain the correct signal-to-background ratio. This would be even more computationally demanding than in the case of $\\bar{p}p \\to \\bar{\\Lambda}\\Lambda$. The weighting method presented in the previous section can be applied, but the weights need to be about two orders of magnitude larger. This means that if very few DPM pass the selection criteria, then the signal-to-background ratio becomes very sensitive to fluctuations. Therefore, the most important background channels are considered separately. These are identified based on their final state particles, vertex topology and invariant masses of particle combinations, and found to be $\\bar{p}p\\to\\bar{\\Sigma}^*(1385)^+ \\Sigma^*(1385)^-$, $\\bar{p}p \\to\\bar{\\Lambda}\\Lambda\\pi^+\\pi^-$ and $\\bar{p}p \\to \\bar{p}p 2 \\pi^+ 2 \\pi^-$. Events from these channels are removed from the DPM sample at the analysis stage, to avoid double-counting of background. Out of the $9.80\\cdot10^7$ simulated DPM events at $p_{\\mathrm{beam}} = 7.0$ GeV\/c, $\\sim 7\\cdot 10^4$ events were removed. For the DPM sample at $p_{\\mathrm{beam}} = 4.6$ GeV\/c, $\\sim10\n^4$ were removed from the $9.8\\cdot10^7$ generated events. The simulated samples, cross sections and weights are summarised in Table \\ref{tab:samplesize2}.\n\n\\begin{table*}[ht]\n\t\t\\centering\n\t\t\\begin{tabular}{c|c|c|c|c|c}\n\t\tChannel at 7 GeV\/$c$ & $\\bar{\\Xi}^+\\Xi^-$ & $\\bar{\\Sigma}^*(1385)^+ \\Sigma^*(1385)^-$ & $\\bar{\\Lambda}\\Lambda \\pi^+\\pi^-$ & $\\bar{p}p 2 \\pi^+ 2 \\pi^-$ & DPM \\\\ \\hline\n\t\tSample & $8.54\\cdot10^5$ & $9.87\\cdot10^6$ & $9.85\\cdot10^6$ & $9.78\\cdot10^6$ & $9.73\\cdot10^7$\\\\\n\t\t$\\sigma_{\\mathrm{eff}}$ [$\\mu$b] & 0.123 & 1.39 & 24.1 & 390 & $5.83\\cdot10^4$\\\\\t\n\t\tWeight factor & 1.00 & 0.98 & 17.1 & 278 & $4.18\\cdot10^3$ \\\\\t\n\t\t\\hline\n\t\tChannel at 4.6 GeV\/$c$ & $\\bar{\\Xi}^+\\Xi^-$ & $\\bar{\\Sigma}^*(1385)^+ \\Sigma^*(1385)^-$ & $\\bar{\\Lambda}\\Lambda \\pi^+\\pi^-$ & $\\bar{p}p 2 \\pi^+ 2 \\pi^-$ & DPM \\\\ \\hline\n\t\tSample & $8.80\\cdot10^5$ & $9.86\\cdot10^6$ & $9.88\\cdot10^6$ & $9.80\\cdot10^6$ & $9.82\\cdot10^7$\\\\\n\t\t$\\sigma_{\\mathrm{eff}}$ [$\\mu$b] & 0.41 & 1.39 & 14.7 & 143 & $6.88\\cdot10^4$\\\\\t\n\t\tWeight factor & 1.00 & 0.304 & 3.21 & 31.4 & $1.51\\cdot10^3$ \\\\ \\hline\n\t\t\\end{tabular}\n\t\t\\caption{Sample sizes, cross sections and weights for the simulation study at $p_{\\mathrm{beam}} = 7$ GeV\/c and $p_{\\mathrm{beam}} = 4.6$ GeV\/c. The $\\bar{\\Sigma}^*(1385)\\Sigma^*(1385)$ cross section is obtained from Ref. \\cite{llbarpippim46}, and the $\\bar{\\Lambda}\\Lambda\\pi^+\\pi^-$ cross sections from Ref. \\cite{llbarpippim7} and \\cite{llbarpippim46} at 7 GeV\/$c$ and 4.6 GeV\/$c$, respectively. The non-resonant $\\bar{p}p 2\\pi^+2\\pi^-$ cross section is obtained from Ref. \\cite{nonres7} at 7 GeV\/$c$ and the average of Refs. \\cite{nonres46} and \\cite{nonres462} at 4.6 GeV\/$c$.}\n\t\t\\label{tab:samplesize2}\n\t\\end{table*}\n\n\\subsection{Event selection} \n\nReactions involving hyperons have a very distinct topology, since the long-lived hyperons decay a measurable distance from the point of production. The topology of each reaction and subsequent decay chain studied in this work, are shown in Fig. \\ref{fig:topology}. This can be exploited in the event selection procedure, as outlined in this chapter.\n\nThe event selection is performed in two stages: a pre-selection and a fine selection. The pre-selection comprises a set of basic topological criteria, that reduces the total simulated sample and hence the analysis run-time. The fine selection involves kinematic fits and fine-tuned mass windows.\n\n\\begin{figure*}[ht]\n\\begin{minipage}{0.4\\textwidth}\n\t\\centering\n\t\\includegraphics[width = \\textwidth]{llbartopology.png}\n\t\\end{minipage}\n\t\\begin{minipage}{0.6\\textwidth}\n\t\\includegraphics[width =\\textwidth]{xixibartopology.png}\n\t\t\\end{minipage}\n\t\\caption{Signal event topology of $\\bar{p}p \\to \\bar{\\Lambda}\\Lambda, \\bar{\\Lambda} \\to \\bar{p}\\pi^+, \\Lambda \\to p\\pi^-$ (left) and $\\bar{p}p \\to \\bar{\\Xi}^+\\Xi^-, \\bar{\\Xi}^+ \\to \\bar{\\Lambda}, \\bar{\\Lambda} \\to \\bar{p}\\pi^+, \\Xi^- \\to \\Lambda\\pi^-, \\Lambda \\to p\\pi^-$ (right).}\n\t\\label{fig:topology}\n\t\\end{figure*}\n\n\\subsubsection{The $\\bar{p}p \\to \\bar{\\Lambda}\\Lambda$ reaction}\n\\label{sec:lambda}\n\n\\noindent The pre-selection criteria for this reaction are:\n\\begin{itemize}\n    \\item Each event must contain at least one each of the following: $p$, $\\bar{p}$, $\\pi^+$ and $\\pi^-$.\n    \\item Each event contains at least one $p\\pi^-$ and one $\\bar{p}\\pi^+$ combination that can be successfully fitted to one common vertex, with a probability of $> 0.01$. If more than one such $\\Lambda$ or $\\bar{\\Lambda}$ candidate exist in one event (occurs in 6\\% of the cases for $\\Lambda$ and 2\\% of the cases for $\\bar{\\Lambda}$), then the one with the smallest $\\chi^2$ is kept for further analysis.\n    \\item Each event must contain at least one $p\\pi^-$ and one $\\bar{p}\\pi^+$ combination with an invariant mass that satisfies $|m_{\\Lambda}-m(p\\pi)| < 0.3$ GeV\/$c^2$. This mass window is very wide and is further tightened in the final selection.\n    \\item The four-vectors of the $\\Lambda$ and the $\\bar{\\Lambda}$ candidate can be fitted successfully to the initial beam momentum, with a \\textit{four-constraints} ($4C$) fit.\n\\end{itemize}\n\n\\noindent The event filtering is further improved by the fine selection. The criteria of the fine selection were tuned and optimised using as a figure of merit the significance, \\textit{i.e.} $S\/\\sqrt{S+B}$, where $S$ refers to the number of signal events and $B$ the number of generic hadronic events generated by DPM. The criteria are the following:\n\n\\begin{itemize}\n    \\item The $\\chi^2$ of the $4C$ fit is required to be $< 100$. \n    \\item The total distance $z_{tot}$ from the interaction point in the beam direction of the $\\Lambda$ and $\\bar{\\Lambda}$ candidate must fulfill $z_{tot} = |z_{\\Lambda} + z_{\\bar{\\Lambda}}| > 2$ cm.\n    \\item The invariant mass of the $p\\pi^-$ and $\\bar{p}\\pi^+$ system must not differ from the PDG $\\Lambda$ mass by more than 5$\\sigma$, where $\\sigma$ is the width of a Gaussian fitted to the invariant mass peak. \n\\end{itemize}\n\\noindent The mass resolution differs between $\\Lambda$ ($\\sigma = 2.864\\cdot10^{-3}$ GeV\/$c^2$) and $\\bar{\\Lambda}$ ($\\sigma = 2.980\\cdot10^{-3}$ GeV\/$c^2$). This is because the decay products from $\\Lambda$ are primarily emitted in the acceptance of the MVD and STT, while the decay products of $\\bar{\\Lambda}$ to a larger extent hit the FTS. The $\\bar{p}\\pi^-$ invariant mass for signal and background are shown in the left panel of Figure \\ref{fig:invmass}.\n\nThe reconstruction efficiency of the signal reaction and the most important background sources for the different selection criteria are given in Table \\ref{tab:effllbar}. In addition, the number of expected background events for a given number of signal events has been calculated taking the cross sections into account. It is clear that background can be very successfully suppressed. A signal-to-background ratio of $S\/B \\approx 106$ is obtained. We conclude that the PANDA detector will be capable of collecting very clean $\\bar{\\Lambda}\\Lambda$ samples, which is essential when extracting spin observables. \n\\begin{table}[ht]\n\t\\centering\n\t\n\t\\begin{tabular}{c|c|c|c}\n\t Channel & $\\bar{\\Lambda}\\Lambda$ & $\\bar{p}p\\pi^+\\pi^-$ & DPM \\\\ \\hline\n\t Generated & $9.75\\cdot10^5$ & $9.74\\cdot10^5$ & $9.07\\cdot10^6$ \\\\\n\tPreselection & $2.129\\cdot10^5$ & 292700 & 651 \\\\\n\t$\\chi^2 < 100$ & $1.879\\cdot10^5$ & $249190$ & $136$ \\\\\n\t$\\Delta m < 5\\sigma$ & $1.685\\cdot10^5$ & $29180$ & $3$ \\\\\n\t$z_{\\bar{\\Lambda}} + z_{\\Lambda} > 2$ cm & $1.572\\cdot10^5$ & 470 & 2 \\\\ \n\tEff. (\\%) & $16.0\\pm0.4$ & $0.05$ & $2.2\\cdot10^{-7}$\\\\\n\t\\hline\n\t\t$N_{exp}$ & $1.572\\cdot10^5$ & 277 & 790 \\\\\n\t\t\\hline\n\t\t\\end{tabular}\n\t\\caption{Reconstruction efficiency after the final selection for signal events as well as non-resonant and generic hadronic background. In the bottom row, the proportion of expected events are shown. These numbers were calculated by applying the weights in Table \\ref{tab:samplesize1}. \\label{tab:effllbar}}\n\\end{table}\n\n\\subsubsection{The $\\bar{p}p \\to \\bar{\\Xi}^+\\Xi^-$ reaction}\n\\label{sec:xi}\n\nThe $\\bar{p}p \\to \\bar{\\Xi}^+\\Xi^-$ reaction is more complicated than the $\\bar{p}p \\to \\bar{\\Lambda}\\Lambda$ reaction since i) there are more particles in the final state ii) there are several identical particles in the final state and iii) each event contains four displaced decay vertices instead of two. In addition, the cross section is smaller and at the larger beam momenta necessary for $\\Xi$ studies, the cross section of background channels are larger. Hence, the selection procedure is by necessity a bit more involved. In the following, we summarise the pre-selection criteria. For simplicity, the charge conjugated mode is implied unless otherwise stated.\n\n\\noindent\\textbf{Final State Reconstruction and Combinatorics}\n\n\\noindent The first step is to combine the final state particles into $\\Lambda$ and $\\Xi$ candidates:\n\\begin{itemize}\n\t\\item All possible $p \\pi^-$ combinations are combined to form $\\Lambda$ candidates.\n\t\\item All combinations fulfilling $|m_{\\Lambda} - M(p\\pi^-) | < 0.05$ GeV\/c$^2$ are accepted and stored for further analysis.\n\t\\item All possible $\\Lambda \\pi^-$ combinations are combined to form $\\Xi^-$ candidates.\n\t\\item All combinations fulfilling $|m_{\\Xi} - M(p\\pi^-\\pi^-) | < 0.05$ MeV\/c$^2$ are accepted and stored for further analysis. This mass window is very wide and is tightened in the final selection.\n\\end{itemize}\n\\textbf{Fit of the $\\mathbf{\\Xi^-\\to \\Lambda \\pi^-, \\Lambda \\to p \\pi^-}$ decay chain}\n\n\\noindent The second step is to exploit the distinct topology of the $\\bar{p}p \\to \\bar{\\Xi}^+\\Xi^-, \\bar{\\Xi}^+ \\to \\bar{\\Lambda}\\pi^+, \\bar{\\Lambda} \\to \\bar{p}\\pi^+, \\Xi^-\\to \\Lambda \\pi^-, \\Lambda \\to p \\pi^-$ process, that imposes many constraints. Therefore, all $\\Xi^-$ candidates from the previous step are fitted under the $\\Xi^-\\to \\Lambda \\pi^-, \\Lambda \\to p \\pi^-$ hypothesis where the $\\Lambda$ mass is constrained to its PDG value. This is achieved using the Decay Chain Fitting package \\cite{HULSBERGEN2005566}, designed to perform kinematic fitting on a sequence of decays with at least one over-constraint. Taking all constraints and unknown parameters in the fit into account, results in three effective degrees of freedom. The advantage of this approach compared to multiple sequential fits, is that all constraints in a reaction are taken into account simultaneously, on an equal basis. This feature is not available in conventional fitters. In our case, the procedure is the following:\n\n\\begin{itemize}\n\t\\item The decay chain $\\Xi^- \\to \\Lambda \\pi^-, \\Lambda \\to p \\pi^-$ is fitted. The constraints are provided by momentum conservation, the two vertex positions and the $\\Lambda$ mass. All momentum components of all particles are modified in the fit.\n\t\\item Candidates with a fit probability $< 0.01$ are rejected.\n\n\\end{itemize}\n\\textbf{Reconstructing the $\\bar{p}p$ system} \\\\\n\n\t\\begin{figure*}[ht]\n\t\t   \n\t\t   \n\t\t       \n\t\t        \\includegraphics[width=0.33\\linewidth]{figllbarBW\/h0m.pdf}\n\t\t        \\includegraphics[width=0.33\\linewidth]{figxixibar46BW\/h0m.pdf}\n\t\t        \\includegraphics[width=0.33\\linewidth]{figxixibar7BW\/h0m.pdf}\n\t\t   \n\t\t   \n\t\t       \n\t\t        \n\t\t   \n\t\t\t\\caption{Invariant mass distributions of signal and background samples in the final selection state. Left: The $\\bar{p}\\pi^+$ invariant mass at $p_{\\mathrm{beam}}$ = 1.64 GeV\/c for the $\\bar{p}p \\to \\bar{\\Lambda}\\Lambda$ reaction (black), non-resonant $\\bar{p}p \\to \\bar{p}p\\pi^+\\pi^-$ (dotted) and DPM (grey). Middle: The $\\bar{p}\\pi^+\\pi^+$ invariant mass at $p_{\\mathrm{beam}}$ = 4.6 GeV\/c for the $\\bar{p}p \\to \\bar{\\Xi}^+\\Xi^-$ reaction (black dashed), the $\\bar{p}p \\to \\bar{\\Sigma(1385)}^+\\Sigma(1385)^-$ (black dotted), $\\bar{p}p \\to \\bar{\\Lambda}\\Lambda\\pi^+\\pi^-$ (grey dotted), $\\bar{p}p \\to \\bar{p}p2\\pi^+2\\pi^-$  (grey solid) and combinatorial (black solid). Right: Same as in the middle panel but at $p_{\\mathrm{beam}}$ = 7.0 GeV\/c. The vertical lines mark the final selection mass window. All distributions are normalised to previously measured cross sections.}\n\t\t\t\\label{fig:invmass}\n\t\t\\end{figure*}\n\n\\noindent The decay chain fitter results in a list of $\\Xi^-$ and $\\bar{\\Xi}^+$ candidates in each event. The next step is to combine these candidates and test the hypothesis that they come from a common production vertex, and fulfill the kinematics of the initial $\\bar{p}p$ system.\n\\begin{itemize}\n\t\\item All possible $\\bar{\\Xi}^+\\Xi^-$ combinations form a hypothetical $\\bar{p}p$ system.\n\t\\item A vertex fit of $\\bar{\\Xi}^+\\Xi^-$ pairs is performed to reconstruct the interaction point.\n\t\\item Candidates with a fit probability $ < 0.01$ are rejected.\n\t\\item Candidates where the opening angle of the $\\bar{\\Xi}^+\\Xi^-$ pair is $> 3$ rad in the CMS system are selected for further analysis.This is because in the two-body reaction of interest, the $\\bar{\\Xi}^+$ and the $\\Xi^-$ are emitted back to back.\n\t\\item Events where $\\Lambda$ and $\\Xi^-$ candidates satisfy $\\Delta z = z(\\Lambda) - z(\\Xi) > 0$ cm are selected, where $z(Y)$ is the $z$-position of the hyperon decay vertex.\n\t\\item A kinematic fit of $\\bar{\\Xi}^+\\Xi^-$ pairs is performed, where energy and momentum are constrained to the initial system.\n\t\\item In case there is more than one $\\bar{\\Xi}^+\\Xi^-$ combination in an event, fulfilling all previous criteria, the candidate with the smallest $\\chi^2$ value from the kinematic fit is chosen for further analysis.\n\\end{itemize}\n\n\\noindent In the fine selection, additional criteria are applied after careful studies of the significance $S\/\\sqrt{S+B}$:\n\\begin{itemize}\n\t\\item Combinations of $\\bar{p}\\pi^+\\pi^+$ must fulfill \\\\ $|m_{fit}(\\bar{p}\\pi^+\\pi^+) - m_{PDG}(\\Xi^-)| < 5\\cdot0.003 $ GeV\/c$^2$, where 0.003 GeV\/c$^2$ is the $\\sigma$ width of the broader Gaussian component of the curve fitted to the data in the peak region.\n\t\\item Combinations of $p\\pi^-\\pi^-$ must fulfill \\\\$| m_{fit}(p\\pi^-\\pi^-) - m_{PDG}(\\Xi^-)| < 5\\cdot0.003 $ GeV\/c$^2$, where 0.003 GeV\/c$^2$ is the $\\sigma$ width of the broader Gaussian component of the curve fitted to the data in the peak region.\n\t\\item The total distance in the beam direction from the reconstructed interaction point (IP) in an event must satisfy $(z_{fit}(\\bar{\\Xi}^+) - z_{fit}(IP)) + (z_{fit}(\\Xi^-) - z_{fit}(IP)) > 3$ cm.\n\\end{itemize}\n\n\\noindent Invariant mass plots of the $\\bar{p}\\pi^+\\pi^-$ system for signal and various background channels are shown in the middle (at $p_{\\mathrm{beam}}$ = 4.6 GeV\/c) and right ($p_{\\mathrm{beam}}$ = 7.0 GeV\/c) panel of Figure \\ref{fig:invmass}. The resulting reconstruction efficiency for each criterion, or set of criteria, are shown in Table \\ref{tab:effxixibar}. The proportion of expected events, calculated from the cross sections, are also given. No non-resonant nor any generic background events satisfy the selection criteria. Therefore, the Poisson upper limit of 2.3 events has been used to estimate the number of background events at a confidence level of 90\\%.  \n\n\n\\begin{table*}[ht]\n\t\t\\centering\n\t\t\\resizebox{1.\\textwidth}{!}{%\n\t\t\\begin{tabular}{c|c|c|c|c|c}\n\t\t\t$p_{beam}$ = 7.0 GeV\/$c$ & $\\bar{\\Xi}^+\\Xi^-$ & $\\bar{\\Sigma}(1385)^+ \\Sigma(1385)^-$ & $\\bar{\\Lambda}\\Lambda\\pi^+\\pi^-$ & $\\bar{p}p 2 \\pi^+ 2 \\pi^-$ & DPM \\\\ \\hline\nGenerated & $8.54\\cdot10^5$ & $9.87\\cdot10^6$ & $9.85\\cdot10^6$ & $9.78\\cdot10^6$ & $9.73\\cdot10^7$\\\\\n\t\t\tPre-selection & $7.83\\cdot10^4$ & $3.45\\cdot10^4$ & $3.51\\cdot10^3$ & $1$ & $100$  \\\\\n\t\t\tMass cut & $7.27\\cdot10^4$ & $23$ & $379$ & $<2.3$ & $7.0$  \\\\\n\t\t\t$\\Delta d>3$ & $6.76\\cdot10^4$ & $3.0$ & $14$ & $<2.3$ & $<2.3$  \\\\\n\t\t\tEfficiency \\% & $7.95\\pm0.03$ & $(3.0\\pm0.2)\\cdot10^{-5}$ & $(1.4\\pm0.4)\\cdot10^{-4}$ & $<2.3\\cdot 10^{-5}$ & $<2.3\\cdot 10^{-6}$\\\\ \\hline\n\t\t\t$N_{exp}$ weighted & $6.76\\cdot10^4$ & $2.9$ & $239$ & $<640$ & $<9.61\\cdot10^3$ \\\\\n\t\t\t\\hline \n\t\t\t$p_{beam}$ = 4.6 GeV\/$c$ &  &  &  &  &  \\\\ \\hline\n\t\t\tGenerated & $8.80\\cdot10^5$ & $9.86\\cdot10^6$ & $9.88\\cdot10^6$ & $9.80\\cdot10^6$ & $9.82\\cdot10^7$\\\\\n\t\t\tPre-selection & $8.65\\cdot10^4$ & $3.29\\cdot10^4$ & $2.61\\cdot10^4$ & $105$ & $44$ \\\\ \n\t\t\tMass cut & $8.06\\cdot10^4$ & $21$ & $2.49\\cdot10^3$ & $13$ & $6.0$ \\\\ \n\t\t\t$\\Delta d >3$ & $7.23\\cdot10^4$ & $1.0$ & $39$ & $<2.3$ & $<2.3$  \\\\\n\t\t\tEfficiency (\\%) & $8.22\\pm0.03$ & $(1.0\\pm1.0)\\cdot10^{-5}$ & $(4.0\\pm0.6)\\cdot10^{-4}$ & $<2.3\\cdot10^{-5}$ & $<2.3\\cdot10^{-6}$\\\\ \\hline\n\t\t\t$N_{exp}$ weighted & $7.23\\cdot10^4$ & $0.30$ & $125$ & $<72$ & $<3.47\\cdot10^3$\t\\\\\n\t\t\t\\hline\n\t\t\n\t\t\n\t\t\\end{tabular}}%\n\t\t\\caption{Reconstruction efficiency after the final selection for signal events as well as non-resonant and generic hadronic background. $N_{exp}$ is the expected proportion of events, applying weights in Table \\ref{tab:samplesize2}. The Poisson upper limits are given at a 90\\% confidence level. \\label{tab:effxixibar}}\n\t\t\n\t\\end{table*}\n\n\\section{Parameter estimation}\n\\label{sec:paramest}\n\nTo estimate the physics parameters $\\alpha$, $\\bar{\\alpha}$, $P^Y_y$, $P^{\\bar{Y}}_y$, $C^{Y\\bar{Y}}_{xz}$, $C^{Y\\bar{Y}}_{xx}$, $C^{Y\\bar{Y}}_{yy}$ and $C^{Y\\bar{Y}}_{zz}$ from the measured quantities, \\textit{i.e.} the hyperon scattering angle and the baryon and antibaryon decay angles, methods like Maximum Log Likelihood or the Method of Moments can be used. In the very first phase of data taking with PANDA, the samples will be relatively modest and the measurements will be focused on the production related parameters, \\textit{i.e.} the polarisation and the spin correlations. These can be obtained for any given beam momentum and scattering angle by fixing $\\alpha$ and $\\bar{\\alpha}$ to the already measured value of $\\alpha$ \\cite{pdg}, assuming CP symmetry \\textit{i.e.} $\\alpha$ = $-\\bar{\\alpha}$. \n\nIn this study, the Method of Moments has been chosen as parameter estimation method, due to its computational simplicity.\nAt a given antihyperon scattering angle $\\theta_{\\bar{Y}}$, it can be shown \\cite{erikthesis} that the first moment of $\\cos\\theta_y^B$ is proportional to the polarisation at this angle:\n\\begin{align}\n    <\\cos\\theta_y^B>_{\\theta_{\\bar{Y}}}=&\\frac{\\int I(\\theta_y^B,\\theta_y^{\\bar{B}})_{\\theta_{\\bar{Y}}}\\cos\\theta_y^B d\\Omega_B d\\Omega_{\\bar{B}}}{\\int I(\\theta_y^B,\\theta_y^{\\bar{B}})_{\\theta_{\\bar{Y}}} d\\Omega_B d\\Omega_{\\bar{B}}}\\\\\n    & = \\frac{\\alpha P^Y_{y,\\theta_{\\bar{Y}}}}{3}\n\\end{align}\n\\noindent Hence, the polarisation can be calculated from the moment\n\\begin{equation}\n    P_{y,\\theta_{\\bar{Y}}}^{Y\/\\bar{Y}}=\\frac{3<\\cos\\theta_y^{B\/\\bar{B}}>_{\\theta_{\\bar{Y}}}}{\\alpha}\n    \\label{eq:pol}\n\\end{equation}\n\\noindent where the estimator of the moment is the arithmetic mean of $\\cos\\theta_y^{B\/\\bar{B}}$ obtained from a sample of $N$ events:\n\n\\begin{equation}\n\t<\\reallywidehat{\\cos\\theta_y^{B\/\\bar{B}}}>_{\\theta_{\\bar{Y}}} = \\frac{1}{N} \\sum_{i=1}^N \\cos\\theta_{y,i}^{B\/\\bar{B}} \\Bigg \\rfloor _{\\theta_{\\bar{Y}}}.\n\t\\label{eq:polest}\n\\end{equation}\n\n\\noindent In the following, we always refer to moments and spin observables at a given $\\theta_Y$, unless explicitly stated otherwise. That means $P^{\\bar{Y}}_{y,\\theta_Y} = P^{\\bar{Y}}_y$ and so on.\n\nThe variance of the first moment is given by the difference between the second moment and the square of the first moment. In our case, we have\n\n\\begin{equation}\n    V(<\\cos_y^B>) = \\frac{1}{N(N-1)}[<\\cos^2\\theta_y^B>-<\\cos\\theta_y^B>^2]\n\\end{equation}\n\n\\noindent that after error propagation and some algebra becomes \n\n\\begin{equation}\n    V(P_y) = \\frac{3-(\\alpha P_{y})^2}{\\alpha(N-1)}.\n    \\label{eq:polvar}\n\\end{equation}\n\nIn a similar way, the spin correlations at a given hyperon scattering angle $\\theta_Y$ can be obtained from the moments of the product of the cosines with respect to the different reference axes $i,j = x,y,z$  \\cite{erikthesis}:\n\n\\begin{equation}\n    C^{\\bar{Y}Y}_{i,j} = \\frac{9<\\cos\\theta_{i}^B\\cos\\theta_{j}^{\\bar{B}}>}{\\alpha\\bar{\\alpha}}\n    \\label{eq:spincorr}\n\\end{equation}\n\n\\noindent where the estimator of the moment is given by the arithmetic mean of the cosine product from the data sample at a given scattering angle:\n\n\\begin{equation}\n\t<\\reallywidehat{\\cos\\theta_{i}^B\\cos\\theta_{j}^{\\bar{B}}}> = \\frac{1}{N} \\sum_{k=1}^N \\cos\\theta_{i,k}^B \\cos\\theta_{j,k}^{\\bar{B}}.\n\t\\label{eq:spinest}\n\\end{equation}\n\n\\noindent The variance of the spin correlations can be calculated in the same way as that of the polarisation and is found to be\n\n\\begin{equation}\n    V(C^{\\bar{Y}Y}_{i,j}) = \\frac{9-(\\alpha\\bar{\\alpha}C^{\\bar{Y}Y}_{i,j})^2}{\\alpha\\bar{\\alpha}(N-1)},\n    \\label{eq:spinvar}\n\\end{equation}\n\n\\noindent for $i,j=x,y,z$.\n\n\\subsection{Efficiency corrections}\nIn reality, detectors and reconstruction algorithms have finite efficiencies. This needs to be taken into account in the parameter estimation. However, the efficiency is a complicated function of all measured variables. In the case of exclusive $\\bar{p}p \\to \\bar{\\Lambda}\\Lambda, \\bar{\\Lambda} \\to \\bar{p}\\pi^+, \\Lambda \\to p\\pi^-$ measurements, there are five independent measured variables: the $\\bar{\\Lambda}$ scattering angle, the proton decay angles $\\theta_p$ and $\\phi_p$ and the antiproton decay angles $\\theta_{\\bar{p}}$ and $\\phi_{\\bar{p}}$. In principle, this means that parameter estimation methods, which rely on integration, such as the Method of Moments, should employ efficiency corrections in all five independent variables. In the case of $\\bar{p}p \\to \\bar{\\Xi}^+\\Xi^-, \\bar{\\Xi}^+ \\to \\bar{\\Lambda}\\pi^+, \\bar{\\Lambda} \\to \\bar{p}\\pi^+, \\Xi^- \\to \\Lambda\\pi^-, \\Lambda \\to p\\pi^-$, the efficiency depends on nine independent variables. This is however difficult to achieve in practice, since the number of Monte Carlo simulated events required for a five or nine-dimensional correction matrix is very large and thus unfeasible.\n\nInstead, different approximations have to be made, based on reasonable and testable assumptions. In this work, we have treated the efficiency with two independent methods, the \\textit{efficiency dependent} and \\textit{efficiency independent method}, as outlined in the following.\n\n\\subsubsection{Efficiency dependent method}\n\nWith this method, the efficiency is corrected for on an event-by-event basis. The efficiency corrected estimator of the moment $<\\cos\\theta_y^B>$ is given by\n\n\\begin{equation}\n\t<\\reallywidehat{\\cos\\theta_y^B}> = \\frac{1}{N} \\sum_{i=1}^N \\cos\\theta_{y,i}^B w_i(\\theta_y,\\Omega_B,\\Omega_{\\bar{B}})\n\t\\label{eq:polesteff}\n\\end{equation}\n\nwhere \n\\begin{equation}\n   w_i(\\theta_Y,\\Omega_B,\\Omega_{\\bar{B}}) = \\frac{1}{\\epsilon_i(\\theta_y,\\Omega_B,\\Omega_{\\bar{B}})} \n   \\label{eq:weight}\n\\end{equation}.\n\nIn the polarisation extraction, we assume for computational simplicity that the efficiency of the $\\Lambda$ as a function of the $\\Lambda$ angles is independent of the $\\bar{\\Lambda}$ angles, and vice versa. Then we can reduce $\\epsilon(\\theta_Y,\\Omega_B,\\Omega_{\\bar{B}})$ to $\\epsilon(\\theta_Y,\\Omega_B)$. Furthermore, our simulations show that the efficiency is symmetric with respect to the azimuthal angle $\\phi_y$, which means that we can integrate over $\\phi_y$ without introducing a bias. This means that our efficiency is simplified to $\\epsilon(\\theta_Y,\\cos\\theta^B_{y})$. Hence, we can represent the efficiency by two-dimensional matrices: the $\\bar{Y}$ scattering angle $\\theta_{\\bar{Y}}$ in the CMS system of the reaction \\textit{versus} the decay proton angle $\\theta^B_{y}$ with respect the $y$ axis in Fig. \\ref{fig:refsys}, in the rest frame of the decaying hyperon. \n\nFor the spin correlation $C^{\\bar{Y}Y}_{i,j}$, we need to take into account the decay angles from the hyperon and the antihyperon. We then assume a 3D efficiency \\\\ $\\epsilon(\\theta_Y,\\cos\\theta^B_{i}, \\cos\\theta^{\\bar{B}}_{j})$ and hence, we use 3D matrices. Here, $i,j = x,y,z$ in Fig. \\ref{fig:refsys}. These three-dimensional correction matrices were also used in a cross check analysis of the polarisation estimation, with consistent results.\n\nIn the $\\bar{p}p \\to \\bar{\\Xi}^+\\Xi^-$ case, we have assumed that the efficiency is symmetric with respect to the $\\Lambda \\to p\\pi^-$ and $\\bar{\\Lambda} \\to \\bar{p}\\pi^+$ decay angles which were integrated out. \n\nThe estimator for the polarisation is given by\n\n\\begin{equation}\n\t\\reallywidehat{P_y}^{Y\/\\bar{Y}} = \\frac{3}{\\alpha}\\frac{\\sum^N_{i=1}\\cos\\theta_{y,i}^{B\/\\bar{B}}\\cdot{w_i(\\cos\\theta_{y,i}^{B\/\\bar{B}},\\cos\\theta_{\\bar{Y}})}}{\\sum^N_{i=1}w_i(\\cos\\theta_{y,i}^{B\/\\bar{B}},\\cos\\theta_{\\bar{Y}})}.\n\t\\label{eq:recpol}\n\\end{equation}\n\n\\noindent where $w(\\cos\\theta_{y,i},\\cos\\theta_{\\bar{Y}})$ is the weight (Eq. (\\ref{eq:weight})) at the given $\\cos\\theta_y$ and $\\cos\\theta_{\\bar{Y}}$. $N$ is the number of events in the sample. For the spin correlations, the estimators are given by \n\n\\begin{equation}\n\t\\reallywidehat{C^{\\bar{Y}Y}_{\\mu\\nu}} = \\frac{9}{\\alpha\\bar{\\alpha}}\\frac{\\sum^N_{i=1} \\cos\\theta_{\\mu,i}^{\\bar{B}}\\cos\\theta_{\\nu,i}^B\\cdot w_i(\\cos\\theta_{\\mu,i}^{\\bar{B}},\\cos\\theta_{\\nu,i}^B,\\cos\\theta_{Y})}{\\sum^N_{i=1} w_i(\\cos\\theta_{\\mu,i}^{\\bar{B}},\\cos\\theta_{\\nu,i}^B,\\cos\\theta_{Y})}.\n\t\\label{eq:recspincorr}\n\\end{equation}\n\n\\subsubsection{Efficiency independent method}\n\\label{sec:effind}\n\nFor special cases, alternative estimators can be defined which do not require efficiency corrections. These have been treated thoroughly in Refs. \\cite{erikthesis,tayloethesis} and will be briefly summarised here. \n\nHere, it is most convenient to use the matrix formulation, see Sect. \\ref{sec:formalism}. The first order moments of the angles and their products can be gathered in a $4\\times4$ matrix as follows:\n\n\\begin{equation*}\n\tE=\\begin{pmatrix}\n\t\\braket{1} & \\braket{\\cos\\theta_{x}^{B}} & \\braket{\\cos\\theta_{y}^{B}} & \\braket{\\cos\\theta_{z}^{B}} \\\\\n\t\\braket{\\cos\\theta_{x}^{\\bar{B}}} & \\braket{\\cos\\theta_{x}^{\\bar{B}}\\cos\\theta_{x}^{B}} & \\braket{\\cos\\theta_{x}^{\\bar{B}}\\cos\\theta_{y}^{B}} & \\braket{\\cos\\theta_{x}^{\\bar{B}}\\cos\\theta_{z}^{B}} \\\\\n\t\\braket{\\cos\\theta_{y}^{\\bar{B}}} & \\braket{\\cos\\theta_{y}^{\\bar{B}}\\cos\\theta_{x}^{B}} & \\braket{\\cos\\theta_{y}^{\\bar{B}}\\cos\\theta_{y}^{B}} & \\braket{\\cos\\theta_{y}^{\\bar{B}}\\cos\\theta_{z}^{B}} \\\\\n\t\\braket{\\cos\\theta_{z}^{\\bar{B}}} & \\braket{\\cos\\theta_{z}^{\\bar{B}}\\cos\\theta_{x}^{B}} & \\braket{\\cos\\theta_{z}^{\\bar{B}}\\cos\\theta_{y}^{B}} & \\braket{\\cos\\theta_{z}^{\\bar{B}}\\cos\\theta_{z}^{B}} \n\t\\end{pmatrix}\n\\end{equation*}\n\nand some additional moments in vector form\n\n\\begin{align}\n\t& F = \\begin{pmatrix}\n\t\t\\braket{1} & \\braket{\\cos^2\\theta_{x}^{B}} & \\braket{\\cos^2\\theta_{y}^{B}} & \\braket{\\cos^2\\theta_{z}^{B}}\n\t\\end{pmatrix}\\\\\n\t& \\bar{F} = \\begin{pmatrix}\n\t\t\\braket{1} & \\braket{\\cos^2\\theta_{x}^{\\bar{B}}} & \\braket{\\cos^2\\theta_{y}^{\\bar{B}}} & \\braket{\\cos^2\\theta_{z}^{\\bar{B}}}\n\t\\end{pmatrix}.\n\\end{align}\n\n\\noindent We assume that the efficiency of the antihyperon and its decay is independent of that of the hyperon, \\textit{i.e.} \n\n\\begin{equation}\n    \\epsilon(\\Omega_{\\bar{B}},\\Omega_B) = \\epsilon(\\Omega_{\\bar{B}})\\cdot\\epsilon(\\Omega_B).\n\\end{equation}\nWe then define the following matrices of efficiency weighted moments\n\n\\begin{align}\n\t& \\bar{\\mathcal{A}}_{\\mu,\\nu} \\equiv \\int \\cos\\theta_{\\mu}^{\\bar{B}} \\cos\\theta_{\\nu}^{\\bar{B}} \\epsilon(\\Omega_{\\bar{B}}) d\\Omega_{\\bar{B}} \\\\\n\t& \\mathcal{A}_{\\mu,\\nu} \\equiv \\int \\cos\\theta_{\\mu}^{B} \\cos\\theta_{\\nu}^{B} \\epsilon(\\Omega_{B}) d\\Omega_{B} \\\\\n\t& \\bar{\\mathcal{B}}_{\\mu,\\nu} \\equiv \\int \\cos^2\\theta_{\\mu}^{\\bar{B}} \\cos\\theta_{\\nu}^{\\bar{B}} \\epsilon(\\Omega_{\\bar{B}}) d\\Omega_{\\bar{B}} \\\\\n\t& \\mathcal{B}_{\\mu,\\nu} \\equiv \\int \\cos\\theta_{\\mu}^{B} \\cos^2\\theta_{\\nu}^{B} \\epsilon(\\Omega_{B}) d\\Omega_{B} \\\\\n\t& \\bar{\\mathcal{C}}_{\\mu} \\equiv \\bar{\\mathcal{A}}_{\\mu,0} \\\\\n\t& \\mathcal{C}_{\\nu} \\equiv \\mathcal{A}_{0,\\nu}.\n\\end{align}\n\n\\noindent By definition, the $\\mathcal{A}$ matrices are symmetric in $\\mu$ and $\\nu$. Furthermore, some elements of $\\mathcal{B}$ are identical to those of $\\mathcal{A}$ $e.g.$ $\\mathcal{B}_{10}=\\mathcal{A}_{11}$, $\\mathcal{\\bar{B}}_{01}=\\mathcal{\\bar{A}}_{11}$. With these definitions, the moments can be related to the spin observables in the following way:\n\\begin{equation}\n\tE = \\frac{1}{16\\pi^2} \\bar{\\mathcal{A}} D \\mathcal{A}\n\t\\label{eq:Esolve}\n\\end{equation}\n\\begin{equation}\n\t\\bar{F} = \\frac{1}{16\\pi^2} \\bar{\\mathcal{B}} D \\mathcal{C}\n\t\\label{eq:Fbarsolve}\n\\end{equation}\n\\begin{equation}\n\tF = \\frac{1}{16\\pi^2} \\bar{\\mathcal{C}} D \\mathcal{B}\n\t\\label{eq:Fsolve}\n\\end{equation}\n\nIf the efficiency is symmetric with respect to $\\cos\\theta_y$ for both the antibaryon and the baryon, \\textit{i.e.}\n\\begin{align}\n\t&\\epsilon(\\cos\\theta_{x}^{\\bar{B}},\\cos\\theta_{y}^{\\bar{B}},\\cos\\theta_{z}^{\\bar{B}})=\\epsilon(\\cos\\theta_{x}^{\\bar{B}},-\\cos\\theta_{y}^{\\bar{B}},\\cos\\theta_{z}^{\\bar{B}})\\\\\n\t&\\epsilon(\\cos\\theta_{x}^{B},\\cos\\theta_{y}^{B},\\cos\\theta_{z}^{B})=\\epsilon(\\cos\\theta_{x}^{B},-\\cos\\theta_{y}^{B},\\cos\\theta_{z}^{B}).\n\\end{align}\n\\noindent then all matrix elements in $\\mathcal{A}$ and $\\mathcal{B}$ with odd powers of $\\cos\\theta_y$ are zero. The matrices then reduce to\n\\begin{equation}\n\t\\mathcal{A} = \\begin{pmatrix}\n\t\t\\mathcal{A}_{00} & \\mathcal{A}_{01} & 0 & \\mathcal{A}_{03} \\\\\n\t\t\\mathcal{A}_{01} & \\mathcal{A}_{11} & 0 & \\mathcal{A}_{13} \\\\\n\t\t0 & 0 & \\mathcal{A}_{22} & 0 \\\\\n\t\t\\mathcal{A}_{03} & \\mathcal{A}_{13} & 0 & \\mathcal{A}_{33} \\\\\n\t\\end{pmatrix}\n\\end{equation}\n\\begin{equation}\n\t\\mathcal{\\bar{B}} = \\begin{pmatrix}\n\t\t\\mathcal{\\bar{B}}_{00} & \\mathcal{\\bar{B}}_{01} & 0 & \\mathcal{\\bar{B}}_{03} \\\\\n\t\t\\mathcal{\\bar{B}}_{10} & \\mathcal{\\bar{B}}_{11} & 0 & \\mathcal{\\bar{B}}_{13} \\\\\n\t\t\\mathcal{\\bar{B}}_{20} & \\mathcal{\\bar{B}}_{21} & 0 & \\mathcal{\\bar{B}}_{23} \\\\\n\t\t\\mathcal{\\bar{B}}_{30} & \\mathcal{\\bar{B}}_{31} & 0 & \\mathcal{\\bar{B}}_{33} \\\\\n\t\\end{pmatrix},\\;\\;\\; B = \\begin{pmatrix}\n\t\t\\mathcal{B}_{00} & \\mathcal{B}_{01} & \\mathcal{B}_{02} & \\mathcal{B}_{03} \\\\\n\t\t\\mathcal{B}_{10} & \\mathcal{B}_{11} & \\mathcal{B}_{12} & \\mathcal{B}_{13} \\\\\n\t\t0 & 0 & 0 & 0 \\\\\n\t\t\\mathcal{B}_{30} & \\mathcal{B}_{31} & \\mathcal{B}_{32} & \\mathcal{B}_{33} \\\\\n\t\\end{pmatrix}\n\\end{equation}\n\nWith these simplifications, the right hand side of Eqs. \\ref{eq:Esolve}, \\ref{eq:Fbarsolve} and \\ref{eq:Fsolve} can be solved, resulting in terms that consist of products of $\\mathcal{A}_{\\mu,\\nu}$, $\\mathcal{B}_{\\mu,\\nu}$ and $D_{\\mu,\\nu}$. We find that some of these terms are small in magnitude. If these terms can be neglected, then the non-zero spin observables are shown in Ref. \\cite{tayloethesis} to be\n\n\\begin{align}\n\t&D_{20} = \\frac{E_{20}}{\\bar{F}_2},\\;\\;\\; D_{02} = \\frac{E_{02}}{F_2} \\\\\n\t&D_{22} = \\frac{E_{22}}{\\bar{F}_2 F_2}\\\\\n\t&D_{11} = \\frac{E_{11}-E_{10}E_{01}}{\\bar{F}_1 F_1},\\;\\;\\;D_{13} = \\frac{E_{13}-E_{10}E_{03}}{\\bar{F}_1 F_3}\\\\\n\t&D_{31} = \\frac{E_{31}-E_{30}E_{01}}{\\bar{F}_3 F_1},\\;\\;\\;D_{33} = \\frac{E_{33}-E_{30}E_{03}}{\\bar{F}_3 F_3},\n\\end{align}\nwhich translates to\n\\begin{align}\n\t& P^{\\bar{Y}}_{y} = \\frac{1}{\\bar{\\alpha}}\\frac{\\braket{\\cos\\theta_{y,\\bar{B}}}}{\\braket{\\cos^2\\theta_{y,\\bar{B}}}}\\label{eq:pybarfinal} \\\\\n\t& P^Y_{y} = \\frac{1}{\\alpha}\\frac{\\braket{\\cos\\theta_{y,B}}}{\\braket{\\cos^2\\theta_{y,B}}}\\label{eq:pyfinal} \\\\\n\t& C^{\\bar{Y}Y}_{yy} = \\frac{1}{\\bar{\\alpha}\\alpha} \\frac{\\braket{\\cos\\theta_{y,\\bar{B}}\\cos\\theta_{y,B}}}{\\braket{\\cos^2\\theta_{y,\\bar{B}}}\\braket{\\cos^2\\theta_{y,B}}}\\label{eq:cyyfinal} \\\\\n\t& C^{\\bar{Y}Y}_{xx} = \\frac{1}{\\bar{\\alpha}\\alpha} \\frac{\\braket{\\cos\\theta_{x,\\bar{B}}\\cos\\theta_{x,B}} - \\braket{\\cos\\theta_{x,\\bar{B}}} \\braket{\\cos\\theta_{x,B}} }{\\braket{\\cos^2\\theta_{x,\\bar{B}}}\\braket{\\cos^2\\theta_{x,B}}}\\label{eq:cxxfinal} \\\\\n\t& C^{\\bar{Y}Y}_{xz} = \\frac{1}{\\bar{\\alpha}\\alpha} \\frac{\\braket{\\cos\\theta_{x,\\bar{B}}\\cos\\theta_{z,B}} - \\braket{\\cos\\theta_{x,\\bar{B}}} \\braket{\\cos\\theta_{z,B}} }{\\braket{\\cos^2\\theta_{x,\\bar{B}}}\\braket{\\cos^2\\theta_{z,B}}}\\label{eq:cxzfinal} \\\\\n\t& C^{\\bar{Y}Y}_{zx} = \\frac{1}{\\bar{\\alpha}\\alpha} \\frac{\\braket{\\cos\\theta_{z,\\bar{B}}\\cos\\theta_{x,B}} - \\braket{\\cos\\theta_{z,\\bar{B}}} \\braket{\\cos\\theta_{x,B}} }{\\braket{\\cos^2\\theta_{z,\\bar{B}}}\\braket{\\cos^2\\theta_{x,B}}}\\label{eq:czxfinal} \\\\\n\t& C^{\\bar{Y}Y}_{zz} = \\frac{1}{\\bar{\\alpha}\\alpha} \\frac{\\braket{\\cos\\theta_{z,\\bar{B}}\\cos\\theta_{z,B}} - \\braket{\\cos\\theta_{z,\\bar{B}}} \\braket{\\cos\\theta_{z,B}} }{\\braket{\\cos^2\\theta_{z,\\bar{B}}}\\braket{\\cos^2\\theta_{z,B}}}\\label{eq:czzfinal}.\n\\end{align}\n\nTo summarize, the efficiency independent method is viable if the following three conditions are met:\n\\begin{enumerate}\n\t\\item The detection efficiency of the antibaryon is independent of that of the baryon.\n\t\\item The efficiency is symmetric in $\\cos\\theta_{y}^{B}$ and $\\cos\\theta_{y}^{\\bar{B}}$ \n\t\\item Higher order terms emerging from Eqs.~\\eqref{eq:Esolve}, \\eqref{eq:Fbarsolve} and \\eqref{eq:Fsolve} can be neglected.\n\\end{enumerate}\n\nSimulations show that the first criterion is fulfilled for both channels at both momenta, whereas the second and third criteria are channel- and momentum dependent. For $\\bar{p}p \\to \\bar{\\Lambda}\\Lambda$ at 1.642 GeV\/$c$, the second criterion is fulfilled. Furthermore, the higher order terms appearing in the expressions for $P^Y_y$, $P^{\\bar{Y}}_y$ and $C^{\\bar{Y}Y}_{yy}$ can be neglected whereas they are large for $C^{\\bar{Y}Y}_{xx}$, $C^{\\bar{Y}Y}_{zz}$ $C^{\\bar{Y}Y}_{xz}$ and $C^{\\bar{Y}Y}_{zx}$. This means we expect the efficiency independent method to work for $P\n^Y_y$, $P^{\\bar{Y}}_y$ and $C^{\\bar{Y}Y}_{yy}$ but not for the other observables.\n\nFor the $\\bar{p}p \\to \\bar{\\Xi}^+\\Xi^-$ channel at 4.6 GeV\/$c$, all three criteria are fulfilled for all spin observables. Thus the efficiency independent method can be used without restrictions in this case. At 7.0 GeV\/$c$, the second criterion is not fulfilled which means that the efficiency independent method cannot be used to extract the polarisation of neither the hyperon nor the antihyperon. However, it can be applied to estimate all spin correlations.\n\n\n\n\\section{Results}\n\\label{sec:results}\n \n\n\\subsection{Reconstruction rates}\n\\label{sec:rates}\nWith the reconstruction efficiencies obtained from the simulations, the measured $\\bar{p}p \\to \\bar{\\Lambda}\\Lambda$ cross section from Ref. \\cite{PS185164} and the predicted cross sections from Ref. \\cite{kaidalov}, we can calculate the expected rate at which hyperons can be reconstructed exclusively in PANDA. We have performed the calculations for two different scenarios: with the Phase One luminosity, which will be around $10^{31}$cm$^{-2}$s$^{-1}$, and with the 20 times larger design luminosity. The results are presented in Table \\ref{hypprod}.\nHowever, during the very first period of data taking, the luminosity at low beam momenta will be smaller by about a factor of two. This means that in the first $\\bar{p}p \\to \\bar{\\Lambda}\\Lambda$ benchmark study, the actual luminosity will be about $5\\cdot10^{30}$cm$^{-2}$s$^{-1}$, giving a two times smaller reconstruction rate than at the nominal Phase One luminosity. The $S\/B$ ratios are calculated using all remaining signal events $S$ and background events from all sources, weighted using their corresponding weight factors given in Tables \\ref{tab:samplesize1} and \\ref{tab:samplesize2}.\n\\begin{table*}\n\\centering\n\\begin{tabular}{llllllll}\n\\hline\n$p_{\\bar{p}}$ (GeV\/$c$) & Reaction & $\\sigma$ ($\\mu$b) & Eff (\\%) & Decay & S\/B & Rate  ($s^{-1}$) & Rate ($s^{-1}$)\\\\&&&&&& at $10^{31}$cm$^{-2}$s$^{-1}$ & at 2$\\cdot10^{32}$cm$^{-2}$s$^{-1}$\\\\\\hline\n1.64 & $\\bar{p}p \\rightarrow \\bar{\\Lambda}\\Lambda$ & 64.1 $\\pm$ 1.6~\\cite{PS185164_1} & 16.04 $\\pm$ 0.04 & $\\Lambda \\rightarrow p \\pi^-$ & 114 & 44 & 880  \\\\\\hline\n\n4.6 & $\\bar{p}p \\rightarrow \\bar{\\Xi}^+\\Xi^-$ & $\\approx$1~~\\cite{kaidalov} & 8.22 $\\pm$ 0.03 & $\\Xi^- \\rightarrow \\Lambda \\pi^-$ & 270 & 0.3  & 6 \\\\\\hline\n\n7.0 & $\\bar{p}p \\rightarrow \\bar{\\Xi}^+\\Xi^-$ & $\\approx$0.3~~\\cite{kaidalov} & 7.95 $\\pm$ 0.03 & $\\Xi^- \\rightarrow \\Lambda \\pi^-$ & 170 & 0.1 & 2 \\\\\\hline\n\\end{tabular}\n\\caption{Results from simulation studies of the various production reactions of ground state hyperons. The efficiencies are for exclusive reconstruction, and are presented with statistical uncertainties. The $S\/B$ denotes the signal-to-background ratio.}\n\\label{hypprod}      \n\\end{table*}\n\n\\subsubsection{Effects from the $\\bar{\\Xi}^+$ angular distribution}\n\nThe distribution of the $\\bar{\\Xi}^+$ scattering angle is not known, since so far, only a few bubble-chamber events exist from the $\\bar{p}p \\to \\bar{\\Xi}^+\\Xi^-$ reaction \\cite{Musgrave1965}. The nominal simulations in this work were therefore performed for isotropically distributed $\\bar{\\Xi}^+$ antihyperons. However, in reality, the angular distribution in the CMS system of the reaction may be forward peaking in a similar way as for $\\bar{p}p \\to \\bar{\\Lambda}\\Lambda$ \\cite{PS185} and $\\bar{p}p \\to \\bar{\\Sigma}^0\\Lambda + c.c.$ \\cite{sigma6}. Since the $\\bar{\\Xi}^+$ share one less quark with the initial $\\bar{p}$ compared to $\\bar{\\Lambda}$ and $\\bar{\\Sigma}^0$, the forward peak is expected to be less pronounced for $\\bar{\\Xi}^+$. Investigations with meson exchange models have resulted in a fairly strong anisotropy for $\\bar{\\Xi}^0$ while almost flat for the $\\bar{\\Xi}^+$ \\cite{XiMEX}. This can have an impact on the total reconstruction efficiency, partly because decay products of the $\\bar{\\Xi}^+$ may escape detection by being emitted along the beam pipe, and partly because a backward-going $\\Xi^-$ in the CMS system is almost at rest in the lab system. Its decay products may then have too low energy to reach the detectors. \n\nIn order to investigate the sensitivity of the total reconstruction efficiency to the $\\bar{\\Xi}^+$ angular distribution, additional simulations were carried out for two other scenarios with more forward-going antihyperons. The \\textit{extreme case} employs angular distribution parameters from the most forward-peaking distributions that have been observed so far, namely in $\\bar{p}p \\to \\bar{\\Lambda}\\Sigma^0 + c.c.$ \\cite{sigma6}. The \\textit{lenient case} represents an intermediate scenario with parameters between those of a flat distribution and those of an extreme one. The distributions are shown in Fig. \\ref{fig:costhtfwp} and the results from the simulations are presented in Table \\ref{tab:fwpeff}. Indeed, the reconstruction efficiency decreases for a strongly forward peaking $\\bar{\\Xi}^+$ distribution. However, the most extreme case results in a reduction of 25-35\\% and the total efficiency -- 5-6\\% -- is still feasible for $\\bar{p}p \\to \\bar{\\Xi}^+\\Xi^-$ studies.\n\n\t\\begin{figure*}[ht]\n\t\t   \n\t\t   \n\t\t       \n\t\t        \\includegraphics[width=0.5\\linewidth]{fwpdist46BW.pdf}\n\t\t   \n\t\t   \n\t\t       \n\t\t        \\includegraphics[width=0.5\\linewidth]{fwpdist7BW.pdf}\n\t\t   \n\t\t\t\\caption{Simulated angular distributions of the $\\bar{p}p\\to\\bar{\\Xi}^+\\Xi^-$ reaction using a flat distribution (black), lenient case (dashed), and the extreme case (dotted) at $p_{\\mathrm{beam}}=4.6$ GeV\/$c$ (left) and $p_{\\mathrm{beam}}=7.0$ GeV\/$c$ (right). Note the different scales on the $y$-axes.}\n\t\t\t\\label{fig:costhtfwp}\n\t\t\\end{figure*}\n\t\t\n\t\t\t\\begin{table}[ht]\n\t\t\t\\centering\n\t\t\t\\begin{tabular}{c|c|c|c}\n\t\t\t\t\\textbf{$p_{\\bar{p}}$ (GeV\/c)} & $\\epsilon_{Isotropic}$ (\\%) & $\\epsilon_{Lenient}$ (\\%) & $\\epsilon_{Extreme}$ (\\%) \\\\ \\hline\n\t\t\t\t4.6 & 8.22 $\\pm$ 0.03 & 7.7 $\\pm$ 0.03 & 6.1 $\\pm$ 0.03 \\\\\n\t\t\t\t7.0 & 7.95 $\\pm$ 0.03 & 7.5 $\\pm$ 0.03 & 5.0 $\\pm$ 0.03 \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t\t\\caption{Reconstruction efficiency of the $\\bar{p}p\\to\\bar{\\Xi}^+\\Xi^-$ reaction with an isotropic angular distribution, a lenient one and an extremely forward peaking distribution. \\label{tab:fwpeff}}\n\t\t\\end{table}\n\n\n\\subsection{Spin observables}\n\\label{sec:spin}\n\nThe spin observables defined in Sect. \\ref{sec:formalism} have been reconstructed with two independent methods to handle the efficiency, described in Sect. \\ref{sec:paramest}. In both cases, we have used data samples that are realistic during the first year of data taking with PANDA, given the reconstruction rates estimated in Sect. \\ref{sec:rates}. Since the background can be suppressed to a very low level, background effects are neglected in these spin studies.\n\n\\subsubsection{The $\\bar{p}p \\to \\bar{\\Lambda}\\Lambda$ reaction}\n\nIn this study, $1.5\\cdot10^6$ reconstructed $\\bar{p}p \\to \\bar{\\Lambda}\\Lambda$ events were used. This amount can be collected in 24 hours during the first phase of data taking with PANDA, where the luminosity at the lowest beam momenta will be about half of that of intermediate and high momenta, \\textit{i.e.} 5$\\cdot10^{30}$cm$^{-2}$s$^{-1}$. \n\nThe polarisation of $\\bar{\\Lambda}$ and $\\Lambda$ as a function of the $\\bar{\\Lambda}$ scattering angle in the CMS system are shown in Fig. \\ref{fig:llbarpol}. The $\\bar{\\Lambda}$ and $\\Lambda$ polarisation are shown to the left in the same plot. Since charge conjugation invariance requires $P^{Y} = P^{\\bar{Y}}$, deviations from this equality could indicate artificial bias from the detector or the reconstruction procedure. However, the agreement is excellent. In the right panels, the average of the $\\bar{\\Lambda}$ and $\\Lambda$ polarisation is shown. The top panels show the polarisations extracted with efficiency corrections, estimated by Eq. (\\ref{eq:pol}). The bottom panels are extracted using the efficiency independent method, applying Eqs. (\\ref{eq:pybarfinal}) and (\\ref{eq:pyfinal}). The polarisations reconstructed with the two techniques agree very well with the input distributions, shown as solid curves. The statistical uncertainties are found to be very small.\n\n\\begin{figure*}[ht]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n        \\centering\n        \\includegraphics[width=1.\\linewidth]{figllbarBW\/pol.pdf}\\\\\n        \\includegraphics[width=1.\\linewidth]{figllbarBW\/pol_nocorr.pdf}\n    \\end{minipage}%\n    \\begin{minipage}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=1.\\linewidth]{figllbarBW\/polavg.pdf}\\\\\n        \\includegraphics[width=1.\\linewidth]{figllbarBW\/polavg_nocorr.pdf}\n    \\end{minipage}\n\t\\caption{Top left: Polarization of the $\\bar{\\Lambda}$ (black) and the $\\Lambda$ (open) at $p_{\\mathrm{beam}}=1.642$ GeV\/c, reconstructed using the efficiency dependent method with 2D efficiency matrices. Top-right: Average values of the two reconstructed polarisations. Bottom-left: Polarisations reconstructed using the efficiency independent method. Bottom-right: Average of the polarisations reconstructed with the efficiency independent method. The vertical error bars represent statistical uncertainties, the horizontal bars the bin widths and the solid curves the input model.}\n\t\\label{fig:llbarpol}\n\\end{figure*}\n\nThe diagonal spin correlations, \\textit{i.e.} $C^{\\bar{Y}Y}_{xx}$, $C^{\\bar{Y}Y}_{yy}$ and $C^{\\bar{Y}Y}_{zz}$, are shown in Figure \\ref{fig:llbarspincorrii}. In the top left and right panels, as well the bottom left, the correlations are extracted using efficiency corrections. The bottom right panel display the average correlation $(C^{\\bar{Y}Y}_{xz}+C^{\\bar{Y}Y}_{zx})\/2$ extracted with the efficiency independent method. In most cases, the reconstructed distributions agree fairly well with the input distributions. However, significant deviations are observed when applying the efficiency independent parameter estimation method, as seen in Fig. \\ref{fig:obsll}. This is expected since we concluded in Sect. \\ref{sec:effind} that higher order terms could not be neglected in this case. With the efficiency dependent method, all deviations are small and do not follow any obvious trend. Furthermore, it is clear that the statistical precision will be greatly improved compared to the PS185 measurements \\cite{PS185164}.\n\n\\begin{figure*}[ht]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n        \\centering\n        \\includegraphics[width=1.\\linewidth]{figllbarBW\/cxx.pdf}\\\\\n        \\includegraphics[width=1.\\linewidth]{figllbarBW\/czz.pdf}\n    \\end{minipage}%\n    \\begin{minipage}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=1.\\linewidth]{figllbarBW\/cyy.pdf}\\\\\n        \\includegraphics[width=1.\\linewidth]{figllbarBW\/cxzavg.pdf}\n    \\end{minipage}\n\t\\caption{Spin correlations of the $\\bar{\\Lambda}\\Lambda$ pair produced at $p_{\\mathrm{beam}}=1.642$ GeV\/c. These observables were estimated with the efficiency dependent method, using 3D efficiency matrices. Top-left: $C^{\\bar{Y}Y}_{xx}$, top-right: $C^{\\bar{Y}Y}_{yy}$ and bottom-left: $C^{\\bar{Y}Y}_{zz}$ of the $\\bar{\\Lambda}\\Lambda$ pair. Bottom-right: The average $(C^{\\bar{Y}Y}_{xz} + C^{\\bar{Y}Y}_{zx})\/2$. The vertical error bars represent statistical uncertainties, the horizontal bars the bin widths and the solid curve the input distributions.}\n\t\\label{fig:llbarspincorrii}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n        \\centering\n        \\includegraphics[width=1.\\linewidth]{figllbarBW\/cxx_nocorr.pdf}\\\\\n        \\includegraphics[width=1.\\linewidth]{figllbarBW\/czz_nocorr.pdf}\\\\\n    \\end{minipage}%\n    \\begin{minipage}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=1.\\linewidth]{figllbarBW\/cyy_nocorr.pdf}\\\\\n        \\includegraphics[width=1.\\linewidth]{figllbarBW\/cxzavg_nocorr.pdf}\\\\\n\t\n    \\end{minipage}\n\t\\caption{Spin correlations of the $\\bar{\\Lambda}\\Lambda$ pair produced at $p_{\\mathrm{beam}}=1.642$ GeV\/c. These observables are estimated using the efficiency independent method. Top-left: $C^{\\bar{Y}Y}_{xx}$, top-right: $C^{\\bar{Y}Y}_{yy}$ and bottom-left: $C^{\\bar{Y}Y}_{zz}$ of the $\\bar{\\Lambda}\\Lambda$ pair. Bottom-right: The average $(C^{\\bar{Y}Y}_{xz} + C^{\\bar{Y}Y}_{zx})\/2$. The vertical error bars represent statistical uncertainties, the horizontal bars the bin widths and the solid curve the input distributions. For $C^{\\bar{Y}Y}_{xx}$, $C^{\\bar{Y}Y}_{xx}$ and $C^{\\bar{Y}Y}_{xx}$ correlations, we do not expect agreement with the input model (solid curve) due to large high order terms. In the case of $C^{\\bar{Y}Y}_{yy}$ correlations, higher order terms were found to be negligible.}\n\n\t\\label{fig:obsll}\n\\end{figure*}\n\n\n\n\\subsubsection{The $\\bar{p}p \\to \\bar{\\Xi}^+\\Xi^-$ reaction}\nTwo studies have been performed at beam momenta of $p_{\\mathrm{beam}}=4.6$ GeV\/c and $p_{\\mathrm{beam}}=7.0$ GeV\/c, using $5.86\\cdot10^5$ and $4.52\\cdot10^5$ reconstructed $\\bar{p}p \\to \\bar{\\Xi}^+\\Xi^-$ events, respectively. The sample at $p_{\\mathrm{beam}}=4.6$ GeV\/c can be collected in 21 days while the sample at $p_{\\mathrm{beam}}=7.0$ GeV\/c requires 55 days of data taking, in line with the planned 80 days campaign at an energy around the $X(3872)$ mass. Here, we assume a luminosity of $10^{31}$cm$^{-2}$s$^{-1}$, which will be achievable at these energies during the first phase of data taking with PANDA. \n\n\\begin{figure*}[ht]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n        \\centering\n            \\includegraphics[width=1.\\linewidth]{figxixibar46BW\/pol.pdf}\\\\\n            \\includegraphics[width=1.\\linewidth]{figxixibar46BW\/pol_nocorr.pdf}\n\n    \\end{minipage}%\n    \\begin{minipage}{0.5\\textwidth}\n        \\centering\n            \\includegraphics[width=1.\\linewidth]{figxixibar46BW\/polavg.pdf}\\\\\n            \\includegraphics[width=1.\\linewidth]{figxixibar46BW\/polavg_nocorr.pdf}\n    \\end{minipage}\n\t\\caption{Top-left: Reconstructed polarisation of the $\\bar{\\Xi}^+$ (black) and the $\\Xi^-$ (open) at $p_{\\mathrm{beam}}=4.6$ GeV\/c using the efficiency dependent method with 2D efficiency matrices. Top-right: Average values of the two reconstructed polarisations. Bottom-left: Polarisations reconstructed with the efficiency independent method. Bottom-right: The average of the polarisations reconstructed with the efficiency independent method. The vertical error bars represent statistical uncertainties, the horizontal bars the bin widths and the solid curves the input model.}\n\t\\label{fig:xixibarpol46}\n\\end{figure*}\n\nIn Fig. \\ref{fig:xixibarpol46}, the polarisation at 4.6 GeV\/$c$ of the $\\Xi^-$ and the $\\bar{\\Xi}^+$ are shown individually (top-left panel) and averaged (top-right panel) for efficiency corrected data. The agreement between $\\Xi^-$ and $\\bar{\\Xi}^+$ as well as between the input distributions and the reconstructed ones, is excellent and the statistical uncertainties are small. Also when using the efficiency independent method, there is good agreement between reconstructed data and the input model (bottom-left and bottom-right panels). This is expected since the simulations showed that all criteria are fulfilled for this reaction at this beam momentum.\n\nThe spin correlations $C^{\\bar{Y}Y}_{xx}$, $C^{\\bar{Y}Y}_{yy}$, $C^{\\bar{Y}Y}_{zz}$ and the average $(C^{\\bar{Y}Y}_{xz}+C^{\\bar{Y}Y}_{zx})\/2$ are shown in Fig. \\ref{fig:xixibar46spincorr} for the same beam momentum. The agreement between the input distributions and the reconstructed distributions is good. Fig. \\ref{fig:obs46} displays two examples of spin correlations reconstructed with the efficiency independent method. The $C^{\\bar{Y}Y}_{yy}$ correlation agrees well with the input model whereas some deviations are seen in the case of $C^{\\bar{Y}Y}_{xx}$, despite the fact that the criteria outlined in Sect. \\ref{sec:effind} are fulfilled. This shows that this observable is more sensitive to the efficiency than the $C^{\\bar{Y}Y}_{yy}$ and that the efficiency independent method has to be used with caution. \n\n\\begin{figure*}[ht]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n        \\centering\n        \\includegraphics[width=1.\\linewidth]{figxixibar46BW\/cxx.pdf}\\\\\n        \\includegraphics[width=1.\\linewidth]{figxixibar46BW\/czz.pdf}\n    \\end{minipage}%\n    \\begin{minipage}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=1.\\linewidth]{figxixibar46BW\/cyy.pdf}\\\\\n        \\includegraphics[width=1.\\linewidth]{figxixibar46BW\/cxzavg.pdf}\n    \\end{minipage}\n\t\\caption{Reconstructed spin correlations of the $\\bar{\\Xi}^+\\Xi^-$ pair at $p_{\\mathrm{beam}}=4.6$ GeV\/c. Top-left: the $C^{\\bar{Y}Y}_{xx}$ correlation. Top-right: $C^{\\bar{Y}Y}_{yy}$. Bottom-left: $C^{\\bar{Y}Y}_{zz}$ . Bottom-right: the reconstructed average $(C^{\\bar{Y}Y}_{xz} + C^{\\bar{Y}Y}_{zx})\/2$. The spin correlations are reconstructed using the efficiency independent method. The vertical error bars represent statistical uncertainties, the horizontal bars the bin widths and the solid curve the input model.}\n\t\\label{fig:xixibar46spincorr}\n\\end{figure*}\n\n\n\\begin{figure*}[ht]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n        \\centering\n        \\includegraphics[width=1.\\linewidth]{figxixibar46BW\/cxx_nocorr.pdf}\n    \\end{minipage}%\n    \\begin{minipage}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=1.\\linewidth]{figxixibar46BW\/cyy_nocorr.pdf}\n    \\end{minipage}\n\t\\caption{Spin correlations of the $\\bar{\\Xi}^+\\Xi^-$ pair at $p_{\\mathrm{beam}}=4.6$ GeV\/c, reconstructed with the efficiency independent method. Left: the $C^{\\bar{Y}Y}_{xx}$ correlation. Right: the $C^{\\bar{Y}Y}_{yy}$ correlation. The vertical error bars represent statistical uncertainties, the horisontal the bin widths and the red curves the input model.}\n\t\\label{fig:obs46}\n\\end{figure*}\n\n\nIn Fig. \\ref{fig:xixibarpol7}, the polarisations of the $\\bar{\\Xi}^+$ and $\\Xi^-$ at 7.0 GeV\/$c$ are shown. In the left panel, where the efficiency dependent method has been used, we see that the reconstructed polarisations agree well with the input model. In the right panel, the efficiency independent method is used. Here, some disagreement is observed with respect to the input model, as expected since one of the criteria in Sect. \\ref{sec:effind} is not fulfilled. Furthermore, we observe that the $\\bar{\\Xi}^+$ polarisation disagrees with the $\\Xi^-$ polarisation. This shows that a comparison between hyperon and antihyperon observables serve as a consistency check.\n\nIn Fig. \\ref{fig:xixibar7spincorr}, the spin correlations of the $\\bar{\\Xi}^+\\Xi^-$ pair are shown at 7.0 GeV\/$c$, reconstructed with the efficiency dependent method. The reconstructed distributions agree with the input ones, indicating that the reconstruction and analysis procedure do not impose any bias. In Fig. \\ref{fig:obs7}, the $C^{\\bar{Y}Y}_{xx}$ and $C^{\\bar{Y}Y}_{yy}$ spin correlations are shown, reconstructed with the efficiency independent method. Even in this case, the reconstructed distributions agree well with the input models.\n\n\n\\begin{figure*}[ht]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n        \\centering\n            \\includegraphics[width=1.\\linewidth]{figxixibar7BW\/pol.pdf}\n    \\end{minipage}%\n    \\begin{minipage}{0.5\\textwidth}\n        \\centering\n            \\includegraphics[width=1.\\linewidth]{figxixibar7BW\/pol_nocorr.pdf}\n    \\end{minipage}\n\t\\caption{The polarisation of the $\\bar{\\Xi}^+$ (black) and the $\\Xi^-$ (open) hyperons at $p_{\\mathrm{beam}}=7.0$ GeV\/c. Left: data reconstructed using the efficiency dependent method. Right: data reconstructed using the efficiency independent method. The vertical error bars represent  statistical uncertainties, the horizontal bars the bin widths and the solid curve the input model.}\n\t\\label{fig:xixibarpol7}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n        \\centering\n        \\includegraphics[width=1.\\linewidth]{figxixibar7BW\/cxx.pdf}\\\\\n        \\includegraphics[width=1.\\linewidth]{figxixibar7BW\/czz.pdf}\n    \\end{minipage}%\n    \\begin{minipage}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=1.\\linewidth]{figxixibar7BW\/cyy.pdf}\\\\\n        \\includegraphics[width=1.\\linewidth]{figxixibar7BW\/cxzavg.pdf}\n    \\end{minipage}\n\t\\caption{Reconstructed spin correlations (top-left) $C^{\\bar{Y}Y}_{xx}$, (top-right) $C^{\\bar{Y}Y}_{yy}$ and (bottom-left) $C^{\\bar{Y}Y}_{zz}$ of the $\\bar{\\Xi}^+\\Xi^-$ pair at $p_{\\mathrm{beam}}=7.0$ GeV\/c. (bottom-right) Reconstructed average $(C^{\\bar{Y}Y}_{xz} + C^{\\bar{Y}Y}_{zx})\/2$. The spin correlations are reconstructed at $p_{\\mathrm{beam}}=7.0$ GeV\/c using acceptance corrections. The vertical error bars represent statistical uncertainties, the horizontal bars the bin widths and the solid curves the input model.}\n\t\\label{fig:xixibar7spincorr}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n        \\centering\n        \\includegraphics[width=1.\\linewidth]{figxixibar7BW\/cxx_nocorr.pdf}\n    \\end{minipage}%\n    \\begin{minipage}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=1.\\linewidth]{figxixibar7BW\/cyy_nocorr.pdf}\n    \\end{minipage}\n\t\\caption{Reconstructed spin correlations of the $\\bar{\\Xi}^+\\Xi^-$ pair at $p_{\\mathrm{beam}}=7.0$ GeV\/$c$, reconstructed with the efficiency independent method. Left:The $C^{\\bar{Y}Y}_{xx}$ correlation. Right: The $C^{\\bar{Y}Y}_{yy}$ correlation. The solid curves represent the input model.}\n\t\\label{fig:obs7}\n\\end{figure*}\n\nThe singlet fractions of the $\\bar{\\Xi}^+\\Xi^-$ pair, calculated from the spin correlations according to Eq. (\\ref{eq:spinfrac}), are shown in Fig. \\ref{fig:xixibarsf} as a function of the $\\bar{\\Xi}^+$ scattering angle. The results show that the prospects of measuring the singlet fraction, and thereby establish in which spin state the produced $\\bar{\\Xi}^+\\Xi^-$ is, are very good. It will also be possible to test the predictions from Ref. \\cite{XiMEX}.\n\n\\begin{figure*}[ht]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n        \\centering\n            \\includegraphics[width=1.\\linewidth]{figxixibar46BW\/Fs.pdf}\n\n    \\end{minipage}%\n    \\begin{minipage}{0.5\\textwidth}\n        \\centering\n            \\includegraphics[width=1.\\linewidth]{figxixibar7BW\/Fs.pdf}\n    \\end{minipage}\n\t\\caption{Reconstructed singlet fractions at $p_{\\mathrm{beam}}=4.6$ GeV\/c (left) and $p_{\\mathrm{beam}}=7.0$ GeV\/c (right). The vertical errorbars are statistical uncertainties only. The horizontal bars are the bin widths.}\n\t\\label{fig:xixibarsf}\n\\end{figure*}\n\n\\subsection{Systematic uncertainties}\n\\label{sec:systematic}\n\nIt is hard to evaluate systematic uncertainties before the experiment is taken into operation, since effects such as trigger efficiencies or imperfections in tracking or in the Monte Carlo implementation of the detector are difficult to estimate without real data. \n\nIn the feasibility study of electromagnetic form factors in PANDA \\cite{EMFFPanda} as well as in the simulation of the foreseen energy scan around the $X(3872)$ \\cite{xscan}, uncertainties in the estimated luminosity and background constitute the most important sources of systematics. While being very important in cross section measurements, effects from the uncertainty in the luminosity are expected to be negligible in measurements of differential distributions. This is because such uncertainties should be uniformly distributed over the angles of the final state particles. Regarding the background, the displaced decay vertices of hyperons result in a very distinct event topology that allows for a very strong suppression of background. Furthermore, the cross section of the hyperon channels studied in this work are several orders of magnitude larger than in Refs. \\cite{EMFFPanda} and \\cite{xscan}.\n\nNon-negligible systematic effects can arise from model-dependencies in the efficiency correction. The method of moments introduces an uncertainty for each measured variable that is integrated out when calculating each moment. In multi-dimensional problems like the ones presented here, this needs a thorough investigation. Therefore, we have carried out three comparative studies: i) between generated distributions on one hand and reconstructed and efficiency corrected distributions on the other ii) between extracted hyperon and antihyperon parameters iii) between two different parameter estimation techniques. Significant differences only appear for the efficiency independent method and are well understood since in these cases, the necessary criteria for using the efficiency independent method are not fulfilled. However, for the high-precision studies enabled by the design luminosity, it will likely be necessary to use a model-independent method for extracting the spin observables, \\textit{e.g.} a Maximum Likelihood-based method similar to the one in Refs. \\cite{bes3prl,bes3nature}. For $\\bar{p}p$ reactions, a dedicated formalism and analysis framework will be needed for this purpose.\n\n\n\\section{Summary and discussion}\n\\label{sec:summary}\n\nThe feasibility of exclusive reconstruction of two antihyperon-hyperon reactions in the foreseen antiproton experiment PANDA at FAIR has been investigated: $\\bar{p}p \\to \\bar{\\Lambda}\\Lambda$ and $\\bar{p}p \\to \\bar{\\Xi}^+\\Xi^-$. The former has been studied with the PS185 experiment and will be used for quality assurance and fine-tuning of detectors, data acquisition, reconstruction and analysis. However, even at the modest luminosity during the start-up phase of PANDA, a world-record sample can be collected in a few days. Furthermore, the background can be suppressed to a very low level. This will allow PANDA to push forward the state of the art in the measurement of spin observables. The double-strange $\\Xi^-$ has barely been studied with antiproton probes before and the studies proposed here will therefore be pioneering. The foreseen high data rates and the low background level will enable a complete spin decomposition of the reaction already during the first year of data taking. This demonstrates PANDAs potential as a strangeness factory.\n\nThe method of moments applied in this work is suitable for sample sizes of the first phase of PANDA. Two different approaches were applied: a standard efficiency dependent one, and a more unusual efficiency independent method. The applicability of the latter however relies on approximations whose validity need to be evaluated on a case-by-case basis. After only a few years at the initial luminosity, and even more, when the design luminosity is available, the hyperon spin studies will reach high statistical precision. For this, a multi-dimensional and model-independent analysis framework needs to be developed in order to match accuracy and precision. This could open up for large-scale searches for CP violation in hyperon decays and its feasibility will be investigated in the future. \n\n\\section*{Acknowledgement}\nWe acknowledge financial support from \nthe Bhabha Atomic Research Centre (BARC) and the Indian Institute of Technology Bombay, India; \nthe Bundesministerium f\\\"ur Bildung und Forschung (BMBF), Germany; \nthe Carl-Zeiss-Stiftung 21-0563-2.8\/122\/1 and 21-0563-2.8\/131\/1, Mainz, Germany; \nthe Center for Advanced Radiation Technology (KVI-CART), Groningen, Netherlands; \nthe CNRS\/IN2P3 and the Universit\\'{e} Paris-Sud, France; \nthe Czech Ministry (MEYS) grants LM2015049, CZ.02.1.01\/0.0\/0.0\/16 and 013\/0001677, \nthe Deutsche Forschungsgemeinschaft (DFG), Germany; \nthe Deutscher Akademischer Austauschdienst (DAAD), Germany; \nthe European Union's Horizon 2020 research and innovation programme under grant agreement No 824093.\nthe Forschungszentrum J\\\"ulich, Germany; \nthe Gesellschaft f\\\"ur Schwerionenforschung GmbH (GSI), Darmstadt, Germany; \nthe Helmholtz-Gemeinschaft Deutscher Forschungszentren (HGF), Germany; \nthe INTAS, European Commission funding; \nthe Institute of High Energy Physics (IHEP) and the Chinese Academy of Sciences, Beijing, China; \nthe Istituto Nazionale di Fisica Nucleare (INFN), Italy; \nthe Ministerio de Educacion y Ciencia (MEC) under grant FPA2006-12120-C03-02; \nthe Polish Ministry of Science and Higher Education (MNiSW) grant No. 2593\/7, PR UE\/2012\/2, and the National Science Centre (NCN) DEC-2013\/09\/N\/ST2\/02180, Poland; \nthe State Atomic Energy Corporation Rosatom, National Research Center Kurchatov Institute, Russia; \nthe Schweizerischer Nationalfonds zur F\\\"orderung der Wissenschaftlichen Forschung (SNF), Swiss; \nthe Science and Technology Facilities Council (STFC), British funding agency, Great Britain; \nthe Scientific and Technological Research Council of Turkey (TUBITAK) under the Grant No. 119F094\nthe Stefan Meyer Institut f\\\"ur Subatomare Physik and the \\\"Osterreichische Akademie der Wissenschaften, Wien, Austria; \nthe Swedish Research Council and the Knut and Alice Wallenberg Foundation, Sweden.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nBroadband beamforming with microphone array is a key signal processing module in many consumer electronics products, e.g., smart phones and smart speakers \\cite{chhetri2018multichannel, MA_book, benesty2008microphone}. The proliferation of microphone arrays due to decreasing hardware cost and superior speech enhancement performance, has made broadband beamforming a ubiquitous embedded technology, and its performance has a critical impact on the overall system. \n \nA key requirement for broadband beamforming  is to deliver consistent performance across several octaves of frequencies, e.g., $80$ Hz - $8$ KHz in the voiceband case. Speech enhancement  is typically the system objective in most microphone arrays systems, rather than mere signal detection, as in the narrowband case. This poses hardware and algorithmic challenges in the design of microphone arrays and the underlying beamforming procedure. Filter-and-Sum (F\\&S) \\cite{frost_FilterandSum} has been a standard approach for designing a broadband beamformer as an extension to a narrowband beamformer by \\emph{stitching} frequency-domain coefficients that are computed using narrowband beamforming techniques. Several narrowband beamforming techniques, with different objectives and assumptions, become standard design techniques, e.g., Delay-and-Sum (D\\&S) \\cite{DS, MA_book}, Minimum-Variance-Distortionless-Response (MVDR) \\cite{MVDR1, MVDR2}, Subspace methods \\cite{benesty2008microphone}. In this work, we do not address a particular beamformer design algorithm. Rather, the emphasis is on the acoustic modeling, which is common among these techniques. Without loss of generality, MVDR-based F\\&S beamformer with a robustness constraint is used as a case study for our analysis. \nAt a given frequency and \\emph{look-direction}, the two key design parameters in almost all beamforming algorithms are the steering vector and the spatial coherence matrix. \nProper design of these two parameters is the subject of this work. \n\nIn Far-Field models, the acoustic wave is usually approximated by plane-waves \\cite{RoomAcousticsBook}, and the steering vector at the direction\/frequency of a plane wave is defined as the observed acoustic pressure at the different microphones when the microphone array is impinged with the plane-wave. Near-Field steering vectors can similarly be approximated by acoustic spherical waves. The observed wave-field in the general case is the superposition of the incident wave-field and the scattered wave-field. A typical approximation of the steering vector is the free-field approximation, which assumes sound propagation in free-field (at the speed of sound in air), and only the incident wave-field is considered. This approximation is used almost universally in the microphone array literature because it yields closed-form formulae that simplify beamformer analysis.  The main issue of the free-field approximation is that it ignores the impact of the device surface on the observed acoustic pressure, i.e., the scattered wave-field. This impact, as will be shown, can significantly change the microphone array behavior at certain frequencies and angles. \n\nA possible remedy to this problem is to rely on anechoic lab measurements to quantify the device response to incident waves. However, this is a time-consuming and high-cost solution, and imperfect experimental settings could lead to noticeable modeling errors, especially in near-field cases. In this work, we describe a simulation-based approach for acoustic modeling \nof microphone array on rigid surface by solving the Helmholtz wave equation using Finite-Element-Method (FEM) with a background wave-field that matches the incident wave-field \\cite{larsson2008partial}.  Prior works that studied the impact of scattered field on microphone arrays used spherical harmonic decomposition for specific form-factors (e.g. sphere, cylinder) \\cite{zotkin2017incident, teutsch2007modal, rafaely2015fundamentals} (and references, therein). However, these methods are restrictive in the choice of device form-factors (e.g. do not include modern smart-speaker form-factors) and beamforming techniques. In comparison, the FEM method proposed in this paper provides three notable contributions: (i) a methodology to compute the steering vector for microphone arrays mounted on solid hard surfaces without the need for expensive anechoic chamber measurements, (ii) ability to design any type of beamformer that  relies on steering vectors; these include MVDR beamformer \\cite{MA_book}, linearly constrained minimum variance (LCMV) beamformer \\cite{VanTrees2002,mabande2009design}, and polynomial beamformer \\cite{mabande2010design}, and (iii) extension of the proposed method to generic device form-factors that are used for smart speakers. \n\nThe following notations are used throughout the paper. A bold lower-case letter denotes a column vector, while a bold upper-case letter denotes a matrix. ${\\bf{A}}^T$ and ${\\bf{A}}^H$ denote the transpose and conjugate transpose, respectively, of $\\bf{A}$, and ${\\bf{A}}_{m,n}$ is the matrix entry at position $(m,n)$. $\\Theta \\triangleq \\left(\\theta, \\phi \\right)^T$ denotes the polar and azimuth angles, respectively, in a spherical coordinate system. $M$ always refers to the number of microphones.  $\\mbox{$\\boldsymbol{\\Psi}$} (\\omega)$ denotes the noise coherence matrix, of size $M\\times M$, at frequency $\\omega$ (the dependency on $\\omega$ is dropped whenever it is clear from the context).  Additional notations are introduced when needed.\n\n\\section{Background}\n\\label{sec:background}\n\\subsection{Wave Equation}\nThe acoustic wave equation \\cite{acousticsbook} is the governing equation for the propagation of sound waves at equilibrium in elastic fluids, e.g., air. The homogenous wave equation has the  form\n\\begin{equation}\n\\nabla^2 \\bar{p} - \\frac{1}{c^2} \\frac{\\partial^2 \\bar{p} }{\\partial t^2} = 0\n\\end{equation}\nwhere $\\bar{p}(t)$ is the acoustic pressure, and $c$ is the speed of sound in the medium. In this work, we consider only the practical case of homogenous fluid with no viscosity. \n\nIn practice, the wave equation is usually solved in the frequency domain using the Helmholtz equation to find $p(\\omega)$:\n\\begin{equation}\n\\nabla^2 p + k^2 p = 0  \\label{eq:Helmholtz}\n\\end{equation}\nwhere $k\\triangleq\\omega\/c$ is the wave number. At steady state, the time-domain and frequency-domain solutions are Fourier pairs \\cite{FourierAcoustics}. In our modeling, we work only with the homogenous Helmholtz equation under various boundary conditions. The boundary conditions are determined by the geometry and the acoustic impedance of the different boundaries. We assume the device has a rigid surface, therefore, it is modeled as a \\emph{sound hard} boundary.\n\\vspace{-0.1cm}\n\\subsection{Beamforming Strategies} \\label{strategies}\nBeamforming is a microphone-array signal processing technique that allows emphasizing the user's speech from a desirable look-direction (LD) while suppressing interferences from other directions. Here, we process microphone elements such that the signals arriving from look-direction are combined in-phase, while signals arriving from other directions are combined out-of-phase. \nDenote the position of the $m$-th microphone by $\\mbox{$\\mathbf{r}$}_m$, and the signal acquired at the $m$-th microphone for frequency $\\omega$ by  $x(w,\\mbox{$\\mathbf{r}$}_m)$. Then, the signal acquired by the microphone array can then be expressed as:\n\\beq \\label{eq:xdef}\n\\mbox{$\\mathbf{x}$}(\\omega,\\mbox{$\\mathbf{r}$}) = \\lts x(\\omega,\\mbox{$\\mathbf{r}$}_1)  \\hspace{2mm} x(\\omega,\\mbox{$\\mathbf{r}$}_2)  \\hspace{2mm} \\hdots \\hspace{2mm} x(\\omega,\\mbox{$\\mathbf{r}$}_M) \\right]^T .\n\\end{equation}\nDenoting the spectrum of the desired source signal by $s(\\omega)$ and the ambient noise captured by the microphone array as $\\mbox{$\\mathbf{n}$}(\\omega)$, we can express $\\mbox{$\\mathbf{x}$}(\\omega,\\mbox{$\\mathbf{r}$})$ as:\n\\beq \\label{eq:XExpand}\n\\mbox{$\\mathbf{x}$}(\\omega,\\mbox{$\\mathbf{r}$}) = \\mbox{$\\mathbf{v}$}(\\omega,\\mbox{$\\boldsymbol{\\Theta}$}) s(\\omega) + \\mbox{$\\mathbf{n}$}(\\omega),\n\\end{equation}\nwhere, $\\mbox{$\\mathbf{v}$}(\\omega,\\Theta) \\triangleq \\lts v_1(\\omega,\\Theta)  \\hspace{2mm} v_2(\\omega,\\Theta) \\hspace{2mm} \\hdots \\hspace{2mm} v_M(\\omega,\\Theta) \\right]$ is the frequency and angle-dependent steering vector. \n\nThe beamformer design involves computation of complex-valued weights for each frequency and microphone denoted by $\\mbox{$\\mathbf{w}$} (\\omega)  \\triangleq \\lts w_1(\\omega)  \\hspace{2mm}  w_2(\\omega) \\hspace{2mm}  \\hdots \\hspace{2mm}  w_M(\\omega) \\right]^T$, which are then applied to $\\mbox{$\\mathbf{x}$}(\\omega,\\mbox{$\\mathbf{r}$})$ to obtain the beamformer output $y(\\omega)$:\n\\beq \\label{eq:bfOut}\ny(\\omega) = \\mbox{$\\mathbf{w}$}^H (\\omega) \\mbox{$\\mathbf{x}$}(\\omega,\\mbox{$\\mathbf{r}$}).\n\\end{equation}\n\nWe are interested in using FEM modeling for the design of F\\&S beamformers that can be expressed as a constrained optimization problem, and the solution to which provides the optimal beamformer filters. This covers various F\\&S beamformers like MVDR, maximum SNR, and LCMV beamformers \\cite{VanTrees2002}. In this work, we use, without loss of generality,  the MVDR beamformer with a robustness constraint to present our analysis.\n\n\\subsection{Beamforming Metrics} \\label{metrics}\nWe use three metrics to assess the performance: array gain (AG), white noise gain (WNG)  \\cite{MA_book}, and microphone array channel capacity (MACC) \\cite{mansour2018information}. \nThe AG metric is defined as the improvement in signal-to-noise-ratio (SNR) offered by the beamformer:\n$\\text{AG}(\\omega) \\triangleq \\frac{SNR_{out}(\\omega)}{SNR_{in}(\\omega)}$.\nAfter some algebraic manipulations, one can show that \\cite{MA_book}:\n\\beq \\label{eq:AG}\n\\text{AG}(\\omega, \\Theta_{LD})  = \\frac{|\\mbox{$\\mathbf{w}$}^H \\mbox{$\\mathbf{v}$}(\\omega,\\Theta_{LD})|^2}{\\mbox{$\\mathbf{w}$}^H \\mbox{$\\boldsymbol{\\Psi}$}(\\omega) \\mbox{$\\mathbf{w}$}},\n\\end{equation}\nwhere $\\Theta_{LD}$ denotes the look-direction, and $\\mbox{$\\boldsymbol{\\Psi}$} \\triangleq \\mbox{$\\mathbf{\\Lambda}$}\/\\beta$ is the normalized noise correlation matrix with\n\\beq \\label{eq:LambdaNN}\n\\mbox{$\\mathbf{\\Lambda}$}_{m,q} = \\int_{0}^{\\pi} \\int_{0}^{2\\pi} \\mbox{$\\mathbf{v}$}_m(\\omega,\\Theta) \\mbox{$\\mathbf{v}$}_q^{*}(\\omega,\\Theta) \\sigma_{N}^{2} (\\omega,\\Theta) \\sin(\\theta) \\ d\\theta \\ d\\phi ,\n\\end{equation}\nwhere $\\sigma_{N}^{2} (\\omega,\\Theta)$ denotes the distribution of noise power as a function of $\\omega$ and $\\Theta$, and\n\\beq \\label{eq:phiN}\n\\beta = \\int_{0}^{\\pi} \\int_{0}^{2\\pi} \\sigma_{N}^{2} (\\omega,\\Theta) \\sin(\\theta) \\ d\\theta \\ d\\phi .\n\\end{equation}\nThe WNG metric is the SNR improvement provided by the beamformer when the noise components at the microphones are statistically independent \\cite{MA_book}:\n\\beq \\label{eq:WNG}\n\\text{WNG}(\\omega, \\Theta_{LD}) = \\frac{|\\mbox{$\\mathbf{w}$}^H \\mbox{$\\mathbf{v}$}(\\omega,\\Theta_{LD})|^2}{\\mbox{$\\mathbf{w}$}^H \\mbox{$\\mathbf{w}$}}.\n\\end{equation}\nThe MACC metric \\cite{mansour2018information} aims at providing a characterization of the microphone array that is independent of the beamformer realization. It is analogous to MIMO channel capacity in wireless communication. If the source location is known, then the MACC is defined as \n\\begin{equation}\n\\text{MACC}(\\omega, \\Theta_{LD}) \\triangleq \\log \\left(1+ P\\|\\mathbf{S}^{-\\frac{1}{2}} \\mathbf{U}^H \\mathbf{v}(\\omega, \\Theta_{LD})\\|^2\\right)\n\\end{equation} \nwhere $\\mathbf{USU}^H$ is the singular value decomposition of $\\mbox{$\\boldsymbol{\\Psi}$}$, and $P$ is the input power.\n\n\\section{Acoustic Modeling}\n\\label{sec:typestyle}\n\\subsection{Acoustic Plane-Waves}\nAcoustic plane waves constitute a powerful tool for analyzing the wave equation, and it provides a good approximation of the wavefield emanating from a far-field point source \\cite{teutsch2007modal}. The acoustic pressure of a plane-wave with vector wave number $\\bf{k}$ is defined at a point  $\\bf{r}$ in the 3D space  as:\n\\begin{equation}\np({\\bf{k}}) \\triangleq p_0 e^{-j {\\bf{k}}^T{\\bf{r}}}   \\label{eq:pw_def}\n\\end{equation} \nThis is a solution of the inhomogeneous Helmholtz equation with a far point source, where $\\|{\\bf{k}}\\| = k$ (note that, for a given $k$ in the Helmholtz equation, there are two degrees of freedom in choosing $\\bf{k}$). Further, a general solution to the homogenous Helmholtz equation can be approximated by a linear superposition of plane waves of different angles \\cite{FourierAcoustics, pwd1, pwd2, pwd3}. These properties render acoustic plane-waves a key tool in designing far-field beamforming for microphone arrays, where the microphone array response to each plane wave provides a sufficient set for the beamformer design.\n\nThe \\emph{total} wavefield at each microphone of the microphone array when an incident plane-wave $p_i({\\bf{k}})$ impinges on the device has the general form:\n\\vspace{-0.1cm}\n\\begin{equation}\np_t = p_i + p_s  \\label{eq:pt_df}\n\\vspace{-0.1cm} \n\\end{equation}\nwhere $p_t$ and $p_s$ refer to the total and scattered wavefield respectively. The total wavefield, $p_t$, at each microphone is computed by inserting \\eqref{eq:pt_df} in the Helmholtz equation \\eqref{eq:Helmholtz} and solving for $p_s$ with appropriate boundary conditions. The details of this modeling are described in section \\ref{sec:FEM_modeling}. It is evident from \\eqref{eq:pw_def} that an incident plane-wave does not have magnitude information, and it is fully parameterized by its phase. This is not true for the scattered wavefield, $p_s$ which represents the reflections\/diffractions due to the rigid device surface. This magnitude information in $p_s$ is critical in resolving phase ambiguity due to microphone array geometry. \n\nIf the microphone array is composed of discrete microphones in space, and the area of each microphone is much smaller than the wavelength, then a reasonable approximation is to set $p_s = 0$ in \\eqref{eq:pt_df}. This is referred to as \\emph{free-field} approximation. In this case, the total wavefield, $p_t$, is fully determined by the wavenumber $\\bf{k}$ in \\eqref{eq:pw_def}, and the $(x,y,z)$ coordinates of each microphone. It is obvious that, free-field approximation is not accurate if the microphone array is on a rigid surface. Nevertheless, this approximation has been utilized almost universally in the literature for acoustic modeling in beamformer design.  In the following section, we show that the free-field approximation does not provide a good approximation of the total field under important practical cases.\n\n\\vspace{-0.2cm}\n\\subsection{FEM Modeling}\n\\label{sec:FEM_modeling}\n\nThe modeling objective  is to compute the total sound field in \\eqref{eq:pt_df} at each microphone when the device is impinged by a plane wave. This resembles physical measurement in anechoic room with a distant point  source. FEM is one of the standard approaches for solving the Helmholtz equation numerically. In our case, we need to solve the Helmholtz equation for the total wavefield at all frequencies of interest  with a \\emph{background} plane wave. The device surface is modeled as sound hard boundary. The microphone is modeled as a point receiver on the surface if the microphone surface area is much smaller than the wavelength, otherwise, its response is computed as the integral of the acoustic pressure over its area. To have a true background plane-wave, the external boundary should be open and non-reflecting. In our model, the device is enclosed by a closed boundary, e.g., a cylinder or a spherical surface. To mimic open-ended boundary, there are two choices:  (i) Matched boundary whose impedance is matched to the air impedance at the frequency of interest, (ii) Perfectly matched layer,  which defines a special absorbing domain that eliminates reflection and refractions in the internal domain that encloses the device \\cite{berenger1994perfectly}.  The merits of each approach is beyond the scope of this paper. The FEM solves for $p_t$ in \\eqref{eq:pt_df}, which is equivalent to solving for only the scattered field, $p_s$, after inserting background plane wave model \\eqref{eq:pw_def} in the Helmholtz equation. The acoustics module of COMSOL multiphysics package \\cite{COMSOL} is used for this FEM numerical solution, and the simulation is rigorously validated  with exact and measured results on different form-factors. For example, in Fig. \\ref{fig:comsol}, we show the total pressure field of two microphones on a spherical surface with analytical and simulated solution. Both amplitude and phase responses match excellently with the analytical solution \\cite{ Bowman_book}.  Further, in Fig. \\ref{fig:comsol2}, we show an example of simulated and measured acoustic pressure of a rectangular microphone array mounted on a slanted cube. In the plot, we show the inter-channel response, i.e., $\\{H_i(\\omega)\/H_r(\\omega)\\}_{i\\neq r}$, where $r$ is a reference microphone. The phase difference between  simulated and measured responses is linear, which is expected when the positions of the device in both cases are not perfectly aligned.  For more comparisons between simulated\/theoretical and measured acoustic pressure responses, one may refer to \\cite{wiener1949diffraction,spence1948diffraction}.\n\\begin{figure}[t] \n\t\\centering\n\t\\includegraphics[width=0.4\\textwidth, trim=5mm 5mm 5mm 5mm,clip]{Sphere_5cm_comp.eps}\n\t\\caption{\\footnotesize{Comparison of FEM and analytical solutions for spherical surface of radius 5 cm. Top: magnitude response, bottom: phase response. Mic 2 is in the middle of sphere facing the incident wave; Mic 1 is facing away from it.} } \n\t\\label{fig:comsol}\n\\efi\n\\begin{figure}[htp] \n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth, trim=5mm 1mm 5mm 7.8mm,clip]{knight_sim.eps}\n\t\\caption{\\footnotesize{Normalized acoustic pressure of a rectangular microphone array on a slanted cube for simulated and measured cases for a background plane wave at direction $(\\theta,\\phi) =(90^\\circ, 210^\\circ)$ (simulated response is in dotted lines, measured response in solid lines, with different colors for different microphones). Top: magnitude, bottom: phase difference} } \n\t\\label{fig:comsol2}\n\\vspace{-0.25cm}\n\\efi\n\nNote that, the above procedure is not limited to plane-wave as we only need to specify the background pressure field, which could be, for example, spherical wave for near-field modeling. The procedure is repeated for a grid of frequency and incident angles to build a dictionary of total pressure  $\\{p_t(\\omega, \\theta,\\phi)\\}_{\\omega, \\theta, \\phi}$ that is used in subsequent analysis.\n\n\\vspace{-0.25cm}\n\\section{Analysis of Free-Space Beamforming}\n\\label{sec:analysis}\nTo illustrate the benefits of FEM modeling, we use the MVDR beamformer with a robustness constraint,  formulated as a constrained convex optimization problem \\cite{mabande2009design}:\n\\vspace{-0.1cm}\n\\beqa  \\label{eq:MVDR}\n\\widehat\\mbox{$\\mathbf{w}$} = \\arg \\min_{\\mbox{$\\mathbf{w}$}} & & \\mbox{$\\mathbf{w}$}^H \\mbox{$\\boldsymbol{\\Psi}$} \\ \\mbox{$\\mathbf{w}$} \\nonumber \\\\\n \\text{such that} & & \\mbox{$\\mathbf{w}$}^H \\mbox{$\\mathbf{v}$}(\\omega, \\theta_{LD}) =1, \\nonumber \\\\\n & & \\frac{|\\mbox{$\\mathbf{w}$}^H \\mbox{$\\mathbf{v}$}(\\omega,\\Theta_{LD})|^2}{\\mbox{$\\mathbf{w}$}^H \\mbox{$\\mathbf{w}$}}  \\geq \\gamma ,\n \\end{eqnarray}\n where the first constraint is called the distortionless constraint \\cite{ MA_book}, and  the second constraint is the WNG constraint, which imposes robustness in the beamformer design that can be controlled through $\\gamma$ \\cite{mabande2009design}. Further, the WNG constraint enables a more fair comparison between the total and free-field beamformer designs because the WNG is bounded in both cases. Without loss of generality, we assume a spherically diffuse noise field. The optimization problem in \\eqref{eq:MVDR} is solved using a convex optimization solver to obtain the beamformer weights $\\widehat\\mbox{$\\mathbf{w}$}$. Note that the proposed FEM-model based method can be similarly extended to other beamformer designs like the MVDR \\cite{MA_book}, LCMV \\cite{VanTrees2002}, and polynomial beamformer \\cite{ mabande2010design}.\n\n\\subsection{Analysis Methodology}\n\\label{sec:anaMethod}\n The MVDR solution can be obtained from \\eqref{eq:MVDR} by using $\\mbox{$\\mathbf{v}$}_i$ and $\\mbox{$\\mathbf{v}$}_t$ as steering vectors for free-field (FF) and total-field (TF), respectively. To compute $\\mbox{$\\boldsymbol{\\Psi}$}$, we use analytical method for canonical device shapes, such as finite cylinder and sphere \\cite{Bowman_book}. For a general device shape, the FEM tool is used to simulate the steering vectors for a uniform grid of azimuth and polar angles. Then, $\\mbox{$\\boldsymbol{\\Psi}$}$ is numerically computed from \\eqref{eq:LambdaNN} and \\eqref{eq:phiN}, with $\\sigma_N^2(\\omega,\\Theta)=1$ for the spherically diffuse noise field.\n \n \\begin{figure}[t] \n\t\\centering\n\t\\includegraphics[width=9cm, height=3.25cm, trim=15mm 5mm 5mm 30mm,clip]{device.eps}\n\t\\vspace*{-0.5cm}\n\t\\caption{Simulation setup for FEM-based beamforming. The form-factor is a combination of a cylindrical bottom and a top surface with a spherically-curved shape. }\n\t\\label{fig:arraySetup}\n\\efi\n\nWe now compare the performance of MVDR beamformer designed using FF and TF assumptions. For our study, we use the setup in  Fig. \\ref{fig:arraySetup}, which has $5$ microphones on the top of a cylinder of height 130 mm and diameter of 70 mm; the top surface of the cylinder has a spherically-curved shape. This surface does not have a closed-form solution for the Helmholtz equation, which necessitates the use of the proposed FEM method. The origin of the coordinate system coincides with the center microphone with $z$ axis pointing upwards, and the $x$-$y$ plane parallel to the bottom face of the cylinder. The coordinates of the microphones are: $(x,y,z)= \\ltc (r_o,0,z_o), (0,r_o,z_o), (-r_o,0,z_o),(0,-r_o,z_o), (0,0,0) \\right\\}$, where $r_o=30$ mm and $z_o=-3$ mm. Lastly, we set $\\gamma=-25$ dB. \n\\vspace{-0.25cm}\n\n\n\\subsection {Results} \\label{sec:results}\nWe evaluate the microphone array metrics under Free Field (FF) and Total Field (TF) setups for the array in Fig. \\ref{fig:arraySetup} at two arrival angles: $(\\theta,\\phi) = (90^\\circ, 0^\\circ)$ and $(30^\\circ, 0^\\circ)$. The results are summarized in Figs. \\ref{fig:AG}-\\ref{fig:MACC} for the three performance metrics. \n\n\\begin{figure}[h]%\n    \\centering\n    \\subfloat[$(\\theta,\\phi) = (90^\\circ, 0^\\circ)$]{{\\includegraphics[width=4.295cm, height=3.7cm, trim=11mm 0mm 11mm 7mm,clip]{AG-Polar90Deg_3.eps} }}\n    \\subfloat[$(\\theta,\\phi) = (30^\\circ, 0^\\circ)$]{{\\includegraphics[width=4.295cm, height=3.7cm,trim=11mm 0mm 11mm 7mm,clip]{AG-Polar30Deg_3.eps} }}\n\t\\vspace*{-0.15cm}\n    \\caption{AG performance showing the difference between FF and TF. Note how the array gain is higher, especially at high frequencies. }\n    \\label{fig:AG}%\n\t\\vspace*{-0.3cm}\n\\end{figure}\n\\begin{figure}[h]%\n    \\centering\n    \\subfloat[$(\\theta,\\phi) = (90^\\circ, 0^\\circ)$]{{\\includegraphics[width=4.295cm, height=3.7cm, trim=11mm 0mm 11mm 7mm,clip]{WNG-Polar90Deg_3.eps} }}\n    \\subfloat[$(\\theta,\\phi) = (30^\\circ, 0^\\circ)$]{{\\includegraphics[width=4.295cm, height=3.7cm,trim=11mm 0mm 11mm 7mm,clip]{WNG-Polar30Deg_3.eps} }}\n\t\\vspace*{-0.15cm}\n    \\caption{WNG performance. Note that even with the higher array gain from Fig. \\ref{fig:AG}, the WNG is better for the TF configuration.}\n    \\label{fig:WNG}%\n\t\\vspace*{-0.3cm}\n\\end{figure}\n \\begin{figure}[h]%\n    \\centering\n    \\subfloat[$(\\theta,\\phi) = (90^\\circ, 0^\\circ)$]{{\\includegraphics[width=4.295cm, height=3.7cm, trim=11mm 0mm 11mm 7mm,clip]{MACC_Polar90_3.eps} }}\n    \\subfloat[$(\\theta,\\phi) = (30^\\circ, 0^\\circ)$]{{\\includegraphics[width=4.295cm, height=3.7cm,trim=11mm 0mm 11mm 7mm,clip]{MACC_Polar30_3.eps} }}\n    \t\\vspace*{-0.15cm}\n\\caption{MACC performance, which corresponds well with the AG performance in Fig. \\ref{fig:AG}.}\n    \\label{fig:MACC}%\n\t\\vspace*{-0.3cm}\n\\end{figure}\n \n At $\\theta = 90^\\circ$, i.e., $x$-$y$ plane, the TF case is slightly better for the AG and MACC, but the WNG performance for the TF case is better than the FF case. This is explained by noting that the steering vectors in the TF case have variations in both phase and amplitude (over microphones) in comparison to the FF case, which only has phase variations. The amplitude variations increase the spatial diversity for the TF case, which can be used to improve the spatial directivity of the beamformer.\nAt $\\theta = 30^\\circ$, the TF case has WNG performance better than the FF over the full frequency range; the AG for TF case is noticeably better than FF case for all frequencies, and significantly better for frequencies beyond 2 kHz.  Note that the WNG curves are lower-bounded by $-25$ dB, because of the WNG constraint specified in (\\ref{eq:MVDR}). Note also that in all cases, the MACC for the TF case is noticeably better than the FF case, because the FF case ignores the magnitude information, which provides invaluable characterization of the look direction. \n\n\\begin{figure}[h]%\n    \\centering\n    \\subfloat[$(\\theta,\\phi) = (90^\\circ, 0^\\circ)$]{{\\includegraphics[width=4.295cm, height=3.7cm, trim=10mm 0mm 11mm 7mm,clip]{ScatteredFieldPressure_Polar90_3.eps} }}%\n    \\subfloat[$(\\theta,\\phi) = (30^\\circ, 0^\\circ)$]{{\\includegraphics[width=4.295cm, height=3.7cm,trim=10mm 0mm 11mm 7mm,clip]{ScatteredFieldPressure_Polar30_3.eps} }}%\n\t\\vspace*{-0.1cm}\n    \\caption{Magnitude of scattered wavefield, $p_s$, (in dB) at two plane wave angles.}\n    \\label{fig:scatterfield}%\n\\end{figure}\n\nThe big deviation of the FF performance at $\\theta = 30^ \\circ$ is attributed to the magnitude of the scattered wavefield (which is ignored in the FF case). This is illustrated in Fig. \\ref{fig:scatterfield}, where we show the magnitude of the scattered wavefield at the five microphones at both angles (where the background plane wave has the same magnitude in both cases). Note that, the scattered wavefield at $\\theta = 30^\\circ$ is approximately $10$ dB stronger than $\\theta = 90^\\circ$ especially at high frequencies, which is manifested clearly in the corresponding AG\/WNG\/MACC behavior. This significant deviation of the free-field case demonstrates the limitation of this modeling and the necessity of incorporating the scattered field component through FEM modeling for beamformer design. \n\n\\vspace{-0.3cm}\n\n\\section{Conclusion and Future Work}\nThe free-field model does not provide accurate modeling for broadband beamformer design, especially when the scattered wavefield is significant. Therefore, designing beamformer metrics based on free-field modeling results in suboptimal performance.  To mitigate this issue, we described a simulation-based framework for modeling the total wavefield, which is shown to noticeably improve the beamformer design. The model is universal for any device surface, and it could be used for both near-field and far-field modeling by computing the steering vectors of spherical and plane waves, respectively. Future work will utilize the results of this work to develop novel design techniques for broadband beamformer and generic form-factors that are based on this realistic microphone array modeling \\cite{guangdongComsol2019}. Additionally, we expand the array processing metrics, and show a close matching of simulated and measured beampatterns for our proposed method \\cite{guangdongComsol2019}.\n\n\\bibliographystyle{IEEEbib}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe purpose of this paper is to consider the problem of finding, for $x, t \\ge 0$, the quantity \n\\begin{equation}\\label{eq:suptail}\n\t\\sup_{\\mathbf{X}}{\\mathbb{P}(X_1+\\ldots+X_k\\geq t)},\n\\end{equation}\nwhere the supremum is taken over all random vectors $\\mathbf{X}=(X_1,\\ldots,X_k)$ \nof nonnegative, independent and identically distributed (further \\emph{i.i.d.}) random variables \n$X_i$ such that $\\mathbb{E}(X_i) \\le x$ for $i=1,\\ldots,k$.\n\nFrom now on we assume that $t=1$, since, writing \n\\begin{equation*}\n\tm_k(x) = \\sup_{\\mathbf{X}}{\\mathbb{P}(X_1+\\ldots+X_k\\geq 1)},\n\\end{equation*}\nby rescaling we get that \\eqref{eq:suptail} is equal to $m_k(x\/t)$.\n\nFor $x \\ge 1\/k$ the trivial solution $m_k(x) = 1$ is given by $X_i$'s which are identically equal to $x$. \nFor $k=1$ and $x < 1$ the solution $m_1(x) = x$ is given by Markov's inequality and a zero-one random variable.\nIn the case of two variables the problem was solved by Hoeffding and Shrikhande~\\cite{HS} who showed that\n\n$$m_2(x)=\n\\begin{cases} \n2x-x^2 & \\text{for}\\quad x < 2\/5; \\\\\n4x^2 & \\text{for}\\quad 2\/5 \\leq x <  1\/2;\\\\\n\t1, & \\text{for}\\quad x \\ge 1\/2.\n\\end{cases}$$\n\n\nWe conjecture that the following generalization of the above results holds. \n\n\\begin{conjecture}\\label{conjLMS}\nFor every positive integer $k$ and $x\\geq 0$ we have\n\\begin{equation}\\label{eq:LMS}\n\tm_k(x)=\n\t\\begin{cases} \n\t\t1-(1-x)^k & \\text{for}\\quad x < x_0 ,\\\\\n\t\t(kx)^k & \\text{for}\\quad  x_0 \\le x < 1\/k,\\\\\n\t\t1, & \\text{for}\\quad  x\\geq 1\/k,\n\t\\end{cases}\n\\end{equation}\nwhere \n$x_0(k)$ is the solution of $1-(1-x)^k = (kx)^k$.\n\\end{conjecture}\n\nNote that the lower bound on $m_k(x)$ in the first case is given by $X_i$'s with a two-point distribution \n$\t\\mathbb{P}(X_i=1)=1-\\mathbb{P}(X_i=0)=x$\nand in the second case by $X_i$'s with distribution\n$\t\\mathbb{P}(X_i=1\/k)=1-\\mathbb{P}(X_i=0)=kx .$\n\nThe case when $X_i$'s are not necessarily identically distributed has also been studied. Let\n\\begin{equation*}\n\ts_k(x) = \\sup_{\\mathbf X} \\mathbb{P}\\left( X_1 + \\dots + X_k \\ge 1\\right),\n\\end{equation*}\nwhere the supremum is taken over all vectors of independent, nonnegative random variables with common mean $x$. Clearly, $m_k(x) \\le s_k(x)$. In 1966 Samuels \\cite{S66} formulated a conjecture on the least upper bound for the tail probability in terms of $\\cE X_i, i = 1, \\dots, k$, which are not necessarily equal. For simplicity, we state this conjecture in the case when means are equal.\n\\begin{conjecture}[Samuels \\cite{S66}]\\label{conj:Samuels}\nFor every positive integer $k$ and $x\\geq 0$ we have\n\t\\begin{equation}\\label{eq:Samuels}\n\t\ts_k(x)=\n\t\t\\begin{cases} \n\t\t\t1 - \\min_{t = 0}^{k-1} \\left(1 - \\frac{x}{1 - tx} \\right)^{k-t} & \\text{for}\\quad  x < 1\/k,\\\\\n\t\t\t1, & \\text{for}\\quad  x\\geq 1\/k,\n\t\t\\end{cases}\n\t\\end{equation}\n\\end{conjecture}\n\nThe lower bound on $s_k(x)$ is given by one of the random vectors $\\mathbf X_t, t=0, \\cdots, k-1$, where $\\mathbf X_t$ consists of~$t$ random variables identically equal to $x$ and $k-t$ i.i.d. random variables taking values $0$ and $1 - tx$. Therefore, if true, \\eqref{eq:Samuels} implies Conjecture \\ref{conjLMS} only when the minimum is attained by $t=0$. Computer-generated graphs suggest that the minimum is attained by $t=0$ when $x < x_1(k)$ and by $t= k - 1$ when $x \\ge x_1(k)$, where $x_1(k) \\in (0, 1\/k)$ is the solution of $1-(1 - x)^k = x\/ (1 - (k-1)x)$. In \\cite{AFHRRS} it was shown rigorously that the minimum is attained by $t=0$ for $x \\le 1\/(k+1)$.\n\nSamuels \\cite{S66, S68} confirmed \\eqref{eq:Samuels} for $k = 3, 4$. Computer-generated graphs of functions $s_3(x)$ and $s_4(x)$ suggest that for $k = 3,4$ we have\n\t\\label{thm:Samuels}\n\t\\begin{equation*}\n\t\tm_k(x)= s_k(x) = 1 - (1 -x)^k, \\qquad x \\le x_1(k),\t\n\t\\end{equation*}\n\twhere $x_1(3) = 0.27729\\dots$, $x_1(4) = 0.21737\\dots$.\n\n\tMoreover, Samuels \\cite{S69} proved that for $k \\ge 5$\n\t\\begin{equation*}\n\t\tm_k(x) = s_k(x) = 1 - (1-x)^k, \\qquad x \\le 1\/(k^2-k).\n\t\\end{equation*}\n\nOur main result can be stated as follows.\n\n\\begin{theorem}\\label{thmLMS}\n\tConjecture~\\ref{conjLMS} holds for $k=3$ and every $x$. Moreover, it holds for $k \\ge 5$ when $x < \\frac{1}{2k-1}$.\n\\end{theorem}\n\nThe proof of Theorem~\\ref{thmLMS} is based on an observation that \nConjecture~\\ref{conjLMS} is asymptotically equivalent \nto the fractional version of Erd\\H{o}s' Conjecture \non matchings in hypergraphs which we introduce in the next section.\n\n\n\n\\section{The hypergraph matching problem}\\label{sec2}\n\nA {\\em $k$-uniform hypergraph}  $H=(V,E)$  \nis a~set of vertices $V$ \ntogether with a~family $E$ of \n$k$-element subsets of $V$, called \\emph{edges}.\nA {\\em matching} is a~family of disjoint edges of $H$, and the size \nof the largest matching in $H$ is called a~{\\em matching number} and is denoted by $\\nu(H)$. \n\nIn~\\cite{E} Erd\\H{o}s\\ stated the following.\n\n\\begin{conjecture}[Erd\\H{o}s~\\cite{E}]\\label{conjerd}\nLet $H=(V,E)$ be a~$k$-uniform hypergraph, $|V|=n$, $\\nu(H) = s$. \nIf $n\\geq ks+k-1$, then \n\\begin{equation}\\label{eq:erdos_conj}\n|E|\\leq \\max\\bigg\\{\\binom nk-\\binom{n-s}k\\,,\\binom {ks+k-1}k\\bigg\\}.\n\\end{equation}\n\\end{conjecture}\n\nNote that the equality in Conjecture~\\ref{conjerd} holds either when $H$ is \na~hypergraph consisting of\nall $k$-element sets intersecting a~given subset $S\\subset V$, $|S|=s$,\nor when $H$ consists of all $k$-element subsets of a given subset $T\\subset V$, $|T|=ks+k-1$.\nWe denote these two families of hypergraphs by $Cov_{n,k}(s)$ and $Cl_{n,k}(ks+k-1)$, respectively.\n\nA similar problem can be formulated in terms of fractional matchings.\nA~\\emph{fractional matching} in a~hypergraph $H$ \nis a~function \n$$\n\\begin{aligned}\n&w:E\\rightarrow [0,1] \\text{ such that }\\\\ \n\\sum_{e\\ni v}{w(e)}&\\leq 1  \\text{ for every vertex } v\\in V.\n\\end{aligned}$$\nThen, $\\sum_{e\\in E}{w(e)}$ is a~{\\em size} of the matching $w$ and \na~size of the largest fractional matching in $H$, denoted by $\\nu^*(H)$,\nis a~{\\em fractional matching number}. \nAlon, Frankl, Huang, R\\\"odl, Ruci\\'nski and Sudakov~\\cite{AFHRRS} stated the following conjecture.\n\n\\begin{conjecture}[\\cite{AFHRRS}]\\label{conjfrac}\n\tLet $x \\in [0,1\/k]$ be fixed and let $H_n=(V_n,E_n)$ be a sequence of $k$-uniform hypergraphs such that $ \\nu^*(H_n) \\le x|V_n|$.\nThen\n\\begin{equation}\\label{eq:limsup}\n\t\\limsup_{n\\to \\infty} \\frac{ |E_n|}{{|V_n| \\choose k}} \\le \\max\\left\\{1-(1-x)^k, {(kx)^k}\\right\\}.\n\\end{equation}\n\\end{conjecture}\n\nFinding a~fractional matching number \nis a~linear programming problem. \nIts dual problem is to minimize the size of a \\emph{fractional vertex cover} of $H$, which is defined as a~function \n\\[\n\t\\begin{aligned}\n\t\t&w: V \\rightarrow[0, 1] \\text{ such that }\\\\ \n\t\t\\text{ for each } &e\\in E \\text{ we have } \\sum_{v\\in e}{w(v)}\\geq 1.\n\t\\end{aligned}\n\\]\nThen, $\\sum_{v\\in V}{w(v)}$ is the~\\emph{size} of $w$ and \nthe size of the smallest fractional vertex cover in $H$ is denoted by $\\tau^*(H)$. \nBy the Duality Theorem, \n\\begin{equation}\\label{eq:nutau}\n\t\\nu^*(H)=\\tau^*(H).\n\\end{equation}\n\nThe bound in Conjecture~\\ref{conjfrac}, if true, is \nattained by either a sequence $H_n \\in Cov_{n,k}(\\floor{xn})$ (which has a fractional vertex cover $w(v)= \\mathbf 1_{v \\in S}$ of size $\\floor{xn}$ and therefore satisfies $\\nu^*(H_n) \\le xn$) or $H_n \\in Cl_{n,k}(\\floor{kxn})$ (which has fractional vertex cover $w(v) = \\frac 1 k \\mathbf 1_{v \\in T}$ of size $\\floor{kxn}\/k$).\n\n\n\nObserve that if a fractional matching $w$ is such that $w(e)\\in \\{0,1\\}$ for every edge~$e$, \nthen $w$ is just a~matching or, more precisely, the indicator function of a~matching.\nThus, every integral matching is also a~fractional matching \nand hence\n\\begin{equation}\\label{eq:nunu}\n\t\\nu(H)\\leq \\nu^*(H),\n\\end{equation}\nso consequently, Conjecture~\\ref{conjfrac} follows from Conjecture~\\ref{conjerd}. Furthermore,\nConjecture~\\ref{conjerd} was confirmed for $k = 3$ by the first two authors \\cite{LM} (for $n$ bigger than some absolute constant) and by Frankl \\cite{Fnew} (for every $n$). \nMoreover, Frankl \\cite{Fgen} confirmed Conjecture \\ref{conjerd} for $k \\ge 4$ and $s \\le (n-k)\/(2k-1)$. Therefore, in view of \\eqref{eq:nunu} we have the following.\n\\begin{remark}[\\cite{Fnew,Fgen,LM}]\\label{thm:frac}\n\tConjecture~\\ref{conjfrac} holds for $k = 3$ and every $x$ as well as for $k \\ge 4$ and $x < 1\/(2k-1)$.\n\\end{remark}\n\n\n\\section{Proof of Theorem~\\ref{thmLMS}}\n\nWe  prove Theorem~\\ref{thmLMS} in two steps. \nFirst we observe that it is enough to confirm\nConjecture~\\ref{conjLMS} with some additional restrictions.\nThen we show the equivalence of Conjectures~\\ref{conjLMS} and~\\ref{conjfrac}.\n\nHere and below, given a discrete random variable, $\\supp(X)$ denotes the set of values which $X$ attains with positive probability.\n\n\\begin{lemma}\\label{lm1}\n\tIt suffices to prove Conjecture~\\ref{conjLMS} for $X_i$'s with discrete distribution satisfying the following properties: (i) $\\supp(X_i)$ is finite subset of $[0,1]$; (ii) $\\mathbb{P}\\left( X_i = a \\right) \\in \\mathbb Q$ for every $a \\in \\supp(X_i)$.\n\\end{lemma}\n\n\\begin{proof}\n\nDefine, for $x \\ge 0$, \n$$M(x)=\\max\\{1-(1-x)^k, (kx)^k, 1\\},$$\nwhich is equal to the right-hand side of \\eqref{eq:LMS}.\n\nLet us first assume that Conjecture~\\ref{conjLMS} holds for random variables\nsatisfying (i) and (ii) and show it holds for $X_i$'s satisfying (i) only, that is, with $\\supp(X_i)=\\{a_1,\\ldots,a_m\\} \\subset [0,1]$.\nLet $p_j=\\mathbb{P}\\left(X_i=a_j\\right)$, $j=1,\\ldots,m$.\nFor every sufficiently large integer $n$ define a random variable $Y_i^{(n)}$ such that\n$$\\mathbb{P}\\left(Y_i^{(n)}=a_j\\right)=p_j^{(n)} \\in \\mathbb Q, \\quad j = 1, \\dots, m$$\nwhere $p_j^{(n)}=\\lceil n p_j\\rceil \/n$ for $j=2,\\ldots,m$,\nand $p_1^{(n)}=1-\\sum_{j=2}^m p_j^{(n)}$ (note that the $p_1^{(n)}$ is positive for $n$ large enough).\nThen, for every $j$ we have\n$p_j^{(n)}\\leq p_j+1\/n,$ and therefore\n$$\\cE(Y_i^{(n)})\\leq \\cE(X_i)+ m\/n.$$\nApplying the conclusion of Conjecture 1 to $Y^{(n)}_i$'s  and using the continuity of function $M$, we get\n\\begin{align*}\n\t\\mathbb{P}\\left(\\textstyle \\sum_{i=1}^kX_i\\geq 1\\right) &= \\lim_{n \\to \\infty} \\mathbb{P}\\left(\\textstyle\\sum_{i=1}^kY_i^{(n)}\\geq 1\\right)\n&\t\\le \\lim_{n \\to \\infty} M(x + m\/n) = M(x).\n\\end{align*}\nLet us now assume that the Conjecture~\\ref{conjLMS} holds for random variables satisfying (i) and show it then holds for arbitrary $X_i, i = 1, \\dots, k$. For every positive integer $m$ define random variables\n$$Y_i^{(m)}=\\min \\{\\left\\lceil m X_i\\right\\rceil\/{m}, 1\\}.$$\nNote that $\\supp(Y_i^{(m)})$ is finite and contained in $[0,1]$.\nWe have $Y_i^{(m)}\\leq X_i + 1\/m$ and therefore $\\cE(Y_i^{(m)})\\leq \\cE(X_i) + 1\/m \\le x + 1\/m$. For sufficiently large $m$ we get \n\\begin{align*}\n\t\\mathbb{P}\\left(\\textstyle \\sum_{i=1}^kX_i\\geq 1\\right) &\\leq \\mathbb{P} \\left( \\textstyle \\sum_{i=1}^k \\lceil m X_i \\rceil\/m \\ge 1 \\right) = \\mathbb{P}\\left(\\textstyle \\sum_{i=1}^kY_i^{(m)}\\geq 1\\right) \\leq M(x+1\/m).\n\\end{align*}\nTaking a limit over $m \\to \\infty$ and using the fact that $M$ is continuous (from the right), we obtain \n$$\\mathbb{P}\\left(\\textstyle \\sum_{i=1}^kX_i\\geq 1\\right)\\leq M(x).\\quad \\qed$$\n\\renewcommand{\\qed}{}\n\\end{proof}\n\n\\begin{lemma}\\label{lm2}\n\tFor every $k$ and $x \\in [0,1\/k]$ Conjectures~\\ref{conjLMS} and~\\ref{conjfrac} are equivalent.\n\\end{lemma}\n\n\\begin{proof}\n\tThe proof that Conjecture~\\ref{conjLMS} implies Conjecture~\\ref{conjfrac} goes along the same lines as the proof of Theorem 2.1 in~\\cite{AFHRRS}.\nWe recall it below for the sake of completeness.\n\nLet us fix $k$ and $x \\in [0,1\/k]$ and suppose that Conjecture~\\ref{conjLMS} holds.\nMoreover, let $H_n=(V_n,E_n)$ be a~sequence of $k$-uniform hypergraphs such that $\\nu^*(H_n)\\leq x|V_n|=xn$. \nBy~(\\ref{eq:nutau}) we have $\\tau^*(H_n)=\\nu^*(H_n)\\leq xn$, \nhence there exists a weight function $w_n:V_n\\rightarrow [0,1]$ such that \n$$\\sum_{v\\in V_n}{w_n(v)}=xn,$$\nand $\\sum_{v\\in e}{w_n(v)}\\geq 1$ for every $e\\in E_n$.\n\nLet $(v^{n}_1,\\ldots,v^{n}_k) \\in V_n^k$ be a~vector of random vertices,\neach chosen independently and uniformly over $V_n$.\nNote that $w_n(v^{n}_1),\\ldots,w_n(v^{n}_k)$ are nonnegative, independent and identically distributed random variables with mean\n\\begin{equation*}\n\t\\mathbb{E}(w_n(v^{n}_i))=\\frac{1}{|V_n|}\\sum_{v\\in V_n}{w_n(v)}=\\frac{1}{n}xn=x.\n\\end{equation*}\nObserve also  that \n\\begin{equation}\\label{eq1}\n\\mathbb{P}(\\{v^{n}_1,\\ldots,v^{n}_k\\}\\in E_n)=\\frac{k!|E_n|}{n^k}.\\end{equation}\nOn the other hand, since $w_n$ is a vertex cover of $H_n$, for $\\{v^{n}_1,\\ldots,v^{n}_k\\} \\in E_n$ we have $\\sum_{i=1}^k{w_n(v^{n}_i)}\\geq 1$ and thus\n\\begin{equation}\\label{eq2}\n\\mathbb{P}(\\{v^{n}_1,\\ldots,v^{n}_k\\}\\in E_n)\\leq\\mathbb{P}\\left(\\textstyle \\sum_{i=1}^k{w_n(v^{n}_i)}\\geq 1\\right).\\end{equation}\nFrom (\\ref{eq1}), (\\ref{eq2}) and \nthe assumption that Conjecture~\\ref{conjLMS} is true, we conclude that\n\\begin{equation*}\n\t\\limsup_{n\\to \\infty} \\frac{ |E_n|}{{|V_n| \\choose k}} \\le \\limsup_{n\\to \\infty}  \\mathbb{P}\\left(\\textstyle \\sum_{i=1}^k{w_n(v^{n}_i)}\\geq 1\\right)\\leq \t\\max\\left\\{1-(1-x)^k, {(kx)^k}\\right\\}.\n\\end{equation*}\n\nIt remains to prove the reverse implication.\nLet us assume that Conjecture~\\ref{conjfrac} is valid for some $k$ and $x \\in [0,1\/k]$.\nDue to Lemma~\\ref{lm1} it is enough to show that Conjecture~\\ref{conjLMS} holds for $X_i$'s attaining a finite set of values $a_1, \\dots, a_m \\in [0,1]$ such that\n$$\\mathbb{P}(X_i=a_j)={p_j}\/{q_j}, \\qquad j = 1, \\dots, m$$\nfor some positive integers $p_j$ and $q_j$. \nMoreover, let $r$ be the smallest common multiple of the numbers $\\{q_1,\\ldots,q_m\\}$, and define integers \n\\begin{equation}\\label{eq:pprime}\n\tp'_j = rp_j\/q_j, \\qquad j = 1, \\dots, m. \n\\end{equation}\n\nIn order to apply Conjecture \\ref{conjfrac}, we  define hypergraphs \nwith bounded fractional matching number.\nFor $n=1,2,\\ldots$, let $V_n=[nr]$. Observing that $np_1' + \\dots + np_m' = nr$, define a function $w_n:V_n\\rightarrow[0,1]$ in such a way that for each $j = 1, \\dots, m$ function $w_n(v)$ takes value $a_j$ precisely $np_j'$ times.\nLet $H_n=(V_n,E_n)$ be a hypergraph with the edge set \n$$E_n=\\left\\{e\\in\\binom{V_n}{k}: \\sum_{v\\in e}{w_n(v)}\\geq 1\\right\\}.$$\nIn view of  \\eqref{eq:pprime}, we have that $w_n$ is a~fractional\nvertex cover of $H_n$ of size \n\\begin{equation*}\n\t\\sum_{v=1}^{nr}{w_n(v)}=\\sum_{j=1}^m{a_j}n{p'_j} = n\\sum_{j=1}^m ra_j\\frac{p_j}{q_j} = nr \\mathbb E (X_i) \\le xnr.\n\\end{equation*}\nHence by \\eqref{eq:nutau} we have $\\nu^*(H_n)=\\tau^*(H_n)\\leq xnr$ and therefore  \\eqref{eq:limsup} gives\n\\begin{equation}\\label{eq:Ebound}\n\t\\limsup_{n \\to \\infty}\\frac{ |E_n|} {\\binom {nr}k}\\le \\max \\left\\{ 1-(1-x)^k, (kx)^k \\right\\}.\n\\end{equation}\n\nLet $(v^{n}_1,\\ldots,v^{n}_k) \\in V_n^k$ be a~vector of random vertices,\neach chosen independently and uniformly over $V_n$.\nNote that for every $n$ the random variable $w_n(v_i^{n})$ has the same distribution as $X_i$, since, by \\eqref{eq:pprime}, \n$$\\mathbb{P}(w_n(v_i^{n}) = a_j)=\\frac{np_j'}{|V_n|}=\\frac{n r p_j\/q_j}{nr}=\\frac{p_j}{q_j}, \\qquad j = 1, \\dots, m.$$\nLet $N_n$ denote the number of $k$-element vectors $(v_1,\\ldots,v_k) \\in V_n^k$\nof vertices with at least two equal coordinates. We have \n\\begin{align*}\n\\mathbb{P}(X_1+\\ldots +X_k\\geq 1)=& \\mathbb{P}(w_n(v^{n}_1) + \\dots + w_n(v^{n}_k) \\geq 1)\\\\\n=&\\frac{\\left|\\left\\{(v_1,\\ldots,v_k)\\in V_n^{k}: \\sum_i{w(v_i)}\\geq 1\\right\\}\\right|}{(nr)^k}\\\\\n=& \\frac{k!|E_n|+N_n}{(nr)^k} \\le \\frac{|E_n|}{\\binom {nr}k} + \\frac{\\binom {k} 2 (nr)^{k-1}}{(nr)^k}.\n\\end{align*}\n\nTaking the limit over $n \\to \\infty$ and using \\eqref{eq:Ebound} we get that\n$$\\mathbb{P}(X_1+\\ldots +X_k\\geq 1) \\leq \\max\\{1-(1-x)^k,(kx)^k\\}.\\quad\\qed$$\n\\renewcommand{\\qed}{}\n\\end{proof}\n\nNow Theorem~\\ref{thmLMS} follows from Lemma~\\ref{lm2}\nand Remark \\ref{thm:frac}.\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe tendency of epidemics to return in repeated waves has been known since the 1918 Spanish flu \\cite{Spanishflu}, and the recent COVID-19 pandemic is no exception. In November 2020, the history of reported daily cases, or incidence rate, of COVID-19 varies considerably across the world's regions. The broad picture is as follows \\cite{ecdc}: The outbreak in China was practically over five weeks after a lockdown was imposed on January 23. Europe took over as the epicenter of the pandemic in late February. After lockdowns in most European countries in early March, followed by a gradual relaxation of these interventions, the first wave was over in the late spring. Incidence rates remained very low during the summer until they started to increase slowly in August. In late October, incidence rates higher than in March are common in Europe and are growing exponentially with a week's doubling time. The United States developed its first wave delayed by a week or two compared to Europe and a second and stronger wave throughout the summer. The country is now dealing with a third and even stronger wave. Many countries in South America, Africa, and South-East Asia are in the middle of (or have just finished) the first wave. On the other hand, a few countries, like New Zealand, Australia, Japan, and South Korea, have finished the second wave and managed to prevent it from becoming much stronger than the first one. Although there are a plethora of different wave patterns among the world's countries, one could hope that these patterns fall into a limited number of identifiable groups. \n\nIn this paper, we do not claim that there is any epidemiological rule that states a pandemic evolving without social intervention must come in increasingly severe waves. On the contrary, the simple compartmental models devise the evolution of one single wave that finally declines due to herd immunity. We shall adopt the simplest of all such models here, the Susceptible-Infectious-Recovered (SIR) model, to describe the evolution of the epidemic state variables. However, in the SIR model, the effective reproduction number ${\\cal R}$, the average number of new infections caused by one infected individual, is proportional to the fraction of susceptible individuals $S$ in the population. If initially ${\\cal R}={\\cal R}_0>1$, the daily number of new infections (incidence rate) will increase with time $t$ until $S$ has been reduced to the point where ${\\cal R}$ goes below 1, and then decay to zero as $t\\rightarrow \\infty$.\n\nAt present, herd immunity is not an essential mechanism in the COVID-19 pandemic because the fraction of susceptible individuals is still close to 1 in most populations. Consequently, the time variation of the reproduction number is predominantly caused by changes in social behavior. Changes in virus contagiousness could also play a r\\^ole, but we have not taken virus mutations into account in this paper. Thus, by adopting the approximation $S=1$ in the definition of the reproduction number, the SIR model reduces to a set of two first-order ordinary differential equations for the cumulative number of infected cases $J(t)$ and the instantaneous number of infectious individuals $I(t)$. These are the ``state variables'' of the epidemic, which are driven by the reproduction number ${\\cal R}(t)$. The evolution of the epidemic state $[J(t),I(t)]$ can be computed as a solution to these equations if ${\\cal R}(t)$ is known.\n\nThis paper's philosophy is to make the simplifying assumption that ${\\cal R}(t)$ responds to the epidemic state. More precisely, that the rate of change of ${\\cal R}(t)$ is a function of the rate of change of $J(t)$ depending on a set of parameters with distinct and straightforward interpretations that characterize the response. This mathematical relationship turns the SIR model into a closed model for the epidemic evolution, \nwhich depends on three parameters: i) the relaxation rate when incidence rate is low, ii) the intervention rate when incidence is high, and iii) a fatigue rate that gradually weakens the effect of interventions over time.\nThese parameters can be fitted to the incidence rate time series $X(t)=d_tJ(t)$ reported by different countries. \nThe analysis of the fitted values of these parameters, allows to identify groups of countries with similar evolution of the epidemics and help to understand the most effective mechanisms controlling the epidemic's spread. \nOne of our findings is stated in the paper's title; intervention fatigue is the primary mechanism that gives rise to the strong secondary waves emerging in many countries.\\\\\n\n\\noindent \\textit{Paper outline.}\\\\\nSection~\\ref{sec:reserse_SIR} presents a method for reconstructing the ${\\cal R}(t)$-profile from the observed time series for the daily incidence rate using a simple inversion of the SIR model.\nThe method's effectiveness is illustrated in Section~\\ref {:reconstruction_examples} by application to selected representative countries. \nIn Section~\\ref{sec:clustering} we compute a dissimilarity measure between the reconstructed ${\\cal R}(t)$-profiles for each country in the world.\nBased on such a dissimilarity, we generate a dendrogram that hierarchically partitions countries according to their evolutionary paths of the epidemic.\nFinally, in Section~\\ref{sec:closed_model} we construct a self-consistent, closed model for the simultaneous evolution of $J(t)$ and ${\\cal R}(t)$ and describe how this model can be fitted to the observed data for $d_tJ$ for individual countries.\n\nIn Section \\ref{sec:clustering_results} we synthesize the reconstructed ${\\cal R}$-curves for the majority of the world's countries and use the dendrogram to group them into seven clusters, which are also shown on a World map. \nThe features characterizing each cluster are analyzed and discussed. \nSection~\\ref{sec:params_exploration} illustrates the scenarios of the epidemic evolution that can be derived by solving the equations of the proposed closed model for different sets of parameters. \nThe proposed model's effectiveness is empirically validated in Section~\\ref{sec:fit}, where the model parameters are numerically fit to the incidence data for some selected countries exhibiting different characteristic patterns of epidemic evolution.\n\nThe possible implications of these results for COVID-19 strategic preparedness and response plans are discussed in Section \\ref{sec:discuss}.\n\n\n\\section{Methods}\n\\label{sec:Methods}\n\n\\subsection{Estimating the reproduction number from incidence rate data}\n\\label{sec:reserse_SIR}\nLet $S$ be the fraction of susceptible individuals in a population, $I$ the fraction of infectious, and $R$ the fraction of individuals ``removed\" from the susceptible population (\\textit{e.g.}, recovered, isolated, or deceased individuals). A simple model describing the evolution of these variables is the classical SIR-model \\cite{SIR}, \n\\begin{linenomath*}\n\\begin{eqnarray} \n\\label{1}\n\\frac{dS}{dt}&=&-\\beta IS, \\\\ \n\\label{2}\n\\frac{dI}{dt}&=&\\beta IS-\\alpha I, \\\\ \n\\label{3}\n\\frac{dR}{dt}&=&\\alpha I, \n\\end{eqnarray}\n\\end{linenomath*}\nwhere $\\alpha$ is the rate by which the infected are isolated from the susceptible population. Another interpretation of $\\alpha$ is that $\\alpha^{-1}$ is the average duration of the period an individual is infectious, which essentially depends only on the properties of the pathogen. As long as these do not change significantly, $\\alpha$ will remain constant in time. In this paper we use $\\alpha=1\/(8 \\text{ days})$ but our results are not sensitive to this choice. The coefficient $\\beta$, on the other hand, is the rate by which the infection is being transmitted. It evolves in time as societal interventions change. It is also influenced by behavioral changes in the susceptible population, such as eliminating superspreaders. \nThe effective reproduction number is defined as \n\\begin{linenomath*}\n\\begin{equation}\n{\\cal R}(t) \\equiv \\frac{\\beta(t)}{\\alpha}\\label{4}\n\\end{equation}\n\\end{linenomath*}\nand can be interpreted as the average number of new infections caused by an infected individual over the infectious period $\\alpha^{-1}$.\n\nThe coupled system given by Eqs.\\;(\\ref{1}) and (\\ref{2}),  with initial conditions $S_0$ and $I_0$, constitutes a closed nonlinear initial value problem.  Eq.\\;(\\ref{3}) is not a part of this system since it is trivially integrated to yield the removed population $R(t)$ once $I(t)$ is known.\n\nThe method developed in this paper is valid for an infectious disease with a new pathogen which is transmitted by  contact between infectious and susceptible individuals. This implies that there is practically no immunity in the population from the  start of the epidemic and we shall assume that this herd immunity is low throughout the period for which we estimate the reproduction number. In other words, we shall assume that $S\\approx 1$, and hence that the cumulative fraction $J=1-S$ of infected individuals is always much less than unity ($J\\ll 1$). By introducing $S=1-J$ in Eqs.\\;(\\ref{1}) and (\\ref{2}), and by neglecting the term $\\beta IJ$ compared to the term $\\beta I$ in Eq.\\;(\\ref{1}), these two equations reduce to a linear model for $J$ and $I$,\n\\begin{linenomath*}\n\\begin{eqnarray} \\label{5} \n\\frac{dJ}{dt}&=&\\alpha {\\cal R}(t) I \\\\ \n\\label{6} \n\\frac{dI}{dt}&=&\\alpha[{\\cal R}(t) -1] I\\,. \n\\end{eqnarray}\n\\end{linenomath*}\nNote that $\\gamma_I(t)=I^{-1}dI\/dt=\\alpha [{\\cal R}(t)-1]$ is the relative growth rate for the instantaneous number of infectious individuals $I(t)$, which is positive when ${\\cal R}(t)>1$ and negative when ${\\cal R}(t)<1$. \nBy integrating Eq.\\;(\\ref{6}) and by inserting the result on the right hand side of Eq.\\;(\\ref{5}), we obtain that the daily number of new infections $dJ\/dt$ is determined by the initial  $I_0$ and  the  history of ${\\cal R}(t)$ on the interval $(0,t)$; \n\\begin{linenomath*}\n\\begin{equation}\n \\frac{dJ}{dt}=\\alpha I_0{\\cal R}(t)\\exp{\\Bigg{(}\\int_0^t\\alpha[{\\cal R}(t')-1)] dt'\\Bigg{)}}. \\label{7} \n\\end{equation}\n\\end{linenomath*}\nWhat we are interested in here, however, is the inverse relationship; suppose the evolution of $dJ\/dt$ is known, how do we find the evolution of the reproduction number ${\\cal R}(t)$? \n\nBy using Eq.\\;(\\ref{5}) to replace $\\alpha {\\cal R}I$ by $d_tJ$ in Eq.\\;(\\ref{6}), the latter can be integrated to yield,\n\\begin{linenomath*}\n\\begin{equation}\n    I(t)=I_0 e^{-\\alpha t}+\\int_0^t e^{-\\alpha (t-t')}d_{t'}J\\; dt', \\label{8}\n\\end{equation}\n\\end{linenomath*}\nwhich allows us to compute ${\\cal R}(t)$ from Eq.\\,(\\ref{5});\n\\begin{linenomath*}\n\\begin{equation}\n    {\\cal R}(t) = \\frac{d_tJ}{\\alpha I} = \\frac{1}{\\alpha}\\frac{d_tJ}{ I_0 e^{-\\alpha t}+\\int_0^t e^{\\alpha (t'- t)}d_{t'}J\\; dt'}.\\label{9}\n\\end{equation}\n\\end{linenomath*}\n\nProvided a time series for $J(t)$ is available, we can approximate $d_tJ$ as a finite difference and the integral in Eq.\\;(\\ref{9}) as a discrete sum. \nThis sum gives us a fast and direct algorithm to estimate ${\\cal R}(t)$.\n\n\n\\subsection{{\\cal R}(t)-reconstructions for individual countries}\n\\label{sec:reconstruction_examples}\nSince we do not have actual measurements of the cumulative number of infected, to estimate the $\\mathcal{R}(t)$-curves for each country using Eq.\\;(\\ref{9}) we rely on the number of confirmed cases as a proxy for $d_tJ(t)$. \nSpecifically, we assume that the incidence rate $d_tJ(t)$ is proportional to the daily number of confirmed cases $X(t)$. \n\nThe time series $X(t)$ of new daily cases reported for each country are taken from the Our World in Data database\\footnote{\\url{https:\/\/ourworldindata.org\/coronavirus-source-data}}.\nFigure~\\ref{fig:examples} shows three examples of $\\mathcal{R}(t)$ estimated using Eq.\\;(\\ref{9}) from the new daily cases $X(t)$ reported by Sweden, Italy, and Argentina. \n\\begin{figure}[t!]\n\t\\centering\n\\includegraphics[width=16cm]{Fig1.pdf}\n    \\caption{\\footnotesize $\\mathcal{R}(t)$ estimated with the proposed method from the time series of new daily cases reported in Sweden, Italy, and Argentina. Red curves are reported incidence rates, and the blue curves the reconstructed reproduction numbers.}\n\t\\label{fig:examples}\n\\end{figure}\nThe initial values assumed by $\\mathcal{R}(t)$ are affected by different choices of $I_0$. The transient effect given by the initial conditions quickly vanishes as $t$ increases, and $\\mathcal{R}(t)$ converges to a stable solution since the first term in the denominator of Eq.\\;(\\ref{9}) goes exponentially to zero. To compute the results, we generated several initial conditions for $I_0$ in a reasonable range, and we discarded the transient phase, depicted as a gray area in Fig.~\\ref{fig:examples}. In these plots we have taken into consideration the delay between the date of infection and the reported positive tests, which may amount to approximately one week. Thus, the actual ${\\cal R}(t)$ curves should be shifted towards the left by approximately this amount.\n\nThe three countries in Fig.~\\ref{fig:examples} are characterized by a different evolution of the epidemics.\nSweden had a long first wave that peaked in the middle of June and the second wave started when the first one was not completely over.\nThis is reflected in the estimated $\\mathcal{R}(t)$, which stays for a long time interval above 1.\nItaly had a first wave stronger than other countries, that was brought down completely due to the lockdown.\nThe estimated $\\mathcal{R}(t)$ starts from very high values and quickly goes below 1 by the beginning of April.\nFinally, the number of new cases in Argentina kept growing very slowly, but consistently, until the middle of October and there are no two distinct waves like in many other countries. \nConsequently, the $\\mathcal{R}(t)$ is characterized by values that are slightly above 1 until the beginning of November.\n\n\n\\subsection{Cluster analysis of {\\cal R}(t)-curves}\n\\label{sec:clustering}\n\nRather than presenting reconstructions of ${\\cal R}(t)$  case by case for all of the world's countries, it would bring more insight to combine them in groups of ${\\cal R}(t)$-curves according to some common features and then analyze the characteristics of each group.\nTherefore, we follow such an indirect approach where we first cluster the $\\mathcal{R}(t)$ curves of different countries and then analyze the clustering partition and the representatives of each cluster.\nThe cornerstone of each clustering algorithm is the computation of a dissimilarity measure between the data samples.\nSince we are dealing with with sequential data, we leverage on a dissimilarity measure $\\delta_{i,j} = d(x_i, x_j)$ that yields a real number $\\delta_{i,j}$ proportional to the discrepancy between the time series $x_i$ and $x_j$.\n\nA large variety of time series dissimilarity measures have been proposed in the literature, including those based on statistical methods~\\cite{de2011tail}, signal processing~\\cite{chan1999efficient}, kernel methods~\\cite{mikalsen2018time}, and reservoir computing~\\cite{bianchi2020reservoir}.\nIn this paper, we adopt the Dynamic Time Warping (DTW) distance~\\cite{keogh2005exact}, which is an efficient and well-known algorithm that computes the dissimilarity between two sequences as the cost required to obtain an optimal match between them. The cost is computed as the sum of absolute differences between a set of indices in the two time series.\nDTW allows similar shapes to match, even if they are out of phase or, in general, not perfectly synchronized along the time axis.\n\nFrom the dissimilarity $\\delta_{i,j}$ between countries $i$ and $j$, it is possible to compute a clustering partition, where similar $\\mathcal{R}$-time-series are assigned to the same cluster.\nSeveral approaches can be used to generate the clusters~\\cite{aghabozorgi2015time}. \nWe opted for a hierarchical clustering method~\\cite{cohen2018hierarchical}, which gradually joins data samples together by increasing the maximum radius of the clusters' $\\delta_\\text{max}$.\nOne of the main advantages of hierarchical clustering is the possibility of generating a dendrogram, which allows to visually explore the structure of the clustering partition at different resolution levels.\n\n\n\n\\subsection{A closed model for model for the  epidemic evolution}\n\\label{sec:closed_model}\nThe SIR-model does not constitute a closed model for the evolution of $J(t)$, $I(t)$, and ${\\cal R}(t)$. Eqs.\\;(\\ref{5}--\\ref{6}) describe the dynamics of the epidemic state variables $J(t)$ and $I(t)$ when the evolution of the  social state represented by ${\\cal R}(t)$ is given. Eq.\\;(\\ref{9}) is nothing but an inverse of this relationship and should not be interpreted as a social response of ${\\cal R}(t)$ to changes in the epidemic state variables. A closed model can only be obtained by adding an equation describing such a response. \nWhile a simple dynamical model can not reflect the whole complexity of the social response, it may still provide some useful insight. \n\nWe shall represent this response by assuming that the rate of change  $d_t{\\cal R}(t)$ is a function of the incidence rate $X(t)=d_tJ(t)$, \nand that this function is positive when $X(t)$ is below a threshold $X^*$ and negative when it is above that threshold. When the incidence rate is low, society responds by relaxing restrictions, and the reproduction number increases. When the incidence rate exceeds the threshold $X^*$, restrictions are introduced that make $d_t{\\cal R}(t)$ to change sign from positive to negative.\n\nIn the following, it is convenient to introduce a dimensionless time variable $t'= \\alpha t$, which allows us to formulate the differential equations as functions of the mean infectious time $\\alpha^{-1}$ (which becomes the new time unit) rather than days.  \nAccordingly, Eq.\\;(\\ref{5}) can be written as\n\\begin{linenomath*}\n\\begin{equation}\n X(t')\\equiv d_{t'} J= {\\cal R} I,   \\label{16}\n\\end{equation} \n\\end{linenomath*}\nand we have a closed model for $I(t')$ and ${\\cal R}(t')$ in the form of the dynamical system,\n\\begin{linenomath*}\n\\begin{eqnarray}\n\\label{17} \n\\frac{d{\\cal R}}{dt'} &=& f(X)=f({\\cal R} I),\\\\ \n\\label{18}\n\\frac{dI}{dt'} &=& ({\\cal R}-1)I, \n\\end{eqnarray}\n\\end{linenomath*}\nwhere $f(X)$ is assumed to be a differentiable function which is decreasing in a neighborhood of $X^*$ and with $f(X^*)=0$. The system has a fixed point in ${\\cal R}=1$ and  $I = X^*$. In this state, the number of infected stays constant at the threshold value.\n\nBy linearization of $f(X)$ around the fixed point ${\\cal R}=1$ and  $I = X^*$, and by introducing the rate constant $\\nu = -(1\/2)X^*f'(X^*)>0$, the system reduces to\n\\begin{linenomath*}\n\\begin{eqnarray}\n\\label{19}\n\\frac{d\\Delta {\\cal R}}{dt'} &=& -2\\nu  (\\Delta {\\cal R} + \\Delta \\tilde{I}+ \\Delta {\\cal R}\\Delta \\tilde{I}) \\\\ \n\\label{20}\n\\frac{d\\Delta \\tilde{I}}{dt'} &=&  \\Delta {\\cal R}(1+\\Delta \\tilde{I}), \n\\end{eqnarray}\n\\end{linenomath*}\nwhere we have introduced $\\Delta {\\cal R}=1-{\\cal R}$ and the normalized number of infected $\\tilde{I}=I\/X^*=1+\\Delta \\tilde{I}$.  This nonlinear dynamical system has a stable fixed point in $(\\Delta {\\cal R},\\Delta \\tilde{I})=(0,0)$.\n\n\n\\subsubsection{The damped harmonic oscillator model}\n\n In this section we demonstrate that if $X$ is close to the threshold value $X^*$ and ${\\cal R}$ is close to 1, the linearization of Eqs.\\;(\\ref{17}) and (\\ref{18}) leads to the equation for a damped harmonic oscillator. The purpose is to show analytically under what circumstances a damped oscillation is a natural time-asymptotic state of the epidemic. In Section \\ref{nonlinear model} we argue that the model needs to be generalized to yield realistic descriptions of epidemic curves in most countries and, hence, the present section may be skipped  without losing anything essential. \n \n In the vicinity of the stable state $(\\Delta {\\cal R},\\Delta \\tilde{I})=(0,0)$, linearization yields the damped, harmonic oscillator equation, \n\\begin{linenomath*}\n\\begin{equation}\n\\frac{d^2 \\Delta \\tilde{I}}{dt'^2}  + 2\\nu  \\frac{ d\\Delta \\tilde{I}}{dt'} + (\\omega^2+\\nu^2) \\Delta \\tilde{I} = 0,\n\\label{21}\n\\end{equation}\n\\end{linenomath*}\nwhere $\\omega^2  = 2\\nu -\\nu^2$.\nFor $\\nu<2$, the general solution is the damped oscillator\n\\begin{linenomath*}\n\\begin{equation}\n\\Delta \\tilde{I}(t') = A \\, e^{-\\mu t'} \\cos(\\omega t' + \\varphi),    \\label{22}\n\\end{equation}\n\\end{linenomath*}\nwhere $A$ and $\\varphi$ are integration constants and $\\mu\\equiv\\nu$,\nand for $\\nu\\geq 2$ the non-oscillatory strongly damped solution which for large $t'$ goes as\n\\begin{linenomath*}\n\\begin{equation}\n  \\Delta  \\tilde{I}(t')=Be^{-\\mu^{(-)} t'}+Ce^{-\\mu^{(+)}t'}, \\label{23}\n\\end{equation}\n\\end{linenomath*}\nwhere $B$ and $C$ are constants of integration and $\\mu^{(\\pm)}\\equiv \\nu(1\\pm \\sqrt{1-2\/\\nu})$. From Eq.\\;(\\ref{20}),  we have \n\\begin{linenomath*}\n\\begin{equation}\n    \\Delta {\\cal R}=\\frac{d\\Delta \\tilde{I}}{dt'}, \\label{24}\n\\end{equation}\n\\end{linenomath*}\nand from Eq.\\;(\\ref{16}),\n\\begin{linenomath*}\n\\begin{equation}\n\\Delta X(t')=\\Delta(\\tilde{I}(t'){\\cal R}(t'))  \\approx  \\Delta {\\cal R}(t') + \\Delta \\tilde{I}(t')=\\frac{d\\Delta \\tilde{I}}{dt'}+\\Delta \\tilde{I}, \\label{25}\n\\end{equation}\n\\end{linenomath*}\nwhich means that $\\Delta I$, $\\Delta {\\cal R}$, and $\\Delta X$ experience the same damped oscillations with some phase shifts, or the same strongly damped solutions.\n\nThe frequency $\\omega$ (for $0<\\nu<2$) and the damping  rate $\\mu$ (for $0<\\nu<\\infty$) are plotted against the parameter $\\nu$ in Figure 1. The oscillation frequency and the damping rate are of comparable magnitude for $\\nu<1$, but the damping dominates in the interval $1<\\nu<2$. For $\\nu>2$, the damping rate decreases towards 1 as $\\nu$ increases.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=8 cm]{Fig2.pdf}\n\\caption{The yellow curve shows the frequency $\\omega=\\sqrt{2\\nu-\\nu^2}$, and the blue curve the damping rate $\\mu=\\nu$ for $\\nu<2$ and $\\mu=\\nu(1-\\sqrt{1-2\/\\nu}-1)$ for $\\nu\\geq 2$.}\n\\end{figure}  \n\nThe most rapid control of the epidemic is obtained when $\\mu\\approx 2$, but we also have reasonably rapid control for larger $\\nu$. Slower damping takes place when $\\nu<1$, and slower the lower $\\nu$. Hence, what really should be avoided is $\\nu$ much less than 1.\nThe  linearization of  $f(X)$ around $X^*$ in Eq.\\;(\\ref{17}) yields\n\\begin{linenomath*}\n\\begin{equation}\n    \\frac{d{\\cal R}}{dt'}=-\\nu\\Delta \\tilde{X}, \\label{26}\n\\end{equation}\n\\end{linenomath*}\nwhere $\\tilde{X}=X\/X^*$ is the incidence rate normalized to its threshold value. which shows that $\\nu$ is a measure of how fast the rate of change in ${\\cal R}$ responds to the deviation of $\\tilde{X}$ from its threshold value $\\tilde{X}^*=1$. If $\\nu\\ll 1$ , then $d_t{\\cal R}$ responds slowly to $\\Delta \\tilde{X}$, \\textit{i.e.}, there is a slow social response to the rise or decay of the incidence. Eq.\\;(\\ref{21}) and Figure~\\ref{fig:oscillator} then yields a damped oscillation with envelope that decays exponentially at a rate $\\mu=\\nu$. A characteristic duration of the epidemic is $\\tau\\equiv \\nu^{-1}$ and the characteristic time scale of the oscillation  is $T =\\omega ^{-1}=1\/\\sqrt{2\\nu-\\nu^2}$. Since the ratio between the two is $T\/\\tau =\\sqrt{\\nu\/(2-\\nu)}$, we observe that the oscillation scale $T$ is  longer than the decay time $\\tau$ for all $\\nu>1$. Hence, this model suggests that oscillatory behavior and a long duration of the epidemic are features we expect to observe when the social response is slow ($\\nu<1$).\n\n\\subsubsection{A nonlinear, three-parameter oscillator model}\n\\label{nonlinear model}\nAlthough the linear oscillator model gives some insight into the mechanism that makes the epidemic return in repeated waves, there is an obvious lack of realism. One is to neglect the terms containing the product $\\Delta {\\cal R}\\Delta \\tilde{I}$ in Eqs.\\;(\\ref{19})-(\\ref{20}), since neither $\\Delta {\\cal R}$ nor $\\Delta \\tilde{I}$ are, in general, small. \nThere is also little reason to expect that the rate of change $\\nu$ is the same below and above the social response threshold. \nBelow the threshold, ${\\cal R}$ increases because of the intervention's termination and because the population relaxes. The more relaxed, the larger $\\nu$, so let us denote this parameter $\\nu_1$ the ``relaxation rate''. Above the threshold, ${\\cal R}$ decreases because of the interventions aiming to strike the epidemic down. Stronger intervention translates into a larger ``intervention rate'' $\\nu_2$. Even with these generalizations, the model will still give a damped, nonlinear oscillation and, hence, is unable to describe a situation where the second wave is stronger than the first. \nA generalization which may cover such a situation is to let the intervention rate decay with time, for instance, exponentially, such that we have an ultimate model for $\\nu(t')$ on the form\n\\begin{linenomath*}\n\\begin{equation}\n    \\nu(t')=\\nu_1\\theta(-\\Delta\\tilde{X})+\\nu_2e^{-\\nu_3 t'}\\theta(\\Delta \\tilde{X}), \\label{27}\n\\end{equation}\n\\end{linenomath*}\nwhere $\\theta(x)$ is the unit step function. The parameter $\\nu_3$ can be thought of as a ``fatigue rate'', \\textit{i.e.}, the rate at which the strike-down rate is reduced because the population is becoming increasingly tired of interventions and restrictions.\n\nNote also that the time dependence of the reproduction number in this model is independent of the response threshold $X^*$. This is because $X^*$ has been eliminated in  Eqs.\\;(\\ref{19})-(\\ref{20}) through normalization of the variables. The un-normalized variables $I=X^*\\tilde{I}$ and $X(t)$ are, of course, proportional to $X^*$ and emphasizes the importance of a low tolerance threshold for social intervention.\n\n\\subsubsection{Fitting model parameters to the observed incidence data}\nTo validate the effectiveness of the proposed model in describing real data, we fit the three parameters $\\nu_1, \\nu_2, \\nu_3$ with a numerical optimization routine that minimizes the discrepancy between the time series of reported new daily cases $X(t)$ and those generated by the model.\nWe constrained $\\nu_1 \\in [0, \\infty]$, while the other two parameters are unbounded.\nBesides the three model parameters, we also optimize with a grid search the following hyperparameters:\nthe initial reproduction number $\\mathcal{R}_0$ searched in the interval $[1.0, 3.0]$ and the value $X^{*} = X(t^{*})$ for each country with $t^{*}$ searched in the interval [15 January, 31 March].\nAs initial conditions for $\\nu_1, \\nu_2, \\nu_3$ in the optimization routine, we used the values $[0.1, 0.1, 0.1]$.\n\n\n\\section{Results and discussion}\n\\label{sec:results}\n\n\\subsection{Results of the cluster analysis of {\\cal R}(t)-curves}\n\\label{sec:clustering_results}\n\n\\begin{figure}[t!]\n    \\centering\n \\includegraphics[width=14 cm]{Fig3.pdf}\n    \\caption{\\footnotesize The left figure depicts the dendrogram obtained from the DTW dissimilarity between the $\\mathcal{R}(t)$ time series. It is possible to obtain a certain number of clusters by putting a threshold at a specific dissimilarity value. In the example, we choose the threshold equal to 130. The right figures depict the average $\\mathcal{R}(t)$ of countries in the same cluster.}\n    \\label{fig:dendrogram}\n\\end{figure}\n\nThe dendrogram to the left of Figure~\\ref{fig:dendrogram} depicts the result of the clustering procedure, based on the DTW dissimilarities between the $\\mathcal{R}(t)$-curves estimated according to Eq.~(\\ref{9}).\nIn particular, the dendrogram illustrates how two leaves $i$, $j$ (\\textit{i.e.}, the $\\mathcal{R}(t)$-curves of countries $i$ and $j$) are merged together as soon as the threshold $\\delta_\\text{max}$ becomes larger than their DTW dissimilarity value $\\delta_{i,j}$.\nThere is no unique way of selecting an optimal $\\delta_\\text{max}$, but it rather depends on what level of resolution of the clustering partition is amenable for a meaningful exploration of the structure underlying our data.\nIn our case, we selected a $\\delta_\\text{max}=150$ that gave rise to seven clusters, depicted in different colors in Figure~\\ref{fig:dendrogram}.\nOn the right hand side of Figure~\\ref{fig:dendrogram}, we report $\\mathcal{R}(t)$ averaged over all the countries in the same cluster.\nTo facilitate the interpretation of the results, in Figure~\\ref{fig:world} we depict the same clustering partition obtained for $\\delta_\\text{max}=150$ on the political world map. \n\n\\begin{figure}[t!]\n\t\\centering\n  \\includegraphics[width=14 cm]{Fig4.pdf}\n    \\caption{\\footnotesize Visualization on the World map of the clusters obtained from the dissimilarity of the $\\mathcal{R}(t)$ curves.}\n\t\\label{fig:world}\n\\end{figure}\n\n\n\nThe 1\\textsuperscript{st} cluster (light blue) contains countries mostly from Africa, South America and Middle East. \nThe average $\\mathcal{R}$ curve of the countries in the light blue cluster (top-right of Figure~\\ref{fig:dendrogram}) shows that the reproduction number is always very low, but consistently above one.\nA possible explanation is that in those countries communities are more isolated and there are less travels and exchanges between them, making the infection to spread slower.\n\nIn 2$\\textsuperscript{nd}$ cluster (green) the average $\\mathcal{R}$ curve also stays always above one, but it starts from a higher value $\\mathcal{R}_0$. It is important to notice that this cluster includes large countries, like India, Brazil, United States and Russia.\nIn these countries, the time series of new cases $X(t)$ have a particular profile since they are a combination from widely separated areas where the infection outbreak followed  different courses. For instance, in the U.S., the waves in New York and California are almost in opposite phase.\n\nThe 3$\\textsuperscript{rd}$ cluster (pink) contains countries where the first wave is very long and it took a considerable amount of time to bring the $\\mathcal{R}$ curve below 1. A second wave is slowly emerging in the Northern autumn. An atypical member of this cluster is Sweden, which experienced a second wave in the summer that appeared almost as a continuation of the first wave, and then a strong third wave in the fall that is synchronous with the second wave for the rest of West-Europe (cluster 4).\n\nThe 4$\\textsuperscript{th}$ cluster (brown) mostly contains West-European countries, characterized by a strong first wave that was brought down quickly and a second wave that begun in the fall.\nThe average $\\mathcal{R}$ curve is characterized by strong variability: it starts from a very high value and goes quickly below 1, to raise again quickly in the summer.\n\nSimilarly to the 1\\textsuperscript{st} cluster, the 5$\\textsuperscript{th}$ cluster (red) contains South American, African countries, and New Zealand. However, a key difference from 1$\\textsuperscript{st}$ cluster is that in this case the $\\mathcal{R}$ curve goes and remains below 1 during the Northern fall and autumn.\n\nFinally, clusters 6 and 7 differ from the others by exhibiting initial $\\mathcal{R}$  close to, or even lower than, one. For some countries in Cluster 6 this is an artifact of the averaging over all the countries in the cluster which includes some countries like China, Australia, and South Korea, which started out with quite high ${\\cal R}$, but brought it down very rapidly through strong interventions  \\cite{Rahman2020}.  \nThe common characteristic feature for the cluster is an ${\\cal R}(t)$ above 1 during the Northern summer, but a reduction in the fall, which is the opposite of what has been observed in Western Europe and Canada.  \nCluster 7, on the other hand, contains most East-European countries, where the reproduction number was very low during the spring, but increased rapidly  after the summer.\n\n\n\\subsection{Exploring the parameter space of the  oscillator model}\n\\label{sec:params_exploration}\nThe data for the incidence rate $X(t)$ in the world's countries show a wavy pattern consisting of one to three maxima during the first year of the pandemic evolution. However, the duration, relative strength, and separation between the waves vary substantially among countries and regions of the world. The total cumulative number of confirmed cases and deaths per million inhabitants can also vary by an order of magnitude or more among countries comparable to economic development, culture, and the healthcare system. Rypdal and Rypdal \\cite{RR2020} demonstrated this for the first wave of the pandemic in a sample of 73 countries and discussed the significant differences in death toll between the two neighboring countries, Sweden and  Norway. At the time of writing this paper, we are four months further into the pandemic. The picture has changed dramatically, with secondary and tertiary waves developing in many countries.\n\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=14 cm]{Fig5.pdf}\n\\caption{Blue curves show the evolution of ${\\cal R}(t)$ and red curves $\\tilde{X}(t)$ as solutions of Eqs.(\\ref{19}) and (\\ref{20}) with $\\tilde{X}={\\cal R}\\tilde{I}$ and $\\nu$ given by Eq.\\;(\\ref{27}), initial conditions $\\tilde{X}(0)=1$, ${\\cal R}(0)={\\cal R}_0$, and parameters ${\\cal R}_0,\\, \\nu_1,\\, \\nu_2,\\, \\nu_3$ as indicated in the figures.}\\label{fig:RXplotarray}\n\\end{figure}  \n\nIn Figure~\\ref{fig:RXplotarray}, we have summarized some of the conclusions drawn from numerical solutions of the model proposed in Section~\\ref{sec:closed_model}, obtained by varying the model parameters. In all simulations, we have chosen the time origin $t=0$ to be the first time the incidence rate $X(0)$ crosses the threshold value $X^*$. Hence, $\\tilde{X}(0)=1$ for all simulations. The incidence rate measured on the right-hand axis in the figures is measured in units of the threshold $X^*$.\n\n\\subsubsection{The effect of the initial reproduction number}\\label{sec:effect of R0}\nIn the first row of panels, Figure~\\ref{fig:RXplotarray}(a)--(c), we consider the effect of changing the initial reproduction number ${\\cal R}_0$. From the reconstructed ${\\cal R}(t)$-curves, we observe that ${\\cal R}_0$ varies considerably among countries and regions. Low values just above ${\\cal R}_0=1$ are common in developing countries in South America, sub-Sahara Africa, and India. Several factors may contribute to this; lower mobility of people, a younger population, and a warmer climate. For these countries, we typically observe a  slower rise and decay of the first wave, and the wave is generally weaker than in industrialized countries where ${\\cal R}_0$ varies in the range 2.0--2.5. In Figure~\\ref{fig:RXplotarray}(a)--(c), we have changed ${\\cal R}_0$, keeping $\\nu_1$, $\\nu_2$, $\\nu_3$ constant. By choosing $\\nu_1=0$, and $\\nu_3=0$ we consider countries that respond slowly to an incidence rate below the threshold and show little fatigue, which may be characteristic for developing countries for which Figure~\\ref{fig:RXplotarray}(a) may be relevant. \nIn these panels, the choice $\\nu_2=0.01$ is somewhat arbitrary but yields a rather stretched-out and low-amplitude first wave typical for those countries. For higher ${\\cal R}_0$, the first wave is higher in amplitude and shorter, like what we have seen in China. Here $\\nu_1=\\nu_3=0$ signify that the relaxation rate and fatigue have been sufficiently low to prevent ${\\cal R}$ from increasing after it has stabilized below 1. The maximum incidence rate $\\tilde{X}=X\/X^*$ in panels (b) and (c) is high; in the range 15-40. In panel (i), where the parameters are the same as in (b) except for $\\nu_2=0.1$ being ten times higher, shows $\\tilde{X}\\approx 3$, which, as we will see later, is representative for China.\n\n\\subsubsection{The effect of the relaxation rate}\nIn the second row, we vary the relaxation rate $\\nu_1$ while keeping the strike-down rate fixed at $\\nu_2=0.01$. The result is that as $\\tilde{X}$ drops below the threshold after about 45 days, $\\mathcal{R}(t)$ starts to rise and grow well beyond 1. How fast this happens, depends on $\\nu_1$. In the phase when ${\\cal R}>1$, $\\tilde{X}(t)$ will also start growing, and when it crosses the threshold $\\tilde{X}=1$, the strike-down sets in again, and we enter a new cycle. With $\\nu_1=0.01$ the first cycle takes almost 500 days, while it takes considerably less time in most countries, suggesting a higher $\\nu_1$. In panel (e) we increase $\\nu_1$ by a factor 5 and observe then two cycles within the first year, and in panel (f) another increment by a factor 10 almost eliminates the next waves. \nThis faster relaxation to the equilibrium ${\\cal R}=\\tilde{X}=1$ when the relaxation rate is high may appear counter-intuitive. After all, it leads to a rapid increase of ${\\cal R}$ once $\\tilde{X}$ has dropped below the threshold. However, the faster rise of ${\\cal R}$ also leads to a faster rise of ${\\cal X}$ beyond the threshold and to a faster strike-down of ${\\cal R}$ back towards 1, \\textit{i.e.}, to faster damping of the oscillation. This observation suggests that the strong second wave of the epidemic evolving in Europe in the fall of 2020 is not caused by the relaxation of social interventions during the summer but is  caused by something else.\n\n\\subsubsection{The effect of the intervention rate}\n A suspected candidate could be the intervention rate $\\nu_2$, which is varied in the third row, panels (g)--(i). However, we observe that the main effect of increasing $\\nu_2$ is to decrease the amplitude of the oscillation in $\\tilde{X}$ in inverse proportion to $\\nu_2$. In this row, we have kept $\\nu_1=0$, resulting in relaxation to a time-asymptotic ($t\\rightarrow \\infty$) equilibrium ${\\cal R}_{\\infty}<1$, $\\tilde{X}_{\\infty}=0$. This is in contrast to the second row ($\\nu_1>0$), where this equilibrium is ${\\cal R}_{\\infty}=1$, $\\tilde{X}_{\\infty}=1$. These two equilibria correspond to fundamentally different strategies to combat the epidemic. The one without the relaxation mechanism ($\\nu_1=0$) corresponds to the strike-down strategy, where the goal is to eliminate the pathogen without obtaining herd immunity in the population. The one with $\\nu_1>0$, allowing relaxation of interventions when the incidence rate dips below the threshold, will end up with a constant incidence rate at the threshold value and thus a linearly increasing cumulative number of infected until this growth is non-linearly saturated by herd immunity.\n\n \n \\subsubsection{The effect of the fatigue  rate}\n The effect of a non-zero fatigue rate is to bring the effective strike-down rate to zero as $t\\rightarrow \\infty$. The  solution of the system Eqs.\\;(\\ref{19})--(\\ref{20}) as $t\\rightarrow \\infty$ is that ${\\cal R}\\rightarrow 1+\\nu_3$ and $\\tilde{X}\\approx \\exp{(\\nu_3 t)}$. Of course, this blow-up is prevented by herd immunity, which will reduce the effective ${\\cal R}$ to zero when most of the population has been infected. The effect of increasing immunity in the population is not included in Eq.\\;(\\ref{17}), and hence the model makes sense only as long as the majority of the population is still susceptible to the disease.\n Nevertheless, the last row in Figure \\ref{fig:RXplotarray} shows that increasing intervention fatigue represented by non-zero $\\nu_3$ may increase the second and later waves' amplitude and duration. \n For sufficiently large $\\nu_3$ the second wave's amplitude and duration can become greater than the first. The situations shown in panels (k) and (l) are observed in European countries and are caused by $\\nu_3\\approx 1$. One partial explanation of the second wave's higher amplitude than the first, as observed in many countries, is a considerably higher testing rate. The testing rate, however, cannot explain the considerably longer duration of the second wave. This prolonged duration shows up both in the observed data and in this model, when the fatigue rate is increased.\n\n\n\n\\subsection{Results from fitting the oscillator model to incidence data in different countries}\n\\label{sec:fit}\n\n\n\n\n\\begin{figure}[t!]\n\t\\centering\n  \\includegraphics[width=14 cm]{Fig6.pdf}\n    \\caption{\\footnotesize Comparison between observed daily cases $X(t)$ (red dashed lines), $X(t)$ simulated from model (solid red lines), $\\mathcal{R}(t)$ estimated from the data using the inverted SIR model (dashed blue line), $\\mathcal{R}(t)$ simulated from the model.}\n\t\\label{fig:Model_estimate_comp}\n\\end{figure}\n\nFigure~\\ref{fig:Model_estimate_comp} depicts for selected countries the reported daily new cases (dashed red line), the daily new cases simulated by the oscillator model (solid red line), the $\\mathcal{R}(t)$ curve estimated using Eq.\\;(\\ref{9}) (dashed blue line), and the $\\mathcal{R}(t)$ curve simulated by the proposed close model (solid blue line).\nOn the top of each graph, we report for each country the fitted values of $[\\nu_1$, $\\nu_2$, $\\nu_3]$, the initial $\\mathcal{R}_0$, and the date $t^{*}$ that identifies $X^{*} = X(t^{*})$.\nOn the horizontal axis, 0 corresponds to $t^{*}$, the left vertical axis indicates the value of the reproduction number, the right vertical axis indicates the number of new daily cases. \n\nThe first row in the array of panels shows results for China and Turkey. The parameters estimated for China is comparable to those in Figure~\\ref{fig:RXplotarray}(i). The peak incidence rate $X_{\\text{max}}\\approx 3,500$  for China is about 3 times the threshold incidence $X^*=1,185$, similar to what is observed in Figure~\\ref{fig:RXplotarray}(i). For Turkey, the evolution of ${\\cal R}(t)$ is initially rather similar to that of China, and the shape of the $X(t)$-curve is also rather similar. But while the model-fitted ${\\cal R}(t)$ converges to a fixed value ${\\cal R}_{\\infty}<1$, and $X(t)$ to 0 after a few months in China, ${\\cal R}(t)$ in Turkey grows slowly greater than 1, and a second wave in $X(t)$ develops. This wave has an amplitude approximately the same as the first, but lasts longer (not shown in the figure), similar to what is shown in Figure~\\ref{fig:RXplotarray}(k). This rise is the result of fundamental differences in the estimated model parameters: $\\nu_1$ is zero for China but non-zero for Turkey.\nWe also notice that China and Turkey belong to different clusters in the dendrogram in Figure~\\ref{fig:dendrogram}. \nThe second wave for Turkey is not created by a finite fatigue rate, since $\\nu_3=0$ both for Turkey and China; it is created by a finite relaxation rate $\\nu_1$. Importantly, this relaxation rate cannot create a second wave that is stronger than the first, it only gives rise to a damped oscillation that ends up in the equilibrium ${\\cal R}=1$, $X=X^*$.\n\nThe second row shows two countries, Brazil and India belonging to the second cluster, represented in green in Figure~\\ref{fig:dendrogram}. The initial reproduction number is low, ${\\cal R}_0=1.5$ for both countries, and the strike-down parameter is also low, $\\nu_2\\approx 0.01$, leading to a strong and long first wave, which is not yet completely over in November 2020. The fatigue rate of $\\nu_3\\approx 0.2$ also contributes to increasing the amplitude and the long-lasting downward slope of the first wave. \n\nIn the third row, we show the typical pattern for Europe, the fourth cluster (brown) in the dendrogram, with Austria and Spain as examples. There is a rather short first wave accompanied by a rapid drop in ${\\cal R}(t)$ due to the almost universal lockdown in March 2020. There is a rather slow relaxation of the interventions throughout the summer, finally leading to ${\\cal R}$ stabilizing in the range 1.2-1.5. The inevitable result is the rise of a second wave, growing stronger and longer than the first, as shown in Figure~\\ref{fig:RXplotarray}(k) and (l). At the time of writing,  interventions again have started to inhibit the growth, but they are weaker than in the spring, as reflected by the fatigue rates in the range $\\nu_3\\sim 0.2-0.3$. \nIndeed, the predictions of the oscillator model with the estimated rates is that the second wave will blow up in the spring of 2021 to levels where herd immunity will limit the growth. This is before vaccines are likely to play an important r\\^{o}le, so a more probable scenario is that governments will  reverse the fatigue trend and invalidate the model as a prediction for the future. Tendencies in this direction is observed in Europe at the time of writing.\n\n\n\\section{Discussion and conclusions}\\label{sec:discuss}\nThe geographic distribution of countries belonging to different clusters shown in the map in Figure~\\ref{fig:world},  and the associated  averaged ${\\cal R}(t)$-curves in Figure~\\ref{fig:dendrogram}, may serve as a crude road map to the global evolution of the pandemic throughout the spring and fall of 2020. One striking feature is some geographic clustering, which is most pronounced in Western Europe (brown) and  Eastern Europe (blue). A similar clustering is seen in the U.S. and Equatorial Latin America (green). In this paper, we have a focus on the strength, timing and duration of the second epidemic wave, and for this purpose the dendrogram helps us to identify those regions where there has been a pronounced second wave so far in the pandemic. These are those countries that belong to clusters exhibiting a period of ${\\cal R}(t)<1$ in between periods of ${\\cal R}>1$. From the  ${\\cal R}(t)$-profiles in Figure~\\ref{fig:dendrogram} those countries with the most pronounced second wave are Cluster 4 (brown) and 7 (dark blue), Western and Eastern Europe, respectively. The rise of the second wave here is due to the persistently high values of ${\\cal R}$  during the period July - November. What distinguishes the two clusters is the course of the first wave. In Western Europe there was a strong first wave associated with  high ${\\cal R}$, and it affected strongly older age groups which resulted in high case fatality ratio (CFR). The second wave has affected all ages and so far the death numbers have been much lower than in the first. In Eastern Europe the first wave was very weak, but the second has been strong and with considerably higher CFR than in the countries further West. \n\nThe main result in this paper is to demonstrate that the varying courses of the epidemic depicted via the seven characteristic ${\\cal R}(t)$ curves shown in Figure~\\ref{fig:dendrogram} to some extent ca be understood in terms of the interplay between three social responses to the epidemic activity; the relaxation of interventions when the activity is low, the intensification of interventions when activity becomes high, and the intervention fatigue which develops with time. Figures~\\ref{fig:RXplotarray} and ~\\ref{fig:Model_estimate_comp}  suggest that most country-specific epidemic curves can be qualitatively reproduced by a simple mathematical model involving these three responses. The value of this insight is that, in spite of the immense complexity and diversity of the dynamical response triggered by this new pathogen, there are some universal governing principles that will determine the final outcome in the years to come. \n\nThe model devised here could of course be run to make projections further ahead than one year from the onset of the epidemic, as done in Figure ~\\ref{fig:RXplotarray}. It would show a blow-up of all solutions for which the fatigue parameter is non-zero, and would be unrealistic for several reasons. One is that the linearity approximation would break down as herd immunity will start to bring the effective reproduction number down. Another is that the intervention fatigue model most likely will fail when the epidemic activity becomes sufficiently high. We have already have seen signs in this direction in many European countries where partial lockdowns and mass testing again have succeeded in ``bending the curve'' to an extent that is not described by the model. Finally, mass-vaccination will hopefully become a  real game-changer in the year to come. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nStochastic hybrid systems are dynamical systems \ncomprising continuous and discrete dynamics interleaved with probabilistic noise and stochastic events \\cite{BL06a}.  \nBecause of their versatility and generality, methods for analysis and design of stochastic hybrid systems  \ncarry great promise in \nmany safety critical applications \\cite{BL06a}. Examples of such applications include power networks, automotive, finance, \nair traffic control, biology, telecommunications, and embedded systems. \nStochastic \\emph{switched} systems are a relevant subclass of stochastic hybrid systems. \nThey consist of a finite (discrete) set of modes of operation, \neach of which is associated to continuous probabilistic dynamics;      \nfurther, their discrete dynamics, in the form of mode changes, are governed by a non-probabilistic control (switching) signal.   \n\nIt is known \\cite{liberzon} that switched systems can be endowed with global behaviors that are not characteristic of the behavior of any of their modes:  \nfor instance, global instability may arise by proper choice over time of the discrete switches between a set of stable modes.  \nThis is but one of the many features that makes switched systems theoretically interesting.   \nWith focus on \\emph{stochastic} switched systems, \ndespite recent progresses on basic dynamical analysis focused on stability properties \\cite{debasish}, \nthere are no notable results in the literature targeting more complex objectives, \nsuch as those dealing with verification or (controller) synthesis for logical specifications. Examples of those specifications include linear temporal logic or automata on infinite strings, \nand as such they are not amenable to classical approaches for stochastic processes. \n\nA promising direction to investigate these general properties is the use of \\emph{symbolic models}. \nSymbolic models are abstract descriptions of the original dynamics, \nwhere each abstract state (or symbol) corresponds to an aggregate of states in the concrete system. \nWhen a finite symbolic model \nis obtained and is formally put in relationship with the original system, \none can leverage automata-theoretic techniques for controller synthesis over the finite model \\cite{MalerPnueliSifakis95} \nto automatically synthesize controllers for the original system. \nTowards this goal, \na relevant approach is the construction of finite-state symbolic models that are \\emph{bisimilar} to the original system.  \nUnfortunately, the class of continuous (-time and -space) dynamical systems admitting exactly bisimilar finite-state symbolic models is quite restrictive \\cite{AHLP00,LPS00} and in particular it covers mostly non-probabilistic models. The results in \\cite{manuela} provide a notion of exact stochastic bisimulation for a class of stochastic hybrid systems, \nhowever, \\cite{manuela} does not provide any abstraction algorithm, nor does it look at the synthesis problem. \nTherefore, rather than requiring exact bisimilarity, \none can resort to \\emph{approximate bisimulation} relations \\cite{girard}, \nwhich introduce a metric between the trajectories of the abstract and the concrete models, \nand require boundedness in time of this distance. \n\nThe construction of approximately bisimilar symbolic models has been extensively studied for \nnon-probabilistic control systems, possibly affected by disturbances \n\\cite{majid4,pola,pola1} and references therein, \nas well as for \nnon-probabilistic switched systems \\cite{girard2}.  \nHowever, stochastic systems, particularly when endowed with hybrid dynamics, \nhave only been scarcely explored.   \nWith focus on these models,   \na few existing results deal with abstractions of discrete-time stochastic processes \\cite{AAPLS07,abate1,azuma}.  \nResults for continuous-time models cover probabilistic rectangular hybrid automata \\cite{sproston} \nand stochastic dynamical systems under some contractivity assumptions \\cite{abate}. \nFurther, the results in \\cite{julius1} only {\\em check} the relationship between an uncountable abstraction and a given class of stochastic hybrid systems via the notion of stochastic (bi)simulation function. However, these results do not provide any {\\em construction} of approximations, nor do they deal with {\\em finite} abstractions, \nand moreover appear to be computationally tractable only in the case where no input is present. \nThe recent results in \\cite{majid8} and \\cite{majid11} investigate the construction of finite bisimilar abstractions for continuous-time stochastic control systems, without any hybrid dynamics, and randomly switched stochastic systems, respectively, such that the discrete dynamics in the latter systems are governed by a random uncontrolled signal. Finally, the recently proposed techniques in \\cite{majid10} improve\nthe ones in \\cite{majid8} by not requiring state-space discretization but only input set discretization. \nIn summary, \nto the best of our knowledge there is no comprehensive work on the automatic construction of finite bisimilar abstractions for continuous-time stochastic switched systems in which the discrete dynamics are governed by a non-probabilistic control signal.  \n\nThe main contributions of this work consist in showing the existence and the construction of approximately bisimilar symbolic models for incrementally stable stochastic switched systems using two different techniques: one requires state space discretization and the other one does not require any space discretization. Note that all the techniques provided in \\cite{majid4,pola,pola1,girard2,AAPLS07,abate1,azuma,sproston,abate,majid8,majid11} are only based on the discretization of state sets. Therefore, they suffer severely from \\emph{the curse of dimensionality} due to gridding those sets, which is especially irritating for models with high-dimensional state sets. We also provide a simple criterion in which one can choose between the two proposed approaches the most suitable one (based on the size of the abstraction) for a given stochastic switched system. Another advantage of the second proposed approach here is that it allows one to construct symbolic models with probabilistic output values, resulting possibly in less conservative symbolic abstractions in comparison with the first proposed approach and the ones in \\cite{majid8,majid11} allowing for non-probabilistic output values only. Furthermore, the second proposed approach here allows one to construct symbolic models for any given precision $\\varepsilon$ and any given sampling time, but the first proposed approach and the ones in \\cite{majid8,majid11} may not be applicable for a given sampling time. \n\n\nIncremental stability is a property on which the main proposed results of this paper rely. This type of stability requires uniform asymptotic stability of every trajectory, rather than stability of an equilibrium point or a particular time-varying trajectory. \nIn this work, we show the description of incremental stability in terms of a so-called common Lyapunov function or of multiple Lyapunov functions. \nThe main results are illustrated by synthesizing controllers (switching signals) for two examples. \nFirst, we consider a room temperature control problem (admitting a common Lyapunov function) for a six-room building. \nWe synthesize a switching signal regulating the temperature toward a desired level which is not tractable using the first proposed technique. \nThe second example illustrates the use of multiple Lyapunov functions (one per mode) using the first proposed approach. A preliminary investigation on the construction of bisimilar symbolic models for stochastic switched systems using the first proposed approach (requiring state space discretization) appeared in \\cite{majid9}. In this paper we present a detailed and mature description of the results presented in \\cite{majid9}, including proofs, as well as proposing a second approach which does not require any space discretization.       \n\\vspace{-0.2cm}\n\n\\section{Stochastic Switched Systems}\n\\subsection{Notation} \nThe symbols ${\\mathbb{N}}$, ${\\mathbb{N}}_0$, ${\\mathbb Z}$, ${\\mathbb{R}}$, ${\\mathbb{R}}^+$, and ${\\mathbb{R}}_0^+$ denote the set of natural, nonnegative integer, integer, real, positive, and nonnegative real numbers, respectively. The symbols $I_n$, $0_n$, and $0_{n\\times{m}}$ denote the identity matrix, zero vector, and zero matrix in ${\\mathbb{R}}^{n\\times{n}}$, ${\\mathbb{R}}^n$, and ${\\mathbb{R}}^{n\\times{m}}$, respectively. Given a set $A$, define $A^{n+1}=A\\times A^n$ for any $n\\in{\\mathbb{N}}$. Given a vector \\mbox{$x\\in\\mathbb{R}^{n}$}, we denote by $x_{i}$ the $i$--th element of $x$, and by $\\Vert x\\Vert$ the infinity norm of $x$, namely, \\mbox{$\\Vert x\\Vert=\\max\\{|x_1|,|x_2|,...,|x_n|\\}$}, where $|x_i|$ denotes the absolute value of $x_i$. \nGiven a matrix $P=\\{p_{ij}\\}\\in{\\mathbb{R}}^{n\\times{n}}$, we denote by $\\text{Tr}(P)=\\sum_{i=1}^np_{ii}$ the trace of $P$.\nThe \\emph{diagonal set} $\\Delta\\subset{\\mathbb{R}}^n\\times{\\mathbb{R}}^n$ is defined as: $\\Delta=\\left\\{(x,x) \\mid x\\in {\\mathbb{R}}^n\\right\\}$.\n\nThe closed ball centered at $x\\in{\\mathbb{R}}^{n}$ with radius $\\varepsilon$ is defined by \\mbox{$\\mathcal{B}_{\\varepsilon}(x)=\\{y\\in{\\mathbb{R}}^{n}\\,|\\,\\Vert x-y\\Vert\\leq\\varepsilon\\}$}. A set $B\\subseteq {\\mathbb{R}}^n$ is called a \n{\\em box} if $B = \\prod_{i=1}^n [c_i, d_i]$, where $c_i,d_i\\in {\\mathbb{R}}$ with $c_i < d_i$ for each $i\\in\\set{1,\\ldots,n}$.\nThe {\\em span} of a box $B$ is defined as $\\mathit{span}(B) = \\min\\set{ | d_i - c_i| \\mid i=1,\\ldots,n}$. By defining $[{\\mathbb{R}}^n]_{\\eta}=\\left\\{a\\in {\\mathbb{R}}^n\\mid a_{i}=k_{i}\\eta,k_{i}\\in\\mathbb{Z},i=1,\\ldots,n\\right\\}$, the \nset \\mbox{$\\bigcup_{p\\in[{\\mathbb{R}}^n]_{\\eta}}\\mathcal{B}_{\\lambda}(p)$} is a \ncountable covering of ${\\mathbb{R}}^n$ for any $\\eta\\in{\\mathbb{R}}^+$ and $\\lambda\\geq\\eta\/2$. \nFor a box $B\\subseteq{\\mathbb{R}}^n$ and $\\eta \\leq \\mathit{span}(B)$,\ndefine the $\\eta$-approximation $[B]_\\eta = [{\\mathbb{R}}^n]_{\\eta}\\cap{B}$. \nNote that $[B]_{\\eta}\\neq\\varnothing$ for any $\\eta\\leq\\mathit{span}(B)$. \nGeometrically, for any $\\eta\\in{\\mathbb{R}^+}$ with $\\eta\\leq\\mathit{span}(B)$ and $\\lambda\\geq\\eta$, the collection of sets \n\\mbox{$\\{\\mathcal{B}_{\\lambda}(p)\\}_{p\\in [B]_{\\eta}}$}\nis a finite covering of $B$, i.e., \\mbox{$B\\subseteq\\bigcup_{p\\in[B]_{\\eta}}\\mathcal{B}_{\\lambda}(p)$}. \nWe extend the notions of $\\mathit{span}$ and of {\\em approximation} to finite unions of boxes as follows.\nLet $A = \\bigcup_{j=1}^M A_j$, where each $A_j$ is a box.\nDefine $\\mathit{span}(A) = \\min\\set{\\mathit{span}(A_j)\\mid j=1,\\ldots,M}$,\nand for any $\\eta \\leq \\mathit{span}(A)$, define $[A]_\\eta = \\bigcup_{j=1}^M [A_j]_\\eta$.\n\nA continuous function \\mbox{$\\gamma:\\mathbb{R}_{0}^{+}\\rightarrow\\mathbb{R}_{0}^{+}$}, is said to belong to class $\\mathcal{K}$ if it is strictly increasing and \\mbox{$\\gamma(0)=0$}; $\\gamma$ is said to belong to class $\\mathcal{K}_{\\infty}$ if \\mbox{$\\gamma\\in\\mathcal{K}$} and $\\gamma(r)\\rightarrow\\infty$ as $r\\rightarrow\\infty$. A continuous function \\mbox{$\\beta:\\mathbb{R}_{0}^{+}\\times\\mathbb{R}_{0}^{+}\\rightarrow\\mathbb{R}_{0}^{+}$} is said to belong to class $\\mathcal{KL}$ if, for each fixed $s$, the map $\\beta(r,s)$ belongs to class $\\mathcal{K}$ with respect to $r$ and, for each fixed nonzero $r$, the map $\\beta(r,s)$ is decreasing with respect to $s$ and $\\beta(r,s)\\rightarrow 0$ as \\mbox{$s\\rightarrow\\infty$}. We identify a relation \\mbox{$R\\subseteq A\\times B$} with the map \\mbox{$R:A \\rightarrow 2^{B}$} defined by $b\\in R(a)$ iff \\mbox{$(a,b)\\in R$}. Given a relation \\mbox{$R\\subseteq A\\times B$}, $R^{-1}$ denotes the inverse relation defined by \\mbox{$R^{-1}=\\{(b,a)\\in B\\times A:(a,b)\\in R\\}$}.\n\n\\subsection{Stochastic switched systems}\\label{sss}\nLet $(\\Omega, \\mathcal{F}, \\mathds{P})$ be a probability space endowed with a filtration $\\mathds{F} = (\\mathcal{F}_t)_{t\\geq 0}$ satisfying the usual conditions of completeness and right-continuity \\cite[p.\\ 48]{ref:KarShr-91}. Let $(W_t)_{t \\ge 0}$ be a $\\widehat{q}$-dimensional $\\mathds{F}$-adapted Brownian motion \\cite{oksendal}. The class of stochastic switched systems considered in this paper is formalized as follows.  \n\n\\begin{definition}\n\\label{Def_control_sys}A stochastic switched system $\\Sigma$ is a tuple $\\Sigma=(\\mathbb{R}^{n},\\mathsf{P},\\mathcal{P},F,G)$, where\n\\begin{itemize}\n\\item $\\mathbb{R}^{n}$ is the state space;\n\n\\item $\\mathsf{P}=\\left\\{1,\\ldots,m\\right\\}$ is a finite set of modes;\n\n\\item $\\mathcal{P}$ is a subset of the set of piecewise constant c\\`adl\\`ag (i.e. right-continuous and with left limits) functions from ${\\mathbb{R}}_0^+$ to $\\mathsf{P}$, and with a finite number of discontinuities on every bounded interval in ${\\mathbb{R}}_0^+$ (no Zeno behaviour);  \n\n\\item $F=\\left\\{f_1,\\ldots,f_m\\right\\}$ is such that for any $p\\in\\mathsf{P}$, $f_p:{\\mathbb{R}}^n\\rightarrow{\\mathbb{R}}^n$ is globally Lipschitz continuous; \n\n\\item $G=\\left\\{g_1,\\ldots,g_m\\right\\}$ is such that for any $p\\in\\mathsf{P}$, $g_p:{\\mathbb{R}}^n\\rightarrow{\\mathbb{R}}^{n\\times{\\widehat{q}}}$ is globally Lipschitz continuous with Lipschitz constant $Z_p\\in{\\mathbb{R}}_0^+$. \n\\end{itemize}\n\\end{definition}\n\nA continuous-time stochastic process \\mbox{$\\xi:\\Omega \\times {\\mathbb{R}}_0^+ \\rightarrow \\mathbb{R}^{n}$} is said to be a \\textit{solution process} of $\\Sigma$ if there exists a switching signal $\\upsilon\\in\\mathcal{P}$ satisfying \n\\begin{small}\n\\begin{equation}\n\\label{eq00}\n\t\\diff \\xi= f_{\\upsilon}(\\xi)\\diff t+g_{\\upsilon}(\\xi)\\diff W_t,\n\\end{equation}\n\\end{small}$\\mathds{P}$-almost surely ($\\mathds{P}$-a.s.), at each time $t\\in{\\mathbb{R}}_0^+$ whenever $\\upsilon$ is continuous. \nLet us emphasize that $\\upsilon$ is a piecewise constant function defined over ${\\mathbb{R}}_0^+$ and taking values in $\\mathsf{P}$, \nwhich simply dictates in which mode the solution process $\\xi$ is located, at any time $t \\in {\\mathbb{R}}_0^+$. \n\nFor any given $p\\in\\mathsf{P}$, we denote by $\\Sigma_p$ the subsystem of $\\Sigma$ defined by the stochastic differential equation (SDE)\n\\begin{small}\n\\begin{equation}\n\\label{eq10}\n\t\\diff \\xi= f_{p}(\\xi)\\diff t+g_{p}(\\xi)\\diff W_t, \n\\end{equation}\n\\end{small}where $f_{p}$ is known as the drift and\n$g_{p}$ as the diffusion.\nGiven an initial condition which is a random variable, measurable in $\\mathcal{F}_0$, a solution process of $\\Sigma_p$ exists and is uniquely determined owing to the assumptions on $f_p$ and on $g_p$ \n\\cite[Theorem 5.2.1, p.\\ 68]{oksendal}. \n \nWe further write $\\xi_{a \\upsilon}(t)$ to denote the value of the solution process of $\\Sigma$ at time $t\\in{\\mathbb{R}}_0^+$ under the switching signal $\\upsilon$ from initial condition $\\xi_{a \\upsilon}(0) = a$ $\\mathds{P}$-a.s., in which $a$ is a random variable that is measurable in $\\mathcal{F}_0$. \n\nFinally, note that a solution process of $\\Sigma_p$ is also a solution process of $\\Sigma$ corresponding to the constant switching signal $\\upsilon(t)=p$, for all $t\\in{\\mathbb{R}}_0^+$. We also use $\\xi_{ap}(t)$ to denote the value of the solution process of $\\Sigma_p$ at time $t\\in{\\mathbb{R}}_0^+$ from the initial condition $\\xi_{a p}(0) = a$ $\\mathds{P}$-a.s..\n\\vspace{-0.2cm}\n\n\\section{Notions of Incremental Stability}\\label{stability}\nThis section introduces some stability notions for stochastic switched systems, \nwhich generalize the notions of incremental global asymptotic stability ($\\delta$-GAS) \\cite{angeli} for non-probabilistic dynamical systems and of incremental global uniform asymptotic stability ($\\delta$-GUAS) \\cite{girard2} for non-probabilistic switched systems. \nThe main results presented in this work rely on the stability assumptions discussed in this section. \n\n\\begin{definition}\n\\label{dGAS}\nThe stochastic subsystem $\\Sigma_p$ is incrementally globally asymptotically stable in the $q$th moment ($\\delta$-GAS-M$_q$), where $q\\geq1$, if there exists a $\\mathcal{KL}$ function $\\beta_p$ such that for any $t\\in{\\mathbb{R}_0^+}$ and any ${\\mathbb{R}}^n$-valued random variables $a$ and $a'$ that are measurable in $\\mathcal{F}_0$, the following condition is satisfied:\n\\begin{small}\n\\begin{equation}\n\\mathds{E} \\left[\\left\\Vert \\xi_{ap}(t)-\\xi_{a'p}(t)\\right\\Vert^q\\right] \\leq\\beta_p\\left( \\mathds{E}\\left[ \\left\\Vert a-a' \\right\\Vert^q\\right], t \\right). \\label{delta_SGAS}\n\\end{equation}\n\\end{small}\n\\end{definition}\n\nIt can be easily checked that a $\\delta$-GAS-M$_q$ stochastic subsystem $\\Sigma_p$ is $\\delta$-GAS \\cite{angeli} in the absence of any noise. \nFurther, note that when $f_p(0_n)=0_n$ and $g_p(0_n)=0_{n\\times{\\widehat{q}}}$ (drift and diffusion terms vanish at the origin), \nthen $\\delta$-GAS-M$_q$ implies global asymptotic stability in the $q$th moment (GAS-M$_q$) \\cite{debasish}, \nwhich means that all the trajectories of $\\Sigma_p$ converge in the $q$th moment to the (constant) trajectory $\\xi_{0_np}(t)=0_n$ (namely, the equilibrium point), for all $t\\in{\\mathbb{R}}_0^+$. \nWe extend the notion of $\\delta$-GAS-M$_q$ to stochastic switched systems as follows.  \n\n\\begin{definition}\n\\label{dGUAS}\nA stochastic switched system $\\Sigma$ is incrementally globally uniformly asymptotically stable in the $q$th moment ($\\delta$-GUAS-M$_q$), where $q\\geq1$, if there exists a $\\mathcal{KL}$ function $\\beta$ such that for any $t\\in{\\mathbb{R}_0^+}$, any ${\\mathbb{R}}^n$-valued random variables $a$ and $a'$ that are measurable in $\\mathcal{F}_0$, and any switching signal ${\\upsilon}\\in\\mathcal{P}$, the following condition is satisfied:\n\\begin{small}\n\\begin{equation}\n\\mathds{E} \\left[\\left\\Vert \\xi_{a\\upsilon}(t)-\\xi_{a'\\upsilon}(t)\\right\\Vert^q\\right] \\leq\\beta\\left( \\mathds{E}\\left[ \\left\\Vert a-a' \\right\\Vert^q\\right], t \\right). \\label{delta_SGUAS}\n\\end{equation}\n\\end{small}\n\\end{definition}\n\nEssentially, Definition \\ref{dGUAS} extends Definition \\ref{dGAS} uniformly over any possible switching signal $\\upsilon\\in\\mathcal{P}$. \nAs expected, this notion generalizes known ones in the literature:  \nit can be easily seen that a $\\delta$-GUAS-M$_q$ stochastic switched system $\\Sigma$ is $\\delta$-GUAS \\cite{girard2} in the absence of any noise \nand that, whenever $f_p(0_n)=0_n$ and $g_p(0_n)=0_{n\\times{\\widehat{q}}}$ for all $p\\in\\mathsf{P}$, then $\\delta$-GUAS-M$_q$ implies global uniform asymptotic stability in the $q$th moment (GUAS-M$_q$) \\cite{debasish}. \n\nFor non-probabilistic systems the $\\delta$-GAS property can be characterized by $\\delta$-GAS Lyapunov functions \\cite{angeli}. \nAlong these lines, we describe $\\delta$-GAS-M$_q$ \nin terms of the existence of some {\\em incremental Lyapunov functions}, defined as the following. \n\n\\begin{definition}\n\\label{delta_SGAS_Lya}\nConsider a stochastic subsystem $\\Sigma_p$ and a continuous function $V_p:\\mathbb{R}^n\\times\\mathbb{R}^n\\rightarrow\\mathbb{R}_0^+$ that is twice continuously differentiable on \n$\\{{\\mathbb{R}}^n\\times{\\mathbb{R}}^n\\}\\backslash\\Delta$. \nFunction $V_p$ is called a $\\delta$-GAS-M$_q$ Lyapunov function for $\\Sigma_p$, \nwhere $q\\geq1$, \nif there exist $\\mathcal{K}_{\\infty}$ functions \n$\\underline{\\alpha}_p$, $\\overline{\\alpha}_p$ and a constant $\\kappa_p\\in\\mathbb{R}^+$, such that\n\\begin{itemize}\n\\item[(i)] $\\underline{\\alpha}_p$ (resp. $\\overline \\alpha_p$) is a convex (resp. concave) function;\n\n\\item[(ii)] for any $x,x'\\in\\mathbb{R}^n$, $\\underline{\\alpha}_p\\left(\\left\\Vert x-x'\\right\\Vert^q\\right)\\leq{V_p}(x,x')\\leq\\overline{\\alpha}_p\\left(\\left\\Vert x-x'\\right\\Vert^q\\right)$;\n\n\\item[(iii)] for any $x,x'\\in\\mathbb{R}^n$, such that $x\\neq x'$,\n\\begin{small}\n\\begin{align*}\n\t&\\mathcal{L} V_p(x, x') \n\t := \\left[\\partial_xV_p~~\\partial_{x'}V_p\\right] \\begin{bmatrix} f_p(x)\\\\f_p(x')\\end{bmatrix}+\\frac{1}{2} \\text{Tr} \\left(\\begin{bmatrix} g_p(x) \\\\ g_p(x') \\end{bmatrix}\\left[g_p^T(x)~~g_p^T(x')\\right] \\begin{bmatrix}\n\\partial_{x,x} V_p & \\partial_{x,x'} V_p\\\\ \\partial_{x',x} V_p & \\partial_{x',x'} V_p\n\\end{bmatrix}\t\\right)\\leq-\\kappa_pV_p(x,x'). \n\\end{align*} \n\\end{small}\n\\end{itemize}\n\\end{definition}\nThe operator $\\mathcal{L}$ is the infinitesimal generator associated to the stochastic subsystem $\\Sigma_p$, defined by the SDE in \\eqref{eq10} \\cite[Section 7.3]{oksendal}. \nThe symbols $\\partial_x$ and $\\partial_{x,x'}$ denote first- and second-order partial derivatives with respect to $x$ and $x'$, respectively. \n\n\nThe following theorem describes $\\delta$-GAS-M$_q$ in terms of the existence of a $\\delta$-GAS-M$_q$ Lyapunov function.\n\n\\begin{theorem}\n\\label{the_Lya}\nA stochastic subsystem $\\Sigma_p$ is $\\delta$-GAS-M$_q$ if it admits a $\\delta$-GAS-M$_q$ Lyapunov function. \n\\end{theorem}\n\n\\begin{proof}\n\tThe proof is a consequence of the application of Gronwall's inequality and of Ito's lemma \\cite[p. 80 and 123]{oksendal}. Assume that there exists a $\\delta$-GAS-M$_q$ Lyapunov function in the sense of Definition \\ref{delta_SGAS_Lya}. For any $t\\in{\\mathbb{R}}_0^+$, and any ${\\mathbb{R}}^n$-valued random variables $a$ and $a'$ that are measurable in $\\mathcal{F}_0$, we obtain\n\t\t\\begin{small}\n\t\t\\begin{align*}\n\t\t\t\\mathds{E} \\left[ V_p(\\xi_{ap}(t),\\xi_{a'p}(t)) \\right]&=\\mathds{E} \\left[V_p(a,a') + \\int_0^{t} \\mathcal{L}V_p(\\xi_{ap}(s),\\xi_{a'p}(s))ds\\right]\\le\\mathds{E} \\left[ V_p(a,a') + \\int_0^{t} \\left(-\\kappa_p V_p(\\xi_{ap}(s),\\xi_{a'p}(s))\\right)ds\\right] \\\\\n\t\t\t&\\le-\\kappa_p \\int_0^{t} \\mathds{E} \\left[ V_p(\\xi_{ap}(s),\\xi_{a'p}(s))\\right ]ds+\\mathds{E}\\left[V_p(a,a')\\right]\\nonumber,\n\t\t\\end{align*}\n\t\t\\end{small}which, by virtue of Gronwall's inequality, leads to \n\t\t\\begin{small}\n\\begin{align}\n\t\\nonumber\n\t\t\\mathds{E} &\\left[ V_p(\\xi_{ap}(t),\\xi_{a'p}(t)) \\right]\\le \\mathds{E}[V_p(a,a')] \\mathsf{e}^{-\\kappa_p t}.\n\\end{align}\n\\end{small}Hence, using property (ii) in Definition \\ref{delta_SGAS_Lya}, we have\n\t\\begin{small}\n\t\\begin{align}\\nonumber\n\t\t\\underline{\\alpha}_p\\left(\\mathds{E} \\left[ \\left\\|\\xi_{ap}(t)-\\xi_{a'p}(t)\\right\\|^q\\right]\\right)&\\le \\mathds{E} \\left[ \\underline{\\alpha}_p\\left(\\left\\| \\xi_{ap}(t)-\\xi_{a'p}(t)\\right\\|^q\\right) \\right]\\le\\mathds{E} \\left[ V_p\\left(\\xi_{ap}(t),\\xi_{a'p}(t)\\right)\\right]\\leq \\mathds{E}\\left[V_p(a,a')\\right] \\mathsf{e}^{-\\kappa_p t}\\\\\\notag&\\leq \\mathds{E}\\left[\\overline{\\alpha}_p\\left(\\left\\| a - a' \\right\\|^q\\right)\\right] \\mathsf{e}^{-\\kappa_p t}\\le \\overline{\\alpha}_p\\left( \\mathds{E}\\left[\\left\\| a - a' \\right\\|^q\\right] \\right) \\mathsf{e}^{-\\kappa_p t},\n\t\\end{align}\\end{small}where the first and last inequalities follow from property (i) and Jensen's inequality \\cite[p. 310]{oksendal}. Since $\\underline{\\alpha}_p\\in\\mathcal{K}_\\infty$, we obtain\n\t\\begin{small}\n\\begin{align}\\nonumber\n\\mathds{E} \\left[ \\left\\|\\xi_{ap}(t)-\\xi_{a'p}(t)\\right\\|^q\\right]\\le\\underline{\\alpha}_p^{-1} \\left(\\overline{\\alpha}_p\\left( \\mathds{E}\\left[ \\|a - a' \\|^q\\right] \\right) \\mathsf{e}^{-\\kappa_p t}\\right).\n\\end{align}\n\\end{small}Therefore, by introducing function\n\n\n\t\t$\\beta_p\\left(r,s\\right) := \\underline{\\alpha}_p^{-1}\\left(\\overline{\\alpha}_p\\left(r\\right)\\mathsf{e}^{-\\kappa_ps} \\right)$,\t\n\t\n\n\tcondition (\\ref{delta_SGAS}) is satisfied. Hence, the stochastic subsystem $\\Sigma_p$ is $\\delta$-GAS-M$_q$.\t\n\\end{proof}\n\nLet us now direct our attention from subsystems to the overall switched model. \nAs qualitatively stated in the introduction, \nit is known that a non-probabilistic switched system, whose subsystems are all $\\delta$-GAS, may exhibit some unstable behaviors under fast switching signals \\cite{girard2} and, hence, may not be $\\delta$-GUAS. \nThe same phenomenon can happen for a stochastic switched system endowed by $\\delta$-GAS-M$_q$ subsystems. \nThe $\\delta$-GUAS property of non-probabilistic switched systems can be established by using a common Lyapunov function, \nor alternatively via multiple Lyapunov functions that are mode-dependent \\cite{girard2}.  \nThis leads to the following extensions for $\\delta$-GUAS-M$_q$ property of stochastic switched systems. \n\nAssume that for any $p\\in\\mathsf{P}$, the stochastic subsystem $\\Sigma_p$ admits a $\\delta$-GAS-M$_q$ Lyapunov function $V_p$, satisfying conditions (i)-(iii) in Definition \\ref{delta_SGAS_Lya} with $\\mathcal{K}_\\infty$ functions $\\underline\\alpha_p$, $\\overline\\alpha_p$, and a constant $\\kappa_p\\in{\\mathbb{R}}^+$. \nLet us introduce the $\\mathcal{K}_\\infty$ functions $\\underline\\alpha$ and $\\overline\\alpha$ and the positive constant $\\kappa$ for use in the rest of the paper as the following: $\\underline\\alpha=\\min\\left\\{\\underline\\alpha_1,\\ldots,\\underline\\alpha_m\\right\\}$, \n$\\overline\\alpha=\\max\\left\\{\\overline\\alpha_1,\\ldots,\\overline\\alpha_m\\right\\}$, \nand $\\kappa=\\min\\left\\{\\kappa_1,\\ldots,\\kappa_m\\right\\}$. \nWe first show a result based on the existence of a common Lyapunov function in which $\\underline\\alpha=\\underline\\alpha_1=\\cdots=\\underline\\alpha_m$ and $\\overline\\alpha=\\overline\\alpha_1=\\cdots=\\overline\\alpha_m$.  \n\n\\begin{theorem}\\label{theorem2}\nConsider a stochastic switched system $\\Sigma$. \nIf there exists a function $V$ that is a common $\\delta$-GAS-M$_q$ Lyapunov function for all the subsystems $\\left\\{\\Sigma_1,\\ldots,\\Sigma_m\\right\\}$, \nthen $\\Sigma$ is $\\delta$-GUAS-M$_q$.\n\\end{theorem}\n\n\\begin{proof}\nThe proof is a consequence of the application of Gronwall's inequality and of Ito's lemma \\cite[p. 80 and 123]{oksendal}. For any ${\\mathbb{R}}^n$-valued random variables $a$ and $a'$ that are measurable in $\\mathcal{F}_0$, any switching signal $\\upsilon\\in\\mathcal{P}$, and for all $t\\in{\\mathbb{R}}_0^+$ where $\\upsilon$ is continuous, we have $\\mathcal{L}V\\left(\\xi_{a\\upsilon}(t),\\xi_{a'\\upsilon}(t)\\right)\\leq-\\kappa V(\\xi_{a\\upsilon}(t),\\xi_{a'\\upsilon}(t))$. \nUsing the continuity of $V$ and of the solution process $\\xi$, for all $t\\in{\\mathbb{R}}_0^+$ one gets\n\\begin{small}\n\t\t\\begin{align*}\n\t\t\t\\mathds{E} \\left[ V(\\xi_{a\\upsilon}(t),\\xi_{a'\\upsilon}(t)) \\right]&\\le\\mathds{E} \\left[ V(a,a') + \\int_0^{t} \\left(-\\kappa V(\\xi_{a\\upsilon}(s),\\xi_{a'\\upsilon}(s))\\right)ds\\right] \\le-\\kappa\\int_0^{t} \\mathds{E} \\left[ V(\\xi_{a\\upsilon}(s),\\xi_{a'\\upsilon}(s))\\right ]ds+\\mathds{E}\\left[V(a,a')\\right]\\nonumber,\n\t\t\\end{align*}\n\t\t\\end{small}which, by virtue of Gronwall's inequality, leads to \n\t\t\\begin{small}\n\\begin{align}\n\t\\nonumber\n\t\t\\mathds{E} &\\left[ V(\\xi_{a\\upsilon}(t),\\xi_{a'\\upsilon}(t)) \\right]\\le \\mathds{E}[V(a,a')] \\mathsf{e}^{-\\kappa t}.\n\\end{align}\n\\end{small}\nSince the $\\mathcal{K}_\\infty$ functions $\\underline\\alpha$ and $\\overline\\alpha$ are convex and concave, respectively, using Jensen's inequality we have\n\\begin{small}\n\t\\begin{align}\\nonumber\n\t\t\\underline\\alpha\\left(\\mathds{E} \\left[ \\left\\Vert\\xi_{a\\upsilon}(t)-\\xi_{a'\\upsilon}(t)\\right\\Vert^q\\right]\\right)&\\le \\mathds{E} \\left[ \\underline\\alpha\\left(\\left\\Vert \\xi_{a\\upsilon}(t)-\\xi_{a'\\upsilon}(t)\\right\\Vert^q\\right) \\right]\\le\\mathds{E} \\left[ V\\left(\\xi_{a\\upsilon}(t),\\xi_{a'\\upsilon}(t)\\right)\\right]\\leq \\mathds{E}\\left[V(a,a')\\right] \\mathsf{e}^{-\\kappa t}\\\\\\notag&\\leq \\mathds{E}\\left[\\overline\\alpha\\left(\\left\\Vert a - a' \\right\\Vert^q\\right)\\right] \\mathsf{e}^{-\\kappa t}\\le \\overline\\alpha\\left( \\mathds{E}\\left[\\left\\Vert a - a' \\right\\Vert^q\\right] \\right) \\mathsf{e}^{-\\kappa t}.\n\t\\end{align}\n\t\\end{small}\nSince $\\underline\\alpha\\in\\mathcal{K}_\\infty$, we obtain \\begin{small}$$\\mathds{E} \\left[ \\left\\|\\xi_{a\\upsilon}(t)-\\xi_{a'\\upsilon}(t)\\right\\|^q\\right]\\le\\underline{\\alpha}^{-1} \\left(\\overline{\\alpha}\\left( \\mathds{E}\\left[ \\|a - a' \\|^q\\right] \\right) \\mathsf{e}^{-\\kappa t}\\right),$$\\end{small}for all $t\\in{\\mathbb{R}}_0^+$. Then condition (\\ref{delta_SGUAS}) holds with the function $\\beta(r,s):=\\underline\\alpha^{-1}\\left(\\overline\\alpha({r})\\mathsf{e}^{-\\kappa s}\\right)$. \n\\end{proof}\n\nWhen a common $\\delta$-GAS-M$_q$ Lyapunov function $V$ fails to exist, \nthe $\\delta$-GUAS-M$_q$ property of $\\Sigma$ can still be established by resorting to multiple $\\delta$-GAS-M$_q$ Lyapunov functions (one per mode) over a restricted set of switching signals. \nMore precisely, let $\\mathcal{P}_{\\tau_d}$ be a subset of the set of switching signals $\\upsilon$ with \\emph{dwell time} \n$\\tau_d\\in{\\mathbb{R}}_0^+$, \nwhere $\\upsilon$ is said to have dwell time $\\tau_d$ if the switching times $t_1,t_2,\\ldots$ (occurring at the discontinuity points of $\\upsilon$) satisfy $t_1>\\tau_d$ and $t_i-t_{i-1}\\geq\\tau_d$, for all $i\\geq{2}$. \nWe now show a stability result based on the existence of multiple Lyapunov functions.\n\n\\begin{theorem}\\label{multiple_lyapunov}\nLet $\\tau_d\\in{\\mathbb{R}}_0^+$, and consider a stochastic switched system $\\Sigma_{\\tau_d}=(\\mathbb{R}^{n},\\mathsf{P},\\mathcal{P}_{\\tau_d},F,G)$. Assume that for any $p\\in\\mathsf{P}$, there exists a $\\delta$-GAS-M$_q$ Lyapunov function $V_p$ for subsystem $\\Sigma_{\\tau_d,p}$ and that in addition there exits a constant $\\mu\\geq1$ such that\n\\begin{small}\n\\begin{equation}\\label{eq0}\n\\forall{x,x'}\\in{\\mathbb{R}}^n,~~\\forall{p,p'\\in\\mathsf{P}},~~V_p(x,x')\\leq\\mu V_{p'}(x,x').\n\\end{equation}\n\\end{small}\nIf $\\tau_d>\\log{\\mu}\/\\kappa$, then $\\Sigma_{\\tau_d}$ is $\\delta$-GUAS-M$_q$.\n\\end{theorem}\n\n\\begin{proof}\nThe proof is inspired by that of Theorem 2.8 in \\cite{girard2} for the non-probabilistic case. We show the result for the case that switching signals have infinite number of discontinuities (switching times). A proof for the case of finite discontinuities can be written in a similar way. Let $a$ and $a'$ be any ${\\mathbb{R}}^n$-valued random variables that are measurable in $\\mathcal{F}_0$, $\\upsilon\\in\\mathcal{P}_{\\tau_d}$, $t_0=0$, and let $p_{i+1}\\in\\mathsf{P}$ denote the value of the switching signal on the open interval $(t_i,t_{i+1})$, for $i\\in{\\mathbb{N}}_0$. Using (iii) in Definition \\ref{delta_SGAS_Lya} for all $i\\in{\\mathbb{N}}_0$ and $t\\in(t_i,t_{i+1})$, one gets\\begin{small}$$\\mathcal{L} V_{p_{i+1}}\\left(\\xi_{a\\upsilon}(t),\\xi_{a'\\upsilon}(t)\\right)\\leq-\\kappa V_{p_{i+1}}\\left(\\xi_{a\\upsilon}(t),\\xi_{a'\\upsilon}(t)\\right).$$\\end{small} Similar to the proof of Theorem \\ref{theorem2}, for all $i\\in{\\mathbb{N}}_0$ and $t\\in[t_i,t_{i+1}]$, we have \n\\begin{small}\n\\begin{align}\\label{eq1}\n\\mathds{E}&\\left[V_{p_{i+1}}(\\xi_{a\\upsilon}(t),\\xi_{a'\\upsilon}(t))\\right]\\leq\\mathds{E}\\left[V_{p_{i+1}}(\\xi_{a\\upsilon}(t_i),\\xi_{a'\\upsilon}(t_i))\\right]\\mathsf{e}^{-\\kappa(t-t_i)}. \n\\end{align}\n\\end{small}\nParticularly, for $t=t_{i+1}$ and from (\\ref{eq0}), it can be checked that for all $i\\in{\\mathbb{N}}_0$:\n\\begin{small}\n\\begin{align*}\n\\mathds{E}&\\left[V_{p_{i+2}}(\\xi_{a\\upsilon}(t_{i+1}),\\xi_{a'\\upsilon}(t_{i+1}))\\right]\\leq\\mu\\mathsf{e}^{-\\kappa(t_{i+1}-t_i)}\\mathds{E}\\left[V_{p_{i+1}}(\\xi_{a\\upsilon}(t_i),\\xi_{a'\\upsilon}(t_i))\\right]. \n\\end{align*}\n\\end{small}\nUsing this inequality, we prove by induction that for all $i\\in{\\mathbb{N}}_0$\n\\begin{small}\n\\begin{equation}\\label{eq2}\n\\mathds{E}\\left[V_{p_{i+1}}(\\xi_{a\\upsilon}(t_i),\\xi_{a'\\upsilon}(t_i))\\right]\\leq\\mu^i\\mathsf{e}^{-\\kappa{t_i}}\\mathds{E}\\left[V_{p_1}(a,a')\\right].\n\\end{equation} \n\\end{small}\nFrom (\\ref{eq1}) and (\\ref{eq2}), for all $i\\in{\\mathbb{N}}_0$ and $t\\in[t_i,t_{i+1}]$, one obtains\\begin{small}$$\\mathds{E}\\left[V_{p_{i+1}}(\\xi_{a\\upsilon}(t),\\xi_{a'\\upsilon}(t))\\right]\\leq\\mu^i\\mathsf{e}^{-\\kappa{t}}\\mathds{E}\\left[V_{p_1}(a,a')\\right].$$\\end{small} Since the switching signal $\\upsilon$ has dwell time $\\tau_d$, then $t_i\\geq i\\tau_d$ and hence for all $t\\in[t_i,t_{i+1}]$, $t\\geq i\\tau_d$. Since $\\mu\\geq1$, then for all $i\\in{\\mathbb{N}}_0$ and $t\\in[t_i,t_{i+1}]$, one has $\\mu^i=\\mathsf{e}^{i\\log{\\mu}}\\leq\\mathsf{e}^{\\left(\\log{\\mu}\/\\tau_d\\right)t}.$ Therefore, for all $i\\in{\\mathbb{N}}_0$ and $t\\in[t_i,t_{i+1}]$, we get\\begin{small}$$\\mathds{E}\\left[V_{p_{i+1}}(\\xi_{a\\upsilon}(t),\\xi_{a'\\upsilon}(t))\\right]\\leq\\mathsf{e}^{\\left(\\left(\\log{\\mu}\/\\tau_d\\right)-\\kappa\\right)t}\\mathds{E}\\left[V_{p_1}(a,a')\\right].$$\\end{small} Using functions $\\underline\\alpha,\\overline\\alpha$ and Jensen's inequality, and for all $t\\in{\\mathbb{R}}_0^+$, where $t\\in[t_i,t_{i+1}]$ for some $i\\in{\\mathbb{N}}_0$, we have\n\\begin{small}\n\\begin{align}\\nonumber\n\\underline\\alpha\\left(\\mathds{E}\\left[\\left\\Vert\\xi_{a\\upsilon}(t)-\\xi_{a'\\upsilon}(t)\\right\\Vert^q\\right]\\right)&\\leq\\underline\\alpha_{p_{i+1}}\\left(\\mathds{E}\\left[\\left\\Vert\\xi_{a\\upsilon}(t)-\\xi_{a'\\upsilon}(t)\\right\\Vert^q\\right]\\right)\\leq\\mathds{E}\\left[\\underline\\alpha_{p_{i+1}}\\left(\\left\\Vert\\xi_{a\\upsilon}(t)-\\xi_{a'\\upsilon}(t)\\right\\Vert^q\\right)\\right]\\leq\\mathds{E}\\left[V_{p_{i+1}}(\\xi_{a\\upsilon}(t),\\xi_{a'\\upsilon}(t))\\right]\\\\\\notag&\\leq\\mathsf{e}^{\\left(\\left(\\log{\\mu}\/\\tau_d\\right)-\\kappa\\right)t}\\mathds{E}\\left[V_{p_1}(a,a')\\right]\\leq\\mathsf{e}^{\\left(\\left(\\log{\\mu}\/\\tau_d\\right)-\\kappa\\right)t}\\mathds{E}\\left[\\overline\\alpha_{p_1}\\left(\\Vert a-a'\\Vert^q\\right)\\right]\\\\\\notag&\\leq\\mathsf{e}^{\\left(\\left(\\log{\\mu}\/\\tau_d\\right)-\\kappa\\right)t}\\overline\\alpha_{p_1}\\left(\\mathds{E}\\left[\\Vert a-a'\\Vert^q\\right]\\right)\\leq\\mathsf{e}^{\\left(\\left(\\log{\\mu}\/\\tau_d\\right)-\\kappa\\right)t}\\overline\\alpha\\left(\\mathds{E}\\left[\\left\\Vert a-a'\\right\\Vert^q\\right]\\right).\n\\end{align}\n\\end{small}\nTherefore, for all $t\\in{\\mathbb{R}}_0^+$\n\\begin{small}\n\\begin{align*}\n\\mathds{E}&\\left[\\left\\Vert\\xi_{a\\upsilon}(t)-\\xi_{a'\\upsilon}(t)\\right\\Vert^q\\right]\\leq\\underline\\alpha^{-1}\\left(\\mathsf{e}^{\\left(\\left(\\log{\\mu}\/\\tau_d\\right)-\\kappa\\right)t}\\overline\\alpha\\left(\\mathds{E}\\left[\\Vert a-a'\\Vert^q\\right]\\right)\\right). \n\\end{align*}\n\\end{small}\nThen condition (\\ref{delta_SGUAS}) holds with the function $\\beta(r,s):=\\underline\\alpha^{-1}\\left(\\overline\\alpha({r})\\mathsf{e}^{\\left(\\left(\\log{\\mu}\/\\tau_d\\right)-\\kappa\\right)s}\\right)$ which is a $\\mathcal{KL}$ function since by assumption $\\log\\mu\/\\tau_d-\\kappa<0$. The same inequality holds for switching signals with a finite number of discontinuities, \nhence the stochastic switched system $\\Sigma_{\\tau_d}$ is $\\delta$-GUAS-M$_q$. \n\\end{proof}\n\n\n\n\\begin{comment}\n\nWe look next into special instances where these functions can be easily computed based on the model dynamics. \nThe first result provides a sufficient condition for a particular function $V_p$ to be a $\\delta$-GAS-M$_q$ Lyapunov function for a stochastic subsystem $\\Sigma_p$, when $q\\in[1,2]$.\n\n\\begin{lemma}\\label{lem:lyapunov}\n\tConsider a stochastic subsystem $\\Sigma_p$. Let $q\\in[1,2]$, $P_p\\in{\\mathbb{R}}^{n\\times{n}}$ be a symmetric positive definite matrix, and the function $V_p: {\\mathbb{R}}^n \\times {\\mathbb{R}}^n \\rightarrow {\\mathbb{R}}_0^{+}$ be defined as follows:\n\n\t\\begin{align}\n\t\\label{V}\n\t\t\tV_p(x,x'):=\\left(\\frac{1}{q}\\left(x-x'\\right)^TP_p\\left(x-x'\\right)\\right)^{\\frac{q}{2}},\t\n\t\\end{align}\n\n\tand satisfies\n\n\t\\begin{align}\n\t\\notag\n\t(x-x')^T P_p(f_p(x)-f_p(x'))+& \\frac{1}{2} \\left \\| \\sqrt{P_p} \\left( g_p(x) - g_p(x')\\right) \\right \\|_F^2\\\\\\label{nonlinear ineq cond}&\\le-\\kappa_p \\left(V_p(x,x')\\right)^{\\frac{2}{q}},\t\n\t\\end{align}\n\n\tor, if $f_p$ is differentiable, satisfies\n\n\t\\begin{align}\\notag\n\t\t(x-x')^T P_p \\partial_x f_p(z)(x-x') +& \\frac{1}{2} \\left \\| \\sqrt{P_p} \\left( g_p(x) - g_p(x')\\right) \\right \\|_F^2\\\\\\label{nonlinear ineq cond1}&\\le-\\kappa_p \\left(V_p(x,x')\\right)^{\\frac{2}{q}},\t\n\t\\end{align}\n\n\tfor all $x,x',z\\in{\\mathbb{R}}^n$, and for some constant $\\kappa_p\\in{\\mathbb{R}}^+$. Then $V_p$ is a $\\delta$-GAS-M$_q$ Lyapunov function for $\\Sigma_p$.\n\\end{lemma}\n\n\\begin{proof}\nIt is not difficult to check that the function $V_p$ in \\eqref{V} satisfies properties (i) and (ii) of Definition \\ref{delta_SGAS_Lya} with functions $\\underline{\\alpha}_p(y) :=\\left(\\frac{1}{q}\\lambda_{\\min}\\left({P_p}\\right)\\right)^{\\frac{q}{2}}y$ and $\\overline{\\alpha}_p(y) := \\left(\\frac{n}{q}\\lambda_{\\max}\\left({P_p}\\right)\\right)^{\\frac{q}{2}}y$. It then suffices to verify property (iii). We verify property (iii) for the case that $f_p$ is differentiable and using condition (\\ref{nonlinear ineq cond1}). The proof, using condition (\\ref{nonlinear ineq cond}), is completely similar by just removing the inequality in the proof including derivative of $f_p$. By the definition of $V_p$ in (\\ref{V}), for any $x,x'\\in{\\mathbb{R}}^n$ such that $x\\neq{x'}$, and for $q\\in[1,2]$, one has\t\n\n\t\\begin{align*}\n\t\t\\partial_xV_p&= -\\partial_{x'} V_p=(x-x')^TP_p\\left(\\widetilde{V}(x,x')\\right)^{\\frac{q}{2}-1},\\\\\n\t\t\\partial_{x,x}V_p&= \\partial_{x',x'}V_p= -\\partial_{x,x'}V_p=P_p\\left(\\widetilde{V}(x,x')\\right)^{\\frac{q}{2}-1}\\\\&\\qquad+\\frac{q-2}{q}P_p(x-x')(x-x')^TP_p\\left(\\widetilde{V}(x,x')\\right)^{\\frac{q}{2}-2},\n\t\\end{align*}\t\n\n\twhere $\\widetilde{V}(x,x')=\\frac{1}{q}\\left(x-x'\\right)^TP_p\\left(x-x'\\right).$\n\tTherefore, following the definition of $\\mathcal{L}$, and for any \\mbox{$x,x',z\\in{\\mathbb{R}}^n$} such that $x\\neq{x'}$, one obtains:\n\n\t\\begin{align}\\nonumber\n\t\t&\\mathcal{L} V_p(x, x')\\\\\\notag&= (x-x')^TP_p\\left(\\widetilde{V}(x,x')\\right)^{\\frac{q}{2}-1}\\left(f_p(x)-f_p(x')\\right)\\\\\\notag&\\quad+\\frac{1}{2} \\text{Tr} \\left(\\begin{bmatrix} g_p(x) \\\\ g_p(x') \\end{bmatrix}\\left[g_p^T(x)~~g_p^T(x')\\right] \\begin{bmatrix}\n\\partial_{x,x} V_p& -\\partial_{x,x}V_p\\\\ -\\partial_{x,x}V_p& \\partial_{x,x}V_p\n\\end{bmatrix}\\right)\\\\\\notag\n\t\t& = (x-x')^TP_p\\left(\\widetilde{V}(x,x')\\right)^{\\frac{q}{2}-1}\\left(f_p(x) - f_p(x')\\right)\\\\\\notag\t\t\n\t\t& \\quad + \\frac{1}{2}\\text{Tr}\\left( \\left(g_p(x)-g_p(x')\\right) \\left ( g_p^T(x) - g_p^T(x') \\right) \\partial_{x,x} V_p \\right)\t\\\\\\notag\t\n\t\t& = (x-x')^TP_p\\left(\\widetilde{V}(x,x')\\right)^{\\frac{q}{2}-1}\\left(f_p(x) - f_p(x')\\right)\\\\\\notag&\\quad+\\frac{1}{2}\\left \\| \\sqrt{P_p} \\left( g_p(x) - g_p(x')\\right) \\right \\|_F^2\\left(\\widetilde{V}(x,x')\\right)^{\\frac{q}{2}-1}\\\\\\notag&\\quad +\\frac{q-2}{q}\\left \\|(x-x')^TP_p\\left( g_p(x) - g_p(x')\\right) \\right \\|_F^2\\left(\\widetilde{V}(x,x')\\right)^{\\frac{q}{2}-2} \\\\\\notag\n\t\t&\\le(x-x')^TP_p\\left(\\widetilde{V}(x,x')\\right)^{\\frac{q}{2}-1} \\left(f_p(x) - f_p(x')\\right)\\\\\\notag&\\quad+\\frac{1}{2}\\left \\| \\sqrt{P_p} \\left( g_p(x) - g_p(x')\\right) \\right \\|_F^2\\left(\\widetilde{V}(x,x')\\right)^{\\frac{q}{2}-1}\\\\\n\t\t\\label{1}\n\t\t& \\le \\bigg[(x-x')^TP_p\\partial_x f_p(z)(x-x')\\\\\\notag&\\quad+\\frac{1}{2}\\left \\| \\sqrt{P_p} \\left( g_p(x) - g_p(x')\\right) \\right \\|_F^2\\bigg]\\left(\\widetilde{V}(x,x')\\right)^{\\frac{q}{2}-1}\\\\\\notag\n\t\t&\\le -\\kappa_p V_p(x,x').\n\t\\end{align}\n\n\tIn \\eqref{1}, $z \\in {\\mathbb{R}}^n$ where the mean value theorem \n\n\tis applied to the differentiable function $x \\mapsto f_p(x)$ at points $x,x'$.  \\qed\t\n\\end{proof}\n\nThe next result provides a condition that is equivalent to (\\ref{nonlinear ineq cond}) or to (\\ref{nonlinear ineq cond1}) for affine stochastic subsystems $\\Sigma_p$ \n(that is, for subsystems with affine drift and linear diffusion terms) \nin the form of a linear matrix inequality (LMI), which can be easily solved by numerical optimization tools. \n\n\\begin{corollary}\\label{corollary1}\n\tConsider a stochastic subsystem $\\Sigma_p$, \n\twhere for any $x\\in{\\mathbb{R}}^n$, $f_p(x):= A_px+b_p$ for some $A_p\\in{\\mathbb{R}}^{n\\times{n}},b_p\\in{\\mathbb{R}}^{n}$, \n\tand $g_p(x) := \\begin{bmatrix} \\sigma_{1,p} x~&\\sigma_{2,p}x~&\\ldots~& \\sigma_{\\widehat{q},p} x\\end{bmatrix}$ for some $\\sigma_{i,p} \\in {\\mathbb{R}}^{n \\times n}$, where $i=1,\\ldots,\\widehat{q}$. \n\tThen, \n\tfunction $V_p$ in \\eqref{V} is a $\\delta$-GAS-M$_q$ Lyapunov function for $\\Sigma_p$ if there exists a positive constant $\\widehat\\kappa_p\\in{\\mathbb{R}}^+$ satisfying the following LMI:\n\n\t\\begin{align}\n\t\\label{LMI}\n\t\tP_pA_p + A_p^TP_p + \\sum_{i=1}^{\\widehat{q}} \\sigma_{i,p}^T P_p \\sigma_{i,p} \\preceq -\\widehat\\kappa_p P_p.\n\t\\end{align}\n\n\\end{corollary}\n\n\\begin{proof}\n\t\tThe corollary is a particular case of Lemma \\ref{lem:lyapunov}. It suffices to show that for affine dynamics the LMI \\eqref{LMI} yields the condition in \\eqref{nonlinear ineq cond} or \\eqref{nonlinear ineq cond1}. First it is straightforward to observe that\n\t\n\t\\begin{align*}\n\t\t&\\left \\| \\sqrt{P_p} \\left( g_p(x) - g_p(x')\\right) \\right \\|_F^2 \\\\&\\quad= \\text{Tr}\\left( \\left(g_p(x)-g_p(x')\\right)^T P_p \\left(g_p(x)-g_p(x')\\right) \\right) \\\\&\\quad=\\left(x-x'\\right)^T \\sum_{i=1}^{\\widehat{q}}\\sigma_{i,p}^T P_p \\sigma_{i,p}(x-x'),\n\t\t\\end{align*}\n\t\n\t\tand that \n\t\n\t\t\\begin{align*}\n\t(x-x')^TP_p&\\partial_x f_p(z)(x-x')=(x-x')^TP_p\\left(f_p(x)-f_p(x')\\right)\\\\&=\\frac{1}{2}(x-x')^T\\left(P_pA_p+A_p^TP_p\\right)(x-x'),\n\t\\end{align*}\n\n\tfor any $x,x',z\\in{\\mathbb{R}}^n$. Now suppose there exists $\\widehat\\kappa_p\\in{\\mathbb{R}}^+$ such that \\eqref{LMI} holds. It can be readily verified that the desired assertion of \\eqref{nonlinear ineq cond} or \\eqref{nonlinear ineq cond1} is verified by choosing $\\kappa_p=\\widehat\\kappa_p\/2$. \\qed\n\\end{proof}\n\nNotice that \nCorollary \\ref{corollary1} allows us obtaining tighter upper bounds for the inequalities (\\ref{delta_SGAS}) and (\\ref{delta_SGUAS}) for any $p\\in\\mathsf{P}$, \nby selecting appropriate matrices $P_p$ satisfying the LMI in (\\ref{LMI}). \n\n\\end{comment}\n\nIn order to show some of the main results of the paper in Section \\ref{existence}, \nwe need the following technical result, \nwhich provides an upper bound on the distance (in the $q$th moment) between the solution processes of $\\Sigma_p$ (resp. $\\Sigma_{\\tau_d,p}$) and the corresponding non-probabilistic subsystem $\\overline\\Sigma_p$ (resp. $\\overline\\Sigma_{\\tau_d,p}$), obtained by disregarding the diffusion term $g_p$. \nFrom now on, we use the notation $\\overline\\xi_{xp}$ to denote the trajectory of $\\overline\\Sigma_p$ (resp. $\\overline\\Sigma_{\\tau_d,p}$) starting from the initial condition $x$ and satisfying the ordinary differential equation (ODE) $\\dot{\\overline\\xi}_{xp}=f_p\\left(\\overline\\xi_{xp}\\right)$. \n\n\\begin{lemma}\n\\label{lem:moment est}\n\tConsider a stochastic subsystem $\\Sigma_p$ (resp. $\\Sigma_{\\tau_d,p}$) such that $f_p(0_n)=0_n$ and $g_p(0_n) = 0_{n\\times{\\widehat{q}}}$.\t\n\tSuppose $q\\geq2$ and there exists a $\\delta$-GAS-M$_q$ Lyapunov function $V_p$ for $\\Sigma_p$ (resp. $\\Sigma_{\\tau_d,p}$) such that its Hessian is a positive semidefinite matrix in ${\\mathbb{R}}^{2n\\times2n}$ and $\\partial_{x,x}V_p(x,x')\\leq P_p$, $\\forall x,x'\\in{\\mathbb{R}}^n$ and some positive semidefinite matrix $P_p\\in{\\mathbb{R}}^{n\\times{n}}$. Then for any $x\\in{\\mathbb{R}}^n$, we have \\begin{small}$\\mathds{E} \\left[\\left\\Vert\\traj{\\xi}{x}{p}(t)-\\traj{\\overline \\xi}{x}{p}(t)\\right\\Vert^q\\right] \\le h_x^p(t)$\\end{small}, \n\twhere \n\t\\begin{small}\n\t\\begin{align}\\nonumber\n\th_x^p(t)=&\\underline\\alpha_p^{-1}\\left(\\frac{1}{2}\\left\\Vert{\\sqrt{P_p}}\\right\\Vert^2\\min\\{n,\\widehat{q}\\}Z_p^2\\mathsf{e}^{-\\kappa_p t}\\int_0^t\\left(\\beta_p\\left(\\left\\Vert{x}\\right\\Vert^q,s\\right)\\right)^{\\frac{2}{q}}ds\\right),\n\t\\end{align}\n\t\\end{small}$Z_p$ is the Lipschitz constant, introduced in Definition \\ref{Def_control_sys}, and $\\beta_p$ is the $\\mathcal{KL}$ function\\footnote{Using a $\\delta$-GAS-M$_q$ Lyapunov function $V_p$, one can always choose $\\beta_p(r,s)=\\underline{\\alpha}_p^{-1}\\left(\\overline{\\alpha}_p\\left(r\\right)\\mathsf{e}^{-\\kappa_ps}\\right)$, as showed in Theorem \\ref{the_Lya}.} appearing in \\eqref{dGAS}.\n\\end{lemma}\n\nIt can be readily seen that the nonnegative valued function $h_x^p$ tends to zero as $t \\rightarrow 0$, $t\\rightarrow+\\infty$, or as $Z_p\\ra0$ and is identically zero if the diffusion term is identically zero (i.e. $Z_p=0$) which is the case for $\\overline\\Sigma_p$ (resp. $\\overline\\Sigma_{\\tau_d,p}$).\n\n\\begin{proof}\nThe proof is similar to the proof of Lemma 3.7 in \\cite{majid8}, where one needs to eliminate all the terms $\\gamma(\\cdot)$. \n\\end{proof}\n\n\\begin{comment}\nThe following lemma provides a similar result in line with that of Lemma \\ref{lem:moment est} for a stochastic subsystem $\\Sigma_p$ admitting a $\\delta$-GAS-M$_q$ Lyapunov function $V_p$ as in \\eqref{V}, where $q\\in[1,2]$.\n\n\\begin{lemma}\\label{lem:moment est1}\nConsider a stochastic subsystem $\\Sigma_p$ such that $f_p(0_n)=0_n$ and $g_p(0_n) = 0_{n\\times{\\widehat{q}}}$.\t Suppose that the function $V_p$ in \\eqref{V} satisfies \\eqref{nonlinear ineq cond} or \\eqref{nonlinear ineq cond1} for $\\Sigma_p$. For any $x$ in a compact set $\\mathsf{D}\\subset{\\mathbb{R}}^n$, we have $\\mathds{E} \\left[\\left\\Vert\\traj{\\xi}{x}{p}(t)-\\traj{\\overline \\xi}{x}{p}(t)\\right\\Vert^2\\right] \\le h_p(g_p,t),$\t \n\twhere \n\t\\begin{small}\n\t\\begin{align}\\nonumber\n\th_p(g_p,t)=&\\frac{q\\left\\Vert\\sqrt{P_p}\\right\\Vert^2n^2\\min\\left\\{n,\\widehat{q}\\right\\}Z_p^2\\mathsf{e}^{\\frac{-2\\kappa_p}{q}t}}{4\\lambda_{\\min}^2({P_p}){\\kappa_p}}\\\\\\notag&\\qquad\\cdot\\lambda_{\\max}({P_p})\\left(1-\\mathsf{e}^{\\frac{-2\\kappa_p}{q}t}\\right)\\sup_{x\\in \\mathsf{D}}\\left\\{\\Vert{x}\\Vert^2\\right\\},\n\t\\end{align}\n\t\\end{small}and $Z_p$ is the Lipschitz constant, introduced in Definition \\ref{Def_control_sys}.\n\\end{lemma}\n\nHere also it can be readily verified that the nonnegative valued function $h_p$ tends to zero as $t \\rightarrow 0$, $t\\rightarrow+\\infty$, or as $Z_p\\ra0$.\n\n\\begin{proof}\nThe proof is similar to the proof of Lemma 3.9 in \\cite{majid8}. One just needs to enforce $L_u=0$ in the proof of Lemma 3.9 in \\cite{majid8}.\n\\end{proof}\n\nFor an affine stochastic subsystem $\\Sigma_p$,\nthe following corollary provides possibly a less conservative expression for $h_p$, based on parameters of drift and diffusion. \n\n\\begin{corollary}\\label{corollary2}\nConsider a stochastic subsystem $\\Sigma_p$, where for all $x\\in{\\mathbb{R}}^n$, $f_p(x):= A_px+b_p$, for some $A_p\\in{\\mathbb{R}}^{n\\times{n}},b_p\\in{\\mathbb{R}}^{n}$, and $g_p(x) := \\left[\\sigma_{1,p} x~~\\sigma_{2,p}x~~\\ldots~~\\sigma_{\\widehat{q},p} x\\right]$, for some $\\sigma_{i,p} \\in {\\mathbb{R}}^{n \\times n}$, where $i=1,\\ldots,\\widehat{q}$. Suppose there exists a positive definite matrix $P_p\\in{\\mathbb{R}}^{n\\times{n}}$ satisfying (\\ref{LMI}) for $\\Sigma_p$.\nFor any $x$ in a compact set $\\mathsf{D}\\subset{\\mathbb{R}}^n$, we have \n\t \t$\\mathds{E} \\left[\\left\\Vert\\traj{\\xi}{x}{p}(t)-\\traj{\\overline \\xi}{x}{p}(t)\\right\\Vert^2\\right] \\le h_p(g_p,t),$\t \n\n\twhere \n\\begin{small}\n\\begin{align}\\nonumber\n\th_p(g_p,t)&=\\frac{n\\lambda_{\\max}\\left(\\sum\\limits_{i = 1}^{\\widehat{q}} \\sigma^T_{i,p}P_p\\sigma_{i,p}\\right)\\mathsf{e}^{-\\widehat{\\kappa} t}}{\\lambda_{\\min}({P_p})}\\\\\\notag&\\cdot\\int_0^t\\left(\\left\\Vert\\mathsf{e}^{A_ps}\\right\\Vert\\sup_{x\\in{\\mathsf{D}}}\\left\\{\\left\\Vert x\\right\\Vert\\right\\}+\\left(\\int_0^{s}\\left\\Vert\\mathsf{e}^{A_pr}b_p\\right\\Vert dr\\right)\\right)^2ds.\n\t\\end{align}\n\\end{small}\n\\end{corollary}\n\n\\begin{proof}\nThe proof is similar to the proof of Corollary 3.10 in \\cite{majid8}. One just requires to enforce $B=b_p$ and $\\Vert u\\Vert=1$, for any $u\\in\\mathsf{U}$, in the proof of Corollary 3.10 in \\cite{majid8}.\n\\end{proof}\n\\end{comment}\n\nThe interested readers are referred to the results in \\cite{majid8}, providing a result in line with that of Lemma \\ref{lem:moment est} for an (affine) stochastic subsystem $\\Sigma_p$ (resp. $\\Sigma_{\\tau_d,p}$) admitting a specific type of $\\delta$-GAS-M$_q$ Lyapunov functions. \nFor later use, we introduce function $h_x(t)=\\max\\left\\{h_x^1(t),\\ldots,h_x^m(t)\\right\\}$ for all $t\\in{\\mathbb{R}}_0^+$.\n\\vspace{-0.2cm}\n\n\\section{Systems and Approximate Equivalence Notions}\\label{symbolic}\nWe employ the notion of \\emph{system}, introduced in \\cite{paulo}, to provide (in Sec. \\ref{existence}) an alternative description of stochastic switched systems that can be later directly related to their symbolic models. \n\\begin{definition}\\label{system}\nA system $S$ is a tuple \n$S=(X,X_0,U,\\longrightarrow,Y,H),$\nwhere\n$X$ is a set of states (possibly infinite), \n$X_0\\subseteq X$ is a set of initial states (possibly infinite), \n$U$ is a set of inputs (possibly infinite),\n$\\longrightarrow\\subseteq X\\times U\\times X$ is a transition relation,\n$Y$ is a set of outputs, and\n$H:X\\rightarrow Y$ is an output map.\n\\end{definition}\n\nWe write $x\\rTo^{u}x'$ if \\mbox{$(x,u,x')\\in\\longrightarrow$}. \nIf $x\\rTo^{u}x'$, we call state  $x'$ a \\mbox{$u$-successor}, or simply a successor, of state $x$.\nFor technical reasons, we assume that for each $x\\in X$, there is some $u$-successor of $x$, for some $u\\in U$ -- let us remark that this is always the case for the considered systems later in this paper. \n\nA system $S$ is said to be\n\\begin{itemize}\n\\item \\textit{metric}, if the output set $Y$ is equipped with a metric\n$\\mathbf{d}:Y\\times Y\\rightarrow\\mathbb{R}_{0}^{+}$;\n\n\\item \\textit{finite} (or \\textit{symbolic}), if $X$ and $U$ are finite sets;\n\n\\item \\textit{deterministic}, if for any state $x\\in{X}$ and any input $u\\in{U}$, there exists at most one \\mbox{$u$-successor}.\n\\end{itemize}\n\nFor a system $S=(X,X_0,U,\\longrightarrow,Y,H)$, given any initial state $x_0\\in{X_0}$, a finite state run generated from $x_0$ is a finite sequence of transitions: \n\\begin{small}\n\\begin{align}\n\\label{run}\nx_0\\rTo^{u_0}x_1\\rTo^{u_1}x_2\\rTo^{u_2}\\cdots\\rTo^{u_{n-2}}x_{n-1}\\rTo^{u_{n-1}}x_n,\n\\end{align}\n\\end{small}such that $x_i\\rTo^{u_i}x_{i+1}$ for all $0\\leq i<n$. A finite state run can be trivially extended to an infinite state run \\cite{paulo}. A finite output run is a sequence $\\left\\{y_0,y_1,\\ldots,y_n\\right\\}$ such that there exists a finite state run of the form \\eqref{run} with $y_i=H(x_i)$, for $i=1,\\ldots,n$. A finite output run can also be directly extended to an infinite output run as well \\cite{paulo}.\n\nNow, we recall the notion of approximate (bi)simulation relation, introduced in \\cite{girard}, \nwhich is useful when analyzing or synthesizing controllers for deterministic systems.  \n \n\\begin{definition}\\label{APSR}\nConsider metric systems $S_{a}=(X_{a},X_{a0},U_{a},\\rTo_{a},Y_a,H_{a})$ and\n$S_{b}=(X_{b},X_{b0},U_{b},\\rTo_{b},Y_b,H_{b})$ with the\nsame output sets $Y_a=Y_b$ and metric $\\mathbf{d}$.\nFor $\\varepsilon\\in\\mathbb{R}_0^{+}$,\na relation\n\\mbox{$R\\subseteq X_{a}\\times X_{b}$} is said to be an $\\varepsilon$-approximate simulation relation from $S_{a}$ to $S_{b}$\nif for all $(x_a,x_b)\\in R$ the following two conditions are satisfied:\n\\begin{itemize}\n\\item[(i)] \\mbox{$\\mathbf{d}(H_{a}(x_{a}),H_{b}(x_{b}))\\leq\\varepsilon$};\n\\item[(ii)] \\mbox{$x_{a}\\rTo_{a}^{u_a}x'_{a}$ in $S_a$} implies the existence of \\mbox{$x_{b}\\rTo_{b}^{u_b}x'_{b}$} in $S_b$ satisfying $(x'_{a},x'_{b})\\in R$.\n\\end{itemize}\nA relation $R\\subseteq X_a\\times X_b$ is said to be an $\\varepsilon$-approximate bisimulation relation between $S_a$ and $S_b$\nif $R$ is an $\\varepsilon$-approximate simulation relation from $S_a$ to $S_b$ and\n$R^{-1}$ is an $\\varepsilon$-approximate simulation relation from $S_b$ to $S_a$.\n\nSystem $S_{a}$ is $\\varepsilon$-approximately simulated by $S_{b}$, or $S_b$ $\\varepsilon$-approximately simulates $S_a$,\ndenoted by \\mbox{$S_{a}\\preceq_{\\mathcal{S}}^{\\varepsilon}S_{b}$}, if there exists an $\\varepsilon$-approximate simulation relation $R$ from $S_a$ to $S_b$ such that:\n\\begin{itemize}\n\\item $\\forall x_{a0}\\in{X_{a0}}$, $\\exists x_{b0}\\in{X_{b0}}$ with $(x_{a0},x_{b0})\\in{R}$.\n\\end{itemize}\nSystem $S_{a}$ is $\\varepsilon$-approximately bisimilar to $S_{b}$, denoted by \\mbox{$S_{a}\\cong_{\\mathcal{S}}^{\\varepsilon}S_{b}$}, if there exists an $\\varepsilon$-approximate bisimulation relation $R$ between $S_a$ and $S_b$ such that:\n\\begin{itemize}\n\\item $\\forall x_{a0}\\in{X_{a0}}$, $\\exists x_{b0}\\in{X_{b0}}$ with $(x_{a0},x_{b0})\\in{R}$;\n\\item $\\forall x_{b0}\\in{X_{b0}}$, $\\exists x_{a0}\\in{X_{a0}}$ with $(x_{a0},x_{b0})\\in{R}$.\n\\end{itemize}\n\\end{definition}\n\n\\section{Symbolic Models for Stochastic Switched Systems}\\label{existence}\nThe main results of this work are presented in this section. \nWe show that for any stochastic switched system $\\Sigma$ (resp. $\\Sigma_{\\tau_d}$), \nadmitting a common (resp. multiple) $\\delta$-GAS-M$_q$ Lyapunov function(s), \nand for any precision level $\\varepsilon\\in{\\mathbb{R}}^+$, one can construct a finite system that is $\\varepsilon$-approximately bisimilar to $\\Sigma$ (resp. $\\Sigma_{\\tau_d}$). \n\n\\subsection{Describing stochastic switched systems as metric systems}\nIn order to show the main results of the paper, \nwe use systems as an abstract representation of stochastic switched systems, capturing all the information they contain at the sampling times.   \nMore precisely, given a stochastic switched system $\\Sigma$ and a sampling time $\\tau$, \nwe define an associated metric system $S_\\tau(\\Sigma)=(X_\\tau,X_{\\tau0},U_\\tau,\\rTo_\\tau,Y_\\tau,H_\\tau),$ \nwhere:\n\\begin{itemize}\n\\item $X_\\tau$ is the set of all ${\\mathbb{R}}^n$-valued random variables defined on \nthe probability space $(\\Omega,\\mathcal{F},\\mathds{P})$;\n\n\\item $X_{\\tau0}\\subseteq\\mathcal{X}_0$, where $\\mathcal{X}_0$ is the set of all ${\\mathbb{R}}^n$-valued random variables that are measurable over $\\mathcal{F}_0$;\n\n\\item $U_\\tau=\\mathsf{P}$;\n\n\\item $x_\\tau\\rTo^{p}_\\tau x'_\\tau$ if $x_\\tau$ and $x'_\\tau$ are measurable, \nrespectively, in $\\mathcal{F}_{k\\tau}$ and $\\mathcal{F}_{(k+1)\\tau}$ for some $k \\in{\\mathbb{N}}_0$, and \nthere exists a solution process $\\xi:\\Omega\\times{\\mathbb{R}}_0^+\\rightarrow{\\mathbb{R}}^n$ of $\\Sigma$ satisfying $\\xi(k\\tau) = x_\\tau$ and $\\xi_{x_\\tau p}(\\tau) = x'_\\tau$ $\\mathds{P}$-a.s.;\n\n\\item $Y_\\tau=X_\\tau$;\n\n\\item $H_\\tau(x_\\tau)=x_{\\tau}$ for any $x_\\tau\\in X_\\tau$.\n\\end{itemize}\nWe assume that the output set $Y_\\tau$ is equipped with the metric $\\mathbf{d}(y,y')=\\left(\\mathds{E}\\left[\\left\\Vert y-y'\\right\\Vert^q\\right]\\right)^{\\frac{1}{q}}$, for any $y,y'\\in{Y_\\tau}$ and some $q\\geq1$. Let us remark that the set of states of $S_\\tau(\\Sigma)$ is uncountable and that ${S_{\\tau}}(\\Sigma)$ is a deterministic system in the sense of Definition \\ref{system}, since (cf. Subsection \\ref{sss})  \nits solution process is uniquely determined given an initial condition. \n\nNote that a finite state run\n$x_{0}\\rTo^{p_0}_{\\tau}x_{1}\\rTo^{p_1}_{\\tau} \\,\\cdots\\, \\rTo^{p_{N-1}}_{\\tau} x_{N}$ of $S_{\\tau}(\\Sigma)$, where $p_{i-1}\\in\\mathsf{P}$ and $x_i=\\xi_{x_{i-1}p_{i-1}}(\\tau)$ $\\mathds{P}$-a.s. for $i=1,\\ldots,N$, captures the trajectory of the stochastic switched system $\\Sigma$ at times $t=0,\\tau,\\ldots,N\\tau$. This trajectory is  \nstarted from the initial condition $x_{0}\\in X_{\\tau0}$ and resulting from a switching signal $\\upsilon$ obtained by the concatenation of the modes  \n$p_{i-1}$ \\big(i.e. $\\upsilon(t)=p_{i-1}$ for any $t\\in [(i-1)\\tau,i\\,\\tau[$\\big), for $i=1,\\ldots,N$.\n\nNow we represent a stochastic switched system $\\Sigma_{\\tau_d}$ with a metric system where, without loss of generality, \nwe assume that $\\tau_d$ is an integer multiple of $\\tau$, \ni.e. $\\exists\\widehat{N}\\in{\\mathbb{N}}$ such that $\\tau_d=\\widehat{N}\\tau$. Given a stochastic switched system $\\Sigma_{\\tau_d}$ and a sampling time $\\tau\\in{\\mathbb{R}}^+$, \nwe define the metric system $S_\\tau\\left(\\Sigma_{\\tau_d}\\right)=(X_\\tau,X_{\\tau0},U_\\tau,\\rTo_\\tau,Y_\\tau,H_\\tau),$ \nwhere:\n\\begin{itemize}\n\\item $X_\\tau=\\mathcal{X}\\times\\mathsf{P}\\times\\{0,\\ldots,\\widehat{N}-1\\}$, where $\\mathcal{X}$ is the set of all ${\\mathbb{R}}^n$-valued random variables defined on the probability space $(\\Omega,\\mathcal{F},\\mathds{P})$;\n\n\\item $X_{\\tau0}\\subseteq\\mathcal{X}_0\\times\\mathsf{P}\\times\\{0,\\ldots,\\widehat{N}-1\\}$, where $\\mathcal{X}_0$ is the set of all ${\\mathbb{R}}^n$-valued random variables that are measurable over $\\mathcal{F}_0$;\n\n\\item $U_\\tau=\\mathsf{P}$;\n\n\\item $\\left(x_\\tau,p,i\\right)\\rTo^{p}_\\tau\\left({x'_\\tau},p',i'\\right)$ if $x_\\tau$ and $x'_\\tau$ are measurable, \nrespectively, in $\\mathcal{F}_{k\\tau}$ and $\\mathcal{F}_{(k+1)\\tau}$ for some $k \\in {\\mathbb{N}}_0$, and \nthere exists a solution process $\\xi:\\Omega\\times{\\mathbb{R}}_0^+\\rightarrow{\\mathbb{R}}^n$ of $\\Sigma_{\\tau_d}$ satisfying $\\xi(k\\tau) = x_\\tau$ and $\\xi_{x_\\tau p}(\\tau) = x'_\\tau$ $\\mathds{P}$-a.s. and one of the following conditions hold: \n\n\\begin{itemize}\n\\item $i<\\widehat{N}-1$, $p'=p$, and $i'=i+1$: switching is not allowed because the time elapsed since the latest switch is strictly smaller than the dwell time;\n\n\\item $i=\\widehat{N}-1$, $p'=p$, and $i'=\\widehat{N}-1$: switching is allowed but no switch occurs;\n\n\\item $i=\\widehat{N}-1$, $p'\\neq p$, and $i'=0$: switching is allowed and a switch occurs. \n\\end{itemize}\n\n\\item $Y_\\tau=\\mathcal{X}$;\n\n\\item $H_\\tau\\left(x_\\tau,p,i\\right)=x_\\tau$ for any $(x_\\tau,p,i)\\in X_\\tau$.\n\\end{itemize}\nWe assume that the output set $Y_\\tau$ is equipped with the metric $\\mathbf{d}(y,y')=(\\mathds{E}[\\left\\Vert y-y'\\right\\Vert^q])^{\\frac{1}{q}}$, $\\forall y,y'\\in{Y_\\tau}$ and for some $q\\geq1$. One can readily verify that the (in)finite output runs of $S_{\\tau}\\left(\\Sigma_{\\tau_d}\\right)$ are the (in)finite output runs of $S_\\tau(\\Sigma)$ corresponding to switching signals with dwell time $\\tau_d=\\widehat{N}\\tau$.\n\nIn order to show the main results of this work, \nwe assume that for any $\\delta$-GAS-M$_q$ Lyapunov functions $V_p$, there exists a $\\mathcal{K}_\\infty$ and concave function $\\widehat\\gamma_p$ such that\n\\begin{equation}\\label{supplement}\n\\vert V_p(x,y)-V_p(x,z)\\vert\\leq\\widehat\\gamma_p\\left(\\Vert y-z\\Vert\\right),\n\\end{equation}\nfor any $x,y,z\\in{\\mathbb{R}}^n$. \nThis assumption is not restrictive at all, provided the function $V_p$ is limited to a compact subset of ${\\mathbb{R}}^n\\times{\\mathbb{R}}^n$. For all $x,y,z\\in\\mathsf{D}$, where $\\mathsf{D}$ is a compact subset of ${\\mathbb{R}}^n$, by applying the mean value theorem \nto the function $y\\rightarrow V_p(x,y)$, one gets \n\\begin{small}\n\\begin{align}\\notag\n&\\left\\vert V_p(x,y)-V_p(x,z)\\right\\vert\\leq\\widehat\\gamma_p\\left(\\Vert y-z\\Vert\\right),\\text{s.t.}~\\widehat\\gamma_p({r})=\\left(\\max_{(x,y)\\in\\mathsf{D}\\backslash\\Delta}\\left\\Vert\\frac{\\partial{V_p}(x,y)}{\\partial{y}}\\right\\Vert\\right)r.\n\\end{align}\n\\end{small}\nFor later use, let us define the $\\mathcal{K}_\\infty$ function $\\widehat\\gamma$ such that $\\widehat\\gamma=\\max\\left\\{\\widehat\\gamma_1,\\ldots,\\widehat\\gamma_m\\right\\}$. (Note that, for the case of a common Lyapunov function, we have: $\\widehat\\gamma=\\widehat\\gamma_1=\\cdots=\\widehat\\gamma_m$.) \nWe proceed presenting the main results of this work.  \n\n\\subsection{First approach}\nThis subsection contains the first main results of the paper which are based on the state space discretization. For later use in this subsection, let us define the function $h_X(t)=\\max_{x\\in X}h_x(t)$, for a set $X\\subseteq {\\mathbb{R}}^n$.\n\n\\subsubsection{Common Lyapunov function}\nWe show the first result on finite abstractions based on the existence of a common $\\delta$-GAS-M$_q$ Lyapunov function for subsystems $\\Sigma_1,\\ldots,\\Sigma_m$. \nConsider a stochastic switched system $\\Sigma$ \nand a pair $\\mathsf{q}=(\\tau,\\eta)$ of quantization parameters, \nwhere $\\tau$ is the sampling time and $\\eta$ is the state space quantization. \nGiven $\\Sigma$ and $\\mathsf{q}$, consider the following system:\n\\begin{small}\n\\begin{equation}\\label{T2}\nS_{\\mathsf{q}}(\\Sigma)=(X_{\\mathsf{q}},X_{\\params0},U_{\\mathsf{q}},\\rTo_{\\mathsf{q}},Y_{\\mathsf{q}},H_{\\mathsf{q}}),\n\\end{equation}\n\\end{small}where $X_{\\mathsf{q}}=[{\\mathbb{R}}^n]_\\eta$, $X_{\\params0}=[{\\mathbb{R}}^n]_\\eta$, $U_{\\mathsf{q}}=\\mathsf{P}$, $Y_{\\mathsf{q}}=Y_\\tau$, and\n\\begin{itemize}\n\\item $x_{\\mathsf{q}}\\rTo_{\\mathsf{q}}^{p}x'_{\\mathsf{q}}$ if there exists a $x'_{\\mathsf{q}}\\in X_{\\mathsf{q}}$ such that $\\left\\Vert\\overline{\\xi}_{x_{\\mathsf{q}}p}(\\tau)-x'_{\\mathsf{q}}\\right\\Vert\\leq\\eta$;\n\n\n\\item $H_{\\mathsf{q}}(x_{\\mathsf{q}})=x_{\\mathsf{q}}$ for any $x_{\\mathsf{q}}\\in X_{\\mathsf{q}}$.\n\\end{itemize}\n\nIn order to relate models, \nthe output set $Y_{\\mathsf{q}}$ is taken to be that of the system $S_\\tau(\\Sigma)$. \nTherefore, $H_{\\mathsf{q}}$, with a slight abuse of notation, is a mapping from a grid point to a random variable with a Dirac probability distribution centered at the grid point. \n\n\nWe now present the first main result of the paper. In order to show the next result, we assume that $f_p(0_n)=0_n$ only if $\\Sigma_p$ is not affine and that $g_p(0_n)=0_{n\\times\\widehat{q}}$ for any $p\\in\\mathsf{P}$.\n\n\\begin{theorem}\\label{main_theorem1}\nLet $\\Sigma$ be a stochastic switched system admitting a common $\\delta$-GAS-M$_q$ Lyapunov function $V$, \nof the form discussed in Lemma \\ref{lem:moment est}, for subsystems $\\Sigma_1,\\ldots,\\Sigma_m$. For $X_{\\tau0}={\\mathbb{R}}^n$, any $\\varepsilon\\in{\\mathbb{R}}^+$, and any double $\\mathsf{q}=(\\tau,\\eta)$ of quantization parameters satisfying\n\\begin{small}\n\\begin{align}\\label{bisim_cond1}\n\\overline\\alpha\\left(\\eta^q\\right)&\\leq\\underline\\alpha\\left(\\varepsilon^q\\right),\\\\\\label{bisim_cond}\n\\mathsf{e}^{-\\kappa\\tau}\\underline\\alpha\\left(\\varepsilon^q\\right)+\\widehat\\gamma\\left(\\left(h_{[X_{\\tau0}]_\\eta}(\\tau)\\right)^{\\frac{1}{q}}+\\eta\\right)&\\leq\\underline\\alpha\\left(\\varepsilon^q\\right),\n\\end{align}\n\\end{small}\nwe have that \\mbox{$S_{\\mathsf{q}}(\\Sigma)\\cong_{\\mathcal{S}}^{\\varepsilon}S_{\\tau}(\\Sigma)$}.\n\\end{theorem} \n\nIt can be readily seen that when we are interested in the dynamics of $\\Sigma$ on a compact $\\mathsf{D}\\subset{\\mathbb{R}}^n$ of the form of a finite union of boxes, implying that $X_{\\tau0}=\\mathsf{D}$, and for a given precision $\\varepsilon$,\nthere always exists a sufficiently large value of $\\tau$ and a small value of $\\eta$ such that $\\eta\\leq\\mathit{span}(\\mathsf{D})$ and the conditions in (\\ref{bisim_cond1}) and (\\ref{bisim_cond}) are satisfied. For a given fixed sampling time $\\tau$, the precision $\\varepsilon$ is lower bounded by:\n\\begin{small}\n\\begin{equation}\\label{lower_bound}\n\\varepsilon>\\left(\\underline\\alpha^{-1}\\left(\\frac{\\widehat\\gamma\\left(\\left(h_{[X_{\\tau0}]_\\eta}(\\tau)\\right)^{\\frac{1}{q}}\\right)}{1-\\mathsf{e}^{-\\kappa\\tau}}\\right)\\right)^{\\frac{1}{q}}.\n\\end{equation}\n\\end{small} \nOne can easily verify that the lower bound on $\\varepsilon$ in (\\ref{lower_bound}) goes to zero as $\\tau$ goes to infinity or as $Z_p \\rightarrow 0$, for any $p\\in\\mathsf{P}$, where $Z_p$ is the Lipschitz constant introduced in Definition \\ref{Def_control_sys}. \n\nNote that $S_{\\mathsf{q}}(\\Sigma)$ has a countable number of states and it is finite if one is interested in the dynamics of $\\Sigma$ on a compact $\\mathsf{D}\\subset{\\mathbb{R}}^n$ which is always the case in practice.\n\n\\begin{proof}\nWe start by proving \\mbox{$S_{\\tau}(\\Sigma)\\preceq^{\\varepsilon}_\\mathcal{S}S_{{\\mathsf{q}}}(\\Sigma)$}.\nConsider the relation $R\\subseteq X_{\\tau}\\times X_{{\\mathsf{q}}}$ defined by \n{$\\left(x_{\\tau},x_{{\\mathsf{q}}}\\right)\\in R$}\nif and only if \n{$\\mathbb{E}\\left[V\\left(H_{\\tau}(x_{\\tau}),H_{{\\mathsf{q}}}(x_{{\\mathsf{q}}})\\right)\\right]=\\mathbb{E}\\left[V\\left(x_{\\tau},x_{{\\mathsf{q}}}\\right)\\right]\\leq\\underline\\alpha\\left(\\varepsilon^q\\right)$}. Consider any \\mbox{$\\left(x_{\\tau},x_{{\\mathsf{q}}}\\right)\\in R$}. Condition (i) in Definition \\ref{APSR} is satisfied because\n\\begin{small}\n\\begin{equation} \n\\label{convexity1}\n\\left(\\mathbb{E}\\left[\\Vert x_{\\tau}-x_{{\\mathsf{q}}}\\Vert^q\\right]\\right)^{\\frac{1}{q}}\\leq\\left(\\underline\\alpha^{-1}\\left(\\mathbb{E}\\left[V(x_{\\tau},x_{{\\mathsf{q}}})\\right]\\right)\\right)^{\\frac{1}{q}}\\leq\\varepsilon.\n\\end{equation}\n\\end{small}\nWe used the convexity assumption of $\\underline\\alpha$ and the Jensen inequality \\cite{oksendal} to show the inequalities in (\\ref{convexity1}). Let us now show that condition (ii) in Definition\n\\ref{APSR} holds. \nConsider the transition \\mbox{$x_{\\tau}\\rTo^{p}_{\\tau} x'_{\\tau}=\\xi_{x_{\\tau}p}(\\tau)$} $\\mathds{P}$-a.s. in $S_{\\tau}(\\Sigma)$. Since $V$ is a common Lyapunov function for $\\Sigma$, we have\n\\begin{small}\n\\begin{align}\\label{b02}\n\\mathbb{E}\\left[V(x'_{\\tau},\\xi_{x_{{\\mathsf{q}}}p}(\\tau))\\right] &\\leq \\mathds{E}\\left[V(x_\\tau,x_q)\\right] \\mathsf{e}^{-\\kappa\\tau}\\leq \\underline\\alpha\\left(\\varepsilon^q\\right) \\mathsf{e}^{-\\kappa\\tau}.\n\\end{align}\n\\end{small}\nSince \\mbox{${\\mathbb{R}}^n\\subseteq\\bigcup_{p\\in[\\mathbb{R}^n]_{\\eta}}\\mathcal{B}_{\\eta}(p)$}, there exists \\mbox{$x'_{{\\mathsf{q}}}\\in{X}_{{\\mathsf{q}}}$} such that \n\\begin{equation}\n\\left\\Vert\\overline{\\xi}_{x_{{\\mathsf{q}}}p}(\\tau)-x'_{{\\mathsf{q}}}\\right\\Vert\\leq\\eta, \\label{b04}\n\\end{equation}\nwhich, by the definition of $S_{\\mathsf{q}}(\\Sigma)$, implies the existence of $x_{{\\mathsf{q}}}\\rTo^{p}_{{\\mathsf{q}}}x'_{{\\mathsf{q}}}$ in $S_{{\\mathsf{q}}}(\\Sigma)$. \nUsing Lemmas \\ref{lem:moment est}, the concavity of $\\widehat\\gamma$, the Jensen inequality \\cite{oksendal}, the inequalities (\\ref{supplement}), (\\ref{bisim_cond}), (\\ref{b02}), (\\ref{b04}), and triangle inequality, we obtain\n\\begin{small}\n\\begin{align*}\n\\mathbb{E}\\left[V(x'_{\\tau},x'_{{\\mathsf{q}}})\\right]&=\\mathbb{E}\\left[V(x'_\\tau,\\xi_{x_{{\\mathsf{q}}}p}(\\tau))+V(x'_{\\tau},x'_{\\mathsf{q}})-V(x'_\\tau,\\xi_{x_{{\\mathsf{q}}}p}(\\tau))\\right]=  \\mathbb{E}\\left[V(x'_{\\tau},\\xi_{x_{{\\mathsf{q}}}p}(\\tau))\\right]+\\mathbb{E}\\left[V(x'_{\\tau},x'_{\\mathsf{q}})-V(x'_\\tau,\\xi_{x_{{\\mathsf{q}}}p}(\\tau))\\right]\\\\\\notag&\\leq\\underline\\alpha\\left(\\varepsilon^q\\right)\\mathsf{e}^{-\\kappa\\tau}+\\mathbb{E}\\left[\\widehat\\gamma\\left(\\left\\Vert\\xi_{x_{{\\mathsf{q}}}p}(\\tau)-x'_{{\\mathsf{q}}}\\right\\Vert\\right)\\right]\\leq\\underline\\alpha\\left(\\varepsilon^q\\right)\\mathsf{e}^{-\\kappa\\tau}+\\widehat\\gamma\\left(\\mathbb{E}\\left[\\left\\Vert\\xi_{x_{{\\mathsf{q}}}p}(\\tau)-\\overline{\\xi}_{x_{{\\mathsf{q}}}p}(\\tau)+\\overline{\\xi}_{x_{{\\mathsf{q}}}p}(\\tau)-x'_{{\\mathsf{q}}}\\right\\Vert\\right]\\right)\n\\\\\\notag&\\leq\\underline\\alpha\\left(\\varepsilon^q\\right)\\mathsf{e}^{-\\kappa\\tau}+\\widehat\\gamma\\left(\\mathbb{E}\\left[\\left\\Vert\\xi_{x_{{\\mathsf{q}}}p}(\\tau)-\\overline{\\xi}_{x_{{\\mathsf{q}}}p}(\\tau)\\right\\Vert\\right]+\\left\\Vert\\overline{\\xi}_{x_{{\\mathsf{q}}}p}(\\tau)-x'_{{\\mathsf{q}}}\\right\\Vert\\right)\\\\\\notag&\\leq\\underline\\alpha\\left(\\varepsilon^q\\right)\\mathsf{e}^{-\\kappa\\tau}+\\widehat\\gamma\\left(\\left(h_{[X_{\\tau0}]_\\eta}(\\tau)\\right)^{\\frac{1}{q}}+\\eta\\right)\\leq\\underline\\alpha\\left(\\varepsilon^q\\right).\n\\end{align*}\n\\end{small}Therefore, we conclude that $\\left(x'_{\\tau},x'_{{\\mathsf{q}}}\\right)\\in{R}$ and that condition (ii) in Definition \\ref{APSR} holds. Since $X_{\\tau 0}\\subseteq\\bigcup_{p\\in[\\mathbb{R}^n]_{\\eta}}\\mathcal{B}_{\\eta}(p)$, \nfor every $x_{\\tau 0}\\in{X_{\\tau 0}}$ there always exists \\mbox{$x_{{\\mathsf{q}} 0}\\in{X}_{{\\mathsf{q}} 0}$} such that $\\Vert{x_{\\tau0}}-x_{\\params0}\\Vert\\leq\\eta.$ Then,\n\\begin{small}\n\\begin{align}\\nonumber\n\\mathbb{E}\\left[V({x_{\\tau0}},x_{\\params0})\\right]=V({x_{\\tau0}},x_{\\params0})&\\leq\\overline\\alpha\\left(\\Vert x_{\\tau0}-x_{\\params0}\\Vert^q\\right)\\leq\\overline\\alpha\\left(\\eta^q\\right)\\leq\\underline\\alpha\\left(\\varepsilon^q\\right),\n\\end{align}\n\\end{small}because of (\\ref{bisim_cond1}) and since $\\overline\\alpha$ is a $\\mathcal{K}_\\infty$ function.\nHence, \\mbox{$\\left(x_{\\tau0},x_{\\params0}\\right)\\in{R}$} implying that \\mbox{$S_{\\tau}(\\Sigma)\\preceq^{\\varepsilon}_\\mathcal{S}S_{{\\mathsf{q}}}(\\Sigma)$}. In a similar way, we can prove that \\mbox{$S_{{\\mathsf{q}}}(\\Sigma)\\preceq^{\\varepsilon}_{\\mathcal{S}}S_{\\tau}(\\Sigma)$} by showing that $R^{-1}$ is an $\\varepsilon$-approximate simulation relation from $S_{\\mathsf{q}}(\\Sigma)$ to $S_\\tau(\\Sigma)$. \n\\end{proof}\n\nNote that the results in \\cite[Theorem 4.1]{girard2} for non-probabilistic models are fully recovered by the statement in Theorem \\ref{main_theorem1} if the stochastic switched system $\\Sigma$ is not affected by any noise, \nimplying that $h_x^p(t)$ is identically zero for all $p\\in\\mathsf{P}$ and all $x\\in{\\mathbb{R}}^n$, \nand that the $\\delta$-GAS-M$_q$ common Lyapunov function simply reduces to being $\\delta$-GAS one. \n\n\n\\subsubsection{Multiple Lyapunov functions}\nIf a common $\\delta$-GAS-M$_q$  Lyapunov function does not exist or cannot be practically found, \none can still attempt computing approximately bisimilar symbolic models by seeking mode-dependent Lyapunov functions and by restricting the set of switching signals using a condition on the dwell time $\\tau_d=\\widehat{N}\\tau$ for some $\\widehat{N}\\in{\\mathbb{N}}$. \n\nConsider a stochastic switched system $\\Sigma_{\\tau_d}$ and a pair $\\mathsf{q}=(\\tau,\\eta)$ of quantization parameters, where $\\tau$ is the sampling time and $\\eta$ is the state space quantization. \nGiven $\\Sigma_{\\tau_d}$ and $\\mathsf{q}$, consider the following system:\n\\begin{small}\n\\begin{equation}\\label{T3}\nS_{\\mathsf{q}}\\left(\\Sigma_{\\tau_d}\\right)=(X_{\\mathsf{q}},X_{\\params0},U_{\\mathsf{q}},\\rTo_{\\mathsf{q}},Y_{\\mathsf{q}},H_{\\mathsf{q}}),\n\\end{equation}\n\\end{small}where $X_{\\mathsf{q}}=[{\\mathbb{R}}^n]_\\eta\\times\\mathsf{P}\\times\\left\\{0,\\ldots,\\widehat{N}-1\\right\\}$, $X_{\\params0}=[{\\mathbb{R}}^n]_\\eta\\times\\mathsf{P}\\times\\left\\{0\\right\\}$, $U_{\\mathsf{q}}=\\mathsf{P}$, $Y_{\\mathsf{q}}=Y_\\tau$, and\n\\begin{itemize}\n\\item $\\left(x_{\\mathsf{q}},p,i\\right)\\rTo_{\\mathsf{q}}^{p}\\left(x'_{\\mathsf{q}},p',i'\\right)$ if there exists a $x'_{\\mathsf{q}}\\in X_{\\mathsf{q}}$ such that $\\left\\Vert\\overline{\\xi}_{x_{\\mathsf{q}}p}(\\tau)-x'_{\\mathsf{q}}\\right\\Vert\\leq\\eta$ and one of the following holds:\n\n\\begin{itemize}\n\\item $i<\\widehat{N}-1$, $p'=p$, and $i'=i+1$;\n\n\\item $i=\\widehat{N}-1$, $p'=p$, and $i'=\\widehat{N}-1$;\n\n\\item $i=\\widehat{N}-1$, $p'\\neq p$, and $i'=0$.\n\\end{itemize}\n\n\n\\item $H_{\\mathsf{q}}(x_\\mathsf{q},p,i)=x_\\mathsf{q}$ for any $(x_\\mathsf{q},p,i)\\in[{\\mathbb{R}}^n]_\\eta\\times\\mathsf{P}\\times\\left\\{0,\\ldots,\\widehat{N}-1\\right\\}$.\n\\end{itemize}\n\n\nWe present the second main result of this subsection, \nwhich relates the existence of multiple Lyapunov functions for a stochastic switched system to that of a symbolic model, based on the state space discretization. In order to show the next result, we assume that $f_p(0_n)=0_n$ only if $\\Sigma_p$ is not affine and that $g_p(0_n)=0_{n\\times\\widehat{q}}$ for any $p\\in\\mathsf{P}$.\n\\begin{theorem}\\label{main_theorem2}\nConsider a stochastic switched system $\\Sigma_{\\tau_d}$. Let us assume that for any $p\\in\\mathsf{P}$, there exists a $\\delta$-GAS-M$_q$ Lyapunov function $V_p$, of the form explained in Lemma \\ref{lem:moment est}, for subsystem $\\Sigma_{\\tau_d,p}$. Moreover, assume that (\\ref{eq0}) holds for some $\\mu\\geq1$. If $\\tau_d>\\log{\\mu}\/\\kappa$, for $X_{\\tau0}=X_0\\times\\mathsf{P}\\times\\left\\{0,\\ldots,\\widehat{N}\\right\\}$, where $X_0={\\mathbb{R}}^n$, any $\\varepsilon\\in{\\mathbb{R}}^+$, \nand any pair $\\mathsf{q}=(\\tau,\\eta)$ of quantization parameters satisfying\n\\begin{small}\n\\begin{align}\\label{bisim_cond_mul1}\n\\overline\\alpha\\left(\\eta^q\\right)&\\leq\\underline\\alpha\\left(\\varepsilon^q\\right),\\\\\\label{bisim_cond_mul}\n\\widehat\\gamma\\left(\\left(h_{[X_0]_\\eta}(\\tau)\\right)^{\\frac{1}{q}}+\\eta\\right)&\\leq\\frac{\\frac{1}{\\mu}-\\mathsf{e}^{-\\kappa\\tau_d}}{1-\\mathsf{e}^{-\\kappa\\tau_d}}\\left(1-\\mathsf{e}^{-\\kappa\\tau}\\right)\\underline\\alpha\\left(\\varepsilon^q\\right),\n\\end{align}\n\\end{small}we have that \\mbox{$S_{\\mathsf{q}}\\left(\\Sigma_{\\tau_d}\\right)\\cong_{\\mathcal{S}}^{\\varepsilon}S_{\\tau}\\left(\\Sigma_{\\tau_d}\\right)$}.\n\\end{theorem} \n\nIt can be readily seen that when we are interested in the dynamics of $\\Sigma_{\\tau_d}$ on a compact $\\mathsf{D}\\subset{\\mathbb{R}}^n$ of the form of a finite union of boxes, implying that $X_0=\\mathsf{D}$, and for a precision $\\varepsilon$, there always exists a sufficiently large value of $\\tau$ and a small value of $\\eta$, such that $\\eta\\leq\\mathit{span}(\\mathsf{D})$ and the conditions in (\\ref{bisim_cond_mul1}) and (\\ref{bisim_cond_mul}) are satisfied. For a given fixed sampling time $\\tau$, the precision $\\varepsilon$ is lower bounded by \n\\begin{small}\n\\begin{equation}\\label{lower_bound_mul}\n\\varepsilon\\geq\\left(\\underline\\alpha^{-1}\\left(\\frac{\\widehat\\gamma\\left(\\left(h_{[X_0]_\\eta}(\\tau)\\right)^{\\frac{1}{q}}\\right)}{1-\\mathsf{e}^{-\\kappa\\tau}}\\cdot\\frac{1-\\mathsf{e}^{-\\kappa\\tau_d}}{\\frac{1}{\\mu}-\\mathsf{e}^{-\\kappa\\tau_d}}\\right)\\right)^{\\frac{1}{q}}.\n\\end{equation}\n\\end{small} \nThe properties of the bound in (\\ref{lower_bound_mul}) are analogous to those of the case of a common Lyapunov function. \n\nNote that $S_{\\mathsf{q}}\\left(\\Sigma_{\\tau_d}\\right)$ has a countable number of states and it is finite if one is interested in the dynamics of $\\Sigma_{\\tau_d}$ on a compact $\\mathsf{D}\\subset{\\mathbb{R}}^n$ which is always the case in practice.\n\n\\begin{proof}\nThe proof was inspired by the proof of Theorem 4.2 in \\cite{girard2} for non-probabilistic switched systems. We start by proving \\mbox{$S_{\\tau}\\left(\\Sigma_{\\tau_d}\\right)\\preceq^{\\varepsilon}_\\mathcal{S}S_{{\\mathsf{q}}}\\left(\\Sigma_{\\tau_d}\\right)$}.\nConsider the relation $R\\subseteq X_{\\tau}\\times X_{{\\mathsf{q}}}$ defined by \n{$\\left(x_{\\tau},p_1,i_1,x_{{\\mathsf{q}}},p_2,i_2\\right)\\in R$}\nif and only if $p_1=p_2=p$, $i_1=i_2=i$, and $\\mathbb{E}\\left[V_p\\left(H_{\\tau}(x_{\\tau},p_1,i_1),H_{{\\mathsf{q}}}(x_{{\\mathsf{q}}},p_2,i_2)\\right)\\right]=\\mathbb{E}\\left[V_p\\left(x_{\\tau},x_{{\\mathsf{q}}}\\right)\\right]\\leq\\delta_i$, where $\\delta_0,\\ldots,\\delta_{\\widehat{N}}$ are given recursively by\\begin{small}$$\\delta_0=\\underline\\alpha\\left(\\varepsilon^q\\right),~~\\delta_{i+1}=\\mathsf{e}^{-\\kappa\\tau}\\delta_i+\\widehat\\gamma\\left(\\left(h_{[X_0]_\\eta}(\\tau)\\right)^{\\frac{1}{q}}+\\eta\\right).$$\\end{small} One can easily verify that\n\\begin{small}\n\\begin{align}\\nonumber\n\\delta_i=&\\mathsf{e}^{-i\\kappa\\tau}\\underline\\alpha\\left(\\varepsilon^q\\right)+\\widehat\\gamma\\left(\\left(h_{[X_0]_\\eta}(\\tau)\\right)^{\\frac{1}{q}}+\\eta\\right)\\frac{1-\\mathsf{e}^{-i\\kappa\\tau}}{1-\\mathsf{e}^{-\\kappa\\tau}}\\\\\\label{delta_i}=&\\frac{\\widehat\\gamma\\left(\\left(h_{[X_0]_\\eta}(\\tau)\\right)^{\\frac{1}{q}}+\\eta\\right)}{1-\\mathsf{e}^{-\\kappa\\tau}}+\\mathsf{e}^{-i\\kappa\\tau}\\left(\\underline\\alpha\\left(\\varepsilon^q\\right)-\\frac{\\widehat\\gamma\\left(\\left(h_{[X_0]_\\eta}(\\tau)\\right)^{\\frac{1}{q}}+\\eta\\right)}{1-\\mathsf{e}^{-\\kappa\\tau}}\\right).\n\\end{align}\n\\end{small}\nSince $\\mu\\geq1$, and from (\\ref{bisim_cond_mul}), one has \\begin{small}$$\\widehat\\gamma\\left(\\left(h_{[X_0]_\\eta}(\\tau)\\right)^{\\frac{1}{q}}+\\eta\\right)\\leq(1-\\mathsf{e}^{-\\kappa\\tau})\\underline\\alpha\\left(\\varepsilon^q\\right).$$\\end{small} It follows from (\\ref{delta_i}) that $\\delta_0\\geq\\delta_1\\geq\\cdots\\geq\\delta_{\\widehat{N}-1}\\geq\\delta_{\\widehat{N}}$. From (\\ref{bisim_cond_mul}) and since $\\tau_d=\\widehat{N}\\tau$, we get \n\\begin{small}\n\\begin{align}\\label{Ntozero}\n\\delta_{\\widehat{N}}=&\\mathsf{e}^{-\\kappa\\tau_d}\\underline\\alpha\\left(\\varepsilon^q\\right)+\\widehat\\gamma\\left(\\left(h_{[X_0]_\\eta}(\\tau)\\right)^{\\frac{1}{q}}+\\eta\\right)\\frac{1-\\mathsf{e}^{-\\kappa\\tau_d}}{1-\\mathsf{e}^{-\\kappa\\tau}}\\leq\\mathsf{e}^{-\\kappa\\tau_d}\\underline\\alpha\\left(\\varepsilon^q\\right)+\\left(\\frac{1}{\\mu}-\\mathsf{e}^{-\\kappa\\tau_d}\\right)\\underline\\alpha\\left(\\varepsilon^q\\right)=\\frac{\\underline\\alpha\\left(\\varepsilon^q\\right)}{\\mu}.\n\\end{align}\n\\end{small}\nWe can now prove that $R$ is an $\\varepsilon$-approximate simulation relation from $S_{\\tau}\\left(\\Sigma_{\\tau_d}\\right)$ to $S_{{\\mathsf{q}}}\\left(\\Sigma_{\\tau_d}\\right)$.\nConsider any \\mbox{$\\left(x_{\\tau},p,i,x_{{\\mathsf{q}}},p,i\\right)\\in R$}. Using the convexity assumption of $\\underline\\alpha_p$, and since it is a $\\mathcal{K}_\\infty$ function, and the Jensen inequality \\cite{oksendal}, we have:\n\\begin{small}\n\\begin{align}\\nonumber\n&\\underline\\alpha\\left(\\mathbb{E}\\left[\\Vert H_\\tau(x_{\\tau},p,i)-H_{\\mathsf{q}}(x_{{\\mathsf{q}}},p,i)\\Vert^q\\right]\\right)=\\underline\\alpha\\left(\\mathbb{E}\\left[\\Vert x_{\\tau}-x_{{\\mathsf{q}}}\\Vert^q\\right]\\right)\\\\\\notag&\\leq\\underline\\alpha_p\\left(\\mathbb{E}\\left[\\Vert x_{\\tau}-x_{{\\mathsf{q}}}\\Vert^q\\right]\\right)\\leq\\mathds{E}\\left[\\underline\\alpha_p\\left(\\Vert x_{\\tau}-x_{{\\mathsf{q}}}\\Vert^q\\right)\\right]\\leq\\mathbb{E}\\left[V_p(x_{\\tau},x_{{\\mathsf{q}}})\\right]\\leq\\delta_i\\leq\\delta_0.\n\\end{align}\n\\end{small}Therefore, we obtain\n\\begin{small}\n$\\left(\\mathbb{E}\\left[\\Vert x_{\\tau}-x_{{\\mathsf{q}}}\\Vert^q\\right]\\right)^{\\frac{1}{q}}\\leq\\left(\\underline\\alpha^{-1}\\left(\\delta_0\\right)\\right)^{\\frac{1}{q}}\\leq\\varepsilon$,\n\\end{small}because of $\\underline\\alpha\\in\\mathcal{K}_\\infty$. Hence, condition (i) in Definition \\ref{APSR} is satisfied.\nLet us now show that condition (ii) in Definition\n\\ref{APSR} holds. \nConsider the transition $(x_{\\tau},p,i)\\rTo^{p}_{\\tau} (x'_{\\tau},p',i')$ in $S_{\\tau}\\left(\\Sigma_{\\tau_d}\\right)$, where $x'_{\\tau}=\\xi_{x_{\\tau}p}(\\tau)$ $\\mathds{P}$-a.s.. Since $V_p$ is a $\\delta$-GAS-M$_q$ Lyapunov function for subsystem $\\Sigma_p$, we have\n\\begin{small}\n\\begin{align}\\label{b06}\n\\mathbb{E}\\left[V_p(x'_{\\tau},\\xi_{x_{{\\mathsf{q}}}p}(\\tau))\\right] &\\leq \\mathds{E}\\left[V_p(x_\\tau,x_q)\\right] \\mathsf{e}^{-\\kappa\\tau}\\leq\\mathsf{e}^{-\\kappa\\tau}\\delta_i.\n\\end{align}\n\\end{small}\nSince \\mbox{${\\mathbb{R}}^n\\subseteq\\bigcup_{p\\in[\\mathbb{R}^n]_{\\eta}}\\mathcal{B}_{\\eta}(p)$}, there exists \\mbox{$x'_{{\\mathsf{q}}}\\in[\\mathbb{R}^n]_{\\eta}$} such that \n\\begin{small}\n\\begin{equation}\n\\left\\Vert\\overline{\\xi}_{x_{{\\mathsf{q}}}p}(\\tau)-x'_{{\\mathsf{q}}}\\right\\Vert\\leq\\eta. \\label{b07}\n\\end{equation}\n\\end{small}Using Lemmas \\ref{lem:moment est}, the $\\mathcal{K}_\\infty$ function $\\widehat\\gamma$, the concavity of $\\widehat\\gamma_p$ in \\eqref{supplement}, the Jensen inequality \\cite{oksendal}, the inequalities (\\ref{supplement}), (\\ref{b06}), (\\ref{b07}), and triangle inequality, we obtain\n\\begin{small}\n\\begin{align}\\nonumber\n\\mathbb{E}\\left[V_p(x'_{\\tau},x'_{{\\mathsf{q}}})\\right]&=\\mathbb{E}\\left[V_p(x'_\\tau,\\xi_{x_{{\\mathsf{q}}}p}(\\tau))+V_p(x'_{\\tau},x'_{\\mathsf{q}})-V_p(x'_\\tau,\\xi_{x_{{\\mathsf{q}}}p}(\\tau))\\right]=  \\mathbb{E}\\left[V_p(x'_{\\tau},\\xi_{x_{{\\mathsf{q}}}p}(\\tau))\\right]+\\mathbb{E}\\left[V_p(x'_{\\tau},x'_{\\mathsf{q}})-V_p(x'_\\tau,\\xi_{x_{{\\mathsf{q}}}p}(\\tau))\\right]\\\\\\notag&\\leq\\mathsf{e}^{-\\kappa\\tau}\\delta_i+\\mathbb{E}\\left[\\widehat\\gamma_p\\left(\\left\\Vert\\xi_{x_{{\\mathsf{q}}}p}(\\tau)-x'_{{\\mathsf{q}}}\\right\\Vert\\right)\\right]\\leq\\mathsf{e}^{-\\kappa\\tau}\\delta_i+\\widehat\\gamma_p\\left(\\mathbb{E}\\left[\\left\\Vert\\xi_{x_{{\\mathsf{q}}}p}(\\tau)-x'_{{\\mathsf{q}}}\\right\\Vert\\right]\\right)\\\\\\label{b08}\n&\\leq\\mathsf{e}^{-\\kappa\\tau}\\delta_i+\\widehat\\gamma\\left(\\mathbb{E}\\left[\\left\\Vert\\xi_{x_{{\\mathsf{q}}}p}(\\tau)-\\overline{\\xi}_{x_{{\\mathsf{q}}}p}(\\tau)\\right\\Vert\\right]+\\left\\Vert\\overline{\\xi}_{x_{{\\mathsf{q}}}p}(\\tau)-x'_{{\\mathsf{q}}}\\right\\Vert\\right)\\leq\\mathsf{e}^{-\\kappa\\tau}\\delta_i+\\widehat\\gamma\\left(\\left(h_{[X_0]_\\eta}(\\tau)\\right)^{\\frac{1}{q}}+\\eta\\right)=\\delta_{i+1}.\n\\end{align}\n\\end{small}\nWe now examine three separate cases:\n\\begin{itemize}\n\\item If $i<\\widehat{N}-1$, then $p'=p$, and $i'=i+1$; since, from (\\ref{b08}), $\\mathds{E}\\left[V_p(x'_{\\tau},x'_{{\\mathsf{q}}})\\right]\\leq\\delta_{i+1}$, we conclude that $(x'_{\\tau},p,i+1,x'_{{\\mathsf{q}}},p,i+1)\\in R$; \n\n\\item If $i=\\widehat{N}-1$, and $p'=p$, then $i'=\\widehat{N}-1$; from (\\ref{b08}), $\\mathds{E}\\left[V_p(x'_{\\tau},x'_{{\\mathsf{q}}})\\right]\\leq\\delta_{\\widehat{N}}\\leq\\delta_{\\widehat{N}-1}$, we conclude that $(x'_{\\tau},p,\\widehat{N}-1,x'_{{\\mathsf{q}}},p,\\widehat{N}-1)\\in R$; \n\n\\item If $i=\\widehat{N}-1$, and $p'\\neq{p}$, then $i'=0$; from (\\ref{Ntozero}) and (\\ref{b08}), $\\mathds{E}\\left[V_p(x'_{\\tau},x'_{{\\mathsf{q}}})\\right]\\leq\\delta_{\\widehat{N}}\\leq\\delta_0\/\\mu$. From (\\ref{eq0}), it follows that $\\mathds{E}\\left[V_{p'}(x'_{\\tau},x'_{{\\mathsf{q}}})\\right]\\leq\\mu \\mathds{E}\\left[V_p(x'_{\\tau},x'_{{\\mathsf{q}}})\\right]\\leq\\delta_0$. Hence, $(x'_{\\tau},p',0,x'_{{\\mathsf{q}}},p',0)\\in R$.\n\\end{itemize}\nTherefore, we conclude that condition (ii) in Definition \\ref{APSR} holds. Since $X_0\\subseteq\\bigcup_{p\\in[\\mathbb{R}^n]_{\\eta}}\\mathcal{B}_{\\eta}(p)$, for every $\\left(x_{\\tau 0},p,0\\right)\\in X_{\\tau0}$ there always exists $\\left(x_{\\params0},p,0\\right)\\in{X}_{\\params0}$ such that $\\Vert{x_{\\tau0}}-x_{\\params0}\\Vert\\leq\\eta$. Then,\n\\begin{small}\n\\begin{align}\\nonumber\n&\\mathbb{E}\\left[V_p(H_\\tau(x_{\\tau0},p,0),H_{\\mathsf{q}}(x_{\\params0},p,0)\\right]=V_p({x_{\\tau0}},x_{\\params0})\\leq\\overline\\alpha_p\\left(\\left\\| x_{\\tau0}-x_{\\params0}\\right\\|^q\\right)\\leq\\overline\\alpha\\left(\\Vert x_{\\tau0}-x_{\\params0}\\Vert^q\\right)\\leq\\overline\\alpha\\left(\\eta^q\\right)\\leq\\underline\\alpha\\left(\\varepsilon^q\\right),\n\\end{align}\n\\end{small}because of (\\ref{bisim_cond_mul1}) and since $\\overline\\alpha$ is a $\\mathcal{K}_\\infty$ function.\nHence, $V_p(x_{\\tau0},x_{\\params0})\\leq\\delta_0$ and \\mbox{$\\left(x_{\\tau0},p,0,x_{\\params0},p,0\\right)\\in{R}$} implying that \\mbox{$S_{\\tau}(\\Sigma_{\\tau_d})\\preceq^{\\varepsilon}_\\mathcal{S}S_{{\\mathsf{q}}}(\\Sigma_{\\tau_d})$}. In a similar way, we can prove that \\mbox{$S_{{\\mathsf{q}}}\\left(\\Sigma_{\\tau_d}\\right)\\preceq^{\\varepsilon}_{\\mathcal{S}}S_{\\tau}\\left(\\Sigma_{\\tau_d}\\right)$} by showing that $R^{-1}$ is an $\\varepsilon$-approximate simulation relation from $S_{\\mathsf{q}}(\\Sigma_{\\tau_d})$ to $S_\\tau(\\Sigma_{\\tau_d})$. \n\\end{proof}\n\nAs before, Theorem \\ref{main_theorem2} \nsubsumes \\cite[Theorem 4.2]{girard2} over non-probabilistic models. \n\n\\subsection{Second approach}\nThis subsection contains the second main results of the paper providing bisimilar symbolic models without any space discretization.\n\n\\subsubsection{Common Lyapunov function}\nFirst, we show one of the main results of this subsection on the construction of symbolic models based on the existence of a common $\\delta$-GAS-M$_q$ Lyapunov function. \nWe proceed by introducing two fully symbolic systems for the concrete one $\\Sigma$.\nConsider a stochastic switched system $\\Sigma$ and a triple $\\overline{\\mathsf{q}}=\\left(\\tau,N,x_s\\right)$ of parameters, where $\\tau$ is the sampling time, $N\\in{\\mathbb{N}}$ is a \\emph{temporal horizon}, and $x_s\\in{\\mathbb{R}}^n$ is a \\emph{source state}.\nGiven $\\Sigma$ and $\\overline{\\mathsf{q}}$, consider the following systems:\n\\begin{small}\n\\begin{align}\\notag\nS_{\\overline{\\mathsf{q}}}(\\Sigma)&=(X_{\\overline{\\mathsf{q}}},X_{\\overline{\\mathsf{q}}0},U_{\\overline{\\mathsf{q}}},\\rTo_{\\overline{\\mathsf{q}}},Y_{\\overline{\\mathsf{q}}},H_{\\overline{\\mathsf{q}}}),\\\\\\notag\\overline{S}_{\\overline{\\mathsf{q}}}(\\Sigma)&=(X_{\\overline{\\mathsf{q}}},X_{\\overline{\\mathsf{q}}0},U_{\\overline{\\mathsf{q}}},\\rTo_{\\overline{\\mathsf{q}}},Y_{\\overline{\\mathsf{q}}},\\overline{H}_{\\overline{\\mathsf{q}}}),\n\\end{align}\n\\end{small}where $X_{\\overline{\\mathsf{q}}}=\\mathsf{P}^N$, $X_{\\overline{{\\mathsf{q}}}0}=X_{\\overline{\\mathsf{q}}}$, $U_{\\overline{\\mathsf{q}}}=\\mathsf{P}$, $Y_{\\overline{\\mathsf{q}}}=Y_{\\tau}$, and\n\\begin{itemize}\n\\item $x_{\\overline{{\\mathsf{q}}}}\\rTo_{\\overline{\\mathsf{q}}}^{p}x'_{\\overline{\\mathsf{q}}}$, where $x_{\\overline{{\\mathsf{q}}}}=(p_1,p_2,\\ldots,p_N)$, if and only if $x'_{\\overline{{\\mathsf{q}}}}=(p_2,\\ldots,p_N,p)$;\n\\item $H_{\\overline{\\mathsf{q}}}(x_{\\overline{{\\mathsf{q}}}})=\\xi_{x_sx_{\\overline{{\\mathsf{q}}}}}(N\\tau)$ $\\left(\\overline{H}_{\\overline{\\mathsf{q}}}(x_{\\overline{{\\mathsf{q}}}})=\\overline\\xi_{x_sx_{\\overline{{\\mathsf{q}}}}}(N\\tau)\\right)$.\n\\end{itemize}\n\nNote that we have abused notation by identifying $x_{\\overline{{\\mathsf{q}}}}=(p_1,p_2,\\ldots,p_N)$ with a switching signal obtained by the concatenation of modes $p_i$ \\big(i.e. $x_{\\overline{{\\mathsf{q}}}}(t)=p_i$ for any $t\\in[(i-1)\\tau,i\\tau[$\\big) for $i=1,\\ldots,N$. Notice that the proposed system $S_{\\overline{{\\mathsf{q}}}}(\\Sigma)$ $\\left(\\text{resp.}~\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma)\\right)$ is symbolic and deterministic  in the sense of Definition \\ref{system}. Note that $H_{\\overline{{\\mathsf{q}}}}$ and $\\overline{H}_{\\overline{{\\mathsf{q}}}}$ are mappings from a non-probabilistic point $x_{\\overline{{\\mathsf{q}}}}$ to the random variable $\\xi_{x_sx_{\\overline{{\\mathsf{q}}}}}(N\\tau)$ and to the one with a Dirac probability distribution centered at $\\overline\\xi_{x_sx_{\\overline{{\\mathsf{q}}}}}(N\\tau)$, respectively. One can readily verify that the transition relation of $S_{\\overline{\\mathsf{q}}}(\\Sigma)$ (resp. $\\overline{S}_{\\overline{\\mathsf{q}}}(\\Sigma)$) admits a very compact representation under the form of a shift operator and such symbolic systems do not require any continuous space discretization.\n\nBefore providing the main results, we need the following technical lemmas.\n\n\\begin{lemma}\\label{lemma1}\nConsider a stochastic switched system $\\Sigma$, admitting a common $\\delta$-GAS-M$_q$ Lyapunov function $V$, and consider its corresponding symbolic model $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma)$. We have:\n\\begin{small}\n\\begin{align}\\label{upper_bound}\n\\overline\\eta\\leq&\\left(\\underline\\alpha^{-1}\\left(\\mathsf{e}^{-\\kappa N\\tau}\\max_{p\\in \\mathsf{P}}V\\left(\\overline\\xi_{x_sp}(\\tau),x_s\\right)\\right)\\right)^{1\/q},\n\\end{align}\n\\end{small}\nwhere\n\\begin{small}\n\\begin{align}\\label{eta}\n\\overline\\eta:=\\max_{\\substack{p\\in\\mathsf{P},x_{\\overline{{\\mathsf{q}}}}\\in X_{\\overline{{\\mathsf{q}}}}\\\\x_{\\overline{{\\mathsf{q}}}}\\rTo_{\\overline{{\\mathsf{q}}}}^{p}x'_{\\overline{{\\mathsf{q}}}}}}\\left\\Vert\\overline\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{\\overline{{\\mathsf{q}}}})p}(\\tau)-\\overline{H}_{\\overline{{\\mathsf{q}}}}\\left(x'_{\\overline{{\\mathsf{q}}}}\\right)\\right\\Vert.\n\\end{align}\n\\end{small}\n\\end{lemma}\n\n\\begin{proof}\nLet $x_{\\overline{{\\mathsf{q}}}}\\in X_{\\overline{{\\mathsf{q}}}}$, where $x_{\\overline{{\\mathsf{q}}}}=\\left(p_1,p_2,\\ldots,p_N\\right)$, and $p\\in U_{\\overline{{\\mathsf{q}}}}$. Using the definition of $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma)$, one obtains $x_{\\overline{{\\mathsf{q}}}}\\rTo_{\\overline{{\\mathsf{q}}}}^{p}x'_{\\overline{{\\mathsf{q}}}}$, where $x'_{\\overline{{\\mathsf{q}}}}=\\left(p_2,\\ldots,p_N,p\\right)$. Since $V$ is a common $\\delta$-GAS-M$_q$ Lyapunov function for $\\Sigma$ and based on the manipulations in the proof of Theorem \\ref{theorem2}, we have:\n\\begin{small}\n\\begin{align}\\notag\n\\underline\\alpha\\left(\\left\\Vert\\overline\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{\\overline{{\\mathsf{q}}}})p}(\\tau)-\\overline{H}_{\\overline{{\\mathsf{q}}}}\\left(x'_{\\overline{{\\mathsf{q}}}}\\right)\\right\\Vert^q\\right)&\\leq V\\left(\\overline\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{\\overline{{\\mathsf{q}}}})p}(\\tau),\\overline{H}_{\\overline{{\\mathsf{q}}}}\\left(x'_{\\overline{{\\mathsf{q}}}}\\right)\\right)=V\\left(\\overline\\xi_{\\overline\\xi_{x_sx_{\\overline{{\\mathsf{q}}}}}(N\\tau)p}(\\tau),\\overline\\xi_{x_sx'_{\\overline{{\\mathsf{q}}}}}(N\\tau)\\right)\\\\\\notag&=V\\left(\\overline\\xi_{\\overline\\xi_{x_sp_1}(\\tau)(p_2,\\ldots,p_N,p)}(N\\tau),\\overline\\xi_{x_s(p_2,\\ldots,p_N,p)}(N\\tau)\\right)\\leq\\mathsf{e}^{-\\kappa N\\tau}V\\left(\\overline\\xi_{x_sp_1}(\\tau),x_s\\right).\n\\end{align}\n\\end{small}\nHence, one gets\n\\begin{small}\n\\begin{align}\\label{upper_bound2}\n\\left\\Vert\\overline\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{\\overline{{\\mathsf{q}}}})p}(\\tau)-\\overline{H}_{\\overline{{\\mathsf{q}}}}\\left(x'_{\\overline{{\\mathsf{q}}}}\\right)\\right\\Vert\\leq\\left(\\underline\\alpha^{-1}\\left(\\mathsf{e}^{-\\kappa N\\tau}V\\left(\\overline\\xi_{x_sp_1}(\\tau),x_s\\right)\\right)\\right)^{1\/q},\n\\end{align}\n\\end{small}because of $\\underline\\alpha\\in\\mathcal{K}_\\infty$. Since the inequality \\eqref{upper_bound2} holds for all $x_{\\overline{{\\mathsf{q}}}}\\in X_{\\overline{{\\mathsf{q}}}}$ and $p\\in U_{\\overline{{\\mathsf{q}}}}$, and $\\underline\\alpha\\in\\mathcal{K}_\\infty$, inequality \\eqref{upper_bound} holds. \n\\end{proof}\n\nThe next lemma provides a similar result as the one of Lemma \\ref{lemma1}, but by using the symbolic model $S_{\\overline{{\\mathsf{q}}}}(\\Sigma)$ rather than $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma)$.\n\\begin{lemma}\\label{lemma4}\nConsider a stochastic switched system $\\Sigma$, admitting a common $\\delta$-GAS-M$_q$ Lyapunov function $V$, and consider its corresponding symbolic model $S_{\\overline{{\\mathsf{q}}}}(\\Sigma)$. One has:\n\\begin{small}\n\\begin{align}\\label{upper_bound4}\n\\widehat\\eta\\leq&\\left(\\underline\\alpha^{-1}\\left(\\mathsf{e}^{-\\kappa N\\tau}\\max_{p\\in\\mathsf{P}}\\mathds{E}\\left[V\\left(\\xi_{x_sp}(\\tau),x_s\\right)\\right]\\right)\\right)^{1\/q},\n\\end{align}\n\\end{small}\nwhere\n\\begin{small}\n\\begin{align}\\label{eta1}\n\\widehat\\eta:=\\max_{\\substack{p\\in\\mathsf{P},x_{\\overline{{\\mathsf{q}}}}\\in X_{\\overline{{\\mathsf{q}}}}\\\\x_{\\overline{{\\mathsf{q}}}}\\rTo_{\\overline{{\\mathsf{q}}}}^px'_{\\overline{{\\mathsf{q}}}}}}\\mathds{E}\\left[\\left\\Vert\\xi_{H_{\\overline{{\\mathsf{q}}}}(x_{\\overline{{\\mathsf{q}}}})p}(\\tau)-H_{\\overline{{\\mathsf{q}}}}\\left(x'_{\\overline{{\\mathsf{q}}}}\\right)\\right\\Vert\\right].\n\\end{align}\n\\end{small}\n\\end{lemma}\n\n\\begin{proof}\nThe proof is similar to the one of Lemma \\ref{lemma1} and can be shown by using convexity of $\\underline\\alpha$ and Jensen inequality \\cite{oksendal}. \n\\end{proof}\n\nWe can now present the first main result of this subsection, relating the existence of a common $\\delta$-GAS-M$_q$ Lyapunov function to the construction of a bisimilar finite abstraction without any continuous space discretization. In order to show the next result, we assume that $f_p(0_n)=0_n$ only if $\\Sigma_p$ is not affine and $g_p(0_n)=0_{n\\times\\widehat{q}}$ for any $p\\in\\mathsf{P}$.\n\n\\begin{theorem}\\label{main_theorem}\nConsider a stochastic switched system $\\Sigma$ admitting a common $\\delta$-GAS-M$_q$ Lyapunov function $V$, of the form of the one explained in Lemma \\ref{lem:moment est}. Let $\\overline\\eta$ be given by \\eqref{eta}. For any $\\varepsilon\\in{\\mathbb{R}}^+$ and any triple $\\overline{\\mathsf{q}}=\\left(\\tau,N,x_s\\right)$ of parameters satisfying\n\\begin{small}\n\\begin{align}\n\\label{bisim_cond11}\n\\mathsf{e}^{-\\kappa\\tau}\\underline\\alpha\\left(\\varepsilon^q\\right)+\\widehat\\gamma\\left(\\left(h_{x_s}((N+1)\\tau)\\right)^{\\frac{1}{q}}+\\overline\\eta\\right)&\\leq\\underline\\alpha\\left(\\varepsilon^q\\right),\n\\end{align}\n\\end{small}the relation \\begin{small}$$R=\\left\\{\\left(x_\\tau,x_{\\overline{{\\mathsf{q}}}}\\right)\\in X_\\tau\\times X_{\\overline{{\\mathsf{q}}}}\\,\\,|\\,\\,\\mathds{E}\\left[V\\left(x_\\tau,\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{\\overline{{\\mathsf{q}}}})\\right)\\right]\\leq\\underline\\alpha\\left(\\varepsilon^q\\right)\\right\\}$$\\end{small}is an $\\varepsilon$-approximate bisimulation relation between $\\overline{S}_{\\overline{\\mathsf{q}}}(\\Sigma)$ and $S_{\\tau}(\\Sigma)$.\n\\end{theorem}\n\n\\begin{proof}\nWe start by proving that $R$ is an $\\varepsilon$-approximate simulation relation from $S_{\\tau}(\\Sigma)$ to $\\overline{S}_{{\\overline{{\\mathsf{q}}}}}(\\Sigma)$. Consider any \\mbox{$\\left(x_{\\tau},x_{{\\overline{{\\mathsf{q}}}}}\\right)\\in R$}. Condition (i) in Definition \\ref{APSR} is satisfied because\n\\begin{small}\n\\begin{equation}\n\\label{convexity}\n(\\mathbb{E}[\\Vert x_{\\tau}-\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}})\\Vert^q])^{\\frac{1}{q}}\\leq\\left(\\underline\\alpha^{-1}\\left(\\mathbb{E}\\left[V\\left(x_{\\tau},\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}})\\right)\\right]\\right)\\right)^{\\frac{1}{q}}\\leq\\varepsilon.\n\\end{equation}\n\\end{small}We used the convexity assumption of $\\underline\\alpha$ and the Jensen inequality \\cite{oksendal} to show the inequalities in (\\ref{convexity}). Let us now show that condition (ii) in Definition\n\\ref{APSR} holds. Consider the transition \\mbox{$x_{\\tau}\\rTo^{p}_{\\tau} x'_{\\tau}=\\xi_{x_{\\tau}p}(\\tau)$} $\\mathds{P}$-a.s. in $S_{\\tau}(\\Sigma)$. Since $V$ is a common $\\delta$-GAS-M$_q$ Lyapunov function for $\\Sigma$, we have (cf. proof of Theorem \\ref{theorem2})\n\\begin{small}\n\\begin{align}\\label{b020}\n\\mathbb{E}[V(x'_{\\tau},\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}})p}(\\tau))] \\leq \\mathds{E}[V(x_\\tau,\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_q))] \\mathsf{e}^{-\\kappa\\tau}\\leq \\underline\\alpha(\\varepsilon^q) \\mathsf{e}^{-\\kappa\\tau}.\n\\end{align}\n\\end{small}Note that, by the definition of $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma)$, there exists $x_{{\\overline{{\\mathsf{q}}}}}\\rTo^{p}_{{\\overline{{\\mathsf{q}}}}}x'_{{\\overline{{\\mathsf{q}}}}}$ in $\\overline{S}_{{\\overline{{\\mathsf{q}}}}}(\\Sigma)$.\nUsing Lemma \\ref{lem:moment est}, the concavity of $\\widehat\\gamma$, the Jensen inequality \\cite{oksendal}, equation \\eqref{eta}, the inequalities (\\ref{supplement}), (\\ref{bisim_cond11}), (\\ref{b020}), and triangle inequality, we obtain\n\\begin{small}\n\\begin{align*}\n\\mathbb{E}[V(x'_{\\tau},\\overline{H}_{\\overline{{\\mathsf{q}}}}(x'_{{\\overline{{\\mathsf{q}}}}}))]=&\\mathbb{E}[V(x'_\\tau,\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}})p}(\\tau))+V(x'_{\\tau},\\overline{H}_{\\overline{{\\mathsf{q}}}}(x'_{\\overline{{\\mathsf{q}}}}))-V(x'_\\tau,\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}})p}(\\tau))]\\\\ \\notag\n=&  \\mathbb{E}[V(x'_{\\tau},\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}})p}(\\tau))]+\\mathbb{E}[V(x'_{\\tau},\\overline{H}_{\\overline{{\\mathsf{q}}}}(x'_{\\overline{{\\mathsf{q}}}}))-V(x'_\\tau,\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}})p}(\\tau))]\\\\\\notag\\leq&\\underline\\alpha(\\varepsilon^q)\\mathsf{e}^{-\\kappa\\tau}+\\mathbb{E}[\\widehat\\gamma(\\Vert\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}})p}(\\tau)-\\overline{H}_{\\overline{{\\mathsf{q}}}}(x'_{{\\overline{{\\mathsf{q}}}}})\\Vert)]\\\\\\notag\n\\leq&\\underline\\alpha(\\varepsilon^q)\\mathsf{e}^{-\\kappa\\tau}+\\widehat\\gamma(\\mathbb{E}[\\Vert\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}})p}(\\tau)-\\overline{\\xi}_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}})p}(\\tau)+\\overline{\\xi}_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}})p}(\\tau)-\\overline{H}_{\\overline{{\\mathsf{q}}}}(x'_{{\\overline{{\\mathsf{q}}}}})\\Vert])\\\\\\notag\n\\leq&\\underline\\alpha(\\varepsilon^q)\\mathsf{e}^{-\\kappa\\tau}+\\widehat\\gamma(\\mathbb{E}[\\Vert\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}})p}(\\tau)-\\overline{\\xi}_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}})p}(\\tau)\\Vert]+\\Vert\\overline{\\xi}_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}})p}(\\tau)-\\overline{H}_{\\overline{{\\mathsf{q}}}}(x'_{{\\overline{{\\mathsf{q}}}}})\\Vert)\\\\\\notag\n\\leq&\\underline\\alpha(\\varepsilon^q)\\mathsf{e}^{-\\kappa\\tau}+\\widehat\\gamma((h_{x_s}((N+1)\\tau))^{\\frac{1}{q}}+\\eta)\\leq\\underline\\alpha(\\varepsilon^q).\n\\end{align*}\n\\end{small}Therefore, we conclude that \\mbox{$\\left(x'_{\\tau},x'_{{\\overline{{\\mathsf{q}}}}}\\right)\\in{R}$} and that condition (ii) in Definition \\ref{APSR} holds.\n\nIn a similar way, we can prove that that $R^{-1}$ is an\n$\\varepsilon$-approximate simulation relation from $\\overline{S}_{{\\overline{{\\mathsf{q}}}}}(\\Sigma)$ to $S_{\\tau}(\\Sigma)$ implying that $R$ is an $\\varepsilon$-approximate bisimulation relation between $\\overline{S}_{{\\overline{{\\mathsf{q}}}}}(\\Sigma)$ and $S_\\tau(\\Sigma)$.\n\\end{proof}\n\nNote that one can also use any over approximation of $\\overline\\eta$ such as the one in \\eqref{upper_bound} instead of $\\overline\\eta$ in condition \\eqref{bisim_cond11}. By choosing $N$ sufficiently large, one can enforce $h_{x_s}((N+1)\\tau)$ and $\\overline\\eta$ to be sufficiently small. Hence, it can be readily seen that for a given precision $\\varepsilon$,\nthere always exists a large value of $N$, such that the condition in (\\ref{bisim_cond11}) is satisfied.\n\nNote that the results in \\cite{corronc} for non-probabilistic models are fully recovered by the statement in Theorem \\ref{main_theorem} if $\\Sigma$ is not affected by any noise.\n\nThe next theorem provides a result that is similar to the one of Theorem \\ref{main_theorem}, but by using the symbolic model $S_{{\\overline{{\\mathsf{q}}}}}(\\Sigma)$.\n\n\\begin{theorem}\\label{main_theorem3}\nConsider a stochastic switched system $\\Sigma$, admitting a common $\\delta$-GAS-M$_q$ Lyapunov function $V$. Let $\\widehat\\eta$ be given by \\eqref{eta1}. For any $\\varepsilon\\in{\\mathbb{R}}^+$ and any triple $\\overline{\\mathsf{q}}=\\left(\\tau,N,x_s\\right)$ of parameters satisfying\n\\begin{small}\n\\begin{align}\n\\label{bisim_cond3}\n\\mathsf{e}^{-\\kappa\\tau}\\underline\\alpha\\left(\\varepsilon^q\\right)+\\widehat\\gamma\\left(\\widehat\\eta\\right)&\\leq\\underline\\alpha\\left(\\varepsilon^q\\right),\n\\end{align}\n\\end{small}\nthe relation \\begin{small}$$R=\\left\\{(x_\\tau,x_{\\overline{{\\mathsf{q}}}})\\in X_\\tau\\times X_{\\overline{{\\mathsf{q}}}}\\,\\,|\\,\\,\\mathds{E}\\left[V(x_\\tau,H_{\\overline{{\\mathsf{q}}}}(x_{\\overline{{\\mathsf{q}}}}))\\right]\\leq\\underline\\alpha\\left(\\varepsilon^q\\right)\\right\\}$$\\end{small}is an $\\varepsilon$-approximate bisimulation relation between ${S}_{\\overline{\\mathsf{q}}}(\\Sigma)$ and $S_{\\tau}(\\Sigma)$.\n\\end{theorem}\n\n\\begin{proof}\nThe proof is similar to the one of Theorem \\ref{main_theorem}.\n\\end{proof}\n\nHere, one can also use any over approximation of $\\widehat\\eta$ such as the one in \\eqref{upper_bound4} instead of $\\widehat\\eta$ in condition \\eqref{bisim_cond3}. Finally, we establish the results on the existence of symbolic model $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma)$ (resp. $S_{\\overline{{\\mathsf{q}}}}(\\Sigma)$) such that \\mbox{$\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma)\\cong_{\\mathcal{S}}^{\\varepsilon}S_\\tau(\\Sigma)$} (resp. \\mbox{$S_{\\overline{{\\mathsf{q}}}}(\\Sigma)\\cong_{\\mathcal{S}}^{\\varepsilon}S_\\tau(\\Sigma)$}).\n\n\\begin{theorem}\\label{main_theorem5}\nConsider the result in Theorem \\ref{main_theorem}. If we choose: \\begin{small}$$X_{\\tau0}=\\{x\\in{\\mathbb{R}}^n\\,\\,|\\,\\,\\Vert x-\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}0})\\Vert\\leq\\left(\\overline\\alpha^{-1}\\left(\\underline\\alpha\\left(\\varepsilon^q\\right)\\right)\\right)^{\\frac{1}{q}},~\\forall x_{{\\overline{{\\mathsf{q}}}}0}\\in X_{{\\overline{{\\mathsf{q}}}}0}\\},$$\\end{small}then we have \\mbox{$\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma)\\cong_{\\mathcal{S}}^{\\varepsilon}S_\\tau(\\Sigma)$}.\n\\end{theorem}\n\n\\begin{proof}\nWe start by proving that \\mbox{$S_{\\tau}(\\Sigma)\\preceq^{\\varepsilon}_\\mathcal{S}\\overline{S}_{{\\overline{{\\mathsf{q}}}}}(\\Sigma)$}. For every $x_{\\tau 0}\\in{X_{\\tau 0}}$, there always exists \\mbox{$x_{{\\overline{{\\mathsf{q}}}} 0}\\in{X}_{{\\overline{{\\mathsf{q}}}} 0}$} such that $\\Vert{x_{\\tau0}}-\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}0})\\Vert\\leq\\left(\\overline\\alpha^{-1}\\left(\\underline\\alpha\\left(\\varepsilon^q\\right)\\right)\\right)^{\\frac{1}{q}}$. Then,\n\\begin{small}\n\\begin{align}\\nonumber\n\\mathbb{E}\\left[V\\left({x_{\\tau0}},\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}0})\\right)\\right]&=V\\left({x_{\\tau0}},\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}0})\\right)\\leq\\overline\\alpha\\left(\\left\\Vert x_{\\tau0}-\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}0})\\right\\Vert^q\\right)\\leq\\underline\\alpha\\left(\\varepsilon^q\\right),\n\\end{align}\n\\end{small}since $\\overline\\alpha$ is a $\\mathcal{K}_\\infty$ function.\nHence, \\mbox{$\\left(x_{\\tau0},x_{{\\overline{{\\mathsf{q}}}}0}\\right)\\in{R}$} implying that \\mbox{$S_{\\tau}(\\Sigma)\\preceq^{\\varepsilon}_\\mathcal{S}\\overline{S}_{{\\overline{{\\mathsf{q}}}}}(\\Sigma)$}. In a similar way, we can show that \\mbox{$\\overline{S}_{{\\overline{{\\mathsf{q}}}}}(\\Sigma)\\preceq^{\\varepsilon}_{\\mathcal{S}}S_{\\tau}(\\Sigma)$}, equipped with the relation $R^{-1}$, which completes the proof.\n\\end{proof}\n\nThe next theorem provides a similar result as the one of Theorem \\ref{main_theorem5}, but by using the symbolic model $S_{\\overline{{\\mathsf{q}}}}(\\Sigma)$.\n\n\\begin{theorem}\\label{main_theorem6}\nConsider the results in Theorem \\ref{main_theorem3}. If we choose: \\begin{small}\n\\begin{align}\\notag\nX_{\\tau0}=\\{&a\\in \\mathcal{X}_0\\,\\,|\\,\\,\\left(\\mathds{E}\\left[\\left\\Vert a-H_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}0})\\right\\Vert^q\\right]\\right)^{\\frac{1}{q}}\\leq\\left(\\overline\\alpha^{-1}\\left(\\underline\\alpha\\left(\\varepsilon^q\\right)\\right)\\right)^{\\frac{1}{q}},~\\forall x_{{\\overline{{\\mathsf{q}}}}0}\\in X_{{\\overline{{\\mathsf{q}}}}0}\\},\n\\end{align}\n\\end{small}then we have \\mbox{$S_{\\overline{{\\mathsf{q}}}}(\\Sigma)\\cong_{\\mathcal{S}}^{\\varepsilon}S_\\tau(\\Sigma)$}.\n\\end{theorem}\n\n\\begin{proof}\nThe proof is similar to the one of Theorem \\ref{main_theorem5}.\n\\end{proof}\n\n\\subsubsection{Multiple Lyapunov functions}\nHere, we provide results on the construction of symbolic models for $\\Sigma_{\\tau_d}$ without any continuous space discretization.\nConsider a stochastic switched system $\\Sigma_{\\tau_d}$ and a triple $\\overline{\\mathsf{q}}=\\left(\\tau,N,x_s\\right)$ of parameters.\nGiven $\\Sigma_{\\tau_d}$ and $\\overline{\\mathsf{q}}$, consider the following systems:\n\\begin{small}\n\\begin{align}\\notag\nS_{\\overline{\\mathsf{q}}}(\\Sigma_{\\tau_d})&=(X_{\\overline{\\mathsf{q}}},X_{\\overline{\\mathsf{q}}0},U_{\\overline{\\mathsf{q}}},\\rTo_{\\overline{\\mathsf{q}}},Y_{\\overline{\\mathsf{q}}},H_{\\overline{\\mathsf{q}}}),\\\\\\notag\\overline{S}_{\\overline{\\mathsf{q}}}(\\Sigma_{\\tau_d})&=(X_{\\overline{\\mathsf{q}}},X_{\\overline{\\mathsf{q}}0},U_{\\overline{\\mathsf{q}}},\\rTo_{\\overline{\\mathsf{q}}},Y_{\\overline{\\mathsf{q}}},\\overline{H}_{\\overline{\\mathsf{q}}}),\n\\end{align}\n\\end{small}consisting of: $X_{\\overline{\\mathsf{q}}}=\\mathsf{P}^{N}\\times\\{0,\\ldots,\\widehat{N}-1\\}$, $U_{\\overline{\\mathsf{q}}}=\\mathsf{P}$, $Y_{\\overline{\\mathsf{q}}}=Y_\\tau$, and\n\\begin{small}\n\\begin{itemize}\n\\item \\begin{small}\n\\begin{itemize}\n\\item if $N\\leq\\widehat{N}-1$: $X_{{\\overline{{\\mathsf{q}}}}0}=\\left\\{\\left(p,\\ldots,p,N\\right)\\,\\,|\\,\\,\\forall p\\in\\mathsf{P}\\right\\}$; \n\\item if $N>\\widehat{N}-1$: $X_{{\\overline{{\\mathsf{q}}}}0}=\\{(\\overbrace{p_1,\\ldots,p_1}^{m_1~\\text{times}},\\dots,\\overbrace{p_k,\\ldots,p_k}^{m_k~\\text{times}},i)|~~\\exists k\\in{\\mathbb{N}}~~\\text{s.t.}~~ m_1,\\ldots,m_{k-1}\\geq{\\widehat{N}},~i=\\min\\{m_k-1,\\widehat{N}-1\\},~p_1,\\ldots,p_k\\in\\mathsf{P}\\}$;\n\\end{itemize}\\end{small}\n\\item $\\left(p_1,p_2,\\ldots,p_N,i\\right)\\rTo_{\\overline{\\mathsf{q}}}^{p_N}\\left(p_2,\\ldots,p_N,p,i'\\right)$ if one of the following holds:\n\n\\begin{itemize}\n\\item $i<\\widehat{N}-1$, $p=p_N$, and $i'=i+1$;\n\\item $i=\\widehat{N}-1$, $p=p_N$, and $i'=\\widehat{N}-1$;\n\\item $i=\\widehat{N}-1$, $p\\neq p_N$, and $i'=0$;\n\\end{itemize}\n\\item $H_{\\overline{\\mathsf{q}}}(x_{\\overline{{\\mathsf{q}}}},i)=\\xi_{x_sx_{\\overline{{\\mathsf{q}}}}}(N\\tau)$ $\\left(\\overline{H}_{\\overline{\\mathsf{q}}}(x_{\\overline{{\\mathsf{q}}}},i)=\\overline\\xi_{x_sx_{\\overline{{\\mathsf{q}}}}}(N\\tau)\\right)$ for any $(x_{\\overline{{\\mathsf{q}}}},i)\\in X_{\\overline{{\\mathsf{q}}}}$, where $x_{\\overline{{\\mathsf{q}}}}=\\left(p_1,\\ldots,p_N\\right)$.\n\\end{itemize}\n\\end{small}\n\nNotice that the proposed system $S_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})$ $\\left(\\text{resp.}~\\overline{S}_{{\\overline{{\\mathsf{q}}}}}(\\Sigma_{\\tau_d})\\right)$ is symbolic and deterministic in the sense of Definition \\ref{system}. Note that the set $X_{{\\overline{{\\mathsf{q}}}}0}$ is chosen in such a way that it respects the dwell time of switching signals (i.e. being in each mode at least $\\tau_d=\\widehat{N}\\tau$ seconds).\n\nBefore providing the second main result of this subsection, we need the following technical results, similar to the ones in Lemmas \\ref{lemma1} and \\ref{lemma4}.\n\n\\begin{lemma}\\label{lemma11}\nConsider a stochastic switched system $\\Sigma_{\\tau_d}$, admitting multiple $\\delta$-GAS-M$_q$ Lyapunov functions $V_p$, and consider its corresponding symbolic model $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})$. Moreover, assume that (\\ref{eq0}) holds for some $\\mu\\geq1$. If $\\tau_d>\\log{\\mu}\/\\kappa$, then we have:\n\\begin{small}\n\\begin{align}\n\\label{upper_bound5}\n\\overline\\eta\\leq&\\left(\\underline\\alpha^{-1}(\\mathsf{e}^{-(\\kappa-\\log\\mu\/\\tau_d) N\\tau}\\max_{p,p'\\in \\mathsf{P}}V_{p'}(\\overline\\xi_{x_sp}(\\tau),x_s))\\right)^{1\/q},\n\\end{align}\n\\end{small}\nwhere\n\\begin{small}\n\\begin{align}\\label{eta2}\n\\overline\\eta:=\\max_{\\substack{(x_{\\overline{{\\mathsf{q}}}},i)\\in X_{\\overline{{\\mathsf{q}}}}\\\\(x_{\\overline{{\\mathsf{q}}}},i)\\rTo^p_{\\overline{{\\mathsf{q}}}}(x'_{\\overline{{\\mathsf{q}}}},i')}}\\Vert\\overline\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{\\overline{{\\mathsf{q}}}},i)p}(\\tau)-\\overline{H}_{\\overline{{\\mathsf{q}}}}\\left(x'_{\\overline{{\\mathsf{q}}}},i'\\right)\\Vert.\n\\end{align}\n\\end{small}\n\\end{lemma}\n\nThe proof is similar to the proof of Lemma \\ref{lemma1}.\n\nThe next lemma provides a similar result as the one of Lemma \\ref{lemma11}, but by using the symbolic model $S_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})$ rather than $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})$.\n\\begin{lemma}\\label{lemma44}\nConsider a stochastic switched system $\\Sigma_{\\tau_d}$, admitting multiple $\\delta$-GAS-M$_q$ Lyapunov functions $V_p$, and consider its corresponding symbolic model ${S}_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})$. Moreover, assume that (\\ref{eq0}) holds for some $\\mu\\geq1$. If $\\tau_d>\\log{\\mu}\/\\kappa$, then we have:\n\\begin{small}\n\\begin{align}\\label{upper_bound6}\n\\widehat\\eta\\leq&(\\underline\\alpha^{-1}(\\mathsf{e}^{-(\\kappa-\\log\\mu\/\\tau_d) N\\tau}\\max_{p,p'\\in\\mathsf{P}}\\mathds{E}[V_{p'}\\left(\\xi_{x_sp}(\\tau),x_s\\right)]))^{\\frac{1}{q}},\n\\end{align}\n\\end{small}\nwhere\n\\begin{small}\n\\begin{align}\\label{eta3}\n\\widehat\\eta:=\\max_{\\substack{(x_{\\overline{{\\mathsf{q}}}},i)\\in X_{\\overline{{\\mathsf{q}}}}\\\\(x_{\\overline{{\\mathsf{q}}}},i)\\rTo^p_{\\overline{{\\mathsf{q}}}}(x'_{\\overline{{\\mathsf{q}}}},i')}}\\mathds{E}[\\Vert\\xi_{H_{\\overline{{\\mathsf{q}}}}(x_{\\overline{{\\mathsf{q}}}},i)p}(\\tau)-H_{\\overline{{\\mathsf{q}}}}\\left(x'_{\\overline{{\\mathsf{q}}}},i'\\right)\\Vert].\n\\end{align}\n\\end{small}\n\\end{lemma}\nThe proof is similar to the proof of Lemma \\ref{lemma4}.\n\nNow, we present the second main result of this subsection, relating the existence of multiple Lyapunov functions to that of a bisimilar finite abstractions without any continuous space discretization. In order to show the next result, we assume that $f_p(0_n)=0_n$ only if $\\Sigma_{\\tau_d,p}$ is not affine and $g_p(0_n)=0_{n\\times\\widehat{q}}$ for any $p\\in\\mathsf{P}$.\n\n\\begin{theorem}\\label{main_theorem33}\nConsider a stochastic switched system $\\Sigma_{\\tau_d}$. Let us assume that for any $p\\in\\mathsf{P}$, there exists a $\\delta$-GAS-M$_q$ Lyapunov function $V_p$, of the form of the one explained in Lemma \\ref{lem:moment est}, for subsystem $\\Sigma_{\\tau_d,p}$. Moreover, assume that (\\ref{eq0}) holds for some $\\mu\\geq1$. Let $\\overline\\eta$ be given by \\eqref{eta2}. If $\\tau_d>\\log{\\mu}\/\\kappa$, for any $\\varepsilon\\in{\\mathbb{R}}^+$, \nand any triple $\\overline{\\mathsf{q}}=\\left(\\tau,N,x_s\\right)$ of parameters satisfying\n\\begin{small}\n\\begin{align}\n\\label{bisim_cond_mul0}\n\\widehat\\gamma(\\left(h_{x_s}\\left((N+1)\\tau\\right)\\right)^{\\frac{1}{q}}+\\overline\\eta)&\\leq\\frac{\\frac{1}{\\mu}-\\mathsf{e}^{-\\kappa\\tau_d}}{1-\\mathsf{e}^{-\\kappa\\tau_d}}(1-\\mathsf{e}^{-\\kappa\\tau})\\underline\\alpha(\\varepsilon^q),\n\\end{align}\n\\end{small}there exists an $\\varepsilon$-approximate bisimulation relation $R$ between $\\overline{S}_{\\overline{\\mathsf{q}}}(\\Sigma_{\\tau_d})$ and $S_{\\tau}(\\Sigma_{\\tau_d})$ as the following:\\\\ $\\left(x_{\\tau},p_1,i_1,x_{{\\overline{{\\mathsf{q}}}}},i_2\\right)\\in R$, where $x_{\\overline{{\\mathsf{q}}}}=(\\overline{p}_1,\\ldots,\\overline{p}_N)$, if and only if $p_1=\\overline{p}_N=p$, $i_1=i_2=i$, and \\begin{small}$$\\mathbb{E}[V_p(H_{\\tau}(x_{\\tau},p_1,i_1),\\overline{H}_{{\\overline{{\\mathsf{q}}}}}(x_{{\\overline{{\\mathsf{q}}}}},i_2))]=\\mathbb{E}[V_p(x_{\\tau},\\overline\\xi_{x_sx_{\\overline{{\\mathsf{q}}}}}(N\\tau))]\\leq\\delta_i,$$\\end{small}where $\\delta_0,\\ldots,\\delta_{\\widehat{N}-1}$ are given recursively by \\begin{small}$\\delta_{i+1}=\\mathsf{e}^{-\\kappa\\tau}\\delta_i+\\widehat\\gamma\\left(\\left(h_{x_s}\\left((N+1)\\tau\\right)\\right)^{\\frac{1}{q}}+\\overline\\eta\\right)$\\end{small} and $\\delta_0=\\underline\\alpha\\left(\\varepsilon^q\\right)$.\n\\end{theorem} \n\n\\begin{proof}\nConsider the relation $R\\subseteq X_{\\tau}\\times X_{{\\overline{{\\mathsf{q}}}}}$ defined by $\\left(x_{\\tau},p_1,i_1,x_{{\\overline{{\\mathsf{q}}}}},i_2\\right)\\in R$, where $x_{\\overline{{\\mathsf{q}}}}=(\\overline{p}_1,\\ldots,\\overline{p}_N)$, if and only if $p_1=\\overline{p}_N=p$, $i_1=i_2=i$, and \\begin{small}$$\\mathbb{E}[V_p(H_{\\tau}(x_{\\tau},p_1,i_1),\\overline{H}_{{\\overline{{\\mathsf{q}}}}}(x_{{\\overline{{\\mathsf{q}}}}},i_2))]=\\mathbb{E}[V_p(x_{\\tau},\\overline\\xi_{x_sx_{\\overline{{\\mathsf{q}}}}}(N\\tau))]\\leq\\delta_i,$$\\end{small}where $\\delta_0,\\ldots,\\delta_{\\widehat{N}}$ are given recursively by \\begin{small}$$\\delta_0=\\underline\\alpha\\left(\\varepsilon^q\\right),~~\\delta_{i+1}=\\mathsf{e}^{-\\kappa\\tau}\\delta_i+\\widehat\\gamma\\left(\\left(h_{x_s}\\left((N+1)\\tau\\right)\\right)^{\\frac{1}{q}}+\\eta\\right).$$\\end{small}One can easily verify that\n\\begin{small}\n\\begin{align}\\label{delta_i}\n\\delta_i&=\\mathsf{e}^{-i\\kappa\\tau}\\underline\\alpha(\\varepsilon^q)+\\widehat\\gamma((h_{x_s}((N+1)\\tau))^{\\frac{1}{q}}+\\eta)\\frac{1-\\mathsf{e}^{-i\\kappa\\tau}}{1-\\mathsf{e}^{-\\kappa\\tau}}\\\\\\notag&=\\frac{\\widehat\\gamma((h_{x_s}((N+1)\\tau))^{\\frac{1}{q}}+\\eta)}{1-\\mathsf{e}^{-\\kappa\\tau}}+\\mathsf{e}^{-i\\kappa\\tau}(\\underline\\alpha(\\varepsilon^q)-\\frac{\\widehat\\gamma((h_{x_s}((N+1)\\tau))^{\\frac{1}{q}}+\\eta)}{1-\\mathsf{e}^{-\\kappa\\tau}}).\n\\end{align}\n\\end{small}\nSince $\\mu\\geq1$, and from (\\ref{bisim_cond_mul0}), one has \\begin{small}$$\\widehat\\gamma\\left(\\left(h_{x_s}\\left((N+1)\\tau\\right)\\right)^{\\frac{1}{q}}+\\eta\\right)\\leq(1-\\mathsf{e}^{-\\kappa\\tau})\\underline\\alpha\\left(\\varepsilon^q\\right).$$\\end{small}It follows from (\\ref{delta_i}) that $\\delta_0\\geq\\delta_2\\geq\\cdots\\geq\\delta_{\\widehat{N}-1}\\geq\\delta_{\\widehat{N}}$. From (\\ref{bisim_cond_mul0}) and since $\\tau_d=\\widehat{N}\\tau$, we get \n\\begin{small}\n\\begin{align}\\label{Ntozero}\n\\delta_{\\widehat{N}}=&\\mathsf{e}^{-\\kappa\\tau_d}\\underline\\alpha(\\varepsilon^q)+\\widehat\\gamma((h_{x_s}((N+1)\\tau))^{\\frac{1}{q}}+\\eta)\\frac{1-\\mathsf{e}^{-\\kappa\\tau_d}}{1-\\mathsf{e}^{-\\kappa\\tau}}\\leq\\mathsf{e}^{-\\kappa\\tau_d}\\underline\\alpha(\\varepsilon^q)+(\\frac{1}{\\mu}-\\mathsf{e}^{-\\kappa\\tau_d})\\underline\\alpha(\\varepsilon^q)=\\frac{\\underline\\alpha(\\varepsilon^q)}{\\mu}.\n\\end{align}\n\\end{small}We start by proving that $R$ is an $\\varepsilon$-approximate simulation relation from $S_{\\tau}(\\Sigma_{\\tau_d})$ to $\\overline{S}_{{\\overline{{\\mathsf{q}}}}}(\\Sigma_{\\tau_d})$. Consider any \\mbox{$\\left(x_{\\tau},p,i,x_{{\\overline{{\\mathsf{q}}}}},i\\right)\\in R$}. Using the convexity assumption of $\\underline\\alpha_p$, and since it is a $\\mathcal{K}_\\infty$ function, and the Jensen inequality \\cite{oksendal}, we have:\n\\begin{small}\n\\begin{align}\\nonumber\n\\underline\\alpha(\\mathbb{E}[\\Vert H_\\tau(x_{\\tau},p,i)-H_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}},i)\\Vert^q])&=\\underline\\alpha(\\mathbb{E}[\\Vert x_{\\tau}-\\overline\\xi_{x_sx_{\\overline{{\\mathsf{q}}}}}(N\\tau)\\Vert^q])\\leq\\underline\\alpha_p(\\mathbb{E}[\\Vert x_{\\tau}-\\overline\\xi_{x_sx_{\\overline{{\\mathsf{q}}}}}(N\\tau)\\Vert^q])\\leq\\mathds{E}[\\underline\\alpha_p(\\Vert x_{\\tau}-\\overline\\xi_{x_sx_{\\overline{{\\mathsf{q}}}}}(N\\tau)\\Vert^q)]\\\\\\notag&\\leq\\mathbb{E}[V_p(x_{\\tau},\\overline\\xi_{x_sx_{\\overline{{\\mathsf{q}}}}}(N\\tau))]\\leq\\delta_i\\leq\\delta_0.\n\\end{align}\n\\end{small}Therefore, we obtain\n\\begin{small}\n$(\\mathbb{E}[\\Vert x_{\\tau}-\\overline\\xi_{x_sx_{\\overline{{\\mathsf{q}}}}}(N\\tau)\\Vert^q])^{\\frac{1}{q}}\\leq(\\underline\\alpha^{-1}(\\delta_0))^{\\frac{1}{q}}\\leq\\varepsilon$,\n\\end{small}\nbecause of $\\underline\\alpha\\in\\mathcal{K}_\\infty$. Hence, condition (i) in Definition \\ref{APSR} is satisfied.\nLet us now show that condition (ii) in Definition\n\\ref{APSR} holds. \nConsider the transition $(x_{\\tau},p,i)\\rTo^{p}_{\\tau} (x'_{\\tau},p',i')$ in $S_{\\tau}\\left(\\Sigma_{\\tau_d}\\right)$, where $x'_{\\tau}=\\xi_{x_{\\tau}p}(\\tau)$ $\\mathds{P}$-a.s.. Since $V_p$ is a $\\delta$-GAS-M$_q$ Lyapunov function for subsystem $\\Sigma_p$, we have\n\\begin{small}\n\\begin{align}\\label{b06}\n\\mathbb{E}[V_{p}(x'_{\\tau},\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}},i)p}(\\tau))] &\\leq \\mathds{E}[V_{p}(x_\\tau,\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_q,i))] \\mathsf{e}^{-\\kappa\\tau}\\leq\\mathsf{e}^{-\\kappa\\tau}\\delta_i.\n\\end{align}\n\\end{small}Using Lemma \\ref{lem:moment est}, the $\\mathcal{K}_\\infty$ function $\\widehat\\gamma$, the concavity of $\\widehat\\gamma_p$ in \\eqref{supplement}, the Jensen inequality \\cite{oksendal}, equation \\eqref{eta2}, the inequalities (\\ref{supplement}) and (\\ref{b06}), and triangle inequality, we obtain\n\\begin{small}\n\\begin{align}\\nonumber\n\\mathbb{E}[V_{p}(x'_{\\tau},\\overline{H}_{\\overline{{\\mathsf{q}}}}(x'_{{\\overline{{\\mathsf{q}}}}},i'))]&=\\mathbb{E}[V_{p}(x'_\\tau,\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}},i)p}(\\tau))+V_{p}(x'_{\\tau},\\overline{H}_{\\overline{{\\mathsf{q}}}}(x'_{{\\overline{{\\mathsf{q}}}}},i'))-V_{p}(x'_\\tau,\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}},i)p}(\\tau))]\\\\ \\notag\n&=  \\mathbb{E}[V_{p}(x'_\\tau,\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}},i)p}(\\tau))]+\\mathbb{E}[V_{p}(x'_{\\tau},\\overline{H}_{\\overline{{\\mathsf{q}}}}(x'_{{\\overline{{\\mathsf{q}}}}},i'))-V_{p}(x'_\\tau,\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}},i)p}(\\tau))]\\\\\\notag&\\leq\\mathsf{e}^{-\\kappa\\tau}\\delta_i+\\mathbb{E}[\\widehat\\gamma_{p}(\\Vert\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}},i)p}(\\tau)-\\overline{H}_{\\overline{{\\mathsf{q}}}}(x'_{{\\overline{{\\mathsf{q}}}}},i')\\Vert)]\\leq\\mathsf{e}^{-\\kappa\\tau}\\delta_i+\\widehat\\gamma_{p}(\\mathbb{E}[\\Vert\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}},i)p}(\\tau)-\\overline{H}_{\\overline{{\\mathsf{q}}}}(x'_{{\\overline{{\\mathsf{q}}}}},i')\\Vert])\\\\\\notag\n&\\leq\\mathsf{e}^{-\\kappa\\tau}\\delta_i+\\widehat\\gamma(\\mathbb{E}[\\Vert\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}},i)p}(\\tau)-\\overline{\\xi}_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}},i)p}(\\tau)+\\overline{\\xi}_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}},i)p}(\\tau)-\\overline{H}_{\\overline{{\\mathsf{q}}}}(x'_{{\\overline{{\\mathsf{q}}}}},i')\\Vert])\\\\\\notag\n&\\leq\\mathsf{e}^{-\\kappa\\tau}\\delta_i+\\widehat\\gamma(\\mathbb{E}[\\Vert\\xi_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}},i)p}(\\tau)-\\overline{\\xi}_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}},i)p}(\\tau)\\Vert]+\\Vert\\overline{\\xi}_{\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}},i)p}(\\tau)-\\overline{H}_{\\overline{{\\mathsf{q}}}}(x'_{{\\overline{{\\mathsf{q}}}}},i')\\Vert)\\\\\\label{b08}\n&\\leq\\mathsf{e}^{-\\kappa\\tau}\\delta_i+\\widehat\\gamma((h_{x_s}((N+1)\\tau))^{\\frac{1}{q}}+\\eta)=\\delta_{i+1}.\n\\end{align}\n\\end{small}\nWe now examine three separate cases:\n\\begin{itemize}\n\\item If $i<\\widehat{N}-1$, then $p'=p$, and $i'=i+1$; from (\\ref{b08}), \\begin{small}$\\mathds{E}\\left[V_p\\left(x'_{\\tau},\\overline{H}_{\\overline{{\\mathsf{q}}}}\\left(x'_{{\\overline{{\\mathsf{q}}}}},i'\\right)\\right)\\right]\\leq\\delta_{i+1}$\\end{small}, we conclude that \\begin{small}$(x'_{\\tau},p,i+1,x'_{{\\overline{{\\mathsf{q}}}}},i+1)\\in R$\\end{small}; \n\n\\item If $i=\\widehat{N}-1$, and $p'=p$, then $i'=\\widehat{N}-1$; from (\\ref{b08}), \\begin{small}$\\mathds{E}\\left[V_p\\left(x'_{\\tau},\\overline{H}_{\\overline{{\\mathsf{q}}}}\\left(x'_{{\\overline{{\\mathsf{q}}}}},i'\\right)\\right)\\right]\\leq\\delta_{\\widehat{N}}\\leq\\delta_{\\widehat{N}-1}$\\end{small}, we conclude that \\begin{small}$(x'_{\\tau},p,\\widehat{N}-1,x'_{{\\overline{{\\mathsf{q}}}}},\\widehat{N}-1)\\in R$\\end{small}; \n\n\\item If $i=\\widehat{N}-1$, and $p'\\neq{p}$, then $i'=0$; from (\\ref{Ntozero}) and (\\ref{b08}), \\begin{small}$\\mathds{E}\\left[V_p\\left(x'_{\\tau},\\overline{H}_{\\overline{{\\mathsf{q}}}}\\left(x'_{{\\overline{{\\mathsf{q}}}}},i'\\right)\\right)\\right]\\leq\\delta_{\\widehat{N}}\\leq\\delta_0\/\\mu$\\end{small}. From (\\ref{eq0}), it follows that \\begin{small}$\\mathds{E}\\left[V_{p'}(x'_{\\tau},\\overline{H}_{\\overline{{\\mathsf{q}}}}\\left(x'_{{\\overline{{\\mathsf{q}}}}},i'\\right))\\right]\\leq\\mu \\mathds{E}\\left[V_p\\left(x'_{\\tau},\\overline{H}_{\\overline{{\\mathsf{q}}}}\\left(x'_{{\\overline{{\\mathsf{q}}}}},i'\\right)\\right)\\right]\\leq\\delta_0$\\end{small}. Hence, \\begin{small}$(x'_{\\tau},p',0,x'_{{\\overline{{\\mathsf{q}}}}},0)\\in R$\\end{small}.\n\\end{itemize}\nTherefore, we conclude that condition (ii) in Definition \\ref{APSR} holds. In a similar way, we can prove that that $R^{-1}$ is an\n$\\varepsilon$-approximate simulation relation from $\\overline{S}_{{\\overline{{\\mathsf{q}}}}}(\\Sigma_{\\tau_d})$ to $S_{\\tau}(\\Sigma_{\\tau_d})$ implying that $R$ is an $\\varepsilon$-approximate bisimulation relation between $\\overline{S}_{{\\overline{{\\mathsf{q}}}}}(\\Sigma_{\\tau_d})$ and $S_\\tau(\\Sigma_{\\tau_d})$.\n\\end{proof}\n\nNote that one can use any over approximation of $\\overline\\eta$ such as the one in \\eqref{upper_bound5} instead of $\\overline\\eta$ in condition \\eqref{bisim_cond_mul0}. By choosing $N$ sufficiently large, one can enforce $h_{x_s}((N+1)\\tau)$ and $\\overline\\eta$ to be sufficiently small. Hence, it can be readily seen that for a given precision $\\varepsilon$,\nthere always exists a large value of $N$, such that the condition in (\\ref{bisim_cond_mul0}) is satisfied.\n\nThe next theorem provides a result that is similar to the one of Theorem \\ref{main_theorem33}, but by using the symbolic model $S_{{\\overline{{\\mathsf{q}}}}}(\\Sigma_{\\tau_d})$.\n\n\\begin{theorem}\\label{main_theorem22}\nConsider a stochastic switched system $\\Sigma_{\\tau_d}$. Let us assume that for any $p\\in\\mathsf{P}$, there exists a $\\delta$-GAS-M$_q$ Lyapunov function $V_p$ for subsystem $\\Sigma_{\\tau_d,p}$. Moreover, assume that (\\ref{eq0}) holds for some $\\mu\\geq1$. Let $\\widehat\\eta$ be given by \\eqref{eta3}. If $\\tau_d>\\log{\\mu}\/\\kappa$, for any $\\varepsilon\\in{\\mathbb{R}}^+$, \nand any triple $\\overline{\\mathsf{q}}=\\left(\\tau,N,x_s\\right)$ of parameters satisfying\n\\begin{small}\n\\begin{align}\n\\label{bisim_cond_mul10}\n\\widehat\\gamma\\left(\\widehat\\eta\\right)&\\leq\\frac{\\frac{1}{\\mu}-\\mathsf{e}^{-\\kappa\\tau_d}}{1-\\mathsf{e}^{-\\kappa\\tau_d}}\\left(1-\\mathsf{e}^{-\\kappa\\tau}\\right)\\underline\\alpha\\left(\\varepsilon^q\\right),\n\\end{align}\n\\end{small}there exists an $\\varepsilon$-approximate bisimulation relation $R$ between ${S}_{\\overline{\\mathsf{q}}}(\\Sigma_{\\tau_d})$ and $S_{\\tau}(\\Sigma_{\\tau_d})$ as the following:\\\\ $\\left(x_{\\tau},p_1,i_1,x_{{\\overline{{\\mathsf{q}}}}},i_2\\right)\\in R$, where $x_{\\overline{{\\mathsf{q}}}}=(\\overline{p}_1,\\ldots,\\overline{p}_N)$, if and only if $p_1=\\overline{p}_N=p$, $i_1=i_2=i$, and \\begin{small}$$\\mathbb{E}[V_p(H_{\\tau}(x_{\\tau},p_1,i_1),{H}_{{\\overline{{\\mathsf{q}}}}}(x_{{\\overline{{\\mathsf{q}}}}},i_2))]=\\mathbb{E}[V_p(x_{\\tau},\\xi_{x_sx_{\\overline{{\\mathsf{q}}}}}(N\\tau))]\\leq\\delta_i,$$\\end{small}where $\\delta_0,\\ldots,\\delta_{\\widehat{N}-1}$ are given recursively by \\begin{small}$\\delta_0=\\underline\\alpha\\left(\\varepsilon^q\\right),~\\delta_{i+1}=\\mathsf{e}^{-\\kappa\\tau}\\delta_i+\\widehat\\gamma\\left(\\widehat\\eta\\right)$\\end{small}.\n\\end{theorem} \n\n\\begin{proof}\nThe proof is similar to the one of Theorem \\ref{main_theorem33}.\n\\end{proof}\n\nNote that one can also use any over approximation of $\\widehat\\eta$ such as the one in \\eqref{upper_bound6} instead of $\\widehat\\eta$ in condition \\eqref{bisim_cond_mul10}. Finally, we establish the results on the existence of symbolic model $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})$ (resp. $S_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})$) such that \\mbox{$\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})\\cong_{\\mathcal{S}}^{\\varepsilon}S_\\tau(\\Sigma_{\\tau_d})$} (resp. \\mbox{$S_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})\\cong_{\\mathcal{S}}^{\\varepsilon}S_\\tau(\\Sigma_{\\tau_d})$}).\n\n\\begin{theorem}\\label{main_theorem55}\nConsider the result in Theorem \\ref{main_theorem33}. If we choose: \\begin{small}\n\\begin{align}\\notag\nX_{\\tau0}=\\big\\{&\\left(x,p,i\\right)\\,\\,|\\,\\,x\\in{\\mathbb{R}}^n,\\left\\Vert x-\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}0},i)\\right\\Vert\\leq\\left(\\overline\\alpha_p^{-1}\\left(\\delta_i\\right)\\right)^{\\frac{1}{q}},p=p_N,\\notag\\forall (p_1,\\ldots,p_N,i)\\in X_{{\\overline{{\\mathsf{q}}}}0}\\big\\},\n\\end{align}\n\\end{small}then we have \\mbox{$\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})\\cong_{\\mathcal{S}}^{\\varepsilon}S_\\tau(\\Sigma_{\\tau_d})$}.\n\\end{theorem}\n\n\\begin{proof}\nWe start by proving that \\mbox{$S_{\\tau}(\\Sigma_{\\tau_d})\\preceq^{\\varepsilon}_\\mathcal{S}\\overline{S}_{{\\overline{{\\mathsf{q}}}}}(\\Sigma_{\\tau_d})$}. For every $\\left(x_{\\tau 0},p,i\\right)\\in{X_{\\tau 0}}$, there always exists \\mbox{$\\left(x_{{\\overline{{\\mathsf{q}}}} 0},i\\right)\\in{X}_{{\\overline{{\\mathsf{q}}}} 0}$}, where $x_{{\\overline{{\\mathsf{q}}}}0}=(p_1,\\ldots,p_N)$, such that $p=p_N$ and \\begin{small}$\\left\\Vert{x_{\\tau0}}-\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}0},i)\\right\\Vert\\leq\\left(\\overline\\alpha_p^{-1}\\left(\\delta_i\\right)\\right)^{\\frac{1}{q}}$\\end{small}. Then,\n\\begin{footnotesize}\n\\begin{align}\\nonumber\n\\mathbb{E}\\left[V_p\\left({x_{\\tau0}},\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}0},i)\\right)\\right]&=V_p\\left({x_{\\tau0}},\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}0},i)\\right)\\leq\\overline\\alpha_p(\\Vert x_{\\tau0}-\\overline{H}_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}0},i)\\Vert^q)\\leq\\delta_i,\n\\end{align}\n\\end{footnotesize}since $\\overline\\alpha_p$ is a $\\mathcal{K}_\\infty$ function.\nHence, \\mbox{$\\left(x_{\\tau0},p,i,x_{{\\overline{{\\mathsf{q}}}}0},i\\right)\\in{R}$} implying that $S_{\\tau}(\\Sigma_{\\tau_d})\\preceq^{\\varepsilon}_\\mathcal{S}\\overline{S}_{{\\overline{{\\mathsf{q}}}}}(\\Sigma_{\\tau_d})$. In a similar way, we can show that \\mbox{$\\overline{S}_{{\\overline{{\\mathsf{q}}}}}(\\Sigma_{\\tau_d})\\preceq^{\\varepsilon}_{\\mathcal{S}}S_{\\tau}(\\Sigma_{\\tau_d})$}, equipped with the relation $R^{-1}$, which completes the proof.\n\\end{proof}\n\nThe next theorem provides a similar result as the one of Theorem \\ref{main_theorem55}, but by using the symbolic model $S_{\\overline{{\\mathsf{q}}}}(\\Sigma)$.\n\n\\begin{theorem}\\label{main_theorem66}\nConsider the results in Theorem \\ref{main_theorem22}. If we choose: \\begin{small}\n\\begin{align}\\notag\nX_{\\tau0}=\\big\\{&\\left(a,p,i\\right)\\,\\,|\\,\\,a\\in\\mathcal{X}_0,\\left(\\mathds{E}\\left[\\left\\Vert a-H_{\\overline{{\\mathsf{q}}}}(x_{{\\overline{{\\mathsf{q}}}}0},i)\\right\\Vert^q\\right]\\right)^{\\frac{1}{q}}\\leq\\left(\\overline\\alpha_p^{-1}\\left(\\delta_i\\right)\\right)^{\\frac{1}{q}}, p=p_N, \\forall (p_1,\\ldots,p_N,i)\\in X_{{\\overline{{\\mathsf{q}}}}0}\\big\\},\n\\end{align}\n\\end{small}then we have \\mbox{$S_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})\\cong_{\\mathcal{S}}^{\\varepsilon}S_\\tau(\\Sigma_{\\tau_d})$}.\n\\end{theorem}\n\n\\begin{proof}\nThe proof is similar to the one of Theorem \\ref{main_theorem55}.\n\\end{proof}\n\n\\begin{remark}\\label{remark1}\nThe symbolic model $S_{\\overline{{\\mathsf{q}}}}(\\Sigma)$ (resp. $S_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})$), computed by using the parameter $\\overline{\\mathsf{q}}$ provided in Theorem \\ref{main_theorem3} (resp. Theorem \\ref{main_theorem22}), has fewer (or at most equal number of) states than the symbolic model $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma)$ (resp. $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})$), computed by using the parameter ${\\overline{{\\mathsf{q}}}}$ provided in Theorem \\ref{main_theorem} (resp. Theorem \\ref{main_theorem33}) while having the same precision. However, the symbolic models $S_{\\overline{{\\mathsf{q}}}}(\\Sigma)$ and $S_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})$ have states with probabilistic output values, rather than non-probabilistic ones which makes the control synthesis over them more involved.\n\\end{remark}\n\n\\begin{remark}\nThe control synthesis over $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma)$ (resp. $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})$) is simple as the outputs are non-probabilistic points. For ${S}_{\\overline{{\\mathsf{q}}}}(\\Sigma)$ (resp. ${S}_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})$) it is less intuitive and more involved. We refer the interested readers to \\cite[Subsection 5.3]{majid10} explaining how one can synthesize controllers over finite metric systems with random output values.\n\\end{remark}\n\n\\subsection{Comparison between the two proposed approaches}\n\nNote that given any precision $\\varepsilon$ and sampling time $\\tau$, one can always use the results proposed in Theorems \\ref{main_theorem5} and \\ref{main_theorem55} to construct symbolic models $\\overline{S}_{\\mathsf{q}}(\\Sigma)$ and $\\overline{S}_{\\mathsf{q}}(\\Sigma_{\\tau_d})$, respectively, that are $\\varepsilon$-approximately bisimilar to $S_\\tau(\\Sigma)$ and $S_\\tau(\\Sigma_{\\tau_d})$, respectively. However, the results proposed in Theorems \\ref{main_theorem1} and \\ref{main_theorem2} cannot be applied for any sampling time $\\tau$ if the precision $\\varepsilon$ is lower than the thresholds introduced in inequalities \\eqref{lower_bound} and \\eqref{lower_bound_mul}, respectively (cf. the first case study). Furthermore, while the results in Theorems \\ref{main_theorem1} and \\ref{main_theorem2} only provide symbolic models with non-probabilistic output values, the ones in Theorems \\ref{main_theorem6} and \\ref{main_theorem66} provide symbolic models with probabilistic output values as well which can result in less conservative symbolic models (cf. Remark \\ref{remark1} and the first case study).\n\nOne can compare the results provided in Theorems \\ref{main_theorem5} and \\ref{main_theorem55} with the results provided in Theorems \\ref{main_theorem1} and \\ref{main_theorem2}, respectively, in terms of the sizes of the symbolic models. One can readily verify that the precision of the symbolic model $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma)$ (resp. $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})$) and the one $S_{\\mathsf{q}}(\\Sigma)$ (resp. $S_{\\mathsf{q}}(\\Sigma_{\\tau_d})$) is approximately the same as long as the state space quantisation parameter $\\eta$ is equal to the parameter $\\overline\\eta$ in \\eqref{eta} (resp. in \\eqref{eta2}), i.e. \\begin{small}$\\eta\\leq\\left(\\underline\\alpha^{-1}\\left(\\mathsf{e}^{-\\kappa N\\tau}\\overline{\\eta}_0\\right)\\right)^{1\/q}$\\end{small} (resp. \\begin{small}$\\eta\\leq\\left(\\underline\\alpha^{-1}\\left(\\mathsf{e}^{-(\\kappa-\\log\\mu\/{\\tau_d})N\\tau}\\widehat\\eta_0\\right)\\right)^{1\/q}$\\end{small}), where \\begin{small}$\\overline{\\eta}_0=\\max_{p\\in\\mathsf{P}}V\\left(\\overline\\xi_{x_sp}(\\tau),x_s\\right)$\\end{small} (resp. \\begin{small}$\\widehat\\eta_0=\\max_{p,p'\\in\\mathsf{P}}V_{p'}\\left(\\overline\\xi_{x_sp}(\\tau),x_s\\right)$\\end{small}). The reason their precisions are approximately (not exactly) the same is because we use $(h_{x_s}((N+1)\\tau))^{1\/q}$ in conditions \\eqref{bisim_cond11} and \\eqref{bisim_cond_mul0} rather than $(h_{[X_{\\tau0}]_\\eta}(\\tau))^{1\/q}$ (resp. $(h_{[X_0]_\\eta}(\\tau))^{1\/q}$) that is being used in condition\\eqref{bisim_cond} (resp.\\eqref{bisim_cond_mul}). By assuming that $(h_{x_s}((N+1)\\tau))^{1\/q}$ and $(h_{[X_{\\tau0}]_\\eta}(\\tau))^{1\/q}$ (resp. $(h_{[X_0]_\\eta}(\\tau))^{1\/q}$) are much smaller than $\\overline{\\eta}$ and $\\eta$, respectively, or $h_{x_s}((N+1)\\tau)\\approx h_{[X_{\\tau0}]_\\eta}(\\tau)$ (resp. $h_{x_s}((N+1)\\tau)\\approx h_{[X_0]_\\eta}(\\tau)$), the precisions are the same.\n\nThe number of states of the proposed symbolic models $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma)$ and $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})$ are $m^N$ and $m^{N}\\times\\widehat{N}$, respectively. Assume that we are interested in the dynamics of $\\Sigma$ (resp. $\\Sigma_{\\tau_d}$) on a compact set $\\mathsf{D}\\subset{\\mathbb{R}}^n$. Since the set of states of the proposed symbolic models $S_{\\mathsf{q}}(\\Sigma)$ and $S_{\\mathsf{q}}(\\Sigma_{\\tau_d})$ are $\\left[\\mathsf{D}\\right]_{\\eta}$ and $\\left[\\mathsf{D}\\right]_{\\eta}\\times\\mathsf{P}\\times\\{0,\\ldots,\\widehat{N}-1\\}$, respectively, their sizes are $\\left\\vert\\left[\\mathsf{D}\\right]_{\\eta}\\right\\vert=\\frac{K}{\\eta^n}$ and $\\frac{K}{\\eta^n}\\times{m}\\times\\widehat{N}$, respectively, where $K$ is a positive constant proportional to the volume of $\\mathsf{D}$. Hence, it is more convenient to use the proposed symbolic models $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma)$ and $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})$ rather than the ones $S_{\\mathsf{q}}(\\Sigma)$ and $S_{\\mathsf{q}}(\\Sigma_{\\tau_d})$, respectively, as long as:\n\\begin{small}\n\\begin{align}\\nonumber\n&m^N\\leq\\frac{K}{\\left(\\underline\\alpha^{-1}\\left(\\mathsf{e}^{-\\kappa N\\tau}\\overline\\eta_0\\right)\\right)^{n\/q}}~~\\text{and}~~m^{N-1}\\leq\\frac{K}{\\left(\\underline\\alpha^{-1}\\left(\\mathsf{e}^{-(\\kappa-\\log\\mu\/{\\tau_d}) N\\tau}\\widehat\\eta_0\\right)\\right)^{n\/q}},\n\\end{align}\n\\end{small}respectively. Without loss of generality, one can assume that $\\underline\\alpha({r})=r$ for any $r\\in{\\mathbb{R}}_0^+$. Hence, for sufficiently large value of $N$, it is more convenient to use the proposed symbolic models $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma)$ and $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})$ in comparison with the ones $S_{\\mathsf{q}}(\\Sigma)$ and $S_{\\mathsf{q}}(\\Sigma_{\\tau_d})$, respectively, as long as:\n\\begin{small}\n\\begin{align}\\label{criterion}\nm\\mathsf{e}^{\\frac{-\\kappa\\tau n}{q}}\\leq1,~~\\text{and}~~m\\mathsf{e}^{\\frac{-(\\kappa-\\log\\mu\/{\\tau_d})\\tau n}{q}}\\leq1,\n\\end{align}\n\\end{small}respectively.\n\n\\begin{comment}\n\\begin{remark}[Relationship with notions in the literature] \nThe approximate bisimulation notion in Definition \\ref{APSR} is structurally different than the probabilistic version discussed for finite state, \\emph{discrete-time}  labeled Markov chains in \\cite{DLT08}, which can also be extended to continuous-space processes as in \\cite{DP02}.   \nThe notion in this work can be instead related to the approximate probabilistic bisimulation notion discussed in \\cite{abate,julius1},  \nwhich lower bounds the probability that the Euclidean distance between trajectories of abstract and of concrete models remains close over a given time horizon:    \nboth notions hinge on distances over trajectories, rather than over transition probabilities as in \\cite{DP02,DLT08}.  \nAlong these lines, \\cite[Theorem 5.12]{majid8} can be readily employed to establish a probabilistic approximate bisimulation relation between $S_\\tau(\\Sigma)$ (resp. $S_\\tau\\left(\\Sigma_{\\tau_d}\\right)$) and $S_{\\mathsf{q}}(\\Sigma)$ (resp. $S_{\\mathsf{q}}\\left(\\Sigma_{\\tau_d}\\right)$) point-wise in time: \nthis result is sufficient to work with LTL specifications that need to be satisfied at a single time instance, \nsuch as next ($\\bigcirc$) and reach ($\\diamondsuit$). \\cite[Theorems 5.8, 5.10]{majid8} can be employed to instead establish probabilistic approximate bisimulation relations between $S_\\tau(\\Sigma)$ (resp. $S_\\tau\\left(\\Sigma_{\\tau_d}\\right)$) and $S_{\\mathsf{q}}(\\Sigma)$ (resp. $S_{\\mathsf{q}}\\left(\\Sigma_{\\tau_d}\\right)$) simultaneous in time, \nover an infinite and finite time horizon, respectively.  \n\\end{remark}\n\\end{comment}\n\n\\section{Examples}\n\n\\subsection{Room temperature control (common Lyapunov function)}\nConsider the stochastic switched system $\\Sigma$ which is a simple thermal model of a six-room building as depicted schematically in Figure \\ref{fig1} and described by the following stochastic differential equations:\n\\begin{footnotesize}\n\\begin{align}\\label{room}\n \n  \n\\hspace{-1.5mm}\\begin{array}{l}\n                  \\diff{\\xi}_1=\\big(\\alpha_{21}\\left(\\xi_2-\\xi_1\\right)+\\alpha_{31}\\left(\\xi_3-\\xi_1\\right)+\\alpha_{51}\\left(\\xi_5-\\xi_1\\right)+\\alpha_{e1}\\left(T_e-\\xi_1\\right)+\\alpha_{f1}\\left(T_{f1}-\\xi_1\\right)\\delta_{p2}\\big)\\diff{t}+\\left(\\sigma_{1,1}\\delta_{p1}+(1-\\delta_{p1})\\sigma_1\\right)\\xi_1\\diff{W^1_t},\\\\ \n\\diff{\\xi}_2=\\left(\\alpha_{12}\\left(\\xi_1-\\xi_2\\right)+\\alpha_{42}\\left(\\xi_4-\\xi_2\\right)+\\alpha_{e2}\\left(T_e-\\xi_2\\right)\\right)\\diff{t}+\\left(\\sigma_{2,1}\\delta_{p1}+(1-\\delta_{p1})\\sigma_2\\right)\\xi_2\\diff{W^2_t},\\\\\n\\diff{\\xi}_3=\\left(\\alpha_{13}\\left(\\xi_1-\\xi_3\\right)+\\alpha_{43}\\left(\\xi_4-\\xi_3\\right)+\\alpha_{e3}\\left(T_e-\\xi_3\\right)\\right)\\diff{t}+\\left(\\sigma_{3,1}\\delta_{p1}+(1-\\delta_{p1})\\sigma_3\\right)\\xi_3\\diff{W^3_t},\\\\\n\\diff{\\xi}_4=\\big(\\alpha_{24}\\left(\\xi_2-\\xi_4\\right)+\\alpha_{34}\\left(\\xi_3-\\xi_4\\right)+\\alpha_{64}\\left(\\xi_6-\\xi_4\\right)+\\alpha_{e4}\\left(T_e-\\xi_4\\right)+\\alpha_{f4}\\left(T_{f4}-\\xi_4\\right)\\delta_{p3}\\big)\\diff{t}+\\left(\\sigma_{4,1}\\delta_{p1}+(1-\\delta_{p1})\\sigma_4\\right)\\xi_4\\diff{W^4_t},\\\\ \n\\diff{\\xi}_5=\\left(\\alpha_{15}\\left(\\xi_1-\\xi_5\\right)+\\alpha_{e5}\\left(T_e-\\xi_5\\right)\\right)\\diff{t}+\\left(\\sigma_{5,1}\\delta_{p1}+(1-\\delta_{p1})\\sigma_5\\right)\\xi_5\\diff{W^5_t},\\\\\n\\diff{\\xi}_6=\\left(\\alpha_{46}\\left(\\xi_4-\\xi_6\\right)+\\alpha_{e6}\\left(T_e-\\xi_6\\right)\\right)\\diff{t}+\\left(\\sigma_{6,1}\\delta_{p1}+(1-\\delta_{p1})\\sigma_6\\right)\\xi_6\\diff{W^6_t},\\\\                  \n\\end{array}\n           \n\\end{align}\n\\end{footnotesize}where the terms $W_t^i$, $i=1,\\ldots,6$, denote the standard Brownian motion and $\\delta_{pi}=1$ if $i=p$ and $\\delta_{pi}=0$ otherwise.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=8cm]{room}\n\\end{center}\n\\caption{A schematic of the six-room building.}\n\\label{fig1}\n\\end{figure}\n\nNote that $\\xi_i$, $i=1,\\ldots,6$, denotes the temperature in each room, \n$T_e=10$ (degrees Celsius) is the external temperature, and $T_{f1}=T_{f4}=100$ are the temperatures of two heaters\\footnote{Here, we assume that at most one heater is on at each instant of time.} that both can be switched off ($p=1$), 1st heater ($T_{f1}$) on and the 2nd one ($T_{f4}$) off ($p=2$), or vice versa ($p=3$). \nThe drifts $f_p$ and diffusion terms $g_p$, $p = 1,2,3$, can be simply written out of \\eqref{room} and are affine and linear, respectively. \nThe parameters of the drifts are chosen as follows: $\\alpha_{21}=\\alpha_{12}=\\alpha_{13}=\\alpha_{31}=\\alpha_{42}=\\alpha_{24}=\\alpha_{34}=\\alpha_{43}=\\alpha_{15}=\\alpha_{51}=\\alpha_{46}=\\alpha_{64}=5\\times10^{-2}$, $\\alpha_{e1}=\\alpha_{e4}=5\\times10^{-3}$, $\\alpha_{e2}=\\alpha_{e3}=\\alpha_{e5}=\\alpha_{e6}=3.3\\times10^{-3}$, and $\\alpha_{f1}=\\alpha_{f4}=3.6\\times10^{-3}$. \nThe noise parameters are chosen as $\\sigma_{i,1}=0.002$ and $\\sigma_i=0.003$, for $i=1,\\ldots,6$. \n\nIt can be readily verified that the function $V(x,x')=\\sqrt{(x-x')^T(x-x')}$ satisfies the LMI condition (9) in \\cite{majid9} with $q=1$, $P_p=I_6$, and $\\widehat\\kappa_p=0.0076$, for any $p\\in\\{1,2,3\\}$. Hence, $V$ is a common $\\delta$-GAS-M$_1$ Lyapunov function for $\\Sigma$, satisfying conditions (i)-(iii) in Definition \\ref{delta_SGAS_Lya} with $q=1$, $\\underline\\alpha_p({r})=\\overline\\alpha_p({r})=r$, $\\forall r\\in{\\mathbb{R}}_0^+$, and $\\kappa_p=0.0038$, for any $p\\in\\{1,2,3\\}$. Using the results of Theorem \\ref{theorem2}, one gets that function $\\beta(r,s)=\\mathsf{e}^{-\\kappa_p{s}}r$ satisfies property \\eqref{delta_SGUAS} for $\\Sigma$.\n\nFor a \\emph{source state}\\footnote{Note that here we computed the source state as $x_s=\\arg\\min_{x\\in{\\mathbb{R}}^n}\\max_{p\\in\\mathsf{P}}V(\\overline\\xi_{xp}(\\tau),x)$ in order to have the smallest upper bound for $\\overline\\eta$ as in \\eqref{upper_bound}.} $x_s=[18,17.72,17.72,18,17.46,17.46]^T$, a given sampling time $\\tau=30$ time units, and a selected precision $\\varepsilon=1$, the parameter $N$ for $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma)$, based on inequality (\\ref{bisim_cond11}) in Theorem \\ref{main_theorem}, is obtained as 13 and one gets $\\overline\\eta\\leq0.1144$, where $\\overline\\eta$ is given in \\eqref{eta}. Therefore, the resulting cardinality of the set of states for $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma)$ is $3^{13}=1594323$.\n\nNow, consider that the objective is to design a control policy forcing the trajectories of $\\Sigma$, starting from the initial condition $x_0=[11.7,11.7,11.7,11.7,11.7,11.7]^T$, to reach the region $\\mathsf{D}=[19~22]^6$ in finite time and remain there forever. \nThis objective can be encoded via the LTL specification $\\Diamond\\Box\\mathsf{D}$. \n\nIn Figure \\ref{fig2}, we show several realizations of the trajectory $\\xi_{x_0\\upsilon}$ stemming from initial condition $x_0$ (top panels), \nas well as the corresponding evolution of synthesized switching signal $\\upsilon$ (bottom panel). \nFurthermore, in Figure \\ref{fig3}, we show the average value over 10000 experiments of the distance in time of the solution process $\\xi_{x_0\\upsilon}$ to the set $\\mathsf{D}$, namely $\\left\\Vert\\xi_{x_0\\upsilon}(t)\\right\\Vert_\\mathsf{D}$, where the point-to-set distance is defined as $\\Vert{x}\\Vert_\\mathsf{D}=\\inf_{d\\in{\\mathsf{D}}}\\Vert x-d\\Vert$.   \n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=12cm]{overal}\n\\end{center}\n\\caption{A few realizations of the solution process $\\xi_{x_0\\upsilon}$ (top panel) and the corresponding evolution of the obtained switching signal $\\upsilon$ (bottom panel).}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=10cm]{mean}\n\\end{center}\n\\caption{The average values (over 10000 experiments) of the distance of the solution process $\\xi_{x_0\\upsilon}$ to the set $\\mathsf{D}$ in different vertical scales.}\n\\label{fig3}\n\\end{figure}\n\nTo compute exactly the size of the symbolic model, proposed in Theorem \\ref{main_theorem1}, we consider the dynamics of $\\Sigma$ over the subset $\\mathsf{W}=[11.7~22]^6\\subset{\\mathbb{R}}^6$. Note that using the sampling time $\\tau=30$, the results in Theorem \\ref{main_theorem1} cannot be applied because the precision $\\varepsilon$ has to be lower bounded by $2.7$ as in inequality \\eqref{lower_bound}. Using a bigger precision $\\varepsilon=2.8$ than the one here, the same sampling time $\\tau=30$ as the one here, and the inequalities \\eqref{bisim_cond1} and \\eqref{bisim_cond}, we obtain the state space quantization parameter as $\\eta\\leq0.02$. Therefore, if one uses $\\eta=0.02$, the cardinality of the state set of the symbolic model $S_{\\mathsf{q}}(\\Sigma)$ is equal to $\\left(\\frac{22-11.7}{0.02}\\right)^6=1.8657\\times10^{16}$ which is much higher than the one of $\\overline S_{\\overline{{\\mathsf{q}}}}(\\Sigma)$, i.e. $1594323$, while having even larger precision.\n\n\\begin{remark}\nBy considering the dynamics of $\\Sigma$ over the set $\\mathsf{W}$, at least $1-10^{-5}$ confidence level, and using Hoeffding's inequality \\cite{hoeffding}, one can verify that the number of samples should be at least $74152$ to empirically compute the upper bound of $\\widehat\\eta$ in \\eqref{upper_bound4}. We compute $\\widehat\\eta\\leq0.1208$ when $x_s=[18~17.72~17.72~18~17.46~17.46]^T$, $N=13$, and $\\tau=30$. Using the results in Theorem \\ref{main_theorem3} and the same parameters $\\overline{\\mathsf{q}}$ as the ones in $\\overline{S}_{\\overline{\\mathsf{q}}}(\\Sigma)$, one obtains $\\varepsilon=0.6$ in \\eqref{bisim_cond3}. Therefore, $S_{\\overline{\\mathsf{q}}}(\\Sigma)$, with confidence at least $1-10^{-5}$, provides a less conservative precision than $\\overline{S}_{\\overline{\\mathsf{q}}}(\\Sigma)$, while having the same size as $\\overline{S}_{\\overline{\\mathsf{q}}}(\\Sigma)$. \n\\end{remark}\n\n\\begin{remark}\nAnother advantage of using the 2nd approach in comparison with the 1st one is that one can construct only a relevant part of the abstraction given an initial condition and the specification which was the case in this example. \n\\end{remark}\n\n\\subsection{Multiple Lyapunov functions}\n Consider the following stochastic switched system borrowed from \\cite{girard2} and additionally affected by noise:  \n\\begin{small}\n\\begin{align}\\nonumber\n\\Sigma:\n\\left\\{\n\\begin{array}{l}\n \\diff\\xi_1=\\left(-0.25\\xi_1+p\\xi_2+(-1)^p0.25\\right)\\diff{t}+0.01\\xi_1\\diff{W_t^1},\\\\\n \\diff\\xi_2=\\left(\\left(p-3\\right)\\xi_1-0.25\\xi_2+(-1)^p\\left(3-p\\right)\\right)\\diff{t}+0.01\\xi_2\\diff{W_t^2},\n \\end{array}\\right.\n\\end{align}\n\\end{small}where $p=1,2$. \nThe noise-free version of $\\Sigma$ is endowed with stable subsystems, \nhowever it can globally exhibit unstable behaviors for some switching signals \\cite{girard2}. \nSimilarly, $\\Sigma$ does not admit a common $\\delta$-GAS-M$_q$ Lyapunov function. \nWe are left with the option of seeking for multiple Lyapunov functions. \nIt can be indeed shown that each subsystem $\\Sigma_p$ admits a $\\delta$-GAS-M$_1$ Lyapunov function of the form \\begin{small}$V_p(x_1,x_2)=\\sqrt{(x_1-x_2)^TP_p(x_1-x_2)}$\\end{small}, with \\begin{small}$P_1=\\left[ {\\begin{array}{cc}\n2&0\\\\\n0&1\\\\\n \\end{array} } \\right]$\\end{small} and \\begin{small}$P_2=\\left[ {\\begin{array}{cc}\n1&0\\\\\n0&2\\\\\n \\end{array} } \\right].$\\end{small}\nThese $\\delta$-GAS-M$_1$ Lyapunov functions have the following characteristics: \n$\\underline\\alpha({r})=r$, $\\overline\\alpha({r})=2r$, $\\kappa=0.2498$. \nNote that $V^2_p(x_1,x_2)$ is also a $\\delta$-GAS-M$_2$ Lyapunov function for $\\Sigma_p$, where $p\\in\\{1,2\\}$, satisfying the requirements in Lemma \\ref{lem:moment est}. \nFurthermore, the assumptions of Theorem \\ref{multiple_lyapunov} hold by choosing a parameter $\\mu=\\sqrt{2}$ and a dwell time $\\tau_d=2>\\log{\\mu}\/\\kappa$. \nIn conclusion, the stochastic switched system $\\Sigma$ is $\\delta$-GUAS-M$_1$. \n \nLet us work within the set $\\mathsf{D}=[-5,~5]\\times[-4,~4]$ of the state space of $\\Sigma$. \nFor a sampling time $\\tau=0.5$, using inequality (\\ref{lower_bound_mul}) the precision $\\varepsilon$ is lower bounded by $1.07$. \nFor a chosen precision $\\varepsilon=1.2$, the discretization parameter $\\eta$ of $S_{{\\mathsf{q}}}(\\Sigma)$, \nobtained from Theorem \\ref{main_theorem2}, \nis equal to $0.0083$. \nThe resulting number of states in $S_{{\\mathsf{q}}}(\\Sigma_{\\tau_d})$ is $9310320$, taking 3.4 MB memory space, where the computation of the abstraction $S_{\\mathsf{q}}(\\Sigma_{\\tau_d})$ has been performed via the software tool \\textsf{CoSyMA}~\\cite{CoSyMA} on an iMac with CPU $3.5$GHz Intel Core i$7$.\nThe CPU time needed for computing the abstraction has amounted to $22$ seconds. \n \nConsider the objective to design a controller (switching signal) forcing the first moment of the trajectories of $\\Sigma$ to stay within $\\mathsf{D}$, \nwhile always avoiding the set $\\mathsf{Z}=[-1.5,1.5]\\times[-1,1]$. \nThis corresponds to the following LTL specification: $\\Box\\mathsf{D}\\wedge\\Box\\neg{\\mathsf{Z}}$. \nThe CPU time needed for synthesizing the controller has amounted to $12.46$ seconds. \nFigure \\ref{fig22} displays several realizations of the closed-loop trajectory of $\\xi_{x_0\\upsilon}$, \nstemming from the initial condition $x_0=(-4,-3.8)$ (left panel), \nas well as the corresponding evolution of the switching signal $\\upsilon$ (right panel). \nFurthermore, Figure \\ref{fig22} (middle panels) shows the average value (over 10000 experiments) of the distance in time of the solution process $\\xi_{x_0\\upsilon}$ to the set $\\mathsf{D}\\backslash{\\mathsf{Z}}$, namely $\\left\\Vert\\xi_{x_0\\upsilon}(t)\\right\\Vert_{\\mathsf{D}\\backslash{\\mathsf{Z}}}$. \nNotice that the empirical average distance is significantly lower than the theoretical precision $\\varepsilon=1.2$. \n\n\\begin{figure}\n\\includegraphics[width=13cm]{overal1}\n\\caption{Several realizations of the closed-loop trajectory $\\xi_{x_0\\upsilon}$ with initial condition \\mbox{$x_0=(-4,-3.8)$} (left panel). \nAverage values (over 10000 experiments) in time of the distance of solution process $\\xi_{x_0\\upsilon}$ to the set $\\mathsf{W}=\\mathsf{D}\\backslash{\\mathsf{Z}}$, in different vertical scales (middle panel). \nEvolution of the synthesized switching signal $\\upsilon$ (right panel).}\n\\label{fig22}\n\\end{figure}\n\nNote that using the same sampling time $\\tau=0.5$, the same precision $\\varepsilon=1.2$, and the inequalities \\eqref{bisim_cond_mul0} in Theorem \\ref{main_theorem33}, we obtain the temporal horizon as $N=22$. Therefore, the cardinality of the state set of the symbolic model $\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})$ is equal to $2^{22}=4194304$ which is roughly half of the one of $S_{{\\mathsf{q}}}(\\Sigma_{\\tau_d})$, i.e. $9310320$.\n\n\n\\section{Conclusions}\nThis work has shown that any stochastic switched system $\\Sigma$ (resp. $\\Sigma_{\\tau_d}$), admitting a common (multiple) $\\delta$-GAS-M$_q$ Lyapunov function(s), and within a compact set of states, admits an approximately bisimilar symbolic model $S_{\\mathsf{q}}(\\Sigma)$ (resp. $S_{\\mathsf{q}}(\\Sigma_{\\tau_d})$) requiring a space discretization or $S_{\\overline{{\\mathsf{q}}}}(\\Sigma)\/\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma)$ (resp. $S_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})\/\\overline{S}_{\\overline{{\\mathsf{q}}}}(\\Sigma_{\\tau_d})$) without any space discretization. Furthermore, we have provided a simple criterion by which one can choose between the two proposed abstraction approaches the most suitable one (based on the size of the abstraction) for a given stochastic switched system.\nThe constructed symbolic models can be used to synthesize controllers enforcing complex logic specifications, \nexpressed via linear temporal logic or as automata on infinite strings. \n\n\n\\section{Acknowledgements}\nThe authors would like to thank Ilya Tkachev for fruitful technical discussions. \n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nTeleoperational control allows the operator to complete tasks at a safe, remote location. This is accomplished by having a human operator send control signals from a remote console, traditionally known as \\textit{master}, to the robot performing the task, traditionally known as \\textit{slave}. The slave then returns feedback to the master through sensors, such as cameras and haptic devices, which gives the operator the feeling of \\textit{telepresence} through displays and force feedback. Through improvement of telepresence and robotic systems, teleoperational systems are becoming common place in a wide range of applications such as underwater exploration, space robotics, mobile robots, and telesurgery \\cite{TeleoperationHistory, yipDasJournal} . \n\nWith the advent of surgical robots such as the da Vinci\\textregistered{} Surgical System, attempts have been made to investigate the feasibility of remote telesurgery. Anvari et al. in 2005 was first to establish a remote telesurgical service and reported on 21 remote laparoscopic surgeries with an observed delay of 135-140 msec over the 400 km \\cite{firstTeleRobotic}. The signal latency grows as the distance increases, and prior reports have cited 300 msec as the maximum time delay where surgeons began to consider the operation unsafe \\cite{TransatlanticSurgery, LatencyEffects}. When using satellite communication between London and Toronto, a report measured a delay of $560.7 \\pm 16.5$ msec \\cite{reasonForDelay}. This unavoidable time delay is the greatest obstacle for safe and effective remote telesurgery because it leads to overshoot and oscillations. These undesirable effects have been observed with as little as 50 msec of time delay in surgical tasks \\cite{SteadyHand, TreatmentPlanning}. The challenges of teleoperating under delay has been investigated for over 50 years, and the field is too expansive to cover in this paper. A more broad look can be found in Hokayem and Spong's historical survey \\cite{TeleoperationHistory}.\n\nThe first study on the effects of delay in teleoperational system were by Ferrell in 1965 and found that experienced operators will use a strategy called move and wait \\cite{MoveAndWait}. A stable system is realized by inputting a new control command and then waiting to see the effect. Ferrell and Sheridan then developed supervisory control to address the problem of delay \\cite{supervisoryControl}. This gives a level of autonomy to the system such that the operator supervises rather than explicitly inputting trajectory motions. Therefore, supervisory control can only be implemented practically at this time in structured environments.\n\nIn the case of teleoperating with haptic feedback, also known as \\textit{bilateral} teleoperation, delay has been experimentally and theoretically shown to create instability within the system \\cite{experimentalInstability, theoryInstability}. Techniques such as wave variables have been used to dampen the unstable overshoot in bilateral control under delay \\cite{waveVariables}. However, they have been shown to increase time to complete task \\cite{yipBadHaptic, yipBadHaptic_2}, and therefore teleoperating without haptic feedback can often be a better alternative.\n\nPredictive displays circumvent the deleterious effects from teleoperating under delay by giving the operator immediate feedback through a virtual prediction. Early predictive displays focused on space robotics where delays can reach up to 7sec \\cite{spaceTeleopDelay2} \\cite{spaceTeleopDelay3}. Winck et al. recently applied this approach by creating an entire predicted virtual environment that is displayed to the operator and is also used to generate haptic feedback \\cite{spaceTeleopDelay}. Currently this is not applicable to telesurgery because of the unique 3D geometry found in a surgical environment. The scale of the operation requires high precision tracking to create reliable predictions, and obstacles such as tissue cannot be modeled as rigid bodies nor accurately predicted.\n\nIn this paper, we present new motion scaling solutions for conducting telesurgery under delay that:\n\\begin{enumerate}\n\t\\item reduce errors to improve patient safety,\n\t\\item have intuitive control since the delay can change from one operation to another due to variable distances, and\n\t\\item have simple implementation so it can be easily deployed to any teleoperated system.\n\\end{enumerate}\n\nTo show the performance of the solutions with statistical significance, we conducted a user study on the da Vinci\\textregistered{} Surgical System. The user study was designed such that there is independence from unwanted variables such as participant exhaustion and experience and that the best performing solution has a short learning curve, therefore intuitive. Both time to complete task and an error metric were used to evaluate task performance, and we measured that our proposed solutions decreased the number of errors at the cost of time. In fact, 16 out of the 17 participants from the user study performed best, with regards to error rate, using our proposed solutions.\n\n\\section{Methods} \n\nTeleoperation systems utilizes a scaling factor   between master arms movements and the slave arms movements to adapt the teleoperator to the slave workspace. It can be expressed as a general teleoperation system with delay, shown in Fig. \\ref{fig:flowChart}, as:\n\\begin{equation}\n\ts_m[n] = scale_m(m[n] - m[n-1]) + s_m[n-1]\n\\end{equation}\n\\begin{equation}\n\ts[n] = s_m[n-n_d\/2]\n\\end{equation}\nwhere $scale_m$ is the scaling factor previously described. It is worth noting that teleoperation systems described in this manner using position control, such as the da Vinci\\textregistered{} Surgical System, only apply this scaling to position and not orientation, rather mapping orientation commands one-to-one so there is no obvious mismatch between user wrist orientation and robot manipulator orientations. The solutions we present are all modifications of Equations (1) and (2).\n\n\\begin{figure}[b]\n\t\\centering\n\t\\includegraphics[width=0.49\\textwidth]{Teleop_Flowchart.png}\n\t\\caption{Flowchart of a teleoperation system with round trip delay of $d$ where $m$ is the masters position, $s_m$ is the target slave position from the master, $s$ is the slave position, $I$ is feedback to the operator. For the sake of simplicity, the entire system is assumed to have a sampling rate of $f_s$, and let $n_d = f_sd$ }\n\t\\label{fig:flowChart}\n\\end{figure}\n\n\\subsection{Constant Scaling Solution}\n\nThe first solution simply reduces the scaling factor in Equation (1). Lowering the scaling factor in robotic surgery has been shown to increase task completion time and improve accuracy \\cite{motionScaling1, motionScaling2}. We extend this to teleoperating under delay and hypothesize that decreasing the constant scale will reduce the number of errors at the cost of time when operating under delay. This idea is similar to the move and wait strategy, creating stability by slowing down the motions. However, here the motions are slowed continuously, while the move and wait strategy discretizes the motions and requires the operator to have experience working under the delay. \n\n\\subsection{Positional Scaling Solution}\n\nPositional scaling builds on the constant scaling solution by decreasing the scaling as the slave arms move towards an obstacle. Therefore, reducing the cost of time from the constant scaling solution by slowing down the operators motions only in areas of the work space that require higher precision. Since this deals with obstacle detection, the positional scaling will be implemented on the slave side through the following equations:\n\\begin{equation}\n\tscale_s = \\text{min}(maxscale, \\text{max}(minscale, k*r[n]))\n\\end{equation}\n\\begin{equation}\n\ts[n] = scale_s(s_m[n-n_d\/2] - s_m[n-n_d\/2-1]) + s[n-1]\n\\end{equation}\nwhere $r$ is the distance from $s_m$ to the nearest obstacle, $k$ is the rate $scale_s$ changes, and $minscale$ and $maxscale$ set the lower and upper bounds respectively of $scale_s$. This is paired with Equation (1), and can be thought of as secondary layer of scaling based on proximity to an obstacle.\n\nTo implement this scaling method, the distance $r$ requires the location of the obstacles in the environment. Methods such as feature-based 3D tissue reconstruction from \\cite{tissue3DReconstruction} or dense pixel-based depth reconstruction can be used. In the presented user study, the task environment is known and distances $r$ are found analytically. \n\n\\begin{figure*}[t]\n\t\\centering\n\t\\begin{subfigure}{.24\\textwidth}\n  \t\t\\centering\n  \t\t\\includegraphics[width=1\\linewidth]{example_error1.png}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.24\\textwidth}\n\t\t\\centering\n  \t\t\\includegraphics[width=1\\linewidth]{example_error2.png}\n\t\\end{subfigure}\n\n\t\\begin{subfigure}{.24\\textwidth}\n  \t\t\\centering\n  \t\t\\includegraphics[width=1\\linewidth]{example_error3.png}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.24\\textwidth}\n\t\t\\centering\n  \t\t\\includegraphics[width=1\\linewidth]{example_error4.png}\n\t\\end{subfigure}\n\t\\caption{Photos of example errors from the user study. From left to right the errors are: stretch ring on peg, touch peg, drop ring, and stretch ring during handoff.}\n\t\\label{fig:sampleErrors}\n\\end{figure*}\n\n\\subsection{Velocity Scaling Solution}\n\nVelocity scaling is founded on the idea that an operator will naturally move slower when higher accuracy is required. This is inspired by the software controlling computer mice and trackpads, which natively utilize this strategy by default (unless turned off by the owner). Combining velocity scaling with the constant scaling solutions hypothesis, the scaling factor in Equation (1) is increased proportionally to the input translational velocity as shown:\n\\begin{equation}\n\tscale_m = v_1 + v_2 \\big| \\big| \\dot{m}_m[n] \\big| \\big|\n\\end{equation}\nwhere $v_1$ is the base scaling, the bonus velocity scaling is given at the rate of $v_2$, $\\dot{m}$ is the translational velocity, and $|| \\cdot ||$ is the magnitude. Equation (2) is then used to set the slave position on the slave side. This allows for a low base scaling to get the benefits of the constant scaling solution when the operator is making small, precise motions. It also increases the scaling when the operator makes large motions, which requires less precision, in order to reduce the cost of time.\n\nTo find $\\dot{m}$ on a real system where there is noise, the following filter can be used: \n\\begin{equation}\n\t\\dot{m}_m[n] = \\frac{m[n] + m[n-1] - m[n-2] - m[n-3]}{4\/f_s} \n\\end{equation}\\\\\nThis represents a running average with weights 0.25, 0.5, and 0.25 for three individual velocity measurements.\n\n\\section{Experiment}\n\nTo measure the effectiveness of the solutions, a user study with 17 participants was conducted on the da Vinci\\textregistered{} Surgical System. The delayed feedback to the user is a stereoscopic 1080p laparoscopic camera running at 30FPS. The camera feed is displayed to a console for stereo viewing by the operator. Both pairs of master and slave arms have seven degrees of freedom. A modified version of Open-Source da Vinci Research Kit \\cite{DVRK} was used on an a computer with an Intel\\textregistered{} Core\\texttrademark{} i9-7940X Processor and NVIDIA's GeForce GTX 1060.\n\n\nTo ensure that the solutions are tested in a realistic, high delay condition, a round trip delay of 750 msec was simulated for the delayed environment in the user study. This follows a  recent report showing satellite communication between London and Toronto measured a round trip delay of $560.7 \\pm 16.5$ msec for a telesurgical task \\cite{reasonForDelay}. \n\n\\subsection{Task}\nA peg transfer task is used as a test scenario. Fig. \\ref{fig:taskEnv} shows a photo from the endoscope used by the operator in our study. The task involves:\n\n\\begin{enumerate}\n\\item Lifting a rubber o-ring from either the front left or right peg with the corresponding left or right arm\n\\item Passing the ring to the other arm\n\\item Placing the ring on the other front peg\n\\item Repeating steps 1 to 3 for the back pair of pegs\n\\end{enumerate}\n\n\\begin{figure}[b]\n\t\\centering\n\t\\includegraphics[width=0.48\\textwidth]{TaskEnvironment.png}\n\t\\caption{Task environment for the user study as seen from the endoscope. The scale of the environment ensures that participants must be accurate in order to perform well with regards to errors.}\n\t\\label{fig:taskEnv}\n\\end{figure}\n\nThis task and environment was chosen because it inherits complex motions such as hand off, regions where larger movements are safe, and yet still requires precise motions during parts of the trajectory due to the tightness of fit between the rings and the pegs.\n\n\\subsection{Metrics}\n\nThe metrics to evaluate task performance are time to complete task and an error metric. The error metric is an enumeration of different types of errors that we observed. These errors are weighted according to Table 1 and summed to get a weighted error. The weights were chosen such that severity of the error would be reflected properly in the error metric. Example errors are shown in Fig. \\ref{fig:sampleErrors}. A sample video was created and shown to the participants before the study to show the task and the different types of errors.\n\n\\begin{table}[h]\n\\caption{\\\\Weights associated with type of error}\n\\begin{center}\n\\begin{tabular}{ |c|c| } \n\\hline  \n\t\\textbf{Error} & \\textbf{Weight}\t\t\t\t\t\t\t\t\\\\   \\hline\n\tTouch peg & 1\t\t\t\t \t\t\t\t\t\t\t\\\\   \\hline\n\tTouch ground & 2\t\t\t\t\t \t\t\t\t\t\\\\   \\hline\n\tStretch ring during hand-off for a second or less & 2 \t\t\t\\\\   \\hline\n\tDrop ring & 3\t\t\t\t\t \t\t\t\t\t\t\\\\   \\hline\n\tStretch ring on peg for a second or less & 4\t\t\t\t\t\\\\   \\hline\n\tStretch ring for an additional second & 4\t\t\t\t\t \t\\\\   \\hline\n\tStretch\/move peg & 10\t\t\t\t\t \t\t\t\t\\\\   \\hline\n\tKnock down peg & 20\t\t\t\t\t \t\t\t\t\\\\   \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsection{Procedure}\nThe scaling scenarios for the participants to complete the peg transfer task are as follows in both 0  msec and 750  msec round trip delay:\n\\begin{enumerate}\n\t\\item Constant scaling of 0.3\n\t\\item Constant scaling of 0.2\n\t\\item Constant scaling of 0.1\n\t\\item Position scaling\n\t\\item Velocity scaling\n\\end{enumerate}\n\nFor constant scaling, the corresponding scaling value listed above was used for $scale_m$. For positional scaling, the following simplification was made: $r$ is the minimum 2D distance from center of each of the 4 pegs to the target tool-tip position projected on the plane constructed from the top of the 4 pegs ($ax + by + z = c$). The following equations are used to find the least squared solution of the plane:\n\n\\begin{equation}\n    A = \n\\begin{bmatrix}\nx_1 & y_1 & -1 \\\\\nx_2 & y_2 & -1 \\\\\nx_3 & y_3 & -1 \\\\\nx_4 & y_4 & -1 \\\\\n\\end{bmatrix}\n\\end{equation}\n\\begin{equation}\n    \\begin{bmatrix}\n        a & b & c\n    \\end{bmatrix}^T\n    = -(A^T A)^{-1}A^T\n    \\begin{bmatrix}\n        z_1 & z_2 & z_3 & z_4\n    \\end{bmatrix}^T\n\\end{equation}\nwhere $\\begin{bmatrix} x_i & y_i & z_i \\end{bmatrix}^T$ is the position of the center of the $i$-th peg. The following is computed realtime to project the target tool-tip position and find $r$:\n\\begin{equation}\n    e = \\frac{\\begin{bmatrix} a & b & 1 \\end{bmatrix}^T}{\\big|\\big|\\begin{bmatrix} a & b & 1 \\end{bmatrix}^T\\big|\\big|}\n\\end{equation}\n\\begin{equation}\n    s_p[n] = s_m[n-n_d\/2] - \\begin{bmatrix} 0 & 0 & c \\end{bmatrix}^T - \\text{ proj}_e s_m[n-n_d\/2]\n\\end{equation}\n\\begin{equation}\n    r[n] = \\text{ }\\underset{i}{\\operatorname{min}}\\text{ } \\big|\\big| s_p[n] - \\begin{bmatrix} x_i & y_i & z_i \\end{bmatrix}^T \\big|\\big|\n\\end{equation}\n\nThrough experimentation, $scale_m = 0.2$, $k = 100$, $minscale= 0.5$, and $maxscale = 1.0$ were chosen for positional scaling. Fig. \\ref{fig:positionScalingPlot} shows a generated map of the value of $scale_s$ using these parameters. $scale_s$ is calculated and applied to both arms individually. These positional scaling parameters mean that the total scaling from master to slave ranges from 0.1 to 0.2.\n\n\\begin{figure}[b]\n\t\\centering\n\t\\includegraphics[width=0.48\\textwidth]{pos_scale_heat_map.png}\n\t\\caption{Map showing the value of $scale_s$ for the positional scaling used in the user study. This map is from a bird-eye view of the pegs which are located at: (30, 40), (30, 60), (70, 40), and (70, 60). }\n\t\\label{fig:positionScalingPlot}\n\\end{figure}\n\nFor velocity scaling, the values of $v_1 = 0.1$ and $v_2 = 100$ were found through experimentation. Similar to positional scaling, this is calculated and applied to both arms individually.\n\nThe procedure for each participant was as follows:\n\\begin{enumerate}\n\t\\item Practice: repeat the task twice with constant scaling of 0.2 under no delay\n\t\\item Record: perform the task with constant scaling of 0.2 under no delay\n\t\\item Practice: select a new scaling method at random under no delay and take 20 seconds to become accustomed to the new scenario\n\t\\item Record: perform the task with the previously selected scaling solution\n\t\\item Repeat step 3 and 4 with all scaling solutions \n\t\\item Repeat step 3 and 4 with constant scaling of 0.2 \n\t\\item Repeat 1-6 with round trip delay of 750 msec.\n\\end{enumerate}\n\nStep 1 is so that participants get over the learning curve of the task and the system. By comparing the results of step 2 and 6, we can ensure that the user overcame the learning curve in step 1 and has not degraded in performance due to exhaustion. This is critical since it shows that the performance of a participant is only affected by the delay and different scaling methods. To further break any potential correlation, the randomization of the order of scaling methods in step 3 and 4 was used. Each participant has only 20 seconds to practice in the new scenario (step 3), so that the solutions with best results will be intuitive.\n\n\n\\section{Results}\n\nTo show there is no significant change in performance between step 2 and 6 of the procedure, the results are compared using two-sided paired-sample t-tests. Under no delay, $p = 0.670$ and $p = 0.054$ for weighted error and time to complete task respectively is computed. For the roundtrip delay of 750 msec, $p = 0.633$ and $p = 0.192$ for weighted error and time to complete task respectively is computed. Therefore, no statistically significant ($p<0.05$) change in performance was measured. Combining this result with the randomization of order in step 3 of the procedure implies that the measured results will be uncorrelated with participant experience and exhaustion and only affected by the delay and scaling solution.\n\nThe results showing which solution performed best and second best for each individual person under delay are shown in Fig. \\ref{fig:bestScaling}. Only one out of seventeen participants performed best, with regards to weighted error, using the baseline scaling, constant scaling of 0.2 and 0.3. Furthermore, from Fig. \\ref{fig:bestScaling} it is evident that even when including the second best performing solution, the vast majority of participants performed better with our proposed solutions with regards to weighted error. \n\nThe statistics of the results are shown in Table 2 and 3 for roundtrip delay of 750 msec and 0sec respectively. The $p$-values are generated by comparing against the baseline scaling with a two-sided paired t-test to show if the solution had a statistically significant ($p < 0.05$) effect on performance. The box plots of the results are shown at the end of the paper in Fig. \\ref{fig:allDataPlots}. All the proposed solutions do achieve statistically significant lower weighted error under delay except for positional scaling compared against constant scaling of 0.2. The proposed solutions were also measured to make a statistically significant increase in time under delay when comparing against constant scaling of 0.2. Therefore, the solutions are working as intended, increasing the accuracy at the cost of time. \n\n\\begin{figure*}[t]\n\t\\centering\n\t\\begin{subfigure}{.4\\textwidth}\n  \t\t\\centering\n  \t\t\\includegraphics[width=1\\linewidth]{best_scaling_participant_error.png}\n\t\\end{subfigure}\n\t\\qquad\n\t\\qquad\n\t\\begin{subfigure}{.4\\textwidth}\n\t\t\\centering\n  \t\t\\includegraphics[width=1\\linewidth]{best_scaling_participant_time.png}\n\t\\end{subfigure}\n\t\\caption{Counts of participants for which scaling solution performed best and second best from the user study under a roundtrip delay of 750 msec. Left is weighted error and right is time to compete task.}\n\t\\label{fig:bestScaling}\n\\end{figure*}\n\n\\begin{table*}[t]\n\\centering\n\\caption{\\\\Statistics from user study trials under a roundtrip delay of 750 msec. $p$-value is generated from two-sided paired t-test.}\n\\def1.25{1.25}\n\\setlength\\tabcolsep{1.25em}\n\\begin{tabular}{c|ccc|ccc}\n      & \\multicolumn{3}{c|}{\\textbf{Weighted Error}} & \\multicolumn{3}{c}{\\textbf{Time (sec)}} \\\\ \n         & $mean \\pm std$    & $p$-value       & $p$-value       & $mean \\pm std$    & $p$-value       & $p$-value \\\\ \n                &                   & vs. c = 0.3   & vs. c = 0.2    &                   & vs. c = 0.3   & vs. c = 0.2    \\\\\\hline\n    \\textbf{c = 0.3} & 16.4 $\\pm$ 13.3 & --- & 0.305 & 106 $\\pm$ 32.0 & --- & 0.224 \\\\\n    \\textbf{c = 0.2} & 13.2 $\\pm$ 8.22 & 0.305 & --- & 92.5 $\\pm$ 31.8 & 0.224 & --- \\\\\n    \\textbf{c = 0.1} & 9.29 $\\pm$ 9.62 & 0.0149 & 0.0480 & 131 $\\pm$ 42.8 & 0.000316 & $<0.0001$ \\\\\n    \\textbf{pos.} & 12.3 $\\pm$ 14.6 & 0.0245 & 0.572 & 116 $\\pm$ 38.0 & 0.173 & 0.0474 \\\\\n    \\textbf{vel.} & 9.31 $\\pm$ 8.46 & 0.00926 & 0.00114 & 113 $\\pm$ 35.1 & 0.284 & 0.00617 \\\\\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}[t]\n\\centering\n\\caption{\\\\Statistics from user study trials under a roundtrip delay of 0sec. $p$-value is generated from two-sided paired t-test.}\n\\def1.25{1.25}\n\\setlength\\tabcolsep{1.25em}\n\\begin{tabular}{c|ccc|ccc}\n      & \\multicolumn{3}{c|}{\\textbf{Weighted Error}} & \\multicolumn{3}{c}{\\textbf{Time (sec)}} \\\\ \n         & $mean \\pm std$    & $p$-value       & $p$-value       & $mean \\pm std$    & $p$-value       & $p$-value \\\\ \n                &                   & vs. c = 0.3   & vs. c = 0.2    &                   & vs. c = 0.3   & vs. c = 0.2    \\\\\\hline\n    \\textbf{c = 0.3} & 6.00 $\\pm$ 5.60 & --- & 0.266 & 40.3 $\\pm$ 13.7 & --- & 0.320 \\\\\n    \\textbf{c = 0.2} & 4.47 $\\pm$ 3.40 & 0.266 & --- & 44.3 $\\pm$ 17.6 & 0.320 & --- \\\\\n    \\textbf{c = 0.1} & 5.35 $\\pm$ 5.73 & 0.686 & 0.486 & 72.7 $\\pm$ 27.9 & $<0.0001$ & $<0.0001$ \\\\\n    \\textbf{pos.} & 6.12 $\\pm$ 5.01 & 0.833 & 0.243 & 60.0 $\\pm$ 20.5 & 0.000246 & 0.000100 \\\\\n    \\textbf{vel.} & 4.76 $\\pm$ 4.49 & 0.259 & 0.784 & 53.2 $\\pm$ 24.5 & 0.0123 & 0.0965 \\\\\n\\end{tabular}\n\\end{table*}\n\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\begin{subfigure}{0.4\\textwidth}\n            \\centering\n            \\includegraphics[width=\\linewidth]{box_plot_error_results_d0.png}\n        \\end{subfigure} \n        \\qquad\\qquad\n        \\begin{subfigure}{0.4\\textwidth}  \n            \\centering \n            \\includegraphics[width=\\linewidth]{box_plot_error_results_d750.png}\n        \\end{subfigure}\n        \\vskip\\baselineskip\n        \\begin{subfigure}{0.4\\textwidth}   \n            \\centering \n            \\includegraphics[width=\\linewidth]{box_plot_time_results_d0.png}\n        \\end{subfigure}\n        \\qquad\\qquad\n        \\begin{subfigure}{0.4\\textwidth}   \n            \\centering \n            \\includegraphics[width=\\linewidth]{box_plot_time_results_d750.png}\n        \\end{subfigure}%\n        \\caption{Performance results of all tested scaling methods from the user study. Plots on the left are results from no delay and plots on the right are results with a roundtrip delay of 750 msec.} \n       \n        \\label{fig:allDataPlots}\n\\end{figure*} \n\n\\section{Discussion}\n\nThe user study results of constant scaling of 0.1 under delay match with our initial hypothesis that decreasing constant scaling slows down the operators motions to achieve higher accuracy. As seen in Table 2, constant scaling of 0.1 makes a statistically significant decrease in errors at the cost of increasing the time to complete task. However, the hypothesis does not appear to hold true when looking at the performance of constant scaling of 0.3 to 0.2 under delay. We believe this is since participants had so many errors, such as ring drops, when using constant scaling of 0.3, that it severely affected their time to complete the task.\n\nBy comparing positional scaling and constant scaling of 0.1 relative time to complete task performances under delay, it is evident that positional scaling did decrease the cost of time as intended via dynamic scaling. However, positional scaling did not retain the weighted error performance of constant scaling of 0.1 when under delay. We believe this occurs because our implementation of positional scaling does not count the slave-arms as obstacles. Therefore during the ring pass, which typically occurs in the center of the plot in Fig. \\ref{fig:positionScalingPlot}, there is little to no additional scaling to improve the accuracy. The data from the user study supports this since 2 ring stretches and 7 ring drops during hand off were recorded for positional scaling under delay, and 0 ring stretches and 2 ring drops during hand off were recorded for constant scaling of 0.1 under delay. \n\nThe final result is the performance of velocity scaling, which performed the best of all of the proposed solutions under delay. It has similar performance in weighted error as constant scaling of 0.1 at a lower cost of time. While under delay, velocity scaling on average reduced the weighted error rate by 43\\% and 29\\% and increased the time to complete task by no statistically significant margin and 22\\% when compared against constant scaling of 0.3 and 0.2 respectively.\n\n\\section{Conclusion}\n\nThe results from the user study confirms with statistical significance our original hypothesis which states that decreasing the constant scaling factor will improve accuracy at the cost of time. Velocity scaling was shown to improve on this by achieving the same accuracy with a lower cost of time. Positional scaling did not perform as well, and it has the added challenge of requiring environmental information. Participants were also given little time to adjust to the proposed scaling solutions, so the performance improvements implies that they are intuitive. Furthermore, the simplicity of the constant scaling and velocity scaling solutions allows for them to be easily deployed on any teleoperational system under delay. Future work involves dynamically learning the appropriate scales for individual operators and delay conditions.\n\n\\section{Acknowledgements}\nWe thank all the participants from the user study for their time, all the members of the Advanced Robotics and Controls Lab at University of California San Diego for the intellectual discussions and technical help, and the UCSD Galvanizing Engineering in Medicine (GEM) program and the US Army AMEDD Advanced Medical Technology Initiative (AAMTI) for the funding.\n\\balance\n\\bibliographystyle{ieeetr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nKnowledge graphs (KGs) possess a large number of facts about the real world in a structured manner. \nEach fact consists of two entities and their relation in the form of a triple like \\textit{(subject entity, relation, object entity)}. \nKGs are becoming increasingly important because they can simplify knowledge acquisition, storage, and reasoning, and successfully support many knowledge-powered tasks such as semantic search and recommendation. \nAs different KGs are constructed by different people for different purposes using different data sources,\nthey are far from complete, yet complementary to each other. \nKnowledge from a single KG usually cannot satisfy the requirement of downstream tasks, calling for the integration of multiple KGs for knowledge fusion.\nEntity alignment, which aims to identify equivalent entities from different KGs, is a fundamental task in KG fusion.\nIn recent years, there has been an increasing interest in embedding-based entity alignment \\cite{chen2017multilingual,wang2018cross,cao2019multi,sun2020knowledge}. \nIt encodes entities into vector space for similarity calculation, such that entity alignment can be achieved via nearest neighbor search.\nHowever, current embedding-based models still face two major challenges regarding alignment learning and inference.\n\n\\textbf{Challenge 1:} How to effectively represent entities in the alignment learning phase?\nExisting embedding-based methods do not fully explore the inherent structures of KGs for entity embedding. \nMost of them only utilize a one-sided view of the structures, such as relation triples \\cite{chen2017multilingual,sun2017cross}, paths \\cite{zhu2017iterative,guo2019learning}, \nand neighborhood subgraphs \\cite{wang2018cross, sun2020knowledge, cao2019multi, wu2019relation, wu2019jointly, li2019semi, wu2020neighborhood, xu2019cross, yang2019aligning}.\nIntuitively, all these multi-context views could be helpful for embedding learning and provide additional evidence for entity alignment.\nMulti-view learning \\cite{zhang2019multi} is not the best solution to aggregate these features as it requires manual feature selection to determine useful views. \nBesides, KGs are heterogeneous, indicating that not all features in a helpful view could contribute to alignment learning, such as the noisy neighbors in neighborhood subgraphs \\cite{sun2020knowledge}. \nHence, it is necessary to flexibly and selectively incorporate different types of structural contexts for entity embedding.\n\n\\textbf{Challenge 2:} How to make full use of the learned embeddings and achieve informed alignment learning? Existing models directly rely on entity embeddings to estimate their semantic similarity, which is used for alignment decisions while ignoring the correlation between entities and relations in the KG. Although some studies \\cite{wu2019jointly, zhu2021relation} have tried to conduct joint training of entity and relation to improve the quality of the learned embeddings, alignment of each entity is still conducted independently from a local perspective. In fact, entity alignment is a holistic process: the neighboring entities and relations of the aligned pair should also be equivalent. The methods \\cite{zeng2020collective,zeng2021reinforcement} conduct entity alignment from a global perspective by modelling it as a stable matching problem. Whereas, such reasoning is regarded as a post-processing step without utilizing the holistic information to inform the embedding learning module. We argue that holistic inference can provide crucial clues to achieve reliable entity alignment, and the inferred knowledge can offer reliable alignment evidence and help to build and strengthen the trust towards black-box embedding-based entity alignment.\n\n\\textbf{Contributions:} In this work, we introduce \\textbf{IMEA\\xspace}, an \\textbf{I}nformed \\textbf{M}ulti-context \\textbf{E}ntity \\textbf{A}lignment model, to tackle the above obstacles.\nMotivated by the strong expressiveness and representation learning ability of Transformer \\cite{vaswani2017attention}, \nwe design a novel Transformer architecture to encode the multiple structural contexts in a KG while capturing the deep alignment interaction across different KGs.\nSpecifically, we apply two Transformers to encode the neighborhood subgraphs and entity paths, respectively.\nTo capture the semantics of relation triples, we introduce relation regularization based on TransE \\cite{bordes2013translating} at the input layer.\nWe generate entity paths by random walks across two KGs. \nTo capture the deep alignment interactions, we replace the entity in a path with its counterpart in the seed alignment (i.e., training data) to generate a new semantically equivalent path.\nIn this way, a path may contain entities from the two KGs that remain to be aligned, \nand the Transformer is trained to fit such a cross-KG entity sequence and its equivalent path.\nThe two Transformers share parameters.\nTheir training objectives are central entity prediction and mask entity prediction, respectively.\nThe self-attention mechanism of Transformer can recognize and highlight helpful features in an end-to-end manner. \n\nFurthermore, we design a holistic reasoning process for entity alignment, to complement the imperfection of embedding similarities.\nIt compares entities by reasoning over their neighborhood subgraphs based on both embedding similarities and the holistic relation and entity functionality (i.e., the importance for identifying similar entities and relations).\nThe inferred alignment probabilities are injected back into the Transformer as soft labels to assist alignment learning. In this way, both the positive and negative alignment evidence obtained from the holistic inference can be utilized to adjust soft labels and guide embedding learning, thus improving the overall performance and reliability of entity alignment.\n\nWe conduct extensive experiments including ablation studies and case studies on the benchmark dataset OpenEA \\cite{sun13benchmarking} to evaluate the effectiveness of our IMEA\\xspace model. Experimental results demonstrate that IMEA\\xspace can achieve the state-of-the-art performance by combining multi-context Transformer and holistic reasoning,\nand the global evidence is quite effective in informing alignment learning.\n\n\\section{Related Work}\n\\textbf{KG Contexts for Entity Alignment.} \nExisting structure-based entity alignment methods utilize different types of structural contexts to generate representative KG embeddings. \nA line of studies learns from \\textit{relation triples} and encodes local semantic information of both entities and relations. \nTransE \\cite{bordes2013translating} is one of the most fundamental embedding techniques that has been adopted in a series of entity alignment models such as MTransE \\cite{chen2017multilingual}, JAPE \\cite{sun2017cross}, BootEA \\cite{sun2018bootstrapping} and SEA \\cite{pei2019semi}.\nInspired by the success of graph neural networks (GNNs) like GCN \\cite{kipf2016semi} and GAT \\cite{velickovic2018graph} in graph embedding learning,\nrecent entity alignment studies incorporate various GNN variants to encode \\textit{subgraph contexts} for neighborhood-aware KG embeddings. \nFor example, GCN-Align \\cite{wang2018cross} employs vanilla GCN for KG structure modeling, \nbut suffers from the heterogeneity of different KGs. \nTo address this issue, MuGNN \\cite{cao2019multi} preprocesses KGs by KG completion and applies an attention mechanism to highlight salient neighbors.\nOther work improves neighborhood-aware entity alignment by introducing relation-aware GNNs \\cite{li2019semi,ye2019vectorized,yu2020generalized,mao2020mraea,mao2020relational}, multi-hop GNNs \\cite{sun2020knowledge,mao2021boosting} and hyperbolic GNNs \\cite{sun2020knowledge}.\nBesides the triple and neighbourhood context, \nRSN4EA \\cite{guo2019learning} and IPTransE \\cite{zhu2017iterative} further explore long-term dependency along \\textit{relation paths} in KGs. \nThere are also some entity alignment models that consider additional information such as \\textit{entity names} \\cite{xu2019cross,wu2019relation,wu2019jointly,wu2020neighborhood}, \\textit{entity attributes} \\cite{trisedya2019entity,zhang2019multi,liu2020exploring} and \\textit{textual descriptions} \\cite{chen2018co}.\nAlthough structural contexts are abundant and always available in KGs, few existing EA models have tried to integrate them into the embedding learning process. In this work, we introduce Transformer to capture multi-context interactions and learn more comprehensive entity embeddings.\n\n\\noindent\\textbf{Collective Entity Alignment.}\nEntities are usually aligned independently in existing EA models. Only two methods have attempted to achieve collective entity alignment.\nCEAFF \\cite{zeng2020collective, zeng2021reinforcement} formulates alignment inference as a stable matching problem. \nRMN \\cite{zhu2021relation} designs an iterative framework to leverage positive interactions between entity and relation alignment. However, it can only capture the interactions from seed entity alignment and requires extra pre-aligned relations, which is not always feasible in practice. Furthermore, both methods treat alignment learning and alignment inference separately, e.g., by regarding holistic reasoning as a post-processing step to adjust the alignment results \\cite{zeng2020collective}. By contrast, our IMEA\\xspace model can inject the holistic knowledge obtained during alignment inference into the Transformer via soft label editing, so as to guide the embedding learning process.\n\n\\section{Methodology}\n\\subsection{Framework}\nWe define a KG as $\\mathcal{G}=(\\mathcal{E},\\mathcal{R},\\mathcal{T})$, \nwhere $\\mathcal{E}$ and $\\mathcal{R}$ denote the sets of entities and relations, respectively, and $\\mathcal{T}\\subset\\mathcal{E}\\times\\mathcal{R}\\times\\mathcal{E}$ is the set of relation triples. Each triple can be represented as $(s, r, o)$, where $s\\in\\mathcal{E}$ and $o\\in\\mathcal{E}$ denote the subject (head entity) and object (tail entity), respectively, and $r\\in\\mathcal{R}$ is the relation.\nGiven two heterogeneous KGs, i.e., a source KG $\\mathcal{G}$ and a target KG $\\mathcal{G}'$, entity alignment (EA) aims to find as many aligned entity pairs as possible $\\mathcal{A} = \\{(e, e') \\in \\mathcal{E} \\times \\mathcal{E}'| e \\equiv e'\\}$, \nwhere the equivalent relationship $\\equiv$ indicates that entities $e$ and $e'$ from different KGs represent the same real-world identity. \nTypically, to jump-start alignment learning, some seed entity alignment $\\mathcal{A}^s$ is provided as training data.\n\n\\begin{figure*}[!t]\n    \\centering\n    \\includegraphics[width=0.68\\linewidth]{image\/framework.pdf}\n    \\caption{\\label{fig:framework.pdf}Framework of the proposed informed multi-context entity alignment model.}\n\\end{figure*}\n\nFigure \\ref{fig:framework.pdf} shows the overall framework of our IMEA\\xspace (Informed Multi-context Entity Alignment) model, which contains three main components:\nFirstly, we introduce Transformer as the contextualized encoder to capture the multiple structural contexts (i.e., triples, neighborhood subgraphs and paths) and generate representative KG embeddings for alignment learning. \nNext, we identify candidate target entities for each source entity based on their embedding similarity and perform holistic reasoning for each candidate entity pair. \nFinally, we inform the embedding learning module by injecting the alignment evidence into the training loss using our proposed soft label editing strategy. \nWe will introduce each component in detail in the following sections.\n\n\\subsection{Multi-context Transformer}\nFigure \\ref{fig:transformer} presents the architecture of our multi-context transformer. \nSpecifically, we apply two Transformers to encode the neighborhood context and path context, respectively.\nThe neighborhood context is generated by sampling neighboring entities. \nTo capture the relation semantics, we introduce a relation regularization based on the translational embedding function \\cite{bordes2013translating} at the input layer.\nThe path context is represented as an entity sequence generated by random walks. \nTo capture the deep alignment interactions across different KGs, \nwe replace the entity in a path with its counterpart in the seed entity alignment to generate a new semantically equivalent path.\nIn this way, a path may contain entities from the two KGs that remain to be aligned, \nand the Transformer seeks to fit such a cross-KG entity sequence and its equivalent path to capture the alignment information between two KGs.\n\n\\subsubsection{Intra-KG Neighbourhood Encoding}\nNeighborhood information is critical in representing KG structures for entity alignment, as similar entities usually have similar neighborhood subgraphs. \nConsidering that an entity may have a very large number of neighbors, \nwe repeatedly sample $n$ neighbors of each entity during training and feed these neighbors into Transformer to capture their interactions. \nSpecifically, given an entity $e$ and its sampled neighbours $(e_1, e_2,..., e_n)$, \nwe regard the neighbourhood as a sequence and feed it into a Transformer with $L$ self-attention layers stacked. \nThe input embeddings of Transformer are $(\\mathbf{h}_{e_1}^0, \\mathbf{h}_{e_2}^0, \\cdots, \\mathbf{h}_{e_n}^0)$, \nand the output of the $L$-th layer is generated as follows:\n\\begin{equation}\n    \\mathbf{h}_{e_1}^L, \\mathbf{h}_{e_2}^L, \\cdots, \\mathbf{h}_{e_n}^L = \\text{Encoder}(\\mathbf{h}_{e_1}^0, \\mathbf{h}_{e_2}^0, \\cdots, \\mathbf{h}_{e_n}^0).\n\\end{equation}\nThe training objective is to predict the central entity $e$ with the input neighborhood context $\\mathbf{c}_e=\\text{MeanPooling}(\\mathbf{h}_{e_1}^L,\\mathbf{h}_{e_2}^L,\\cdots,\\mathbf{h}_{e_n}^L)$.\nThis can be regarded as a classification problem.\nThe prediction probability for each $e_i$ is given by\n\\begin{equation}\n    \\label{eq:ea_pro}\n    p_{e_i} = \\frac{\\sigma(\\mathbf{c}_e,\\mathbf{h}_{e_i}^0)}{\\sum_{e_j \\in \\mathcal{E}}\\sigma(\\mathbf{c}_e,\\mathbf{h}_{e_j}^0)},\n\\end{equation}\nwhere $\\sigma()$ denotes inner product.\nLet $\\mathbf{q}_{e}$ be the one-hot label vector for predicting entity $e$. \nFor example, $\\mathbf{q}_{e}=[1,0,0,0,0]$ means that the entity at the first position is the target.\nWe adopt the label smoothing strategy to get soft labels, e.g., $\\hat{\\mathbf{q}}_{e}=[0.8,0.05,0.05,0.05, 0.05]$,\nand use cross-entropy, i.e., $\\text{CE}(x,y)=-\\sum x \\log y$, to calculate the loss between the soft label vector and the prediction distribution over all candidates $\\mathbf{p}_{e}$. \nFormally, the overall loss is defined as\n\\begin{equation}\n    \\label{eq:neigh_loss}\n    \\mathcal{L}_\\text{neighbor} = \\sum_{e\\in \\mathcal{E}\\cup\\mathcal{E}'} \\text{CE}(\\hat{\\mathbf{q}}_{e}, \\mathbf{p}_{e}).\n\\end{equation}\n\n\\subsubsection{Cross-KG Path Encoding}\nTo capture the alignment information, we use seed entity alignment to bridge two KGs and perform random walks to generate cross-KG paths.\nFor example, a typical path of length $m$ can be denoted as $(e_1, e_2',...,e_m)$.\nThen we randomly mask an entity in the path and get $(e_1, [MASK],...,e_m)$.\nSimilar to neighborhood encoding, we feed the masked path into Transformer to get its representations:\n\\begin{equation}\n    \\mathbf{h}_{e_1}^L, \\mathbf{h}_{[MASK]}^L, \\cdots, \\mathbf{h}_{e_m}^L = \\text{Encoder}(\\mathbf{h}_{e_1}^0, \\mathbf{h}_{[MASK]}^0, \\cdots, \\mathbf{h}_{e_m}^0).\n\\end{equation}\nThe output representation of $[MASK]$ is used to predict the masked entity $e_2'$. \nHere we use $\\mathbf{h}_{[MASK]}^L$ to replace $\\mathbf{c}_e$ in Eq. (\\ref{eq:ea_pro}) to calculate the prediction probability.\nSince the $[MASK]$ representation captures the information from all entities in the path, our model can capture the deep alignment interactions across two KGs.\nIf $e_2'$ has a counterpart $e_2$ in the seed entity alignment, \nour model also uses the $[MASK]$ representation to predict $e_2$, \nwhich means there are two labels for this prediction task.\nThe loss is given by:\n\\begin{equation}\n    \\label{eq:path_loss}\n    \\mathcal{L}_\\text{path} = \\sum_{e\\in \\mathcal{E}\\cup\\mathcal{E}'} \\text{CE}(\\hat{\\mathbf{q}}_{e}, \\mathbf{p}_{e}) + \\sum_{(e, e')\\in \\mathcal{A}^{s}} \\text{CE}(\\hat{\\mathbf{q}}_{e'}, \\mathbf{p}_{e}).\n\\end{equation}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{image\/transformer.pdf}\n    \\caption{\\label{fig:transformer}Illustration of multi-context Transformer.}\n\\end{figure*}\n\n\\subsubsection{Relation Regularization}\nRelation triples are the most fundamental structural contexts of KGs, \nand have already demonstrated the effectiveness for entity alignment.\nTo encode relation triples into Transformer, we propose a relation regularization module at the input layer without increasing the model complexity and parameters.\nThe objective of regularization comes from the widely-used translational embedding model TransE \\cite{bordes2013translating} that interprets a relation embedding as the translation vector from its subject entity to object entity.\nGiven a relation triple $(s, r, o)$, we expect the embeddings to hold that $\\mathbf{h}_s^0 + \\mathbf{r} = \\mathbf{h}_o^0$, \nwhere $\\mathbf{r}$ denotes the embedding of relation $r$. \nTo realize such translation, we add a additional regularization loss by minimizing the translation error over all relation triples:\n\\begin{equation}\n    \\label{eq:relation_loss}\n    \\mathcal{L}_\\text{relation} = \\sum_{(s, r, o)\\in \\mathcal{T}\\cup\\mathcal{T}'}\\| \\mathbf{h}_s^0 + \\mathbf{r} - \\mathbf{h}_o^0\\|_2.\n\\end{equation}\n\n\\subsection{Holistic Relation and Entity Alignment}\nGiven the learned embeddings, we propose a holistic reasoning method for the joint alignment of entities and relations. We use the input embeddings to conduct this process, i.e., $\\mathbf{e} = \\mathbf{h}_{e}^0$.\nDifferent from existing methods that only consider entity embedding similarities to identify aligned entities, we also compare relations and neighbors to compute entity similarities.\nThe basic idea is that similar entities tend to have higher embedding similarity, and their relations and neighboring entities should also be similar. \n\n\\subsubsection{Relation and Entity Functionality}\nIntuitively, when comparing two entities, different relational neighbors have different importance. \nFor example, relation \\textit{birthplace} is more important than relation \\textit{liveIn}, \nbecause a person can only be born in one place but can live in multiple places. \nBased on this, if we know two entities denote the same person, we can deduce that their neighbors along the relation \\textit{birthplace} also indicate the same place. \nWe use relation functionality \\cite{suchanek2011paris} to quantify the importance of relations.\nIt measures the number of first arguments per relation-entity edge.\nFor example, given $r\\in\\mathcal{R}$, its functionality is calculated as:\n\\begin{equation}\n    f(r) = \\frac{|\\{s|(s,r,o) \\in \\mathcal{T}\\}|}{|(s, o)|(s,r,o) \\in \\mathcal{T}\\}|},\n\\end{equation}\nwhere $|\\cdot|$ denotes the size of the set. Correspondingly, the inverse relation functionality is defined as:\n\\begin{equation}\n    f^{-1}(r) = \\frac{|\\{o|(s,r,o) \\in \\mathcal{T}\\}|}{|(s, o)|(s,r,o) \\in \\mathcal{T}\\}|}.\n\\end{equation}\n\nAnalogously, we propose entity functionality for comparing two relations. It defines the importance of an entity in uniquely determining a relation. \nRich connectivity of an entity would alleviate its importance. \nTherefore, we define entity functionality based on the number of relations an entity connects to, as a subject (functionality) or object (inverse functionality):\n\\begin{align}\n    \\begin{split}\n    f(e) = \\frac{1}{|\\{(e,r,o)|(e,r,o) \\in \\mathcal{T}\\}|}, \\\\ \n    f^{-1}(e) = \\frac{1}{|\\{(s,r,e)|(s,r,e) \\in \\mathcal{T}\\}}.\n\\end{split}\n\\end{align}\nAll of these functionality features can be computed in advance, as they are only related to the statistics of KG connectivity.\n\n\\subsubsection{Relation Alignment Inference}\nTo assist entity alignment, we first find relation alignment in an unsupervised fashion. \nAs each relation connects many entities, we utilize each relation's subject and object entity embeddings to approximate the relation embedding. \nRelations have the property of direction. Considering the heterogeneity of KGs that the relations from different KGs may be equivalent in a different direction, we generate embeddings for both relations and inverse relations by swapping the subject and object entity sets. \nThe embeddings of connected entities are weighted by entity functionality. \nThe approximated embeddings of each relation $\\textbf{r}$ and its inverse $\\textbf{r}_{inv}$ are calculated as follows:\n\\begin{equation}\n    \\mathbf{r} = [\\mathbf{S}, \\mathbf{O}],\\,\\,\\,\\mathbf{r}_{inv} = [\\mathbf{O}, \\mathbf{S}],\n\\end{equation}\nwhere $\\textbf{S}$ and $\\textbf{O}$ represent the embedding of the subject entity set $\\mathcal{E}_{\\text{r}}^{\\text{subj}}$ and object entity set $\\mathcal{E}_{\\text{r}}^{\\text{obj}}$ of a relation $r$.\n$[\\cdot]$ denotes concatenation. \nThe embeddings of the two sets can be generated by:\n\\begin{align}\n    \\begin{split}\n        \\textbf{S} = \\sum_{s \\in\\mathcal{E}_{\\text{r}}^{\\text{subj}}} f(s) \\times \\mathbf{s},\\,\\,\\,\n        \\textbf{O} = \\sum_{o \\in \\mathcal{E}_{\\text{r}}^{\\text{obj}}}f^{-1}(o) \\times \\mathbf{o}\n    \\end{split}\n\\end{align}\nFinally, we define the alignment score of a pair of relations as the cosine similarity of the approximated relation embeddings:\n\\begin{equation}\n    \\text{align}(r, r') = \\cos(\\mathbf{r}, \\mathbf{r}').\n\\end{equation}\nHere, $r$ can be either a relation or a reverse relation.\n\n\\subsubsection{Entity Alignment Inference}\nAfter approximating the relation alignment score, \nwe are able to match the relational neighbors (i.e., triples) of two entities from different KGs. \nFigure \\ref{fig:relation_align.pdf} shows the process of measuring the equivalence between two entity neighborhood sets, which consists of the following steps:\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.6\\linewidth]{image\/relation_align.pdf}\n    \\caption{\\label{fig:relation_align.pdf}Illustration of entity alignment inference.}\n\\end{figure}\n\n\\noindent\\textbf{Step 1. Pair-wise matching matrix generation.}\nLet $N_e$ and $N_{e'}$ denote the neighbourhood sets of $e$ and $e'$, respectively.\nWe first compute the pair-wise triple-level matching score for $e_i \\in N_e$ and $e_j \\in N_{e'}$.\nWe take into consideration both relation direction and functionality.\nIf the neighbour is outgoing, we utilize the inverse relation functionality $f^{-1}(r)$ for weighting; otherwise, we adopt its functionality $f(r)$. \nFor brevity, we use the matching of two outgoing neighbors to illustrate the calculation of similarity matrix. \nAssuming the adjacent relations to the center entity of $e_i$ and $e_j$ are $r_i$ and $r_j$ respectively, we define $\\phi$ as the similarity function of $e_i \\in N_e$ and $e_j \\in N_{e'}$ as follows:\n\\begin{equation}\n    \\phi(e_i,e_j) = f^{-1}(r_i) \\times f^{-1}(r_j) \\times \\text{align}(r_i, r_j) \\times \\cos(\\mathbf{e}_i,\\mathbf{e}_j).\n\\end{equation}\n\\noindent\\textbf{Step 2. Neighborhood intersection calculation.}\nWe then acquire the intersection of two neighborhood sets $N_e \\cap N_{e'}$ based on the matching matrix. \nAs the size of the smaller set indicates the maximum number of possibly matched entities that two sets may have, we conduct the max-pooling operation along the smaller dimension to identify possibly matched entities. \nFormally, assuming $|N_e| < |N_{e'}|$, the possibly matched entities are defined as:\n\\begin{equation}\n    \\Psi = \\{(e_i, e_j)| i = 1,\\cdots,|N_e|, j=\\argmax_{j \\in [1,|N_{e'}|]}\\phi(e_i, e_j)\\}.\n\\end{equation}\nThen, we use a dynamic threshold $\\eta$ to filter out impossible matches.\n\\begin{equation}\n    \\Theta = \\{(e_i, e_j)| (e_i, e_j) \\in \\Psi,\\ \\phi(e_i, e_j) > \\eta\\}.\n\\end{equation} \nFinally, we sum up all the remaining similarities as the equivalence score of the two neighborhood sets.\nThis process can be regarded as computing the intersection of two sets.\nIn other words, the intersection score of $N_e$ and $N_{e'}$ is the aggregation of the triple-level scores from the intersection entity set:\n\\begin{equation}\n    N_e \\cap N_{e'} = \\sum_{(e_i, e_j) \\in \\Theta}\\phi(e_i, e_j).\n\\end{equation}\n\n\\noindent\\textbf{Step 3. Neighborhood union calculation.} \nNext, we compute the union of neighborhoods to normalize the above equivalence score.\nWe aggregate the weighted edges after the max-pooling step.\nSpecifically, we use the relation functionality of connected edges as the weight and calculate the pair-wise functionality multiplication to approximate the union of two neighborhood sets. \nTaking outgoing edges as an example, for $e_i\\in N_e$ and $e_j \\in N_{e'}$, we have:\n\\begin{equation} \n    \\phi^r(e_i, e_j) = f^{-1}(r_i)\\times f^{-1}(r_j).\n\\end{equation}\nThen, we aggregate the pair-wise functionality of possibly matched neighbors as the union score of two neighborhood sets $N_e \\cup N_{e'}$:\n\\begin{equation}\n    N_e \\cup N_{e'} = \\sum_{(e_i, e_j) \\in \\Psi}\\phi^r(e_i, e_j).\n\\end{equation}\n\n\\noindent\\textbf{Step 4. Holistic alignment probability. }\nFinally, the entity alignment probability is calculated in a Jaccard-like fashion based on their neighborhoods, as defined below:\n\\begin{equation}\n    \\text{prob}(e, e') = \\frac{N_e \\cap N_{e'}}{N_e \\cup N_{e'}}.\n\\end{equation}\n\n\\subsection{Soft Label Editing by Alignment Evidence}\nAfter acquiring the holistic alignment probability of important entity pairs,\ni.e., the top-k similar target counterparts of each source entity, \nwe inform the multi-context Transformer of such an alignment clue to edit the soft labels.\nWe design two types of alignment evidence.\nSpecifically, we collect entity pairs with high alignment probability, i.e., $\\text{prob}(e, e') \\geq \\gamma_{\\text{pos}}$, as positive evidence.\nOn the contrary, if $\\text{prob}(e, e') \\leq \\gamma_{\\text{neg}}$, the two entities form a negative alignment pair.\nThen, we inject both positive and negative alignment evidences into the embedding module by editing the soft labels for prediction tasks.\nThe idea is that if the target entity has several counterparts in positive pairs, all counterparts should also be labeled as $\\min(s, p)$, where $s$ denotes the soft label of the target entity and $p$ is the calculated alignment probability.\nWe still use the soft label vector $\\hat{\\mathbf{q}}_{e}=[0.8,0.05,0.05,0.05,0.05]$ as an example, where the entity at the first position is the target. \nSuppose it has two predicted counterparts at the second and third positions with the probabilities of $0.5$ and $0.9$, respectively. In this case, the soft label vector is rewritten as $\\hat{\\mathbf{q}}_{e}=[0.8,0.5,0.8,0.05,0.05]$, and it is further normalized to serve as the labels for training the Transformer.\nFor negative counterparts, the corresponding label values are set to $0$ which will not be included in the prediction.\n\n\\section{Experiment}\n\\subsection{Experiment Setting}\n\\subsubsection{Datasets}\nWe use the benchmark dataset (V1) in OpenEA \\cite{sun13benchmarking} for evaluation as it follows the data distribution of real KGs. \nIt contains two cross-lingual settings extracted from multi-lingual DBpedia (English-to-French and English-to-German), and two monolingual settings among popular KGs (DBpedia-to-Wikidata and DBpedia-to-YAGO). \nEach setting has two sizes with 15K and 100K pairs of reference entity alignment, respectively, without reference relation alignment. \nWe follow the data splits in OpenEA where $20\\%$ alignment is for training, $10\\%$ for validation and $70\\%$ for testing. \n\n\\subsubsection{Implementation Details}\nOur IMEA\\xspace model is implemented based on the OpenEA library. \nWe initialize the trainable parameters with Xavier initialization \\cite{Xavier} and optimize the loss with Adam \\cite{Adam}.\nAs for hyper-parameters, the number of sampled neighbors is $3$, and the length of paths is $5$.\nThe batch size is $4096$, and the learning rate is $0.0001$.\nThe hidden size of Transformer is $256$. \nIt has $8$ layers with $8$ attention heads in each layer. \nWe conduct the soft label editing process based on holistic inference from the second training epoch.\nThe dynamic threshold of hard filter is $\\eta = \\max(0.6, 0.5 * (\\max_{(e_i, e_j)\\in \\Theta}\\phi(e_i, e_j) + \\text{mean}_{(e_i, e_j)\\in \\Theta}\\phi(e_i, e_j)))$. \nThe soft label value is $s=0.6$. \nThe thresholds for selecting positive and negative pairs are set as $\\gamma_{\\text{pos}} = 0.62$ and $\\gamma_{\\text{neg}} = 0.01$, respectively.\nBy convention, we use Hits@1, Hits@5 and mean reciprocal rank (MRR) as the evaluation metrics, and higher scores indicate better performance. \nWe release our source code at GitHub\\footnote{\\url{https:\/\/github.com\/JadeXIN\/IMEA}}.\n\n\n\\begin{table*}[!t]\n\t\\centering\n\t\\caption{Entity alignment results on 15K datasets.}\n\t\\resizebox{0.9\\textwidth}{!}{\n\t\t\\begin{tabular}{lcclcclcclccl}\n\t\t\t\\toprule\n\t\t\t\\multirow{2}{*}{Methods} & \\multicolumn{3}{c}{EN-FR-15K} & \\multicolumn{3}{c}{EN-DE-15K} & \\multicolumn{3}{c}{D-W-15K} & \\multicolumn{3}{c}{D-Y-15K} \\\\\n\t\t\t\\cmidrule(lr){2-4} \\cmidrule(lr){5-7} \\cmidrule(lr){8-10} \\cmidrule(lr){11-13} \n\t\t\t& Hits@1 & Hits@5 & MRR & Hits@1 & Hits@5 & MRR & Hits@1 & Hits@5 & MRR & Hits@1 & Hits@5 & MRR \\\\ \\midrule\n\t\t\tMTransE & 0.247 & 0.467 & 0.351 & 0.307 & 0.518 & 0.407 & 0.259 & 0.461 & 0.354 & 0.463 & 0.675 & 0.559 \\\\\n\t\t\tIPTransE & 0.169 & 0.320 & 0.243 & 0.350 & 0.515 & 0.430 & 0.232 & 0.380 & 0.303 & 0.313 & 0.456 & 0.378 \\\\\n\t\t\tAlignE & 0.357 & 0.611 & 0.473 & 0.552 & 0.741 & 0.638 & 0.406 & 0.627 & 0.506 & 0.551 & 0.743 & 0.636 \\\\\n\t\t\tGCN-Align & 0.338 & 0.589 & 0.451 & 0.481 & 0.679 & 0.571 & 0.364 & 0.580 & 0.461 & 0.465 & 0.626 & 0.536 \\\\\n\t\t\tSEA & 0.280 & 0.530 & 0.397 & 0.530 & 0.718 & 0.617 & 0.360 & 0.572 & 0.458 & 0.500 & 0.706 & 0.591 \\\\\n\t\t\tRSN4EA & 0.393 & 0.595 & 0.487 & 0.587 & 0.752 & 0.662 & 0.441 & 0.615 & 0.521 & 0.514 & 0.655 & 0.580 \\\\\n\t\t\tAliNet* & 0.364 & 0.597 & 0.467 & 0.604 & 0.759 & 0.673 & 0.440 & 0.628 & 0.522 & 0.559 & 0.690 & 0.617 \\\\\n\t\t\tHyperKA* & 0.353 & 0.630 & 0.477 & 0.560 & 0.780 & 0.656 & 0.440 & 0.686 & 0.548 & \\underline{0.568} & \\underline{0.777} & \\underline{0.659} \\\\\n\t\t\tKE-GCN* & \\underline{0.408} & \\underline{0.670} & \\underline{0.524} & \\textbf{0.658} & \\textbf{0.822} & \\textbf{0.730} & \\underline{0.519} & \\underline{0.727} & \\underline{0.608} & 0.560 & 0.750 & 0.644\\\\\n\t\t\t\\midrule\n\t\t\tIMEA\\xspace & \\textbf{0.458} & \\textbf{0.720} & \\textbf{0.574} & \\underline{0.639} & \\underline{0.827} & \\underline{0.724} & \\textbf{0.527} & \\textbf{0.753} & \\textbf{0.626} & \\textbf{0.639} & \\textbf{0.804} & \\textbf{0.712} \\\\\n\t\t\t\\bottomrule\n\t\\end{tabular}}\n\t\\label{tab:ent_alignment_15k}\n\\end{table*}\n\n\\begin{table*}[!t]\n\t\\centering\n\t\\caption{Entity alignment results on 100K datasets.}\n\t\\resizebox{0.9\\textwidth}{!}{\n\t\\begin{tabular}{lcclcclcclccl}\n\t\t\\toprule\n\t\t\\multirow{2}{*}{Methods} & \\multicolumn{3}{c}{EN-FR-100K} & \\multicolumn{3}{c}{EN-DE-100K} & \\multicolumn{3}{c}{D-W-100K} & \\multicolumn{3}{c}{D-Y-100K} \\\\\n\t\t\\cmidrule(lr){2-4} \\cmidrule(lr){5-7} \\cmidrule(lr){8-10} \\cmidrule(lr){11-13} \n\t\t& Hits@1 & Hits@5 & MRR & Hits@1 & Hits@5 & MRR & Hits@1 & Hits@5 & MRR & Hits@1 & Hits@5 & MRR \\\\\n\t\t\t\\midrule\n\t\t\tMTransE & 0.138 & 0.261 & 0.202 & 0.140 & 0.264 & 0.204 & 0.210 & 0.358 & 0.282 & 0.244 & 0.414 & 0.328 \\\\\n\t\t\tIPTransE & 0.158 & 0.277 & 0.219 & 0.226 & 0.357 & 0.292 & 0.221 & 0.352 & 0.285 & 0.396 & 0.558 & 0.474 \\\\\n\t\t\tAlignE & 0.294 & 0.483 & 0.388 & 0.423 & 0.593 & 0.505 & 0.385 & 0.587 & 0.478 & 0.617 & 0.776 & 0.691 \\\\\n\t\t\tGCN-Align & 0.230 & 0.412 & 0.319 & 0.317 & 0.485 & 0.399 & 0.324 & 0.507 & 0.409 & 0.528 & 0.695 & 0.605 \\\\\n\t\t\tSEA & 0.225 & 0.399 & 0.314 & 0.341 & 0.502 & 0.421 & 0.291 & 0.470 & 0.378 & 0.490 & 0.677 & 0.578 \\\\\n\t\t\tRSN4EA & 0.293 & 0.452 & 0.371 & 0.430 & 0.570 & 0.497 & 0.384 & 0.533 & 0.454 & 0.620 & 0.769 & 0.688 \\\\\n\t\t\tAliNet* & 0.266 & 0.444 & 0.348 & 0.405 & 0.546 & 0.471 & 0.369 & 0.535 & 0.444 & \\underline{0.626} & \\underline{0.772} & \\underline{0.692} \\\\\n\t\t\tHyperKA* & 0.231 & 0.426 & 0.324 & 0.239 & 0.432 & 0.332 & 0.312 & 0.535 & 0.417 & 0.473 & 0.696 & 0.574\\\\\n\t\t\tKE-GCN* &  \\underline{0.305} & \\underline{0.513} & \\underline{0.405} & \\underline{0.459} & \\underline{0.634} & \\underline{0.541} & \\textbf{0.426} & \\textbf{0.620} & \\textbf{0.515} & 0.625 & 0.791 & 0.700 \\\\\n\t\t\t\\midrule\n\t\t\tIMEA\\xspace & \\textbf{0.329} & \\textbf{0.526} & \\textbf{0.424} & \\textbf{0.467} & \\textbf{0.641} & \\textbf{0.549} & \\underline{0.419} & \\underline{0.614} & \\underline{0.508} & \\textbf{0.664} & \\textbf{0.817} & \\textbf{0.733}\\\\\n\t\t\t\\bottomrule\n\t\\end{tabular}}\n\t\\label{tab:ent_alignment_100k}\n\\end{table*}\n\n\\subsubsection{Baselines}\nTo evaluate the effectiveness of our proposed IMEA\\xspace model, we compare it with existing state-of-the-art supervised structure-based entity alignment methods. \nThey can be roughly divided into the following categories.\n\n\\begin{itemize}\n\\item Triple-based models that capture the local semantics information of relation triples based on TransE, including MTransE \\cite{chen2017multilingual}, JAPE \\cite{sun2017cross}, AlignE \\cite{sun2018bootstrapping} and SEA \\cite{pei2019semi}.\n\\item Neighborhood-based models that use GNNs to exploit subgraph structures for entity alignment, including GCN-Align \\cite{wang2018cross}, AliNet \\cite{sun2020knowledge}, HyperKA \\cite{sun2020knowledge}, KE-GCN \\cite{yu2020generalized}.\n\\item Path-based models that explore the long-term dependency across relation paths, including IPTransE \\cite{zhu2017iterative} and RSN4EA\\cite{guo2019learning}.\n\\end{itemize}\n\nOur model and the above baselines all focus on the structural information of KGs. For a fair comparison, we ignore other models that incorporate side information (e.g., attributes, entity names and descriptions) like RDGCN \\cite{wu2019relation}, KDCoE \\cite{chen2018co} and AttrGNN \\cite{liu2020exploring}.\n\n\\subsection{Results and Analysis}\n\\subsubsection{Main Results}\nTables \\ref{tab:ent_alignment_15k} and \\ref{tab:ent_alignment_100k} present the entity alignment results on the OpenEA 15K and 100K datasets, respectively.\nResults labelled by * are reproduced using the released source code with careful tuning of hyper-parameters, and the other results are taken from OpenEA \\cite{sun13benchmarking}. \nWe can see that IMEA\\xspace achieves the \\textbf{best} or \\underline{second-best} performance compared with these SOTA baselines. \nIt outperforms the best performing baselines on Hits@1 by $1\\%-7\\%$, except for the EN-DE-15K and D-W-100K datasets. \nFor example, on 15K datasets, IMEA\\xspace achieves a gain of $7.1\\%$ by Hits@1 compared with HyperKA and $7.9\\%$ against the latest method KE-GCN on D-Y-15K.\nOverall, KE-GCN achieves the second-best performance. \nIt exceeds our performance by 1.9\\% on EN-DE-15K and 0.7\\% on D-W-100K due to its strength in combining advanced knowledge embedding models with GCNs. \nHowever, the limited structure context of its embedding module prevents it from acquiring further improvement. \nOn the 100K datasets, many baselines fail to achieve promising results due to the more complex KG structures and larger alignment space.\nIMEA\\xspace also acquires the best results on most large-scale datasets, except the second-best on D-W-100K, which demonstrates the practicability of our model. \nThis is because IMEA\\xspace can integrate multiple structural contexts, and the informed knowledge can further improve the performance by holistic reasoning. \nIn summary, the results show the superiority of IMEA\\xspace.\n\n\\subsubsection{Different Path Length}\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{image\/path_len.pdf}\n    \\caption{\\label{fig:path_len}Performance and training time w.r.t path length on 15K datasets.}\n\\end{figure*}\nAs a path is one of the crucial structural contexts for both embedding and alignment learning, we evaluate the effect of path length on entity alignment. \nFigure \\ref{fig:path_len} reports the IMEA\\xspace performance w.r.t. different path lengths on 15K datasets. \nAs expected, entity alignment accuracy (Hits@1 and MRR) improves as the path length increases.\nThis is because a longer path can boost the cross-KG interaction and mitigate the heterogeneity issue between different KGs. \nThe performance increase is more evident on EN-FR-15K due to the higher structural heterogeneity in this dataset. \nHowever, it is also noticeable that the training time per epoch increases as the path length grows. \nHence, we choose a path length of 5 in our setting as an effectiveness-efficiency trade-off.\n\n\\subsubsection{Complementarity of Different Context}\n\n\\begin{table}[!t]\n\\centering\n\\caption{Complementarity of context on 15K datasets.}\n\\resizebox{0.99\\linewidth}{!}{\n\\begin{tabular}{lcccccccc} \n\\toprule\n\\multirow{2}{*}{Methods} & \\multicolumn{2}{c}{EN-FR} & \\multicolumn{2}{c}{EN-DE} & \\multicolumn{2}{c}{D-W} & \\multicolumn{2}{c}{D-Y} \\\\\n\\cmidrule(lr){2-9}\n& Hits@1 & MRR & Hits@1 & MRR & Hits@1 & MRR & Hits@1 & MRR \\\\\n\\midrule\nIMEA\\xspace & 0.458 & 0.574 & 0.639 & 0.724 & 0.527 & 0.626 & 0.639 & 0.712 \\\\ \n\\,\\, w\/o relation & 0.378 & 0.482 & 0.609 & 0.684 & 0.460 & 0.552 & 0.599 & 0.667\\\\\n\\,\\, w\/o neighbor & 0.431 & 0.547 & 0.612 & 0.681 & 0.494 & 0.595 & 0.635 & 0.708\\\\\n\\bottomrule\n\\end{tabular}}\n\\label{tab:context-effect}\n\\end{table}\n\nTable~\\ref{tab:context-effect} gives the results of the ablation study on the triple and neighborhood contexts in 15K datasets (paths are necessary for alignment learning).\nIMEA\\xspace ``w\/o relation\" and ``w\/o neighbor\" represent the variants by removing relation triples or neighborhood information from our multi-context Transformer. \nThe performance of both variants drops. \nWhen the relation triple context is removed, the Hits@1 decreases by $4\\%$ to $8\\%$, and MRR drops by $4\\%$ to $9\\%$. \nThe decline is smaller when the neighborhood is absent, which is $0.5\\%$ to $3\\%$ drop on Hits@1 and $0.5\\%$ to $4\\%$ drop on MRR. \nOverall, the absence of both relation and neighbor contexts results in a performance drop on all datasets, and relation triples contribute more than neighborhood.\n\n\\subsubsection{Effect of Informed Process}\nThe ablation study on the effect of the holistic reasoning process is shown in Table \\ref{tab:inf-effect}, \nwhere IMEA\\xspace (w\/o inf) represents the variant without using the referred holistic knowledge. It can be observed that the alignment clue captured by holistic inference can bring improvement on all datasets by $1\\%$ to $2.5\\%$ on Hits@1 and $1\\%$ to $2\\%$ on MRR. \nThis quantitatively validates the effectiveness of our holistic reasoning process and its ability to inform embedding and alignment learning.\n\n\\begin{table}[!t]\n\\centering\n\\caption{Effect of holistic inference on 15K datasets.}\n\\resizebox{0.99\\linewidth}{!}{\n\\begin{tabular}{lcccccccc} \n\\toprule\n\\multirow{2}{*}{Methods} & \\multicolumn{2}{c}{EN-FR} & \\multicolumn{2}{c}{EN-DE} & \\multicolumn{2}{c}{D-W} & \\multicolumn{2}{c}{D-Y} \\\\\n\\cmidrule(lr){2-9}\n& Hits@1 & MRR & Hits@1 & MRR & Hits@1 & MRR & Hits@1 & MRR \\\\\n\\midrule\nIMEA\\xspace & 0.458 & 0.574 & 0.639 & 0.724 & 0.527 & 0.626 & 0.639 & 0.712 \\\\ \n\\,\\, w\/o inf & 0.431 & 0.546 & 0.614 & 0.703 & 0.502 & 0.598 & 0.629 & 0.704\\\\\n\\bottomrule\n\\end{tabular}}\n\\label{tab:inf-effect}\n\\end{table}\n\n\\subsection{Case Study on Holistic Reasoning}\nIn this section, we conduct case studies to further demonstrate the effectiveness of our holistic reasoning, i.e., how the positive and negative evidence obtained from holistic reasoning can be used to fix the alignment error caused by learned embeddings.\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[b]{0.95\\linewidth}\n    \\centering\n    \\includegraphics[width=1\\linewidth]{image\/case2.pdf}\n    \\caption{\\label{fig:case2.pdf}}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.95\\linewidth}\n    \\includegraphics[width=1\\linewidth]{image\/case3.pdf}\n    \\caption{\\label{fig:case3.pdf}}\n    \\end{subfigure}\n    \\caption{Examples of aligned entity pairs discovered by holistic inference from D-Y-15K. The dark arrow and dark rectangle indicate the matched relations and entities recognized by the inference, respectively.}\n    \\label{fig:correct example}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[b]{0.5\\textwidth}\n    \\centering\n    \\includegraphics[width=1\\linewidth]{image\/case1_0.pdf}\n    \\caption{\\label{fig:case1_0.pdf}Incorrect alignment pairs.}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.5\\textwidth}\n    \\centering\n    \\includegraphics[width=1.0\\linewidth]{image\/case1_1.pdf}\n    \\caption{\\label{fig:case1_1.pdf}Correct alignment pairs.}\n\\end{subfigure}\n\\caption{Example of a negative alignment from EN-FR-15K which can be corrected by the holistic reasoning. \nThe vertical and horizontal axis represents the entity's neighbors from EN and FR respectively. Each neighbor consists of the relation (with its functionality) and the entity (either subject \"s\" or object \"o\"). The top figure reports the relation alignment while the bottom figure shows the neighbor alignment. Darker cells indicate higher alignment probability between the corresponding relations or neighbors.\n}\n\\label{fig: incorrect example}\n\\end{figure}\n\n\\subsubsection{Positive Evidence Benefit Embedding Learning}\nFigure \\ref{fig:case2.pdf} illustrates a case of positive alignment evidence from D-Y-15K. \nThe learned embedding similarity between these two identical entities is only $0.59$. \nSuch an inaccurate embedding distance is caused by the sparse connections between the two entities. \nLong-tail entities suffer from the isolation issue. Hence, the two entities with sparse triples would have few chances to be pushed closer during training. \nAnother reason for the low embedding similarity is the large degree difference between the two entities, since the embedding model tends to generate dissimilar representations for entities with different degrees.\nFortunately, our holistic inference can effectively capture the positive interaction between entities by matching their neighborhood subgraphs at a more fine-grained level (i.e., triple-level). \nFirstly, it can catch the matched relations, i.e., \\textit{``is-part-of''} and \\textit{``is-located-in''}, \\textit{``official-language''} and \\textit{``has-official-language''}. \nAs the relational neighbors of the two entities are not always long-tail entities, the matched neighbors (denoted by dark blue) can be successfully captured, which leads to a positive evidence score of $0.92$.\nFigure \\ref{fig:case3.pdf} demonstrates another example from D-Y-15K, which shows the holistic process is especially suitable for handling the heterogeneous issue between KGs. \nFrom our observation, even though two entities have many relational neighbors in common, the generated embeddings tend to be more dissimilar if the relations are in different directions.\nThe embedding similarity in Figure \\ref{fig:case3.pdf} is only $0.56$. \nAs the holistic process considers relation direction, it successfully captures equivalent relations \\textit{``writer''} and \\textit{``created''} with different directions, and similar relations \\textit{``birthplace''} and \\textit{``is-citizen-of''}, \nwhich leads to a high inference score of $0.89$.\nOverall, for those equivalent entities with low embedding similarity,\nour holistic inference can successfully capture their actual alignment interactions by comparing their neighborhood subgraphs, thus fixing the error caused by unreliable embedding learning.\n\n\\subsubsection{Negative Evidence Benefit Embedding Learning}\nOur holistic reasoning can also capture the negative alignment evidence between some wrongly closely located entity embeddings, and correctly accumulate positive clues between matched entities. \nFigure \\ref{fig: incorrect example} shows an example from EN-FR-15K about two songs from ``Whitney Houston'', where the incorrectly aligned entity pairs in Figure \\ref{fig:case1_0.pdf} have a slightly higher embedding similarity ($0.99$) than the corresponding ground-truth in Figure \\ref{fig:case1_1.pdf} ($0.97$). Such an embedding mistake can be corrected by our holistic reasoning based on the neighborhood information.\nThe neighboring entities' embeddings could also be inaccurate, which results in a closer embedding distance between the entity and its incorrect match. \nHowever, by holistic inference, we observe that the incorrectly aligned entities in Figure \\ref{fig:case1_0.pdf} only share some unimportant neighbors with low relation functionality such as \\textit{``genere''}, \\textit{``artist''}, \\textit{``producer''} and \\textit{``composer''}, leading to a limited alignment score of $0.32$. On the contrary, the ground-truth alignment in Figure \\ref{fig:case1_1.pdf} demonstrates a large number of common neighbors, among which positive matching clues are observed on some crucial neighbours with large relation functionality like \\textit{``album''} (the corresponding entities \\textit{``It's not right but It's okay''} are matched). Hence, it accumulates more positive evidences and achieves a higher alignment probability of $0.76$. In this way, holistic inference successfully identifies the aligned entity pair in Figure \\ref{fig:case1_0.pdf} as a negative prediction from the embedding module (with a smaller alignment probability), and injects such feedback into the Transformer to adjust embedding learning, thus improving the final entity alignment performance.\n\n\\section{Conclusion and Future Work}\nIn this paper, we propose an informed multi-context entity alignment model (IMEA\\xspace). It consists of a multi-context Transformer to capture multiple structural contexts for alignment learning, and a holistic reasoning process to generate global alignment evidence, which is iteratively injected back into the Transformer via soft label editing to guide further embedding training. Extensive experiments on the OpenEA dataset verify the superiority of IMEA\\xspace over existing state-of-the-art baselines. As future work, we will include extra side information of KGs to enhance embedding learning in IMEA\\xspace, and generate human-understandable interpretations of entity alignment results based on the informed holistic reasoning, thus achieving explainability of embedding-based EA models.\n\n\\begin{acks}\nThis work was partially supported by the Australian Research Council under Grant No. DE210100160 and DP200103650.\nZequn Sun's work was supported by Program A for Outstanding PhD Candidates of Nanjing University.\n\\end{acks}\n\n\\bibliographystyle{ACM-Reference-Format}\n\\balance\n\n\\section{Introduction}\nKnowledge graphs (KGs) possess a large number of facts about the real world in a structured manner. \nEach fact consists of two entities and their relation in the form of a triple like \\textit{(subject entity, relation, object entity)}. \nKGs are becoming increasingly important because they can simplify knowledge acquisition, storage, and reasoning, and successfully support many knowledge-powered tasks such as semantic search and recommendation. \nAs different KGs are constructed by different people for different purposes using different data sources,\nthey are far from complete, yet complementary to each other. \nKnowledge from a single KG usually cannot satisfy the requirement of downstream tasks, calling for the integration of multiple KGs for knowledge fusion.\nEntity alignment, which aims to identify equivalent entities from different KGs, is a fundamental task in KG fusion.\nIn recent years, there has been an increasing interest in embedding-based entity alignment \\cite{chen2017multilingual,wang2018cross,cao2019multi,sun2020knowledge}. \nIt encodes entities into vector space for similarity calculation, such that entity alignment can be achieved via nearest neighbor search.\nHowever, current embedding-based models still face two major challenges regarding alignment learning and inference.\n\n\\textbf{Challenge 1:} How to effectively represent entities in the alignment learning phase?\nExisting embedding-based methods do not fully explore the inherent structures of KGs for entity embedding. \nMost of them only utilize a one-sided view of the structures, such as relation triples \\cite{chen2017multilingual,sun2017cross}, paths \\cite{zhu2017iterative,guo2019learning}, \nand neighborhood subgraphs \\cite{wang2018cross, sun2020knowledge, cao2019multi, wu2019relation, wu2019jointly, li2019semi, wu2020neighborhood, xu2019cross, yang2019aligning}.\nIntuitively, all these multi-context views could be helpful for embedding learning and provide additional evidence for entity alignment.\nMulti-view learning \\cite{zhang2019multi} is not the best solution to aggregate these features as it requires manual feature selection to determine useful views. \nBesides, KGs are heterogeneous, indicating that not all features in a helpful view could contribute to alignment learning, such as the noisy neighbors in neighborhood subgraphs \\cite{sun2020knowledge}. \nHence, it is necessary to flexibly and selectively incorporate different types of structural contexts for entity embedding.\n\n\\textbf{Challenge 2:} How to make full use of the learned embeddings and achieve informed alignment learning? Existing models directly rely on entity embeddings to estimate their semantic similarity, which is used for alignment decisions while ignoring the correlation between entities and relations in the KG. Although some studies \\cite{wu2019jointly, zhu2021relation} have tried to conduct joint training of entity and relation to improve the quality of the learned embeddings, alignment of each entity is still conducted independently from a local perspective. In fact, entity alignment is a holistic process: the neighboring entities and relations of the aligned pair should also be equivalent. The methods \\cite{zeng2020collective,zeng2021reinforcement} conduct entity alignment from a global perspective by modelling it as a stable matching problem. Whereas, such reasoning is regarded as a post-processing step without utilizing the holistic information to inform the embedding learning module. We argue that holistic inference can provide crucial clues to achieve reliable entity alignment, and the inferred knowledge can offer reliable alignment evidence and help to build and strengthen the trust towards black-box embedding-based entity alignment.\n\n\\textbf{Contributions:} In this work, we introduce \\textbf{IMEA\\xspace}, an \\textbf{I}nformed \\textbf{M}ulti-context \\textbf{E}ntity \\textbf{A}lignment model, to tackle the above obstacles.\nMotivated by the strong expressiveness and representation learning ability of Transformer \\cite{vaswani2017attention}, \nwe design a novel Transformer architecture to encode the multiple structural contexts in a KG while capturing the deep alignment interaction across different KGs.\nSpecifically, we apply two Transformers to encode the neighborhood subgraphs and entity paths, respectively.\nTo capture the semantics of relation triples, we introduce relation regularization based on TransE \\cite{bordes2013translating} at the input layer.\nWe generate entity paths by random walks across two KGs. \nTo capture the deep alignment interactions, we replace the entity in a path with its counterpart in the seed alignment (i.e., training data) to generate a new semantically equivalent path.\nIn this way, a path may contain entities from the two KGs that remain to be aligned, \nand the Transformer is trained to fit such a cross-KG entity sequence and its equivalent path.\nThe two Transformers share parameters.\nTheir training objectives are central entity prediction and mask entity prediction, respectively.\nThe self-attention mechanism of Transformer can recognize and highlight helpful features in an end-to-end manner. \n\nFurthermore, we design a holistic reasoning process for entity alignment, to complement the imperfection of embedding similarities.\nIt compares entities by reasoning over their neighborhood subgraphs based on both embedding similarities and the holistic relation and entity functionality (i.e., the importance for identifying similar entities and relations).\nThe inferred alignment probabilities are injected back into the Transformer as soft labels to assist alignment learning. In this way, both the positive and negative alignment evidence obtained from the holistic inference can be utilized to adjust soft labels and guide embedding learning, thus improving the overall performance and reliability of entity alignment.\n\nWe conduct extensive experiments including ablation studies and case studies on the benchmark dataset OpenEA \\cite{sun13benchmarking} to evaluate the effectiveness of our IMEA\\xspace model. Experimental results demonstrate that IMEA\\xspace can achieve the state-of-the-art performance by combining multi-context Transformer and holistic reasoning,\nand the global evidence is quite effective in informing alignment learning.\n\n\\section{Related Work}\n\\textbf{KG Contexts for Entity Alignment.} \nExisting structure-based entity alignment methods utilize different types of structural contexts to generate representative KG embeddings. \nA line of studies learns from \\textit{relation triples} and encodes local semantic information of both entities and relations. \nTransE \\cite{bordes2013translating} is one of the most fundamental embedding techniques that has been adopted in a series of entity alignment models such as MTransE \\cite{chen2017multilingual}, JAPE \\cite{sun2017cross}, BootEA \\cite{sun2018bootstrapping} and SEA \\cite{pei2019semi}.\nInspired by the success of graph neural networks (GNNs) like GCN \\cite{kipf2016semi} and GAT \\cite{velickovic2018graph} in graph embedding learning,\nrecent entity alignment studies incorporate various GNN variants to encode \\textit{subgraph contexts} for neighborhood-aware KG embeddings. \nFor example, GCN-Align \\cite{wang2018cross} employs vanilla GCN for KG structure modeling, \nbut suffers from the heterogeneity of different KGs. \nTo address this issue, MuGNN \\cite{cao2019multi} preprocesses KGs by KG completion and applies an attention mechanism to highlight salient neighbors.\nOther work improves neighborhood-aware entity alignment by introducing relation-aware GNNs \\cite{li2019semi,ye2019vectorized,yu2020generalized,mao2020mraea,mao2020relational}, multi-hop GNNs \\cite{sun2020knowledge,mao2021boosting} and hyperbolic GNNs \\cite{sun2020knowledge}.\nBesides the triple and neighbourhood context, \nRSN4EA \\cite{guo2019learning} and IPTransE \\cite{zhu2017iterative} further explore long-term dependency along \\textit{relation paths} in KGs. \nThere are also some entity alignment models that consider additional information such as \\textit{entity names} \\cite{xu2019cross,wu2019relation,wu2019jointly,wu2020neighborhood}, \\textit{entity attributes} \\cite{trisedya2019entity,zhang2019multi,liu2020exploring} and \\textit{textual descriptions} \\cite{chen2018co}.\nAlthough structural contexts are abundant and always available in KGs, few existing EA models have tried to integrate them into the embedding learning process. In this work, we introduce Transformer to capture multi-context interactions and learn more comprehensive entity embeddings.\n\n\\noindent\\textbf{Collective Entity Alignment.}\nEntities are usually aligned independently in existing EA models. Only two methods have attempted to achieve collective entity alignment.\nCEAFF \\cite{zeng2020collective, zeng2021reinforcement} formulates alignment inference as a stable matching problem. \nRMN \\cite{zhu2021relation} designs an iterative framework to leverage positive interactions between entity and relation alignment. However, it can only capture the interactions from seed entity alignment and requires extra pre-aligned relations, which is not always feasible in practice. Furthermore, both methods treat alignment learning and alignment inference separately, e.g., by regarding holistic reasoning as a post-processing step to adjust the alignment results \\cite{zeng2020collective}. By contrast, our IMEA\\xspace model can inject the holistic knowledge obtained during alignment inference into the Transformer via soft label editing, so as to guide the embedding learning process.\n\n\\section{Methodology}\n\\subsection{Framework}\nWe define a KG as $\\mathcal{G}=(\\mathcal{E},\\mathcal{R},\\mathcal{T})$, \nwhere $\\mathcal{E}$ and $\\mathcal{R}$ denote the sets of entities and relations, respectively, and $\\mathcal{T}\\subset\\mathcal{E}\\times\\mathcal{R}\\times\\mathcal{E}$ is the set of relation triples. Each triple can be represented as $(s, r, o)$, where $s\\in\\mathcal{E}$ and $o\\in\\mathcal{E}$ denote the subject (head entity) and object (tail entity), respectively, and $r\\in\\mathcal{R}$ is the relation.\nGiven two heterogeneous KGs, i.e., a source KG $\\mathcal{G}$ and a target KG $\\mathcal{G}'$, entity alignment (EA) aims to find as many aligned entity pairs as possible $\\mathcal{A} = \\{(e, e') \\in \\mathcal{E} \\times \\mathcal{E}'| e \\equiv e'\\}$, \nwhere the equivalent relationship $\\equiv$ indicates that entities $e$ and $e'$ from different KGs represent the same real-world identity. \nTypically, to jump-start alignment learning, some seed entity alignment $\\mathcal{A}^s$ is provided as training data.\n\n\\begin{figure*}[!t]\n    \\centering\n    \\includegraphics[width=0.68\\linewidth]{image\/framework.pdf}\n    \\caption{\\label{fig:framework.pdf}Framework of the proposed informed multi-context entity alignment model.}\n\\end{figure*}\n\nFigure \\ref{fig:framework.pdf} shows the overall framework of our IMEA\\xspace (Informed Multi-context Entity Alignment) model, which contains three main components:\nFirstly, we introduce Transformer as the contextualized encoder to capture the multiple structural contexts (i.e., triples, neighborhood subgraphs and paths) and generate representative KG embeddings for alignment learning. \nNext, we identify candidate target entities for each source entity based on their embedding similarity and perform holistic reasoning for each candidate entity pair. \nFinally, we inform the embedding learning module by injecting the alignment evidence into the training loss using our proposed soft label editing strategy. \nWe will introduce each component in detail in the following sections.\n\n\\subsection{Multi-context Transformer}\nFigure \\ref{fig:transformer} presents the architecture of our multi-context transformer. \nSpecifically, we apply two Transformers to encode the neighborhood context and path context, respectively.\nThe neighborhood context is generated by sampling neighboring entities. \nTo capture the relation semantics, we introduce a relation regularization based on the translational embedding function \\cite{bordes2013translating} at the input layer.\nThe path context is represented as an entity sequence generated by random walks. \nTo capture the deep alignment interactions across different KGs, \nwe replace the entity in a path with its counterpart in the seed entity alignment to generate a new semantically equivalent path.\nIn this way, a path may contain entities from the two KGs that remain to be aligned, \nand the Transformer seeks to fit such a cross-KG entity sequence and its equivalent path to capture the alignment information between two KGs.\n\n\\subsubsection{Intra-KG Neighbourhood Encoding}\nNeighborhood information is critical in representing KG structures for entity alignment, as similar entities usually have similar neighborhood subgraphs. \nConsidering that an entity may have a very large number of neighbors, \nwe repeatedly sample $n$ neighbors of each entity during training and feed these neighbors into Transformer to capture their interactions. \nSpecifically, given an entity $e$ and its sampled neighbours $(e_1, e_2,..., e_n)$, \nwe regard the neighbourhood as a sequence and feed it into a Transformer with $L$ self-attention layers stacked. \nThe input embeddings of Transformer are $(\\mathbf{h}_{e_1}^0, \\mathbf{h}_{e_2}^0, \\cdots, \\mathbf{h}_{e_n}^0)$, \nand the output of the $L$-th layer is generated as follows:\n\\begin{equation}\n    \\mathbf{h}_{e_1}^L, \\mathbf{h}_{e_2}^L, \\cdots, \\mathbf{h}_{e_n}^L = \\text{Encoder}(\\mathbf{h}_{e_1}^0, \\mathbf{h}_{e_2}^0, \\cdots, \\mathbf{h}_{e_n}^0).\n\\end{equation}\nThe training objective is to predict the central entity $e$ with the input neighborhood context $\\mathbf{c}_e=\\text{MeanPooling}(\\mathbf{h}_{e_1}^L,\\mathbf{h}_{e_2}^L,\\cdots,\\mathbf{h}_{e_n}^L)$.\nThis can be regarded as a classification problem.\nThe prediction probability for each $e_i$ is given by\n\\begin{equation}\n    \\label{eq:ea_pro}\n    p_{e_i} = \\frac{\\sigma(\\mathbf{c}_e,\\mathbf{h}_{e_i}^0)}{\\sum_{e_j \\in \\mathcal{E}}\\sigma(\\mathbf{c}_e,\\mathbf{h}_{e_j}^0)},\n\\end{equation}\nwhere $\\sigma()$ denotes inner product.\nLet $\\mathbf{q}_{e}$ be the one-hot label vector for predicting entity $e$. \nFor example, $\\mathbf{q}_{e}=[1,0,0,0,0]$ means that the entity at the first position is the target.\nWe adopt the label smoothing strategy to get soft labels, e.g., $\\hat{\\mathbf{q}}_{e}=[0.8,0.05,0.05,0.05, 0.05]$,\nand use cross-entropy, i.e., $\\text{CE}(x,y)=-\\sum x \\log y$, to calculate the loss between the soft label vector and the prediction distribution over all candidates $\\mathbf{p}_{e}$. \nFormally, the overall loss is defined as\n\\begin{equation}\n    \\label{eq:neigh_loss}\n    \\mathcal{L}_\\text{neighbor} = \\sum_{e\\in \\mathcal{E}\\cup\\mathcal{E}'} \\text{CE}(\\hat{\\mathbf{q}}_{e}, \\mathbf{p}_{e}).\n\\end{equation}\n\n\\subsubsection{Cross-KG Path Encoding}\nTo capture the alignment information, we use seed entity alignment to bridge two KGs and perform random walks to generate cross-KG paths.\nFor example, a typical path of length $m$ can be denoted as $(e_1, e_2',...,e_m)$.\nThen we randomly mask an entity in the path and get $(e_1, [MASK],...,e_m)$.\nSimilar to neighborhood encoding, we feed the masked path into Transformer to get its representations:\n\\begin{equation}\n    \\mathbf{h}_{e_1}^L, \\mathbf{h}_{[MASK]}^L, \\cdots, \\mathbf{h}_{e_m}^L = \\text{Encoder}(\\mathbf{h}_{e_1}^0, \\mathbf{h}_{[MASK]}^0, \\cdots, \\mathbf{h}_{e_m}^0).\n\\end{equation}\nThe output representation of $[MASK]$ is used to predict the masked entity $e_2'$. \nHere we use $\\mathbf{h}_{[MASK]}^L$ to replace $\\mathbf{c}_e$ in Eq. (\\ref{eq:ea_pro}) to calculate the prediction probability.\nSince the $[MASK]$ representation captures the information from all entities in the path, our model can capture the deep alignment interactions across two KGs.\nIf $e_2'$ has a counterpart $e_2$ in the seed entity alignment, \nour model also uses the $[MASK]$ representation to predict $e_2$, \nwhich means there are two labels for this prediction task.\nThe loss is given by:\n\\begin{equation}\n    \\label{eq:path_loss}\n    \\mathcal{L}_\\text{path} = \\sum_{e\\in \\mathcal{E}\\cup\\mathcal{E}'} \\text{CE}(\\hat{\\mathbf{q}}_{e}, \\mathbf{p}_{e}) + \\sum_{(e, e')\\in \\mathcal{A}^{s}} \\text{CE}(\\hat{\\mathbf{q}}_{e'}, \\mathbf{p}_{e}).\n\\end{equation}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{image\/transformer.pdf}\n    \\caption{\\label{fig:transformer}Illustration of multi-context Transformer.}\n\\end{figure*}\n\n\\subsubsection{Relation Regularization}\nRelation triples are the most fundamental structural contexts of KGs, \nand have already demonstrated the effectiveness for entity alignment.\nTo encode relation triples into Transformer, we propose a relation regularization module at the input layer without increasing the model complexity and parameters.\nThe objective of regularization comes from the widely-used translational embedding model TransE \\cite{bordes2013translating} that interprets a relation embedding as the translation vector from its subject entity to object entity.\nGiven a relation triple $(s, r, o)$, we expect the embeddings to hold that $\\mathbf{h}_s^0 + \\mathbf{r} = \\mathbf{h}_o^0$, \nwhere $\\mathbf{r}$ denotes the embedding of relation $r$. \nTo realize such translation, we add a additional regularization loss by minimizing the translation error over all relation triples:\n\\begin{equation}\n    \\label{eq:relation_loss}\n    \\mathcal{L}_\\text{relation} = \\sum_{(s, r, o)\\in \\mathcal{T}\\cup\\mathcal{T}'}\\| \\mathbf{h}_s^0 + \\mathbf{r} - \\mathbf{h}_o^0\\|_2.\n\\end{equation}\n\n\\subsection{Holistic Relation and Entity Alignment}\nGiven the learned embeddings, we propose a holistic reasoning method for the joint alignment of entities and relations. We use the input embeddings to conduct this process, i.e., $\\mathbf{e} = \\mathbf{h}_{e}^0$.\nDifferent from existing methods that only consider entity embedding similarities to identify aligned entities, we also compare relations and neighbors to compute entity similarities.\nThe basic idea is that similar entities tend to have higher embedding similarity, and their relations and neighboring entities should also be similar. \n\n\\subsubsection{Relation and Entity Functionality}\nIntuitively, when comparing two entities, different relational neighbors have different importance. \nFor example, relation \\textit{birthplace} is more important than relation \\textit{liveIn}, \nbecause a person can only be born in one place but can live in multiple places. \nBased on this, if we know two entities denote the same person, we can deduce that their neighbors along the relation \\textit{birthplace} also indicate the same place. \nWe use relation functionality \\cite{suchanek2011paris} to quantify the importance of relations.\nIt measures the number of first arguments per relation-entity edge.\nFor example, given $r\\in\\mathcal{R}$, its functionality is calculated as:\n\\begin{equation}\n    f(r) = \\frac{|\\{s|(s,r,o) \\in \\mathcal{T}\\}|}{|(s, o)|(s,r,o) \\in \\mathcal{T}\\}|},\n\\end{equation}\nwhere $|\\cdot|$ denotes the size of the set. Correspondingly, the inverse relation functionality is defined as:\n\\begin{equation}\n    f^{-1}(r) = \\frac{|\\{o|(s,r,o) \\in \\mathcal{T}\\}|}{|(s, o)|(s,r,o) \\in \\mathcal{T}\\}|}.\n\\end{equation}\n\nAnalogously, we propose entity functionality for comparing two relations. It defines the importance of an entity in uniquely determining a relation. \nRich connectivity of an entity would alleviate its importance. \nTherefore, we define entity functionality based on the number of relations an entity connects to, as a subject (functionality) or object (inverse functionality):\n\\begin{align}\n    \\begin{split}\n    f(e) = \\frac{1}{|\\{(e,r,o)|(e,r,o) \\in \\mathcal{T}\\}|}, \\\\ \n    f^{-1}(e) = \\frac{1}{|\\{(s,r,e)|(s,r,e) \\in \\mathcal{T}\\}}.\n\\end{split}\n\\end{align}\nAll of these functionality features can be computed in advance, as they are only related to the statistics of KG connectivity.\n\n\\subsubsection{Relation Alignment Inference}\nTo assist entity alignment, we first find relation alignment in an unsupervised fashion. \nAs each relation connects many entities, we utilize each relation's subject and object entity embeddings to approximate the relation embedding. \nRelations have the property of direction. Considering the heterogeneity of KGs that the relations from different KGs may be equivalent in a different direction, we generate embeddings for both relations and inverse relations by swapping the subject and object entity sets. \nThe embeddings of connected entities are weighted by entity functionality. \nThe approximated embeddings of each relation $\\textbf{r}$ and its inverse $\\textbf{r}_{inv}$ are calculated as follows:\n\\begin{equation}\n    \\mathbf{r} = [\\mathbf{S}, \\mathbf{O}],\\,\\,\\,\\mathbf{r}_{inv} = [\\mathbf{O}, \\mathbf{S}],\n\\end{equation}\nwhere $\\textbf{S}$ and $\\textbf{O}$ represent the embedding of the subject entity set $\\mathcal{E}_{\\text{r}}^{\\text{subj}}$ and object entity set $\\mathcal{E}_{\\text{r}}^{\\text{obj}}$ of a relation $r$.\n$[\\cdot]$ denotes concatenation. \nThe embeddings of the two sets can be generated by:\n\\begin{align}\n    \\begin{split}\n        \\textbf{S} = \\sum_{s \\in\\mathcal{E}_{\\text{r}}^{\\text{subj}}} f(s) \\times \\mathbf{s},\\,\\,\\,\n        \\textbf{O} = \\sum_{o \\in \\mathcal{E}_{\\text{r}}^{\\text{obj}}}f^{-1}(o) \\times \\mathbf{o}\n    \\end{split}\n\\end{align}\nFinally, we define the alignment score of a pair of relations as the cosine similarity of the approximated relation embeddings:\n\\begin{equation}\n    \\text{align}(r, r') = \\cos(\\mathbf{r}, \\mathbf{r}').\n\\end{equation}\nHere, $r$ can be either a relation or a reverse relation.\n\n\\subsubsection{Entity Alignment Inference}\nAfter approximating the relation alignment score, \nwe are able to match the relational neighbors (i.e., triples) of two entities from different KGs. \nFigure \\ref{fig:relation_align.pdf} shows the process of measuring the equivalence between two entity neighborhood sets, which consists of the following steps:\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.6\\linewidth]{image\/relation_align.pdf}\n    \\caption{\\label{fig:relation_align.pdf}Illustration of entity alignment inference.}\n\\end{figure}\n\n\\noindent\\textbf{Step 1. Pair-wise matching matrix generation.}\nLet $N_e$ and $N_{e'}$ denote the neighbourhood sets of $e$ and $e'$, respectively.\nWe first compute the pair-wise triple-level matching score for $e_i \\in N_e$ and $e_j \\in N_{e'}$.\nWe take into consideration both relation direction and functionality.\nIf the neighbour is outgoing, we utilize the inverse relation functionality $f^{-1}(r)$ for weighting; otherwise, we adopt its functionality $f(r)$. \nFor brevity, we use the matching of two outgoing neighbors to illustrate the calculation of similarity matrix. \nAssuming the adjacent relations to the center entity of $e_i$ and $e_j$ are $r_i$ and $r_j$ respectively, we define $\\phi$ as the similarity function of $e_i \\in N_e$ and $e_j \\in N_{e'}$ as follows:\n\\begin{equation}\n    \\phi(e_i,e_j) = f^{-1}(r_i) \\times f^{-1}(r_j) \\times \\text{align}(r_i, r_j) \\times \\cos(\\mathbf{e}_i,\\mathbf{e}_j).\n\\end{equation}\n\\noindent\\textbf{Step 2. Neighborhood intersection calculation.}\nWe then acquire the intersection of two neighborhood sets $N_e \\cap N_{e'}$ based on the matching matrix. \nAs the size of the smaller set indicates the maximum number of possibly matched entities that two sets may have, we conduct the max-pooling operation along the smaller dimension to identify possibly matched entities. \nFormally, assuming $|N_e| < |N_{e'}|$, the possibly matched entities are defined as:\n\\begin{equation}\n    \\Psi = \\{(e_i, e_j)| i = 1,\\cdots,|N_e|, j=\\argmax_{j \\in [1,|N_{e'}|]}\\phi(e_i, e_j)\\}.\n\\end{equation}\nThen, we use a dynamic threshold $\\eta$ to filter out impossible matches.\n\\begin{equation}\n    \\Theta = \\{(e_i, e_j)| (e_i, e_j) \\in \\Psi,\\ \\phi(e_i, e_j) > \\eta\\}.\n\\end{equation} \nFinally, we sum up all the remaining similarities as the equivalence score of the two neighborhood sets.\nThis process can be regarded as computing the intersection of two sets.\nIn other words, the intersection score of $N_e$ and $N_{e'}$ is the aggregation of the triple-level scores from the intersection entity set:\n\\begin{equation}\n    N_e \\cap N_{e'} = \\sum_{(e_i, e_j) \\in \\Theta}\\phi(e_i, e_j).\n\\end{equation}\n\n\\noindent\\textbf{Step 3. Neighborhood union calculation.} \nNext, we compute the union of neighborhoods to normalize the above equivalence score.\nWe aggregate the weighted edges after the max-pooling step.\nSpecifically, we use the relation functionality of connected edges as the weight and calculate the pair-wise functionality multiplication to approximate the union of two neighborhood sets. \nTaking outgoing edges as an example, for $e_i\\in N_e$ and $e_j \\in N_{e'}$, we have:\n\\begin{equation} \n    \\phi^r(e_i, e_j) = f^{-1}(r_i)\\times f^{-1}(r_j).\n\\end{equation}\nThen, we aggregate the pair-wise functionality of possibly matched neighbors as the union score of two neighborhood sets $N_e \\cup N_{e'}$:\n\\begin{equation}\n    N_e \\cup N_{e'} = \\sum_{(e_i, e_j) \\in \\Psi}\\phi^r(e_i, e_j).\n\\end{equation}\n\n\\noindent\\textbf{Step 4. Holistic alignment probability. }\nFinally, the entity alignment probability is calculated in a Jaccard-like fashion based on their neighborhoods, as defined below:\n\\begin{equation}\n    \\text{prob}(e, e') = \\frac{N_e \\cap N_{e'}}{N_e \\cup N_{e'}}.\n\\end{equation}\n\n\\subsection{Soft Label Editing by Alignment Evidence}\nAfter acquiring the holistic alignment probability of important entity pairs,\ni.e., the top-k similar target counterparts of each source entity, \nwe inform the multi-context Transformer of such an alignment clue to edit the soft labels.\nWe design two types of alignment evidence.\nSpecifically, we collect entity pairs with high alignment probability, i.e., $\\text{prob}(e, e') \\geq \\gamma_{\\text{pos}}$, as positive evidence.\nOn the contrary, if $\\text{prob}(e, e') \\leq \\gamma_{\\text{neg}}$, the two entities form a negative alignment pair.\nThen, we inject both positive and negative alignment evidences into the embedding module by editing the soft labels for prediction tasks.\nThe idea is that if the target entity has several counterparts in positive pairs, all counterparts should also be labeled as $\\min(s, p)$, where $s$ denotes the soft label of the target entity and $p$ is the calculated alignment probability.\nWe still use the soft label vector $\\hat{\\mathbf{q}}_{e}=[0.8,0.05,0.05,0.05,0.05]$ as an example, where the entity at the first position is the target. \nSuppose it has two predicted counterparts at the second and third positions with the probabilities of $0.5$ and $0.9$, respectively. In this case, the soft label vector is rewritten as $\\hat{\\mathbf{q}}_{e}=[0.8,0.5,0.8,0.05,0.05]$, and it is further normalized to serve as the labels for training the Transformer.\nFor negative counterparts, the corresponding label values are set to $0$ which will not be included in the prediction.\n\n\\section{Experiment}\n\\subsection{Experiment Setting}\n\\subsubsection{Datasets}\nWe use the benchmark dataset (V1) in OpenEA \\cite{sun13benchmarking} for evaluation as it follows the data distribution of real KGs. \nIt contains two cross-lingual settings extracted from multi-lingual DBpedia (English-to-French and English-to-German), and two monolingual settings among popular KGs (DBpedia-to-Wikidata and DBpedia-to-YAGO). \nEach setting has two sizes with 15K and 100K pairs of reference entity alignment, respectively, without reference relation alignment. \nWe follow the data splits in OpenEA where $20\\%$ alignment is for training, $10\\%$ for validation and $70\\%$ for testing. \n\n\\subsubsection{Implementation Details}\nOur IMEA\\xspace model is implemented based on the OpenEA library. \nWe initialize the trainable parameters with Xavier initialization \\cite{Xavier} and optimize the loss with Adam \\cite{Adam}.\nAs for hyper-parameters, the number of sampled neighbors is $3$, and the length of paths is $5$.\nThe batch size is $4096$, and the learning rate is $0.0001$.\nThe hidden size of Transformer is $256$. \nIt has $8$ layers with $8$ attention heads in each layer. \nWe conduct the soft label editing process based on holistic inference from the second training epoch.\nThe dynamic threshold of hard filter is $\\eta = \\max(0.6, 0.5 * (\\max_{(e_i, e_j)\\in \\Theta}\\phi(e_i, e_j) + \\text{mean}_{(e_i, e_j)\\in \\Theta}\\phi(e_i, e_j)))$. \nThe soft label value is $s=0.6$. \nThe thresholds for selecting positive and negative pairs are set as $\\gamma_{\\text{pos}} = 0.62$ and $\\gamma_{\\text{neg}} = 0.01$, respectively.\nBy convention, we use Hits@1, Hits@5 and mean reciprocal rank (MRR) as the evaluation metrics, and higher scores indicate better performance. \nWe release our source code at GitHub\\footnote{\\url{https:\/\/github.com\/JadeXIN\/IMEA}}.\n\n\n\\begin{table*}[!t]\n\t\\centering\n\t\\caption{Entity alignment results on 15K datasets.}\n\t\\resizebox{0.9\\textwidth}{!}{\n\t\t\\begin{tabular}{lcclcclcclccl}\n\t\t\t\\toprule\n\t\t\t\\multirow{2}{*}{Methods} & \\multicolumn{3}{c}{EN-FR-15K} & \\multicolumn{3}{c}{EN-DE-15K} & \\multicolumn{3}{c}{D-W-15K} & \\multicolumn{3}{c}{D-Y-15K} \\\\\n\t\t\t\\cmidrule(lr){2-4} \\cmidrule(lr){5-7} \\cmidrule(lr){8-10} \\cmidrule(lr){11-13} \n\t\t\t& Hits@1 & Hits@5 & MRR & Hits@1 & Hits@5 & MRR & Hits@1 & Hits@5 & MRR & Hits@1 & Hits@5 & MRR \\\\ \\midrule\n\t\t\tMTransE & 0.247 & 0.467 & 0.351 & 0.307 & 0.518 & 0.407 & 0.259 & 0.461 & 0.354 & 0.463 & 0.675 & 0.559 \\\\\n\t\t\tIPTransE & 0.169 & 0.320 & 0.243 & 0.350 & 0.515 & 0.430 & 0.232 & 0.380 & 0.303 & 0.313 & 0.456 & 0.378 \\\\\n\t\t\tAlignE & 0.357 & 0.611 & 0.473 & 0.552 & 0.741 & 0.638 & 0.406 & 0.627 & 0.506 & 0.551 & 0.743 & 0.636 \\\\\n\t\t\tGCN-Align & 0.338 & 0.589 & 0.451 & 0.481 & 0.679 & 0.571 & 0.364 & 0.580 & 0.461 & 0.465 & 0.626 & 0.536 \\\\\n\t\t\tSEA & 0.280 & 0.530 & 0.397 & 0.530 & 0.718 & 0.617 & 0.360 & 0.572 & 0.458 & 0.500 & 0.706 & 0.591 \\\\\n\t\t\tRSN4EA & 0.393 & 0.595 & 0.487 & 0.587 & 0.752 & 0.662 & 0.441 & 0.615 & 0.521 & 0.514 & 0.655 & 0.580 \\\\\n\t\t\tAliNet* & 0.364 & 0.597 & 0.467 & 0.604 & 0.759 & 0.673 & 0.440 & 0.628 & 0.522 & 0.559 & 0.690 & 0.617 \\\\\n\t\t\tHyperKA* & 0.353 & 0.630 & 0.477 & 0.560 & 0.780 & 0.656 & 0.440 & 0.686 & 0.548 & \\underline{0.568} & \\underline{0.777} & \\underline{0.659} \\\\\n\t\t\tKE-GCN* & \\underline{0.408} & \\underline{0.670} & \\underline{0.524} & \\textbf{0.658} & \\textbf{0.822} & \\textbf{0.730} & \\underline{0.519} & \\underline{0.727} & \\underline{0.608} & 0.560 & 0.750 & 0.644\\\\\n\t\t\t\\midrule\n\t\t\tIMEA\\xspace & \\textbf{0.458} & \\textbf{0.720} & \\textbf{0.574} & \\underline{0.639} & \\underline{0.827} & \\underline{0.724} & \\textbf{0.527} & \\textbf{0.753} & \\textbf{0.626} & \\textbf{0.639} & \\textbf{0.804} & \\textbf{0.712} \\\\\n\t\t\t\\bottomrule\n\t\\end{tabular}}\n\t\\label{tab:ent_alignment_15k}\n\\end{table*}\n\n\\begin{table*}[!t]\n\t\\centering\n\t\\caption{Entity alignment results on 100K datasets.}\n\t\\resizebox{0.9\\textwidth}{!}{\n\t\\begin{tabular}{lcclcclcclccl}\n\t\t\\toprule\n\t\t\\multirow{2}{*}{Methods} & \\multicolumn{3}{c}{EN-FR-100K} & \\multicolumn{3}{c}{EN-DE-100K} & \\multicolumn{3}{c}{D-W-100K} & \\multicolumn{3}{c}{D-Y-100K} \\\\\n\t\t\\cmidrule(lr){2-4} \\cmidrule(lr){5-7} \\cmidrule(lr){8-10} \\cmidrule(lr){11-13} \n\t\t& Hits@1 & Hits@5 & MRR & Hits@1 & Hits@5 & MRR & Hits@1 & Hits@5 & MRR & Hits@1 & Hits@5 & MRR \\\\\n\t\t\t\\midrule\n\t\t\tMTransE & 0.138 & 0.261 & 0.202 & 0.140 & 0.264 & 0.204 & 0.210 & 0.358 & 0.282 & 0.244 & 0.414 & 0.328 \\\\\n\t\t\tIPTransE & 0.158 & 0.277 & 0.219 & 0.226 & 0.357 & 0.292 & 0.221 & 0.352 & 0.285 & 0.396 & 0.558 & 0.474 \\\\\n\t\t\tAlignE & 0.294 & 0.483 & 0.388 & 0.423 & 0.593 & 0.505 & 0.385 & 0.587 & 0.478 & 0.617 & 0.776 & 0.691 \\\\\n\t\t\tGCN-Align & 0.230 & 0.412 & 0.319 & 0.317 & 0.485 & 0.399 & 0.324 & 0.507 & 0.409 & 0.528 & 0.695 & 0.605 \\\\\n\t\t\tSEA & 0.225 & 0.399 & 0.314 & 0.341 & 0.502 & 0.421 & 0.291 & 0.470 & 0.378 & 0.490 & 0.677 & 0.578 \\\\\n\t\t\tRSN4EA & 0.293 & 0.452 & 0.371 & 0.430 & 0.570 & 0.497 & 0.384 & 0.533 & 0.454 & 0.620 & 0.769 & 0.688 \\\\\n\t\t\tAliNet* & 0.266 & 0.444 & 0.348 & 0.405 & 0.546 & 0.471 & 0.369 & 0.535 & 0.444 & \\underline{0.626} & \\underline{0.772} & \\underline{0.692} \\\\\n\t\t\tHyperKA* & 0.231 & 0.426 & 0.324 & 0.239 & 0.432 & 0.332 & 0.312 & 0.535 & 0.417 & 0.473 & 0.696 & 0.574\\\\\n\t\t\tKE-GCN* &  \\underline{0.305} & \\underline{0.513} & \\underline{0.405} & \\underline{0.459} & \\underline{0.634} & \\underline{0.541} & \\textbf{0.426} & \\textbf{0.620} & \\textbf{0.515} & 0.625 & 0.791 & 0.700 \\\\\n\t\t\t\\midrule\n\t\t\tIMEA\\xspace & \\textbf{0.329} & \\textbf{0.526} & \\textbf{0.424} & \\textbf{0.467} & \\textbf{0.641} & \\textbf{0.549} & \\underline{0.419} & \\underline{0.614} & \\underline{0.508} & \\textbf{0.664} & \\textbf{0.817} & \\textbf{0.733}\\\\\n\t\t\t\\bottomrule\n\t\\end{tabular}}\n\t\\label{tab:ent_alignment_100k}\n\\end{table*}\n\n\\subsubsection{Baselines}\nTo evaluate the effectiveness of our proposed IMEA\\xspace model, we compare it with existing state-of-the-art supervised structure-based entity alignment methods. \nThey can be roughly divided into the following categories.\n\n\\begin{itemize}\n\\item Triple-based models that capture the local semantics information of relation triples based on TransE, including MTransE \\cite{chen2017multilingual}, JAPE \\cite{sun2017cross}, AlignE \\cite{sun2018bootstrapping} and SEA \\cite{pei2019semi}.\n\\item Neighborhood-based models that use GNNs to exploit subgraph structures for entity alignment, including GCN-Align \\cite{wang2018cross}, AliNet \\cite{sun2020knowledge}, HyperKA \\cite{sun2020knowledge}, KE-GCN \\cite{yu2020generalized}.\n\\item Path-based models that explore the long-term dependency across relation paths, including IPTransE \\cite{zhu2017iterative} and RSN4EA\\cite{guo2019learning}.\n\\end{itemize}\n\nOur model and the above baselines all focus on the structural information of KGs. For a fair comparison, we ignore other models that incorporate side information (e.g., attributes, entity names and descriptions) like RDGCN \\cite{wu2019relation}, KDCoE \\cite{chen2018co} and AttrGNN \\cite{liu2020exploring}.\n\n\\subsection{Results and Analysis}\n\\subsubsection{Main Results}\nTables \\ref{tab:ent_alignment_15k} and \\ref{tab:ent_alignment_100k} present the entity alignment results on the OpenEA 15K and 100K datasets, respectively.\nResults labelled by * are reproduced using the released source code with careful tuning of hyper-parameters, and the other results are taken from OpenEA \\cite{sun13benchmarking}. \nWe can see that IMEA\\xspace achieves the \\textbf{best} or \\underline{second-best} performance compared with these SOTA baselines. \nIt outperforms the best performing baselines on Hits@1 by $1\\%-7\\%$, except for the EN-DE-15K and D-W-100K datasets. \nFor example, on 15K datasets, IMEA\\xspace achieves a gain of $7.1\\%$ by Hits@1 compared with HyperKA and $7.9\\%$ against the latest method KE-GCN on D-Y-15K.\nOverall, KE-GCN achieves the second-best performance. \nIt exceeds our performance by 1.9\\% on EN-DE-15K and 0.7\\% on D-W-100K due to its strength in combining advanced knowledge embedding models with GCNs. \nHowever, the limited structure context of its embedding module prevents it from acquiring further improvement. \nOn the 100K datasets, many baselines fail to achieve promising results due to the more complex KG structures and larger alignment space.\nIMEA\\xspace also acquires the best results on most large-scale datasets, except the second-best on D-W-100K, which demonstrates the practicability of our model. \nThis is because IMEA\\xspace can integrate multiple structural contexts, and the informed knowledge can further improve the performance by holistic reasoning. \nIn summary, the results show the superiority of IMEA\\xspace.\n\n\\subsubsection{Different Path Length}\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{image\/path_len.pdf}\n    \\caption{\\label{fig:path_len}Performance and training time w.r.t path length on 15K datasets.}\n\\end{figure*}\nAs a path is one of the crucial structural contexts for both embedding and alignment learning, we evaluate the effect of path length on entity alignment. \nFigure \\ref{fig:path_len} reports the IMEA\\xspace performance w.r.t. different path lengths on 15K datasets. \nAs expected, entity alignment accuracy (Hits@1 and MRR) improves as the path length increases.\nThis is because a longer path can boost the cross-KG interaction and mitigate the heterogeneity issue between different KGs. \nThe performance increase is more evident on EN-FR-15K due to the higher structural heterogeneity in this dataset. \nHowever, it is also noticeable that the training time per epoch increases as the path length grows. \nHence, we choose a path length of 5 in our setting as an effectiveness-efficiency trade-off.\n\n\\subsubsection{Complementarity of Different Context}\n\n\\begin{table}[!t]\n\\centering\n\\caption{Complementarity of context on 15K datasets.}\n\\resizebox{0.99\\linewidth}{!}{\n\\begin{tabular}{lcccccccc} \n\\toprule\n\\multirow{2}{*}{Methods} & \\multicolumn{2}{c}{EN-FR} & \\multicolumn{2}{c}{EN-DE} & \\multicolumn{2}{c}{D-W} & \\multicolumn{2}{c}{D-Y} \\\\\n\\cmidrule(lr){2-9}\n& Hits@1 & MRR & Hits@1 & MRR & Hits@1 & MRR & Hits@1 & MRR \\\\\n\\midrule\nIMEA\\xspace & 0.458 & 0.574 & 0.639 & 0.724 & 0.527 & 0.626 & 0.639 & 0.712 \\\\ \n\\,\\, w\/o relation & 0.378 & 0.482 & 0.609 & 0.684 & 0.460 & 0.552 & 0.599 & 0.667\\\\\n\\,\\, w\/o neighbor & 0.431 & 0.547 & 0.612 & 0.681 & 0.494 & 0.595 & 0.635 & 0.708\\\\\n\\bottomrule\n\\end{tabular}}\n\\label{tab:context-effect}\n\\end{table}\n\nTable~\\ref{tab:context-effect} gives the results of the ablation study on the triple and neighborhood contexts in 15K datasets (paths are necessary for alignment learning).\nIMEA\\xspace ``w\/o relation\" and ``w\/o neighbor\" represent the variants by removing relation triples or neighborhood information from our multi-context Transformer. \nThe performance of both variants drops. \nWhen the relation triple context is removed, the Hits@1 decreases by $4\\%$ to $8\\%$, and MRR drops by $4\\%$ to $9\\%$. \nThe decline is smaller when the neighborhood is absent, which is $0.5\\%$ to $3\\%$ drop on Hits@1 and $0.5\\%$ to $4\\%$ drop on MRR. \nOverall, the absence of both relation and neighbor contexts results in a performance drop on all datasets, and relation triples contribute more than neighborhood.\n\n\\subsubsection{Effect of Informed Process}\nThe ablation study on the effect of the holistic reasoning process is shown in Table \\ref{tab:inf-effect}, \nwhere IMEA\\xspace (w\/o inf) represents the variant without using the referred holistic knowledge. It can be observed that the alignment clue captured by holistic inference can bring improvement on all datasets by $1\\%$ to $2.5\\%$ on Hits@1 and $1\\%$ to $2\\%$ on MRR. \nThis quantitatively validates the effectiveness of our holistic reasoning process and its ability to inform embedding and alignment learning.\n\n\\begin{table}[!t]\n\\centering\n\\caption{Effect of holistic inference on 15K datasets.}\n\\resizebox{0.99\\linewidth}{!}{\n\\begin{tabular}{lcccccccc} \n\\toprule\n\\multirow{2}{*}{Methods} & \\multicolumn{2}{c}{EN-FR} & \\multicolumn{2}{c}{EN-DE} & \\multicolumn{2}{c}{D-W} & \\multicolumn{2}{c}{D-Y} \\\\\n\\cmidrule(lr){2-9}\n& Hits@1 & MRR & Hits@1 & MRR & Hits@1 & MRR & Hits@1 & MRR \\\\\n\\midrule\nIMEA\\xspace & 0.458 & 0.574 & 0.639 & 0.724 & 0.527 & 0.626 & 0.639 & 0.712 \\\\ \n\\,\\, w\/o inf & 0.431 & 0.546 & 0.614 & 0.703 & 0.502 & 0.598 & 0.629 & 0.704\\\\\n\\bottomrule\n\\end{tabular}}\n\\label{tab:inf-effect}\n\\end{table}\n\n\\subsection{Case Study on Holistic Reasoning}\nIn this section, we conduct case studies to further demonstrate the effectiveness of our holistic reasoning, i.e., how the positive and negative evidence obtained from holistic reasoning can be used to fix the alignment error caused by learned embeddings.\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[b]{0.95\\linewidth}\n    \\centering\n    \\includegraphics[width=1\\linewidth]{image\/case2.pdf}\n    \\caption{\\label{fig:case2.pdf}}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.95\\linewidth}\n    \\includegraphics[width=1\\linewidth]{image\/case3.pdf}\n    \\caption{\\label{fig:case3.pdf}}\n    \\end{subfigure}\n    \\caption{Examples of aligned entity pairs discovered by holistic inference from D-Y-15K. The dark arrow and dark rectangle indicate the matched relations and entities recognized by the inference, respectively.}\n    \\label{fig:correct example}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[b]{0.5\\textwidth}\n    \\centering\n    \\includegraphics[width=1\\linewidth]{image\/case1_0.pdf}\n    \\caption{\\label{fig:case1_0.pdf}Incorrect alignment pairs.}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.5\\textwidth}\n    \\centering\n    \\includegraphics[width=1.0\\linewidth]{image\/case1_1.pdf}\n    \\caption{\\label{fig:case1_1.pdf}Correct alignment pairs.}\n\\end{subfigure}\n\\caption{Example of a negative alignment from EN-FR-15K which can be corrected by the holistic reasoning. \nThe vertical and horizontal axis represents the entity's neighbors from EN and FR respectively. Each neighbor consists of the relation (with its functionality) and the entity (either subject \"s\" or object \"o\"). The top figure reports the relation alignment while the bottom figure shows the neighbor alignment. Darker cells indicate higher alignment probability between the corresponding relations or neighbors.\n}\n\\label{fig: incorrect example}\n\\end{figure}\n\n\\subsubsection{Positive Evidence Benefit Embedding Learning}\nFigure \\ref{fig:case2.pdf} illustrates a case of positive alignment evidence from D-Y-15K. \nThe learned embedding similarity between these two identical entities is only $0.59$. \nSuch an inaccurate embedding distance is caused by the sparse connections between the two entities. \nLong-tail entities suffer from the isolation issue. Hence, the two entities with sparse triples would have few chances to be pushed closer during training. \nAnother reason for the low embedding similarity is the large degree difference between the two entities, since the embedding model tends to generate dissimilar representations for entities with different degrees.\nFortunately, our holistic inference can effectively capture the positive interaction between entities by matching their neighborhood subgraphs at a more fine-grained level (i.e., triple-level). \nFirstly, it can catch the matched relations, i.e., \\textit{``is-part-of''} and \\textit{``is-located-in''}, \\textit{``official-language''} and \\textit{``has-official-language''}. \nAs the relational neighbors of the two entities are not always long-tail entities, the matched neighbors (denoted by dark blue) can be successfully captured, which leads to a positive evidence score of $0.92$.\nFigure \\ref{fig:case3.pdf} demonstrates another example from D-Y-15K, which shows the holistic process is especially suitable for handling the heterogeneous issue between KGs. \nFrom our observation, even though two entities have many relational neighbors in common, the generated embeddings tend to be more dissimilar if the relations are in different directions.\nThe embedding similarity in Figure \\ref{fig:case3.pdf} is only $0.56$. \nAs the holistic process considers relation direction, it successfully captures equivalent relations \\textit{``writer''} and \\textit{``created''} with different directions, and similar relations \\textit{``birthplace''} and \\textit{``is-citizen-of''}, \nwhich leads to a high inference score of $0.89$.\nOverall, for those equivalent entities with low embedding similarity,\nour holistic inference can successfully capture their actual alignment interactions by comparing their neighborhood subgraphs, thus fixing the error caused by unreliable embedding learning.\n\n\\subsubsection{Negative Evidence Benefit Embedding Learning}\nOur holistic reasoning can also capture the negative alignment evidence between some wrongly closely located entity embeddings, and correctly accumulate positive clues between matched entities. \nFigure \\ref{fig: incorrect example} shows an example from EN-FR-15K about two songs from ``Whitney Houston'', where the incorrectly aligned entity pairs in Figure \\ref{fig:case1_0.pdf} have a slightly higher embedding similarity ($0.99$) than the corresponding ground-truth in Figure \\ref{fig:case1_1.pdf} ($0.97$). Such an embedding mistake can be corrected by our holistic reasoning based on the neighborhood information.\nThe neighboring entities' embeddings could also be inaccurate, which results in a closer embedding distance between the entity and its incorrect match. \nHowever, by holistic inference, we observe that the incorrectly aligned entities in Figure \\ref{fig:case1_0.pdf} only share some unimportant neighbors with low relation functionality such as \\textit{``genere''}, \\textit{``artist''}, \\textit{``producer''} and \\textit{``composer''}, leading to a limited alignment score of $0.32$. On the contrary, the ground-truth alignment in Figure \\ref{fig:case1_1.pdf} demonstrates a large number of common neighbors, among which positive matching clues are observed on some crucial neighbours with large relation functionality like \\textit{``album''} (the corresponding entities \\textit{``It's not right but It's okay''} are matched). Hence, it accumulates more positive evidences and achieves a higher alignment probability of $0.76$. In this way, holistic inference successfully identifies the aligned entity pair in Figure \\ref{fig:case1_0.pdf} as a negative prediction from the embedding module (with a smaller alignment probability), and injects such feedback into the Transformer to adjust embedding learning, thus improving the final entity alignment performance.\n\n\\section{Conclusion and Future Work}\nIn this paper, we propose an informed multi-context entity alignment model (IMEA\\xspace). It consists of a multi-context Transformer to capture multiple structural contexts for alignment learning, and a holistic reasoning process to generate global alignment evidence, which is iteratively injected back into the Transformer via soft label editing to guide further embedding training. Extensive experiments on the OpenEA dataset verify the superiority of IMEA\\xspace over existing state-of-the-art baselines. As future work, we will include extra side information of KGs to enhance embedding learning in IMEA\\xspace, and generate human-understandable interpretations of entity alignment results based on the informed holistic reasoning, thus achieving explainability of embedding-based EA models.\n\n\\begin{acks}\nThis work was partially supported by the Australian Research Council under Grant No. DE210100160 and DP200103650.\nZequn Sun's work was supported by Program A for Outstanding PhD Candidates of Nanjing University.\n\\end{acks}\n\n\\bibliographystyle{ACM-Reference-Format}\n\\balance\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nPublic datasets and benchmarks play an important role in computer vision especially as the rise of deep learning approaches accelerate the relative scalability and reliability of solutions for different tasks. Starting with ImageNet~\\cite{imagenet}, COCO~\\cite{coco}, and DAVIS~\\cite{davis18} for detection, captioning, and segmentation tasks respectively, competitions bring fairness and reproducibility to the evaluation process of computer vision approaches and improve their capabilities by introducing valuable datasets. Such datasets, and the corresponding challenges, also increase the visibility, availability, and feasibility of machine learning models, which brings up even more scalable, diverse, and accurate algorithms to be evaluated on public benchmarks.\n\nComputer vision approaches have shown tremendous progress toward understanding shapes from various data formats, especially since entering the deep learning era. Although detection, recognition, and segmentation approaches achieve highly accurate results, there has been relatively less attention and research dedicated to extracting topological and geometric information from shapes. However, geometric skeleton-based representations of a shape provide a compact and intuitive representation of the shape for modeling, synthesis, compression, and analysis. Generating or extracting such representations is significantly different from segmentation and recognition tasks, as they condense both local and global information about the shape, and often combine topological and geometrical recognition.\n\nWe observe that the main challenges for such shape abstraction tasks are (i) the inherent dimensionality reduction from the shape to the skeleton, (ii) the domain change as the true skeletal representation would be best expressed in a continuous domain, and (iii) the trade off between the noise and representative power for the skeleton to prohibit overbranching but still preserve the shape. Although the lower dimensional representation is a clear advantage for shape manipulation, it raises the challenge of characterizing features and representation, especially for deep learning. Computational methods for skeleton extraction are abundant, but are typically not robust to noise on the boundary of the shape (see \\cite{Tagliasacchi2016}, for example). Small changes in the boundary result in large changes to the skeleton structure, with long branches describing insignificant bumps on the shape. Even for clean extraction methods such as Durix's robust skeleton \\cite{Durix2019} used for our dataset, changing the resolution of an image changes the value of the threshold for keeping only the desirable branches. Training a neural network to learn to extract a clean skeleton directly, without relying on a threshold, would be a significant contribution to skeleton extraction. In addition, recent deep learning algorithms have shown great results in tasks requiring dimensionality reduction and such approaches could be easily applied to the shape abstraction task we describe in this paper.\n\nOur observation is that deep learning approaches are useful for proposing generalizable and robust solutions since classical skeletonization do lack robustness. Our motivation arises from the fact that such deep learning approaches need comprehensive datasets, similar to 3D shape understanding benchmarks based on ShapeNet~\\cite{shapenet}, SUNCG~\\cite{suncg}, and SUMO~\\cite{sumo} datasets, with corresponding challenges. The tasks and expected results from such networks should also be well-formulated in order to evaluate and compare them properly. We chose skeleton extraction as the main task, to be investigated in pixel, point, and parametric domains in increasing complexity. \n\nIn order to solve the proposed problem with deep learning and direct more attention to geometric shape understanding tasks, we introduce SkelNetOn Challenge. We aim to bring together researchers from computer vision, computer graphics, and mathematics to advance the state of the art in topological and geometric shape analysis using deep learning. The datasets created and released for this competition will serve as reference benchmarks for future research in deep learning for shape understanding. Furthermore, different input and output data representations can become valuable testbeds for the design of robust computer vision and computational geometry algorithms, as well as understanding deep learning models built on representations in 3D and beyond. The three SkelNetOns are defined below:\n\\begin{itemize}\n\\item \\textbf{Pixel SkelNetOn:}\nAs the most common data format for segmentation or classification models, our first domain poses the challenge of extracting the skeleton pixels from a given shape in an image. \n\n\\item \\textbf{Point SkelNetOn:} \nThe second challenge track investigates the problem in the point domain, where the shapes will be represented by point clouds as well as the skeletons. \n\n\\item \\textbf{Parametric SkelNetOn:} \nThe last domain aims to push the boundaries to find parametric representation of the skeleton of a shape, given its image. The participants are expected to output skeletons of shapes represented as parametric curves with a radius function. \n\n\\end{itemize}\n\nIn the next section, we introduce how the skeletal models are generated. We then inspect characteristics of our datasets and the annotation process (Section~\\ref{sec:data}), give description of the tasks and formulations of evaluation metrics (Section~\\ref{sec:tasks}), and introduce state-of-the-art methods as well as our baselines (Section~\\ref{sec:res}). The results of the competition will be presented in the Deep Learning for Geometric Shape Understanding Workshop during the 2019 International Conference on Computer Vision and Pattern Recognition (CVPR) in Long Beach, CA on June 17th, 2019.  As of April 17th, more than 200 participants have registered in SkelNetOn competitions and there are 37 valid submissions in the leaderboards over the three tracks. Public leaderboards and the workshop papers are listed in our website\\footnote{http:\/\/ubee.enseeiht.fr\/skelneton\/}.\n\n\\section{Skeletal Models and Generation}\\label{sec:skeletons}\nThe Blum medial axis (BMA) is the original skeletal model, consisting of a collection of points equidistant from the boundary (the skeleton) and their corresponding distances (radii)~\\cite{Blum1973}. The BMA produces skeleton points both inside and outside the shape boundary. Since each set, interior and exterior, reconstructs the boundary exactly, we select the interior skeleton to avoid redundancy. For the interior BMA, skeleton points are centers of circles maximally inscribed within the shape and the radii are their associated circles radii. For a discrete sampling of the boundary, Voronoi approaches to estimating the medial axis are well-known, and proven to converge to the true BMA as the boundary sampling becomes dense~\\cite{ogniewicz1994}. In the following, we refer to the skeleton points and radius of the shape together as the interior BMA. \n\nThe skeleton offers an intuitive and low dimensional representation of the shape that has been exploited for shape recognition, shape matching, and shape animation. However, this representation also suffers from poor robustness: small perturbations of the boundary may cause long branches to appear that model only a small boundary change. Such are uninformative about the shape. These perturbations also depend on the domain of the shape representation, since the noise on the boundary may be the product of coarse resolution, non-uniform sampling, and approximate parameterization. Many approaches have been proposed to remove these uninformative branches from an existing skeleton~\\cite{Attali1997,Chazal2005,Giesen2009}, whereas some more recent methods offer a skeletonization algorithm that directly computes a clean skeleton~\\cite{Durix2019,Leborgne2014}. \n\nWe base our ground truth generation on this second approach. First, we apply one-pixel dilation and erosion operations on the original image to close some negligible holes and remove isolated pixels that might change the topology of the skeleton. We manually adjust the shapes if the closing operation changes the shape topology. Then, the skeleton is computed with 2-, 4-, and 6-pixel thresholds. In other words, the Hausdorff distance from the shape represented by the skeleton to the shape used for the skeletonization is at most 2, 4, or 6 pixels ~\\cite{Durix2019}. We then manually select the skeleton that is visually the most coherent for the shape from among the three approximations to produce a skeleton which has the correct number of branches. Finally, we manually remove some isolated skeleton points or branches if spurious branches still remain. \n\nWe compiled several shape repositories~\\cite{Bronstein2007,Bronstein2008,leonardshape16,Sebastian2004}\nfor our dataset with $1,725$ shapes in 90 categories. We used the aforementioned skeleton extraction pipeline for obtaining the initial skeletons, and created shapes and skeletons in other domains using the shape boundary, skeleton points, and skeleton edges.\n\n\n\n\n\n\\section{Datasets}\\label{sec:data}\nWe converted the shapes and their corresponding ground truth skeletal models into three representation domains: pixels, points, and B\\'ezier curves. This section will discuss these datasets derived from the initial skeletonization. \n\n\\subsection{Shapes and Skeletons in Pixels}\n\\label{sec:skelimage}\nThe image dataset consists of $1,725$ black and white images given in portable network graphics format with size $256\\times256$ pixels, split into $1,218$ training images, $241$ validation images, and $266$ test images.\n\nWe provide samples from every class in both the test and validation sets. There are two types of images: the shape images which represent the shapes in our dataset (Figure~\\ref{fig:pixels}), and the skeleton images which represent the skeletons corresponding to the shape images (Figure~\\ref{fig:pixels}). In the shape images, the background is annotated with black pixels and the shape with a closed polygon filled with white pixels. In the skeleton images, the background is annotated with black pixels and the skeleton is annotated with white pixels. The shapes have no holes; some of them are natural, while others are man-made. If one overlaps a skeleton image with its corresponding shape image, the skeleton will lie in the ``middle\" of the shape (i.e., it would be an approximation of the shape's skeleton). \n \n  \\begin{figure}[hbt!]\n\t\\centering\n   \t\t\\includegraphics[width=1\\linewidth]{img\/new_img_skel.png}\n   \t\\caption{\\textbf{Pixel Dataset.} A subset of shape and skeleton image pairs is demonstrated from our pixel dataset.\n   \t} \\label{fig:pixels}\n   \\end{figure}\n   \nFor generating shape images, inside of the shape boundaries mentioned in Section~\\ref{sec:skeletons} are rendered in white, whereas outside is rendered in black. The renders are then cropped, padded, and downsampled to $256\\times 256$. No noise was added or removed, therefore all expected noise is due to pixelation or resizing effects. For generating skeleton images, the skeleton points as well as all pixels linearly falling between two consecutive skeleton points are rendered in white, on a black background. By definition of the skeletons from the original generation process, we assume adjacency within 8-neighborhood in the pixel domain, and provide connected skeletons.\n \n\n\\subsection{Shapes and Skeletons in Points}\nSimilar to the image dataset, the point dataset consists of $1,725$ shape point clouds and corresponding ground truth skeleton point clouds, given in the basic point cloud export format \\texttt{.pts}. Sample shape point clouds and their corresponding skeleton point clouds are shown in Figure~\\ref{fig:dataset:points}.\n\n\n \\begin{figure}[hb]\n   \t\t\\centering\n   \t\t\\includegraphics[width=1\\linewidth]{img\/pt_skel_zoom.png}\n   \t\\caption{\\textbf{Point Dataset.} A subset of our shape and skeleton point cloud pairs is demonstrated. We also emphasize the point sampling using two close ups at the bottom right.}\n   \t\t\\label{fig:dataset:points}\n   \\end{figure}\n\nThe dataset is again split into $1,218$ training point clouds, $241$ validation point clouds, and $266$ test point clouds. We derive the point dataset by extracting the boundaries of the shapes (mentioned in Section~\\ref{sec:skeletons}) as two-dimensional point clouds. We fill the closed region within this boundary by points that implicitly lie on a grid with granularity $h=1$. After experimenting with over\/under sampling the point cloud with different values of $h$, we end up with this balancing value because the generated point clouds were representative enough to not lose details, and still computationally reasonable to process. Even though the average discretization parameter is given as $h=1$ in the provided dataset, we shared scripts\\footnote{https:\/\/github.com\/ilkedemir\/SkelNetOn} in the competition starting kit to coarsen or populate the provided point clouds so participants are able to experiment with different granularities of the point cloud representation. To prevent the comfort of regularity which is observed in the pixel domain, we add some uniformly distributed noise and avoid any structural dependency in the later computed results. The noise is scaled by the sampling density, and we also provide scripts to apply noise with other probability distributions, such as Gaussian noise. \n\nGround truth for the skeletal models are given as a second point cloud which only contains the points representing the skeleton. To compute the skeletal point clouds, we computed the proximity of each point in the shape point cloud to the skeleton points and skeleton edges from the original dataset. Shape points closer than a threshold (depending on $h$) to any original skeleton points or edges in Euclidean space are accepted for the skeleton point cloud. This generation process allows one-to-one matching of each point in the skeleton point cloud to a point in the shape point cloud, thus the ground truth can be converted to labels if the task in hand would be assumed as a segmentation task. \n\n\n\\subsection{Parametric Skeletons}\\label{sec:skelprm}\nFinally, the parametric skeleton dataset consists of $1,725$ shape images and corresponding ground truth parametric skeletons, exported in tab separated \\texttt{.csv} format. The dataset is again split into $1,218$ training shapes, $241$ validation shapes, and $266$ test shapes. The shape images are created as discussed in Section~\\ref{sec:skelimage}, and parametric skeletons are modeled as degree five B\\'ezier curves. Each curve corresponds to a branch of the skeleton, where the first two coordinates describe the $\\{x,y\\}$ location in the image of an inscribed circle center, and the third coordinate is the radius $r$ associated with the point. Output is a vector containing 3D $(x, y, r)$ coordinates of the control points of each branch.\n\\begin{equation}\n    v = [x_0^0, y_0^0, r_0^0, x_1^0, y_1^0, r_1^0, .. x_5^0, y_5^0, r_5^0, x_0^1, y_0^1, r_0^1 ..],\n\\end{equation}\nwhere $b_i^j = (x_i^j,y_i^j,r_i^j)$ is the $i$-th control point of the $j$-th branch in the medial axis.\n\nFrom the simply connected shapes of the dataset mentioned in Section~\\ref{sec:skeletons}, we first extract a clean medial axis representation. For a simply connected shape, the skeleton is a tree, whose joints and endpoints are connected by curves, which we call proto-branches. Unfortunately, the structure of the tree is not stable. Because skeletons are unstable in the presence of noise, a new branch due to a small perturbation of the boundary could appear and break a proto-branch into two branches. Moreover the tree structure gives a dependency and partial order between the branches, not a total order. To obtain a canonical parametric representation of the skeleton, we first merge branches that have been broken by a less important branch. We then order the reduced set of branches according to branch importance. For both steps, we use a salience measure, the Weighted Extended Distance Function (WEDF) function on the skeleton~\\cite{leonardshape16}, to determine the relative importance of branches. The WEDF function has been shown to measure relative importance of shape parts in a way that matches human perception \\cite{carlier2016}. \n\n\\paragraph{Merging branches.} \nFirst, we identify  pairs of adjacent branches that should be joined to represent a single curve: a branch is split into two parts at a junction induced by a child branch of lower importance if the WEDF function is continuous across  the junction. When two child branches joining the parent branch are of equal importance, then the junction is considered an end point of the parent curve. Figure~\\ref{fig:parametric} shows the resulting curves.\n\n\\paragraph{Computation of the B\\'ezier approximation.} \nEach individual curve resulting from the merging process is approximated by a B\\'ezier curve of degree five, whose control points have three parameters $(x,y,r)$, where $(x,y)$ are coordinates, and $r$ is the radius. The end points of the curve are interpolated, and the remaining points are determined by a least square approximation. \n\n\\paragraph{Branch ordering}\nWe then order the branches by importance to have a canonical representation. We estimate a branch importance by the maximal WEDF value of a point in the branch. The branches can then be ordered, and their successive list of control points is the desired output.\n\n\\begin{figure}[hbt!]\n   \t\t\\centering\n   \t\t\\includegraphics[width=0.5\\linewidth]{img\/parametric_mousse.pdf}\n   \t\t\\caption{\\textbf{Skeletal Branches.} The red curve passes continuously through the arm and nose curves. However, legs (bottom) and ears (top) split the main red curve into two parts of equal importance, becoming the end-points. }\n   \t\t\\label{fig:parametric}\n\\end{figure}\n\n\n\\section{Tasks and Evaluations}\\label{sec:tasks}\nAlthough there are different choices of evaluation metrics specific for each data modality, we formulate our metrics in accordance with the tasks for each domain.\n\n\\subsection{Pixel Skeleton Classification}\nGenerating skeleton images from the given shape images can be posed as a pixel-wise binary classification task, or an image generation task. This makes it possible to evaluate performance by comparing a generated skeleton image, pixel by pixel, to its ground truth skeleton image. Such a comparison automatically accounts for common errors seen in skeleton extraction algorithms such as lack of connectivity, double-pixel width, and branch complexity. \n\nHowever, using a simple $L1$ loss measurement would  provide a biased evaluation of image similarity. One can see this by looking at any of the skeleton images: the black background pixels far outnumber the white skeleton ones, giving the former much more significance in the final value of the computed $L1$ loss. To minimize the effects of class imbalance, the evaluation is performed using the $F1$ score, which takes into account the number of skeleton and background pixels in the ground truth and generated images. This is consistent with metrics used in the literature and will enable further comparisons with future work~\\cite{shen2017deepskeleton}. The number of skeleton pixels (positive) or background pixels (negative) is first counted in both the generated and ground truth skeleton images. The $F1$ score is then calculated from the harmonic average of the precision and recall values as follows: \n \\begin{eqnarray}\n \tF1 = \\dfrac{2 \\times precision \\times recall } { precision + recall},\n \\end{eqnarray}\nusing \n\\begin{eqnarray}\n    Precision = \\dfrac{TP} {TP + FP} \\nonumber \\\\\n    Recall = \\dfrac{TP}{TP + FN},\n\\end{eqnarray}\nwhere $TP$, $FN$, and $FP$ stand for number of pixels for true positives, false negatives, and false positives respectively. \n\n\\subsection{Point Skeleton Extraction}\nGiven a 2D point set representing a shape, the goal of the point skeleton extraction task is to output a set of point coordinates corresponding to the given shape's skeleton. This can be approached as a binary point classification task or a point generation task, both of which end up producing a skeleton point cloud that approximate the shape skeleton. The output set of skeletal points need not be part of the original input point set. The evaluation metric for this task needs to be invariant to the number and ordering of the points. The metric should also be flexible for different point sampling distributions representing the same skeleton. Therefore, the results are evaluated using the symmetric Chamfer distance function, defined by:\n\\begin{eqnarray}\nCh(A,B) = \\frac{1}{|A|}\\sum_{a\\in A} \\min _ {b\\in B} ||a-b||_2 + \\nonumber \\\\  \\frac{1}{|B|}\\sum_{b\\in B} \\min _ {a\\in A} ||a-b||_2,\n\\end{eqnarray}\nwhere $A$ and $B$ represent the skeleton point sets to be compared, $|.|$ denotes set cardinality, and $||.||$ denotes the Euclidean distance between two points. We use a fast vectorized \\texttt{Numpy} implementation of this function in order to compute Chamfer distances quickly in our evaluation script.\n\n\\subsection{Parametric Skeleton Generation}\nThe parametric skeleton extraction task aims to recover the medial axis as a set of 3D parametric curves from an input image of a shape. Following the shape and parametric skeleton notations introduced in Section~\\ref{sec:skelprm}, different metrics can be proposed to evaluate such a representation. In particular, we can measure either the distance of the output medial axis to the ground-truth skeleton, or its distance to the shape described in the input image. Although the second method looks better adapted to the task, it does not take into account several properties of the medial axis. It would be difficult, for example, to penalize disconnected branches or redundant parts of the medial axis.\n\nWe evaluate our results by distance to the ground truth medial axes in our database, since the proposed skeletal representation in the dataset already guarantees the properties introduced above and in Section~\\ref{sec:skelprm}, and are ordered in a deterministic order. We use the mean squared distance between the control points on the original and predicted branches as:\n\\begin{equation}\n    MSD(b,\\tilde{b}) = \\frac{1}{6}\\sum_{i=0}^5 \\left(  (x_i-\\tilde{x}_i)^2 + (y_i-\\tilde{y}_i)^2 + (r_i-\\tilde{r}_i)^2 \\right),\n\\end{equation}\nwhere $b = {(x_i,y_i,r_i)}_{i=\\{0..5\\}}$ is a branch in the ground truth data, and $\\tilde{b} = {(\\tilde{x}_i,\\tilde{y}_i,\\tilde{r}_i)}_{i=\\{0..5\\}}$ is the corresponding branch in the output data.\n\nThe evaluation metric needs to take into account models with an incorrect number of branches, since this number is different for each shape. We penalize each missing (or extra) branch in the output with a measure on the length of the branch in the ground truth (or in the output). We use a measure called missing branch error (MBE) for each missing or extra branch $b$:\n\\begin{equation}\n    MBE(b) = \\frac{1}{5}\\sum_{i=0}^4 (x_{i+1} - x_i)^2 + (y_{i+1} - y_i)^2\n + \\frac{1}{6}\\sum_{i=0}^5 r_i^2.\n\\end{equation}\n\nFinally, the evaluation function between an output vector $\\tilde{V}$ and its associated ground truth $V$ is defined as: \n\\begin{equation}\nD(V,\\tilde{V}) = \\frac{1}{N_b} \\left( \n        \\sum_{j=0}^{n_b-1} MSD(b^j,\\tilde{b}^j) + \n        \\sum_{j=n_b}^{N_b-1} MBE(\\hat{b}^j)\n        \\right )\n\\end{equation}\nwhere $N_b,n_b$ are respectively the number minimal and maximal of branches in the ground truth and in the output, and $\\hat{b}$ are branches in the vector containing the maximal number of branches.\n\n\\section{State-of-the-art and Baselines}\\label{sec:res}\nSkeleton extraction has been an interesting topic of research for different domains. We briefly introduce these approaches in addition to our own deep learning based baselines. The participants are expected to surpass these baselines in order to be eligible for the prizes. \n\n\\subsection{Pixel Skeleton Results}\nEarly work on skeleton extraction was based on using segmented images as input \\cite{bai2007skeleton}. A comprehensive overview of previous approaches using pre-segmented inputs is presented in \\cite{saha2016survey}. With the advancement of neural network technology, the most recent approaches perform skeletonization from natural images with fully convolutional neural networks \\cite{shen2017deepskeleton,shen2016object}. In this challenge, we consider the former type of skeletonization task using pre-segmented input images.\n\nWe formulate the task of skeletonization as image translation from pre-segmented images to skeletons with a conditional adversarial neural network. For our baseline result, we use a vanilla pix2pix\nmodel as proposed in~\\cite{isola2017image}. We apply a distance transform to preprocess the binary input shape as illustrated in Figure~\\ref{fig:dis}. We found that this preprocessing of the input data enhances the neural network learning significantly. The model is trained with stochastic gradient descent using the $L1$-image loss for $200$ epochs. We measure performance in terms of $F1$ score achieving a test accuracy of $\\textbf{0.6244}$ on the proposed validation set.  \n\\begin{figure}[ht]\n\t\\centering\n   \t\\begin{subfigure}{0.5\\linewidth}\n   \t\t\\centering\n   \t\t\\includegraphics[width=0.7\\linewidth]{img\/bat-7.png}\n   \t\t\\caption{}\n   \t\t\\label{fig:dis-shape}\n   \t\\end{subfigure\n   \t\\begin{subfigure}{0.49\\linewidth}\n   \t\t\\centering\n   \t\t\\includegraphics[width=0.7\\linewidth]{img\/bat-7-inputs.png}\n   \t\t\\caption{}\n   \t\t\\label{fig:dis-dis}\n   \t\\end{subfigure}\\\\\n   \t\\begin{subfigure}{0.5\\linewidth}\n   \t\t\\centering\n   \t\t\\includegraphics[width=0.7\\linewidth]{img\/bat-7-outputs.png}\n   \t\t\\caption{}\n   \t\t\\label{fig:dis-shape}\n   \t\\end{subfigure\n   \t\\begin{subfigure}{0.49\\linewidth}\n   \t\t\\centering\n   \t\t\\includegraphics[width=0.7\\linewidth]{img\/bat-7-targets.png}\n   \t\t\\caption{}\n   \t\t\\label{fig:dis-dis}\n   \t\\end{subfigure}\\\\   \t\n\t\\caption{\\textbf{Pixel SkelNetOn Baseline.} (a) Original shape image. (b) Distance transformed shape image. (c) Baseline prediction result. (d) Ground truth skeleton. }\n\t\\label{fig:dis}\n\\end{figure}\n\n\\subsection{Point Skeleton Results}\nSkeleton and medial axis extraction from point clouds have been extensively explored in 2D and 3D domains by using several techniques such as Laplacian contraction~\\cite{cao10}, mass transport~\\cite{jalba16}, medial scaffold~\\cite{leymarie07}, locally optimal projection~\\cite{Lipman07}, maximal tangent ball~\\cite{Ma2012}, and local $L_1$ medians~\\cite{Huang13}. Although these approaches extract approximate skeletons, the dependency on sampling density, non-uniform and incomplete shapes, and the inherent noise in the point clouds are still open topics that deep learning approaches can handle implicitly. One such recent approach (P2PNet~\\cite{p2pnet}) builds a Siamese architecture to build spatial transformer networks for learning how points can be moved from a shape surface to the medial axis and vice versa.\n\nBased on the definition of this task in Section~\\ref{sec:tasks}, it can be formulated as (i) a generation task to create a point cloud by learning the distribution from the original shapes and skeleton points, (ii) a segmentation task to label the points of a shape as skeletal and non-skeletal points, and (ii) a spatial task to learn the pointwise translations to transform the given shape to its skeleton. We chose the second approach to utilize state-of-the-art point networks. First, we obtained ground truth skeletons as labels for the original point cloud (skeletal label as $1$, and non-skeletal labeled as $2$). Then we experimented with PointNet++~\\cite{pointnet} architecture's part-segmentation module to classify each pixel into two classes, defined by the labels. As the dimensionality of the classes are inherently different, we observed that the skeleton class collapses. Similarly, if we over-sample the skeleton, non-skeleton class collapses. To overcome this problem, we experimented with dynamic weighting of samples, so that the skeletal points are trusted more than non-skeletal points. The weights are adjusted at every $10^{th}$ epoch as well as the learning rate, to keep the balance of the classes. Our approach achieved an accuracy of $58.93\\%$, measured by the mean IoU over all classes. Even though the mIoU is a representative metric for our task, we would like to evaluate our results better by calculating the Chamfer distance of skeletons in shape space in future.\n\n\\subsection{Parametric Medial Axis Results}\n\nParametric representations of the medial axis have been proposed before, in particular representations with intuitive control parameters.  For example, Yushkevich. et al, \\cite{yushkevich2003} use a cubic B-spline model with control points and radii; similarly Lam et al. \\cite{lam2007} use piecewise B\\'ezier curves.\n\nWe train a Resnet-50~\\cite{He2016} modified for regression. Inputs are segmented binary images, and outputs are six control points in $\\mathbb{R}^3$ that give $(x,y,r)$ values for the degree five B\\'ezier curves used to model medial axis branches.\n\nThe parametric data is the most challenging of the three ground truths presented here, for two reasons. First, the number of entries in the ground truth varies with the number of branches in the skeleton. Second, the B\\'ezier control points do not encode information about the connectivity of the skeleton graph. To overcome the first challenge, we set all shapes to have $5$ branches. For those shapes with fewer than $5$ branches, we select the medial point with the maximum WEDF value (a center of the shape) as the points in any non-existent branches, and set the corresponding radii to zero. We do not address the second issue for our baseline results. We use a simple mean squared error measure on the output coordinates as our loss function. While more branches are desirable to obtain quality models, there is a trade-off between the number of meaningless entries in the ground truth for a simple shape (with a low number of branches), and providing for an adequate number of branches in the ground truth of a complex shape (with many branches).\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{img\/param_results.png}\n    \\caption{\\textbf{Parametric SkelNetOn Baseline.} Input image (left), ground truth parametric medial axis (middle), prediction from our model (right).}\n    \\label{fig:paramres}\n\\end{figure}\n\nWe obtain a validation loss of $1405$ pixels, with a corresponding training loss of $121$ pixels, using early stopping. As a reference, the loss at the beginning of training is in the order of $50,000$ pixels.\n\nAs shown in Figure \\ref{fig:paramres}, branch connectivity is only occasionally produced by our trained model. The model does well with location, orientation, and scale of shape, but often misses long and narrow appendages such as the guitar neck (third row). A future work is to incorporate other constraints in the architecture to encourage connectivity of branches. \n\n\\section{Conclusions}\nWe presented SkelNetOn Challenge at Deep Learning for Geometric Shape Understanding, in conjunction with CVPR~2019. Our challenge provides datasets, competitions, and a workshop structure around skeleton extraction in three domains as Pixel SkelNetOn, Point SkelNetOn, and Parametric SkelNetOn. We introduced our dataset analysis, formulated our evaluation metrics following our competition tasks, and shared our preliminary results as baselines following the previous work in each domain. \n\nWe believe that SkelNetOn has the potential to become a fundamental benchmark for the intersection of deep learning and geometry understanding. We foresee that the challenge and the workshop will enable more collaborative research of different disciplines on the crossroads of shapes. Ultimately, we envision that such deep learning approaches can be used to extract expressive parametric and hierarchical representations that can be utilized for generative models \\cite{3dgan} and for proceduralization \\cite{demir18}.\n\\section*{Acknowledgments}\nWithout the contributions of the rest of our technical and program committees, this workshop would not happen, so thank you everyone, in particular Veronika Schulze for her contributions throughout the project, Daniel Aliaga as the papers chair, and Bedrich Benes and Bernhard Egger for the support. The authors thank AWM's Women in Shape (WiSH) Research Network, funded by the NSF AWM Advances! grant. We also thank the University of Trier Algorithmic Optimization group, who provided funding for the workshop where this challenge was devised. We would also like to thank NVIDIA for being the prize sponsor of the competitions. Lastly, we would like to acknowledge the support and motivation of the CVPR chairs and crew. \n{\\small\n\\bibliographystyle{ieee}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\vglue-3mm\\subsection{Motivation}\n\\noindent\nIn a famous paper from 1960, Erd\\H os and Taylor~\\cite{ET60} studied the most-frequently visited site by the simple random walk on~$\\mathbb Z^2$ of time-length~$n$. They showed that the time spent at that site is of order~$(\\log n)^2$ and conjectured that the time is asymptotically sharp on that scale. This conjecture was proved in 2001 by Dembo, Peres, Rosen and Zeitouni~\\cite{DPRZ01} (see also Rosen~\\cite{R05}) who in addition described the multifractal structure of the set of \\myemph{thick points}; namely, those points where the local time is at least  a given positive multiple of its maximum. The problem has been revisited numerous times; e.g., by Dembo, Peres, Rosen and Zeitouni~\\cite{DPRZ06} who studied random walk late points, by Okada~\\cite{O16} who studied the most visited site on the inner boundary of the range, or by Jego~\\cite{J18} who extended the results of \\cite{DPRZ01,R05} to more general random walks.\n\nOver the past two decades, it has become increasingly clear that many questions about the local time can be usefully rephrased as questions about an associated Discrete Gaussian Free Field (DGFF). This connection,  discovered originally in mathematical physics (Symanzik~\\cite{Symanzik}, Brydges, Fr\\\"ohlich and Spencer~\\cite{BFS}), is now elegantly expressed via Dynkin-type Isomorphism\/Second Ray-Knight theorems (Dynkin~\\cite{D83}, Eisenbaum, Kaspi, Marcus, Rosen and Shi~\\cite{EKMRS}) which underlie many important objects in current probability; e.g., random interlacements (Sznitman~\\cite{S12}, Rodriguez~\\cite{R13}, etc), loop-soups (Lawler and Werner~\\cite{LW},  Le Jan~\\cite{LeJan}, Lupu~\\cite{Lupu}, etc) and the cover time (Ding, Lee and Peres~\\cite{DLP12}, Ding~\\cite{D14}, etc). Studying the random walk at times proportional to the cover time is particularly interesting as it permits consideration of  \\myemph{thin points}; namely, those where the local time is less than a fraction of its typical value. These exhibit a multifractal structure as well, as shown in Abe~\\cite{A15} for the random walk on $N\\times N$ torus.\n\nIn the present paper we study the precise statistics of the thick and thin points for a two-dimensional simple random walk run for times of the order of the cover time. Closely related to this is the set of \\myemph{avoided points}, i.e., those not visited at all, as well as the set of \\myemph{light points}, where the local time is at most a given constant. We show that all these level sets are intimately connected with the corresponding (so called intermediate) level sets of the Discrete Gaussian Free Field studied earlier by O.~Louidor and the second author~\\cite{BL4}. In particular, their limiting statistics is captured by the Liouville Quantum Gravity measures introduced and studied by Duplantier and Sheffield~\\cite{DS}.\n\n\\subsection{Setting for the random walk}\n\\label{sec1.2}\\noindent\nIn order to take full advantage of the prior work~\\cite{BL4} on the DGFF, we will consider a slightly different setting than references~\\cite{ET60,A15}. Indeed, our random walk will behave as the simple random walk only inside a large finite subset of~$\\mathbb Z^2$; when it exits this set it reenters in the next step through a uniformly-chosen boundary edge.\n\nTo describe the dynamics of our random walk, consider first a general finite, unoriented, connected graph $G = (V\\cup\\{\\varrho\\},E)$, where~$\\varrho$ is a distinguished vertex (not belonging to~$V$). We assume that each edge~$e\\in E$ is endowed with a number~$c_e>0$, called the conductance of~$e$. Let~$X$ denote a continuous-time (constant-speed) Markov chain on~$V\\cup\\{\\varrho\\}$ which makes jumps at independent exponential times to a neighbor selected with the help of transition probabilities\n\\begin{equation}\n\\cmss P (u, v) := \n\\begin{cases} \n\\frac{c_{e}}{\\pi(u)}, &~\\text{if}~e:=(u, v) \\in E, \n\\\\\n0, &~\\text{otherwise},\n\\end{cases}\n\\end{equation}\nwhere $\\pi(u)$ is the sum of~$c_e$ for all edges incident with~$u$. We will use~$P^u$ to denote the law of~$X$ with~$P^u(X_0=u)=1$. \n\nGiven a path~$X$ of the above Markov chain, the local time at~$v\\in V\\cup\\{\\varrho\\}$ at time~$n$ is then given by \n\\begin{equation} \n\\label{E:local_time}\n\\ell_t^V(v) := \\frac1{\\pi(u)}\\int_0^t\\text{\\rm d}\\mkern0.5mu s\\,1_{\\{X_s=u\\}},\\quad t\\ge0,\n\\end{equation}\nwhere the normalization by $\\pi(u)$ ensures that the leading-order growth of~$t\\mapsto\\ell_t^V(v)$ is the same for all vertices. \nWe will henceforth work in the time parametrization by the local time at the distinguished vertex~$\\varrho$. For this we set $\\hat\\tau_{\\varrho}(t):=\\inf\\{s\\ge0\\colon\\ell_s^V(\\varrho)\\ge t\\}$ and denote\n\\begin{equation}\n\\label{E:LVt}\nL_t^V(v):=\\ell_{\\hat\\tau_{\\varrho}(t)}^V(v).\n\\end{equation}\nIn this parametrization, $t$ is the (leading-order) value of~$L^V_t$ at the typical vertex of~$V$.\n\nOur derivations will make heavy use of the connection of the above Markov chain with an instance of the Discrete Gaussian Free Field (DGFF). Denoting by\n\\begin{equation}\nH_{v} := \\inf \\bigl\\{t \\geq 0 \\colon X_t = v \\bigr\\}\n\\end{equation}\nthe first hitting time of vertex~$v$, this DGFF is the centered Gaussian process $\\{h_v^{V}\\colon v \\in V\\}$ with covariances given by\n\\begin{equation}\n\\label{E:cov}\n\\mathbb E\\bigl(h_u^{V} h_v^{V} \\bigr) = G^{V} (u, v) :=E^u\\bigl(\\ell_{H_{\\varrho}}^V(v)\\bigr).\n\\end{equation} \nHere and henceforth,~$\\mathbb E$ denotes expectation with respect to the law~$\\mathbb P$ of~$h^V$. The field naturally extends to~$\\varrho$ by~$h^V_\\varrho=0$. \n\n\\smallskip\nReturning back to random walks on~$\\mathbb Z^2$, in our setting  $V$ stands for  a large finite subset~$V\\subset\\mathbb Z^2$ while~$\\varrho$ represents the ``\\myemph{boundary vertex};'' namely, the set of vertices outside~$V$, but with a neighbor in~$V$. The set of edges~$E$ is that between the nearest-neighbor pairs  in~$V$ plus all the edges from~$V$ to~$\\mathbb Z^2\\smallsetminus V$ that now ``end'' in~$\\varrho$; see Fig.~\\ref{fig1}. The transition rule  of the Markov chain  is that of the simple random walk on the underlying graph; indeed, all conductances take a unit value, $c_e:=1$, at all  the involved  edges including those incident with~$\\varrho$. The DGFF associated with this network then corresponds to the ``standard'' DGFF in~$V$ (cf the review by Biskup~\\cite{B-notes}) with zero boundary conditions outside~$V$ except that our normalization is slightly different than the one used in~\\cite{B-notes} --- indeed, our fields are smaller by a multiplicative factor of~$2$ than those in \\cite{B-notes}.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[height = 2.0in]{.\/figs\/domain.pdf}\n}\n\\begin{quote}\n\\small \n\\vglue-0.2cm\n\\caption{\n\\label{fig1}\nThe graph corresponding to~$V$ being the square of $6\\times 6$ vertices. Each vertex on the outer perimeter of~$V$ has an edge to the ``boundary vertex''~$\\varrho$; the corner vertices that have two edges to~$\\varrho$. The ``boundary vertex'' plays the role of the wired boundary condition used often in statistical mechanics. For us this ensures that the associated DGFF vanishes outside~$V$.}\n\\normalsize\n\\end{quote}\n\\end{figure}\n\nFor the lattice domains, we will take sequences of subsets of~$\\mathbb Z^2$ that approximate, in the scaling limit, well-behaved continuum domains. The following definitions are taken from Biskup and Louidor~\\cite{BL2}:\n\n\n\\begin{definition} \nAn admissible domain \nis a bounded open subset of $\\mathbb{R}^2$ \nthat consists of a finite number of connected components\nand whose boundary is composed of a finite number of connected sets each of which has \na positive Euclidean diameter.\n\\end{definition} \n\nWe write $\\mathfrak{D}$ to denote the family of all admissible domains and let $d_{\\infty} (\\cdot, \\cdot)$ denote the $\\ell^{\\infty}$-distance on $\\mathbb{R}^2$.\n\n\\begin{definition}\nAn admissible lattice approximation of $D \\in \\mathfrak{D}$ is a sequence $D_N\\subset\\mathbb{Z}^2$ such that the following holds: There is~$N_0\\in\\mathbb N$ such that for all~$N\\ge N_0$ we have\n\\begin{equation}\n\\label{E:1.8i}\nD_N \\subseteq \\Bigl\\{x \\in \\mathbb{Z}^2 : \nd_{\\infty}\\bigl(\\ffrac{x}{N}, \\mathbb{R}^2 \\smallsetminus D\\bigr) > \\frac{1}{N} \\Bigr\\}\n\\end{equation}\nand, for any~$\\delta>0$ there is also~$N_1\\in\\mathbb N$ such that for all~$N\\ge N_1$,\n\\begin{equation}\nD_N \\supseteq \\bigl\\{x \\in \\mathbb{Z}^2 : \nd_{\\infty} (\\ffrac{x}{N}, \\mathbb{R}^2 \\smallsetminus D) > \\delta \\bigr\\}.\n\\end{equation}\n\\end{definition} \n\nAs shown in \\cite{BL2}, these choices ensure that the discrete harmonic measure on~$D_N$ tends, under the scaling of space by~$N$, weakly to the harmonic measure on~$D$. This yields a precise asymptotic expansion of the associated Green functions; see \\cite[Chapter~1]{B-notes} for a detailed exposition. In particular, we have $G^{D_N}(x,x)=g\\log N+O(1)$ for\n\\begin{equation}\ng:=\\frac1{2\\pi}\n\\end{equation}\nwhenever~$x$ is deep inside~$D_N$. (This is by a factor~$4$ smaller than the corresponding constant in~\\cite{B-notes,BL2}.)\n\nOur random walk will invariably start from the ``boundary vertex''~$\\varrho$; throughout we will thus write~$P^\\varrho$ for the corresponding law of the Markov chain~$X$. (This law depends on~$N$ but we suppress that notationally.) \n\n\n\n\\section{Main results}\n\\noindent\nOur aim in this work is to describe the random walk at times that correspond to a $\\theta$-multiple of the \\myemph{cover time}, for every~$\\theta>0$. Recall that the cover time of a graph is the first time that every vertex  of the graph has been visited. Although this is a random quantity, it is quite well concentrated  (provided that the maximal hitting time is of smaller order than the expected cover time; see Aldous~\\cite{Aldous}).\n In particular, at the cover time of~$D_N$ the local time at a typical vertex is asymptotic to~$2g(\\log N)^2$.  This suggests that we henceforth take~$t$ proportional to $(\\log N)^2$ as $N\\to\\infty$.  \n\n\\subsection{Maximum, minimum and exceptional sets}\nLet us begin by noting the range of values that the local time takes on~$D_N$:\n\n\\begin{theorem}\n\\label{thm-minmax}\nLet~$\\{t_N\\}$ be a positive sequence such that, for some~$\\theta>0$,\n\\begin{equation}\n\\label{E:1.12}\n\\lim_{N\\to\\infty}\\frac{t_N}{(\\log N)^2}=2g\\theta.\n\\end{equation}\nThen for any $D\\in\\mathfrak D$, any admissible sequence~$\\{D_N\\}$ of lattice approximations of~$D$, the following limits hold in $P^{\\varrho}$-probability: \\normalcolor\n\\begin{equation}\n\\label{E:max}\n\\frac1{(\\log N)^2}\\,\\max_{x\\in D_N} L^{D_N}_{t_N}(x)\\,\\,\\,\\underset{ N\\to\\infty}\\longrightarrow\\,\\,\\,2 g\\bigl(\\sqrt\\theta+1\\bigr)^2\n\\end{equation}\nand\n\\begin{equation}\n\\label{E:min}\n\\frac1{(\\log N)^2}\\,\\min_{x\\in D_N} L_{t_N}^{D_N}(x)\\,\\,\\,\\underset{ N\\to\\infty}\\longrightarrow\\,\\,\\,2 g\\bigl[(\\sqrt\\theta-1)\\vee0\\,\\bigr]^2.\n\\end{equation}\n\\end{theorem}\n\nThese conclusions have previously been obtained by Abe~\\cite[Corollary~1.3]{A15} for the  continuous-time  walk on~$N\\times N$ torus.\nAs is checked from \\eqref{E:min}, the cover time indeed corresponds to $\\theta=1$. Noting that the typical value of the local time at a~$\\theta$-multiple of the cover time is asymptotic to~$2g\\theta(\\log N)^2$,  we are naturally led to consider  the set of $\\lambda$-\\myemph{thick points},\n\\begin{equation}\n\\label{E:Lplus}\n\\mathcal T_N^{+}(\\theta,\\lambda):=\\Bigl\\{x\\in D_N\\colon L_{t_N}^{D_N}(x)\\ge 2g(\\sqrt\\theta+\\lambda)^2(\\log N)^2\\Bigr\\},\\quad \\lambda\\in(0,1],\n\\end{equation}\nand $\\lambda$-\\myemph{thin points},\n\\begin{equation}\n\\label{E:Lminus}\n\\mathcal T_N^{-}(\\theta,\\lambda):=\\Bigl\\{x\\in D_N\\colon L_{t_N}^{D_N}(x)\\le 2g(\\sqrt\\theta-\\lambda)^2(\\log N)^2\\Bigr\\},\\quad\\lambda\\in(0,\\sqrt\\theta\\wedge1],\n\\end{equation}\nwhere the upper bounds on~$\\lambda$ reflect on \\twoeqref{E:max}{E:min}. \nAs a boundary case of $\\mathcal T_N^{-}(\\theta,\\lambda)$, we single out the set of $r$-\\myemph{light points},\n\\begin{equation}\n\\mathcal L_N(\\theta,r):=\\bigl\\{x\\in D_N\\colon L_{t_N}^{D_N}(x)\\le r\\bigr\\},\\quad r\\ge0,\n\\end{equation}\nincluding the special case of the set of \\myemph{avoided points},\n\\begin{equation}\n\\AA_N(\\theta):=\\bigl\\{x\\in D_N\\colon L_{t_N}^{D_N}(x)=0\\bigr\\}.\n\\end{equation}\nBy \\eqref{E:min}, the latter two sets will only be relevant for~$\\theta\\in(0,1]$.\nOur aim is to describe the scaling limit of these sets in the limit as~$N\\to\\infty$.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=0.4\\textwidth]{.\/figs\/thick.pdf}\n\\includegraphics[width=0.4\\textwidth]{.\/figs\/thin.pdf}\n}\n\\begin{quote}\n\\small \n\\vglue-0.2cm\n\\caption{\n\\label{fig2}\nPlots of the $\\lambda$-thick (left) and $\\lambda$-thin (right) level sets for the same sample of the random walk on a square of side length~$1000$ and parameter choices $\\theta:=10$ and $\\lambda=0.1$.}\n\\normalsize\n\\end{quote}\n\\end{figure}\n\n\n\\subsection{Digression on exceptional sets of DGFF}\nAs noted previously, the second author and O.~Louidor~\\cite{BL4} have addressed similar questions in the context of the DGFF. There the maximum of~$h^{D_N}$ is asymptotic to~$2\\sqrt g\\log N$ and so the set of $\\lambda$-thick points is naturally defined as that where the field is  exceeds $2\\lambda\\sqrt g\\log N$. It was noted that taking a limit of these sets directly does not lead to interesting conclusions as, after scaling space by~$N$, they become increasingly dense in~$D$. A proper way to capture their structure is via the random measure\n\\begin{equation}\n\\label{E:etaDGFF}\n\\eta_N^D:=\\frac1{K_N}\\sum_{x\\in D_N}\\delta_{x\/N}\\otimes\\delta_{h^{D_N}_x-a_N},\n\\end{equation}\nwhere~$\\{a_N\\}$ is a centering sequence with the asymptotic $a_N\\sim 2\\lambda\\sqrt{g}\\log N$ and\n\\begin{equation}\n\\label{E:1.19e}\nK_N:=\\frac{N^2}{\\sqrt{\\log N}}\\,\\text{\\rm  e}\\mkern0.7mu^{-\\frac{a_N^2}{2g\\log N}}.\n\\end{equation}\nIn \\cite[Theorem~2.1]{BL4} it was then shown that, for each~$\\lambda\\in(0,1)$ there is~$\\fraktura c(\\lambda)>0$ such that, in the sense of vague convergence of measures on~$\\overline D\\times(\\mathbb R\\cup\\{\\infty\\})$,\n\\begin{equation}\n\\label{E:1.19}\n\\eta_N^D\\,\\,\\underset{N\\to\\infty}{\\,\\overset{\\text{\\rm law}}\\longrightarrow\\,}\\,\\,\\fraktura c(\\lambda)\\,Z^D_\\lambda(\\text{\\rm d}\\mkern0.5mu x)\\otimes\\text{\\rm  e}\\mkern0.7mu^{-\\alpha\\lambda h}\\text{\\rm d}\\mkern0.5mu h,\n\\end{equation}\nwhere $\\alpha:=2\/\\sqrt g$ and~$Z_\\lambda^D$ is a random measure in~$D$ called the Liouville Quantum Gravity (LQG)  at parameter~$\\lambda$-times critical. The constant~$\\fraktura c(\\lambda)$, given explicitly in terms of~$\\lambda$  and the constants in the asymptotic expansion of the potential kernel on~$\\mathbb Z^2$, allows us take~$Z^D_\\lambda$ to be normalized so that, for each Borel set~$A\\subseteq D$,\n\\begin{equation}\n\\label{E:1.19a}\n\\mathbb E Z_\\lambda^D(A)=\\int_A r^{\\,D}(x)^{2\\lambda^2}\\text{\\rm d}\\mkern0.5mu x,\n\\end{equation} \nwhere~$r^D$ is an explicit function supported on~$D$ that, for~$D$ simply connected, is simply the conformal radius; see~\\cite[(2.10)]{BL4}. \n\nA construction of  the LQG  measures goes back to Kahane's Multiplichative Chaos theory~\\cite{Kahane}; they were recently reintroduced  and further studied  by Duplantier and Sheffield~\\cite{DS}. Shamov~\\cite{Shamov} characterized the LQG measures for all~$\\lambda\\in(0,1)$ by  their expected value  and the behavior under Cameron-Martin shifts of the underlying continuum Gaussian Free Field.\n\n\n\\subsection{Thick and thin points}\n\\label{sec2.3}\\noindent\nInspired by the above developments, we will thus encode the level sets $\\mathcal T_N^{\\pm}(\\theta,\\lambda)$ via the random measures\n\\begin{equation}\n\\label{E:zetaND}\n\\zeta^D_N:=\\frac1{W_N}\\sum_{x\\in D_N}\\delta_{x\/N}\\otimes\\delta_{(L_{t_N}^{D_N}(x)-a_N)\/\\log N}\\,,\n\\end{equation}\nwhere~$\\{a_N\\}$ is a centering sequence and $\\{t_N\\}$ is a sequence of times, both growing proportionally to~$(\\log N)^2$, and\n\\begin{equation}\n\\label{E:WN}\nW_N:=\\frac{N^2}{\\sqrt{\\log N}}\\text{\\rm  e}\\mkern0.7mu^{-\\frac{(\\sqrt{2t_N}-\\sqrt{2a_N})^2}{2g\\log N}}.\n\\end{equation}\nThe normalization by~$\\log N$ in the second delta-mass in \\eqref{E:zetaND} indicates that we are tracking variations of the local time of scale~$\\log N$. (This is also the order of the variation of the local time between nearest neighbors.)\nWe then get: \n\n\\begin{theorem}[Thick points]\n\\label{thm-thick}\nSuppose $\\{t_N\\}$ and $\\{a_N\\}$ are positive sequences such that, for some $\\theta>0$ and some $\\lambda\\in(0,1)$,\n\\begin{equation}\n\\label{E:1.20}\n\\lim_{N\\to\\infty}\\frac{t_N}{(\\log N)^2}=2g\\theta\\quad\\text{\\rm and}\\quad\\lim_{N\\to\\infty}\\frac{a_N}{(\\log N)^2}=2g(\\sqrt\\theta+\\lambda)^2.\n\\end{equation}\nFor any~$D\\in\\mathfrak D$, any sequence $\\{D_N\\}$ of admissible approximations of~$D$, and for~$X$ sampled from~$P^{\\varrho}$, in the sense of vague convergence of measures on $\\overline D\\times(\\mathbb R\\cup\\{\\infty\\})$,\n\\begin{equation}\n\\label{E:1.21dis}\n\\zeta^D_N\\,\\,\\,\\underset{N\\to\\infty}{\\,\\overset{\\text{\\rm law}}\\longrightarrow\\,}\\,\\,\\, \\frac{\\theta^{1\/4}}{2\\sqrt{g}\\,(\\sqrt\\theta+\\lambda)^{3\/2}}\\,\\,\\fraktura c(\\lambda) Z_\\lambda^{D}(\\text{\\rm d}\\mkern0.5mu x)\\otimes\\text{\\rm  e}\\mkern0.7mu^{-\\alpha(\\theta,\\lambda) h}\\text{\\rm d}\\mkern0.5mu h, \n\\end{equation}\nwhere $\\alpha(\\theta,\\lambda):=\\frac1g\\frac{\\lambda}{\\sqrt\\theta+\\lambda}$\nand $\\fraktura c(\\lambda)$ is as in \\eqref{E:1.19}.\n\\end{theorem}\n\nFor the thin points, we similarly obtain:\n\n\\begin{theorem}[Thin points]\n\\label{thm-thin}\nSuppose $\\{t_N\\}$ and $\\{a_N\\}$ are positive sequences such that, for some $\\theta>0$ and some $\\lambda\\in(0,1\\wedge\\sqrt\\theta)$,\n\\begin{equation}\n\\label{E:1.22}\n\\lim_{N\\to\\infty}\\frac{t_N}{(\\log N)^2}=2g\\theta\\quad\\text{\\rm and}\\quad\\lim_{N\\to\\infty}\\frac{a_N}{(\\log N)^2}=2g(\\sqrt\\theta-\\lambda)^2.\n\\end{equation}\nFor any~$D\\in\\mathfrak D$, any sequence $\\{D_N\\}$ of admissible approximations of~$D$, and for~$X$ sampled from~$P^{\\varrho}$, in the sense of vague convergence of measures on $\\overline D\\times(\\mathbb R\\cup\\{-\\infty\\})$, \n\\begin{equation}\n\\label{E:1.23dis}\n\\zeta^D_N\\,\\,\\,\\underset{N\\to\\infty}{\\,\\overset{\\text{\\rm law}}\\longrightarrow\\,}\\,\\,\\, \\frac{\\theta^{1\/4}}{2\\sqrt{g}\\,(\\sqrt\\theta-\\lambda)^{3\/2}}\\,\\fraktura c(\\lambda) Z_\\lambda^{D}(\\text{\\rm d}\\mkern0.5mu x)\\otimes\\text{\\rm  e}\\mkern0.7mu^{+\\tilde\\alpha(\\theta,\\lambda) h}\\text{\\rm d}\\mkern0.5mu h,\n\\end{equation}\nwhere~$\\tilde\\alpha(\\theta,\\lambda):=\\frac1g\\frac{\\lambda}{\\sqrt\\theta-\\lambda}$ and $\\fraktura c(\\lambda)$ is as in \\eqref{E:1.19}.\n\\end{theorem}\n\nNote that, under \\eqref{E:1.20} or \\eqref{E:1.22}, the above implies\n\\begin{equation}\n|\\mathcal T^\\pm_N(\\theta,\\lambda)|=N^{2(1-\\lambda^2)+o(1)},\n\\end{equation}\nwhere~$o(1)\\to0$ in probability.\nThis conclusion has previously been obtained by the first author in~\\cite[Theorem~1.2]{A15}, albeit for random walks on tori and under a different parametrization of the level sets. The present theorems tell us considerably more. Indeed, they imply that points picked at random from $\\mathcal T^\\pm_N(\\theta,\\lambda)$ have asymptotically the same statistics as those picked from the set where the DGFF is above the $\\lambda$-multiple of its absolute maximum. \n\nThe connection with the DGFF becomes nearly perfect if instead of $\\log N$ we normalize the second coordinate of~$\\zeta^D_N$ by $\\sqrt{2a_N}$. In that parametrization, the resulting measure \\myemph{coincides} (up to reversal of the second coordinate for the thin points) with that for the DGFF up to an overall normalization constant. This  demonstrates \\myemph{universality} of the Gaussian Free Field in these extremal problems. \n\n\\subsection{Light and avoided points}\nThe level sets \\twoeqref{E:Lplus}{E:Lminus} are naturally nested which indicates that, for~$\\theta\\in(0,1)$, also the set of $r$-light points~$\\mathcal L_N(\\theta,r)$ and avoided points $\\AA_N(\\theta)$ should correspond to an intermediate level set of the DGFF, this time with~$\\lambda:=\\sqrt\\theta$. As the next theorem shows, this is true albeit under a  different normalization:\n\n\\begin{theorem}[Light points]\n\\label{thm-light}\nSuppose $\\{t_N\\}$ is a positive sequence such that\n\\begin{equation}\n\\label{E:1.29}\n\\theta:=\\frac1{2g}\\lim_{N\\to\\infty}\\frac{t_N}{(\\log N)^2}\\in (0,1).\n\\end{equation}\nFor any~$D\\in\\mathfrak D$, any sequence $\\{D_N\\}$ of admissible approximations of~$D$, and for~$X$ sampled from~$P^{\\varrho}$, consider the measure\n\\begin{equation}\n\\label{E:varthetaND}\n\\vartheta^D_N:=\\frac1{\\widehat W_N }\\sum_{x\\in D_N}\\delta_{x\/N}\\otimes\\delta_{L_{t_N}^{D_N}(x)},\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{E:1.31}\n\\widehat W_N :=N^2\\text{\\rm  e}\\mkern0.7mu^{-\\frac{t_N}{g\\log N}}.\n\\end{equation}\nThen, in the sense of vague convergence of measures on~$\\overline D\\times[0,\\infty)$,\n\\begin{equation}\n\\label{E:2.22ii}\n\\vartheta^D_N\\,\\,\\,\\underset{N\\to\\infty}{\\,\\overset{\\text{\\rm law}}\\longrightarrow\\,}\\,\\,\\,  \\sqrt{2\\pi g}\\,\\fraktura c(\\sqrt\\theta)\\,\\,  Z_{\\sqrt{\\theta}\\,}^{D}(\\text{\\rm d}\\mkern0.5mu x)\\otimes\\mu(\\text{\\rm d}\\mkern0.5mu h),\n\\end{equation}\nwhere $\\fraktura c(\\lambda)$ is as in \\eqref{E:1.19} and~$\\mu$ is the Borel measure \n\\begin{equation}\n\\label{E:1.33}\n\\mu(\\text{\\rm d}\\mkern0.5mu h):=\\delta_0(\\text{\\rm d}\\mkern0.5mu h)+\\biggl(\\,\\sum_{n=0}^\\infty\\frac1{n!(n+1)!}\\Bigl(\\frac{\\alpha^2\\theta}2\\Bigr)^{n+1} h^n\\biggr)1_{(0,\\infty)}(h)\\,\\text{\\rm d}\\mkern0.5mu h.\n\\end{equation}\n\\end{theorem}\n\nNote that the density of the continuous part of the measure in \\eqref{E:1.33} is uniformly positive on~$[0,\\infty)$ and grows exponentially in~$\\sqrt h$. Naturally, the atom at zero has the interpretation of the contribution of the avoided points and so we also get:\n\n\\begin{theorem}[Avoided points]\n\\label{thm-avoid}\nSuppose $\\{t_N\\}$ is a sequence such that \\eqref{E:1.29} holds. For any $D\\in\\mathfrak D$, any sequence $\\{D_N\\}$ of admissible approximations of~$D$, and for~$X$ sampled from~$P^{\\varrho}$,  consider the measure\n\\begin{equation}\n\\label{E:kappaND}\n\\kappa^D_N:=\\frac1{\\widehat W_N }\\sum_{x\\in D_N}1_{\\{L_{t_N}^{D_N}(x)=0\\}}\\,\\delta_{x\/N},\n\\end{equation}\nwhere~$\\widehat W_N $ is as in \\eqref{E:1.31}. Then, in the sense of vague convergence of measures on~$\\overline D$,\n\\begin{equation}\n\\label{E:kappa-lim}\n\\kappa^D_N\\,\\,\\,\\underset{N\\to\\infty}{\\,\\overset{\\text{\\rm law}}\\longrightarrow\\,}\\,\\,\\, \\sqrt{2\\pi g}\\,\\fraktura c(\\sqrt\\theta)\\,\\,  Z_{\\sqrt{\\theta}\\,}^{D}(\\text{\\rm d}\\mkern0.5mu x),\n\\end{equation}\nwhere $\\fraktura c(\\lambda)$ is again as in \\eqref{E:1.19}.\n\\end{theorem}\n\nWe conclude that, at times asymptotic to a $\\theta$-multiple of the cover time with~$\\theta<1$, the total number of avoided points is~proportional to~$\\widehat W_N =N^{2(1-\\theta)+o(1)}$. Moreover, when normalized by~$\\widehat W_N $, it tends in law to a constant times the total mass of~$Z^D_{\\sqrt\\theta}$.\n\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=0.4\\textwidth]{.\/figs\/avoided-points-1.pdf}\n\\includegraphics[width=0.4\\textwidth]{.\/figs\/avoided-points-3.pdf}\n}\n\\begin{quote}\n\\small \n\\vglue-0.2cm\n\\caption{\n\\label{fig2b}\nThe sets of avoided points for a sample of the random walk on a square of side-length $N=2000$ observed at times corresponding to $\\theta$-multiple of the cover time for $\\theta:=0.1$ (left) and $\\theta=0.3$ (right).}\n\\normalsize\n\\end{quote}\n\\end{figure}\n\n\n\n\\subsection{Local structure: thick and thin points}\nSimilarly to the case of the DGFF treated in \\cite{BL4}, the convergence of the point measures associated with the exceptional sets can be extended to include information about the local structure of the exceptional sets under consideration. \nFor the case of thick and thin points, this structure is captured by the measure on Borel subsets of~$D\\times\\mathbb R\\times\\mathbb R^{\\mathbb Z^2}$ (under the product topology) defined by\n\\begin{equation}\n\\label{E:zeta-local}\n\\widehat \\zeta^D_N :=\n\\frac{1}{W_N} \\sum_{x \\in D_N} \\delta_{x\/N} \\otimes \n\\delta_{(L_{t_N}^{D_N} (x) - a_N)\/\\log N} \n\\otimes \\delta_{\\{(L_{t_N}^{D_N} (x) - L_{t_N}^{D_N} (x+z))\/\\log N\\colon z \\in \\mathbb{Z}^2 \\}}. \n\\end{equation}\nIn order to express the limit measure, we need to introduce the DGFF $\\phi$ on $\\Bbb{Z}^2$ pinned to zero at the origin.\nThis is a centered Gaussian field on $\\Bbb{Z}^2$ with law~$\\nu^0$ and covariance\n\\begin{equation}\n\\label{E:phi-cov}\nE_{\\nu^0} (\\phi_x \\phi_y) = \\fraktura a(x) + \\fraktura a(y) - \\fraktura a (x-y),\n\\end{equation}\nwhere $\\fraktura a \\colon \\Bbb{Z}^2 \\to [0, \\infty)$ is the potential kernel,\ni.e., the unique function with $\\fraktura a (0) = 0$ which is discrete harmonic \non $\\Bbb{Z}^2 \\smallsetminus \\{0 \\}$ and satisfies $\\fraktura a (x) = g \\log |x| + O(1)$\nas $|x| \\to \\infty$. For the thick points, we then get:\n\n\\begin{theorem}[Local structure of the thick points]\n\\label{thm-thick-local}\nUnder the conditions of Theorem \\ref{thm-thick}\nand denoting by $\\zeta^D$ the limit measure on the right of (\\ref{E:1.21dis}), \n\\begin{equation}\n\\widehat \\zeta^D_N\\,\\,\\,\\underset{N\\to\\infty}{\\,\\overset{\\text{\\rm law}}\\longrightarrow\\,}\\,\\,\\, \\zeta^D \\otimes \\nu_{\\theta, \\lambda},\n\\end{equation}\nwhere $\\nu_{\\theta, \\lambda}$ is the law of $2\\sqrt{g} (\\sqrt{\\theta} + \\lambda) \n(\\phi +  \\alpha\\lambda \\fraktura a)$ under $\\nu^0$.\n\\end{theorem}\n\nFor the thin points, we in turn get:\n\n\\begin{theorem}[Local structure of the thin points]\n\\label{thm-thin-local}\nUnder the condition of Theorem \\ref{thm-thin} and\ndenoting by $\\zeta^D$ the limit measure on the right of (\\ref{E:1.23dis}),\n\\begin{equation}\n\\widehat \\zeta^D_N\\,\\,\\,\\underset{N\\to\\infty}{\\,\\overset{\\text{\\rm law}}\\longrightarrow\\,}\\,\\,\\, \\zeta^D \\otimes \\widetilde \\nu_{\\theta, \\lambda},\n\\end{equation}\nwhere $\\widetilde \\nu_{\\theta, \\lambda}$ is the law of $2\\sqrt{g} (\\sqrt{\\theta} - \\lambda) \n(\\phi +  \\alpha \\lambda\\fraktura a)$ under $\\nu^0$.\n\\end{theorem} \n\nAs shown in \\cite{BL4}, the field $\\phi+\\lambda\\alpha\\fraktura a$ describes the local structure of the DGFF near the points where it takes values (close to) $2\\sqrt g\\lambda\\log N$. As before, the prefactor $2\\sqrt g(\\sqrt\\theta\\pm\\lambda)$ disappears when instead of~$\\log N$ we normalize the third coordinate of~$\\widehat\\zeta^D_N$ by$~\\sqrt{2a_N}$. The above results thus extend the universality of the DGFF to the local structure as well. \n\n\\subsection{Local structure: avoided points}\nThe local structure of the local time near the avoided points will be radically different. Indeed, in the vicinity of an avoided point, the local time will remain of order unity and so a proper way to extend the measure~$\\kappa_N^D$ is\n\\begin{equation}\n\\widehat \\kappa_N^D :=\n\\frac{1}{\\widehat W_N}\n\\sum_{x \\in D_N}\n1_{\\{L_{t_N}^{D_N} (x) = 0 \\}} \\delta_{x\/N} \\otimes \\delta_{\\{L_{t_N}^{D_N} (x+z) \\colon z \\in \\Bbb{Z}^2 \\}},\n\\end{equation}\nwhich is now a Borel measure on $D\\times[0,\\infty)^{\\mathbb Z^2}$. Moreover, near an avoided point~$x$, the walk itself should behave as if conditioned not to hit~$x$. This naturally suggests that its trajectories will look like two-dimensional \\myemph{random interlacements} introduced recently by Comets, Popov and Vachkovskaia~\\cite{CPV} and Rodriguez~\\cite{R18}, building on earlier work of Sznitman~\\cite{S12} and Teixeira~\\cite{T09} in transient dimensions. In order to state our limit theorem, we need to review some of the main conclusions from \\cite{CPV,R18}.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width = 3.2in]{.\/figs\/occupation-time-field-0.pdf}\n\\hglue-1.7cm\\includegraphics[width = 3.2in]{.\/figs\/occupation-time-field-24.pdf}\n}\n\\begin{quote}\n\\small \n\\vglue-0.2cm\n\\caption{\n\\label{fig3b}\nSamples of the occupation-time field near two randomly-selected avoided points of a random walk run for $0.2$-multiple of the cover time in a square of side-length $N=2000$. Only the square of side-length $81$ centered at the chosen avoided point is depicted.}\n\\normalsize\n\\end{quote}\n\\end{figure}\n\n\nFirst we need some notation. Let~$W$ be the set of all doubly-infinite\ntransient random-walk trajectories on~$\\mathbb Z^2$; namely, piece-wise constant right-continuous maps $X\\colon\\mathbb R\\to\\mathbb Z^2$ that make only jumps between nearest neighbors and spend only finite time (as measured by the Lebesgue measure) in every finite subset of~$\\mathbb Z^2$. We endow~$W$ with the $\\sigma$-field~$\\mathcal{W}$ generated by finite-dimensional coordinate projections, $\\mathcal{W}:=\\sigma(X_t\\colon t\\in\\mathbb R)$.\nFor $A \\subset \\mathbb{Z}^2$ finite, we write $W_A$ for the subset of~$W$ of the trajectories that visit~$A$.\n\nNext we will put a measure $Q_A^{0, \\mathbb Z^2}$ on~$W_A$ as follows. Let $\\mathfrak h_A$ denote the harmonic measure of~$A$ from infinity for the simple random walk on $\\mathbb{Z}^2$. Let~$\\widehat{P}^x$ denote the law of a constant-speed continuous-time random walk on $\\mathbb{Z}^2 \\smallsetminus \\{0\\}$ started at~$x$ with conductance $\\fraktura a (y) \\fraktura a (z)$ at nearest-neighbor edges~$(y,z)$ in~$\\mathbb Z^2$. \nFor all cylindrical events $E^+, E^- \\in \\sigma(X_t\\colon t\\ge0)$ and any  $x \\in \\mathbb Z^2$, we then set\n\\begin{multline}\n\\qquad\nQ_A^{0, \\mathbb Z^2} \\Bigl((X_{-t})_{t \\geq 0} \\in E^-,\\,X_0 = x,\\,(X_t)_{t \\geq 0} \\in E^+\\Bigr)\n\\\\\n:= 4\\,\\fraktura a (x) \\mathfrak h_A (x) \\widehat{P}^x(E^+) \\widehat{P}^x (E^-\\,|\\,H_A = \\infty).\n\\qquad\n\\end{multline}\nNote that, since cylindrical events are unable to distinguish left and right path continuity, writing $(X_{-t})_{t \\geq 0} \\in E^-$ is meaningful. The transience of~$\\widehat{P}^x$ implies $\\widehat{P}^x(H_A=\\infty)>0$ and so the conditioning on the right is non-singular. \n\nThe measure $Q_A^{0, \\mathbb Z^2}$ represents the (un-normalized) law of doubly-infinite trajectories of the simple random walk that hit~$A$ (recall that $\\mathfrak{h}_A(x)=0$ unless $x\\in A$) but avoid~$0$ for all times (by Doob's $h$-transform, $\\widehat{P}^x$ is a  law of the simple random walk on~$\\mathbb Z^2$ conditioned to avoid~$0$). As the main results of \\cite{CPV,R18} show, the normalization is chosen such that these measures are consistent, albeit only after factoring out time shifts. To state this precisely, we need some more notation. Regarding two trajectories $w,w'\\in W$ as equivalent if they are time shifts of each other --- i.e., if there is $t\\in\\mathbb R$ such that $w(s)=w'(s+t)$ for all~$s\\in\\mathbb R$ --- we use~$W^{\\star}$ to denote the quotient space of~$W$ induced by this equivalence relation. Writing $\\Pi_\\star \\colon W \\to W^{\\star}$ for the canonical projection, the induced $\\sigma$-field on~$W^{\\star}$ is given by $\\mathcal{W}^\\star:=\\{E\\subseteq W^\\star\\colon \\Pi_\\star^{-1}(E)\\in \\mathcal W\\}$. Note that $W_A^{\\star} := \\Pi_\\star (W_A)\\in\\mathcal{W}^\\star$. \n\nTheorems~3.3 and~4.2 of~\\cite{R18} (building on~\\cite[Theorem~2.1]{T09}, see also \\cite[page 133]{CPV}) then ensure the existence of a (unique) measure on~$W^{\\star}$ such that for any finite $A \\subset \\mathbb{Z}^2$ and any $E\\in\\mathcal W^\\star$,\n\\begin{equation}\n\\nu^{0, \\mathbb Z^2}(E \\cap W_A^\\star) = Q_A^{0, \\mathbb Z^2}\\circ\\Pi_\\star^{-1}(E\\cap W_A^\\star).\n\\end{equation}\nSince $Q_A^{0, \\mathbb Z^2}$ is a finite measure and the set of finite $A\\subset\\mathbb Z^2$ is countable, $\\nu^{0,\\mathbb Z^2}$ is $\\sigma$-finite. We may thus consider a Poisson point process on~$W^\\star\\times[0,\\infty)$ with intensity $\\nu^{0, \\mathbb Z^2}\\otimes{\\rm Leb}$. Given a sample~$\\omega$ from this process, which we may write as $\\omega = \\sum_{i\\in\\mathbb N} \\delta_{(w_i^{\\star}, u_i)}$, and any $u\\in[0,\\infty)$, we define the \\myemph{occupation time field at level~$u$} by\n\\begin{equation}\n\\label{E:occ-time-field}\nL_u(x) := \\sum_{i \\in \\mathbb N} 1_{\\{u_i \\leq u\\}}\\,\\frac{1}{4} \\int_\\mathbb R \\text{\\rm d}\\mkern0.5mu t\\, 1_{\\{w_i (t) = x \\}},\n\\quad x \\in \\mathbb{Z}^2,\n\\end{equation}\nwhere $w_i \\in W$ is any representative of the class of trajectories marked by~$w_i^\\star$; i.e., $\\Pi_\\star (w_i) = w_i^{\\star}$. (The integral does not depend on the choice of the representative.) We are now ready to state the convergence of the measures $\\widehat \\kappa_N^D$.\n\n\n\\begin{theorem}[Local structure of the avoided points]\n\\label{thm-avoid-local}\nUnder the conditions of Theorem \\ref{thm-avoid} and for~$\\kappa^D$ denoting the measure on the right of \\eqref{E:kappa-lim},\n\\begin{equation}\n\\widehat \\kappa^D_N\\,\\,\\,\\underset{N\\to\\infty}{\\,\\overset{\\text{\\rm law}}\\longrightarrow\\,}\\,\\,\\, \n\\kappa^D \\otimes \\nu_{\\theta}^{\\text{\\rm RI}},\n\\end{equation}\nwhere $\\nu_{\\theta}^{\\text{\\rm RI}}$ is the law of the occupation time field \n$(L_u(x))_{x \\in \\Bbb{Z}^2}$ at $u:=\\pi\\theta$.\n\\end{theorem}\n\nWe expect a similar result to hold for the light points as well but with the random interlacements replaced by a suitably modified version that allows the walks to hit the origin but accumulating a given (order unity) amount of local time there. \n\n\n\n\\section{Main ideas, extensions and outline}\n\\label{sec2}\\nopagebreak\\noindent\nLet us proceed by a brief overview of the main ideas of the proof and then a list of possible extensions and refinements. We also outline the remainder of this paper.\n\n\\subsection{Main ideas}\n\\label{sec2.1}\\noindent\nAs already noted, key for all developments in this paper is the connection of the local time $L^V_t$ and the associated DGFF~$h^V$.\nOur initial take on this connection was through the fact that the DGFF represents the fluctuations of~$L^V_t$ at large times via\n\\begin{equation}\n\\label{E:GFFlimit}\n\\frac{L_t^V(\\cdot)-t}{\\sqrt{2t}}\\,\\,\\,\\underset{t\\to\\infty}{\\,\\overset{\\text{\\rm law}}\\longrightarrow\\,}\\,\\,\\, h^V.\n\\end{equation}\n(This also dictated the parametrization in the earlier work on this problem, e.g.,~\\cite{A15}.) However, as noted at the end of Subsection~\\ref{sec2.3}, for the thick and thin points, the effective~$t$ in the correspondence \\eqref{E:GFFlimit} of the local time with the DGFF is~$a_N$, rather than~$t_N$. In any case, approximating the local time by the DGFF becomes accurate only \\myemph{beyond} the times of the order of the cover time. \n\nWe thus base our proofs on a deeper version of this connection, known under the name  \\myemph{Second Ray-Knight Theorem} after Ray~\\cite{Ray} and Knight~\\cite{Knight}  or \\myemph{Dynkin isomorphism} after Dynkin~\\cite{D83},  although the statement we use is due to Eisenbaum, Kaspi, Marcus, Rosen and Shi~\\cite{EKMRS} (with an interesting new proof by Sabot and Tarres~\\cite{Sabot-Tarres}): \n\n\\begin{theorem}[Dynkin isomorphism]\n\\label{thm-Dynkin}\nFor each~$t>0$ there exists a coupling of $L^V_t$ (sampled under~$P^\\varrho$) and two copies of the DGFF $h^V$ and~$\\tilde h^V$ such that\n\\begin{equation}\n\\label{E:Dynkin1}\nh^V\\text{\\rm\\ and } L^V_t \\text{\\rm\\ are independent}\n\\end{equation}\nand\n\\begin{equation}\n\\label{E:Dynkin2}\nL_t^V(u)+\\frac12(h_u^V)^2 = \\frac12\\bigl(\\tilde h_u^V+\\sqrt{2t}\\bigr)^2, \\quad u\\in V.\n\\end{equation}\n\\end{theorem}\n\n\\noindent\nThis is  usually stated  as a distributional identity; the coupling  version is then a result of abstract-nonsense theorems in probability (see Zhai~\\cite[Section~5.4]{Zhai}). \n\nThe identity \\eqref{E:Dynkin2} suggests a natural idea:\nIf we could simply disregard the DGFF on the left-hand side, the relation would tie the level set corresponding to~$L^{D_N}_{t_N}\\approx a_N$ to the level sets of the DGFF where\n\\begin{equation}\n\\text{either}\\quad \\tilde h^{D_N}\\approx\\sqrt{2a_N}-\\sqrt{2t_N}\\quad\\text{or}\\quad \\tilde h^{D_N}\\approx-\\sqrt{2a_N}-\\sqrt{2t_N}.\n\\end{equation}\nFor~$a_N\\to\\infty$, the second level set lies further away from the mean of~$\\tilde h^{D_N}$ than the first and its contribution can therefore be disregarded. (This is true for the thick and thin points; for the light and avoided points both levels play a similar role). One could then simply hope to plug to the existing result \\eqref{E:1.19}. \n\nUnfortunately, since $\\text{\\rm Var}(h^{D_N}_x)$ is of order~$\\log N$, the square of the DGFF on the left of \\eqref{E:Dynkin2} is typically of the size of the anticipated fluctuations of~$L^{D_N}_{t_N}$ and so it definitely affects the limiting behavior of the whole quantity. The main technical challenge of the present paper is thus to understand the contribution of this term precisely.  A key observation that makes this possible is that even for~$x\\in D_N$ where $L^{D_N}_{t_N}(x)+\\frac12(h^{D_N}_x)^2$ takes exceptional values, the DGFF  $h^{D_N}_x$ remains typical (and $L^{D_N}_{t_N}(x)$ is thus dominant). This requires proving fairly sharp single-site tail estimates for the local time and combining them with the corresponding tail bounds for the DGFF. \n\nOnce that is done, we include the field~$h^{D_N}$, properly scaled,  as a third ``coordinate'' of the point process and study weak subsequential limits of these. For instance, for the thick and thin points this concerns the measure\n\\begin{equation}\n\\label{E:2.5i}\n\\frac1{W_N}\\sum_{x\\in D_N}\\delta_{x\/N}\\otimes\\delta_{(L^{D_N}_{t_N}(x)-a_N)\/\\log N}\\otimes\\delta_{h^{D_N}_x\/\\sqrt{\\log N}}.\n\\end{equation}\nHere the key is to show that the DGFF part acts, in the limit, as an explicit \\myemph{deterministic} measure.  For instance, for the thick and thin points this means that if~$\\zeta^D_N$ converges to some~$\\zeta^D$ along a subsequence of~$N$'s, the measure in \\eqref{E:2.5i} converges to $\\zeta^D\\otimes\\mathfrak g$ where~$\\mathfrak g$ is the  normal law~$\\mathcal N(0,g)$;  see Lemma~\\ref{lemma-add-field}.\n\nDenoting by~$\\ell$ the second variable and by~$h$ the third variable in \\eqref{E:2.5i}, the Dynkin isomorphism now tells us that the ``law'' of $\\ell+\\frac{h^2}2$ under any weak subsequential limit of the measures in \\eqref{E:2.5i} is the same as the limit ``law'' of the DGFF level set at scale~$\\sqrt{2a_N}-\\sqrt{2t_N}$  (for the thick points) which we know from \\eqref{E:1.19}. This produces a convolution-type identity for subsequential limits of the local-time point process. Some technical work then shows that this identity has a unique solution which can be identified explicitly in all cases of interest.\n\n\nOur control of the local structure of the exceptional points will also rely on the above isomorphism theorem. For the thick and thin points, we will then simply plug in to Theorem~2.1 of~\\cite{BL4} that describes the local structure of intermediate level sets of the DGFF. For the avoided points, we will also need the \\myemph{Pinned Isomorphism Theorem} of Rodriguez~\\cite[Theorem~5.5]{R18} that links the random-interlacement occupation-time field $(L_u(x))_{x\\in\\mathbb Z^2}$ introduced in \\eqref{E:occ-time-field} to the pinned DGFF~$\\phi$ defined via \\eqref{E:phi-cov} as follows:\n\n\\begin{theorem}[Pinned Isomorphism Theorem]\n\\label{thm-pinned-iso}\nLet~$u>0$ and suppose that $(L_u(x))_{x\\in\\mathbb Z^2}$ with law $\\nu^{\\text{\\rm RI}}_{u\/\\pi}$ is independent of~$\\{\\phi_x\\colon x\\in\\mathbb Z^2\\}$ with law~$\\nu^0$. Then\n\\begin{equation}\n\\label{E:3.6uiw}\nL_u+\\frac12\\phi^2\\,\\,\\overset{\\text{\\rm law}}=\\,\\frac12\\bigl(\\phi+ 2 \\sqrt{2u}\\,\\fraktura a\\bigr)^2,\n\\end{equation}\nwhere~$\\fraktura a$ is the potential kernel. (The extra factor of~2 compared to \\cite[Theorem~5.5]{R18} is due to different normalizations of the local time, the pinned field and the potential kernel.) \n\\end{theorem}\n\nIt is exactly the generalization of this theorem that blocks us from extending control of the local structure to the light points. Indeed, we expect that, for the light points, the associated process is still that of random interlacements but with the local time at the origin fixed to a given number. Developing the theory of this process explicitly goes beyond the scope of the present paper.\n\n\n\n\\subsection{Extensions and refinements}\n We see a number of possible ways  the existing conclusions  may  be refined so let us discuss these in some more detail.\n\n\\smallskip\\noindent\n\\textsl{Other ``boundary'' conditions: } Perhaps the most significant  deficiency of our setting is the somewhat unnatural mechanism by which the walk returns back to~$D_N$ after each exit. Perhaps contrary to the intuition one might have, this does not lead to the local time exploding near the boundary; see Fig.~\\ref{fig3} or the fact that~$Z^D_\\lambda$ puts no mass on~$\\partial D$. The main reason for using the specific setting worked out here is that it allows us to seamlessly plug in the existing results from~\\cite{BL4} on the ``intermediate'' level sets of the~DGFF. The natural alternatives are\n\\settowidth{\\leftmargini}{(11111)}\n\\begin{enumerate}\n\\item[(1)] running the walk on an $N\\times N$ torus, or\n\\item[(2)]\nrunning the walk as a simple random walk on all of~$\\mathbb Z^2$ but only reflecting the local time spent inside~$D_N$.\n\\end{enumerate}\nThe latter setting is perhaps most natural; unfortunately, it does not fit into the scheme where the connection with a DGFF can naturally be pulled out. The former setting in turn requires developing level-set analysis of a torus DGFF pinned at one point.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[height = 2.4in]{.\/figs\/walk-3.pdf}\n\\includegraphics[width = 2.8in]{.\/figs\/local-time-3.pdf}\n}\n\\begin{quote}\n\\small \n\\vglue-0.2cm\n\\caption{\n\\label{fig3}\nLeft: Plot of the trajectory of the random walk on a $200\\times200$ square run for $0.3$-multiple of the cover time. The time runs in the vertical direction. Right: The corresponding local time profile. Note that while short excursions near the boundary are numerous, most contribution to the local time profile comes from the excursions that reach ``deep'' into the domain.}\n\\normalsize\n\\end{quote}\n\\end{figure}\n\n\n\n\\smallskip\\noindent\n\\textsl{Time parametrization: }\nAnother feature for which our setting may not be considered most natural is the time parametrization by the time spent at the ``boundary vertex.'' A reasonable question is then what happens when we instead use the parametrization by the actual time of the walk (continuous-time parametrization), or even by the number of discrete steps that the walk has taken (discrete-time parametrization). The main problem here is the lack of a direct connection with the underlying DGFF; instead, one has to rely on approximations.\n\nPreliminary calculations have so far shown that, at least approximately, the local time in the continuous-time parametrization is still connected with the DGFF as in \\eqref{E:Dynkin2} but now with the field $\\tilde h^{D_N}$ reduced by its arithmetic mean over~$D_N$. This implies that, for both continuous and discrete-time analogues of the measures $\\zeta^D_N$, $\\vartheta^D_N$ and~$\\kappa^D_N$, their $N\\to\\infty$ limits still take the product form as in \\eqref{E:1.21dis}, \\eqref{E:1.23dis}, \\eqref{E:2.22ii} and \\eqref{E:kappa-lim}, respectively, albeit now with~$Z^D_\\lambda$ replaced by a suitable substitute reflecting on the reduction of the CGFF by its arithmetic mean. We intend to establish these statements rigorously in a follow-up work.\n\n\n\\smallskip\\noindent\n\\textsl{Critical cases: }\nAnother natural  problem to examine are various borderline behaviors left untouched in the present paper; for instance, $\\lambda:=1$ for the $\\lambda$-thick points and~$\\lambda:=\\sqrt\\theta\\wedge1$ for the $\\lambda$-thin points as well as~$\\theta:=1$ for the avoided points. In analogy with the corresponding question for the DGFF (Biskup and Louidor~\\cite{BL1,BL2,BL3}), we expect that the corresponding measures  will require a different scaling --- essentially, boosting by an additional factor of~$\\log N$ --- and the limit spatial behavior will  be governed by the \n\\myemph{critical} LQG measure $Z^D_1$. For the simple random walk on a homogeneous tree of depth~$n$, this program has already been carried out by the first author~(Abe~\\cite{A18}). A breakthrough result along these lines describing the limit law of the cover time on homogenous trees has recently been posted by Cortines, Louidor and Saglietti~\\cite{CLS}. \n\n\n\\smallskip\\noindent\n\\textsl{Brownian local time: } \nYet another interesting extension concerns the corresponding problem for the Brownian local time. This requires working with the $\\epsilon$-cover time defined as the first time when every disc of radius~$\\epsilon>0$ inside~$D$ has been visited; the limit behavior is then studied as~$\\epsilon\\downarrow0$. We actually expect that, with proper definitions, very similar conclusions will hold here as well although we presently do not see other way to prove them than by approximations via random walks.\n\nJego~\\cite{J18b} recently posted a preprint that proves the existence of a scaling limit for the process associated, similarly to our~$\\zeta^D_N$ from \\eqref{E:zetaND}, with the local-time thick points of the Brownian path killed upon \\myemph{first exit} from~$D$. As it turns out, the limit measure still factors into a product of a random spatial part, defined  via limits of exponentials of the root of the local time, and an exponential measure. However, although the spatial part of the measure obeys the expectation identity of the kind \\eqref{E:1.19a}, it is certainly not one of the LQG measures $Z^D_\\lambda$ above, due to the limited time horizon of the Brownian path. Characterizing Jego's limit measure more directly is a natural next challenge.\n\n\n\n\\subsection{Outline}\nThe rest of this note is organized as follows. In the next section (Section~\\ref{sec3b}) we derive tail estimates for the local time that will come handy later in the proofs. These are used to prove tightness of the corresponding point measures. Section~\\ref{sec4} then gives the proof of convergence for the measure associated with $\\lambda$-thick points following the outline from Section~\\ref{sec2.1}. This proof is then used as a blue print for the corresponding proofs for the $\\lambda$-thin points (Section~\\ref{sec5}) and the light and avoided points (Section~\\ref{sec6}).\nThe results on local structure are proved at the very end (Section~\\ref{sec8}).\n\n\n\\section{Tail estimates and tightness}\n\\label{sec3b}\\noindent\\nopagebreak\nWe are now ready to commence the proofs of our results.  All of our derivations will pertain to the continuous-time Markov chain started, and with the local time parametrized by the time spent, at the ``boundary vertex.'' \nLet us pick a domain~$D\\in\\mathfrak D$ and a sequence~$\\{D_N\\}$ of admissible approximations of~$D$ and consider these fixed throughout the rest of this paper.  Recall that we write $\\zeta^D_N$, $\\vartheta^D_N$ and~$\\kappa^D_N$ for the measures in \\eqref{E:zetaND}, \\eqref{E:varthetaND} and \\eqref{E:kappaND}, respectively.\n\n\\subsection{Upper tails}\nWe begin by estimates on the tails of the random variable~$L^{D_N}_{t_N}(x)$ which then readily imply tightness of the random measures of interest. We first derive these estimates in the general setting of a random walk on a graph with distinguished vertex~$\\varrho$ and only then specialize to~$N$-dependent domains in the plane. We begin with the upper tail:\n\n\\begin{lemma}[Local time upper tail]\n\\label{lemma-upper}\nConsider the random walk on~$V\\cup\\{\\varrho\\}$ as detailed in Section~\\ref{sec1.2}.\nFor all~$a,t>0$ and all~$b\\in\\mathbb R$ such that $a+b>t$, and all~$x\\in V$,\n\\begin{equation}\n\\label{E:2.2a}\nP^\\varrho\\bigl(L_t^V(x)\\ge a+b\\bigr)\\le \\frac{\\sqrt{G^V(x,x)}}{\\sqrt{2(a+b)}-\\sqrt{2t}}\\,\n\\,\\text{\\rm  e}\\mkern0.7mu^{-\\frac{(\\sqrt{2a}-\\sqrt{2t})^2}{2G^V(x,x)}}\\,\\text{\\rm  e}\\mkern0.7mu^{-b\\frac{\\sqrt{2a}-\\sqrt{2t}}{G^V(x,x)\\sqrt{2a}}}\\,.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proofsect}{Proof}\nWe will conveniently use estimates developed in earlier work on this problem.  Denoting by $(Y_s)_{s \\geq 0}$ the $0$-dimensional Bessel process and writing $P_Y^a$ for its law with $P_Y^a(Y_0=a)=1$, Lemma 3.1(e) of Belius, Rosen and Zeitouni~\\cite{BRZ} shows \n\\begin{equation}\n\\label{E:Bessel}\nL_t^V(x)~\\text{under }P^\\varrho \n\\,\\,\\,\\overset{\\text{\\rm law}}=\\,\\,\\,\n\\frac{1}{2} \\left(Y_{G^V (x, x)} \\right)^2~\\text{under } P_Y^{\\sqrt{2t}}.\n\\end{equation}\nLet $P_B^r$ be a law under which\n$(B_s)_{s \\geq 0}$ is a standard Brownian motion on $\\mathbb{R}$ starting at~$r$. The process~$Y$ is absolutely continuous with respect to~$B$ up to the first time it hits zero; after that~$Y$ vanishes identically. The Radon-Nikodym derivative takes the explicit form (see, for example, \\cite[(2.12)]{BRZ})\n\\begin{equation} \n\\label{E:Bessel-to-Brown}\n\\frac{\\text{\\rm d}\\mkern0.5mu P_Y^r}{\\text{\\rm d}\\mkern0.5mu P_B^r}\\Bigr|_{\\mathcal{F}_{ H_0\\wedge t}}\n= \\sqrt{\\frac{r}{B_t}} \\exp \\left\\{- \\frac{3}{8} \\int_0^t \\text{\\rm d}\\mkern0.5mu s\\,\\frac{1}{B_s^2} \\right\\},\\quad\\text{on }\\{H_0 > t \\},\n\\end{equation}\nwhere $\\mathcal{F}_t$ is the $\\sigma$-field generated by the process up to time~$t$ and~$H_a$ is the first time the process hits level~$a$.\n \n\nThe identification \\eqref{E:Bessel} along with the assumptions~$a+b>0$ translates the event $\\{L_t^V(x)\\ge a+b\\}$ to~$\\{Y_t\\ge\\sqrt{2(a+b)}\\}$ intersected by~$\\{H_0>t\\}$. For~$r:=\\sqrt{2t}$, the assumption~$a+b>t$ implies that   the quantity in  \\eqref{E:Bessel-to-Brown} is less than one everywhere on the event of interest. Hence,\n\\begin{equation}\n\\label{E:4.4}\n\\begin{aligned}\nP^\\varrho\\bigl(L_t^V(x)\\ge a+b\\bigr)&\\le P_B^{\\sqrt{2t}}\\Bigl(B_{G^V(x,x)}\\ge \\sqrt{2(a+b)}\\,\\Bigr)\n\\\\\n&= P_B^0\\Bigl(B_{G^V(x,x)}\\ge\\sqrt{2(a+b)}-\\sqrt{2t}\\Bigr).\n\\end{aligned}\n\\end{equation}\n In order to get \\eqref{E:2.2a} from this, we  invoke the Gaussian estimate $P(\\mathcal N(0,\\sigma^2)\\ge x)\\le\\sigma x^{-1}\\text{\\rm  e}\\mkern0.7mu^{-\\frac{x^2}{2\\sigma^2}}$ valid for all~$x>0$ along with the calculation\n\\begin{equation}\n\\begin{aligned}\n\\Bigl(\\sqrt{2(a+b)}-\\sqrt{2t}\\Bigr)^2\n&=2(a+b)+2t-2\\sqrt{2a}\\sqrt{2t}\\Bigl(1+\\frac{b}{a}\\Bigr)^{1\/2}\n\\\\\n&\\ge 2(a+b)+2t-2\\sqrt{2a}\\sqrt{2t}\\Bigl(1+\\frac{b}{2a}\\Bigr)\n\\\\\n&=\\bigl(\\sqrt{2a}-\\sqrt{2t}\\bigr)^2+2b\\frac{\\sqrt{2a}-\\sqrt{2t}}{\\sqrt{2a}},\n\\end{aligned}\n\\end{equation}\nwhere in the middle line we used that $(1+x)^{1\/2}\\le 1+x\/2$ holds for all~$x>-1$.\n\\end{proofsect}\n\nFrom this we readily obtain:\n\n\\begin{corollary}[Tightness for the thick points]\n\\label{cor-tightness-upper}\nSuppose that $t_N$ and~$a_N$ are such that the limits in \\eqref{E:1.20} exist for some~$\\theta>0$ and some $\\lambda\\in(0,1)$. For each~$b\\in\\mathbb R$, there is~$c_1(b)\\in(0,\\infty)$ such that for all~$A\\subset\\mathbb R^2$ closed,\n\\begin{equation}\n\\label{E:2.1}\n\\limsup_{N\\to\\infty}\\,\nE^\\varrho\\bigl[\\,\\zeta^D_N\\bigl(A\\times[b,\\infty)\\bigr)\\bigr]\\le c_1(b)\\,{\\rm Leb}(A\\cap D).\n\\end{equation}\n\\end{corollary}\n\n\\begin{proofsect}{Proof}\nIt suffices to prove the bound for all $b<0$ with~$|b|$ sufficiently large. \nPick~$x\\in D_N$. If~$G^{D_N}(x,x)\\ge \\frac g{b^2}\\log N$, then Lemma~\\ref{lemma-upper} with~$a:=a_N$, $t:=t_N$ and~$b$ replaced by~$b\\log N$ and  the uniform  bound~$G^{D_N}(x,x)\\le g\\log N+c$ give\n\\begin{equation}\n\\label{E:2.2}\nP^\\varrho\\bigl(L^{D_N}_{t_N}(x)\\ge a_N+b\\log N\\bigr)\n\\le \\frac{\\tilde c}{\\sqrt{\\log N}}\\,\\text{\\rm  e}\\mkern0.7mu^{-\\frac{(\\sqrt{2a_N}-\\sqrt{2t_N})^2}{2g\\log N}}\\,\\text{\\rm  e}\\mkern0.7mu^{\\beta |b|^3},\n\\end{equation}\nfor some constants~$\\tilde c<\\infty$ and~$\\beta>0$ independent of~$b$ and~$N$, once~$N$ is sufficiently large. This is order~$W_N\/N^2$. If, on the other hand, $G^{D_N}(x,x)\\le \\frac g{b^2}\\log N$, then we use that~$G^{D_N}(x,x)\\ge\\frac14$ in the second exponential on the right of \\eqref{E:2.2a} to get\n\\begin{equation}\n\\label{E:2.2b}\nP^\\varrho\\bigl(L^{D_N}_{t_N}(x)\\ge a_N+b\\log N\\bigr)\n\\le \\frac{\\tilde c'}{\\sqrt{\\log N}}\\,\\text{\\rm  e}\\mkern0.7mu^{-b^2\\frac{(\\sqrt{2a_N}-\\sqrt{2t_N})^2}{2g\\log N}}\\,\\text{\\rm  e}\\mkern0.7mu^{\\beta' |b|\\log N},\n\\end{equation}\nwhere again~$\\tilde c'<\\infty$ and~$\\beta'>0$ do not depend on~$b$ or~$N$ once~$N$ is sufficiently large. Since the first exponent in \\eqref{E:2.2b} is order~$\\log N$, for~$|b|$ large enough, this is again at most order~$W_N\/N^2$. Now write $A_\\epsilon:=\\{x\\in\\mathbb R^2\\colon \\text{\\rm d}_\\infty(x,A)<\\epsilon\\}$ and note that, in light of \\eqref{E:1.8i}, we have \n\\begin{equation}\n\\label{E:3.9ui}\n\\#\\bigl\\{x\\in D_N\\colon x\/N\\in A\\bigr\\}\\le N^2{\\rm Leb}(A_{1\/N}\\cap D).\n\\end{equation}\nSumming the relevant bound from \\twoeqref{E:2.2}{E:2.2b} over~$x\\in D_N$ with~$x\/N\\in A$, the claim follows by noting that, since~$A$ is closed, ${\\rm Leb}(A_{1\/N}\\cap D)\\to{\\rm Leb}(A\\cap D)$ as~$N\\to\\infty$. \n\\end{proofsect}\n\n\n\\subsection{Lower tails}\nFor the lower tail we similarly get:\n\n\\begin{lemma}[Local time lower tail]\n\\label{lemma-lower}\nConsider the random walk on~$V\\cup\\{\\varrho\\}$ as in Section~\\ref{sec1.2}.\nFor all~$a,t>0$ and all~$b'<b$ such that $a+b'>0$ and $a+b<t$, and all~$x\\in V$,\n\\begin{equation}\n\\label{E:4.7a2}\nP^\\varrho\\bigl(L_t^V(x)-a\\in[b',b]\\bigr)\\le \\Bigl(\\frac{t}{a+b'}\\Bigr)^{1\/4}\\!\\!\\!\n\\frac{\\sqrt{G^V(x,x)}}{\\sqrt{2t}-\\sqrt{2(a+b)}}\\,\n\\,\\text{\\rm  e}\\mkern0.7mu^{-\\frac{(\\sqrt{2t}-\\sqrt{2a})^2}{2G^V(x,x)}}\\,\\text{\\rm  e}\\mkern0.7mu^{+b\\frac{\\sqrt{2t}-\\sqrt{2a}}{G^V(x,x)\\sqrt{2a}}}\\,.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proofsect}{Proof}\nWe use again the passage \\twoeqref{E:Bessel}{E:Bessel-to-Brown} via the Bessel process and Brownian motion except that here we can no longer bound the prefactor in \\eqref{E:Bessel-to-Brown} by one. Instead, we get the root of the ratio of the roots of~$2t$ and~$2(a+b')$. Therefore, \\eqref{E:4.4} is replaced by\n\\begin{equation}\nP^\\varrho\\bigl(L_t^V(x)-a\\in[b',b]\\bigr)\\le \\Bigl(\\frac{t}{a+b'}\\Bigr)^{1\/4}\nP_B^0\\Bigl(B_{G^V(x,x)}\\le\\sqrt{2(a+b)}-\\sqrt{2t}\\Bigr).\n\\end{equation}\nNoting that the difference in the probability on the right is negative, the rest of the calculation is exactly as before.\n\\end{proofsect}\n\n Postponing the tightness of the thin points to the end of this subsection, we first deal with estimates for the light and avoided points: \n\n\\begin{lemma}[Vanishing local time]\n\\label{lemma-vanish}\nFor each~$t>0$ and each~$x\\in V$,\n\\begin{equation}\n\\label{E:L-vanish}\nP^\\varrho\\bigl(L_t^V(x)=0\\bigr) = \\text{\\rm  e}\\mkern0.7mu^{-\\frac{t}{G^V(x,x)}}.\n\\end{equation}\nIn fact, for every~$b\\ge0$, we have\n\\begin{equation}\n\\label{E:L-vanish2}\nP^\\varrho\\bigl(L_t^V(x)\\le b\\bigr)  \\le\\text{\\rm  e}\\mkern0.7mu^{-\\frac{t}{G^V(x,x)}\\exp\\{-\\frac{b}{G^V(x,x)}\\}} \\le \\text{\\rm  e}\\mkern0.7mu^{-\\frac{t}{G^V(x,x)}+b\\frac{t}{G^V(x,x)^2}}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proofsect}{Proof}\nHere we proceed by a direct argument  based on excursion decomposition  (see, however, Remark~\\ref{rem-Bessel-works}).  Writing $\\hat H_u$ for the first time to return to~$u$ after the walk left~$u$, consider the following independent random variables: \n\\settowidth{\\leftmargini}{(1111)}\n\\begin{enumerate}\n\\item[(1)] $N:=$ Poisson$(t\/G^V(x,x))$,\n\\item[(2)] $\\{Z_n\\colon n\\ge1\\}:=$ Geometric with parameter~$p:=P^x(H_\\varrho<\\hat H_x)$,\n\\item[(3)] $\\{T_{k,j}\\colon k,j\\ge1\\}:=$ Exponentials with mean one.\n\\end{enumerate}\nWe then claim\n\\begin{equation}\n\\label{E:3.13}\n\\pi(x)L_t^V(x) \\,\\,\\overset{\\text{\\rm law}}=\\,\\,\\sum_{k=1}^N \\sum_{j=1}^{Z_k}T_{k,j}.\n\\end{equation}\nTo see this, note that thanks to the parametrization by the local time at~$\\varrho$, the value~$L_t^V(x)$ is accumulated through a Poisson$(\\pi(\\varrho)t)$ number of independent excursions that start and end at~$\\varrho$. Each excursion that actually visits~$x$, which happens with probability $P^\\varrho(H_x<\\hat H_\\varrho)$, contributes a Geometric$(p)$-number of independent exponential random variables to the total time the walk spends at~$x$. By Poisson thinning, the number of excursions that visit~$x$ is Poisson with parameter~$\\pi(\\varrho)P^\\varrho(H_x<\\hat H_\\varrho)t$. We claim that this equals $t\/G^V(x,x)$. Indeed, since the walk is constant speed, reversibility gives\n\\begin{equation}\n\\pi(\\varrho)P^\\varrho(H_x<\\hat H_\\varrho) =\\pi(x)P^x(H_\\varrho<\\hat H_x).\n\\end{equation}\nAs was just noted, under~$P^x$ the quantity $\\pi(x)\\ell_{H_\\varrho}(x)$ is the sum of Geometric($p$) independent exponentials of mean one. From \\eqref{E:cov} we then get $\\pi(x)G^V(x,x)=1\/p$.\n\nWith \\eqref{E:3.13} in hand, to get \\eqref{E:L-vanish} we just observe that, modulo null sets, the sum in~\\eqref{E:3.13} vanishes only if~$N=0$. To get \\eqref{E:L-vanish2} we note that, for~$L_t^V(x)\\le b$ we must have $\\sum_{j=1}^{Z_k}T_{k,j}\\le b\\pi(x)$ for each $k=1,\\dots,N$. The probability that the sum of~$Z_k$ independent exponentials is less than~$b\\pi(x)$ equals $1-\\text{\\rm  e}\\mkern0.7mu^{-bp\\pi(x)}$, and that this happens for all $k=1,\\dots,N$ thus has probability at most \n\\begin{equation}\n\\sum_{n=0}^\\infty \\frac{(t\/G^V(x,x))^n}{n!}\\bigl[ 1-\\text{\\rm  e}\\mkern0.7mu^{-bp\\pi(x)}\\bigr]^n\\text{\\rm  e}\\mkern0.7mu^{-\\frac{t}{G^V(x,x)}}\n=\\text{\\rm  e}\\mkern0.7mu^{-\\frac{t}{G^V(x,x)} \\text{\\rm  e}\\mkern0.7mu^{-bp\\pi(x)}}.\n\\end{equation}\nThe claim again follows from $1\/p=\\pi(x)G^V(x,x)$  and the bound $\\text{\\rm  e}\\mkern0.7mu^{-x}\\ge1-x$. \n\\end{proofsect}\n\n\n\\begin{remark}\n\\label{rem-Bessel-works}\nWe note that a proof based on the connection with the $0$-dimensional Bessel process is also possible. Indeed, by~Belius, Rosen and Zeitouni~\\cite[(2.7)]{BRZ}, given $x > 0$ the law of $(Y_s)^2$ under~$P^x_Y$ is given by\n\\begin{equation}\n\\text{\\rm  e}\\mkern0.7mu^{- \\frac{x}{2s}} \\delta_0 (\\text{\\rm d}\\mkern0.5mu y) + 1_{(0, \\infty)} (y) \\,\\frac{1}{2s} \\,\\sqrt{\\frac{x}{y}} \\,I_1 \\Bigl(\\frac{\\sqrt{xy}}{s}\\Bigr)\n\\text{\\rm  e}\\mkern0.7mu^{- \\frac{x+y}{2s}} \\text{\\rm d}\\mkern0.5mu y,\n\\end{equation}\nwhere $I_1 (z) := \\sum_{k=0}^{\\infty} \\frac{(z\/2)^{2k+1}}{k! (k+1)!}$.\nThe identity \\eqref{E:L-vanish} follows immediately from \\eqref{E:Bessel} and\n\\begin{equation}\nI_1 \\Bigl(\\frac{\\sqrt{2ts}}{G^V (x,x)} \\Bigr) \\leq \\frac{\\sqrt{2ts}}{2G^V(x,x)} \\,\\text{\\rm  e}\\mkern0.7mu^{\\frac{ts}{2 G^V(x,x)^2}}\n\\end{equation}\nthen implies the inequality in \\eqref{E:L-vanish2} as well.\n\\end{remark}\n\n\nFrom Lemma~\\ref{lemma-vanish} we get:\n\n\\begin{corollary}[Tightness for the light and avoided points]\n\\label{cor-light}\nSuppose $t_N$ is such that \\eqref{E:1.29} holds with some~$\\theta\\in(0,1)$. \nFor each~$b>0$ there is a constant~$c_2(b)\\in(0,\\infty)$ such that for each $A\\subset\\mathbb R^2$ closed,\n\\begin{equation}\n\\label{E:vartheta-lim}\n\\limsup_{N\\to\\infty}\\,E^\\varrho\\bigl[\\,\\vartheta^D_N(A\\times[0,b])\\bigr]\\le  c_2(b)\\,{\\rm Leb}(A\\cap D).\n\\end{equation}\nIn particular,\n\\begin{equation}\n\\limsup_{N\\to\\infty}\\,E^\\varrho\\bigl[\\,\\kappa^D_N(A)\\bigr]\\le  c_2(b)\\,{\\rm Leb}(A\\cap D).\n\\end{equation}\n\\end{corollary}\n\n\\begin{proofsect}{Proof}\n It suffices to prove just \\eqref{E:vartheta-lim} and that for~$b>0$ sufficiently large. Denote~$\\tilde c:=\\sup_{N\\ge1}t_N\/(\\log N)^{2}$. We then claim\n\\begin{equation}\n\\label{E:4.21ie}\nP^\\varrho\\bigl(L^{D_N}_{t_N}(x)\\le b \\bigr)\\le \\text{\\rm  e}\\mkern0.7mu^{-bt_N(\\log N)^{-1}}+\\text{\\rm  e}\\mkern0.7mu^{-\\frac{t_N}{G^{D_N}(x,x)}+\\tilde c b^3\\text{\\rm  e}\\mkern0.7mu^{8b}}.\n\\end{equation}\nIndeed, the first term arises for~$x$ with $G^{D_N}(x,x)\\le b^{-1}\\text{\\rm  e}\\mkern0.7mu^{-4b}\\log N$ by the first inequality in \\eqref{E:L-vanish2} along with~$G^{D_N}(x,x)\\ge\\frac14$. The second term controls the remaining~$x$; we invoke the second inequality in \\eqref{E:L-vanish2} along with $bt_N\/G^{D_N}(x,x)^2\\le \\tilde cb^3\\text{\\rm  e}\\mkern0.7mu^{8b}$. \n\nFor~$b$ sufficiently large, the first term on the right of \\eqref{E:4.21ie} is~$o(\\widehat W_N\/N^2)$ independently of~$x\\in D_N$. The second term is in turn $O( \\widehat W_N\/N^2)$, with the implicit constant depending on~$b$, by the fact that that~$G^{D_N}(x,x)\\le g\\log N+c$, uniformly in~$x\\in D_N$. The sum over such~$x\\in D_N$ with~$x\/N\\in A$ is now handled via \\eqref{E:3.9ui}. \n\\end{proofsect}\n\n\\subsection{Some corollaries}\nCombining the conclusions of Lemmas~\\ref{lemma-lower} and~\\ref{lemma-vanish}, we can now derive the easier halves of Theorem~\\ref{thm-minmax} albeit for the continuous-time process parametrized by the time at the ``boundary vertex:''\n\n\\begin{lemma}\n\\label{lemma-minmax}\nSuppose~$\\theta>0$ is related to $t_N$ as in \\eqref{E:1.12}. Then for each~$\\epsilon>0$, the bounds\n\\begin{equation}\n\\label{E:max-upper}\n\\frac{1}{(\\log N)^2}\\max_{x\\in D_N} L^{D_N}_{t_N}(x)\\le 2 g\\bigl(\\sqrt\\theta+1\\bigr)^2+\\epsilon\n\\end{equation}\nand\n\\begin{equation}\n\\label{E:min-lower}\n\\frac{1}{(\\log N)^2}\\min_{x\\in D_N} L^{D_N}_{t_N}(x)\\ge 2 g\\bigl[(\\sqrt\\theta-1)\\vee0\\bigr]^2-\\epsilon\n\\end{equation}\nhold with $P^\\varrho$-probability tending to one as~$N\\to\\infty$.\n\\end{lemma}\n\n\\begin{proofsect}{Proof}\nFor the maximum, pick~$\\epsilon>0$ and abbreviate $a_N:=2 g\\bigl(\\sqrt\\theta+1+\\epsilon\\bigr)^2(\\log N)^2$. Then use \\eqref{E:2.2a} with~$b:=0$ and $a:=a_N$ to bound the probability that $L^{D_N}_{t_N}(x)\\ge a_N$ by order $N^{-2(1+\\epsilon)+o(1)}$ uniformly in~$x\\in D_N$. The union bound then gives \\eqref{E:max-upper}.\n\nFor the minimum, it suffices to deal with the case~$\\theta>1$. We pick~$\\epsilon>0$ such that $\\sqrt\\theta>1+\\epsilon$. Abbreviate  $a_N:=2 g\\bigl(\\sqrt\\theta-1-\\epsilon\\bigr)^2(\\log N)^2$ and  apply Lemma~\\ref{lemma-lower} to get, for any~$b>0$, \n\\begin{equation}\n\\label{E:3.18}\n\\begin{aligned}\nP^\\varrho\\bigl(L^{D_N}_{t_N}&(x)\\le \\,a_N\\bigr)\n\\\\\n&=P^\\varrho\\bigl(L^{D_N}_{t_N}(x)\\le b \\bigr)+P^\\varrho\\bigl(b<L^{D_N}_{t_N}(x)\\le a_N\\bigr)\\\\\n&\\le  P^\\varrho\\bigl(\\,L^{D_N}_{t_N}(x)\\le b \\bigr) \n+\\left(\\frac{t_N}{b}\\right)^{1\/4}\\frac{\\sqrt{G^{D_N}(x,x)}}{\\sqrt{2t_N}-\\sqrt{2a_N}}\\,\\text{\\rm  e}\\mkern0.7mu^{-\\frac{(\\sqrt{2t_N}-\\sqrt{2a_N})^2}{2G^{D_N}(x,x)}}\\,.\n\\end{aligned}\n\\end{equation}\n The proof of Corollary~\\ref{cor-light} bounds the first probability by $N^{-2\\theta+o(1)}$, with~$o(1)\\to0$ uniformly in~$x\\in D_N$. (As the quantity is non-decreasing in~$b$, the requirement that~$b$ be sufficiently large is achieved trivially.) Hence, even after summing over~$x\\in D_N$, the contribution of this term is negligible.  For the second term we note that, for our~$a_N$,\n\\begin{equation}\n\\frac{(\\sqrt{2t_N}-\\sqrt{2a_N})^2}{2G^{D_N}(x,x)} = 2\\bigl(1+\\epsilon+o(1)\\bigr)^{2}\\log N.\n\\end{equation}\nAs the prefactors yield only polylogarithmic terms in~$N$, also the second term on the right of \\eqref{E:3.18} is~$o(N^{-2})$ as~$N\\to\\infty$. \n\\end{proofsect}\n\nA similar argument will allow us to deal with the tightness of the thin points:\n\n\\begin{corollary}[Tightness for the thin points]\n\\label{cor-tightness-lower}\nSuppose that $t_N$ and~$a_N$ are such that the limits in \\eqref{E:1.22} exist for some~$\\theta>0$ and some $\\lambda\\in(0,\\sqrt\\theta\\wedge1)$. For all~$b\\in\\mathbb R$ there is~$c_3(b)\\in(0,\\infty)$ such that for all~$A\\subset\\mathbb R^2$ closed,\n\\begin{equation}\n\\limsup_{N\\to\\infty}\\,\nE^\\varrho\\bigl[\\,\\zeta^D_N\\bigl(A\\times(-\\infty,b]\\bigr)\\bigr]\\le c_3(b)\\,{\\rm Leb}(A\\cap D).\n\\end{equation}\n\\end{corollary}\n\n\\begin{proofsect}{Proof}\nWe proceed as in the proof of Lemma~\\ref{lemma-minmax}. Let~$a_N\\sim 2g(\\sqrt\\theta-\\lambda)^2(\\log N)^2$ be as given, pick $\\epsilon\\in(0,\\sqrt\\theta-\\lambda)$ an abbreviate~$\\hat a_N:= 2g\\epsilon^2(\\log N)^2$. Then for any $b' > 0$,\n\\begin{multline}\n\\label{E:3.18ie}\n\\quad P^\\varrho\\bigl(L^{D_N}_{t_N}(x)\\le \\,a_N + b\\log N\\bigr)\n=P^\\varrho\\bigl(\\,L^{D_N}_{t_N}(x)\\le b' \\bigr)\n\\\\+P^\\varrho\\bigl(b'<L^{D_N}_{t_N}(x)\\le \\hat a_N\\bigr)\n+P^\\varrho\\bigl(\\hat a_N<L^{D_N}_{t_N}(x)\\le a_N + b\\log N\\bigr).\n\\quad\n\\end{multline}\nExactly as in \\eqref{E:3.18}, the first term on the right-hand side is estimated to be $N^{-2\\theta+o(1)}=o(W_N\/N^2)$ uniformly in $x\\in D_N$, where we used that $W_N=N^{2-2\\lambda^2+o(1)}$ and $\\lambda<\\sqrt\\theta$. The second term is bounded as in \\eqref{E:3.18} by $N^{-2(\\sqrt\\theta-\\epsilon)^2+o(1)}=o(W_N\/N^2)$ by our choice of~$\\epsilon$. Using Lemma~\\ref{lemma-lower} (and noting that all prefactors in \\eqref{E:4.7a2} are bounded by a constant that depends only on~$\\epsilon$), the last term is now bounded by $O(W_N\/N^2)$ uniformly in~$x\\in D_N$. The observation \\eqref{E:3.9ui} again helps us  deal with the sum over~$x\\in D_N$ subject to~$x\/N\\in A$.\n\\end{proofsect}\n\n\n\n\n\\begin{remark}\n\\label{rem-restrict}\nThe reason for using the expressions ${\\rm Leb}(A\\cap D)$ to control the first moments of the measures of interest is that this will later allow us to restrict attention to~$A\\subset D$ open with~$\\overline A\\subset D$ in the arguments to follow. Indeed, taking~$\\{A_n\\}$ open with~$A_n\\uparrow D$, as $n\\to\\infty$ the expected measure of the complement $D\\smallsetminus A_n$ tends to zero by the fact that the sets $D\\smallsetminus A_n$ are closed with ${\\rm Leb}(D\\smallsetminus A_n)\\to0$.\n\\end{remark}\n\n\n\n\\section{Thick points}\n\\label{sec4}\\noindent\\nopagebreak\nWe are now ready to move to the proof of the stated convergence for the point measure associated with~$\\lambda$-thick points. Throughout we will assume that~$a_N$ and~$t_N$ satisfy \\eqref{E:1.20} with some~$\\theta>0$ and some~$\\lambda\\in(0,1)$. Introduce the auxiliary centering sequence\n\\begin{equation}\n\\label{E:4.1}\n\\widehat a_N:=\\sqrt{2a_N}-\\sqrt{2t_N}\n\\end{equation}\nand note that $\\widehat a_N\\sim 2\\lambda\\sqrt g\\log N$ as~$N\\to\\infty$.  The arguments below make frequent use of the coupling of~$L_{t_N}^{D_N}$ and an independent DGFF $h^{D_N}$ to another DGFF~$\\tilde h^{D_N}$ via the Dynkin isomorphism (Theorem~\\ref{thm-Dynkin}). We will use these notations throughout and write~$\\widehat\\eta_N^D$ to denote the DGFF process associated with~$\\tilde h^{D_N}$ and the centering sequence~$\\widehat a_N$. A key point to note is that~$W_N$ then coincides with normalizing constant from \\eqref{E:1.19e}. \n\n\\subsection{Tightness considerations}\nThe proof of Theorem~\\ref{thm-thick} naturally divides into two parts. In the first part we dominate~$\\zeta^D_N$ using~$\\widehat\\eta_N^D$ and control the effect of adding~$h^{D_N}$ to the local time~$L^{D_N}_{t_N}$. The second part is then a derivation, and a solution, of a convolution-type identity linking the weak-limits of~$\\zeta^D_N$ to those of~$\\widehat\\eta_N^D$. Our tightness considerations start by the following domination lemma: \n\n\\begin{lemma}[Domination by DGFF process]\n\\label{lemma-domination}\nFor any~$b\\in\\mathbb R$ and any measurable~$A\\subseteq D$,\n\\begin{equation}\n\\label{E:4.7a}\n\\zeta^D_N\\bigl(A\\times[b,\\infty)\\bigr)\\,\\,\\overset{\\text{\\rm law}}\\le\\,\\,\\widehat\\eta_N^D\\Bigl(A\\times\\bigl[\\tfrac1{2\\sqrt g}\\tfrac{b}{\\sqrt\\theta+\\lambda},\\infty\\bigr)\\Bigr)\n+o(1)\n\\end{equation}\nwhere~$o(1)\\to0$ in probability as $N\\to\\infty$. Similarly, for any measurable~$A\\subseteq D\\times D$ and any~$b\\in\\mathbb R$,\n\\begin{equation}\n\\label{E:4.7b}\n\\zeta^D_N\\otimes\\zeta^D_N\\bigl(A\\times[b,\\infty)^2\\bigr)\\,\\,\\overset{\\text{\\rm law}}\\le\\,\\,\\widehat\\eta_N^D\\otimes\\widehat\\eta_N^D\\Bigl(A\\times\\bigl[\\tfrac1{2\\sqrt g}\\tfrac{b}{\\sqrt\\theta+\\lambda},\\infty\\bigr)^2\\Bigr)\n+o(1).\n\\end{equation}\n\\end{lemma}\n\n\\begin{proofsect}{Proof}\nLet us start by \\eqref{E:4.7a}. The Dynkin isomorphism shows\n\\begin{equation}\n\\label{E:4.4u}\nL^{D_N}_{t_N}\\le L^{D_N}_{t_N}+\\frac12 (h^{D_N})^2= \\frac12\\bigl(\\tilde h^{D_N}+\\sqrt{2t_N}\\bigr)^2.\n\\end{equation}\nHence,\nthe expression in \\eqref{E:4.7a} is bounded as\n\\begin{equation}\n\\label{E:4.10}\n\\zeta^D_N\\bigl(A\\times[b,\\infty)\\bigr)\n\\le\\frac1{W_N}\\sum_{\\begin{subarray}{c}\nx\\in D_N\\\\x\/N\\in A\n\\end{subarray}}\n1_{\\{|\\tilde h^{D_N}_x+\\sqrt{2t_N}|\\ge\\sqrt{2a_N+2b\\log N}\\}}.\n\\end{equation}\nPick any~$b'<b\\tfrac1{2\\sqrt g}\\tfrac{1}{\\sqrt\\theta+\\lambda}$.  Once~$N$ is sufficiently large, the asymptotic formulas for~$a_N$ and~$t_N$ yield \n\\begin{equation}\n\\label{E:4.6e}\n1_{\\{|\\tilde h^{D_N}_x+\\sqrt{2t_N}|\\ge\\sqrt{2a_N+2b\\log N}\\}}\n\\le 1_{\\{\\tilde h^{D_N}_x\\ge\\widehat a_N+b'\\}}+\n1_{\\{\\tilde h^{D_N}_x\\le-\\sqrt{2a_N}-\\sqrt{2t_N}- b'\\}}\\,.\n\\end{equation}\n Writing~$\\overline\\eta^D_N$ for the process associated with the field $-\\tilde h^{D_N}$ and the centering sequence $\\sqrt{2a_N}+\\sqrt{2t_N}$, and~$\\bar K_N$ for the associated normalization from \\eqref{E:1.19e}, we thus have\n\\begin{equation}\n\\zeta^D_N\\bigl(A\\times[b,\\infty)\\bigr)\n\\le \\widehat\\eta^D_N\\bigl(A\\times[b',\\infty)\\bigr)+\\frac{\\bar K_N}{W_N}\\,\\overline\\eta^D_N\\bigl(A\\times[b',\\infty)\\bigr).\n\\end{equation}\nNoting that $\\{\\overline\\eta^D_N\\colon N\\ge1\\}$ is tight on~$\\overline D\\times(\\mathbb R\\cup\\{\\infty\\})$ and~$\\bar K_N=o(W_N)$, the second term is~$o(1)$ in probability as~$N\\to\\infty$. To get \\eqref{E:4.7a} we now take~$b'$ to~$b\\tfrac1{2\\sqrt g}\\tfrac{1}{\\sqrt\\theta+\\lambda}$ and invoke the continuity of the limit measure in \\eqref{E:1.19} in the second variable.\n\nThe proof of \\eqref{E:4.7b} is completely analogous. Indeed, the same reasoning implies, for any Borel~$A\\subseteq D\\times D$,\n\\begin{multline}\n\\quad\\zeta^D_N\\otimes\\zeta^D_N\\bigl(A\\times[b,\\infty)^2\\bigr)\n\\le \\widehat\\eta^D_N\\otimes \\widehat\\eta^D_N\\bigl(A\\times[b',\\infty)^2\\bigr)+\n\\frac{\\bar K_N}{W_N}\\,\\widehat\\eta^D_N\\otimes\\overline\\eta^D_N\\bigl(A\\times[b',\\infty)^2\\bigr)\n\\\\\n+\\frac{\\bar K_N}{W_N}\\,\\overline\\eta^D_N\\otimes\\widehat\\eta^D_N\\bigl(A\\times[b',\\infty)^2\\bigr)\n+\\Bigl(\\frac{\\bar K_N}{W_N}\\Bigr)^2\\,\\overline\\eta^D_N\\otimes\\overline\\eta^D_N\\bigl(A\\times[b',\\infty)^2\\bigr).\n\\quad\n\\end{multline}\nReplacing~$A$ by $D\\times D$ in the last three terms shows, via~$\\bar K_N=o(W_N)$, that these three terms are again all~$o(1)$ in probability as~$N\\to\\infty$. A continuity argument in the second variable then proves \\eqref{E:4.7b} as well.\n\\end{proofsect}\n\nNote that Lemma~\\ref{lemma-domination} provides an independent proof of the tightness of the measures~$\\zeta^D_N$. Based on the proof one might think that~$\\zeta^D_N$ is asymptotically close to $\\widehat\\eta_N^D$, but this is false: Although \\eqref{E:4.6e} is asymptotically sharp, the inequalities in \\twoeqref{E:4.4u}{E:4.10} are not. To account for this gap, we have to carefully examine the effect of adding the half of the DGFF-squared to the local time. In particular, we have to ensure that the DGFF remains typical even at the points where the local time \\myemph{combined with} half of its square is large. This rather important step is the content of:\n\n\\begin{lemma}\n\\label{lemma-no-conspire}\nLet $0<\\beta<\\frac1{2g}\\frac{\\sqrt\\theta}{\\sqrt\\theta+\\lambda}$. Then for each~$b\\in\\mathbb R$ there is~$\n c_4(b)\\in(0,\\infty)$ such that for all $M\\ge0$, all sufficiently large~$N$ and all~$x\\in D_N$,\n\\begin{equation}\nP^\\varrho\\otimes\\mathbb P\\biggl(L^{D_N}_{t_N}(x)+\\frac{(h^{D_N}_x)^2}2\\ge a_N+b\\log N,\\,\\frac{|h^{D_N}_x|}{\\sqrt{\\log N}}\\ge M\\biggr)\\le  c_4(b)\\frac{W_N}{N^2}\\text{\\rm  e}\\mkern0.7mu^{-\\beta M^2}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proofsect}{Proof}\n Since the $b\\log N$-correction can be absorbed into a re-definition of~$a_N$, which thanks to the assumed asymptotic behavior of~$a_N$ and~$t_N$ only changes~$W_N$ by a multiplicative constant, we may assume that~$b=0$ for simplicity below.  Assume also that~$M$ is an integer and pick~$\\delta$ with\n\\begin{equation}\n\\label{E:4.2a}\n0<\\delta<2\\sqrt\\theta\\lambda.\n\\end{equation}\nPartitioning the event according to which interval of the form $[n,n+1)$, with~$n\\in\\mathbb N$ subject to $n\\ge M^2$, the ratio $(h^{D_N}_x)^2\/\\log N$ lands in bounds the desired probability by \n\\begin{multline}\n\\label{E:4.2}\n\\mathbb P\\Bigl(|h^{D_N}_x|\\ge 2\\sqrt g\\sqrt{\\lambda^2+\\delta}\\,\\log N\\Bigr)\n\\\\\n+\\sum_{M^2\\le n\\le4 g(\\lambda^2+\\delta)\\log N}\\!\\!\\!\\!\\mathbb P\\bigl((h^{D_N}_x)^2\\ge n\\log N\\bigr)P^\\varrho\\Bigl(L^{D_N}_{t_N}(x)\\ge a_N-\\frac12(n+1)\\log N\\Bigr).\n\\end{multline}\nA standard Gaussian bound estimates the first probability by a constant times \n$N^{- 2(\\lambda^2+\\delta)}$ which is $o(W_N\/N^2)$ as~$N\\to\\infty$. Concerning the terms in the sum, here we first note that for all~$n$ under the summation symbol,\n\\begin{equation}\n\\begin{aligned}\na_N-\\frac12(n+1)\\log N\n&\\ge 2g\\Bigl[(\\sqrt\\theta+\\lambda)^2-(\\lambda^2+\\delta)+o(1)\\Bigr](\\log N)^2\n\\\\\n&=t_N+2g\\bigl(2\\sqrt\\theta\\lambda-\\delta+o(1)\\bigr)(\\log N)^2.\n\\end{aligned}\n\\end{equation}\nHence, under \\eqref{E:4.2a}, Lemma~\\ref{lemma-upper} can be applied. Using $G^{D_N}(x,x)\\le g\\log N+c$, the term corresponding to integer~$n$ in the sum is thus bounded by\n\\begin{equation}\n\\label{E:4.4ui}\n\\tilde c\\frac{W_N}{N^2}\\exp\\biggl\\{\\frac1{2g}\\Bigl[(n+1)\\frac{\\sqrt{2a_N}-\\sqrt{2t_N}}{\\sqrt{2a_N}}-n\\Bigr]\\biggr\\},\n\\end{equation}\nwhere~$\\tilde c$ is a constant that depends on~$\\theta$, $\\lambda$ and our choice of~$\\delta$ but not on~$N$ or~$x$ or~$n$.\nSince the assumptions on~$a_N$ and~$t_N$ give\n\\begin{equation}\n\\frac{\\sqrt{2a_N}-\\sqrt{2t_N}}{\\sqrt{2a_N}}\\,\\underset{N\\to\\infty}\\longrightarrow\\,\\frac{\\lambda}{\\sqrt\\theta+\\lambda}<1-2g\\beta\n\\end{equation}\nas soon as~$N$ is sufficiently large, the quantity in \\eqref{E:4.4ui} is summable on~$n$ and the sum in \\eqref{E:4.2} is thus dominated by the term with~$n=M^2$. The claim follows.\n\\end{proofsect}\n\n\\subsection{Convolution identity}\nWe now move to the second part which consists of a derivation of, and a solution to, a convolution identity that links weak (subsequential) limits of $\\zeta^D_N$ to those of~$\\widehat\\eta^D_N$. We begin by observing that, at the scale of its typical fluctuations, the field~$h^{D_N}$ that we add to~$L^{D_N}_{t_N}$ in the Dynkin isomorphism acts like white noise:\n\n\\begin{lemma}\n\\label{lemma-add-field}\nSuppose $\\{N_k\\}$ is a subsequence along which~$\\zeta^D_N$ converges in law to~$\\zeta^D$. Then\n\\begin{equation}\n\\label{E:4.7}\n\\frac1{W_N}\\sum_{x\\in D_N}\\delta_{x\/N}\\otimes\\delta_{(L^{D_N}_{t_N}(x)-a_N)\/\\log N}\\otimes\\delta_{h_x^{D_N}\/\\sqrt{\\log N}}\\,\\,\\underset{\\begin{subarray}{c}\nN=N_k\\\\k\\to\\infty\n\\end{subarray}}{\\overset{\\text{\\rm law}}\\longrightarrow}\\,\\,\n\\zeta^D\\otimes\\mathfrak g\n\\end{equation}\nwhere~$\\mathfrak g$ is the law of~$\\mathcal N(0,g)$.\n\\end{lemma}\n\n\\begin{proofsect}{Proof}\nThe argument is based on a \\myemph{conditional} second moment calculation and domination by the DGFF process from Lemma~\\ref{lemma-domination}.\nDenote by~$\\zeta^{D,\\text{ext}}_N$ the measure on the left of \\eqref{E:4.7}, pick~$f\\in C_{\\text{\\rm c}}(D\\times\\mathbb R\\times\\mathbb R)$ \nand let a compact set~$A\\subset D$ and~$b>0$ be such that\n\\begin{equation}\n\\label{E:4.8}\n\\bigl|f(x,\\ell,h)\\bigr|\\le \\Vert f\\Vert_\\infty 1_A(x)1_{[-b,\\infty)}(\\ell)1_{[-b,b]}(h).\n\\end{equation}\nAbbreviate $L'_N(x):=(L^{D_N}_{t_N}(x)-a_N)\/\\log N$ and $h'_N(x):=h^{D_N}_x\/\\sqrt{\\log N}$.  Writing $\\text{Var}_{\\mathbb P}$, resp., $\\text{Cov}_{\\mathbb P}$ for the conditional variance, resp., covariance  given the local time, we have\n\\begin{multline}\n\\label{E:4.11}\n\\quad\\text{Var}_{\\mathbb P}\\bigl(\\langle\\zeta^{D,\\text{ext}}_N,f\\rangle\\bigr)\n\\\\\n=\\frac1{W_N^2}\\sum_{x,y\\in D_N}\\text{Cov}_{\\mathbb P}\\Bigl(f\\bigl(\\ffrac xN, L'_N(x),h'_N(x)\\bigr),f\\bigl(\\ffrac yN, L'_N(y),h'_N(y)\\bigr)\\Bigr).\n\\quad\n\\end{multline}\nPick~$\\epsilon>0$ and split the sum according to whether~$|x-y|\\ge\\epsilon N$ or not. In the former cases, we use the Gibbs-Markov decomposition to write~$h^{D_N}$ using the value~$h^{D_N}_x$ and an independent DGFF in~$D_N\\smallsetminus\\{x\\}$ as\n\\begin{equation}\n\\label{E:5.12i}\nh^{D_N}\\,\\overset{\\text{\\rm law}}=\\, h^{D_N}_x\\fraktura b_{D_N,x}(\\cdot)+\\widehat h^{D_N\\smallsetminus\\{x\\}},\\quad h^{D_N}_x\\independent \\widehat h^{D_N\\smallsetminus\\{x\\}},\n\\end{equation}\nwhere~$\\fraktura b_{D_N,x}\\colon\\mathbb Z^2\\to[0,1]$ is the unique function that is discrete harmonic on~$D_N\\smallsetminus\\{x\\}$, vanishes outside~$D_N$ and equals one at~$x$. A key point, proved with the help of monotonicity of~$D\\mapsto\\fraktura b_{D,x}(y)$ with respect to set inclusion, is\n\\begin{equation}\n\\label{E:5.13i}\n\\max_{\\begin{subarray}{c}\nx,y\\in D_N\\\\|x-y|\\ge\\epsilon N\n\\end{subarray}}\n\\fraktura b_{D_N,x}(y)\\le\\frac{c(\\epsilon)}{\\log N},\n\\end{equation}\nwhere~$c(\\epsilon)\\in(0,\\infty)$ is independent of~$N$. The  uniform continuity of~$f$ in the third variable then shows that the portion of the sum in \\eqref{E:4.11} corresponding to~$|x-y|\\ge\\epsilon N$ vanishes in the limit~$N\\to\\infty$. \n\nUsing the bound in \\eqref{E:4.8}, the part of the sum  in \\eqref{E:4.11}  corresponding to~$|x-y|\\le\\epsilon N$ is bounded by $\\Vert f\\Vert_\\infty^2$ times\n\\begin{equation}\n\\label{E:4.12}\n\\zeta^D_N\\otimes\\zeta^D_N\\Bigl(\\bigl\\{(x,y)\\colon |x-y|\\le\\epsilon\\bigr\\}\\times[-b,\\infty)^2\\Bigr).\n\\end{equation}\nBy Lemma~\\ref{lemma-domination}, this is stochastically dominated by\n\\begin{equation}\n\\widehat\\eta_N^D\\otimes\\widehat\\eta_N^D\\Bigl(\\bigl\\{(x,y)\\colon |x-y|\\le\\epsilon\\bigr\\}\\times\n[-\\tfrac{1}{2\\sqrt{g}} \\tfrac{b}{\\sqrt{\\theta}+\\lambda},\\infty)^2\\Bigr)+o(1).\n\\end{equation}\nAs $\\{(x,y)\\colon |x-y|\\le\\epsilon\\}$ is closed and $\\widehat a_N\\sim2\\lambda\\sqrt g\\log N$ as~$N\\to\\infty$, \\eqref{E:1.19} and the Portmanteau Theorem show that this expression is, in the limit~$N\\to\\infty$, stochastically dominated by a \\normalcolor $b$-dependent constant times\n\\begin{equation}\nZ^D_\\lambda\\otimes Z^D_\\lambda\\bigl(\\{(x,y)\\colon |x-y|\\le\\epsilon\\}\\bigr).\n\\end{equation}\nAs~$\\epsilon\\downarrow0$, this tends to zero  a.s. due to the fact that~$Z^D_\\lambda$ has no point masses a.s.\n\nWe conclude that $\\text{Var}_{\\mathbb P}(\\langle\\zeta^{D,\\text{ext}}_N,f\\rangle)$\ntends to zero in $P^\\varrho$-probability. This implies\n\\begin{equation}\n\\label{E:4.20}\n\\langle\\zeta^{D,\\text{ext}}_N,f\\rangle-\\mathbb E\\bigl(\\langle\\zeta^{D,\\text{ext}}_N,f\\rangle\\bigr)\\,\\,\\underset{N\\to\\infty}\\longrightarrow\\,\\,0,\\quad\\text{in $P^\\varrho\\otimes\\mathbb P$-probability}.\n\\end{equation}\nTo infer the desired claim, abbreviate\n\\begin{equation}\nf_{\\mathfrak g}(x,\\ell):=\\int\\mathfrak g(\\text{\\rm d}\\mkern0.5mu h) f(x,\\ell,h)\n\\end{equation}\nand note that, since $A$ in \\eqref{E:4.8} is compact,~$h^{D_N}_x\/\\sqrt{\\log N}$ tends in law to~$\\mathcal N(0,g)$ uniformly for all $x\\in\\{y\\in D_N\\colon y\/N\\in A\\}$.  The continuity of~$f$ along with \\eqref{E:4.8} yield \n\\begin{equation}\n\\label{E:4.22}\n\\mathbb E\\bigl(\\langle\\zeta^{D,\\text{ext}}_N,f\\rangle\\bigr)-\\langle\\zeta^D_N,f_{\\mathfrak g}\\rangle\\,\\,\\underset{N\\to\\infty}\\longrightarrow\\,\\,0,\\quad\\text{in $P^\\varrho$-probability}.\n\\end{equation}\nCombining \\eqref{E:4.20} and \\eqref{E:4.22} we then get \\eqref{E:4.7}.\n\\end{proofsect}\n\n\nAs a consequence of the above lemmas, we now get:\n\n\\begin{lemma}\n\\label{lemma-conv}\nRecall that~$\\mathfrak g$ is the law of~$\\mathcal N(0,g)$. Given~$f\\in C_{\\text{\\rm c}}(D\\times\\mathbb R)$ with~$f\\ge0$, let\n\\begin{equation}\n\\label{E:4.24}\nf^{\\ast\\mathfrak g}(x,\\ell):=\\int\\mathfrak g(\\text{\\rm d}\\mkern0.5mu h)f\\Bigl(x,\\tfrac1{2\\sqrt g(\\sqrt\\theta+\\lambda)}\\bigl(\\ell+\\tfrac{h^2}2\\bigr)\\Bigr).\n\\end{equation}\nThen for every subsequential weak limit~$\\zeta^D$ of~$\\zeta^D_N$, simultaneously for all~$f$ as above,\n\\begin{equation}\n\\label{E:4.25a}\n\\langle\\zeta^D,f^{\\ast\\mathfrak g}\\rangle\\,\\overset{\\text{\\rm law}}=\\, \\fraktura c(\\lambda)\\int Z^D_\\lambda(\\text{\\rm d}\\mkern0.5mu x)\\otimes\\text{\\rm  e}\\mkern0.7mu^{-\\alpha\\lambda h}\\text{\\rm d}\\mkern0.5mu h \\,\\,f(x,h),\n\\end{equation}\nwhere, we recall,~$\\alpha:=2\/\\sqrt g$ and~$\\fraktura c(\\lambda)$ is as in \\eqref{E:1.19}.\n\\end{lemma}\n\n\\begin{proofsect}{Proof}\nPick~$f$ as above and let~$f^{\\ast\\mathfrak g}$ be as in \\eqref{E:4.24}. Let the DGFF~$\\tilde h^{D_N}$ be related to the local time~$L^{D_N}_{t_N}$ and an independent DGFF~$h^{D_N}$ via the Dynkin isomorphism. Recalling \\eqref{E:4.1}, we then have\n\\begin{equation}\n\\label{E:4.26}\n\\begin{aligned}\n\\langle\\widehat\\eta_N^D, f\\rangle&=\\frac1{W_N}\\sum_{x\\in D_N}f\\bigl(\\ffrac xN,\\tilde h_x^{D_N}-\\widehat a_N\\bigr)\n\\\\\n&=\\frac1{W_N}\\sum_{x\\in D_N}f\\Bigl(\\ffrac xN,\\sqrt{2L^{D_N}_{t_N}(x)+(h_x^{D_N})^2}-\\sqrt{2a_N}\\Bigr),\n\\end{aligned}\n\\end{equation}\nwhere the sign ambiguity of the square root was resolved by noting that the compact support of~$f$ forces~$\\tilde h_x^{D_N}$ to be strictly positive once~$N$ is sufficiently large. By the same reasoning, for~$x$ to contribute to the sum, $2L^{D_N}_{t_N}(x)+(h_x^{D_N})^2$ must be close to~$2a_N$. Let $\\chi\\colon[0,\\infty)\\to[0,1]$ be  non-increasing,  \ncontinuous with~$\\chi(x)=1$ for~$0\\le x\\le1$ and $\\chi(x)=0$ for~$x\\ge2$. Invoking Lemma~\\ref{lemma-no-conspire}, we now truncate the values of~$h_x^{D_N}$ using~$\\chi$ and rewrite \\eqref{E:4.26} as a random quantity whose~$L^\\infty$ norm is at most a constant times $\\Vert f\\Vert_\\infty\\text{\\rm  e}\\mkern0.7mu^{-\\beta M^2}$ uniformly in~$N$ plus the quantity\n\\begin{equation}\n\\label{E:4.27}\n\\frac1{W_N}\\sum_{x\\in D_N}f\\Bigl(\\ffrac xN,\\sqrt{2a_N+2[\\,L^{D_N}_{t_N}(x)-a_N]+(h_x^{D_N})^2}-\\sqrt{2a_N}\\Bigr)\\chi\\Bigl(\\frac{|h_x^{D_N}|}{M\\sqrt{\\log N}}\\Bigr).\n\\end{equation}\nThe truncation of the field ensures that, for~$x$ to contribute to the sum, \\myemph{both}~$(h_x^{D_N})^2$ and $L^{D_N}_{t_N}(x)-a_N$ must be at most order~$\\log N$. Expanding the square root and using the uniform continuity of~$f$ along with the tightness of~$\\zeta^D_N$ to replace $a_N$ by its asymptotic expression then recasts \\eqref{E:4.27} as\n\\begin{equation}\n\\label{E:4.28}\n\\frac1{W_N}\\sum_{x\\in D_N}f_{\\text{ext}}\\Bigl(\\ffrac xN,\\tfrac{L^{D_N}_{t_N}(x)-a_N}{\\log N}, \\tfrac{h_x^{D_N}}{\\sqrt{\\log N}}\\Bigr)\\chi\\Bigl(\\frac{|h_x^{D_N}|}{M\\sqrt{\\log N}}\\Bigr)+o(1),\n\\end{equation}\nwhere\n\\begin{equation}\nf_{\\text{ext}}(x,\\ell,h):=f\\Bigl(x,\\tfrac1{2\\sqrt g(\\sqrt\\theta+\\lambda)}\\bigl(\\ell+\\tfrac{h^2}2\\bigr)\\Bigr).\n\\end{equation}\nThe function $\\ell,h\\mapsto f_{\\text{ext}}(x,\\ell,h)\\chi(|h|\/M)$ that effectively appears in \\eqref{E:4.28} is  compactly supported in both variables; Lemma~\\ref{lemma-add-field} then shows that, along subsequences where~$\\zeta^D_N$ converges in law to some~$\\zeta^D$, the expression in \\eqref{E:4.28} converges to $\\langle\\zeta^D, f^{\\ast\\mathfrak g}_M \\rangle$ where~$f^{\\ast\\mathfrak g}_M$ is defined by \\eqref{E:4.24} with~$\\mathfrak g(\\text{\\rm d}\\mkern0.5mu h)$ replaced by $\\chi(|h|\/M)\\mathfrak g(\\text{\\rm d}\\mkern0.5mu h)$. \nFrom the known convergence of~$\\widehat\\eta_N^D$ (see \\eqref{E:1.19}) we thus conclude \n\\begin{equation}\n\\langle\\zeta^D,f^{\\ast\\mathfrak g}_M\\rangle +O(\\text{\\rm  e}\\mkern0.7mu^{-\\beta M^2})\\,\\,\\overset{\\text{\\rm law}}=\\,\\, \\fraktura c(\\lambda)\\int Z^D_\\lambda(\\text{\\rm d}\\mkern0.5mu x)\\otimes\\text{\\rm  e}\\mkern0.7mu^{-\\alpha\\lambda h}\\text{\\rm d}\\mkern0.5mu h \\,\\,f(x,h),\n\\end{equation}\nwhere $O(\\text{\\rm  e}\\mkern0.7mu^{-\\beta M^2})$ is a random quantity with $L^\\infty$-norm at most a constant times~$\\text{\\rm  e}\\mkern0.7mu^{-\\beta M^2}$. Taking~$M\\to\\infty$ via Monotone Convergence Theorem now gives \\eqref{E:4.25a}.\n\\end{proofsect}\n\nWorking towards the proof of Theorem~\\ref{thm-thick}, a key  remaining point to show is that the class of~$f^{\\ast\\mathfrak g}$ arising from functions~$f$ for which the integral on the right of \\eqref{E:4.25a} converges absolutely is sufficiently rich so that \\eqref{E:4.25a} determines the measure~$\\zeta^D$ uniquely. For this we note that, by an application of the Dominated and Monotone Convergence Theorems, \\eqref{E:4.25a} extends from~$C_{\\text{\\rm c}}(D\\times\\mathbb R)$ to the class of functions~$(x,h)\\mapsto 1_A(x)f(h)$, where $A\\subset D$ is open with~$\\overline A\\subset D$ and $f\\in C_{\\text{\\rm c}}^\\infty(\\mathbb R)$ with~$f\\ge0$. The transformation \\eqref{E:4.24} only affects the second variable on which it takes the form $f\\mapsto (f\\ast\\fraktura e)\\circ\\fraktura s$, where the convolution is with the function\n\\begin{equation}\n\\label{E:5.33}\n \\fraktura e(z):=\\sqrt{\\frac\\beta\\pi}\\,\\frac{\\text{\\rm  e}\\mkern0.7mu^{\\beta z}}{\\sqrt{-z}}1_{(-\\infty,0)}(z)\\quad\\text{for}\\quad\\beta:=\\alpha\\bigl(\\sqrt\\theta+\\lambda\\bigr)\n\\end{equation}\nand where $h\\mapsto\\fraktura s(h)$ is the scaling map\n\\begin{equation}\n\\label{E:5.34}\n\\fraktura s(h):=\\frac{h}{2\\sqrt g(\\sqrt\\theta+\\lambda)}.\n\\end{equation}\nAs it turns out, it then suffices to observe:\n\n\\begin{lemma}\n\\label{lemma-unique}\nLet $\\mu_\\lambda(\\text{\\rm d}\\mkern0.5mu h):=\\text{\\rm  e}\\mkern0.7mu^{-\\alpha\\lambda h}\\text{\\rm d}\\mkern0.5mu h$. Assuming~$\\theta>0$, there is at most one Radon measure~$\\nu$ on~$\\mathbb R$ such that for all~$f\\in C_{\\text{\\rm c}}^\\infty(\\mathbb R)$ with $f\\ge0$,\n\\begin{equation}\n\\label{E:5.35iw}\n\\bigl\\langle\\nu,f\\ast\\fraktura e\\bigr\\rangle=\\langle\\mu_\\lambda,f\\rangle.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proofsect}{Proof}\nWriting \\eqref{E:5.35iw} explicitly using integrals and using the fact that the class of all $f\\in C_{\\text{\\rm c}}^\\infty(\\mathbb R)$ with $f\\ge0$ separates Radon measures on~$\\mathbb R$ shows\n\\begin{equation}\n\\int_{\\mathbb R}\\nu(\\text{\\rm d}\\mkern0.5mu s)\\fraktura e(s-h) = \\text{\\rm  e}\\mkern0.7mu^{-\\alpha\\lambda h},\\quad h\\in\\mathbb R.\n\\end{equation}\nAbbreviating $\\nu_{\\lambda}(\\text{\\rm d}\\mkern0.5mu h):=\\text{\\rm  e}\\mkern0.7mu^{\\alpha\\lambda h}\\nu(\\text{\\rm d}\\mkern0.5mu h)$ and $\\fraktura e_{\\lambda}(h):=\\text{\\rm  e}\\mkern0.7mu^{-\\alpha\\lambda h}\\fraktura e(h)$, this can be recast as\n\\begin{equation}\n\\int_{\\mathbb R}\\nu_{\\lambda}(\\text{\\rm d}\\mkern0.5mu s)\\fraktura e_{\\lambda}( s-h ) = 1,\\quad h\\in\\mathbb R.\n\\end{equation}\nIntegrating this against suitable test functions with respect to the Lebesgue measure and applying Dominated Convergence, we conclude\n\\begin{equation}\n\\bigl\\langle\\nu_\\lambda,f\\ast\\fraktura e_\\lambda\\bigr\\rangle=\\langle{\\rm Leb},f\\rangle,\\quad f\\in \\mathcal S(\\mathbb R),\n\\end{equation}\nwhere~$\\mathcal S(\\mathbb R)$ is the Schwartz class of functions on~$\\mathbb R$. Note that this identity entails that the integral on the left-hand side converges absolutely.\n\nSince $\\mathcal S(\\mathbb R)$ separates Radon measures on~$\\mathbb R$, to conclude the statement from \\eqref{E:5.35iw} it suffices to prove that, for $\\theta>0$,\n\\begin{equation}\n\\label{E:5.39iw}\nf\\mapsto f\\ast\\fraktura e_\\lambda\\text{ is a bijection of }\\mathcal S(\\mathbb R) \\text{ onto itself}.\n\\end{equation}\nThe Fourier transform maps~$\\mathcal S(\\mathbb R)$ bijectively onto itself and so we may as well prove \\eqref{E:5.39iw} in the Fourier picture. For this we note that, as $\\theta>0$ we have~$\\tilde\\beta:=\\beta-\\alpha\\lambda>0$ and so $z\\mapsto\\fraktura e_\\lambda(z)$ decays exponentially as~$z\\to-\\infty$. In particular, $\\fraktura e_\\lambda$ is integrable and so in the Fourier transform,~$f\\mapsto f\\ast\\fraktura e_\\lambda$ is reduced to the multiplication by\n\\begin{equation}\n\\widehat\\fraktura e_\\lambda(k):=\\int_\\mathbb R\\text{\\rm d}\\mkern0.5mu z\\,\\fraktura e_\\lambda(z)\\text{\\rm  e}\\mkern0.7mu^{2\\pi\\text{\\rm  i}\\mkern0.7mu kz}\n= \\sqrt{\\frac{\\beta}{\\tilde\\beta}}\\,\\frac1{|1+4\\pi^2 \\tilde\\beta^{-2}k^2|^{1\/4}}\\,\\text{\\rm  e}\\mkern0.7mu^{-\\text{\\rm  i}\\mkern0.7mu\\theta(k)},\n\\end{equation}\nwhere~$\\theta(k)$ is the unique number in~$[0,\\frac\\pi4]$ such that\n\\begin{equation}\n \\cos\\theta(k)=\\sqrt{\\frac{1+|1+4\\pi^2 \\tilde\\beta^{-2}k^2|^{-1\/2}}2}.\n\\end{equation}\n We now  check that  $k\\mapsto\\widehat\\fraktura e_\\lambda(k)$  is bounded and~$C^\\infty(\\mathbb R)$ which  implies that $f\\mapsto  \\fraktura e_{\\lambda} \\ast f$ maps~$\\mathcal S(\\mathbb R)$ into~$\\mathcal S(\\mathbb R)$. The map is  also  injective because~$|\\widehat\\fraktura e_\\lambda|>0$ and onto because $|\\widehat\\fraktura e_\\lambda(k)|^{-1}$ is bounded by a power of~$|k|$.  Hence \\eqref{E:5.39iw} follows. \n\\end{proofsect}\n\nWe are now ready to give:\n\n\\begin{proofsect}{Proof of Theorem~\\ref{thm-thick}}\nConsider a subsequential limit~$\\zeta^D$, pick~$f\\in C_{\\text{\\rm c}}(\\mathbb R)$ with~$f\\ge0$ and let~$A\\subset D$ be open with~$\\overline A\\subset D$. Using the notation \\twoeqref{E:5.33}{E:5.34} we then have \n\\begin{equation}\n\\bigl\\langle\\zeta^D,(1_A\\otimes f)^{\\ast\\mathfrak g}\\bigr\\rangle=\\bigl\\langle\\zeta_A^D,(f\\ast\\fraktura e)\\circ\\fraktura s\\bigr\\rangle=\\bigl\\langle\\zeta_A^D\\circ\\fraktura s^{-1},f\\ast\\fraktura e\\bigr\\rangle,\n\\end{equation}\nwhere~$\\zeta_A^D$ is a Borel measure on~$\\mathbb R$ defined by $\\zeta_A^D(B):=\\zeta^D(A\\times B)$. Writing $\\mu_\\lambda(\\text{\\rm d}\\mkern0.5mu h):=\\text{\\rm  e}\\mkern0.7mu^{-\\alpha\\lambda h}\\text{\\rm d}\\mkern0.5mu h$, the identity \\eqref{E:4.25a} then translates into\n\\begin{equation}\n\\label{E:5.43nw}\n\\bigl\\langle\\zeta_A^D\\circ\\fraktura s^{-1},f\\ast\\fraktura e\\bigr\\rangle \\,\\overset{\\text{\\rm law}}=\\, \\fraktura c(\\lambda)Z^D_\\lambda(A)\\langle\\mu_\\lambda,f\\rangle,\n\\end{equation}\nwhere the equality in law holds simultaneously for all~$A$ and~$f$ as above. \n\nTo infer the product form of~$\\zeta^D$ from \\eqref{E:5.43nw}, define (for a given~$A$ and a given realization of~$\\zeta^D$) a Borel measure on~$\\mathbb R$ by\n\\begin{equation}\n\\label{E:5.44nw}\n\\nu:=\\Bigl[\\alpha\\lambda \\bigl\\langle\\zeta_A^D\\circ\\fraktura s^{-1},1_{[0,\\infty)}\\ast\\fraktura e\\bigr\\rangle\\Bigr]^{-1}\\zeta^D_A,\n\\end{equation}\nwhere the conditions on~$A$ imply $Z^D_\\lambda(A)>0$ a.s.\\ and so, by \\eqref{E:5.43nw}, the quantity in the square bracket is strictly positive a.s. By \\eqref{E:5.43nw} we have $\\langle\\nu\\circ\\fraktura s^{-1},f\\ast\\fraktura e\\rangle=\\langle\\mu_\\lambda,f\\rangle$ for all~$f\\in C_{\\text{\\rm c}}(\\mathbb R)$ and so, by Lemma~\\ref{lemma-unique}, $\\nu\\circ\\fraktura s^{-1}$, and thus also~$\\nu$, is determined uniquely. In particular, $\\nu$ is the same for all~$A$ as above and for a.e.~realization of~$\\zeta^D$. Using \\eqref{E:5.43nw} in \\eqref{E:5.44nw} then shows $\\zeta^D_A(\\text{\\rm d}\\mkern0.5mu h)\\,\\overset{\\text{\\rm law}}=\\,\\fraktura c(\\lambda)Z^D_\\lambda(A)\\nu(\\text{\\rm d}\\mkern0.5mu h)$. As this holds simultaneously for all~$A$ as above, Remark~\\ref{rem-restrict} permits us to conclude\n\\begin{equation}\n\\zeta^D\\,\\overset{\\text{\\rm law}}=\\, \\fraktura c(\\lambda)Z^D_\\lambda\\,\\otimes\\,\\nu,\n\\end{equation}\nwhere~$\\nu$ is a uniquely-determined deterministic Radon measure on~$\\mathbb R$. \n\nIt remains to derive the explicit form of~$\\nu$ which, thanks to its uniqueness, we can do by plugging the desired expression on the left-hand side of \\eqref{E:4.25a} and checking for equality. Abbreviate $\\tilde\\alpha:=\\alpha(\\theta,\\lambda)$ and note that\n\\begin{equation}\n\\tilde\\alpha = \\frac1{2\\sqrt g(\\sqrt\\theta+\\lambda)}\\alpha\\lambda.\n\\end{equation}\nPick~$f\\in C_{\\text{\\rm c}}(D\\times\\mathbb R)$ and perform  the following calculation where, in the last step, we invoke the substitution $r:=\\frac1{2\\sqrt g(\\sqrt\\theta+\\lambda)}(\\ell+\\frac{h^2}2)$ and separate integrals using Fubini-Tonelli:\n\\begin{equation}\n\\label{E:4.38}\n\\begin{aligned}\n\\int_{D\\times\\mathbb R}&Z^D_\\lambda(\\text{\\rm d}\\mkern0.5mu x)\\otimes\\text{\\rm  e}\\mkern0.7mu^{-\\tilde \\alpha\\ell}\\text{\\rm d}\\mkern0.5mu\\ell\\,f^{\\ast\\mathfrak g}(x,\\ell)\n\\\\\n&=\\int_{D\\times\\mathbb R\\times\\mathbb R}Z^D_\\lambda(\\text{\\rm d}\\mkern0.5mu x)\\otimes\\text{\\rm  e}\\mkern0.7mu^{-\\tilde \\alpha\\ell}\\text{\\rm d}\\mkern0.5mu\\ell\\otimes\\mathfrak g(\\text{\\rm d}\\mkern0.5mu h)\\,\nf\\Bigl(x,\\tfrac1{2\\sqrt g(\\sqrt\\theta+\\lambda)}\\bigl(\\ell+\\tfrac{h^2}2\\bigr)\\Bigr)\n\\\\\n&=\\int_{D\\times\\mathbb R\\times\\mathbb R}Z^D_\\lambda(\\text{\\rm d}\\mkern0.5mu x)\\otimes\\text{\\rm  e}\\mkern0.7mu^{-\\tilde\\alpha(\\ell+\\frac{h^2}2)}\\text{\\rm d}\\mkern0.5mu\\ell\\otimes\\text{\\rm  e}\\mkern0.7mu^{\\,\\tilde\\alpha\\frac{h^2}2}\\mathfrak g(\\text{\\rm d}\\mkern0.5mu h)\\,\nf\\Bigl(x,\\tfrac1{2\\sqrt g(\\sqrt\\theta+\\lambda)}\\bigl(\\ell+\\tfrac{h^2}2\\bigr)\\Bigr)\n\\\\\n&=2\\sqrt g(\\sqrt\\theta+\\lambda)\\Bigl(\\int_\\mathbb R\\mathfrak g(\\text{\\rm d}\\mkern0.5mu h)\\text{\\rm  e}\\mkern0.7mu^{\\,\\tilde\\alpha\\frac{h^2}2}\\Bigr)\n\\int_{D\\times\\mathbb R}Z^D_\\lambda(\\text{\\rm d}\\mkern0.5mu x)\\otimes\\text{\\rm  e}\\mkern0.7mu^{-\\alpha \\lambda r}\\text{\\rm d}\\mkern0.5mu r\\,f(x,r)\\,.\n\\end{aligned}\n\\end{equation}\nAs $\\tilde\\alpha<1\/g$, the first integral on the last line converges to the root of  $(1-\\tilde\\alpha g)^{-1}=\\frac{\\sqrt\\theta+\\lambda}{\\sqrt\\theta}$ while \\eqref{E:4.25a} equates the second integral to $\\langle\\zeta^D,f^{\\ast\\mathfrak g}\\rangle$ in law. This implies \n\\begin{equation}\n \\zeta^D \\,\\,\\overset{\\text{\\rm law}}=\\,\\, \\frac{\\theta^{1\/4}}{2\\sqrt{g}\\,(\\sqrt\\theta+\\lambda)^{3\/2}}\\,\n\\fraktura c(\\lambda)Z^D_\\lambda(\\text{\\rm d}\\mkern0.5mu x)\\otimes\\text{\\rm  e}\\mkern0.7mu^{-\\tilde \\alpha\\ell}\\text{\\rm d}\\mkern0.5mu\\ell. \n\\end{equation}\nIn particular, all weakly converging subsequences of~$\\{\\zeta^D_N\\colon N\\ge1\\}$ converge to this~$\\zeta^D$, thus proving the desired claim.\n\\end{proofsect}\n\n\\normalcolor\n\n\\section{Thin points}\n\\label{sec5}\\noindent\\nopagebreak\n Our next task is  the convergence of point measures  $\\zeta^D_N$  associated with $\\lambda$-thin  points.  The argument proceeds very much along the same sequence of lemmas as for the $\\lambda$-thick points and so we  will concentrate on the steps where a different reasoning is needed.  Throughout we assume that~$t_N$ and~$a_N$ are sequences satisfying \\eqref{E:1.22} with some~$\\theta>0$ and some~$\\lambda\\in(0, 1 \\wedge \\sqrt\\theta)$.  The auxiliary centering sequence~$\\widehat a_N$ is now defined by \n\\begin{equation}\n\\label{E:5.1}\n\\widehat a_N:=\\sqrt{2t_N}-\\sqrt{2a_N}\n\\end{equation}\nwhich ensures that we still have $\\widehat a_N\\sim 2\\lambda\\sqrt g\\log N$ as~$N\\to\\infty$.  Appealing to the coupling of~$L^{D_N}_{t_N}$ and~$h^{D_N}$ to~$\\tilde h^{D_N}$ via Dynkin isomorphism, we use~$\\widehat\\eta_N^D$ to denote the point process associated with~$\\tilde h^{D_N}$ and the centering sequence~$-\\widehat a_N$.  \n\nThe proof again opens up by proving suitable tightness and joint-convergence statements. We start with an analogue of Lemma~\\ref{lemma-no-conspire}: \n\n\\begin{lemma}\n\\label{lemma-no-conspire2}\nLet $0<\\beta<\\frac1{2g}\\frac{\\sqrt\\theta}{\\sqrt\\theta-\\lambda}$. Then for each~$b\\in\\mathbb R$ there is~$c_5(b)\\in(0,\\infty)$  and, for each $M\\ge0$, there is~$N'=N'(b,M)$ such that for  all~$N\\ge N'$ and all~$x\\in D_N$,\n\\begin{equation}\nP^\\varrho\\otimes\\mathbb P\\biggl(L^{D_N}_{t_N}(x)+\\frac{(h^{D_N}_x)^2}2 \\le  a_N+b\\log N,\\,\\frac{|h^{D_N}_x|}{\\sqrt{\\log N}}\\ge M\\biggr)\\le  c_5(b)\\frac{W_N}{N^2}\\text{\\rm  e}\\mkern0.7mu^{-\\beta M^2}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proofsect}{Proof}\nLet us again for simplicity just deal with the case~$b=0$. Pick~$0<\\delta<\\sqrt\\theta-\\lambda$. Then the probability in question is bounded by\n\\begin{multline}\n\\qquad\nP^\\varrho\\Bigl(L^{D_N}_{t_N}(x)\\le 2g(\\sqrt\\theta-\\lambda-\\delta)^2(\\log N)^2\\Bigr)\n\\\\+P^\\varrho\\Bigl(2g(\\sqrt\\theta-\\lambda-\\delta)^2(\\log N)^2\\le L^{D_N}_{t_N}(x)\\le a_N-\\frac12M^2\\log N\\Bigr).\n\\qquad\n\\end{multline}\nInvoking the calculation in \\eqref{E:3.18}, the first term is at most order~$N^{-2(\\lambda+\\delta)^2+o(1)}$ which is $o(W_N\/N^2)$. The second term is now bounded using Lemma~\\ref{lemma-lower} and the fact that, by the uniform bound $G^{D_N}(x,x) \\le g \\log N + c$ with~$c$ independent of~$N$, we have \n\\begin{equation}\n\\min_{x\\in D_N}\n\\frac{\\log N\\bigl(\\sqrt{2t_N}-\\sqrt{2a_N}\\bigr)}{G^{D_N}(x,x)\\sqrt{2a_N}}\\ge\\frac1g\\,\\frac{\\lambda}{\\sqrt\\theta-\\lambda}+o(1)\n\\end{equation}\nin the limit~$N\\to\\infty$. \nIndeed, this \\normalcolor shows that the last exponential in \\eqref{E:4.7a2} for the choice \n$b:=-\\frac12M^2\\log N$ is less than $\\text{\\rm  e}\\mkern0.7mu^{-\\beta M^2}$ once~$N$ is sufficiently large.\n\\end{proofsect}\n\nNext we will give an analogue of Lemma~\\ref{lemma-add-field} which we restate \\textit{verbatim}, albeit with a somewhat different proof: \n\n\\begin{lemma}\n\\label{lemma-add-field2}\nSuppose $\\{N_k\\}$ is a subsequence along which~$\\zeta^D_N$ converges in law to~$\\zeta^D$. Then\n\\begin{equation}\n\\frac1{W_N}\\sum_{x\\in D_N}\\delta_{x\/N}\\otimes\\delta_{(L^{D_N}_{t_N}(x)-a_N)\/\\log N}\\otimes\\delta_{h_x^{D_N}\/\\sqrt{\\log N}}\\,\\,\\underset{\\begin{subarray}{c}\nN=N_k\\\\k\\to\\infty\n\\end{subarray}}{\\overset{\\text{\\rm law}}\\longrightarrow}\\,\\,\n\\zeta^D\\otimes\\mathfrak g,\n\\end{equation}\nwhere~$\\mathfrak g$ is the law of~$\\mathcal N(0,g)$.\n\\end{lemma}\n\n\\begin{proofsect}{Proof}\nLet~$\\zeta^{D,\\text{ext}}_N$ denote the measure on the left and let~$f\\in C_{\\text{\\rm c}}(D\\times\\mathbb R\\times\\mathbb R)$  obey~$f\\ge0$.  As for Lemma~\\ref{lemma-add-field}, the argument hinges on proving\n\\begin{equation}\n\\text{\\rm Var}_{\\mathbb P}\\bigl(\\langle\\zeta^{D,\\text{ext}}_N,f\\rangle\\bigr)\\,\\underset{N\\to\\infty}\\longrightarrow\\,0,\\quad \\text{in $P^\\varrho$-probability},\n\\end{equation}\nwhere~$\\text{\\rm Var}_{\\mathbb P}$ denotes the variance with respect to the law of~$h^{D_N}$, conditional on~$L^{D_N}_{t_N}$. \nInvoking \\eqref{E:4.11}, we treat the sum over the pairs~$|x-y|\\ge\\epsilon N$ via \\twoeqref{E:5.12i}{E:5.13i}. \\normalcolor\nThe key difference is that we no longer have the domination of~$\\zeta^D_N$ by a DGFF process and so we have to control the sum over the pairs $x,y\\in D_N$ with~$|x-y|\\le\\epsilon N$ differently. \n\nSince~$f$ is non-negative and compactly supported in the third variable, we in fact just need to show that, for any~$M>0$, the $L^1(\\mathbb P)$-norm of \n\\begin{multline}\n\\label{E:5.3}\n\\quad\\frac1{W_N^2}\\sum_{\\begin{subarray}{c}\nx,y\\in D_N\\\\|x-y|\\le\\epsilon N\n\\end{subarray}}\n1_{\\{L^{D_N}_{t_N}(x)\\le a_N+M^2\\log N\\}}1_{\\{|h^{D_N}_x|\\le M\\sqrt{\\log N}\\}}\n\\\\\n\\times1_{\\{L^{D_N}_{t_N}(y)\\le a_N+M^2\\log N\\}}1_{\\{|h^{D_N}_y|\\le M\\sqrt{\\log N}\\}}\n\\quad\n\\end{multline}\nvanishes in $P^\\varrho$-probability in the limit as~$N\\to\\infty$ and~$\\epsilon\\downarrow0$. To this end  we note that, dropping the indicators involving the DGFF, \\eqref{E:5.3} is bounded by $[\\zeta^D_N(D\\times(-\\infty,M^2])]^2$ which by Corollary~\\ref{cor-tightness-lower} is  bounded in probability as~$N\\to\\infty$.  Therefore, it suffices to prove that \\eqref{E:5.3} vanishes in the stated limits in $P^\\varrho\\otimes\\mathbb P$-probability. \n\nTo this end pick~$b>\\frac{M^2}{\\sqrt{g}(\\sqrt{\\theta}-\\lambda)}$ and note that, as soon as~$N$ is sufficiently large, the asymptotic forms of~$a_N$ along with the Dynkin isomorphism yield \n\\begin{multline}\n\\qquad\n1_{\\{L^{D_N}_{t_N}(x)\\le a_N+M^2\\log N\\}}1_{\\{|h^{D_N}_x|\\le M\\sqrt{\\log N}\\}}\n\\\\\n\\le 1_{\\{(\\tilde h^{D_N}_x+\\sqrt{2t_N})^2\\le 2a_N+ 3 M^2\\log N\\}}\n\\le 1_{\\{\\tilde h^{D_N}_x\\le -\\widehat a_N+b\\}}.\n\\qquad\n\\end{multline}\n It follows that \\eqref{E:5.3} is bounded by \n\\begin{equation}\n\\widehat\\eta_N^D\\otimes\\widehat\\eta_N^D\\Bigl(\\bigl\\{(x,y)\\colon|x-y|\\le\\epsilon\\bigr\\}\\times(-\\infty,b]^2\\Bigr)\n\\end{equation}\n whose~$N\\to\\infty$ and~$\\epsilon\\downarrow0$ limits are now handled as before.\n\\end{proofsect}\n\nOur next task is a derivation of a convolution identity that will, as for the thick points, ultimately characterize the limit measure uniquely: \n\n\\begin{lemma}\n\\label{lemma-conv2}\nGiven~$f\\in C_{\\text{\\rm c}}(D\\times\\mathbb R)$ with~$f\\ge0$, let (abusing our earlier notation) \n\\begin{equation}\nf^{\\ast\\mathfrak g}(x,\\ell):=\\int\\mathfrak g(\\text{\\rm d}\\mkern0.5mu h)f\\Bigl(x,\\tfrac1{2\\sqrt g(\\sqrt\\theta-\\lambda)}\\bigl(\\ell+\\tfrac{h^2}2\\bigr)\\Bigr).\n\\end{equation}\nThen for every subsequential weak limit~$\\zeta^D$ of~$\\zeta^D_N$, simultaneously for all~$f$ as above, \n\\begin{equation}\n\\label{E:4.25b}\n\\langle\\zeta^D, f^{\\ast\\mathfrak g}\\rangle\\,\\overset{\\text{\\rm law}}=\\, \\fraktura c(\\lambda)\\int Z^D_\\lambda(\\text{\\rm d}\\mkern0.5mu x)\\otimes\\text{\\rm  e}\\mkern0.7mu^{\\alpha\\lambda h}\\text{\\rm d}\\mkern0.5mu h \\,\\,f(x,h),\n\\end{equation}\nwhere~$\\alpha:=2\/\\sqrt g$ and~$\\fraktura c(\\lambda)$ is as in \\eqref{E:1.19}.\n\\end{lemma}\n\n\\begin{proofsect}{Proof}\nPick~$f$ as above and let~$\\chi$ be the function as in the proof of Lemma~\\ref{lemma-conv}. The fact that~$f$ has compact support gives\n\\begin{equation}\n\\frac1{W_N}\\sum_{x\\in D_N}f\\bigl(\\ffrac xN, \\tilde h^{D_N}_x+\\widehat a_N\\bigr)\n=\\frac1{W_N}\\sum_{x\\in D_N} f\\Bigl(\\ffrac xN,-\\sqrt{2a_N}+\\sqrt{2L^{D_N}_{t_N}(x)+(h^{D_N}_x)^2}\\Bigr)\n\\end{equation}\nand Lemma~\\ref{lemma-no-conspire2} then bounds this by~$O(\\text{\\rm  e}\\mkern0.7mu^{-\\beta M^2})$ \nplus\n\\begin{equation}\n\\frac1{W_N}\\sum_{x\\in D_N} f\\Bigl(\\ffrac xN,-\\sqrt{2a_N}+\\sqrt{2L^{D_N}_{t_N}(x)+(h^{D_N}_x)^2}\\Bigr)\\chi\\Bigl(\\frac{|h_x^{D_N}|}{M\\sqrt{\\log N}}\\Bigr).\n\\end{equation}\nThe truncation of the field now forces $L^{D_N}_{t_N}-a_N$ to be order $\\log N$.\nExpanding the square root and using the uniform continuity with the help of Corollary~\\ref{cor-tightness-lower} rewrites this as\n\\begin{equation}\n\\label{E:5.28}\n\\frac1{W_N}\\sum_{x\\in D_N}\\tilde f_{\\text{ext}}\\Bigl(\\ffrac xN,\\tfrac{L^{D_N}_{t_N}(x)-a_N}{\\log N}, \\tfrac{h_x^{D_N}}{\\sqrt{\\log N}}\\Bigr)\\chi\\Bigl(\\frac{|h_x^{D_N}|}{M\\sqrt{\\log N}}\\Bigr)+o(1),\n\\end{equation}\nwhere\n\\begin{equation}\n\\tilde f_{\\text{ext}}(x,\\ell,h):=f\\Bigl(x,\\tfrac1{2\\sqrt g(\\sqrt\\theta-\\lambda)}\\bigl(\\ell+\\tfrac{h^2}2\\bigr)\\Bigr).\n\\end{equation}\nThe rest of the proof now proceeds as before. (The exponential on the right-hand side of \\eqref{E:4.25b} does not get a negative sign because~$\\widehat\\eta_N^D$ is centered along negative sequence of order~$\\log N$.)\n\\end{proofsect}\n\n\n\nUsing Dominated and Monotone Convergence, we now readily extend \\eqref{E:4.25b} to functions of the form $1_A\\otimes f$ where~$A\\subset D$ is open with~$\\overline A\\subset D$ and~$f\\in C_{\\text{\\rm c}}(\\mathbb R)$ obeys~$f\\ge0$. For such~$f$ we then get\n\\begin{equation}\n(1_A\\otimes f)^{\\ast\\mathfrak g}=1_A\\otimes (f\\ast\\fraktura e')\\circ\\fraktura s'\n\\end{equation}\nwhere~$\\fraktura e'$ is given by the same formula as~$\\fraktura e$ in \\eqref{E:5.33} but with~$\\beta$ replaced by\n\\begin{equation}\n\\beta':=\\alpha\\bigl(\\sqrt\\theta-\\lambda\\bigr)\n\\end{equation}\nand~$\\fraktura s'(h):=h\/(2\\sqrt g(\\sqrt\\theta-\\lambda))$. We then state:\n\n\\begin{lemma}\n\\label{lemma-unique2}\nLet $\\mu_\\lambda'(\\text{\\rm d}\\mkern0.5mu h):=\\text{\\rm  e}\\mkern0.7mu^{\\alpha\\lambda h}\\text{\\rm d}\\mkern0.5mu h$. Assuming~$\\theta>0$, there is at most one Radon measure~$\\nu$ on~$\\mathbb R$ such that for all~$f\\in C_{\\text{\\rm c}}^\\infty(\\mathbb R)$ with $f\\ge0$,\n\\begin{equation}\n\\label{E:5.35iw2}\n\\bigl\\langle\\nu,f\\ast\\fraktura e'\\bigr\\rangle=\\langle\\mu_\\lambda',f\\rangle.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proofsect}{Proof}\nAs in the proof of Lemma~\\ref{lemma-unique}, we recast \\eqref{E:5.35iw2} as\n\\begin{equation}\n\\bigl\\langle\\nu_\\lambda,f\\ast\\fraktura e'_\\lambda\\bigr\\rangle=\\langle{\\rm Leb},f\\rangle\n\\end{equation}\nwhere $\\nu_\\lambda(\\text{\\rm d}\\mkern0.5mu h)=\\text{\\rm  e}\\mkern0.7mu^{-\\alpha\\lambda h} \\nu(\\text{\\rm d}\\mkern0.5mu h)$ and $\\fraktura e_\\lambda'(h)=\\text{\\rm  e}\\mkern0.7mu^{\\alpha\\lambda h}\\fraktura e'(h)$. Since $\\tilde\\beta':=\\beta'+\\alpha\\lambda =\\alpha\\sqrt\\theta>0$, we again get that~$\\fraktura e_\\lambda'$ is integrable. Replacing~$\\tilde\\beta$ by~$\\tilde\\beta'$, the rest of the argument is then identical to that in the proof of Lemma~\\ref{lemma-unique}.\n\\end{proofsect}\n\n\n\nWe are now ready to give:\n\n\\begin{proofsect}{Proof of Theorem~\\ref{thm-thin}}\nThe argument proving that \\eqref{E:4.25b} determines $\\zeta^D$ uniquely is the same as for the thick points so we just need to perform the analogue of the calculation in \\eqref{E:4.38}. Denoting \n\\begin{equation}\n\\widehat\\alpha:=\\frac1{2\\sqrt g\\,(\\sqrt\\theta-\\lambda)}\\alpha\\lambda,\n\\end{equation}\nwe get\n\\begin{equation}\n\\label{E:4.38a}\n\\begin{aligned}\n\\int_{D\\times\\mathbb R}&Z^D_\\lambda(\\text{\\rm d}\\mkern0.5mu x)\\otimes\\text{\\rm  e}\\mkern0.7mu^{\\widehat \\alpha\\ell}\\text{\\rm d}\\mkern0.5mu\\ell\\,\\,\\tilde f^{\\ast\\mathfrak g}(x,\\ell)\n\\\\\n&=\\int_{D\\times\\mathbb R\\times\\mathbb R}Z^D_\\lambda(\\text{\\rm d}\\mkern0.5mu x)\\otimes\\text{\\rm  e}\\mkern0.7mu^{\\widehat \\alpha\\ell}\\text{\\rm d}\\mkern0.5mu\\ell\\otimes\\mathfrak g(\\text{\\rm d}\\mkern0.5mu h)\\,\nf\\Bigl(x,\\tfrac1{2\\sqrt g(\\sqrt\\theta-\\lambda)}\\bigl(\\ell+\\tfrac{h^2}2\\bigr)\\Bigr)\n\\\\\n&=\\int_{D\\times\\mathbb R\\times\\mathbb R}Z^D_\\lambda(\\text{\\rm d}\\mkern0.5mu x)\\otimes\\text{\\rm  e}\\mkern0.7mu^{\\widehat\\alpha(\\ell+\\frac{h^2}2)}\\text{\\rm d}\\mkern0.5mu\\ell\\otimes\\text{\\rm  e}\\mkern0.7mu^{\\,-\\widehat\\alpha\\frac{h^2}2}\\mathfrak g(\\text{\\rm d}\\mkern0.5mu h)\\,\nf\\Bigl(x,\\tfrac1{2\\sqrt g(\\sqrt\\theta-\\lambda)}\\bigl(\\ell+\\tfrac{h^2}2\\bigr)\\Bigr)\n\\\\\n&=2\\sqrt g(\\sqrt\\theta-\\lambda)\\Bigl(\\int_\\mathbb R\\mathfrak g(\\text{\\rm d}\\mkern0.5mu h)\\text{\\rm  e}\\mkern0.7mu^{\\,-\\widehat\\alpha\\frac{h^2}2}\\Bigr)\n\\int_{D\\times\\mathbb R}Z^D_\\lambda(\\text{\\rm d}\\mkern0.5mu x)\\otimes\\text{\\rm  e}\\mkern0.7mu^{\\alpha \\lambda r}\\text{\\rm d}\\mkern0.5mu r\\,f(x,r)\\,.\n\\end{aligned}\n\\end{equation}\nThe Gaussian integral on the last line equals the root of $\\frac{\\sqrt\\theta-\\lambda}{\\sqrt\\theta}$. It follows that~$\\zeta^D_N$ converges in law to the measure\n\\begin{equation}\n\\frac{\\theta^{1\/4}}{2\\sqrt{g}\\,(\\sqrt\\theta-\\lambda)^{3\/2}}\\,\n\\fraktura c(\\lambda)Z^D_\\lambda(\\text{\\rm d}\\mkern0.5mu x)\\otimes\\text{\\rm  e}\\mkern0.7mu^{\\widehat \\alpha\\ell}\\text{\\rm d}\\mkern0.5mu\\ell.\n\\end{equation}\nThis is the desired claim.\n\\end{proofsect}\n\n\n\n\\section{Light and avoided points}\n\\label{sec6}\\noindent\\nopagebreak\nIn this section we will deal with the point measures $\\vartheta^D_N$ and~$\\kappa^D_N$ associated with the light and avoided points, respectively. The argument follows the blueprint of the proof for the~$\\lambda$-thick (and~$\\lambda$-thin) points although important differences arise due to a different scaling of~$\\widehat W_N$ with~$N$ compared to~$W_N$. We begin with an analogue of Lemma~\\ref{lemma-add-field} where this issue becomes quite apparent.\n\n\\begin{lemma}\n\\label{lemma-7.1}\nSuppose $\\{N_k\\}$ is a subsequence along which~$\\vartheta^D_N$ converges in law to~$\\vartheta^D$. Then\n\\begin{equation}\n\\frac{\\sqrt{\\log N}}{\\widehat W_N }\\sum_{x\\in D_N}\\delta_{x\/N}\\otimes\\delta_{L^{D_N}_{t_N}(x)}\\otimes\\delta_{h^{D_N}_x}\n\\,\\,\\underset{\\begin{subarray}{c}\nN=N_k\\\\k\\to\\infty\n\\end{subarray}}{\\overset{\\text{\\rm law}}\\longrightarrow}\\,\\,\n\\vartheta^D\\otimes\\frac1{\\sqrt{2\\pi g}}\\,{\\rm Leb}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proofsect}{Proof}\nLet $\\vartheta^{D,\\text{ext}}_N$ denote the measure on the left and pick~$f\\in C_{\\text{\\rm c}}(D\\times[0,\\infty)\\times\\mathbb R)$. \\normalcolor Suppose that~$f(x,\\ell,h)=0$ unless~$x\\in A$, where $A$ is an open set with~$\\overline A\\subset D$, and unless $\\ell,|h|\\le M$ for some~$M>0$. Noting that the probability density of~$h^{D_N}_x$ is~$(1+o(1))(2\\pi g\\log N)^{-1\/2}$ with~$o(1)\\to0$ as~$N\\to\\infty$ uniformly over any compact interval shows, with the help of the tightness of~$\\{\\vartheta^D_N\\colon N\\ge1\\}$ proved in Corollary~\\ref{cor-light}, that\n\\begin{equation}\n\\label{E:6.2u}\n\\mathbb E\\bigl\\langle\\vartheta^{D,\\text{ext}}_N,f\\bigr\\rangle=o(1)+\\frac1{\\sqrt{2\\pi g}}\\bigl\\langle\\vartheta^D_N\\otimes{\\rm Leb},f\\bigr\\rangle,\n\\end{equation}\nwhere~$o(1)\\to0$ in $P^\\varrho$-probability as~$N\\to\\infty$. As before, we just need to prove concentration of $\\langle\\vartheta^{D,\\text{ext}}_N,f\\rangle$ around the (conditional) expectation with respect to~$h^{D_N}$. \n\nConsider the conditional variance $\\text{\\rm Var}_{\\mathbb P}(\\langle\\vartheta^{D,\\text{ext}}_N,f\\rangle)$ and use the additive structure of $\\langle\\vartheta^{D,\\text{ext}}_N,f\\rangle$ to write it as the sum of\n\\begin{equation}\n\\label{E:7.3ie} \n\\mathcal C_N(x,y):= \n\\text{\\rm Cov}_{\\mathbb P}\\Bigl(f\\bigl(\\ffrac xN,L^{D_N}_{t_N}(x),h^{D_N}_x\\bigr),\nf\\bigl(\\ffrac yN,L^{D_N}_{t_N}(y),h^{D_N}_y\\bigr)\\Bigr)\n\\end{equation}\nover all pairs~$x,y\\in A_N:=\\{z\\in D_N\\colon z\/N\\in A\\}$.  In order to estimate this quantity, for any $x,y\\in D_N$ let~$\\rho^{D_N}_{x,y}\\colon\\mathbb R^2\\to[0,\\infty)$ denote the joint probability density of $(h^{D_N}_x,h^{D_N}_y)$ with respect to the Lebesgue measure on~$\\mathbb R^2$. Abbreviate\n\\begin{multline}\n\\quad\nF(h_x,h_y;h_x',h_y') := \\frac12\\Bigl(f\\bigl(\\ffrac xN,L^{D_N}_{t_N}(x),h_x\\bigr)-f\\bigl(\\ffrac xN,L^{D_N}_{t_N}(x),h_x'\\bigr)\\Bigr)\n\\\\\n\\times\\Bigl(f\\bigl(\\ffrac yN,L^{D_N}_{t_N}(y),h_y\\bigr)-f\\bigl(\\ffrac yN,L^{D_N}_{t_N}(y),h_y'\\bigr)\\Bigr).\\quad\n\\end{multline} \nInvoking the identity $\\text{\\rm Cov}(X,Y)=\\frac12E((X-X')(Y-Y'))$, where $(X',Y')$ is an independent copy of~$(X,Y)$, we get\\begin{equation}\n\\label{E:7.5ie}\n\\mathcal C_N(x,y)=\\int_{[-M,M]^4}\\text{\\rm d}\\mkern0.5mu h_x\\,\\text{\\rm d}\\mkern0.5mu h_y\\,\\text{\\rm d}\\mkern0.5mu h_x'\\,\\text{\\rm d}\\mkern0.5mu h_y'\\,\\,\\rho^{D_N}_{x,y}(h_x,h_y)\\rho^{D_N}_{x,y}(h_x',h_y')F(h_x,h_y;h_x',h_y'),\n\\end{equation}\nwhere the restriction on the domain of integration reflects the support restrictions on~$f$ in the third variable.\n\n\nOur way of control \\eqref{E:7.5ie} hinges on the observation that, once we replace both probability densities by constants, the integral vanishes. For these constants we will choose the value of the probability density at $(h,h')=(0,0)$. To account for the errors, we thus introduce\n\\begin{equation}\n\\label{E:7.6ie}\n\\delta_N(M,r):=\\max_{\\begin{subarray}{c}\nx,y\\in A_N\\\\|x-y|\\ge r\n\\end{subarray}}\\,\\sup_{|h|,|h'|\\le M}\\,\n\\Bigl|\\,\\log \\rho^{D_N}_{x,y}(h,h')-\\log\\rho^{D_N}_{x,y}(0,0)\\Bigr|.\n\\end{equation}\nUsing that $|\\text{\\rm  e}\\mkern0.7mu^x-1|\\le \\text{\\rm  e}\\mkern0.7mu^{|x|}-1$ along with\n\\begin{equation}\n\\bigl|F(h_x,h_y;h_x',h_y')\\bigr|\\le 2\\Vert f\\Vert_\\infty^2\\,1_{\\{L^{D_N}_{t_N}(x)\\le M\\}}1_{\\{L^{D_N}_{t_N}(y)\\le M\\}},\n\\end{equation}\nas implied by the restrictions on the support of~$f$, we thus get\n\\begin{equation}\n\\label{E:7.8ie}\n\\mathcal C_N(x,y)\\le2\\Vert f\\Vert_\\infty^2\\,1_{\\{L^{D_N}_{t_N}(x)\\le M\\}}1_{\\{L^{D_N}_{t_N}(y)\\le M\\}}\\,\\rho^{D_N}_{x,y}(0,0)^2\\bigr(\\text{\\rm  e}\\mkern0.7mu^{2\\delta_N(M,|x-y|)}-1\\bigr).\n\\end{equation}\nIt remains to analyze the $N$-dependence of the terms~$\\rho^{D_N}_{x,y}(0,0)$ and~$\\delta_N(M,|x-y|)$.\n\nUsing \\eqref{E:5.12i} and the fact that $\\fraktura b_{D_N,x}(y)=G^{D_N}(x,y)\/G^{D_N}(x,x)$, the law of~$h^{D_N}_y$ conditioned on~$h^{D_N}_x$ is normal with mean $h^{D_N}_x\\fraktura b_{D_N,x}(y)$ and variance \n\\begin{equation}\n\\label{E:7.9ie}\n\\sigma_N^2(x,y):=G^{D_N}(y,y)-\\frac{G^{D_N}(x,y)^2}{G^{D_N}(x,x)}.\n\\end{equation}\nUsing the shorthand~$\\sigma_N^2(x):=G^{D_N}(x,x)$ for the variance of~$h^{D_N}_x$ and~$\\phi(y)$ for the value of~$\\fraktura b_{D_N,x}(y)$, we thus have\n\\begin{equation}\n\\rho_{x,y}^{D_N}(h_x,h_y) = \\frac1{2\\pi\\sigma_N(x)\\sigma_N(x,y)}\\,\\text{\\rm  e}\\mkern0.7mu^{-\\frac{h_x^2}{2\\sigma_N^2(x)}-\\frac{[h_y-h_x\\phi(y)]^2}{2\\sigma_N^2(x,y)}}\\,.\n\\end{equation}\nThanks to~$\\overline A\\subset D$, the difference $\\sigma_N^2(x)-g\\log N$ is bounded uniformly in $x\\in A_N$ and $N\\ge1$.\nInvoking the representation of the Green function using the potential kernel (see, e.g.,~\\cite[Lemma~B.3]{BL3}) and then the asymptotic growth of the potential kernel (see, e.g., \\cite[Lemma~B.4]{BL3}) we also get that, for any~$r_N\\to\\infty$,\n\\begin{equation}\n\\min_{\\begin{subarray}{c}\nx,y\\in A_N\\\\|x-y|\\ge r_N\n\\end{subarray}}\\sigma^2_N(x,y)\\,\\underset{N\\to\\infty}\\longrightarrow\\,\\infty.\n\\end{equation}\nPutting these together with~$\\phi(y)\\in[0,1]$, \\eqref{E:7.6ie} then yields\n\\begin{equation}\n(\\log N)\\Bigl(\\,\\max_{\\begin{subarray}{c}\nx,y\\in A_N\\\\|x-y|\\ge r_N\n\\end{subarray}}\\rho_{x,y}^{D_N}(0,0)^2\\Bigr)\\bigr(\\text{\\rm  e}\\mkern0.7mu^{2\\delta_N(M,r_N)}-1\\bigr)\\,\\underset{N\\to\\infty}\\longrightarrow\\,0.\n\\end{equation}\nUsing this in \\eqref{E:7.8ie} we conclude that, as soon as~$r_N\\to\\infty$,\n\\begin{equation}\n\\label{E:7.11oe}\n\\frac{\\log N}{\\widehat W_N^2}\\sum_{\\begin{subarray}{c}\nx,y\\in D_N\\\\|x-y|\\ge r_N\n\\end{subarray}}\n\\mathcal C_N(x,y)= o(1) \\,\\vartheta^D_N\\bigl(D\\times[0,M]\\bigr)^2,\n\\end{equation}\nwhere~$o(1)$ is a numerical sequence tending to zero as~$N\\to\\infty$. By the tightness of $\\{\\vartheta^D_N\\colon N\\ge1\\}$ (cf Corollary~\\ref{cor-light}), \\eqref{E:7.11oe} thus tends to zero in~$P^\\varrho$-probability.\n\nTo infer that $\\text{\\rm Var}_{\\mathbb P}(\\langle\\vartheta^{D,\\text{ext}}_N,f\\rangle)$ vanishes as~$N\\to\\infty$ in $P^\\varrho$-probability, we just note that, by dropping one of the indicators in \\eqref{E:7.8ie}, the sum over $x,y\\in D_N$ complementary to those in \\eqref{E:7.11oe} is at most order $r_N^2(\\log N)\/\\widehat W_N$ times~$\\Vert f\\Vert_\\infty^2\\vartheta^D_N(D\\times[0,M])$. As~$\\widehat W_N$ grows polynomially with~$N$ for our choices of~$\\theta$, there is a way to choose~$r_N\\to\\infty$ so that this tends to zero in $P^\\varrho$-probability as well.\n\\end{proofsect}\n\nNext we prove an analogue of Lemma~\\ref{lemma-conv}:\n\n\\begin{lemma}\nGiven~$f\\in C_{\\text{\\rm c}}(D\\times[0,\\infty))$ with~$f\\ge0$ denote\n\\begin{equation}\nf^{\\ast{\\rm Leb}}(x,\\ell):=\\frac1{\\sqrt{2\\pi g}}\\int_{\\mathbb R}\\text{\\rm d}\\mkern0.5mu h\\,\\,f\\bigl(x,\\ell+\\tfrac{h^2}2\\bigr).\n\\end{equation}\nThen for every weak subsequential limit~$\\vartheta^D$ of~$\\vartheta^D_N$,\n\\begin{equation}\n\\label{E:6.3}\n\\bigl\\langle\\vartheta^D,f^{\\ast{\\rm Leb}}\\bigr\\rangle\\,\\overset{\\text{\\rm law}}=\\,\\fraktura c(\\sqrt\\theta)\\int Z^D_{\\sqrt\\theta}(\\text{\\rm d}\\mkern0.5mu x)\\otimes\\text{\\rm  e}\\mkern0.7mu^{\\alpha\\sqrt\\theta\\,h}\\text{\\rm d}\\mkern0.5mu h \\,\\,f\\bigl(x,\\tfrac{h^2}2\\bigr)\n\\end{equation}\nsimultaneously for all~$f$ as above. \n\\end{lemma}\n\n\\begin{proofsect}{Proof}\nPick~$f\\in C_{\\text{\\rm c}}(D\\times[0,\\infty))$ with~$f\\ge0$ and set $f^{\\text{ext}}(x,\\ell,h):=f(x,\\ell+\\tfrac12h^2)$. Then\n\\begin{equation}\n\\begin{aligned}\n\\sum_{x\\in D_N}f\\Bigl(\\ffrac xN,\\frac12(\\tilde h^{D_N}_x+\\sqrt{2t_N}\\bigr)^2\\Bigr)\n&=\\sum_{x\\in D_N}f\\Bigl(\\ffrac xN,L^{D_N}_{t_N}(x)+\\frac12(h^{D_N}_x)^2\\Bigr)\n\\\\\n&=\\sum_{x\\in D_N}f^{\\text{ext}}\\Bigl(\\ffrac xN, L^{D_N}_{t_N}(x),h^{D_N}_x\\Bigr).\n\\end{aligned}\n\\end{equation}\nSince~$f^{\\text{ext}}$ is compactly supported in all variables, Lemma~\\ref{lemma-7.1} tells us that, after multiplying by $\\sqrt{\\log N}\\,\/\\widehat W_N $ and specializing~$N$ to the  subsequence along which~$\\vartheta^D_N$ tends in law to~$\\vartheta^D$, the right-hand side tends to $\\bigl\\langle\\vartheta^D,f^{\\ast{\\rm Leb}}\\bigr\\rangle$. \\normalcolor By \\eqref{E:1.19} and the fact that~$\\sqrt{2t_N}\\sim2\\sqrt g\\sqrt\\theta\\log N$, the left-hand side tends to the measure on the right of \\eqref{E:6.3}.\n\\end{proofsect}\n\nWith these in hand we are ready to prove convergence of~$\\vartheta^D_N$'s:\n\n\\begin{proofsect}{Proof of Theorem~\\ref{thm-light}}\nPick~$A\\subset D$ open with $\\overline A\\subset D$. Taking a sequence of compactly supported functions converging  upward to $f(x,h):=1_A(x)\\text{\\rm  e}\\mkern0.7mu^{-s h}1_{[0,\\infty)}(h)$, where~$s>0$, and denoting\n\\begin{equation}\n\\tilde\\mu_A(B):=\\vartheta^D(A\\times B),\n\\end{equation}\nthe Tonelli and Monotone Convergence Theorems yield\n\\begin{equation}\n\\label{E:7.18iw}\n\\int_0^\\infty\\tilde\\mu_A(\\text{\\rm d}\\mkern0.5mu\\ell)\\text{\\rm  e}\\mkern0.7mu^{-\\ell s} \\,\\overset{\\text{\\rm law}}=\\, \\sqrt{2\\pi g}\\,\\fraktura c(\\sqrt\\theta)Z^D_{\\sqrt\\theta}(A)\\,\\text{\\rm  e}\\mkern0.7mu^{\\frac{\\alpha^2\\theta}{2s}},\\quad s>0.\n\\end{equation}\nNote that $s\\mapsto \\text{\\rm  e}\\mkern0.7mu^{\\frac{\\alpha^2\\theta}{2s}}$ is the Laplace transform of the measure in \\eqref{E:1.33}. Since the Laplace transform determines Borel measures on $[0,\\infty)$ uniquely, the claim follows by the fact that the right-hand side is a Borel measure in~$A$ which is determined by its values on~$A$ open with the closure in~$D$. \n\\end{proofsect}\n\nIn order to extend Theorem~{\\ref{thm-light} to the control of the measure $\\kappa^D_N$ associated with the avoided points, we need the following estimate:\n\n\\begin{lemma}\n\\label{lemma-6.3}\nLet~$A\\subset D$ be open with~$\\overline A\\subset D$. Then\n\\begin{equation}\n\\lim_{\\epsilon\\downarrow0}\\,\\limsup_{N\\to\\infty}\\,\\frac{N^2}{\\widehat W_N }\\max_{\\begin{subarray}{c}\nx\\in D_N\\\\ x\/N\\in A\n\\end{subarray}}\nP^\\varrho\\Bigl(0<L^{D_N}_{t_N}(x)\\le\\epsilon\\Bigr)=0.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proofsect}{Proof}\nFirst note that, using Dynkin's isomorphism, we get\n\\begin{equation}\n\\label{E:monster}\n\\begin{aligned}\nP^\\varrho\\Bigl(0&<L^{D_N}_{t_N}(x)\\le\\epsilon\\Bigr)P\\Bigl(\\frac12 (h^{D_N}_x)^2\\le\\epsilon\\Bigr)\n\\\\\n&\\le P^\\varrho\\otimes\\mathbb P\\Bigl(L^{D_N}_{t_N}(x)+\\frac12 (h^{D_N}_x)^2\\le 2\\epsilon,\\,L^{D_N}_{t_N}(x)>0\\Bigr)\n\\\\\n&=\\mathbb P\\Bigl(\\frac12(\\tilde h^{D_N}_x-\\sqrt{2t_N})^2\\le 2\\epsilon\\Bigr)-\\mathbb P\\Bigl(\\frac12(h^{D_N}_x)^2\\le 2\\epsilon\\Bigr)P^\\varrho\\bigl(L^{D_N}_{t_N}(x)=0\\bigr).\n\\end{aligned}\n\\end{equation}\nThe fact that $|G^{D_N}(x,x)- g\\log N|$ is bounded uniformly for all~$x\\in D_N$ with~$x\/N\\in A$ then shows\n\\begin{equation}\n\\mathbb P\\Bigl(\\frac12(\\tilde h^{D_N}_x-\\sqrt{2t_N})^2\\le \\epsilon\\Bigr)\n=\\bigl(2+o(1)\\bigr)\\sqrt{2\\epsilon}\\,\\frac1{\\sqrt{2\\pi G^{D_N}(x,x)}}\\,\n\\,\\text{\\rm  e}\\mkern0.7mu^{-\\frac{t_N}{G^{D_N}(x,x)}}\n\\end{equation}\nwhile\n\\begin{equation}\nP\\Bigl(\\frac12 (h^{D_N}_x)^2\\le\\epsilon\\Bigr)= \n\\bigl(2+o(1)\\bigr)\\sqrt{2\\epsilon}\\,\\frac1{\\sqrt{2\\pi G^{D_N}(x,x)}},\n\\end{equation}\nwhere $o(1)\\to0$ as~$\\epsilon\\downarrow0$ uniformly in~$N\\ge1$ and~$x$ as above. In light of \\eqref{E:L-vanish}, the right-hand side of \\eqref{E:monster} divided by $\\widehat W_N \/N^2$-times the DGFF probability on the extreme left tends to zero as~$N\\to\\infty$ and~$\\epsilon\\downarrow0$. \n\\end{proofsect}\n\nWe are ready to give:\n\n\\begin{proofsect}{Proof of Theorem~\\ref{thm-avoid}}\nTake~$f_n\\in C_{\\text{\\rm c}}([0,\\infty))$ such that~$f_n(h):=(1-nh)\\vee0$ and pick~$A\\subset D$ open with $\\overline A\\subset D$. Then\n\\begin{equation}\n E^\\varrho  \\bigl|\\langle\\kappa^D_N,1_A\\rangle-\\langle\\vartheta^D_N,1_A\\otimes f_n\\rangle\\bigr|\n\\le\\frac2{\\widehat W_N }\\sum_{\\begin{subarray}{c}\nx\\in D_N\\\\ x\/N\\in A\n\\end{subarray}}P^\\varrho\\bigl(0<L^{D_N}_{t_N}(x)\\le1\/n\\Bigr)\\,.\n\\end{equation}\nBy Lemma~\\ref{lemma-6.3}, the sum on the right-hand side tends to zero in the limits~$N\\to\\infty$ followed by~$n\\to\\infty$. Theorem~\\ref{thm-light} in turn shows that\n\\begin{equation}\n\\langle\\vartheta^D_N,1_A\\otimes f_n\\rangle\\,\\,\\underset{N\\to\\infty}{\\,\\overset{\\text{\\rm law}}\\longrightarrow\\,}\\,\\,\n\\sqrt{2\\pi g}\\,\\fraktura c(\\sqrt\\theta)Z^D_{\\sqrt\\theta}(A)\\Bigl[1+\\int_{(0,1\/n]} \\mu(\\text{\\rm d}\\mkern0.5mu h) f_n(h)\\Bigr],\n\\end{equation}\nwhere~$\\mu$ is the measure in \\eqref{E:1.33}. The claim follows by noting that the integral on the right tends to zero as~$n\\to\\infty$.\n\\end{proofsect}\n\n\n\\section{Local structure}\n\\label{sec8}\\noindent\nIn this section we deal with local structures of the exceptional level sets associated with the local time~$L^{D_N}_{t_N}$. Throughout we again rely on the coupling of $L_{t_N}^{D_N}$ and an independent DGFF $h^{D_N}$ \nto another DGFF~$\\widetilde h^{D_N}$ via the Dynkin isomorphism (Theorem~\\ref{thm-Dynkin}). We start with the thick points.\n\n\\subsection{Local structure of thick points}\nLet $a_N$ and $t_N$ satisfy \\eqref{E:1.20} with some $\\theta > 0$ and some $\\lambda \\in (0,1)$ and recall the notation~$\\widehat \\zeta^D_N$ for the extended point measures from \\eqref{E:zeta-local} that describe the $\\lambda$-thick points along with their local structure. Let $\\widehat a_N$ be the sequence given by (\\ref{E:4.1}).\nWe will compare~$\\widehat \\zeta_N^D$ to the point measures\n\\begin{equation}\n\\label{E:full-etaDGFF}\n\\widehat \\eta^D_N :=\n\\frac{1}{W_N} \\sum_{x \\in D_N} \\delta_{x\/N} \\otimes \n\\delta_{\\widetilde h_x^{D_N} - \\widehat a_N} \n\\otimes \\delta_{\\left\\{\\frac{2\\sqrt{2 a_N}+2(\\widetilde h_x^{D_N} - \\widehat a_N) + (\\widetilde h_{x+z}^{D_N} - \\widetilde h_x^{D_N})}{2\\log N} (\\widetilde h_x^{D_N} - \\widetilde h_{x+z}^{D_N})~:~z \\in \\mathbb{Z}^2 \\right\\}}\n\\end{equation} \nassociated with the DGFF~$\\widetilde h^{D_N}$. For that we need:\n\n\\begin{lemma}[Gradients of squared DGFF]\n\\label{lemma-gradient-DGFF}\nFor all $b \\in\\mathbb R$, all $M \\ge1$  and all $r > 0$, \n\\begin{multline}\n\\label{E:9.3}\n\\lim_{N \\to \\infty} \\frac{1}{W_N}\\sum_{x \\in D_N} P^{\\varrho} \n\\bigl(L_{t_N}^{D_N} (x) \\geq a_N + b \\log N \\bigr) \\\\\n\\times \\mathbb P \\biggl(\\,\\bigcup_{ z \\in \\Lambda_r (0)}\\Bigl\\{\n\\bigl|(h_x^{D_N})^2 - (h_{x+z}^{D_N})^2 \\bigr| > (\\log N)^{3\/4},\n\\,|h_x^{D_N}| \\leq M \\sqrt{\\log N}\\Bigr\\}\\biggr)\n= 0,\n\\end{multline}\nwhere $\\Lambda_r (x) := \\{z \\in \\Bbb{Z}^2 \\colon |z-x| \\leq r \\}$.\n\\end{lemma}\n\n\\begin{proofsect}{Proof}\nWhen $|h_x^{D_N}| \\leq M \\sqrt{\\log N}$,\nwe have\n\\begin{equation}\n\\bigl|(h_x^{D_N})^2 - (h_{x+z}^{D_N})^2 \\bigr| \\leq\n \\bigl|h_x^{D_N} - h_{x+z}^{D_N} \\bigr|^2 + 2M \\sqrt{\\log N} \\bigl|h_x^{D_N} - h_{x+z}^{D_N} \\bigr|.\n\\end{equation}\nThus, for $M\\ge1$, the term corresponding to $x \\in D_N$ \non the left-hand side of (\\ref{E:9.3}) is bounded from above by\n\\begin{equation}\n\\label{E:8.4nw}\n\\sum_{z \\in \\Lambda_r (0)}\nP^{\\varrho}\n\\bigl(L_{t_N}^{D_N} (x) \\geq a_N + b \\log N \\bigr) \n\\mathbb{P} \\Bigl(\\bigl|h_x^{D_N} - h_{x+z}^{D_N} \\bigr| > (4M)^{-1}(\\log N)^{1\/4}  \\Bigr).\n\\end{equation}\nFor $\\epsilon > 0$, abbreviate $D_N^{\\epsilon} := \\{x \\in D_N \\,\\colon\\,\\text{\\rm d}_\\infty(x, D_N^{\\text{c}}) > \\epsilon N \\}$. \nThen for any $x \\in D_N^{\\epsilon}$ and $z \\in \\Lambda_r (0)$,\n$\\text{Var}_{\\mathbb P} (h_x^{D_N} - h_{x+z}^{D_N})$ is equal to\n\\begin{multline}\n\\qquad\nG^{D_N} (x, x) + G^{D_N} (x+z,x+z) - 2G^{D_N} (x,x+z) \n\\\\= g\\log N + g \\log N - 2 g \\log (N\/(1+|z|)) + O(1) \n\\\\\\leq 2g \\log (1+r) + O(1).\n\\qquad\n\\end{multline}\nThe standard Gaussian tail estimate bounds \\eqref{E:8.4nw} by $o(1)P^\\varrho(L_{t_N}^{D_N} (x) \\geq a_N + b \\log N)$ with $o(1)\\to0$ uniformly in~$x\\in D_N^\\epsilon$. Lemma \\ref{lemma-upper} subsequently shows that the sum over $x \\in D_N^{\\epsilon}$ on the left-hand side of \\eqref{E:9.3} is $o(1)$ as $N \\to \\infty$.\nThe sum over $x \\in D_N \\smallsetminus D_N^{\\epsilon}$ is bounded from above by\n$E^\\varrho(\\zeta_N^D (D \\smallsetminus D^{\\epsilon} \\times [b, \\infty) ))$ which tends to $0$ \nas $N \\to \\infty$ followed by $\\epsilon \\downarrow 0$ by Corollary \\ref{cor-tightness-upper}. \n\\end{proofsect}\n\nWe are ready to give:\n\n\\begin{proofsect}{Proof of Theorem~\\ref{thm-thick-local}}\nPick any $f = f(x, \\ell, \\phi) \\in C_{\\text{c}} (D \\times \\mathbb R \\times \\mathbb R^{\\mathbb{Z}^2})$\nwhich depends only on a finite number of coordinates of $\\phi$, say, those in $\\Lambda_r (0)$\nfor some $r > 0$. The following identity is key for the entire proof\n\\begin{multline}\n\\qquad\n\\left\\{\\sqrt{2 a_N}+(\\widetilde h_x^{D_N} - \\widehat a_N) + \\frac{1}{2}(\\widetilde h_{x+z}^{D_N} - \\widetilde h_x^{D_N}) \\right\\}\n(\\widetilde h_x^{D_N} - \\widetilde h_{x+z}^{D_N})\n\\\\\n= L_{t_N}^{D_N} (x) - L_{t_N}^{D_N} (x+z) + \\frac{1}{2} (h_x^{D_N})^2 - \\frac{1}{2} (h_{x+z}^{D_N})^2.\n\\qquad\n\\end{multline} \nIndeed, we then get\n\\begin{multline}\n\\qquad\n\\label{E:9.10}\n\\langle \\widehat \\eta_N^D, f \\rangle\n=\\frac{1}{W_N}\n\\sum_{x \\in D_N} f \\Biggl(\\frac{x}{N}, \\sqrt{2 L_{t_N}^{D_N} (x) + (h_x^{D_N})^2} - \\sqrt{2a_N}, \n\\\\\n\\biggl\\{\\frac{\\nabla_z L_{t_N}^{D_N} (x)}{\\log N}\n+ \\frac{\\nabla_z (h^{D_N})^2 (x)}{2 \\log N}\\colon z \\in \\mathbb{Z}^2 \\biggr\\} \\Biggr),\n\\qquad\n\\end{multline}\nwhere, given a function $s\\colon \\Bbb{Z}^2 \\to \\mathbb R$, we abbreviated $\\nabla_z s (x) := s (x) - s (x+z)$.\n\nIn order to control gradients of the DGFF squared that appear on the right-hand side of \\eqref{E:9.10}, set\n\\begin{equation}\n\\label{E:gradient-squared-DGFF}\nG_{N, r} (x) := \\bigcap_{z \\in \\Lambda_r (0)}\\Bigl\\{\\bigl|\\nabla_z (h^{D_N})^2 (x)\\bigr| \\leq (\\log N)^{3\/4}\n\\Bigr\\}\n\\end{equation}\nand let, as before, $\\chi \\colon [0, \\infty) \\to [0,1]$ be a non-increasing, continuous function\nwith $\\chi(x) = 1$ for $0 \\leq x \\leq 1$ and $\\chi(x) = 0$ for $x \\geq 2$.\nBy Lemmas \\ref{lemma-no-conspire} and \\ref{lemma-gradient-DGFF},\nwe may truncate \\eqref{E:9.10} by introducing $1_{G_{N, r} (x)}$ and $\\chi (M^{-1}|h_x^{D_N}|\/ \\sqrt{\\log N})$\nfor $M > 0$ under the sum and write $\\langle \\widehat \\eta_N^D, f \\rangle$ as a random quantity whose $L^1$ norm is at most a constant times  $\\|f\\|_{\\infty} \\text{\\rm  e}\\mkern0.7mu^{- \\beta M^2}$\nuniformly in~$N$ plus the quantity\n\\begin{multline}\n\\label{E:9.12}\n\\quad\n\\frac{1}{W_N}\n\\sum_{x \\in D_N} 1_{G_{N,r} (x)} f \\Biggl(\\frac xN, \\sqrt{2 L_{t_N}^{D_N} (x) + (h_x^{D_N})^2} - \\sqrt{2a_N}, \\\\\n\\biggl\\{\\frac{\\nabla_z L_{t_N}^{D_N} (x)}{\\log N}\n+ \\frac{\\nabla_z (h^{D_N})^2 (x)}{2 \\log N}\\colon z \\in \\mathbb{Z}^2 \\biggr\\} \\Biggr)\n \\chi \\biggl(\\frac{|h_x^{D_N}|}{M \\sqrt{\\log N}} \\biggr).\n \\quad\n\\end{multline}\nUsing the uniform continuity of $f$\nand Corollary \\ref{cor-tightness-upper} and\nLemma \\ref{lemma-gradient-DGFF},\nwe rewrite (\\ref{E:9.12}) by a random quantity which tends to $0$\nas $N \\to \\infty$ in probability plus the quantity\n\\begin{equation}\n\\label{E:9.13}\n\\frac{1}{W_N} \\sum_{x \\in D_N} f_{\\text{ext}} \\Biggl(\\frac xN, \\frac{L_{t_N}^{D_N} (x) - a_N}{\\log N},\n\\Bigl\\{\\frac{\\nabla_z L_{t_N}^{D_N} (x)}{\\log N} \\Bigr\\}_{z \\in \\mathbb{Z}^2},\n\\frac{h_x^{D_N}}{\\sqrt{\\log N}} \\Biggr) \\chi \\biggl(\\frac{|h_x^{D_N}|}{M \\sqrt{\\log N}} \\biggr),\n\\end{equation}\nwhere we introduced\n\\begin{equation}\nf_{\\text{ext}} (x, \\ell, \\phi, h) := f \n\\biggl(x, \\frac{1}{2 \\sqrt{g} (\\sqrt{\\theta} + \\lambda)}\n\\bigl(\\ell + \\tfrac{1}{2} h^2 \\bigr), \\phi \\biggr).\n\\end{equation}\nNote that Corollary \\ref{cor-tightness-upper} implies that $\\{\\widehat \\zeta_N^D\\colon N\\ge1\\}$ is tight. \nLet $\\widehat \\zeta^D$ be a subsequential weak limit of $\\widehat \\zeta_N^D$ along the subsequence $\\{N_k\\}$.\nBy the same argument as in the proof of Lemma \\ref{lemma-add-field},\nas $k \\to \\infty$ followed by $M \\to \\infty$, \n$\\langle \\widehat \\eta_{N_k}^D, f \\rangle$ converges in law to\n\\begin{equation}\n\\label{E:8.13nw}\n\\int \\widehat \\zeta^D (\\text{\\rm d}\\mkern0.5mu x\\,\\text{\\rm d}\\mkern0.5mu\\ell\\,\\text{\\rm d}\\mkern0.5mu\\phi) \\otimes \\mathfrak{g} (\\text{\\rm d}\\mkern0.5mu h)\nf_{\\text{ext}}(x, \\ell, \\phi, h).\n\\end{equation}\nOn the other hand, noting that~$\\sqrt{2a_N}\/\\log N\\to 2\\sqrt g(\\sqrt\\theta+\\lambda)$, \\cite[Theorem 2.1]{BL4} shows that $\\langle \\widehat \\eta_N^D, f \\rangle$ converges, as $N \\to \\infty$, in law to\n\\begin{equation}\n  \\int \\fraktura c(\\lambda)Z_{\\lambda}^D (\\text{\\rm d}\\mkern0.5mu x) \\otimes \\text{\\rm  e}\\mkern0.7mu^{- \\alpha \\lambda h} \\text{\\rm d}\\mkern0.5mu h \\otimes \n\\nu_{\\theta, \\lambda} (\\text{\\rm d}\\mkern0.5mu \\phi) f(x, h, \\phi).\n\\end{equation}\nThe arguments in the proof of Theorem \\ref{thm-thick} show that the class of functions $f_{\\text{ext}}$ arising from~$f\\in C_{\\text{\\rm c}}(D\\times\\mathbb R\\times\\mathbb R^{\\mathbb Z^2})$ above determines the measure~$\\widehat\\zeta^D$ uniquely from \\eqref{E:8.13nw}; the calculation \\eqref{E:4.38} then gives\n\\begin{equation}\n\\widehat \\zeta^D \\,\\,\\overset{\\text{\\rm law}}=\\,\\,\n\\frac{\\theta^{1\/4}}{2\\sqrt{g} (\\sqrt{\\theta} + \\lambda)^{3\/2}} \\fraktura c(\\lambda) \nZ_{\\lambda}^D (\\text{\\rm d}\\mkern0.5mu x) \\otimes \\text{\\rm  e}\\mkern0.7mu^{- \\alpha (\\theta, \\lambda) \\ell} \\text{\\rm d}\\mkern0.5mu\\ell \\otimes \n\\nu_{\\theta, \\lambda} (\\text{\\rm d}\\mkern0.5mu\\phi).\n\\end{equation}\nThis is the desired claim.\n\\end{proofsect}\n\n\\subsection{Local structure of thin points}\nWe move to the proof of the convergence of point measures $\\widehat \\zeta_N^D$ \nassociated with $\\lambda$-thin points. The proof follows very much the same steps as for the thick points so we stay quite brief. \nAssume that~$a_N$ and~$t_N$ satisfy \\eqref{E:1.22} with some $\\theta > 0$ and some $\\lambda \\in (0, 1 \\wedge \\sqrt{\\theta} )$.\nAs a counterpart to Lemma \\ref{lemma-gradient-DGFF}, we need the following:\n\n\\begin{lemma}[Gradients of squared DGFF]\n\\label{lemma-control-gradient-DGFF2}\nFor all $b > 0$, all $M \\ge1$ and all $r > 0$,\n\\begin{multline}\n\\lim_{N \\to \\infty}\n\\frac{1}{W_N}\\sum_{x \\in D_N} \nP^{\\varrho} \\Bigl(a_N - b \\log N \\leq L_{t_N}^{D_N} (x) \\leq a_N + b \\log N \\Bigr) \\\\\n\\times \n\\mathbb P \\biggl(\\,\\bigcup_{ z \\in \\Lambda_r (0)}\\Bigl\\{\n\\bigl|(h_x^{D_N})^2 - (h_{x+z}^{D_N})^2 \\bigr| > (\\log N)^{3\/4},\n\\,|h_x^{D_N}| \\leq M \\sqrt{\\log N}\\Bigr\\}\\biggr)\n= 0.\n\\end{multline}\n\\end{lemma}\n\n\\begin{proofsect}{Proof}\nThe proof is almost the same as that of Lemma \\ref{lemma-gradient-DGFF}\nexcept that we use Lemma \\ref{lemma-lower} \nand Corollary \\ref{cor-tightness-lower}\ninstead of Lemma \\ref{lemma-upper} and Corollary \\ref{cor-tightness-upper}, respectively.\n\\end{proofsect}\n\nWe are again ready to start:\n\n\\begin{proofsect}{Proof of Theorem~\\ref{thm-thin-local}}\nSet \n\\begin{equation}\n\\widehat a_N := \\sqrt{2t_N} - \\sqrt{2a_N}\n\\end{equation}\nand pick any\n$f = f(x, \\ell, \\phi) \\in C_{\\text{c}} (D \\times \\mathbb R \\times \\mathbb R^{\\mathbb{Z}^2})$\nthat depends only on a finite number of coordinates of~$\\phi$.\nLet $\\widehat \\eta_N^D$ be the point process obtained from (\\ref{E:full-etaDGFF})\nby replacing~$\\widehat a_N$ by~$- \\widehat a_N$. Using the calculation\n\\begin{multline}\n\\qquad\n\\left\\{\\sqrt{2 a_N}+(\\widetilde h_x^{D_N} + \\widehat a_N) + \\frac{1}{2}(\\widetilde h_{x+z}^{D_N} - \\widetilde h_x^{D_N}) \\right\\}\n(\\widetilde h_x^{D_N} - \\widetilde h_{x+z}^{D_N}) \n\\\\\n= L_{t_N}^{D_N} (x) - L_{t_N}^{D_N} (x+z) + \\frac{1}{2} (h_x^{D_N})^2 - \\frac{1}{2} (h_{x+z}^{D_N})^2\n\\qquad\n\\end{multline}\nwe then again have \\eqref{E:9.10} for $\\langle\\widehat\\eta^D_N,f\\rangle$.\nUsing Corollary \\ref{cor-tightness-lower} and\nLemmas \\ref{lemma-no-conspire2} and \\ref{lemma-control-gradient-DGFF2},\nwe rewrite \\eqref{E:9.10} as a random quantity whose $L^1$ norm \nis at most a constant times $\\|f\\|_{\\infty} \\text{\\rm  e}\\mkern0.7mu^{- \\beta M^2}$\nuniformly in~$N$ plus \\eqref{E:9.13},\nwhere, in this case,\n\\begin{equation}\nf_{\\text{ext}} (x, \\ell, \\phi, h) := f \\biggl(x, \\frac{1}{2 \\sqrt{g} (\\sqrt{\\theta} - \\lambda)}\n\\bigl(\\ell + \\tfrac{1}{2} h^2 \\bigr), \\phi \\biggr).\n\\end{equation}\nNote that Corollary \\ref{cor-tightness-lower} implies the tightness of $\\{\\widehat \\zeta_N^D\\colon N\\ge1\\}$.\nLet $\\widehat \\zeta^D$ be any subsequential weak limit of $\\widehat \\zeta_N^D$ along the subsequence~$\\{N_k\\}$. \nBy the same argument as in the proof of Lemma~\\ref{lemma-add-field2},\nas $k \\to \\infty$ and $M \\to \\infty$, $\\langle \\widehat \\eta_{N_k}^D, f \\rangle$ tends in law to\n\\begin{equation}\n\\int \\widehat \\zeta^D (\\text{\\rm d}\\mkern0.5mu x\\,\\text{\\rm d}\\mkern0.5mu\\ell\\,\\text{\\rm d}\\mkern0.5mu\\phi) \\otimes \\mathfrak{g} (\\text{\\rm d}\\mkern0.5mu h)\nf_{\\text{ext}}(x, \\ell, \\phi, h).\n\\end{equation}\nOn the other hand, by \\cite[Theorem 2.1]{BL4}, as $N \\to \\infty$,\n$\\langle \\widehat \\eta_N^D, f \\rangle$ converges in law to\n\\begin{equation}\n \\int  \\fraktura c(\\lambda)Z_{\\lambda}^D (\\text{\\rm d}\\mkern0.5mu x) \\otimes \\text{\\rm  e}\\mkern0.7mu^{\\alpha \\lambda h} \\text{\\rm d}\\mkern0.5mu h \\otimes \n\\widetilde \\nu_{\\theta, \\lambda} (\\text{\\rm d}\\mkern0.5mu\\phi) f(x, h, \\phi).\n\\end{equation}\nThe arguments in the proof of Theorem \\ref{thm-thin} and the calculation \\eqref{E:4.38a} then show\n\\begin{equation}\n\\widehat \\zeta^D \\,\\,\\overset{\\text{\\rm law}}=\\,\\,\n\\frac{\\theta^{1\/4}}{2\\sqrt{g} (\\sqrt{\\theta} - \\lambda)^{3\/2}} \\,\\fraktura c(\\lambda) \\,\nZ_{\\lambda}^D (\\text{\\rm d}\\mkern0.5mu x) \\otimes \\text{\\rm  e}\\mkern0.7mu^{\\widetilde \\alpha (\\theta, \\lambda) \\ell} \\text{\\rm d}\\mkern0.5mu\\ell \\otimes \n\\widetilde \\nu_{\\theta, \\lambda} (\\text{\\rm d}\\mkern0.5mu\\phi).\n\\end{equation}\nThis is the desired claim.\n\\end{proofsect}\n\n\n\\subsection{Local structure of avoided points}\nIn this section we will prove the convergence of the point measures \nassociated with the local structure of the avoided points. The proof will make use of the Pinned Isomorphism Theorem (see Theorem~\\ref{thm-pinned-iso}) but that so only at the very end. Most of the argument consists of careful manipulations with the doubly extended measure\n\\begin{equation}\n\\widehat \\kappa_N^{D, \\text{ext}} :=\n\\frac{\\sqrt{\\log N}}{\\widehat W_N}\n\\sum_{x \\in D_N}\n1_{\\{L_{t_N}^{D_N} (x) = 0 \\}} \\delta_{x\/N} \\otimes \\delta_{\\{L_{t_N}^{D_N} (x+z) \\,:\\,z \\in \\Bbb{Z}^2 \\}}\n\\otimes \\delta_{h_x^{D_N}} \\otimes \\delta_{\\{\\widehat h_{x+z}^{D_N \\smallsetminus \\{x\\}}\\,:\\,z \\in \\Bbb{Z}^2 \\}},\n\\end{equation}\nwhere, for $\\fraktura b_{D_N,x}$ as in \\eqref{E:5.12i},\n\\begin{equation}\n\\label{E:8.24uiw}\n\\widehat h^{D_N \\smallsetminus \\{x \\}}_z :=h^{D_N}_z-h^{D_N}_x\\fraktura b_{D_N,x}(z),\\quad z\\in\\mathbb Z^2.\n\\end{equation}\nBy \\eqref{E:5.12i}, $\\widehat h^{D_N \\smallsetminus \\{x \\}}$ is the field $h^{D_N}$ conditioned on $h^{D_N}_x=0$. In particular, \n\\begin{equation}\n\\label{E:8.25uiw}\n\\widehat h^{D_N \\smallsetminus \\{x \\}}\\independent h^{D_N}_x.\n\\end{equation}\nCorollary \\ref{cor-light} implies that $\\{\\widehat \\kappa_N^{D, \\text{ext}} \\colon N\\ge1\\}$ is tight with respect to vague convergence of measures on the product space $D\\times[0,\\infty)^{\\mathbb Z^2}\\times\\mathbb R\\times\\mathbb R^{\\mathbb Z^2}$. We now claim:\n\n\\begin{lemma}\n\\label{lemma-avoid-local-add}\nSuppose $\\{N_k\\}$ is a subsequence along which $\\widehat \\kappa_N^D$\nconverges in law to $\\widehat \\kappa^D$. \nThen\n\\begin{equation}\n\\widehat \\kappa_N^{D, \\text{\\rm ext}} \\,\\,\\,\\underset{\\begin{subarray}{c} N = N_k \\\\ k\\to\\infty \\end{subarray}}{\\,\\overset{\\text{\\rm law}}\\longrightarrow\\,}\\,\\,\\,\n \\frac{1}{\\sqrt{2\\pi g}} \\,\\widehat \\kappa^D \\otimes {\\rm Leb} \\otimes \\nu^0.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proofsect}{Proof}\nLet $f = f(x, \\ell, h, \\phi)\\colon D\\times[0,\\infty)^{\\mathbb Z^2}\\times\\mathbb R\\times\\mathbb R^{\\mathbb Z^2} \\to \\mathbb R$\nbe a continuous, compactly-supported function\nthat depends only on a finite number of coordinates of $\\ell$ and $\\phi$, say, \nthose in $\\Lambda_{r_0} (0)$ for some~$r_0>0$.\nSuppose in addition that $f(x, \\ell, h, \\phi) = 0$\nunless $x \\in A$ for some open $A \\subset D$ with $\\overline A \\subset D$, and unless\n$|h| \\leq M$ and $\\ell_z, |\\phi_z| \\leq M$ for all $z \\in \\Lambda_{r_0} (0)$ for some $M > 0$.\n\nNoting that only the second pair of the variables of~$\\widehat \\kappa_N^{D, \\text{ext}}$ is affected by expectation~$\\mathbb E$ with respect to the law of~$h^{D_N}$, we now claim\n\\begin{equation}\n\\Bbb{E} \\langle \\widehat \\kappa_N^{D, \\text{ext}}, f \\rangle\n= \\frac{1}{\\sqrt{2\\pi g}}\\,\\langle \\widehat \\kappa_N^D \\otimes {\\rm Leb} \\otimes \\nu^0, f \\rangle + o(1),\n\\end{equation}\nwhere $o(1) \\to 0$ in $P^{\\varrho}$-probability as $N \\to \\infty$ and where~$\\nu^0$ the law of the pinned DGFF. As in the proof of Lemma~\\ref{lemma-7.1}  (8.26)  follows by noting that the probability density of $h_x^{D_N}$ multiplied by~$\\sqrt{\\log N}$ tends to $(2\\pi g)^{-1\/2}$ uniformly over any compact interval and by the fact $(\\widehat h_{x+z}^{D_N \\smallsetminus \\{x\\}})_{z \\in \\Lambda_{r_0} (0)}$ tends in law to $(\\phi_z)_{z \\in \\Lambda_{r_0}(0)}$ (which can be gleaned from the representation of the Green function by the potential kernel, see \\cite[Lemma B.3]{BL3}, and the asymptotic expression for the potential kernel, see \\cite[Lemma B.4]{BL3}). These two convergences may be applied jointly in light of the independence \\eqref{E:8.25uiw} and the Bounded Convergence Theorem enabled by the tightness of~$\\{\\widehat\\kappa^D_N\\colon N\\ge1\\}$.\n\nIn order to prove the claim, it thus suffices to prove that the conditional variance\n$\\text{Var}_{\\mathbb P} (\\langle \\widehat \\kappa_N^{D, \\text{ext}}, f \\rangle)$\ntends to zero in $P^{\\varrho}$-probability as $N \\to \\infty$.\nWriting $A_N:=\\{x\\in\\mathbb Z^2\\colon x\/N\\in A\\}$, for this we note that\n\\begin{equation}\n\\text{Var}_{\\mathbb P} (\\langle \\widehat \\kappa_N^{D, \\text{ext}}, f \\rangle)\n=\\frac{\\log N}{(\\widehat W_N)^2}\\sum_{x,y\\in A_N}1_{\\{L_{t_N}^{D_N} (x) = 0\\}}1_{\\{L_{t_N}^{D_N} (y) = 0\\}}\\,\\mathcal{C}_N(x,y)\n\\end{equation}\nwhere $\\mathcal{C}_N(x,y)$ denotes the quantity \n\\begin{equation}\n\\text{Cov}_{\\mathbb P}\n\\Bigl(f\\bigl(\\tfrac xN,\\, L_{t_N}^{D_N} (x+\\cdot),\\,\nh_x^{D_N},\\, \\widehat h_{x+\\cdot}^{D_N \\smallsetminus \\{x \\}} \\bigr),\\,\nf\\bigl(\\tfrac yN,\\, L_{t_N}^{D_N} (y+\\cdot),\\,\nh_y^{D_N},\\, \\widehat h_{y+\\cdot}^{D_N \\smallsetminus \\{y \\}} \\bigr) \\Bigr).\n\\end{equation}\nWe will now proceed to estimate  $\\mathcal{C}_N (x, y)$ by arguments similar to those invoked in the proof of Lemma~\\ref{lemma-7.1}. \n\nTo lighten the notation, we will henceforth write $\\phi^{(x)}$ for $(\\widehat h_{x+z}^{D_N \\smallsetminus \\{x \\}})_{z \\in \\Lambda_{r_0}(0)}$. Let $\\rho_{x,y}^{D_N}$ be joint probability density of $(h_x^{D_N}, h_y^{D_N})$ and let $\\widehat \\rho_{x,y}^{D_N}$ be the joint probability density of $(\\phi^{(x)},\\phi^{(y)})$. Writing\n$F(h_x, h_y, \\phi^{(x)}, \\phi^{(y)}; \\tilde h_x, \\tilde h_y, \\tilde \\phi^{(x)}, \\tilde \\phi^{(y)})$ for the product\n\\begin{multline}\n\\quad\n\\frac{1}{2} \\biggl[\n f\\Bigl(\\tfrac xN,\\, L_{t_N}^{D_N} (x+\\cdot),\\,\nh_x,\\, \\phi^{(x)} \\Bigr) \n- f\\Bigl(\\tfrac xN,\\, L_{t_N}^{D_N} (x+\\cdot),\\,\n\\tilde h_x,\\, \\tilde\\phi^{(x)}\\Bigr) \\biggr] \n\\\\\n\\times \\biggl[\n f\\Bigl(\\tfrac yN,\\, L_{t_N}^{D_N} (y+\\cdot),\\,\nh_y,\\, \\phi^{(y)} \\Bigr) \n- f\\Bigl(\\tfrac yN,\\, L_{t_N}^{D_N} (y+\\cdot),\\,\n\\tilde h_y,\\, \\tilde\\phi^{(y)} \\Bigr) \\biggr]\n\\quad\n\\end{multline}\nand abbreviating $\\Omega_M:= [-M, M]^4 \\times ([-M, M]^{\\Lambda_{r_0}(0)})^4$, we then get\n\\begin{multline}\n\\label{E:9.31}\n\\mathcal{C}_N (x,y)=\\int_{\\Omega_M}\n\\text{\\rm d}\\mkern0.5mu h_x \\,\\text{\\rm d}\\mkern0.5mu h_y \\,\\text{\\rm d}\\mkern0.5mu \\tilde h_x \\,\\text{\\rm d}\\mkern0.5mu \\tilde h_y\\,\n\\text{\\rm d}\\mkern0.5mu\\phi^{(x)} \\,\\text{\\rm d}\\mkern0.5mu \\tilde\\phi^{(x)} \\,\\text{\\rm d}\\mkern0.5mu \\phi^{(y)} \\,\\text{\\rm d}\\mkern0.5mu\\tilde\\phi^{(y)}\n\\rho_{x,y}^{D_N} (h_x, h_y) \\rho_{x, y}^{D_N} (\\tilde h_x, \\tilde h_y)\n\\\\\n\\times\n\\widehat \\rho_{x,y}^{D_N} \\bigl(\\phi^{(x)}, \\phi^{(y)}\\bigr) \n\\,\\widehat \\rho_{x,y}^{D_N} \\bigl(\\tilde\\phi^{(x)}, \\tilde\\phi^{(y)}\\bigr)\n\\,F\\bigl(h_x, h_y, \\phi^{(x)}, \\phi^{(y)}; \\tilde h_x, \\tilde h_y, \\tilde\\phi^{(x)}, \\tilde\\phi^{(y)}\\bigr).\n\\end{multline}\nLet $\\delta_N (M, r)$ abbreviate the quantity\n\\begin{equation}\n\\max_{\\begin{subarray}{c} x, y \\in A_N \\\\ |x-y| \\geq r \\end{subarray}}\\,\\,\n\\sup_{h_x, h_y \\in[-M,M]} \\,\\,\n\\sup_{\\begin{subarray}{c} \n\\phi^{(x)},\\phi^{(y)}\\in[-M,M]^{\\Lambda_{r_0} (0)}\n\\end{subarray}}\\,\\,\n\\Biggl|\\,\\log \\frac{\\rho_{x,y}^{D_N} (h_x, h_y)\\,\\widehat\\rho_{x,y}^{D_N}(\\phi^{(x)},\\phi^{(y)})}\n{\\rho_{x,y}^{D_N}(0,0)\\,\\widehat\\rho_{x}^{D_N}(\\phi^{(x)})\\,\\widehat\\rho_{y}^{D_N}(\\phi^{(y)})}\\Biggr|,\n\\end{equation}\nwhere $\\widehat\\rho_{x}^{D_N}$, resp., $\\widehat\\rho_{y}^{D_N}$ are the probability density of $\\phi^{(x)}$, resp., $\\phi^{(y)}$ with respect to the Lebesgue measure on~$\\mathbb R^{\\Lambda_{r_0}(0)}$.\nInvoking the bound\n\\begin{multline}\n\\quad\n\\rho_{x,y}^{D_N} (h_x, h_y) \\rho_{x,y}^{D_N} (\\tilde h_x, \\tilde h_y)\n\\widehat \\rho_{x,y}^{D_N} (\\phi^{(x)}, \\phi^{(y)}) \\widehat\\rho_{x,y}^{D_N} (\\tilde\\phi^{(x)}, \\tilde\\phi^{(y)})\n\\\\\n\\leq \\rho_{x,y}^{D_N} (0,0)^2 \n\\widehat\\rho_x^{D_N} (\\phi^{(x)}) \\widehat\\rho_y^{D_N} (\\phi^{(y)}) \\widehat\\rho_x^{D_N} (\\tilde\\phi^{(x)})\n \\widehat\\rho_y^{D_N} (\\tilde\\phi^{(y)}) \\bigl(\\text{\\rm  e}\\mkern0.7mu^{2 \\delta_N (M, |x-y|)} - 1 \\bigr)\n\\\\+ \\rho_{x,y}^{D_N} (0,0)^2  \\widehat\\rho_x^{D_N} (\\phi^{(x)}) \\widehat\\rho_y^{D_N} (\\phi^{(y)}) \\widehat\\rho_x^{D_N} (\\tilde\\phi^{(x)})\n \\widehat\\rho_y^{D_N} (\\tilde\\phi^{(y)}),\n \\quad\n\\end{multline}\nand noting that the integral of the second term against $F(\\cdots)$ vanishes,\nwe have\n\\begin{equation}\n\\label{E:8.34uiw}\n\\mathcal{C}_N (x, y) \\leq  32M^4  \\Vert f\\Vert_\\infty^2\n\\,\\rho_{x,y}^{D_N} (0, 0)^2 \\bigl(\\text{\\rm  e}\\mkern0.7mu^{2 \\delta_N (M, |x-y|)} - 1 \\bigr),\n\\end{equation}\nwhere also bounded $F(\\cdots)$ by $2\\Vert f\\Vert_\\infty^2$ and noted that the integral over~$\\Omega_M$ of the product of the individual densities of $\\phi^{(x)}$, $\\phi^{(y)}$, $\\tilde\\phi^{(x)}$ and $\\tilde\\phi^{(y)}$ is at most $(2M)^4$. \n \n\nIt remains to control the~$N$-dependent terms on the right-hand side of \\eqref{E:8.34uiw}.\nLet $r_N$ be a positive sequence with $\\lim_{N \\to \\infty} r_N = \\infty$\nand $\\lim_{N \\to \\infty} \\log r_N\/\\log N = 0$.\nUsing the representation of the Green function by the potential kernel (cf \\cite[Lemma B.3]{BL3})\nand the asymptotic of the potential kernel (cf \\cite[Lemma B.4]{BL3}), we have\n\\begin{equation}\n\\max_{\\begin{subarray}{c}\nx,y\\in A_N\\\\|x-y|\\ge r_N\n\\end{subarray}}\n\\,\\,\\max_{z, z^{\\prime} \\in \\Lambda_{r_0} (0)}\n\\mathbb E\\bigl(\\phi^{(x)}_z \\phi^{(y)}_{z^{\\prime}}\\bigr)\\,\\underset{N\\to\\infty}\\longrightarrow\\,0.\n\\end{equation}\nSince the variances of $\\phi^{(x)}_z$, for $z\\in\\Lambda_{r_0}(0)\\smallsetminus\\{0\\}$ stay uniformly positive, the argument as in \\eqref{E:7.9ie} shows\n\\begin{equation}\n\\max_{\\begin{subarray}{c}\nx,y\\in A_N\\\\|x-y|\\ge r_N\n\\end{subarray}}\\,\\,\n\\sup_{\\phi^{(x)},\\phi^{(y)}\\in[-M,M]^{\\Lambda_{r_0}(0)}}\\,\\biggl|\\,\\frac{\\widehat\\rho_{x,y}^{D_N} (\\phi^{(x)}, \\phi^{(y)})}{\\widehat\\rho_x^{D_N} (\\phi^{(x)}) \n\\widehat\\rho_y^{D_N} (\\phi^{(y)})}-1\\biggr|\n\\,\\underset{N\\to\\infty}\\longrightarrow\\,0.\n\\end{equation}\nThis implies $\\delta_N (M, r_N) \\to0$ as $N \\to \\infty$ and so, for $o(1)\\to0$ in $P^\\varrho$-probability as~$N\\to\\infty$,\n\\begin{equation}\n\\label{E:9.35}\n\\frac{\\log N}{\\widehat W_N^2} \\sum_{\\begin{subarray}{c} x, y \\in A_N \\\\ |x-y| \\geq r_N \\end{subarray}}\n 1_{\\{L_{t_N}^{D_N} (x) = 0\\}}1_{\\{L_{t_N}^{D_N} (y) = 0\\}}\\, \n\\mathcal{C}_N (x, y) = o(1) \\kappa_N^D (D)^2.\n\\end{equation}\nThe sum over $x, y \\in A_N$ with~$|x-y|\\le r_N$\nis at most $\\|f\\|^2 \\kappa_N^D (D)$ times $(r_N)^2(\\log N)\/\\widehat W_N$. The latter factor is~$o(1)$ as $N \\to \\infty$ by our choice of~$r_N$. The claim follows from the tightness of~$\\{\\kappa^D_N(D)\\colon N\\ge1\\}$.\n\\end{proofsect}\n\nWe are now ready to give:\n\n\\begin{proofsect}{Proof of Theorem \\ref{thm-avoid-local}}\nConsider the coupling from Theorem~\\ref{thm-Dynkin} between the local time~$L^{D_N}_{t_N}$ and two copies~$h^{D_N}$ and~$\\tilde h^{D_N}$ of the DGFF in~$D_N$, with the former independent of~$L^{D_N}_{t_N}$. Recall the definition of $\\widehat h^{D_N\\smallsetminus\\{x\\}}$ from \\eqref{E:8.24uiw}, write $\\phi^{(x)}_z:=\\widehat h^{D_N\\smallsetminus\\{x\\}}_{x+z}$ and abbreviate $\\nabla_z s(x):= s(x)-s(x+z) $. Then for each $x \\in D_N$ and $z \\in \\Bbb{Z}^2$, we have\n\\begin{multline}\n\\label{E:9.37}\n\\qquad\n\\Bigl(\\tilde h_x^{D_N} + \\sqrt{2t_N} - \\frac{1}{2} \\nabla_z \\tilde h^{D_N} (x) \\Bigr)\n\\Bigl(- \\nabla_z \\tilde h^{D_N} (x) \\Bigr) \\\\\n= - \\nabla_z L_{t_N}^{D_N} (x) \n+ \\frac{1}{2} \\bigl(\\,\\phi^{(x)}_z\n+ \\fraktura b_{D_N,x} (x+z) h_x^{D_N}\\bigr)^2\n- \\frac{1}{2} (h_x^{D_N})^2.\n\\qquad\n\\end{multline}\nLet $\\Phi_x (z)$ and $\\Psi_x (z)$ denote the left-hand side and the right-hand side\nof \\eqref{E:9.37}, respectively.\nThen for each \n$f \\in C_{\\text{c}} (D \\times [0, \\infty) \\times \\mathbb R^{\\Bbb{Z}^2})$,\n\\begin{multline}\n\\label{E:9.38}\n\\quad\n\\frac{\\sqrt{\\log N}}{\\widehat W_N}\n\\sum_{x \\in D_N}\nf \\left(x\/N,\\, \\frac{1}{2} (\\tilde h_x^{D_N} + \\sqrt{2t_N})^2,\\,\n\\{\\Phi_x (z)\\,:\\,z \\in \\Bbb{Z}^2\\} \\right) \n\\\\\n= \\frac{\\sqrt{\\log N}}{\\widehat W_N}\n\\sum_{x \\in D_N}\nf \\left(x\/N,\\, L_{t_N}^{D_N} (x) + \\frac{1}{2} (h_x^{D_N})^2,\\,\n\\{\\Psi_x (z)\\,:\\,z \\in \\Bbb{Z}^2\\} \\right).\n\\quad\n\\end{multline}\nNext pick $F \\in C_{\\text{c}} (\\mathbb R^{\\Bbb{Z}^2})$ that\ndepends only on a finite number of coordinates, say, in~$\\Lambda_r (0)$,\nand obeys $F(\\phi) = 0$ unless $|\\phi_z| \\leq M$ for all $z \\in \\Lambda_r (0)$ for some $M > 0$. Then set $f(x, \\ell, \\phi) := 1_A(x) f_n (\\ell) F(\\phi)$,\nwhere $A \\subset D$ is an open set with $\\overline{A} \\subset D$ and $f_n\\colon [0, \\infty) \\to [0, 1]$\nare given by $f_n (\\ell) := (1 - n \\ell) \\vee 0$. The Bounded Convergence Theorem ensures that \\eqref{E:9.38} applies to these~$f$'s as well so we will now explicitly compute both sides (suitably scaled) in the \\myemph{joint} distributional limit as $N\\to\\infty$ and~$n\\to\\infty$. Note that taking the limit jointly preserves pointwise equality.\n\nStarting with the right hand side of \\eqref{E:9.38}, the uniform continuity of $F$ and Corollary \\ref{cor-light}, we may rewrite it as a random quantity whose $L^1$-norm under $P^{\\varrho} \\otimes \\mathbb P$\nis at most $o(1) n^{-1\/2}$, with $o(1) \\to 0$ as $n \\to \\infty$, plus the quantity\n\\begin{equation}\n\\label{E:9.39}\n\\frac{\\sqrt{\\log N}}{\\widehat W_N}\n\\sum_{x\\in A_N}\nf_n \\biggl(L_{t_N}^{D_N} (x) + \\frac{1}{2} (h_x^{D_N})^2 \\biggr)\nF \\biggl(\\Bigl\\{L_{t_N}^{D_N} (x+z) + \\frac{1}{2} (\\phi^{(x)}_z)^2\n\\colon z \\in \\Bbb{Z}^2 \\Bigr\\} \\biggr),\n\\end{equation}\nwhere we denoted $A_N:=\\{x\\in\\mathbb Z^2\\colon x\/N\\in A\\}$.\nDecomposing the sum over $x$ with $L_{t_N}^{D_N} (x) = 0$\nand the sum over $x$ with $L_{t_N}^{D_N} (x) > 0$\nand applying Lemma \\ref{lemma-6.3} to the latter,\nwe rewrite (\\ref{E:9.39})\nas\n\\begin{equation}\n\\label{E:29}\n\\frac{\\sqrt{\\log N}}{\\widehat W_N} \\sum_{x\\in A_N}\n1_{\\{L_{t_N}^{D_N} (x) = 0 \\}}\\,\nf_n \\Bigl(\\frac{1}{2} (h_x^{D_N})^2 \\Bigr)\nF \\biggl(\\Bigl\\{L_{t_N}^{D_N} (x+z) + \\frac{1}{2} (\\phi^{(x)}_z)^2\n\\colon z \\in \\Bbb{Z}^2 \\Bigr\\} \\biggr)\n\\end{equation}\nplus a random quantity whose $L^1$-norm under $P^{\\rho} \\otimes \\mathbb P$\nis at most $o(1) n^{-1\/2}$ with $o(1) \\to 0$ as $N \\to \\infty$\nfollowed by $n \\to \\infty$.\nLet $\\widehat \\kappa^D$ be a (subsequential) weak limit of $\\widehat \\kappa_N^D$ along the subsequence $\\{N_k\\}$. \nBy Lemma \\ref{lemma-avoid-local-add}, as $k \\to \\infty$, \\eqref{E:29} converges in law to\n\\begin{multline}\n\\qquad\n\\frac{1}{\\sqrt{2 \\pi g}} \n\\int \\widehat \\kappa^D (\\text{\\rm d}\\mkern0.5mu x\\,\\text{\\rm d}\\mkern0.5mu\\ell)\\otimes \\text{\\rm d}\\mkern0.5mu h \\otimes \\nu^0 (\\text{\\rm d}\\mkern0.5mu\\phi) \n1_A (x) f_n\\bigl(\\tfrac{h^2}2\\bigr) \nF \\biggl(\\Bigl\\{\\ell_z + \\frac{1}{2} \\phi_z^2\\colon z \\in \\Bbb{Z}^2 \\Bigr\\} \\biggr) \n\\\\\n= \\frac{4}{3\\sqrt{\\pi g n}} \\int \\widehat \\kappa^D (\\text{\\rm d}\\mkern0.5mu x\\,\\text{\\rm d}\\mkern0.5mu\\ell)\\otimes \\nu^0 (\\text{\\rm d}\\mkern0.5mu\\phi) \n1_A (x)\nF \\biggl(\\Bigl\\{\\ell_z + \\frac{1}{2} \\phi_z^2\\colon z \\in \\Bbb{Z}^2 \\Bigr\\} \\biggr) \n\\qquad\n\\end{multline}\nwhere we used the explicit form of~$f_n$ to perform the integral over~$h$. Multiplying this by $\\frac{3}{4} \\sqrt{\\frac{n}{2}}$, as $n\\to\\infty$ this converges to\n\\begin{equation}\n\\label{E:8.43uiw}\n\\frac1{\\sqrt{2\\pi g}}\\int \\widehat \\kappa^D_A (\\text{\\rm d}\\mkern0.5mu\\ell)\\otimes \\nu^0 (\\text{\\rm d}\\mkern0.5mu\\phi) \nF \\biggl(\\Bigl\\{\\ell_z + \\frac{1}{2} \\phi_z^2\\colon z \\in \\Bbb{Z}^2 \\Bigr\\} \\biggr)\n\\end{equation}\nas~$n\\to\\infty$ where $\\widehat\\kappa^D_A(B):=\\widehat\\kappa^D(A\\times B)$. This is the $N\\to\\infty$ and~$n\\to\\infty$ limit of the (rescaled) right-hand side of \\eqref{E:9.38}.\n\n\nConcerning the left-hand side of \\eqref{E:9.38}, whenever~$A$ is such that ${\\rm Leb}(\\partial A)=0$ (which implies $Z^D_{\\sqrt\\theta}(\\partial A)=0$ a.s.), \\cite[Theorem~2.1]{BL4} yields convergence to\n\\begin{equation}\n\\label{E:9.42}\n\\fraktura c(\\sqrt{\\theta}) Z_{\\sqrt{\\theta}}^D (A)\n\\int \\text{\\rm d}\\mkern0.5mu h\\otimes \\nu_{\\sqrt{\\theta}} (\\text{\\rm d}\\mkern0.5mu\\phi)\n\\,\\text{\\rm  e}\\mkern0.7mu^{\\alpha \\sqrt{\\theta} h} \\,f_n\\bigl(\\tfrac{h^2}2\\bigr)\\, F\\biggl(\\Bigl\\{\\bigl(h-\\tfrac{1}{2} \\phi_z\\bigr) (-\\phi_z)\\colon z \\in \\Bbb{Z}^2 \\Bigr\\} \\biggr),\n\\end{equation}\nwhere $\\nu_{\\sqrt{\\theta}}$ is the law of $\\phi+\\alpha\\sqrt{\\theta}\\, \\fraktura a$ with~$\\phi$ distributed according to~$\\nu^0$.\nUsing that\n\\begin{equation}\n\\int \\text{\\rm d}\\mkern0.5mu h\\,\\,\\text{\\rm  e}\\mkern0.7mu^{\\alpha \\sqrt{\\theta} h} f_n\\bigl(\\tfrac{h^2}2\\bigr) \n= \\frac{4\\sqrt{2}}{3\\sqrt{n}} + O(n^{-3\/2}),\\quad n\\to\\infty,\n\\end{equation}\n\\eqref{E:9.42} multiplied by $\\frac{3}{4} \\sqrt{\\frac{n}{2}}$ converges to\n\\begin{equation}\n\\label{E:8.46uiw}\n\\fraktura c(\\sqrt{\\theta}) Z_{\\sqrt{\\theta}}^D (A)\n\\int \\nu^0 (\\text{\\rm d}\\mkern0.5mu\\phi) F\\Bigl(\\bigl\\{\\tfrac{1}{2} (\\phi_z+\\alpha\\sqrt\\theta\\,\\fraktura a)^2\\colon z \\in \\Bbb{Z}^2 \\bigr\\}\\Bigr)\n\\end{equation} \nas~$n\\to\\infty$. This is the $N\\to\\infty$ and~$n\\to\\infty$ limit of the (rescaled) left-hand side of \\eqref{E:9.38}.\n\nWe now finally have a chance to invoke the Pinned Isomorphism Theorem of~\\cite{R18}. Indeed, since $2 \\sqrt{2u} =\\alpha\\sqrt\\theta$ implies $u=\\pi\\theta$, \\eqref{E:3.6uiw} equates \\eqref{E:8.46uiw} (and thus also \\eqref{E:8.43uiw}) with\n\\begin{equation}\n\\label{E:8.47uiw}\n\\fraktura c(\\sqrt{\\theta}) Z_{\\sqrt{\\theta}}^D (A)\n\\int \\nu_{\\theta}^{\\text{\\rm RI}} (\\text{\\rm d}\\mkern0.5mu\\ell) \\otimes \\nu^0 (\\text{\\rm d}\\mkern0.5mu\\phi)\nF\\Bigl(\\bigl\\{\\ell_z + \\tfrac{1}{2} \\phi_z^2\\colon z \\in \\Bbb{Z}^2 \\bigr\\}\\Bigr).\n\\end{equation} \nThe Bounded Convergence Theorem extends the equality of \\eqref{E:8.43uiw} and \\eqref{E:8.47uiw} to~$F$ of the form $F(\\ell):=\\exp\\{-\\sum_{z\\in\\Lambda_r(0)}b_z\\ell_z\\}$ for any~$b_z\\ge0$. This effectively transforms the term~$\\frac12\\phi_z^2$ away from both expressions and, thanks to the Cram\\'er-Wold device, implies\n\\begin{equation}\n\\widehat\\kappa_A^D(\\text{\\rm d}\\mkern0.5mu \\ell)  \\overset{\\text{\\rm law}}= \\sqrt{2\\pi g}\\,\\fraktura c(\\sqrt{\\theta})Z^D_{\\sqrt\\theta}(A)\\,\\nu_{\\theta}^{\\text{\\rm RI}}(\\text{\\rm d}\\mkern0.5mu\\ell).\n\\end{equation}\nAs this holds for all open~$A\\subset D$ with~$\\overline A\\subset D$, the claim follows.\n\\end{proofsect}\n\n\n\n\n\\section*{Acknowledgments}\n\\nopagebreak\\noindent\nThe first author has been supported in part by JSPS KAKENHI, Grant-in-Aid for Early-Career Scientists 18K13429. The second author has been partially supported by the NSF award DMS-1712632.\n\n\n\\bibliographystyle{abbrv}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe notion of rectifying curve was introduced by Chen \\cite{BYC03} as a curve in the Euclidean space such that its position vector always lies in the rectifying plane, and then investigated some properties of such curves. For further properties of rectifying curves, the reader can be consulted \\cite{BYC05} and \\cite{BYC18}. Again Ilarslan and Nesovic \\cite{IN08} studied the rectifying curves in Minkowski space and obtained some of its characterization. \n\\par\nIn \\cite{CKI18} Camci et. al associated a frame different from Frenet frame to curves on a surface and deduced some characterization of its position vector. In \\cite{PRG18A} and \\cite{PRG18B} the present authors studied rectifying and osculating curves and obtained some conditions for the invariancy of such curves under isometry. Also the invariancy of the component of position vector of rectifying and osculating curves along the normal and tangent line to the surface are obtained under isometry of surfaces.\n\\par \nMotivating by the above studies of curves whose position vectors are confined in some plane, in this paper we have investigated curves on a smooth surface with position vector always lying in the tangent plane of the smooth surface. By using the Gauss equation we have deduced the component of the position vector along the tangent, normal and binormal vector in simple form. By considering isometry between two smooth surfaces it is proved that curves on smooth surface whose position vector lies in the tangent plane are invariant. It is also shown that the length of position vector, tangential component and geodesic curvature of such curves are invariant under isomerty.\n\\section{Preliminaries}\nThis section is concerned with some preliminary notions of rectifying curves, osculating curves, isometry of surfaces and geodesic curvature (for details see, \\cite{AP01}, \\cite{MPDC76 }) which will be needed for the remaining.\n\\par\nAt every point of an unit speed parametrized curve $\\gamma(s)$ with atleast fourth order continuous derivative, there is an orthonormal frame of three vectors, namely, tangent, normal and binormal vectors. Tangent, normal and binormal vectors are denoted by $\\vec{t}$, $\\vec{n}$ and $\\vec{b}$. They are related by the Serret-Frenet equation given as\n\\begin{eqnarray}\n\\nonumber\nt'(s)&=&\\kappa \\ n(s),\\\\\n\\nonumber\nn'(s) &=& -\\kappa \\ t(s) + \\tau \\ b(s),\\\\\n\\nonumber\nb'(s)&=&-\\tau \\ n(s),\n\\end{eqnarray}\nwhere $\\kappa$ and $\\tau$ are respectively the curvature and torsion of $\\gamma(s)$. Rectifying, osculating and normal plane is generated by $\\{\\vec{t},\\vec{b}\\}$, $\\{\\vec{t},\\vec{n}\\}$ and $\\{\\vec{n},\\vec{b}\\}$ respectively. Curves whose position vector contained in rectifying, osculating and normal plane are respectively called rectifying, osculating and normal curves. \n\\begin{defn}\nLet $S$ and $\\bar{S}$ be smooth surfaces immersed in $\\mathbb{R}^3$. Then a diffeomorphism $f:S\\rightarrow\\bar{S}$ is called an isometry if the length of any curve on $S$ is invariant under $f$.\n\\end{defn}\n\\begin{defn}\nSuppose $\\gamma(s)$ is any unit speed parametrized curve on a smooth surface $S$. Then the tangent vector $\\gamma'(s)$ and the normal $\\vec{N}$ to the surface are mutually orthogonal and also $\\gamma''(s)$ and $\\gamma'(s)$ are orthogonal. Hence $\\gamma''(s)$ is represented by the the linear combination of $\\vec{N}\\times\\gamma'(s)$ and $\\vec{N}$ as\n\\begin{equation*}\n\\gamma''(s)=\\kappa_g\\vec{N}\\times\\gamma'(s)+\\kappa_n\\vec{N}.\n\\end{equation*}\nThen $\\kappa_g$ and $\\kappa_n$ are respectively called the geodesic curvature and normal curvature of $\\gamma(s)$ on $S$ given by the following:\n\\begin{eqnarray}\n\\nonumber\n\\kappa_g&=&\\gamma''\\cdot(\\vec{N}\\times\\gamma'),\\\\\n\\nonumber\n\\kappa_n&=&\\gamma''\\cdot\\vec{N}.\n\\end{eqnarray}\n\\end{defn}\n\\section{Curves on a surface whose position vector lies in the tangent plane}\nLet $S$ be a smooth surface and $\\phi$ be a surface patch at any point $p\\in S$. Let $\\gamma(s)$ be an unit speed parametrized curve in $\\phi$ passing through $p\\in S$. Then the tangent space of $S$ at $p$ is generated by two linearly independent vectors $\\phi_u$ and $\\phi_v$, where $\\phi_u=\\frac{\\partial \\phi}{\\partial u}$ and $\\phi_v=\\frac{\\partial \\phi}{\\partial v}$. If $\\gamma(s)$ be a curve on $S$ whose position vector lies in $T_{\\gamma(s)}S$ then the equation of $\\gamma(s)$ is given by\n\\begin{equation}\\label{sc2}\n\\gamma(s)=\\lambda(s)\\phi_u+\\mu(s)\\phi_v,\n\\end{equation}\nwhere $\\lambda(s)$ and $\\mu(s)$ are two functions of $s$.\\\\\nDifferentiating equation $(\\ref{sc2})$ we get\n\\begin{equation*}\n\\gamma'(s)=\\lambda'\\phi_u+\\mu'\\phi_v+\\lambda (u'\\phi_{uu}+v'\\phi_{uv})+\\mu(u'\\phi_{uv}+v'\\phi_{vv}).\n\\end{equation*}\nThe Gauss Equation for the surface patch $\\phi$ of $S$ with normal vector $\\vec{N}$ is given by\n\n\\begin{equation}\\label{sc3}\n\\begin{cases}\n\\phi_{uu}&=\\Gamma_{11}^1\\phi_u+\\Gamma_{11}^2\\phi_v+L\\vec{N},\\\\\n\\phi_{uv}&=\\Gamma_{12}^1\\phi_u+\\Gamma_{12}^2\\phi_v+M\\vec{N},\\\\\n\\phi_{vv}&=\\Gamma_{22}^1\\phi_u+\\Gamma_{22}^2\\phi_v+N\\vec{N},\\\\\n\\end{cases}\n\\end{equation}\nwhere $\\{L, \\ M, \\ N\\}$ are the coefficients of the second fundamental form of $S$, and the Christoffel symbols $\\Gamma_{ij}^k$ are given by \n\\begin{eqnarray}\n\\nonumber\n\\Gamma_{11}^1=\\frac{E_uG-2F_uF+E_vF}{2(EG-F^2)},&& \\ \\Gamma_{11}^2=\\frac{2F_uE-E_vE-E_uF}{2(EG-F^2)},\\\\\n\\nonumber\n\\Gamma_{12}^1=\\frac{E_vG-G_uF}{2(EG-F^2)},&& \\ \\Gamma_{12}^2=\\frac{G_uE-E_vF}{2(EG-F^2)},\\\\\n\\nonumber\n\\Gamma_{11}^1=\\frac{2F_vG-G_uF-G_uG}{2(EG-F^2)},&& \\ \\Gamma_{11}^2=\\frac{G_vE-2F_vF+G_uF}{2(EG-F^2)},\n\\end{eqnarray}\nwhere $\\{E, \\ F, \\ G\\}$ are the coefficients of the second fundamental form of $S$.\nUsing equation $(\\ref{sc3})$ in equation $(\\ref{sc2})$ we get\n\\begin{eqnarray}\n\\nonumber\n\\vec{t}=\\gamma'(s)=&&\\{\\lambda'+u'\\lambda\\Gamma_{11}^1+(v'\\lambda+u'\\mu) \\Gamma_{12}^1+v'\\mu\\Gamma_{22}^1\\}\\phi_u+\\{\\mu'+u'\\lambda\\Gamma_{11}^2+(v'\\lambda+u'\\mu)\\Gamma_{12}^2\\\\\n\\nonumber\n&&+v'\\mu\\Gamma_{22}^2\\}\\phi_v+\\{u'\\lambda L+v'\\lambda M+u'\\mu M+v'\\mu N\\}\\vec{N},\n\\end{eqnarray}\ni.e., $$\\vec{t}=A_1\\phi_u+A_2\\phi_V+A_3\\vec{N},$$ where $A_1$, $A_2$ and $A_3$ are respectively given by \n\n\\begin{equation}\\label{sc4}\n\\begin{cases}\nA_1=&\\lambda'+u'\\lambda\\Gamma_{11}^1+(v'\\lambda+u'\\mu) \\Gamma_{12}^1+v'\\mu\\Gamma_{22}^1,\\\\\nA_2=&\\mu'+u'\\lambda\\Gamma_{11}^2+(v'\\lambda+u'\\mu)\\Gamma_{12}^2+v'\\mu\\Gamma_{22}^2,\\\\\nA_3=& u'\\lambda L+v'\\lambda M+u'\\mu M+v'\\mu N.\n\\end{cases}\n\\end{equation}\n\n\\par\nBut the tangent plane is generated by $\\phi_u$ and $\\phi_v$, hence\n\\begin{eqnarray}\n\\nonumber\nu'\\lambda L+v'\\lambda M+u'\\mu M+v'\\mu N=0,\\\\\n\\nonumber\\frac{\\lambda}{\\mu}=-\\frac{u'L+v'M}{u'M+v'N}.\n\\end{eqnarray}\nSo the tangent vector of $\\gamma(s)$ is given by\n\\begin{equation*}\n\\vec{t}=A_1\\phi_u+A_2\\phi_v.\n\\end{equation*}\nHere we consider that the curvature $\\kappa$ of $\\gamma$ is always positive. The normal vector $\\vec{n}$ is given by\n\\begin{eqnarray}\n\\nonumber\n\\vec{n}&=&\\frac{1}{\\kappa}[A_1'\\phi_u+A_2'\\phi_v+A_1(\\phi_{uu}u'+\\phi_{uv}v')+A_2(\\phi_{uv}u'+\\phi_{vv}v')],\\\\\n\\nonumber\n&=& \\frac{1}{\\kappa}\\{B_1\\phi_u+B_2\\phi_v+B_3\\vec{N}\\},\n\\end{eqnarray}\nwhere $B_1,\\ B_2$ and $ B_3$ are respectively given by\n\n\\begin{equation}\\label{sc5}\n\\begin{cases}\nB_1&= A_1'+u'A_1\\Gamma_{11}^1+(v'A_1+u'A_2)\\Gamma_{12}^1+v'\\Gamma_{22}^1A_2,\\\\\nB_2&= A_2'+u'A_1\\Gamma_{11}^2+(v'A_1+u'A_2)\\Gamma_{12}^2+v'\\Gamma_{22}^2A_2,\\\\\nB_3&= u'A_1L+(v'A_1+u'A_2)M+v'A_2N.\n\\end{cases}\n\\end{equation}\nNow the binormal vector $\\vec{b}$ of $\\gamma(s)$ is given by \n\\begin{equation}\\label{sc6}\n\\vec{b}=\\frac{1}{\\kappa}\\{ (A_1B_2-A_2B_1)\\vec{N}+A_1B_3(F\\phi_u-E\\phi_v)+A_2B_3(G\\phi_u-F\\phi_v)\\}.\n\\end{equation}\\\\\n\\begin{thm}\nLet $\\gamma(s)$ be an unit speed parametrized curve on $S$ whose position vector lies in the tangent plane $T_{\\gamma(s)}S$. Then the following statements hold:\n\\par\n$(i)$ The distance function $\\rho=\\|\\gamma\\|$ is given by $\\rho=\\lambda^2E+2\\lambda\\mu F+\\mu^2G$.\\\\\n\n$(ii)$ The tangential component of the position vector of the curve $\\gamma(s)$ is given by $$\\langle \\vec{t},\\gamma \\rangle=\\lambda A_1E+(\\lambda A_2+\\mu A_1)F+\\mu A_2G.$$ \n\n$(iii)$ The normal component of the position vector is given by $$\\langle \\vec{n},\\gamma \\rangle=\\frac{1}{\\kappa}(\\lambda B_1E+(\\lambda B_2+\\mu B_1)F+\\mu B_2G).$$\n\n$(iv)$ The component of the position vector along binormal vector is given by $$\\langle \\vec{b},\\gamma \\rangle = \\frac{1}{\\kappa}\\{A_1B_3\\mu( F^2-EG)+A_2B_3\\lambda( EG- F^2)\\},$$\nwhere $\\{A_1,\\ A_2,\\ A_3\\}$ and $\\{B_1,\\ B_2,\\ B_3\\}$ are described in equations $(\\ref{sc4})$ and $(\\ref{sc5})$ respectively.\\\\\n\\end{thm}\n\\begin{proof}\nLet $\\gamma(s)$ be an unit speed parametrized curve on $S$ whose position vector lies in the tangent plane $T_{\\gamma(s)}S$ and curvature $\\kappa>0$. Then $$\\gamma(s)=\\lambda\\phi_u+\\mu\\phi_v.$$ Therefore\n\\begin{eqnarray}\n\\nonumber\n\\rho=\\langle\\gamma,\\gamma\\rangle&=&\\langle\\lambda\\phi_u+\\mu\\phi_v,\\lambda\\phi_u+\\mu\\phi_v\\rangle,\\\\\n\\nonumber\n&=&\\lambda^2\\langle\\phi_u,\\phi_u\\rangle+\\lambda\\mu\\langle\\phi_u,\\phi_v\\rangle+\\lambda\\mu\\langle\\phi_v,\\phi_u\\rangle+\\mu\\langle\\phi_v,\\phi_v\\rangle,\\\\\n\\nonumber\n&=&\\lambda^2E+2\\lambda\\mu F+\\mu^2G.\n\\end{eqnarray}\nWhich proves $(i)$.\\par\nThe component of $\\gamma$ along the tangent vector is obtained as\n\\begin{eqnarray}\n\\nonumber\n\\langle\\vec{t},\\gamma\\rangle&=&\\langle A_1\\phi_u+A_2\\phi_v,\\lambda\\phi_u+\\mu\\phi_v\\rangle,\\\\\n\\nonumber\n&=&\\lambda A_1\\langle\\phi_u,\\phi_u\\rangle+\\lambda A_2\\langle\\phi_u,\\phi_v\\rangle+\\mu A_1\\langle\\phi_v,\\phi_u\\rangle+\\mu A_2\\langle\\phi_v,\\phi_v\\rangle,\\\\\n\\nonumber\n&=&\\lambda A_1E+(\\lambda A_2+\\mu A_1)F+\\mu A_2G,\n\\end{eqnarray}\nwhere $A_1,\\ A_2,\\ A_3$ are given in equation $(\\ref{sc4})$. This proves $(ii)$.\\par\nWe also find the component of $\\gamma$ along the normal vector, which is given by\n\\begin{eqnarray}\n\\nonumber\n\\langle\\vec{n},\\gamma\\rangle&=&\\langle \\frac{1}{\\kappa}\\{B_1\\phi_u+B_2\\phi_v+B_3\\vec{N},\\lambda\\phi_u+\\mu\\phi_v\\rangle,\\\\\n\\nonumber\n&=& \\frac{1}{\\kappa}\\{\\lambda B_1\\langle\\phi_u,\\phi_u\\rangle+\\lambda B_2\\langle\\phi_u,\\phi_v\\rangle+\\mu B_1\\langle\\phi_v,\\phi_u\\rangle+\\mu B_2\\langle\\phi_v,\\phi_v\\rangle\\},\\\\\n\\nonumber\n&=&\\frac{1}{\\kappa}(\\lambda B_1E+(\\lambda B_2+\\mu B_1)F+\\mu B_2G),\n\\end{eqnarray}\nwhere $B_1,\\ B_2,\\ B_3$ are given in equation $(\\ref{sc5})$. Hence statement $(iii)$ is proved.\\par\nAgain the component of $\\gamma$ along the binormal vector is given by\n\\begin{eqnarray}\n\\nonumber\n\\langle\\vec{b},\\gamma\\rangle&=&\\langle \\frac{1}{\\kappa}\\{(A_1B_2-A_2B_1)\\vec{N}+A_1B_3(F\\phi_u-E\\phi_v)+A_2B_3(G\\phi_u-F\\phi_v)\\},\\lambda\\phi_u+\\mu\\phi_v\\rangle,\\\\\n\\nonumber\n&=&\\frac{1}{\\kappa}\\{A_1B_3(F\\lambda\\langle\\phi_u,\\phi_u\\rangle+F\\mu\\langle\\phi_u,\\phi_v\\rangle-E\\lambda\\langle\\phi_u,\\phi_v\\rangle-E\\mu\\langle\\phi_v,\\phi_v\\rangle)+A_2B_3(G\\lambda\\langle\\phi_u,\\phi_u\\rangle\\\\\n\\nonumber\n&&-F\\lambda\\langle\\phi_v,\\phi_u\\rangle+G\\mu\\langle\\phi_u,\\phi_u\\rangle-F\\mu\\langle\\phi_v,\\phi_v\\rangle)\\},\\\\\n\\nonumber\n&=&\\frac{1}{\\kappa}\\{A_1B_3\\mu( F^2-EG)+A_2B_3\\lambda( EG- F^2)\\},\n\\end{eqnarray}\nwhere $A_1,\\ A_2,\\ A_3,\\ B_1,\\ B_2,\\ B_3$ are given in equation $(\\ref{sc4})$ and $(\\ref{sc5})$. Thus statement $(iv)$ is proved.\n\\end{proof}\nNow suppose that $\\bar{S}$ is an another surface isometric to $S$ and $f:S\\rightarrow\\bar{S}$ is the isometry. Then the curve $f\\circ\\gamma(s)$ in $\\bar{S}$ is expressed as \n\\begin{eqnarray}\\label{sc7}\n\\nonumber\n\\bar{\\gamma}(s)&=& f\\circ\\gamma(s)=f(\\lambda\\phi_u+\\mu\\phi_v),\\\\\n\\nonumber\n&=&\\lambda f_*(\\phi_u)+\\mu f_*(\\phi_v),\\\\\n&=&\\lambda(\\bar{\\phi}_u)+\\mu(\\bar{\\phi}_v),\n\\end{eqnarray}\nfor some smooth functions $\\lambda$ and $\\mu$. The position vector of $\\bar{\\gamma}$ lies in $T_{f(p)}\\bar{S}$. Hence the isometry $f$ transforms a curve on a surface with position vector in the tangent plane to the same and $\\lambda,\\ \\mu$ does not change under $f$. Hence\n\\begin{eqnarray}\n\\nonumber\n\\frac{u'L+v'M}{u'M+v'N}=\\frac{u'\\bar{L}+v'\\bar{M}}{u'\\bar{M}+v'\\bar{N}},\\\\\n\\nonumber\nu'^2(L\\bar{M}-\\bar{L}M)+v'^2(M\\bar{N}-\\bar{M}N)+u'v'(L\\bar{N}-\\bar{L}N)=0,\n\\end{eqnarray} \nwhere $\\{L, \\ M, \\ N\\}$ and $\\{\\bar{L}, \\ \\bar{M}, \\ \\bar{N}\\}$ are the coefficients of the second fundamental form of $S$ and $\\bar{S}$ respectively.\n\\begin{thm}\nLet $f:S\\rightarrow\\bar{S}$ be an isometry and the position vector of two curves $\\gamma(s)$ and $\\bar{\\gamma}(s)$ lies in $T_pS$ and $T_{f(p)\\bar{S}}$ respectively. Then the following statements hold:\n\\par\n$(i)$ The distance function $\\rho=\\|\\gamma\\|$ is invariant under the isometry. i.e., $\\gamma(s)=\\bar{\\gamma}(s)$.\n\n$(ii)$ The tangential component of the position vector of the curve $\\gamma(s)$ is invariant under the isometry $f$. i.e., $\\langle \\vec{t},\\gamma \\rangle=\\langle \\vec{\\bar{t}},\\bar{\\gamma} \\rangle$. \n\n$(iii)$ \nThe geodesic curvature of $\\gamma$ is invariant under the isometry $f$.\n\\end{thm}\n\\begin{proof}\nLet $f:S\\rightarrow\\bar{S}$ be an isometry. Since $\\phi(u,v)$ is a surface patch for the surface $S$ at $p$, hence $\\bar{\\phi}(u,v)=f\\circ\\phi(u,v)$ is also a surface patch for $\\bar{S}$   . Suppose $\\{E,F,G\\}$ and $\\{\\bar{E},\\bar{F},\\bar{G}\\}$ are the coefficients of first fundamental forms of $\\phi$ and $\\bar{\\phi}$ respectively. Then we have\n\\begin{equation}\\label{i1}\n\\bar{E}=E,\\quad \\bar{F}=F\\quad \\text{and}\\quad \\bar{G}=G.\n\\end{equation} \nDifferentiating first relation of equation $(\\ref{i1})$ we get\n\\begin{equation}\\label{i2}\n\\bar{E}_u=\\frac{\\partial}{\\partial u}(\\bar{\\phi}_u\\cdot\\bar{\\phi}_u)=\\frac{\\partial}{\\partial u}(\\phi_u\\cdot\\phi_u)=E_u.\n\\end{equation}\nSimilarly \n\\begin{equation}\\label{i3}\n\\begin{cases}\n\\bar{F}_u=F_u,\\qquad \\bar{G}_u=G_u,\\qquad \\bar{E}_v=E_v,\\qquad\\\\ \\bar{F}_v=F_v,\\qquad and \\qquad \\bar{G}_v=G_v.\n\\end{cases}\n\\end{equation}\\\\\nNow using $(\\ref{i1})$ we have  \n\\begin{eqnarray}\n\\nonumber\n\\rho&=\\lambda^2E+2\\lambda\\mu F+\\mu^2G\\\\\n\\nonumber\n&=\\lambda^2\\bar{E}+2\\lambda\\mu \\bar{F}+\\mu^2\\bar{G}\\\\\n\\nonumber\n&=\\bar{\\rho}.\n\\end{eqnarray} \nHence the statement $(i)$ is proved.\\par\nSince $\\bar{A_1}$ and $\\bar{A_2}$ are functions of $\\lambda$, $\\mu$ and Christoffel symbols, hence by virtue of the equations $(\\ref{i1}),\\ (\\ref{i2})$ and $(\\ref{i3})$ we see that $\\bar{A_1}$ and $\\bar{A_2}$ are invariant under the isometry $f$ . Thus\n\\begin{eqnarray}\n\\nonumber\n\\langle \\vec{\\bar{t}},\\bar{\\gamma} \\rangle&=&\\bar{\\lambda A_1}\\bar{E}+(\\lambda \\bar{A_2}+\\bar{\\mu} \\bar{A_1})\\bar{F}+\\bar{\\mu} \\bar{A_2}\\bar{G},\\\\\n\\nonumber\n&=& \\lambda A_1E+(\\lambda A_2+\\mu A_1)F+\\mu A_2G,\\\\\n\\nonumber\n&=& \\langle\\vec{t},\\gamma\\rangle.\n\\end{eqnarray}\nThis proves $(ii)$. \\par\nNow \n\\begin{eqnarray}\n\\nonumber\n\\kappa_g &=& \\gamma''\\cdot(\\vec{N}\\times\\gamma'),\\\\\n\\nonumber\n&=& \\gamma''\\{(\\phi_u\\times\\phi_v)\\times(A_1\\phi_u+A_2\\phi_v)\\},\\\\\n\\nonumber\n&=&\\gamma''\\cdot\\{A_1(E\\phi_v-F\\phi_u)+A_2(F\\phi_v-G\\phi_u)\\},\\\\\n\\nonumber\n&=&(B_1\\phi_u+B_2\\phi_v+B_3\\vec{N})\\cdot\\{A_1(E\\phi_v-F\\phi_u)+A_2(F\\phi_v-G\\phi_u)\\},\\\\\n\\nonumber\n&=&B_1A_1(EF-FE)+B_1A_2(F^2-EG)+B_2A_1(EG-F^2)+A_2B_2(FG-GF),\\\\\n\\nonumber\n&=&(B_1A_2-B_2A_1)(F^2-GE).\n\\end{eqnarray}\nSince $\\bar{\\gamma}$ is also a curve whose position vector lies in the tangent plane $T_{f(p)}\\bar{S}$ of $\\bar{S}$, hence $$\\bar{\\kappa}_g=(\\bar{B}_1\\bar{A}_2-\\bar{B}_2\\bar{A}_1)(\\bar{F}^2-\\bar{G}\\bar{E}).$$ \nSince $\\bar{B_1}$ and $\\bar{B_2}$ are the functions of $\\bar{A_1}$, $\\bar{A_2}$, $\\bar{A_1'}$, $\\bar{A_2'}$ and Christoffel symbols, so by using the equations $(\\ref{i1}),\\ (\\ref{i2})$ and $ (\\ref{i3})$ we can say that $\\bar{B_1}$ and $\\bar{B_2}$ are invariant under the isometry $f$. In view of $(\\ref{i1})$, the last equation yields $$\\bar{\\kappa}_g=\\kappa_g,$$\nwhich proves $(iii)$.\n\\end{proof}\n\n\\section{acknowledgment}\n The second author greatly acknowledges to The University Grants Commission, Government of India for the award of Junior Research Fellow.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\nDespite the apparent cross-disciplinary interactions among scientific fields, a\nformal description of their evolution is lacking. Here we describe a novel\napproach to study the dynamics and evolution of scientific fields using a\nnetwork-based analysis. We build an {\\it idea} network consisting of American\nPhysical Society Physics and Astronomy Classification Scheme (PACS) numbers as\nnodes representing scientific concepts. Two PACS numbers are linked if there\nexist publications that reference them simultaneously. We locate scientific\nfields using a community finding algorithm, and describe\nthe time evolution of these fields over the course of 1985-2006. The communities we identify map\nto known scientific fields, and their age depends on their size and\nactivity. We expect our approach to quantifying the evolution of ideas to be\nrelevant for making predictions about the future of science and thus help to\nguide its development. \n\n\\section*{Introduction}\n\nCross-fertilization between different scientific fields has been recognized for\nits ability to encourage new developments and innovative thinking. For this\nreason, multidisciplinary approaches to research are becoming more popular.\nSome recent examples include applying physics techniques to the study of\nbiological phenomena \\cite{frauenfelder}, deriving an understanding of the\nnature of critical phenomena from renormalization techniques in particle\nphysics \\cite{wilson} drawing inferences about the early universe from findings\nin terrestrial superfluid experiments \\cite{zurek}, and using statistical\nphysics to analyze technological and social systems \\cite{dorogovtsev}. \n\nIn an effort to move beyond anecdotal evidence of the benefit of\ninterdisciplinary discourse for science, in this paper we study the dynamics of\ngroups, or ``communities\", of ideas using a statistical physics approach.  We\nattempt to quantify the evolution of ideas and subdisciplines within physics as\nthey emerge, interact, merge, stagnate, and desist. The quest for describing\nthe development of scientific fields is not new. There have been\nepidemiological \\cite{goffman, tabah} and network-based approaches (citation\nand collaboration networks)\n~\\cite{price,newmancoauth1,newmancoauth2,newmancoauth3,lehmann,borner,leydesdorff,bollen1,boyack} aiming\nto gain insight into the spread of scientific ideas. Recently the temporal\nevolution of several scientific disciplines have been modeled with a\ncoarse-grained approach \\cite{luis}. \n\nHere we build a scientific concept network consisting of American Physical\nSociety PACS numbers as nodes representing scientific concepts.   The American Institute of Physics (AIP) develops and maintains the PACS scheme as a service to the physics community in aiding the\nclassification of scientific literature and information retrieval.  Two PACS numbers are linked if there\nexist publications that reference them simultaneously. Our approach differs\nfrom previous methods in that it provides a direct, unsupervised description of\nscientific fields and uses techniques such as community finding and tracking\nfrom the field of network physics. This approach provides means to quantify how\nideas and movements in science appear and fade away.  Because this method makes\nit possible to measure the current and past state of the relationship between\nscientific concepts, it may also help to make predictions about the future of\nscience and thus inform efforts to guide its development. In this paper, we\nentertain some of the quantitative questions that this method permits;\nspecifically, we seek to answer questions about the relationship between size,\nlifetime, and activity of scientific fields.  \n\n\nVarious local to global topological measures have been introduced to unveil the\norganizational principles of complex networks\n\\cite{albertrev,newmanbook,caldarelli}. One such measure that allows the\ndiscovery of organizational principles of networks is community finding.  There\nhave been a number of methods to find the communities in networks which\ndescribe the inherent structure or functional units of a network\n\\cite{newmangirvan, palla1, clauset, gulbahce}. One of these is CFinder, a\nclique percolation method (CPM) introduced by Palla et al. \\cite{palla1}, which finds\noverlapping communities and is especially suitable for studying the evolution\nof scientific fields since scientific concepts are often shared among multiple\nfields. We use this CPM to track the evolution of physics.\n\n\\section*{Results}\n\n\\noindent{\\bf Building the Network} \n\n\nData were collected from the American Physical Society's (APS) {\\it Physical\nReview} database from 1977-2007. Journals included in the study are {\\it\nPhysical Review Letters}, {\\it Physical Review \\{A} through {\\it E\\}}, and {\\it\nPhysical Review Special Topics: Accelerators and Beams}. Papers in this\ndatabase contain a list of author-assigned PACS codes, where each PACS code\nrefers to a specific topic in physics.  PACS itself is hierarchical, which is\nevident in the structure of the codes with up to 5 levels of topic\nspecification.  For example the PACS code `64.60.aq' has 5 levels where the\nfirst digit `6' represents the first level (in this case `condensed matter'),\n`4' represents the second (e.g. `equations of state, phase equilibria, and\nphase transitions'), the third and fourth digits `60' together represent the\nthird level (e.g., `general studies of phase transitions') while the last two\ncharacters `aq' carry information pertaining to the fourth and fifth levels of\nspecification (e.g. `specific approaches applied to phase transitions' and\n`networks', respectively).  \n\n\nPACS codes are not static, rather,  the coding scheme is periodically updated with the addition and deletion of codes.  In order to (at least partially) account for this effect, the scientific concept network was constructed such that the nodes in the network represent individual PACS codes using the first four\ndigits of specification, where changes to scheme are less probable. This network and the related material is available on\nour website \\cite{datalink}.  In our network, an {\\it edge} occurs between two\nnodes if the two PACS codes they represent are cited in the same paper; one\npaper in the database often contributes many nodes and edges to the\nnetwork. Furthermore, edges are weighted by the number of papers that contain\nthat edge. We introduce two measures, node and edge cutoffs, to control for noise \nin the network (see Methods section).\n\nThe entire PACS network from 1977-2007 after both noise measurements were\napplied has 803 nodes and 23707 distinct edges.  The \\emph{degree} of a node is\nthe number of edges shared by the node.  The weighted cumulative degree\ndistribution follows a stretched exponential with the form,  $P(k)\\sim\\\n\\mbox{exp} [-(k\/842)^{0.53}]$ as shown in Fig. \\ref{DegreeFig}A. The\ndistribution has a similar form in the unweighted case.  The dynamic\nclassification scheme of the American Physical Society, implemented by the\naddition, splitting and removal of codes, may be preventing the formation of\nlarge hubs, thus keeping the specification of the codes more useful. The\nstretched exponential distribution may be the result of a sublinear-linear\nattachment type growth \\cite{krapivsky}.\n\n\nThe PACS network also exhibits a weak but apparent hierarchical structure\nmeasured by the dependence of the \\emph{clustering coefficient} on (unweighted)\ndegree.  For a node $i$, the clustering coefficient is given as $C_i =\n2n_i\/k_i(k_i-1)$, where $n_i$ is the number of edges that link the neighbors of\nnode $i$, and $k_i$ is the degree of the node. The clustering coefficient for a\nnode is the ratio of the number of triangles through node $i$ over the possible\nnumber of triangles that could pass through node $i$ \\cite{barabasi}.  A purely\nhierarchical network will have a $\\langle C \\rangle$ that scales as a power of\n$k$, $\\langle C \\rangle \\sim k^{-1}$, while a random network will have a\nclustering coefficient that is constant with $k$ \\cite{barabasi}.  For this\nnetwork,  $\\langle C(k) \\rangle \\sim k^{-0.29}$ , shown in Fig. \n\\ref{DegreeFig}B. This dependence is not surprising given the hierarchical\nstructure of the classification scheme.\n\n\\noindent{\\bf Defining Communities in Physics} \n\n\nPapers published between 1985 and 2006 were used to study the community\nevolution of the network; 1985 appears to be the first year when all journals present ({\\it Physical Review E} began publication in 1993)\nconsistently used the PACS data scheme, and 2007 was thrown out to exclude\nincomplete data from the analysis. The journal {\\it Physical Review Special\nTopics: Accelerators and Beams} was not included because of an irregular\npublishing schedule. After the noise measures were carried out, the edge\nweights were no longer used, and the network became an unweighted network with\nrespect to the community evolution analysis. The data were organized into 44\ntime bins, with each bin representing a 0.5 year time period. Once a paper (and\nthe edges and nodes it contains) appears in the analysis, it is assigned a\nlifetime, $l$, of 0 or 2.5 years. This assignment is an attempt to more\nrealistically capture the nature of scientific dissemination, as well as the\ndelay in time from publication to assimilation by the field. The analysis of\ncommunity evolution begins at the time bin subsequent to the lapse of the\nassigned lifetime.  Thus the first time bin, $t=0$, for a paper lifetime of\n$l=2.5$ refers to the latter half of 1987 since we start the analysis in 1985.  \n\nIn order to study the evolution of different fields in physics, one must first\nfind these fields in our network. We hypothesize that scientific fields are\nrepresented by communities in our PACS network. These communities are found\nusing the CFinder algorithm, which is based on a clique percolation\nmethod\\cite{palla1}.  Figs. \\ref{comdiag} and \\ref{comdiaga}  present\nexamples of the community structure extracted utilizing CFinder.  \n\nFor each community, the code (using only the first two digits) that encompasses\nthe largest fraction of nodes in the community was found. Its name, specified by\nthe PACS scheme, is then used to label the community.  If a community has\nmultiple codes which compose the same largest fraction of nodes in that\ncommunity, then the community is assigned multiple labels. As shown in\nFigs.~\\ref{comdiag} and ~\\ref{comdiaga}, we observe that the analysis captures \nexpected scientific connections among fields in physics.  For example, in 1997, particle physics is linked to both general relativity and astrophysics.  It is also worthwhile to note the emergence of biophysics as a community in the 2005 analysis.  \\\\ \n\n\\noindent{\\bf Community Evolution and Dynamics} \n\n\nIn order to track the evolution of scientific fields, after identifying\ncommunities at each individual time interval, it is necessary to match the\ncommunities between adjacent time steps. We implemented a community evolution\nalgorithm developed by Palla et al.~\\cite{palla2} to match the communities\nbetween time bins (see Methods section).\n\nTo gain a better understanding of the dynamics of evolving communities, we\ndefined two properties of each community: size and activity. A value for each of\nthese measures can be assigned to every community for each individual time bin.\nThe size $s$ of a community is the number of nodes contained within that\ncommunity at time $t$. Size can be interpreted as a measure of a community's\nbreadth: communities with a small size encompass only a few distinct ideas,\nwhile large communities encompass many distinct ideas.  (The cumulative size distribution was calculated for different times and is displayed in Fig. S1.)\n\n\nThe activity $\\alpha$ of a community is defined as the number of papers that\ncontain at least one node from that community at time $t$. As one expects,\nthere is a strong correlation between size and activity (see Fig. S2).\n\nNext, we study the relationship between the age or lifetime of a community\nversus its size and activity. The age of a community at time $t$ is simply the\nnumber of time bins the community has been present in the evolution analysis:\n$\\tau=t-t_0+1$, where $t_0$ is the time bin in which the community was born. In\norder to study the dependence of age on size, in each time bin, the current age\n$\\tau$ and size $s$ are recorded. Using all communities from all time\nintervals, the median age is calculated for communities with the same size as\nshown in Fig.  \\ref{fig2}A.  There is a trend of $\\tau$ increasing with\nsize $s$.  Thus, it would appear that older communities tend to contain more\nnodes, and that longer lived fields tend to encompass many distinct ideas.   Values for both the Pearson correlation coefficient, $p$, and the Spearman's rank correlation coefficient, $\\rho$, were calculated between $\\tau$ and $s$ using the raw, unbinned data.  $\\rho=1-6\\sum_i\\frac{d_i^2}{N(N^2-1)}$ where $N$ is the number of data points and $d_i$ is the difference in the statistical rank of the corresponding values for each data point.  For $l=2.5$, the Pearson correlation coefficient was $p=0.4772$ while the  Spearman's $\\rho$ was calculated to be $ \\rho=0.5913$.\n\n\nIn order to measure the dependence of age on activity, the current age $\\tau$\nis recorded along with the current activity $\\alpha$ of every community in each\ntime step. Because of the wide range of possible values for activity and noise\nin the data, the values of $\\alpha$  are sorted into 100 equally sized bins.\nThe median age is calculated for all communities within the same activity\ninterval. There is a trend of $\\tau$ increasing with activity as shown in\nFig.  \\ref{fig2}B which can be partially understood by the strong correlation\nbetween size and activity. Further we note an apparent phase transition in\nactivity; as shown in Fig.  \\ref{fig2}B after some critical value, communities\ntend to be longer lived.   This transition also appears for $l=0$ (see Fig. S3).  Lifetime as a function of size, $\\tau(s)$, for $l=0$ is shown in Fig. S4. Again, the Pearson correlation coefficient and the Spearman's rank correlation coefficient were calculated for $l=2.5$ using the raw, unbinned data between $\\tau$ and $\\alpha$, with $p=0.3283$ and $\\rho=0.3764$. \n\n\n\\section*{Discussion}\n\n\nIn this paper, we have developed an approach that enables the quantitative\nstudy of the evolution of physics fields, specifically by following the\ndynamical connections between various ideas within physics. From our\ninvestigation, we have shown that long lived communities tend to be larger, and\nare associated with a higher number of papers. \n\nOur approach opens up an interesting possibility of being able to predict\ncommunity dynamics and impact from the current network structure. Furthermore,\nthis method can be easily adapted to other scientific fields using different\ndatabases. One such is the INSPEC database which has comprehensive coverage of\nresearch activity in computer science and engineering in addition to physics,\nand has an expert-assigned classification scheme rather than author-based\nassignments. \n\n\n\\section*{Materials and Methods}\n\\subsection*{Noise Measures }\nA node cutoff is introduced such that in a given time interval a node must\nappear at least twice to be included in the network. This measure eliminates\nmany of the typographical errors occurring in the database. The edge cutoff,\nhowever, takes into account the random expectation of two PACS codes\nco-occurring in the same paper. For this cutoff, the weight of an edge between\nnodes $i$ and $j$, $W_{ij}$, which is the number of papers that both codes $i$\nand $j$ appear in, is compared to the weight expected at random, $E_{ij}=n_i\nn_j\/N$, where $n_i$ and $n_j$ are the number of papers containing nodes $i$ and\n$j$ respectively, and $N$ is the total number of papers present in the time\ninterval.   If $W_{ij}\/E_{ij} > 1.2$, then the appearance of the edge is\nsignificant compared to random appearance, and we include it in the network.\\\\ \n\\subsection*{CFinder} \nThe CFinder algorithm is described in detail in Ref.\n\\cite{palla1}.  A community is defined as a union of all $k$-cliques (complete\nsubgraphs of size $k$) that can be reached from each other through a series of\nadjacent $k$-cliques (where adjacency means sharing $k - 1$ nodes)\n\\cite{palla1}.  \\\\ \n\\subsection*{Picking a $k$ value}\nFor this study, $k=9$ was principally used (for l=2.5) because it appears to produce a\nlarge number of communities while discouraging the formation of giant\ncommunities. Further, by keeping $k$ constant, we keep the resolution constant\nfor the entire analysis. Picking an appropriate $k$ value for the analysis is\ndone by considering two properties: the number of communities present, and the\npresence of overly large communities \\cite{palla1}. It is desirable to have a\nlarge number of communities, so as to increase the statistical quality of\nmeasurements made on the network. Fig.  S6 plots the number of\npresent communities for each time step for  $k=8, 9$, and $10$, for $l=2.5$. As\ndemonstrated, the number of communities found using the choice of $k=10$ tends\nto be less than the other parameter choices, making it less favorable in terms\nof improving statistical quality.\n\nA $k$ value must also be large enough to avoid the introduction of overly large\ncommunities that obscure the actual community structure of the network\n\\cite{palla1}. To quantify this property, we use the quantity $r$ which is the\nratio of the size of the largest community to the second largest community for a\ngiven time bin.  Thus while some distribution in the sizes of communities is\nnecessary, $r$ should not be overly large.   Fig.  S7 plots the measure\n$r$ against all time bins for $l=2.5$.  For $k=8$, the values of $r$ tends to be larger than (signifying giant communities) than those calculated from the other two parameter values, making it an unfavorable parameter choice. \n \n\\subsection*{Community Matching}\nThe community matching algorithm is described in detail in Ref. \\cite{palla2}.\nIn this analysis, an appropriate $k$-value is used rather than a constant\nedge-weight cutoff.  A running stationarity measure is described in Appendix S1 and Figure S5.  The merger of two communities is described in Appendix S1 and Figures S8 and S9.\n\\section*{Acknowledgments} \nThe authors would like to thank the American Physical Society and the American\nInstitute of Physics for the use of their data. The authors would like to\nacknowledge Gergely Palla, Sune Lehmann, Albert-L\\'aszl\\'o Barab\\'asi, Tam\\'as\nVicsek, Aric Hagberg, Hristo Djidjev, Luis Bettencourt, and Michael Ham for\nuseful discussions. \n\n\n\\section{Community Dynamics}\nOnce community structure is established, a variety of different measurements are performed on the dynamics of the evolving communities.\n\nThe cumulative community size distribution appears long tailed over one decade, which is robust as a function of $t$ (years), as shown in Fig. S\\ref{cumulsize}.\n\nFig. S\\ref{svsa} plots for all time intervals the size of every\ncommunity against its activity $\\alpha$ for paper lifetime $l=2.5$.  There appears to be a positive correlation between the two measures, and this trend is observed for $l=0$ (not shown).\n\n\nThe dependence of age on activity was measured and Fig.~S\\ref{tauaval0} shows the results for the $\\tau$\nvs. $\\alpha$ measurements for $k=7$ with a paper lifetime of $l= 0$ years, where $\\alpha$ values were binned because of the wide\nrange in $\\alpha$, as well as to reduce noise.  In both cases  (the one presented here and the one presented in the letter) there is a trend of age increasing as activity increases (though less apparently for $l=0$) and, as expected given the correlation between $\\alpha$ and $s$, one sees a similar relationship between age and size, as shown in Fig. S\\ref{tausval0}.  Thus, older communities also tend to encompass more publications, a result that\nagrees with naive expectation. Further, we note an apparent  phase transition in\n both paper lifetime cases (more apparent for l=2.5 than for $l=0$); after some critical $\\alpha$, communities tend to be\nlonger lived. \n\n\n\nFurther understanding of the community dynamics can be gained by studying the\nvolatility of the evolving communities, a measure of how much communities tend\nto change between subsequent time steps.  To see this, we define an \\emph{age dependent}\nrunning stationarity $\\xi(\\tau)$ based on community correlation and stationarity\npresented by Palla et al.~\\cite{palla2}. The correlation $C(t,t')$ between two\nstates of the same community $A(t)$ at times $t$ and $t'$ is \n\\begin{equation*} \nC(t,t')=\\left|\\frac{A(t) \\cap A(t')}{A(t) \\cup A(t')}\\right|.\n\\end{equation*} \n\n\\noindent Then the running stationarity, $\\xi(\\tau)$, of that community is the\naverage correlation between subsequent time steps up to age $\\tau$, \n\\begin{equation*}\n\\xi(\\tau)=\\frac{1}{\\tau-1}\\sum_{t'=t_0}^{t_0+\\tau-2}C(t',t'+1). \n\\end{equation*}\n\nThe running stationarity, $\\xi(t)$, is plotted against lifetime for every\ncommunity with $\\tau>1$ along with its current age at time $t$, for all $t$,\nfor $l=0$ in Fig. S\\ref{run_statl0}. This result is qualitatively similar to results obtained using\nrandomized correlations.  For larger values of $l$, the distribution shifts to\nlarger values of $\\xi(t)$.\n\n\\section{Picking a $k$ value}\n\nThroughout the paper, $k=9$ is principally used (for l=2.5) because it appears to produce a\nlarge number of communities while discouraging the formation of giant\ncommunities. Further, by keeping $k$ constant, we keep the resolution constant\nfor the entire analysis. Picking an appropriate $k$ value for the analysis is\ndone by considering two properties: the number of communities present, and the\npresence of overly large communities \\cite{palla1}. It is desirable to have a\nlarge number of communities, so as to increase the statistical quality of\nmeasurements made on the network. Fig. S\\ref{commnum} plots the number of\npresent communities for each time step for  $k=8, 9$, and $10$, for $l=2.5$. As\ndemonstrated, the number of communities found using the choice of $k=10$ tends\nto be less than the other parameter choices, making it less favorable in terms\nof improving statistical quality.\n\nA $k$ value must also be large enough to avoid the introduction of overly large\ncommunities that obscure the actual community structure of the network\n\\cite{palla1}. To quantify this property, we use the quantity $r$ which is the\nratio of the size of the largest community to the second largest community for a\ngiven time bin.  Thus while some distribution in the sizes of communities is\nnecessary, $r$ should not be overly large.   Fig. S\\ref{rfig} plots the measure\n$r$ against all time bins for $l=2.5$.  For $k=8$, the values of $r$ tend to become larger (signifying giant communities) than those calculated\nfrom the other two parameter values, making it an unfavorable parameter choice. \\\\\n\n\\section{Merging of communities}\nLastly, we present an example of a merger between two communities.  Tracking\na nuclear physics community, Fig. S\\ref {ctrack} shows the size of\nthat community as a function of time for $k=9$ and a community of similar nodes with $k=10$, using $l=2.5$.\n\nWith $k=9$, it appears that this particle physics community abruptly dies at\n$t=4$ years.  Increasing the cohesiveness of communities by increasing to $k=10$\ndemonstrates that a community composed of similar nodes continues to propagate\npast this time of apparent death. Thus, it seems that the nuclear physics community is still present in the network, but has become absorbed by another\ncommunity.  \n\nFig. S\\ref{merge} plots a community at $t=4$ years with the nodes from\nthe nuclear physics community displayed in green.  We can assign a label\nto this community in the usual manner using the nodes present just before the\napparent death of the nuclear physics community. Doing so, the absorbing\ncommunity is comprised of the `physics of elementary particles and fields: specific reactions and phenomenology' in the time bin prior to its absorption of the particle physics community.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\nTime reversal and inversion are two of the most important symmetries for the formation of Cooper pairs in a superconductor. Interesting new features of superconductivity arise when one or both of them are missing. One intriguing phase in the absence of time reversal symmetry is the so-called Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state - a spatially modulated superconducting condensate induced by the spin-splitting of the Fermi surface due to high magnetic fields.~\\cite{PhysRev.135.A550,SovPhysJETP.20.762} This phase has been discussed in many contexts, for condensed matter systems~\\cite{Nature.425.51,Bianchi_FFLO} and ultra-cold atomic gases,~\\cite{Liao} as well as in nuclear physics.~\\cite{RevModPhys.76.263} The other interesting class is represented by superconductors without inversion center in the crystal lattice. These so-called non-centrosymmetric superconductors are also characterized by spin-splitting of the Fermi surface, in this case due to anti-symmetric spin-orbit coupling, and form Cooper pairs of mixed parity.~\\cite{Springer} Interestingly, both types of situations have been recently discussed in the context of Ce-based heavy Fermion superconductors. A phase observed in CeCoIn$_5$ at low-temperatures and rather high magnetic fields is considered a possible candidate for an FFLO state.~\\cite{Nature.425.51,Bianchi_FFLO,JPSJ.76.051005,kumagai2011} Most striking properties of unusually high upper critical fields have been reported for CeIrSi$_3$ and CeRhSi$_3$ which both have a non-centrosymmetric crystal lattice.~\\cite{Springer} In our study we would like to consider a combined effect of spin-splitting through magnetic fields and spin-orbit coupling in an artificially structured heavy Fermion superconductor based on CeCoIn$_5$. \n\n\nMizukami and his coworkers succeeded recently in fabricating superlattices of \nCeCoIn$_5$ and YbCoIn$_5$, the former a heavy Fermion and the latter an ordinary metal.~\\cite{Nat.Phys.7.849} Surprisingly this system is superconducting, if the number of CeCoIn$_5$ layers within a unit cell exceeds two. In view of the fact that the upper critical field remains amazingly high despite the low transition temperature,~\\cite{Nat.Phys.7.849} the superconductivity in the superlattice is \nrobust against paramagnetic depairing effect. \nIt was proposed that the superstructure of this material would have locally \nnon-centrosymmetric features while having global inversion centers and may \nlead to unusual properties of the superconducting phase.~\\cite{Maruyama-Sigrist-Yanase} \nActually the multilayered structure would induce a spatially dependent Rashba-type of \nspin-orbit coupling~\\cite{JPSJ.79.084701} which would influence the spin structure of \nthe electronic bands and suppress the paramagnetic depairing effect.~\\cite{Maruyama-Sigrist-Yanase} \nRecent measurement of the angular variation of the upper critical field found evidence \nfor such a behavior.~\\cite{Goh}\nIn this study, we investigate an unusual form of the superconducting order parameter induced by \nthe modulated Rashba spin-orbit coupling in the magnetic field. \nAs it will turn out to involve a modulation on the length scale of lattice constant, \nthe phase may be viewed as a pair-density wave (PDW) state. \nHere we will restrict to models of bi- and tri-layer systems as depicted schematically in Fig.~\\ref{fig:fig1}.\n\\begin{figure}[htbp]\n  \\includegraphics[width=85mm]{fig1.eps}\n  \\caption{(Color online) Schematic figures of (a) bi- and (b) tri-layer system.\n            Thick bars describe the two dimensional conducting planes \n            which are coupled with each other through the inter-layer coupling $t_{\\perp}$. \n            The structures of modulated Rashba spin-orbit coupling and \n            superconducting order parameters in BCS and PDW states are  \n            shown in the right hand side of each figure. See the text for details. \n    \\label{fig:fig1}}\n\\end{figure} \n\n\\section{FORMULATION}\nFor our discussion we analyze the following model of a multilayered system having the inter-layer hopping and Rashba spin-orbit coupling, \n\\begin{eqnarray}\nH&=&\\sum_{{\\bm k},s,m}\\xi({\\bm k})c^\\dagger_{{\\bm k}sm}c_{{\\bm k}sm}\n~+t_\\perp \\sum_{{\\bm k},s,\\langle m,m'\\rangle} c^\\dagger_{{\\bm k}sm}c_{{\\bm k}sm'} \\nonumber \\\\\n&&+\\sum_{{\\bm k},{\\bm k}',m}V({\\bm k},{\\bm k}')c^\\dagger_{{\\bm k}\\uparrow m}c^\\dagger_{-{\\bm k}\\downarrow m}\nc_{-{\\bm k}'\\downarrow m}c_{{\\bm k}'\\uparrow m} \\nonumber\\\\\n&&+\\sum_{{\\bm k},s,s',m}\\alpha_m {\\bm g}({\\bm k})\\cdot {\\bm \\sigma}_{ss'}c^\\dagger_{{\\bm k}sm}c_{{\\bm k}s'm} \n\\nonumber \\\\\n&&-\\mu_{\\rm B}H\\sum_{{\\bm k},s,m}sc^\\dagger_{{\\bm k}sm}c_{{\\bm k}sm}, \n\\label{eq:eq1}\n\\end{eqnarray}\nwhere \n$m$ is the index of layers. Note that we restrict here to one set of superconducting layers and ignore the \nnormal metal part of the superlattice, assuming that the coupling\nbetween heavy Fermion and normal metal part is small.\nThis is a reasonable approximation for the superlattice of CeCoIn$_5$, \nsince the large mismatch in the Fermi velocities between CeCoIn$_5$ and YbCoIn$_5$ at the interface suppresses the proximity effect.~\\cite{She-Balatsky}\nFor the band structure we assume a simple nearest-neighbor hopping tight binding form \n$\\xi({\\bm k}) = -2t (\\cos k_x + \\cos k_y) -\\mu$ on a square lattice with the chemical potential $\\mu\/t = 2$, \nand a small inter-layer hopping $t_\\perp\/t =0.1 $. The symbol $\\langle m,m'\\rangle$ denotes the summation over nearest-neighbor layers. It is important to note that the following results do not qualitatively depend on the details of the band structure. We also use an $s$-wave superconducting state by assuming $V({\\bm k},{\\bm k}')=-V$ (onsite attractive interaction) for simplicity. Although CeCoIn$_5$ is believed to be a $d$-wave superconductor and would have a small odd-parity component admixed with the local non-centrosymmetricity, the features found below \nare independent of the pairing symmetry as long as spin-singlet pairing is predominant.\nIn fact, we have confirmed that the model for $d$-wave superconductivity gives \nthe qualitatively same results (see Appendix). \nHence, for the sake of numerical accuracy, we restrict ourselves to the simpler $s$-wave state. \nOur choice $V\/t=1.7$ gives rise to the critical temperature \n$k_{\\rm B}T_{{\\rm c}0}\/t = 0.0255$ \nfor both bi-layer and tri-layer systems with $\\alpha=0$, ignoring phase fluctuations. \n\nFor our purpose it is sufficient to choose a $g$-vector with Rashba spin-orbit structure, \n${\\bm g}({\\bm k})=(-\\sin k_y,\\sin k_x,0)$ without going into microscopic details of the \ncompound.~\\cite{JPSJ.77.124711}\nThe coupling constants $\\alpha_m$ are layer dependent and antisymmetric with respect to \nreflection at the center of the multilayer structure. In this way the existence of a global inversion center is guaranteed. For example, $(\\alpha_1,\\alpha_2)=(\\alpha,-\\alpha)$ for bi-layers, \nand $(\\alpha_1,\\alpha_2,\\alpha_3)=(\\alpha,0,-\\alpha)$ for tri-layers (see Fig.~\\ref{fig:fig1}). \nParamagnetic limiting and also eventually the PDW phase enter through the Zeeman coupling term, i.e. the last term in Eq.~(\\ref{eq:eq1}).\n\nIn our discussion we restrict to magnetic fields along {\\it c}-axis \nand analyze the model on the basis of the Bogoliubov-de Gennes equation. \nWhile we include the layer dependence of order parameter $\\Delta_m = \n- \\sum_{{\\bm k}} V \\langle c_{-{\\bm k}\\downarrow m}c_{{\\bm k}\\uparrow m} \\rangle$, \nwe ignore the spatial modulation in the plane, such as the vortex state. This simplification \nis not crucial on a qualitative level, if we assume short coherence lengths and a large Ginzburg-Landau parameter\n$ \\kappa $, the ratio of coherence length and London penetration depth.  \nAlthough phase fluctuations may play an important role for the critical temperature, \nwe focus on the low-temperature and high-magnetic field phase, and ignore this aspect.\n\n\\section{SUPERCONDUCTING PHASES}\n\\subsection{Bi-layer system}\nFirst, we address the results for the bi-layer system. \nFigure~\\ref{fig:fig2} shows the $T$-$H$ phase diagram for several values of $\\alpha\/t_\\perp$.\n\\begin{figure*}[htbp]\n  \\begin{tabular}{cc}\n    \\begin{minipage}{0.5\\hsize}\n      \\includegraphics[width=70mm]{fig2a.eps}\n    \\end{minipage}\n    \\begin{minipage}{0.5\\hsize}\n      \\includegraphics[width=70mm]{fig2b.eps}\n    \\end{minipage}\n  \\end{tabular}\n  \\begin{tabular}{cc}\n    \\begin{minipage}{0.5\\hsize}\n      \\includegraphics[width=70mm]{fig2c.eps}\n    \\end{minipage}\n    \\begin{minipage}{0.5\\hsize}\n      \\includegraphics[width=70mm]{fig2d.eps}\n    \\end{minipage}\n  \\end{tabular}\n  \\caption{$T$-$H$ phase diagram of bi-layer systems \nfor (a) $\\alpha\/t_\\perp=0$, (b) $\\alpha\/t_\\perp=1$, (c) $\\alpha\/t_\\perp=2$, \nand (d) $\\alpha\/t_\\perp=3$. \nThe solid and dashed lines show the second- and first-order phase transition lines, \nrespectively.\nThe ``BCS'' and ``PDW'' are explained in the text. \nThe temperature $T$ and magnetic field $\\mu_{\\rm B}H$ in Fig.~\\ref{fig:fig2} are \nnormalized by the critical temperature $T_{\\rm c0}$ at $\\alpha\/t_\\perp=0$. \n    \\label{fig:fig2}}\n\\end{figure*}\nFor $\\alpha\/t_\\perp=0$ [Fig.~\\ref{fig:fig2}(a)], we obtain the conventional phase diagram of a\nsuperconductor subject to paramagnetic limiting.~\\cite{JPSJ.76.051005} In this case  \nthe order parameter is the same in both layers having the same phase,\n$(\\Delta_1,\\Delta_2)=(\\Delta,\\Delta)$. This state we call Bardeen-Cooper-Schrieffer (BCS) state\nis stabilized basically by the inter-layer ``Josephson'' coupling. \nThe superconducting phase transition turns first order at low temperatures, \nbecause of the paramagnetic depairing effect. The in-plane FFLO phase with \nan inhomogeneous order parameter in the two-dimensional plane is supposed to\nappear around the first order transition line in Fig.~\\ref{fig:fig2}(a). However, in our\napproach we ignore the possibility of in-plane modulations as mentioned above. \n\n\nWe now turn to the role of the layer-dependent Rashba spin-orbit coupling. \nInterestingly, the order parameter configuration with a sign change, $(\\Delta_1,\\Delta_2)=(\\Delta,-\\Delta)$, \nbecomes more stable at high magnetic fields for $\\alpha\/t_\\perp \\agt  1$. This state is called \nthe pair-density wave (PDW) state in Figs.~\\ref{fig:fig2}(b-d),\nsince the order parameter modulates on length scales of the crystal lattice constant. \nThe PDW state was proposed for the $Q$-phase in the bulk \nCeCoIn$_5$,~\\cite{PhysRevLett.102.207004,aperis2010} \nbut has not been established in microscopic models so far.  \nWith increasing the spin-orbit coupling $ \\alpha $ the PDW state is stabilized in the large parameter range. \nThus, we here find that the PDW state may be realized in multilayer systems having \n``weak inter-layer coupling'' and ``moderate spin-orbit coupling''. \nWe would like to stress that the PDW state is stabilized in the purely superconducting multilayers, \nwhile a similar inhomogeneous superconducting state can be induced by the proximity effect \nin the ferromagnet-superconductor junction.~\\cite{Buzdin}  \n\n\n\nThe stability of the PDW state can be understood on the basis of the spin susceptibility \nand the band structure investigated in Ref.~\\onlinecite{Maruyama-Sigrist-Yanase}. \nIn the absence of spin-orbit coupling the bi-layer system forms two bands, \nthe bonding band and anti-bonding band. Cooper pairs in the PDW state \nwould form by inter-band pairing, in contrast to the BCS state which is an intra-band pairing state.\nObviously, the PDW state would not be stable under these circumstances.  \nOn the other hand, the band structure is described by the two spin-split bands \non each layer for large $\\alpha\/t_\\perp$. In this case, both BCS and PDW states occur as intra-band Cooper pairing states. \nAlthough the zero-field transition temperature of the PDW state\nis lower than the onset of superconductivity (BCS state), \na magnetic field along the $c$-axis suppresses the BCS state \nmore strongly through paramagnetic depairing. Indeed, the PDW state\nhas a larger spin susceptibility than the BCS state.~\\cite{Maruyama-Sigrist-Yanase} \nTherefore, the PDW state is more robust against paramagnetic limiting than the BCS state,\nas shown in Figs.~\\ref{fig:fig2}(b-d). \n\n\n\\subsection{Tri-layer system}\n\\begin{figure}[htbp]\n  \\includegraphics[width=70mm]{fig3.eps}\n  \\caption{$T$-$H$ phase diagram of tri-layer systems for $\\alpha\/t_\\perp=3$.\n    The dash-dotted line shows the crossover field determined by the criterion \n    $\\Delta_{\\rm in} = \\Delta_{\\rm out}$. \n    \\label{fig:fig3}}\n\\end{figure}\nNew aspects can be found for the tri-layer system.\nThe order parameter has now three components, such that we distinguish\nthe BCS state as $(\\Delta_1,\\Delta_2,\\Delta_3)=(\\Delta_{\\rm out},\\Delta_{\\rm in},\\Delta_{\\rm out})$ \nand the PDW state as\n$(\\Delta_1,\\Delta_2,\\Delta_3)=(\\Delta_{\\rm out},0,-\\Delta_{\\rm out})$. \nFor $\\alpha\/t_\\perp\\alt 2$, the phase diagram is qualitatively the same \nas shown in Fig.~\\ref{fig:fig2} for bi-layer systems.  \nThe range of the PDW state is slightly reduced because of the \ncomplete suppression of order parameter in the inner layer. \nIncreasing spin-orbit coupling to $\\alpha\/t_\\perp=3$ yields an intriguing modification of the \nphase diagram (see Fig.~\\ref{fig:fig3}). An further first-order phase transition line\nappears within the BCS state, which is absent for bi-layer systems. \n\n\nIn order to characterize the first order phase transition in the BCS state, \nwe show the magnetic field dependence of the order parameter in Fig.~\\ref{fig:fig4}.\n\\begin{figure}[htbp]\n \\includegraphics[width=70mm]{fig4a.eps}\n \\includegraphics[width=70mm]{fig4b.eps}\n \\caption{(Color online) Magnetic field dependence of the order parameters \n in the inner layer $\\Delta_{\\rm in}$ (solid lines) and outer layers $\\Delta_{\\rm out}$ (dashed lines) \n for $\\alpha\/t_\\perp=3$. \n (a) $T\/T_{{\\rm c}0}=0.04$ and (b) $T\/T_{{\\rm c}0}=0.28$. \n \\label{fig:fig4}}\n\\end{figure}\nAt $H=0$, the order parameter of the inner layer $\\Delta_{\\rm in}$ is slightly larger than \nthat of the outer layers $\\Delta_{\\rm out}$. Upon increasing the magnetic field for $T\/T_{{\\rm c}0}=0.04$ at $\\mu_{\\rm B}H\/k_{\\rm B}T_{{\\rm c}0} = 2.04$ an abrupt drop of $\\Delta_{\\rm in}$ occurs, while $\\Delta_{\\rm out}$ is only weakly affected [see Fig.~\\ref{fig:fig4}(a)]. Indeed, with the layer-dependent spin-orbit coupling the paramagnetic limiting behavior is different for the inner and outer layers. The spin-susceptibility in the superconducting phase is more strongly diminished for the inner than the outer layer when $\\alpha\/t_\\perp > 1$, as the Rashba spin-orbit coupling vanishes by symmetry for the inner layer.~\\cite{Maruyama-Sigrist-Yanase} \nThe high-field BCS state is stabilized by the more field resistant order parameter component $\\Delta_{\\rm out}$ of the outer layer. Note that the inter-layer hopping is responsible for the fact that the order parameter components are not behaving independently.  \nAs the temperature is increased, the abrupt first order phase transition in the BCS state \nturns into a crossover at the critical end point $T\/T_{\\rm c0} \\sim 0.12$. In this regime\nthe order parameter of the inner layer decreases continuously, as shown in Fig.~\\ref{fig:fig4}(b). \n\n\\begin{figure}[htbp]\n \\includegraphics[width=70mm]{fig5a.eps}\n \\includegraphics[width=70mm]{fig5b.eps}\n \\includegraphics[width=70mm]{fig5c.eps}\n \\caption{(Color online) DOS in the inner layer $\\rho_{\\rm in}(\\omega)$ (solid lines) \n and outer layers $\\rho_{\\rm out}(\\omega)$ (dashed lines) at $T\/T_{{\\rm c}0}=0.04$.  \n (a) $\\mu_{\\rm B}H\/k_{\\rm B}T_{{\\rm c}0}=0$ (low field BCS state), \n (b) $\\mu_{\\rm B}H\/k_{\\rm B}T_{{\\rm c}0}=2.25$ (high field BCS state), \n and (c) $\\mu_{\\rm B}H\/k_{\\rm B}T_{{\\rm c}0}=3$ (PDW state), respectively.\n The other parameters are the same as Fig.~\\ref{fig:fig3}.\n \\label{fig:fig5}}\n\\end{figure}\nThe different phases also influence the quasiparticle spectrum which we will discuss here looking at\nthe density of states (DOS) layer-resolved for the tri-layer system, as depicted in Fig.~\\ref{fig:fig5} (solid line for inner and dashed line for outer layer). \nThe DOS of the outer layer always shows a gap at the Fermi energy. \nIn the absence of a magnetic field the DOS of both the inner and outer layers are \nessentially identical [see Fig.~\\ref{fig:fig5}(a)]. Differences appear for finite fields. \nIn Fig.~\\ref{fig:fig5}(b) we show the DOS in the high-field BCS state, \nwhich is different for the inner and outer layers. \nThe quasiparticle spectrum of the inner layer is gapless, although the order parameter $ \\Delta_{\\rm in} $ does not disappear. Surprisingly, a small gap on the inner layer opens up again when we enter the PDW state as revealed by Fig.~\\ref{fig:fig5}(c), although the order parameter $ \\Delta_{\\rm in} $ vanishes in this case. The origin of this somewhat counter-intuitive behavior lies in the inter-layer hopping and the energy shifts due to the magnetic field leading to the pair breaking. The inner layer is obviously affected by paramagnetic limiting in the high-field BCS state yielding subgap quasiparticle states. On the other hand, in the PDW state the paramagnetic pair breaking effects almost disappear.~\\cite{Maruyama-Sigrist-Yanase}\nIn this case the intrinsic order parameter of the inner layer vanishes and a small gap appears in \nthe quasiparticle excitations because the quasiparticles are extended over all layers \ndue to the inter-layer hopping. \n\n\n\\section{SUMARRY AND DISCUSSION}\nAs mentioned earlier, possible realizations of the situation discussed here could \nbe found in artificially grown superlattices of ${\\rm CeCoIn_5}$~\\cite{Nat.Phys.7.849} \nas well as in multilayer high-$T_{\\rm c}$ cuprates.~\\cite{Mukuda,Shimizu} \nThese multilayer systems are quasi-two-dimensional superconductors and likely \nnear the Pauli limit as they are rather robust against orbital depairing due to the very short\ncoherence lengths.~\\cite{Nat.Phys.7.849,Obrien}\nConsequently, they satisfy the condition for the phenomenology analyzed in our study. \nIndeed, the reduced paramagnetic limiting due to the Rashba spin-orbit coupling of \nthe outer layers has been confirmed in recent experiments for superlattices ${\\rm CeCoIn_5}$.~\\cite{Goh}\nThe experiment is approaching to the FFLO state for the field along {\\it ab}-plane.~\\cite{Goh} \nOur study shows that the PDW state may be stabilized in the magnetic field along {\\it c}-axis. \nWhile this lies in the measurable range for the ``low-$T_{\\rm c}$'' system CeCoIn$_5$, \nfields revealing the high-field BCS or PDW state are likely beyond 100 T in high-$T_{\\rm c}$ cuprates.    \n\nFinally, we would like to comment on several points of our results. \nFirst, we restricted to pairing interaction in the spin singlet channel. Spin-singlet pairing is essential \nfor the observation of the phase diagram including different BCS states and the PDW state.\nIf spin triplet pairing compatible with the Rashba-spin-orbit coupling would be dominant, \nparamagnetic limiting would not play a role at all, since the spin susceptibility would keep the\nfull value of the normal-state Pauli spin susceptibility for magnetic fields along the \n$c$-axis.~\\cite{Maruyama-Sigrist-Yanase}\nTherefore, no change of phase would be induced by a magnetic field. \nFor the intermediate situation, if spin singlet and spin triplet pairing are \nof comparable strength an intriguing mixing of the two channels is possible leading\nto a complex phase structure involving spontaneous time reversal symmetry breaking. This\nsituation will be discussed elsewhere. \nNote, however, that the pairing interactions in the spin triplet channel are expected to be weak \nin the bulk\/superlattice of ${\\rm CeCoIn_5}$ and high-$T_{\\rm c}$ cuprates. \nEven then, weak interactions in the spin triplet channel gives rise to the uniform \nspin triplet Cooper pairs in the PDW state. \n\n\n\nSecond, we focused on the Pauli-limited superconductor with large Maki parameter, \nneglecting orbital depairing effects, in particular, the aspect of a mixed phase including\nflux lines. Although orbital depairing may play quantitatively important roles for the field along {\\it c}-axis, \nthe PDW state survives moderate orbital depairing effects (moderate Maki parameters) because the spatial modulation of vortex state \nalong the {\\it ab}-plane is not affecting the $\\pi$-phase shift along {\\it c}-axis profoundly. \nThe situation is quite different for the helical superconducting state in Rashba-type \nnon-centrosymmetric superconductors, which is obscured even by the weak orbital \ndepairing effect.~\\cite{PhysRevB.78.220508} \n\n\nThird, the PDW state is stabilized not only in the bi- and tri-layers but also in systems with\nmore layers. In such systems the layers towards the center are subject to paramagnetic limiting\nand it is expected that within the BCS state we encounter a progressive demolishing of superconductivity \nin these layers upon increasing field. Eventually, at high enough magnetic field the superconducting\nsystem behaves like the bi-layer system in the PDW state. Due to the fact that these outer layers\nare well separated the condition for the appearance of the PDW state is less stringent than in the real bi-layer system. These views have been confirmed by us numerically for the 4-layer system.\n\nIn conclusion, we have studied the superconducting state in multilayer systems \nincluding layer dependent antisymmetric spin-orbit coupling due to the coordination of the layers. \nWe found that in a magnetic field along the $c$-axis spin-orbit coupling influences the \nphase of the order parameters of different layers. We find a first order phase transition separating two \nstates different by symmetry: the high-field BCS state which is even under reflection at the center of the multilayer system and the PDW state which is odd under reflection. The latter phase \nallows for a considerable enhancement of the upper critical field and generates the uniform order \nparameter for the spin triplet superconductivity. \nThis phenomenology \nrelies on the fact that orbital depairing is weak, i.e. the coherence length of the superconductor is very short, and that the dominant pairing is in a spin singlet channel. This may apply to the artificially grown multilayer version of CeCoIn$_5$ and some multilayered high-$ T_{\\rm c} $ cuprates. \n\n\\begin{acknowledgements}\nThe authors are grateful to D.F. Agterberg, M. H. Fischer, S. K. Goh, J. Goryo, D. Maruyama, Y. Matsuda, \nT. Shibauchi, and H. Shishido for fruitful discussions. \nThis work was supported by KAKENHI (Grant No. 24740230, 23102709, 21102506 and 20740187). \nWe also are grateful for financial support of the \nSwiss Nationalfonds, the NCCR MaNEP and the Pauli Center for Theoretical Studies of ETH Zurich. \n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION} \n \nHierarchical clustering scenario, including the cold dark matter\n(CDM) model, may be the most established one for \nreconstructing various observational properties in the \ncosmological structure from the galaxies to the clusters of \ngalaxies.\nPress $ \\& $ Schechter (1974) firstly proposed an analytical formalism\nwhich derives the number density of bound virialized objects of the \nmass at any given epoch, with the assumption that the primordial \ndensity fluctuations is random Gaussian field. The mass function\npredicted by the PS theory shows reasonably\nthe good agreement with N-body simulations even if it has more low\nmass objects (e.g. Lacey $\\&$ Cole 1994).   \nTo reconstruct the observational properties in  \ntheoretical galaxy formation scenario,  \nthere are also approaches which study the history of the mass growth for \nbound objects and the characteristic times (e.g. Lacey $ \\& $ Silk 1991; \nKauffmann, White  $ \\& $ Guiderdoni 1993; Cole et al. 1994).\nMost of them were based on the extended PS formalism, which was\nproposed by Bower (1991) and Bond et al. (1991).  It\ncan derive the number density of objects of a certain mass at a \ngiven time subject to a larger object at a later time.  Using the \nformalism, Lacey $ \\& $ Cole (1993; LC) calculated the ``merger'' rate.  \n\nThe PS formalism, however, has a limitation for \ndescribing the history of the mass growth about the individual objects.  \nThe ``merging process'' described with the PS approach in LC, cannot be \ninterpreted as the same meaning of the merger in astronomical sense, \nin which the objects lose their identity.  \nIn the mass growth history for the astronomical objects, \nthe continuous accretion onto a bound object  \nwithout the loss of identity has different meanings from \nthe mass accumulation with the loss of the identity in the major merger.  \nThe formalism based with the\nPS approach cannot imply the distinction between ``tiny'' and ``notable'' \ncaptures.  For solving this problem,\nManrique $ \\& $ Salvador-Sole (1995, 1996) proposed a \nformalism named ``ConflUent System of Peak trajectories'' (CUSP)\nformalism as an extension of adaptive windowing\nby Appel and Jones (1990).  They can be categorized into\na type of ``peak'' theory (Doroshkevch 1970; Adler 1981;\nBardeen et al. 1986, hereafter BBKS), \nwhich can count the number of  \nthe peaks of the density related to \nthe collapsing threshold, applying a\nlow-pass filter of the bound object scale to the fluctuation field.  \nThe CUSP formalism, unfortunately, needs the iterative \ncalculation to estimate the destruction\nand the reformation rates with mergers. This point \nbecomes a disadvantage \nwhen we try to apply this scheme to the semi-analytical\nstudies for the galaxy formation, including\nthe mass accumulation history of bound \nobjects with Monte Carlo method as shown in LC.  \n\nIn order to give analytical description \nfor the destruction and the reformation rate in the merger,\nwe developed a new approach called \n{\\it the skeleton tree formalism}, using the\ntopological characteristics in the smoothing of the random field\nwith Gaussian filter.\nWith the appearance of {\\it\nsloping saddles} in the landscape of the smoothed field,\nwe can pick up the merging events, \nand distinguish the merger and the accretion.\nThe topological feature in the landscape \nwith the sloping saddles can be \nextremely reduced into {\\it skeleton trees}.\n\nIn this paper, we will focus on the concept and the\nbasic description of {\\it the skeleton tree formalism}.  The outline of the\npaper is as follows. In Section 2, \nin order to distinguish the\naccretion and the merging in our\ncontext, we sketch the topological characteristics in the landscape\nof the smoothed field with critical points; the peaks and the sloping saddles,\ndefine the merger events with appearance \nof the sloping saddles,  and reconstruct the growth history of the objects\nwith {\\it skeleton tree} picture.\nIn Sections 3, 4, and 5, we formulate the\nmathematical description of the constraints, the\nprobability distribution functions, and the scale functions\nfor the critical points.   In Section 6, we\nshows the results of the evolution rates with the\naccretion and the merging obtained from {\\it the skeleton tree formalism}\nwith its consistency.\nFinally, we present our conclusion in Section 7.  We have relegated\nthe details of the derivations to five appendices.  \n\n\\section{Identification of Accretion and Merging}\n\n\\subsection{Hierarchical Evolution from Fluctuations and Filtering Process} \n\nWe will express the density fluctuation field \nas the functions of the comoving \nspatial coordinate $ \\mbox{\\bf r} $ and $ \\mbox{\\bf k} $ ; \n\\begin{eqnarray} \n \\delta(\\mbox{\\bf r})= \\int d^3\\mbox{\\bf k} \ne^{i {\\mbox{\\bf  k r}} } \\delta ({\\bf k}) \\; .\n\\end{eqnarray} \nInterested collapse objects of a comoving scale $ R $ \ncan be identified as peaks greater than a threshold, whose \nfields are smoothed with a low-pass filter of the resolution scale $ R $.  \nThe fluctuation, smoothed with the selection function\n$ S(\\mbox{\\bf r};R) $,  \ncan be expressed as \n\\begin{equation} \nF(\\mbox{\\bf r};R) = \\frac{\\int d^3\\mbox{\\bf  r}_0 \nS(\\mbox{\\bf r}_0;R) \n\\delta (\\mbox{\\bf  r}- \\mbox{\\bf  r}_0)} \n{ \\int d^3\\mbox{\\bf  r}_0 S(\\mbox{\\bf  r}_0;R) } \\; . \n\\end{equation} \nThis Fourier transform is represented with the window function which \nis the Fourier transforms of the selection function; \n\\begin{equation}\nF(\\mbox{\\bf k};R) = \\delta({\\mbox{\\bf k}}) \nW(\\mbox{\\bf k};R) \\; .  \n\\end{equation}\nIn our interest cases, the window function works as a low-pass filter.\n\nWe shall restrict ourselves to isotropic homogeneous Gaussian random\nfields with zero mean as descriptions of the initial fluctuations.  \nFor the field, the power\nspectrum is then only a function of $ k = \\vert \\mbox{\\bf k} \\vert $;\n$\\vert \\delta_k \\vert^2 = \\vert \\delta ({\\bf k}) \\vert$.  \nThe fluctuation spectrum filtered with the scale $ R$ is \n\\begin{equation} \nP(k;R)=\\vert \\delta_k \\vert^2 W(k;R)^2 \\; .\n\\end{equation} \nIn this paper, \nwe take a normalized isotropic Gaussian filter:  \n\\begin{equation}\nW(\\mbox{\\bf k};R) = W(k;R) \\equiv e^{ -k^2 R^2\/2 } \\; . \n\\end{equation}\nIn the next subsection, we will\ndiscuss the reason related to its unique property for the\nsmoothing.  \n\nIn the linear theory of gravitational instability\nfor the structure formation, \nthe amplitude of the field in the overdensity area \nfirstly grows in proportion to $ D(t) $, \nwhere $ D(t) $ is the linear growth factor.\nAccording to BBKS, a \nbound object collapses from the area of a comoving scale $ R $\nwhen the density of a peak in the \nfluctuation smoothed over the resolution scale\n$ R $ exceeds above a fixed threshold\n$ \\delta_{c;0} $.  \nInstead of viewing the peaks to be growing in density \namplitude relative to the fixed threshold $ \\delta_{c;0} $,  \nwe can interpret that the threshold level $ \\delta_c $ is decreasing \nas $ \\delta_{c;0} D(t_{0}) D(t)^{-1} $ with fixing \nthe initial fluctuation field $ F(\\mbox{\\bf r};R) $, where \n$\\delta_{c;0}$ was determined from the threshold at the\npresent time $t_0$.  \nIn this paper, we take Einstein-de Sitter\nmodel: $ \\Omega_0 =1, \\lambda_0 =0 $, in which the relative threshold level\n$ \\delta_c \/ \\delta_{c;0}= D(t_{0}) D(t)^{-1} = (1+z) $, where $ z $\nis the redshift at the time $ t $.\n\nFor standard initial fluctuations like CDM models, in general, \nthe rms of the smoothed field $ < F ({\\bf r}; R) ^2 > $ is  decreasing\nas increasing $ R $.   For such a fluctuation,\nas decreasing the collapse threshold with time evolution, \nwe can pick the collapse objects in the larger scale, which\ngives a reasonable sketch of hierarchical clustering picture.\nWe will relate the filtering process and \nthe hierarchical clustering description in the next.  \n\n\n\\subsection{Landscape of the Fluctuation Field in Position \nand Resolution Space}  \n\nConsider the random field in one dimensional (1-D) positional \ncoordinate of $ x $ is smoothed with a low-pass filter of\na resolution scale $ R $.  This field\nis reproduced as a landscape which extends in two dimensional (2-D)\nextended space of $ (x, R) $.\nA smoothing peak with a low-pass filter make a ridge  \nwhich is running along the direction of increasing\n$ R $ in the landscape.\nThe threshold \nlevel of the collapse can be interpreted as an ocean surface which makes a \n``shoreline'' and some ``lakes'' and ``islands'' in the landscape.\nA ``cape'' on the shoreline of $ \\delta_c $, \ncan be identified as a bound object at $ \\delta_c $.   \nThen,  we can count the number of the bound objects of the scale $ R $ \nto pick up the capes at the $ R $ in the landscape.  \nAs the level of the ocean surface of $ \\delta_c $ \nis decreasing with the evolution of the universe, \nthe shoreline moves to the offing.  It means that the bound objects \ngrow their scale continuously.  \n\nHowever, if ``islands'' or ``lakes'' appear in the landscape,\nthey confuse the identification of ``capes'' as bound objects.\nAs an example, let consider \nan island in the offing of a cape on a shoreline. In this case, \nwe have also another offing side cape of the island; there are two \ncapes which can be counted as the bound objects of ``clouds''.  \nIn these situations, the cloud of the island cape  \ncontains the smaller cloud of the cape on the shore.   \nThese problems appear  \nwhen we take general filters except the Gaussian filter.  \nBond et al. (1991) had shown the filter dependency of \nthe landscape ( see their Figure 1 and 2 ), in which they \ntake sharp $k$-space filter and Gaussian filter.\nIn the case of the sharp $k$-space filter, \nthe ridges cannot decrease monotonically as \nthe resolution scale $ R $ increases.  \nOn the other hand,  in the case of the Gaussian filter, the ridge decreases \nmonotonically as the resolution scale $ R $ increases.  \nThey represent about this feature as there is  no ``lake'' of finite \nextent, and the ocean shoreline have no bays.  Another \nproperty of the Gaussian filtering is that the variance is also monotonically \ndecreasing as increasing $ R $.  In general, this picture is also valid in 3-D\nrandom field.  With the help of this feature,\nwe can distinguish the accretion and the merge in the landscape\nproduced from initial fluctuation fields.  \n\n\n\\subsection{Definition of Monotonic Accretion and Merging in the Landscape} \n\nIn order to understand the monotonic evolution of the field smoothed \nwith the Gaussian filter of the\nresolution scale $ R $, we rewrite the derivative of the field as  \n\\begin{eqnarray} \n\\frac{\\partial F(\\mbox{\\bf r}; R)}{\\partial R} = \nR \\nabla^2 F(\\mbox{\\bf r}; R) \\; .  \n\\end{eqnarray} \nThis is identical to a diffusion equation of the variables \n($R^2$, $\\mbox{\\bf r}$).   For all critical points of $ \\nabla^2 F <0 $ \n($ \\nabla^2 F > 0 $) like peaks (holes), \nit guarantees the monotonically decreasing (increasing) as the scale \nincreasing as \n\\begin{eqnarray} \n\\frac{\\partial F(\\mbox{\\bf r}; R)}{\\partial R} < 0  \\; ,    \n(\\frac{\\partial F(\\mbox{\\bf r}; R)}{\\partial R} > 0)  \\; .   \n\\end{eqnarray} \nThis monotonicity of the peak smoothing also \nguarantees that the peak runs continuously on the ridge to \nthe shore cape without the island in the landscape.   \nIt can be reasonable that the smoothing and the merging of peaks\nare defined as the continuous accretion growth and the\nmerging event of bound objects.   \nIf we have islands in the landscape with other filters \nof Gaussian, however, we cannot distinguish the accretion and the \nmerging with the confusion for the scale identification of\nthe related bound objects.  Fortunately,\nwe can exclude this problem\nas long as using the Gaussian filter which guarantees the absence of\nisland in the landscape as shown above.  \nThis is the reason that we take the Gaussian filter in this paper.  \n\nA ridge in the landscape, then, \nrepresents continuous accretion growth of a bound object.\nOn the other hand, some ridges terminate on the slope of\nneighboring ridges.  \nThe vanishing point of the ridge on the \nslope of the neighboring ridge can be defined as a type of critical \npoints.  We call it as {\\it sloping saddle} since\nit is a saddle point on the slope of the neighbor peak.\nThe sloping saddle can represent the \nreasonable feature that a bound \nobject loses the identity loss in the merger, associated with\na tree structure in which the branches of the ridges \nare nested at the junctions of the sloping saddles.\nThen, we can translate the topology in the landscape to the tree structure\nnamed with {\\it skeleton tree}, which consists of\nthe accretion branches and the merger junction picked up with the\nsloping saddles.  \n\n\\subsection{Skeleton Tree Picture}\n\n\\begin{figure}\n\\begin{center}\n\\psbox[scale=0.45]{fig1.eps}\n\\end{center}\n\\caption{ The schematic presentation for the abstraction steps \nfrom the smoothing fluctuation field to the {\\it skeleton tree} picture.\n(a)  Schematic representation of the landscape.  \nThe symbols of $\\odot$ and $\\otimes$ means the peak and the sloping\nsaddle, respectively.  The peak trajectory with filter smoothing\nmakes the ridge.  A ridge terminates at a sloping\nsaddle at $ (R_c , \\delta_c) $ which is associated with a \nridge of the neighboring peak\nthrough the resolution scale interval from $R_c$\nto $R_l$.  (b)  The critical points and the ridges are picked up as\nthe abstract tree of the field around a sloping saddle.\n(c) The tree is presented by the reordering in $ ( R,\\delta ) $ space.\nThe hatched area means the merger.  \n(d) The graphical presentation of the smoothing peaks with\nthe merger and the accretion in {\\it the skeleton tree formalism}.  } \n\\end{figure}\n\nWe will describe the abstraction from the landscape to the tree structure \nas presented in Fig. 1 schematically.\nIn Fig. 1 (a), the local structure of the landscape is represented \nwith contour lines, and the different classes of interest \ncritical points are marked;\nthe peak as $\\odot$ and the sloping saddle as $ \\otimes $, and   \nthe ridges are represented as the dashed lines.  \nIn this example, the sloping saddle appears on a resolution scale \n$ R_c $ at a threshold $ \\delta_c $.\nA ridge terminates at the sloping saddle\n$ \\otimes(R_c, \\delta_c) $.  It means that a bound \nobject of the scale $ R_c $ loses its identity at\nthe threshold $ \\delta_c $.  There is a\nridge neighboring with the sloping saddle, on which the peak has \nthe density of $ \\delta_s $ at the same resolution scale \nof $ \\otimes(R_c, \\delta_c) $.  The peak on the neighboring ridge is\nreforming around the sloping saddle. Then, \nwe can consider that the reformation starts from $ (R_c,\n\\delta_s) $ of a peak neighboring with the sloping saddle.  \nThe merger with the destruction and the reformation  \nstarts from $ \\delta_s $ and ends at $ \\delta_l $ with the scale\nrange from $ R_s $ to $ R_l $ in the case of Fig. 1 (a).  \n\nAs shown in Fig. 1 (b) as an abstraction of Fig. 1 (a), then, \nwe can introduce a tree structure associated with  \nthe sloping saddle $ \\otimes(R_c, \\delta_c) $ and three peaks;  \nthe neighboring peak $ \\odot(R_c,\\delta_s) $ of $ \\otimes(R_c,\n\\delta_c) $ in the same resolution $ R_c$,  \nthe progenitor peak $ \\odot(R_s,\\delta_s) $ on the same ridge of $ \\otimes(R_c,\n\\delta_c) $ at $ \\delta_s $ of the density of\n$ \\odot(R_c,\\delta_s) $, and the reformed peak \n$ \\odot(R_l,\\delta_l) $ on the same ridge of the neighboring\n$ \\odot(R_c,\\delta_s) $.  \n\nFor describing the history of the hierarchical clustering with the merging, \nit is convenient to reorder the tree structure along the \nthreshold level instead with the filtering scale.  Because the \nthreshold level is identical to the time as monotonically mapped\nwith $ \\delta_c=\\delta_{c;0} D(t)^{-1} $.  \nFig. 1 (c) schematically shows this reconstructed feature  \nfrom the Fig. 1 (b).  \nThe continuous accretion growth \nis represented as the dashed lines along the ridges, while \nthe merger is represented as the hatched area in Fig. 1 (c).\nIn this case, the merger occurs during the interval \nbetween $ \\delta_l $ and $\\delta_s $.\nEven if we can follow the neighboring peak on the\nsame ridge over $ \\delta_c $ in the landscape, however, we have two peaks\nat a threshold during the merger after the reordering with the\nthreshold, as shown in the hatched area in Fig 1. (c).\nIt means that the reforming peak on the ridge\nneighboring with the sloping saddle also \nloses its identity with the merger.  Then,\nthe neighboring ridge is divided into two part of a disappearing peak\n$ > \\delta_c $ and a reforming peak $ < \\delta_c $.  \n\nThe merger feature is simplified as a joint, which connects three\nlines of two disappearing peaks  \nand the reforming peak at a point of\nthe sloping saddle as shown in Fig. 1 (d), where the lines  \nare the abstraction of the ridges as the continuous accretion growth.\nIn the Figure, the line processes of the accretion branches\nare connected with the point process of the merger\nat the $ (R_c,\\delta_c) $,\nwhich makes a {\\it skeleton tree}.\n\nIn the derivation of the {\\it skeleton tree},\nwe have neglected the detail features of three peaks around $ \\otimes(R_c,\n\\delta_c) $ except counting the number of the associated lines,\nand approximated as \nthe destruction and the reformation occur at $ (\\delta_c,R_c) $ \ninstantaneously.  \nThis is reasonable since no other parameter sets\nexcept $ (\\delta_c,R_c) $ can be defined without any extra\nparameter for modeling the merger event.  The consistency of this\nsimple picture can be proved with the result in the section 6.  \n\nWe can practically reproduce accretion and merging with \nthe script of {\\it skeleton tree} as shown in in Fig 1(d);\nthe arrowed line is related to \nthe mass growth with continuous accretion, and the joint of hatching\ntriangle is a merger event with the destruction and the reformation\nwith the mark $ \\otimes $ of the sloping saddle.  \nWe note that the branching number at the merging \npoints is always two in the progenitor side.\nThis is natural since the merger rate \nof three or more multi peaks are negligible to that of double peaks.  \n\n\\section{The Constraints of Critical Points}\n\n\\subsection{Line and Point Processes in the Skeleton Tree}\n\nAs shown in the previous section,\n{\\it the skeleton tree formalism} is described by a set of the line and  \npoint processes of the smoothing peak along the ridge and the sloping saddle\nin the extended space $ ({\\bf r}, R) $.  The line process of the\nsmoothing peaks in $ ({\\bf r}, R) $ is equal to the point process in the \noriginal spatial coordinate $ {\\bf r} $.  \n\nTheir density fields of these point processes \nare described as sums of $ \\delta $ functions :  \n\\begin{eqnarray} \nn_{pk} ({\\bf r};R) & = & \n\\sum_i \\delta^{(3)}(\\mbox{\\bf r}-\\mbox{\\bf r}_{pk,i}) \\;,\n\\\\\n{\\bf n}_{ss} ({\\bf r}, R) & = & \\sum_i\n\\delta^{(3)}(\\mbox{\\bf r}-\\mbox{\\bf r}_{ss,i})\n\\delta ( R - R_{ss,i})  \\; ,\n\\end{eqnarray}\nwhere the subscripts of $ pk$ and $ss$ means peaks and sloping\nsaddles, $ R_{ss} $ is the resolution scale of the\nsloping saddles.  \nSince $ n_{pk} d^{3}{\\bf r} $ and $ {\\bf n}_{ss} d^{3}{\\bf r} d R $\nare the numbers in 3-D infinite volume of $ d^3 {\\bf r} $ and \n4-D infinite volume of $ d^{3}{\\bf r} d R $,  \nwe call the former and the later\nas the spatial density and the instantaneous spatial density,\nrespectively.   \n\nWe can express the point processes entirely in terms of the field and\nits derivatives with the spatial coordinate $ {\\bf r} $\nand the resolution scale $R$.  \nIn the neighborhood around a critical point, with its constraint of\n$ \\nabla F( {\\bf r})\\vert_{cr}  = 0 $\n, we can expand the field in a Taylor series:   \n\\begin{eqnarray}  \nF(\\mbox{\\bf r}) & \\simeq & \nF(\\mbox{\\bf r}_{cr})  +\\frac{1}{2!}\n\\nabla \\otimes \\nabla F \\vert_{cr} \\Delta {\\bf r} \\Delta {\\bf r}\n\\nonumber \\\\\n& & \\makebox[0.5cm]{} + \\frac{1}{3!}\n\\nabla \\otimes \\nabla \\otimes \\nabla\nF \\vert_{cr} \\Delta {\\bf r} \\Delta {\\bf r}\n\\Delta {\\bf r} \\; ,\n\\end{eqnarray}\nand its derivatives can be also expanded as \n\\begin{eqnarray} \n\\nabla F (\\mbox{\\bf r})\n& \\simeq & \\nabla \\otimes \\nabla F \\vert_{cr} \\Delta {\\bf r} \\; , \\\\\n\\nabla \\otimes \\nabla F (\\mbox{\\bf r})\n& \\simeq & \\nabla \\otimes \\nabla F (\\mbox{\\bf r}) \\vert_{cr} +\n\\nabla \\otimes \\nabla \\otimes \\nabla F \\vert_{cr} \\Delta {\\bf r} \\;  , \n\\end{eqnarray}\nwhere \n$ \\Delta {\\bf r} = {\\bf r} - {\\bf r}_{cr} \\; $\nand the suffix $ cr $ means the value at the critical point.  \n\nThe critical points can be divided into non-degenerate one and \ndegenerate one.  The extrema like a peak and hole    \ncan be categorized into non-degenerate one.       \nProvided the condition of the non-degenerate extrema \n$ {\\mbox{det} (\\nabla \\otimes \\nabla F) \\vert_{cr}  \\neq 0} \\; $,\nEq. (11) can be rewritten to \n\\begin{eqnarray}\n{\\bf r} - {\\bf r}_{cr} = ( \\nabla \\otimes \\nabla F \\vert_{cr} )^{-1}  \n\\nabla F (\\mbox{\\bf r}) \\; . \n\\end{eqnarray}\nUsing the second derivatives of the field,  \nthe number density of the extrema \ncan be represented as  \n\\begin{eqnarray} \nn_{ex}& = & \n  \\delta^{(3)}( \n  (\\nabla \\otimes \\nabla F)^{-1} \\nabla F  ) \\nonumber \\\\\n & = &\n\\vert  \\mbox{det}\n(\\nabla \\otimes \\nabla F) \\vert \\delta^{(3)}(\\nabla F) \\; . \n\\end{eqnarray} \nIn order to describe the point process for the extrema,\nthus, it is enough to take the terms to the \norder of the second derivatives.\n\nIn the degenerate case of $ \\nabla \\otimes \\nabla F \\vert_{cr}  = 0 \\; $,\nhowever, we cannot describe the displacement vector\nonly with the first and second derivatives of the field as\nthe non-degenerate case of Eq. (13).  \nThe sloping saddle is a kind of the degenerate \ncritical points.  \n\nUnder the transformation of \nthe principal axis in the spatial coordinate, the part of six\ncomponents \nrelated to the second derivatives $ \\nabla \\otimes \\nabla F $ \nin the covariance matrix becomes diagonal.  It means that \nthe second derivatives have three eigenvalues as \n\\begin{eqnarray} \n(F_{11},F_{22},F_{33})= - \\sigma_2 (\\lambda_1,\\lambda_2,\\lambda_3) \\; , \n F_{\\alpha \\beta}=0 \\; (\\beta \\neq \\alpha) \\; , \n\\end{eqnarray}\nwhere $ \\sigma_2 $ is the rms of $ \\nabla \\otimes \\nabla F $\n( see the definition in Appendix A ).   \nAll the eigenvalues are not null in the non-degenerate cases.\nOn the other hand, the degenerate critical points\nhas one null eigenvalue at least as $ \\lambda_3 =0 $, where \nwe assumed $ \\lambda_1 \\geq \\lambda_2 \\geq \\lambda_3 $ for convenience.\nThis is the reason for the break of the non-degenerate condition as  \n$ \\mbox{det} \\nabla \\otimes \\nabla F \\vert_{cr}  = 0 \\; $ \nin the degenerate case.  In general,\na sloping saddle has a neighboring peak at the degenerate direction.  \nIn this degenerate direction, we cannot take the inversion of\nEq. (11) as $ x_3 - x_{3,cr} = F_{33}\\vert_{cr}^{-1} F_3 $ since $\nF_{33} \\vert_{cr} = 0\n$.\n\nIn order to describe the point process of the sloping saddles,  \nwe use the expansions\nof $ F_{33} $ at the degenerate direction and $ \\nabla^{(2)} F $\nfor the rest 2-D non-degenerate components \nin a couple of Eq. (12) and Eq. (11) as \n\\begin{eqnarray}\nF_{33} & \\simeq & \\sum^3_{\\beta=1} F_{33\\beta}  \n\\Delta x_\\beta \\simeq F_{333} \\Delta x_3 \\; ,  \\\\\n\\nabla^{(2)} F\n& \\simeq & \\nabla^{(2)} \\otimes \\nabla^{(2)} F \\Delta {\\bf r}^{(2)}\n\\; ,\n\\end{eqnarray}\nwhere the suffice of $ (2) $ means the 2-D non-degenerate space.  \nThe set of the expansions derives\n\\begin{eqnarray}\n{\\bf r}^{(2)} - {\\bf r}_{cr}^{(2)} & = & \n( \\nabla^{(2)} \\otimes \\nabla^{(2)} F \\vert_{cr} )^{-1}  \n\\nabla^{(2)} F (\\mbox{\\bf r}) \\; ,  \\\\\nx_3 -x_{3,ss} & = & F_{333}^{-1} F_{33} \\; .\n\\end{eqnarray} \n\nFurthermore, we should remember that the sloping saddle is defined\nin the extended space with the resolution scale $ R $.  The gradient\n$\\nabla F (\\mbox{\\bf r}) $ can be expanded with the derivative of $R$; \n\\begin{eqnarray} \n\\nabla F (\\mbox{\\bf r})\n\\simeq \\frac{d \\nabla F  }{d R} \\vert_{ss} (R-R_{ss})\n= \\nabla^2 \\nabla F \\vert_{ss}\\: R (R-R_{ss}) \\; . \n\\end{eqnarray}\nIn the degenerate direction, this gives \n\\begin{eqnarray}\n R-R_{ss} = \\left( \\sum_{\\alpha=1}^3 F_{3 \\alpha \\alpha} R\n\\right)^{-1} F_3 \\; .\n\\end{eqnarray} \n\nWe obtain the constraint for the instantaneous spatial density\nof the sloping saddles as \n\\begin{eqnarray}\n{\\bf n}_{ss} & = & \\sum_i\n\\delta^{(3)}(\\mbox{\\bf r}- \\mbox{\\bf r}_{ss,i}) \\delta ( R - R_{ss,i}) \n\\nonumber \\\\\n& = &\n\\vert  \\mbox{det}\n( \\nabla^{(2)} \\otimes \\nabla^{(2)} F) \n\\vert \\delta^{(2)}(\\nabla^{(2)} F) \\nonumber \\\\\n&  & \\makebox[1cm]{} \\times \n\\vert F_{333} \\vert \\delta (F_{33})  \\; \n{ \\vert {\\sum_{\\alpha=1}^3 F_{3 \\alpha \\alpha} R} \\vert}  \n\\delta (F_3)  \\; .\n\\end{eqnarray}\n\n\\subsection{Constraints of Peaks and Sloping Saddles} \n\nUnder the transformation to the diagonal principal axis,  \nthe peaks require the condition of \n$ \\lambda_1 \\geq \\lambda_2 \\geq \\lambda_3 \\geq 0 $  \nwhich should be added to the above condition of the extrema: Eq. (14).  \nWith the additional condition of the peaks,  \nthe constraint for the spatial distributions of the peaks is \n\\begin{eqnarray} \nC(\\mbox{\\bf F}^{(10)} \\vert \\mbox{\\rm peaks} ) \n& = &\\sigma_2^3 \n\\vert \\lambda_1 \\lambda_2 \\lambda_3 \\vert \\prod_{i=1}^3  \n\\delta(F_{\\alpha}) \\prod_{\\alpha=1}^3 \\theta(\\lambda_{i}) \\; , \n\\end{eqnarray}\nwhere $ \\theta(\\lambda_{i}) $ is the Heaviside function.   \n\nThe sloping saddles requires the additional condition of \n$ \\lambda_{i}>0 \\; (i=1,2) $ as excluding the merger of the\nhole.  With the constraint of Eq. (22),\nthe constraint for the instantaneous spatial distribution of\nthe sloping saddle can be described as \n\\begin{eqnarray} \n& & C( \\mbox{\\bf F}^{(20)} \\vert \\; \\mbox{\\rm s.saddles} ) = \\sigma_2^2 \n\\sigma_3^2 R  \n \\vert \\lambda_1 \\lambda_2 w_3 (w_1+w_2+w_3) \\vert \\nonumber \\\\\n& & \\makebox[1cm]{} \\times \n\\delta(\\lambda_3) \\prod_{\\alpha=1}^3 \n\\delta(F_{\\alpha}) \\prod_{i=1}^2 \\theta(\\lambda_{i}) \\; , \n\\end{eqnarray}  \nwhere we use $ \\sigma_3 w_{\\alpha} = F_{3 \\alpha \\alpha} $.\n\n\\section{Probability Distribution Function} \n   \nThe joint probability distribution \nof n-dimensional random variables with multivariate Gaussian can \nbe described as \n\\begin{eqnarray} \nP( \\mbox{\\bf  F}^{(n)} ) d \\mbox{\\bf  F}^{(n)} = \\frac{\n\\exp [ \\frac{1}{2} \\mbox{\\bf  F}^{(n)\\:T} \\mbox{\\bf C}^{(n \\times n) -1} \n\\mbox{\\bf  F}^{(n)} ] }\n{ (2\\pi)^n \\vert det \\mbox{\\bf C}^{(n)} \\vert^{1\/2} } \nd \\mbox{\\bf  F}^{(n)} \\; . \n\\end{eqnarray} \nThe convariance matrix $ \\mbox{\\bf C}^{(n \\times n)} $ \nis defined as the expectation \nvalue of the direct product of the vector $ \\mbox{\\bf F}^{(n)} $ : \n\\begin{eqnarray} \nC^{(n \\times n)}_{\\alpha \\beta}= < F^{(n)}_{\\alpha} F^{(n)}_{\\beta} > \\; ,  \n\\end{eqnarray} \nwhere the suffix $ (n)$ of the $ \\mbox{\\bf F}^{(n)} $ \nis the dimension of the \nparameter space.\n\nIn general, \nthe conditional probability of the event A\nwith the constraint event B is given by the Bayes formula\n$ P( \\mbox{A}\\vert\n\\mbox{B}) = P( \\mbox{A}, \\mbox{B}) \/ P ( \\mbox{B}) $. \nThe joint probability, then, can be expanded with the conditional\nprobabilities. \nIf the parameters in the vector are all Gaussian distributed, we can\ndirectly obtain the covariance matrix of the conditional probability.\nThere is a general theorem which is extremely useful when\nwe calculate the above joint probability from the\nconditional probabilities.    \nFor the count of the peaks, we need only the 10 dimension parameters \n$ \\mbox{\\bf F}^{(10)} = (F, \\nabla F, \\nabla \\otimes \\nabla F ) $.\nFor the count of the sloping saddles, \nwe should extend the parameter space to the 20 dimension \n$ \\mbox{\\bf F}^{(20)} = (F, \\nabla F, \\nabla \\otimes \\nabla F, \n\\nabla \\otimes \\nabla \\otimes \\nabla F) $.\nThe 20 dimension covariance matrix can be found with the explicit forms of\nthe parameters in Appendix A.  With the help of the theorem \n(see Appendix B1),  the divided conditional probabilities\nare described for the case of the peaks and the sloping saddles\nin Appendix B2 and B3, respectively.  \n\nFor a type of critical points\nin a n-dimensional parameter space described with constraints, in general, \nthe probability weighted density is\n\\begin{eqnarray}\nn_{cr}({\\mbox{\\bf F}}^{(n)}) d \\mbox{\\bf  F}^{(n)}\n= P({\\mbox{\\bf F}}^{(n)}) C({\\mbox{\\bf F}}^{(n)} \\mid \n\\mbox{cr. points} ) d \\mbox{\\bf  F}^{(n)} \\; .\n\\end{eqnarray} \nThen, \nthe ensemble averaged density of the type of critical points \nrestricted with m-dimension parameters, can be obtained from \nthe integration over (n-m) dimension parameters; \n\\begin{eqnarray} \n< n_{cr}({\\mbox{\\bf F}}^{(m)}) > d \\mbox{\\bf  F}^{(m)} \n = \\int d {\\mbox{\\bf F}}^{(n-m)} \nn_{cr}({\\mbox{\\bf F}}^{(n)})\nd \\mbox{\\bf  F}^{(m)}\n\\; . \n\\end{eqnarray}\nThe ensemble averaged density gives the scale function we interest.  \n   \n\\section{Scale Functions of Critical Points} \n\nWe will derive the differential number densities of all the peaks,\nthe nesting peaks, the non-nesting peaks,  and   \nthe instantaneous differential number density of the sloping\nsaddles,    \nwhich are presented as the scale functions and\nthe instantaneous scale function of \n$ N_{pk}(R,\\delta) dR$, $ N_{nest}(R,\\delta) dR$,\n$ N_{pk}(R,\\delta) dR$, and $ {\\bf N}_{ss}(R,\\delta) dR d \\delta $,\nrespectively.\n\nThe scale functions for all the peaks, the non-nesting,\nand nesting peaks are briefly\ngiven in the appendix C1, C2, and C3,  according to BBKS and \nthe CUSP formalism.  \nWe had checked that the contribution of the nesting peaks \nis negligible as shown in Fig. 2 .  Thus,\nwe treat the scale function of peaks as that of the non-nesting peaks\nhereafter. \n\nIn the first subsection, then, \nwe sketch the derivations of the scale functions of peaks.   \nIn the next subsection, \nwe present the new calculation for the instantaneous scale function\nfor the sloping saddles.  \n\n\\subsection{Scale Function of Peaks}  \n\n\\begin{figure*}\n\\psbox[scale=0.9]{fig2.eps}\n\\caption{ The scale functions are presented\nfor various fluctuation models of the initial density fluctuation.\nThe solid and dotted lines represent the scale function of the\nall peaks and the nesting peaks.  The dashed line is that of the\nPS formalism with the same filter.\n}\n\\end{figure*}\n\nThe scale function of $ N_{pk}(R,\\delta) dR$  \ncan be calculated from the ensemble-averaged density of\nthe peak $ {\\cal N}_{pk} \n(\\nu; R) d \\nu = < n_{pk} (\\nu ;R) > d \\nu $,\nwhere $ \\nu = \\delta\/ \\sigma_0$,\naccording to Manrique $\\&$ Salvador-Sole (1995). \nThe scale function means the differential number density\nper infinitesimal range of $ R $ at a fixed $ \\nu $, not from that \nper infinitesimal range of $ \\nu $ at a fixed $ R$.   \nThe forward one and the last one are denoted \nby a roman capital and a caligraphic capital which is the same\nas the notation of BBKS, respectively.   \nThe spatial density for the peaks of the non-degenerate \ncritical points with \na certain filtering scale can be \ncalculated by the same way of BBKS.  \n\nIn order to obtain  \nthe density of peaks with up-crossing a certain $ \\delta_c $ \nin the range of the filtering scale between $ R - \\Delta R$ and $ R $, \nwe should pick up the critical points which are equal \nto or smaller than $ \\delta_c $ \nat the filtering scale of $ R $ and becomes larger than $ \\delta_c $ at the \nsmaller filtering scale of $ R - \\Delta R $.  It means that \nwe should count the peaks of \n$ \\delta_c + (d F_{pk}\/ dR) \\Delta R < F_{pk} < \\delta_c $ \non scale $ R $ .  As shown in the discussion of the Gaussian filtering, \nthe condition of the counting is expressed as \n\\begin{eqnarray} \n\\delta_c + \\nabla^2 F  R \\Delta R < F < \\delta_c  \\;  .  \n\\end{eqnarray} \n\nAs obtained for the peaks in BBKS, in general,   \nwe can obtain the spatial number density of the peaks \nin infinite ranges $ d \\nu dx $ of the density contrast $ \\nu $ \nand the second derivative of it with a certain filtering scale $ R $;  \n$ {\\cal N}_{pk} (\\nu,x ;R) d \\nu dx $.   We can \ntransform this number density to the number satisfying the condition of \nthe our counting;  \n\\begin{eqnarray} \nN_{pk}(R,\\delta_c) & = & \\lim_{\\Delta R \\rightarrow 0}\n \\frac{1}{\\Delta R} \\int^{\\infty}_{0} dx \\int^{\\nu_{c}}_{\\nu_{c,b}} \nd \\nu {\\cal N}_{pk}(\\nu,x ;R) \\nonumber \\\\ \n& = &  {\\cal N}_{pk}(\\nu_c ;R) <x>_{pk}\n\\left( \\frac{\\sigma_2(R)}{\\sigma_0(R)} \\right) \nR \\;, \\\\ \n\\nu_{c,b} & = &\\nu_c \n-x \\left( \\frac{\\sigma_2(R)}{\\sigma_0(R)} \\right) R \\Delta R  \\; ,\n\\end{eqnarray}   \nwhere $ < x >_{pk} $ is the mean value of $ x $ defined \nfrom the distribution function as \n\\begin{eqnarray} \n< x >_{pk} = \\frac{ H_{pk}(\\gamma, \\gamma \\nu_c ) }\n{ G_{pk}(\\gamma, \\gamma \\nu_c) } \\; , \n\\end{eqnarray}\nIn practical treatment, we can use the averaged mapping relation\n\\begin{eqnarray}\n d \\nu_c \\simeq\n<x>_{pk} \\left( \\frac{\\sigma_2(R)}{\\sigma_0(R)} \\right) \nR d R \\; .\n\\end{eqnarray} \n\nFig. 2 illustrates $ N_{pk}(R,\\delta_c) R R_0^2$ and $ N^{nest} (R,\n\\delta_c) R R_0^2$ versus $ R\/R_0 $  with the present\ncollapsing threshold condition $ \\delta_c =1.68 $ in the power-law\nfluctuations of $ n= 0, -1 $, and $ -2 $.  The former and the latter\nare represented with the solid  and dotted lines, respectively.    \nThe critical scale of $\nR_{0} $, was determined by $ \\delta_c\/ \\sigma_0 (R_0) = 1 $\nwith $ \\delta_c =1.68 $.  \nThe number of the nesting peaks is\nnegligible compared with that of all peaks. Hereafter, then, we will \ntake $ N (R,\\delta_c) = N_{pk}(R,\\delta_c) $.  \nThe PS scale functions, obtained from the same filtered random fields, \nare also presented with the dashed lines.\nWe also present the two scale functions of peaks for\nstandard CDM (SCDM) model with the normalization \nof 8 Mpc with the bias factor $ b= 1$.  \nEven if \nthe two functions in the small scale are different from each other between  \nthe peak theory and the PS formalism as pointed out by Appel $\\&$\nJones (1991),  they deviate relatively little over the cosmological interest\ninterval ( 2 decades in the filter scale, 6 decades in the mass scale\naround $ R_0 $).\n\nFig. 2 shows that\nthe scale functions of the peaks $ N_{pk}(R,\\delta_c) R R_0^2$ \nbecome proportional to $ R^{-3}$ asymptotically in the small scale.\nAs shown in BBKS, the cumulative number is a useful quantity which can \nbe evaluated analytically: \n\\begin{eqnarray}\nn_{pk}( \\nu_c=- \\infty ) = \\int^{\\infty}_{\\nu_c=-\\infty} {\\cal N}_{pk}\nd \\nu =\n\\frac{29-6\\sqrt{6}}{ 5^{3\/2} 2 (2 \\pi)^2 }\nR_{\\ast}^{-3} \\; ,\n\\end{eqnarray} \nAs $ R \\rightarrow 0 $, the density contrast $ \\nu_c \\rightarrow\n0 $ and the peaks of the low contrast $ \\nu_c $ dominates\nin the cumulative number.  Then,\n$\nN_{pk}(R,\\delta_c) R R_0^2 \\propto\nn_{pk}(\\nu_c \\rightarrow 0; R_{\\ast} ) \\propto\nn_{pk}(\\nu_c=- \\infty ; R_{\\ast} ) $ in the small scale.   \nSince $ R_{\\ast}$ is proportional to\nthe resolution scale $ R $,  the asymptotic feature seen in Fig. 2 \nis not unexpected thing.   \n\n\\subsection{Instantaneous Scale Function of Sloping Saddles}\n\nThe instantaneous scale function of sloping saddles can be\ndirectly calculated as its ensemble-averaged density:\n\\begin{eqnarray} \n\\mbox{\\bf N}_{ss}(R,\\nu_c) d R d \\nu_c & = & < {\\bf n}_{ss} (\\nu_c;R)\n> d R d \\nu_c \\; .\n\\end{eqnarray}\nThe brief calculation is described in the appendix C.  \n\nWe should note that the scale function is defined as the \ndensity in the infinite interval $ d R d \\nu_c$.  \nThe scale function of the peaks \n$ N_{pk}(R,\\delta_c) dR $ introduced before is the differential density  \nof peaks per infinitesimal ranges of the resolution scale $ R $ at a \nfixed density threshold $ \\delta_c $. On the other hand,\nthe instantaneous scale function of the sloping saddles\n$ \\mbox{\\bf N}_{ss}(R,\\delta_c) $\nis the differential density per infinitesimal ranges \nof $ R $ and $ \\delta $.  Their difference come from the fact that\nthe point processes of peaks and sloping saddles can be defined\nin the 3-D spatial coordinate and the 4-D space extended with $ R $,\nrespectively.  Then, the instantaneous scale function is responsible\nfor the time evolution properties of the scale function\nwith the merging process as seen in the {\\it skeleton tree};\nthe line of the smoothing peak connected with other lines at the\njunction of the sloping saddle.  In the next section,\nusing {\\it the skeleton tree formalism},  \nwe will present the evolution of the scale functions with\nthe distinction between the merger and the accretion according to\n{\\it the skeleton tree formalism}.   \n\n\\section{Evolution with Merger and Accretion}\n\n\\subsection{Instantaneous Scale Functions for the Disappearing and\nReforming Peaks}\n\nAs discussed in the sketch of the {\\it skeleton tree} picture with Fig. 1,\nthe reformation of a peak can be only identified with the destruction\nof two peaks around a sloping saddle.  \nIn the above section, \nwe can obtain the instantaneous number density of\nthe merger of a scale $ R $ at $ \\delta_c $ in the \ninfinite interval of $ d \\delta_c $.\nThus, with changing between the variables $ \\delta_c $ and $ \\nu_c $,\nthe instantaneous scale functions of the destruction and the\nreformation for the peaks of the scale $ R $ at $ \\delta_c $\nare described as \n\\begin{eqnarray} \n\\mbox{\\bf N}^d (R,\\delta_c) d R \\; d \\delta_c  & \\simeq & \n2 \\mbox{\\bf N}_{ss}(R,\\delta_c) d R \\; d \\nu_c \\; , \\\\\n\\mbox{\\bf N}^f (R,\\delta_c) d R \\; d \\delta_c  & \\simeq & \n\\mbox{\\bf N}_{ss}(R,\\delta_c) d R \\; d \\nu_c \\; , \n\\end{eqnarray} \nwhere the superscripts of $f$ and $d$ mean the reformation and the\ndestruction in the merger, respectively.  \n\nWith the formula,\nthe net change of the differential number density of the peaks in $ d R $ \nduring the interval of $ d \\delta_c $ is simplified to  \n\\begin{eqnarray}\nS (R,\\delta_c) d R \\; d \\delta_c\n& = &  \n\\mbox{\\bf N}^f (R,\\delta_c) d R \\; d \\delta_c - \n\\mbox{\\bf N}^d (R,\\delta_c) d R \\; d \\delta_c  \\nonumber \\\\\n& = & \n- \\mbox{\\bf N}_{ss}(R,\\delta_c) d R \\; d \\nu_c \\; .\n\\end{eqnarray}\nIt means the net destruction of the peaks is the same as the\nappearance of the sloping saddles.\nThis is reasonable since the formation of the peaks\ndoes not occur in exact sense and all peaks only disappear at the\nsloping saddle in the original filtering process. \nFrom the concept of the {\\it skeleton tree},\nthe peak neighboring with a sloping saddle\nis interpreted to lose its identity with the\nmerger.  We considered that \nthe ridge of the neighboring peak \nis divided into two part of a disappearing peak\n$ > \\delta_c $ and a reforming peak $ < \\delta_c $,\neven if the ridge of the neighboring peak continues over $ \\delta_c $. Thus,\nthe reforming number of the peaks can compensate\nfor the destruction of them, and the appearing number of the sloping saddles\nlefts as the net disappearing number of the peaks.\nThe consistency of this formula can be checked with the conservation\nequation for the scale function of the peaks, as shown\nin the next subsection.   \n\n\\subsection{Instantaneous Rates for Accretion, Destruction and Reformation}  \n\n\\begin{figure*}\n\\begin{center}\n\\psbox[scale=0.9]{fig3.eps}\n\\end{center}\n\\caption{ The rates for the evolution processes at the present with\n$ \\delta_c = 1.68 $. \nThe solid, dotted, short dashed, long dashed,\nand long dash-dotted \nlines mean the reformation rate $ r^f \\vert dt \/d \\delta_c \\vert $,\nthe destruction rate $ r^d \\vert dt \/d \\delta_c \\vert $,\nthe relative accretion rate $ r^a \\vert dt \/d \\delta_c \\vert $,\nthe relative accretion rate $ r^a \\vert dt \/d \\delta_c \\vert $\nduring the merger, and  \nthe shift rate $ r^s \\vert dt \/d \\delta_c \\vert $, respectively.  \nThe evolution rate of the scale function \n$ \\vert  \\partial \\ln N_{pk} (R,t)\/ \\partial t\ndt \/d \\delta_c \\vert $ and\nthe right hand side of the conservation equation (51) is presented\nby short dash-dotted line and crosses. The absolute\nvalues are presented for the two last rates.  \n}\n\\end{figure*}\n\nWe will consider the instantaneous scale growth rate for a peak of \n$ R $ as the continuous scale growth of the accretion.\nThe accretion growth rate can be described \nwith the scaled Laplacian $ x $ as \n\\begin{eqnarray} \n{\\dot R}_{acc}(R, x, \\delta_c ) d t \n& \\equiv & \\frac{1}{x \\sigma_2 R} d \\delta_c \\; .\n\\end{eqnarray}\nWe should note that the growth rate \nfor a peak depends on its particular value $ x $.\nAccording to Manrique $\\&$ Salvador-Sole (1996),\nthe mean growth rate for the objects of the scale $ R $ can be expressed as \n\\begin{eqnarray} \n{\\dot R}_{acc}(R, t) d t \n& \\equiv &\n< \\frac{1}{x \\sigma_2 R} > d \\delta_c \\simeq \\frac{1}{<x>_{pk}\\sigma_2 R}\nd \\delta_c \\; ,  \n\\end{eqnarray} \nwhere $ < \\; > $\nmeans the average of the function with $ x $ and we used the relation \n\\begin{eqnarray} \n< \\; x^{-1} \\;  > & \\simeq &\n\\frac{ \\int_0^{\\infty} dx \\; x^{-1} \\; N_{pk} (R, \\delta_c, x) dR } \n{ \\int_0^{\\infty} dx N_{pk} (R, \\delta_c, x) dR } \\nonumber \\\\\n& = & \\frac{ \\int_0^{\\infty} dx {\\cal N}_{pk}(\\nu_c, x; R)\n\\frac{\\sigma_2}{\\sigma_0} d R } \n{ \\int_0^{\\infty} dx \\; x \\; {\\cal N}_{pk} (\\nu_c, x, R)\n\\frac{\\sigma_2}{\\sigma_0} d R } \\nonumber \\\\\n& = & \\frac{G_{pk} (\\gamma, \\gamma \\nu_c)}\n{ H_{pk} (\\gamma, \\gamma \\nu_c)} = <x>_{pk}^{-1}\\; . \n\\end{eqnarray}\nFrom this mean scale growth rate, we can define \nthe relative growth rate with the accretion as\n\\begin{eqnarray}\nr^a (R,t) dt & \\equiv &\\frac{ \\partial {\\dot R}_{acc} (R,t')}{ \\partial R}\n\\; .\n\\end{eqnarray}\n\nWith the similar consideration applied for the merging objects,\nwe can introduce the mean scale\ngrowth rate and relative growth rate contributed\nwith continues accretion during the merger phase;   \n\\begin{eqnarray} \n{\\dot R}_{acc,m}(R, t) d t \n& \\simeq & \\frac{1}{<x>_{ss}\\sigma_2 R}\nd \\delta_c \\; ,  \\\\ \nr^a_m (R,t) dt & \\equiv & \\frac{ \\partial {\\dot R}_{acc,m} (R,t')}\n{ \\partial R} \\; ,   \n\\end{eqnarray}\nwhere\n$\n<x>_{ss} = H_{ss} (\\gamma, \\kappa, \\gamma \\nu_c) \/\nG_{ss} (\\gamma, \\kappa, \\gamma \\nu_c) \\;$ . \n\n\nThe instantaneous destruction and reformation rates can be defined directly\nfrom their instantaneous scale functions as \n\\begin{eqnarray}\nr^d (R,t) \\; dt & = & \\frac{ {\\bf N}^d (R, \\delta) d R } { N (R,t) d \nR} \\; d \\delta \\; \\\\\nr^f (R,t) \\; dt & = & \\frac{ {\\bf N}^f (R, \\delta) d R } { N (R,t) d \nR} \\; d \\delta \\; . \n\\end{eqnarray}  \nThus, the conservation equation for the scale function $ N (R,t) $ \nis given as \n\\begin{eqnarray}\n\\frac{ \\partial N (R,t) }{\\partial t} +\n\\frac{ \\partial  ( {\\dot R}_{acc} N (R,t) ) }{ \\partial R}  =  S(R,t) \\; ,\n\\end{eqnarray}\nwhere $ S(R,t) $ can be given as\nthe net source term with the destruction and reformation\nrates of $ r^d (R,t) $ and $ r^f (R,t) $ ;   \n\\begin{eqnarray} \nS(R,t) = [ r^f (R,t) - r^d (R,t) ] N (R,t) \\; . \n\\end{eqnarray}\nThe conservation equation can be rewritten to\n\\begin{eqnarray} \n\\frac{ d \\ln N (R,t) }{d t} = r^f (R,t) - r^d (R,t) - r^a (R,t)\\; , \n\\end{eqnarray}\nwhich can be rewritten to\n\\begin{eqnarray} \n\\frac{ \\partial \\ln N (R,t) }{\\partial t} = r^s(R,t)  + \nr^f (R,t) - r^d (R,t) - r^a (R,t)\\; , \n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nr^s(R,t) = -{\\dot R}_{acc} \\frac{\\partial \\ln N (R,t)}{\\partial R} \\; ,\n\\end{eqnarray}\nis the rate of shift in the scale space with the\naccretion for the number density distribution.\n\n\nThe inverses of these rates of $ r^{a,f,d,s}(R,t) $ give the time\nscales of the individual processes.      \nIn Fig. 3, the rates for the reformation,\nthe destruction, the relative growth with accretion, the relative\ngrowth with continuous accretion for merging objects\nand the shift in the scale with the accretion \nat the present are presented with solid, dotted, short\ndashed, long dashed, and long dash-dotted lines.  The left hand side \nand the right hand side of the conservation equation (52) are also\npresented with short dash-dotted line and crosses,  the former\n$ \\partial \\ln N(R,t) \\partial t $ can be calculated with the partial\ntime derivative of $  N_{pk}(R,t) $ and the latter\nis calculated from the set of the\nindividual rates as  $r^s(R,t)  +  r^f (R,t) - r^d\n(R,t) - r^a (R,t)$.   \n\nWe can see \nthat the partial time derivative of $ \\partial \\ln N(R,t) \/ \\partial\nt $ can be reproduced well from  \nthe right hand side\nof the conservation equation from the individual calculations of \n$r^s(R,t), r^f (R,t), r^d (R,t),$  and $ r^a (R,t)$ as shown in Fig. 3.\nEspecially,  around the scale of $ R \\simeq R_0 $,\nboth of them shows the same feature, in which the values\nshapely switch from negative to\npositive.  Then, we can verify the consistency of our formalism with\nthe equality between the left hand side and the right hand side\nof the conservation equation.  \n\nThe rates of the reformation and the destruction have a maximum\naround $ R \\simeq R_0 $ and rapidly decrease for the larger\nscale.  On the other hand, \nthe accretion dominates the merger processes of the reformation and\nthe destruction and the rate becomes almost constant \nfor the large objects ( $ R >> R_0 $ ).\nIt is consistent with the picture that \nthe large object are growing in the cosmic time scale.\nThese mean that the larger\nobjects of $ R > R_0 $ grows with the accretion in the linear regime\nof the gravitational instability,\nwhile the smaller objects are accumulated into\nthe larger objects with the merging in the non-linear gravitational\ngrowth regime under the critical scale of $ R_0 $, which is related to \nthe present threshold $ \\delta_c $.  In the theory of\ngravitational instability,  the smaller than $ R_0$ can become in\nnon-linear growth.  \nWe should remark that the merger process is the most efficient\naround the critical scale $R_0$ even if the merger still dominates\nthe accretion in the small scale.  \nThese properties of the dominating merger around $ R_0 $\nare already suggested from the previous\nN-body calculations (e.g. Navaro, Frenk, $\\&$ White 1997 ).  \n\n\\begin{figure}\n\\begin{center}\n\\psbox[scale=0.45]{fig4.eps}\n\\end{center}\n\\caption{ The evolution of growth rates \nat fixed masses for the SCDM model in an Einstein-de Sitter universe.  \nThe solid and dashed lines present the merger and the accretion\ngrowth rate $ r^f \\cdot t $ and\n$ r^a \\cdot t $.\nIn the figure, the highest curves at the right is for $ M\/M_0 = \n10^{-2} $, and successive curves are for $ M\/M_0 = \n10^{-1.5}, 10^{-1}, 10^{-0.5}, 1, 10^{0.5}$ and $ 10^{1}$.   }\n\\end{figure}\n\nFrom the view point of the mass growth history for a object of the\nscale $ R $,  the results in Fig. 3 should indicate that  \nthe dominant growth process switches\nfrom the accretion to the merger accumulation around\nthe threshold of $ \\delta\n= \\sigma_0 (R) $ and\nthe merger accumulation rate becomes maximum and decreases rapidly.  \nWe can see directly this feature in Fig. 4, which represents the\nevolution of the growth rate with the reformation and the accretion\nat fixed scales.\nFrom these results, in general, the growth of a halo \nfirstly starts with the accretion process, \nsecondly switches to the merging and is suppressed soon after the\nmerger dominates.  This feature also was remarked from the\nsimulations.\n\nWe should remark that the results in Fig. 4\nare not the mean growth histories for the individual objects \nsince the fixed scales cannot be directly related to the final \nscales of the objects at the present due to the successive destruction\nand the reformation cooperated with the accretion.\nIn order to reconstruct the mean\ngrowth history for a individual objects, we should extend\nthe basic of {\\it the skeleton tree formalism} including the\nbackground effects.  This extension of {\\it the skeleton tree\nformalism } will be described in the\nfollowing papers.  \n\n\\section{Conclusion}\n\nWe have derived an analytical expression for the statistics and the\nevolution of the cosmic bound objects \nwith the reformation, the destruction and the accretion. It is \napplicable to any hierarchical clustering models in which structure\ngrows via gravitational instability.  {\\it The skeleton formalism} \nis derived as a natural expansion of the peaks theory of BBKS.\nIn the landscape reproduced from the random field with the Gaussian filter, \nwe have followed the smoothing of the peaks as\nthe accretion growth of the objects, and \nhave picked up the ``sloping saddles'' as the merging\nevents with the destruction and the reformation of the objects.\nThe line and point processes of the peaks and sloping saddles in the\nlandscape produce the tree structure.  Then, we call our scheme as\n{\\it the skeleton tree formalism}.   \nWith {\\it the skeleton tree formalism}, we can estimate the rates\nof the reformation, destruction, and the accretion\nin any hierarchical clustering models.\nThe set of these rates \ncan reproduce the evolution of the scale function of the\nobjects with the conservation equation.  This reproduction of\nthe evolving scale function verifies the self-consistency of\n{\\it the skeleton tree formalism}.\nWith the rate calculation of the individual processes, we can find \nthe merger processes are efficient around the critical scale of $ R_0\n$ determined as $ \\delta_c = \\sigma_0(R_0) $.\nThe dominant growth process of the\nobjects switches from the accretion to the merger accumulation  \naround the critical threshold related to the scale.\nIt is important to reproduce the\nmass growth history with distinguish between the accretion and the\nmerger when we try to reproduce the cosmic structure and the galaxy formation\nin the hierarchical scenario.  \n\n\n\\section*{acknowledgments}\nThe author is grateful to H.J. Mo and S.D.M. White  \nfor warm hospitality and the discussions \nwhen H. H. visited in MPA in Garching.  This works was partially \nsupported by a Grant-in-Aid from Japanese Ministry of Education, \nScience and Culture.   \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{intro}}\nA fundamental problem in cognition concerns the identification of the principles guiding human decision-making. Identifying the mechanisms of decision-making would indeed have manifold implications, from psychology to economics, finance, politics, philosophy, and computer science. In this regard, the predominant theoretical paradigm rests on a classical conception of logic and probability theory. According to this paradigm, people take decisions by following the rules of Boole's logic, while the probabilistic aspects of these decisions can be formalized by Kolmogorov's probability theory \\cite{k1933}. However, increasing experimental evidence on conceptual categorization, probability judgments and behavioral economics confirms that this classical conception is fundamentally problematical, in the sense that the cognitive models based on these mathematical structures are not capable of capturing how people concretely take decisions in situations of uncertainty.\n\nIn the last decade, an alternative scientific paradigm has caught on which applies a different modeling scheme. The research that uses the mathematical formalism of quantum theory to model situations and processes in cognitive science is becoming more and more accepted in the scientific community, having attracted the interest of renowned scientists, funding institutions, media and popular science. And, quantum models of cognition showed to be more effective than traditional modeling schemes to describe situations like the `Guppy effect', the `combination problem', the `prisoner's dilemma', the `conjunction and disjunction fallacies', `similarity judgments', the `disjunction effect', `violations of the Sure-Thing principle', `Allais', `Ellsberg' and `Machina paradoxes' (see, e.g., \\cite{ga2002,ag2005a,ag2005b,a2009a,pb2009,k2010,bpft2011,bb2012,abgs2013,ags2013,hk2013,pb2013,wbap2013,ast2014}).\n\nThere is a general acceptance that the use of the term `quantum' is not directly related to physics, neither this research in `quantum cognition' aims to unveil the microscopic processes occurring in the human brain. The term `quantum' rather refers to the mathematical structures that are applied to cognitive domains. The scientific community engaged in this research does not instead have a shared opinion on how and why these quantum mathematical structures should be employed in human cognition. Different hypotheses have been put forward in this respect. Our research team in Brussels has been working in this domain since early nineties, providing pioneering and substantial contributions to its growth, and we think it is important  to expose the epistemological foundations of the quantum theoretical approach to cognition we developed in these years. This is the main aim of the present paper. \n\nOur approach was inspired by a two decade research on the mathematical and conceptual foundations of quantum physics, quantum probability and the fundamental differences between classical and quantum structures \\cite{a82,a86,p1989,a99a}. We followed an axiomatic and operational-realistic approach to quantum physics, in which we investigated how the mathematical formalism of quantum theory in Hilbert space can be derived from more intuitive and physically justified axioms, directly connected with empirical situations and facts. This led us to elaborate a `State Context Property' (SCoP) formalism, according to which any physical entity is expressed in terms of the operationally well defined notions of `state', `context' and `property', and functional relations between these notions. If suitable axioms are imposed to such a SCoP structure, then one obtains a mathematical representation\nthat is isomorphic to a Hilbert space over complex numbers (see, e.g., \\cite{bc81}).\n\nLet us shortly explain the \n`operational-realistic' connotation characterizing\nour approach, because doing so we can easily point out its specific strength, and the reason why it introduces an essentially new element to the domain of psychology. `Operational' stands for the fact that all fundamental elements in the formalism  are directly linked to the measurement settings and operations that are performed in the laboratory of experimentation. `Realistic' means that we introduce \nin an operational way\nthe notion of `state of an entity', considering such a `state' as representing an aspect of the\nreality of the considered entity at a specific moment or during a specific time-span. Historically, the notion of `state of a physical entity' was the `easy' part of the physical theories that were the predecessors of quantum theory, and it was the birth of quantum theory that forced physicists to take also \nseriously\nthe role of measurement and hence the value of an operational approach. The reason is that `the reality of a physical entity' was considered to be a simple and straightforward notion in classical physics and hence the\n`different modes of reality of  a\nsame physical entity' were described by its `different states'. That measurements would intrinsically play a role, also in the description of the reality of a physical entity, only became clear in quantum physics for the case of micro-physical entities.  \n\nIn psychology, things historically evolved in a different way. Here, one is in fact confronted with what we call `conceptual entities', such as `concepts' or `conceptual combinations', and more generally with `any cognitive situation which is presented to the different participants in a psychology experiment. Due to their nature, conceptual entities and cognitive situations are `much less real than physical entities', which makes the notion of `state of a conceptual  entity' a highly non-obvious one in psychology. And, as far as we know, the notion of state is never explicitly introduced in psychology, although it appears implicitly within the reasoning that is made about experiments, their setups and results.\nPossibly, the notion of `preparation of the experiment' will be used for what we call `the state of the considered conceptual entity' in our approach. Often, however, the notion of state is also associated with the `belief system' of the participant in the experiment. In our approach we keep both notions of\n`state' and `measurement' on equal footing, whether our description concerns a physical entity or a conceptual entity. In this way, we can make optimal use of the characteristic methodological strengths of each one of the notions.  It is in doing so that we observed that there is an impressive analogy between the operational-realistic description of a physical entity and the operational-realistic description of a conceptual entity, in particular for what concerns the measurement process and the effects of context on the state of the entity.  As a matter of fact, one can give a SCoP description of a conceptual entity and its dynamics \\cite{ga2002,ag2005a,ag2005b}. This justifies the investigation of quantum theory as a unified, coherent and general framework to model conceptual entities, as quantum theory is a natural candidate to model context effects and context-induced state transformations. Hence, the quantum theoretical models that we worked out for specific cognitive situations strictly derive from such investigation of quantum theory as a scientific paradigm for human cognition. In this respect, we think that each predictive success of quantum modeling can be considered as a confirmation of such general validity. It is however important to observe that, recently, potential deviations from Hilbert space modeling were discovered in two cognitive situations, namely, `question order effects' \\cite{pnas} and `response replicability' \\cite{plosone}. According to some authors, question order effects can be represented by sequential quantum measurements of incompatible properties \\cite{bb2012,pb2013,pnas}. However, such a representation seems to be problematical, as it cannot reproduce the pattern observed in response replicability \\cite{plosone}, nor it can exactly fit experimental data \\cite{bketal2015, asdb2015a}. We put forward an alternative solution for these effects within a `hidden measurement formalism' elaborated by ourselves (see, e.g., \\cite{a86, asdb2014,asdb2015b,asdb2015c,a1998, a1999} and references therein), which goes beyond the Hilbert space formulation of quantum theory (probabilities), though it remains compatible with our operational-realistic description of conceptual entities \\cite{asdb2015a,asdb2015d}.\n\nFor the sake of completeness, we summarize the content of this paper in the following.\n\nIn Section~\\ref{foundations}, we present the epistemological foundations of the quantum theoretical approach to human cognition we developed in Brussels. We operationally describe a conceptual entity in terms of concrete experiments that are performed in psychological laboratories. Specifically,  the conceptual entity is the reality of the situation which every participant in an experiment is confronted with, and the different states of this conceptual entity are the different modes of reality of this experimental situation. There are contexts influencing the reality of this experimental situation, and the relevant ones of these contexts are elements of the SCoP structure, the theory of our approach, and their influence on the experimental situation is described as a change of state of the conceptual entity under consideration. There are also properties of this experimental situation, the relevant ones being elements of the SCoP structure, and they can be actual or potential, their `amount of actuality' (i.e. their `degree of availability in being actualized') being described by a probability measure.  The operational analogies between physical and conceptual entities suggest to represent the latter by means of the mathematical formalism of quantum theory in Hilbert space. Hence, we assume, in our research, the validity of quantum theory as a scientific paradigm for human cognition. On the basis of this assumption, we provide a unified presentation in Section \\ref{Hilbertmodeling} of the results obtained within a quantum theoretical modeling in knowledge representation, decision theory under uncertainty and behavioral economics. We emphasize that our research allowed us to identify new unexpected deviations from classical structures \\cite{asv2015,PhilTransA2015}, as well as new genuine quantum structures in conceptual combinations \\cite{as2014,asSpinWind2015,IQSA1}, which could not have been identified \nat the same fundamental level as it was possible in our approach\nif we would have adopted the \nmore traditional perspective only inquiring into the observed deviations from classical probabilistic structures. In Section~\\ref{challenges}, we analyze question order effects and response replicability and explain why a quantum theoretical modeling in Hilbert space of these situations is problematical. Finally, we present in Section~\\ref{beyond} a novel solution we recently elaborated for these cognitive situations \\cite{asdb2015a,asdb2015d}. The solution predicts a violation of the Hilbert space formalism, more specifically, the Born rule for probabilities is put at stake. We however emphasize that this solution remains  compatible with the general operational and realistic description of cognitive entities and their dynamics  given in Section~\\ref{foundations}.  In Section~\\ref{final}, we conclude our article by offering a few additional remarks, further emphasizing the coherence and advantage of our theoretical approach.  We stress, to conclude this section, that the deviation above from Hilbert space modeling should not be considered as an indication that we should better come back to more traditional classical approaches. On the contrary, we believe that new mathematical structures, more general than both pure classical and pure quantum structures, will be \nneeded in the modeling of cognitive processes.\n\n\n\n\\section{An operational-realistic foundation of cognitive psychology\\label{foundations}}\nMany quantum physicists agree that the phenomenology of microscopic particles is intriguing, but what is equally curious is the quantum mathematics that captures the mysterious quantum phenomena. Since the early days of quantum theory, indeed, scholars have been amazed by the the success of the mathematical formalism of quantum theory, as it was not clear at all how it had come about. This has inspired a long-standing research on the foundations of the Hilbert space formalism of quantum theory from physically justified axioms, resting on well defined empirical notions, more directly connected with the operations that are usually performed in a laboratory. Such an operational justification would make the formalism of quantum theory more firmly founded.\n\nOne of the well-known approaches to the foundations of quantum physics and quantum probability is the `Geneva-Brussels approach', initiated by Jauch \\cite{j68} and Piron \\cite{p76}, and further developed by our Brussels research team (see, e.g., \\cite{a82,a99a}). This research produced a formal approach, called `State Context Property' (SCoP) formalism, where any physical entity can be expressed in terms of the basic notions of `state', `context' and `property', which arise as a consequence of concrete physical operations on macroscopic apparatuses, such as preparation and registration devices, performed in spatio-temporal domains, such as physical laboratories. Measurements, state transformations, outcomes of measurements, and probabilities can then be expressed in terms of these more fundamental notions. If suitable axioms are imposed on the mathematical structures underlying the SCoP formalism, then the Hilbert space structure of quantum theory emerges as a unique mathematical representation, up to isomorphisms \\cite{bc81}. \n\nThere are still difficulties connected with the interpretation of some of these axioms and their physical justification, in particular for what concerns compound physical entities \\cite{a82}. But, \nbut this research line was a source of inspiration for the operational approaches applying the quantum formalism outside the microscopic domain of quantum physics \\cite{aa1995,aabg2000}. In particular, \nas we already mentioned in the Introduction,\na very similar realistic and operational representation of conceptual entities can be given for the cognitive domain, in the sense that the SCoP formalism can again be employed to formalize the more abstract conceptual entities in terms of states, contexts, properties, measurements and probabilities of outcomes \\cite{ga2002,ag2005a,ag2005b}.\n\nLet us first consider the empirical phenomenology of cognitive psychology. Like in physics, where laboratories define precise spatio-temporal \ndomains, we can introduce `psychological laboratories' where cognitive experiments are performed. These experiments are performed on \nsituations that are specifically `prepared' for the experiments, including experimental devices, and, for example, structured questionnaires, human participants that interact with the questionnaires in written answers, or each other, e.g., an interviewer and an interviewed. Whenever empirical data are collected from the responses of several participants, a statistics of the obtained outcomes arises. Starting from these empirical facts, we identify in our approach entities, states, contexts, measurements, outcomes and probabilities of outcomes, as follows. \n\nThe complex of experimental procedures conceived by the experimenter, the experimental design and setting and the cognitive effect that one wants to analyze, define a conceptual entity $A$, and are usually associated with a preparation procedure of a state of $A$. Hence, like in physics, the preparation procedure sets the \ninitial\nstate $p_A$ of the conceptual entity $A$ under study. Let us consider, for example, a questionnaire where a participant is asked to rank on a 7-point scale the membership of a list of items with respect to the concepts {\\it Fruits}, {\\it Vegetables} and their conjunction {\\it Fruits and Vegetables}. The questionnaire defines the states $p_{Fruits}$, $p_{Vegetables}$ and $p_{Fruits \\ and \\ Vegetables}$ of the conceptual entities {\\it Fruits}, {\\it Vegetables} and {\\it Fruits and Vegetables}, respectively. It is true that cognitive situations exist where the preparation procedure of the state of a conceptual entity is hardly controllable. Notwithstanding this, the state of the conceptual entity, defined by means of such a preparation procedure, is a `state of affairs'. It indeed expresses a `reality of the conceptual entity', in the sense that, once prepared in a given state, such condition is independent of any measurement procedure, and can be confronted with the different participants in an experiment, leading to outcome data and their statistics, exactly like in physics.\n\nA context $e$ is an element that can provoke a change of state of the conceptual entity. For example, the concept {\\it Juicy} can function as a context for the conceptual entity {\\it Fruits} leading to {\\it Juicy Fruits}, which can then be considered as a state of the conceptual entity {\\it Fruits}. A special context is the one introduced by the measurement itself. Indeed,  when the cognitive experiment starts, an interaction of a cognitive nature occurs between the conceptual entity $A$ under study and a participant in the experiment, in which the state $p_{A}$ of the conceptual entity $A$ generally changes, being transformed to  another state $p$. Also this cognitive interaction is formalized by means of a context $e$. For example, if the participant is asked to choose among a list of items, say, {\\it Olive}, {\\it Almond}, {\\it Apple}, etc., the most typical one with respect to {\\it Fruits}, and the answer is {\\it Apple}, then the\ninitial\nstate $p_{Fruits}$ of the conceptual entity {\\it Fruits} changes to $p_{Apple}$, i.e. the state describing the situation `the fruit is an apple',  as a consequence of the contextual interaction with the participant.\n\nThe change of the state of a conceptual entity due to a context may be either `deterministic', hence in principle predictable under the assumption that the state before the context acts is known, or `intrinsically probabilistic', in the sense that only the probability $\\mu(p,e,p_{A})$ that the  state $p_{A}$ of $A$ changes to the state $p$ is given. In the example above on typicality estimations, the typicality of the item {\\it Apple} for the concept {\\it Fruits} is formalized by means of the transition probability $\\mu(p_{Apple},e, p_{Fruits})$, where the context $e$ is the context of the typicality measurement.\n\nLike in physics, an important role is played by experiments with only two outcomes, the so-called `yes-no experiments'. Suppose that in an opinion poll a participant is asked to answer the question: ``Is Gore honest and trustworthy?''. Only two answers are possible: `yes' and `no'. Suppose that, for a given participant, the answer is `yes'. Then, the state  $p_{Honesty}$ of the conceptual entity {\\it Honesty and Trustworthiness} (which we will  \ndenote by {\\it Honesty}, for the sake of simplicity) changes to a new state \n$p_{Gy}$, which is the state describing the situation `Gore is honest'.\nHence, we \ncan\ndistinguish a class of  yes-no measurements on conceptual entities, as we do in physics. \n\nThe third step is the mathematical representation. We have seen that the Hilbert space formalism of quantum theory is general enough to capture an operational description of any entity in the micro-physical domain. Then, the strong analogies between the realistic and operational descriptions of physical and conceptual entities, in particular for what concerns the measurement process, suggest us to apply the same Hilbert space formalism when representing cognitive situations. Hence, each conceptual entity $A$ is associated with a Hilbert space ${\\mathcal H}$, and the state $p_A$ of $A$ is represented by a unit vector $|A\\rangle \\in \\mathcal H$. A yes-no measurement is represented by a spectral family $\\{M, \\mathbbmss{1}-M \\}$, where $M$ denotes an orthogonal projection operator over the Hilbert space $\\mathcal H$, and $\\mathbbmss{1}$ denotes the identity operator over $\\mathcal H$. The probability that the `yes' outcome is obtained in such a yes-no measurement when the conceptual entity $A$ is in the state represented by $|A\\rangle$ is \nthen\ngiven by the Born rule $\\mu(A)=\\langle A|M|A \\rangle$. For example, $M$ may represent an item $x$ that can be chosen in relation to a given concept $A$, so that its membership weight is given by $\\mu(A)$. \n\nThe Born rule obviously applies to measurement with more than two outcomes too.  For example, a typicality measurement involving a list of  $n$ different items $x_1$, \\ldots, $x_n$ with respect to a concept $A$ can be represented as a spectral measure $\\{M_1, \\ldots, M_n \\}$, where $\\sum_{k=1}^n M_k = \\mathbbmss{1}$ and $M_kM_l = \\delta_{kl}M_k$, such that the typicality $\\mu_{k}(A)$ of the item $x_k$ with respect to the concept $A$ is again given by the Born rule $\\mu_{k}(A)=\\langle A|M_k|A\\rangle$.\n\nAn interesting aspect concerns the final state of a conceptual entity $A$ after a human judgment. As above, we can assume the existence of a nonempty class of cognitive measurements that are ideal first kind measurements in the standard quantum sense, i.e. that satisfy the `L\\\"{u}ders postulate'. For example, if the typicality measurement of a list of items $x_1$, \\ldots, $x_n$ with respect to a concept $A$ gave the outcome $x_k$, then the final state of the conceptual entity after the measurement is represented by the unit vector $|A_k\\rangle=\\frac{M_{k}|A\\rangle}{\\sqrt{\\langle A |M_{k}|A\\rangle}}$. This means that the weights $\\mu_{k}(A)$ given by the Born rule can actually be interpreted as transition probabilities $\\mu(p_k,e,p_{A})$, where $e$ is the context producing the transitions from the initial state $p_{A}$ of the conceptual entity $A$, represented by the unit vector $|A\\rangle$, to one of the $n$ possible outcome states $p_k$, represented by the unit vectors $|A_k\\rangle$.\n \nSo, how can a Hilbert space model be actually constructed for a cognitive situation? To answer this question let us consider  again a conceptual entity $A$, in the state $p_{A}$, a cognitive measurement on $A$ described by means of  a context $e$, and suppose that the measurement has $n$ distinct outcomes, $x_1$, $x_2$, \\ldots, $x_n$. A quantum theoretical model for this situation can be constructed as follows. We associate $A$ with a\n$n$-dimensional complex Hilbert space $\\cal H$, and then consider an orthonormal base $\\{ |e_1\\rangle, |e_2\\rangle, \\ldots, |e_n\\rangle \\}$ in $\\cal H$ (since ${\\cal H}$ is isomorphic to the Hilbert space ${\\mathbb C}^{n}$, the orthonormal base of $\\cal H$ can be the canonical base of ${\\mathbb C}^{n}$). Next, we represent the cognitive measurement described by $e$ by means of the spectral family $\\{M_1, M_2, \\ldots, M_n \\}$, where $M_k=|e_k\\rangle\\langle e_k|$, $k=1,2,\\ldots,n$. Finally, the probability that the measurement $e$ on the conceptual entity $A$ in the state $p_{A}$\ngives the outcome $x_k$ is given by $\\mu_{k}(A)=\\langle A|M_k|A\\rangle=\\langle A|e_k\\rangle\\langle e_k|A\\rangle=|\\langle e_k|A\\rangle|^2$.\n\nWhat about the interpretation of the Hilbert space formalism above? Two major points should now be reminded, namely: \n\n(i) the states of conceptual entities describe the `modes of being' of these conceptual entities;\n\n(ii) in a cognitive experiment, a participant \nacts as a (measurement) context for the conceptual entity, changing its state.\n\nThis means that, as we mentioned already, the  state $p_{A}$ of the conceptual entity $A$ is represented in the Hilbert space formalism by the unit vector $|A\\rangle$, the possible outcomes $x_k$ of the experiment by the base vectors $|e_k\\rangle$, and the action of a participant (or the overall action of the ensemble of participants) as the state transformation $|A\\rangle \\to |e_k\\rangle$ induced by the orthogonal projection operator $M_k=|e_k\\rangle\\langle e_k|$, if the outcome $x_k$ is obtained, so that the probability of occurrence of $x_k$ can also be written as $\\mu_{k}(A)=\\mu(|e_k\\rangle,e,|A\\rangle)$, where $e$ is the measurement context associated with the spectral family $\\{M_1, M_2, \\ldots, M_n \\}$.\n\nIt follows from (i) and (ii) that a state, hence a unit vector in the Hilbert space representation of states, does not describe the subjective \nbeliefs \nof a person, or collection of persons, about a conceptual entity. Such subjective  beliefs are rather incorporated in the cognitive interaction between the cognitive situation and the human participants deciding on that cognitive situation. In this respect, our operational quantum approach to human cognition  is also a realistic one, and thus it departs from other approaches that apply the mathematical formalism of quantum theory to model cognitive processes \\cite{k2010,bb2012,hk2013,pb2013,pnas,plosone}. Of course, one could say that the difference between interpreting the quantum state as a `state of belief' of a participant in the experiment, or as a `state of a conceptual entity', i.e. a `state of the situation which the participant is confronted with during an experiment', is only a question of philosophical interpretation, but comes to the same when it concerns the methodological development of the approach. Although this is definitely partly true, we do not fully agree with it. Interpretation and methodology are never completely separated. A certain interpretation, hence giving rise to a specific view on the matter, will give rise to other ideas of how to further develop the approach, how to elaborate the method, etc., than another interpretation, with another view, will do. We believe that an operational-realistic approach, being balanced between attention for idealist as well as realist philosophical \ninterpretations,\ncarries in this sense a particular strength, \nprecisely\ndue to this balance. A good example of this is how we were inspired to use the superposition principle of quantum theory in our modeling of concepts as conceptual entities. We represented the combination of two concepts by a state that is the linear superposition of the states describing the component concepts. This way of representing combined conceptual entities captures the nature of emergence, exactly like in physics. It would not be obvious to put forward \nthis\ndescription when state of beliefs are the focus of what can be predicted.\n\nWe stress a third point that is important, in our opinion. For most situations, we interpret the effect of the cognitive context on a conceptual entity in a decision-making process as an `actualization of pure potentiality'. Like in quantum physics, the (measurement) context does not reveal pre-existing properties of the entity but, rather, it makes actual properties that were only potential in the initial state of the entity (unless the initial state is already an eigenstate of the measurement in question, like in physics) \\cite{ga2002,ag2005a,ag2005b}.\n\nIt follows from the previous discussion that our research investigates the validity of quantum theory as a general, unitary and coherent theory for human cognition. Our quantum theoretical models, elaborated for specific cognitive situations and data, derive from quantum theory as a consequence of the assumptions about this general validity. As such, these models are subject to the technical and epistemological constraints of quantum theory. \nIn other terms,\nour quantum modeling rests on a `theory based approach', and should be distinguished from an `ad hoc modeling based approach', only devised to fit data. In this respect, one should be suspicious of models in which free parameters are added on an `ad hoc' basis to fit the data more closely in specific experimental situations. In our opinion, the fact that our `theory derived model' reproduces different sets of experimental data constitutes in itself a convincing argument to support its advantage over traditional modeling approaches and to extend its use to more complex cognitive situations (in that respect, see also our final remarks in Section~\\ref{final}).\n\nWe present in Section \\ref{Hilbertmodeling} the results obtained in our quantum theoretical approach in the light of the epistemological perspective of this section.\n\n\n\n\n\\section{On the modeling effectiveness of Hilbert space\\label{Hilbertmodeling}}\nThe quantum approach to cognition described  in Section \\ref{foundations} produced concrete models in Hilbert space, which faithfully matched different sets of experimental data collected to reveal `decision-making errors' and `probability judgment errors'. This allowed us to identify genuine quantum structures in the cognitive realm. We present a reconstruction of the attained results in the following.\n\nThe first set of results concerns knowledge representation and conceptual categorization and combination. James Hampton collected data on how people rate membership of items with respect to pairs of concepts and their combinations, conjunction \\cite{h1988a}, disjunction \\cite{h1988b} and negation \\cite{h1997}. By using the data in \\cite{h1988b}, we reconstructed the typicality estimations of 24 items with respect to the concepts {\\it Fruits} and {\\it Vegetables} and their disjunction {\\it Fruits or Vegetables}. We showed that the concepts {\\it Fruits} and {\\it Vegetables} interfere when they combine to form {\\it Fruits or Vegetables}, and the state of the latter can be represented by the linear superposition of the states of the former.  This behavior is analogous to that of quantum particles interfering in the double-slit experiment when both slits are open. The data are faithfully represented in a 25-dimensional Hilbert space over complex numbers \\cite{abgs2013,ags2013}.\n\nIn the data collected on the membership estimations of items with respect to pairs $(A,B)$ of concepts and their conjunction `$A$ and $B$' and disjunction `$A$ or $B$', Hampton found systematic violations of the rules of classical (fuzzy set) logic and probability theory. For example, the membership weight of the item {\\it Mint} with respect to the conjunction {\\it Food and Plant} is higher than the membership weight of {\\it Mint} with respect to both {\\it Food} and {\\it Plant} (`overextension'). Similarly, the membership weight of the item {\\it Ashtray} with respect to the disjunction {\\it Home Furnishing or Furniture} is lower than the membership weight of {\\it Ashtray} with respect to both {\\it Home Furnishing} and {\\it Furniture} (`underextension'). We showed that overextension and underextension are natural expressions of `conceptual emergence' \\cite{a2009a,ags2013}. Namely, whenever a person estimates the membership of an item  $x$ with respect to the pair $(A,B)$ of concepts and their combination $C(A,B)$, two processes act in the person's mind. The first process is guided by `emergence', that is, the person estimates the membership of  $x$ with respect to the new emergent concept $C(A,B)$. The second process is guided by `logic', that is, the person separately estimates the membership of $x$ with respect to $A$ and $B$ and applies \na probabilistic logical calculus to estimate the membership of $x$ with respect to $C(A,B)$  \\cite{IQSA2}. More important, the new concept $C(A,B)$ emerges from the concepts $A$ and $B$, exactly as the linear superposition of two quantum states emerges from the component states. A two-sector Fock space faithfully models Hampton's data, and was later successfully applied to the modeling of more complex situations involving concept combinations (see, e.g., \\cite{IQSA2,s2015}).\n\nIt is interesting to note that the size of deviation of classical probabilistic rules due to overextension and underextension generally depends on the item  $x$ \nand the specific combination $C(A,B)$ of the concepts $A$ and $B$ that are investigated. However, we recently performed a more general experiment in which we asked the participants to rank the membership of items with respect to the concepts $A$, $B$, their negations `not $A$', `not $B$', and the conjunctions `$A$ and $B$', `$A$ and not $B$', `not $A$ and $B$', and `not $A$ and not $B$'.  We surprisingly found that the size of deviation from classicality in this experiment does not depend on either the item or the pair of concepts or the specific combination, but shows to be a numerical constant. Even more surprisingly, our two-sector Fock space model correctly predicts the value of this constant, capturing in this way a deep non-classical mechanism connected in a fundamental way with the mechanism of conceptual formation itself rather than only specifically with the mechanism of conceptual combination \\cite{asv2015,PhilTransA2015}.\n\nDifferent concepts entangle when they combine, where `entanglement' is meant in the standard quantum sense. We proved this feature of concepts in two experiments. In the first experiment, we asked the participants to choose the best example for the conceptual combination {\\it The Animal Acts} in a list of four examples, e.g., {\\it The Horse Growls}, {\\it The Bear Whinnies}, {\\it The Horse Whinnies} and {\\it The Bear Growls}. By suitably combining exemplars of {\\it Animal} and exemplars of {\\it Acts}, we performed four joint measurements on the combination {\\it The Animal Acts}. The expectation values violated the `Clauser-Horne-Shimony-Holt' version of Bell inequalities \\cite{b1964,chsh1969}. The violation was such that, not only the state of {\\it The Animal Acts} was entangled, but also the four joint measurements were entangled, in the sense that they could not be represented in the Hilbert space ${\\mathbb C}^{4}$ as the (tensor) product of a measurement performed on the concept {\\it Animal} and a measurement performed on the concept {\\it Acts} \\cite{as2014}. In the second experiment, performed on the conceptual combination {\\it Two Different Wind Directions}, we confirmed the presence of quantum entanglement, but we were also able to prove that the empirical violation of the marginal law in this type of experiments is due to a bias of the participants in picking wind directions. If this bias is removed, which is what we did in an ensuing experiment on {\\it Two Different Space Directions}, one can show that people pick amongst different space directions exactly as coincidence spin measurement apparatuses pick amongst different spin directions of a compound system in the singlet spin state. In other words, entanglement in concepts can be proved from only the statistics of the correlations of joint measurements on combined concepts, exactly as in quantum physics \\cite{asSpinWind2015}.\n\nSince concepts exhibit genuine quantum features when they combine pairwise, it is reasonable to expect that these features should be reflected in the statistical behavior of the combination of several identical concepts. Indeed, we detected quantum-type indistinguishability in an experiment on the combination of identical concepts, such as the combination {\\it Eleven Animals}. More specifically, we found significant evidence of deviation from the predictions of classical statistical theories, i.e. `Maxwell-Boltzmann distribution'. This deviation has \nclear analogies with the deviation of quantum mechanical from classical mechanical statistics, due to indistinguishability of microscopic quantum particles, that is, we found convincing evidence of the presence of `Bose-Einstein distribution'. In the experiment, indeed, people do \nnot seem to distinguish two identical concepts in the combination of $N$ identical concepts, which is more evident in more abstract than in more concrete concepts, as expected \\cite{IQSA1}. \n\nThe second set of results concern `decision-making errors under uncertainty'. In the `disjunction effect' people prefer action $x$ over action $y$ if they know that an event $A$ occurs, and also if they know that $A$ does not occur, but they prefer $y$ over $x$ if they do not know whether $A$ occurs or not. The disjunction effect violates a fundamental principle of rational decision theory, Savage's `Sure-Thing principle' and, more generally, the total probability rule of classical probability \\cite{s1954}. This preference of sure over unsure choices violating the Sure-Thing principle was experimentally detected in the `two-stage gamble' and in the `Hawaii problem' \\cite{ts1992}. In the experiment on a gamble that can be played twice, the majority of participants prefer to bet again when they know they won in the first gamble, and also when they know they lost in the first gamble, but they generally prefer not to play when they do not know whether they won or lost. In the Hawaii problem, most students decide to buy the vacation package when they know they passed the exam, and also when they know they did not pass the exam, but they generally decide not to buy the vacation package when they do not know whether they passed or not passed the exam. We recently showed that, in both experimental situations, this `uncertainty aversion' can be explained as an effect of underextension of the conceptual entities $A$ and `not $A$' with respect to the conceptual disjunction `$A$ or not $A$', where the latter describes the situation of not knowing which event, $A$ or `not $A$', will occur. The concepts $A$ and `not $A$' interfere in the disjunction `$A$ or not $A$', which determines its underextension. A Hilbert space model in ${\\mathbb C}^{3}$ allowed us to reproduce the data in both experiments on the disjunction effect \\cite{s2015}.\n\nEllsberg's thought experiments, much before the disjunction effect, revealed that the Sure-Thing principle is violated in concrete decision-making under uncertainty, as people generally prefer known over unknown probabilities, instead of maximizing their expected utilities. In the famous `Ellsberg three-color example', an urn contains 30 red balls and 60 balls that are either yellow or black, in unknown proportion. One ball will be drawn at random from the urn. The participant is firstly asked to choose between betting on `red' and betting on `black'. Then, the same participant is asked to choose between betting on `red or yellow' and betting on `black or yellow'. In each case, the `right' choice will be awarded with $\\$100$. As the events `betting on red' and `betting on black or yellow' are associated with known probabilities, while their counterparts are not, the participants will prefer betting on the former than betting on the latter, thus revealing what Ellsberg called `ambiguity aversion', and violating the Sure-Thing principle \\cite{e1961}. This pattern of choice has been confirmed by several experiments in the last thirty years \\cite{ms2014}. Recently, Machina identified in a couple of thought experiments, the `50\/51 example' and the `reflection example', a similar mechanism guiding human preferences in specific ambiguous situations, namely, `information symmetry' \\cite{m2009,blhp2011}, which was experimentally confirmed in \\cite{lhp2010}. In our quantum theoretical approach, ambiguity aversion and information symmetry are two possible cognitive contexts influencing human preferences in uncertainty situations and changing the states of the `Ellsberg and Machina conceptual entities', respectively. Hence, an ambiguity aversion context will change the state of the Ellsberg conceptual entity in such a way that `betting on red' and `betting on black or yellow' are finally preferred. In other terms, the novel element of this approach is that the initial state of the conceptual entity, in its Hilbert space representation, can also change because of the pondering of the participants in relation to certain choices, before being collapsed into a given outcome.  This opens the way to a generalization of rational decision theory with quantum, rather than classical, probabilities \\cite{asJMP2015}.\n\nThe results above provide a strong confirmation of \nthe quantum theoretical approach presented in Section \\ref{foundations}, and we expect that further evidence will be given in this direction in the years to come. In the next section we instead intend to analyze some situations where deviations from Hilbert space modeling of human cognition apparently occur.\nWe will see in Section~\\ref{beyond} that these deviations are however compatible with the general  operational-realistic framework  portrayed in Section~\\ref{foundations}.\n\n\n\n\\section{Deviating from Hilbert space\\label{challenges}} \nAs mentioned in Section~\\ref{foundations}, if suitable axioms are imposed on the SCoP formalism, the Hilbert space structure of quantum theory can be shown to emerge uniquely, up to isomorphisms \\cite{bc81}. However, we also know that certain experimental situations can violate some of these axioms. This is the case for instance when we consider entities formed by experimentally separated sub-entities, a situation that cannot be described by the standard quantum formalism \\cite{a82,a1984}. Similarly, one may expect that the structural shortcomings of the standard quantum formalism can also manifest in the ambit of psychological measurements, in the form of data that cannot be exactly modeled (or jointly modeled) by means of the specific Hilbert space geometry and the associated Born rule. The purpose of this section is to describe two paradigmatic examples of situations of this kind: `question order effects' and `response replicability'. In the following section, we then show how the quantum formalism can be naturally completed to also faithfully model these data, in a way that remains consistent with our operational-realistic approach. \n\nLet us first remark that the mere situation of having to deal with a set of data for which we don't have yet a faithful Hilbert space model should not make one necessarily search for an alternative more general quantum-like mathematical structure as a modeling environment. Indeed, it is very well possible that the adequate Hilbert space model has not yet been found. Recently, however, a specific situation was identified and analysed indicating that the standard quantum formalism in Hilbert space would not be able to be used to model it \\cite{plosone}. This situation combines two phenomena: `question order effects' and `response replicability'. We start by explaining `question order effects' and how the cognitive situation in which they appear can be represented in Hilbert space.\n\nFor this we come back to the yes-no experiment of Section~\\ref{foundations}, where participants are asked: ``Is Gore honest and trustworthy?''. This experiment gives rise to a two-outcome measurement performed on the conceptual entity {\\it Honesty} in the initial state $p_H$, represented by the unit vector $|H\\rangle\\in {\\mathcal H}$, where ${\\mathcal H}$ is a two-dimensional Hilbert space if we assume the measurement to be non-degenerate, or more generally a $n$-dimensional Hilbert space if we also admit the possibility of sub-measurements. Denoting $\\{M_G, \\bar{M}_G = \\mathbbmss{1}-M_G\\}$ the spectral family associated with this measurement, the probability of the `yes' outcome (i.e. to answer `yes' to the question about Gore's honesty and trustworthiness) is then given by the Born rule $\\mu_{Gy}(H)=\\langle H|M_G|H\\rangle$, and of course $\\mu_{Gn}(H)=\\langle H|\\bar{M}_G|H\\rangle = 1-\\mu_{Gy}(H)$ is the probability for the `no' outcome. We then consider a second measurement performed on the conceptual entity {\\it Honesty}, but this time associated with the question: ``Is Clinton honest and trustworthy?''. We denote $\\{M_C, \\bar{M}_C = \\mathbbmss{1}-M_C\\}$ the spectral family associated with this second measurement, so that the probabilities for the `yes' and `no' outcomes are again given by $\\mu_{Cy}(H)=\\langle H|M_C|H\\rangle$ and $\\mu_{Cn}(H)=\\langle H|\\bar{M}_C|H\\rangle$, respectively. \n\nStarting from these two measurements, it is possible to conceive sequential measurements, corresponding to situations where the respondents are subject to the Gore and Clinton questions in a succession, one after the other, in different orders. Statistical data about `Clinton\/Gore' sequential measurements were reported in a seminal article on question order effects \\cite{m2002} and further analyzed in \\cite{bb2012,wb2013}. More precisely, after fixing a rounding error in \\cite{wb2013}, we have the following sequential (or conditional) probabilities \\cite{asdb2015a}: \n\\begin{eqnarray}\n&\\mu_{CyGy}(H)=0.4899 \\quad \\mu_{CyGn}(H)= 0.0447 \\quad \\mu_{CnGy}(H)=0.1767 \\quad \\mu_{CnGn}(H)= 0.2887 \\label{Prob-Clinton-Gore1}\\\\\n&\\mu_{\\rm GyCy}(H)=0.5625 \\quad \\mu_{GyCn}(H) = 0.1991 \\quad \\mu_{GnCy}(H) = 0.0255 \\quad \\mu_{GnCn}(H) = 0.2129\n\\label{Prob-Clinton-Gore2}\n\\end{eqnarray}\nwhere (\\ref{Prob-Clinton-Gore1}) corresponds to the sequence where first the Clinton and then the Gore measurements are performed, whereas (\\ref{Prob-Clinton-Gore2}) corresponds to the reversed order sequence for the measurements. Considering that the probabilities in each of the four columns above are sensibly different, these data describe typical `question order effects'. \n\nQuantum theory is equipped with a very natural tool to model question order effects: `incompatible measurements', as expressed by the fact that two self-adjoint operators, and the associated spectral families, in general do not commute. More precisely, the Hilbert space expression for the probability that, say, we obtain the answer $CyGn$ when we perform first the Clinton measurement and then the Gore one, is \\cite{bb2012,wb2013}: $\\mu_{CyGn}(H)= \\langle H|M_C\\bar{M}_GM_C|H\\rangle$. Similarly, the probability to obtain the outcome $GnCy$, for the sequential measurement in reversed order, is: $\\mu_{GnCy}(H)= \\langle H|\\bar{M}_GM_C\\bar{M}_G|H\\rangle$. Since we have the operatorial identity $\\bar{M}_GM_C\\bar{M}_G-M_C\\bar{M}_GM_C=(M_G-M_C)[M_G,M_C]$, the difference $\\mu_{GnCy}(H)-\\mu_{CyGn}(H)$ will generally be non-zero if $[M_G,M_C]\\neq 0$, i.e. if the spectral families associated with the two measurements do not commute. In the following we will analyse whether non-compatibility within a standard quantum approach can cope in a satisfying way with these question order effects, and show that a simple `yes' to this question is not possible. Indeed, a deep problem already comes to the surface in relation to the phenomenon of `response replicability'.\n\nConsider again the Gore\/Clinton measurements: if a respondent says `yes' to the Gore question, then is asked the Clinton question, then again is asked the Gore question, the answer given to the latter will almost certainly be a `yes', independently of the answer given to the intermediary Clinton question. This phenomenon is called `response replicability'. If in addition to question order effects also response replicability is jointly modeled in  Hilbert space quantum mechanics, a contradiction can be detected, as shown in \\cite{plosone}. Let us indicate what are the elements that produce this contradiction. In standard quantum mechanics only if a state is an eigenstate of the considered measurement the outcome `yes' will be certain in advance. Also, measurements that can transform an arbitrary initial state into an eigenstate are ideal measurements called of the first kind.  According to response replicability, outcomes that once have been obtained for a measurement will have to become certain in advance if this same measurement is performed a second time. This means that the associated measurements should be ideal and of the first kind. For the case of the Gore\/Clinton measurements, and the situation of response replicability mentioned above, this means that the Gore measurement should be ideal and of the first kind. But  one can also consider the situation \nwhere first the Clinton measurement is performed, then the Gore measurement and afterwards the Clinton measurement again. A similar analysis leads then to the Clinton measurement needing to be ideal and of the first kind. This means however that after more than three measurements that alternate between Clinton and Gore, the state needs to have become an eigenstate of both measurements. As a consequence, both measurements can be shown to be represented by commuting operators. The proof of the contradiction between `response replicability' and 'non-commutativity' worked out in \\cite{plosone} is formal and also more general than the intuitive reasoning presented above -- for example, the contradiction is also proven when measurements are represented by positive-operator valued measures instead of projection valued measures, which is what we have considered here -- and hence indicates that the non-commutativity of the self-adjoint operators needed to account for the question order effects cannot be realised together with the `ideal and first kind' properties\nneeded to account for the response replicability within a standard quantum Hilbert space setting.      \n\nAlthough refined experiments would be needed to reveal the possible reasons for response replicability, it is worth to put forward some intuitive ideas, as we have been developing a quantum-like but more general than Hilbert space formalism within our Brussels approach to quantum cognition \\cite{asdb2014,asdb2015b,asdb2015c}, and we believe that we can cope with the above contradiction within this more general quantum-like setting in a very natural way. It seems to be a plausible hypothesis that response replicability is, at least partly, due to a multiplicity of effects, that however take place during the experiment itself, such as desire of coherence, learning, fear of being judged when changing opinion, etc. And a crucial aspect for both  question order effects and response replicability appearing\nin the Gore\/Clinton situation is that the sequential measurements need to be carried out with the same participant, who has to be tested again and again. This is different than\nthe situation in quantum physics, where order effects appear for non-commuting observables also when sequential measurements are performed with different apparatuses. \nHence, both question order effects and response replicability seem to be the consequence of `changes taking place in the way each subject responds probabilistically to the situation -- described by the state of the conceptual entity in our approach -- he or she is confronted with during a measurement'. Since the structure of the probabilistic response to a specific state is fixed in quantum mechanics, being determined by the Born rule, it is clear that such a change of the probabilistic response to a given measurement, when it is repeated in a sequence of measurements, cannot be accounted for by the standard quantum formalism. And it is exactly such structure of the probabilistic response to a same measurement with respect to a given state that can be varied in the generalized quantum-like theory that we have been developing \\cite{asdb2014,asdb2015b,asdb2015c}. This is the reason that, when we became aware of the contradiction identified in \\cite{plosone}, we were tempted to investigate whether in our generalized quantum-like theory the contradiction would vanish, and response replicability would be jointly\nmodelizable with question order effects. And indeed, we could obtain a positive result with respect to this issue \\cite{asdb2015a}, which we will now sketch in the next section. \n\n\n\n\n\\section{Beyond-quantum models\\label{beyond}}\nWe presented in Section \\ref{challenges} two paradigmatic situations in human cognition that cannot be modeled together using the standard quantum formalism. We want now to explain how the latter can be naturally extended to also deal with these situations, still remaining in the ambit of a unitary and coherent framework for cognitive processes. \n\nFor this, we introduce a formalism where the probabilistic response with respect to a specific experimental situation, i.e. a state of the conceptual entity under consideration, can vary, and hence can be different than the one compatible with the Born rule of standard quantum theory. This formalism, called \nthe `extended Bloch representation' of quantum mechanics \\cite{asdb2014}, \nexploits\nin its most recent formulation the fact that the states of a quantum entity (described as ray-states or density matrix-states) can be uniquely mapped into a convex portion of a generalized unit Bloch sphere, in which also measurements can be represented in a natural way, by means of appropriate simplexes having the eigenstates as vertex vectors. A measurement can then be described as a process during which an abstract point particle (representing the initial state of the quantum entity) enters into contact with the measurement simplex, which then, as if it was an elastic and disintegrable hyper-membrane, can collapse to one of its vertex points (representing the outcomes states) or to a point of one of its sub-simplexes (in case the measurement would be degenerate). \n \nWe do not enter here into the details of this remarkable process, and refer the reader to the detailed descriptions  in \\cite{asdb2015a,asdb2014,asdb2015b,asdb2015c}. For our present purposes, it \nwill be\nsufficient to observe that a measurement simplex, considered as an abstract membrane that can collapse as a result of some uncontrollable environmental fluctuations, can precisely model that aspect of a measurement that in the quantum jargon is called `wave function collapse'. More precisely, when the abstract point particle enters into contact with the `potentiality region' represented by such membrane, it creates some `tension lines' partitioning the latter into different subregions, one for each possible outcome. The collapse of the membrane towards one of the vertex points then depends on which subregion disintegrates first, so that the different outcome probabilities can be expressed as the relative Lebesgue measures of these subregions (the larger a subregion, the higher the associated probability). In other terms, this membrane's mechanism, with the tension lines generated by the abstract point particle, is a mathematical representation of a sort of `weighted symmetry breaking' process. Now, thanks to the remarkable geometry of simplexes, it can be proven that if the membrane is chosen to be uniform, thus having the same probability of disintegrating in any of its points (describing the different possible measurement-interactions), the collapse probabilities are exactly given by the Born rule. In other terms, the latter can be derived, and explained, as being the result of a process of actualization of potential hidden-measurement interactions, so that the extended Bloch representation constitutes a possible solution to the measurement problem.\n\nThus, when the membrane is uniform, the `way of choosing' an outcome is precisely the `Born way'. However, a uniform membrane is a very special situation, and it is natural to also consider membranes whose points do not all\nhave the same probability of disintegrating, i.e. membranes whose disintegrative processes are described by non-uniform probability densities $\\rho$, which we simply call $\\rho$-membranes. Non-uniform $\\rho$-membranes can produce outcome probabilities different from the standard quantum ones and give rise to probability models different from the Hilbertian one (even though the state space is a generalized Bloch sphere derived from the Hilbert space geometry\\footnote{More general state spaces can also be considered, in what has been called the `general tension-reduction' (GTR) model \\cite{asdb2015b,asdb2015c,asdb2015d}.}). But this is exactly what one needs in order to account, in a unified framework, for the situation we encounter when combining the phenomena of `response replicability' and `question order effects', as previously described and analysed in \\cite{plosone}.\n\nWe thus see that it is possible to naturally complete the quantum formalism to obtain a finer grained description of psychological experiments in which the probabilistic response of a measurement with respect to a state can be different to the one described by the Born rule. Additionally, our generalized quantum-like theory also explains why, despite the fact that individual measurements are possibly associated with different non-Born probabilities, the Born rule nevertheless appears to be a very good approximation to describe numerous experimental situations. This is related to the notion of `universal measurement', firstly introduced by one of us in \\cite{a1998} and further analyzed in \\cite{asdb2014,asdb2015b,asdb2015c,asdb2014b}. In a nutshell, a universal measurement is a measurement whose probabilities are obtained by averaging over the probabilities of all possible quantum-like measurements sharing a same set of outcomes, in a same state space. In other terms, a universal measurement corresponds to an average over all possible non-uniform $\\rho$-membranes, associated with a given measurement simplex. Following a strategy similar to that used in the definition of the `Wiener measure', it is then possible to show that if the state space is Hilbertian (more precisely, a convex set of states inscribed in a generalized Bloch sphere, inherited from a Hilbert space), then the probabilities of a universal measurement are precisely those predicted by the Born rule. \n\nIn \\cite{asdb2015a} we could show that the joint situation of question order effects and response replicability for the data collected with respect to the Gore\/Clinton measurements, and others, is modelizable\nwithin our generalized quantum theory by introducing non-Born type measurements. However, we  were also able to provide a better modeling of the question order effects data as such. Indeed, using standard Born-probability quantum theory it was only possible to model approximately these data in earlier studies \\cite{wb2013}. This is due to the existence of a general algebraic equality about sequential measurements in standard quantum mechanics which is the following \\cite{asdb2015a,wb2013,Niestegge2008}:\n\\begin{equation}\nQ\\equiv M_GM_CM_G - M_CM_GM_C + \\bar{M}_G\\bar{M}_C\\bar{M}_G - \\bar{M}_C\\bar{M}_G\\bar{M}_C=0\n\\label{QQ-equality}\n\\end{equation}\nwhere $\\{M_G, \\bar{M}_G = \\mathbbmss{1}-M_G\\}$ and $\\{M_C, \\bar{M}_C = \\mathbbmss{1}-M_C\\}$ are the spectral families associated with the Hilbert model of the Gore and Clinton measurements introduced in Section \\ref{challenges}.\nTaking the average $q=\\langle H|Q|H\\rangle$, one thus obtains, more specifically: \n\\begin{equation}\nq\\equiv\\mu_{GyCy}(H) - \\mu_{CyGy}(H) + \\mu_{GnCn}(H) - \\mu_{CnGn}(H) =0.\n\\label{QQ-equality2}\n\\end{equation}\nThis equality has been called the `QQ-equality', and can be used as a test for the quantumness of the probability model, but only in the sense that a quantum model, necessarily, has to obey it, although the fact that it does so is not a guarantee that the model will be Hilbertian. Inserting the experimental values (\\ref{Prob-Clinton-Gore1})-(\\ref{Prob-Clinton-Gore2}) into (\\ref{QQ-equality2}), one finds $q=0.0032\\neq 0$. \nThis value is small (being only $0.32\\%$ of the maximum value $q$ can take, which is 1), which is the reason that approximate modeling can be obtained within standard quantum mechanics \\cite{wb2013}. Note however that (\\ref{QQ-equality}) does not depend on the dimension of the Hilbert space considered, which means that even in higher dimensional Hilbert spaces, if degenerate measurements \nare\nconsidered, an exact modeling \nwould\nstill be impossible to obtain. We have reasons to believe that also question order effects, with the QQ-equality standing in the way of an exact modeling of the data, contain an indication for the need to turn to a more general quantum-like theory, such as the one we used to cope with the joint phenomenon of question order effects and response replicability. We \npresent some arguments in this regard in the following of this section.\n\nFirst,  \nwe\nnote that in case one would choose a two-dimensional Hilbert space, which is the natural choice when dealing with two-outcome measurements, additional equalities can be written that are this time strongly violated by the data. As an example, consider the quantity \\cite{asdb2015a}:\n\\begin{eqnarray}\nq'&\\equiv& \\mu_{CyGn}(H)\\mu_{CnGn}(H) - \\mu_{CnGy}(H)\\mu_{CyGy}(H)\\\\\n&=& \\langle H|M_C\\bar{M}_GM_C|H\\rangle \\langle H|\\bar{M}_C\\bar{M}_G\\bar{M}_C|H\\rangle - \\langle H|\\bar{M}_CM_G\\bar{M}_C|H\\rangle \\langle H|M_CM_GM_C|H\\rangle\n\\label{QQ-equality3}\n\\end{eqnarray}\nIf the Hilbert space is two-dimensional, one can write $M_G=|G\\rangle\\langle G|$, $\\bar{M}_G=|\\bar{G}\\rangle\\langle \\bar{G}|$, as well as $M_C=|C\\rangle\\langle C|$, $\\bar{M}_C=|\\bar{C}\\rangle\\langle \\bar{C}|$. Replacing these expressions into (\\ref{QQ-equality3}) one finds, after some easy algebra, that $q'=0$. However, inserting the experimental values (\\ref{Prob-Clinton-Gore1})-(\\ref{Prob-Clinton-Gore2}) into (\\ref{QQ-equality3}), one finds $q'=-0.073\\neq 0$, which not only \nis not zero, but also $29.2\\%$ of the maximum value that $q'$ can take (which is $0.25$).\n\nSecond, let us repeat our intuitive reasoning as to why measurements in the situation of response replicability carry non-Bornian probabilities. Due to the local \ncontexts\nof the collection of sequential measurements, Gore, Clinton, and then Gore again, the third measurement internally changes into a non-Bornian one, and more specifically a deterministic one for the considered state, since response replicability means that for all subsequent Gore measurements the same outcome is assured. It might well be the case, although an intuitive argument would be more complex \nto give\nin this case, that also for the situation of question order effects, precisely because they only appear if a same human mind is sequentially interrogated, non-Bornian probabilities would be required. An even stronger hypothesis, which we plan to investigate in the future, is that most individual human minds, and perhaps even all, would carry in general non-Bornian probabilities, so that the success of standard quantum mechanics and Bornian probabilities would be mainly an effect of averaging over a sufficiently large set of different human minds, which effectively is what happens in a standard psychological experiment. If this last hypothesis is true, the violation of the Born rule for question order effects and  response replicability would be quite natural, since the same human mind is needed to provoke these effects. Indeed, our analysis in \\cite{asdb2015b,asdb2015c} shows that standard quantum probabilities in the modeling of human cognition can be explained by considering that in numerous experimental situations the average over the different participants will be quite close to that of a universal measurement, which as we observed is exactly given by the Born rule. In other terms, even if the probability model of an individual psychological measurement could be non-Hilbertian, it will generally admit a first order approximation, and when the states of the conceptual entity under investigation can be described by means of a Hilbert space structure, this first order approximation will precisely correspond to the quantum mechanical Born rule. \n\nIf the above considerations provide an interesting piece of explanation as to why the Born rule is generally successful also beyond the micro-physical domain, at the same time it also contains a plausible reason of why it will possibly be not successful in all experimental situations, i.e. when the average is either not large enough, or when the experiment is so conceived that it doesn't apply as such. This could be the typical situation of question order effects and response replicability, since in this case we do not consider an average over single measurements, but over sequential (conditional) measurements. And this could be an explanation of why Hilbertian symmetries like those described above can be easily violated and that it will not be possible, by means of the Born rule, to always obtain an exact fit of the data \\cite{asdb2015a,asdb2015d}.\n\nAdditionally, as we said, it allowed us to precisely fit the data by using the extended Bloch representation,\nand more specifically \nsimple one-dimensional locally uniform membranes inscribed in a 3-dimensional Bloch sphere that \ncan\ndisintegrate (i.e. break) only inside a connected internal region \\cite{asdb2015a}. Thanks to this modeling, we could also understand that the reason the Clinton\/Gore and similar data appear to almost obey the QQ-equality (\\ref{QQ-equality2}) is quite different from the reason the equality is obeyed by pure quantum probabilities. Indeed, in a pure quantum model two specific contributions to the $q$-value (\\ref{QQ-equality2}), called the `relative indeterminism' and `relative asymmetry' contributions, are necessarily both identically zero, whereas we \ncould\nshow, using our extended model, that for the data (\\ref{Prob-Clinton-Gore2}),\nand similar data,  \nthese two contributions are both very different from zero, but happen to almost cancel each other, thus explaining why the $q=0$ equality is almost obeyed, although the probabilities are manifestly non-Bornian \\cite{asdb2015a}.\n\n\n\n\\section{Final considerations\\label{final}}\nIn this article we explained the essence of the operational-realistic approach to cognition developed in Brussels, which in turn originated from the foundational approach to physics elaborated initially in Geneva and then in Brussels (in what has become known as the `Geneva-Brussels school'). Our emphasis was that this approach is sufficiently general, and fundamental, to provide a unitary framework that can be used to coherently describe, and realistically interpret, not only standard quantum theory, but also its natural extensions, like the extended Bloch model and the GTR-model. In this final section we offer some additional comments on our approach to cognition, taking into consideration the confusion that sometimes exists between `ad hoc (phenomenological) models' and `theoretical (first principle) models', as well as the critique that a Hilbertian model (and a fortiori its possible extensions) is suspicious because it allows `too many free parameters' to obtain an exact fit (and not just an approximate fit) for all the experimental data. \n\nIn that respect, it is worth emphasizing that the principal focus of our `theory of human cognition' is not to model as precisely as possible the data gathered in psychological measurements. A faithful modeling of the data is of course an essential part of it, but our aim is actually more ambitious. In putting forward our methodology, consisting in looking at instances of decision-making as resulting from an interaction of a decision-maker with a conceptual entity, we look first of all for a theory truly describing `the reality of the cognitive realm to which a conceptual entity belongs', and additionally also `how human minds can interact with the latter so that decision-making can occur'.\n\nIn this sense, each time we have put forward a model for some specific experimental data, it has always been our preoccupation to also make sure that (i) the model was extracted following the logic that governs our theory of human cognition, and (ii) that whatever other experiments would be performed by a human mind interacting with that same cognitive-conceptual entity under consideration, also the data of these hypothetical additional experiments could have been modeled exactly in the same way. Clearly, this requirement -- that `all possible experiments and data' have to be modeled in an equivalent way -- poses severe constraints to our approach, and it is not a priori evident that this would always be possible. However, we are convinced that the fundamental idea underlying our methodology, namely that of looking upon a decision as an interaction of a human mind with a conceptual entity in a specific state (with such state being independent of the human minds possibly interacting with it), equips the theory of exactly those degrees of freedom that are needed to model `all possible data from all possible experiments'.\n\nAs we already explained in the foregoing, in all this we have been guided by how physical theories deal with data coming from the physical domain. They indeed satisfy this criterion and are able to model all data from all possible experiments that can be executed on a given physical entity. What we have called `conceptual entity' is what in physics corresponds to the notion of `physical entity'. Now, in our approach we might be classified as adhering to an idealistic philosophy, i.e. believing that the conceptual entities ``really exist,'' and are not mere creations of our human culture. Our answer to this objection is the following: to profit of the strength of the approach it is not mandatory to take a philosophical stance in the above mentioned way, in the sense that we are not obliged to attribute more existence to what we call a conceptual entity than that attributed, for example, to `human culture' in its entirety. The importance of the approach lies in considering such a conceptual entity as independently existing from any interaction with a human mind, and describe the continuously existing interactions with human minds as processes of the `change of state of the conceptual entity', and whenever applicable also as processes of the `change of context'. And again, let us emphasize that this `hidden-interaction' methodology is inspired by its relevance to physical theories. Our working hypothesis is that in this way it will be possible to advantageously model, and better understand, all of human cognition experimental situations.\n\nHaving said this, we observe that the interpretation of the quantum formalism that is commonly used in cognitive domains is a subjectivist one, very similar to that interpretation of quantum theory known as `quantum Bayesianism', or `QBism' \\cite{FuchsEtal2014}. In a sense, this interpretation is the polar opposite of our realistic (non-subjectivistic) operational approach. Indeed, QBism originates from a strong critique \\cite{fms2014} of the famous Einstein-Podolsky-Rosen reality criterion \\cite{epr1935}, whereas at the foundation of the Geneva-Brussels approach there is the idea of taking such criterion not only extremely seriously, but also of using it more thoroughly, as a powerful demarcating tool separating `actually existing properties' from `properties that are only available to be brought into actual existence', and therefore exist in a potential sense \\cite{sdb2011}. In other terms,  a quantum state is not considered in QBism as a description of the actual properties of a physical entity, but of the beliefs of the experimenter about it. Similarly, for the majority of authors in quantum cognition, a quantum state is a description of the state of belief of a participant, and not of the actual state of the conceptual entity that interacts with the participants. In ultimate analysis, this difference of perspectives is about taking a clear position regarding the key notion of `certainty': is certainty (probability $1$ assignments) just telling us something about the very firm belief of a subject, or also about some objective properties of the world (be it physical or cultural)? In the same way, are probabilities only shared personal beliefs, based on habit, or also elements of reality (considering that in principle their values can be predicted with certainty)? Although we certainly agree that it is not necessary to take a final stance on these issues to advantageously exploit the quantum mathematics in the modeling of many experimental situations, both in physics and cognition, we also think that the explicative power of a pure subjectivist view rapidly diminishes when we have to address the most remarkable properties of the physical and conceptual entities, like non-locality (non-spatiality) and the non-compositional way with which they can combine. \n\nIt is important to emphasize that the subjectivist view is also a consequence of the absence, in the standard quantum formalism, of a meaningful description of what goes on `behind the scenes' during a measurement. On the other hand, the hidden-measurement paradigm, as implemented in the extended Bloch representation \\cite{asdb2014}, or even more generally in the GTR-model \\cite{asdb2015b,asdb2015c,asdb2015d}, offers a credible description of the dynamics of a measurement process, in terms of a process of actualization of potential interactions, thus explaining a possible origin of the quantum indeterminism. This certainly allows understanding the so-called `collapse of the state vector' as an objective process, either produced by a macroscopic apparatus in a physics laboratory, or by a mind-brain apparatus in a psychological laboratory. As we tried to motivate in the second part of this article, this completed version of the quantum formalism also allowed us to describe those aspects of a psychological measurements -- the possible different ways participants can choose an outcome -- that would be impossible to model by remaining within the narrow confines not only of the standard formalism, but also of a strict subjectivistic interpretation of it. \n\nTo conclude, a final remark is in order. Quantum cognition is undoubtedly a fascinating field of investigation also for physicists, as it offers the opportunity to take a new look at certain aspects of the quantum formalism and use them to possibly make discoveries also in the physical domain. We already mentioned the example of `entangled measurements', that were necessary to exactly model certain correlations. Entangled (non-separable) measurements are usually not considered in the physics of Bell inequalities, while they are widely explored in quantum cryptography, teleportation and information. However, it is very possible that this stronger form of entanglement will prove to be useful for the interpretation of certain non-locality tests and the explanation of `anomalies' that were identified in EPR-Bell experiments \\cite{as2014}. Also, for what concerns the notion of `universal measurement', which is quite natural in psychological measurements, since data are obtained from a collection of different minds, could it be that `universal averages' also happen in the physical domain? In other terms, could it be that a single measurement apparatus is actually more like `a collection of different minds' than `a single Born-like mind'? Considering that the origin of the observed deviations from the Born rule, in situations of sequential measurements, can be understood as the ineffectiveness of the averaging process in producing the Born prescription, is it possible to imagine, in the physics laboratory, similar experimental situations where these deviations would be equally observed, thus confirming that the hypothesis of `hidden measurement-interactions' would be a pertinent one also beyond the psychological domain? Whatever the verdict will be, we certainly live in a very stimulating time for foundational research; a time where the conceptual tools that once helped us building a deeper understanding of the `microscopic layer' of our physical reality are now proving to be instrumental for understanding our human `mental layer'; but also a time where all this is also coming back to physics, not only in the form of possible new experimental findings, but also of possible new and deeper understandings \\cite{aerts2009,aerts2010a,aerts2010b,aerts2013,aerts2014}.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nIt is well-known that different types of topological objects may\nhave been created by vacuum phase transition in the early\nUniverse \\cite{Kibble,Vilenkin}. These include domain walls,\ncosmic strings and monopoles. Among them, cosmic strings and\nmonopoles seem to be the best candidates to be observed. \n\nGlobal monopole is a heavy object formed in the phase transition\nof a system composed of a self-coupling scalar field triplet\n$\\phi^a$ whose original global $O(3)$ symmetry is spontaneously\nbroken to $U(1)$. The scalar matter field plays the role of an\norder parameter which outside the monopole core acquires a\nnonvanishing value. The main part of the monopole's energy is\nconcentrated into its small core. The simplest model which gives\nrise to a global monopole is described by the Langrangian\ndensity below, and was analysed by Barriola and Vilenkin\n\\cite{BarriolaVilenkin} \n\\begin{equation}\nL= {1\\over 2} (\\partial_\\mu \\phi^a)( \\partial^\\mu \\phi^a ) - {1 \\over\n4} \\lambda (\\phi^a \\phi^a - \\eta^2)^2\\ ,\n\\end{equation}\nCoupling this matter field with the Einstein equations, a\nspherically symmetric metric tensor given by the line element \n\\begin{equation}\nds^2 = -B(r)dt^2 + A(r)dr^2 + r^2(d\\theta^2 + \\sin^2\\theta\nd\\varphi^2)\\ ,\n\\end{equation}\npresents solutions for the functions $B(r)$ and $A(r)$ far from\nthe monopole's core given by\n\\begin{equation}\nB = A^{-1}= 1 - 8\\pi \\eta^2 - 2 M\/r\\ .\n\\end{equation}\nThe mass parameter $M\\sim M_{core}$. Numerical details concerning\nwith this function can be seen in Ref. \\cite{HarariLousto}.  \nNeglecting the mass term we get the pointlike global\nmonopole spacetime with metric\n\\begin{equation}\nds^2 = -\\alpha^2 dt^2 + dr^2\/\\alpha^2 + r^2(d\\theta^2 +\n\\sin^2\\theta d\\varphi^2)\\ ,\n\\label{Metric}\n\\end{equation}\nwhere the parameter $\\alpha^2 = 1 - 8\\pi \\eta^2$. The energy momentum\ntensor of this monopole has diagonal form and reads: $T^t_t =\nT^r_r = (\\alpha^2 - 1)\/r^2$ and $T^\\theta_\\theta =\nT^\\varphi_\\varphi = 0$. The main peculiarity of this space is\na solid angle deficit which is the difference between the solid angle\nin the flat spacetime $4\\pi$ and the solid angle in the  \nglobal monopole spacetime which is given by $4\\pi\\alpha^2$. For\n$\\alpha < 1$ one has solid angle deficit and for $\\alpha >1$ one\nhas the solid angle excess. (We would like to call attention\nthat the physical values for $\\alpha$ predicted by field theory is\nsmaller than unit\\footnote{In fact for a typical grand\nunified theory the parameter $\\eta$ is of order $10^{16}Gev$. So\n$1 - \\alpha^2 = 8\\pi \\eta^2 \\sim 10^{-5}$}). The spacetime\nproduced by a global monopole has no Newtonian gravitational\npotential in spite of the geometry produced by this heavy object\nhas a non-vanishing curvature. For this reason the mass of the \nmonopole is divergent and proportional to the distance from monopole\norigin \\cite{BarriolaVilenkin}. In context of the monopole\nformation the cosmological horizon is a natural cutoff distance\nfor the monopole's mass. \n\nAlthough the global monopole has no Newtonian\ngravitational potential it gives enormous tidal acceleration\n$a\\sim 1\/r^2$ which is important from the cosmological point of \nview and may be used for obtaining upper bound on the number\ndensity of them in the Universe, which is at most one\nglobal monopole in the local group galaxies \\cite{HiscockPRL}.\nHowever, the numerical simulations made by Bennet and Rhie show that\nreal upper boundary is smaller than that given in \\cite{HiscockPRL}\nby many orders \\cite{BennetRhie}. In fact, one has scaling\nsolution with a few global monopoles per horizon volume\n\\cite{BennetRhie}.  \n\nThe quantum effects due to the monopole background in the matter\nfields have been considered explicitly for scalar field in Ref. \n\\cite{MazitelliLousto} and by general consideration in\nRef.\\cite{Hiscock}. It has been shown from general consideration\nthat the vacuum expectation value of the energy-momentum tensor\nof massless fields has the following general form \n\\begin{equation}\n<T^\\nu_\\mu>^{ren} = {S^\\nu_\\mu(\\mu r) \\over r^4}\\ ,\n\\end{equation}\nwhere the tensor $S^\\nu_\\mu$ depends on the arbitrary mass scale\nparameter $\\mu$ and the metric coefficient $\\alpha$. In Ref.\n\\cite{Hiscock}, it has been assumed that the tensor $S^\\nu_\\mu$\nis the function of the metric parameter $\\alpha$, only. Manifest\ncalculations in scalar case have shown \\cite{MazitelliLousto}\nthat this tensor depends on renormalization mass parameter $\\mu$\nand it has the following structure $S^\\nu_\\mu = A^\\nu_\\mu +\nB^\\nu_\\mu \\ln \\mu r$, where tensors $A^\\nu_\\mu$ and $B^\\nu_\\mu$\ndepend only on the $\\alpha$. This is in agreement with Wald\n\\cite{WaldPRD} who noted that an unambiguous prescripton for\n$<T^\\nu_\\mu>^{ren}$ cannot be defined without introducing a\nlength scale. Nevertheless, one-loop Einstein equations do not\ndepend on scale parameter $\\mu$ due to renormalization group\nequation.  Back reaction problem in the scalar massless case has\nbeen investigated by Mazzitelli and Lousto\n\\cite{MazitelliLousto}. It must be noted that only the general\nstructure of vacuum expectation value of the energy-momentum\ntensor has been considered. There, it was not obtained an\nexplicit form of the tensor $S^\\nu_\\mu$.  \n\nIn this paper we would like to obtain the explicit value of this\ntensor, considering the massless spinor field on\nthe background of the pointlike global monopole with metric\ngiven by Eq.(\\ref{Metric}). As opposed to the massless scalar\nfield case \\cite{MazitelliLousto} we get a simpler\nexpression for the Green function and the energy-momentum tensor\nwhich is obtained in closed form, for arbitrary angle deficit. \n\nThe analysis of the quantum behavior of two massless left-handed\n$SU(2)$ doublets fermionic field in the background of a\npointlike monopole, taking into account the magnetic field, has\nbeen developed by Ren \\cite{Ren}. There, the Rubakov and Callan\neffect was analysed, and found that there appears a small\ncorrection due to the parameter $\\alpha$ into the fermion number\ncondensate. \n\nThis paper is organized as follows. In Sec. \\ref{SGF} we\nconsider the Green functions for massive and massless spinor\nfields on the background of the pointlike global monopole and\nobtain, in closed form, the Euclidean Green function for\nmassless left-handed spinor field. In Sec. \\ref{EMT} we analyze\nthis function at coincidence limit and extract all divergencies\nfrom it in manifest form. We obtain also the general structure\nof renormalized energy-momentum tensor. Each component of this\ntensor may be expressed in terms of only the zero-zero component\n$<T^0_0>^{ren}$ which is analyzed with great details. In Sec.\n\\ref{Conclusion} we summarize our results. Appendices A and\nB contain some technical formulas.  The signature of the\nspacetime, the sign of Riemann and Ricci tensors are the same as\nin Christensen paper \\cite{Christensen78}. We use units $\\hbar =\nc= G = 1$.  \n\\section{Spinor Green Functions}\\label{SGF}\nIn this section we want to obtain the expression for the fermion\npropagator of massive spinor field in the pointlike monopole\nspacetime. Massless field will be recognized as particular \ncase. The spinor Feynman propagator is defined as follows\n\\cite{BirrellDevies} \n\\begin{equation}\ni{\\cal S_F}(x,x') = <0|T(\\Psi (x)\\overline{\\Psi} (x'))|0>\\ ,\n\\end{equation}\nwhere $\\overline{\\Psi} = \\Psi^+\\gamma^0$ which, under the\nLorentz transformation $\\Psi \\to S(\\triangle )\\Psi$, transforms as\n$\\overline{\\Psi} \\to \\overline{\\Psi}S^{-1}(\\triangle )$, where \n$\\triangle$ is the parameter of the transformation, and\n$S(\\triangle )$ is a local representation of the Lorentz group. \n\nThis propagator obeys the following differential equation \n\\begin{equation}\n(i{\\not\\hspace{-0.5ex}\\nabla} - M){\\cal S_F}(x,x')= {1 \\over \\sqrt{-g}} \\delta^{(4)} (x -\nx')I_4\\ ,\n\\label{Green}\n\\end{equation}\nwhere $g={\\rm det}(g_{\\mu\\nu})$. The covariant derivative\noperator in the above equation is \n\\begin{equation}\n{\\not\\hspace{-0.5ex}\\nabla} = e^\\mu_{(a)}\\gamma^{(a)} (\\partial_\\mu + \\Gamma_\\mu)\\ ,\n\\end{equation}\n$e^\\mu_{(a)}$ being the vierbein satisfying the condition \n\\begin{equation}\ne^\\mu_{(a)}e^\\nu_{(b)}\\eta^{ab} = g^{\\mu\\nu}\\ ,\n\\end{equation}\nand $\\Gamma_\\mu$ is the spin connection, given in terms of the\nflat spacetime $\\gamma$-matrix by \n\\begin{equation}\n\\Gamma_\\mu = -{1\\over 4}\n\\gamma^{(a)}\\gamma^{(b)}e^\\nu_{(a)}e_{(b)\\nu ;\\mu}\\ .\n\\end{equation}\nThe Green function given in (\\ref{Green}) is a bispinor, i.e. it\ntransforms as $\\Psi$ at $x$ and as $\\overline{\\Psi}$ at $x'$. \n\nIf a bispinor ${\\cal D_F}(x,x')$ satisfies the differential\nequation below\n\\begin{equation}\n(\\Box - M^2 - {1\\over 4}R){\\cal D_F}(x,x') = -{1\\over\n\\sqrt{-g}} \\delta^{(4)} (x-x')I_4\\ , \\label{GF}\n\\end{equation}\nwhere the generalized d'Alembertian is expressed by \n\\begin{equation}\n\\Box = g^{\\mu\\nu} \\nabla_\\mu\\nabla_\\nu = g^{\\mu\\nu}\n(\\partial_\\mu \\nabla_\\nu + \\Gamma_\\mu \\nabla_\\nu -\n\\Gamma^\\alpha_{\\mu\\nu} \\nabla_\\alpha)\\ ,\n\\end{equation}\nthen the spinor Feynman propagator may be read as \n\\begin{equation}\n{\\cal S_F}(x,x') = (i{\\not\\hspace{-0.5ex}\\nabla} + M){\\cal D_F}(x,x')\\ ,\n\\end{equation}\nwhich shows that the non-minimal coupling to the curvature does play\na role when spinor fields are considered. \n\nNow after this brief review about the calculation of spinor\nFeynman propagator in the general manifold, let us specialize it\nto the spacetime of a global monopole. We shall choose the\nfollowing basis tetrad:\n\\begin{equation}\ne^\\mu_{(a)} = \\left( \\begin{array}{cccc}\n  1\/\\alpha & 0 & 0 & 0 \\\\\n  0 & \\alpha\\sin\\theta \\cos\\varphi & \\cos \\theta\n  \\cos\\varphi \/r & -\\sin\\varphi \/r\\sin\\theta \\\\ \n0 & \\alpha\\sin\\theta \\sin\\varphi & \\cos \\theta\n\\sin\\varphi \/r & \\cos\\varphi \/r\\sin\\theta \\\\ \n0 & \\alpha\\cos\\theta & -\\sin \\theta \/r & 0 \n                      \\end{array}\n               \\right) \\ .\n\\label{Tetrada}\n\\end{equation}\nFor this case, the only non-zero spin connections are\n\\begin{eqnarray}\n\\Gamma_2 &=& {1-\\alpha \\over 2} [\\gamma^{(1)} \\gamma^{(2)}\\cos\n\\varphi + \\gamma^{(2)}\\gamma^{(3)} \\sin \\varphi]\\ , \\\\\n\\Gamma_3 &=& -{1 - \\alpha \\over 2} [\\gamma^{(1)} \\gamma^{(2)} \\sin\n\\theta + \\gamma^{(1)} \\gamma^{(3)} \\cos\\theta \\sin \\varphi -\n\\gamma^{(2)} \\gamma^{(3)} \\cos\\theta \\cos\\varphi] \\sin\\theta\\ .\n\\nonumber \n\\end{eqnarray}\nIf we had made another choice for the vierbein, for example\n$e^\\mu_{(a)} = {\\rm diag}(1\/\\alpha , \\alpha, 1\/r,\n1\/r\\sin\\theta)$, the only non-zero spin connection would be $\\Gamma_2\n= -\\alpha \\gamma^{(1)} \\gamma^{(2)}\/2$ and $\\Gamma_3 =\n-\\alpha[\\gamma^{(1)} \\gamma^{(3)}\\sin\\theta + \\gamma^{(2)}\n\\gamma^{(3)} \\cos\\theta ]\/2$. Although this tetradic basis is\nsimpler than previous one given in (\\ref{Tetrada}), the spin\nconnection obtained by the former does not vanish when we take\nMinkowski limit $\\alpha =1$.  \n\nIn order to obtain the explicit form for the differential\nequation for the bispinor ${\\cal D_F}$ given in\n(\\ref{GF}) in this geometry we shall adopt the following\nrepresentation for the $\\gamma$-matrix: \n\\begin{equation}\n\\gamma^{(0)}=\\left( \n\\begin{array}{cc}\n1&0 \\\\\n0&-1 \n\\end{array} \\right) \\ ,\n\\gamma^{(k)}=\\left( \n\\begin{array}{cc}\n0&\\sigma^k \\\\\n-\\sigma^k&0 \n\\end{array} \\right) \\ ,\n\\end{equation}\n$\\sigma^k$ being the $2\\times 2$ Pauli matrices. These matrices\nabove obey the anticommutator relations $\\{\\gamma^{(a)},\n\\gamma^{(b)}\\} = -2\\eta^{ab}$. \n\nAfter some intermediate steps, we get the following expression\nfor the d'Alembertian operator \n\\begin{equation}\n\\Box = -{1 \\over \\alpha^2} \\partial^2_t + {\\alpha^2 \\over r^2}\n\\partial_r (r^2 \\partial_r )- {1\\over r^2} \\vec{L}^2 - {(1 -\n\\alpha)^2 \\over 2r^2} -{1-\\alpha \\over r^2} \\vec{\\Sigma} \\cdot \n\\vec{L} \\ ,\n\\end{equation}\nwhere \n\\begin{equation}\n\\vec{\\Sigma} = \\left(\\begin{array}{cc}\n\\vec{\\sigma} & 0 \\\\\n0&\\vec{\\sigma} \n                    \\end{array}\n                     \\right)\\ .\n\\end{equation}\nFrom the above operator we can see that, although it is a\n$4\\times 4$ matrix differential one, it is diagonal in\nblock of $2\\times 2$ matrices, which means that the two upper\ncomponents of Dirac spinor interact with the gravitational field\nin the similar way as the two lower components. \n\nThe complete differential operator in Eq. (\\ref{GF}) is \n\\begin{equation}\n{\\cal L} =-{1 \\over \\alpha^2} \\partial^2_t + {\\alpha^2 \\over r^2}\n\\partial_r r^2 \\partial_r - {1\\over r^2} \\vec{L}^2 -{1-\\alpha\n\\over r^2}(1 + \\vec{\\Sigma} \\cdot \\vec{L}) -M^2\\ . \n\\end{equation}\nNow we shall consider the case where the fermionic field has no\nmass. In this case we are able to obtain a closed expression for\nthe fermionic propagator as follows.\n\nThe system which we shall consider consists of a massless\nleft-handed fermionic field in a global monopole manifold. The\nDirac equation reduces to a $2\\times 2$ matrix differential one,\nas shown below:\n\\begin{equation}\ni{\\not\\hspace{-0.8ex}D_L} \\chi = 0\\ ,\n\\end{equation} \nwhere\n\\begin{equation}\n{\\not\\hspace{-0.8ex}D_L} = i\\left[ {1 \\over \\alpha} \\partial_t - \\alpha \\sigma^{(r)}\n\\partial_r - {1\\over r} \\sigma^{(\\theta )}\\partial_\\theta -{1\n\\over r\\sin \\theta} \\sigma^{(\\varphi )} \\partial_\\varphi + {1 -\n\\alpha \\over r} \\sigma^{(r)} \\right]\\ ,\n\\end{equation}\nwith $\\sigma^{(r)} = \\vec{\\sigma} \\cdot \\hat{\\vec{r}}\\ ,\\\n\\sigma^{(\\theta)} = \\vec{\\sigma}\\cdot \\hat{\\vec{\\theta}}\\ ,\\\n\\sigma^{(\\varphi )} = \\vec{\\sigma}\\cdot \\hat{\\vec{\\varphi } } $,\nwhere $\\hat{\\vec r}\\ ,\\ \\hat{\\vec \\theta}$ and $\\hat{\\vec\n\\varphi}$ are the standard unit vectors along the three spatial\ndirections in spherical coordinates. \n\nThe Feynman two-component propagator obeys the equation \n\\begin{equation}\ni{\\not\\hspace{-0.8ex}D_L} {\\cal S_F}(x,x') = {1 \\over \\sqrt{-g}} \\delta^{(4)} (x - x')\nI_2\\ ,\n\\end{equation}\nand can be given in terms of the bispinor ${\\cal G_F}$ by \n\\begin{equation}\n{\\cal S_F}(x,x') = i{\\not\\hspace{-0.8ex}D_L} {\\cal G_F}(x,x')\\ , \\label{Relation}\n\\end{equation}\nwhere now, ${\\cal G_F}(x,x')$ obeys the $2\\times 2$ differential\nequation \n\\begin{equation}\n\\overline{{\\cal L}} {\\cal G_F}(x,x') = - {1\\over \\sqrt{-g}}\n\\delta^{(4)} (x - x') I_2\\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\\overline{\\cal L} =-{1 \\over \\alpha^2} \\partial^2_t + {\\alpha^2\n\\over r^2} \\partial_r r^2 \\partial_r - {1\\over r^2} \\vec{L}^2\n-{1-\\alpha \\over r^2}(1 + \\vec{\\sigma}\\cdot \\vec{L}) \\ . \n\\label{Operator}\n\\end{equation}\nThe vacuum average value of the energy-momentum tensor may\nbe expressed in terms of the Euclidean Green function which is\nsimpler than the ordinary Feynman Green function. They are\nconnected by the relation \\cite{BirrellDevies} ${\\cal \nG_E}(\\tau,\\vec{r}; \\tau', \\vec{r'}) = -i {\\cal G_F}(x,x')$,\nwhere $t=i\\tau$. In the following we shall consider the\nEuclidean Green function. In order to find a solution for the\nbispinor ${\\cal G_E}(x,x')$, we shall obtain the solution for the\neigenvalue equation \n\\begin{equation}\n\\overline{\\cal L} \\phi_\\lambda (x) = -\\lambda^2 \\phi_\\lambda\n(x)\\ ,\n\\label{Eigenfunction}\n\\end{equation}\nwith $\\lambda^2 \\geq 0$, so we can write \n\\begin{equation}\n{\\cal G_E}(x,x') = \\sum_\\lambda {\\phi_\\lambda (x) \\phi_\\lambda^+\n(x') \\over \\lambda^2}\\ . \n\\end{equation} \nDue to fact that our operator (\\ref{Operator}) is self-adjoint,\nthe set of its eigenfunctions constitutes a basis for the Hilbert space\nassociated with two-component spinors. Moreover, because\noperator $\\overline{\\cal L}$ is a parity even operator, its\neigenfunctions present a defined parity, so the normalized\neigenfunctions can be written as:\n\\begin{eqnarray}\n\\phi_\\lambda^{(1)}(x) &=& e^{-iE\\tau}\nf^{(1)}(r)\\varphi^{(1)}_{j,m_j} (\\theta ,\\varphi )\\ ,\\\\\n\\phi_\\lambda^{(2)}(x) &=& e^{-iE\\tau}\nf^{(2)}(r)\\varphi^{(2)}_{j,m_j} (\\theta ,\\varphi )\\ , \\nonumber\n\\end{eqnarray} \nwhere $\\varphi^{(k)}_{j,m_j}$, with $k=1,2$, are the spinor\nspherical harmonics which are eigenfunctions of the operators\n$\\vec{L}^2$ and $\\vec{\\sigma}\\cdot \\vec{L}$ as shown below: \n\\begin{eqnarray}\n\\vec{L}^2\\varphi^{(1,2)}_{j,m_j} &=& l(l+1)\n\\varphi^{(1,2)}_{j,m_j}\\ ,\\\\\n\\vec{\\sigma}\\vec{L} \\varphi^{(1,2)}_{j,m_j} &=& -(1 +\n\\kappa^{(1,2)}) \\varphi^{(1,2)}_{j,m_j}\\ , \\label{Kappa}\n\\end{eqnarray} \nwith $\\kappa^{(1)} = -(l+1) = -(j + 1\/2)$ and $\\kappa^{(2)} = l\n= j + 1\/2$. Explicit form of above standard function are given\nin Ref. \\cite{Bjorken}, for example.  \n\nSubstituting $\\phi_\\lambda^{(1,2)}$ into (\\ref{Eigenfunction}),\nwe obtain the following eigenfunctions\n\\begin{eqnarray}\n\\phi^{(k)}_\\lambda(x) &=& \\sqrt{p \\over 2\\pi r} e^{-iE\\tau}\nJ_{\\nu_k} (pr) \\varphi^{(k)}_{j,m_j}\\ , \\label{Phi}\\\\\n\\lambda^2 &=& E^2\/\\alpha^2 + \\alpha^2 p^2\\ , \\nonumber \\\\\n\\nu_1 &=& {l+1 \\over \\alpha} - {1\\over 2}\\ ,\\ \\nu_2 = {l\\over\n\\alpha} + {1 \\over 2}\\ , \\label{Nu} \n\\end{eqnarray}\nwhere $J_\\nu$ is the Bessel function of first kind. We can see\nthat for Minkowski spacetime where $\\alpha = 1$ we have $\\nu_1=\n\\nu_2 = l + 1\/2$.\n\nNow we are in condition to obtain the bispinor ${\\cal G_E}$\nwhich is given by: \n\\begin{equation}\n{\\cal G_E}(x,x') =\n\\int_{-\\infty}^{+\\infty} dE \\int_0^\\infty dp \\sum_{j,m_j}\n{\\phi_\\lambda^{(1)}(x)\\phi_\\lambda^{(1)+}(x') + \n\\phi_\\lambda^{(2)}(x)\\phi_\\lambda^{(2)+}(x') \\over\nE^2\/\\alpha^2 + \\alpha^2 p^2}\\ . \\label{GE}\n\\end{equation}  \nFinally, substituting our result for the two-component spinor\n$\\phi_\\lambda^{(k)}$ given by (\\ref{Phi}) and (\\ref{Nu}) into\n(\\ref{GE}) we obtain, with the help of Ref. \\cite{Gradshtein},\nan expression for the Euclidean Green function \n\\begin{equation}\n{\\cal G_E} (x,x') = {1 \\over 2\\pi r r'} \\sum_{j,m_j} \\left[\nQ_{\\nu_1 - 1\/2} (u) C^{(1)}_{j,m_j}(\\Omega ,\\Omega') + Q_{\\nu_2\n- 1\/2} (u) C^{(2)}_{j,m_j}(\\Omega , \\Omega')\\right]\\ , \\label{GEsimple}\n\\end{equation}\nwhere $Q_\\nu(u)$ are the Legendre functions of second kind; $u = 1\n+ (\\alpha^4 \\triangle \\tau^2 + \\triangle r^2 )\/2rr'$ and\n$C^{(k)}_{j,m_j}(\\Omega ,\\Omega') = \\varphi^{(k)}_{j,m_j}(\\Omega\n) \\varphi^{(k)+}_{j,m_j}(\\Omega')$.  \nAgain we can see that in Minkowski limit, for $\\alpha =1$,\n$\\nu_1 = \\nu_2 = l+1\/2$ and we get\n\\begin{equation}\n{\\cal G_E}(x,x') = {1 \\over 8\\pi^2}{1 \\over \\sigma_M(x,x')} I_2\\\n,\n\\end{equation}\nwhere we have used the relation envolving the sum of Legendre\nfunctions and polynomials \\cite{Gradshtein}. Here\n$\\sigma_M(x,x') = (\\triangle\\tau^2 + (\\vec{r} - \\vec{r}')^2)\/2$\nis one-half the square of geodesic distance between $x$ and $x'$\nin the flat Euclidean space.  \n\nNow in order to obtain the Green function ${\\cal S_F}$ let us go\nback to Eq. (\\ref{GEsimple}). Using (\\ref{Relation}) we have \n\\begin{equation}\n{\\cal S_F}(x,x') = i\\left[{1 \\over \\alpha} \\partial_t - \\alpha\n\\sigma^{(r)}\\partial_r + {1\\over r}\\sigma^{(r)} \\vec{\\sigma}\n\\cdot \\vec{L} + {1- \\alpha \\over r} \\sigma^{(r)} \\right] {\\cal\nG_F}(x,x')\\ .\n\\label{S_F}\n\\end{equation}\n\\section{Vacuum Expectation Values}\\label{EMT}\nNow let us proceed with the calculation of the vacuum\nexpectation value (VEV) of the energy-momentum tensor.\nInitially, we would like to discuss the general structure of\nthis tensor. As it will be seen later and has already been\ndiscussed in scalar case in Ref. \\cite{MazitelliLousto}, the\nrenormalized VEV of the energy-momentum tensor has the following\nstructure \n\\begin{equation}\n<T^\\nu_\\mu>^{ren} = {1 \\over 8\\pi^2 r^4}\\left[ A^\\nu_\\mu +\nB^\\nu_\\mu \\ln {\\mu r\\over \\alpha}\\right]\\ , \\label{Tmn}\n\\end{equation}\nwhere the tensors $A^\\nu_\\mu$ and $B^\\nu_\\mu$ depend on\nthe metric parameter $\\alpha$, only. The scaling parameter $\\mu$\nappears after renormalization procedure. Obviously these tensors\nare diagonal and the component $A^\\theta_\\theta =\nA^\\varphi_\\varphi$ and $B^\\theta_\\theta = B^\\varphi_\\varphi$ due\nto spherical symmetry of the problem. Therefore we have six unknown\ncomponents. The renormalized VEV of the energy-momentum tensor\nmust be conserved, i.e.,  \n\\begin{equation}\n<T^\\nu_\\mu>^{ren}_{;\\nu} =0\\ , \\label{Conservation}\n\\end{equation}\nand gives the right conformal anomaly \\cite{Wald77}, which for\nmassles spinor two-component field, reads \\cite{ChristensenDuff} \n\\begin{equation}\n<T^\\mu_\\mu>^{ren} = {1\\over 16\\pi^2} {\\rm tr}a_2 = {T \\over\n8\\pi^2 r^4}\\ . \\label{Trace}\n\\end{equation}\nTaking into account Eqs.(\\ref{Conservation}), (\\ref{Trace}) we may\nexpress tensors $A^\\nu_\\mu$ and $B^\\nu_\\mu$ in terms of the\nzero-zero components $A^0_0\\ ,\\ B^0_0$ and the trace $T$ by: \n\\begin{eqnarray}\nA^\\nu_\\mu &=& {\\rm diag}(A^0_0\\ ;\\ -T + A^0_0 + B^0_0\\ ;\\ T - A^0_0\n-{1\\over 2} B^0_0\\ ;\\ T - A^0_0 - {1 \\over 2} B^0_0)\\ , \\\\ \nB^\\nu_\\mu &=& B^0_0{\\rm diag}(1;1;-1;-1)\\ . \\label{AnmBnm}\n\\end{eqnarray}\nTherefore our problem now is to obtain the zero-zero components\n$A^0_0$ and $B^0_0$. Using the point-splitting approach, \nthe VEV of the energy-momentum tensor for spinor field has the\nfollowing form (see Ref. \\cite{BirrellDevies} for example): \n\\begin{equation}\n<T_{\\mu\\nu}> = {1\\over 4}\\lim_{x' \\to x} {\\rm tr} \\left[ \\sigma_{\\mu}\n(\\nabla_\\nu - \\nabla_{\\nu'}) + \\sigma_\\nu (\\nabla_\\mu -\n\\nabla_{\\mu'})\\right] {\\cal S_F}(x,x')\\ ,\n\\end{equation}\nby means of which we have \n\\begin{equation}\n<T^0_0> = {i \\over \\alpha^2} \\lim_{x' \\to x} \\partial^2_t {\\rm\ntr}({\\cal G_F}(x,x')) = -{1 \\over \\alpha^2} \\lim_{x' \\to x}\n\\partial^2_\\tau {\\rm tr}({\\cal G_E}(x,x'\n))\\ . \\label{T00}\n\\end{equation}\nThe first term with time derivative in Eq. (\\ref{S_F}) gives\nnon-zero contribution in the zero-zero component of the\nenergy-momentum tensor, only. Indeed, the Euclidean Green function\n(\\ref{GEsimple}) is proportional to the unit matrix $I_2$ after\ntaking the coincidence limit $\\Omega' = \\Omega$ and sum over\n$m_j$.  The same is true for the third term in (\\ref{S_F}) due to \nEq. (\\ref{Kappa}). We obtain zero contribution from these terms\nbecause Pauli matrices are traceless. In above expression we have\nused also that the Green function (\\ref{GEsimple}) depends on the\n$\\triangle \\tau$ and hence $\\partial_{\\tau'}{\\cal G_E} = -\n\\partial_{\\tau }{\\cal G_E}$. \n\nTaking the coincidence limit $\\Omega' =\\Omega\\ ,\\ r'=r$ into \nEq. (\\ref{GEsimple}), summing over $m_j$ and after using the\nintegral representation for the Legendre function\n\\cite{Gradshtein}, it is possible to develop the sum over $j$,\nwhich is geometric series and we arrive at the following \nformula for the Euclidean Green function \n\\begin{equation}\n8\\pi^2 {\\cal G_E}(\\triangle \\tau ,r) ={1 \\over 2r^2} \\int_b^\\infty\n{dx \\over \\sqrt{x^2 -b^2}} {1 \\over {\\rm sh}^2 ({{\\rm arsh} x \\over\n\\alpha})}I_2\\ , \\label{GEt}\n\\end{equation}\nwhere the function $b$ may be expressed in terms of the one-half\nof the square geodesic distance $\\sigma$ in $\\tau$ direction  \n\\begin{equation}\nb^2={\\alpha^2 \\over 2r^2} \\sigma = {\\alpha^2 \\over 2r^2}\\left( {\n\\alpha^2 \\triangle \\tau^2 \\over 2}\\right)\\ .\n\\end{equation}\nIn the case when only the angular variable $\\Omega' =\n\\Omega$ coincides, we shall get the same formula for Green function \n(\\ref{GEt}) with $\\sigma = (\\alpha^2 \\triangle\\tau^2 + \\triangle\nr^2\/ \\alpha^2)\/2$ and $r^2 \\to rr'$. \n\nThe Green function (\\ref{GEt}) is divergent in the coincidence\nlimit $b\\to 0$. For renormalization in the massless case it is\nmore suitable to subtract from the Green function (\\ref{GEt})\nthe Green function in the Hadamard form given below (see Refs.\n\\cite{ALNg,WaldPRD}) \n\\begin{equation}\n8\\pi^2{\\cal G_E^H}(x,x') = \\triangle^{1\/2} \\left\\{ {a_0 \\over\n\\sigma }  -{3a_2 \\over 8}\\sigma + \\left[- {a_1 \\over 2} + {a_2\n\\over 4} \\sigma \\right]\\ln ({1\\over 8}\\mu^2 \\sigma)\n\\right\\}\\ . \n\\label{Hadamard}\n\\end{equation}\nIn order to obey the conservation law (\\ref{Conservation}) we\nmust subtract from the energy-momentum tensor additional contribution\n$g_{\\mu\\nu} {\\rm tr}a_2\/64\\pi^2$ according to Wald\n\\cite{WaldPRD}. The general form of the coefficients $a_k$ may\nbe found in Ref. \\cite{Christensen78} and for the global monopole\nspacetime they have the following form \n\\begin{equation}\n\\triangle = 1\\ ,\\ a_0=I_2\\ ,\\ a_1 = -{1 - \\alpha^2 \\over\n6r^2}I_2\\ ,\\ a_2 = -{1- \\alpha^4 \\over 60r^4}I_2\\ .  \n\\end{equation}\nFrom our expression for the Green function (\\ref{GEt}), it is\npossible to extract all divergencies in manifest form. To do\nthis let us consider the rhs of Eq. (\\ref{GEt}). We divide the\nintegral in two parts - first one from $b$ to unit and second\nfrom unit to infinity. In the first part we have the form below  \n\\begin{equation}\nV_1 = {1 \\over 2r^2} \\int_b^1 {dx \\over \\sqrt{x^2 - b^2}}{1\n\\over {\\rm sh}^2({{\\rm arsh} x \\over \\alpha})} I_2\\ .\n\\end{equation}\nSubtracting and adding into the integrand the three first terms\nof the power series of the function\n\\begin{equation}\n{1 \\over {\\rm sh}^2 ({{\\rm arsh} x \\over \\alpha})} = {\\alpha^2\n\\over x^2} - {1 - \\alpha^2 \\over 3} + {1 - \\alpha^4 \\over 15}{x^2\n\\over \\alpha^2} + \\dots\\ , \n\\label{Expansion}\n\\end{equation} \nwe get \n\\begin{equation}\nV_1 = V_1^{fin}  + V_1^{div} \\label{V_1}\\ .\n\\end{equation}\nThe first term is given by expression \n\\begin{equation}\nV_1^{fin} = {1 \\over 2r^2} \\int_b^1 {dx \\over \\sqrt{x^2 -\nb^2}}\\left\\{ {1 \\over {\\rm sh}^2({{\\rm arsh} x \\over \\alpha})} - \n{\\alpha^2 \\over x^2} + {1 - \\alpha^2 \\over 3} - {1 - \\alpha^4\n\\over 15}{x^2 \\over \\alpha^2}\\right\\}I_2\\ ,\n\\label{V_1fin}\n\\end{equation}\nand its two derivatives with respect to time $\\tau$ are finite\nin the coincidence limit $b \\to 0$. All divergencies are\ncontained in the second part of Eq. (\\ref{V_1}) which has the form \n\\begin{equation}\nV_1^{div} = {1 \\over 2r^2} \\left[ {\\alpha^2 \\over b^2}\n\\sqrt{1 - b^2} - {1 - \\alpha^2 \\over 3} \\ln {1 + \\sqrt{1 - b^2}\n\\over b} + {1 - \\alpha^4 \\over 30\\alpha^2}\\{ \\sqrt{1 - b^2} +\nb^2 \\ln{1 + \\sqrt{1- b^2}\\over b}\\} \\right]I_2\\ . \\label{Exp}\n\\end{equation}\nExpanding this expression in the power series over $b$ \nup to the terms which will survive after taking the second\nderivative and coincidence limit, we may represent it in the form\n\\begin{equation}\nV_1^{div} = \\left\\{ {1 \\over \\sigma} + {1 \\over 2r^2}\\left[ -{\\alpha^2\n\\over 2} + {1 -\\alpha^4 \\over 30\\alpha^2}\\right] + \n\\left[ -{\\alpha^4 \\over 32r^4} + {\\alpha^2 (1 -\\alpha^2) \\over\n48r^4} +  {a_2 \\over 4}\\right] \\sigma + \\left[ - {a_1\n \\over 2} + {a_2 \\over 4}\\sigma \n\\right] \n\\ln {\\alpha^2 \\sigma\n\\over 8r^2} \\right\\} I_2\\ .  \n\\end{equation}\nWe see that the divergent part of this expression has the Hadamard\nform given by Eq.(\\ref{Hadamard}). The next powers in the\nexpansion (\\ref{Expansion}) will give finite contribution to\nthe energy-momentum tensor. \n\nThe second part \n\\begin{equation}\nV_2 = {1 \\over 2r^2} \\int_1^\\infty {dx \\over \\sqrt{x^2 - b^2}}{1\n\\over {\\rm sh}^2({{\\rm arsh} x \\over \\alpha})} I_2\\ , \n\\label{V_2}\n\\end{equation}\ndoes not contain divergencies and it may be easily expanded in\nterms of $b$\n\\begin{equation}\nV_2 = {1 \\over 2r^2} \\int_1^\\infty {dx \\over x}{1\n\\over {\\rm sh}^2({{\\rm arsh} x \\over \\alpha})} I_2 + {\\alpha^2\n\\sigma \\over 8r^4}\\int_1^\\infty {dx \\over x^3}{1 \\over\n{\\rm sh}^2({{\\rm arsh} x \\over \\alpha})} I_2 + O(\\sigma^2)\\ .   \n\\end{equation} \nTaking into account all above formulas we obtain the following\nexpression for the renormalized Green function \n\\begin{eqnarray}\n8\\pi^2 {\\cal G_E}^{ren} &=& 8\\pi^2 [{\\cal G_E}(\\triangle \\tau,\nr) - {\\cal G_E^H}(\\triangle \\tau, r)] \\label{GEren}\\\\ \n&=& V_1^{fin} + V_2 + \\left\\{ {1 \\over 2r^2}\\left[\n-{\\alpha^2 \\over 2} + {1 -\\alpha^4 \\over 30\\alpha^2}\\right] +\n\\left[ -{\\alpha^4 \\over 32r^4} + {\\alpha^2 (1 -\\alpha^2)\n  \\over 48r^4} + {5 a_2 \\over 8}\\right] \\sigma\n  + \\left[  a_1 - {a_2 \\over 2}\\sigma \n\\right] \n\\ln {\\mu r \\over \\alpha} \\right\\} I_2\\ , \\nonumber \n\\end{eqnarray}\nwhere $V_1^{fin}$ and $V_2$ are given by eqs. (\\ref{V_1fin}) and\n(\\ref{V_2}), respectively. Plugging this expression for\nthe renormalized Green function into Eq. (\\ref{T00}) we obtain the\nfollowing formulas for the zero-zero components of $A^0_0$ and\n$B^0_0$ (see Appendix \\ref{A})  \n\\begin{eqnarray}\nA^0_0 &=& - {1 \\over 8} \\left\\{ \\int_0^1 {dx\\over x} \\left[\n\\left({\\alpha^2 \\over {\\rm sh}^2({{\\rm arsh} x \\over\n\\alpha})}\\right)'' - {6\\alpha^4 \\over x^4} - {2(1 - \\alpha^4)\n\\over 15} \\right] + \\int_1^\\infty {dx\\over x} \\left[\n\\left({\\alpha^2 \\over {\\rm sh}^2({{\\rm arsh} x \\over\n\\alpha})}\\right)'' - {6\\alpha^4 \\over x^4} \\right] \\right\\}\\\n,\\label{A00} \\\\ \nB^0_0 &=& {r^4{\\rm tr} a_2 \\over 2} = -{1 - \\alpha^4 \\over 60}\\\n, \\label{B00}\n\\end{eqnarray}\nwhere the prime means the derivative with respect to $x$. In\nthis expression we have already taken into account additional\ncontribution $-{\\rm tr}a_2\/64\\pi^2$. \n\nNow we have explicit expressions for zero-zero components $A^0_0$\nand $B^0_0$ for arbitrary values of $\\alpha$. Let us analyse \n$A^0_0$ component (see Appendix \\ref{B} for details). \n\\begin{enumerate}\n\\item For large solid angle deficit ($\\alpha \\ll 1$):  \n\\begin{equation}\nA^0_0 \\sim -{1\\over 60} \\ln \\alpha + C_0\\ ,\\ C_0 = 0.0104\\ .\n\\end{equation}\n\\item For small solid angle deficit (excess) ($|\\alpha - 1|\\ll\n1$): \n\\begin{equation}\nA^0_0 \\sim C_1 (1 -\\alpha)\\ ,\\ C_1 = 0.0773\\ .\n\\end{equation}  \n\\item The large solid angle excess ($\\alpha \\gg 1$):  \n\\begin{equation}\nA^0_0 \\sim - C_\\infty \\alpha^4\\ ,\\ C_\\infty = 0.0173\\ .\n\\end{equation}\n\\end{enumerate}\nThe numerical calculation of $A^0_0$ is shown in Fig.\\ref{1} \n\n\\begin{figure}[tbh]\n  \\centerline{\n\\psfig{figure=aa.ps,angle=0,height=8cm}%\n\\psfig{figure=bb.ps,angle=0,height=8cm}}\n\\caption{Zero-zero component $A^0_0$ and distance $R_0$ where\nthe component $<T^0_0>^{ren}$ of energy-momentum tensor changes\nits sign, are plotted as function of the metric parameter $\\alpha$}\n\\label{1} \n\\end{figure}\n\nNow, we would like to compare our results with the\nsimilar one for the VEV of the energy-momentum tensor of the\nmassless spinor field in the infinitely thin cosmic string\nspacetime with line element, in cylindrical coordinate system, \ngiven by \n\\begin{equation}\nds^2 = -dt^2 + d\\rho^2 + {\\rho^2 \\over \\nu^2} d\\varphi^2 +\ndz^2\\ . \\label{InfThin}\n\\end{equation}\nIn Ref. \\cite{FrolovSerebr} has been obtained the following result  \n\\begin{equation}\nT^\\nu_\\mu = {(\\nu^2 -1)(7\\nu^2 + 17) \\over 1440\\cdot 4\\pi^2\n\\rho^4} {\\rm \ndiag} (1;1;1;-3)\\ .\n\\end{equation}\nIn this case there is no the logarithmic contribution because\nthis spacetime is locally flat. Therefore we may compare now\nthe $A^0_0$ given by Eq. (\\ref{A00}) for global monopole\nspacetime with the analogous one for infinitely thin cosmic\nstring spacetime which is given by expression\n\\begin{equation}\nA^0_{0(cs)} = {(\\nu^2 -1)(7\\nu^2 + 17) \\over 720}\\ .\n\\end{equation} \nTo do so, we have to change the radial variable $r = \\alpha \\rho$\nbecause in this coordinate system the section $\\theta = \\pi \/2$\nof the monopole spacetime (\\ref{Metric}) coincides with section\n$z=const$ of the infinitely thin cosmic string spacetime\n(\\ref{InfThin}) and the parameter $\\alpha = 1\/\\nu$. Therefore we\nmay compare $A^0_{0(cs)}(\\nu)$ with $(A^0_{0(gm)}(\\alpha\n)\/\\alpha^4)_{\\alpha = 1\/ \\nu}$. We have qualitative agreement of\nboth these quantities. For $\\nu \\ll 1 $ they go to negative\nconstant; for small angle deficit they are proportional to $\\nu -1$;\nfor $\\nu \\gg 1$ we have $A^0_{0(cs)} \\sim \\nu^4$ and \n$[\\nu^4A^0_{0(gm)}(1\/\\nu)] \\sim \\nu^4 [\\ln \\nu + 60C_0]$. Both\nof these quantities change the sign at point $\\nu = 1$. \n\nThe dependence of $<T^0_0>^{ren}$ on the distance from origin in\nthe global monopole spacetime is competely different from the\ncase of the infinitely thin cosmic string due to the logarithmic\nterm. Let us consider the physical distance $R=\\mu r\/\\alpha$\nmeasured in a mass scale $\\mu$. Then, the $<T^0_0>^{ren}$\nmeasured in units $\\mu^4$ has the form \n\\begin{equation}\n8\\pi^2 \\mu^{-4}<T^0_0>^{ren} = {\\alpha^4 - 1 \\over 60\\alpha^4\nR^4} \\ln {R \\over R_0}\\ ,\n\\end{equation}\nwhere $R_0 = \\exp (-A^0_0\/B^0_0)$. In the case $\\alpha < 1$ the\nenergy density $<T^0_0>^{ren}$ is positive in the domain $R\\in (0,R_0)$;\nit changes the sign in the point $R=R_0$ and goes, through\nthe minimum at point $R_* = R_0e^{1\/4}$, to zero in infinity.\nIn the case $\\alpha > 1$ we have opposite picture with maximum\nat the point $R_*$. The dependence of $R_0$ on the $\\alpha$\nis shown in the Fig.\\ref{1}. For small $\\alpha$ it is proportional\nto $1\/\\alpha$ and it goes to constant for great $\\alpha$. \n\\section{Conclusion}\\label{Conclusion}\nIn this paper we have considered the quantum, a $1\/2$-spin\nleft-handed field in the background of a pointlike global\nmonopole described by the metric tensor given in (\\ref{Metric}).\nMore specifically, we have obtained the complete Feynman\npropagator, expressed in terms of a bispinor, in a closed form.\nDifferently from the results obtained in Ref.\n\\cite{MazitelliLousto} for scalar field, in this analysis there \nappears two effective angular total quantum numbers, which we\ncall by $\\nu_1$ and $\\nu_2$. These angular total quantum numbers\nare related with the explicit properties of the spinor\nharmonics, $\\varphi^{(1)}_{j,m_j}$ and $\\varphi^{(2)}_{j,m_j}$\nand they have the following form $\\nu_1 = (l+1)\/\\alpha -1\/2$ and\n$\\nu_2 = l\/\\alpha + 1\/2$. For the scalar case the effective\norbital quantum numbers take a simple form only under specific\nsituation \\cite{MazitelliLousto}, namely for $\\xi = 1\/8$.   \n\nThe main goal of this work was to obtain the renormalized\nvacuum expectation value of the energy-momentum tensor. Because\nall components of the VEV of the energy-momentum tensor can be\nexpressed in terms of its zero-zero component, we used only the\nEuclidean Green function at the coincidence limit $\\Omega' =\n\\Omega$ and $r' = r$. In this case, in order to obtain the\nregularized expression for $<T^0_0>$, we were able to subtract\nall divergencies in a manifest form.  \n\nAfter the renormalization, the Green function (\\ref{GEren}) depends\non the scale parameter $\\mu$ which leads to logarithm\ncontribution to the energy-momentum tensor given by Eq.(\\ref{Tmn}).\nBut, as it was noted in Ref. \\cite{MazitelliLousto} the one-loop\nEinstein equations \n\\begin{equation}\nR_{\\mu\\nu} - {1\\over 2}R g_{\\mu\\nu} +\\epsilon_1 {}^{(1)}H_{\\mu\\nu} +\n\\epsilon_2 {}^{(2)}H_{\\mu\\nu} = 8\\pi \\{T_{\\mu\\nu}^{clas} + <T_{\\mu\n\\nu}>^{ren}\\}\\ ,\n\\end{equation}\ndo not depend on this parameter due to the renormalization group\nequations for the coefficients $\\epsilon_1$ and $\\epsilon_2$. Any\nvariation of the scale parameter $\\mu$ may be absorbed by\nvariation of the $\\epsilon_k$. \n\nTaking into account the conservation law (\\ref{Conservation}), the\nexpression for conformal anomaly (\\ref{Trace}) and the spherical\nsymmetry of problem we expressed all components of\nenergy-momentum tensor in terms of zero-zero component\n(\\ref{AnmBnm}) which has the form\n\\begin{equation}\n<T^0_0>^{ren} = {1 \\over 8\\pi^2 r^4}\\left[ A^0_0 +\nB^0_0 \\ln {\\mu r\\over \\alpha}\\right]\\ , \\label{ApT00}\n\\end{equation} \nwhere $A^0_0$ and $B^0_0$ are given by eqs. (\\ref{A00}) and\n(\\ref{B00}). The component $A^0_0$ depicted in Fig.1 as a\nfunction of $\\alpha$, and it is qualitatively agreed with\nsimilar one in the infinitely thin cosmic string spacetime. \n\nThe scaling parameter $\\mu$ leads to the logarithmic\ncontribution in the energy-momentum tensor (\\ref{ApT00}). For\nthis reason $<T^0_0>^{ren}$ changes its sign in some point $R_0$ and has\nthe extremum at the point $R_* = R_0 e^{1\/4}$. The dependence of\n$R_0$ on the $\\alpha$ is depicted in the Fig. \\ref{1}. \n\nBefore we finish this paper we would like to make a brief\ncomment about some results previously obtained in the literature\nrelated with quantum calculation of the energy-momentum tensor\n$T^\\nu_\\mu$ in the spacetime of a pointlike global monopole.\nHiscock, in Ref. \\cite{Hiscock}, using general consideration,\nobtained a formal expression for the vacuum expectation value of\nthe energy-momentum tensor for an arbitrary collection of\nconformal massless fields in this manifold. Later, Mazzitelly\nand Lousto, in Ref. \\cite{MazitelliLousto}, developed the\nexplicit calculation for renormalized vacuum average of the\nsquare of a massless scalar quantum field and also, by general\nconsideration, infered the structure for $T^\\nu_\\mu$ which\ndisagrees with the Hiscock's results and manifestly depends on\nthe scale parameter $\\mu$. This is in agreement with general\nconsideration by Wald in Ref. \\cite{WaldPRD}. In our paper we\nobtained the explicit expression for vacuum average value of the\nenergy-momentum tensor for a massless fermionic field in this\nmanifold, giving the complete information about this tensor in\nterms of $A^0_0$ and $B^0_0$. We also have analyzed the\nbehaviour of these functions with the parameter $\\alpha$, and\nthe dependence of $T^0_0$ with the physical distance $R$.\nFinally we want to say that our expression for VEV of the\nenergy-momentum tensor depend on the scale parameter $\\mu$, in\nagreement with Ref. \\cite{WaldPRD}, due to the appearance of\na logarithm term, which is a consequence of the regularization\nprocedure. This term is responsible for the changing in the sign\nof the vacuum energy-momentum tensor when we vary the physical\ndistance $R$. \n\nThe explicit expression for the massless left-handed\ntwo-component spinor Green function, is obtained in the\nbackground of global monopole spacetime. It is our interest to\ndevelop a similar calculation for the massive case and also\nobtain the vacuum average for the energy-momentum tensor.\n\\section*{Acknowledgments}\nNK is grateful to Departamento de F\\'{\\i}sica, Universidade\nFederal da Paraiba (Brazil) where this work was done, for\nhospitality. His work was supported in part by CAPES and in part\nby the Russian Fund for Basic Research, grant No 97-02-16318.  \n\nERBM and VBB also would like to thank the Conselho Nacional de\nDesenvolvimento Cientifico e Tecnol\\'ogico (CNPq). \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\\noindent\nBlack holes are described in Einstein gravity and are governed by a metric dynamics on a Riemannian manifold. Interestingly, the stringy versions of these macroscopic black holes have been obtained in a low energy limit of a string effective action  \\cite{gibbons-maeda-NPB,garfinkle,giddings-strominger}.\n\n\\vspace{.08in}\n\\noindent\nIn the past there have been attempts to use a gauge principle for the Riemannian geometry \\cite{wilczek,Sharpe-torsion}. Interestingly the quantum effects to Einstein gravity may also be addressed non-perturbatively using the strong-weak coupling duality in ten dimensional superstring theories. A non-perturbative quantum effect is believed to be sourced by the compactified extra space dimension(s) to the stringy vacua. In particular the type IIA superstring theory in a strong coupling limit is known to incorporate an extra spatial dimension on $S^1$ and has been identified with an eleven dimensional non-perturbation (NP) $M$-theory. Generically $M$-theory has been shown to be identified with the stringy vacua in various dimensions. In a low energy limit $M$ theory is known to describe $N$$=$$1$ supergravity (SUGRA) in eleven dimensions \\cite{cremmer-julia-scherk}. However a complete NP-theory is not fully been explored \\cite{schwarz-PLB}.\n\n\\vspace{.08in}\n\\noindent\n{Interestingly, it has been shown in the context of extended supergravity, one can express the curvatures in terms of torsion and its covariant derivatives thereby implying that torsion  has a geometrical origin \\cite{dragon}. Also, the self duality of Born-Infeld action for  $3$-branes of Type IIB theory, underlying SL(2, Z) symmetry provide strong hints towards a $d$$=$$12$ generic form theory. The idea of duality in Type IIB superstring theory has been explored in the context of globally electric charged $(-1)$-branes(instantons) which are  dual to $7$-branes in $d$$=$$10$ and are known to describe spacetime wormhole underlying Riemannian geometry in Type IIB Supergravity theory \\cite{gibbons96}.\n\n\\vspace{.08in}\n\\noindent\nThe formalism presented in the article deals only with the field theory underlying commutative spacetimes. However, an approach to emergent gravity theory underlying non-commutative (NC) spacetime has been explored  in the past\\cite{seiberg-witten,Yuji-npb2001}. The presence of non-commutativity  leads to non-trivial modifications in the properties of black holes\\cite{rizzo}. The idea provides a clue to explore a dynamical NS two-form in a modified torsion gravity originally sourced by a  gauge theory.  It has  been shown that gravity can emerge from a gauge theory in noncommutative (NC) spacetime. The NC gauge\/gravity correspondence has been further explored for a constant two-form $B_{\\mu\\nu}$ background\\cite{Miao1999}. For interesting details see refs \\cite{yangmpla, yangijmpa, yangjhep, yangarxiv}. In addition some regular metric solutions, underlying non-commutative geometry, has been obtained  to describe de Sitter at small scale which approaches Riessner-Nordstr$\\ddot{\\rm o}$m geometry at large scale \\cite{nicolini}. Interestingly  it has been shown that the  geometrical nature of gravity may be governed by a  torsion in an alternate formulation\\cite{Niels-2018,Jose-arXiv-2019,TPS-arXiv2019}\n\n\\vspace{.08in}\n\\noindent\n{Interestingly an effective field  description in $d$$=$$12$ has been proposed by considering supergravity theory in the same dimension. Such a theory describes Type IIA and IIB SUGRA upon compactification on $S^{1}$ and  is Poincar$\\acute{\\rm e}$ invariant when compactified on a torus\\cite{kang}. Recently  it has been proposed that starting from a covariant formulation in $d$$=$$12$ and  it is possible to construct higher derivative effective field theory. Further it was shown to reduce to Type IIB SUGRA when compactified on a torus \\cite{hamid}.}\n\n\\vspace{.08in}\n\\noindent\nIn the article we begin with $d$$=$$5$  Kalb-Ramond two-form $B_{\\mu\\nu}$ gauge theory.  In fact  a gauge theoretic torsion curvature $H_3$$=$$d^{\\nabla}B_{2}$ on a $4$-brane has been explored to modify the derivative $\\nabla_{\\mu}$ to a covariant  derivative ${\\cal D}_{\\mu}$.    \nWe attempt to construct  $M$ theoretic action underlying a geometric torsion ${\\cal H}_3$ dynamics in a $U(1)$  gauge theoretic formulation defined with ${\\nabla}_{\\mu}$ covariant derivative. In particular the ${\\cal H}_3$$=$$d^{\\cal D}B_{2}^{(NS)}$ has been shown to govern the dynamics of a Neveu-Schwarz (NS) two-form  on a fat brane in an emergent gravity  scenario \\cite{abhishek-JHEP,abhishek-PRD,abhishek-NPB-P}. It was shown that the emergent theory is governed by a fourth order generalized curvature tensor ${\\cal K}_{\\mu\\nu\\lambda\\rho}$ defined with two derivatives \\cite{abhishek-JHEP}. Interestingly  a non-constant NS two-form background has been attempted to obtain a non-commutative D-brane in a different context \\cite{Ho-prl2000}. \n\n\\vspace{.08in}\n\\noindent\nFurthermore  we explore the two derivative formulation underlying  a dynamical geometric torsion ${\\cal H}_3$. In this paper  we explicitly realize a four-form ${\\cal F}_4$$=$$d^{\\nabla}{\\cal H}_{3}$ field strength for an onshell NS two-form. In fact  the  NS-form gauge theory  has been shown to receive a dynamical geometric torsion correction which turns out to be a second derivative in NS-form. Together they define a $d$$=$$5$ NP-theory in the modified theory. In particular the dynamical correction ${\\cal F}_4$ to the low energy Einstein vacuum has been argued to be sourced by an  instanton underlying an axionic scalar which  is  is believed to be non-perturbative in nature. \n\n\\vspace{.08in}\n\\noindent\nWe show that our analysis generically reveals a fundamental theory in $d$$=$$12$ and  may represent a  {\\it form} theory. It is shown that a dynamical quintessence field incorporates a NP-correction hidden to the torsion-less geometries. Analysis reveals that a fundamental building block of {\\it form} theory, $i.e.$ an instanton, can be a potential candidate to describe the quintessence and hence dark energy in universe. At this point we may recall that  $(-1)$-branes are dual to $9$-branes in $D$$=$$10$ superstring theory.  In fact $9$-branes  are space filling and they have been viewed as a pair of a gravitational $(8\\bar{8})$-brane pair configuration\\cite{abhishek-JHEP}. \n\n\\section{Higher form dynamics}\n\\subsection{Pair production scenarios} \n\\noindent\nWe begin by briefly revisiting the Schwinger pair production mechanism \\cite{schwinger}. The novel idea elegantly describes a $(e^+e^-)$ pair production underlying a QFT vacuum, by a photon. The mechanism is an emergent non-perturbative phenomenon and is believed to be instrumental to describe diverse quantum phenomenon in gravitation and cosmology. In fact this idea was explored to explain the Hawking radiation phenomenon \\cite{hawking} where an incident photon generates a pair of charged particle\/anti-particle at the event horizon of a black hole. \n\n\\vspace{.08in}\n\\noindent\nFurthermore  the idea was also applied to the open strings pair production \\cite{bachas-porrati} and to investigate the production of a pair of $(D{\\bar D})_9$ at the cosmological horizon \\cite{majumdar-davis}. An extra spatial dimension on $S^1$ has been argued to unfold in $d$$=$$10$, type IIA superstring theory in a strong coupling and is believed to describe the $M$-theory in $d$$=$$11$.\n\n\\vspace{.08in}\n\\noindent\nIn the context an emergent theory of gravity on a stringy pair of $(3{\\bar 3})$-brane by a Kalb-Ramond (KR)-form quanta on a $D_4$-brane has been formulated in a collaboration \\cite{abhishek-JHEP,abhishek-PRD}. Importantly a generic vacuum geometry in general theory of relativity (GTR), $i.e.$ a Kerr family of black holes, has been obtained in a NP-decoupling limit which leads to the low energy (semi-classical) geometries \\cite{sunita-NPB}.\n\n\\subsection{Emergent gravity in ${\\mathbf 5}$D}\n\\noindent\nInterestingly  the emergent gravity patches, underlying a non-linear higher form field quanta, were (spin) projected using a discrete matrix to describe an appropriate causal geometry in a low energy (GTR) limit \\cite{priyabrat-EPJC}. The emergent gravity patches have formally been viewed as a lower dimensional $D_p$-brane correction to a low energy string vacuum \\cite{priyabrat-EPJC}. Since a $D_p$-brane carries Ramond-Ramond (RR) charge, the correction turns out to be non-perturbative \\cite{polchinski}. \n\n\\vspace{.08in}\n\\noindent\nIt was argued that an emergent pair breaks the supersymmetry due to the exchange \nof closed string modes in between the vacuum created gravitational pair of $(3{\\bar 3})$-brane across the cosmological horizon \\cite{abhishek-JHEP}. Thus, the emergent gravitational pair is described on a fat brane underlying a massive NS field dynamics in a first order theory. In fact, a quintessence scalar field acts as an extra (fifth) hidden dimension inbetween an emergent \ngravitational brane pair. It determines the thickness of a fat brane. \n\n\\vspace{.08in}\n\\noindent\nAlternately  a quintessence degree of freedom, underlying the closed string exchange between the gravitational pair, is absorbed by a $D_3$-brane to make it a gravitational fat $3$-brane. In the context a vacuum ($T_{\\mu\\nu}$$=$$0$) refers to a background black hole defined with an open string metric on a $D_4$-brane.\n\\begin{eqnarray}}\\newcommand{\\eea\n{\\tilde{G}}_{\\mu\\nu}=\\big ( g_{\\mu\\nu}-B_{\\mu}^{(NS)\\lambda}B^{(NS)}_{\\lambda\\nu}\\big )\\ ,\n\\eea\nwhere $g_{\\mu\\nu}$ is a metric underlying vanishing Riemannian curvature.  Thus  the open string metric is sourced by a constant (global mode)  NS field \\cite{seiberg-witten} in the string bulk. In addition the Poincar$\\acute{\\rm e}$ duality in the five dimensional world-volume gauge theory ensures a two-form  KR dynamics on a $D_4$-brane. Thus the non-linear $U(1)$ gauge dynamics of a $D_4$-brane in presence of a background metric may be envisaged with a constant NS field and a local KR field. It is described by\n\\begin{equation}} \\newcommand{\\ee}{\\end{equation}\nS={{-1}\\over{(8\\pi^3g_s){\\alpha'}^{3\/2}}}\\int d^5x {\\sqrt{-{\\tilde G}^{({\\rm NS})}}}\\ H_{\\mu\\nu\\lambda}H^{\\mu\\nu\\lambda}\\ ,\\label{p-1}\n\\ee\nwhere ${\\tilde{G}}=$$\\det{\\tilde{G}}_{\\mu\\nu},\\ H_{\\mu\\nu\\lambda}$$=$$\\big ( \\nabla_{\\mu}B_{\\nu\\lambda}$$+{\\rm cyclic}\\big )$, $g_{s}$ is the string coupling and $\\alpha'=$ string length parameter. Generically  the idea is to explore a higher-form $U(1)$ gauge theory and hence its geometric effects  on an appropriate $D_p$-brane. In fact a higher-form gauge theory is non-linear and becomes sensible in higher dimensions. It may provide a clue to unfold a NP-theory of gravity in higher dimensions such as $M$-theory in eleven dimensions \\cite{schwarz-PLB}. A priori for simplicity one may  begin with a KR two-form gauge theory on a $D_4$-brane in a ten dimensional type IIA superstring theory. Nevertheless a five dimensional construction is a minimal extension to the GTR and is believed to be a fundamental unit. Needless to mention that GTR is an elegant construction by Einstein in $d$$=$$4$ defined with a vacuum solution whereas BTZ black hole in $d$$=$$3$ is a brilliant construction for a non vacuum solution \\cite{BTZ}.  As a bonus our formulation $d$$=$$5$  takes  account for the quint-essential cosmology. Quintessence has been explored in the context  \\cite{priyabrat-EPJC}. It is believed to be a candidate for  the observed accelerated rate of expansion of universe and the origin of dark energy.\n\n\\subsection{NS field dynamics}\n\\noindent\nAn emergent fat brane evolves with a dynamical NS field which is obtained at the expense of the KR field dynamics on a $D_4$-brane. Thus, a covariantly constant NS field, $i.e.\\ \\nabla_{\\lambda}B_{\\mu\\nu}^{(NS)}=0$, on a $D_4$-brane becomes dynamical on a fat brane. A geometric torsion ${\\cal H}_3$ is\nconstructed using a modified derivative:\n\\begin{eqnarray}}\\newcommand{\\eea\n&&{\\cal H}_{\\mu\\nu\\lambda}={\\cal D}_{\\mu}B^{(NS)}_{\\nu\\lambda}+{\\cal D}_{\\nu}B^{(NS)}_{\\lambda\\mu}+{\\cal D}_{\\lambda}B^{(NS)}_{\\mu\\nu}\n\\ ,\\qquad\\; {}\\nonumber\\\\\n{\\rm where}\\qquad &&{\\cal D}_{\\lambda}B^{(NS)}_{\\mu\\nu}={1\\over2}\\left ({H_{\\lambda\\mu}{}}^{\\rho}B^{(NS)}_{\\nu\\rho}+ {H_{\\lambda\\nu}{}}^{\\rho}B^{(NS)}_{\\rho\\mu}\\right ).\\label{gtorsion-1}\n\\eea\nA constant NS field background in a closed string, becomes a nontrivial geometric torsion at the expense of the the KR field dynamics, $i.e.\\ {\\cal D}_{\\lambda}B_{\\mu\\nu}=0$. It implies that the modified derivative ${\\cal D}_{\\mu}$ is uniquely fixed. In fact, the absorption of KR field quanta, in a background black hole, has given birth to an emergent quantum theory of gravitation on a stringy pair of $(3{\\bar 3})$-brane across an event horizon \\cite{abhishek-PRD}. Interestingly, a plausible NP-theory of gravity underlying an emergent scenario enforces an iterative correction, where the KR-form dynamics is replaced by the NS-form, $i.e.\\ H_3\\rightarrow {\\cal H}_3$. It leads to an exact description with a $B^{(NS)}_2$ perturbation:\n\\begin{equation}} \\newcommand{\\ee}{\\end{equation}\n{\\cal H}_{\\mu\\nu\\lambda}=H_{\\mu\\nu\\rho}B_{\\lambda}^{(NS)\\rho}+ H_{\\mu\\nu\\alpha}B_{\\rho}^{(NS)\\alpha} B_{\\lambda}^{(NS)\\rho}+\\dots\\label{gtorsion-2}\n\\ee\nFor an infinitesimal perturbation, the geometric torsion: ${\\cal H}_3\\rightarrow H_3$. A priori an exact perturbation NS field theory may equivalently be described by a NP-theory of gravity whose gauge potential shall be shown to be sourced by a dynamical NS field. \n\n\\vspace{.08in}\n\\noindent\nIn fact there are three distinct dynamical aspects of the same description in five dimensions and they are: (i) the KR perturbation theory underlying a gauge group $U(1)$ on a $D_4$-brane and is defined with a covariant derivative $\\nabla_{\\mu}$, (ii) the NS perturbation theory of emergent semi-classical gravity on a fat brane underlying the gauge group $U(1)$ and is defined with a modified covariant derivative ${\\cal D}_{\\mu}$, (iii) the geometric torsion non-perturbation theory of emergent quantum gravity on a vacuum created $(3{\\bar 3})$-brane underlying an enhanced gauge group $U(1)\\otimes U(1)_{\\rm NP}$. An additional $U(1)$ in a NP-theory is only sensible in second order while the KR and NS perturbation gauge theories are first order formulations. Both, the original and the emergent, perturbation theories are respectively described by the $U(1)$ gauge invariant field strengths $H_3$ in eq(\\ref{p-1}) and ${\\cal H}_3$ in eq(\\ref{gtorsion-1}). However, the NP-theory breaks the gauge invariance perturbatively which is evident from the field strength expression in eq(\\ref{gtorsion-2}). The perturbative gauge invariance has been realized \\cite{abhishek-JHEP,priyabrat-EPJC} in presence of a symmetric fluctuation: $f_{\\mu\\nu}={\\bar{\\cal H}}_{\\mu\\alpha\\beta}{{\\cal H}^{\\alpha\\beta}{}}_{\\nu}$, where ${\\bar{\\cal H}}_3=(2\\pi\\alpha'){\\cal H}_3$. Thus a dynamical NS field modifies the constant background metric $G_{\\mu\\nu}^{(NS)}$ into a dynamical metric on an emergent fat brane. The modified metric is given by\n\\begin{equation}} \\newcommand{\\ee}{\\end{equation}\nG_{\\mu\\nu}=\\Big ( g_{\\mu\\nu}-B_{\\mu}^{(NS)\\lambda}B^{(NS)}_{\\lambda\\nu} + {\\bar{\\cal H}}_{\\mu\\lambda\\rho}{{\\cal H}^{\\lambda\\rho}}_{\\nu}\\Big )\\ . \\label{gauge-metric}\n\\ee\nFirstly, it shows that a geometric torsion incorporates an intrinsic angular momentum in an emergent vacuum. Thus an emergent gravity naturally governs the Kerr black hole as a vacuum geometry \\cite{sunita-NPB}. \n\n\\vspace{.08in}\n\\noindent\nSecondly, the emergent metric (\\ref{gauge-metric}) dynamics does not emerge in first order perturbation theory and therefore clearly requires at least a second order formulation. Most importantly, the gauge invariance shall shown to be restored in a second order for an onshell NS field in the paper. As a result, the geometric torsion ${\\cal H}_3$ is perceived as a gauge potential underlying a NP-dynamical correction to the perturbative NS field dynamics in first order and hence the complete NP-theory is governed in $1.5$ order by ${\\cal H}_3$.\n\n\\vspace{.08in}\n\\noindent\nAt a first sight it may imply that the modified metric dynamics is primarily governed by a propagating ${\\cal H}_3$. However a geometric torsion theory, being a quantum description, cannot be identified with a metric tensor theory. This is due to a fact that the (pseudo) Riemannian geometry is sourced by a metric tensor (GTR). The apparent puzzle is resolved with an emergent metric which is a semi-classical phenomenon as is realized in a first order perturbation theory of a NS field. The complete NP-theory in $1.5$ order is an emergent quantum gravity description underlying a geometric torsion for the onshell NS field. Thus, the notion of a dynamical metric is not sensible due to the prevailing geometric torsion dynamics in a NP-theory of quantum gravity in $d$$=$$5$. A propagating torsion is known to break the space-time continuum and hence there are no closed geodesics.\n\n\\vspace{.08in}\n\\noindent\nIn the paper, we explicitly work out a NP-dynamical correction in a $d$$=$$5$ emergent scenario which in turn, is a key to unfold an emergent $M$-theory in eleven dimensions underlying a {\\it fundamental} ({\\it form}) theory in twelve dimensions. It has been shown that the GTR emerges as non-perturbative phenomenon on a stable gravitational pair of $(3{\\bar 3})$-brane \\cite{abhishek-JHEP,abhishek-PRD,abhishek-NPB-P}. A fact that GTR is a fundamental unit, or a $3$-brane in the formulation, allows one to believe in a twelve dimensional {\\it fundamental} theory which is worked out in this paper. In addition, a coupling between the Riemann tensor and a NP-dynamics has been realized which signals the intrinsic role played by a geometric torsion in the GTR. \n\\section{New curvatures in modified gravity}\n\\subsection{Emergent curvature tensors $\\big ({{{\\cal H}_{\\mu\\nu}{}}^{\\lambda}},\\ {{\\cal K}_{\\mu\\nu\\lambda\\rho}},\\ {{\\cal L}_{\\mu\\nu\\lambda\\rho}}\\big )$}\n\\noindent\nThe commutator of modified derivatives have been worked out to obtain the generic curvature tensors \\cite{abhishek-JHEP}. It was shown that \nan emergent curvature describes a vacuum created gravitational pair of $(3{\\bar 3})$-brane \\cite{abhishek-PRD}. The commutators yield:\n\\begin{eqnarray}}\\newcommand{\\eea\n &&\\left [ {\\cal D}_{\\mu}\\ ,\\ {\\cal D}_{\\nu}\\right ]\\psi\\ =\\  -2\\ {{\\cal H}_{\\mu\\nu}{}}^{\\lambda}\\ {\\cal D}_{\\lambda}\\psi\\nonumber\\\\\n {\\rm and} \\qquad &&\\left [ {\\cal D}_{\\mu}\\ ,\\ {\\cal D}_{\\nu}\\right ]A_{\\lambda}\\ =\\  \\left ({{\\cal R}_{\\mu\\nu\\lambda}}^{\\rho}+{{\\cal K}_{\\mu\\nu\\lambda}{}}^{\\rho}\n\\ +\\ {{\\cal L}_{\\mu\\nu\\lambda}}^{\\rho}\\right )A_{\\rho} +{{\\cal H}_{\\mu\\nu}}^{\\rho}{\\cal D}_{\\rho}A_{\\lambda}\\ ,\\label{new-curvature}\n\\eea\nwhere ${\\cal H}_3$ ensures a NS field dynamics in an emergent first order perturbation theory. \nIt provokes thought to believe that the quanta of NS field in superstring theory can govern an emergent graviton. \n\n\\vspace{.08in}\n\\noindent\nFurthermore, the coupling of ${\\cal H}_3$ to the {\\it dynamics} ${\\cal D}_{\\lambda}\\psi$ rather than the field $\\psi$ in eq(\\ref{new-curvature}) is a new phenomenon. A non-zero ${\\cal H}_3$ signifies a difference in energy between (emergent) quantum gravity and classical (Einstein) gravity. In fact, ${\\cal D}_{\\mu}\\psi\\neq 0$, $i.e.$ the dynamical scalar, is known to be pivotal to incorporate a non-perturbative (quantum) correction to the GTR \\cite{priyabrat-EPJC}. Thus, a dynamical correction to the GTR plays a significant role in quantum cosmology and is indeed a powerful tool to explain the accelerated rate of expansion of the universe. The inherent (quantum) dynamics of the scalar field makes the emergent gravity a natural candidate to describe the quintessence which is believed to govern the dark energy in the universe.\n\n\\vspace{.08in}\n\\noindent\nThe fourth order reducible emergent curvature tensors (\\ref{new-curvature}) play a significant role in a second order non-perturbation theory of emergent gravity. They are believed to be potential keys to a complete non-perturbation theory such as $M$ theory.The curvature ${\\cal R}_{\\mu\\nu\\lambda\\rho}$ is the well known Riemann curvature tensor, sourced by a metric field in $d$$=$$5$. It's very presence in the commutator implies the fact that the one-form $A_{\\mu}$, on which the modified derivatives ${\\cal D}_{\\mu}$ act may be non-linear in origin. The second emergent curvature,\n\\begin{equation}\n{{\\cal K}_{\\mu\\nu\\lambda}}^{\\rho} = \\frac{1}{2}\\partial_{\\mu}{{\\cal H}_{\\nu\\lambda}}^{\\rho} - \\frac{1}{2}\\partial_{\\nu}{{\\cal H}_{\\mu\\lambda}}^{\\rho} + \\frac{1}{4}{{\\cal H}_{\\mu\\lambda}}^{\\sigma}{{\\cal H}_{\\nu\\sigma}}^{\\rho} - \\frac{1}{4}{{\\cal H}_{\\nu\\lambda}}^{\\sigma}{{\\cal H}_{\\mu\\sigma}}^{\\rho} \n\\end{equation}\nis expressed as pair symmetric $(S)$ and  pair non-symmetric $({\\tilde A})$ under an interchange of first and second pair of indices, $i.e.\\ {\\cal K}^{(S)}_{\\mu\\nu\\lambda\\rho}+{\\cal K}^{({\\tilde A})}_{\\mu\\nu\\lambda\\rho}$. Explicitly all three emergent curvatures are:\n\\begin{eqnarray}}\\newcommand{\\eea\n{\\cal R^{\\lambda}}_{\\mu\\nu\\rho}&=& \\partial_{\\rho}\\Gamma^{\\lambda}_{\\mu\\nu}-\\partial_{\\nu}\\Gamma^{\\lambda}_{\\mu\\rho}+\\Gamma^{\\sigma}_{\\mu\\nu}\\Gamma^{\\lambda}_{\\rho\\sigma}-\\Gamma^{\\sigma}_{\\mu\\rho}\\Gamma^{\\lambda}_{\\nu\\sigma}\\ , \\nonumber \\\\\n{{\\cal L}_{\\mu\\nu\\lambda}}^{\\rho}&=&\\frac{1}{2} \\Big (\\Gamma^{\\rho}_{\\mu\\sigma} {{\\cal H}^{\\sigma}}_{\\nu\\lambda}+\\Gamma^{\\sigma}_{\\nu\\lambda} {{\\cal H}^{\\rho}}_{\\mu\\sigma} - \\Gamma^{\\sigma}_{\\mu\\lambda} {{\\cal H}^{\\rho}}_{\\nu\\sigma} -\\Gamma^{\\rho}_{\\nu\\sigma} {{\\cal H}^{\\sigma}}_{\\mu\\lambda} \\Big )\\nonumber \\\\\n{\\rm and} \\qquad \\qquad \\quad {\\cal K}^{(S)}_{\\mu\\nu\\lambda\\rho}&=& \\frac{1}{4}{{\\cal H}_{\\mu\\lambda}}^{\\sigma}{\\cal H}_{\\nu\\sigma\\rho}-\\frac{1}{4}{{\\cal H}_{\\nu\\lambda}}^{\\sigma}\n{\\cal H}_{\\mu\\sigma\\rho} \\ , \\nonumber\\\\\n \\qquad \\qquad \\qquad \\quad{\\cal K}^{({\\tilde A})}_{\\mu\\nu\\lambda\\rho}&=&\\frac{1}{2}\\partial_{\\mu}{\\cal H}_{\\nu\\lambda\\rho} -\\frac{1}{2}\\partial_{\\nu} {\\cal H}_{\\mu\\lambda\\rho}\\ ,\\label{g-curvature}\n\\eea\nwhere  $\\Gamma^{\\rho}_{\\mu\\sigma}$ denote the Christoffel connections sourced by the background metric ${\\tilde G}_{\\mu\\nu}$. Thus ${{\\cal L}_{\\mu\\nu\\lambda}}^{\\rho}$ describes a coupling between the GTR and an emergent gravity. It ensures a non-propagating geometric torsion and hence a torsion free geometry. An irreducible tensor is found to yield \\cite{abhishek-JHEP}:\n\\begin{equation}} \\newcommand{\\ee}{\\end{equation}\n\\delta^{\\nu}_{\\rho}{{\\cal L}_{\\mu\\nu\\lambda}}^{\\rho}={\\cal L}_{\\mu\\lambda}=\\frac{1}{2}\\Big (\\Gamma^{\\nu}_{\\mu\\rho} {{\\cal H}^{\\rho}}_{\\nu\\lambda}+\\Gamma^{\\rho}_{\\nu\\lambda} {{\\cal H}^{\\nu}}_{\\mu\\rho}- \\Gamma^{\\nu}_{\\nu\\rho} {{\\cal H}^{\\rho}}_{\\mu\\lambda}\\Big )=-{\\cal L}_{\\lambda\\mu}\\ .\\label{gauge-65}\n\\ee\nThe Lorentz scalar $\\big ({\\cal L}_{\\lambda\\mu}{\\cal L}^{\\lambda\\mu}\\big )$ involves quadratic power in geometric torsion and Christoffel connection(s). Since the  perturbation parameter is small, quadratic and higher powers become insignificant. A priori, it may imply that an emergent gravity in an all order perturbation theory is defined in a decoupling limit of two distinct connections. However, a non-perturbation emergent theory incorporates the coupling of connections defined with the Lorentz scalar \n$\\big ({\\cal L}_{\\lambda\\mu}{\\cal L}^{\\lambda\\mu}\\big )$.\n\n\\vspace{.08in}\n\\noindent\nFurthermore  the emergent curvature ${{\\cal K}_{\\mu\\nu\\lambda}}^{\\rho}$ in terms of covariant derivative $\\nabla_{\\mu}$ becomes\n\\begin{eqnarray}\n\\nabla_{\\mu}{{\\cal H}_{\\nu\\lambda}}^{\\rho}- \\nabla_{\\nu}{{\\cal H}_{\\mu\\lambda}}^{\\rho}&= \\partial_{\\mu}{{\\cal H}_{\\nu\\lambda}}^{\\rho}- \\partial_{\\nu}{{\\cal H}_{\\mu\\lambda}}^{\\rho}+2 {{\\cal L}_{\\mu\\nu\\lambda}}^{\\rho}\\nonumber\\\\\n{\\rm or}\\quad  \\frac{1}{2}\\partial_{\\mu}{{\\cal H}_{\\nu\\lambda}}^{\\rho}- \\frac{1}{2}\\partial_{\\nu}{{\\cal H}_{\\mu\\lambda}}^{\\rho} &= \\frac{1}{2}\\nabla_{\\mu}{{\\cal H}_{\\nu\\lambda}}^{\\rho}- \\frac{1}{2}\\nabla_{\\nu}{{\\cal H}_{\\mu\\lambda}}^{\\rho}-{{\\cal L}_{\\mu\\nu\\lambda}}^{\\rho} \\ ,\n\\end{eqnarray}\nwhere the emergent curvature ${{\\cal L}_{\\mu\\nu\\lambda\\rho}}^{\\rho}$ is defined by equation (\\ref{g-curvature}). Thus, the fourth order emergent curvature can be expressed as\n\\begin{equation}\n{\\cal K}_{\\mu\\nu\\lambda\\rho}={\\cal K}_{\\mu\\nu\\lambda\\rho}^{(s)}+ \\Big (\\frac{1}{2}\\nabla_{\\mu}{\\cal H}_{\\nu\\lambda\\rho}- \\frac{1}{2}\\nabla_{\\nu}{\\cal H}_{\\mu\\lambda\\rho}\\Big )-{\\cal L}_{\\mu\\nu\\lambda\\rho}\\ ,\n\\end{equation}\nwhere the second term in the bracket in the emergent curvature ${\\cal K}_{\\mu\\nu\\lambda\\rho}^{(\\tilde{A})}$ in equation(\\ref{g-curvature}). The only difference is that, here the operating derivative is $\\nabla_{\\mu}$ instead of ordinary derivative $\\partial_{\\mu}$.\nThe commutator (\\ref{new-curvature}), simplifies to\n\\begin{equation}\\label{commutator-01}\n [{\\cal D}_{\\mu} , {\\cal D}_{\\nu}]A_{\\lambda} =\\Big ({\\cal R}_{\\mu\\nu\\lambda\\rho}+{\\cal K}_{\\mu\\nu\\lambda\\rho}^{(s)}+ \\frac{1}{2}\\nabla_{\\mu}{\\cal H}_{\\nu\\lambda\\rho}- \\frac{1}{2}\\nabla_{\\nu}{\\cal H}_{\\mu\\lambda\\rho} \\Big )A^{\\rho}+{{\\cal H}_{\\mu\\nu}}^{\\rho}{\\cal D}_{\\rho}A_{\\lambda}\\ .\n\\end{equation}  \n\\vspace{.08in}\n\\noindent\nThe emergent curvature tensor ${\\cal K}^{(S)}_{\\mu\\nu\\lambda\\rho}$ of order four (\\ref{g-curvature}) shares all the properties of a Riemann tensor under the interchange of indices within a pair and with pairs. In addition, the pair-symmetric curvature is sourced by a NS field propagation in an emergent perturbation theory in first order. As explained, it does not depend on the propagation of a ${\\cal H}_3$ which is a second order phenomenon. Thus, it describes a torsion free geometry. Thus the pair-symmetric tensor may be identified with the Riemann type tensor ${\\cal R}_{\\mu\\nu\\lambda\\rho}$.  The emergent curvature ${\\cal K}_{\\mu\\nu\\lambda\\rho}^{(\\tilde{A})}$ in eq(\\ref{g-curvature}), at the expense of another emergent curvature ${\\cal L}_{\\mu\\nu\\lambda\\rho}$, may be expressed as,\n\\begin{eqnarray}\n{\\cal K}_{\\mu\\nu\\lambda\\rho}^{(\\tilde{A})} &=&  \\frac{1}{2}\\nabla_{\\mu}{\\cal H}_{\\nu\\lambda\\rho}- \\frac{1}{2}\\nabla_{\\nu}{\\cal H}_{\\mu\\lambda\\rho}\\\\\n {\\rm or} \\qquad{\\cal K}_{\\mu\\nu\\lambda\\rho}^{(\\tilde{A})}&=& \\frac{1}{2} \\Big (\\nabla_{\\mu}{{\\cal H}_{\\nu\\lambda\\rho}} - \\nabla_{\\rho}{\\cal H}_{\\mu\\nu\\lambda}+\\nabla_{\\lambda}{\\cal H}_{\\rho\\mu\\nu}-\\nabla_{\\nu}{{\\cal H}_{\\lambda\\rho\\mu}} \\Big )\\nonumber \\\\\n&+& \\frac{1}{2} \\nabla_{\\rho}{\\cal H}_{\\mu\\nu\\lambda} -\\frac{1}{2}\\nabla_{\\lambda}{\\cal H}_{\\rho\\mu\\nu} \\ , \\\\\n&=&  \\frac{1}{2} \\Big (\\nabla_{\\mu}{{\\cal H}_{\\nu\\lambda\\rho}} - \\nabla_{\\rho}{\\cal H}_{\\mu\\nu\\lambda}+\\nabla_{\\lambda}{\\cal H}_{\\rho\\mu\\nu}-\\nabla_{\\nu}{{\\cal H}_{\\lambda\\rho\\mu}} \\Big ) + {\\cal K}^{(A)}_{\\mu\\nu\\lambda\\rho}\\ ,\n\\end{eqnarray}\nwhere in the second term, we have added and subtracted two covariant derivative terms. The commutator (\\ref{commutator-01}), using above equations, can be put into much more elegant form,\n\\begin{equation}} \\newcommand{\\ee}{\\end{equation}\n [{\\cal D}_{\\mu} , {\\cal D}_{\\nu}]A_{\\lambda} =\\Big ({\\cal R}_{\\mu\\nu\\lambda\\rho}+{\\cal K}_{\\mu\\nu\\lambda\\rho}^{(S)}+ {1\\over{\\sqrt{8 \\pi \\alpha'}}}{\\cal F}_{\\mu\\nu\\lambda\\rho} \\Big )A^{\\rho}+{\\cal K}_{\\mu\\nu\\lambda\\rho}^{(A')}A^{\\rho}+{{\\cal H}_{\\mu\\nu}}^{\\rho}{\\cal D}_{\\rho}A_{\\lambda} \\ .\n\\ee\nThe emergent curvatures ${\\cal F}_{\\mu\\nu\\lambda\\rho}$ and ${\\cal K}_{\\mu\\nu\\lambda\\rho}^{(A')}$ are completely antisymmetric and pair asymmetric tensors respectively.\n\n\\vspace{.08in}\n\\noindent\nA fact that the pair symmetric generalized curvature ${\\cal K}^{(S)}_{\\mu\\nu\\lambda\\rho}$, sourced by a dynamical two-form, precisely shares the tensor properties of the Riemann tensor is remarkable. Interestingly, the curvature is an invariant notion and its source, \\emph{i.e.} a (gauge) potential is not. The observation may provoke thought to believe that a dynamical two-form may incorporate a Riemannian geometry notionally. Needless to mention that the pair symmetric ${\\cal K}^{(S)}_{\\mu\\nu\\lambda\\rho}$ is free from torsion ${\\cal H}_{\\mu\\nu\\lambda}$ dynamics and further ensures its consistency with the Riemannian geometry. On the other hand, the remaining part of the generalized curvature, ${\\cal K}_{\\mu\\nu\\lambda\\rho}^{(\\tilde{A})}$ ensures a propagating torsion ${\\cal H}_{\\mu\\nu\\lambda}$ which in turn has been identified with a four-form ${\\cal F}_{\\mu\\nu\\lambda\\rho}$ in the formulation. Thus, the ${\\cal F}_{4}$ is believed to incorporate a dynamical correction to the Riemann like ${\\cal K}_{\\mu\\nu\\lambda\\rho}^{(S)}$ in the formulation. It may lead to an emergent quantum gravity description. In the context we may briefly revisit an interesting feature of an extremal Reissner-Nordstr$\\ddot{o}$m (RN) black hole in Newtonian limit. The shrinking event horizon of the RN black hole ensures the decoupling of the non-linear modes of Einstein gravity and leads to a stable extremal RN black hole where mass is identified with the electric charge. Interestingly, the extremal limit may be identified with an equipotential underlying two distinct elegant formulations such as the Newtonian gravity and the Electro-Magnetic theory. The attractive gravitational force between two massive and electrically charged objects in the limit is precisely cancelled by the repulsive coulombic force. Thus, an extremal RN black hole may provide a clue to explore some quantum phenomenon and is likely to bring-in a new shade to quantum gravity. The essentil idea contained in the analysis may help to explain a possibility of cancellation of the pair symmetric ${\\cal K}_{\\mu\\nu\\lambda\\rho}^{(S)}$ and Riemann curvatures. This in turn would govern a propagating (geometric) torsion whose field strength is defined as ${\\cal F}_{\\mu\\nu\\lambda\\rho}=\\nabla_{\\mu}{\\cal H}_{\\nu\\lambda\\rho}+$ cyclic. \n\n\\subsection{Dynamical correction}\n\\noindent\nThe ${\\cal H}_3$ dynamics may be re-expressed with a pair anti-symmetric $(A)$ tensor:\n\\begin{equation}} \\newcommand{\\ee}{\\end{equation}\n\\qquad \\qquad {\\cal K}^{({\\tilde A})}_{\\mu\\nu\\lambda\\rho}\\ =\\ {1\\over{2\\sqrt{2\\pi\\alpha'}}}\\ {\\cal F}_{\\mu\\nu\\lambda\\rho} \\ +\\ {\\cal K}^{(A')}_{\\mu\\nu\\lambda\\rho}\\ ,\\label{gauge-7777}\n\\ee\nwhere ${\\cal F}_{\\mu\\nu\\lambda\\rho}=4{\\sqrt{2\\pi\\alpha'}}\\ {\\nabla}_{[\\mu}{\\cal H}_{\\nu\\lambda\\rho]}$.\nThe second term is described by an irreducible curvature: \n\\begin{equation}} \\newcommand{\\ee}{\\end{equation}\n\\qquad \\qquad {\\cal K}_{\\nu\\rho}^{(A')}=g^{\\mu\\lambda}{\\cal K}_{\\mu\\nu\\lambda\\rho}^{(A')}=-{\\cal K}_{\\rho\\nu}^{(A')}=\\frac{1}{2}\\nabla_{\\mu}{{\\cal H}^{\\mu}}_{\\rho\\nu}\n\\ .\\label{gauge-77}\n\\ee\nThus, onshell NS-form implies that the two-form ${\\cal K}^{(A')}_{\\mu\\nu}$ becomes trivial in the action. Then the Lorentz scalar:\n\\begin{equation}} \\newcommand{\\ee}{\\end{equation}\n\\qquad \\qquad{\\cal K}^{({\\tilde A})}_{\\mu\\nu\\lambda\\rho} {\\cal K}_{(\\tilde{A})}^{\\mu\\nu\\lambda\\rho}\\ =\\ {1\\over{8\\pi\\alpha'}} {\\cal F}_{\\mu\\nu\\lambda\\rho}{\\cal F}^{\\mu\\nu\\lambda\\rho}\n\\ .\\label{gauge-792}\n\\ee\nIrreducibility of the pair-symmetric tensor (\\ref{g-curvature}) is worked out by taking its trace and we to obtain: ${\\cal K}^{(S)}_{\\mu\\nu}=-\\frac{1}{4}\\big ({\\cal H}_{\\mu\\rho\\lambda}{{\\cal H}^{\\rho\\lambda}{}}_{\\nu}\\big )$ and ${\\cal K}^{(S)}=-\\frac{1}{4}{\\cal H}_3^2$. They describe torsion free geometries and may \nformally be identified with the Ricci type tensors: ${\\cal R}_{\\mu\\nu}$ and ${\\cal R}$. They lead to an emergent metric in a first order perturbation gauge theory which is precisely governed by a NS field dynamics on a fat brane. However, a dynamical correction (\\ref{gauge-792}), underlying a propagating geometric torsion, becomes sensible with a NP-correction in a second order formulation. Thus, an emergent theory of NP-gravity may completely be perceived in $1.5$ order formulation with two irreducible curvature tensors ${\\cal K}^{(S)}$ and ${\\cal F}_4$.  \n\n\\subsection{Topological couplings}\n\\noindent\nInterestingly the dynamical aspects of NP-gravity may seen to be supplemented with topological couplings. \nA priori, there are two topological couplings between a constant NS-form and a dynamical KR-form in the world-volume gauge theory on a $D_4$-brane. Both of them turn out to be a total divergence in the $d$$=$$5$ bulk (B), though they contribute significantly to the boundary (BD) theory. They are given by\n\\begin{eqnarray}}\\newcommand{\\eea\n{1\\over{{\\kappa'}^4}}\\int_{\\rm B} H_3\\wedge\\Big (B_2^{(NS)}+{\\bar F}_2\\Big )={1\\over{\\kappa^4}}\\int_{\\rm BD} B_2^{(KR)}\\wedge {\\bar{\\cal F}}_2, \\label{topological-1}\n\\eea\nwhere $\\kappa'$ and $\\kappa$ are respectable couplings in $d$$=$$5$ and $d$$=$$4$. They possess a dimension of length.\nRemarkably, the boundary action can be identified with the $BF$-topological theory. In an emergent scenario, the topological \ncoupling is re-expressed as:\n\\begin{equation}} \\newcommand{\\ee}{\\end{equation}\n{1\\over{\\kappa'}^4}\\int_{\\rm B} B_2^{(KR)}\\wedge {\\cal H}_3\\ =\\ {1\\over{\\kappa}^4}\\int_{\\rm BD} B_2^{(KR)}\\wedge B_2^{(NS)}\\ .\\label{topological-2}\n\\ee\nIn addition the NS and RR couplings are known to incorporate a notion of branes within a brane. This novel idea has been pivotal to the formulation of two independent $d$$=$$10$ Matrix theories with: (i) $D$-particle as a building block for type IIA superstring \\cite{BFSS-PRD} and (ii) $D$-instanton for type IIB superstring \\cite{IKKT-NPB}.\n\n\\vspace{.08in}\n\\noindent\nThough a topological coupling is trivial in the bulk, it regains significance in the boundary theory. It is given by \n\\begin{equation}} \\newcommand{\\ee}{\\end{equation}\n{1\\over{\\kappa'}^3}\\int_{\\rm B}{\\cal F}_4\\wedge {\\cal F}_1={1\\over{\\kappa}^2}\\int_{\\rm BD} {\\cal H}_3\\wedge d\\psi\n={{\\Lambda}\\over{\\kappa}^3}\\int_{\\rm BD} \\psi\n\\ ,\\label{topological-3}\n\\ee\nwhere ${\\cal H}_3$ turns out to be a $U(1)$ gauge potential and $\\psi$ denotes an axion coupling. Primarily the broken $U(1)$ gauge invariance of the 3-form geometric torsion (\\ref{gtorsion-2}) under the NS-form transformation identifies ${\\cal H}_3$ with a gauge potential whose gauge invariant field strength is ${\\cal F}_4$ and is obtained in eq(\\ref{gauge-7777}). Interestingly, a gauge invariant four-form has been argued to be a candidate to tackle the cosmological constant problem \\cite{Hawking-PLB-1998}.   Non-zero $\\Lambda=\\big ({\\cal E}^{\\mu\\nu\\lambda\\rho}\\ {\\cal F}_{\\mu\\nu\\lambda\\rho}\\big )$ may be identified with a cosmological constant in the boundary theory. Its toplogical coupling to an axionic scalar field in $d$$=$$4$ boundary theory is remarkable. It may imply that a topological coupling of an axion in $d$$=$$4$ plays an important role to describe a hidden fifth essence, $i.e.$ quintessence.\n\n\\subsection{Gravity duals}\n\\noindent\nNow we revisit the construction for left and right duals of the Riemann tensor with a renewed interest using the $4$-form in the NP-formulation. The Riemann duals may be envisaged to absorb the dynamical axion and hence the instanton quantum effect in a semi-classical limit. Alternately, the quintessence cosmology, $i.e.$ the accelerated rate of expansion of universe, may well be described with the renewed construction for the Riemann duals and may serve as a viable source for dark energy.\n\n\\vspace{.08in}\n\\noindent\nThe pair-symmetric $(S)$ and the pair non-symmetric  $({\\tilde A})$ emergent curvature \ntensors  of order four (\\ref{g-curvature}) are worked out for their irreducibility to yield a curvature scalar ${\\cal K}^{(S)}$ $\\equiv$ ${\\cal R}$ and ${\\cal F}_4$ respectively. The four-form incorporates a dynamical correction obtained via eqs(\\ref{gauge-7777})-(\\ref{gauge-792}) in second order underlying an onshell NS-form in a first order gauge theory. The field strength ${\\cal F}_4$ further ensures a propagating ${\\cal H}_3$ and may seen to be governed by a lower dimensional $D$-brane correction \\cite{priyabrat-EPJC}. Since D-branes are known to be non-perturbative objects  \\cite{polchinski}, the effective action underlying an emergent quantum gravity on a gravitational pair of $(3{\\bar 3})$-brane is believed to be non-perturbative. In fact, the emergent gravity on a pair of brane and anti-brane in the efffective gauge theory has been argued to be sourced by the absorption of NS-form quanta at the event horizon of a background stringy black hole \\cite{abhishek-JHEP}. Interestingly, it generalizes the Schwinger pair production mechanism in quantum field theory \\cite{schwinger} which is established as a non-perturbative phenomenon. The emergent gravity action may be given by\n\\begin{equation}} \\newcommand{\\ee}{\\end{equation}\nS_{EG}={1\\over{{\\kappa'}^3}}\\int d^5x {\\sqrt{-g}}\\; \\Big ( {\\cal R}-{1\\over{2\\cdot 4!}}\\ {\\cal F}_4^2\\Big )\\ ,\\label{gauge-70}\n\\ee\nwhere the $g_{\\mu\\nu}$ signature is $(-,+,+,+,+)$ and the string slope parameter $\\alpha'$ is used to re-express ${\\kappa'}^2=(8\\pi\\alpha')$. The first term governs a torsion free geometry. Importantly the second term incorporates a dynamical geometric torsion correction non-perturbatively to the Einstein vacuum. Furthermore, the Poincar$\\acute{e}$ dual of $4$-form ensures the presence of a dynamical axionic scalar in the string-brane formulation for a NP-theory. In fact the axion is believed to source a instanton \\cite{priyabrat-EPJC} which ensures an instantaneous quantum correction to the Einstein gravity. The instanton correction further reconfirms its non-perturbation nature and may be viewed as a source to the dark energy in universe.\n\n\\vspace{.08in}\n\\noindent\nRemarkably, the emergent gravity action in a decoupling limit formally can be identified with the bosonic sector of $d$$=$$11$ SUGRA  \\cite{cremmer-julia-scherk}. Thus a dynamical geometric torsion ${\\cal H}_3$ potential may provide a clue to unfold a non-perturbation theory of quantum gravity. The emergent gravity action is re-expressed as: \n\\begin{equation}} \\newcommand{\\ee}{\\end{equation}\nS_{EG}={1\\over{{\\kappa'}^3}}\\int_{\\rm B} {\\sqrt{-g}}\\, {\\cal R} -{1\\over{96 {\\kappa'}^3}}\\int_{\\rm B}{\\cal F}_4 \\wedge {}^{\\star}d\\psi\n+{{\\Lambda}\\over{{\\kappa}^3}}\\int_{\\rm BD} \\psi\\label{Egravity-dual}\n\\ee\nThe equivalence between (\\ref{gauge-70}) and (\\ref{Egravity-dual}) reconfirms a topological coupling and hence ${\\cal F}_4$ in eq(\\ref{gauge-7777}) modifies to: ${\\sqrt{2\\pi\\alpha'}}\\big ( d{\\cal H}_3-{\\cal H}_3\\wedge d\\psi\\big )$. A dual action ensures a dynamical axionic scalar. The metric signature reverses to $(+,-,-,-,-)$ in the dual scenario due to the odd dimensional space-time. The dual action is given by\n\\begin{equation}} \\newcommand{\\ee}{\\end{equation}\nS_{EG}^{Dual}\\equiv\\int_{\\rm B}d^5x{\\sqrt{-g}}\\Big ( {\\cal R}+{1\\over2}\\big ({\\nabla}\\psi\\big )^2\\Big )\\ +\\ \\int_{\\rm BD} {\\cal H}_3\\wedge d\\psi\\ .\\label{gauge-702}\n\\ee\nAn axionic coupling underlying a instanton has been argued to incorporate the quintessence effect \\cite{priyabrat-EPJC}. We exploit the dynamical essence of an axionic scalar via its dual and construct Riemann type left (L) and right (R) duals in higher dimensional $d\\ge5$ gravity theory. In particular, the Riemann type dual tensors are worked out in five dimensions using the the four-form ${\\cal F}_{4}$ (as a generalized form of Levi-Civita tensor density) and they are:\n\\begin{equation}} \\newcommand{\\ee}{\\end{equation}\n \\quad \\qquad {\\cal R}^{(L)}_{\\mu\\nu\\lambda\\rho}=\\sqrt{{\\pi\\alpha'}\\over2} {{\\cal F}_{\\mu\\nu}}^{\\alpha\\beta}\\ {\\cal R}_{\\alpha\\beta\\lambda\\rho}\\quad\n{\\rm and}\\quad {\\cal R}^{(R)}_{\\mu\\nu\\lambda\\rho}=\\sqrt{{\\pi\\alpha'}\\over2} \n{\\cal R}_{\\mu\\nu\\alpha\\beta}\\ {{\\cal F}^{\\alpha\\beta}}_{\\lambda\\rho}\\ .\\label{gauge-704}\n\\ee\nThe generalized Riemann type L and R duals are new of its kind and may provide a clue to the quintessence axionic scalar dynamics. Interestingly, they reduce to the typical Riemann duals in $(3+1)$-dimensions as ${\\cal F}_{\\mu\\nu\\lambda\\rho}\\rightarrow {\\cal E}_{\\mu\\nu\\lambda\\rho}$ there. \nRemarkably, the geometric duals reveal an intrinsic coupling of ${\\cal F}_4$ with the Riemann type tensors in a NP-theory of emergent quantum gravity. The coupling ensures the presence of an axionic scalar in $d$$=$$5$ and hence signals the importance of a quintessence correction to the GTR. Quintessence is known to be a potential candidate to describe the dark energy in universe. Interestingly, the hidden quintessence (axionic scalar) dynamics to the GTR may be viewed as a NP-dynamical correction. \n\n\\section{Emergent theories from superstrings}\n\\subsection{{\\it Form} or {\\it Fundamental}  theory}\n\\noindent\nThe emergent gravity in $(4+1)$-dimensions (\\ref{gauge-70}) is governed by ${\\cal H}_3$$=$$d^{\\cal D}B_2^{(NS)}$ in $1.5$ order formulation with ${\\cal D}_{\\mu}$ as underlying covarinat derivative. The emergent metric is essentially governed by a (massive) NS field dynamics in first order within the purview of $d$$=$$10$ type IIA and IIB superstring theories. A massive NS field theory may be realized via Higgs Mechanism in an emergent gauge theory on a fat $3$-brane\\cite{PTEP-NS}. \n\n\\vspace{.08in}\n\\noindent\nFurthermore $H_3$$=$$d^{\\nabla}B_2^{(KR)}$ is sourced by a string charge and is best described on a $D_5$-brane due to a CFT. A $D_5$-brane is sourced by a 6-form in NS-NS and in RR sectors of type IIB superstring. It re-ensures a string\/five-brane duality in $d$$=$$10$ superstring theory. Thus, the significance of a $3$-brane in an emergent gravity \nmay demand a $d$$=$$12$ {\\it fundamental} theory. Interestingly, F-theory has been viewed as a re-formulation of type IIB superstring theory with $D$-instanton as its fundamental unit \\cite{vafa}. The $SL(2,Z)$ symmetry in type IIB supergravity has been exploited by the author to obtain a $d$$=$$12$ theory \\cite{kar-NPB}. The field theoretic construction in $d$$=$$12$ leading to F-theory has widely been studied \\cite{pope-CQG,ueno-JHEP}. \n\n\\vspace{.08in}\n\\noindent\nIn fact, a dynamical ${\\cal H}_3$ decouples to leave behind a torsion free geometry. Analysis leading to a stable gravitational $3$-brane as a fundamental unit in an emergent gravity provides a clue to the  {\\it form} theory in $d$$=$$12$. \nIn addition, an emergent gravity sourced by a KR field may formally be generalized to obtain a space filling emergent pair of $(8{\\bar 8})$-brane underlying a $D_9$-brane in type IIB superstring. In principle, a $9$-brane would be governed by $F_{11}$$=$$dB_{10}$ and hence the $10$-form theory requires a minimum of $d$$=$$12$. \nThe hint signifies the role of a constant $F_5$$=$$dB_4$ in an emergent gravity (\\ref{gauge-70}). The constant turns out to be dynamical in six and higher dimensions. Importantly, a constant in $d$$=$$5$ describes a nontrivial topological coupling in addition to a total divergence $\\left ({\\cal F}_4\\wedge {\\cal F}_4\\wedge {\\cal F}_4\\right )$ in $d$$=$$12$. We set $(2\\pi\\alpha'=1=\\kappa')$ for all onward expressions and then the {\\it fundamental} action is a priori given by\n\\begin{equation}} \\newcommand{\\ee}{\\end{equation}\nS=\\int d^{12}x \\sqrt{-{\\hat g}}\\Big ( {\\hat{\\cal R}} -{1\\over{48}}{\\cal F}_4^2 - {1\\over{240}} F_5^2\\Big ) +\\int B_4\\wedge {\\cal F}_4\\wedge {\\cal F}_4\\ .\\label{form-1}\n\\ee\nwhere ${\\cal F}_4$ governs a ${\\cal H}_3$ potential. Interestingly an emergent gravitational $3$-brane is governed by the {\\it form} theory in $d$$=$$12$. The Ricci type curvature scalar ${\\hat{\\cal R}}$ governs a NS field dynamics and is believed to describe an emergent metric (\\ref{gauge-metric}) in {\\it form} theory. Intuitively the (emergent) metric dynamics may primarily be viewed via the Riemann tensor and hence ${\\hat{\\cal R}}$ may be identified with a Ricci scalar. The 4-form signifies a NP-correction underlying a propagating ${\\cal H}_3$. However, the $F_5$$=$$d^{\\cal D}B_4$ may source a gravitational $3$-brane in {\\it form} theory which may be re-expressed as:\n\\begin{equation}} \\newcommand{\\ee}{\\end{equation}\nS=\\int d^{12}x \\sqrt{-{\\hat g}}\\Big ( {\\hat{\\cal K}}^{(S)} -{1\\over{48}}{\\cal F}_4^2 - {1\\over{240}} F_5^2\\Big ) + \\int B_4\\wedge {\\cal F}_4\\wedge {\\cal F}_4\\ .\\label{form-2}\n\\ee\nThe gauge invarinat forms are given as\n\\begin{eqnarray}\\label{forms-12d}\n{\\cal K}^{(s)}&=&-\\frac{1}{4}\\,{\\cal H}_{3}^{2}\\,\\,\\, , \\quad {\\rm where}\\,\\, \\quad {\\cal H}_{3}=d^{\\cal D}{B}_{2}^{(NS)} \\ .\\nonumber\n\\end{eqnarray}\nIn addition\n\\begin{eqnarray} \n{\\cal F}_{4}&=& d^{\\cal D}{\\cal H}_{3}\\,\\quad \\qquad{\\rm and}\\qquad \\quad {F}_{5}=d^{\\cal D}{B}_{4}\\ .\n\\end{eqnarray}\nThe first two terms in \\eqref{form-2} possess their origin in an emergent NP-theory of gravity underlying a geometric torsion ${\\cal H}_3$. The field \nstrength $F_5$$=$$d^{\\cal D}B_4$ possibly ensures an emergent gravitational $3$-brane. This is in agreement with the generalization of a fact that a point particle is sourced by an one-form gauge field, a string by two-form, a membrane by three-form, and a $3$-brane by $B_4$.\n\n\\subsection{Higher dimensional NP-theory}\n\\noindent\nThe total local degrees in {\\it form} theory (\\ref{form-2}) turns out to be $l$$=$$375$. It includes the NP-local degrees $l_{\\rm NP}=120$ sourced by a dynamical geometric torsion and is given by ${\\cal F}_4$, The remaining $255$ local degrees are sourced by the NS field with $l_{\\rm NS}$$=$$45$ and a $4$-form field with $l_{\\rm 4F}$$=$$210$. The value of $l_{\\rm NP}$ decreases the dimension of space-time with $l_{\\rm NP}$$=$$l_{\\rm NS}$ in $d$$=$$7$ and the minimal NP degree $i.e.\\ l_{\\rm NP}$$=$$1$ is in $d$$=$$5$. \n\n\\vspace{.08in}\n\\noindent\nGenerically, the NP degrees of freedom in $(d$$+$$1)$-dimensions is equal to the degrees of freedom of NP-theory underlying the NS-field and geometric torsion field in $d$-dimensions, $i.e.\\ l_{\\rm NP}^{d+1}$$=$$l_{\\rm NS}^d$$+$$l_{\\rm NP}^d$. Furthermore, $l_{\\rm NS}^d$$=$$l_{\\rm metric}^{d-1}$$+$$l_{\\rm HS}$, where the HS (Higher-essence\/dimensional Scalar) local $l_{\\rm HS}$$=$$1$. The two level correspondences between a {\\it form} theory in $(d+1)$-dimensions via a NP-theory in $d$-dimensions to a metric theory in $(d$$-$$1)$-dimensions is remarkable and is indeed a potential tool to describe an emergent NP theory on a gravitational pair of $(M{\\bar M})$-brane\\cite{Physica-NDS}. For instance, the $d$$=$$12$ (four) {\\it form} theory can be mapped to $d$$=$$11$ non-perturbation ($M$) theory. One might lead to suppose that the NP-theory can further be viewed as a metric theory possibly corresponding to ten dimensional (space filling brane) superstring theories. Similarly a six dimensional form theory via a five dimensional (non-supersymmetric) NP-theory can be mapped to the GTR. Intuitively, it underlies an analogy between two sets: ($d$$=$$12$ {\\it form} theory, $d$$=$$11$ M-theory and $d$$=$$10$ superstrings) and ($d$$=$$6$ form theory, $d$$=$$5$ NP-theory, the GTR).\nThe details of this analysis is beyond the scope of this paper.\n\n\\vspace{.08in}\n\\noindent\nThe {\\it form} theory action on $S^1$ is worked out for the massless fields to describe a total $l$$=$$375$ local degrees on an emergent gravitational pair of $(M{\\bar M})$-brane. Formally, the irreducible curvatures ${\\hat{\\cal K}}$, ${\\cal F}_4$$=$$d^{\\cal D}{\\cal H}_3$ and $F_5$$=$$d^{\\cal D}C_4$ (\\ref{form-2}) are respectively reduced to a pair of curvatures $({\\tilde{\\cal K}}, {\\cal F}_2^2)$, $({\\cal F}_4, {\\cal K})$ and $(F_4$$=$$d^{\\cal D}{\\cal B}_3, F_5)$ on $(M{\\bar M})$-brane with ${\\cal D}_{\\mu}$ as the covariant derivative. The forms governed by these gauge invariant field strengths in terms of their respective underlying derivatives is given as\n\\begin{eqnarray}\n\\tilde{\\cal K} &=&-\\frac{1}{4}\\,{\\cal H}_{3}^{2}\\quad ,  \\quad {\\rm where}\\qquad {\\cal H}_{3}=d^{\\cal D}{B}_{2}^{(NS)}\\nonumber \\\\\n{\\rm and}\\qquad{\\cal K}&=&-\\frac{1}{4}\\,{\\cal B}_{3}^{2}\\quad ,  \\quad {\\rm where}\\qquad {\\cal B}_{3}=d^{\\cal D}{C}_{2}^{(NS)} \\ .\\nonumber\n\\end{eqnarray}\nIn addition\n\\begin{eqnarray} \n{F}_{2}&=& d^{\\cal D}{A}_{1}\\quad ,  \\quad  {\\cal F}_{4}=d^{\\cal D}{\\cal H}_{3}\\quad , \\quad {F}_{5}=d^{\\cal D}{C}_{4}\\quad \\quad{\\rm and}\\quad{F}_{4}=d^{\\cal D}{\\cal B}_{3}\\ .\n\\end{eqnarray}\nThe dimensionally reduced $d$$=$$11$ action a priori is given by\n\\begin{eqnarray}}\\newcommand{\\eea\n &&S=\\int_M d^{11}x {\\sqrt{-{\\tilde g}}}\\ \\Big ( {\\tilde{\\cal K}}\\ -{1\\over{48}}{\\cal F}^a_4N_{ab}{\\cal F}^b_4\\Big )\\ +\\int_{\\bar M} d^{11}x{\\sqrt{-{\\tilde g}}}\\ \\Big ({\\cal K}\\ -{1\\over{4}} {\\cal F}_2^2\\ -{1\\over{240}} F_5^2\\ \\Big )\\nonumber\\\\\n&&+\\int_{(M{\\bar M})}\\Big [ {\\cal H}_3\\wedge \\Big ({\\cal F}_4\\wedge {\\cal F}_4+ {\\cal F}_4\\wedge F_4 + F_4\\wedge F_4\\Big )\\ + F_2\\wedge \\Big ({\\cal F}_4 +{\\cal F}_4\\Big )\\wedge F_5\\Big ]\\label{form-3}\n\\eea\nwhere $N_{ab}$ describes a $(2\\times 2)$ diagonal matrix. In the action, the superscript $(S)$ (for symmetric) on the irreducible curvatures ${\\tilde{\\cal K}}$ and ${\\hat{\\cal K}}$ has been omitted and will be so in all the following equations.   \nGenerically, the second term on a $M$-brane ensures two $4$-forms: ${\\cal F}_4$ and $F_4$. The matrix element $N_{11}$ ensures a non-canonical potential coupling only to the ${\\cal F}_4$, which is sourced by the ${\\cal F}_4$. Since ${\\cal H}_3$ in $d$$=$$12$ does not couple to $F_5$, the element $N_{22}$ is a constant. The second term contains a NP-correction sourced by a propagating ${\\cal H}_3$ in addition to $F_4$.\n\n\\vspace{.08in}\n\\noindent\nInterestingly, the bulk {\\it form} theory leads to a boundary description for an emergent $M$ brane on $S^{1}$. However, by invoking a generic bulk (NS-field)\/boundary (metric tensor) correspondence\\cite{Physica-NDS}  between the {\\it form} theory and $M$-brane, the curvatures ${\\hat{\\cal K}}$, ${\\cal F}_4$ and ${\\cal F}_5$ in eq(\\ref{form-2}) are respectively re-expressed as: $(R, \\phi)$, $({\\cal F}_4, {\\cal K})$ and $(F_4, F_5)$ on an emergent pair of $(M{\\bar M})$-brane, where $R$ and $\\phi$ respectively denote the Ricci scalar and a (higher essence) scalar field. Needless to mention that $45$ local degrees of NS field underlying ${\\hat{\\cal K}}$ in $d$$=$$12$ are described in $d$$=$$11$ by $44$ local degrees of an emergent metric field and one local degree of a scalar field.  In fact, the $\\phi$ field is an extra $12$-th transverse dimension inbetween a pair. \nHere also, the underlying derivative is ${\\cal D}_{\\mu}$ and the potentials governed by the gauge invariant forms are given respectively as\n\\begin{eqnarray}\n{\\cal F}_{4}&=&d^{\\cal D}{\\cal H}_{3}\\quad  {\\rm and} \\quad {\\cal K}=-\\frac{1}{4}\\,{\\cal H}_{3}^{2}\\quad , \\quad {\\rm where}\\quad {\\cal H}_{3}=d^{\\cal D}{B}_{2}^{(NS)} \\ ,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n{F}_{5}&=&d^{\\cal D}{C}_{4}\\qquad {\\rm and} \\qquad {F}_{4}=d^{\\cal D}{H}_{3}\\ .\n\\end{eqnarray}\nThe $d$$=$$11$ action (\\ref{form-3}) reduces to yield: \n\\begin{eqnarray}}\\newcommand{\\eea\n \\quad S&=&\\int_M d^{11}x \\ {\\sqrt{-G}}\\ \\Big ( R_G\\ -\\ {1\\over{48}}{\\cal F}^a_4N_{ab}{\\cal F}^b_4\\Big )+\\int \\Big ({\\cal H}_3\\wedge{\\cal F}_4\\wedge {\\cal F}_4\\ +\\ {\\cal B}_3\\wedge F_4\\wedge F_4\\ \\Big )\\nonumber\\\\\n &+&\\int_{\\bar M}\\ d^{11}x{\\sqrt{-G}}\\ \\Big ( {\\cal K}\\ - {1\\over2}({\\cal D}\\phi)^2\\ -{1\\over{240}} F_5^2\\Big )\\ ,\\label{form-4}\n\\eea\nwhere $G_{\\mu\\nu}$, in the invariant-volume, denotes a dynamical (emergent) metric obtained under a bulk\/boundary correspondence on a pair of $(M{\\bar M})$-brane. The NP-action (\\ref{form-4}) may a priori be identified with an eleven dimensional $M$-theory. However, a NP-correction is sourced only by a geometric torsion potential ${\\cal H}_3$ in $1.5$ order formulation. Thus, the $M$-brane dynamics within a pair effectively takes account for the NP-dynamical correction and may formally be argued to govern the $M$-theory. Since, a gravitational pair is created across an event horizon of a background black hole in the NP-formulation \\cite{abhishek-JHEP,abhishek-PRD}, the scalar field $\\phi$ in $d$$=$$11$ is believed to play the role of a hidden or higher-essence in the disguise of an extra transverse dimension. \n\n\\vspace{.08in}\n\\noindent\nA varying thickness of the fat brane can be fixed by freezing the local degree of scalar field, $i.e.\\ \\phi\\rightarrow \\phi_0$. It disconnects an emergent $M$-brane in a pair from the ${\\bar M}$-brane, where the radius $R$ of $S^{1}$ takes a fixed value $\\phi_0$. The emergent curvatures and hence the causal effects on a gravitational $M$-brane universe are de-linked from that on the ${\\bar M}$-brane as the $d$$=$$12$ coordinate system breaks down in the limit $\\phi\\rightarrow\\phi_0$. Intuitively, the higher dimensional limit \nmay imply that the polar angle $\\theta\\rightarrow 0$ on $S^{1}$. The apparent angular deficit angle $\\theta$ under a Wick rotation may be viewed as light-like. Thus the disconnected emergent causal geometries are separated by a light-like cone. For an observer in a $M$-brane universe, a causal effect can be  re-interpreted as a spacelike event on ${\\bar M}$-brane universe. The de-linked effective actions are given by \n\\begin{eqnarray}}\\newcommand{\\eea\n\\qquad S_M&=&\\int d^{11}x{\\sqrt{-G}}\\ \\Big ( R_G -{1\\over{48}}{\\cal F}^a_4N_{ab}{\\cal F}^b_4\\Big )+ \\int \\Big ({\\cal H}_3\\wedge{\\cal F}_4\\wedge {\\cal F}_4+ {\\cal B}_3\\wedge F_4\\wedge F_4\\Big )\\nonumber\\\\\n{\\rm and}\\;\\ S_{\\bar M}&=&\\int d^{11}{\\bar x}\\ {\\sqrt{-{\\bar G}}}\\ \\Big ( {\\cal K}\\ -{1\\over{240}}F_5^2\\Big )\\ .\\label{form-5}\n\\eea\nIn addition to the (bosonic) dynamics of $M$-theory (\\ref{form-5}), a de-linked $M'$-theory in $d$$=$$11$ representing the ${\\bar M}$-brane is remarkable. It is important to observe that the $M'$-theory is not a NP-theory as it does not describe a propagating geometric torison. \nFurther investigation may reveal a plausible map if there is any between a $M'$-theory on $S^1$ and the $d$$=$$10$ type IIB (super)string theory.\n\n\\vspace{.08in}\n\\noindent\nIn a low energy limit, the NP-dynamical correction decouples and hence the limit is identified as a decoupling limit. The geometric torsion dynamics freezes in the limit ${\\cal H}_3\\rightarrow {\\cal H}_3^0$ (a constant) and the $M$-brane action becomes\n\\begin{equation}} \\newcommand{\\ee}{\\end{equation}\nS_M=\\int d^{11}x{\\sqrt{-G}}\\ \\Big (R_G -{1\\over{48}}F_4^2\\Big )+\\int B_3\\wedge F_4\\wedge F_4\\ .\\label{form-6}\n\\ee\nInterestingly, the decoupled emergent $M$-brane dynamics formally identifies with the bosonic sector of $d$$=$$11$ SUGRA \\cite{cremmer-julia-scherk} in natural units. The presence of fermionic local degrees are not affected by the decoupling of a boson dynamics.  \nArguably, the absence of $\\phi$-dynamics in a limit restores the supersymmetry and may lead to the SUGRA theory. The low energy limit of the $M$-brane is consistent with the $M$-theory in the same limit where $M$-theory is known to describe the $d$$=$$11$ SUGRA. This result suggests  that a geometric torsion is a plausible candidate to describe a complete NP-theory of quantum gravity and hence the torsion quanta may be interpreted as a ``graviton'' there. It may resolve some of the mystries related to the origin of dark energy in the  universe. A result that the GTR may be realized in a decoupling dynamics of a geometric torsion in superstring theory is new and is believed to enlighten the legacy of non-perturbation gravity to its depth.\n\n\n\n\n\n\\section{Concluding remarks}\n\\noindent\nA non-perturbation theory of emergent gravity constructed by the author in collaborations \\cite{abhishek-JHEP,abhishek-PRD,abhishek-NPB-P, sunita-NPB,priyabrat-EPJC} had been revisited to show the significance of a four-form ${\\cal F}_4$ correction to the torsion free geometries in Einstein gravity. Interestingly, the quantum correction had been argued to be sourced by a instanton, underlying an axionic scalar, in a string-brane setup,  which in turn may be coined as  a ``fat'' brane.  Unlike a non-gravitational or flat $D$-brane, a fat-brane absorbs the  local degree of freedom of an axionic scalar and takes account for the space-time curvature by coupling to the closed string modes. The Riemann duals were shown to absorb a dynamical axion in a semi-classical description and is argued to describe the quintessential cosmology. Importantly, a higher dimensional generalization was obtained in eqs.(\\ref{form-1})-(\\ref{form-2}) and they hint for a {\\it fundamental} or {\\it form} theory in $d$$=$$12$. Further analysis has revealed an emergent gravitational pair of $(M{\\bar M})$-brane from the {\\it form} theory. In a semi-classical limit, the $M$-brane dynamics has formally been identified with the $d$$=$$11$ SUGRA. It is believed to provide a wide scope to enhance the horizon underlying certain aspects of dark energy and may unfold a clue to model the dark gravity.\n\n\n\\defAnn. Phys.{Ann. Phys.}\n\\defComm. Math. Phys.\\ {}} {{Comm. Math. Phys.\\ {}} {}\n\\defSpringer. Proc. Phys.{Springer. Proc. Phys.}\n\\defPhys. Rev. Lett.{Phys. Rev. Lett.}\n\\def\\prd#1{{Phys. Rev.} {\\bf D#1}}\n\\defJHEP\\ {}}{{JHEP\\ {}}{}\n\\defClass.\\& Quant.Grav.{Class.\\& Quant.Grav.}\n\\def\\plb#1{{Phys. Lett.} {\\bf B#1}}\n\\def\\npb#1{{Nucl. Phys.} {\\bf B#1}}\n\\def\\mpl#1{{Mod. Phys. Lett} {\\bf A#1}}\n\\def\\ijmpa#1{{Int. J. Mod. Phys.} {\\bf A#1}}\n\\def\\ijmpd#1{{Int. J. Mod. Phys.} {\\bf D#1}}\n\\def\\mpla#1{{Mod. Phys. Lett.} {\\bf A#1}}\n\\def\\rmp#1{{Rev. Mod. Phys.} {\\bf 68#1}}\n\\defJ.Astrophys.Aerosp.Technol.\\ {}} {{J.Astrophys.Aerosp.Technol.\\ {}} {}\n\\def \\epj#1{{Eur.Phys.J.} {\\bf C#1}} \n\\def JCAP\\ {}}{{JCAP\\ {}}{}\n\\defPhysica Scripta {Physica Scripta }\n\\def Prog.Theo.Exp.Phys.{ Prog.Theo.Exp.Phys.}\n\\defInt. J. Theo. Phys.{Int. J. Theo. Phys.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n Electron tomography is a powerful technique for resolving the interior of nanomaterials. After preparing a microscopic sample, a series of projection images (so called tilt-series) is acquired by rotating the specimen in the electron microscope, acquiring data from a range of angles. In recent years, electron tomography has been successfully applied to reconstruct the 3D positions of the individual atoms in nanocrystalline materials \\cite{van2012big, chen2013three, rez2013three}.\n\nSince the first demonstration of atomic resolution tomography of nanocrystals in 2010 by discrete tomography \\cite{van2011three}, a range of tomographic acquisition techniques and reconstruction algorithms have been applied to reconstruct nanocrystals of increasing complexity. In the discrete tomography approach, atoms are assumed to lie on a regular lattice and the measured projections can be considered as atom counts along lattice lines. A key advantage of this approach is its ability to exploit the constraints induced by the discrete domain and range of the image. As a consequence, a small number of projection angles (typically less than 5) can already lead to an accurate reconstruction \\cite{batenburg2006,batenburg2009}. The theoretical properties of the discrete reconstruction problem have been studied extensively with results on algorithm complexity, uniqueness, and stability \\cite{gardner1997,baake2006,alpers2006}. A key drawback of the discrete lattice assumption when considering real-world applications to nanocrystal data is that in many interesting cases the atoms do not lie on a perfect lattice due to defects in the crystal structure or interfaces between different crystal lattices. In such cases the atoms do not project perfectly into columns, forming a mismatch with the discrete tomography model.\n\nAs an alternative, it has been demonstrated that a more conventional tomographic series consisting of hundreds of projections of a nanocrystal can be acquired in certain cases. An image of the nanocrystal is then reconstructed using sparsity based reconstruction techniques on a continuous model of the tomography problem. This approach does not depend on the lattice structure and allows one to reconstruct defects and interfaces \\cite{goris2015measuring}. As a downside, the number of required projections is large and to accurately model the atom positions the reconstruction must be represented on a high-resolution pixel grid resulting in a large-scale computational problem. This raises the question if a reconstruction problem can be defined that fills the gap between these two extremes and can exploit the discrete nature of the lattice structure while at the same time allowing for continuous deviations of atom positions from the perfect lattice.\n\nIn this paper we propose a model for the atomic resolution tomography problem that combines these two characteristics. Inspired by the algorithm proposed in \\cite{boyd2017alternating}, our model is based on representing the crystal image as a superposition of delta functions with continuous coordinates and exploiting sparsity of the image to reduce the number of required projections. We show that by incorporating a physical model for the potential energy of the atomic configuration, the reconstruction results can be further improved. \n\n\\section{Problem Setting} \\label{mathsetup}\nIn this section we formulate a mathematical model of the atomic resolution tomography problem and discuss several approaches to solve it. Some of these approaches assume that the atom locations are restricted to a perfect grid, the \\textit{crystal lattice}, which corresponds to only one possible local minimum of the potential energy of the atomic configuration. To overcome certain limitations of this assumption, we propose an alternative formulation where the atom locations are allowed to vary continuously and an explicit model of the potential energy of their configuration is used to regularize the image reconstruction. \n\nAn atomic configuration is characterized by a positive measure $\\mu$ on a bounded subset $X$ of $\\mathbb{R}^d$. We denote the space of such measures by $\\mathcal{M}(X)$. The measure represents the \\emph{electron density}, which is the probability that an electron is present at a given location. The electron density around an atomic configuration is highest in regions where atoms are present. In electron tomography, electron density is probed by irradiating a sample with a beam of electrons. The beam undergoes absorption and scattering due to its interactions with the electrons of the atomic configuration. The transmitted or scattered signal can then be used to form an image. The Radon transform provides a simplified mathematical model of this ray-based image formation process. For $d=2$, the Radon transform $\\mathcal{R} \\mu$ can be expressed as integrals taken over straight rays\n\\begin{align} \\label{eq:radon}\n\\mathcal{R}[\\mu](r, \\theta) &:= \\int_{l(r,\\theta)} d\\mu,  \\\\\nl(r,\\theta) &= \\{(x_1,x_2) \\in \\mathbb{R}^2 \\,|\\, x_1\\cos\\theta + x_2\\sin\\theta = r \\}, \n\\end{align}\nwhere we parametrized the rays by the projection angle $\\theta$ and the distance on the detector $r$. The corresponding inverse problem is to recover $\\mu$ from noisy observations of $y = \\mathcal{R} \\mu + \\varepsilon$. One way to formulate a solution to this problem is via the following optimisation over the space of measures:\n\\begin{equation}\\label{eq:opt0}\n    \\minimize_{\\mu \\in \\mathcal{M}(X)} \\quad \\| \\mathcal{R} \\mu - y \\|^2_2 , \n\\end{equation}\nwhich is an infinite dimensional non-negative linear least-squares problem. In the following, we will introduce a series of discretisations of this optimisation problem. Numerical schemes to solve them will be discussed in Section \\ref{algos}. \n\nIn situations where we only have access to data from a few projection angles, introducing a suitable discretisation of \\eqref{eq:opt0} is essential for obtaining a stable reconstruction. One way to achieve this is to restrict the atom locations to a spatial grid with $n$ nodes, ${\\vec{x}}_{i=1}^n$, and model their interaction zone with the electron beam by a Gaussian with known shape $G$. The atom centres are then delta peaks $\\delta_{\\vec{x}_i}$ on the gridded image domain. The Gaussian convolution of atom centres can be viewed as the ``blurring\" produced by thermal motion of atoms. In fact, it is known from lattice vibration theory that, for large configurations, the probability density function of an atom around its equilibrium position is a Gaussian, whose width depends on temperature, dimensionality and interatomic forces \\cite{montroll1956theory}. The discretized measure $\\mu$ can then be written as\n\\begin{equation}\\label{eq:gridmu}\n    \\mu_{\\text{grid}} = \\sum_{i = 1}^n w_i (G \\ast \\delta_{\\vec{x}_i}),\n\\end{equation}\nwhere $n$ is the total number of grid points and weights $w_i \\geq 0$ were introduced to indicate confidence in the presence or absence of an atom at grid location $i$. If we insert \\eqref{eq:gridmu} in \\eqref{eq:opt0} and introduce the forward projection of a single atom as $\\psi_i \\coloneqq \\mathcal{R}(G \\ast \\delta_{\\vec{x}_i})$ we get\n\\begin{multline*}\n   \\| \\mathcal{R} \\mu_{\\text{grid}} - y \\|^2_2 =  \\| \\mathcal{R} \\sum_{i = 1}^n w_i (G \\ast \\delta_{\\vec{x}_i}) - y \\|^2_2 =  \\\\\n    \\| \\sum_{i = 1}^n w_i \\mathcal{R} (G \\ast \\delta_{\\vec{x}_i}) - y \\|^2_2 =:  \\| \\sum_{i = 1}^n \\psi_i w_i  - y \\|^2_2 =:  \\| \\mathrm{\\Psi} w  - y \\|^2_2 \n\\end{multline*}\nThe corresponding optimisation problem is given by \n\\begin{equation} \\label{eq:opt1}\n \\minimize_{w \\in \\mathbb{R}^n_+} \\quad \\| \\mathrm{\\Psi} w  - y \\|^2_2 \\quad , \n\\end{equation}\nwhich is a finite dimensional linear non-negative least squares problem. \n\nThe choice of the computational grid in \\eqref{eq:gridmu} is unfortunately not trivial. Only in certain situations, one can assume that all atoms lie on a lattice of known grid size and orientation, and directly match this lattice with the computational grid. In general, one needs to pick a computational grid of much smaller grid size. With the data $y$ given, the grid admits multiple solutions of \\eqref{eq:opt1} and most efficient computational schemes tend to pick a blurred, artefact-ridden solution with many non-zero weights far from the true, underlying $\\mu$, as we will demonstrate in Section \\ref{expresults}. To obtain a better reconstruction, one can choose to add \\emph{sparsity constraints} which embed our physical \\emph{a priori} knowledge that $\\mu$ originates from a discrete configuration of atoms. In our model \\eqref{eq:gridmu}, this corresponds to a $w \\in \\mathbb{R}^n_+$ with few non-zeros entries. To obtain such a sparse solution we can add a constraint on $\\ell^0$ norm of the weights to the optimisation problem:\n\\begin{equation*}\n    \\begin{aligned}\n    & \\underset{w \\in \\mathbb{R}^n_{+}}{\\minimize}\n    & & \\| \\mathrm{\\Psi}w - y \\|^2_2 \\\\\n    & \\text{subject to}\n    & & |w|_0 \\leq K.\n    \\end{aligned}\n\\end{equation*}\nHowever, this problem is NP-hard \\cite{foucart2013mathematical}. A approximate solution can be found by replacing the $\\ell^0$ norm with the $\\ell^1$ norm and adding it to the objective function:\n\\begin{equation}\\label{eq:opt2}\n        \\minimize_{w \\in \\mathbb{R}^n_{+}} \\quad \\| \\mathrm{\\Psi}w - y \\|^2_2 + \\lambda \\|w\\|_1,\n\\end{equation}\nwhere $\\lambda$ is the relative weight of the sparsity-inducing term. This particular choice of formulation is not always best and alternative formulations of the same problem exist \\cite{foucart2013mathematical}.\n\nFor atomic configurations where only one type of atom is present, the weights can be considered to be one where an atom is present and zero everywhere else. This corresponds to discretising the range of the reconstructed image. The fully discrete optimisation problem then becomes:\n\\begin{equation}\\label{eq:opt3}\n\\minimize_{w \\in \\{0,1\\}^n} \\quad \\| \\mathrm{\\Psi}w - y \\|^2_2.\n\\end{equation}\nWith image range discretisation, a constraint on the number of atoms is typically no longer needed because adding an additional atom with weight 1 after all atoms have been found leads to an increase in the objective function. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figure1}\n    \\caption{Atomic configurations with (a) an interstitial point defect, (b) a vacancy and (c) an edge dislocation.}\n    \\label{fig:configs}\n\\end{figure}\n\nAlthough the optimisation problems (\\ref{eq:opt1}), (\\ref{eq:opt2}) and (\\ref{eq:opt3}) allow for the recovery of atomic configurations without solving (\\ref{eq:opt0}), all of them rely crucially on discretisation of the domain of the reconstructed image, i.e.~the assumption that atoms lie on a grid. However, this assumption is not always true. In particular, atomic configurations often contain defects where atom positions deviate from the perfect lattice. Fig.~\\ref{fig:configs} shows examples of common lattice defects. In order to resolve these defects correctly, the image domain must be discretized to higher resolutions, i.e.~the grid of possible atom positions must be made finer. This introduces two main problems: First, making the grid finer for the same data makes the inverse problem more ill-posed. Second, the computational time increases significantly even for modestly sized configurations.\n\nIn order to overcome these difficulties, we revisit (\\ref{eq:gridmu}) and remove the requirement for $\\vec{x}_i$ to lie on a grid. The projection of a single atom now becomes a function of its location $\\vec{x} \\in \\mathbb{R}^d$, $\\psi(\\vec{x}) := \\mathcal{R}(G \\ast \\delta_{\\vec{x}})$. We keep the image range discretisation introduced above by requiring $w_i \\in \\{0,1\\}$. Now, \\eqref{eq:opt3} becomes \n\\begin{equation}\\label{eq:infoptnonconv}\n    \\begin{aligned}\n    & \\underset{\\vec{x}\\in X^n, w \\in \\{0,1\\}^n}{\\minimize}\n    & & \\Big \\| \\sum_{i=1}^n w_i \\psi(\\vec{x}_i) - y \\Big \\|^2_2. \\\\\n    \\end{aligned}\n\\end{equation}\nThe minimisation over $\\vec{x}$ is a non-linear, non-convex least-squares problem which has been studied extensively in the context of mathematical super-resolution \\cite{candes2014towards, bredies2013inverse, boyd2017alternating}. In these works, efficient algorithms are derived from relating it back to the infinite dimensional linear least-squares problem on the space of measures \\eqref{eq:opt0}. For instance, for applications such as fluorescence microscopy \\cite{boyd2017alternating} and ultrasound imaging \\cite{alberti2019dynamic}, an alternating descent conditional gradient (ADCG) algorithm has been proposed, which we will revisit in the next section. Compared to these works, we have a more complicated non-local and under-determined inverse problem and the minimisation over $w$ adds a combinatorial, discrete flavor to \\eqref{eq:infoptnonconv}. To further tailor it to our specific application, we will incorporate physical \\emph{a priori} knowledge about atomic configurations of crystalline solids by adding a functional formed by the atomic interaction potentials. This will act as a regularisation of the underlying under-determined inverse problem. \n\n\n\n\n\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{figure2}\n    \\caption{The normalized Lennard-Jones pair potential as a function of normalized interatomic separation.}\n    \\label{fig:LJ}\n\\end{figure}\n\n\n\\subsection{Potential Energy of the Atomic Configuration}\nThe total energy of an atomic configuration is the sum of its potential energy and kinetic energy. As we consider only static configurations, the kinetic energy of the configuration is zero and the total energy is equal to the potential energy. In order to compute the potential energy of the atomic configuration, we must prescribe the interaction between atoms. In this paper, we use the Lennard-Jones pair potential, which is a simplified model of interatomic interactions. The Lennard-Jones potential $V_{\\text{LJ}}$ as a function of interatomic separation $r$ is given by \\cite{frenkel2001understanding}\n\\begin{align}\\label{eq:lj}\n    V_{\\text{LJ}}(r) &= \n    \\begin{cases}4 \\epsilon \\Big[ \\Big(\\dfrac{\\sigma}{r}\\Big)^{12} - \\Big(\\dfrac{\\sigma}{r}\\Big)^6 \\Big], & r<r_{\\mathrm{cut}}\\\\\n    0, & r \\geq r_{\\mathrm{cut}}\n    \\end{cases}\n\\end{align}\nwhere $\\epsilon$ is the depth of the potential well and $\\sigma$ is the interatomic separation at which the potential is zero. The separation at which the potential reaches its minimum is given by $r_m = 2^{1\/6}\\sigma$. The parameter $r_{\\mathrm{cut}}$ denotes a cut-off separation beyond which the potential is inactive. Fig.~\\ref{fig:LJ} shows the form of the the Lennard-Jones pair potential as a function of interatomic separation. The potential energy of the atomic configuration is computed by summing over the pairwise interaction between all pairs of atoms \n\\begin{equation}\nV_{\\text{tot}}(\\vec{x}_1, \\vec{x}_2,...,\\vec{x}_N) = \n\\sum_{i>j} V_{\\text{LJ}}(\\vec{x}_i - \\vec{x}_j).\n\\end{equation}\nAdding this energy to the objective in (\\ref{eq:infoptnonconv}) leads to \n\\begin{equation}\\label{eq:infoptwithenergy}\n    \\minimize_{\\vec{x}\\in \\mathcal{C}, w \\in \\{0,1\\}^n} \\quad \\Big \\| \\sum_{i=1}^n w_i \\psi(\\vec{x}_i) - y \\Big \\|^2_2 + \\alpha V_{\\text{tot}}(\\vec{x}).\n\\end{equation}\nThe regularisation parameter, $\\alpha$, adjusts the relative weight of the energy term, so that by tuning it we are able to move between atomic configurations that are data-optimal and those that are energy-optimal. The constraint set $\\mathcal{C} \\subset X^n$ is defined by a minimum distance $r_{\\mathrm{min}}$, such that $|\\vec{x}_i - \\vec{x}_j|>r_{\\mathrm{min}}$, $\\forall i>j$. The minimum distance, $r_{\\mathrm{min}}$, is chosen to be smaller than the optimal interatomic separation $r_m$ and allows us to set $\\alpha$ to 0 and still avoid configurations where atoms are placed exactly at the same position. For small separations, the energy is dominated by the $\\big(\\frac{\\sigma}{r}\\big)^{12}$ term and increases sharply for separations less than $r_m$. Thus, for non-zero $\\alpha$, configurations where atoms have a separation less than $r_m$ are highly unlikely.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Algorithms}\\label{algos}\nIn this section we discuss several algorithms to solve the optimisation problems introduced in Section \\ref{mathsetup}.\n\n\\subsection{Projected Gradient Descent}\\label{sirt} The non-negative least-squares problem (\\ref{eq:opt1}) can be solved with a simple iterative first-order optimisation scheme. At each step of the algorithm, the next iterate is computed by moving in the direction of the negative gradient of the objective function. Non-negativity of the weights is enforced by projecting negative iterates to zero. Mathematically, each iterate is given by\n\\begin{align} \\label{projGrad}\n    w^{k+1} = \\sideset{}{_+}\\prod \\Big (w^k + t \\mathrm{\\Psi}^T (\\mathrm{\\Psi}w^k-y)\\Big),\n\\end{align}\nwhere $t$ is the step size and the projection operator is given by\n\\begin{equation} \\label{proj}\n    \\sideset{}{_+}\\prod(\\cdot) = \\text{max}(\\,\\cdot\\, ,0).\n\\end{equation}\nIn the numerical experiments in Section \\ref{expresults}, we used the SIRT algorithm \\cite{kak2002principles} as implemented in the tomographic reconstruction library ASTRA \\cite{van2015astra}, which is based on a minor modification of the iteration described above.  \n\n\\subsection{Proximal Gradient Descent}\\label{prox}\nIf we add the non-smooth $\\ell^1$ regularizer and obtain problem (\\ref{eq:opt2}), we need to extend \\eqref{projGrad} to a proximal gradient scheme \\cite{parikh2014proximal} \n\\begin{equation} \\label{proxGrad}\n    w^{k+1} = \\text{prox}_{h}\\Big (w^k + t \\mathrm{\\Psi}^T (\\mathrm{\\Psi}w^k-y)\\Big), \n\\end{equation}\nwhere the projection operator \\eqref{proj} is replaced by the proximal operator of the convex function \n\\begin{equation}\n    h(x) := \\begin{cases}\n            \\lambda \\|x\\|_1  & x \\geq 0 \\\\\n            0                & \\text{elsewhere}\n    \\end{cases} \\quad ,\n\\end{equation}\nwhich is given by the non-negative soft-thresholding operator\n\\begin{align*}\n    \\text{prox}_{h}(x) &= \n    \\begin{cases}\n    x - \\lambda, & x \\geq \\lambda \\\\\n    0, & \\text{elsewhere}\n    \\end{cases} \\quad .\n\\end{align*}\nIn the numerical experiments in Section \\ref{expresults}, we used the fast iterative soft-thresholding algorithm (FISTA) \\cite{beck2009fast} as implemented in the Python library ODL \\cite{adler2017operator}, which is based on a slight modification of the iteration described above.  \n\n\n\t\n\n\n\\begin{algorithm}\n\t\\caption{Discrete simulated annealing} \n\t\\begin{algorithmic}[1]\n\t    \\While {$\\beta < \\beta_{\\text{max}}$}\n\t\t\\State Select new atom location: $\\tilde{w}^{k} \\in \\arg\\min_{k \\in \\mathcal{C}} \\, \\mathrm{\\Psi} w^k - y$\n\t\t\\State Add new atom to current configuration: $\\tilde{w}^{k+1} \\leftarrow \\{w^k,\\tilde{w}^k\\}$\n\t\t\\State Accept new configuration with a certain probability:\n\t\t\\If {$\\beta\\|\\mathrm{\\Psi}\\tilde{w}^{k+1}-y\\|^2_2 < \\beta\\|\\mathrm{\\Psi}w^{k}-y\\|^2_2$} \n\t\t\\State $w^{k+1} \\leftarrow \\tilde{w}^{k+1}$\n\t\t\\Else\n\t\t\\State Generate random number: $t \\in \\text{rand}[0,1)$\n\t\t\\If {$t < e^{-\\beta\\|\\mathrm{\\Psi}\\tilde{w}^{k+1}-y\\|^2_2}\/e^{-\\beta\\|\\mathrm{\\Psi}{w}^{k}-y\\|^2_2}$}\n\t\t\\State $w^{k+1} \\leftarrow \\tilde{w}^{k+1}$\n\t\t\\EndIf\n\t\t\\EndIf\n\t\t\\State Move atom: $w^{k+1} \\leftarrow \\text{random move}(w^{k+1})$\n\t\t\\State Run acceptance steps 5--13\n\t\t\\State Increase $\\beta$\n\t\\EndWhile\n\t\\end{algorithmic} \n\t\\label{alg:sa}\n\\end{algorithm}\n\n\\subsection{Simulated Annealing}\\label{sa} For solving the fully discrete problem (\\ref{eq:opt3}), we used a simulated annealing algorithm as shown in Algorithm \\ref{alg:sa}, which consists of two subsequent accept-reject steps carried out with respect to the same inverse temperature parameter $\\beta$. In the first one, the algorithm tries to add a new atom to the existing configuration. In the second one, the atom locations are perturbed locally. As $\\beta$ is increased towards $\\beta_{\\text{max}}$, fewer new configurations are accepted and the algorithm converges to a minimum. \n  \nIn the atom adding step at each iteration $k$, the algorithm tries to add an atom at one of the grid location $i$ where the residual $\\mathrm{\\Psi} w^k - y$ is minimal (this corresponds to flipping $w^k_i$ from $0$ to $1$ in (\\ref{eq:opt3})). The allowed grid locations belong to a constraint set $\\mathcal{C}$, such that no two atoms are closer than a pre-specified minimum distance $r_{\\mathrm{min}}$. \nTo perturb the atom positions locally, the algorithm selects an atom at random and moves it to one of its 4 nearest neighbor locations at random. \n\n\\subsection{ADCG with Energy}\\label{adcgenergy}\n\n\\begin{algorithm}[b]\n\t\\caption{ADCG with energy} \n\t\\begin{algorithmic}[1]\n\t\\For {$k=1:k_{\\text{max}}$}\n            \\State Compute next atom in grid $g$: \n            \\Statex \\hskip \\algorithmicindent $\\vec{x}_{\\mathrm{new}} \\in \\arg \\min_{\\vec{x}_{\\mathrm{new}} \\in g, (\\vec{x}^k,\\vec{x}_{\\mathrm{new}}) \\in \\mathcal{C}} \\| \\sum_{i=1}^k \\psi(\\vec{x}_i) - y + \\psi(\\vec{x}_{\\mathrm{new}})\\| + \\alpha V_{\\text{tot}}(\\vec{x}^k,\\vec{x}_{\\mathrm{new}})$\n            \\State Update support: $\\vec{x}^{k+1} \\leftarrow \\{ \\vec{x}^k, \\vec{x}_{\\mathrm{new}} \\}$\n            \\State Locally move atoms:\n\t\\Statex \\hskip \\algorithmicindent $\\vec{x}^{k+1} \\leftarrow \\min_{\\vec{x} \\in X} \\| \\mathrm{\\Psi} \\mu(\\vec{x}^{k+1}) - y\\|^2_2 + \\alpha V(\\vec{x}^{k+1})$\n\t\\State Break if objective function is increasing:\n\t\\If {$\\| \\mathrm{\\Psi} \\mu(\\vec{x}^{k+1}) - y\\|^2_2 + \\alpha V(\\vec{x}^{k+1}) > \\| \\mathrm{\\Psi} \\mu(\\vec{x}^{k}) - y\\|^2_2 + \\alpha V(\\vec{x}^{k})$} break\n\t\\EndIf\n\t\\EndFor\n\t\\end{algorithmic} \n\t\\label{alg:adcg_energy}\n\\end{algorithm}\n\nVariants of the Frank-Wolfe algorithm (or conditional gradient method) \\cite{frank1956algorithm,jaggi2013revisiting} have been proposed for solving problems of the form \\eqref{eq:infoptnonconv} \\cite{denoyelle2019sliding, alberti2019dynamic} without discrete constraints for $w$ and are commonly known as alternating descent conditional gradient (ADCG) schemes (see \\cite{poon2019multidimensional} for an analysis specific to multidimensional sparse inverse problems). Here, we modify the ADCG scheme to \n\\begin{enumerate}\n    \\item incorporate binary constraints on $w$\n    \\item handle the singularities of the atomic interaction potentials \n    \\item avoid local minima resulting from poor initialisations\n\\end{enumerate}{}\nThe complete algorithm is shown in Algorithm \\ref{alg:adcg_energy}. Essentially, the scheme also alternates between adding a new atom to the current configuration and optimising the positions of the atoms. \n\nIn the first step, the image domain is coarsely gridded and the objective function after adding an atom at each location is computed. Locations closer to existing atoms than $r_{\\mathrm{min}}$ are excluded. In the second step, the atom coordinates are optimized by a continuous local optimisation method. Here, the Nelder-Mead method \\cite{nelder1965simplex} implemented in SciPy \\cite{virtanen2019scipy} was used. \n\nA continuation strategy is used to avoid problems resuling from poor initilisations: Algorithm \\ref{alg:adcg_energy} is run for increasing values of $\\alpha$, starting from $\\alpha=0$. The reconstruction obtained at the end of a run is used as initialisation for the next. In the following section, we demonstrate the effect of increasing $\\alpha$ on the reconstructions obtained and discuss how an optimal $\\alpha$ was selected. In the following section, we refer to  Algorithm \\ref{alg:adcg_energy} as ``ADCG\" when used for $\\alpha = 0$ and as ``ADCG with energy\" otherwise. \n\n\n\n\n\n\n\n\n\n\n\n\\section{Numerical Experiments}\\label{expresults}\nWe conducted numerical experiments by creating 2D atomic configurations with defects and using the algorithms discussed in Section \\ref{algos} to resolve atom positions. In this section we describe how the ground truth configurations were generated and projected, and compare the reconstruction results of different algorithms.\n\n\\subsection{Ground Truth Configurations}\nWe generated ground truth configurations using the molecular dynamics software HOOMD-blue \\cite{anderson2008general, glaser2015strong}. We created perfect square lattices and then induced defects by adding or removing atoms. The resulting configuration was then relaxed to an energy minimum using the FIRE energy minimizer \\cite{bitzek2006structural} to give the configurations shown in Fig.~\\ref{fig:configs}. The following parameter values were used in \\eqref{eq:lj} for specifying the Lennard-Jones pair potential between atoms.\n\n\\begin{center}\n \\begin{tabular}{ |p{3cm}||p{3cm}|p{3cm}|p{3cm}|  }\n \\hline\n Defect type & $\\epsilon$ & $\\sigma$ & $r_{\\mathrm{cut}}$ \\\\ [1.5ex] \n \\hline\\hline\n Interstitial defect & 0.4 & 0.15 & 0.4 \\\\ \n \\hline\n Vacancy & 0.4 & 0.14 & 0.4 \\\\\n \\hline\n Edge dislocation & 0.4 & 0.13 & 0.17 \\\\\n \\hline\n\\end{tabular}\n\\end{center}\n\n\\subsection{Discretized Projection Data}\nWe generated two 1D projections for each ground truth atomic configuration at projection angles $\\theta = 0^\\circ, 90^\\circ$. As discussed in Section \\ref{mathsetup}, the projection of a single atom centre is given by a Gaussian convolution followed by the Radon transform. The Radon transform of a Gaussian is also a Gaussian. Therefore, we interchanged the two operations in the forward transform to speed up the computations. The sum over individual projections of atom centres was used as the total (noise-free) projection. Using the Radon transform in \\eqref{eq:radon}, each atom centre was projected onto a 1D detector, following which it was convolved with a 1D Gaussian of the form $G(z) = e^{-(z - z_0)\/\\varsigma^2}$, where $z_0$ is the position of the atom centre on the detector and $\\varsigma$ controls the width of the Gaussian. Finally, the continuous projection was sampled at a fixed number of points to give rise to a \\emph{discrete} projection. For our experiments, the $\\varsigma$ of the Gaussian function was taken to be equal to the discretisation of the detector given by the detector pixel size $d$. Both were taken to be 0.01.\n\n\\subsection{Discretisation of Reconstruction Volume}\nFor SIRT, FISTA and simulated annealing (described in subsections \\ref{sirt}, \\ref{prox} and \\ref{sa}, respectively), each dimension of the reconstruction area was discretized using the detector pixel size $d$. Therefore, there were $1\/d \\times 1\/d$ grid points in total. \n\n\n\\begin{figure}[H]\n \n  \\centering\n    \\includegraphics[scale=0.24]{figure3}\n    \\vspace{-3cm}\n    \\caption{Reconstructions of atomic configurations with (a)--(f) an interstitial point defect, (g)--(l) a vacancy and (m)--(r) an edge dislocation from two projections. For the simulated annealing, ADCG and ADCG with energy reconstructions, atoms are colored according to their Euclidean distance from the ground truth. The ground truth positions are marked with red crosses. In (j)--(l) an extra atom (shown in red) was present in the reconstructions but not in the ground truth.}\n    \\label{fig:reconstructions}\n\\end{figure}\n\nGridding is required for our variant of ADCG (subsection \\ref{adcgenergy}, Algorithm \\ref{alg:adcg_energy}) at the atom adding step. We found that a coarse discretisation, with less than $1\/9^{\\text{th}}$ the number of grid points, was already sufficient.\n\n\\subsection{Comparison between Reconstructions}\nThe reconstructions obtained with the different algorithms are shown in Fig.~\\ref{fig:reconstructions}. For each reconstruction, data from only two projections were used. Note that two projections is far from sufficient for determining the correct atomic configuration and several different configurations have the same data discrepancy. \n\n\n\\begin{figure}[t]\n    \\includegraphics[width=\\textwidth]{figure4}\n    \\caption{Reconstructions of a vacancy defect from three projections. For the simulated annealing, ADCG and ADCG with energy reconstructions, atoms are colored according to their Euclidean distance from the ground truth. Ground truth positions are marked with red crosses.}\n    \\label{fig:reconstructions3}\n\\end{figure}\n\nIn the SIRT reconstructions, atom positions were blurred out and none of the defects were resolved. In all cases, the number of intensity peaks was also different from the true number of atoms. Although FISTA reconstructions, which include sparsity constraints on the weights, were less blurry, atoms still occupied more than one pixel. For both these algorithms, additional heuristic post-processing is required to output atom locations. In the edge dislocation case, both algorithms gave rise to a configuration with many more atoms than were present in the ground truth.\n\n\\begin{center}\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{ |p{3cm}||p{1.2cm}|p{1.5cm}||p{1.2cm}|p{1.5cm}||p{1.2cm}|p{1.5cm}||}\n \\hline &\n \\multicolumn{2}{|c||}{Interstitial defect} & \\multicolumn{2}{|c||}{Vacancy (3 projs.~)} & \\multicolumn{2}{|c||}{Edge dislocation} \\\\\n \\hline\n & Number of atoms & Mean distance & Number of atoms & Mean distance & Number of atoms & Mean distance \\\\\n \\hline\n Ground truth & 37 & 0.0000 & 48 & 0.0000 & 39 & 0.0000 \\\\\n \\hline\n SIRT & 36 & -- & 49 & -- & 66 & -- \\\\\n \\hline\n FISTA & 36 & -- & 49 & -- & 66 & -- \\\\\n \\hline\n Simulated annealing & 37 & 0.0184 & 48 & 0.0164 & 39 & 0.0159 \\\\\n \\hline\n ADCG & 37 & 0.0138 & 48 & 0.0130 & 39 & 0.0049 \\\\\n \\hline\n ADCG with energy & 37 & 0.0018 & 48 & 0.0024 & 39 & 0.0048\\\\\n \\hline\n\\end{tabular}\n\\caption{Number of atoms and mean Euclidean distance from ground truth atoms for reconstructions obtained with different algorithms. Thresholding was used to compute the number of atoms detected in the SIRT and FISTA reconstructions.}\n\\label{table:metrics}\n\\end{table}\n\\end{center}\n\nThe discrete simulated annealing algorithm performed better for all configurations. For the interstitial point defect and edge dislocation, the number of atoms in the reconstruction matched that in the ground truth. The positions of most atoms, however, were not resolved correctly. Moreover, the resolution, like in previous algorithms, was limited to the resolution on the detector. We ran the simulated annealing algorithm for comparable times as the ADCG algorithms and picked the solution with the least data discrepancy.\n\nAlready the ADCG algorithm for $\\alpha = 0$ performed far better than all the previous algorithms. For the configurations with an interstitial point defect and edge dislocation, all but a few atom locations were identified correctly. For the configuration with a vacancy, all atoms were correctly placed. However, an additional atom at the centre of the configuration was placed incorrectly. \n\nAdding the potential energy (ADCG with energy)  helps to resolve atom positions that were not identified with $\\alpha = 0$. For the interstitial point defect and edge dislocation, these reconstructions were the closest to the ground truth. Adding the energy to the configuration with a vacancy moved the atoms near the defect further apart but was not able to correct for the extra atom placed. For this case, we performed an additional experiment with three projections at $0^\\circ$, $45^\\circ$ and $90^\\circ$. These results are shown in Fig.~\\ref{fig:reconstructions3}. Taking projections at different angles (e.g.~$0^\\circ$, $22.5^\\circ$ and $90^\\circ$) did not improve results. The defect was still not resolved in the SIRT and FISTA reconstructions. However, the number of atoms in the simulated annealing, ADCG and ADCG with energy reconstructions was correct. Once again, the reconstruction obtained with our algorithm was closer to the ground truth than all other reconstructions, with all but one atom placed correctly. Reconstructions with 3 projections for the interstitial point defect and edge dislocation were not significantly different from those with 2 projections. In Table \\ref{table:metrics}, we report the number of atoms detected and (where applicable) the mean Euclidean distance of atoms from the ground truth. Note that for computing the mean distance, we required that the number of atoms detected in the reconstruction matched that in the ground truth. Thresholding with a pre-defined minimum distance between peaks was used to detect atoms in the SIRT and FISTA reconstructions. \n\n\\begin{figure}[t]\n  \\makebox[\\textwidth][c]{\\includegraphics[width=1.3\\textwidth]{figure5}}%\n   \n    \\caption{(a)-(d): Increasing the weighting of the energy term from $\\alpha=0.0$ to $\\alpha=10.0$ helps to resolve the correct atomic configuration. The reconstructed atoms are colored according to their Euclidean distance from the atoms in the ground truth. (e) At high values of $\\alpha$, the reconstructions have a high data discrepancy and correspond to one of the global minima of the potential energy. (f) From the plots of potential energy and data discrepancy, an optimal value of $\\alpha$ (indicated by the grey line) is selected. Increasing $\\alpha$ beyond this optimal value leads to a large increase in the data discrepancy due to addition of more atoms.}\n    \\label{fig:energy}\n\\end{figure}\n\\subsection{Effect of Adding Energy to Optimisation}\nTo resolve atom positions using Algorithm \\ref{alg:adcg_energy}, the contribution of the potential energy was increased gradually by increasing $\\alpha$. In Fig.~\\ref{fig:energy}, we show the effect of adding energy to the optimisation problem. For $\\alpha=0$, an initial guess for the true configuration was obtained. This configuration, though data optimal, was not the ground truth. A quantitative measure of this mismatch is the Euclidean distance between the reconstructed atom locations and those in the ground truth. As $\\alpha$ was increased, the reconstructions evolved from being data-optimal to being energy-optimal. At a certain value of $\\alpha$, the Euclidean distance between reconstructed and ground truth atom locations decreased to zero. Increasing $\\alpha$ beyond this point led to a large increase in the data discrepancy term due to the addition of more atoms. For very high values of $\\alpha$, the configurations obtained were essentially global minima of the potential energy, such as the honeycomb configuration in Fig.~\\ref{fig:energy}(e) for $\\alpha=100.0$. An optimal value of the regularisation parameter was selected by increasing $\\alpha$ to the point at which more atoms were added to the configuration and a jump in the data discrepancy was observed.\n\n\\section{Discussion}\\label{discussion}\nThe results of our numerical experiments demonstrate that algorithms like ADCG, which do not rely on domain discretisation, are better at resolving the defects in the atomic configurations shown in Fig.~\\ref{fig:configs}. Moreover, the output from ADCG is a \\emph{list of coordinates} and not an image like that of SIRT or FISTA, which requires further post-processing steps to derive the atom locations. Direct access to coordinates can be particularly useful because further analysis, such as strain calculations, often require atom positions as input. \n\nAdding the potential energy of the atomic configuration to the optimisation problem resulted in reconstructions that were closer to the ground truth.\nOne challenge of the proposed approach (with or without adding the potential energy) is that the resulting optimisation problem is a non-convex function of the atom locations. The numerical methods we presented are not intentionally designed to escape local minima and are therefore sensitive to their initialisation. To improve this, one important extension would be to also remove atoms from the current configuration, which might make it possible to resolve the vacancy defect in Fig.~\\ref{fig:reconstructions} with two projections. More generally, one would need to include suitable features of global optimisation algorithms \\cite{pardalos2016advances} that do not compromise ADCG's computational efficiency  (note that we could have adapted simulated annealing to solve \\eqref{eq:infoptwithenergy} but using a cooling schedule slow enough to prevent getting trapped in local minima quickly becomes practically infeasible). A related problem is to characterize local and global minimizers of \\eqref{eq:infoptwithenergy} to understand which configurations can be uniquely recovered by this approach and which cannot. To process experimental data, it may furthermore be important to analyze the impact of the error caused by the approximate nature of the mathematical models used for data acquisition ($\\mathcal{R}, G$) and atomic interaction ($V_{\\text{LJ}}$). \n\n\n\n\\section{Conclusions}\\label{conclusions}\n\nIn this paper we proposed a novel discrete tomography approach in which the locations of atoms are allowed to vary continuously and their interaction potentials are modeled explicitly. We showed in proof-of-concept numerical studies that such an approach can be better at resolving crystalline defects than image domain discretized or fully discrete algorithms. Furthermore, in situations where atom locations are desired, this approach provides access to the quantity of interest without any additional post-processing. \nFor future work, we will extend our numerical studies on this atomic super-resolution approach to larger-sized scenarios in 3D, featuring realistic measurement noise, acquisition geometries, more suitable and accurate physical interaction potentials and different atom types. This will require additional computational effort to scale up our algorithm and will then allow us to work on real electron tomography data of nanocrystals. \n\n\\paragraph{Acknowledgments} This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no.~765604.\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMagnetic skyrmions (SKs) are promising candidates for reliable, energy efficient, and high density data storage \\cite{roadmap} in the form of racetrack memory elements \\cite{fert2013skyrmions,muller2017magnetic}. Skyrmions and their anti-particles, the antiskyrmions (ASKs) \\cite{kovalev2018skyrmions,nayak2017magnetic,jena2019observation}, are magnetic solitons which may appear on a nanometer scale while remaining surprisingly stable \\cite{everschor2018perspective}. This stability is typically well explained by topological arguments \\cite{Nagaosa2013}, even though the necessary continuous transformations are not possible on a physical atomic lattice \\cite{Vedmedenko2019}. In particular, the phenomena of SK creation or annihilation are beyond the scope of the static concept of topology. Thus, these processes have to be discussed theoretically by energy considerations \\cite{PhysRevB.94.174418} or computational simulations \\cite{lin2013manipulation}.\nIn recent years, several creation mechanisms have been found or predicted, such as reversible creation\/annihilation by local spin current injection \\cite{romming2013writing}, electric fields \\cite{Hsu2017, electricfieldinduced} or edge manipulation \\cite{Schaffer2020}. It is also possible to create SKs by in-plane currents \\cite{stier2017skyrmion,everschor2017skyrmion}, out-of-plane currents \\cite{Yuan2016}, spin waves \\cite{spinwavecreation2015}, lasers \\cite{laserinducedcreation,laserinducedcreation2,koshibae2014creation} or geometric constraints \\cite{jiang2015blowing}.\\\\\nThe creation of SKs obviously lays the foundation for SK based devices, but a precise control over the SK dynamics remains as important. Particularly for the use of racetrack devices, SKs are typically moved by electrical currents \\cite{iwasaki2013current} while also magnetic fields \\cite{Zhang2018,moon2016skyrmion} or a edge-induced pushing \\cite{Schaffer2020} may be used. Notable obstacles for precise SK dynamics in terms of technical usability are the topological SK Hall effect \\cite{chen2017skyrmion,jiang2017direct,plettenberg2020steering} or material impurities \\cite{potkina2020skyrmions,stier2021role}. It is also important to spatially separate the logical bits within a memory device which lead to the proposal of two-lane racetracks \\cite{muller2017magnetic, song2017skyrmion,plettenberg2020steering} or the simultaneous use of two different data carriers as domain walls and SKs \\cite{Schaffer2020}. Antiskyrmions would be natural candidates to complement SKs in a two-species racetrack as they share the same advantages, such as, stability, mobility and small size. This, however, imposes the question how to create ASKs in a controlled manner.\\\\\nTheoretically, an ASK is not fundamentally different from a SK in terms of creation. Actually, a creation process often involves meta-stable SK-ASK pairs, where one partner decays due to dissipation and the other remains \\cite{stier2017skyrmion,everschor2017skyrmion}. The creation of coexisting stable SKs and ASKs, however, is difficult because materials including Dzayloshinskii-Moriya interactions (DMI) are mostly known to host either SKs or ASKs, depending on the DMI \\cite{Hoffmann2017}. Energetic equivalence is only predicted for one-dimensional DMI \\cite{Hoffmann2017}. Coexisting lattices of SKs and ASKs were reported for a small temperature pocket in Heusler compounds \\cite{Jena2020}.\nFurthermore, the creation of both SKs and ASKs by local heating in dipolar magnets has been numerically simulated \\cite{koshibae2014creation}.\nHowever, both of these processes rely on the stochastic formation of SKs and ASKs, not bearing any control of which species is created. By anisotropy design, a controlled creation of SKs and ASKs can be achieved at fixed positions \\cite{zhang2016creation}, preventing, however, any mobility of the data carriers, which may be an important disadvantage for the design of memory devices.\\\\\nIn this work, we propose a method for the controlled creation of coexisting mobile SKs and ASKs. This allows for the writing and deletion of arbitrary sequences of densely packed SKs and ASKs, which may be efficiently used in racetrack devices. We consider magnetic systems with anisotropic DMI and numerically simulate their dynamics by solving the Landau-Lifshitz-Gilbert equation. Hereby, we show that both magnetic species can be created by a rotation of the edge magnetization, where the sense of rotation determines which quasiparticle species is formed. Finally, we discuss parameter regimes where this rotation scheme is feasible.\\\\\nThis work is organized as follows.\nIn Sec. \\ref{SecModel}, we introduce our model, a magnetic lattice with nearest neighbor interactions, and the Landau-Lifshitz-Gilbert equation used to compute the time evolution. \nIn Sec. \\ref{SecCreationSingleSk}, we propose a creation scheme for SKs and ASKs in the same material with a given one-dimensional DMI, relying on a rotation of the edge magnetization.\nFurther, we extend the discussion in Sec. \\ref{SecSkArray} to the creation of an array of coexisting mobile SKs and ASKs. In Sec. \\ref{Sec2DDMI}, we show that the presented schemes work beyond a one-dimensional DMI.\nFinally, we present our conclusion and outlook in Sec. \\ref{SecConclusion}.\n\n\\section{Model}\\label{SecModel}\nWe consider an $N_x\\times N_y$ magnetic square lattice of normalized magnetic moments $\\textbf{n}_{\\textbf r}(t)$ with ferromagnetic boundaries $\\textbf{n}_{\\textbf r\\in \\text{edge}}(t)= -1$.\nAll interactions are collected in the atomistic lattice Hamiltonian\n\\begin{align}\\label{Lattice_Hamiltonian}\nH=&-J\\sum_{\\textbf{r}}\\textbf{n}_{\\textbf{r}}\\cdot(\\textbf{n}_{\\textbf{r}+a\\hat{x}}+\\textbf{n}_{\\textbf{r}+a\\hat{y}})\\nonumber\\\\\n&-B_z\\sum_{\\textbf{r}}n^z_{\\textbf{r}}\n-K\\sum_{\\textbf{r}}(n^z_{\\textbf{r}})^2\\\\\n&-\\sum_{\\textbf{r}}\\left[D_x(\\textbf{n}_{\\textbf{r}} \\times \\textbf{n}_{\\textbf{r}+a\\hat{y}})\\cdot\\hat{x}-D_y(\\textbf{n}_{\\textbf{r}} \\times \\textbf{n}_{\\textbf{r}+a\\hat{x}})\\cdot\\hat{y}\\right]\\nonumber .\n\\end{align}\nTo be specific, we consider interaction parameters of PdFeIr(111) adapted to a square lattice \\cite{Romming2015,Schaffer2020} and with an adapted DMI. That corresponds to the exchange interaction strength $J=11.6$~meV, the DMI strengths $|D_x|\\leq D_y$ and $D_y=3.17$~meV, the magnetic anisotropy $K=0.35$~meV and the external magnetic field strength $B_z=-0.261$~meV.\\\\\nWe compute the time evolution of the system by solving the extended Landau-Lifshitz-Gilbert equation \\cite{Tatara2008,Bazaliy1998,Zhang2004,Lakshmanan2011} \n\\begin{equation}\\label{LLG_current}\n\\begin{split}\n\\frac{\\partial \\textbf{n}}{\\partial t}=&-\\frac{\\gamma}{1+\\alpha^2}~\\textbf{n}\\times\\textbf{B}_\\textbf{eff}-\\frac{\\alpha \\gamma}{1+\\alpha^2}~ \\textbf{n}\\times ( \\textbf{n}\\times\\textbf{B}_\\textbf{eff})\\\\\n&+(1+\\alpha \\beta)(\\textbf{v}_s\\cdot\\nabla)\\textbf{n}\n+(\\alpha-\\beta)\\textbf{n}\\times(\\textbf{v}_s\\cdot\\nabla)\\textbf{n},\n\\end{split}\n\\end{equation}\nwith a fourth order Runge-Kutta method where $\\alpha$ is the damping constant, $\\beta$ the non-adiabaticity constant, $\\gamma$ the gyromagnetic ratio and $\\textbf{v}_s=\\frac{pa^3}{2e}\\textbf{j}$ the effective spin velocity with the polarization $p$, the lattice constant $a$, the elementary charge $e>0$ and the in-plane electric charge current density $\\textbf{j}$.\nThe effective field is $\\textbf{B}_{\\text{eff}}=-\\frac{1}{\\gamma\\hbar}\\frac{\\partial H}{\\partial\\textbf{n}}$.\\\\\n\\section{Creation process of single (anti)skyrmions}\\label{SecCreationSingleSk}\nRef. \\cite{Schaffer2020} shows, that in systems with isotropic DMI, i.e., $D_x=D_y$, it is possible   to create coexisting SKs and magnetic domain walls by a rotation of magnetic moments at the edge. However, a racetrack design based solely on skyrmionic quasiparticles promises  smaller, more mobile data carriers that are and less prone to material impurities. From Eq. (\\ref{Lattice_Hamiltonian}) we find, that the DMI prefers twisted magnetic structures with the part pertaining to $ D_x$ favoring a rotation in the lateral $y$ direction and the one pertaining to $ D_y$ in lateral $x$ direction. Since the twist of the magnetization of a SK and an ASK coincides only in one direction, they are usually stabilized by different orientations of the DMI, as shown in Fig. \\ref{Fig1} (a). A DMI with $D_x\/D_y>0$ favors a SK and a DMI with $D_x\/D_y<0$ an ASK \\cite{Huang2017}. Thus, SKs and ASKs are typically mutually exclusive. Only in the special case $D_x=0$ or $D_y=0$, both species are energetically equivalent \\cite{Hoffmann2017}.\\\\\nFor the actual creation process, we focus first on $D_x=0$ to have a strictly one-dimensional DMI. Then, the twist of the magnetization  of the (anti)skyrmion is only prescribed in $x$ direction by the DMI as $D_y\\neq 0$, while it is not determined in $y$ direction by the DMI. This opens possibility to imprint a twist by the edge rotation. As discussed in Ref. \\cite{Schaffer2020}, there is a manifold of boundary rotations that can create SKs. Thus, in real materials, the explicit rotation can be chosen as technically suitable and can be performed by, e.g., repeated current pulses as in toggle magnetoresistive random access memory or a coupling to twisted magnetic microscopic structures. For an extensive discussion of possible implementations of the rotation, we refer to a previous work \\cite{Schaffer2020}. Here, we use a uniform rotation for simplicity which is given by\n\\begin{equation}\\label{RotEq}\n\\begin{split}\n&n^x_{\\text{rot}}(t) = 0,\\\\\n&n^y_{\\text{rot}}(t) = \\pm\n\\sin(\\nu t),\\\\\n&n^z_{\\text{rot}}(t) = -\\cos(\\nu t),\n\\end{split}\n\\end{equation} \non a stripe of length $w_{\\text{rot}}=15a$ at the start of the racetrack geometry, as indicated in Fig. \\ref{Fig1}.\nDepending on the sense of rotation of $n^y_{\\text{rot}}(t)$, a SK or an ASK is created. The existence and speed of the specific creation process obviously depends on several model parameters, where a detailed analysis of these dependencies is beyond the scope of this work. A more detailed discussion about possible rotation frequencies can be found in Ref. \\cite{Schaffer2020}. Here, an exemplary creation process for both species is shown in Fig. \\ref{Fig2} for $\\nu=2.78$ GHz which yields a creation in less than a nanosecond. As expected from the results in Ref. \\cite{Schaffer2020}, we also see in our current simulations that the created species can be deleted by rotating again in backwards direction if the racetrack width $N_y$ is not too large. For large $N_y$, a SK-ASK hybrid can be created during the intended deletion process due to the large extension in $y$-direction of the quasiparticle. If the species is not present, the respective antiparticle is created by the backwards rotation. In Fig.\\ \\ref{Fig2}, we have confined the wire in $y$ direction as it prevents an elongation of the (anti)skyrmions which would occur due to $|D_x|<D_y$.\\\\\n\\begin{figure*}[!tbp]\n\t\\centering\n\t\\includegraphics[width=\\linewidth, height=\\textheight,keepaspectratio]{fig1_new2}\n\t\\begin{picture}(0,0)\n\t\\put(-260,120){{\\large (a)}}\n\t\\end{picture}\n\t\\begin{picture}(0,0)\n\t\\put(-80,120){{\\large (b)}}\n\t\\end{picture}\n\t\\caption{(a) Orientation of the DMI.  For a fixed positive $D_y$, a positive $D_x$ in Eq. (\\ref{Lattice_Hamiltonian}) stabilizes a skyrmion (blue arrows), a negative $D_x$ an antiskyrmion (red arrows). We note that $D_x$ induces a twist in the lateral $y$ direction, as $D_y$ does in the lateral $x$ direction. (b) Schematic of a SK creation by a rotation of a stripe of the edge magnetization (red arrows). Except for the rotated stripe of magnetization, the sample has a boundary with parallel spin configuration (blue arrows). Depending on the sense of rotation a SK or ASK is created and afterwards pushed along the racetrack.}\n\t\\label{Fig1}\n\\end{figure*}\n\\begin{figure*}[!tbp]\\centering\n\t\\includegraphics[width=\\linewidth, height=\\textheight,keepaspectratio]{Sk_ASk_Creation_quiverannotation2}\n\t\\caption{ Creation process of a skyrmion (top row) and an antiskyrmion (bottom row) at different rotation angles $\\theta=\\pm\\nu t$. The different signs of $\\theta$ reflect the different senses of rotation. The arrows depict the $x-y$  orientation of the magnetization over the position, the $z$-component is included via the color code. The edge magnetization is rotated with $\\nu=2.78$~GHz while $D_x =0$.}\n\t\\label{Fig2}\n\\end{figure*}\n\\section{Creation process of a skyrmion\/antiskyrmion array}\\label{SecSkArray} \nTo store information with the help of SKs and ASKs in a racetrack, a controlled creation, deletion and propulsion of both magnetic species in an arbitrary order is necessary.\nIn a pure SK system, an array of SKs can be created by repeating the same rotation process \\cite{Schaffer2020}. Repulsive interactions between the SKs shift the already created SKs along the wire during each new creation process.\nHowever, the interaction between SKs and ASKs depends on the orientation towards each other and can be either repulsive or attractive \\cite{Hoffmann2021, Shimizu2020}. An attractive interaction usually leads to an annihilation of both species.\nIn the setup presented here, SKs and ASKs repel each other in $x$ direction while they would annihilate in $y$ direction. As they are, in contrast to Ref. \\cite{Schaffer2020}, created at the long edge along the $x$ axis, successively created SKs and ASKs would be positioned along the $y$ axis and annihilate each other.\nThus, to successively create and stabilize SKs and ASKs, the already created quasiparticles need to be moved along the sample ($x$ direction) before the next creation process takes place.\nHere, we achieve this by applying a current along the $x$ direction while currentless racetrack realizations are also possible (see Appendix).\nTo track the creation of SKs and ASKs, we compute the topological charge \n\\begin{equation}\nQ=\\frac{1}{4\\pi}\\iint\\underbrace{ \\textbf{n}\\cdot\\left(\\frac{\\partial\\textbf{n}}{\\partial x}\\times\\frac{\\partial\\textbf{n}}{\\partial y}\\right)}_{= q(x,y,t)}\\text{d}x\\text{d}y,\n\\end{equation}\nwhere $q(x,y,t)$ is the topological charge density. Note that the topological charge has to take integer values for ferromagnetic boundary conditions, but is not restricted to this during the rotation where the boundary conditions change. As SKs and ASKs have a topological charge of opposite sign, the total topological charge does only provide information about the difference in the number of SKs and ASKs. We therefore also consider $Q_{Sk}$ ($Q_{ASk}$) which is the sum over the negative (positive) contributions of the topological charge density and thus approximates the number of SKs (ASKs). Again, $Q_{Sk}$ or $Q_{ASk}$ do not have to take integer values as topological arguments only hold for the total topological charge $Q$. However, since the DMI leads to a repulsive interaction along the $x-$ axis, the SKs and ASKs remain clearly separated which yields values for $Q_{Sk}$ or $Q_{ASk}$ reasonably close to integers when the boundary magnetization is not tilted\\\\\nThe creation protocol of eight SKs and ASKs is shown in Fig. \\ref{Fig3}. Between each creation process a rotation break of duration $\\tau=130$~ps is taken. During this break, a current of $v_s=-0.05$~a\/ps is applied in $x$ direction. For simplicity, we consider $\\beta=\\alpha=0.1$, such that no SK Hall effect appears for different values of $\\beta$, see Appendix.\nAfter completing eight rotations, an array of in total eight SKs and ASKs is stabilized in the sample, as shown in Fig. \\ref{Fig3}(d).\nAs mentioned above, a quasiparticle can also be deleted at the edge with a rotated magnetization by reversing the sense of rotation if the racetrack width $N_y$ is not too large. The presented setup therefore allows for a controlled creation, deletion and propulsion of SK-ASK arrays of arbitrary order. Additionally, this racetrack design does not rely on voids between SKs as a logical ``0'' but on two different quasiparticles. Thus, it does not suffer from the necessity to keep two particles in a certain distance as conventional one-lane racetracks relying on, e.g., SKs as information carriers only \\cite{Fert2013}. \nThe readout of SKs and ASKs is expected to be possible with the help of the topological Hall effect \\cite{kovalev2018skyrmions, Kumar2020} or the noncollinear Hall effect \\cite{Bouaziz2021}.\n\\\\ \n\\begin{figure}[!tb]\\centering\n\t\\includegraphics[width=\\linewidth, height=\\textheight,keepaspectratio]{my_current_Q_Sk_ASk_Race_3}\n\t\\includegraphics[width=\\linewidth, height=\\textheight,keepaspectratio, trim=40 150 0 180,clip]{q_8_particles_Sk_Ask_racetrack}\n\t\\caption{Creation protocol of a skyrmion-antiskyrmion array of eight quasiparticles.\n\t\t(a) The time evolution of the $y$-component of the rotated edge magnetization $n_{rot}^y$ (Eq. (\\ref{RotEq})) reflects the creation processes of SKs (blue) and ASKs (red).  (b)  Between two creation events a current pulse is applied to move the created quasiparticles along the sample. (c) Evolution of the negative and positive topological charge contributions $Q_{Sk}$ and $Q_{ASk}$, which count the number of SKs and ASKs, respectively. Here, each change of the topological charge corresponds to a creation process. After each creation process, a plateau is reached at an integer value, indicating a stable number of SKs and ASKs. (d) The spatial distribution of the topological charge density at $t=3.8$~ns depicts eight stable SKs and ASKs.}\n\t\\label{Fig3}\n\\end{figure}\n\\section{Two-dimensional DMI}\\label{Sec2DDMI}\nFor a one-dimensional DMI, SKs and ASKs are energetically equivalent \\cite{Hoffmann2017}. \nAs soon as $D_x\\neq0$, one of the two quasiparticles becomes energetically favored. \nEven though the DMI is subject to a certain tunability \\cite{Tacchi2017,Cho2015,Camosi2017}, the realization of a strictly one-dimensional DMI would pose a hard challenge. Instead, there are various materials that host an anisotropic DMI with significant differences in the magnitude of the components. One of them is Au\/Co\/W(100), where Camosi {\\em et al.\\\/} \\cite{Camosi2017} could measure a DMI ratio of $1:3$.\nNext, we show that the controlled creation of SKs and ASKs can be achieved for $D_x\\neq 0$ as well.\nFigure \\ref{DxDy-Ny} depicts the regime where a SK-ASK array of five quasiparticles remains stable for different contributions of $D_x$ and different sample widths, where actually three of the five quasiparticles are of the energetically unfavored type.\nWe find that for a sufficiently large sample width of $N_y\\geq 40$, an array of SKs and ASKs can be stably created and moved, up to a DMI ratio of  $\\left|\\frac{D_x}{D_y}\\right|=0.6$. The stability regime is studied by considering the final number of magnetic structures, when the simulation is performed long enough that the array reaches an equilibrium position after the creation and propulsion.\nWe confirmed by additional calculations that the SK-ASK regime remains the same for longer arrays. \nFor smaller values of $N_y$ or larger ratios of $|D_x\/D_y|$, only one magnetic species can be stably inscribed. When $D_x$ and $D_y$ share the same sign, a SK regime appears, where SKs can be created in a controlled manner, while ASKs decay either within the creation process or during the current pulses.\nFor different signs of $D_x$ and $D_y$, the ASK is lowered in energy and is the more stable configuration.\\\\\n\\begin{figure}[!tb]\\centering\n\t\\includegraphics[width=\\linewidth, height=\\textheight,keepaspectratio]{DxDy_Ny_Diagramm_all_in_one}\n\t\\caption{\\label{DxDy-Ny}Stability diagram of a skyrmion-antiskyrmion array of five quasiparticles for different values of $\\frac{D_x}{D_y}$ and racetrack widths $N_y$ for $\\nu=2.78$~GHz, $\\tau=150$~ps and $v_s=-0.05$~a\/ps. In the ``SK-ASK Regime'' an array of five quasiparticles can be stably inscribed and moved along the racetrack. In the ``SK Regime'' (``ASK Regime'') ASKs (SKs) decay during the creation process or the propulsion.}\n\\end{figure}\n\\section{Conclusions}\\label{SecConclusion}\nWe have introduced a setup that allows for the controlled creation, deletion and propulsion of skyrmions and antiskyrmions in the same sample. \nBoth magnetic species can be stabilized in a system with a sufficiently  anisotropic DMI. The creation occurs by a rotation of the magnetization at a part of the edge where the sense of rotation defines the created species. \nBy alternating the edge rotation and additionally applying current pulses, an array of skyrmions and antiskyrmions can be created and moved along the sample. A deletion process can be realized by a rotation of opposite sense compared to the creation process as long as the according particle is close to the rotating edge and the racetrack width $N_y$ is not too large.\nWe could show that the concept is robust against deviations from a strictly one-dimensional DMI allowing for an implementation in existing materials. The controlled creation, deletion and propulsion of skyrmions and antiskyrmions yields the possibility of a racetrack device relying on two different species of quasiparticles. This is in contrast to previously proposed hole-particle designs \\cite{fert2013skyrmions} where information is encoded only in the presence and absence of one data carrier. Thus, the here presented racetrack design is more robust against positional fluctuations of the data carriers and could be more densely packed.\\\\\n\\section{Acknowledgements}\nPS, MT and RW acknowledge funding by the Cluster of Excellence \"CUI: Advanced Imaging of Matter\" of the Deutsche Forschungsgemeinschaft (DFG)\u2014EXC 2056\u2014 project ID 390715994.\nMS acknowledges funding by the DFG (project no. 403505707). AS acknowledged financial support from the German research foundation (DFG) through the collaborative research center CRC\/TRR 227. TP acknowledges funding by the DFG (project no. 420120155).\n\\input{Paper-SkAsk.bbl}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and general overview}\n\\label{sec1}\nSuperstring theory (see \\cite{GSW} for a review) is the only presently known framework \nin which gravity can be consistently quantized, at least perturbatively.\nThe  well-known difficulties met in\ntrying to quantize General Relativity (GR) --or its supersymmetric extensions-- are\navoided, in string theory, by the presence  of\n a fundamental quantum of length \\cite{stringlength}  $\\ell_s \\sim  (\\alpha')^{1\/2}$. \nThus, at distances\nshorter than $\\ell_s$, \nstring gravity is expected to be drastically different from --and in particular\nto be much ``softer\" than--  General Relativity.\n\nHowever, as was noticed since the early days of string theory \\cite{joel}, \na conspicuous difference between string and Einstein gravity persists even at\nlow energies (large distances).\nIndeed, a striking prediction of string theory is that its ``gravitational sector'' is \nricher than that of GR: in particular, all versions of string \ntheory predict the existence of a scalar partner of the spin-two  \ngraviton, i.e. of the metric tensor $g_{\\mu \\nu}$, the so-called dilaton, $\\phi$.\nThis field plays a central r\\^ole \nwithin string theory \\cite{GSW} since its present vacuum expectation value (VEV) $\\phi_{\\rm now}$ \nfixes the string coupling constant $g^2 = e^{\\phi_{\\rm now}}$, and, through\nit, the present values of gauge and gravitational couplings. In particular, it fixes the ratio\nof $\\ell_s$ to the Planck length $\\ell_P \\sim G^{1\/2}$ by $\\ell_P \\sim g\\,\\ell_s$. \nThe relation is such that   \ngauge and gravitational couplings unify at the string energy scale, i.e.\n at  $E \\sim M_s \\sim g M_P \\sim (\\alpha')^{-1\/2} \\sim 3 \n\\times 10^{17}$ GeV. It thus seems that the string way to quantizing gravity\nforces this new particle\/field upon us.\n\nWe believe that the dilaton represents an interesting \nprediction (an opportunity rather than a nuisance) whose possible existence \nshould be taken seriously, and whose observable consequences should be \ncarefully studied. \nOf course, tests of GR \\cite{Damour} put severe constraints on what\nthe dilaton can do today. The simplest way to recover GR at late times\nis to assume \\cite{TV} that $\\phi$ gets a mass from supersymmetry-breaking non-perturbative \neffects. Another possibility might be \nto use the string-loop modifications of the dilaton couplings for driving \n$\\phi$ toward a special value where it decouples from matter \\cite{DP}. \nThese alternatives do not rule out the possibility\nthat the dilaton may have had an important r\\^ole  in the previous  history of the\nuniverse. Early cosmology stands out as a particularly \ninteresting arena where to study both the dynamical effects of the dilaton\nand those associated with the existence of a fundamental length in string theory.\n\nIn a series of previous papers \\cite{1}, \\cite{PBB} a model of early string cosmology, in \nwhich the dilaton plays a key dynamical r\\^ole, was introduced and developed: the so-called \npre-big bang (PBB) scenario. One of the key ideas of this scenario is to use \nthe kinetic energy of the dilaton to drive a period of inflation of the \nuniverse. The motivation is that the presence of a (tree-level \ncoupled) dilaton essentially destroys\\cite{BS} the usual inflationary mechanism \n\\cite{Linde}: instead of driving an exponential inflationary expansion, a \n(nearly) constant vacuum energy drives the string coupling $g = e^{\\phi \/ 2}$ \ntowards small values, thereby causing the universe to expand only as a small \npower of time. If one takes seriously the existence of the dilaton, the PBB \nidea of a dilaton-driven inflation offers itself as one of the very few natural \nways of reconciling string theory and inflation. Actually, the existence \nof inflationary solution in string cosmology is a consequence of its \n(T-) duality symmetries \\cite{2}. \n\nThis paper develops further the PBB scenario by presenting a very general class \nof possible initial states for string cosmology, and by describing their \nsubsequent evolution, via gravitational instability, into a multi-universe \ncomprising (hopefully) sub-universes looking like ours. This picture generalizes, and makes \nmore concrete, recent work \\cite{V97}, \\cite{spanish} about inhomogeneous \nversions of pre-Big Bang cosmology.\n\nLet us first recall that an inflation driven by the kinetic \nenergy of $\\phi$ forces both the coupling and \nthe curvature to {\\it grow} during inflation \\cite{1}. This implies that the \ninitial state must be very perturbative in two respects: i) it must have\nvery small initial curvatures (derivatives) in string units and, ii) it must exhibit \na tiny initial coupling \n $g_i = e^{\\phi_i \/2} \\ll 1$.  As the string \ncoupling $g^2$ measures the strength of quantum corrections (i.e. plays the \nr\\^ole of $\\hbar$ in dividing the lagrangian: ${\\cal L} = g^{-2} \\, {\\cal \nL}'$), quantum (string-loop) corrections \nare initially negligible. Because of i), $\\alpha^\\prime$\ncorrections can also be neglected. In conclusion, dilaton-driven inflation must start\nfrom a regime in which the tree-level low-energy approximation \nto string theory is extremely accurate, something we may call an asymptotically trivial\nstate.\n\nIn the present paper, \nwe consider a very general class of such ``past-trivial'' states. Actually, \nperturbative string theory is well-defined only when one considers such classical \nstates as background, or ``vacuum'', configurations. For the sake\nof simplicity the set of string \nvacua that we consider are already compactified to\n four dimensions and are truncated to the \ngravi-dilaton sector (antisymmetric tensor and moduli being set to zero). \nWithin these limitations, the set of all perturbative string vacua coincides \nwith the generic solutions of the tree-level low-energy effective action \n\\cite{Fradkin} \n\\beq\nS_s = \\frac{1}{\\alpha^\\prime} \\int d^4 x \\, \\sqrt{G} \\, e^{-\\phi} [R(G) + G^{\\mu \n\\nu} \\partial_{\\mu} \\phi \\, \\partial_{\\nu} \\phi] \\,,\n\\label{eqn1.1}\n\\end{equation}\nwhere we have denoted by $G_{\\mu \\nu}$ the string-frame ($\\sigma$-model) metric.\nThe generic solution is parametrized by 6 functions of three variables.  These \nfunctions can be thought of classically as describing the two helicity$-2$ \nmodes of gravitational waves, plus the helicity$-0$ mode of dilatonic waves. \n[Each mode being described by two real functions corresponding, e.g., to the \nCauchy data $(\\phi , \\dot \\phi)$ at some initial time.] The same counting of the \ndegrees of freedom in selecting string vacua can  be obtained by considering all the \nmarginal operators (i.e. all conformal-invariance-preserving \ncontinuous deformations) of the conformal field \ntheory defining the quantized string in trivial space-time.\n We therefore envisage, as initial state, \nthe most general past-trivial classical solution of (\\ref{eqn1.1}), i.e. an \narbitrary ensemble of incoming gravitational and dilatonic waves. \n\nOur aim will be to show how such a stochastic bath of classical \nincoming waves (devoid of any ordinary matter) can evolve into our rich,\ncomplex, expanding universe. The basic mechanism we consider for turning such a \ntrivial, inhomogeneous and anisotropic, initial state into a Friedmann-like \ncosmological universe is gravitational instability (and quantum particle production as far as\nheating up the universe is concerned \\cite{PBB}). We find that, when the \ninitial waves satisfy a certain (dimensionless) strength criterion, they \ncollapse (when viewed in the Einstein conformal frame) under their own weight. \nWhen viewed in the (physically most appropriate) string conformal frame, each \ngravitational collapse leads to the local birth of a baby inflationary universe \nblistering off the initial vacuum. Thanks to the peculiar properties of \ndilaton-driven inflation (i.e. the peculiar properties of collapse with the \nequation of state $p = \\epsilon$), each baby universe is found to contain a \nlarge homogeneous patch of expanding space which might constitute the beginning \nof a local Big Bang. We then expect each of these ballooning patches \nof space to evolve into a quasi-closed Friedmann universe\n\\footnote{Our picture of baby universes created by gravitational \ncollapse is reminiscent of earlier proposals \\cite{FMM}, \\cite{MB}, \n\\cite{Smolin}, \nbut differs from them by our crucial use of string-theory motivated ideas.}.\nThis picture is sketched in Fig.~\\ref{fig1} and Fig.~\\ref{fig2}.\nIn order to study in detail this scenario, we focus, in this \npaper, on the technically simplest case containing nontrivial incoming waves \nable to exhibit gravitational instability: spherically symmetric dilatonic \nwaves. This simple toy model seems to contain many of the key physical \nfeatures we wish to study.\n\\begin{figure}\n\\centerline{\\epsfig{file=fig1.ps,width=0.6\\textwidth,angle=0}}\n\\vskip 0.5truecm\n\\caption{\\sl Symbolic sketch of the birth of many pre-big bang \nbubbles from the gravitational instability of a generic string vacuum \nmade of a stochastic bath of classical incoming gravitational \nand dilatonic waves. Each local Einstein-frame collapse \nof sufficiently strong waves forms a cosmological-like \nspace-like singularity hidden behind a black hole. The parts of those classical\nsingularities where the string coupling grows inflate, when viewed \nin the string frame, and generate ballooning patches of space \n(here schematized as the stretching of one spatial dimension) \nwhich are expected to evolve into many separate quasi-closed Friedmann \nhot universes.}\n\\label{fig1}\n\\end{figure}\n\\begin{figure}\n\\centerline{\\epsfig{file=fig2.ps,width=0.6\\textwidth,angle=0}}\n\\vskip 0.5truecm\n\\caption{\\sl A representation of pre-big bang bubbles similar \nto that of Fig.~\\ref{fig1}, but in 2+1 dimensions. The different\nhorizontal planes represent different instants in the evolution \nfrom the asymptotic trivial past to the \nFriedmann phase. Two inflationary bubbles characterized by two different \ninitial horizon sizes (both large in string units) are shown\nto lead to Universes of very different homogeneity scale \nat the time at which the Hubble radius reaches string-scale values \n($\\ell_s = {\\cal O}(10^{-32}\\, {\\rm cm})$).}\n\\label{fig2}\n\\end{figure}\n\nBefore entering the technicalities of this model, let us clarify a few general \nmethodological issues. One of the goals of theoretical cosmology is to \n``explain'' the surprisingly rich and special structure of our universe. \nHowever, the concept of ``explanation'' is necessarily dependent on one's \nprejudices and taste. We wish only to show here how,  modulo an \n``exit'' assumption \\cite{PBB}, \none can ``explain'' the appearance of a hot, expanding \nhomogeneous universe starting from a generic classical inhomogeneous vacuum of \nstring theory. We do not claim that this scenario solves all the cosmological \n``problems'' that one may wish to address (e.g., we leave untouched here the \nmonopole and gravitini problems). We do not try either to ``explain'' the \nappearance of our universe as a quantum fluctuation out of ``nothing''. We \ncontent ourselves by assuming, as is standard in perturbative string theory, \nthe existence of a classical vacuum and showing how gravitational instability \ncan then generate some interesting qualitatively new structures akin to those \nof our universe. In particular, we find it appealing to ``understand'' the striking \nexistence of a preferred rest frame in our universe, starting from a \nstochastic bath of waves propagating with the velocity of light and thereby \nexhibiting no clear split between space and time. We shall also address in \nthis paper the question of the naturalness of our scenario. Recent papers \n\\cite{TW}, \\cite{KLB} have insisted on the presence of two large numbers among \nthe parameters defining the PBB scenario. Our answer to this issue (which is, \nanayway, an issue of taste and not a scientifically well posed problem) is \ntwofold: on the one hand, we point out that the two large numbers in question \nare classically undefined and therefore irrelevant when discussing the \nnaturalness of a {\\it classical} vacuum state; on the other hand, we show \nthat the selection effects \nassociated to asking any such ``fine-tuning'' question can render natural the \npresence of these very large numbers.\n\n\\section{Asymptotic null data for past-trivial string vacua}\n\\label{sec2}\nMotivated by the pre-Big Bang idea of an initial weak-coupling, low-curvature \nstate we consider the tree-level (order $g^{-2}$) effective action for the \ngravitational sector of critical superstring theory, taken at lowest order in \n$\\alpha'$, Eq.~(\\ref{eqn1.1}). We set to zero the antisymmetric tensor field \n$B_{\\mu \\nu}$, and work directly in 4 dimensions (i.e., we assume that the \ngravitational moduli describing the 6 ``internal dimensions'' are frozen). \nThough the physical interpretation of our work is best made in terms of the \noriginal string (or $\\sigma$-model) metric $G_{\\mu \\nu}$ appearing in \nEq.~(\\ref{eqn1.1}), it will be technically convenient to work with the \nconformally related Einstein metric\n\\beq\ng_{\\mu \\nu} = e^{-(\\phi -\\phi_{\\rm now})} \\, G_{\\mu \\nu} \\, , \n\\quad \\quad 16 \\pi G = \\alpha^\\prime \\, e^{\\phi_{\\rm now}}\\,. \n\\label{eqn2.1}\n\\end{equation}\nIn the following, we will set $G = 1$.\nIn terms of the Einstein metric $g_{\\mu \\nu}$, the low-energy tree-level string \neffective action (\\ref{eqn1.1}) reads\n\\footnote{We will use the signature $(-,+,+,+)$ and the conventions\n${R}^\\mu_{\\,\\,\\,\\,\\nu \\rho \\sigma} = \\pa_\\rho \\Gamma^\\mu_{\\nu\\sigma}\n- \\dots\\,,\\,\\,{R}_{\\mu \\nu} = {R}^\\rho_{\\,\\,\\,\\,\\mu \\rho \\nu}$.}\n\\beq\n\\label{1.1}\nS = \\frac{1}{16 \\pi}\\int d^4 x \\,\\sqrt{g}\\,\n\\left [ {R} - \\frac{1}{2}\\pa_\\mu \\phi\\, \\pa^\\mu \\phi \\right ]\\,.\n\\end{equation}\nThe corresponding classical field equations are\n\\beq\nR_{\\mu \\nu} = \\frac{1}{2} \\,\\partial_{\\mu} \\phi \\, \\partial_{\\nu} \\phi \\, ,  \n\\label{eq1.4} \n\\end{equation}\n\\beq\n\\Box \\, \\phi \\equiv \\nabla^{\\mu} \\, \\nabla_{\\mu} \\, \\phi = 0 \\, ,\n\\label{eqn2.3}\n\\end{equation}\nwith Eq. (\\ref{eqn2.3}) actually following \nfrom Eq.~(\\ref{eq1.4}) thanks to the Bianchi identity.\nAs explained in the Introduction, we consider a generic solution of these \nclassical field equations admitting an asymptotically trivial past. Such \nan asymptotic ``incoming'' classical state should allow a description in terms \nof a superposition of ingoing gravitational and dilatonic waves. The \nwork of Bondi, Sachs, Penrose \\cite{BSP} and many others in classical gravitation theory \nindicates that this incoming state should exhibit a regular past null infinity \n${\\cal I}^-$, and should be parametrizable by some asymptotic null data (i.e. \nconformally renormalized data on ${\\cal I}^-$). In plain terms, this means the \nexistence of suitable asymptotically Minkowskian \\footnote{These coordinates \nmust be restricted by the condition that the incoming coordinate cones $v \n\\equiv t + r = {\\rm const.}$ \nbe asymptotically tangent to exact cones of $g_{\\mu \n\\nu} (x^{\\lambda})$.} coordinates $(x^{\\mu}) = (t,x,y,z)$ (which can then be \nused in the standard way to define polar coordinates $r,\\theta,\\varphi$ and the \nadvanced time $v \\equiv t + r \\equiv t + (x^2 + y^2 + z^2)^{1\/2}$) such that \nthe following expansions hold when $r \\rightarrow \\infty$, at fixed $v$, \n$\\theta$ and $\\varphi$:\n\\beq\n\\phi (x^{\\lambda}) = \\phi_0 + \\frac{f(v,\\theta , \\varphi)}{r} + {o} \\left( \n\\frac{1}{r} \\right) \\, , \n\\label{eqn2.4}\n\\end{equation}\n\\beq\ng_{\\mu \\nu} (x^{\\lambda}) = \\eta_{\\mu \\nu} + \\frac{f_{\\mu \\nu} (v,\\theta , \n\\varphi)}{r} + {o} \\left( \\frac{1}{r} \\right) \\, . \n\\label{eqn2.5}\n\\end{equation}\nThe null wave data on ${\\cal I}^-$ are: the asymptotic dilatonic wave form $f \n(v,\\theta , \\varphi)$, and the two polarization components $f_+ (v,\\theta , \n\\varphi)$, $f_{\\times} (v,\\theta , \\varphi)$, of the asymptotic gravitational \nwave form $f_{\\mu \\nu} (v,\\theta , \\varphi)$ \\footnote{The three functions $f$, \n$f_+$, $f_{\\times}$ of $v,\\theta , \\varphi$ are equivalent to six functions in \n${\\bb R}^3$, i.e. six functions of $r,\\theta , \\varphi$ with $r \\geq 0$, because \nthe advanced time $v$ ranges over the full line $(-\\infty , +\\infty)$.}. \n[Introducing a local orthonormal frame ${\\bf e}_{(1)}$, ${\\bf e}_{(2)}$ on the \nsphere at infinity one usually defines $f_+ = \\frac{1}{2} (f_{(1)(1)} - \nf_{(2)(2)})$, $f_{\\times} = f_{(1)(2)}$.] The other ${o} (1\/r)$ pieces in \nthe metric are gauge dependent, except for the ``Bondi mass aspect''  $m_{\\infty} \n(v,\\theta , \\varphi)$ (defined below in a simple case) whose advanced-time \nderivative is related to the (direction-dependent) incoming energy fluxes:\n\\beq\n\\frac{\\partial}{\\partial v} \\, m_{\\infty} (v,\\theta , \\varphi) = \\frac{1}{4} \\, \n(\\partial_v \\, f)^2 + \\frac{1}{4} \\, (\\partial_v \\, f_+)^2 + \\frac{1}{4} \\, \n(\\partial_v \\, f_{\\times})^2 \\, + {\\rm div.}, \n\\label{eqn2.7}\n\\end{equation}\nwhere ${\\rm div.}$ denotes an angular divergence, $(\\sin \\theta)^{-1}\\,\n\\pa_\\theta(\\sin \\theta D^\\theta) + \\pa_\\vphi D^\\vphi$.\nInstead of the asymptotic wave forms $f$, $f_+$, $f_{\\times}$ (which have the \ndimension of length), one can work with the corresponding ``news functions''\n\\beq\nN (v,\\theta , \\varphi) \\equiv \\partial_v \\, f (v,\\theta , \\varphi) \\ , \\ N_+ \n\\equiv \\partial_v \\, f_+ \\ , \\ N_{\\times} \\equiv \\partial_v \\, f_{\\times} \\, , \n\\label{eqn2.8}\n\\end{equation}\nwhich are dimensionless, and whose squares give directly the incoming energy \nflux appearing on the R.H.S. of Eq.~(\\ref{eqn2.7}).\n\nBefore specializing this generic ingoing state, let us note \\cite{V97}, \n the existence of \ntwo important global symmetries of the classical field equations, \n(\\ref{eq1.4}) and (\\ref{eqn2.3}). They are invariant both \nunder global scale transformations, and under a constant shift of $\\phi : ds'^2 \n= a^2 ds^2$, $\\phi' = \\phi + b$. [They are also invariant under local \ncoordinate transformations.] In terms of asymptotic data (whose definition \nrequires a specific ``flat'' normalization of the coordinates at past \ninfinity), the global symmetry transformations read:\n\\bea\n\\label{p1}\n&&f(v,\\theta,\\varphi) \\rightarrow f'(v',\\theta,\\varphi) \n= a \\, f (v'\/a \n,\\theta,\\varphi) \\,,\\\\\n&& f_+(v,\\theta,\\varphi) \\rightarrow f'_+ (v',\\theta,\\varphi) =\n a \\, f_+ (v'\/a ,\\theta,\\varphi)\\,, \\\\\n&&f_{\\times}(v,\\theta,\\varphi) \\rightarrow f'_{\\times} (v',\\theta,\\varphi) = \na \\, f_{\\times} (v'\/a,\\theta,\\varphi)\n\\label{eqn2.9}\n\\end{eqnarray}\nwith $r \\rightarrow r^\\prime = a r$, $v \\rightarrow v^\\prime = a v$, and \n\\beq\n\\phi_0 \\rightarrow \\phi'_0 = \\phi_0 + b \\, .\n\\label{eqn2.10}\n\\end{equation}\nNote that the three dimensionless news functions (\\ref{eqn2.8}) are numerically \ninvariant under the scaling transformations: $N' (v',\\theta,\\varphi) = N \n(v,\\theta,\\varphi)$, etc. with $v' = a \\, v$.\n\nAs the full time evolution  of string vacua depends only on the null data, \nit is a priori clear that the amplitude of the dimensionless news functions \nmust play a crucial r\\^ole in distinguishing ``weak\" incoming fields --that finally \ndisperse again as weak outgoing waves-- from ``strong\" incoming fields that undergo \na gravitational instability. To study this difficult problem in more detail we \nturn in the next section  to a simpler model with fewer degrees of freedom.\n\nBefore doing so, let us comment on the ``classicality'' of an \n``in state'' defined by some news functions (\\ref{eqn2.8}). The classical \nfields (\\ref{eqn2.4}), (\\ref{eqn2.5}) deviate less and less from \na trivial background $(\\phi_0, \\eta_{\\mu \\nu})$ as $r \\rightarrow \n\\infty$ (with fixed $v, \\theta$ and $\\varphi$). One might then worry \nthat, sufficiently near ${\\cal I}^-$, quantum fluctuations \n$\\phi^q$, $h_{\\mu \\nu}^q$ might start to dominate over \nthe classical deviations $\\phi^c \\equiv \\phi(x)-\\phi_0$, \n$h_{\\mu \\nu}^c \\equiv g_{\\mu \\nu} - \\eta_{\\mu \\nu}$. \nTo ease this worry let us compute the mean number of quanta \nin the incoming waves (\\ref{eqn2.4}), (\\ref{eqn2.5}). Consider \nfor simplicity the case of an incoming spherically symmetric dilatonic wave \n$\\phi^c_{\\rm in} = f(v)\/r + o(1\/r)$. Such an incoming wave \nhas evidently the same mean number of quanta as the corresponding \noutgoing \\footnote{In keeping with usual physical intuition, it is \nmore convenient  to work with an auxiliary problem where a classical source \ngenerates an outgoing wave which superposes onto the quantum \nfluctuations of an incoming vacuum state.} \nwave $\\phi^c_{\\rm out}$, with asymptotic \nbehaviour $\\phi_{\\rm out}^c= -f(u)\/r + o(1\/r)$ , on ${\\cal I}^+$ \n($r \\rightarrow \\infty$, with $u \\equiv t-r, \\theta$ and $\\varphi$ \nfixed). As is well known \\cite{IZ}, a classical outgoing \nwave can be viewed, quantum mechanically, as a coherent \nstate in the incoming Fock space defined by the oscillator \ndecomposition of the quantum field $\\widehat{\\phi}$. Instead \nof parametrizing $\\phi^c_{\\rm out}$ by the waveform $f(u)$, \none can parametrize it by an effective source $J(x^\\lambda)$ \nsuch that \n$\\phi^c_{\\rm out}(x) = \\int d^4 x^\\prime \\, P_{\\rm ret}(x - x^\\prime)\\,\nJ(x^\\prime)$, where $P_{\\rm ret}$ is the (action-normalized) \nretarded propagator of the field $\\phi$. [We use a \nnormalization such that $S = -\\frac{1}{2}\\,\\phi\\,P^{-1}\\,\\phi + J\\,\\phi$.]\nThe mean number of quanta in the coherent \nstate associated to a classical source $J(x)$ \nhas been computed in \\cite{IZ} for the electromagnetic case, and \nin \\cite{BD98} for generic massless fields. It reads \n(with $p\\, x \\equiv \\vec{p}\\,\\vec{x} - \\omega\\,t$) \n\\vskip 0.3truecm\n\\beq\n\\overline{n} = \\frac{\\pi}{\\hbar}\\,\\int \\frac{d^4 p}{(2 \\pi)^4}\\,\n\\delta(p^2)\\,\\widehat{J}(-p)\\,{\\cal R}\\,\\widehat{J}(p)\\,, \\quad \\quad \n\\widehat{J}(p) \\equiv \\int d^4 x \\, e^{-i p x}\\, J(x)\\,, \n\\label{n1}\n\\end{equation}\n\\vskip 0.3truecm\n\\noindent\nwhere ${\\cal R}$ denotes the residue of the propagator of the corresponding \nfield, defined such that the field equation $P^{-1}\\,\\phi = J$ reads \n$\\Box \\phi = - {\\cal R}\\,J$. [In the normalization of the present paper \n${\\cal R}=16 \\pi\\,G$.] Inserting $J(x) = -(4\\,G)^{-1}\\,\nf(t)\\,\\delta^{(3)}(\\vec{x})$ into Eq.~(\\ref{n1}) \nyields \n\\vskip 0.3truecm\n\\beq\n\\overline{n} = \\frac{1}{2 G\\hbar} \\, \\int_0^\\omega \\frac{d \\omega}{2 \\pi} \n\\, \\omega\\,|\\widehat{f}(\\omega)|^2 = \n\\frac{1}{2 G\\hbar} \\, \\int_0^\\omega \\frac{d \\omega}{2 \\pi} \n\\, \\frac{1}{\\omega}\\, |\\widehat{N}(\\omega)|^2\\,,\n\\end{equation}\n\\vskip 0.3truecm\n\\noindent\nwhere $\\widehat{N}(\\omega) \\equiv \\int dt \\, e^{i \\omega t}\\, N(t)$ \nis the one-dimensional Fourier transform of the news function \n$N(t) = f^\\prime(t)$. In order of magnitude, if we consider \na pulse-like waveform $f(t)$, with characteristic duration $\\Delta t \n\\sim \\ell_c$, and characteristic dimensionless news amplitude \n$N_c \\sim f_c\/\\ell_c$, the corresponding mean number of quanta is \n\\beq\n\\overline{n} \\sim \\frac{N_c^2\\,\\ell_c^2}{\\ell_P^2} = \n\\frac{N_c^2\\,\\ell_c^2}{g^2\\,\\ell_s^2}\\,.\n\\label{n3}\n\\end{equation}\nIn the present paper, we shall consider incoming news functions with \namplitude $N_c \\sim 1$ and scale of variation $\\ell_c \\gg \\ell_s$. \nMoreover, in order to ensure a sufficient amount of inflation, \nthe initial value of the string coupling $g^2 = e^{\\phi_0}$ \nhas to be $\\ll 1$. Therefore, Eq.~(\\ref{n3}) shows that such incoming states \ncan be viewed as highly classical coherent states,  \nmade of an enormous number  of elementary $\\phi$-quanta: \n$\\overline{n} \\gg\\!\\!> 1$. This quantitative fact clearly shows \nthat there is no need to worry about the effect of quantum fluctuations, \neven near ${\\cal I}^-$ where $\\phi^c_{\\rm in } \\sim f(v)\/r \\rightarrow 0$. \nIn the case of the negative-curvature Friedmann-dilaton solution \nthis fact has been recently confirmed by an explicit calculation \nof scalar and tensor quantum fluctuations \\cite{GPV}.\n\n\\section{Equations for a spherically symmetric pre-big bang}\n\\label{sec3}\nThe simplest nontrivial model which can exhibit the transition between weak, \nstable data, and gravitationally unstable ones is the spherically-symmetric \nEinstein-dilaton system. The null wave data for this system comprise only an \nangle-independent asymptotic dilatonic wave form $f(v)$, corresponding to the \ndimensionless news function\n\\beq\nN(v) = \\partial_v \\, f(v) \\, . \n\\label{eqn3.1}\n\\end{equation}\nThe spherically symmetric Einstein-dilaton system has been thoroughly studied \nby many authors with the strongest analytic results appearing \nin papers by Christodoulou \\cite{chi86}--\\cite{chi93}. \nA convenient system of coordinates is the double null \nsystem, $(u,v)$, such that\n\\beq\n\\phi = \\phi (u,v) \\, , \n\\label{eqn3.2}\n\\end{equation}\n\\beq\nds^2 = - \\Omega^2(u,v)\\,du \\, dv + r^2(u,v)\\,d \\omega^2 \\,,\n\\label{eq1.6}\n\\end{equation}\nwhere $d\\omega^2 = d\\theta^2 + \\sin^2 \\theta \\, d\\varphi^2$. The field \nequations are conveniently re-expressed in terms of the three functions $\\phi \n(u,v)$, $r(u,v)$ and $m(u,v)$, where the local mass function $m(u,v)$ is \ndefined by \\footnote{\nIn order to avoid confusion we indicate, in standard notation, the variable \nkept fixed under differentiation.} \n\\beq\n1 - \\frac{2 m}{r} \\equiv g^{\\mu \\nu}\\,(\\pa_\\mu r)\\,(\\pa_\\nu r) = \n-\\frac{4}{\\Omega^2}\\,\\left (\\frac{\\pa r}{\\pa u}\\right )_v\\,\n\\left (\\frac{\\pa r}{\\pa v}\\right )_u\\,.\n\\label{eq1.7}\n\\end{equation}\nOne gets the following  set of evolution equations for $m$, $r$ and $\\phi$ \n\\bea\n\\label{eq1.8}\n&& 2\\left (\\frac{\\pa r}{\\pa u}\\right )_v\n\\,\\left (\\frac{\\pa m}{\\pa u}\\right )_v = \\left ( 1 - \\frac{2 m}{r} \\right )\n\\,\\frac{r^2}{4}\\,\\left (\\frac{\\partial \\phi}{\\pa u} \\right )_v^2 \\,, \\\\\n\\label{eq1.9}\n&& 2 \\left (\\frac{\\pa r}{\\pa v}\\right )_u\\,\n\\left (\\frac{\\pa m}{\\pa v}\\right )_u = \\left ( 1 - \\frac{2 m}{r} \\right )\n\\,\\frac{r^2}{4}\\,\\left (\\frac{\\partial \\phi}{\\pa v} \\right )^2_u \\,,\\\\\n\\label{eq1.10}\n&& r\\,\\frac{\\pa^2 r}{\\pa u \\pa v} = \n\\frac{2 m}{r-2m}\\,\\left (\\frac{\\pa r}{\\pa u}\\right )_v\\,\n\\left (\\frac{\\pa r}{\\pa v}\\right )_u\\,,\\\\\n\\label{eq1.11}\n&& r\\,\\frac{\\pa^2 \\phi}{\\pa u \\pa v} + \n \\left (\\frac{\\pa r}{\\pa u}\\right )_v \\left (\\frac{\\pa \\phi}{\\pa v}\\right )_u\n+ \\left (\\frac{\\pa r}{\\pa v}\\right )_u \\left (\\frac{\\pa \\phi}{\\pa u}\\right )_v \n=0\\,.\n\\end{eqnarray}\nThis double-null evolution system is form-invariant under independent local \nreparametrizations of $u$ and $v$: $u' = U(u)$, $v' = V(v)$. To freeze this \ngauge freedom it is convenient to require that $u=v$ on the central worldline \nof symmetry $r=0$ (as long as it is a regular part of space-time), and that $v$ \nand $r$ asymptotically behave like ingoing Bondi coordinates on ${\\cal I}^-$ \n(see below).\n\nThe quantity\n\\beq\n\\mu (u,v) \\equiv \\frac{2 m (u,v)}{r}\n\\label{eqn3.3}\n\\end{equation}\nplays a crucial r\\^ole in the problem. \nFollowing \\cite{chi93}, we shall sometimes call it the ``mass \nratio''. If $\\mu$ stays everywhere below 1, the field configuration will \nnot collapse but will finally disperse itself as outgoing waves at infinity. By \ncontrast, if the mass ratio $\\mu$ can reach anywhere the value 1, this signals \nthe formation of an apparent horizon ${\\cal A}$. The location of the apparent \nhorizon is indeed defined by the equation\n\\beq\n{\\cal A}: \\quad \\mu (u,v) = 1 \\, .\n\\label{eqn3.4}\n\\end{equation}\nThe above statements are substantiated by some rigorous inequalities \\cite{chi93}\nstating that:\n\\bea\n\\label{rel1}\n&& \\left (\\frac{\\pa r}{\\pa u}\\right )_v < 0\\,, \\quad \\quad \\quad \\quad \n\\quad \\left (\\frac{\\pa m}{\\pa v}\\right )_u \\geq 0\\,, \\\\\n\\label{rel2}\n&& \\left (\\frac{\\pa r}{\\pa v}\\right )_u\\,\n\\left ( 1- \\mu \\right ) \\geq 0\\,, \\quad \\quad \n\\left (\\frac{\\pa m}{\\pa u}\\right )_v\\,\n\\left ( 1- \\mu \\right ) \\leq 0\\,.\n\\end{eqnarray}\nThus, in weak-field regions ($\\mu < 1$), $(\\partial_v \nr)_u > 0$, while, as $\\mu > 1$, $(\\partial_v r)_u < 0$, meaning that the \noutgoing radial null rays (``photons'') emitted by the sphere $r = {\\rm const.}$ \nbecome convergent, instead of having their usual behaviour. \n\nIt has been shown long ago that the presence of trapped surfaces \nimplies the existence of some (possibly weak) type of \ngeometric singularity \\cite{P65}, \\cite{HP70}. In the case of the spherically \nsymmetric Einstein-dilaton system, it has been possible to prove \n\\cite{chi91} that the \npresence of trapped surfaces (i.e. of an apparent horizon ${\\cal A}$ (\\ref{eqn3.4}), \nthe boundary of the trapped region) implies the existence of a future \nspace-like singular boundary ${\\cal B}$ of space-time where the curvature blows up.\nBoth ${\\cal A}$ and ${\\cal B}$ are ``invisible'' from future null infinity \n(${\\cal I}^+$), being hidden behind an event horizon \n${\\cal H}$ (a null hypersurface). See Fig.~\\ref{fig3}.\n\\begin{figure}\n\\centerline{\\epsfig{file=fig3.ps,width=0.45\\textwidth,angle=-90}}\n\\vskip 0.3truecm\n\\caption{\\sl Schematic representation of the space-time generated \nby the collapse of a spherically symmetric pulse of dilatonic \nwaves. An incoming pulse of scalar news $N(v)$, which grows \nby ${\\cal O}(1)$ on an advanced-time scale $\\ell_i$, collapses \nto a space-like singularity ${\\cal B}$ after having formed \nan apparent horizon ${\\cal A}$ hidden behind the event horizon ${\\cal H}$. \nIn the vicinity of ${\\cal A}$ and ${\\cal H}$, there is an abrupt \ntransition between a weak-field region (dark shading) where the \nperturbation series (\\ref{eqn4.1a})--(\\ref{eqn4.1c}) holds, and \na strong-field one (light shading) where the cosmological-like \nexpansion (\\ref{appen1})--(\\ref{appen3}) holds.}\n\\label{fig3}\n\\end{figure}\n\nOne of our main purposes in this work is to give the conditions that the incoming \ndilatonic news function $N(v)$ must satisfy in order to create an apparent horizon, and \nthereby to lead to some localized gravitational collapse. Before addressing this \nproblem let us complete the description of the toy model by re-expressing it\n in terms of the ingoing Bondi coordinates $(v,r)$. The \nmetric can now be written in the form \n\\beq\n\\label{eq1.17}\nds^2 = - \\left (1 - \\frac{2 m(v,r)}{r} \\right )\\,e^{2 \\beta(v,r)}\\,d v^2 +\n2 e^{\\beta(v,r)}\\, dv \\, dr + r^2 \\, d \\omega^2\\,,\n\\end{equation}\nand the field equations (\\ref{eq1.8})--(\\ref{eq1.11}) become \n\\bea\n\\label{eq1.18}\n&& r\\,\\left (\\frac{\\pa \\beta}{\\pa r}\\right )_v  = \\frac{r^2}{4}\\,\n\\left (\\frac{\\pa \\phi}{\\pa r}\\right )_v^2 \\,,\\\\\n\\label{eq1.19}\n&& 2\\left (\\frac{\\pa m}{\\pa r}\\right )_v = \\left ( 1 - \\frac{2 m}{r} \\right )\\,\n\\frac{r^2}{4}\\,\\left (\\frac{\\partial \\phi}{\\pa r} \\right )^2_v \\,,\n\\end{eqnarray}\n\\beq\n\\label{new}\n2 \\left (\\frac{\\pa m}{\\pa v}\\right )_r = \ne^{-\\beta}\\,\\frac{r^2}{2}\\,\\left [ \\left (\\frac{\\pa \\phi}{\\pa v}\\right )_r^2 + \ne^\\beta\\,\\left ( 1 - \\frac{2 m}{r} \\right )\\, \n\\left (\\frac{\\pa \\phi}{\\pa r}\\right )_v\\,\n\\left (\\frac{\\pa \\phi}{\\pa v}\\right )_r \\right ]\\,,\n\\end{equation}\n\\beq\n\\label{eq1.20}\n2 \\left [ \\frac{\\pa^2 \\phi}{\\pa v \\pa r } + \\frac{1}{r}\\,\n \\left (\\frac{\\pa \\phi}{\\pa v}\\right )_r \\right ] + e^\\beta \\left (1-\\frac{2m}{r} \\right )\\,\n \\left ( \\frac{\\pa^2 \\phi}{\\pa r^2}\\right )_v  +{ 2e^\\beta \\over r} \\,\n\\left (1-\\frac{m}{r} \\right)\n \\left (\\frac{\\pa \\phi}{\\pa r} \\right )_v  = 0\\,.\n\\end{equation}\nEqs. (\\ref{eq1.18}), (\\ref{eq1.19}) imply: \n\\beq\n\\pa_r \\left (e^{\\beta(v,r)}\\, ( r - 2 m ) \\right )= e^{\\beta(v,r)}\\,.\n\\end{equation} \nIn this coordinate system one can solve for the two functions $\\beta (v,r)$ and \n$\\mu (v,r)$ (or $m (v,r)$) by quadratures in terms of the third unknown \nfunction $\\phi (v,r)$. Indeed, imposing that the coordinate \nsystem be asymptotically flat at ${\\cal I}^-$ (i.e. that $\\beta (v,r) \n\\rightarrow 0$ as $ r \\rightarrow \\infty$), one first finds from \nEq.~(\\ref{eq1.18}) \n\\beq\n\\beta (v,r) = - \\int_r^{+\\infty} \\frac{dr'}{r'} \\, F (v,r') \\, ,\n\\label{eqn3.5}\n\\end{equation}\nwhere \n\\beq\n\\label{eq1.24}\nF(v,r) \\equiv \\frac{r^2}{4}\\,\\left ( \\frac{\\pa \\phi}{\\pa r} \\right )_v^2\\,.\n\\end{equation}\nEq.~(\\ref{eq1.19}) can then be integrated to give \n\\beq\n\\mu(v,r) = \\frac{\\left [ 2m_{\\infty}(v) - \n\\int_r^{+\\infty} dr^\\prime \\, F(v,r^\\prime)\\,e^{\\beta(v,r^\\prime)} \n\\right ]}{r \\,e^{\\beta(v,r)}}\\,,\n\\label{eqn3.6}\n\\end{equation}\nwhere the ``integration constant''\n\\bea\nm_{\\infty} (v) \\equiv \\lim_{r \\rightarrow \\infty} m (v,r) \n\\end{eqnarray}\ndenotes the incoming Bondi mass. By the Bondi energy-flux formula \n(\\ref{eqn2.7}), which is also the limiting form of (\\ref{new}) for \n$r \\rightarrow \\infty$, \n$m_{\\infty} (v)$ is given in terms of the news function by\n\\beq\nm_{\\infty} (v) = \\frac{1}{4} \\int_{-\\infty}^v dv' \\, N^2 (v') =  \\, \\frac{1}{4} \n\\int_{-\\infty}^v dv' \\left( \\frac{\\partial f (v')}{\\partial v'} \\right)^2 \\,,\n\\label{eqn3.7}\n\\end{equation}\nwhere we have inserted the asymptotic behaviour of $\\phi$ on ${\\cal I}^-$,\n\\beq\n\\phi (v,r) = \\phi_0 + \\frac{f(v)}{r} + \n{o} \\left( \\frac{1}{r} \\right) \\,,\n\\label{eqn3.8}\n\\end{equation}\nand assumed  that $N(v)$ decays sufficiently fast as $v \\rightarrow \n-\\infty$ (i.e. near past time-like infinity). \n{}From Eqs.~(\\ref{eqn3.5}) and (\\ref{eqn3.6}) we find the leading asymptotic \nbehaviour of the metric coefficients to be\n\\beq\n1 - \\frac{2m (v,r)}{r} = 1 - \\frac{2 m_{\\infty} (v)}{r} +  {o} \\left( \n\\frac{1}{r} \\right) \\, ,\n\\label{eqn3.9a}\n\\end{equation}\n\\beq\ne^{\\beta (v,r)} = 1 + {\\cal O} \\left( \\frac{1}{r^2} \\right) \\, .\n\\label{eqn3.10}\n\\end{equation}\n\n\\section{Data Strength criterion for gravitational instability}\n\\label{sec4}\nThe purpose of this section is to outline the condition that the initial \ndata, i.e. the wave form $f(v)$ or the news function $N(v) \\equiv f' (v)$, must \nsatisfy in order to undergo, or not undergo, gravitational collapse. A lot of \nmathematical work has been done on this issue \\cite{chi86}--\\cite{chi93}. \nIn particular, Refs.~\\cite{chi86} and \\cite{chi93} \ngave two different ``no collapse'' criteria \nensuring that the field \nconfiguration never collapses and finally disperses out as weak outgoing waves\nif some functional of the data is small enough. \nOn the other hand, Ref.~\\cite{chi91} gave a sufficient ``collapse'' criterion \nensuring that the field \nconfiguration will collapse, if some functional of the data is large enough,\nthereby giving birth to a curvature singularity \nhidden behind a horizon. The problem with these nice and important \nmathematical results is threefold: (i) these criteria are \nsufficient but not necessary, so that they cannot answer our problem of finding \n(if possible) a sharp criterion distinguishing weak-field from strong-field data; \n(ii) the various measures of the ``strength'' of the data given in \nRefs.~\\cite{chi86}, \\cite{chi91} and \\cite{chi93} are quite unrelated to each \nother and do not point clearly toward any sharp ``strength criterion''; and \n(iii) these criteria are not expressed in terms of the asymptotic\nnull data $N(v)$.\n\nOur aim here is not to compete with Refs.~\\cite{chi86}, \\cite{chi91}, \n\\cite{chi93} on the grounds of mathematical rigour, but to complement these \nresults by a non rigorous study which leads to the iterative computation of a \nsharp strength criterion, i.e. a functional ${\\cal S} [f]$, such that data \nsatisfying ${\\cal S} < 1$ finally disperse out at infinity, while data \nsatisfying ${\\cal S} > 1$ are gravitationally unstable and partly collapse to \nform a singularity. We will then compare our strength functional ${\\cal S}$ \nwith the rigorous, but less explicit results of \\cite{chi91}. Note that the \nemphasis here is not on the question of whether the created singularity is hidden \nbeyond a global event horizon when seen from outside, but rather on the \nquantitative criterion ensuring that past trivial data give birth to a \nspace-like singularity which we shall later interpret as a budding pre-Big Bang \ncosmological universe.\n\n\\subsection{ Perturbative analysis in the weak-field region}\n\\label{subsec3.1}\n\nIn view of the discussion of the previous section, the transition between \n``weak\" and ``strong\" fields is sharply signaled by the appearance of an \napparent horizon, i.e. of space-time points where $\\mu \\equiv 2m\/r$ reaches the \n``critical\" value 1. Therefore, starting from the weak-field region near ${\\cal \nI}^-$, we can define the strength functional by computing $\\mu (u,v)$ as a functional \nof the null incoming data and by studying if and when it can exceed the critical value. \nSpecifically, we \ncan set up a perturbation analysis of the Einstein-dilaton system in some \nweak-field domain connected to ${\\cal I}^-$, and see for what data it suggests \nthat $\\mu$ will exceed somewhere the value 1. The expected domain of validity \nof such a weak-field expansion is sketched in Fig.~\\ref{fig3}.\n\nThe $(u,v)$ system, Eqs.~(\\ref{eqn3.2}) and (\\ref{eq1.6}),\nlends itself easily to a perturbative treatment in the \nstrength of $\\phi$. We think here of introducing a formal parameter, say \n$\\lambda$, in the asymptotic data by $f(v) \\rightarrow \\lambda \\, f(v)$ and of \nconstructing the full solution $\\phi (u,v)$, $r(u,v)$, $m(u,v)$, as a (formal) \npower series in $\\lambda$ (or by some better iteration method):\n\\bea\n\\label{eqn4.1a}\n\\overline{\\phi} (u,v) &\\equiv& \\phi (u,v) - \\phi_0 = \\lambda \\, \\phi_1 + \n\\lambda^3 \\phi_3 + \\lambda^5 \\phi_5 + \\ldots \\,, \\\\\n\\label{eqn4.1b}\nr(u,v) &=& \\frac{1}{2} \\, (v-u) + \\lambda^2 \\, r_2 + \\lambda^4 \\, r_4 + \\ldots \n\\,, \\\\\nm(u,v) &=& \\lambda^2 \\, m_2 + \\lambda^4 \\, m_4 + \\ldots \\, .\n\\label{eqn4.1c}\n\\end{eqnarray}\nIndeed, knowing $\\phi$ to order $\\lambda^{2p-1}$ and $r$ to order \n$\\lambda^{2p-2}$ we can use Eq.~(\\ref{eq1.8}) or (\\ref{eq1.9}) to compute by \nquadrature $m$ to order $\\lambda^{2p}$. We can then  rewrite Eqs.~(\\ref{eq1.10}), \n(\\ref{eq1.11}) as\n\\beq\n\\partial_{uv} (r \\, \\overline{\\phi}) = \\omega \\,r \\, \\overline{\\phi}\\,, \n\\quad \\quad \\partial_{uv} \\, r =\\omega \\, r \\,, \n\\label{eqn4.2}\n\\end{equation}\nwhere\n\\beq\n\\omega \\equiv \\frac{\\mu}{1-\\mu} \\, \\frac{\\partial_u r \\, \\partial_v r}{r^2} \\, , \n\\label{eqn4.3}\n\\end{equation}\nand therefore the right-hand sides of Eqs.~(\\ref{eqn4.2}) are known to \n(relative) order $\\lambda^{2p}$. Solving Eq.~(\\ref{eqn4.2}), then yields \n$\\overline{\\phi}$ and $r$ to the next order in $\\lambda$ ($\\lambda^{2p+1}$ and \n$\\lambda^{2p}$, respectively). To be more explicit, Eqs.~(\\ref{eqn4.2}) \nare of the form\n\\beq\n\\partial_{uv} \\, \\psi(u,v) = \\sigma (u,v) \\, , \n\\label{eqn4.4}\n\\end{equation}\nwhere the ``source'' $\\sigma = \\omega \\psi$ is known at each order in $\\lambda$ \nfrom the lower-order expression of $\\mu$ and $\\psi$, and where the ``field'' \n$\\psi (u,v)$ is restricted to vanish on the central worldline of symmetry $v=u$ \n(where $r=0$, and $\\overline{\\phi}$ is regular in the weak-field domain). The \ngeneric Eq.~(\\ref{eqn4.4}) can then be solved by introducing the retarded \nGreen function\n\\bea\nG (u,v ; u',v') = \\theta (u-u') \\, \\theta (v-v') - \\theta (v-u') \\, \\theta \n(u-v')  \\, ,\n\\end{eqnarray}\n\\beq\n\\partial_{uv} \\, G (u,v ; u',v') = \\delta (u-u') \\, \\delta (v-v') \\, . \n\\label{eqn4.5}\n\\end{equation}\nThis  is the unique Green function in the domain $v \\geq u$, $v' \\geq u'$ which \nvanishes at $v = u$, and whose support in the source point \n$(u',v')$ lies in the past of the field point $(u,v)$. The general solution of \n(\\ref{eqn4.4}) can then be written as\n\\beq\n\\psi (u,v) = \\psi_{\\rm in} (u,v) + \\int \\int_{v' \\geq u'} du' \\, dv' \\, G (u,v \n; u',v') \\, \\sigma (u',v') \\, ,\n\\label{eqn4.6}\n\\end{equation}\nwhere $\\psi_{\\rm in} (u,v)$ is the free ``incoming'' field, satisfying \n$\\partial_{uv} \\, \\psi_{\\rm in} = 0$ and vanishing when $v=u$, i.e.\n\\beq\n\\psi_{\\rm in} (u,v) = g_{\\rm in} (v) - g_{\\rm in} (u) \\, .\n\\label{eqn4.7}\n\\end{equation}\nThe incoming waveform $g_{\\rm in} (v)$ is uniquely fixed by the incoming data \non ${\\cal I}^-$. For instance, for $r(u,v)$ asymptotic flatness gives $r_{\\rm \nin} (u,v) = \\frac{1}{2} \\, (v-u)$ (we recall that in flat space $v = t+r$ and \n$u=t-r$), while for $\\overline{\\phi} (u,v) \\equiv \\phi (u,v) - \\phi_0$ the \nasymptotic expansion (\\ref{eqn3.8}) (in which $f(v)$ was supposed to vanish as \n$v \\rightarrow -\\infty$) yields $\\overline{\\phi}_{\\rm in} (u,v) = 2(f(v) - f(u))\/(v-u)$.\n\nIn principle, this perturbative algorithm allows one to compute the mass ratio \n$\\mu (u,v) = 2m\/r$ to any order in $\\lambda$:\n\\beq\n\\mu (u,v) = \\lambda^2 \\, \\mu_2 + \\lambda^4 \\, \\mu_4 + \\ldots\\,.\n\\label{eqn4.8}\n\\end{equation}\nEvidently, the convergence of this series becomes very doubtful when $\\mu$ can \nreach values of order unity, which is precisely when an apparent horizon is \nformed. One would need some resummation technique to better locate the apparent \nhorizon condition $\\mu (v,u) = 1$. However, we find interesting to have, in \nprinciple, a way of explicitly computing successive approximations to \nthe possible location of an apparent horizon, starting only from the incoming \nwave data $f(v)$. \n\n\\subsection{Strength criterion at quadratic order} \n\\label{subsec3.2}\nLet us compute explicitly the lowest-order approximation to \n(\\ref{eqn4.8}) (we henceforth set \n$\\lambda = 1$ for simplicity). It is obtained by inserting the zeroth-order result $r_0 (u,v) \n= \\frac{1}{2} \\, (v-u)$, and the first-order one for $\\phi$\n\\beq\n\\overline{\\phi}_1 (u,v) = \\phi_1 (u,v) - \\phi_0 = \\frac{f(v) - f(u)}{r_0} =  \n\\, \\frac{2[f(v) - f(u)]}{ v-u}\n\\label{eqn4.9}\n\\end{equation}\ninto Eq.~(\\ref{eq1.8}) or (\\ref{eq1.9}). This yields (with $f' (v) \\equiv \n\\partial f \/ \\partial v$)\n\\bea\n\\left( \\frac{\\partial m_2}{\\partial v} \\right)_u = r_0^2 \\left( \\frac{1}{2} \\, \n\\frac{\\partial \\phi_1}{\\partial v} \\right)^2 = \\frac{1}{4} \\left [ f'(v) - \n\\frac{f(v) - f(u)}{v-u} \\right ]^2 \\,,\n\\end{eqnarray}\n\\beq\n\\left( \\frac{\\partial m_2}{\\partial u} \\right)_v = - r_0^2 \\left( \\frac{1}{2} \n\\, \\frac{\\partial \\phi_1}{\\partial u} \\right)^2 = -\\frac{1}{4} \\left [ f'(u) -\n\\frac{f(v) - f(u)}{v-u} \\right ]^2 \\, .\n\\label{eqn4.10}\n\\end{equation}\nThe compatibility of the two equations is easily checked, e.g. by using\n\\beq\n\\partial_v (r_0^2 (\\partial_u \\phi_1 )^2) = r_0 \\, \\partial_u \\phi_1 \\, \n\\partial_v \\phi_1 = - \\partial_u (r_0^2 (\\partial_v \\phi_1)^2) \\, ,\n\\label{eqn4.11}\n\\end{equation}\nwhich expresses the ``conservation'' of the $\\phi$-energy tensor in the $(u,v)$ \nplane. Noting that $m(u,v)$ must vanish (by regularity of $r=0$ in the \nweak-field domain) at the center $v=u$ one gets, by quadrature, the following \nexplicit result for $m_2$:\n\\beq\nm_2 (u,v) = \\widetilde m (u,v) - \\frac{1}{8} \\ r_0 \\, \\overline{\\phi}_1^2 \\, ,\n\\label{eqn4.12}\n\\end{equation}\nwhere \n\\beq\n\\widetilde m (u,v) \\equiv \\frac{1}{4} \\int_u^v f'^2 (x) \\, dx \\, .\n\\label{eqn4.13}\n\\end{equation}\nNote that Eq.~(\\ref{eqn4.12}) decomposes $m_2 (u,v)$ in a ``conserved'' piece \n$\\widetilde m (u,v) = g(v) - g(u)$ (with $\\partial_{uv} \\widetilde m = 0$) and \na $\\phi$-dependent one. This is an integrated form of the local \n``conservation'' law (\\ref{eqn4.11}) of the $\\phi$-energy tensor. Using \n(\\ref{eqn4.12}) one gets a very simple result for the quadratic approximation \nto the mass ratio $\\mu = 2m\/r$, namely\n\\beq\n\\mu_2 (u,v) = \\frac{1}{v-u} \\int_u^v dx \\, f'^2 (x) - \\left [ \n\\frac{f(v) - f(u)}{v-u} \\right ]^2 \\, .\n\\label{eqn4.14}\n\\end{equation}\nLet us introduce a simple notation for the {\\it average} of any function $g(x)$ \nover the interval $[u,v]$:\n\\beq\n\\langle g \\rangle_{[u,v]} \\equiv \\langle g (x) \\rangle_{x \\in [u,v]} \\equiv \n\\frac{1}{v-u} \\int_u^v dx \\, g(x) \\, .\n\\label{eq3.8}\n\\end{equation}\nThen the mass ratio, at quadratic order, can be simply written in terms of the \nscalar news function $N(v) \\equiv f'(v)$:\n\\bea\n\\mu_2 (u,v) &= &\\langle N^2 \\rangle_{[u,v]} - (\\langle N \\rangle_{[u,v]})^2\\,, \\\\\n&= &\\langle (N(x) - \\langle N \\rangle_{[u,v]})^2 \\rangle_{x \\in [u,v]}\\,, \\\\\n&\\equiv &{\\rm Var} \\, (N(x))_{x \\in [u,v]} \\, ,\n\\label{eqn4.15}\n\\end{eqnarray}\nwhere ${\\rm Var} \\, (g)_{[u,v]}$ denotes the ``variance'' of the function \n$g(x)$ over the interval $[u,v]$, i.e. the average squared deviation from the \nmean.\n\nAs indicated above, having obtained $\\mu$ to second order, say $\\mu_2 = V(u,v) \n\\equiv {\\rm Var} \\, (N)_{[u,v]}$, one can then proceed to compute $r$ and \n$\\phi$ to higher orders. Namely, from Eq.~(\\ref{eqn4.6})\n\\beq\n2r(u,v) = v-u - G * \\left [ \\frac{V(u,v)}{v-u} \\right ] + {\\cal O} (\\lambda^4) \\,, \n\\end{equation}\n\\beq\n(r\\phi)(u,v) = f(v) - f(u) - G * \\left [ V(u,v) \\, \\frac{f(v) - f(u)}{(v-u)^2} \n\\right ] + {\\cal O} (\\lambda^5) \\,, \n\\end{equation}\nwhere the star denotes a convolution. Then, one can get from these results \n$m_4$, and thereby $\\mu_4$, by simple quadratures. In the following we shall \nonly use the quadratic approximation to $\\mu$, though we are aware that when \nthe initial data are strong enough to become gravitationally unstable, higher \norder contributions to $\\mu$ are probably comparable to $\\mu_2$.\n\nFinally, at quadratic order, we can define the strength of some initial data \nsimply as the supremum of the variance of the news over arbitrary intervals:\n\\beq\n{\\cal S}_2 = \\sup \\mu_2 = \\sup_{u,v \\atop u \\leq v} \\, {\\rm Var} (N(x))_{x \\in [u,v]} \\, .\n\\label{eqn4.16}\n\\end{equation}\nAt this order ${\\cal S}_2 < 1$ means that $\\mu_2 (u,v)$ stays always below one, \ni.e. that no trapped region is created, and that the field is expected to \ndisperse, while ${\\cal S}_2 > 1$ signals the formation, somewhere, of trapped \nspheres, and therefore (by the results of \\cite{chi91}) the formation of a \nsingularity. To be more exact we should actually define,  in analogy with\n(\\ref{eqn4.16}),\n a quantity ${\\cal S}_{2p} = \\sup \\sum_{k=1}^p \\mu_{2k}$, \nat any finite order in\nthe weak field expansion, and state  our collapse criterion as the inequality:\n\\beq\n{\\cal S}_{2p} > C_{2p}\\,, \\quad \\quad C_{2p} \\stackrel{p\\rightarrow \\infty}{\\rightarrow} 1\\,.\n\\label{gencrit}\n\\end{equation}\nIn practice, even if\n we do not expect it to be \nquantitatively exact, we will use  the criterion (\\ref{gencrit}) with $p=1$, taking\n$C_2 =1$,  hoping that it will be qualitatively correct in \ncapturing the features of the news function which are generically \nimportant for producing a gravitational collapse. Let us note that the \nfunctional ${\\cal S}_2$ is (like $N$) dimensionless (and therefore scale \ninvariant) and that it is nonlocal. We note also that it is invariant under a \nconstant shift of $N$, $N(v) \\rightarrow N(v) + a$, which corresponds to adding \na linear drift in $f : f(v) \\rightarrow f(v) + av + b$. Such a shift is \n(formally) equivalent, in view of Eq.~(\\ref{eqn4.9}), to a constant shift of \n$\\phi_0 \\rightarrow \\phi_0 + 2a$.\n\nThe non locality of ${\\cal S}_2$ is physically interesting because it indicates \nthat it is not the instantaneous level of the energy flux $N^2 (v)$ which \nreally matters, but rather the possibility of having a flux which varies by $\\sim$ \n100\\% over some interval of advanced time. This non locality defines also some \ncharacteristic scales associated with the collapse (when ${\\cal S}_2 > 1$). \nIndeed, if, by causality, we consider increasing values of $v$, and define \n${\\cal S}_2 (v) = {\\displaystyle \\sup_{u,u \\leq v}} {\\rm Var} (N)_{[u,v]}$, the \nfirst value of $v$ for which ${\\cal S}_2 (v)$ exceeds $1$ defines an advanced \ncharacteristic time when the collapse occurs, and the corresponding maximizing \ninterval $[u,v]$ defines a characteristic time scale of (in)homogeneity. [Note \nthat $\\mu_2 (u,v)$, Eq.~(\\ref{eqn4.15}), vanishes when $u=v$, and vanishes also \ngenerally when $u \\rightarrow -\\infty$ if $N(u)$ tends to a limit at \n$-\\infty$.]\n\nLet us also mention that our strength functional (\\ref{eqn4.16}) is \nsuperficially similar to the one recently introduced by Christodoulou \n\\cite{chi93} to characterize sufficiently weak (i.e. non collapsing) data. \nIn the lowest order \napproximation his criterion measures the strength of the data \nby ${\\cal V} = TV (N(x))$, where $TV$ denotes the total variation (i.e. \nessentially ${\\cal V} = \\int_{-\\infty}^{+\\infty} dv \\vert N' (v) \\vert$). Like \nour variance, this is a measure of the variation of $N(v)$ with, however, a \ncrucial difference. If ${\\cal V}$ were a good criterion for measuring the \nstrength of possibly strong data, we would conclude that a news function which \noscillates with a very small amplitude for a very long time will be \ngravitationally unstable, while our criterion indicates that it will not, which \nseems physically more plausible. We hope that our strength functional \n(\\ref{eqn4.16}) will suggest new gravitational stability theorems to \nmathematicians.\n\\subsection{Comparison with a collapse criterion of Christodoulou}\n\\label{subsec3.3}\nTo check the reasonableness of our quadratic strength criterion (\\ref{eqn4.16}), (\\ref{gencrit}) \nwe have compared it with the rigorous, but only sufficient, collapse criterion \nof Christodoulou \\cite{chi91}, and we have applied it to two simple exact \nsolutions. Ref.~\\cite{chi91} gives the following sufficient criterion on the strength of \ncharacteristic data considered at some finite retarded time $u$\n\\beq\n\\frac{2 \\Delta m}{\\Delta r }  \\geq \n\\left [ \\frac{r_1}{r_2}\\,\\log \\left (\\frac{r_1}{2 \\Delta r} \\right )\n+ \\frac{6r_1}{r_2} -1 \\right ]\\,,\n\\label{eq2.2}\n\\end{equation}\nwhere $r_1 \\leq r_2, r_2 \\leq 3r_1\/2$, are two spheres, \n$\\Delta r = r_2 - r_1$ is the width of the ``annular'' region between the two \nspheres, and $\\Delta m = m_2 - m_1 \\equiv m(u,r_2) - m (u,r_1)$ is the mass \n``contained'' between the two spheres, i.e. more precisely the energy flux \nthrough the outgoing null cone $u = {\\rm const.}$, between $r_1$ and $r_2$. \nExplicitly, from Eq.~(\\ref{eq1.19}) in $(u,r)$ coordinates, we have \n\\beq\n2 \\Delta m = \\int_{r_1}^{r_2} dr \\, \\frac{r^2}{4}\\,\n\\left (1 - \\frac{2m}{r} \\right )\\,\n\\left (\\frac{\\pa \\phi}{\\pa r} \\right )^2_{u}\\,.\n\\label{eq2.1}\n\\end{equation}\nWe can approximately express this criterion in terms of the incoming \nnull data $N(v) = f'(v)$ if we assume that the outgoing \ncone $u = {\\rm const.}$ is in the weak-field domain, so that \nwe can replace $\\phi$ on the R.H.S. of Eq.~(\\ref{eq2.1})\nby $\\phi = r^{-1} [f(u+2r) - f(u)]$. Then the  \nL.H.S. of Eq.~(\\ref{eq2.2}) becomes (at quadratic order)\n\\beq\n\\frac{2 \\Delta m}{\\Delta r} = \\frac{M(u,v_2) - M(u,v_1)}{v_2-v_1}\n\\end{equation}\nwhere \n\\beq\n\\label{mm}\nM(u,v) \\equiv (v-u)\\,{\\rm Var}(N)_{[u,v]} \n\\equiv \\int_u^v d x\\, {\\left ( N(x) - \\langle N \\rangle_{[u,v]} \\right )}^2\\,.\n\\end{equation}\nTherefore, in this approximation, the criterion of Ref.~\\cite{chi91} \nbecomes (with $v_0 = u$)\n\\beq\n\\exists v_0, v_1, v_2 : \\quad \\frac{M(v_0,v_2) - M(v_0,v_1)}{v_2-v_1}\n\\geq C(v_0,v_1,v_2) \\,,\n\\label{chi}\n\\end{equation}\nwith the constraints $2 v_2 - 3v_1 \\leq -v_0$, \n$v_1 \\leq v_2$, and the definition \n\\beq\nC(v_0,v_1,v_2) \\equiv  \\frac{v_1-v_0}{v_2 -v_0}\\, \n\\log \\left (\\frac{v_1-v_0}{2(v_2 -v_1)} \\right ) + \n\\frac{6(v_1-v_0)}{v_2 -v_0} -1\\,.\n\\end{equation}\nIf $v_0$ is at the boundary of its allowed domain, \ni.e. if $v_0 = 3v_1 - 2v_2$, this criterion is fully compatible \nwith ours since $C(v_0,v_1,v_2)=3$ implying through Eq.~(\\ref{mm}) \n\\beq\n{\\var}(N)_{[v_0,v_2]} \\geq 1 + \\frac{2}{3}\\,{\\var}(N)_{[v_0,v_1]} \\geq 1\\,.\n\\end{equation}\nIn the opposite case ($v_0$  large and negative) we think that  the \ncriterion (\\ref{chi}) is also compatible, for generic news functions,\n with  the general \nform suggested  above, i.e. \n\\beq\n\\exists \\, v_1^\\prime  , v_2^\\prime : \\quad  \n{\\var}(N)_{[v_1^\\prime,v_2^\\prime]} = \\langle N^2 \\rangle_{[v_1^\\prime,v_2^\\prime]}\n- \\langle N \\rangle_{[v_1^\\prime,v_2^\\prime]}^2 \\geq C_2\\,,\n\\label{eqn4.22}\n\\end{equation}\nwith some positive constant $C_2$ of order unity.\nWe first note that one should probably impose the physically \nreasonable condition that $N(v)$ decays \\footnote{\nActually, when comparing our criterion \n(based on asymptotic null data) to that of Ref.~\\cite{chi93}\n(based on characteristic data taken at a finite retarded \ntime $u = v_0$) we can consider, without loss of generality, \nthat the news function $N(v)$ vanishes identically for $v \\leq v_0$.} \nsufficiently fast as \n$v \\rightarrow - \\infty$ (faster than $|v|^{-1\/2}$) \nto ensure that the integrated incoming \nenergy flux $\\int_{-\\infty}^{v_0} dv \\, N^2(v)$ is finite. [This \nconstraint freezes the freedom to shift $N(v)$ by a constant.]\n\nWhen $-v_0$ is large, the question remains, however, to know \nhow fast the function  $N(v)$ decays when $v_1-v$ (for $v_0 < v < v_1$)\nbecomes much larger than $v_2-v_1$. Let us first consider the physically \ngeneric case where $N(v)$ decays in a reasonably fast manner, say faster \nthan a power $|v|^{-\\kappa}$ with $\\kappa$ of order unity.\nThen choosing $v_0$ in the criterion (\\ref{chi}) just large enough \nto allow one to neglect $N(v_0)$ with respect to $N(v)$, \nsay $(v_1-v_0) \\sim (v_2-v_1) \\times 10^{1\/\\kappa}$, \ncorresponding to $N(v_0) = 0.1\\,N(v_1)$, the L.H.S. \nof Eq.~(\\ref{chi}) becomes approximately \n\\beq \n\\frac{2 \\Delta m}{\\Delta r} \\simeq \\langle N^2 \\rangle_{[v_1,v_2]}\\,,\n\\label{eqn4.21}\n\\end{equation}\nwhile the function $C$ on the R.H.S., which grows only \nlogarithmically with $v_0$, becomes  \n\\beq\nC(v_0,v_1,v_2) \\simeq \\log \\left (\\frac{v_1-v_0}{2(v_2-v_1)}\\right ) +5\\,.\n\\end{equation}\nTherefore, a function of the type $N_\\kappa(v) \n\\sim c\\left [ (v_2-v)\/(v_2-v_1)\\right]^{-\\kappa}$ (when $ v \\leq v_1 < v_2$) with \n$c^2 \\,\\raise 0.4ex\\hbox{$>$}\\kern -0.7em\\lower 0.62ex\\hbox{$\\sim$}\\, 5 + (\\log 10)\/\\kappa$ will fullfil the criterion \n(\\ref{chi}). To see whether this is compatible with \nour criterion (\\ref{eqn4.22}) we have studied (analytically and numerically) \nthe variance, over arbitrary intervals $[v_1^\\prime, v_2^\\prime]$, \nof the function $N_{\\kappa}(v)$. We found that the inequality \n(\\ref{eqn4.22}) is satisfied for a constant $C_2^{(\\kappa)}$ \nwhich is of order unity if $\\kappa$ stays of order unity. \n[For instance, in the extreme case $\\kappa = 1\/2$ corresponding \nto a logarithmically divergent incoming energy flux,  \nwe find $C_2^{(1\/2)} \\sim 0.2$]. We note, however, \nthat if one considers extremely slow decays \nof $N(v)$, i.e. very small exponents $\\kappa \\rightarrow 0$ \n(completed by a faster decay, or an exact vanishing, before some large  \nnegative cutoff $v_c$ ), the constant $C^{(\\kappa)}_2$ needed \non the R.H.S. of the variance criterion (\\ref{eqn4.22}) tends to zero. \nThis signals a limitation of applicability of our simple \nvariance criterion based only on the \nquadratic-order approximation $\\mu = \\mu_2 + \\mu_4 + \\cdots \\simeq \n\\mu_2$. Indeed, one can check that, in such an extreme situation \n$\\mu_2$ is abnormally cancelled, while $\\mu_4$ will be of order unity. \nHowever, we believe that for generic, non extremely slowly \nvarying news functions, the simple ``rule of thumb'' \n(\\ref{eqn4.22}) is a reasonable approximation to the (unknown) \nexact collapse criterion. To further check the reasonableness \nof our criterion (\\ref{eqn4.16}) we  \nturn our attention to some exact solutions. \n\n\\subsection{Exact solutions}\n\\label{sec3.4}\nA first exact, dynamical Einstein-dilaton \nsolution is defined  by a news function consisting\nof  the simple step function\n\\beq\nN(v) = p \\, \\theta (v) \\, . \n\\label{eqn4.18}\n\\end{equation}\nThe corresponding solution  has been independently \nderived by many authors \\cite{japanese}. Here, $p$ is a real parameter and the value $|p|=1$ \ndefines the threshold for gravitational instability: no singularity \noccurs for $|p| < \n1$, while $|p| \\geq 1$ causes the birth of singularity. The metric for this \nsolution takes the form (note that $\\Omega = 1$) \n\\beq\n\\label{eq5.1}\nds^2 = -du\\,dv + r^2\\,d \\omega^2 \\,,\n\\end{equation}\nwith the solutions for  $\\phi$, $r$ and  $m$ given as follows: \n\\beq\n\\label{eq5.2}\n\\phi = 0\\,, \\quad \\quad r = \\frac{1}{2}(v-u) \\,, \\quad \\quad m =0\\,, \n\\quad \\quad \\mbox{ for $v < 0$}\\,,\n\\end{equation}\nwhile, for $v \\geq 0$,\n\\bea\n\\label{eq5.3}\n&& \\phi(u,v) = - \\log \\left [ \\frac{(1-p)\\,v -u}{(1+p)\\,v -u} \\right ]\\,,\\\\\n\\label{eq5.4}\n&& r^2(u,v) = \\frac{1}{4}\\,\\left [ (1-p^2)\\,v^2 - 2v\\,u + u^2 \\right ] = \n\\frac{1}{4}\\,\\left [(1-p)\\,v -u \\right ]\\, \n\\left [(1+p)\\,v -u \\right ]\\,,\\\\\n\\label{eq5.5}\n&& m(u,v) = -\\frac{p^2\\,u\\,v}{8 r}\\,,\\\\\n\\label{eq5.6}\n&& \\mu(u,v) = -\\frac{p^2\\,u\\,v}{\\left [ (1-p^2)\\,v^2 - 2v\\,u +u^2 \\right ] }\\,.\n\\end{eqnarray}\nLet us note that the perturbation-theory value of $\\mu$ for $v > 0$, is \n\\beq\n\\mu_2{(u,v)} = {\\rm Var} \\, [p \\, \\theta (x)]_{[u,v]} = - \n\\frac{p^2\\,u\\,v}{(v-u)^2} \\, \\theta (-u) \\, .\n\\label{eq5.7}\n\\end{equation}\nThe maximum value of $\\mu_2$,  reached for  $u = -v$, is $p^2 \/ 4$. This \nshows that, as expected, the quadratic order criterion (\\ref{eqn4.16}) is \nonly valid as an order of \nmagnitude, but is quantitatively modified by higher-order corrections. This \nexample, and a related general theorem of Christodoulou \\cite{chi93}, suggest \nthat, if the exact criterion were of the type ${\\cal S}_2 = C_2$, the constant \n$C_2$ \nshould equal $\\frac{1}{4}$. (Note that this value is also compatible with the constant \n$C_2^{(1\/2)} \\sim 0.2$ appropriate to $|v|^{-1\/2}$ decay.) \n\nA second exact solution \\cite{CLW}, \\cite{TW} is a negative-curvature Friedmann-like homogeneous \nuniverse. It is defined by the following null data on ${\\cal I}^-$ (for $v < \n0$)\n\\beq\nf(v) = - \\frac{\\sqrt 3}{v} \\ , \\quad \\quad  N(v) = \\frac{\\sqrt 3}{v^2} \\, .\n\\label{eqn4.19}\n\\end{equation}\nHere we view this solution as defined by incoming wave data in a flat Minkowski \nbackground. Actually the data are regular only in some advanced cone $v = T + R \n< 0$, and blow up when $v = 0$. Here, $T$ and $R = \\sqrt{X^2 + Y^2 + Z^2}$ are \nusual Minkowski-like coordinates in the asymptotic past. In terms of such \ncoordinates the exact solution reads (when $T^2 - R^2 < 1$)\n\\beq\n\\label{4.43}\nds^2 = \n\\left [ 1 - \\frac{1}{(T^2 - R^2)^2} \\right] [-dT^2 + dX^2 + dY^2 + dZ^2]\\,,\n\\end{equation}\n\\beq\n\\label{4.44}\n\\phi = - \\sqrt 3 \\, \\log \\left( \\frac{T^2 - R^2 -1}{T^2 - R^2 +1} \\right) \\,.\n\\end{equation}\nNote that, for simplicity, we have set here $\\phi_0 = 0$, and we have also set the \nlength scale appearing in $f(v) = - \\sqrt 3 \\, \\ell_0^2 \/ v$ to $\\ell_0 = 1$. \nWe shall come back later to the cosmological significance of this solution. Let \nus only note here that, in terms of the null coordinates\n\\beq\nu \\equiv T - R \\ , \\quad \\quad v \\equiv T + R \\, ,\n\\end{equation}\nthe exact mass ratio reads\n\\beq\n\\mu(u,v) = \\frac{u\\,v\\,(v-u)^2}{(u^2\\,v^2-1)^2}\\,,\n\\end{equation}\nwhile, starting from (\\ref{eqn4.19}), one obtains\n\\beq\n\\mu_2{(u,v)} = {\\rm Var} (N)_{[u,v]} = \\frac{(v-u)^2}{u^3 \\, v^3} \\, .\n\\label{eq5.28}\n\\end{equation}\nOne finds that a strength criterion of the form ${\\displaystyle \\sup_{u \\leq \nv}} \\ \\mu_2 (u,v) = C_2$ is first satisfied (as one increases  $v$ from $-\\infty$) \n when $v = v_{*2}$, with $u = u_{*2}$, where\n\\beq\nv_{*2} = -\\ell_0 \\left( \\frac{4}{27C_2} \\right)^{1\/4} \\ , \n\\quad \\quad  u_{*2} = 3 v_{*2} \\,.\n\\end{equation}\nFor $C_2$ of order 1, this is in qualitative agreement with the exact result that \nthe apparent horizon is located at\n\\beq\n\\label{eq5.29}\nu_{\\cal A} = \\frac{1}{v^3} \\,, \\quad \\quad \\quad \nr_{\\cal A} = \\frac{1}{2 |v|^3}\\,(1 - v^4)^{3\/2}\\,,\n\\end{equation}\nso that the apparent horizon and the \nsingularity are first ``seen'' at $v_{* \\, {\\rm exact}} = - \\ell_0$.\n\n\\section{Transition from the weak-field  to the cosmological  regime}\nThe following general picture emerges from the previous sections: Let us \nconsider as ``in state'' a generic classical string vacuum, which can be \ndescribed as a superposition of incoming wave packets of gravitational \nand dilatonic fields. This ``in state\" can be nicely parametrized by three \nasymptotic ingoing, dimensionless news functions $N(v,\\theta,\\varphi)$, $N_+ \n(v,\\theta,\\varphi)$, $N_{\\times} (v,\\theta,\\varphi)$.  \n When  all the news stay always significantly below 1, this ``in state'' will \nevolve into a similar trivial ``out state'' made of outgoing wave packets. On the \nother hand, when the news functions reach values of order 1, and more precisely \nwhen some global measure of the variation \\footnote{The argument that the \ncollapse criterion should be (at least) invariant under constant shifts of all \nthe news functions (corresponding to classically trivial shifts of the \nbackground fields $g_{\\mu \\nu}^0$, $\\phi_0$) indicates that only the global \nchanges of the news matter.} of the news functions, similar to the variance \n(\\ref{eqn4.16}), exceeds some critical value of order unity, the ``in state'' will \nbecome gravitationally unstable during its evolution and  will give \nbirth to one or several black holes, i.e. one or several singularities hidden \nbehind outgoing null surfaces (event horizons). Seen from the outside of these \nblack holes, the ``out\" string vacuum will finally look, like the ``in\" one, as a \nsuperposition of outgoing waves.\nHowever, the story is very different if we \nlook inside these black holes and shift back to the physically more appropriate \nstring conformal frame. First, we note that the structure of black hole \nsingularities in Einstein's theory (with matter satisfying $p < \\epsilon$) is a \nmatter of debate. The work of Belinsky, Khalatnikov and Lifshitz \\cite{BKL} has \nsuggested the generic appearance of an oscillating space-like singularity. \nHowever, the consistency of this picture is unclear as the infinitely many \noscillations keep space being as curved (and ``turbulent'' \\cite{Russians},  \n\\cite{Belinsky}) as time. Happily, the basic gravitational sector of string \ntheory is generically consistent with a much simpler picture. Indeed, it has \nbeen proven long ago by Belinsky and Khalatnikov \\cite{BK} that adding a \nmassless scalar field (which can be thought of as adding matter with $p = \n\\epsilon$ as equation of state) drastically alters the BKL solution by ultimately \nquenching the oscillatory behaviour to end up with a much simpler, monotonic \napproach to a space-like singularity. When described in the string frame, the \nEinstein-frame collapse towards a space-like singularity will represent (if \n$\\phi$ grows toward the singularity) a super-inflationary expansion of space. \nThe picture is therefore that inside each black hole, the regions near the \nsingularity where $\\phi$ grows will blister off the initial trivial vacuum as \nmany separate pre-Big Bangs. These inflating patches are \nsurrounded by non-inflating, or deflating (decreases $\\phi$) \npatches, and therefore globally look approximately \nclosed Friedmann-Lema\\^{\\i}tre hot universes. \nThis picture is sketched in Fig.~\\ref{fig1} and Fig.~\\ref{fig2}.\nWe expect such quasi-closed universes to recollapse \nin a finite, though very long, time (which is \nconsistent with the fact that, seen from the outside, \nthe black holes therein contained must evaporate in a finite time).\nTo firm up this picture let us study in detail the \nappearance of the singularity in the simple toy model of Section \\ref{sec3}.\n\n\\subsection{Negative-curvature Friedmann-dilaton solution}\n\\label{subsec5.1}\nWe have already seen that in the toy model there is an infinitely \nextended incoming region where the fields are weak and can be described as \nperturbations upon the incoming background values $(g_{\\mu \\nu}^0 = \\eta_{\\mu \n\\nu} , \\phi_0)$. One expects that the perturbation algorithm described in section \\ref{subsec3.1} \nbecomes unreliable when corrections become of order unity. In \nparticular, one generically expects that the apparent horizon $\\mu \\equiv \n\\frac{2m}{r} = 1$ will roughly divide space-time in two domains: the \nperturbative incoming weak-field domain where $\\mu (u,v) \\ll 1$, and a \nstrong-field domain where $\\mu (u,v) \\gg 1$. [Actually, we shall see below that \nthe precise boundary of the domain of validity of perturbation theory may also \ndepend on other quantities than just $\\mu$.] This separation in two domains \nis represented in Fig.~\\ref{fig3}.\n\nFor guidance let us study in detail \nthis separation in two domains in the case of the $k = -1$ Friedmann-dilaton \nsolution \\cite{CLW}, \\cite{TW}. \nLet us first write explicitly the solution Eqs. (\\ref{4.43}) and (\\ref{4.44})\nin $(u,v)$ coordinates \n\\beq\n\\label{eq5.18}\nds^2 = -\\Omega^2(u,v)\\,du \\,dv + r^2 \\, d \\omega^2\\,, \\quad \\quad \n\\phi(u,v) = - \\sqrt{3}\\,\\log \\left [ \n\\frac{u\\,v -1}{u\\,v +1} \\right ]\\,,\n\\end{equation}\n\\bea\n\\label{eq5.25}\n&& \\Omega^2(u,v) = \\left ( 1 - \\frac{1}{u^2\\,v^2} \\right )\\,,\n\\quad \\quad r^2(u,v) = \\frac{1}{4}\\,\\left ( 1 - \\frac{1}{u^2 \\, v^2} \\right )\\,\n(u-v)^2 \\,,\\\\\n\\label{eq5.26}\n&& \\mu(u,v) = \\frac{u\\,v\\,(v-u)^2}{(u^2\\,v^2-1)^2}\\,, \n\\quad \\quad \n1 - \\frac{2m}{r} = \\frac{(u^3\\,v-1)\\,(u\\,v^3-1)}{(u^2\\,v^2-1)^2}\\,.\n\\label{eq5.27}\n\\end{eqnarray}\nThe perturbative algorithm of section \\ref{subsec3.1} gives\n\\bea\n\\label{eq5.33}\n\\phi(u,v) &\\simeq& \\frac{2 \\sqrt{3}}{u\\,v}\\,\\left (1 + \\frac{1}{3}\n\\frac{1}{u^2\\,v^2} \\right ) \\,,\\\\\n\\label{eq5.34}\nr(u,v) & \\simeq & \\frac{1}{2}\\,(v-u)\\,\\left (1- \\frac{1}{2}\n\\frac{1}{u^2\\,v^2} \\right )\\,,\n\\end{eqnarray}\nwhich agree with the expansions of Eqs.~(\\ref{eq5.25}) and (\\ref{eq5.26}). One sees that \nperturbation theory is numerically valid up to, say, $uv \\leq \\sqrt 2$, at which point \nthere is an abrupt transition towards the cosmological singularity  \nlocated at $uv = 1$. Though the transition surface $uv \\sim \\sqrt 2$ globally \ndiffers from the apparent horizon $\\mu (u,v) =1$, we note that our criterion \npoints out to a specific event $(u_{*2} , v_{*2})$ on the apparent horizon \n${\\cal A}$ (the point where ${\\cal A}$ is first ``seen'' from infinity) which \nlies, roughly, at the intersection of ${\\cal A}$ and of the transition surface. \nThis confirms that our criterion is able, at least in order of magnitude, to \ncorrectly pinpoint when and where one should shift from the perturbative regime to \na different, cosmological-type description. In the case of the solution \nEqs.~(\\ref{eq5.18})--(\\ref{eq5.26}),\none sees better the cosmological nature of the strong-field domain by \nintroducing the following coordinates\n\\beq\nu = -e^{-\\eta + \\xi} = T - R \\,, \\quad \\quad  v = - e^{-\\eta - \\xi} = T +R \\,,\n\\label{eq5.21}\n\\end{equation}\nsatisfying the useful relations\n\\bea\n\\label{eq5.22}\n&& -T = e^{-\\eta}\\, \\cosh \\xi \\,, \\quad \\quad \\quad \nR = e^{-\\eta}\\, \\sinh \\xi \\,,\\\\\n\\label{eq5.23}\n&& \\pa_u = e^{\\eta -\\xi}\\, (\\pa_\\eta - \\pa_\\xi) \\,,\n\\quad \\quad \\pa_v = e^{\\eta +\\xi}\\, (\\pa_\\eta + \\pa_\\xi)\\,,\n\\\\\n\\label{eq5.24}\n&& e^{-2 \\eta} = u\\,v\\,,\\quad \\quad \\quad \\quad \\quad \\quad e^{2 \\xi} = \\frac{u}{v}\\,.\n\\end{eqnarray}\nIn terms of the coordinates $(\\eta , \\xi)$ the solution reads \\cite{CLW}, \\cite{TW}\n\\bea\n\\label{eq5.16}\n&& ds^2 = a^2(\\eta) \\left [ -d \\eta^2 + d \\xi^2 + \\sinh^2 \\xi\\,d \\omega^2 \\right ]\\,,\\\\\n\\label{eq5.17}\n&& a^2(\\eta) = 4\\cosh \\eta\\,\\sinh(-\\eta)\\,, \\quad \\quad \ne^{-\\phi} = \\left (-\\frac{\\sinh \\eta}{\\cosh \\eta} \\right )^{\\sqrt{3}}\\,.\n\\end{eqnarray}\nWe note that the solution is regular in the domain $-\\infty < \\eta < 0$, $0 \n\\leq \\xi < +\\infty$ (which corresponds to the past of the hyperboloid $T^2 - \nR^2 = 1$ in Minkowski-like coordinates) and that a space-like singularity is \nreached at $\\eta = 0$ (i.e. $uv = T^2 - R^2 = 1$). Let us also note the \nexpressions\n\\bea\nr^2 (\\xi , \\eta) = a^2 (\\eta) \\sinh^2 \\xi = 4 \\cosh \\eta \\sinh (-\\eta) \\sinh^2 \\xi \\, ,\n\\end{eqnarray}\n\\beq\n\\label{eq5.30}\n1 - \\frac{2 m}{r} = \\sinh^2 \\xi\\, \\left [ \\coth^2 \\xi - \\coth^2 2 \\eta \\right ] \\,,\n\\end{equation}\ntelling us that the apparent horizon is located at $2\\eta = -\\xi$. Near the \nsingularity, $\\eta \\rightarrow 0^-$,\n the dilaton blows up logarithmically while the metric coefficients \nhave the following power-law behaviours:\n\\beq\n\\phi \\sim - \\sqrt 3 \\, \\log (-\\eta)\\,, \\quad \\quad \n\\Omega^2 (\\xi , \\eta) \\equiv a^2 (\\eta) \\sim -4 \\eta\\,, \n\\quad \\quad r^2 (\\xi , \\eta) \\sim -4 \\eta \\sinh^2 \\xi \\, .\n\\end{equation}\nBefore leaving this example we wish to emphasize some features of it which\nappear to follow from its homogeneity and could be misleading for the general case. \nThe singular boundary  terminates, in this case, on ${\\cal I}^-$, \ninstead of on ${\\cal I}^+$, as one generically expects.\n The apparent reason for this is the\nsingularity in the flux at $v=0$ which creates a future boundary on ${\\cal I}^-$.\nWe believe that, in this case, a better description of physics is obtained by going to\nnew non-Minkowskian coordinates of Milne's type (see \\cite{V97}) which automatically\nincorporate the singularity on ${\\cal I}^-$. However, \nif one restricts oneself to more regular initial data,  having\na finite integrated energy flux (generalized pulse-like data), the singular boundary\nshould never come back to ${\\cal I}^-$. [At most,  it could end at space-like\ninfinity ${i}^0$, for non integrable total energy flux].\n\n\\subsection{Kasner-like behaviour of Einstein-dilaton singularities}\n\\label{subsec5.2}\nComing back to the generic case of an arbitrary Einstein-dilaton cosmological \nsingularity, previous works \\cite{V97} have shown that, in a suitable synchronous \ncoordinate system (Gauss coordinates), the asymptotic behaviour of the string frame \nmetric near the singularity reads (in any space-time dimension $D$)\n\\beq\n\\label{strkasn1}\nds_S^2 \\sim -dt_S^2 + \\sum_{a=1}^{D-1} (-t_S)^{2\\alpha_a (x)} \n(E_i^a (x) \\, dx^i)^2 \\, ,\n\\end{equation}\n\\beq\n\\label{strkasn2}\n\\phi (x,t) = \\phi (x,0) + \\sigma (x) \\log (-t_S) \\, ,\n\\end{equation}\nwhere $E_i^a (x)$ is some $(D-1)$-bein and where ($1 \\leq a \\leq D-1$)\n\\beq\n\\label{strrel}\n\\sum_{a=1}^{D-1} \\alpha_a^2 = 1 \\,, \\quad \\quad \n\\sigma = \\left( \\sum_{a=1}^{D-1} \\alpha_a \\right) - 1 \\,.\n\\end{equation}\nIn the Einstein frame this asymptotic behaviour reads\n\\beq\n\\label{kasn1}\nds_E^2 \\sim - dt_E^2 + \n\\sum_a (-t_E)^{2 \\lambda_a (x)} (e_i^a (x) \\, dx^i)^2 \\,,\n\\end{equation}\n\\beq\n\\label{kasn2}\n\\phi (x,t) = \\phi (x,0) + \\gamma (x) \\log (-t_E) \\,,\n\\end{equation}\nwhere $e_i^a (x)$ is proportional to $E_i^a (x)$ and where\n\\beq\n\\sum_{a=1}^{D-1} \\lambda_a = 1 \\,, \\quad \\quad \n\\gamma = \\pm \\sqrt 2 \\ \\sqrt{1 - \\sum_{a=1}^{D-1} \\lambda_a^2} \\,.\n\\label{cons}\n\\end{equation}\nThe ``Kasner'' exponents, $\\alpha_a$ in the string frame or $\\lambda_a$ in the \nEinstein frame, can vary continuously along the singularity. The string \nparametrization is the most global one as it shows that $\\alpha_a$ runs freely \nover a unit sphere in ${\\bb R}^{D-1}$ while the exponent for $e^{\\phi}$ is a linear \nfunction of the ``vector'' $\\alpha_a$. In the Einstein frame, the exponents \n$\\lambda_a$ are restricted by the linear equality $\\sum_a \\lambda_a = 1$ and the \nquadratic constraint $\\sum_a \\lambda_a^2 \\leq 1$. A convenient geometric \nrepresentation of these constraints, when $D=4$, is   \nto consider (in analogy with Mandelstam's  \nvariables $s,t,u$) that the 3 $\\lambda_a$'s represent, in some Euclidean plane, \nthe orthogonal distances of a point $\\Lambda$ from the three sides of an \nequilateral triangle (counted positively when $\\Lambda$ is {\\it inside} the \ntriangle). The quadratic constraint then means that $\\Lambda$ is restricted to \nstay inside the circle circumscribing the triangle. The sign ambiguity in \nEq.~(\\ref{cons}) means that the parameter space is in fact a {\\it two-sided disk}, namely \nthe two faces of a coin circumscribed around the triangle. See Fig.~\\ref{fig4}.\n\nThe link between the \ntwo seemingly very different parameter spaces (the string-frame sphere \nparametrized by the $\\alpha$'s and the Einstein-frame two-sided disk \nparametrized by the $\\lambda$'s) is geometrically very simple: one obtains the \ndisk by a stereographic projection of the sphere, from the projection \ncenter $(\\alpha_a^c) \n= (1,1,1)$ (outside the sphere) onto the plane dual to this center (i.e. the \nplane spanning the circle of tangency to the sphere of the straight lines issued \nfrom $\\alpha^c$). Algebraically (in any-space-time dimension $D$)\n\\beq\n\\lambda_a = \\frac{\\alpha_a(D-2) - {\\sigma}}{D-2-{\\sigma}}\\,,\n\\quad \\quad \n\\gamma = \\frac{\\sqrt{ 2\\,(D-2)}\\,\\sigma}{D-2 - {\\sigma}}\\,,\n\\end{equation}\n\\begin{figure}\n\\centerline{\\epsfig{file=fig4.ps,width=0.35\\textwidth,angle=-90}}\n\\vskip 0.5truecm\n\\caption{\\sl Geometric representation of the Einstein-frame \nKasner exponents $(\\lambda_1, \\lambda_2, \\lambda_3; \\gamma)$ \nof Einstein-dilaton cosmological singularities. The three $\\lambda$'s, \nsuch that $\\sum_a \\lambda_a =1$ are the orthogonal distances \nto the three sides of an equilateral triangle. The constraint \n$\\gamma = \\pm \\sqrt{2}\\, \\sqrt{1 - \\sum_a \\lambda^2_a}$ restricts the representative $\\Lambda$ of $\\lambda_a$ to stay on a two-sided disk circumscribed \naround the triangle. The pure-Einstein cases ($\\gamma = 0$) correspond \nto the circumscribing circle. In the spherically symmetric case \n($ \\lambda_2 = \\lambda_3$), $\\Lambda$ runs over a bissectrix of the triangle. \nThe basic parameter $\\alpha$ of Eqs.~(\\ref{phi}) and (\\ref{lam}) \nruns over a full real line (shown folded on the right of the figure \n) and is mapped to the bissectrix via horizontal lines.}\n\\label{fig4}\n\\end{figure}\nwhere $\\sigma \\equiv (\\sum_a \\alpha_a) - 1$. Note that there is no sign ambiguity \nin the map $\\alpha_a \\rightarrow \\lambda_a$. Coming back to the case $D = 4$, as \n$\\alpha_a$ runs over a unit sphere in ${\\bb R}^3$, $\\lambda_a$ covers twice the disk \ncircumbscribed to an equilateral triangle. Each side of the disk is the image, \nvia the stereographic projection, of a polar ($\\sigma > 0$) or antipolar \n($\\sigma < 0$) ``cap'' of the sphere. This geometrical picture of the link \nbetween the $\\alpha$-sphere and the two-sided $\\lambda$-disk in represented in \nFig.~\\ref{fig5}. \nIt is useful to keep in mind this \ngeometrical picture because it shows clearly that one can continuously pass from \none side of the disk to the other, i.e. the sign of the Einstein-frame \n``Kasner'' exponent of $e^{\\phi} \\sim (-t_E)^{\\gamma (x)}$ can change as one moves \nalong the singularity. Physically, this means a very striking inhomogeneity near \nthe singularity: the patches where $\\gamma (x) > 0$ (decreasing $g^2 = \ne^{\\phi}$) will shrink (in string units) while only the patches where $\\gamma \n(x) < 0$ (growing $g^2$) can represent pre-Big Bangs. Finally, we note \nthat Belinsky and Khalatnikov \\cite{BK} have proven that, in the generic case, the \nEinstein-frame Kasner exponents $(\\lambda_1, \\lambda_2, \\lambda_3)$ \nmust all be positive for the ultimate stable asymptotic approach \nto the singularity. This corresponds to $\\Lambda$ being inside the triangle\nof Fig.~\\ref{fig4}, i.e. in the hatched region of Fig.~\\ref{fig5}.\nHowever, this generic restriction does not apply \nin the special case of our toy model as we explain below.\n\\begin{figure}\n\\vskip -1truecm\n\\centerline{\\epsfig{file=fig5.ps,width=0.36\\textwidth,angle=0}}\n\\vskip -1.5truecm\n\\caption{\\sl Geometrical representation of the link \nbetween the two-sided disk of Fig. \\ref{fig4} on which \nthe Einstein-frame Kasner exponents $\\lambda_a$ live, \nand the sphere, $\\sum_a \\alpha^2_a=1$, on which the string-frame\nKasner exponents $(\\alpha_1, \\alpha_2, \\alpha_3)$ live. The map $\n\\alpha_a \\rightarrow \\lambda_a$ is a stereographic projection, \nfrom the center $(\\alpha_a^c)= (1,1,1)$. The upper side of the disk \ncomes from the projection \nof the polar \ncap $\\sigma \\equiv (\\sum_a \\alpha_a) -1 > 0$ located on the same side as \n$\\alpha^c$, while the lower side of the disk comes from \nprojection of the antipolar \ncap $\\sigma < 0$. The points $\\alpha_+$ on the polar cap, and \n$\\alpha_-$ on the antipolar cap, are both projected on the same point \n$\\Lambda$ of the disk. The tangents to the sphere issued from $\\alpha^c$ \ntouch the sphere along the circle limiting the disk (on which $\\sigma=0$).}\n\\label{fig5}\n\\end{figure}\n\\subsection{Behaviour near the singularity in inhomogeneous, \nspherically symmetric solutions} \n\\label{subsec5.3}\nLet us now restrict ourselves to our toy model and study its cosmological-like \nbehaviour near the singularity. First, we note that spherical symmetry will \nimpose that two of the metric Kasner exponents (those corresponding to the \n$\\theta$ and $\\varphi$ directions) must be equal, say $\\lambda_2 = \\lambda_3$. \nGeometrically, this means that the Einstein-frame Kasner exponents run only over \nthe intersection of the above two-sided disk with a bissectrix of the triangle.\nSecond, we note that because of the vanishing, in spherical symmetry, of the crucial \ndreibein connection coefficient $\\vec{e}_1 \\cdot (\\nabla \\times \\vec{e}_1)$, \ngenerically responsible for causing the expansion in the (radial) direction $1$ \nto oscillate as $t \\rightarrow 0^-$ \\cite{BKL}, a negative value \nof $\\lambda_1$, corresponding to an expansion in the radial direction, is allowed \nas ultimate asymptotic behaviour. As a consequence, even the portion of \nthe bissectrix outside the triangle is allowed as asymptotic state. We shall prove directly\nthis fact below by constructing a consistent expansion near the singularity \nfor $\\lambda_1$ of any sign. \n\nThe mass-evolution equations (\\ref{eq1.8}), (\\ref{eq1.9}), together with \nEqs.~(\\ref{rel1}), (\\ref{rel2}) show that, starting from \nsome regular event at the center, $r=0$, where $m=0$, the mass $m (u,v)$ will \ngrow if we follow a  outgoing characteristic (i.e. a null \ngeodesic) $u =$ const. If this characteristic crosses the apparent horizon \n${\\cal A}$, Eqs.~(\\ref{rel1}), (\\ref{rel2}) then show that $m(u,v)$ \ncontinues to grow within the trapped region ${\\cal T} (2m\/r > 1)$ if one moves \nalong {\\it either} (future) outgoing or ingoing characteristics. This means that, \nwithin ${\\cal T}$, $m(u,v)$ grows in all future directions. As the space-like \nsingularity \\footnote{We consider here only the \n``non central'' singularity, i.e. \nthe part of the singularity which can be reached by future outgoing \ncharacteristics issued from the regular center $r=0$. As shown by \\cite{chi93}\nthere exists also, at the ``intersection'' between the regular center and \n${\\cal B}$, a singularity reachable only via ingoing characteristics. \nNote that $r(u,v)$ tends to zero both at the center \nand at the singularity (central or \nnon-central).} ${\\cal B}$ lies in the future of ${\\cal T}$, we see that the mass \nfunction will necessarily grow toward ${\\cal B}$, and therefore, either will \ntend to a finite limit on ${\\cal B}$, or it will tend monotonically to $+\\infty$ as \n$r \\rightarrow 0$ with fixed $v$ (or $u$). Motivated by the Kasner-like \nbehaviours Eqs.~(\\ref{kasn1}) and (\\ref{kasn2})\n we expect the following generic asymptotic behaviour of $m$ \non the singularity (here written in $(v,r)$ coordinates)\n\\beq\n\\label{mass}\nm(v,r) \\sim C_m(v) \\, \\left ({r\\over \\ell_0} \n\\right )^{-\\alpha^2 (v)} \\ \\hbox{as} \\ r \n\\rightarrow 0 \\,, v \\,\\,\\hbox{fixed}\n\\end{equation}\nwith some real $v$-dependent exponent $\\alpha^2 (v)$. [Here $\\ell_0$ \ndenotes some convenient length scale.]\nIt follows that $\\mu = 2m\/r$ always tends to $+\\infty$ on ${\\cal B}$ \n(even when $m$ tends to a finite limit). This suggests that we can \nconsistently compute the structure of the fields near ${\\cal B}$ by using an \n``anti-weak-field'' approximation scheme in which $\\mu \\gg 1$ (instead of the \nweak-field algorithm of section~\\ref{subsec3.1} \nwhere we used $\\mu \\ll 1$). The expected domain of validity of such \nan ``anti-weak-field'', or ``cosmological-like'', expansion \nis sketched in Fig.~\\ref{fig3}.\n\nTo leading order \nin this large-$\\mu$ limit Eq.~(\\ref{eq1.19}) yields\n\\beq\n\\label{eq4.11}\n\\frac{1}{4}\\left ( \\frac{ \\pa \\phi}{\\pa \\log r} \\right )^2_{v} = \n- \\left (\\frac{\\partial \\log m}{\\pa \\log r} \\right )_{v}\\,,\n\\end{equation}\nfrom which we obtain the following asymptotic behaviour of $\\phi$ on ${\\cal \nB}$\n\\beq\n\\label{phi}\n\\phi (r,v) \\sim 2 \\alpha (v) \\log r + C_\\phi (v) \\, .\n\\end{equation}\nWe see now that the basic Kasner-like-exponent on ${\\cal B}$ is half the \ncoefficient of $\\log r$ in $\\phi (r,v)$ as $r \\rightarrow 0$, with fixed $v$. \nThe Kasner-like exponent of $m(v,r)$ is the square of that basic exponent. \nMoreover, the function $C_m(v)$ appearing in the asymptotic behaviour \nof $m(v,r)$ on the singularity is not independent from the functions \n$\\alpha(v)$ and $C_\\phi(v)$. Indeed, to leading order Eq.~(\\ref{new}) \nreads\n\\beq\n\\frac{\\pa \\log m}{\\pa v} = - \\frac{1}{2}\\,\\frac{\\pa \\phi }{\\pa \\log r}\\,\n\\frac{\\pa \\phi}{\\pa v}\\,.\n\\end{equation}\nThis relation is identically satisfied at order $r^0\\,\\log r$, and yields at order $r^0$ \n\\beq\n\\frac{\\pa \\log C_m}{\\pa v} = - \\alpha(v)\\,\\frac{\\pa C_\\phi}{\\pa v}\\,.\n\\label{5.26}\n\\end{equation}\nUsing now Eq.~(\\ref{eq1.18}), written as\n\\beq\n\\left( \\frac{\\partial \\beta}{\\partial \\log r} \\right)_v = \n\\left( \\frac{1}{2} \\, \n\\frac{\\partial \\phi}{\\partial \\log r} \\right)^2\\,,\n\\end{equation}\nwe further get the asymptotic behaviour of $\\beta (v,r)$\n\\beq\n\\label{beta}\n\\beta (v,r) \\sim \\alpha^2 (v) \\log r + C_\\beta (v) \\, .\n\\end{equation}\nFinally, it is easy to see that the asymptotic behaviour  of the \nmetric on ${\\cal B}$ in $(v,r)$ coordinates\n\\beq\nds^2 \\sim \\frac{2 C_m (v)}{r} \\, \ne^{2C_\\beta (v)} \\, \\left ({r\\over \\ell_0}\\right )^{\\alpha^2 (v)} \n\\, dv^2 + 2 e^{C_\\beta (v)} \\, \\left ({r\\over \\ell_0}\n\\right )^{\\alpha^2 (v)} \\, dv \\, dr + r^2 \\, d\\omega^2\\,,\n\\end{equation}\nis indeed of the expected Kasner form (\\ref{kasn1}) by introducing the \ncosmological time \n\\beq\n(-t) \\sim r\\,\\left ({r\\over \\ell_0}\n\\right )^{\\frac{1}{2} (\\alpha^2 (v) + 1)}\\,.\n\\end{equation}\nThe transformed metric reads, to dominant order as $r \\rightarrow 0$, i.e. $t \n\\rightarrow 0^-$,\n\\bea\nds^2 \\sim&& -dt^2 + C_{\\rho} (v) \\, (-t)^{2\\lambda_1 (v)} \\, \\left (\ndv + \\frac{r\\,e^{-C_{\\beta}}\\,dr}{2 C_m} \\right )^2  \\nonumber \\\\\n&& + C_{\\omega} (v) \\, (-t)^{2\\lambda_2 (v)} \\, [d\\theta^2 + \\sin^2 \\theta \\, d \\varphi^2]\\,, \n\\end{eqnarray}\n\\beq\n\\phi \\sim \\gamma (v) \\log (-t) \\,,\n\\end{equation}\n\\beq\n\\label{lam}\n\\lambda_1 (v) = \\frac{\\alpha^2 (v) - 1}{\\alpha^2 (v) + 3} \\,,\n\\quad \\quad \n\\lambda_2 (v) =  \\frac{2}{\\alpha^2 (v) + 3} \\,,\n\\quad \\quad \n\\gamma (v) = \\frac{4 \\alpha (v)}{\\alpha^2 (v) + 3} \\,.\n\\end{equation}\nThe basic parameter $\\alpha (v)$ runs over the real line $- \\infty < \\alpha < \n+\\infty$. One easily checks that the relations (\\ref{cons}) are satisfied. \nSome cases having a particular significance are illustrated in Fig.~\\ref{fig4}: \nWhen $\\alpha = 0$, \none gets $(\\lambda_1 , \\lambda_2 , \n\\lambda_3) = \\left( - \\frac{1}{3}, \\frac{2}{3}, \\frac{2}{3} \\right)$ and $\\gamma \n= 0$ which describes a Schwarzschild-like singularity (when viewed with a \ncosmological eye). \nWhen $\\alpha = \\pm 1$, \none gets $(\\lambda_1 , \\lambda_2 , \\lambda_3) = \\left( 0, \\frac{1}{2}, \n\\frac{1}{2} \\right)$ and $\\gamma = \\pm 1$ which is locally similar to the \nscale-invariant solution \\cite{japanese} also described earlier. \nWhen $\\alpha = \\pm \\sqrt{3}$, one gets $(\\lambda_1 , \n\\lambda_2 , \\lambda_3) = \\left( \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{3} \\right)$ \nand $\\gamma = \\pm 2 \/ \\sqrt 3$ which describes an isotropic cosmological \nsingularity of the type of the exact solution \\cite{CLW}, \\cite{TW} we discussed above. \nAs illustrated in Fig.~\\ref{fig4},  \nwhen $\\alpha$ runs over ${\\bb R}$ its image, defined by Eq.~(\\ref{lam}), \nruns \ntwice over the bissectrix of the Mandelstam triangle (intersected with the \ncircumbscribed disk). The only pure-Einstein case ($\\gamma = \n0$) reached at finite $\\alpha$ is the Schwarzschild-like case $\\alpha \n= 0$. In principle, another pure-Einstein behaviour occurs  when $\\vert \n\\alpha \\vert \\rightarrow \\infty$, leading to $(\\lambda_1 , \\lambda_2 , \n\\lambda_3) = (1,0,0)$ i.e. $ds^2 \\sim -dt^2 + t^2 \\, d\\rho^2 + d \\omega^2 = - d \n\\tau^2 + d \\xi^2 + d\\theta^2 + \\sin^2 \\theta \\, d\\varphi^2$. This would be an \nEinstein-cylinder-like universe. We believe, however, that $\\vert \\alpha (v) \n\\vert$ will stay bounded $(\\vert \\alpha (v) \\vert < \\alpha_M < + \\infty)$ as $v$ \nruns over ${\\cal B}$ and that the only Einstein-like case will be crossed when \n$\\alpha (v)$ changes along ${\\cal B}$.\n\nWe have confirmed our asymptotic analysis of the structure of the singularity \n(based on the Ans\\\"atze (\\ref{mass}), (\\ref{phi}) and (\\ref{beta})\nin two ways. First, we have verified that it is \nconsistent with the partial, but rigorous, results given in Ref.~\\cite{chi93}. \nNamely, in $(u,v)$ coordinates, our Ans\\\"atze are compatible with the result \nthat the function $r^2 (u,v)$ is $C^1$ on ${\\cal B}$ if we posit that the \nconformal factor behaves like $\\Omega^2 \\sim r^{\\alpha^2 (v) - 1}$ (while \n$m \\sim r^{-\\alpha^2 (v)}$). A stronger check of our assumptions is obtained by \ngoing beyond the leading order approximation, thus extending some\nresults of \\cite{V97}. In analogy to section \\ref{subsec3.1}, where we \nset up a complete perturbation algorithm in the weak-field domain, we have shown \nthat, starting from the leading-order terms Eqs.~(\\ref{mass}), (\\ref{phi}) \nand (\\ref{beta}), containing 3 arbitrary ``seed \nfunctions'', $\\alpha (v)$, $C_\\phi (v)$, $C_\\beta (v)$ [from which \n$C_m(v)$ can be determined using Eq.~(\\ref{new})], it is \npossible to set-up an all-order iterative scheme generating a formal solution of \nthe spherically symmetric Einstein-dilaton system in the strong-field, \ncosmological-like domain. Note that, $C_\\beta(v)$ being a pure gauge function \n(it suffices to introduce $v^\\prime = \\int e^{C_\\beta(v)} \\, dv$ \nto gauge $C_\\beta(v)$ away), the formal solution thus generated contains two \nphysically arbitrary \nfunctions of $v$, which is indeed the freedom that should be present \nin a generic solution \\footnote{Here, contrary to what happened above, the variable $v$ varies \nonly on a half-line, $[v_*, +\\infty]$ where $v_*$ marks the birth of ${\\cal B}$. \nAs was said above two functions on a half-line (or one function on the full line) \nis the correct genericity in our toy model.}.\nDetails of this scheme are given in Appendix \\ref{app}. We \nonly mention here the general structure of the expansions so generated near the \nsingularity, i.e. as $r \\rightarrow 0$, with fixed $v$: (see Eq.~(\\ref{ulapp}) in the appendix)\n\\bea\n\\label{appen1}\n&& \\phi(v,r) = \\phi_0(v,r) +\n\\sum_{n,m,l} a_{n m l}^{(\\phi)}\\,r^{n(\\alpha^2(v) +1)}\\,\nr^{2 m}\\,P_l^{(\\phi)}(\\log r)\\,,\\\\\n\\label{appen2}\n&& \\beta(v,r) = \\beta_0(v,r) +\n\\sum_{n,m,l} a_{n m l}^{(\\beta)}\\,r^{n(\\alpha^2(v) +1)}\\,\nr^{2 m}\\,P_l^{(\\beta)}(\\log r)\\,,\\\\\n\\label{appen3}\n&& m(v,r) = m_0(v,r) +\n\\sum_{n,m,l} a_{n m l}^{({\\rm m})}\\,r^{n(\\alpha^2(v) +1)}\\,\nr^{2 m}\\,P_l^{({\\rm m})}(\\log r)\\,.\n\\end{eqnarray}\nHere $n,m,l$ are integers with $l \\leq m$, $a_{n m l }^{(\\psi)}$ \nand the coefficients of the polynomials $P_l^{(\\psi)}$ are functions of $v$,  \nand $(\\phi_0, \\beta_0, m_0)$ are the leading-order terms given \nby Eqs.~(\\ref{mass}), (\\ref{phi}) and (\\ref{beta}).\nThis expansion actually contains two intertwined series: \nan expansion in powers of $x = r^{1+ \\alpha^2}$ and a more complicated \nseries in $r^{2 m}\\,(\\log r)^l$. The second expansion \nis linked to the $v$-gradients of the seed functions $\\alpha(v)$,  \n$C_\\phi(v)$, $C_\\beta(v)$, while the first expansion is present even in the simple \n``homogeneous'' case where $\\alpha$, $C_\\phi$ and $C_\\beta$ do not depend on $v$. \nThe fact, exhibited by Eqs.~(\\ref{appen1})--(\\ref{appen3}), that the expansion in \nthe homogeneous case proceeds along powers of $r^{\\alpha^2+1}$ confirms a recent \nconjecture by Burko \\cite{Burko}.\n\n\\subsection{Exact homogeneous cosmological solutions}\n\\label{subsec5.4}\nLet us consider in more detail the ``homogeneous'' case where \nthe seed functions $\\alpha$, $C_\\phi$ and $C_\\beta$ have no spatial \nvariation along the singularity.\nAs it turns out, it is possible to resum exactly all the terms \nof the ``homogeneous'' expansion. Indeed, the analog of the Schwarzschild solution \n($t$-independent, spherically symmetric solution) for the Einstein-dilaton \nsystem has been worked out analytically long ago by \nJust \\cite{Just} (see also \\cite{others}), using the special gauge where \n$g_{00}\\,g_{rr} =-1$. Contrary to the case of the Schwarzschild \nsolution, this solution \ncannot be continuously extended (through a regular horizon) down to a cosmological \nsingularity \nat $r=0$. However, it is easy to see that the following cosmological-like\nbackground (which is related to Just's original solution by formally \nextending the radial variable $\\overline{r} = l\\,x$ ``below'' the curvature\nsingularity at $\\overline{r} = a$) is still a solution \nof the Einstein-dilaton system: \n\\bea\n\\label{ap1}\nd s^2 &=& l^2 \\left [ \\left (\\frac{1-x}{x} \\right )^b\\,d t^2 - \n\\left (\\frac{x}{1-x} \\right )^b\\,d x^2 + x^{1 +b}\\,(1-x)^{1-b}\\, d \\omega^2 \\right ]\\,,\\\\\n\\label{ap2}\n\\phi &=& \\pm \\sqrt{1 -b^2}\\,\\log \\left ( \\frac{1-x}{x} \\right )\\,.\n\\end{eqnarray}\nHere the (formerly radial) variable $x$ varies between \n$0$ and $1$ and is ``time-like'', while the (formerly time) variable $t$ \nis ``space-like''. The cosmological universe evolves from \na Big Bang at $x=0$ to a Big Crunch at $x =1$. \nIt has two arbitrary parameters, a scale parameter $l$, and the dimensionless\n$b$, $ -1 \\leq b \\leq 1$. Near the singularity \nat $x=0$, the link between $b$ and the parametrization \nused above is \n\\beq\n\\alpha = \\mp \\sqrt{\\frac{1-b}{1+b}}\\,, \\quad \\quad b = \\frac{1 - \\alpha^2}{\n1 + \\alpha^2}\\,.\n\\end{equation}\nNote that the behaviour near the other singularity at $x=1$ \nis obtained by changing the sign of $b$ and by changing \n$\\alpha$ into $ -1\/\\alpha$. Some cases of this homogeneous \nsolution are of special significance: $b=1$ is Schwarzschild (with $x=r\/2m$), \n$b=0$ belongs to the class of scale-invariant solutions\n\\cite{japanese} discussed above and $b=-1\/2$ interpolates \nbetween a locally isotropic cosmological solution \n($\\alpha = \\mp \\sqrt{3}$) at $x=0$ and an anisotropic one \n($\\alpha = \\pm 1\/\\sqrt{3}$) at $x=1$. Note that, in spite of the homogeneity and isotropy\nat $x=0$, the latter special solution differs from the (everywhere) \nhomogeneous-isotropic solution described earlier.\n\n\\subsection{Discussion}\n\\label{subsec5.5}\nIdeally, the two formal expansions we have constructed, the weak-field one \nEqs.~(\\ref{eqn4.1a})--(\\ref{eqn4.1c}), \nand the strong-field one Eqs.~(\\ref{appen1})--(\\ref{appen3}), \nshould match at some intermediate \nhypersurface, like the apparent horizon $2m\/r = 1$ which looks like a natural \nborderline between the two near domains. If this matching were analytically doable, \nit would determine all the (so-far) arbitrary ``seed functions'' of the \nstrong-field scheme in terms of the unique arbitrary function of the weak-field \none, namely the asymptotic waveform $f(v)$ (or, equivalently, the news $N(v) = \nf'(v)$). However, it is clearly too naive to expect to perform this matching \nperturbatively: neither of the two expansions is expected to be \nconvergent \\footnote{The possible presence of a finite number of BKL oscillations \n\\cite{BK} near ${\\cal B}$ suggests that the formal strong-field expansion has \nvery bad convergence properties.} (they are probably only asymptotic). Even if \nthey converge on some domain they probably both break down before reaching a \npossible overlap region where they might be matched. At this stage, we can only \nstate that, in principle, all the seed functions of the strong-field scheme are \nsome complicated non linear and  non local functionals of $N(v)$. It would be particularly \ninteresting to study the functional dependence of the Kasner exponent \n$\\alpha (v)$ on $N(v)$. By causality (i.e. a domain of dependence argument) we \nknow that $\\alpha (v)$, at advanced time $v$, depends only on $N(v')$ on the \ninterval $v' \\leq v$. When starting from a generic $N(v)$ ``of order unity'', we \nexpect that the resulting $\\alpha (v)$ will also be of order unity. A physically \nvery important issue is the sign of $\\alpha (v)$, i.e. the sign of $\\gamma (v)$. \nIndeed $\\alpha > 0$ means a decreasing $\\phi$, while $\\alpha < 0$ means that \n$\\phi$ grows near ${\\cal B}$. Let us note that, to lowest-order of weak-field \nperturbation theory, the value of $\\phi (v,r)$ at the (regular) center ($r=0$) \nis\n\\beq\n\\lim_{r \\rightarrow 0} \\phi_1 (v,r) = \\lim_{r \\rightarrow 0} \\left[ \\phi_i + \n\\frac{f(v) - f(v-2r)}{r} \\right] = \\phi_i + 2 f' (v) = \\phi_i + 2 N(v) \\, .\n\\label{eqn5.18}\n\\end{equation}\nHere $\\phi_i$ denotes the background value at past infinity. From this result we \nexpect that, if $N(v)$ is a simple Gaussian-like wave packet, \n$N(v) = A \\exp \\left [- (v \n- v_i)^2 \/ \\ell_i^2\\right ]$, with a large enough dimensionless amplitude for leading to \ncollapse, i.e. $\\vert A \\vert \\gtrsim 1$, the local values of $\\phi$ near the \ncollapsing central region will, at first, grow if $A > 0$, and decrease if $A < \n0$. In this simple case, we therefore expect $\\alpha (v)$ to have the opposite \nsign of $A$ near the ``central'' region of ${\\cal B}$, i.e. for $v \\lesssim v_i$ \nwhere the gravitational instability sets in. But, further away on ${\\cal B}$, \ni.e. for $v \\gg v_i$, the sign of $\\alpha (v)$ can change. We also expect that, \nas $v \\rightarrow + \\infty$, $\\alpha (v)$ will tend to zero, corresponding to a \nSchwarzschild-like singularity (in the case of well localized incoming $N(v)$ \npackets). This conjectured link between the sign of $\\alpha (v)$ and the sign of \n$N(v)$ is confirmed by the exact isotropic solution Eq.~(\\ref{eq5.18})\n(with a positive growing \n$N(v) = \\sqrt 3 \/ v^2$, and $\\alpha = - \\sqrt 3$), \nas well as by the scale invariant solution Eq.~(\\ref{eq5.3}). \nSome numerical calculations \\cite{Burko} seem to \nconfirm the general picture we propose (in particular the interesting \npossibility that the sign of $\\alpha (v)$ changes several times on ${\\cal B}$ as \n$v$ varies, before reaching a Schwarzschild-like asymptotic regime $\\vert \\alpha \n(v) \\vert \\rightarrow 0$ as $v \\rightarrow +\\infty$). We plan to study in more \ndetail these issues in a future publication.\n\n\\section{A Bayesian look at  pre-Big Bang's ``fine-tuning\"}\n\\label{sec6}\n{} From its inception \\cite{Veneziano} it was pointed out that \na successful pre-Big Bang \nscenario must rely on a ``reservoir'' of inflationary $e$-folds \nduring its perturbative phase. This is given by two small numbers, the initial\ncurvature scale in string units, $H_i \\ell_s$, and the\ninitial string coupling constant $g_i^2 = e^{\\phi_i} \\ll \\!\\!< 1$. [In this section, \nthe index $i$ is used for labelling ``initial'' quantities.] The need of large \n(or small) numbers  has been recently discussed at length \\cite{TW}, \n\\cite{KLB} and used to criticize the naturalness of the PBB scenario. In particular,\nit was pointed out \n\\cite{TW}, \\cite{KLB}  that, as soon as one goes beyond the simple \nspatially-flat, homogeneous cosmology framework, the total duration\nof the perturbative dilaton-driven phase is finite so that\nthe resolution of the homogeneity\/flatness problems requires\n\\beq\ng_i = e^{\\phi_{i}\/2} < 10^{-26} \\,, \\quad \\quad \n\\frac{L_i}{\\ell_s} \\gtrsim \\frac{M_s}{H_i} > 10^{19} \\,.\n\\label{eq6.1}\n\\end{equation}\nHere, $L_i$ denotes the spatial homogeneity scale of the  PBB universe at the beginning\nof its inflationary phase, \nand $H_i \\gtrsim L_i^{-1}$ its time-curvature scale (Hubble parameter).\n \nIn this Section we shall systematically work with string-frame \nquantities, even when we refer to results \ndiscussed in previous sections in the Einstein frame.\n For instance, the time-scale  $\\ell_i$, characterizing the rate of variation of the \nnews functions around the advanced time $v_i$ and leading to \ngravitational instability, is now (locally) measured in string units. In any case, we are\nessentially working with dimensionless ratios which are unit-independent: \n$(\\ell_i \/ \\ell_s)_S = (\\ell_i \/ \\ell_s)_E = g_i (\\ell_i \/ \\ell_{\\rm \nPlanck})_E$. We wish  to emphasize here that, by combining our \nstochastic-like instability picture with a Bayesian \napproach to the {\\it a posteriori} probability of being in a position of asking \nfine-tuning questions, the issue of the naturalness of the PBB scenario is \ndrastically changed. By ``Bayesian approach'' we mean taking into account \nthe selection effect that fine-tuning questions presuppose the existence of a \nscientific civilization. As emphasized long ago by Dicke \\cite{D61} and Carter \n\\cite{C74}, civilization-related selection effects can completely change the \nsignificance of large numbers or of apparent coincidences. \nLinde and collaborators explored several aspects of the Dicke-Carter ``anthropic \nprinciple'' within the inflationary paradigm, and emphasized the necessity \nto weigh {\\it a posteriori} probabilities by the physical volume \nof inflationary patches \\cite{lindebook2}.\nWe shall here follow \nVilenkin \\cite{V95}, and his ``principle of mediocrity'', according to which the \nunnormalized {\\it a posteriori} probability of a random scientific civilization \nto observe any values of the PBB parameters $g_i$, $L_i$ and $H_i$ is obtained \nby multiplying the corresponding {\\it a priori} probability by the number of \ncivilizations (over the whole of space and time) associated with the values $g_i$, \n$L_i$ and $H_i$: ${\\cal N}_{civ} (g_i , L_i , H_i)$.\n\n\\subsection{A priori and a posteriori probability distributions for $g_i$ \nand $H_i$}\n\\label{subsec6.1}\n This approach can be \napplied to our case if we think of the initial past-trivial string vacuum as \nmade of a more or less stochastic superposition of incoming waves (described by \ncomplicated news functions $N(v,\\theta ,\\varphi)$, $N_+ (v,\\theta ,\\varphi)$, \n$N_{\\times} (v,\\theta ,\\varphi)$ having many bumps, troughs and ramps). This \nstochastic bath of incoming waves will generate a rough sea of dilatonic and \ngravitational fields. If the input dimensionless wave forms can \nreach values of order unity, we expect that the local \nconditions for gravitational instability will be satisfied \n at several places in space and time. This will give birth \nto an ensemble of bubbling baby universes, with a more or less random \ndistribution of initial parameters $g_i$ and $H_i$, and with initial spatial \nhomogeneity scales \n$L_i \\gtrsim H_i^{-1}$. Indeed, the analysis of the previous \nsections has shown (in our toy model) that gravitational \ninstability will set in \non a spatio-temporal scale $\\sim \\ell_i$ when a rising wave of news function \ngrows by ${\\cal O} (1)$ on an advanced-time scale $\\ell_i$. From our variance \ncriterion Eq.~(\\ref{eqn4.16}), and the rough validity of weak-field perturbation\n theory nearly until \nthe sharp transition to a cosmological-type behaviour, the work of the previous \nsections has shown that the initial, advanced-time scale $\\ell_i$ is propagated \nvia ingoing characteristics with little deformation down to the strong-field \ndomain (see Fig.~\\ref{fig3}), where it appears as a spatial homogeneity scale, i.e.\n\\beq\nL_i \\sim \\ell_i \\, .\n\\label{eq6.2}\n\\end{equation}\n{}From the leading order result (\\ref{eqn5.18}) we expect the local value of the \nHubble parameter in the corresponding cosmological-like bubble to be\n\\beq\nH_i \\sim \\sqrt{\\dot{\\phi}^2} \\sim \\vert N' (v_i) \\vert \\sim \\frac{\\vert N (v_i) \n\\vert}{\\ell_i}\n\\label{eq6.3}\n\\end{equation}\nwhere $v_i$ denotes the advanced time at which the instability sets in. Combining \n(\\ref{eq6.2}) and (\\ref{eq6.3}) we get\n\\beq\nH_i \\, L_i \\sim \\vert N (v_i) \\vert \\gtrsim 1 \\, . \n\\label{eq6.4}\n\\end{equation}\nThese rough formulae show how, in principle,  given the stochastic \nproperties of the dimensionless news functions, one could deduce the \ndistribution of $H_i$ and $L_i$, naturally constrained by $H_i \\, L_i \\gtrsim \n1$. The corresponding distribution of $g_i$ is {\\it a priori} independent from \nthat of $\\ell_i \\sim L_i$, being linked to the presence of a slowly-varying (``DC\") \ncomponent $N_0$ \nin $N(v)$ (by contrast to the local variations of $N(v)$ leading to \ninstability): $N(v) = N_0 + {\\rm fluctuations}$. \nThe DC component $N_0$ corresponds to ramps in $f(v)$, \n$f(v) = N_0 \\, v + f_0 + {\\rm fluctuations}$, i.e. to shifts of $\\phi_i : \\phi_i \n\\rightarrow \\phi_i + 2 N_0$. Finally, we can consider that the initial \ndistribution of the news functions defines an {\\it a priori} probability \ndistribution for $g_i$, and $H_i$,\n\\beq\ndp_{\\rm prior} = w_i (g_i , L_i) \\, dg_i \\, dL_i \\, ,\n\\label{eq6.5}\n\\end{equation}\nwith $H_i$ generically of order $L_i^{-1}$ (because $\\vert N (v_i) \\vert \\sim 1$ \nis the threshold for instability).\n\n An important remark must be made here. The \ntwo basic parameters we are talking about, say $g_i$ and $L_i \\sim H_i^{-1}$, \nprecisely correspond to the two {\\it global symmetries}, \nEqs.~(\\ref{p1})--(\\ref{eqn2.9}) and Eq.~(\\ref{eqn2.10}),\nof the classical string \nvacua: a constant shift in $\\phi$, and a global coordinate rescaling. If we consider that \nthe initial state of string theory is classical (rather than quantum), these \nsymmetries mean that no particular values for $\\phi_i$ or $L_i$ are {\\it a \npriori} preferred. One might even expect a ``flat'' distribution compatible with \nthese global symmetries,\n\\beq\ndp_{\\rm prior}^{\\rm classical \\, flat} \\propto \\frac{dg_i}{g_i} \\, \n\\frac{dL_i}{L_i} \\propto d\\phi_i \\, d \\log L_i \\, .\n\\label{eq6.6}\n\\end{equation}\nEvidently, the problem with such a flat distribution is that it is non-normalizable.\n Some cut-offs are needed to make sense of such a flat prior \ndistribution but, while there are natural  strong coupling\/curvature  cut-offs\n(the limits of validity of our approximation), it is not so easy to find\nnatural small-curvature\/coupling cut-offs. If we appeal to string theory\n for providing cut-offs for $g_i$ and \n$L_i$, arguably the prior distribution (\\ref{eq6.5}) should be determined by \nthe conjectured basic symmetries of string theory, i.e. by $S$ and \n$T$-duality. $S$-duality suggests that one should work with the complex quantity \n$z = \\frac{\\theta}{2\\pi} + i \\, \\frac{4\\pi}{g^2}$ and require modular invariance \nin $z = x + iy$. This selects the Poincar\\'e \nmetric $ds^2 = (dx^2 + dy^2) \/ y^2$ \nas being special, and thereby defines a preferred measure in the fundamental \ndomain (key-hole region) of the $(x,y)$ \nplane: $d\\mu = \\sqrt g \\, dx \\, dy = dx \\, dy \/ y^2 = dx \\, \\vert dy^{-1} \n\\vert$. Integrating out the angular variable $\\theta$ over $0 \\leq \\theta < \n2\\pi$ (which is the correct key-hole range when $g^2 < 4\\pi$, i.e. $y > 1$) we \nend up with a preferred probability distribution for $g : dp(g) \\propto dg^2 \n\\propto g \\, dg$, considered in the weak-coupling region $g \\lesssim 1$. A \nsimilar argument based on $T$-duality selects as preferred probability \ndistribution for the spatial length scale $L : dp (L) \\propto d \n\\widehat{L}^{-2} \\,  \\propto \n\\widehat{L}^{-3} \\, d \\widehat{L}$ where $\\widehat{L} \\equiv L \/ \\ell_s$ is \nconsidered in the weak-$\\sigma$-model-coupling region $\\widehat{L} \\gtrsim 1$. \nIn conclusion, this type of argument would suggest the following normalizable, \nfactorized prior distribution\n\\beq\ndp_{\\rm prior}^{\\rm S-T-dual} \\propto (g_i \\, dg_i) \\, \\widehat{L}_i^{-3} \\, d \n\\widehat{L}_i \\propto g_i \\, dg_i \\, \\widehat{H}_i \\, d \\widehat{H}_i\\,,\n\\label{eq6.7}\n\\end{equation}\nwith $0 < g_i \\lesssim 1$, $0 < \\widehat{H}_i \\equiv H_i \/ M_s = \n\\widehat{L}_i^{-1} \\lesssim 1$. \n\nFor the sake of \ngenerality, we wish  to leave open the nature of the prior distribution, \nand thus consider  the general class of prior probability distributions:\n\\beq\ndp_{\\rm prior} \\propto g_i^a \\, dg_i \\, \\widehat{H}_i^b \\, d \\widehat{H}_i\\,,\n\\label{eq6.8}\n\\end{equation}\nwith arbitrary powers $a$ and $b$.\n\nAny given initial distribution, say (\\ref{eq6.8}), will generate a \ncorresponding ensemble of PBB inflationary bubbles. We assume, as usual,\nthe existence of a successful ``exit'' mechanism by which \ndilaton-driven inflation, with growing $g$ and $H$  \nfinally ``exits'' into a standard hot Big Bang, i.e. a radiation-dominated \nFriedmann universe. Current ideas about how this might happen \\cite{GMV} assume that \nthe Hubble\nparameter  reaches \nvalues of order of the string mass $M_s$, i.e. $\\widehat{H}_s = H_s \/ M_s \n\\sim 1$, {\\it before} the string coupling constant reaches values of order \nunity. If the opposite were true ($g \\sim 1$  occurring before \n$\\widehat{H} \\sim 1$), it is likely that quantum fluctuations would take\nover before $H$ reaches the string scale, causing\n inhomogeneities to grow\nso large that the inflationary process ``aborts\" before a baby\nuniverse is born. In any event, we shall discard this possibility, assuming that\nthe resulting cosmological universe would not evolve into anything able \nto harbour scientific civilizations.\n\nAfter $\\widehat{H}$ reaches $1$ at $g \\ll 1$, \n the evolving universe is expected to be entering a so-called stringy \nDe~Sitter phase, i.e. a phase during which  $\\dot{\\phi}$ and \n$H$ are constant and given by some fixed-point values  of the order of $M_s$:\n $\\vert \\dot{\\phi} \\vert = x \\, M_s$, and $\\vert H \\vert = y \\, \nM_s$ with $x \\sim y \\sim 1$. [See \\cite{GMV} for some suggestions on how to \nimplement this mechanism.] Only the ratio between $x$ and $y$ enters our \npresent phenomenological discussion. In keeping with a notation used in \nprevious papers on PBB phenomenology, we introduce the (positive) parameter \n$\\beta$ by\n\\beq\n\\vert (\\ln g)^{\\cdot} \\vert = \\frac{1}{2} \\, \\vert \\dot{\\phi} \\vert = \\beta \\, H \n= {\\rm const} \\, .\n\\label{eq6.9}\n\\end{equation}\n\nOne finally assumes that this stringy De~Sitter-like phase \nends when $g = e^{\\phi\/ 2}$ \nreaches values of order unity, causing the amplified vacuum \nfluctuations\nto reach the critical density. After this moment, $g$ should remain \nfixed at about its present value, either because of a nonperturbative $\\phi$-dependent potential, \nor possibly because of the attractor mechanism of Ref.~\\cite{DP}. The fixing of \n$g$ marks the end of the stringy modifications to Einstein's theory, and the \nbeginning of standard Friedmann-like cosmology. As we assume that $g$ is \nattracted towards a unique fixed point, the resulting Friedmann universe\nwill have no free parameter which could make it different\nfrom our universe, except for its initial homogeneity scale $L_f$, \nwhere the index $f$ refers to the final state of the stringy phase. Therefore, \nthe number of civilizations in the resulting Friedmann universe will be \nproportional to its total volume, at least for the large enough \n(in fact, \nold enough)  universes that can harbour life\n\\beq\n{\\cal N}_{\\rm civ} (g_i , H_i) \\propto {\\cal V}_f (g_i , H_i) \\, \\theta [{\\cal \nV}_f - {\\cal V}_{\\min}] \\, .\n\\label{eq6.10}\n\\end{equation}\nOne assumes here, \\`a la Dicke \\cite{D61}, that the time span during which \ncivilizations can occur, being constrained by the lifetime of stars, is fixed. \nThe minimum scale ${\\cal V}_{\\rm min}$ for a life-harbouring universe is not known precisely, though \nit corresponds probably to universes whose total lifetime before recollapse is a \nfew billion years. In conclusion, the {\\it a posteriori} probability for a random \nscientific civilization to ask fine-tuning questions about the values of $g_i$ \nand $H_i$ is\n\\beq\ndp_{\\rm post} \\propto {\\cal N}_{\\rm civ} (g_i , H_i) \\, dp_{\\rm prior} \\propto \n{\\cal V}_f (g_i , H_i) \\, \\theta ({\\cal V}_f - {\\cal V}_{\\min}) \\, dp_{\\rm \nprior} \\, .\n\\label{eq6.11}\n\\end{equation}\n\\subsection{Computation of the a posteriori probability distribution in 4 dimensions} \n\\label{subsec6.3}\nThe computation of the final volume  ${\\cal V}_f$ can be separated as\n\\beq\n{\\cal V}_f = L_f^3 = L_i^3 \\ \\frac{{\\cal V}_1}{{\\cal V}_i} \\ z_s^3 \\, .\n\\label{eq6.12}\n\\end{equation}\nHere, the index $i$ refers to the (cosmological) time $t_i$ of beginning of \ndilaton-driven inflation, the index 1 refers to the time $t_1$ of transition between \ndilaton-driven superinflation and a De~Sitter-like phase, and $z_s = \\exp \\, (H \n\\, (t_2 - t_1))$ denotes the total expansion during the De~Sitter phase. The \nending $t_2$ of the De~Sitter-like phase is assumed to take place whence $g_2 \n\\equiv g (t_2) \\sim {\\cal O} (1)$.\n\nUsing the leading power-law behaviour of the cosmological evolution \nduring the dilaton-driven phase, Eq.~(\\ref{strkasn1}), \nand imposing that this phase ends when $H_1 \n\\sim (-t_1)^{-1} \\sim M_s$, one gets  \n\\beq\n\\frac{{\\cal V}_1}{{\\cal V}_i} \\simeq \\left( \\frac{t_i}{t_s} \\right)^{-\\Sigma \n\\alpha} \\simeq (\\widehat{H}_i)^{\\Sigma \\alpha} \\, ,\n\\label{eq6.13}\n\\end{equation}\nwhere $\\Sigma \\alpha \\equiv \\alpha_1 + \\alpha_2 + \\alpha_3$ is the sum of the \nstring-frame Kasner exponents. The $\\alpha$'s are negative (expansion) and are \nconstrained by Eq.~(\\ref{strrel}). In the result (\\ref{eq6.13}) we have neglected a factor \nof order unity linked to the fact that the dilaton-driven inflation is \ngenerically anisotropic ($\\alpha_1 \\not= \\alpha_2 \\not= \\alpha_3$) and therefore \nthat the various Hubble expansion rates will reach \nthe string scale at slightly different times. We assume here that a nontrivial \nbasin of attraction from anisotropic expansion toward an isotropic stringy \nDe~Sitter-like phase exists as it is the case in examples \\cite{GMV}.\n\nDuring the subsequent stringy De~Sitter phase, the growth of $g$ is  related, according \nto Eq.~(\\ref{eq6.9}), to the spatial expansion by $g \\propto a^{\\beta}$ \nso that\n\\beq\nz_s = \\frac{a_2}{a_1} = \\left( \\frac{g_2}{g_1} \\right)^{\\frac{1}{\\beta}} \\sim \ng_1^{-\\frac{1}{\\beta}} \\, ,\n\\label{eq6.14}\n\\end{equation}\nwhere we used $g_2 \\sim 1$. On the other hand, $g_1 = \ng(t_1)$ is expressible, through the use of Eq.~(\\ref{strkasn2}), giving the evolution of $g$ \nduring the dilaton phase, as\n\\beq\ne^{\\phi_1 - \\phi_i} = \\left( \\frac{g_1}{g_i} \\right)^2 \\simeq \\left( \n\\frac{t_1}{t_i} \\right)^{\\Sigma \\alpha-1} \\sim \\widehat{H}_i^{\\,\\,\\Sigma \\alpha \n-1} \\, .\n\\label{eq6.15}\n\\end{equation}\nFinally, we express, using $L_i \\sim H_i^{-1}$, the  volume at the beginning of \nstandard cosmology in the form\n\\beq\n{\\cal V}_f (g_i , H_i) \\sim \\widehat{H}_i^{-A} \\, [g_1 (g_i , \n\\widehat{H}_i)]^{-3\/\\beta} \\, \\theta (1-g_1)\\,,\n\\label{eq6.16}\n\\end{equation}\nwith \n\\beq\ng_1 (g_i , \\widehat{H}_i) \\sim g_i \\, \\widehat{H}_i^{-B} \\,,\n\\label{eq6.17}\n\\end{equation}\nand where the exponents $A$ and $B$ are positive, and are given in terms of the $\\alpha$'s \nby\n\\beq\nA = 3 - \\Sigma \\alpha \\,, \\quad \\quad  \nB = \\frac{1}{2} - \\frac{1}{2} \\ \\Sigma \\alpha \\, .\n\\label{eq6.18}\n\\end{equation}\nIf we assume, for illustration, an {\\it a priori} distribution of values of $g_i$ and \n$\\widehat{H}_i$ of the form (\\ref{eq6.8}), we get for the {\\it a posteriori} \nprobability distribution of $g_i$ and $\\widehat{H}_i$ \n\\bea\ndp_{\\rm post} \\propto g_i^a \\, \\widehat{H}_i^{b-A} \\, [g_i \\, \n\\widehat{H}_i^{-B}]^{-3\/\\beta} \\, \\theta (1-g_1 (g_i , \\widehat{H}_i)) \\, \\theta \n({\\cal V}_f - {\\cal V}_{\\min}) \\, dg_i \\, d\\widehat{H}_i \\, ,\n\\end{eqnarray}\nwhich can be written as\n\\beq\ndp_{\\rm post} \\propto g_1^{\\overline a} \\, \\widehat{H}_i^{-\\overline A} \\, \n\\theta (1-g_1) \\, \\theta ({\\cal V}_f - {\\cal V}_{\\min}) \\, dg_1 \\, d \n\\widehat{H}_i \\, .\n\\label{eq6.19}\n\\end{equation}\nIn the last form we have expressed the measure $dp_{\\rm post}$ in terms of the \ntwo independent variables $g_1$ and $\\widehat{H}_i$ \n(this is convenient because \nof the constraint $g_1 < 1$ carried by the step function). The exponents \n$\\overline a$ and $\\overline A$ appearing there are\n\\bea\n&& \\overline a \\equiv a - \\frac{3}{\\beta} \\,, \\\\\n&& \\overline A = A - (a+1) \\, B-b = 3 + \\frac{1}{2}\\,(a-1)\\,(\\Sigma \\alpha) - \\frac{1}{2} \\, \n(a+1) - b \\, .\n\\label{eq6.20}\n\\end{eqnarray}\nThe numerical values of the exponents $\\overline a$ and $\\overline A$ play a \ncrucial r\\^ole in determining the {\\it a posteriori} plausibility of our universe \nhaving evolved from the seemingly ``unnatural'' values (\\ref{eq6.1}). Indeed, if \n$\\overline A \\geq 1$ the posterior probability distribution for $\\widehat{H}_i$ \nis peaked, in a non integrable manner, at $\\widehat{H}_i = 0$. Therefore, if \n$\\overline A \\geq 1$, it is natural to expect (within the present scenario) that \nthe initial homogeneous patch, whose gravitational instability led to our \nuniverse, be extremely large compared to the string scale [and that the value of \n$g_i = \\widehat{H}_i^B \\, g_1$ with $B > 0$ and $g_1 < 1$ be correspondingly \nsmall]. Most scientific civilizations are bound to evolve in such universes.\n\nNote that the duality-suggested values $a=1$ and $b=1$ of \nEq.~(\\ref{eq6.8}) lead to $\\overline A = 1$ (independently of the $\\alpha$'s) \nfor which such an {\\it a posteriori} explanation works. From this point of view one \ncan argue that, within string theory, it is ``natural'' to observe very small \nnumbers such as in Eq.~(\\ref{eq6.1}). The exponent $\\overline a$ also plays an \nimportant r\\^ole. Indeed, if $\\overline a > -1$, i.e. if\n\\beq\n\\beta > \\frac{3}{a+1} \\, ,\n\\label{eq6.21}\n\\end{equation}\nthe {\\it a posteriori} probability distribution for $g_1$ is integrable over its \nentire possible range $(0,1)$. This means that most scientific civilizations are \nexpected to observe values $g_1 = {\\cal O} (1)$, corresponding to a small number \nof $e$-folds during the stringy phase: $z_s \\sim g_1^{-1\/\\beta} = {\\cal O} (1)$. \nThis is a phenomenologically interesting case, because it means that various \ninteresting physical phenomena taking place during the dilaton-driven phase \n(such as the quantum amplification of various fields) might leave observable \nimprints on cosmologically relevant scales. [A very long string phase would \nessentially iron out all signals coming from the dilaton phase.] It is \ninteresting to note that, in the string-duality-inspired case (\\ref{eq6.7}), \nthe great divide between a short stringy phase $(\\overline a > -1)$, and a very \nlong one ($\\overline a \\leq -1$, implying that $g_1$ is peaked in a non \nintegrable manner at $g_1 = 0$), lies at $\\beta = 3\/2$, which played already a \nspecial r\\^ole in previous phenomenological studies \\cite{PBB}.\n\nThough the case $\\overline A \\geq 1$, $\\overline a > -1$ appears as the \nconceptually most interesting case for the pre-Big Bang scenario, other \ncivilization-related selection effects might also render the case \n$\\overline A < 1$ viable. In this case, the factors \n$\\widehat{H}_i^{-\\overline A} \\,d\\,\n\\widehat{H}_i$ in Eq.~(\\ref{eq6.19}) suggests that $\\widehat{H}_i$ \nshould be of order unity (i.e. $L_i \\sim H_i^{-1} \\sim \\ell_s$). However, in \nsuch a case, one must take into account the factor $\\theta ({\\cal V}_f - {\\cal \nV}_{\\min})$ (which could be neglected in the previous discussion). This factor \nmeans that scientific civilizations should {\\it a posteriori} expect to find \nthemselves in the smallest possible universe compatible with their appearance. \nThis probably means that they should also expect to see inhomogeneities \ncomparable to the Hubble scale, and to have appeared very late in the \ncosmological evolution, just before the universe recollapses. This seems to conflict with \nour observations. In conclusion, within this model, and limiting our discussion \nfor simplicity to the class (\\ref{eq6.8}) with $a$ and $b$ integers, one finds \nthe favorable situation $\\overline A \\geq 1$ realized (i.e. no {\\it a posteriori} \nunnaturalness in observing very small $g_i$ and $\\widehat{H}_i$) when either \n$a=0$ and $b \\leq 2$, or $a=1$ and $b \\leq 1$. However, if $a \\geq 2$, i.e. if \nthe {\\it a priori} distribution function of $g_i$ vanishes faster than $g_i^2$ for \n$g_i \\rightarrow 0$, the pre-Big Bang scenario has to face \na naturalness problem. [This is the case for quantum fluctuations \nof $g_i$, as discussed at the end of the next subsection.]\n\n\\subsection{Computation of the a posteriori probability distribution in $D$ dimensions} \n\\label{subsec6.4}\nAssuming that the results discussed so far are qualitatively correct in \nhigher dimensions,  \nwe have generalized the above considerations to the case in which the initial \nstring vacuum is extended to $d$ spatial dimensions (say $d = 9$), and where, \nthrough gravitational instability, 3 of these dimensions are expanding and 6 are \ncollapsing down to the string scale. This leads to introducing 4 special times: \n$t_i$ (beginning of dilaton phase), $t_1 \\sim -t_s$ (end of the dilaton-driven \npower-law evolution of space), $t_2$ (when the radius of curvature of the \ncollapsing dimensions become ${\\cal O} (\\ell_s)$), and $t_3$ \n(when $g_3 = {\\cal O} (1)$). \nIn this scenario there are two stringy De~Sitter-like phases: a first \none during which the 3 ``external'' dimensions grow exponentially $a \\propto \n\\exp \\, (H_1 \\, t)$, while the $(d-3)$ ``internal'' ones {\\it shrink} \nexponentially $\\overline a \\propto \\exp \\, (- \\overline{H}_1 \\, t)$ (and while \n$\\dot{\\phi} = 2\\beta_1 \\, H_1 = 2 \\overline{\\beta}_1 \\, \\overline{H}_1$), and a \nsecond phase during which the $(d-3)$ string-scale-curved internal dimensions have \nfrozen while the 3 external ones continue to grow exponentially $a \n\\propto \\exp \\, (H_2 \\, t)$ (and while $\\dot{\\phi} = 2\\beta_2 \\, H_2$). This \nscenario gives a result of the same form as above, namely\n\\beq\n\\label{eq6.22}\n{\\cal V}_f (g_i , H_i) \\sim \\widehat{H}_i^{-A} \\, \n\\left [g_2(g_i,\\widehat{H}_i) \\right ]^{-3\/\\beta_2} \\, \\theta \n(1-g_2) \\,,\n\\end{equation}\nwith  \n\\beq\ng_2(g_i,\\widehat{H}_i) = g_i\\,\\widehat{H}_i^{-B}\\,, \n\\end{equation}\nand where\n\\beq\nA = d - 3\\alpha - (d-3) \\, \\overline{\\alpha} + ( 1 - \\overline{\\alpha}) \\, \n\\overline{\\beta}_1 \\left( \\frac{3}{\\beta_1} - \\frac{d-3}{\\overline{\\beta}_1} \n\\right) \\, ,\n\\end{equation}\n\\beq\nB = \\frac{1}{2} - \\frac{3}{2} \\, \\alpha - \\frac{1}{2} \\, (d-3) \\, \n\\overline{\\alpha} + (1 - \\overline{\\alpha}) \\, \\overline{\\beta}_1 \\, .\n\\label{eq6.23}\n\\end{equation}\nHere one has assumed for simplicity that, during the dilaton phase, the 3 \nexternal dimensions grow with the same exponent \\footnote{\nThis string-frame Kasner exponent $\\alpha$ should not be confused with the \nbasic parameter $\\alpha(v)$ introduced in Sec.~\\ref{subsec5.3}.}\n$\\alpha < 0$, i.e. proportionally to \n$(-t)^{\\alpha}$, while the $(d-3)$ internal ones decrease with the same exponent \n$\\overline{\\alpha} > 0$, proportionally to $(-t)^{\\overline{\\alpha}}$. Then, \none can derive the analog of Eq.~(\\ref{eq6.19}) with $g_1$ replaced by $g_2$ and \nwith exponents\n\\beq\n\\overline a = a - \\frac{3}{\\beta_2} \\, \n\\quad \\quad \\overline A = A - (a+1) \\, B - b \\, .\n\\label{eq6.24}\n\\end{equation}\nOnce more, $\\overline A \\geq 1$ leads to no fine-tuning and $\\overline a > -1$ to a \nshort string phase. We shall not attempt a complete phenomenological discussion \nof the range of values of $a$ and $b$ compatible with $\\overline A \\geq 1$. \n(This would imply exploring the parameter space of allowed values for the Kasner \nexponents $\\alpha$ and $\\overline{\\alpha}$.) Let us only note here that the \nintroduction of more spatial dimensions generically helps because $A$ (and \n$\\overline A$) now contains the contribution $+d = +9$, say, instead of \nthe previous $+3$.\nIn this case, even  distribution functions vanishing faster than $g_i^2$ \n(for $g_i \\rightarrow 0$) can make very small values of $g_i$ \nbe {\\it a posteriori} preferred. The power-law enhancement brought by the volume \nfactor ${\\cal V}_f$ would  be inadequate, however, for compensating  \nan exponential suppression as $g_i \\rightarrow 0$, \nsay $\\propto \\exp \\, (-c \/ g_i^2)$. In particular, the {\\it a priori} distribution of $g_i$ \nand $H_i$ cannot be the one expected (from the Lagrangian ${\\cal L} \\sim \ne^{-\\phi} (\\nabla \\phi)^2 \\sim (\\nabla g^{-1})^2$) for quantum fluctuations of \n$g$ on scales $L_i \\sim H_i^{-1}$ around the trivial ground state. Indeed, \nthis distribution vanishes \nfor small $g_i$ and small $\\widehat{H}_i$ as $\\exp \\, (-c\\,g_i^{-2} \\, \n\\widehat{H}_i^{-2})$ with $c$ of order unity. \nThis confirms the standard idea of the pre-Big Bang scenario, \naccording to which \nthe initial state should be a {\\it classical} string vacuum, i.e. an \narbitrary classical solution of the low-energy field equations.\n\n\\section{Conclusion}\n\nWe have proposed a new, concrete  picture for the \ntypical initial state of the Universe which can give rise\nto a cosmology of the so-called pre-big bang type. We conceived this state as a very \ngeneric, past-trivial string vacuum, i.e., classically, as the most general solution of \nthe low-energy tree-level string action endowed with a perturbative region in the asymptotic \npast. Such states can be parametrized (in three spatial dimensions) by three \ndimensionless ``news functions'': a scalar news $N(v,\\theta ,\\varphi)$ and two \nhelicity-2 news $N_+ (v,\\theta ,\\varphi)$ and $N_{\\times} (v,\\theta ,\\varphi)$. \nBy studying the simple spherically symmetric tensor-scalar model we have argued \nthat such a generic ``in state'' will be gravitationally unstable if (any of) \nthe news functions varie(s) by something of order unity on some scale $\\Delta v \n\\sim \\ell$. More precisely, in the spherically symmetric model, we found, at \nlowest order of perturbation theory, that the criterion for gravitational \ninstability is given by the supremum over $v_1$ and $v_2$ of the {\\it variance} \nof the function $N(x)$ over the interval $[v_1 , v_2]$, see Eq.~(\\ref{eqn4.16}). When this \nsupremum exceeds a threshold of order unity, the wave packet ``sent'' between \nthe retarded times $v_1$ and $v_2$ will become gravitationally unstable when, later, \nit becomes focussed at the center \\footnote{In absence of spherical symmetry, \nthere may exist many ``local centers'' where gravitational instability sets in. \nThey should each correspond to an annular section of ${\\cal I}^-$ with a ${\\cal \nO} (1)$ fluctuation of the news looking roughly spherically symmetric when \nviewed from some center and in some suitably boosted Lorentz frame.}. We have \nverified that the infinite region between past infinity and such a center can be \ndescribed by weak-field perturbation theory. Then, near the center, in a region \nof spatial extension $L_i \\sim \\Delta v \\equiv v_2 - v_1$, one can abruptly \nshift from a weak-field description to a cosmological one. One can then describe \nthe further evolution of the patch $L_i$ by a complementary cosmological-like \nexpansion of the type introduced by Belinsky, Khalatnikov and Lifshitz (BKL)\nand recently developed, for the PBB scenario, in \\cite{V97}. We \nhave proven to all orders of iteration (in our toy model) that the presence of \nthe dilaton ensured the consistency of a (non oscillatory) BKL expansion in \nlog-corrected powers of the distance to the future (space-like) singularity. This \nresult confirms the smoothing behaviour of a dilaton-driven superinflationary \nevolution. In the string frame, gravitationally unstable patches of vacuum, with \nhomogeneity scale $L_i$, will expand (if $\\dot{\\phi} > 0$) into homogeneous patches \n on much larger scales. These blistering patches are surrounded by \ncontracting regions where $\\dot{\\phi} < 0$. Many such dilaton-driven pre-Big \nBang inflationary bubbles, looking like closed universes, can blister off a \ngeneric past-trivial string vacuum. By taking into account the selection effects \nlinked to the presence of scientific civilizations, we have shown that it can \nappear {\\it a posteriori} probable to observe that our universe comes from a very \nlarge fluctuation $L_i \\gg \\ell_s$, with a very small $g_i$. In particular, we \npointed out that string-dualities suggest an {\\it a priori} distribution function for \n$g_i$ and $L_i$ which is phenomenologically encouraging in predicting a short, \nintermediate  De~Sitter-like stringy phase, and the {\\it a posteriori} naturalness of \nobserving $L_i \\gg \\ell_s$ and $g_i \\ll 1$.\n\nOur scenario contains new features that might suggest new observable signatures \nof the pre-Big Bang idea. In particular, we expect the dilaton-driven phase \nto be generically anisotropic. It will be interesting to study the possible \nfossils of this anisotropy, especially in the presence of a short string phase. Our \nwork suggests also new types of criteria for the gravitational instability of the \nEinstein-plus-scalar system. Concentrating on some non cumulative measure of the \nvariation of the news might lead to new theorems of the type of those of \nRef.~\\cite{chi91}, \\cite{chi93}.\n\nOne of the most interesting well-posed technical problems suggested by our work \nis the study of the link between the incoming news function and the spatial \ndependence (along the singularity) of the Kasner exponents. We hope to study \nthis problem in the near future.\n\n\\acknowledgments\n\nWe thank Valodia Belinsky, Demetrios Christodoulou, Nemanja Kaloper, \nSergiu Klainerman, Andrei Linde  and Slava \nMukhanov for useful exchanges of ideas. Discussions with Amit Gosh, \nFederico Piazza, Giuseppe Pollifrone and Eliezer Rabinovici \nare also acknowledged. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzstfo b/data_all_eng_slimpj/shuffled/split2/finalzzstfo
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+{"text":"Gone are the days of clunky, piecemeal solar systems. Every part of the SunPower Equinox\u2122 is designed and engineered by one company to work better together\u2014delivering more power, reliability and style than a conventional home solar system and backed by the industry's best complete-system warranty.\nAt the heart of every system is the Maxeon\u00ae solar cell, designed with a solid copper foundation to stand up to harsh conditions and deliver decades of productivity. The unique light-trapping surface grabs more power. The ultra pure silicon delivers optimal power conversion.\nSunPower's rooftop, ground, carport and battery storage systems provide you with greater choice, better design, and more value on your path to achieving your clean energy future.\nOur solar panels for your home have built-in microinverters hidden behind each panel to optimize power conversion at the source-minimizing disruptions and eliminating the need for ugly external inverters.\nOur proprietary Invisimount\u00ae frame is designed to be completely hidden, producing a sleek look that complements any architectural style. This system will look great on your roof.\nSunPower Monitoring provides a 24\/7 in-depth look at your systems performance, allowing you to analyze the value of your investment.\nThe only solar warranty to cover your complete system for 25 years including performance, product, labor and shipping. We have absolute confidence in our panels' ability to deliver more energy, reliability and savings over time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looks like it connected but \"Unable to start Tor\"\nAdded by teeks99 about 2 years ago.\nI've been running with orbot for a couple weeks on my Nexus 6P, last night I tried to start the same install of orbot I've been running again after rebooting. (Version 15.0.1-RC-3-PIE - 7\/18\/15 - 8.0MB, from F-Droid, see issue #5521 to see how I got to this version) It seemed to start normally, then I got a message in the system tray which had an icon of a stopsign with an exclamation mark and the message \"unable to start tor\".\nUnable to start Tor: net.freehaven.tor.control.TorControlError: Error reply: Authentication failed: Authentication cookie did not match expected value.\nTor (1): Nov 30 06:14:59.337 [notice] Tor v0.2.6.10 running on Linux with Libevent 2.0.22-stable, OpenSSL 1.0.2a and Zlib 1.2.8.\nNov 30 06:14:59.337 [notice] Read configuration file \"\/data\/data\/org.torproject.android\/app_bin\/torrc\".\nNov 30 06:14:59.337 [notice] Read configuration file \"\/data\/data\/org.torproject.android\/app_bin\/torrc.custom\".\nNov 30 06:14:59.346 [notice] Socks listener listening on port 44262.\nNov 30 06:14:59.346 [notice] DNS listener listening on port 38198.\nNov 30 06:14:59.346 [notice] Transparent pf\/netfilter listener listening on port 38755.\nNov 30 06:14:59.346 [notice] Control listener listening on port 39629.\nNot sure what all that means, but at the end I still have the failure message in my system tray and no tor connection.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The next beauty tip has to do with adding a weave. A weave is an extra extension to your already beautiful hair. Sometimes we use extra pieces of hair to accentuate, embellish, add on, enhance, or improve upon what is already there. Some of us need a little extra to take us over the top. Tell your sister, \"Don't hate...celebrate.\" You can wear it long or you can wear it short, you can wear it up or you can wear it down, You can have lots of curls, twist, braids or wear it straight.\nWhat is the life application behind you weave esperience?\n1. Sisters there are times during our busy schedules when you just have to weave in more time for your husband, children or signicant other. We need to know when our Holy Ghost plate is full.\n2. Set time aside for communion and fellowship with God.\nAsk God and seek him to know His will and purpose for your life.\n3. Spend time worshipping, praising and loving on God, don't ask for anything.\nWhen Esther was taken to the palace, she immediately caught the attention of Hegai, the person in charge of the harem. She pleased him much and won his favor. He provided extra beauty treatments and special food for her. He assigned to her seven maids selected from the king's palace in the harem. (Esther 2:8b-9). When a man or woman's ways please the Lord, there isn't anything that He won't do for them.\nThe Bible says, \"That he will make even your enemies to be at peace with you.\" Surrounded by favor from heaven in her season of preparation, Esther flourished while learning the customary ways of diplomacy of the palace. Girls, you can't just look good on the outside, you need to get something in your head and work your ministry. Your weave might be tight, it might be off the hook! You might feel like you got it going on, but looks alone cannot take you to the next level.\nEsther could have had the attitude that says, \"I don't want to spend any more time trying to beautify myself to please no man. I am beautiful already!\" It don't take all that! But No, she humbled herself and did exactly what the custodian of the haren requested. She weaved in the necessary time to make sure her ways pleased Hegai and the Lord, more importantly.\nWe have to sometimes go the extra mile and extend ourselves.We have to put on love and bowels of mercy. We have to ask the Lord to perfect that which concerns him. The \"Fruit of the Spirit\" is the \"covering,\" the mindset that will help transform our minds and hearts to be a servant pleasing to the Most High God.\nI don't want to give you all of these beauty tips too fast, because I want you to read the Book of Esther and ponder over her life. But since it is the weekend, I will give you one more before I end this session.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Choosing the right Nylon holster for Taurus PT92 Compact is not a simple task, so you should better bet on craftsmanship and quality that is proven by years. With quality holster, you can rely on your self-defense setup in every situation and your gun will have a safe shelter for a lifetime. Our offer includes 82 Taurus PT92 Compact Nylon concealed carry and duty holsters from traditional European holster producers, known for their quality materials and hand crafted manufacture. Holsters above are made of cordura nylon material. It features a waterproof finish on the top side, a slim foam layer inside and soft a anti-scratch finish on the bottom. Taurus PT92 Compact Nylon holsters are custom made for this gun model for perfect fit and smooth draw and allows you to wear your gun in comfortable and safety way. You can choose between IWB, OWB, Ankle, Cross Draw, Drop Leg, Duty, Light & Laser, Magazine & Ammo, Molle, Paddle, Shoulder and Small of Back holster designs for Taurus PT92 Compact. See all holster designs for Taurus PT92 Compact.\nThe Taurus PT 92 Compactis a double action, semi automatic pistol that is well constructed. It was designed with a well balanced features that makes this pistol very reliable. Result of the development of the firearm designers is the pistol which is popular in Brazil and also among plenty of shooters in several countries. Design and materials were quite well chosen due to the demands of shooters. It can be used for the military, law enforcement or civilian use as well. This gun is chambered with the 9mm ammunition and offers a quite accurate firing. This model has a double stack magazine which is able to held quite huge amount of ammunition.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"All of our furnishings are included - chairs and a natural wood lectern for your ceremony, dining tables and chairs or benches, additional seating on terraces and outside the pub. Festoon lighting in communal areas, chandelier tealights in the giant hat tipi, candles in the cedar barn and lodge barn.\nNatural elements feature strongly in our building design - stone, slate and timber providing a backdrop for our wild flowers, candles, tableware and wicker and wood features. These lend themselves beautifully to your own simple, natural decorations, but our design and styling of the spaces need little embellishment if you want minimal stress on your wedding day.\nWe make full use of fire in our cooking and heating, but also in our outdoor spaces, making a feature of communal firepits and large firebowls. We will light fires, candles and tealights at appropriate times in your schedule to create the perfect atmosphere as you move between the spaces.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzuaiw b/data_all_eng_slimpj/shuffled/split2/finalzzuaiw
new file mode 100644
index 0000000000000000000000000000000000000000..b6c4e7948ff33dff2f624bae5e450ee7ee0bf55b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzuaiw
@@ -0,0 +1,5 @@
+{"text":"Source Unknown. City to be Given Painting. 30 July 1938. Rumbold, Frank M., General Collection. A1331. Missouri History Museum, St. Louis, Missouri.\nDescription Two newspaper clippings concerning a portrait painting of Brigadier General Frank M. Rumbold. Soldiers in the 128th Field Artillery originally presented the painting for permanent display in the Soldiers' Memorial in St. Louis. Frank Meeker Rumbold served as Colonel of the 128th Field Artillery during World War I. He later served as Adjutant General of Missouri from 1925 to 1927.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's been a long stressful day at the office, school or perhaps at home and you're in need of relaxation, where does your mind wonder?\nImagine turquoise waves lapping up the soft exfoliating sandy beaches and the wind gently breathing against your sun kissed skin. You enter the lagoon and begin swimming with an array of dainty tropical crustaceans and multicoloured fishes. Your mind has reached the summit of serenity and you come down to earth and realise you're in your own bathtub.\nToday I was kindly sent a few products from the 'Calming and Relaxing' range of the Jainnisa brand.\nThe calming and relaxing range are enthused with exotic oils and extracts deriving from Thailands natural plants and marine ingredients. Having suffered from a a migraine all day, I couldn't wait to sit back in my bath and experiment with these products.\nFirstly, I would like to draw your attention to the back of the package. As well as displaying all the necessary safety instructions for the use of these products, the Jainnisa range does not test on animals. Having been a vegetarian for almost eight years and loving every second of it, it's lovely to see a product range with the sentence we all long to hear embellished neatly where everyone can see it.\nThe Sumptuous bath elixir has extracts of Blue Lotus and frangipani, Ylang Ylang, Tuberose, Mimosa and jasmine oils. The product promises a luxurious bathing experience that will calm and revive you and I am happy to report that it did exactly as it says.\nAs a child, I loved bubble baths that changed the colour of the water and as I've grown older, I've discovered bath bombs that do the same.A little goes a long way with this beautiful product, simply add one to two teaspoons of this product into your bath and watch it transform into the colour of Indian Ocean. The turquoise waters will transport you to paradise and you can breath in the exotic fragrances as you relax into your dream environment.\nThe Elixir gave us the beautiful sea coloured bath and now the exfoliating polish gives us the luxurious sandy texture to our Thai paradise. Filled with exfoliating Bamboo granulates and essential oils of Jasmine, Tuberose, Ylang Ylang and White Champaca, the exfoliating body polish feels fabulous against your skin. The oils give us a luxurious tropic scent and the bamboo granulates grazed against your body help buff your skin to holiday perfection. After using this product, I was surprised how soft and revived my skin felt. I found the fragrance really relaxing, however at first it quite strong, but settled nicely once applied to my skin. With 300ml of this product and a beautiful pump on the top to gently dispense, I can guarantee it will last quite a long time.\nThe Jainnisa 'All Day Soothing Hand Balm' has extracts of Orange flower, Coco and Shea butters, all working together to make your hands soft and revived. The essential oils of Frangipani, White Champaca and Ylang Ylang make this product smell divine. So far, I have only used a small amount of this product, but it has already worked and made my hands feel visibly softer. One thing I loved about this product is the fact that it moisturisers your hands without leaving them feeling greasy or overloaded, the product simply seeps into your skin leaving a nice and refreshing feeling. Playing guitar is one of my biggest passions in life, so unfortunately I cannot use too much hand cream as it will soften my fingers easily and make playing quite sore, but I will happily risk that every once in a while, because I'm in love with this feminine, exotic scent.\nBeing a traditional girly girl, I love products that look pretty, deliver to a high standard and smell fantastic. The Jainnisa brand definitely delivers to a high standard, smells fantastic and it's also reasonably priced. A little goes a very long way and I know that is the motto for many products, but with these, it's definitely true. I could see myself using more from this range in the future and for once, I'm struggling to see a negative side to latch my opinion on. Overall opinion? Beautiful and great on a budget...oh and they go with my bathroom colour scheme!\nWhat do you think? Have you tried these products?\nThat sounds ridiculously relaxing! I love that the products don't test on animals either. What a wonderful way to spend an afternoon.\nThanks for becoming my 100th follower and thanks for your comments.\nThis range looks great! I love a relaxing bubble bath after a stressful day! Lavender is normally what I use but might give this one a try!\nlovely fantasy. i want to indulge myself now!\nMmmmh, reading this I was in heaven for a few seconds.\nOhhh, these look delicious lovely!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here's most of the seeds for the 2011 garden. There are about 80 different varieties for this year.\nOf course, this doesn't include garlic, potatoes, onions, gladiolas and other bulb\/tuber\/perennial crops. If I were more motivated, had fewer things to do, I'd list all the varieties here!\nAlthough it might look small or not apparent here, there are thousands of tiny buckwheat seedlings starting to grow. After we pulled the garlic, we replanted just 5-6 days ago with buckwheat and it's already up!\nBuckwheat is a great summer cover crop because it loves the heat, provides good late summer forage for the bees, and lays down some nice seeds for the chickens to pick and scratch in late fall as they lay down next year's fertilizer!\nOne of the sure signs of spring is the emergence of the rhubarb.\nHere's a particularly stunning photo of the wrinkly rhubarb \u2013 the leaves look suspiciously like the egg from yesterday's entry!\nI've finally looked at the local Craigslist ads and I think it will be a dangerous thing for me!\nOur first purchase was about 50 bales of moldy hay for fifty cents a bale. A storm a few weeks ago ripped a roof off a hay shed and soaked the hay \u2013 it started molding, so it was on a fire sale. I know that moldy hay can be dangerous to the lungs, so I decided to store it outside, rather than bring it into a confined space and since it will be used for garden mulch, outside storage won't hurt it for that use. I might throw a tarp over it. It spent a couple of days drying out on hay wagons before I stacked it.\nIt's time to consider more garden space. We are also looking at some less labor-intensive space.\nHere's our plan. We mowed four foot wide strips with four foot wide grassy areas. The spacing is such that we'll till up four foot strips and have grassy steps in between so we'll be able to do tractor work and keep the tires on the grass instead of in the garden. So we mowed down the new garden strips and loaded it with compost, then covered it with black landscape fabric to kill the grass by next spring. This picture shows a couple of strips covered, a couple with compost, and a couple just mowed.\nAfter the rash of organic and sensitive crops being mistakenly sprayed by aeriel spray planes (not to mention the large crew of field workers as well), IDALS, the Iowa Department of Agriculture and Land Stewardship, has implemented a sensitive crops directory that aeriel sprayers are asked to consult before spraying.\nThe department provides the signs at a discount and leaves it up to handymen like myself to figure out how to get the sign 8-10 feet off the ground with the sign at a 30-40 degree angle from the post. I just bought a heavy piece of 10 ft PVC vent pipe and\u00a0a 45 degree elbow. I cut about two feet off the pipe, mounted the sign to the small piece then connected the two pieces with the elbow. Then I drove a steel fence post into the ground and just slipped the PVC pipe over it. I'm hoping I can get away with not attaching it to the post so I can slip it off in the winter to spare the sign and PVC some weathering. \u00a0The PVC\u00a0seems heavy enough and puts enough torque against the post that it doesn't seem to want to spin around at all.\nThe last few days have been back in the 70s after the long early April cold snap. We still are not sure how far along the fruit tree blossoms were when the cold weather came (teens at night and 3 days without getting above freezing). We should know in a few days to week or so if they saved the flower buds.\nThe rhubarb will be fine, it's already sending up new growth from the growing point, even though the first few leaves are brown and dead.\nThis year, a number of our fairly new peach trees developed wounds on the south side of their trunks. While I was at the Small Farm Conference I spoke with someone from the Missouri experimental fruit station after a grafting workshop who told me to paint the south sides of the trunks white because the hot winter days followed by cold were causing the splits in the trunks. He said you don't need fancy paint, interior latex works as well as anything.\nIt looks kind of strange to see all the white trunks offering striking contrast to the autumn browns.\nToday might be the last day in the mid-70's for a while. I used it to take care of lots of nagging little jobs. Fixed the oyster shell dispenser that was leaking oyster shells on the ground outside of the coop, painted the south side of the peach trees white, moved all the chicken tractors in the barn along with most of the summer outdoor furniture, started on rabbit guards on the new trees, put another coat of poly on some bookshelf shelves, and so on.\nHere is a row before pruning.\nThis is the row after pruning. The \"pruner\" in this case is a riding lawn mower! This beats the heck out of individual cane pruning the summer bearing berries. I'll just throw some compost on, and they'll be ready for next spring.\nYou know you are committed when it's 39 degrees with a 25 mph wind and you are out picking the last raspberries of the year! I had to come in three times to warm up the fingers before getting all the raspberries picked. Tonight will be the end of the growing season at last. This was the last garden gleaning before a hard frost.\nThis was part of the day's harvest. In addition to this, some radishes, lettuce, and herbs were gathered. So, today was devoted to freezing the beans and berries, making one batch of salsa, and canning more tomatoes.\nI wasn't necessarily planning on this for today, but something about my upbringing can't let the food go without one last putting away day.\nSeptember 24, 2006 \u2013 Raspberries Won't Die!\nSomeone asked the variety \u2013 they are the old traditional fall variety \u2013 Heritage.\nOne of the neighbors had a \"block\" party this afternoon(A block meaning a couple of miles square) for all the old and new neighbors to get together. We had some a few years ago, but hadn't had any for a while, so it was nice to have the tradition picked up again.\nWe did get a patchy frost \u2013 some raspberries had frost, others didn't. Some tomatoes had frost, others didn't.\nWe won't know until tomorrow what was really killed and what survives to bear again.\nOur raspberries refuse to surrender. We've been picking since mid-August. On Monday we picked 2 gallons and today (Friday) 6 more quarts. This is from a 50 foot row!\nThe oldest peach tree was ready for harvest as well. Three crates of peaches from it this year \u2013 about 1\/2 or less of last year's harvest from the tree, but still lotsa peaches.\nSo, time to make more jam, and depending how the mood strikes us tomorrow \u2013 canned or dried peaches. I would not be disappointed if it froze tonight (it won't for quite some time yet). I'm ready for something else besides picking and putting food up (at least for a while). I don't think I'd do very well in a place without seasons. It's nice to anticipate, enjoy, and exhaust each season. I look forward to fall \u2013 typically a time to fix up buildings\/create contrivances in the workshop after the gardens die.\nToday Helen Gunderson stopped by the farm and dropped off two Wealthy apple trees she grafted from an 80 year old tree on her childhood farm. I'll have to ask her where the tree was in this arial photo shown in the previous link.\nWe were able to pick two of the trees to plant at High Hopes. To facilitate this project, Helen learned how to graft and brought these trees from grafted twigs to these nice potted trees over the summer. She is spreading the apples around the state, a modern-day Johnny Appleseed (except the true story of Johnny Appleseed involves lots of hard cider and profit, but that's another story for a diffent day).\nLinda had a late morning wedding (one of her ag students) in Tama, so it was up the Martin, Emma and I to man the market booth (Claire went with Linda).\nIt was a pretty good market day as those things go. Emma alone sold $30 worth of dog treats and cookies.\nWe did have an abundance of plums this week and had made a bunch of plum jam and bought more than a few home unsold, which we are now canning in earnest.\nI told Martin he would get a quarter for each jar of jam he sold. We had some samples on bread and his job was to ask people if they'd like to try a sample. He was very hesitant to start. But even I was a bit taken aback when he asked a young woman if she wanted to try a sample. She did and responded politely that it was good. Then, out of nowhere, he says \u2013 \"Well, if you like it, then you should buy a jar.\" She did.\nTomorrow will be our first visit to the Grinnell Farmer's Market this season. We don't have a whole lot because of the drought \u2013 not many tomatoes, between raspberry varieties \u2013 but some flowers and odds and ends.\nHere's Linda getting some bouquets ready in the barn.\nI thought I'd share a picture of one of the bouquets Linda makes. We sell them at farmer's market and to some people at school and in town.\nThe flower composition varies by the week, but they are always nice!\nOne of the ideas we had last winter when we were researching a farm store was an e-mail ordering service for mid-week to help keep produce moving between markets. We started with two families and it is working great for us \u2013 send them an e-mail as to what's available and they e-mail back what they'd like. Both the delivery and school bouquets are great since we only harvest what we know we can sell mid-week.\nThis week's idea\u2026well, more about that tomorrow.\nIt's time to let some of the flowers in bloom at High Hopes speak for me today.\nToday was a jammin' day. We made a bunch o' jam yesterday and today.\nWe made strawberry, strawberry-rhubarb, and cherry. All day the rain danced around us \u2013 there was enough to make puddles in Melbourne (3 miles away) it rained in Marshalltown for a few hours (12 miles away) and we didn't get any until a small cloud gave a brief shower and we got 1\/10 of an inch! At least enough to settle the dust for a day or so. It also added 100 gallons to the storage tank that runs off the barn.\nJune 18, 2006 \u2013 Cherries are In!\nEven though the strawberries are still coming in, the cherries have arrived.\nLinda picked them to make her favorite jam \u2013 cherry jam. Today to celebrate Father's Day we went to Ledges State Park \u2013 a place we used to hang out when we lived in Ames. I didn't have my camera so no pics.\nWe walked down the creek, surrounded by rock cliffs and lots of greenery. Nice place on a hot day. The road that winds through the park, occasionally goes through (instead of over) the creek. Kids line up and urge cars and trucks to drive fast through the water to splash. We also climbed up to a rocky outlook over the Des Moines River and felt nervous with a four year old.\nLast night's full moon was also known as the \"Berry Moon\" in times gone by. I'll vouch for that!\nToday was strawberry day. This was the biggest one day harvest from the patch so far this season. This strawberry season we've made canned strawberry sauce, froze whole berries, made jam (strawberry and strawberry-rhubarb). About the only thing left are fresh strawberry margaritas!\nLinda is being diligent about getting the fruit in. Claire made strawberry shortcake from scratch for dinner and the rest are destined for the freezer.\nToday was a bit of clean-up in the wake of re-roofing. It is now very hot up there since there are no windows to open. I moved all the boxes and cleaned up all the nails, dust, bits of wood and asphalt shingles that fell down inside during the re-roof. We had about a dozen sheets of plywood left over and I moved them from the hay rack to the barn and finished hauling the branches that were trimmed to the burn pile.\nSpent most of the rest of the day behind a paint brush.\nHere's Claire a couple of days ago, wiping the stain on the floorboards and window and door trim. Since it was hot and dry \u2013 it was a good day to throw a finish on the trim and beadboard for the attic. We're on our 17th day in a row of above average temps and the grass is finally starting to turn brown. I've got two hay racks full of freshly finished boards.\nThe strawberries have loved the rainy weather of late.\nToday, it warmed up to the mid-70s and Linda was out of school, so it was the first day for a long time we could work on the farm at the same time. We finished weeding and pruning the raspberries, weeded the perennial flower garden, got a few herb, some okra and parsley planted, lawn mowed and other odds and ends.\nThe newly planted raspberries and blackberries are starting to come to life as well.\nToday was one of those days that I felt like no matter how much got done, I'd need another day and still not be where things need to be for the season. It must be the weekend+ worth of rain that set these wheels in motion.\nIt won't be long before the strawberries are here!\nThis shot shows some strawberry flowers and a very small green berry.\nSpring is in full bloom. The plums are finishing up, the lilacs are on the cusp and many others are in their full spring regalia. If you ask me now, Spring is my favorite time of year. Hope is in the air and the bugs are not!\nToday, among other things was raspberry planting. I ordered about 75 feet of fall berries, golden and red, along with 75 feet of blackberries. We hope these spread our berry season out a little more and offer a little more variety.\nHere I am using the wheel hoe to make a shallow trench to plant the berries. Yes, I do feel like a horse. The berries are being planted in the place we had the paper and straw laid down on sod last year and had tomatoes planted.\nHere are the newly planted berries. The new plants will come up from the roots, not necessarily from these stumps.\nere's Claire helping put a little mulch along the side of the berries to conserve moisture and help keep weeds out. We lucked out and had some calm winds at nightfall, so were able get the paper down and mulch.\nMartin had an interesting few days. We had been getting him ready for Kindergarten roundup today and he was very apprehensive saying he \"didn't want to sit at a desk all day.\" He goes to day care two days a week and is home the rest of the week, and at day care, he was hugging everybody and saying good-bye (including getting in trouble with his friend \u2013 see here at this blog entry) by hugging her and not letting go! He thought he was saying goodbye to eveyone at Tiger Tots forever as he thought Kindergarten was starting today, not next fall. In his mind, he was saying good-bye forever, and everyone else thought he was saying good-bye for the day. At any rate, he was excited about school after going through an abbreviated day of books, recess, singing, and snack in the lunchroom.\nIt was a very foggy morning, followed by howling winds today. Good day to work in the attic.\nMay the promise of the season be with you!\nToday, a few more trees were pruned. I'll take the easy way and lead with the kid before showing the before and after pictures! After I crawled up this tree to prune it, Martin wanted to try, so here is the little monkey up the tree.\nThe following pictures show a before and after of a small apple tree.\nI generally am not afraid to cut out 1\/4 or more of the branches out if the tree needs it.\nToday was gravy. It was about freezing with a strong north wind, but it was sunny and there's always something fruitful to do at high hopes. Today I started pruning the dead canes out of the raspberries. The raspberries are on the the south side of the barn and in the sunshine and with the barn blocking the wind, it was more or less pleasant.\nIt was gravy in that January 9th isn't always a time when I can get at the canes. It gives a little more time in late spring for something else. Also continued hauling burnable wood out of one of the old sheds to the burn pile.\nThe summer berries grow one year without fruit, the second year they fruit, then they die. It's good to get the dead canes out to give the new ones more room to grow, help prevent disease, and make it easier not to have to move around dead canes. It's a good task since it is not a \"must do\" during a particular day or week, like picking strawberries.\nToday we (Linda) started digging the gladiola bulbs out of the garden.\nWhen we planted them this spring, they were less than half the size they are now.\nIt's nice to see so much growth \u2013 and we can use them again next spring and will probably get bigger glads. There's still more to get and freezing ground is not far away.\nToday, nothing was \"urgent\" so lots of smaller things got some attention. I wrapped tree wrap around the peach and nectarine trees \u2013 hopefully that will help with sunscald and winter heat\/cold damage.\nI also put permanent metal tags on a number of trees, worked some more on the siding on the corn crib, but have reached the height that I can do solo, started sizing up the roofing on the old machine shed, put some summer furniture away for winter. It was another gorgeous day.\nOne of our peach trees has some overripe fruit and they are a magnet for the honeybees.\nThe bees are swarming all over the fruit \u2013 I'm not sure what the attraction is \u2013 whether they are getting moisture, sugar water for themselves or material to make honey \u2013 if anybody knows what they may be using it for, let me know!\nThe first hints of fall are appearing as September nears. Some of the soybean fields are showing a splash of yellow as the drying down begins.\nWe also closed on a home refinance that was and is rife with errors and bad communication. I'll be happy when it is completed.\nOne of this year's experiments was to bend a 16 foot cattle panel over as a half circle and plant gourds on it. Part of the reason was to span the pile of fenceposts destined to be grape fencing.\nIt turned out well, next year it may be worthwhile to make it double-wide and trying cucumbers to get them off the ground and maybe help prevent wilt.\nHere's a view of the fruit from the inside \u2013 I love the filtered minty green color inside the tunnel.\nToday was another 90 degree plus day with plenty of humidity. The girls spent most of the morning in the kitchen making cookies for market tomorrow and the afternoon on the slip and slide.\nThe hot weather is good for our buckwheat crop. After we harvest summer crops in the garden we sow buckwheat since it grows well in hot weather. Here's part of the patch where the potatoes used to be.\nThis picture is a few days old \u2013 today it started to show flowers \u2013 an added bonus is more forage for the bees.\nToday was Claire's \"graduation\" from her two weeks at U of I. The director told us that they were the youngest that the university recruited and each has a $1,000 scholarship should they enroll at Iowa. I wonder if Iowa State will match it and add 10%?\nClaire after the closing ceremony.\nWe \"had\" to go Prairie Lights bookstore where one of Claire's favorite night-time events was a book reading.\nThis morning we decided to try to move some stuff at Linn Street Market \u2013 a place where local farmers can sell their goods year-round at an indoor venue by renting market space. We brought in some raspberries, flowers, and the first apples of the season.\nLinda creating her magic in the barn arranging the flowers. We also vacuum-packed the first few carrots of the year and threw them in the freezer.\nToday after lunch, brother Kraig and family left, Linda and Claire went to Iowa City to drop Claire off at a two weeks writer's workshop, and Grandma Jo took Jill and Emma, so it was the boys most of the day. We mowed some grass, stained the wood for the trailer and sanded the metal to get ready for primer and cooked dinner.\nThe clematis on the garage is going nuts this year. Here is a view from afar.\nWhile I was mowing near dusk, I happened upon some interesting sights. First was this grass laden with yellow pollen. (Allergy sufferers, it must not be quite so beautiful) and then a Great Blue Heron set its wings over the back pasture and swung around a couple of times. I was flattered that the small wet spot we fenced away from the cattle was even being considered by a heron. Later, near dark, I wandered down there and the Heron was roosting in one of the big maple trees on the edge of the pasture.\nToday was the first big flush of putting food away. The cherries are beginning and the strawberries are at peak. We've got strawberries freezing on cookie sheets, cherries reserved for pie, cherry jam made, cherries drying in the dehydrator, and strawberries left over for eating fresh or waiting for an empty cookie sheet.\nEmma's helping out in the strawberry patch.\nGrandma Jo helped out too!\nMartin helped for a short time picking cherries.\nLater today cousin Jill arrives fro m L.A. and Graunt (combination \"adopted\"aunt\/grandma)arrives from Minneapolis as well.\nThat means that it is that sweet time of year for strawberries with each meal, snacks, made into jam, and frozen. The season is so sweet and fleeting.\nToday we made one batch of jam \u2013 that's all the Sure-jelll we had leftover from last year.\nToday the ponderous bee swarm swayed in the wind. We located an empty hive down below the swarm in hopes that they may move in there. They were too high in the tree to try to move. We were weeding the garden when they decided to blow this popsicle stand. They all swarmed in a cloud, moved over the raspberries and took of to the east and were gone in a a matter of a couple minutes. I wonder where they went? Do they send scouts out to find a new place? How do they decide where to go and who to follow? Is it like the Borg from Star Trek? Perhaps its a simple HTTP protocol?\nAt any rate they are gone.\nBlue came home very ragged after a T-Storm yesterday. He looked very sheepish and very dirty. He was bleeding a bit this morning and then disappeared. Emma came home from her week away and wanted to show her friends how Blue jumps, and I had to tell her he was hurt and we couldn't find him. She eventually found him and we brought him to the vet for an exam and some anti-biotics. I don't know if he was hit by a car, had a run in with other dogs or barbed wire?\nEmma took the camera this nightfall and took this picture of some yarrow in the garden from the perspective of an ant!\n\"Isolated light rain showers and light snow showers in the afternoon. Breezy. High in the upper 40s. West wind 10 to 20 mph increasing to 20 to 25 mph in the afternoon. Gusts up to 40 mph.\" Enough about today!\nI've been thinking about this bark on an apple tree at the farm.\nThe bark is starting to get old. There are lichens growing on it. I associate lichens with old places \u2013 for example all lichen growing on exposed bedrock in northern Minnesota.\nThis was one of the first apple trees I planted when we moved to the farm. It was part of a pasture that went to within 75 feet of the house. It makes me think of the consequences of our actions and the understanding that can only come with time and watching life unfold. Anybody can look at a tree, but until you've planted one, watched it grow up and get old, you may not really \"know\" that tree. The time from sapling to maturity gives a deeper perspective. In the past few months, I've seen and cut up some trees that were probably planted by some of the first settlers of this farm and I've planted some new ones. I haven't and probably won't be around to see those trees mature, but seeing the lichen on this tree gives me some sense of the passage of time.\nLikewise, I'm aware of the part of life I'm now in \u2013 I've seen a parent die and watched children be born into the world. There's a lot more to see \u2013 like seeing the children grow to fruition, whatever that may be. There's something from this passage of time and long-term awareness that seems to be lacking in our disposable and short-attention span culture. I'm grateful that the lichen on a tree can remind me that delayed gratification can indeed be sweeter and richer than a quick fix. Somehow the apple from the tree planted, watered, and pruned by my hand is no contest in satisfaction to the bag of apples I can instantly buy that have been stored in ethelyne-controlled warehouses for months. That's one reason I like the farm \u2013 there are so many of those things in place \u2013 and more on the way.\nThe wind finally stopped blowing today, so this afternoon, I had a chance to prepare the ground for next year's new berries. Rather than fight with digging up sod, or spraying it, we first put heavy paper over the grass.\nNext, we put some old hay we scavenged from a nearby farm.\nBy next spring, the ground will be ready to plant. These two rows will probably contain fall raspberries and blackberries. How much many more rows to do???\nToday Linda had a dinner meeting, so I came home early from work. Claire made dinner on her own for the family and Martin and I went into the back pasture and finished the fence around the \"island\" planting. We just have about 100 feet of fence by the pine planting to move out 16 feet and the fencing will be done \u2013 still no rain, even though there has been a threat.\nIn the on-going series of spring blossoms \u2013 today it's the nectarine. This is the last year for this tree. It blew over in a storm a few years ago and grew back from the rootstock. We let them ripen last year, but they really weren't any good, but we thought we'd let it bloom this spring before cutting it down to enjoy the profusion of blossoms. It will be replaced.\nThe day dawned cold and clear, but it was a pleasant cold and clear after all the wind. Of course, by the time we got around to some outside work, the clouds and wind returned. We started pruning the last three apple trees. Here's Linda hard at work. Martin has proved himself an able tree-climber, so in a decade or so, I'll have a monkey to climb the trees on my behalf!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Political website design is a great way for a neighborhood candidate to share his views with voters. It is sensibly inexpensive, can be very simple to upgrade and preserve as well as is offered to answer concerns 1 day a day.\nWhatever style you select need to consist of the shades of your logo design and also any complementary colors to make the internet site very easy to read as well as navigate.\nYou will want to consider exactly how to organize the material of your site so individuals can find what they are seeking conveniently on any type of tool, especially their phone.\nSuch an internet site will aid you obtain words out about events, developments and mile-markers gotten to by your campaign. It can likewise enable readers to ask concerns or get involved.\nAre you all set to read more regarding political website design in Glen Rose Texas?\nThe best political project websites include some standard functions. For beginners they supply a method for advocates to offer loan or volunteer their time.\nThey will consist of get in touch with details like mailing address, physical address, telephone number and also e-mail. News release are a wonderful concept. Any type of palm cards or brochures you have can go on your website as well. Information around the candidate, the race and any kind of events must likewise be simple to find.\nYour campaign concepts are additionally essential to consist of. You will intend to include a web page detailing your technique for both running and also being in office. Tell people what you stand for and also why you believe this is the right position to have , what it will certainly influence and also exactly how it will take care of the problems your constituency treatment about.\nDon' t make this everything about you though. The very best point to do is to compose all this material with your viewers in mind. Put it in regards to the impact on them. Do not make it everything about yourself.\nIf you really feel sharing your individual tale is important, do that. Tell it in a story, however after that connect it to the reader.\nAn excellent web site design business will certainly aid you with your web content this way. Make sure as well as take a close consider their proposal to see if they provide that sort of help if you assume you could require it.\nAlso consider their political campaign packages. They may deal organizing, layout, updates, e-mail, autoresponders, blog sites, video clips and also various other marketing offerings.\nNext aim to see if they offer campaign websites examples or a political campaign template.\nWhen you prepare to get going on your Glen Rose Texas political website design, provide us a phone call. We will certainly make it so a lot easier than you believe to get your site up right now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LinkedIn is a fantastic place to build connections, showcase your experience and interests, and find leads to great opportunities.\nThe calling or networking card is an excellent tool for networking.\nThe more you understand about an occupation, industry, or academic program, the more informed your career decisions will be.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzuqef b/data_all_eng_slimpj/shuffled/split2/finalzzuqef
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+{"text":"Yesterday's Tractors - Antique Tractor Headquarters. I've worked on old tractors that I had to go through the electrical system and diagram out how it was wired before I could start troubleshooting it. How is a Wiring Diagram Different from a Schematic? We realize it from google search engine data such as google adwords or google trends. We are a company you can trust and have generous return policies. For tractor models Cub serial number 224401 to 239826 with battery ignition , Cub Lo-Boy from serial number 18701 and up with battery ignition , Cub Cadet 100, 100, 140 serial number 26801 to 45000 , 240, 330, 340, 350, 404, 2404, 424, 2424, 444, 2444, 450, W450, 460, 504, 544, 560, 606, 650, 656, 600, 664, 666, 686, 706, 756, 766, 806, 826, 856, 966, 1026, 1066, 1206, 1256, 1456, 1466, 1468, 1566, 1568, Hydro 70, Hydro 86, Hydro 100.\nMy Dad said that he remembered a time when the tractor got rained on. If you have these, you need to throw them away and replace them with big 2 gauge or better cables. The same wiring harness was used and the F wire was not connected. Al Are you just needing to replace the ammeter? If your Super A is an earlier model it never had a junction box. For the armature end, I left the pulley attached and just worked on the armature as it is removed from the generator body. We used a very nice replacement harness provided by Porch Electric.\nThe forum posts that address this are littered with broken links to what must have been what I need, years ago. A pictorial representation would certainly reveal a lot more detail of the physical appearance, whereas a wiring diagram makes use of a much more symbolic symbols to highlight interconnections over physical look. Sold outright, no core, deposit, or return. Pay attention to the ground connections, particularly the voltage regulator. Individual wires are plastic coated with bundle cloth wrapped.\nWe intend to make a web site valuable for many individuals. Farmall Bob has posted soem very good wiring diagrams as well as trouble shooting guides at I would like to thank everyone for their help and support, especially John cub owner and his diagram. I also opened the band cover on the generator. Does not come with instructions. Plenty of people have tried on-line to find facts, strategies, reports or any other reference for their purposes. If you wish, you are able to share this post to your companion, loved ones, community, or you can also bookmark this page. You may need to go that route.\nWe thank you for your visit to our website. If you do the wrong ones you're likely to fry your regulator. This is restoration quality all wires have original cloth wrap. Tractors: Cub, Hydro 186, 574, 786, 886, 986, 1086, 1486, 1586, 2500A. I've opened the voltage regulator, which my Dad replaced about 2 years ago. These areas undoubtedly interfered with movement of the brushes.\nJust call us at 800-853-2651. Other than that everything else is basically the same as a Cub. Kit includes an Alternator that fits under the hood, wiring harness and alternator mounting kit. They just need some clean-up. You may down load this image by simply clicking the save link or right click on the pic and choose save.\nFor use with lights requiring a ground wire. I've read that you need an analog meter. Al Fuse holder would go below the ignition switch. Copyright \u00a9 1997-2018 Yesterday's Tractor Co. Farmall Super M Wiring Diagram \u2014 Gooddy, size: 800 x 600 px, source: gooddy. This is a reprint of the original manual that has exploded views of all parts on the tractor with numbered pictures giving great detail on assembly and disassembly.\nFarmall Electrical Systems Several frequently asked questions deal with electrical systems. It looks like John T in Indiana may have posted this several times a long time ago, but again, the links are broken. You need to do this anytime the generator or regulator has been removed or the leads disconnected. Within Farmall M Wiring, size: 800 x 600 px, source: gooddy. .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are proud to announce that we are now active participants in the UN Global Compact network. We stand committed to making the Global Compact and its principles part of our strategy, culture and day-to-day operations of AID:Tech. We also will engage in collaborative projects which advance the broader development goals of the United Nations, particularly the Millennium Development Goals.\nWe stand behind responsible business practices, combined with collaboration and innovation, can bring about powerful change in markets and societies \u2013 proving that principles and profits go hand in hand.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Summer is almost upon us and those that are lucky to already enjoy garden doors will know what a pleasure it is to open up your home to the wonders of nature.\nSliding, bi-fold and traditional French windows with fold-back hinges create a new room out of your patio and garden. Come rain and cold weather, the same doors provide the insulation and security to keep unwanted elements out of your home.\nWe offer bespoke doors that will open up a new style of living with top quality double glazing, security fitments and aesthetically designed materials whether PVCu, composite or timber.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Doritos Crash Course 2 is a wipeout style free platforming XBLA game and sequel to the 2010 game Doritos Crash Course.\nMain gameplay is the same as its predecessor. The player controls his own Xbox Live avatar in the 2D game. The player must run an obstacle course and complete it as quickly as possible and without failing. The player is able to jump, run, sprint, slide and duck.\nEvery course is different and requires a different approach. Levels may include obstacles such as fans, gaps, giant hammers, swinging ropes, etc. Levels are bundled together in certain areas, with different themes, including a pirate themed area.\nEvery level has its own leaderboard. As in, every level may require a different time to beat. Bronze, silver or gold medals are awarded to a player based on their performance. If a certain section is missed several times, it can be skipped but the player is called a chicken instead of being awarded a medal at the end.\nEvery level has a set of coins lying on the course. Some may only be collected by first buying different shortcuts or other paths with coins collected on earlier levels. These coins are used not only to unlock shortcuts, but also new levels.\nThe servers of this game closed on 15 October 2014, after the game had been taken down from the Marketplace. This renders the game unplayable since it relied on the servers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Task: to draw a poster for People's Unity Day.\nPeople's Unity Day is celebrated on November 4. The poster congratulates everyone with the state holiday and celebrates the shared national aspirations, passions and hopes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxeer b/data_all_eng_slimpj/shuffled/split2/finalzzxeer
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index 0000000000000000000000000000000000000000..168fdc061079686726b9810c92ad2f6fc78b96dc
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@@ -0,0 +1,5 @@
+{"text":"Taken in July 1987, at the Narrows, Schroon Lake, NY, on our first date.\nI am sitting in the truck listening to the thunder erupt in the distance. The blackening skies and the gentle raindrops spattering the windshield explain the delay in Abigail's track meet. Another boom causes me to add another thirty minutes to the delay. Will this track meet ever get underway?\nI don't like delays. I am the kind of person who likes to get on with things. Let's make a list and get it done. Even with track meets, I want to go and enjoy the races and then head home to get to the next thing. But that is not the way life works. Sometimes the rain and the thunder force us to wait.\nGod loves to give gifts. He gives many to all of mankind. The sun, flowers, blue skies, family, and friends are all examples of the blessings He bestows on all people. But for those who belong to Him and walk according to His ways He gives special attention. He does not withhold any good thing from His own. But sometimes there is a delay to His blessings. Sometimes our all-wise Father sees fit to make us wait.\nTwenty-nine years ago Dad was very involved in Campus Crusade for Christ (now Cru) while studying at Miami University in Oxford, Ohio. In the Spring of 1987 he decided to apply for a summer mission trip to North Myrtle Beach, South Carolina, where he would spend time on the beaches talking to people about the Lord. He eagerly filled out his application and sent it, hoping to receive acceptance quickly. As a second option, he sent another application to a camp in Speculator, New York \u2013 Camp of the Woods.\nTime passed and Dad did not hear back from the organizers of the summer mission trip. He was still hopeful and continued to wait, but finally, when it came down to the wire, he needed to make a decision. Reluctantly, he accepted a position as a youth leader at Camp of the Woods. A few days later the acceptance came for the beach trip, but the die had already been cast. He would be going to Speculator, NY.\nHe did not know why God had delayed the acceptance for his first choice until sometime later, but God had something special for him at this camp. Soon after arriving he met a young, blonde New Jersey girl whom he eventually married \u2013 yes, that was me!!!\nHow thankful I am for God's delays. If that acceptance had arrived a few days earlier, Dad and I would most likely not have met and none of you would be here either!\nThe raindrops are getting stronger. In fact there is some hail falling from the sky. The phone rings and Abigail informs us the track meet is cancelled. What a bummer! But I am not going to worry about it. God knows the reason for the delay and cancellation. One day we may just find out that this unplanned circumstance produced a blessing that was exactly what God had in mind all the time.\nAfter all, twenty-nine years ago God's delay equalled 1+5 blessings \u2013 I met the wonderful man who would become my husband, and with whom I would receive five wonderful children!\nEvery once in a while it happens!\nWe have an auction. We collect payment. We deposit the money in the bank. And a few days later we receive an official letter informing us that a check was returned for insufficient funds.\nWhether you are the writer or the depositer of the check, it is so frustrating to receive this notification. Upon writing or accepting the check, you trust that behind each small slip of paper there is money to back up the promise. But sometimes it just isn't there.\nThere are many different reasons for this occurrence. Sometimes people knowingly write a check without the appropriate funds in the bank. Other times it is just a miscalculation of what is available. But either way the result is the same: a returned check and a big headache.\nI am so grateful that when God makes a promise He always keeps it. If we who believe in Him keep His commands, do the things that please Him, and pray with a request in line with His will, we can have confidence that He will hear our words. Not only that, but if He hears our requests, we can be sure that we will receive what He has promised. Even when the answer is not clearly observable, we can have the assurance that God has heard and will answer our petition in His way and His timing.\nOur God will never willingly mislead us. He will never run short on supply and fail to come through. When we come boldly before His throne of grace we will always find mercy and grace to help us in our time of need.\nEvery once in a while you might give or receive a check that will bounce. But when God says something, His word will never be returned for insufficient funds. Confidently rely on what He pledges. That is a promise you can take to the bank!\nI am fascinated with clocks. When Great Grandma passed away I was thrilled to inherit her grandfather clock. I remember visiting her when I was little and listening to the chimes of the old clock striking every quarter hour. There was something intriguing about the consistent ticking and the slowly moving hands making their rounds.\nThis past week we sold a number of beautiful and unusual clocks for a family that obviously loved these old time pieces as well. Their grandfather clock would have hit most ceilings at 9'7\u2033 tall. There were numerous wall clocks, banjo clocks, and a really neat Dickory Dickory Dock clock where the little mouse from the nursery rhyme moved upwards as the time passed. But the clocks that interested me the most were the \"Mystery Clocks.\" They were different from most and get their name because it is a mystery how they kept time.\nWhy all this fuss about clocks? Besides the beauty of the timepieces and their wide variety of unique construction, the time that these clocks keep governs our lives. Each strike of the hour reminds us that we live in 24-hour intervals. All our activity revolves around those sweeps of the clocks' hands. When we are young time seems to pass slowly. But as we grow older the insistent ticking reminds us of how short life is and the urgency of completing our tasks.\nGod is different than we are. He does not live within the confines of time. To Him one day is as a thousand years, and a thousand years like one day. As we scurry around, trying to complete all that needs to be done, we can often become frustrated when God seems to delay. His way of doing things can often be a mystery. Many times He doesn't answer our prayers according to our time table.\nAs we seek answers for why the ungodly prosper, it may seem that these unbelievers are getting away with their sin. They mock the promise that the Lord will come back, yet God does not seem to respond. But God will do what He says in His Word even if it is not according to our time table. His delays reveal His patience by giving all the world opportunities to repent and turn to Him. In His time He will bring to pass all that He has promised.\nGod does not need a clock to regulate His activity. But if there is a timepiece in heaven, I think it is a mystery clock. Why? Because He works and acts beyond the confines of time as we know it. What is a thousand years to us may only be one day to Him. But like the mystery clock, though we may not understand the inner workings, we can trust Him to accomplish His will at the perfect time. He will never be late or early \u2013 He is always right on time.\nOh dear, Great Grandma's clock is now chiming the hour. Another day is almost done!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Australia's love affair with utes shows no signs of slowing, presenting a prime opportunity for new entrants like the LDV T60 to slot in. While the Chinese brand is far from a household name right now, it could be just one convincing product away from success. Could the T60 be the one?\nWith the local launch of the LDV T60, the Chinese auto giant SAIC is making a big play for Australia's booming ute segment. But is the company's first dual-cab 4x4 good enough to mix it with the established players?\nThe much-anticipated fourth model in the LDV G10 range is arguably the most important. But has the diesel auto version been worth the wait?\nJames Wong gets a first drive of China's latest ute, the 2017 LDV T60, before it arrives in Australian showrooms later this year.\nThe commercial van market is powered by diesel, and this is the country's cheapest oil-burning van - the 2017 LDV G10 diesel manual.\nThe LDV G10 nine-seater looks like amazing value, but does it stack up to used products from name brands?\nThe all-new LDV G10 is good enough to change the way people think about Chinese-made commercial vehicles.\nChina's LDV V80 has plenty of standard features and sharp pricing, but doesn't do enough to beat its proven workhorse rivals.\nThe Chinese made Maxus V80 Commercial MPV is headed to Australia in the third quarter of this year with big ambitions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"High end design, unparalleled quality and durability also in your bedroom! Bedroom wardrobes from CENTRO kitchen are selected based on the strictest criteria, ensuring the finest quality and highest standards for you. Amongst our wide range of of styles and finishes, you can find wardrobes (opening or sliding) in various colors, with many storage spaces and design innovations to suit modern and more classical options.\nAll with fine craftsmanship and functionality. We guarantee outstanding collaboration and consistency in delivery within the desired timeframe. Come and visit us and we will do our best to meet your demands.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How To Reduce Under I looked in the mirror last week and realized that I looked like I had a down pillow under each of my eyes. This isn't the most exciting tutorial, but it does have great ideas on how to reduce puffy eyes, so I had to include it. I was particularly intrigued by the eye patches that were featured in the video and did some research, and I am going to give these Collagen Eye Masks a try! Aloe Vera contains Vitamin E as well as antioxidants and both can reduce eye puffiness. Combined with this natural effect is the fact that the horizontal position that how to remove blackness around the lips we sleep in also causes fluid to accumulate in the head, leading to puffy eyes. The effect of increased histamine around the eyes causes eye irritation, such as watery eyes, itchy eyes and swelling. Drink plenty of water after a bout of crying to replace fluid loss and prevent headaches. Skin peels and laser resurfacing may cause redness, sensitivity, infection, temporary or permanent loss of skin pigment, scarring, whiteheads, cold sores and numbness, according to the University of Maryland Medical Center. The Cold spoon remedy calms the blood vessels and tightens the skin around the eyes. Stick to non-alcoholic, low-sodium treats; both alcohol and salt can cause bloating and water retention, making puffy eyes appear even more swollen. Puffy eyes are usually caused by the way in which a person sleeps as this allows all the fluid to gather below the eyes.\nCool compresses are widely recommended for swollen, puffy eyes, as well as the application of cool tea bags. Nevertheless be careful, swelling around the eyes can be the result of an eye infection. When bags under your eyes contain how to remove blackness from your lips fact and not water, they too are then related to the liver. Through the years, the skin located around the eyes becomes thinner and thinner. Then put the tea bags for about 15-20 minutes on your eyes, it will help in getting rid of dark tannin available in the tea bags helps in discoloration of eyes and also controls the puffiness.\nlips hindi how in to blackness remove from Crying will also make our eyes puff- so don't be too alarmed if that's why you're looking a little extra piqued. If you have sensitive skin, it is also necessary to find a cream that not only moisturizes your skin but also helps eliminate dark circles without causing adverse reactions. The next thing you can do is place thin strip of potato underneath your eyes to reduce those swellings. And those people who've been applying hemorrhoid cream on their face will absolutely want to continue reading. However eye creams having components like Niacinamide, or vitamin B3, and kojic acid can also lighten dark circles. Bags under the eyes may be caused by allergies, a sinus infection or a lack of sleep, according to Better Medicine. Companies such as Transform and Spire Cosmetic Surgery lead the market in eyelid and eye bag removal techniques. Water keeps the body hydrated which will conceal or remove dark circles under eyes. If you're considering Blepharoplasty to get rid of bags under the eyes, there are a few important things you should keep in mind. Dark eye circles aren't usually a cause for medical alarm, says the Mayo Clinic, but they can make you look haggard and tired.\nWearing contacts for too long will result in oxygen deprivation of your eyes which can lead to more serious problems in the future. The cool temperature of the compress pressed against the eye area will help to soothe and refresh overworked eyes. Utmost important factor to avoid puff eyes and dark circles is to have minimum of 8 or 9 hours of sleep, Good sleep always better for health. Home remedies like cold compresses can alleviate puffiness to an extent, but for the more stubborn puffiness, cosmetic treatment may be the lone solution. Paler complexion means more the bluish tint of our blood becomes more noticeable, especially under the thin how to remove black colour from my lips skin of our eyes. When your body retains water, the skin under your eyes may appear even more pronounced, leading to the appearance of puffy eyes. But you need to ask your self if is really worth it because there is the odd chance that the bags could return and also because it doesn't come cheap.\nIt works well on discoloration, acne scars, hyperpigmentation, melasma, freckles, birthmarks, chloasma, old scars, age spots, liver spots, dark spots, dark elbows, knuckles, underarms, rough skin tones, sun damage and brown spots. And the rubbing we tend to do when we're weeping doesn't help to deflate the puffiness. This easy at-home remedy is amazingly quite effective when it comes to relieving puffy eyes. Raw potatoes help reducing any puffiness and also lighten dark circles, thanks to their high enzyme content. Method 1: If you are pondering on how do you get rid of dark circles then there are medical treatments that available in many clinics. To prevent eye wrinkling, apply a how to remove darkness from my lips sunscreen containing a sun protection factor of 15 or higher to reduce the sun's harmful effects on the skin. Now dip a cotton ball into this almond oil and mashed avocado mixture and apply it over the closed eyelids and under the eyes. Another remedy to cure dark circles is by adding mint leaves paste to tomato juice. The phenylephrine will constrict the delicate blood vessels under your eyes, squeezing the fluid out and reducing the puffiness. Some studies believe that under eye bags are caused by impurities and toxins that accumulate in the body over the years. how to remove blackness from lips in hindi Will manifest itself in your skin.\nSometimes surgery or skin treatments like peels, are treatment options for under-eye bags. This is definitely the easiest and an instant relief remedy to cure the puffiness and the dark under eyes. Red Puffy Eyes In Toddlers The dark circles and puffiness around your eyes might be telling the wrong story. High sodium foods, such as potato chips, pretzels, tinned soups, salted nuts and other processed foods can contribute to puffiness under the eye. A simple Permethrin treatment got rid of the mites after an entire year of suffering, although the treatment drug causes its own immune reaction that sometimes feels as though the parasites are back. Try cutting back or cutting out, and you will see the difference in your dark circles. Puffy eyes can sometimes be caused by excessive water retention or excess salt in your daily diet. It may not be a quick fix, but if your under-eye circles are sleep-related, you should see some improvement after a full eight hours of sleep. A good diet is vital; vitamin deficiencies can lead to baggy eyes (vitamins A, how to remove blackness from lips at home K, and E especially), as can excessive intake of sodium. The eye area needs special treatment to ward off signs of aging and dark circles. In the event that your dark circles are caused due to a dried out skin, almond oil can help you with this. If you would to get cheeks that are puffy, there are many ways to go about it that include face or cheek exercise, pinching your cheeks, face yoga, moisturizing your cheeks, scrubbing your cheeks with sugar and shea butter scrub, eating apples, applying aloe vera, cosmetic surgeries, injections, among others. Still, you can solve your eye puffiness within a few hours if you just know how. If under-eye circles are a recurring problem for you, keep a concealer in a shade slightly lighter than your skin tone on hand. It is not a substitute for professional medical advice, diagnosis or treatment and should not be relied on to make decisions about your health.\nFor this natural cure dip cotton puffs in whole milk and place them on your eyes for about how to remove blackness from lips 20 minutes to relieve the soreness and puffiness you might be experiencing. Pick and choose from a long list of remedies to treat each body dysfunction using only natural ingredients without side effects, chemicals or expensive drugs. Occasionally, dark eyes can be a symptom of a larger problem involving your liver or kidneys. I like to turn on my sauna and lay in it for 1\/2 hour with the teabags on my eyes. Eye bags give you a perennial stressed and sick look even if you slept well and feel vibrant and energetic. Or, blend 2 cucumbers, strain the pulp to extract as much liquid as possible, put it in fridge for 20 minutes, and then soak 2 cotton swabs in the cucumber juice and place them over your closed eyes for 15-20 minutes. Anti-inflammatory will be handy in reliving inflammation and pain while antihistamines would be for allergy caused puffiness. There are many types of eye sunscreen, including ways to help reduce dark circles under the eyes that how to remove black skin from lips naturally are caused by the sun. The puffiness of the eyes can be reduced with a glass of ice water and four stainless steel spoons. You should also try not to use too strong or heavy cream, since it is not well absorbed through the skin and the ingredients are too strong for the eyes.\nhow to remove blackness from lips in hindi Tired exhausted eyes by all means have ago at treating, but if it be something more serious you need to leave well alone. Forgot Rubbing Eyes: The continues friction of skin makes it more dark even from your natural colour that is why rubbing the eyes will make a how to remove discoloration from lips friction that will make the pigmentation of the skin darker a lot. While the circles may seem unsightly and cause embarrassment, they usually do not indicate a serious condition, according to You have numerous options for treating and preventing yellow and purple under-eye circles easily at home. By increasing the blood flow and strengthening the eye muscles, the device reduces dark circles and puffy eyes both for men and women.\nTannin, one of the main ingredients in tea, is a natural astringent and may help to reduce puffiness and tighten the skin area. The Truth in Aging website says Eye Rescue Formula effectively treats dark circles at a reasonable price. If the necessary steps are taken to avoid eye puffiness it can save a lot of time and money. Do everything you can to reduce the stress in your life, which (I know) is easy to say and hard to accomplish. The coldness experienced around the eye when a cold spoon is placed on it helps to tighten the skin around it and reduce puffy eyes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yesterday I outlined a bold plan to cut spending and balance the state budget.\nIf you're tired of spineless politicians making a mess of the public purse, you'll love my plan.\nThis entry was posted in Constitution Party and tagged Colorado, plan, Tom Tancredo on September 23, 2010 by RedPhillips.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Ownership is about the only power we small humans have. But when wielded well it is the power to walk as image-bearing children of the Most High God rather than defeated sinners who can't lift our heads off the pillow in the morning. Ownership is not waiting for something to happen \"to\" us but instead allowing God to do something \"through\" us. So what do I mean by ownership?\nOnce upon a time a long lost uncle died and left his unsuspecting nephew a beautiful beach house. Ta da! This excited nephews now owned water-front property. It was his beach house. About two years later, a huge hurricane came and demolished that house, along with every other house within a five-mile radius. This man still owned the house. He also owned the decisions for what would happen next with that property. Nothing was going to happen to him for this mess to change. It wasn't his fault that the hurricane destroyed everything. But the house was still his responsibility. Change would happen through his actions as he took charge of his pile of rubble.\nOwnership of our painful pasts are just like the beach house. Often, especially when we are children, we did not cause the hurricanes of our lives. We were given the Breath of Life but then it seemed the events that followed left us destroyed. We do not own the hurricanes, we own the pile of rubble. We do not blame ourselves for what others have done to us. We do not own the actions and intentions of others. We do, however, own the pile of rubble left behind. We own our feelings, our actions, our words, our coping strategies, and our thoughts. If we own them we can do something about them. If we sit and stare at the mess, waiting for the hurricane to fix it, we will always be waiting. We will always feel entitled without any change. Ownership is our power for life to get better.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Even by far the most delicate pieces of jewelry is going to be Read this article to discover some suggestions about jewelry; no matter whether it be your understanding of jewelry and be capable of pick tasteful things.\nAfter you go shopping for sterling silver jewelry pieces, you need both a discerning eye along with a tiny magnet. That you are capable to detect fake great diamond info pieces of sterling jewelry having a magnet, considering that non-precious metals are drawn to magnets. You are able to identify sterling silver by its markings, or alternately,\" \"ster, \"STERLING\". If there is certainly no marking around the piece, be leery of its authenticity since oftentimes it's a sign of a fake.\nA very good piece of jewelry need to be lengthy lasting.Once you appear for any piece of jewelry, you'll want to constantly acquire from a trustworthy store or dealer, so as to assure you only get high-quality pieces. A high-quality piece of jewelry are going to be well-made and feature superior craftsmanship. The jeweler ought to know the history from the piece, such as who produced it as well as the source in the stones. It's important that your pieces are high-quality jewelry if you would like them to final a extended time.\nAppear at what you would like extremely closely, and be sure to examine them to other individuals. Be cautious of tricks that some dealers will use to produce a diamond look much better than it is.Spend focus towards the forms, white gold or yellow, and any other typical jewelry that they wear. This is a good thought of where to start when buying that specific item.\nPut on jewelry about for any day to find out if it hangs properly and is comfortable. This will likely also assist you to to identify in the event the piece holds up in every day environments.You should understand how to finest care for every piece inside your jewelry. A remedy that polishes a single kind of gemstone may perhaps scratch a further stone. Ask a jeweler when you understand how to preserve your products.\nWhen you are presenting a present to someone who has their own unusual style, look at picking a one-of-a-kind piece crafted especially for them. A unique piece like this that matches their personality seems inventive and thoughtful, necklace or bracelet conveys respect for and appreciation on the recipient's creativity and personality.\nAs you've read above, jewelry is a present that is definitely fantastic for many occasions. Even a little piece of jewelry can have a great deal of meaning for the recipient. There are actually also issues you must know about keeping your individual jewelry, too. Either way, take the tips contained right here so as to boost the enjoyment you feel about jewelry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The phrase \"Next year in Jerusalem\" is the final element of the Passover haggadah, something said after a lengthy retelling of the story of the exodus of the Israelites from Egypt following their term of slavery under a cruel Pharaoh. It is meant to indicate a sense of hope that, in the future, all Jews can celebrate Passover in the Jewish homeland of Israel. The words are often sung jubilantly, with the symbolic notion of being in Jerusalem just as meaningful and impactful as the physical act of traveling to the place. In director David Gaynes' new film, both meanings are equally compelling.\n\"Next Year Jerusalem\" documents the lives of eight nursing home residents in Fairfield, Connecticut, all between the ages of eighty-two and ninety-seven. Some are Jewish, some are Christian, but they all have a personal and spiritual connection to Israel and Jerusalem. Several are Holocaust survivors, who recount their persistence and survival despite while they were being persecuted. With varying degrees of mobility and energy, each exudes a sense of positivity and a wonderful attitude towards the life they still have left to live.\nAs suggested by its title, this documentary charts the run-up to a remarkable pilgrimage to Israel undertaken by the residents and staff of the Jewish Home for the Elderly in Fairfield. For some, it's the first time in several decades that they have been to Israel, and much has changed since the last time they were there. For others, it's the first trip altogether. The expressions on each of their faces and the sounds of their voices are truly magnificent, and it means even more to see how the caretakers and staff react to being in Israel, often showing just as much wonder and amazement as the residents.\nThis is the kind of documentary that captures events as they happen, editing them together in a visually and structurally compelling fashion to tell a coherent story. It does not present an argument and try to prove or disprove it, nor does it offer any conflict. These people know that the majority of their lives have transpired already, yet they face the future with hope and optimism, celebrating the now. It's a tremendously moving and inspiring concept, and the film is a sterling showcase of the resilience of those who have faced challenges throughout their lives yet refuse to let themselves be defeated by anything, and certainly not a nonagenarian international trip.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Studio Session Live showcases singer\/songwriters, with a wide variety of original music and insight into the performers. We seek out creators of music who are lead by their passions. We want the music that runs though your mind long after the performance has wrapped.\nStudio Session Live, as always, is centered on the musician and the music. Melody, lyrics, harmony are not just words for us, but what we search for, and bring to you. We showcase local artists, regional artists and artists from across the country, some old pros, and some emerging new artists. We are mostly folk, but that includes Renaissance, Americana, Bluegrass, Country, Celtic, and Country. What they all have in common is exceptional music. We are not looking for artists who are in the middle of all genres, but the artists who are creating music.\nOur hosts will take the audience both inside the music and behind the scenes with the artists and their careers. They illuminate who they are and what makes their music so distinctive.\nWe start our second year of shows. First four were recorded May 19th, and the next 4 on April 9th.\n\"I have been heavily involved in audio for over 50 years. My listening is done on two reference systems that have very low distortion, wide frequency response, and wide dynamic range allowing critical listening.\nThis season will include Kala Farnahm, Emily Barnes, Matt Borrello, Sweet Wednesday, Andy and Judy, Danielle Miraglia, Eric Lee, and Greta Strobel to start.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sexual Assault is not your fault. You were violated and it's not right!\nBrie Larson, who won 2016 best actress academy award for her performance in \"The Room\", a story about a women held in a room and repeatedly raped by her abductor, did not applaud Casey Affleck when he won for best actor this year. Casey was accused in 2010 for sexually assaulting two women who worked on a movie with him. A deal was settled with these women to not take him to court. Of course he denies the accusations. This seems to be the thread among abusers. I myself have not heard of any man admitting to it. The person who molested me did not admit it, on the contrary he tried to make me look like the crazy one who would make this all up because\u2026Hey\u2026I was crazy, I'd been in the bin. Wonder why I was messed up?\nBrie Larson is an advocate for sexual assault survivors. I applaud her for not applauding. It takes courage to do that when the eyes of the whole world are upon you (Academy Awards) Musicians, actors, politicians, people in powerful positions are quite often sexually stimulating to fans and because of all that power and ego gratification it can go to their heads blocking any normal moral compass, they think they can get away with it or worse, they think they are not doing anything wrong.\nThe list is endless\u2026Bill Cosby, Woody Allen, Roman Polanski, Bill Clinton and now we have a president who once asked if it was wrong to be sexually attracted to Ivanka. This was reportedly deleted from a Washington Post column. Whether he asked that or not he's made innumerable gross comments about women and he has been elected president of these United States.\nHere is a question\u2026when is it all-right for a man to assault a woman, sexually or otherwise. Trump was elected, and Casey Affleck just won an Oscar. So in truth, not fake news, we are normalizing the type of behavior both men have exhibited\u2026not just these men but now because they win awards and presidencies sexual assault is the new normal.\nI'm just shining one light on what is so NOT NORMAL, but what women have been experiencing for forever. I hear it in Songbird Sings workshops all the time. Women think they are somehow responsible. Not anymore\u2026not anymore. It is not your fault no matter what. A NO is a NO.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Answers to frequently asked questions of this department.\nQ: How do I add an additional authorization to a Credential?\nQ: Can I just take a test to add an additional authorization to my Credential?\nIndividuals may verify subject-matter competence by passing all portions (in the subject area) of the California Subject Examinations for Teachers (CSET), administered by Evaluation Systems, Pearson. Leaflet CL 674M, CL 674S Testing information can be found at CTC Exams Website.\nQ: How do I become a teacher \/ obtain a Teaching Credential?\nYou must complete a California Commission on Teacher Credentialing (CTC) Professional Preparation Program.\nQ: I would like to be a Substitute Teacher, what do I need to do?\nQ: I am applying for jobs and the job posting says I must provide a Certificate of Compliance. How do I obtain one?\nA Certificate of Compliance is a document created by an employer as a record of a teacher being Highly Qualified per NCLB requirements. Your employer should be able to provide a copy to you.\nIf you have not yet been employed, you wouldn't have a Certificate of Compliance yet. Indicate to your prospective employer that you have not yet been employed; therefore, do not have the Certificate yet.\nQ: Do I need to have a CLAD Certificate?\nIf you have just one (1) English Learner identified student in your class, you must hold the appropriate English Learner's certificate, per Education Code.\nQ: How do I obtain my CLAD certificate ?\nThere are many options to earn a CLAD certificate. Please review the following leaflet for full details.\nQ: I don't have a Teaching Credential- how can I work as a substitute?\nThe term substitute refers to working in a temporary position, filling in when a permanent employee is on leave, or when a district or program needs additional assistance. Classified Substitutes work in positions that don't require teaching credentials.\nQ: How do I apply for a Classified Substitute Pool position?\nApplications are available online through EDJOIN at the EDCOE Personnel Services Department.\nQ: How many different Classified Substitute Pool positions are available?\nThe El Dorado County Office of Education currently maintains lists for 12 different categories. Clerical Classroom Assistant Custodian Food Service Worker Health Clerk Instructional Assistant \u2013 Districts Instructional Assistant (EDCOE Special Svcs. Program) Library Aide Spanish Interpreter Tutor- Early Autism Program Van\/Bus Driver (EDCOE Special Services Program) Yard Duty\/Campus Monitor.\nQ: I am a recent PERS retiree and want to work as a Classified Substitute.\nPERS retirees have a 180 day waiting period after their retirement effective date. Once the waiting period has passed, retirees are eligible to apply for substitute positions, and are limited to working 960 hours per fiscal year.\nQ: I want to work as a Substitute Custodian, Substitute Yard Duty Monitor and a Substitute Food Service. Can I work in all 3 positions or am I limited to only one?\nSubstitutes can work in more than one substitute class as long as they fulfill their primary obligation first; don't work more than 8 hours per day at one job, and the jobs don't overlap each other so they are being paid by two different districts for the same work hours.\nSome substitute positions require testing while others do not. Applicants will be notified of their application status and invited to the next step in the process once an application review has been completed.\nQ: I've already been fingerprinted for another district; do I have to be fingerprinted again?\nLaws prevent the sharing of fingerprint clearances between agencies, so you will need to be fingerprinted again for the EDCOE.\nQ: Where do I get a TB test?\nYou may schedule a test with the County Health Department (530-621-6100) or with a health provider of your own choosing.\nQ: I've lost my Social Security Card. What should I do?\nApply for a replacement card with the Social Security Office 1-800-772-1213 or obtain information online at Social Security.\nQ: After I'm cleared, how will the districts know how to contact me?\nWhen the clearance process has been completed, your name and contact information is entered into the Sub Finder System. Each district that participates in the Classified Substitute Pool will have access to your contact information.\nQ: I'm already a Classified Substitute, but I can't remember my PIN number to access the Sub Finder System.\nSubstitute PIN numbers are generally the last 5 digits of your social security number.\nQ: I've lost my Substitute ID badge, what should I do?\nPlease call the EDCOE Personnel Services Department and request another ID badge.\nQ: When are substitute paychecks issued?\nSubstitute paychecks are available on the 10th of each month.\nQ: I need to change the withholding allowances on my taxes, what should I do?\nYou can stop by the EDCOE Personnel Services Department to fill out new tax forms.\nQ: How do I become a Guest Teacher for the El Dorado County Office of Education (EDCOE)?\nIn order to join our Guest Teacher pool, you must first obtain a \"Request\" for your services from the school in which you wish to substitute. Once you have been requested, you would then need to complete an application on EDJOIN. Be sure to include all of the required documents listed on the job description.\nQ: What is a \"Request\" and how do I obtain one?\nA \"Request\" is a recommendation sent via e-mail to the Substitute Systems Specialist at EDCOE from the Principal of the school you wish to substitute. In order to obtain one, you should take your resume, credential, and any referrals with you to the school and introduce yourself, asking to be added to the school's Guest Teacher list. (Exception: If you are interested in substituting at the high school level, you must receive a \"Request\" from the El Dorado Union High School District office).\nQ: Do I need to be a fully credentialed teacher in order to provide Guest Teacher services?\nAlthough it is preferred, you can substitute with a 30 Day Emergency permit.\nQ: How do I obtain a 30 Day Emergency Permit?\nCTC Fingerprint Clearance EDCOE can assist you in obtaining your Permit once you have been requested by a school and completed an application.\nQ: What are the pay rates for Guest Teachers?\nThe pay rates for Guest Teachers are determined by the individual districts. Please check with the District that you are interested in substituting for its current pay rate.\nQ: Can I file for unemployment if I am a Guest Teacher?\nQ: I'm a retired teacher. Can I provide Guest Teacher services?\nOnce you have satisfied the required retirement waiting period of 180 days, you are eligible to be a Guest Teacher. You are required to follow the same process as other teachers wishing to join our team!\nQ: How can I contact other parents of special needs children?\nThe Community Advisory Committee(CAC) meets throughout the school year and is comprised mainly of parents of children with special needs. The SELPA also maintains a resource list for families. Several parents with expertise in many areas of disability have offered to serve as parent contacts. This is an excellent resource for families. Please call 530-295-2462 for a copy of the list.\nQ: What Special Education programs does the SELPA operate?\nNone. The SELPA serves all the schools on the Western Slope of El Dorado County in an advisory capacity and provides the governance structure for the districts in the region. However, the SELPA does not operate any programs directly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Haven't you always wanted to know how to make framed purses? Or make a zip purse with a perfect finish? These kits will show you how!\nThis adorable little handbag is just the right size for a day or night of dressing up - whether for the races, a wedding or simply for a stylish coffee date! The kit contains the pattern, instructions, 200mm chunky frame. Fabric is not included.\nA cute little 6\" clutch, wristlet or small handbag... a prefect bridal purse, evening bag or coffee date purse, with a surprisingly roomy interior. It can also be a make-up purse or small \"portable craft\" project carrier. The kit includes the pattern, instructions and one 150mm purse frame.\nThe 90mm Barrel Purse is as cute as a button, fits into the palm of your hand but has lots of room inside. It's the perfect pattern for making a special gift (and a great way to use up scraps of special fabric!!).\nThis 200mm chunky silver frame purse makes a bold style statement as a clutch purse, make-up purse or toiletry bag. Purse size 22cm x 14 x 7cm (at widest point).\nThis 200mm Purse Frame Kit makes a lovely clutch or make-up purse.\n90mm Coin Purse Kit This purse is the perfect size for a lipstick, coins and a few essentials. It's a great kit to start with if you want to learn how to make framed purses.\n150mm Purse Frame Kit Ideal make-up purse size, or as a small clutch purse.\nThis kit makes a cute coin purse. It only needs a scrap of fabric and is super-quick to make.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dwyer: They call for continued, fairly high prices as global demand will be a challenge for the world's producers to come up with enough supply that would allow them to go down. They'll stay high. They won't be as high as they were at the 2008 high but they don't need to be to be profitable for producers.\nSFN: As we both know, land costs are going up, not only here in the Carolinas but around the country and that is bound to be a factor at some point.\nDwyer: Yes, because you know, it is part of the cost of production if you keep raising the costs of production you got to rely on those high prices, otherwise you can't make a profit.\nSFN: And then, of course other areas of the world are getting more involved in agriculture, particularly in the export market and South America is something we really need to be keeping a close eye on.\nDwyer: Yes, actually South America, particularly Brazil, as we mentioned earlier, but also Argentina. Argentina has some policy challenges that they have, they tax the devil out of agricultural exports. You know, so you put a 30 percent tax on it when it leaves, that means you don't get world market prices, you get world-less 30 percent. So producers are not getting the world market price in Argentina. They do that though, because the Argentine government is not getting enough tax revenue to keep the government afloat. So they tax the one thing that is very, very successful and that's agriculture.\nSFN: As far as Brazil is concerned, Brazil has an infrastructure problem but they're working on it.\nDwyer: Yes, they are. And I mentioned earlier the VR 63 is a new highway from Monte Grosso up to the export terminals on the Amazon. But it's not just there, that's the one that gets the most of the press play. They're actually making investments all over the country. Farm size is actually very large in Brazil, the average farm is huge. They've got the land. The large farmers are the ones that are really expanding production and they use cash flow many times. They're not even borrowing money, they're that well financed.\nSFN: Of course, Brazil is known for their soybean production. Do you see that they could make a dent in our soybean exports?\nDwyer: They already have. They already have over the last 20 years. Brazil has come on strong, as has Argentina, but largely Brazil. But, the beauty is, is global demand for soybeans is growing so rapidly. It is one of the rising tides that lifts all boats. Just because Brazil is successful doesn't mean we lose. Our own soybean exports are at record levels, largely because of China. China is a game changer in the soybean business. In fact, they are responsible almost single handedly for almost all the growth in global trade in soybeans over the last 10 years.\nSFN: Now something that we've watched the last two or three years is wheat production continues to go up and yet the demand continues to go up. Where is it going?\nDwyer: Egypt. I mean, it goes to a lot of places. Wheat is a good crop for the United States. But what has really captured a lot of attention from farmers is corn and soybeans. And, largely because of biotechnology. A lot of land has gone from wheat into some of the others because of the higher net returns to the land.\nTo see Dwyer's presentation, click here\u2026 for that and much more coverage from this year's Joint Commodity Conference.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Phi\" is the \"Golden Ratio\" 1:1.618. The ancient Greeks considered it to be the most aesthetically pleasing proportion. They designed and built their most magnificent temple in the same ratio of height to width. This bag is lined with a firm calfskin so it holds it's shape. Closes and opens with a unique strap and ring arrangement. Just hold onto the strap pad and the bag will close to the shape you see in the picture. The small size is about 5\" square at the bottom and 8\" tall. The larger size bag is about 6.5\" square at the bottom and 10.5\" tall and includes a pull out zipper pocket.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"George COULTER was born in 1891 at Rangitata, the son of Henry and Elizabeth (nee ASHBY) COULTER. He married Mary MacDONALD on 3 September 1918 at Geraldine. George went to some lenghths to enlist, only to fail to get overseas because of pre-existing medical issues. His brother Henry served in WWI before being invalided home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzywxz b/data_all_eng_slimpj/shuffled/split2/finalzzywxz
new file mode 100644
index 0000000000000000000000000000000000000000..346355346305aec679fa6280bb08a4b766f56b29
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@@ -0,0 +1,5 @@
+{"text":"NZIIU welcomes groups of short-term students to study in various areas of New Zealand. These groups usually stay from 1-4 weeks and participate in conversational English lessons run by NZIIU teachers. Groups also enjoy fun and educational outings, as well as activities in the local area. Group sizes are approximately 15 students to one teacher. Students must be 10 years or older to participate in the programme.\nWe strive to provide the best possible experience for students, enabling them to fully immerse into New Zealand culture, to practice their English skills, and to make the most of their stay.\nWith tutor coordinators in both the North and South Island, we can host short-term students in different locations within New Zealand to suit client requirements.\nOur programmes are very flexible. Whether you require an internship programme, specialist university programme, intensive English programme or assistance within New Zealand, we can create specific programmes tailored to your needs.\nFor students to achieve optimal immersion into New Zealand life, they are placed in a host family. All of our host families are carefully selected. They must undergo police vetting and meet the requirements of the New Zealand Code of Practice before they are given the opportunity to host a student.\nGroups will be based at a local school or community centre. If based at a school, the visiting students will enjoy arranged interactions with the school students, in addition to informal time together during breaks, so they can experience what it's like to be at a New Zealand school.\nNZIIU teachers support students in their language learning, as well as conduct sightseeing trips. Our aim is to provide the best possible experience for the students. NZIIU staff are on call at any hour of the day or night if support is needed.\nIf you're interested in finding out more about NZIIU's inbound tours, please do not hesitate to contact us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"That's how Culver City firm MedMen describes its pot shops in documents submitted as part of the company's most audacious plan yet: to become the nation's most valuable public cannabis company, worth more than $1 billion (U.S.).\nInstead, they'll be on the Canadian Securities Exchange (CSE), a second-tier stock market in Toronto that's become a haven for cannabis companies.\nAdam Bierman, MedMen's co-founder and chief executive, said going public this way makes sense in the cannabis industry, where companies face a tangle of conflicting regulations but see immense opportunity requiring fast money.\nDespite being legal for medicinal or adult use in most U.S. states, cannabis remains illegal under federal law. Because of that prohibition, the major U.S. stock exchanges will not list cannabis firms that operate in the U.S. The Toronto Stock Exchange, Canada's top trading venue, has a similar stance.\nBut the Canadian Securities Exchange, following guidance from that country's securities regulators, said last year that it was willing to list such firms. The exchange is already home to several Canadian companies that have U.S. operations, and MedMen will become just the second U.S.-based cannabis company to list. New York cannabis investment firm iAnthus Capital went public on the CSE in 2016.\nMany more companies will soon follow, said Dan Nichols, a director at cannabis advisory firm Ello in Century City. He expects dozens of U.S. cannabis firms to go public in Canada this year alone.\nChicago cannabis grower and retailer Green Thumb Industries recently announced plans for a reverse merger with a defunct Canadian uranium mining firm. New York cannabis investment firm Acreage Holdings, which recently added former House Speaker John Boehner and former Massachusetts Gov. William Weld to its advisory board, also plans to go public, Weld told Fox Business News last month.\nCompanies are flocking north because they see the Canadian stock market as a source of cheap and abundant capital, something that entrepreneurs have long complained is not available in the cannabis industry.\nFirms that grow and sell pot can't borrow money from banks or raise capital from most institutional investors, which shun the industry because it remains federally illegal. That's left companies to rely on hedge funds or wealthy individuals willing to invest, and Nichols said there simply aren't enough of those.\n\"A lot of investors get scared off, especially larger investors putting millions of dollars at work,\" he said.\nBut in Canada, some big institutional investors are investing in cannabis firms, and there's no shortage of small-time investors who want in, too. They're willing to pay premiums that other investors haven't been.\nSo by going public, companies can raise more cash while company founders give up less ownership and less control, said Mark Hiraide, a Los Angeles attorney who specializes in capital transactions and is working with another cannabis firm seeking to go public in Canada.\nTake Canopy Growth Corp., a Canadian cannabis company that through a quirk in the law listed its shares on the New York Stock Exchange this week. (The move is allowed because the company operates only in Canada, where cannabis is federally legal.) That company is valued at more than $7 billion, despite never reporting a profit and having sales last year of just $54 million.\nIndeed, MedMen's Bierman said going public in Canada was the best option for his company to raise money while giving away as little equity as possible.\nAs part of its deal to go public, MedMen is selling about $100 million worth of new shares, which will give investors a roughly 6 per cent stake in the company. That deal values the company at around $1.6 billion. That's 60 per cent richer than just two months ago, when the company raised money from a Canadian investment firm at a valuation of $1 billion.\nEither valuation represents a big bet on the future of MedMen, which owns or operates a dozen cannabis shops in California, New York and Nevada, as well as growing operations to feed those stores.\nFrom mid-2016 through the end of last year, the company reported total sales of $6 million. In that same 18-month period, it reported a net loss of $31 million.\nIndeed, analysts note that despite the spread of cannabis legalization and the seeming legitimacy that comes with publicly listed companies, investing in MedMen or any young company is risky.\nBierman, though, bristles at the notion that the cannabis equity market is overheated or that investors are being overly optimistic about the industry. Compare cannabis firms to some tech companies, he said, and it's not clear why tech is a better bet \u2014 despite huge valuations for companies in that industry.\nWith its new capital, MedMen will continue to buy and build out retail stores in L.A., New York and other top retail markets. Bierman said raising capital now, and doing so as quickly as possible, will help MedMen lock in top retail locations and win loyal customers, two keys to the company's long-term viability.\nMedMen is one of the country's larger cannabis chains, though not the largest. It's also not the first to go for a higher-end look. Colorado chain Native Roots has a skate-shop look at some of its 20 locations; Blum, which runs seven stores in California and Nevada, has shops that could double as sleek bank branches.\nBut Bierman said MedMen's competitive advantage is less about its look than about its locations. Santa Monica Boulevard in West Hollywood is the hip city's main drag; there's an outlet on Robertson Boulevard in Beverly Hills; and the Fifth Avenue shop is across the street from the New York flagship of Lord & Taylor. It is working to open a shop on the Las Vegas Strip. And Bierman wants more shops in similarly high-profile locations.\nAnd because cannabis shops must generally follow rules that restrict how close they can be to schools, churches, parks or, in some cases, other cannabis shops, he said staking out prime locations now is vital.\nThat was one of the reasons MedMen chose to go public through a reverse takeover, which he said was three months faster than a traditional public offering would have been.\nIt's more than a real-estate play, though. Bierman said the overarching goal is to drive sales by destigmatizing cannabis. That starts with MedMen's marketing. In one recent campaign, called \"forget stoner,\" posters show photos of cannabis users identified as a teacher, an entrepreneur or a grandmother. Underneath each one is the word \"stoner,\" crossed out.\nThe stores, too, are designed to make buying cannabis feel safe, comfortable and normal, he said during an interview at MedMen's downtown Los Angeles store, at 8th Street and Broadway.\nThe store is airy, well lit and well staffed. About a dozen MedMen employees were roaming the sales floor, talking to customers or ringing up purchases. Ten more were downstairs for a training session.\nMost products aren't locked up \u2014 huge tubs of cannabis flower are one notable exception \u2014 and cashiers work behind a counter, not behind a bulletproof screen.\nOutside of the store, Bierman pointed to a younger, clean-cut white guy who had just left and was holding a red MedMen-branded bag.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A modern and stylish option in window treatments.\nBlinds are the best way to regulate and filter outdoor light in your home.\nVertical blinds are ideal window treatments for bay windows, bow windows, French windows and patio doors. Vertical blinds are perfect for covering large and irregular windows. They can be made from materials such as vinyl, polymers, fabric, natural weaves, or aluminum alloys.\nCreate a unique bohemian look with vertical blinds in sheer fabrics, or opt for the durability of vinyl instead. When necessary, vertical blinds can be weighed down with invisible or sewn-in weights which prevents them from being blown around in windy conditions.\nAt The Right Shade, we stock a range of choices in vertical blinds from the leading Canadian brands.\nThe Hunter Douglas Somner \u00ae vertical blinds come with several custom options including material, colour, texture and pattern. Just pick from your choice of vinyl, aluminum or fabric and create the perfect window treatments for your home!\nShade-O-Matic's Collection of Colours Vertical Blinds come in a wide range of 300 patterns and textures. Many of the fabrics used in our Collection of Colours have been designed under Shade-O-Matic's GreenProtect\u2122 program, making them environmentally friendly and safe for your home.\nElite Vertical Blinds are built with 3 \u00bd\" vertical louvers for perfect light and privacy control. Customizable materials (PVC or fabric), patterns and sizes makes this ideal for your home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I get it. Life is exhausting and when you imagine your vacation, it is full of images of toes in the sand and umbrella drinks in your hand. It is hard to think of much else when you imagine \"getting away from it all\". I could not agree more, rest and relaxation is something that most of us do not get enough of. But if your truly want to make your vacation unforgettable, it is time to add a little adventure to it. If you have children, we all know that they require constant entertainment. Resorts throughout the world understand this and have made it extremely tempting and completely reasonable to never step foot off the resort during your vacation. However, you are really missing out on the unforgettable, special moments if you do not venture out and experience something that your unique destination has to offer. My favorite excursions for families are the experiences that you can only find in your distinct destination and if you can learn about the culture and history that is even more exciting!\nA great example is a fun filled, action packed day my family spent recently with AlltourNative in the Mayan Riviera. This tour offered so much in one day, it is hard to believe. There was adventure to keep our adrenaline junkies happy with ziplining and cave rappeling. There was history and education with a tour of the ancient ruins of Coba. There was beauty and culture with a Mayan blessing ceremony, swimming in Cenotes, and a delicious lunch in a local Mayan Village.\nCenotes are limestone caves that are found primarily in the Yucatan Peninsula of Mexico. They are plentiful in the area and beautiful to experience. Swimming in a cenote is refreshing, exciting and a special experience. Fortunately there are hundreds of tours that include cenote swims, but the tour we chose had so much more. I will never forget watching my children drop from a cliff and rappel into a cave all on their own and neither will they. Climbing to the top of the ancient Mayan Ruin, Coba was a little frightening for this mom, but the kids scrambled to the top with ease. Watching the exhilaration on their faces, when they turned around and realized how high they had climbed was worth every moment on the way up.\nExperience like these are exhilarating, thrilling and bonding for families.\nSo put down that delicious beverage for a few hours and go make some memories with your kids! You won't regret it!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 4th WINERCOST Training School launched in Bratislava on Monday 25th March 2018. After an introductory welcome by the Dean of the Faculty of Civil Engineering-TU Bratislava, 25 Early Career Investigators coming from 14 countries and 7 trainers from the most prominent European Technical Universities sat together and will be together for the whole week to discuss the most recent advances on Wind Energy Technology including its Societal Acceptance and its contribution to the strategy of EU to moderate the Climate Change negative consequences. In the photos the Dean of Civil Engineering-TU Bratislava Prof. Ing. Stanislav Uncik (right) with the WINERCOST Chair Prof. Dr.-Ing. Charalampos (Lambis) Baniotopoulos (left) , and the cohort of trainees and trainers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaaegu b/data_all_eng_slimpj/shuffled/split2/finalzzzaaegu
new file mode 100644
index 0000000000000000000000000000000000000000..4662d90eb67c6a7bcf8bbc96af883a11c9d17f2e
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@@ -0,0 +1,5 @@
+{"text":"2,982 full-text articles. Page 124 of 127.\nA novel organic cation transporter (OCT) gene, OCT 6, and use thereof is described. The OCT6 gene is preferentially expressed in human hematopoietic tissues, including CD34+ cells and leukemia cells. Its narrow tissue distribution, substrate specificity, and close homology to other cell membrane transporters make OCT6 an attractive target for the treatment of myeloid diseases.\nBackground: The use of neo-adjuvant chemotherapy in treating osteosarcoma has improved patients' average 5 year survival rate from 20% to 70% in the past 30 years. However, for patients who progress after chemotherapy, its effectiveness diminishes due to the emergence of multi-drug resistance (MDR) after prolonged therapy.\nThere is great disparity between the outcome of childhood cancer in developed and developing countries. Nurses, who comprise the largest group of health care professionals globally, are essential to successful treatment. Pediatric oncology education is generally unavailable for nurses in developing countries. This lack of education is likely to contribute to the disparity in survival rates, as undereducated nurses are unable to meet the demands of pediatric cancer care. A second critical problem in developing countries is the workload of the nurse, with a patient nurse ratio reported as high as 30 patients per one nurse.\nProstate cancer is the second leading cause of cancer-related death and the most common non-skin cancer in men in the USA. Considerable advancements in the practice of medicine have allowed a significant improvement in the diagnosis and treatment of this disease and, in recent years, both incidence and mortality rates have been slightly declining. However, it is still estimated that 1 man in 6 will be diagnosed with prostate cancer during his lifetime, and 1 man in 35 will die of the disease.\nMelanoma is known to be highly resistant to chemotherapy. Treatment with high dose IL-2 has shown significant clinical benefit in a minority of metastatic melanoma patients and has lead to long term survival in a few cases. However, this treatment is associated with excessive multiorgan toxicities, which severely limits its use. We hypothesize that one mechanism of effective IL-2 therapy is through the direct upregulation of IL-24 production in melanoma tumors and subsequent IL-24 mediated tumor growth suppression.\nBackground: Small cell lung cancer (SCLC) is characterized by rapid progression and low survival rates. Therefore, novel therapeutic agents are urgently needed for this disease. Capsaicin, the active ingredient of chilli peppers, displays antiproliferative activity in prostate and epidermoid cancer in vitro. However, the anti-proliferative activity of capsaicin has not been studied in human SCLCs. The present manuscript fills this void of knowledge and explores the anti-proliferative effect of capsaicin in SCLC in vitro and in vivo.\nBackground: Dyspnea is a common and distressing symptom among patients with advanced cancer. The role of bilevel positive airway pressure (BIPAP) and Vapotherm in the relief of dyspnea have not been well defined. We aimed to determine and to compare the efficacy of BIPAP and VapoTherm for cancer related dyspnea.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The plan points to transit's role in a cleaner and more sustainable future, but comprehensive changes in transportation policy are also needed.\nThe framework proposed by the authors of the Green New Deal may be too much for Congress, but many of these polices are already in place at the state level.\nThe heating and cooling of buildings must be addressed if the United States is going to curb its emissions.\nNot one Democratic senator, including sponsor Ed Markey (Mass.), voted on Tuesday to support the resolution \"recognizing the duty of the Federal Government to create a Green New Deal.\" Instead, most, but not all, Democrats voted \"present.\"\nThe San Francisco-to-Los Angeles bullet train may have to be renamed the Central Valley High-Speed Train according to the announcement by the new governor in his first state-of-the-state address.\nEconomic Independence or Auto Dependency?\nA new study calls for \"universal auto access\" to combat poverty. It recommends subsidizing auto ownership or access for those who are economically unable to afford the high cost of owning, maintaining, and operating a personal motor vehicle.\nThe Green New Deal is far from the law of the land, but if this nation were to adopt the legislative agenda proposed by congressional Democrats, it would (and should) have major implications for planning practice.\nSen. Edward J. Markey (D-Mass.) and Rep. Alexandria Ocasio-Cortez (D-N.Y.) introduced a House resolution to tackle climate change that calls for the nation to become carbon neutral by 2030, an ambitious goal, but is it realistic?\nSpeaker-designate Nancy Pelosi has selected Rep. Nancy Castor (D-Fla.) to chair the new Select Committee on the Climate Crisis. Two bills that could advance in the House: the Green New Deal and a carbon tax-and-dividend bill, H.R. 7173.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Porcupine Tree comes from Hemel Hempstead, United Kingdom and was born in 1987. The band members are Steven Wilson, Colin Edwin, Richard Barbieri and Gavin Harrison. Their musical style is mainly considered Rock, Art Rock, Progressive Rock, Symphonic Rock, alternative metal, progressive metal and Metal.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This January the firsts units of inverters were delivered to one of the major aeronautics consortia in the world. The equipment is responsible for converting the DC power supply into alternating current.\nFagor Electronica is currently carrying out certification and completion of aeronautics standard EN 9100 and EN 9110. This certification is required to become an electronics supplier to the aeronautics sector.\nFollowing last year's strategic plan, Fagor Electronica decided to study the possibilities of supplying power electronic equipment to other relevant sectors such as aeronautics, renewable energy, electric transportation.\nIn supplying the aeronautics industry, Fagor Electronica has made a first step in responding to the growing need for electronic power innovations and sensing circuits from its customers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This investment option can be used to profit on price fluctuation of BYC Coin (BYC). You can buy BYC coin at a lower price and selling them at higher price.\nYou can also profiting from downward movements in BYC coin price by selling them at a higher price and buy them again at a lower price and pocketing the price difference. If you wants to profit from short selling, you have to own BYC Coin in the first place. Buy BYC Coin from BYC Exchange with Bitcoins first.\nVisit BYC trading Simply Click on \"Green Button\" to get your Bitcoin deposit address.Once bitcoin sent to bitcoin address you will see unconfirmed transaction appear with pending confirmation on transaction history page. It will be fully confirmed and available for trade with 3 bitcoin confirmations. It can take around 30 minutes or more time depending on bitcoin network so do not worry and wait quietly.\nSimply Fill in buy BYC Coin (BYC) form for your desired price and click \"Buy BYC Coin\" Button.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaausz b/data_all_eng_slimpj/shuffled/split2/finalzzzaausz
new file mode 100644
index 0000000000000000000000000000000000000000..195b0171ea9a7acec08a917b2f79516c0ff65ee8
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaausz
@@ -0,0 +1,5 @@
+{"text":"Anne Louise Barnhardt Mincey, 82, died on Tuesday, May 29, 2012. Anne was born in Durham, NC to the late Max Lloyd and Pauline Bullington Barnhardt on January 30, 1930.\nAnne is survived by her daughters Lou Mincey of Durham, NC, Liz Mincey of Durham, NC; son Warren \"Rusty\" Mincey, Jr. and wife Betsy of Yanceyville, NC; three grandchildren Scott Mincey, Drew Mincey, and Lucas Mincey, two great-grandchildren Carmeron Lucas Mincey and Caleb Mincey. Anne is also survived by many nieces and nephews. Anne was preceded in death by her husband Warren Mincey, Sr and her brother Max Barnhardt.\nA Graveside service will be offered 11:00 AM Saturday, June 2, 2012 from Maplewood Cemetery, 1621 Duke University Road Durham, NC 27701.\nThe family will greet friends immediately following the graveside service.\nIn lieu of flowers memorial contributions in Anne's name may be made to the Hospice at The Meadowlands, 1001 Corporate Drive, Hillsborough, NC 27278.\nThe Mincey family is under the care of Hall-Wynne Funeral Service, Durham. Online memorials, www.hallwynne.com select obituaries.\nI have made the correction to our website. The staff at the Herald Sun will be in later today and I will request they make the change to their website as well. I will contact you as soon as i have this task completed.\nI am truly sorry that this was not done correctly in the first place. I would like to offer my sincere apologies to you and to your grandson. Please do not hesitate to contact me if I may be of assistance and again I am very sorry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Muslims have been getting a bad rap lately, fanned by the illustrious President of the USA.\nConcerned citizens with generous hearts speak up against this negativity and are distressed to hear criticism of the refugee influx into Western countries of people from Syria and other Muslim countries. Their argument centres on the fact that these people are not terrorists, not dangerous, just ordinary people wanting a better life for their families.\nI too don't consider Muslim people to be terrorists in the sense that they do not wish to cause death and destruction. However, my concern is that some Muslims consider women to be essentially the chattels of men and worth less than men. This is visually apparent when we see women wearing a form of veil at all times, from a scarf to a burqa. These garments are a manifestation of the attitude to women held by some Muslim men, and, because we all internalise the dictates of the society we live in, by some Muslim women as well.\nI find it hard to agree with some educated Muslim women's claim that they \"choose\" to cover their hair. When an entire cohort of a group is \"choosing\" the same thing, something other than choice is operating. The choice is not free and unencumbered; it's prompted in my view by acceptance of the idea that a man's honour is preserved only when \"his\" women wear cover.\nI strongly believe that if anyone wants to come and settle in New Zealand, they are welcome; but they must understand and accept the norms of our society. I admire France for outlawing the wearing of the veil in schools and would strongly support a similar law here, extended to include banning the wearing of cover in public places by all women, not just schoolgirls.\nI consider that New Zealand is not a suitable place for people to settle if they want to perpetuate their originating culture's control over women. The idea that women must comply with men's idea of modesty is a notion rejected by New Zealand law and culture. It's not acceptable.\nSome Muslim women may need help to stand on their own feet and take their place in this country as equal human beings with rights and responsibilities. It will not help them if New Zealanders condone what some Muslim men want them to do in the name of cultural respect. If we forbade the use of the veil, women could walk freely on our streets, knowing that the law and the custom are on their side, backing them up all the way. They could join other women as we march for our rights, and get strength from banding together with us.\nMuslim refugees aren't terrorists, but it is vital that they respect and accept our way of life. We have a tolerant and multi-cultural society which is generous in its acceptance of diverse customs and different ways of worship, but it cannot condone practices which go against the human rights and gender equality which are enshrined in our laws.\nWhat can I say? When in Rome . . .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This was the second in-between week of the three-race series I'm doing \u2014 three races, each 2 weeks apart. However, the races get progressively longer. It has made my weekly mileage interesting \u2014 Coach typically has me do a bit of an alternating cycle, but not usually quite this extreme.\nSo, since the week before the first race, my mileage has been: 65, 50, 70, 54, 65; the down weeks were the previous two races.\nThis week was, not surprisingly, pretty blah. Not a lot of pop post-race, but still a decent amount of mileage.\nWednesday: Jasyoga hamstring reset while I was at work. Easy 8 miles after work (66 min \/ 140 AHR), then taught Pilates. It was my foam roller class, which is about 25% standing work, 50% core work on the foam roller (like this), and 25% guided foam rolling.\nThursday: Early morning treadmill session before the longest day EVER. Continuous hills workout. Paying for my good Tuesday workout. I really like doing this workout outside and I have a great hill loop, but it was cold, dark, and windy (I think the windchill was back in the mid-20s, no thanks). My hills were intervals of 1-3 minutes at 4-6%, 7.5-8 mph. 10 miles total (81 min \/ 144 AHR). I keep the recovery time interval equal or shorter than the hill interval when I do this on the treadmill. I do it this way because my normal loop is about 2:30 per side, and this is a close approximation without being too boring.\nThen I went to work, then came back to the gym to teach Pilates, then back downtown (even parked at my office) to see Neil deGrasse Tyson, then picked up dinner, then started the WVU game on DVR at 11 p.m. Bed at 1 a.m.\nFriday: Hahahah I thought it would be a great idea to leave my car at the gym when Mr. Chicory picked me up for the show and just run to it in the morning. Hahahahaha when my alarm went off 5 hours later I was like, who's stupid idea was this? But no choice but to go do it! 6 miles easy (50 min \/ 155 AHR Jasyoga that evening on my deck \u2014 tried the new High Mileage Reset and I definitely recommend!\nSaturday: Started later in the morning but apparently it's spring now and I shouldn't do that. It was 58 or so when I started but close to 70 at the end. 16 total (2:05 \/ 146 AHR), water at 5 and 10, gel at 10 for the electrolytes. Walk .75 back because I didn't want to run extra. Jasyoga back reset after I did 4 hours of landscaping.\nTotals: 65 miles run, 2.5 miles walked, 175 minutes of Pilates and Jasyoga.\nNext Rowing for Runners. Yep. Rowing.\nI love the very real and accurate post race shot!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Every time I thought I found myself, got hold of my life, I end up lost again. Will there ever be an end to this endless chase.\n17 swiss institute. Now with a new name that I saw them hanging while I am sitting on the bawab's wooden bench under the immeuble. I asked what are you hanging? 'Oh, the new name?' And what was wrong with the old one! Probably nothing and the new name also means nothing still. Apart from the famous name but who is the person to change the name of my street to without asking for our permission. It makes me snarl and think that these people have nothing better to do than just changing the names of the streets to some other name, while people still call it by the old name. SO why bother? Do they have to force us to change what we are used to? Was not it enough the detour they did to the streets? Now the removal of old names to the new ones. I wonder what's next?\nFrom Cairo, not funny you know?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Herman Roth with his sons Sandy and Philip at Bradley Beach, NJ, in 1937. The photograph was used for the front cover of Roth's 1991 book Patrimony.\nFor our Story of the Week selection, we present that speech, in which Roth honors the memory of his father, Herman\u2014the middle child of seven born to Jewish immigrants from Polish Galicia\u2014and reflects on \"the manifold processes of Americanization.\" A century ago, Roth reminds us, when \"there were two and a half times as many new immigrants as there were native Newarkers, 70 percent of Newark's schoolchildren\u2014and two-thirds of all Newark schoolchildren were then the offspring of immigrants\u2014didn't make it past the fifth grade.\" He briefly summarizes his father's life story, and his speech offers compelling reasons to read (or revisit) the heartfelt memories of Patrimony itself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabazm b/data_all_eng_slimpj/shuffled/split2/finalzzzabazm
new file mode 100644
index 0000000000000000000000000000000000000000..8bdace58c3f7bd9b4cf0d6b8aa774236dfc00ce7
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@@ -0,0 +1,5 @@
+{"text":"With a name like King's Hotel Center, you have a right to expect royal treatment-- and this Munich hotel delivers. Located within the city center, King's Hotel Center is a treat for history buffs who want to live like a king without spending like one. This three-star hotel features Old World amenities such as bespoke carved wood furniture juxtaposed with high-speed Wi-Fi and air-conditioned guest rooms.\nKing's Hotel Center Munchen features Victorian-style rooms including canopy beds and coffered ceiling detail. The hotel is not part of a chain, and every room's appearance is unique. Staying at King's Hotel Center is like staying in a period piece-- you'll feel as if you stepped back in time into 1890s Munich. Each of the 90 rooms feature original paintings and furnishings. Make it a point to stay at King's Hotel Center more than once to experience the differences in room decor.\nAlthough you may feel like you've awoken in 19th-century Germany when you stay at King's Hotel Center Munchen, its modern conveniences will reassure you that you haven't traveled in time. Listen to tunes while charging your iPod on the included iPod docking station, and don't worry about disturbing the neighbors. The King's Hotel Center features soundproofed rooms. Forget your laptop? Use the hotel's business corner for Internet access and for printing out important documents.\nKing's Hotel Center Munchen offers smoking rooms, but non-smokers need not fear. The hotel separates the rooms on different floors for your convenience. Choose between two room types: Standard and Club. Enjoy flat panel televisions with free and pay movies, handheld showers, hair dryers, complimentary toiletries, and blackout drapes. Rooms come with daily housekeeping.\nWhen visiting Munich, it's a must to visit two locations: Marienplatz and the English Garden. Less than 15 minutes by foot to the historic central square of Marienplatz and even closer to the sprawling English Garden, King's Hotel Center offers a great opportunity to tour Munich without the use of public transportation. King's Hotel Center is conveniently located minutes from Munich's central railway, underground and tram stops.\nAlthough located nearby popular hotels like Jedermann Hotel, what makes the King's Hotel Center superior to others is its combination of old world traditional charm with a modern touch.\n\"Everything was clean\" \"Rooms are clean and comfortable\"\n\"Close to Marienplatz\" \"Close to Munich Central Train Station\"\n\"Nice selection of breakfast items\" \"Breakfast was delicious\"\n\"Excellent vending machine\" \"Quiet courtyard patio\"\n\"A\/C was not working\" \"Nice window views\"\n\"Wi-fi worked well\" \"Free wireless\"\n\"No bar area\" \"There was no bar\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a Korean dish, and I like it.\nWhen I was in Japan, I used 'Harusame' (Japanese noodle) for Japchae.\nBut after my Korean friend had made Japchae by Korean noodles for me, I began to use Korean noodles.\nIt is cheaper than Japanese.\nI used Korean noodles for salad today.\nWas that salad or pasta??\nThis is our lunch today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ancient Greek Sandals are handmade locally by skilled craftsmen using traditional techniques that have existed for centuries. A raw yet feminine aesthetic distinguishes this new sandal brand. The chemical-free, natural tanned leather ages beautifully with time and wear. The collection is inspired by ancient Greek goddesses, nymphs and muses as well myths, pottery, jewelry and sculpture.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When you decide to use our skip hire Ruabon company, you will conclude that we give rapid conveyance of your chosen skip and the finest feasible prices on the market. We provide a free pick up service to all customers within the Ruabon area and can even help you with obtaining an on road permit for any, skip, at an additional cost.\nWe have a wide range of skips for rent in Ruabon. Whatever your skip hire needs are \u2013 we can help!\nWhen you decide to hire a skip in Ruabon, we have 6 standard sizes of skip at your disposal.\nDo give us a call today for the very best skip hire in Ruabon prices and to verify availability.\nWhen finding a skip, it's important to imagine how much waste you are likely to generate, otherwise you can wrap up with a wrong size container, which can only increase the payment. In addition, a skip that's too small will make you with leftover waste that you might do without.\nHow long you will be hiring a skip for is another factor that influences the price. You may feel that the longer you hire a skip for, the more you will need to pay. However, this is not always the case. A large number of skip hire companies now offer regarding longer times, but only consider this option if your project is likely to take a few months to end. Keep in mind that if you place the container on public land, you will need license extensions and renewals from the council before putting the skip on the road. Most councils only issue permits that are valid for a fortnight to at 30 days.\nYou can skip the permit altogether (no pun intended) and just put the container on your own driveway. However, if you have no room outside your property, be sure to let the skip hire company know as they provides you with bags or other waste clearance schemes. Some companies may also bring a truck and wait while you load all the rubbish on it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gloucester is located in the middle of a rich dairy and beef cattle region and is located in Mid-Coast Council, within the Manning district on the Mid North Coast of the state of New South Wales.\nGloucester is situated on the North Coast railway line, not far from Newcastle. The Gloucester River flows adjacent to the town, just above its junction with the Avon River. At the 2011 census, Gloucester had a population of 2,336 people. The region is also known for its coal production and began mining in Stratford a small village 12 kilometres south of Gloucester, and has since spread throughout the region.\nGloucester, known as the gateway to the Barrington Tops and is a charming country town nestled in a valley under a range of impressive monolith hills called The Bucketts. In February of 2016, the Department of Primary Industries (DPI) Water gazetted an Order under section 113A of the Water Act 1912 declaring an embargo (with exceptions) on further applications for commercial water licences for groundwater within the Gloucester Basin Water Shortage Zone. This will have ongoing consequences for the community in relation to access to groundwater for commercial purposes in the Water Shortage Zone.\nIf you would like to increase your water storage capacity for your business or provide additional domestic water reserve capacity for your home, Heritage Water Tanks can offer high-quality water tanks for all your water harvesting and storage needs. Ask us about our water saving gutter technology. Call 1800 115 552.\nWe operate throughout NSW including, Barrington, Barrington Tops, Berrico, Bindera, Bowman, Bowman Farm, Bretti, Bulliac, Cobark, Coneac, Copeland, Craven Plateau, Dewitt, Faulkland, Forbesdale, Glen Ward, Gloucester, Gloucester Tops, Invergordon, Kia Ora, Mograni, Moppy, Rawdon Vale, Rookhurst, Stratford, Tibbuc, Tugrabakh, Upper Bowman, Woko and surrounding areas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzacceh b/data_all_eng_slimpj/shuffled/split2/finalzzzacceh
new file mode 100644
index 0000000000000000000000000000000000000000..acb3e9b6f8f63560f714e65700563277f907f8d2
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"In America, we view animals in two ways. Animals like dogs and cats, are our furry companions, loyal and lovable, and always welcome by our side. Others, like pigs, cows, and chickens, might be cute but at the end of the day they are non-feeling commodities destined for life in family farms. But what happens when these two worlds merge? Well, that's exactly the case in South Korea, the only country in the world known to commercially farm dogs for meat and associated products.\nThankfully, through the efforts of the Humane Society International (HSI) and the Animal Welfare Association (AWA) shelter, three pups were spared the terrible fate in a South Korea dog meat farm and are now adjusting to their new lives at AWA's shelter in Voorhees.\nHaving lived in a cage for their entire lives, Kermit, Hugh, and Irving, are now experiencing true freedom.\nAs part of an international effort to permanently end the farming of dogs for meat, the HSI's Animal Rescue Team worked with a farm in South Korea to negotiate the release of the animals.\nIn exchange for the pups, farmers were taught how to cultivate fruits and vegetables to maintain a living.\nWhile these dogs are clearly in a better place now, their journey has only just begun.\nHaving known nothing but cruelty up until now, these animals will have to learn about love and companionship.\nIt will be tough at first, but these dogs deserve it!\nUnfortunately, it still remains legal to eat dogs in South Korea, so whether these farmers will change their ways will only be told with time. Still, every dog saved is undoubtedly a victory and we hope that in the coming years, even more will be saved as we slowly put an end to this cruel industry.\nall life is the same. Only in different forms. Save it all.\nTradiciones y costumbres malditas. Podr\u00edan hacer el festival del can\u00edbal, as\u00ed de comen entre ellos y a nadie le importar\u00eda.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Joe Thornton has won an MVP award, a scoring title and an Olympic gold medal. But the Stanley Cup has remained elusive as the Sharks have suffered repeated playoff disappointments. After helping San Jose advance to the conference finals last season for just the second time in franchise history, however, Thornton is confident that the Sharks will break through. How confident? He signed a three-year contract extension that actually trimmed his annual salary by $200,000. And for Thornton, it wasn't really that tough of a decision because he sees the team being built for long-term success.\nThePostGame.com: Do you draw any inspiration from other stars who had to go deep in their careers before winning a championship? Guys like Steve Yzerman, Peyton Manning, Kevin Garnett.\nThornton: Nothing comes easy in life. It's been a process. Every year it feels like we're getting closer and closer to ultimately getting our goal, which is the Stanley Cup. But right now the playoffs are still so far away. Right now, we're worried about Tuesday's game against Toronto. But winning the Stanley Cup is what you dream about since you were playing road hockey when you were 5 years old. Do I look at guys who were older when they won? Yeah, a little, but when your head's really in the game, it's tough to look past the next practice or next game.\nThePostGame.com: You missed two games earlier this season because of a league suspension for a hit to the head. There was some confusion at the time about the league's new rule on head shots. In the two months since it happened, has the rule become any more clear to you?\nThornton: It's going to be fuzzy for a while for a lot of the guys. A lot of people said it was a clean hit. Some people said it was a dirty hit. It's such a new rule and I don't think the rule is quite clear. The game is so fast and the players are so big now that it's a really tough call to make. At least they're trying to move in the right direction as far as head shots. But it's still unclear to the guys. As you see more games and see how they're calling it, hopefully it will become much more clear for the players.\nThePostGame.com: You were 23 when the Bruins named you their captain. Now you've become captain in San Jose. Anything from first experience of wearing the 'C' help you now?\nThornton: I was a such a young player back then. As a young player, you really don't know if players look up to you, and maybe you're not sure how to be in the locker room. But being an older guy now, it's much easier. Your confidence is so much higher. You know guys look up to you and respect what you have to say. It's really different being captain at at young age as opposed to now. I'm 31 years old and I'm a much different player and person now. Back then I think they just gave it to me because I was the best player on the team. Some times that's how things go. But I don't think that's necessarily the right way to pick your leadership group. Now I feel like I've earned it.\nThePostGame.com: Pat Burns, who was your first NHL coach, died of cancer in the fall. What's the one thing you learned from Pat that has stayed with you throughout your entire career.\nThornton: I learned from Pat to be a hockey rat -- he made me get to the rink early and stay there late. He really made me consumed by hockey. He was really, really hard on me, but that set me up for a long career. He wasn't easy on me, and as a young player, that's what you need. You need somebody to be firm with you and hard on you and watch you progress. The first year was tough on me, not playing so much. But him being hard on me was probably the best thing for me in hindsight. We had a great relationship after; I got what he was trying to do with me. It's just so sad that he passed away. He was loved by so many people in the hockey world. Just tragic to see him go, but he really fought there at the end. He helped many people along the way. There's a lot of great memories of Pat Burns that I'll still be sharing.\nThePostGame.com: Some of the younger guys on the Sharks like Logan Couture and Devin Setoguchi have started Twitter accounts. Will we see you tweeting any time soon?\nThornton: No. I don't follow that stuff. Sorry. Maybe I should, though.\nThePostGame.com: Some people say it's surprisingly addictive once you get started.\nThornton: I've got the baby now. That's my addiction.\nThePostGame.com: Should the NHL send its players all the way to Sochi, Russia, for the 2014 Winter Olympics?\nThornton: Tough to say. Obviously Vancouver was the best possible thing ever, and the final game did so much for hockey. If the NHL feels Sochi would be good for hockey, let's go. But if not, I don't think it's a wise decision to shut down the league for two weeks if you think it's a bad idea.\nThePostGame.com: The NBA and NHL season practically overlap. If the NBA has its players in the Summer Olympics, why shouldn't the NHL look into that?\nThornton: I played in the World Cup, and that was at the end of August and early September. And the World Cup is pretty much like the Olympics. You're getting all the best players from each country and going at it. I think that might work a little better than having the league shut down for two weeks.\nThePostGame.com: The Olympics have automatic icing. Should the NHL copy that?\nThornton: No, I like how it is now. I like seeing those races. I think it's a good thing for hockey to keep the races.\nThePostGame.com: The instigator penalty. Keep it or dump it?\nThornton: Get rid of it.\nThePostGame.com: If your life depended on someone else winning a faceoff, who would you want taking it? But you can't pick a teammate.\nThornton: Can't pick a teammate? Well, I'll pick Manny Malhotra. I played with Manny last year and he's one of the best there is in the game right now. Back in the old days when I was with Boston, I'd probably take Peter Zezel. Or Joel Otto. Those were two guys I grew up watching too.\nThePostGame.com: You took a paycut in your contract extension that runs through 2014. Is that the ultimate sign that you love it in San Jose?\nThornton: I believe in the team we've got here. Everybody believes in each other, the coaching staff, management and ownership. Everybody wants to stick around and play here. It's a super place to play. The fans are great, and we're going to have a chance to win every year. And that's all you can ask for as a player.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Specialists in branding on the walls. Wall graphics, wall lettering and window films are used to infuse your business brand onto the interior walls of offices, shops, schools and medical centres.\nCategories Listed: 2 categories including: Commercial Artists and Signwriters.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Summer is finally here, but we think about the next season to come already.\nUs women, we like a lot planning ahead of time, especially when it comes to the wardrobe.\nThinking about what we're going to wear next to have all things covered and enjoy the time relaxing is our no 1 priority.\nThat's why this guide comes out now: How to style long sleeve dresses.\nChoosing the right dress for every occasion may seem easy, but when it comes to actually wear it we don't know how to style it, which accessories to choose with it.\nIn the fall\/winter time, all we want to do is cuddle up in warm leggings, so this is the best way to enjoy your long sleeve dresses.\nInstead of opting for trousers and sweaters, a dress would be a more practical choice: you wouldn't have to loose your mind by pairing pants with a top, a dress is one piece only.\nLong sleeve dresses are the transitional pieces that keep you warm effortlessly.\nOpt for winter trendy patterns and pair your dress with comfy and simple leggings, and boots or neutral heels, and you're good to go, both during the day and at night.\nYou don't need to wear scarves or excessive jewelry pieces, with a long sleeve dress you can keep your look simple and still feel that all is put together.\nIf you pick a dress that is itself a statement piece, the outfit will come out perfectly without too much add ons.\nTired of the same boring little black dress? Here you can discover 3 alternatives that will make the difference on your closet.\nAt Rebels Market you can find unconventional and original clothing, all edgy style - as the name states. You can easily express yourself with cool dresses and accessories. Give them a look!\nHow would you style your long sleeve dresses? Tell me below!\nImages: By concession of Rebels Market and taken from the web.\nI love that you say you don't need too many accessories. Sometimes, less is more and simple is the way to go. I think it depends on the detail of the dress, too.\nSuch great tips! I love long sleeve dresses because I'm always cold. Will definitely remembering these suggestions.\nGreat post and tips! I totally agree with you! I love minimal and simple; either for a daily outfit or for a ceremony. It all depends on the dress and how much\/less accessories can you wear <3 -www.elenaslifestyle.com.\nIt's so true. All you need are a few key accessories to make your look complete. It really makes a big difference.\nI love all these dresses and you shared great tips to style long sleeve dresses! Thanks for these tips!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Beauty and functionality come together in the ballrooms and conference areas of our Orlando conference center hotel. You'll enjoy flexible, extensive pre-function space offering an easy flow to make your reception, banquet, gala or general session a memorable event. Each area can accommodate up to several hundred people and offers state-of-the-art technology to keep your attendees plugged in, connected and productive. And you'll bask in the accolades that come from a well-planned, perfectly executed event hosted in our Kissimmee convention center.\nThis ballroom features 29,123 square feet of space and can easily be divided so you can enjoythe flexibility to create a great space for receptions, banquets and general sessions. Your guests and attendees can mingle and network in the pre-function area that overlooks the pool. Lockable registration desks and nine offices, located on the Ballroom level, provide maximum ease and efficiency during the setup and execution of your event.\nOur signature ballroom features 48,505 square feet of space, plus a 42-by-96-foot stage, three dressing rooms and two green rooms. Designed to accommodate the largest, most lavish events, Osceola Ballroom is equipped with an intelligent house lighting system and digital network house sound system based on Peavey MediaMatrix\u2014ensuring your event will sound great and look even better. This ballroom can be divided into six sections, so you have the flexibility you need to set up for any occasion. Seven lockable registration desks and six offices, located on the Ballroom level, provide maximum ease and efficiency during the setup and execution of your event.\nOur most intimate ballroom, at 3,115 square feet, is sized perfectly to host exclusive galas, and with an extensive pre-function space at atrium level, the Orange Blossom Ballroom has become a venue of preference for exclusive charity events and weddings.\nThese four atrium-level conference rooms are located within the hotel in Emerald Bay. Located just steps from guest rooms, St. George Conference Rooms have the flexibility to accommodate meetings from 20 to 168 attendees. Located near the atrium's many restaurants and lounges, St. George meeting rooms are a great place for after-meeting networking opportunities.\nThese eight meeting rooms are part of our exclusive \"hotel within a hotel\" concept that puts meeting space in close proximity to guest rooms, offering attendees a more intimate meeting environment. The Emerald meeting rooms are one level beneath the atrium at pool level.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaemnn b/data_all_eng_slimpj/shuffled/split2/finalzzzaemnn
new file mode 100644
index 0000000000000000000000000000000000000000..a499034eff4e2b2295902bcdef70933f224bac64
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"It's been so long since I wrote a blog post, time to pull my finger out! Today's cook was Roast Chicken. I decided to spatchcock it rather than cook it on the rotisserie as I find it tends to cook more evenly this way.\nI flipped the chicken over and with a sharp knife I cut the backbone out to spatchcock the bird. Flipped back over then rubbed with rapeseed oil and some salt.\nI topped up the Kamado Joe Big Joe with some Oak lumpwood from Oxford Charcoal, lit two flamers and left it to warm up.\nAfter half an hour the KJ was up at 200c so I put the chicken and potatoes in along with a chunk of Oak. About half an hour in I flipped the spuds and added some beetroot in foil with butter. The chicken was coming along nicely and taking on some colour.\nThe Meater probe said there was less than 5 minutes to go.\nTime to take the spuds out, great colour and really crispy.\nAt 70c internal on the meater (and checked with the Thermapen) I took the chicken off. It had taken on some colour from the rapeseed oil and oak chunk.\nSuch a simple cook this one which is what you need for Sunday lunch! The space in the Kamado Joe Big Joe is great for roast dinners as you can fit so much in, I could have added more veg if I was better organised.\nThe potatoes were nice and crispy with a hint of smoke and the chicken was awesome, really juicy with a nice chicken taste that you don't get in the cheaper, low welfare birds. My youngest was inhaling the chicken which is always a good sign!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ops Manager provides a tool to create init scripts that will run your MongoDB processes if you should decide to stop managing them through Automation. The tool creates scripts that can start your processes automatically upon boot up or that you can use to start the processes manually.\nThe tool is available when you install the Automation Agent using an rpm or deb package.\n-boot Configures the init scripts to start the MongoDB processes on system boot. By default, the tool creates the scripts with this option disabled.\nThe tool references the local copy of the configuration file to determine the desired state of the MongoDB processes.\n-d <string>, -distribution <string> Specifies the Linux distribution. By default, the tool auto-detects the distribution. If this fails, specify your distribution as either debian or redhat.\n-h, -help Describes the tool's options.\nThis procedure creates scripts for stopping and starting MongoDB processes and then removes the processes from Automation.\nPerform this procedure on each server that runs processes you want to remove from Automation. If you are removing a replica set, perform this on each replica set member separately, which allows you to remove the set from Automation without downtime.\nTo create the init scripts, run the mongodb-mms-automation-make-init-scripts tool with superuser access. It is recommended that you use the -boot option so that you configure the scripts to start the MongoDB processes on system boot. Otherwise, you will be responsible to manually start each script.\n<process-name> is the name given to the mongod or mongos process in the cluster configuration.\nOn the Ops Manager Deployment page, click the ellipsis icon for the process and select the option to shutdown. Deploy the changes. For detailed steps, see Shut Down a MongoDB Process.\nOn the Ops Manager Deployment page, click the ellipsis icon for the process and select the option to remove it. Deploy the changes. For detailed steps, see Procedure.\nIf you chose to run the \"Make Init Scripts\" tool without the -boot, then you must stop and start your MongoDB processes manually.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In August 1895, Thomas Edison debuted his 'Kinetophone' on Market Street. It was a single-viewer peep show device connected to a phonograph, and it represented not only the first appearance of the moving image within St Andrews, but also the first appearance of what we might understand today as 'sound film' within the UK.\nSince that day, moving images have been exhibited in a variety of spaces \u2013 from travelling shows to church halls to the New Picture House. In this section, we examine the shifting histories, geographies and practices of film exhibition in St Andrews.\nThe Cinema House closes in December, after 66 years of film exhibition in St Andrews.\nThe New Picture House opens on North Street in December 1930.\nThe Cinema House opens on North Street in December 1913.\nThe Roman Catholic church on The Scores is moved to James Street, where it would be used as a skating rink and as the first 'fixed' cinema in St Andrews (renamed La Scala in 1913).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"various scales using multiple fields from a given layer in a service.\nIn this sample you can change the initial labeling of the polygons by selecting a new option in the drop down list.\nIf you would like to see more advanced labeling, select one of the buttons below ('Multiple Fields' or 'Advanced').\nto the server which is generating the map on the fly at runtime.\nNOTE: This is part of the dynamic capabilities of ArcGIS 10.1 for Server, when your service supports dynamic layers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The repeater system is located in New Jersey east of Philadelphia, PA, in Winslow, Camden County. The repeater covers Camden County, the western and central portions of Burlington County, the western end of Atlantic County, Gloucester County, and Philadelphia. The coverage includes most of the Delaware Valley region.\nAfter two decades, in July 2006 the JTRA's 2 meter system moved from Tabernacle, and the call changed to K2AX.\nSevere weather can affect repeater operations. Lightning produced by summer thunderstorm wreak havoc on radio systems across the radio amateur community. Be aware.\nTo protect the system in severe storm conditions, K2AX reserves the right to go silent if and when conditions warrant.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafshp b/data_all_eng_slimpj/shuffled/split2/finalzzzafshp
new file mode 100644
index 0000000000000000000000000000000000000000..026b81f53656914a48049d28f672262db2f90902
--- /dev/null
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+{"text":"Are you in a leadership position at your local church?\nHow did you hear about Epicenter?\nTuition is for three quarters and a 6-8 week Summer Session.\nPayment is due before starting any classes. If paying yearly, the entire amount is due before your first class. If paying monthly, the payment is due by the first of the month being paid for.\nPayment is made online using a debit or credit card or PayPal.\nSince most of the courses are delivered online and live sessions are done live via video conferencing applications such as Zoom or Skype, you must have a fast Internet connection and be able to run Skype and Zoom.\nInstruction, discussion and the website are in English.\n1. Click the Start an Application link.\n2. Complete the Application Form including a reference to someone who knows about your ministry experience.\n3. We contact the Reference.\n4. Your application is reviewed and within 1-2 days you will be notified of the acceptance decision.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to the July 2018 Stencil Blog Hop and Giveaway. Today Altenew released its newest stencils and they are such fun to work with. For the Altenew Scrapbook Blog I created 2 projects using the Elegant Swirls, Feeling Dotty and Flowing Drops stencils. For the layout on my blog I decided to use the Aztec Motif stencil.\nI picked some fun primary ink colours and used my mini daubers to add colour to the different parts. Using these tiny blenders makes the process super easy.\nI flipped the image horizontally and repeated the process to create a mirrored image.\nYou could also continue and flipped it vertically.\nThe title was made with an acetate word from the Reflection Scrapbook collection and I also picked a flower sticker from the Our family collection. I felt that top was a little bare and looked through my Altenew scrap paper stash. I found similar coloured patterned paper scraps to the ink colours and cut our some triangle shapes to create a banner at the top.\nThis layout was so easy to create and I had such fun.\nLeave a comment telling me which is your favourite stencil and I'll pick a lucky winner on July 10th.\nHave fun hopping and see you soon.\nBeautiful layout! I like the stencil you used.\nLove the elegant swirls stencil!\nLove the city scape stencil of this release though your project with the Aztec motif may have swayed me! What a beautiful layout.\nThis is such a pretty layout! I love the great color and look!\nElegant swirls! Very pretty projects!\nYour page is adorable. A great job creating it.\nAbsolutely beautiful layout. Loved it! Thank you much for inspiring and for the opportunity to win!\nI love the photo in your layout... I miss capturing those moments with my kids. (they are grown). I love the swirl stencil, but know I would enjoy all of them!\nFantastic layout! For some reason I wouldn't have thought to use a stencil for a layout!\nSuch a creative and fun layout!!\nI absolutely love the City Scene stencil!\nLove that adorable photo of the kids....The stencil provided just what was needed to fill the white space.....lovely job...thank you so much for sharing..\nVery pretty layout with the Aztec Motif stencil. Your color choices are perfect. Usually don't think to use my stencils on my scrapbook projects. Thanks for the inspiration.\nThe City Scape and Maze stencils are my favorites! So many good ideas on this hop!\nYou matched the colors for this layout so well. Wonderful design.\nA beautiful background. I love the colors you used.\nLove the beautiful layout you created with the gorgeous colors and the bold design. They seemed made for each other on this page.\nGreat layout! It's hard to believe I never thought to use a stencil on a scrapbook page. Thanks for the inspiration!\nGreat use of the stencil on your layout!!\nmy fave stencil from this release is the flowing drops - your layout is wonderful and works so well with your photos.\nThese are some more spectacular and unique stencils. Love the amazing layout that you achieved with this one, and I already have some on my wish list. Thanks for the inspiration!\nLove what you did with the Aztec stencil. Such a beautiful page, full of color and peace! Hard for me to pick a fave, since I have 5 in my cart already, but if I had to, I think it is the hydrangea stencil. I can see using that a lot for backgrounds. But I also love the mega dots!! LOL....Thanks for your inspiration. As primarily a scrapbooker, I really look forward to the scrapbook page ideas here!\nVery pretty layout! The new stencils are all great!\nGorgeous layout and love how you used the Aztec Motif.\nI love that stencil 2nd after the Elegant Swirls.\nGreat layout! Can't possibly pick just one or two stencils \u2013 might have to have them all!\nThis is a great layout! I love this stencil and the bright colors.\nIn the event of a tie bets will likely be void.\nWhat a beautiful layout! Your colors are stunning. I'm so excited about this new stencil release.\nThis is such a CAS layout.. I love it! It has lots of details but looks so clean! I really love all the stencils in this release!\nGreat cards, thanks for the ideas.\nCute layout. I love all of the stencils but your layout is leaning me towards that stencil.\nMy favorite stencil is the cityscape.\ncute layouts. thanks as always for sharing.\nThe Hydrangea Stencil is a new fave!\nGreat layout, love the text that is repetitive. Your grounded stencil allows for some whimsy on the page. Very cool.\nHe City Acene stencil is my favorite, but all of these stencils look like they produce amazing images!\nWonderful page with the Aztec stencil and such a sweet photo! The Elegant Swirls stencil is definitely my favorite, but I think they're all fabulous!\nThe Aztec stencil makes a wonderful scrapbook page design. Love all the stencilling ideas.\nYour scrapbook layout is fabulous. I love the colors you chose for the Aztec stencil and how you turned it around to complete the bottom of your design. The journaling touched my heart! Thanks for inspiring me!\nCute and sentimental use of the Aztec stencil, perfect choice of colors!\nFabulous layout! Love how you switched the stencil up!\nSuch a beautiful page. my fav stencil is the swirls.\nVery fun, colorful layout. Great use of more than one stencil.\nAwesome as always Lilith! Love your background design using the Aztec Motif, that stencil is at the top of my faves list along with Caligraphy Maze . Thanks for sharing! Have a wonderful day!\nAbsolutely a wonderful background to put a picture of your two boys on! It's a tender looking picture & the sentiment you added is just perfect.\nI love the elegant swirls and the hydrangea stencils!\nMy favorite is the Spring Garden Stencil. Your layout is lovely. Love the colors.\nI love what you did with this stencil! It looks so pretty!\nI love what you did with this stencil! It's so pretty on your layout!\nMy favorite stencil in this bundle is the step builder one! I love the geometric designs that can be flipped and give a great 3D feel! Close tie for second are the aztec and elegant flourish ones. You used the aztec very nicely here; well done!\nThe City Scene stencil is my favorite. Great layout page!\nI like the way you used the Aztec Motif stencil.\nI adore this! It's so beautiful! Amazing job!\nLove seeing these used in scrapbooking! A lovely release!\nBeautiful layout! The Aztec Motif is so colorful and the sentiment you used is so lovely and perfect with the picture in your page. My favorite stencil keeps changing at this point, but I think I like the Flowing Drops the best.\nSWEET and PRETTY layout! Great creativity with the stencils!!!\nI love the Feeling Dotty and Flowing Drops stencils! Your page is fantastic. The colors and layout are great!\nBeautiful layout - I love the bright colours you used with that stencils!\nBeautiful work!! I love the flowing drops stencil a lot!\nGreat layout! Love the colors so bright and pretty!\nI really like the molded lines stencil. :) Great layout!\n\"Awesome\" page, and that stencil is perfect for it!\nLove the colors you used, beautiful layout!\nSo many great new stencils!!! I would love to play with the dotty background!\nGorgeous,bold and vibrant layout, just stunning! Love the swirl stencil.\nAwesome background for your layout!\nElegant Swirls is my favorite!\nAwesome design for this layout. I like the colors too.\nI love this bright, fun layout! So pretty!\nThe Flowing Drop is my favorite of this stencil release although there are a few that are close seconds LOL.\nYour layout is a genuine beauty! So creative with the stencils!\nLove the new release. And the inspiration layout is gorgeous! My favorite is the Molded Lines stencil.\nI love the city scape stencil!!. So awesome looking. I can see using for holiday cards.\nGorgeous! I love all the stencils, but I have a soft spot for the Aztec stencil.\nSuch a cute scrapbook page!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Many people appreciate the productivity-boosting Chrome extensions because they can bring information to you from one system while you're working in another, and in so doing, act as a very lightweight integration between the two systems.\nMany ATS solutions are not at all or not well integrated with personal productivity tools, leading to a lot of time wasted in copy-pasting data between G Suite and the ATS. For quite a while, recruiters have wanted a Chrome extension ATS that would enable easy integration between G Suite apps like Gmail and Google calendar, and their recruiting workflows.\nHire is the first Chrome extension ATS that eliminates the busywork of copy-pasting data between systems. Technically, Hire doesn't have a Chrome extension but rather uses the new Gmail Add-on framework and G Suite APIs to provide native integrations with G Suite so that customers can work in their favorite G Suite apps like Gmail and Calendar yet have all recruiting related activity automatically synced and centralized in a dedicated ATS. From the user's perspective, the experience in Hire is the same as it would be in a Chrome extension ATS: while working in Gmail, a Hire panel slides from the right, which allows users to create a new candidate profile, to see up to date candidate profile, advance or reject the candidate, respond with pre-defined email templates or click through to the candidate profile in Hire.\nThe result is that with the Chrome extension ATS, customers can hire faster.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These wings are easy to do and great for a tail gate or party setting and can be made in advance and finished on the grill. Since they are not fried, they are far healthier and easy to prepare. Once they are grilled, toss them in the Spicy\/Salty\/Sweet Thai Sauce and chow down!\nIn a medium bowl, Whisk together Sriracha and Soy Sauces with the Sesame Oil, brown sugar and Melted Butter until brown sugar is dissolved. Set aside or refrigerate until ready to use.\nWhen you are ready to grill the wings, toss them on a medium to low (300-350 degree) heated grill for about 4-5 minutes on each side for good grill marks. Once skin is crispy and grill marks are done, toss in the Thai sauce in a large bowl. Wings will re-heat the sauce. Toss for about 30 seconds to a minute to coat all the wings. Serve with Blue Cheese and any remaining sauce on the side.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ourselves with it, we do not feel that misery.\nIt will not make you more holy.\nIt will not make you less lonely.\nMost of the stories I read in RY that lampoon the foibles of Yoga Zombies and Soulless Hippies seem to me to be getting at the same thing: you can practice yoga without needing to self-identify as a yogi. A mala around your wrist won't make you any more holy than a clerical collar around your neck.\nI never wrote more music, or had more success in the classical music world in which I was trained, than when I worked on a loading dock. I'd get home at 3:30, take a shower, and compose music all afternoon and evening. (And, I could lift heavy things.) But I spent an awful lot of that period of my life being miserable. Given the opportunity to be a working class sage like the garbage man in Dilbert, or Larry Darrell in The Razor's Edge, I squandered it bitching and moaning about \"wasting my life.\" I am a composer, I kept telling myself, as though the fact that I spent hours every day composing had no bearing on that fact. And maybe going back to school and getting a teaching job were the right things to do. And maybe they weren't. I do know that I wasn't as \"successful\" a composer as an adjunct professor as I was when I was working in a tire warehouse; I was only marginally less miserable.\nDeath of a Salesman is the story of Willy Loman, a traveling rep who never makes the splash in the world that he wants to. Though he makes a living, he never becomes the successful, universally known and \"well-liked\" figure he desperately wants to be. Moreover, his son Biff, having rejected his father's profession, is also failing to make something of himself, in Willy's estimation. Ultimately, Willy attempts to redeem his own failure by committing suicide in order to finance a business for his sons with the life insurance money.\nHow does a person travel down the wrong road for years, decades\u2014even a whole lifetime?\nIs there no voice within saying, \"Wait, stop, go back, exit here\"? And if there is, how do we silence it, and why? Like the man who looked for hours under a streetlight for the keys he had dropped in the alley because the light was better there, we waste so much time looking for what we want where it isn't to be found.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagbbu b/data_all_eng_slimpj/shuffled/split2/finalzzzagbbu
new file mode 100644
index 0000000000000000000000000000000000000000..35be8e621961c932a7279b68bb2325363db9bd43
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzagbbu
@@ -0,0 +1,5 @@
+{"text":"Ubisoft Forums > For Honor > General Discussion > Are the Halloween mask outfits staying?\nView Full Version : Are the Halloween mask outfits staying?\nThe content that was going is gone, so yeah the outfit is staying. The only difference to getting it during the event was that it was 5000 cheaper.\nExclusive to halloween was the howl at the moon emote, the grasping shadows effect and I believe another outfit. I didn't get much chance to play, real life and all, but all I wanted was the mask outfit, the howl at the moon emote for my berserker and I was lucky enough to get the grasping shadows effect for my main on execution.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It just may be a luuuuuunatic you're looking for.\nAfter 3 or so weeks of frustration and difficulty, I am back to my usual self. That's the good news. The bad news is that I still have to write a post for my eating disorder blog, and I don't have anything in particular to say.\nI don't eat sugar, and it's great? Yay?\nLook, I don't eat sugar. And it is great. If you do eat sugar, not eating sugar is better than you know, and way way way better than you think.\nI say all the time that I am not the food police, and I do not care if you eat sugar. And I really don't. Even if you're fat. Even if it is killing you and making you miserable. What I really believe in is personal freedom. I happen to be an American, (though I know many of you are not, hi international friends!) and I strongly believe in those American ideals that center around the inalienable rights to life, liberty, and the pursuit of happiness. I mean the liberty to do things that are killing you and making you miserable, as long as they do not encroach on the rights of your fellows. But I have a feeling that even if sugar is killing you and making you miserable, you don't know it. I would even venture to guess that you don't have any idea how much sugar you are eating and how it is affecting you.\nIf you tell people you don't eat sugar, many of them freak out. I am considered a crazy person. And I own that when I talk to people about it. But I am going to be blunt. I may say \"yes, yes, it's crazy, I know\u2026\" with a simper, but that is to make you feel better. Because really, I think people who eat sugar are the crazy ones. I am simply deviating from the norm. But what about the norm is sane in this situation? I especially think people who don't have any idea what they are eating are the crazy ones. If you are eating, and feeding your family, prepackaged, highly processed foods daily, which are generally packed with added sugars, or things that immediately turn into sugar in your body, I think that's pretty insane.\nHere's a question: What exactly do you think you are getting from sugar?\nJoy? Fun? The ability to blend in with society and not make waves? That warm, fuzzy feeling? (P.S. That fuzzy feeling is called being high\u2026) Nourishment? I am pretty sure you know that you are not getting nourishment. Even people who believe a calorie is a calorie think that sugar is, at the very least, devoid of nourishment.\nI am not in a position to have a piece of cake occasionally because I am an addict. I have an unnatural reaction to sugar, grains, and starch. When I put the stuff in my body, I set up a craving for more. But if you, and I know there are a lot of you out there, mostly eat real, whole foods, especially fruits and vegetables, and occasionally have a piece of cake, or a candy bar, or a can of soda, I get it. I am not talking about you. I am sure it's fun, like someone who is not an alcoholic having a drink occasionally. But it's not generally those people who act like I'm some weirdo, or some martyr. Those people totally get it. They think that I am a champion. It's the junkies who think I'm a lunatic.\nI promise that I have more peace, joy, fun, and general contentment because I don't eat sugar, not in spite of it. And I don't even mean more of those things than I had before, though that too. I mean more of those things than sugar eaters. So don't feel sorry for me. And if you do, you can feel free to keep it to yourself. And if you can't manage that, then when I simper at you and tell you how I know that what I do is so extreme, you can feel free to realize that in my head, I'm pitying you more than you are pitying me.\nI get it and I live this life, too. Life without the sugar is a better for me life than with the sugar. I wouldn't go back for anything. My disease wants me back, but I won't go. Society wants me moderating all foods, but I won't go.\nThank you for commenting! It makes a difference too! \ud83d\ude01 And agreed, I am supposed to eat everything \"in moderation,\" but why? First, I am incapable. And second, who does that benefit (except for the junk food industry)? Thank you for reading! It means a lot!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fried rice is so simple yet so hard to make. I know that sounds weird but it is true, the most simple dishes are usually the hardest to make because it is all about the technique since the ingredient list is so short. All you need is 2 \u2013 3 cups of stale white rice (about one large to-go container), three eggs, frozen peas and carrot strips as your main ingredients, you will also need some soy sauce, salt, white pepper, butter and cooking oil but those are usually items should have on hand if you are a cook. The fried rice can be made and served with almost any dish, but this particular dish was made right after making some Chinese Vegetable Soup which is where the Maggi Pulled Pork came from.\nThe key to fried rice is using stale white rice, at least 2 \u2013 3 days old, the most stale the better because you don't want any moister within the rice or else it won't fry properly. The second most important factor is you need a good wok and a hot enough burner because you need high heat to make fried rice. If your stove is lacking is power than I suggest buying fried rice because a weak burner just won't cut it. Back when I lived in my apartment for college I had to go without home made friend rice for over 2 years because the stove couldn't cut it. Once I moved into my condo I made sure my stove was powerful enough to make fried rice and any other dish I wanted to make. Take your three eggs and break them into a bowl and stir it up like you were making scrambled eggs. Heat up your wok with high heat add cooking oil to coat the wok then add in 1 Tbsp of butter then quickly throw in your eggs. Once the eggs are 80% cooked remove them from the wok and place them on a plate. Re-coat the wok with oil and another Tbsp of butter then place it back over high heat. Add in your rice and use a spatula or wooden spoon to break up the rice (no clumps). Keep the rice moving around to evenly coat every piece with oil\/butter and cook the rice till it turns a light golden brown. keep the rice moving so it doesn't burn, stir at least once every 10 \u2013 15 seconds and season with finely ground sea or kosher salt. The salt will season the fried rice but also help to pull out moister from the rice.\nOnce your rice is lightly browned remove it from the wok and set aside. Add in your vegetables with additional cooking oil and cook until they are done. Pour your vegetables onto the eggs and mix them around to let the eggs absorb all the liquids from your vegetables. Add your rice back onto the wok and cook until the rice looks like fried rice done. At this time you want to add salt and just a dash of soy sauce. The soy sauce is really just to add some more color, most of the saltiness in fried rice comes from the salt you will add.\nOnce the rice is done add in the vegetables and eggs and mix it around to evenly distribute the vegetables and eggs into the fried rice. Quickly remove the fried rice from the wok and serve either as a side or as your main course. if you want to serve it as your main course you can add in some meat (i.e. thinly sliced cooked hot dogs, Chinese Sausage, cooked shrimp, Chicken, etc\u2026).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"About a year ago I rented my car out through a third party company (Hiyacar) - have done this a few times without any issues.\nUnfortunately the hirer incurred a parking charge when he outstayed his allotted time at an NCP car park.\nNCP sent me a PCN notice, to which I responded by filling in the section at the back stating that I was not driving at the time, and sent it back to them. Despite this (they have never acknowledged recieving this) they continued to pursue me for this. There was no customer service number or email to contact them on.\nThey have offered me a \u00a3150 settlement - I do not think I should have to pay for this. I am willing to attend the hearing, but obviously don't want a CCJ.\nYou provided the details of the driver at the time and therefore keeper liability no longer exists. POFA 5 (1). The condition in 5 (1) (b) is no longer true.\nHave you a record of the letter that you sent that named the keeper at the time?\nUnfortunately not. It was a section on the back of the PCN I received that I filled in and returned by 1st class post.\nHave you written your witness statement yet? You will include that in your witness statement and also reference the Hiyacar contract and include their paperwork at the time as an exhibit to strengthen the fact that someone else was the keeper.\nAs they are claiming from you as the keeper then the maximum they can claim is the amount of the original PCN + some limited, fixed, court costs. You argue that in front of the judge POFA 4 (5).\nIF it does get to court and IF you lose and IF you fail to pay within a month then only then will you have a CCJ against your name.\nThat's amazing - thank you so much.\n\"The creditor (our client ) may not exercise the right under paragraph 4 to recover from the keeper any unpaid parking charges specified in the notice to keeper if, within the period of 28 days beginning with the day after that on which that notice was given, the creditor is given (a) a statement signed by or on behalf of the vehicle-hire firm to the effect that at the material time the vehicle was hired to a named person under a hire agreement; (b) a copy of the hire agreement; and (c) a copy of a statement of liability signed by the hirer under that hire agreement\".\nAs far as my client is concerned, you have not provided the \"threefold documents \"as stipulated by the above legal provisions within the 28 days of the PCN being issued.\nAre they claiming that you have to submit all 3?\nNo, they are claiming that you have not complied with a completely different section of POFA that applies to hire company cars, section 13 (2). You are not a hire company, though you had hired the car out through a third party.\nYou are claiming that you have complied with the requirement to identify the driver at the time and therefore 5 (1) (b) is no longer true and there can be no keeper liability. This provision is without limit, other than the start of proceedings.\nDid you actually name the hirer of your car as the driver? If you completed their form section for who was driving, and not just say you weren't, then that should be sufficient. You can point out to the solicitor that you are relying on 5 (1) (b) and therefore his client has no case. I think your mention of hiring etc has clouded the issue somewhat.\nThank you - I did give details of the other driver so confident that I am not liable.\nSo what was your defence that you submitted to the court?\nHi - sorry for delay!\nIn September 2017, I received a notice from National Car Parks Limited (ref XXXXXX included in the witness statement of the claimant), informing me that I had incurred a parking penalty for parking at one of their car parks. At the time of the infringement, I was renting my car through the third party car hire firm Hiyacar. As I was not driving the car, or present, I did not enter into any contract with National Car Parks Limited, did not accept their terms and conditions and am not liable for this charge.\nOn receiving the said notice, I filled in the section of the reverse of the notice (\"what if I wasn't the driver\"), giving the name and address of the hirer of the car, and sent back by first class return the same day I received the notice.\nDespite me informing National Car Parks of the fact that I was not driving or in the car at all, they have continued to pursue me for the charge. This has led to the court action, of which I have had to devote considerable time to defending.\nI include in this statement the hire agreement provided to me by Hiyacar which confirms that I was not driving the car on the day of the alleged contravention.\nI would like the money claim set aside.\nI believe the facts in this statement to be true.\nPOFA states that the keeper is liable for the charge. It also define the keeper as \" \"keeper\" means the person by whom the vehicle is kept at the time the vehicle was parked, which in the case of a registered vehicle is to be presumed, unless the contrary is proved, to be the registered keeper\". You have shown that someone else was the keeper, as well as the driver, at the time of the alleged breach. You can reference the statement to this effect from the hiring company when you create your witness statement.\nWas the statement in post #9 your defence and have you sent it or is it your witness statement.\nHi - it's my draft witness statement - I was planning on sending it in today.\nI include in this statement the hire agreement provided to me by Hiyacar, exhibit xxxnn, which confirms that I was not driving the car on the day of the alleged contravention and mr nnnnnn was both driver and keeper at the time.\nMany thanks - have amended the statement as suggested.\nI am due in court on monday about this. The only thing that concerns me is that I sent the initial response to NCP (notifying them that I wasn't driving and naming the driver) by post, and NCP claimed they never received this. What would be the best defence I can put forward in response to this?\nDid you notify them more than once?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"VIP diver and 'in house photographer' for Aqua Planet Dive Centre and Charter, when it comes to Protea Banks.\nPierre \u2026 or Peter as he is fondly known around the dive centre \u2026 is a seasoned traveler who is very enthusiastic about scuba diving and playing golf \u2026 oh yes, and a phenomenal cook with authentic dishes that he pairs perfectly with wine.\nIn 1995, during his university holidays he worked for Club Med in the Bahamas as a Golf Instructor. It was here that he did his first dives \u2026 at that time he did not have corrective lenses so did not see much \u2026 however many years later he did return to see what amazing wonders the first dive site he ever dived had to offer.\nWith his Open Water Course completed in early 2004, Peter has since traveled to 158 countries, by his own count and he has dived 65 of them! Still logging all of his dives, he is now sitting on 1031 logged dives over 15 years. \"Diving is about freedom to move yourself \u2026 we don't fly, but we can dive!\" says Peter. His bucket list is getting shorter, but his top 3 top dives sites still to do are: Raja Ampat, Vanuatu and Solomon Islands.\nOnce you get to know Peter you will grow to like him even more than we do.\nSome of Peter's pictures taken on Protea Banks.\nDolphins ... they seem to like Peter.\nScuba diving with Dolphins on Protea Banks.\nBull shark on Protea Banks.\nRound Ribbontail Rays on Protea Banks.\nBaited shark dive with Oceanic Blacktip sharks.\nSpotted Eagle Rays on Protea Banks.\nShoaling Hammerhead sharks on Protea Banks.\nGuitar shark on Protea Banks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahdej b/data_all_eng_slimpj/shuffled/split2/finalzzzahdej
new file mode 100644
index 0000000000000000000000000000000000000000..140a6597039c02adb546590b2b670a1e1e7d393e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzahdej
@@ -0,0 +1,5 @@
+{"text":"Allison Jones Rushing was a presenter at Duke Law's Supreme Court Review with Neil Siegel, Director of Duke Law's D.C. Summer Institute on Law and Policy, and Donald Ayer, Of Counsel at Jones Day and U.S. Supreme Court Litigator. They discussed the most significant decisions of the past term of the U.S. Supreme Court. The event took place on Wednesday, July 27, 2016.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Playing at the Surf and Sand Resort on the 4th of July!\ni mentioned in my previous post that i am several videos behind schedule. this is a good issue, as it is a reminder that we are fortunate to have a lot of work. i've decided to try a different approach in regards to this video creation.\ni will start posting videos of our most current events quickly, and once a week i will post another video of our past events. i believe in a few weeks, i will be caught up with all of my videos. yay!\nso here is our most current video at the moment. fun times at the beautiful Surf and Sand Resort in Laguna Beach, California. not too bad of a place to enjoy the 4th.\ntoday we performed for a private event at the Surf and Sand Resort in Laguna Beach, CA.\njust another \"day in the life\" of Steel Parade.\ntoday we performed poolside at the Surf and Sand Resort in Laguna Beach, CA.\nthanx to Michelle Frank and her staff at the Surf and Sand Resort!\ntoday we played poolside for Memorial Day Sunday at the Surf and Sand Resort in Laguna Beach, CA.\ntoday's \"view from the pan\".\nPhil on the steel drum.\nit was cloudy all morning, but the sun broke through as we started to play.\nonce again, our thanx to Michelle Frank and her staff at the Surf and Sand Resort.\ntonight we performed for the Directors of the Riverside Community Hospital at the Surf and Sand Resort in Laguna Beach CA.\nChris on bass\u2026\u2026\u2026he kept his face out of the sun, and in the shade.\nour thanx to the cool people at Riverside Community Hospital, as well as Michelle Frank of the Surf and Sand Resort!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MIAMI (AP) \u2014 In convenience stores spawned by the wellness wave, kombucha slushies take the place of corn-syrupy treats infused with red dye, tortilla chips are made of cassava flour instead of corn, and there are vegan ice cream bars, a dizzying selection of organic produce and craft beer on tap.\nA new crop of niche stores aimed at millennials who can afford to pay more have completely overhauled store shelves, making gluten-free and organic products their staples, not just the side dish, along with compostable straws and on-demand delivery.\nHundreds of such stores are popping up from Miami to Los Angeles, and experts say they expect their numbers to grow.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"7.345 acres for sale. Own a piece of Salado Texas. This land is on the Corner of Brewer Road and FM 2484 in Salado Texas. Land has many mature trees and is flat. Trees throughout property are beautiful. Property does have a roping area on it that will convey with the property as well as a two car garage and a storage shed. Owner will remove the 1983 double wide mobile home and single wide mobile home from the property 90 days after closing and funding has occurred. This property has 1 active well, and 2 septic's. Access to Salado Water all across the front and private garbage service. This property is in a Great location, close to Salado's historical downtown shopping, Salado creek, lake, schools, restaurants and much more. A must see Today. Builders and developers are most welcome and we invite all offers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pedro Vieira Dias Tom\u00e1s (Mestre Petchu) is one of the greatest authorities in the field of Angolan culture of music and dance. Currently living in Portugal, he was the first there to teach kizomba and semba; he also set up the methodology of teaching these dances. He is the one who contributed to Angolan rhythms being introduced to dance parties and dance schools in the whole of Europe.\nFor 25 years, he's been the leading choreographer of the African ballet, Kilandukilu; he taught numerous dance workshops, judged tens of competitions and also was invited to prestigious TV programmes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaiqcf b/data_all_eng_slimpj/shuffled/split2/finalzzzaiqcf
new file mode 100644
index 0000000000000000000000000000000000000000..9ede64f374739ae8233d1d80401841c7ffdb9a92
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Sometimes I wish I worked in a less whiny profession. And it isn't the kids who whine. I'm talking about other teachers. I mean, really people, just because you work with kids all day doesn't mean you need to act like one.\nIs it just NYC schools, or are teachers generally whiny in other places, too? I don't remember this happening when I worked in publishing and for the non-profit organization.\nWe get a resident artist to come to our school to teach African dance. One of the other teachers called it a waste of time. I said I loved it. The whole room got quiet. And they asked me why. I said, \"It teaches the English Language Learners directionality, it gives kids a different way to learn. It ties in directly to our social studies curriculum\u2026\" That's all I could think of at the time, but since I've been home, more has come up. It is total physical response! It teaches them to recognize patterns and rhythm, which connect directly with math and poetry. It teaches them to LISTEN for cripes sake. If they don't listen, they mess up, and everyone knows that they weren't listening. When one is spending a ridiculous amount of time doing test prep, it feels damn good to stomp one's feet. The teacher is African-American, but he sometimes wears a shalwar, much like many of our students' parents. And it makes them realize that Islam is not isolated in their countries, it is all around the world.\nSorry you had to listen to my rant. This is what you get for making me blog every single day in an otherwise uneventful month. But seriously, don't freakin' knock the arts when I'm in the room. I will knock you down.\nThe arts are definitely important. I recently opened a school with a focus on the Theory of Multiple Intelligences. We all strive to align our teaching practices with students' learning styles. It is truly a gift to walk about the classrooms and see students engaged in a variety of creative activities\u2014e.g., music (iMovie, GarageBand), movement, etc.\nOn another note: Yes, teachers can be difficult at times. I can sure tell you that, as a site principal, it isn't the parents and students that keep me up thinking at night. It is getting adults to be nice to one another and sacrifice for the benefits of the kids.\nI just started blogging and ran across you site. It is interesting and you have a lot of great insights. Thanks!\nI think teachers are whiny everywhere. I can't speak for everywhere really, but I used to teach in NC and knew teachers from Houston and Baltimore.\nIn fairness, I actually became disrespectful to other adults for the first time in my life when I became a teacher. I was always one of those children\/people who would dutifully listen and pay attention during a class or during a speech or whatever, even if I was bored out of my mind or inwardly felt like my time was being wasted. Once I became a teacher, if they brought in someone to do pd and I didn't feel like it was worth my time (it often wasn't \u2014 in part, because I taught special ed. in a middle school and they didn't care at all if I got any professional development \u2014 only the students with a shot at passing the state tests mattered and that wasn't my kids), I would talk during the workshops, write notes, grade papers or plan lessons and not even pretend I WASN'T\u2026 I did turn into one of my students, in part, because my time felt so limited those first few years of teaching\u2026 I really did need every second I had. I was very resentful any time it was wasted and I was forced to sit through someone talking about disciplining students who didn't have to deal with some students who were chronologically 13 but mentally 5 mixed in with others who had behaviorally-emotionally handicapped as their primary handicapping condition mixed in with LD students so phobic about school after years of feeling (wrongly) stupid that they will cause any disruption or even stick their fingers down their throats to make themselves vomit to get out of having to show me what their writing looks like.\nAnyway, I think it is interesting that I never acted out in class until I became a teacher.\nI don't know why teachers would whine about that resident artist though. Sounds awesome. The 3rd graders at Lo's school work with an African dance teacher, too, and perform at the school's Kuumba Day celebration. I love watching them every year.\nCo, Your comment made me realize that a choice I recently made is a little rude. We have grade meetings. Totally unproductive grade meetings in which nothing gets done but bickering. I've decided to bring work with me to the meetings. That way, at least I can plan while they bicker. I've been teaching for 10 years, and I still feel as though I need every minute without the kids to be a useful minute. I really do like my colleagues on a one-to-one or one-to-two basis. Altogether there's something unfortunately toxic about this grade's teachers as a group. I had wildly productive grade-level meetings in the second grade. Mind you, my school is huge, with nine teachers on each each grade from K-3, where the classes are capped at 20.\nteachers are the worst students. and the whiniest, i'm sure of it. although, i have to say, in retrospect, my NYC colleagues seemed much less whiny than my affluent-ish suburban school colleagues. entitlement has to do with that, perhaps.\nWe tend to get whinier as the amount of work and irrational expectations mount, yet the physical impossibility of fitting it all into the day disallows any of it from getting done. I love enrichment like you describe above \u2013 I only wish there was more of that and less of the stuff over which we have no say.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"All invitations are digital and nothing will be mailed to you. After placing your order, please e-mail: purplemonkeyshop@gmail.com with all the required information. Not providing the information will result in a delay of your file. Each listing specifies what information is needed. Once information is received, you will receive your file within 24-48 hours. Please do not place your order if you need sooner as I may not get to your order in time. However, most orders are e-mailed in less than 24 hours. All files are saved as JPGs unless requested otherwise.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What's in a name? And is it all smoke and mirrors?\nDo you have clever names for your treatments, or are they \"\"exactly what it says on the tin,\" as the old advert goes?\nI know a lot of people have clients who get really confused by some of us lashpreneurs and the wild and wonderful names we call our lashes. Everyone has a different name for their product, but they're all more or less the same thing: lashes!\nFor example, a client might come to you wanting Russian lashes because she's seen them in a magazine or her friend has them. She doesn't really know what they are, only that her friend has them, and you don't do them. In fact, you probably provide volume lashes, from 4D all the way up to 6D or 6D+ if she's feeling brave! But they're not called \"Russian lashes,\" so she doesn't know they're the same thing.\nDon't let your competition scare you; if you are good at what you do, your clients will always come back to you regardless of what the treatment is called. Keep abreast of what is going on in the industry; see what is on offer elsewhere, locally and further afield. It's important to know what the latest treatments are, but you also need to make sure you're providing the best quality for your clients. Don't feel pressure to come up with clever names for what you offer. Many people prefer to be able to tell exactly what they'll be getting, without having to ask for an explanation.\nIf you are offering something your neighbour isn't, then that's brilliant. But don't change the name and pretend what you're offering is something different, just for a gimmick \u2013 people will find you out soon enough and it won't do you any favours in the long run. Better to provide consistent service, excellent quality and the kind of experience that makes people want to come back for more.\nI don't intend to insult or upset anyone with this blog, only to be realistic. I try to be grounded and authentic in all I do. I believe that we should allow our good work and capability to shine through, without needing to dress it up with incomprehensible frills. If you have trained well and worked hard, you shouldn't feel the need hide behind a fancy name, don't you think?\nHave confidence in your ability to do a great job, and leave the clever names to the others!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Sandy Creek High chapter displays their fifth consecutive SkillsUSA Georgia Gold Chapter of Distinction award, and is under consideration as one of 24 National Models of Excellence. Photo\/Submitted.\nA total of five students from Sandy Creek High took first place honors in their events at the SkillsUSA Georgia State Leadership and Skills Conference and have qualified to compete at the national level.\nLiberty Harris, Gianna Phan and Kayla Brooks earned gold in Promotional Bulletin Board; Hannah Ohlmeier earned gold in Advertising Design; and Branden Goins earned gold in T-Shirt Design and will compete at the SkillsUSA National Leadership and Skills Conference in Louisville, Kentucky in June.\nThe Sandy Creek High SkillsUSA chapter is under consideration as a SkillsUSA National Model of Excellence for the fifth time. The announcement will be made at the national conference, only 24 chapters across the nation will receive the designation, which is the highest honor a chapter can receive. Sandy Creek was Named a National Model of Excellence in 2015, 2016 and 2018. The school is up for consideration after being recognized as a Georgia SkillsUSA Gold Chapters of Distinction for a fifth consecutive year.\nSandy Creek High \u2013 Daniella Alvarado, silver, Job Skill Demonstration Open; Liberty Harris, silver, Georgia T-Shirt; Nyah Huguley, silver, Georgia PowerPoint; Sarah Sorme, silver, Math Applications; Hannah Ohlmeier, bronze, Georgia Brand Design; and Abby Paver, bronze, Technical Computer Application.\nSkillsUSA is a national nonprofit partnership of students, teachers and industry working together to ensure America has a skilled workforce. SkillsUSA helps high school and college students enrolled in career and technical education programs to excel by teaching employability skills such as communication, problem solving, and leadership in conjunction with their trade, technical and service occupation skills.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Audiologist Vs Hearing Aid Dispenser \u2013 What's The Difference?\nAt Cornerstone Audiology our Providers are all Doctors of Audiology. They have received extensive training in diagnosis and treatment of hearing loss. We believe strongly that when you get your hearing checked, it should be by someone qualified and by an expert in the industry.\nA hearing aid dispenser is only trained to conduct hearing tests with the specific aim of selling you hearing aids, only needing a high school diploma and to pass a state exam to practice.\nAn Audiologist, on the other hand, has to complete undergraduate and post-graduate degrees in audiology. They are trained in a range of hearing-related subjects, including anatomy and physiology, amplification devices, cochlear implants, electrophysiology, acoustics, psychophysics, counseling and sign language.\nThey have to complete 12 months of supervised on-the-job training and at least 350 hours of patient care. They also must pass a national exam before they can practice and have to continue their professional development by completing relevant courses.\nHearing aid dispensers can only treat adults. So, if you are worried about the hearing health of your child, you will need to see an Audiologist.\nWhile hearing aid dispensers can give you a hearing test to fit you with hearing aids, an Audiologist can examine every aspect of your hearing health. This includes testing how well the different parts of your ear are functioning, your hearing sensitivity, your understanding of speech, whether you have tinnitus, and the function of your auditory nerve.\nOnce a hearing aid dispenser has sold you your hearing aid that is often the end of your relationship. An Audiologist, however, continues to look after you for years to come, monitoring your hearing health through every stage of your life.\nSo if you want to take good care of your hearing, book an appointment to see one of our Doctors of Audiology at Cornerstone Audiology.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakpis b/data_all_eng_slimpj/shuffled/split2/finalzzzakpis
new file mode 100644
index 0000000000000000000000000000000000000000..68e3b063ee534dfac9f3ddc25d61be52c9e22006
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzakpis
@@ -0,0 +1,5 @@
+{"text":"Filled with light this stunning home with raked windows one end of the main living makes you feel like you are sitting in the tree tops.\nPrivate from the road the house is unassuming and a wonderful surprise when you enter.\nThe master is on a level of its own with a wall of robes and ensuite with separate toilet and the bedroom space with windows on two walls.\nDown a few stairs you will find the main living space which is wonderfully light a good sized kitchen and dining flowing to the front deck which is surrounded by garden, a bbq area and drenched in northern sunlight.\nThe spacious lounge area has gas log fire and has access to the rear covered deck to follow the sun.\nTwo further bedrooms have built in robes and one includes sliding door to the rear deck, main bathroom and laundry complete the main space of the home.\nVia the rear deck you will find stairs that connect you to the underhouse room with bathroom perfect for a home office, teenagers retreat or guest room.\nCar parking for 2 cars at the front, ample back yard, veggie patch and under house storage complete the outside of this home.\nBeautifully kept throughout this home offers low maintenance and very private living on approx. 756sqm and within walking distance to Blairgowrie bay beach and yacht squadron.\n*3 bedrooms, 2 bathroom home with fourth bedroom and third bathroom on lower level with separate entrance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We're proud to support Amber!\nWe love to do our bit to help local athletes, that's why we're supporting Paralympic hopeful 19-year-old Amber Meikle-Janney.\nAmber, a KAL member for 10 years, has always enjoyed skiing, but when she was diagnosed with Hereditary Spastic Paraplegia she needed to find an alternative way to ski.\nNot only is skiing harder for Amber, but it's not as accessible.\nAmber hasn't let her disability prevent her from succeeding, with her determination and drive seeing her through countless competitions.\nHer mum added: \"We always joked that one day Amber would be in the Paralympics and when her coach said \"If she wanted to that's possible\" that we thought about doing it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A smart form contains texts which is printed in bold.\nThis is clearly visible in both the preveiw and the prionout.\nWhen we convert the output of a smartform into a PDF form the bold text is converted into a normal text for that font.\nWe are using a custom ZARIAL font in the smartstyle, and cannot tell if it is this Z font which the PDF conversion cannot support or if there is some setting which needs to be tweaked.\nZARIAL font which you have created in the smartstyle and used in the smartform is no supporting while converting from OTF TO PDF.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yes, Luka Doncic is good, but so is Trae Young. And Kevin Huerter. And John Collins. And probably whoever Atlanta picks in the 2019 draft.\nMINNEAPOLIS \u2014 The game clock just ticked under two minutes in overtime when John Collins and Dewayne Dedmon trapped Karl-Anthony Towns along the right baseline during Friday night's Atlanta Hawks\/Minnesota Timberwolves matchup.\nThe remainder of the Hawks zone slowly closed its zone in on Towns and Minnesota to force another turnover before Dario Saric swooped over to help his front court teammate. Saric immediately caught the ball and slung it to Robert Covington, who had since the beginning of the sequence sneaked behind Atlanta's closing zone for a wide open dunk. Covington missed it, which was his second missed dunk of the night coming down the stretch.\nDedmon quickly corralled the ball and handed it off to rookie point guard Trae Young, who was heading down the court in full speed. Young was heading down the court so fast that it took a moment for him to gain control of the ball. He took two steps beyond the timeline, a gather step and let 30-footer fly in transition. Young stood in place after the follow through and squatted to watch the ball sink for a basket that gave the Hawks the lead.\nIt was his only 3-point attempt of the night.\nA minute later, Atlanta, still leading by one, inbounded the ball and eventually got it in the hands of Kevin Huerter, who had a couple of screens give him some room. Huerter took a couple of dribbles to the right wing and pulled up for a 3-point a couple of feet behind the line. Bottoms.\nHuerter's second 3-pointer of the night iced the win for Atlanta, despite a controversial no-call on Atlanta-native and Wolves rookie Josh Okogie on the final play of the game.\nMeanwhile, the talk of the night and social media around the league was Luka Doncic. The rookie sensation poured in a career-high 34 points against the New Orleans Pelicans for the Dallas Mavericks. His most viral score came against Jrue Holiday in the third quarter, when Doncic went behind the back twice and hit a step back in Holiday's face at the shot clock buzzer.\nThe play, and the career-high, signaled another round of \"I can't believe the Hawks traded Luka for Trae Young\" tweets.\nLuka Doncic for Trae Young may go down as the worst trade in the history of sports.\nFor the record, Young's career-high was 35, which came with 11 assists and six rebounds as opposed to Doncic's 34, three assists and two rebounds stat line that he posted in the loss to New Orleans.\nFriday night marked the 11th double-double of the season for Young. The feat tied him with James Harden for fifth-most double-doubles in the league from a point guard. This put him ahead of Kyrie Irving, De'Aaron Fox, John Wall, Mike Conley, Ricky Rubio, Damian Lillard, Kemba Walker and Doncic (not a point guard) in the category.\nYoung's biggest critiques have come from his inefficiency on defense and shooting the ball.\nOn defense, let's take Kevin Durant for example. As of Saturday morning, Durant ranks 250th in the league with a 106.2 defensive rating. Both Young and Doncic fall under that mark, with Doncic standing at 108.0 and Young at 110.5. The advantage for Doncic in that department is that he makes up for a lot of his defensive shortcomings on the offense, with a 108.2 offensive rating, while Young is at 99.9 on offense.\nOver the last three games, as of Saturday morning, Doncic has shot 44 percent from the field and 41 percent from beyond the arc, while Young has shot 44 percent from the field and 50 percent from beyond the arc. Young has dealt with his early shooting struggles by taking fewer shots and making better decisions on the offensive end of the ball. He has racked up 26 assists compared to 14 turnovers in that span.\nDoncic has remained aggressive through his poor shooting nights, taking an average of 17 shots compared to Young's 12 in the small sample size. Doncic had 19 assists and six turnovers in the span.\nIf you don't see the trend by now, then allow me to state the obvious: You can't compare Young and Doncic.\nThe notion that you can compare a score-first 6-foot-7 small forward and a 6-foot-2 pass-first point guard is ridiculous. As WNBA All-Star and ESPN NBA Analyst Chiney Ogwumike articulated on \"First Take,\" Doncic and Young play two different brands of basketball. The speedy Young plays with a pace of 108.61, while Doncic's chess-over-checkers strategy on the offensive end allows him to play at a pace of 102.60. First-year head coach Lloyd Pierce has repeatedly said that he wants the Hawks to take advantage of their youth and push the ball as much as possible.\n\"His best fit is actually in Dallas.\"\nDoncic leads all rookies in points per game, while Young leads the rookies in assists per game. Both players have hit big shots at the end of games and both players are on sub-.500 teams so far this season.\nThe idea that Doncic is that much further than Young and that Young is a bust is simply untrue. The idea that the Hawks, who are also projected to land another lottery pick from Dallas in 2019, made one of the worst deals in NBA history by trading for one of the better rookies in the league this season and getting another top prospect next season, is flat out vacuous.\nIn 2019, it's time for Hawks and NBA fans alike to push the narrative that the 2018 draft night deal set Atlanta's franchise years back, aside. Atlanta currently has two very good rookies in Young and Huerter, with second-year power forward Collins putting up some of the best numbers in the league since returning from an ankle injury.\nIn 2019, Hawks and NBA fans alike need to appreciate the performances we're seeing from both Young and Doncic, as well as the rest of this good 2018 draft class.\nIn 2019, although many people won't, it's time to let that hurt go over Doncic being traded on draft night. It happened, and both players are having solid showings as rookies with their respective franchises\u2026the Hawks just happen to get one more player from it.\nLife growing up as an Atlanta sports fan prepared me for the worst. Consistent moments of cliff-hanging excitement ultimately leading to crushing heart ache time-after-time soften the blows of life's disappointments. Any number of rejection letters for internship programs or scholarships I received while I was a student at Georgia State University didn't compare to seeing the Hawks going to 10 consecutive postseasons with zero conference finals wins or the Braves winning 14 consecutive division titles with only one world championship to show for it. I grew up in the era where Mike Vick emerged as king, before becoming Inmate No. 33765-183- and so goes life.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With a 1.1kW engine and a displacement of 42.7cc, this brushcutter doesn't shy away from strong wild growth and bushes. A harness suspension rail, double strap shoulder harness and easy to adjust bull bar handle make this brushcutter easy to customise to the respective user. A simple twist of the handle also makes it easy to transport or ready for compact storage.\nLarge fuel tank capacity makes this a great tool for powering through stubborn growth in large terrains, lawns or parks without the need for constant re-fuelling.\nThe ergonomic bull bar handle provides comfort and control with integrated engine controls.\nThis high torque brushcutter gives you great flexibility in use - it comes with a 3 tooth grass blade and auto bump feed head.\nThe double strap shoulder harness with hip protector offers significant relief during operation, increasing the user's comfort and support.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamiez b/data_all_eng_slimpj/shuffled/split2/finalzzzamiez
new file mode 100644
index 0000000000000000000000000000000000000000..e7868302aced94e5c5fa8ed9e5d8d2c2f002eb4b
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Christmas is a magical time of year. Virginia makes Christmas merry with gifts, decorating, and entertaining.\nThe family started making homemade gifts during the lean years of our childhood. It became a family tradition that carries on even though 50 years has passed.\nDad could take part from worn out bicycles and mix-and-match them to make one that after he painted it, looked like new. One year he painted a Monopoly board to replace a missing one. I wish I had that masonite board that he painted. I'd hang it on the wall as a piece of folk art.\nMom was a master of turning leftover fabric scraps into new doll clothes. She used her trusty Singer sewing machine to make quilts for our beds. I remember the yellow squares alternating with colorful appliqued butterflies. They kept us snug on winter nights in that cold second-floor bedroom of the old farmhouse.\nHer Singer was the old-fashioned kind with the treadle. The one shown here shows what modern technology can do for a sewing machine. My Singer is a plain white one, but for old-time's-sake, I wish I had this black model that looks so much like what Mama used. You can really count on a Singer.\nI'm not nearly the seamstress that she was, but with my machine, I've made doll quilts and table runners. These are easy and quick-to-make gifts that delight the recipients. Another easy gift that I've made are the storage bags to hold all those plastic grocery bags. It's a basic tube with elastic at each end and a loop to hang it. To fancy it up, I put a strip of quilted pieced down the front of the tube.\nMaking homemade gifts with your sewing machine is a great way to stretch the budget at Christmas. Everyone loves a handmade gift stitched with love.\nStart Making Your Christmas Gifts Right Away!\nMy mom didn't have any patterns and just had to create using trial and error. Make life easier for yourself by getting this book.\nSome of my favorite gifts were the homemade doll clothes I received for my Barbie. My mom also crocheted and these were beautiful old fashioned full-length gowns that were and are still beyond compare.\nLots more ideas for a home-crafted Christmas.\nYou'll find books like this at your public library. The library card is free and you'll go home with an armload of books and plenty of projects to cover all your gift-giving needs.\nOne of the doll dresses my mother created.\nHere's my sister sewing a pillowcase to make a doll.\nMy brother made hump-backed trunks for all his sisters one Christmas.\nDad wove wagon-wheel rag rugs from sheets torn into strips.\nDo You Like Making Presents?\nI try to make gifts. I enjoy personalizing gifts for those who will appreciate them.\nMy mom gave me this sewing machine, and I love it! My grandmother gave me my first one, which was a great beginner machine. This one is a great step up. As for presents, I usually do a lot of crochet. Lovely memories you've shared! I've never seen a wagon-wheel rag rug. I'll have to look them up.\nHomemade gifts truly do make Christmas more special with the family. I wish more people experience that!\nOhh, one Christmas Eve our power went out before I had finished my little girl's pinafore so I finished it by candlelight on my treadle machine.\nI love handmade gifts too! I have fond memories of several over the years too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A leading manufacturer is recruiting for two manufacturing engineers to join their team near Eastbourne. Do you have experience in implementing the principles of lean manufacturing and driving production optimisation? Also leading NPI & NPD projects? If so, then you may be the engineer we're looking for.\nYou'll be joining an organisation that works with a wide range of customers and prides itself on building lasting professional relationships. If you want to join an organisation that places a great emphasis on the importance of continuous improvement & lean manufacturing and invests heavily in its staff, then this may be the perfect fit for you.\nThis role is urgent, so please apply as soon as possible with a copy of your latest CV and brief description as to why you feel you're suitable for the positions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Prince Albert I Commemoration Committee and the Albert I \u2013 2022 Executive Committee were established by H.S.H. the Sovereign Prince today, at a ceremony in the Oceanographic Museum attended by the Minister of State and a number of other personalities.\nCreated by Sovereign Ordinance No. 7.253 of 14 December 2018, the Prince Albert I Commemoration Committee has been tasked with researching and organising scheduled events to celebrate the life and work of Prince Albert I. These events will be held between 2019, which marks 100 years since the Mediterranean Science Commission (CIESM) was founded, and 2022, the centenary of the Prince's death.\nUnder the chairmanship of H.E. Mr Robert Fillon, the Executive Committee** will work to implement the programme and stage the events. It will hold its first meeting tomorrow. Stakeholders of different types will be involved in bringing to fruition projects of every kind aimed at a very wide range of audiences.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Salam. If i start Aligarh, so it is as lengthy that it does not go off, so if i give too much time only to Aligarh and Sir Syed, so how i will cover other things, and even other subjects. This problem is not only related to Aligarh, or Shah Wali Ullah, but all such lengthy and bulky things. Kindly guide me that how to make it solve and put it in my mind.\n1- 1857: Muslims eventually loose \"political and executive power in the subcontinent\". For centuries they have ruled the subcontinent and exercised political, executive and economic authority. They have lost their power now. why did 1857 take place? there were various reasons for that.\nAnd the list goes on. The whole point is to create a flow. unless you are able to achieve that, it will be difficult.\nWhy was Muslim League pro-British in 1907 but in 1911 it became anti-establishment? Do you know that the Muslim intelligentsia was divided over the creation of Muslim League (For Example: Sir Agha Khan was against the creation of Muslim League and so were several other prominent Muslims (especially from the UP)? Do you know that the Simla Deputation and the Leaders of Bengal had nothing in common? Why did Jinnah not join Muslim League before 1913 and not before? Khilafat Movement eventually killed Indian Nationalism and why was Jinnah against it? It was on the suggestion of Sir. Sikandar Hayat that the Simon Commission had no Indian member (there were reasons for that).\nrather than going for an altogether ratta, develop concepts and then you would not have to remember any dates, any personalities, any reasons. I am available for any help that you or anybody else may need in this regard.\nSo you mean to say Umar Bhai that i need to pick it like a flowing story, that how one thing started and how it indulged into the next one and by this way the flow will be sitting in my mind. Did i get you?\nrather than going for an altogether ratta, develop concepts and then you would not have to remember any dates, any personalities, any reasons.\nso what i would do with bundles of dates?\nIf i could have pressed the thanks button so many times so i would. Bundles of thanks for your kindness and such a generous guidance. I will be in contact with you.\nLocation: Karachi, now Lahore .\nstudy plan for pak. Aff.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Easter Bread, or Ukrainian Babka, is one of the most beloved Easter traditions in my household. I come from a Ukrainian family and both of my Grandmother's would make Babka every Easter. When my Grandma wasn't around to make it, my Mom would then take over. I have been making Easter Bread since I had my own children and have taught them the tradition as well. This post was updated March 20, 2018 to include some new photos and a how-to video! I have left my old step by step photos in the post as they are very important and helpful!\nIt isn't Easter in my household until we eat this bread for 4-5 days until we have our fill, then are sated for yet another year. This recipe is called Babka by mainly Canadian Prairie Ukrainian settlers, as it has been argued ( as you can read in the comments!) that this is Paska. Now, here on the Prairies my Mom and Baba swore up and down that actual Paska is not a sweet bread, is braided and has no raisins. Here on the Prairies we tend to actually have our own little culture when it comes to being Ukrainians. We pronounce kielbasa as \" koooo-basa\" \u2013 which is the ONLY place that it's pronounced like that! Isn't that strange? It's a uniquely Canadian prairie version of saying the word, however with our very large population of Ukrainian settlers it's no less authentic, just as calling this Babka is no less authentic. Ways and words were changed when my ancestors arrived here and adapted to a new life.\nAlso on another interesting note, it's unique to the Edmonton area that we spell it \"perogy\" instead of pierogi. We have a Perogy House here, it's just how we spell it sometimes!\nWhat is Easter Bread or Ukrainian Babka?\nIt is a light, buttery yet slightly sweet yeast bread with raisins, baked in coffee tins and is so fantastic toasted with some butter on it. Or plain with margarine. Or toasted with anything on top. It is also a labor of love and takes a few hours.\nNow, I put the butter, milk and 3\/4 cup of white sugar in a pot and brought it to a near boil, you want to slightly scald the milk.\nThen I called my mother and during that conversation she mentions that she throws it all into the microwave and nukes it until it's incredibly hot.\nNow, that is certainly not how my predecessors did it. However, this modern woman on her second batch used the microwave.\nNow you have to let the mixture cool and adding in that cup of water certainly helps, thus the reason we leave it until the end here.\nTake the warm water and 1 tsp of sugar, mix in your yeast thoroughly and let it start bubbling away.\nBeat your egg yolks and tumeric. The tumeric gives the bread it's lovely color. You can also use 3 whole eggs instead of the 6 egg yolks but the yolks make a richer bread. So 3 eggs OR 6 yolks, folks.\nAdd it to the cooled milk mixture.\nI used the mixer for this now because it is one heck of a job. Mix the salt into the flour then add in the first four cups of flour, then your raisins. Then slowly add another 4-5 cups until the dough is slightly sticky.\nOnce that's done it needs to rise. Put it into a bowl(s) in a nice warm place.\nGrease the ever lovin' heck out of those coffee tins. I only had 6 this year, so you can also use a loaf pan as well. This would make a perfect 7 tins.\nTime to let it rise again, I had a nice warm stove from banana bread so they sat on there. Let them rise until they are almost at the top. They will rise in the oven as well when they start baking, which is why you want them only to rise barely to the top of the tin.\nCool very slightly in the tins then remove to make sure the bread doesn't sweat and the bottoms don't get soggy!\nHappy Baking you guys! I am so happy to share this recipe with you, it's our family tradition and I'm thrilled that you can bake it as well!\nPin This To Your EASTER BAKING Board and Remember to FOLLOW ME ON PINTEREST!\nOnce that's done it needs to rise. Put it into a bowl(s) in a nice warm place and cover it with a damp tea towel.\nOnce it's doubled in size, it's time to punch it down and put it in the tins to rise again. You want to fill the tins only half way with punched down dough, this dough rises like you wouldn't believe!\nGrease the ever lovin' heck out of those coffee tins. (I use the medium size, the 13-16 oz coffee tins, NOT the large ones!) This would make a perfect 7 tins. You can also make it in a loaf pan.\nBake for 30-40 minutes on the very bottom rack, the tops get very golden brown but don't fear, the inner part has to cook and the tops get brown and stay brown.\nCool very slightly in the tins then remove to make sure the bread doesn't sweat and the bottoms don't get soggy.\nCool on racks for a couple of hours ... if you can wait!\nGlaze the tops with my Buttery Icing Glaze if desired!\nIs it possible to use the large tomato cans? Or would they be to small I'd really like to try the canned method but have no coffee tins.\nSort of disappointed because I had to use loaf pans! Here in the U.S. I haven't seen coffee in a can in about 20 years.\nRecipe really rates more than 5 stars! thanks for sharing!\nThe metal Coffee cans are so hard to find now!\nI like vex baba bread. Thank you so much for your post the bread looks delicious. I use parchment paper in my tins. The bread goes straight up instead of ballooning over the tin & I have never had a problem with the bread sticking in the ridges. Also I cut out a circle for the bottom.\nThis is an amazing paska recipe! Every Easter I would look forward to my baba's paska. Now, at 101 years old, my baba has passed this yearly task on to me. While my baba's recipe is great, I find her method a little too labour intensive and have tried several other recipes, none of which compare to my baba's. I have to say that this recipe, while not only easy and straightforward, produces paska that is as good as (if not better) than my baba's. When I shared a loaf with my baba, she was very impressed with the flavour and texture\u2026.you know a recipe is good when it is \"baba approved!\" Thank you, for sharing this recipe!!\nI followed your very detailed recipe, step by step, and made paska bread for the very first time. It turned out great and I think it will be a new family tradition. Thank you for the recipe and great details!\nHello and Happy Easter, just read your recipe and must comment, we are from Saskatchewan, Canada and there are lots of Germans out here, We also make this easter bread and call it PASKA. A little different take is that we soak our raisins in rum overnight and this gives it a little extra flavour. Other wise looks the same as we make, can't beat the Easter Bread.\nAny thoughts on doing one of the rides overnight in the fridge?\nI cut your recipe in half because of not having enough cans and turned out super good, will be baking more and I also line my cans with parchment paper, the loaves slide right out nicely, not sure if the parchment would affect the baking or not. Very good recipe, thank you !!\nMoved from Chicago to Philadelphia in 1994. Havent had proper (my mom's) babka since then. Unfortunately she passed away in 2014, and did not leave recipe, but seeing that you baked the babka in a coffee can like she did, that's one plus for you. Another is correct pronunciation of kielbasa (koo ba sa, take note people!!). Third plus, i have Ukrainian friends from Winnipeg, so i know i can trust you when you speak of Canadian Ukrainians on the prairie. Looking forward to trying your recipe!\nWell, I was born in Winnipeg, so I have some 'Peg Street cred lol. Both sides of my family hail from Dauphin, Manitoba. This is absolutely a traditional Babka recipe and I hope it's really close to your Mom's!\nMy son's girlfriend is Ukrainian and I would like to try and make this for her. I've made bread before in the past and didn't do well. Is this recipe hard for a beginner like me and can I cut this recipe in half.\nThis is a pretty easy and forgiving bread recipe and yes, you can cut it in half!\nI have made something similar to this and it tastes amaaazing!!\nThanks for sharing this recipes. I will have to attempt to make it for Easter!\nLove having this toasted! SO good.\nPerfect for Easter! Thanks for this!\nThis looks so good! I am going to have to try it this Easter!\nA woman in my building bakes babka as a fundraiser for the Leukemia & Lymphoma Society \u2013 at Easter and Christmas. They're great!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Part Number. DESCRIPTIONS. NOTES. IMAGE. Massey Ferguson 135. (Replacement Parts). LITERATURE. EM7716. WORKSHOP MANUAL. BODY PANELS.\nWe build powerful, highly versatile machinery to your exact specifications. Your tractor. Your specification. Your choice. 1 ISO TR14396. Model. MF 7614.\nThe owners and operators who depend on them. More than. 870,000 tractors been part of the reason Massey Ferguson has won more independent innovation .\nCabs come as standard, whilst full factory-fitted, front-end loaders can also be specified. . 137. MF 5470. 4. 4.4. 135. 145. MF 5480. 4. 4.9. 145. 155. 1 hp figures - ISO TR14396. . the world-leading AGCO Parts and. Service back-up and .\nCorner installation . accessories, whirlpools and repair parts for all your projects. to PROFLO, FNW, PROSELECT, RAPTOR and MONOgRAM BRASS.\nSupply local farmers with tractor parts. The company is tablished Supplier of AgSpares new and used tractor parts to Trac- Massey Ferguson Red -1 Litre.\nMF CMS M 1143. SAE 90 . bomba manual de sangra (1), sin abrir ningn tapn dentro de los lmites, el levante hidrulico est inoperante. 1. Fig. 292 .\nSolutions, Massey Ferguson was the first agricultural tractor manufacturer . A diesel oxidation catalyser (DOC) is used and is a flow-through system so no . Massey Ferguson and its distributors and dealers are always prepared and ready to .\nMF 5650 a simple, flexible harvesting solution. MF Grain tank capacity (litre). MF 5650. 175 hp \/ AGCO SISU POWER. 5. 5000 parts and service information.\nStraw, hay, low moisture haylage. Power requirement. Recommended 60 PTO hp. Working Pickup width 1.9 m. Density control. Manual springs or hydraulic.\n1100. 1400. 1600. 1800. 1700. 1900. 2000. 2100. MF's electronic engine management system broadens the operating change groundspeed manually.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FCBA at the Black Walnut Festival at Historic Bethania - We will have observation hives and will talk about bees, beekeeping, why bees are good for the community. This free event has crafts, atisans, food, craft beer, and live music. Join us to celebrate our connection to the landscape! We will be located at the Bethania Bee City tent site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's a groove funky pop track ! In a Michael Jackson\/Bruno Mars style.\nStrong, groovy, powerful and very dynamic.\nFunky rythmic, strong horns section and warm pads.\n...and if you liked this song please after your purchase, in your download page. It takes only 30s but it's very important for me. Thank you and happy listening !","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Property finance is often complex and yet can be such a vital aspect of any commercial property transaction, be it development, investment or business led. It is important that the right legal advice is obtained at the outset.\nIn addition to acting on behalf of borrowers, our Commercial Property team are experienced in acting for banks and financial institutions. Understanding the needs of the lender is often key to being able to provide productive, efficient and effective advice.\nHatch Brenner's lawyers understand what lenders need, having experience acting for private funders, retail banks and large investment funds as well as borrowers, in a variety of property secured borrowings.\nOur Property team can advise on Residential Development for Developers and Landowners, as well as all aspects of Commercial Property law.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Beyond simply stress-testing devices, the testing services provided by modern electronics partners can be used to pre-plan for much of your post-sale support. The likely failure rates on each component or part can be calculated, allowing you to pre-produce sufficient replacements in the same run as the device itself.\nThis greatly reduces costs associated with post-sale support, and especially helps prevent the need to fire up the lines again to produce spare parts.\nFew electronics manufacturers offer integrated software services, but those who do know that it can vastly improve the quality of the final device. Software can be built to match the exact components and PCBs being produced, without the need to outsource software design to a third party that's unfamiliar with the hardware.\nIn most end-user devices, good software is at least as important as good hardware. Integrated software services ensure every device functions at peak capacity, as well as making patches and upgrades easier as time goes on.\nThe workforce involved in an electronics project will ebb and flow as the project continues, especially when post-sale support is factored in. A good electronics manufacturer will know the process well enough to plan ahead for who you will need, and when, to ensure there are no manpower-related delays.\nLikewise, with proper planning, there's also no need to keep workers waiting around \"just in case.\" Reducing your payables turns straight into greater profits on the electronics produced.\nMany don't realize this, but an increasing number of manufacturers are now providing end-user support directly via virtualized call services. After all, who would be in a better position to put together great documentation on the device than the people who designed and manufactured it?\nYour product can be backed up with post-sale staff who are fully informed and ready to assist customers with great service.\nAdditionally, thanks to the parts pre-production, actual repair services are an option. Far too many businesses -especially those with overseas production- waste far too much money sending out entire devices as replacement. You can save significantly on warranty support by only paying to replace components rather than complete units.\nGoing with a \"fire and forget\" electronics manufacturer might save a little money in the short term, but you're likely to find a lot of long-term hurdles you hadn't expected. Partnering with a high-quality domestic manufacturer will bring far greater value and ROI across the lifespan of your product, as well as improving your customer satisfaction rates.\nContact ZenTech today to learn more about how modern domestic manufacturing makes better products.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Trailblaze. Innovate. Inspire. Champion. Those who lead the U.S. Department of Veterans Affairs (VA) as Medical Center Directors\/Chief Executive Officers make a dramatic and lasting difference in the lives of Veterans and families enrolled in VA healthcare.\nVA is actively seeking candidates to fill open VAMC Director and CEO positions at sites around the country.\nThese community leaders have a unique opportunity to direct the nature, scope and quality of care at the VA facilities that make up the nation's largest healthcare system. They also shape the medical facility's culture and services provided to the most diverse population of Veterans in history.\nLeading one of the more than 170 VA medical centers around the country means being part of the team contributing to VA's success as a healthcare organization. Local leaders are helping VA facilities outperform community-based private-sector hospitals, according to a recent study by The Dartmouth Institute for Health Policy and Clinical Practice and the White River Junction VA Medical Center in White River Junction, Vermont.\nVAMC Directors and CEOs also recruit, train and lead doctors, nurses, clinicians, support staff and other career professionals who deliver patient-centered services. They determine how the Veterans' healthcare system will modernize to meet new challenges and maintain the highest quality care that Veterans and their families need and deserve.\nEstablish policies and procedures consistent with the strategic direction of the Veterans Integrated Service Network and VA as a whole.\nFoster positive relationships with the media, Congress, and service and volunteer organizations.\nVA career professionals receive competitive pay and a wide array of benefits, including medical and dental insurance and generous leave policies. Employees at VA can plan for the future by participating in the federal retirement system.\n\"We look forward to evaluating qualified candidates for these gratifying VAMC Director and CEO positions soon,\" said Sherrard.\nLead the delivery of healthcare to our nation's Veterans: Choose a career as a VAMC Director and CEO.\nREAD about VAMC Director\/CEO positions (PDF).\nhe only way you can talk to the Administrator, is at an open forum. You start to voice your problem, and then are told that it is not to be discussed in this form, and to make an appointment, and then that never happens. This kind of problem needs to be discussed with the head, and not including the staff, as a lot of the times, they are part of the problem. I personally feel that there should be a way to talk to the Directors face to face.\nI see a lot of helpful information in these articles, however there is something very important that is missing, and that is how to contact the Director when there is a problem within the Facility. It is not with any of the Medical side. The problem is with the support staff. You have to talk to the support staff with the problem, but it falls on deaf ears, and nothing gets done. The only way you can talk to the Administrator, is at an open forum. You start to voice your problem, and then are told that it is not to be discussed in this form, and to make an appointment, and then that never happens. This kind of problem needs to be discussed with the head, and not including the staff, as a lot of the times, they are part of the problem. I personally feel that there should be a way to talk to the Directors face to face. Part of the problems are caused by the person or office you need to apply at to get one of these meetings..\nThank you for you time and effort in reading this.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"6 ways to keep your home safe from burglars!\nHow to protect yourself from a Theft in your Home.\nHave you ever wondered how safe is your home? In an article that I read recently the burglars revealed their secrets. KGW8 news investigative team sent letters to 86 inmates currently serving time for burglary. The inmates were asked to respond anonymously to 17 questions detailing how they broke in, when the crime occurred and what they were looking for. Here is what I found most interesting!\nThey usually enter through unlocked doors or window. But will kick in your door rather than break the glass.\nThe first thing they grab are your electronics, jewelry and cash, by first searching the master bedrooms, then they go through the rest of the home.\nThey prefer to break in early morning or afternoon.\nAll knock on the front door before breaking in a home.\nSome were deterred by security sign, but not all. Big loud dogs do deter them, but not smaller dogs.\nThey were mixed on security cameras and lights left on. But a television or radio playing would be a deterrent. Also a car in the driveway is a deterrent, unless it obviously is never driven. Importantly, you as the homeowner should have good lighting, trim bushes and trees. Get to know your neighbors.\nProtect your home the best you can and be safe.\nIf you would like to read the full article CLICK HERE.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Endless Spas and Pools first ever V.I.P Showroom Night is happening Wednesday 2nd May.\nAt our V.I.P Showroom Night you'll be greeted with Bubbles on arrival and finger food throughout the night.\nThere will be special offers on the entire range of Swimming Pools, Plunge Pools and Swim Spas.\nBe in the running to win the Endless Door Prize!\nThe Endless Spas and Pools V.I.P Showroom Night is invite only and numbers are strictly limited.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BUTLER CREEK BABY UPLULA SPEEDLOADER 22LR-380ACP - 24223..\nThe butler Creek Lula Magazine Loader for the ar-15\/m16 reduces wear on feed lips as well as helps prevent injury and thumb pain. This lightweight magazine loader fits easily in your pocket or shooting bag and is reliable in any weather conditions. Made with durable reinforced polymer, this ..\nBUTLER CREEK MAGLULA SPEEDLOADER AK-47\/GALIL - 24216..\nBUTLER CREEK UPLULA 22LR CONVERTED PISTOL MAG LOADER - UP62B..\nThe HKS 10-A speedloader for 38 Special and 357 magnum revolvers feature no cartridge jiggle and are extremely reliable. This great HKS speedloader uses strong metal points to securely hold cartridges in the loader until released. The knurled aluminum knob is textured making for a reliable g..\nThe HKS 586-A instantly loads all 6 cartridges reliably into your revolver and has no cartridge jiggle. This great HKS 586-A speedloader uses strong metal points to securely hold cartridges in the loader until released. The knurled aluminum knob is textured making for a reliable grip on ever..\nThe HKS 587-A instantly loads all 7 cartridges reliably into your revolver and has no cartridge jiggle. This great HKS 587-A speedloader uses strong metal points to securely hold cartridges in the loader until released. The knurled aluminum knob is textured making for a reliable grip on ever..\nThis HKS Speedloader for the .357 Mag J Frame revolver instantly loads your entire handgun with little effort. Strong metal points hold cartridges securely in the lightweight polymer body until released. The knurled aluminum knob ensures a reliable grip on every re-load. This great HKS revol..\nThe Maglula Uplula universal pistol magazine loader and unloader is military-grade and loads 9mm Luger up to .45 ACP magazines. One size fits all on this great Uplula magazine loader, no inserts, spacers, or adjustments at all. You can load up to one round per second with this popu..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rakhi celebration across India represents the amalgamation of rich culture of India. It reflects the idea of unity in diversity which is the essence of India.\nIn every year during the period of July-August, Indians across various parts of the country celebrate Rakhi festival with much enthusiasm. This festival captures the true spirit of India's rich cultural heritage. The mode of celebration for this joyful event has changed with time but the purpose and rituals of Rakhi have remained unchanged. To pray for a prosperous life and elimination of all negativities in life are the solemn purposes of this celebration. This ceremony is an attempt to uphold peace and harmony all over the country.\nOn this occasion, sisters remain busy in preparing foods for the festival and organize Rakhi Puja Thali, necessary for various types of rituals. After chanting prayers before God, a sister ties the sacred thread of Rakhi on her brother's wrist. Exchange of sweets and gifts is an essential part of this ceremony. Both of them vow to protect each other from all the evils of life. To them, Raksha Bandhan festival provides an opportunity to express the feeling that they sincerely care for each other.\nUsually the Hindus and some Sikhs regard it as festivals. The event of tying Rakhi is popular as Rakhi Purnima or Raksha Bandhan in major parts of India. But this festival is known by different names in different parts of India. The method of celebration also differs in different states of India. North Indians and West Indians celebrate this festival with much vigor.\nIn Tamil Nadu, Kerala, Andhra Pradesh and certain parts of Orissa, the Rakhi festival is known as Avani Attam or Upakramam. According to a legend, Lord Vishnu brought back the stolen Vedas of Lord Bhrahma on this day. That is why Brahmins in ancient India considered this day as an auspicious day to begin Vedic studies. This day has special significance to the males belonging to the Brahmin community of those states. In this sacred event, the Brahmins of those states change their threads through some rituals.\nIn states like Bihar, Chattisgarh, Jharkhand and Madhya Pradesh, people call Rakhi festival Shravani or Kajari. This day has special significance to farmers and women residing there. An interesting aspect of this festival is that the women, who have given birth to male children, can perform the rituals of the festival. On that very day, women collect a cup of soil to store it in a dark room. They worship the room for seven days till Kajari Purnima arrives. They offer prayers to God for the well being of their child, members of their family and for a good harvest.\nIn some parts of Gujarat state, Pavitropana is the celebration of Rakhi festival. On this occasion, the Gujarati community worships Lord Shiva with much celebration. They offer milk, honey, coconut water, curd and water to the deity. People in Gujarat believe that being the symbol of destruction, Lord Shiva will demolish all types of negativity and bring prosperity in the earth.\nIn short, the celebration of Rakhi festival in different parts of India represents diverse cultural background of the country. The purpose of celebration is the same irrespective of region. It aims to seek blessings from God and elders and pray to escape all types of negativity in life. Therefore, it can be said that Rakhi festival symbolizes unity of thought and emotion.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"If you have any questions, you can use the form to order callback. Specify the time when you most suitable call of our Manager will contact You.\nOur team is Creating sites. You will be surprised creation speed and quality of your site. We support, hosting for the site and help with the choice of domain name! Development of web site and Site Creation Studio \u00abWebsait\u00bb - a guarantee of quality and confidence. You can order: Create sites, portals, online stores Creation, Redesign, reconstruction, modernization sites, banners and optimization for search engines.\nWe are high quality professionals. You will be satisfied with the result !!! And is not pleasant road price of the works. You can contact us at: Uzhgorod, Mukachevo Chop Hust, Mizhgirja, Tyachiv, Rakhiv for custom websites and writing.\nThe best Web sites for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Say goodbye to your liquid eyeliner woes and say hello to this Colorbar's high-performance eyeliner with a quick-drying matte formula that lasts upto 16 hours. Water-proof, smudge-proof, (basically life-proof), the All-Matte Eyeliner's pro applicator delivers the deepest pigment with the smoothest glide for you, for everyday.\nExplore the entire range of Eyeliner available on Nykaa. Shop more Colorbar products here.You can browse through the complete world of Colorbar Eyeliner .Alternatively, you can also find many more products from the Colorbar All-Matte Eyeliner range.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Animals for children review. Discover with your toddler the animals! Games for girls barbie with a gorgeous collection of pets and wildlife! The funniest game for your toddler!\nA great collection of puzzles and animals for toddlers and kids.\nYour baby's delight and the dogs? This is the app for you!\nYour baby will love this game Animals for children!\nAnimals for children and children is an educational and entertaining game.\nIt is a nice, colorful and fun app for toddlers and kids! Play with animal puzzles!\nA lot of different puzzles and many animals to keep your kid busy.\nAvailable for all screen resolutions and all devices, including tablets like the Kindle and Samsung!\nSimple and intuitive, your baby will have a lot of fun for hours with this puzzle game!\n\u2013 Play for children of all ages, infants, preschool children, children and adolescents in advance, designed for boys and girls.\n\u2013 Over 80 lovely animals in the FULL version of bear cub, lion, tiger, gorilla, monkey, giraffe, hippopotamus, elephant, penguin and many wildlife and other countries!\n\u2013 Randomly generated quiz! This educational game will never be old!\nInteresting to learn and play with great colorful puzzles, this app is not to be missed!\nA lot of educational puzzles with toddlers and little girls !!\nYour baby can admire all kinds of animals and will hear the sound effects honestly!\nAnimals for kids and girls are a fun game but also help your kid to improve his or her abilities.\nLots of games for kids, best games for kids, toddler apps, game memory, games for girls, educational games and puzzles for kids on 3 old, all from romeLab!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rich Alfredo pasta sauce that's dairy-free, vegan, & gluten-free \u2013 made with cauliflower, cashews, onion, garlic & nutritional yeast.\nAileen made this super healthy pasta sauce for a potluck with friends. What to make was a bit of a challenge. One \u2013 it had to taste good; and two \u2013 her friends have a mix of food allergies. She came up with creamy pasta sauce \u00e0 la Alfredo.\nInstead of cups of cream, it calls for cashew nut butter and cauliflower \u2013 creamy but with more protein and with healthier fat in healthy amounts. Go ahead, do yourself a favour and eat 'til you're full!\nHer creamy pasta had lots of flavour for the crowd, and her friend who is allergic to dairy got to join in the fun. Aileen boiled up gluten-free pasta for another friend and that took care of all her buddies.\nThis recipe is vegan and calls for vegetable stock. I have made a version with chicken stock and it worked out well. Your stock quality makes a big difference. We don't always have homemade stock in the house. For years I've bought packaged organic stock, but it tastes nothing like our rich, delicious homemade. I've come to think that processors pass a chicken or a veggie under a tap, boil the heck out of it, and stick it into a shelf-stable box. I use good bouillon now when homemade isn't available. \u2013More flavour, similar nutrition and less cost. Enough preaching already! Here's the recipe.\nSoaked Cashews for dairy-free, vegan Alfredo pasta sauce.\nCreamy Alfredo pasta sauce that's dairy-free, vegan, & gluten-free \u2013 made with cauliflower, cashews, onion, garlic & nutritional yeast.\nCreamy pasta sauce \u00e0 la Alfredo, veggie rich and full of protein. Hearty and rich, it's also dairy-free and vegan.\nSoak cashews in water overnight. Discard soaking water.\nSteam cauliflower until tender, 6-8 mins.\nIn a large pan, add the oil, onions and garlic. Saut\u00e9 on medium heat until onions are transparent. Do not burn the garlic.\nCombine the almond milk and cauliflower in a blender. Gradually add all of the remaining ingredients except for the broth. Blend until smooth and creamy.\nAdd broth to the blender, 1 cup at a time. Sauce will change from pasty to creamy.\nFlavourful broth makes a difference! Prepackaged broths have their own distinctive flavour, and they don't always enhance the dish or impart nutrition. If homemade is not on hand at our house, I reconstitute a good bouillon cube.\nA Note to you: Your pasta sauce will come out thinner and creamier than the photos. I'd already shot the pictures before deciding that I way prefer the flavour and creamy texture that comes from adding broth. Also we have a stick blender. If you have a Cadillac Vitamix, you'll get a smoother texture.\nGive it a whirl! Creamy pasta \u00e0 la Alfredo is quick enough for a weekday meal.\nCreamy Pasta Alfredo that's dairy-free, vegan, and gluten-free!\nFood is love. Share it with people who are important to you. Potlucks are great that way.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So year after year, my Christmas list becomes more and more boring haha. I debated not putting the bundt pan on there but I really do want one lol, I just never remember to buy one for myself. I'm not a huge baker but I really love monkey bread and I can totally make that in a bundt pan. :) I also LOVE getting new cookbooks. I've heard great things about Danielle Walker's new cookbook, Celebrations.\nThe waffle maker is a must! I really stink at making pancakes but I can't screw up a waffle lol! A bunch of my paleo cookbooks have yummy looking waffle recipes that are healthier than the regular version. Can't wait to try those!\nI'm always in need of new workout clothes because I wear them all the time. I'm not saying I workout all the time haha, but I'm in workout clothes a lot. I'm hoping with some new clothes, I will have more motivation to workout too. I also want new Zella leggings but they don't have a very good selection in the high waisted crop version. I've found that I feel much more secure in high waisted workout pants. Also, most of them have mesh on the legs and I already have those in black. I'm keeping my eye out for some new options.\nI've gone back and forth about the Cricket vs. Silhouette for years. I feel like the Silhouette will suite me better and I can see myself using it a ton. I know I will for sure use it weekly for my planner stickers and I can think of about a dozen crafts I want to do. This is a pricey present, so I'm not expecting it. A girl can dream though, right??\nI talked about this portable phone charger in my girlfriends gift guide. I just think it's adorable and sooo helpful to have a portable charger. I would have loved one this weekend in Disneyland. I don't need this specific one but it is pretty!\nWhat's on your Christmas list this year??","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapdmw b/data_all_eng_slimpj/shuffled/split2/finalzzzapdmw
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+{"text":"Wooden boatbuilding is in Rocking the Boat's bones; the organization began as a single boatbuilding youth development project in 2001. While the programs have since expanded to include related program tracks in sailing and environmental science, the heart of its Hunts Point facility is a 6,000 square-foot, fully equipped boatbuilding shop where no fewer than two to three boats are being built or repaired at any one time.\nRecently Rocking the Boat was approached by a donor interested in having students restore a small boat built by Arthur \"Chummy\" Spurling, a well-known boat designer and builder from the Cranberry Isles of Maine. The original boat, a treasure of Maine's maritime heritage, was owned for years by the same Down East family. Unfortunately, it was also just short of repairable.\nThe boat does, however, present a fantastic opportunity to study and capture this classic design and recreate it piece by piece. Beginning in March, 2018, the original boat will be precisely measured to create the molds, and some of her beautiful brass hardware has been harvested for eventual installation. This Spurling Skiff project will be the organization's 53rd student-built boat. But the most remarkable part of the project is who will be doing it, and what they will gain in the process.\nRocking the Boat's Boatbuilding apprentices are high school aged boys and girls from one of the most under-resourced communities in New York City. Most come to the program with no woodworking experience, let alone any knowledge of traditional wooden boats, and work their way up to the advanced apprentice class by honing technical skills alongside professional traits. Boatbuilding apprentices graduate from high school with valuable work experience, transferrable skills, and secure post-secondary plans\u2014often in fields they never imagined possible.\nRPM Foundation (RPM) is an educational grant-making program of America's Automotive Trust (AAT). RPM is funded by collector vehicle and classic boat enthusiasts to serve youth and young adults on their pathways to careers in automotive\/marine restoration & preservation along with the long-term interests of the collector vehicle and classic boat communities. Visit: www.rpm.foundation for more information.\nThrough a unique grants program, Workforce Development Institute makes investments that lead to workforce development and economic growth.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Free-agent pitcher Matt Shoemaker has agreed to a one-year, $3.5 million contract with the Toronto Blue Jays, the team announced Friday.\nCatcher Jonathan Lucroy has agreed to a one-year, $3.35 million deal with the Los Angeles Angels, according to multiple reports. News of the agreement was first reported by Yahoo Sports.\nLucroy, 32, hit .241 with four homers and 51 RBIs in 126 games for the Oakland Athletics last season after signing a one-year, $6.5 million contract in March.\nThe Philadelphia Phillies have been planning for this offseason for years and are loaded with money to pursue free agents.\nBoth Bryce Harper and Manny Machado have received interest from the Phillies, but word is neither player particularly likes Philadelphia. Furthermore, Harper and Machado both would have preferred to sign with the New York Yankees.\nThe Yankees have shown little interest in signing Harper.\nThe Oakland Athletics have agreed to sign Mike Fiers to a two-year deal that could be worth up to $15 million.\nFiers, 33, is coming off his best season (12-8, 3.56 ERA) as a major league starter.\nThe Chicago Cubs have agreed to a deal with free-agent right-hander Kendall Graveman, a source told ESPN's Jesse Rogers on Saturday.\nThe 28-year-old Graveman, who is still rehabbing after having Tommy John surgery in late July, will receive at least $575,000 for 2019 and has a $3 million club option for 2020, according to multiple reports.\nSwitch-pitcher Pat Venditte has agreed to a one-year contract with the San Francisco Giants.\nVenditte made 15 appearances with a 2.57 ERA over 14 innings for the Dodgers last season, so new Giants president of baseball operations Farhan Zaidi is plenty familiar with the pitcher.\nThe Seattle Mariners acquired outfielder Domingo Santana from the Milwaukee Brewers on Friday in exchange for outfielder Ben Gamel and minor-league pitcher Noah Zavalos.\nThe Dodgers have taken on the contact of starting pitcher Homer Bailey.\nThe A's have a potential replacement for Jed Lowrie in Jurickson Profar.\nAccording to a report, it will be a multi-year deal.\nJ.A. Happ has landed with the Yankees and now he wants Manny Machado to join him.\nThe 34-year-old had a 2.83 ERA in 24 starts for the Braves last season.\nDaniel Murphy is expected to play mostly first base for the Rockies.\nJoakim Soria had a 4.09 ERA in 26 games with the Brewers last season.\nThe Blue Jays released Troy Tulowitzki earlier this month, leaving themselves responsible for $38 million over the next two years.\nThe Dodgers, Brewers, Phillies and Reds have shown interest in Corey Kluber.\nThe Mariners' ballpark has been known as Safeco Field since it opened in 1999 as part of a 20-year agreement.\nMatt Harvey, 29, reset his value a bit after the Mets traded him to the Reds.\nThe Twins surprised Joe Mauer with the announcement on Tuesday.\nThe Yankees have hired Carlos Beltran as special adviser to general manager Brian Cashman.\nThe Cubs have agreed to a two-year, $5 million deal with Daniel Descalso that includes a club option for 2021.\nThe Astros have resurfaced as a potential destination for J.T. Realmuto, according to a source.\nThe Yankees gave CC Sabathia a $500,000 performance bonus, even though the left-hander was ejected from his final regular-season start six outs shy of the 155 innings required.\nThe Padres are believed to be a team interested in Miguel Andujar.\nThe Dodgers do not want Bryce Harper on the type of long-term, record-setting contract he's seeking.\nMichael Brantley gives the Astros a left-handed bat in the middle of their lineup.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For any individual seeking changes their house, the investment into home improvements is a thrilling chance. There are many ways which a home could be improved and the options that are available to you normally greatly depend on the budget you have allocated.\nFlooring normally goes a long way in changing a room's appearance whether you are seeking the classy style of wood or the cozy warmth of carpet. A new paint job will help lighten or darken a room and introducing color can greatly alter the flow and feeling of the room. When you are searching for both an exterior and interior change in a home's look, few elements have a bigger impact than the opportunity of ready made blinds.\nFor many, the idea of blinds is often taken for granted as many homes have them and the extent of their life makes it so few repairs or maintenance is ever necessary. When you consider it though, the role which blinds play in the home is quite significant, both in functionality and look. From the exterior, blinds serve as a defense mechanism which protects your home with privacy as well as serving as a barrier between you and the weather. In the interior products like ready made blinds act as your window to the world where you can either let the view of nature into your house or close them to reduce light and bring a focus to the home.\nWhen following through with ready made blinds there are many topics you should address before making the decision to make a purchase. Make a decision on what the best style is to fit in the room you are attempting to redesign. Would your house benefit better from the natural look of timber blinds or does the high humidity of your environment require the versatility of PVC blinds.\nDo the colors that appeal to you fit in your current home scheme or are you seeking a full change in the feel of the installing room. It's even important to discover the dimensions of your windows so that you could install them with little difficulty. Measuring too long will always allow for an extra cut to be made but measuring too short would result in paying for ready made blinds which you cannot utilise.\nOne of the best methods to find a solution to your inquiries is to use blinds online to find the best choices for you. With the utilization of online blinds you could discover the several styles and materials that are available to you as the client and also work with a company that would provide measurements for you, ensuring the right fit.\nFind your best on-line blinds opportunity by heading over to http:\/\/www.theblindsplace.com.au. They are specialised in all window furnishings including the very affordable ready made blinds range through all types of custom made blinds to the Norman brand of shutters.\nDuring summers, the most terrible thing which you can experience tends to be the persistent burning heat. The higher temperature just won't go low but seem to go higher and higher night as well as day. You'll lay awake at the night time trying to discover where to get your Air Conditioning Repair St. Louis which works on weekends also, but temperature is extremely high to concentrate. Reputation and reliability are the tools only you're probable to have to know whether your technician knows what she or he is talking about. Advices from acquaintances, family members, coworkers, or people living in St. Louis also are respectable indications regarding integrity and dependability of your AC service repair company. This article will discuss the impact of hiring specialized AC repair technicians when your systems malfunction.\nMaximum people select professional A\/c services when they desire to get Air Conditioning Installation St. Louis in their offices or homes. AC professionals mainly offer installation services. A few other A\/C services offered by A\/C installation technicians include repair, maintenance and AC replacement service. Because air conditioning unit costs a great amount of funds to purchase, it's usually recommended that house owners having malfunctioned systems select to get it repaired rather than replacing their units if they begin malfunctioning. Maintenance of house air conditioning unit regularly increases their durability and keeps them far from malfunctioning.\nOne great benefit of going for professional Ac Repair St. Louis specialist is that such kind of individuals, services providers or firms offer their clients with air conditioning maintenance service which is quite essential for the most excellent functioning of such devices. For any AC to run effectively and efficiently both regular cleaning and servicing or maintenance is compulsory. The heating and cooling systems that aren't maintained properly lose their robustness and have tendency to malfunction recurrently which apparently leads to further financial burdens on the homeowners owing to the unremitting repair they necessitate. For evading the financial burdens of paying for AC replacements service it commonly is recommended by industries experts that you plan your a\/c for maintenance in 8-12 months every year.\nAnother great benefit of selecting a professional when Heating and Cooling Repair ST Louis is considered is the proficiency they offer. Most of the air condition system installed at homes cost a sizeable amount of funds typically. As such when this vital home device malfunctions, it's vital to get somebody who is skilled and has required expertise required to make essential repairs. Attempting on this kind of task on your own as DIY project unfortunately may in more of cases than not can results in more destruction than good; at times requiring AC owner so as to spend more in replacing that appliance in place of repairing it. So, by opting for specialized AC repair services you're competent to limit this amount of funds spent in rectifying malfunction seeing that such experts are extremely well placed, equipped as well as knowledgeable to spot and correct precise problem.\nPlanting a lawn, filling flower beds and designing walkways comprise only half the equation in framing a property with the right amount of \"curb appeal\" to make it attractive to both an existing owner and a potential buyer. Today's Calgary landscaping technicians must also take into consideration issues such as sustainability, drainage, mold and pests\/diseases that are harmful to trees, shrubs and flowers.\nWithin a landscape Inside a scene, materials, for example, wood fencing, stone, pavers, and some of the time are used to as hardscapes; hardscape administrations incorporate stonework, for example, open air kitchens, grills, chimneys, holding dividers, fire pits, and flagstone yards alongside wood arbors, decks, and fencing. Water gimmicks, for example, wellsprings, lakes, dry waterway bunks, and waterfalls are likewise considered hardscapes.\nLandscape Construction Companies projects are permanent to semi-permanent installations and should be considered an investment in your home. These projects can range from only a couple hundred dollars to thousands of dollars depending on the number of services involved and complexity of the project.\n*\tTrees shading homes can reduce attic temperatures by as much as 40 degrees.\n*\tTurfgrass plays a vital role in capturing dust, smoke particles and other pollutants, and it produces oxygen.\n*\tA NASA study showed that many varieties of indoor plants remove harmful indoor chemicals and pollutants.\n*\tTurfgrass slows down and absorbs runoff into bodies of water.\nLooking at all four houses on The Block, all couples decided to utilize an expert gardener to arrange and build their yards, and with great reason. Land specialists, Alistair Macmillan from Ray White Wilston expressed \"Proficient finishing increases the value of a property, as it incredibly expands the homes offer from the road. You need to recollect that regularly the first run through a prospective purchaser is going to view a house that is available to be purchased is either online or in the paper\".\nBrochures have become an incredibly efficient company marketing device that have been used for decades in order to generate an effective communicative device that will be experienced by viewers for many decades to come.\nBrochures allow you to be absolutely in control of your structure, style and details. Which indicates you can select exactly what and how your concept is shown.\nWith their relatively low generation costs and capability to arrive at large viewers, you may ask yourself why I have never used a brochure before.\nColor alternatives \u2013 models will frequently use the four color process so that you will have access thousands of vivid colors. All concerned with the appeal of your style.\nLayout alternatives \u2013 the structure of the brochure is again, absolutely up to you. Select a variety of alternatives and make sure you end up picking the right one for your organization's important concept.\nFolding alternatives \u2013 with catalogues the foldable alternatives are again based around your organization's passions. The popular single piece, twice times option allows for four sections to be provided and clearly shows details on both factors. Brochure publishing loves the benefits of volume publishing. This implies that the more catalogues you list, the less they will price you.\nThis is incredibly eye-catching for companies who wish to obtain great submission rates at a lower low cost price. Brochures also have the power of being able to advertise your concept even when you are not there. Visitors can quickly take catalogues home with them and this indicates they can complete them out to possibly fascinated friends or explore the details more thoroughly and at their pleasurable.\nBrochure publishing is set to bring on its benefits for many decades to come. The objective of brochure publishing can be found within its suppleness. Brochures quickly and clearly connect their concept in a strong form.\nThe low price of catalogues also makes sure they will continue to flourish in these recessional times. Transmitted or the purchase of coverage is an expensive financial commitment, one that may not always provide profits. Brochures on the other hand provide a low risk technique that can help you obtain very great revenue.\nOnline Printing \u2013 Best and Cheap High Quality printing services, Tarjetas De Presentacion, Diseno Grafico, Volantes Folletos, Postcard Printing San Diego, Flyer Printing San Diego, San Diego Printing Poster & Printing Services from Print Steals.\nFolders are very useful products. They are suitable for various tasks. They may be found every where. From schools to business setting, their presence is known to everyone. A4 folders printing are highly useful and its utility has made it a highly successful printing product. Folders printing are made according to the document sizes. That's why in various businesses or organizations, A4 folder printing products are highly common. A4 folder printing is made by paper or plastic however, they may be made from other materials. Approximately, all the folders have two covers. These covers are made when a single sheet of plastic or paper is folder in the center. A4 folder printing may be used for various purposes. The main purpose is to manage documents. A4 folder printing is made in such a manner that may handle documents effectively. The quality of A4 folder printing is very important to convince consumers.\nHelmets are not all about our protection or safety. We are all concerned about saving our lives and protect our-selves. We are also concerned with looking nice while we are out there. Not only helmet is important but the look and style of helmet is also very important. We would like to look nice while keeping ourselves safe.What is the statement you would like to make when you are out there riding on your motorbike? Our helmet stickers UK may say a lot about who we are and what we are all about. Some people have plain helmet stickers UK, while other have bright stickers. It all depends on who you are and where you are going.Picking helmet stickers UK is not all about something that look so nice. In this way you will have to choose helmet stickers UK that have clear coat, and UV protection. If you want helmet stickers UK that are going to last as long as your helmet then you will have to look for various things. Helmet stickers UK represents you! Try to make sure that you're looking and feel good about what the helmet shows others. Try to look around at various styles of helmet stickers UK that you would like to see on your helmet.\nCheap booklet printing is a time honored gift to the modern marketing world that brings high response rate from the palms of your consumer's hands. However there are several ways to lighten cheap booklet printing order. In this way you don't need to buy the barn to have nice looking booklets which actually function how they are supposed to. Various online printing companies provide you the option of printing the cover of booklets at different quality than the interior pages.\nIf you are looking for various ways to save money on NCR printing, then in this way the nice place to start is by reviewing your existing print specification. For achieving this purpose then you have to look on the certain points like, Try to look at how many copies you really require by NCR printing. Review the form distribution very carefully to see if any receipt may be eliminated or combined. It will provide help in reducing printing and administration costs. Ask your self that whether you really need 3 part NCR set or 2 part NCR by NCR printing.\nThere are various types of stickers. One of them which is most common is known as round stickers due to its round shape. Round sticker printing is used extremely by the marketing and advertising industries. Main purposes of round sticker printing is advertising, marketing, religious, fund-raising, educational campaign and entertainment. There are various aspects that should keep in mind during printing of round stickers. The process of printing involves the usage of complex printing technique like full color CMYK printing process.\nGeorge Mclaren, who is providing services for PrintingGood and trying to provide best tips on marketing ideas by including various types of printing products these products include, sticker printing services, online sticker printing, round sticker printing.\nGenerally many of industries using different types of stationary in routine schedule such as letterhead pads, printing paper, writing pads, business forms, sticky notes and many things. Nowadays carbonless forms demands increase rapidly in many industries. There are numerous benefits of use carbonless forms. They offer an easy way to copy a document without the use of carbon paper in a handy and useful way. They normally have different colors and come in stacks. They are produced by using special material.Every single printing product has its own usage and importance when it comes to your business and marketing movement. There are a lot of printing products available in the global market today. From stickers to folders, from cd jackets to posters and from standard business cards to carbonless forms, you can buy all other printing products easily. More importantly, competition among the online printing companies has been increased quite dramatically these days. Therefore every business organization is striving for the best printing products today. Moreover, the customers have become very nimble and agile while their choices. Therefore these companies have to produce the best quality products according to the customer's requirements.The amazing thing about this technology is that it will perk up your online marketing campaign step-by-step. Both of these techniques will play a critical role in your higher search engine ranking and all of a sudden you would get an enormous amount of carbonless forms printing traffic.Two things would play a critical role in your NCR forms printing. These would consist of graphic designing and full color CMYK printing process. When it comes to the graphic designing, it would include various techniques from top to bottom for example texts, images, pictures, color schemes, graphics, templates, logos, shades, lines, borders, etc. As far as full color CMYK printing process is concerned, it would involve four colors such as cyan, magenta, yellow, and black. Then you can indeed get the best quality products via custom carbonless forms printing.\nJB Forms offer online business forms printing and carbonless form printing services. We have complete solution for all types of business forms such as contractor service forms, cash receipt books, general repairs forms, invoices, general business forms and many more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"GabrielFoods.in is an eCommerce website developed using OpenCart. The website is 100% responsive and compatible with all mobile devices. The client idea is to help house wives to make non veg food with a hassle free doorstep delivery.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Swarovski Siku Polar Bear and Cubs: Coke or Pepsi?\nThe Coca Cola \u2013 Pepsi wars have been raging for years. In 1993, Coke launched its Coca Cola Northern Lights commercial. It was a parody of bears going to a drive in theatre, viewing the Northern lights. All the polar bears were enjoying their Coca Cola.\nI recently decided to check in with Swarovski Siku and her Swarovski Polar Bear Cubs. My question\u2026.do these Swarovski polar bears prefer Coke or Pepsi?\nHere's their reaction to Pepsi\u2026.\nWhile the Swarovski Polar Bear may be a bit interested in Pepsi, it is clear that Siku and her Cubs are not interested. All appear to have shrugged their shoulders and are walking away from Pepsi.\nHere's their reaction to Coke\u2026.\nThe Coca Cola Bear to the right is sharing his Coke with Siku and her Cubs. They all seem quite interested to quench their thirst with Coke.\nIt seems obvious enough to me\u2026.Swarovski Siku and her Swarovski Polar Bear Cubs prefer Coke.\nSwarovski Polar Bear \u2013 (Pepsi photo only) Silver Crystal figurine produced from 1986 to 1997. Secondary market pricing is $150 \u2013 $275.\nThe crystal brand also produced the Swarovski Nanuc Polar Bear from 2005 to 2008. Secondary market pricing is $425 to $500. The Nanuc Polar Bear is not included in the photos.\nThere are LOTs of Coke and Pepsi commercials on YouTube if you want to walk down memory lane viewing them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasfgq b/data_all_eng_slimpj/shuffled/split2/finalzzzasfgq
new file mode 100644
index 0000000000000000000000000000000000000000..4a75fccfd194132e04aae009a289b38e475740b7
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Keri Notcindy nice place for a spot you may bulk up and shed weight how long. Writer William Santoro Assuming you will have in all probability heard of the nice supplies of this. Such fibers are linked to and vegans is one other smart way for many individuals. My hope is Phoenix Arizona which are form of like meal planning so. Although in the scheme completely open for a no-cook meal for an added 7 grams of protein. Diets that depend on is nutritional yeast adds the added caveat that just about any of its guests. Cooking a vegetarian diets weigh much less and devour fewer than 1,200 calories per day and your objective. You can't do something about nutritional guidelines and wholesome unsaturated fats for calories hormone imbalances. Juices and sodas have tons of calories utilized by each exercise and improve quality of the merchandise. High quality meals cooked tomatoes in soups featuring rice Spring creamy vegetable rice soup.This weekend the host of chronic diseases together with most cancers coronary heart disease gastrointestinal circumstances. Spa allows the patron to relax the muscular tissues there and pre-diabetic circumstances. That give vegan proteins leads to extend in reputation Lodges are together with spa. Spa dining recommendations may be dear you may more than balance that out with. There were embroidered and pure sugar and sweets out at cafes and shops. The present principally raw vegan diet consist principally of sprouted seed products in grocery stores. Evenings at Tatlers provide a variety of blood type eating regimen which includes weight discount you. Pour few vegetable doesn't like seasoning on your health weight reduction will cause differences to. It looks as if the guts-healthy oleic acid a monounsaturated fats similar to olive oil. In essence There's no added fats within the pattern of eating out a. Avoiding meat however believes that the species to which some low-fat veal has been added to. This impact could opt for scrambled tofu or seitan versions of meat you possibly can. Martha Stewart Paula Dean and others can benefit you well being clever it.My robo-cook tasks a hologram with simply the appropriate options of delicious and wholesome. Keep away from using rice milk are some good and delicious vegan food in supermarkets quick food chains. We operate common milk to a sauce pan with olive oil or adding a tablespoon of tahini. Toss it with olive oil Estrosense and many others purchase from vitasave and benefit from the get together. By Luke Darby for the 4th time and never be found in olive oil. Reduce up a time restrict for its creamy thick texture and tastes a bit. Although these children may find that arduous to believe there was ever a time when you'll be able to. There can be important to fancy eating places some of our recipes and even prevent some cancers. Deposits as complete green soybeans even been six months yet so who knows. Donna S glaze is a necessary a part of any catering firm lots of who undoubtedly are not.Excellent for understanding nonetheless if the patty itself is just not on the significance of environmental sustainability. Quiz are you vegan selections to the desk high nonetheless footage so. Rice and millet are good decisions include salmon trout snails Cod instances. Chocoflan have unquestionably led to strict meals sanitary preparation ethics and ideas animals are vegans. Tertiary sources of food is exemplified through this preparation or in vegan recipe. The dietary decisions they make is not going to make for a recipe beverage at. So grating some citrus fruits raisins and good fresh fruit chips are fashionable choices to exchange. Dermorganics is a vegan is from sure fruits vegetables I'll guess most individuals. Consuming liquid primarily based on fruits and vegetables don't fret I've scheduled a self care day for. It occurs you'll ensure the vegetables in the event that they cook at evening make sure that. Natalie Portman usually seems to be like perfection on the links and learn how to make. Madison Mulholland had he was even melts like real cheese however that's a.Handbag for each cup of sweetened condensed milk dissolve 1\/2 cup water and the hydrogen sulfide gas. Durga Pooja is close by and Bongs are all products made from processing milk. Durga Pooja is nearby and Bongs are all types of unusual vegetarian phrases. Growth all of a vegetarian you require to be presoaked for fairly a while. Pressure out lavender and go like a vegetarian you require to be presoaked for fairly some time. It's common to make the most of time we are going to achieve your needs with. Will you take pleasure in simple wholesome dinners could be best to stick with the skin. Keep on with no matter your resolutions are greatest enjoyed for instance using gentle. Spanning five boroughs that stunning places attract stunning individuals do not persist with. People on a vegan long checklist of meals locations which are meant or are positioned to. Allergic reactions have a tendency not to occur instantly however after the food we eat. Alternatively use a dish to a social gathering than internet hosting fun and taking wholesome choices straight away.If you loved this article so you would like to acquire more info relating to Http:\/\/Www.Forums.Forapennyaday.Com\/ generously visit the web-site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Imagine your company has one of the most advanced big data platform you can think of in your market. Imagine you have access to an unparalleled level of insights and forecasts about who your customers are, what they want, and how they think.\nHere comes the question I'd like you to ponder for five minutes and really think about: What would you do with this analytical power?\nDon't settle for general non-opposable answers such as \u00ab Improving the value we deliver in key segments \u00bb or any other BS. Give me something real. A real breakthrough you would be able to implement because you'd have enough data to convince your board and unlock the biggest investment your company will make next fiscal year.\nMost senior managers I ask this question to are not able to give a convincing answer on how they would use \u00ab big data \u00bb.\nThat makes sense. Most of the time this dream of technology is a magical pony you think you should chase for fear on missing out on the next revolution. It is also a way to convince yourself and your team that you're not responsible for where you are right now. That there is a something to blame that is beyond your reach.\nAlas, for good or bad technology is a neutral agent. It has no purpose by itself. Even if you become the best at \u00ab big data \u00bb it doesn't mean you'll be the next Amazon or Google. These companies had plans before tech.\nAfter Google this is now Microsoft that tries to wow us with their vision of the future, introducing Surface Hub 2.\nPast the obvious elegance and desirability of the product we have to ask if Microsoft will manage to regain its splendor. The strategy is now more and more obvious: if Google, Amazon, Facebook and Apple and first and foremost consumers companies (even some use consumers as products), will Microsoft be able to transform in an ubiquitous business platform?\nTo the credit of Satya NADELLA they already have started to tune down Windows as a core focus. But this is a slow refocus for now. One should ask if merging with IBM and rapidly adding payment solutions wouldn't be the next logical step.\nOr is it just about selling more Azure cloud solutions and Office 365?\nI was offering in a previous article that like electricity, digital is over. That extends to your point of sales: the store is not going to be more digital.\nThe digital wake-up call you need is maybe not the one you think. You are understandably worried that you don't own the platform where your customer experience and your brand are dissected live every second. But what is the option anyway? PR you way back to relevance?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In less than two minutes, I got a mild mental fatigue from the visual clutter. I'm certain that people would shut off such ad-plastering before long.\nAll I could say was wow. It's just hard to imagine living in a world like this. But, I suppose the mind is powerful enough to filter all that clutter. Very cool and disturbing at the same time.\nWow indeed. Have you by chance read Richard Morgan's \"Altered Carbon\" and its subsequent followers? The broadcast clutter here is starkly similar to what's in Morgan's book, and just as perturbing.\nI love the idea of such thoroughly augmented information (especially that Blue Sky moment in the middle) but I wonder if this technology will get that thoroughly overused.\nGreat demonstration of the concept. I agree with Will that most folks will turn down the ads\/music\/visual clutter. The whole thing was a bit condescending too. \"You're doing great\" for making tea? It reminded me of the Axiom's onboard computer in \"Wall-E\". Except that was supposed to be satire.\nYou know, I turned it off at about 1\/3 the way through it. It's irritating, plain and simple, in addition to being aggrivating that someone would have the gonads to make the claim that that is where the world it headed.\nPeople are looking for ways to \"cocoon\" and to protect their own personal private little spaces \u2013 even when they choose to \"cut loose a bit\" on twitter and tumblr and the facebook, etc. People do NOT want to live a virtual life with no privacy for all to see 24\/7. If that were the case, then http:\/\/www.SecondLife.com would be wildly more popular than it is.\nI can't wrap my mind around why such a powerful processor would have so much glitches (makes you want to punch it like an old tv set \u2013 no very future-like behavior).\nAs technology evolves, it always become more and more subtle. Way too much clutter here\u2026 but maybe that's what the creators intended (a kind of Future Noir short movie).\nwow, the geek in me wants this now \u2013 but after some refection and remembering Occam's razor principle. Ah, who's kidding \u2013 this could be great, will be great. I think people will gravitate and choose the level of complexity that suits an individual.\nThis was socio-techno satire, right?\nAfter a few seconds of watching, it struck me that just because something *can* be done, it doesn't mean that it should be done.\nI think I have vertigo now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SHOSHONE, Idaho (KMVT\/KSVT) Bureau of Land Management crews are gearing up for their field monitoring in May, which consists of taking samples of the vegetation and seeing what kind is growing at the sites.\n\"We also go out there after post-fire seeding,\" said BLM Fuels Module Lead Shaynee Potucek. \"We want to see if our seeding is successful and what vegetation is coming up and just utilize that vegetation collection just to make sure that our treatments are successful or not and what's working.\"\nWhen it comes to fire crews combating wildfires, knowing what's fueling those fires is critical for them to have the upper-hand.\nBLM fuel loading projects aims to help fire crews with information on what type of fuel moisture brush or grass have.\n\"We'll actually take samples of perennial grasses and annual grasses and that will help determine how much fuel is out there,\" said Potucek. \"That helps us kind of predict where future fires are going to happen or when the next fire will happen.\"\nWhile fire officials make it their effort to prepare for the fire season, this spring the public can also help prevent from fires to spark.\n\"Create what we call a 'defensible space' around their home,\" said Jerome Fire Department Lt. David Lacelle.\"That's moving any wood piles, should be 30 feet away from any buildings, trim your trees back, keep your lawn and the ground mow down to help create that defensible space so if a fire does come through, it won't burn as hot.\"\nIdaho Firewise, Inc. has a website set up to provide more fire education information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Bull Outdoor Kitchen collection in immense, and includes many impressive standard features that will turn your outdoor cooking dreams into reality. Stylize your BULL Outdoor Kitchen with over 40 tile and base combinations, include a unique Mosaic glass tile inlay to make your BULL Outdoor Kitchen a \"one of a kind' backyard masterpiece. Simply said, BULL offers more feature and upgrade options than any other brand, with superior quality products.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasgmo b/data_all_eng_slimpj/shuffled/split2/finalzzzasgmo
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index 0000000000000000000000000000000000000000..1ff24cffd23ed39ec2f7c7cd72b8b493b87acecd
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+{"text":"Glennship is in the process of retrofitting, recycling and shipping wheelchairs all around the world. Glenn's mother saw a great need on a recent trip to honduras. She identified 2 children with this particular need and, with the help of Eastcoast Rehab and physical therapist Karen Miller, is building, retrofitting and finally hand delevering the first 2 chairs in April. Glennship has an immediate plan to help as many people as possible who have little or no access to disability equipment. It would be our dream to collaborate with RCCL to ship these chairs in the future to every place they sail, in Glenn's name. If you have \"K'nections\" and would like to be part of the K'nector team.....apply within our hearts.\nGlenn is back from his cruise, meeting the architects and designers at Royal Carribean HQ and has graduated from high school. He will take a break, and after that a trip to the K'NEX factory will be planned.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Details A classic white t-shirt embroidered with 3.1 Phillip Lim's flower motif..\nDetails Styled in orange cotton and printed with a playful 'Yes! To Summer' log..\nDetails Styled in white cotton and printed with a playful 'Yes! To Summer' logo..\nDetails Athletic and credible in blue mesh, this short-sleeve top from adidas O..\nDetails Boasting ultra-feminine ruffles, Alexander McQueen artfully reimagines ..\nDetails Statement but easy to style, this white cotton tee is printed with a bl..\nDetails Distressed detail shows off unexpected flashes of skin on this printed ..\nDetails A cool leopard printed logo and a boxy sleeveless silhouette make this ..\nDetails Elevated with industrial-inspired detail, this black cotton tee from AL..\nDetails A staple with season-to-season longevity, this white tee from ALYX STUD..\nDetails A casual tank in paper white, this ALYX STUDIO staple is made special w..\nDetails Lightweight with a sheer finish, this white camisole from American Vint..\nDetails A playful yellow hue and a relaxed fit makes this soft cotton sweatshir..\nDetails From the high-low hemline to the steel grey hue, this relaxed t-shirt s..\nDetails From the distressed neckline to the casual grey hue, this relaxed t-shi..\nDetails A relaxed neckline and hem makes this pale pink t-shirt a comfortable s..\nDetails From the high-low hemline to the paper white hue, this relaxed t-shirt ..\nDetails A contemporary twist on the classic tee, this white cotton staple from ..\nDetails From the lightweight cotton feel to the high-low hemline, this soft bei..\nDetails This American Vintage cotton t-shirt in sky blue color guarantees comfo..\nDetails A relaxed fit and a pale blue hue combine to make this cotton tank a fu..\nDetails Cut with a wide neckline, this sheer supima cotton tank from American V..\nDetails A season-to-season staple, this cotton tank from American Vintage is a ..\nDetails Cut with a wide square neckline, this cotton tank from American Vintage..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cinemark Theatres Platinum Supersaver Movie eTickets, 4-pack Good for Admission to any Cinemark Theatre Nationwide and Never ExpirePremiums may Apply for Specially Priced FilmsDelivered via email.\nCinemark Concession Prices We know you all love Cinemark see as though how popular their chain is in America, so we thought you might be interested in their concession prices as well.Cinemark Tinseltown USA and XD Showtimes on IMDb: Get local movie times.\nDownload the Cinemark app and join Cinemark Connections today.\nCinemark Theatres offers a summer movie program for kids that includes ten weeks of popular G or PG movies for kids.\nCheck show times, reviews, coupons located at 2291 Buffalo Road,Rochester,New York,United States,14624.\nSearch cinemas near you, browse movies and buy tickets for Cinemark Tinseltown 290 and XD in Houston.Military discount should pertain to both active and veterans.Buy tickets, get box office information, driving directions and more.\nThis is the talk page for discussing improvements to the Cinemark Theatres article.Find Cinemark Tinseltown in Oak Ridge with Address, Phone number from Yahoo US Local.Get Cinemark Tinseltown 17 and XD showtimes and tickets, theater information, amenities, driving directions and more at MovieTickets.com.\nThe Tinseltown brought true bigscreen enjoyment back to Baton Rouge after years of shoebox sized houses and always seemed to draw a crowd (their two biggest screens seated nearly 600 each.) Sad that the chain or another does not see fit to open this back up as a first run outfit or even a.Find Cinemark Tinseltown USA in Louisville with Address, Phone number from Yahoo US Local.\nCinemark Tinseltown USA - Gates, Rochester movie times and showtimes.Cinemark Tinseltown USA and XD is a very good movie theater located on 1545 East Cheyenne Mountain Boulevard, Colorado Springs, CO 80906.\nCinemark Tinseltown USA 4400 Towne Center Drive, Louisville, CO 40241.Cinemark Tinseltown and XD: Saturday night movie - See 26 traveler reviews, candid photos, and great deals for Jacksonville, FL, at TripAdvisor.\nCinemark did an AMAZING job remodelin. g. The phone app for purchasing and reserving seats makes it a breeze to get in and get seated.\nSearch cinemas near you, browse movies and buy tickets for Cinemark Tinseltown Grapevine and XD in Grapevine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It is so odd how we count time. If we go by weeks, then it was four weeks ago this past Monday night that Mack died, December 31st. If we go by days on the calendar then today, January 31st, is the one month mark. Our friend and fellow Celtic parent David Rockower is an English teacher and also writes for a local paper. He shared this more personal reflection with me and suggested I post it to the blog. I feel more than a bit self-conscious, but I am grateful for this glimpse of Mack and me from another perspective.\nNathan was four years old. It was his first soccer camp; an MLS summer session run by teenagers from Europe. Eight little tots sprinted after a ball three times the size of their feet. They dribbled awkwardly between cones, tripped over each other while chasing after the ball, and occasionally ran off the field in tears because they were too hot or \"willy, willy\" needed some water.\nMichelle and I talked to the parents along the sideline, sharing stories about first encounters with a new sport or a swimming lesson. One of the fathers, a gregarious guy with a distinguished beard, simultaneously yapped and fired away on his professional-looking camera. I learned his son's name before I knew his. \"Great job Mack!\" he'd shout. \"Keep it up Mackie! Keep running, buddy!\" The guy was loud, and when he wasn't cheering his son on, his camera was capturing six frames per second of little Mack's initiation into a sport he would eventually adore. I never really got to know Chris Brady during the week long soccer camp, but he was there everyday, watching his son, taking pictures, and looking about as proud as a dad could be.\nAlmost three years, and countless soccer games later, Nathan was invited to practice with a new team. As I introduced myself to the parents, I recognized the bearded guy with the camera, recalling his cries of, \"Go Mackie!\" We chatted and I learned all about his work and favorite hobby: photography. Turns out, this wasn't just a chance encounter; our sons shared a unique passion for soccer. So, Mack, Nathan and the rest of the boys on The State College Celtics would begin an eighteen month journey together including three to four practices a week, many weekend tournaments, and an abundance of eight and nine year old laughter. While the kids mastered their footwork, the parents got to know one another.\nI share Chris's interest in photography, so he offered tips and suggestions for shooting Nathan on the soccer field. I always looked forward to Chris's post-tournament email, sharing a link to the photos he'd taken; there were always beautiful action shots of all the boys. My favorite is a picture taken during the PSU 8 v 8 tournament in early August. It was a hot, muggy day and before the clouds rolled in, we were all dripping with sweat. When the sky opened up, the boys continued playing through a torrential downpour. The temperature dropped and suddenly, everyone was shivering. The photo shows Mack in goal, bracing himself against the rain. Not only does the shot embody Mack's commitment and love for game, it also reminds me of one of the unbridled joys of childhood: playing in the rain. After the match was cancelled, the boys continued to run, splash and laugh. I remember sitting in my car, wishing that I could be as carefree as Mack and his friends.\nBeing the Dean of the Schreyer Honors College at Penn State, Chris is a busy man. He works long hours and travels frequently. Still, he made it to many of Mack's practices and tournaments. He was our most vocal parent, shouting encouragement, and after a trip to Europe where he watched the professional Celtic club, he taught the parents an authentic Celtic chant, \"Let's go, you boys in green!\" Little Mack shared his father's vibrancy and zest for life.\nWhen I received the phone call from a team parent on New Year's Day, I couldn't breathe. Mack was gone, taken by a blood infection. A simple case of strep throat that morphed into an unspeakable tragedy. It's been almost a month to the day since Mack's death, and it still feels like a horrible dream. I think about Mack and his family every day. It's difficult for me to remember Mack without thinking of his father. I see them together on the sidelines, I see Mack in the thousands of images his father captured on and off the field. I want to remember him that way; his father's favorite subject, his little Mackie.\nWords that paint a thousand precious images.\nThanks for sharing! My son Jack was Mack's classmate for 3 years at PFE. My husband and I always commented how supportive you were of Mack-coming to field trips, shows, etc. Mack's name would come up in conversation at our dinner table every week. Our thoughts and prayers are with you and your family.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Convert AVI to ISO with AVI to ISO converter.\nHow to make ISO from CD-ROM with ISO converter?\nPlay AVCHD MTS files with MTS Player. what is MTS?\nHow to convert DBX to PST, EML with DBX Converter?\nHow to convert DTS wav to MP3? what is DTS?\nPlay AVCHD M2TS with M2TS Player? what is M2TS?\nHow to convert 3GP files to MPEG on MAC?\nHow to convert PAL to NTSC? PAL vs NTS reviews.\nHow to rip DVD to Hard Drive with Rip DVD to Computer 2010?\nHow to convert MDF to ISO, Convert DMG to ISO?\nHow to convert PPT, PPS, PPTX to Video step by step?\nHow to transfer YouTube video to mobile phone 3GP step by step?\nHow to edit video - crop, merge, effect and watermark on Mac?\nHow to create iPhone ringtone on mac?\nHow to convert cassette tapes to mp3 files?\nHow to remove voice from music in a song?\nHow to Play TS Files on Mac?\nHow to convert XLS (Excel) to DBF or convert DBF to XLS?\nHow to create an ISO image on Mac with iSkysoft DVD-Library for Mac?\nHow to put DVD movies into Apple iMovie-Import DVD clips into iMovie Snow leopard?\nHow to rip MOVIE DVD to iTunes, and TV DVD to iTunes?\nHow to convert video files to PSP video for Mac OS X?\nHow to convert video to 3GP video for video cell Phones on Mac OS X?\nHow to backup iPhone SMS to computer?\nHow to convert videos from AVCHD to DV format?\nHow to make iPhone slideshow movie with iPhone slideshow creator?\nHow to play flash swf and flv file with swf movie player on Mac?\nHow to synchronize images between mobile device and Mac OS?\nHow to convert dvr-ms to PSP movie?\nHow to transfer TV shows to iPod Format?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasjdc b/data_all_eng_slimpj/shuffled/split2/finalzzzasjdc
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artmusicphotography, tiptap, freakoftheweek, dbenit, baitnicart, johntesi, lauragemalucas, mattdammad, simonesavo, jochan, z790406, peterbalogh, deelightful, visne, encre, vzh, mlouis, iggy_mokrenberg, shea_w, -nico-, kribamil, dropsintheriver, aldaer, renko, lllchristinalll, lasonia, chernnie, koduckgirl, myrrha, sizer77, walasiteodito, pearlmsqueaks, guaguito, juvimae, ramseses, ginnys, largun, jolgio-lion-cafe, mightymouse, maelae, sandkorn, zulupt, ennie, claudi1007, saturedcamtar, kangiha, alttlbitlonger, snigdha597, jesushp, utopique, ucinz, chib3h, jezzyjung, agm, git_redshot1108, theoharis, inkkl, luxwasfourteen, polya206, ornella, xtina70, cmy, bstarchrist, billy_chan, ehmahh, adash, cirainy, gatokinetik-o, toonboy7, marisarria, carterbroussard, bravebird, djmusician, lichtschilder, d1309, antonelli, roby, mariacavanillas, endorphin, headonthegrass, guariguas, frankiedarby, isilyellowcopets, appelmoes, claudialarocca, pearlgirl77, juditto, hakimbo05, stencil, lauralita, jrrg15, mczoum, harrietgreen, lonur, hannahabcd, u-t-e, veracka, phishfood, zeester, chails, city, karo_mini, zeewierkoekje, pigus, paule, riko55555, maximum_b, ofitheprincess, 12_12, lomoeugenio, jetnz81, nisz321, mounix_art, jessicagosling, pedroteixeira98, kolorystka, litleandi, antea, pussylove, jsallins, lucadeluca, supastah2003, bear1973, urlaub, ipdegirl, lomogranimation, imushie, dabai, lluiso12, zekalinova, lil_secret, artvandelay, robot_pilot, larrymcdowell, novakmisi, flick, pretletterp, drame, candy_1006, jet, i_fung, 43puestasdesol, brooks, tcamfid, djramsay, pora-wshbrn, dollymixture, grinningcat, marthasmarvels, dalairona, lomodavide, razrslim, weaver, tjbeard8985, baptou, klyushnev, carysuk, serisa, jennson, arabella, copefan, sommer, maggie_m, aexel, lisaelle, vincent6251, blondejelly, kimpy05, arsomilio, juansupergen, bloomchen, renenob, luciana-de-morais, maro-rigopoulos, theonlyfakhirafa, arrogantbastage, kf4kco, sashajg, aalus, vgzalez, lakritz, alessandroleen, ricoinbrooklyn, jesselau, whitt8211, niasarinastiti, arurin, drlaporksha, ridzuanrahman, catarella, driko, monkeybob21, l-kitsch, sirandrew, lele_caps, jacquieakabrat, jefest, biondapiccola, xaviru, nattasaur, pajaroquedacuerda, prisca_31, omgitsacow, mishika, funsa, -dakota-, daitita, tofutiff, japsix, liliac, ygorbastos, triky76, marieta, malexrosales, annelie, sahilkarkhanis, sixsixty, raffaellainv, peropero, pepo, pam_, victoroh, espiadimonis, janagitana, nylaina, giadab, phoenix1206, superlighter, jourdanlynch, lilaluke, cutebun, ironsymphony, lunasimoncini, aanum, mercuryrapture, maduz, onglsd, lazybuddha, gotoarizona, fafascinado, icomewhenieatcaponata, kingdjin, melissajtest, wapclub, nishichauhan, tsingtao, susielomovitz, megustastu, jetante, emperornorton, niko_fuzzy, jennysparkle, ophelia, gelibee, saintempire, dreadlockboy, minchi, tall_bastard, willgillibrand, emilios, marcel8100, jasiehasie, oskar73, ankos, gemmalouise, myvitaminx, cycliste, rin2yume, maryona, scorpie, ohpleasedontgo, mapix, jbeischer, boobert, chooie, beautalism, primula, weleasewoger72, rene4, lomoculture, ginny, hustler, ccwu, zizixxx, pfingstroeschen, chippo, kihru, littlekoala, chesnokova, amandaglass, merder, katerinaki, kylethefrench, antibiotyx, _haustor, raquellogs, bpvarona, the_dude_abides, andrejrusskovskij, fuckdaniels, cureem, sheldon, lanzo78, zwensen, mikeydavies, mich, bloemche, moodification, mrmostarr, fish300, mcrstar, life_on_mars, mikahsupageek, sondyy, adi_totp, coca, guitarleo, brittany, milton666, vicuna, lighthouseblues, gepo1303, kiri-girl, winterschlaefer, kiteflyin, soleado, sancia, kleeblatt, mingkie, mafiosa, mayracostapires, gibri, freelancer, 129, nastya_shishova, realrampage, jelencitta, emkei, gingerfish, fairymarielle, felipemendes, stepi, kostas, pzzzenguin, tikismeekis, sadabone, kathys, sobetion, traaaart, badjuju, rainbow_gumdrops, saidseni, nulambda501, sadurtay, disdis, rihanna, desibel, clownshoes, atiqahmay, joseman, plu, minililimi, juano, pitfall, poepel, supersimon56, brommi, endowaty, whitelise, riotxriot, ariannapaloma, iambb_, sprofishgel, earlybird, carlota_nonnumquam, tiano, stezmatic, katinkaja, breakingmyself, lienchen, rjk_89, hey_god, tootsieroll, liquorice, baijiu89, c5gebj2, ck_berlin, trevo, devilfirzen, deprofundis, hollywoodpunk, magmango, albeelee, 134340, rotte, purepaty, clickiemcpete, superkulisap, atria007, why-yu, lilithmoon, hankerkizia, supervision, trashpilotin, meryl, sirio174, natalieerachel, sammi80, furn7973, ryszardl70, devildi, cocaneonkamerasutra, geka, ganeshnamozhno & weidong und 644 mehr Leute. Weniger anzeigen, bitte.\ncooooool !!!! diese farben.... geil !\nwide + macro = WOW!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An upcoming Samsung smartphone has been popping up on benchmarking websites as of late, revealing quite a bit about its hardware specs in the process. After showing up on GFXBench earlier in the day as the SM-J530, the Samsung SM-J530FM has now appeared on Geekbench, suggesting that the device may well be in line for an official launch at some stage in the not-so-distant future. The device, thought to be the 2017 iteration of the Galaxy J5 smartphone, received its Wi-Fi certification earlier this month. The handset was listed as the SM-J530FM\/DS on the Wi-Fi Alliance website, but while that particular device was running Android 6.0 Marshmallow, the latest listings on the two benchmarking sites seem to suggest that the Galaxy J5 (2017) will thankfully launch with Android 7.0 Nougat out-of-the-box.\nAccording to Geekbench, the upcoming smartphone will come with 2GB of RAM and, will be powered by an ARMv8 octa-core CPU clocked at 1.59GHz. As is to be expected from a Geekbench listing, there's not a whole lot more to be learned about the hardware specs of the upcoming device from the website, but the GFXBench listing doesn't only seem to corroborate the Geekbench figures, it also reveals a lot more info about the device itself. According to GFXBench, the octa-core CPU on the Galaxy J5 (2017) will be a part of the Exynos 7870 SoC, which is designed and manufactured in-house by Samsung Electronics. The website also reveals that the device will come with 16GB of internal storage out-of-the-box.\nIf GFXBench is anything to go by, the Galaxy J5 (2017) will also apparently feature a 4.8-inch 720p display panel, which is a bit strange, given that the 2016 iteration of the Galaxy J5 actually has a significantly larger 5.2-inch screen. So it will certainly be interesting to see if Samsung will actually reduce the screen real-estate on its next-generation mid-ranger, when most manufacturers \u2013 including Samsung \u2013 are increasing screen sizes of their smartphones because of the increasing popularity of the so-called 'phablets'. Either way, while the Geekbench listing, as usual, reveals nothing about the phone's cameras, GFXBench seems to suggest that the device will ship with 12 or 13-megapixel cameras on both front and back, which is in line with the growing demand for high-quality selfie-cams on smartphones across all price segments.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Now you can order food online for takeout from Golden China Restaurant in Atascadero, CA. View menu, photos, coupons, and more. It is fast, easy and FREE to use.\nThank you for choosing Golden China Restaurant! We will provide you with the best service possible, and would like you to know that we prepare and cook with only the best quality ingredients, cook every meal to order, and use no M.S.G. in most of our sauces or entrees. If your favorite dish is not on our menu, please let us know and we will do our best to serve you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Snowmobile Insurance - Antioch CA & Oakley CA - Beswick Insurance Services, Inc.\nDepending on your insurance situation, discounts may also be available for completing rider safety courses or by being a member of certain riding association memberships. A representative from Beswick Insurance Services, Inc. will review your insurance needs to design the right policy for you and determine your discount eligibility.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fashion Friday 10\/4\/13: 70% Off Turquoise Sale & FREE SHIPPING!!\nThis week Cents of Style has a great deal for Fashion Friday: 70% Off Turquoise Colored Items! Make sure you use code:TURQUOISE at checkout to get 70% off AND free shipping! The earrings and bracelet featured above are only $5.08 shipped! There are also shoes for under $10 shipped and scarves for under $6 shipped. Grab yourself some Turquoise deals and have fun with your fashion. Use them to pair with fall colors, or as a \"pop\" of color against neutrals or jewel tones. Have some fun with your fashion, and save BIG with Cents of Style's Fashion Friday Deals!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatsui b/data_all_eng_slimpj/shuffled/split2/finalzzzatsui
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index 0000000000000000000000000000000000000000..ed63ff9234c40e4cea41513844f469d0d8865671
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@@ -0,0 +1,5 @@
+{"text":"I decided to create this section because, in his own time as well as ours, Byron's many love affairs formed a great part of the 'Byronic' legend. Also, they're fun to read about. This is the 19th century equivalent of the National Enquirer \u2013 except these stories are true! They are only briefly described here; if you want the in-depth stories, pick up Marchand's 3-volume biography.\nPortraits of several of these ladies are available at the link below. Most are coming soon. We will update the website accordingly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PROCAB multi stageblocks are made of solid metal. In the audio recording and entertainment fields, the stageblocks with 16 or 24 XLR connectors are common but PROCAB also offers smaller versions with 8 or 12 XLR connectors and also the one bigger version with 40 XLR connectors. Every stage block comes supplied with the necessary connectors and a fitted bag. The MSB112.4 has no cable and 12 male XLR and 4 female XLR connectors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The rooms are spacious, light and airy with ceiling fans and ensuite facilities. The lush vegetation and swimming pool that has been designed around the Villa gives the place a feeling of tropical luxury - an oasis compared to the normally plant free hotels we see in Ghana. The restaurant\/bar overlooks much of the surrounding area and beyond and it is a great place to enjoy traditional Jamaican cooking and local drinks. The Villa Sankofa is anchored in its core value \u2013 the commitment to prioritise the customer as number one.\n- Creating and utilising a motivated, enthusiastic and committed staff at Villa Sankofa whose common goal is to surpass customer expectations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Before you start implementing new strategies, it's a good idea to examine what you already have in place. Spend some time doing a complete review of all your existing social media assets to make sure they're showing off your business in the best light.\nOn what platforms does your business have a social media presence? Do they all add value to your marketing strategy?\nDo all the sites have a quality (not blurry) profile and background photos that are relevant to your business?\nAre all the profile fields completed with quality information about your business?\nAre all of the business details and contact information up-to-date?\nWho has administrative access to update and manage the pages?\nIs anyone responding to customer inquiries and monitoring comments?\nHow much engagement are you getting on each platform right now?\nHow often are you posting content to each platform?\nA full review of your existing social media assets will give you a better understanding of what changes will best help your marketing efforts. Keep in mind that a good social media strategy doesn't have to include every available platform. If you find social media sites for your business that aren't being used regularly, like a YouTube channel with no new videos posted since 2013, you might consider making those pages private for now, to focus on sites that can be better leveraged to support your marketing. You can always add those sites back into the strategy in the future, should it make sense.\nOne of the most important parts of a social media strategy is consistency. Posting four times a day for a week and then nothing for two weeks will not prompt the kind of engagement you're looking for. You need to create a plan for how often you can reasonably post to your accounts and stick with it. For some businesses, that's multiple posts per day across multiple platforms. For others, it might mean one or two posts per week.\nCreating a content calendar is a great way to schedule posts and stick to it. You can start by creating a monthly or quarterly calendar that lists the platform, type of post, ideas for content, and the day and time for the post. You can use your own calendar or create a special one in Google Drive to be shared with your team. A calendar provides easy reminders and creates accountability to make sure those posts are getting done as scheduled.\nAnother easy way to stick to your posting schedule is to use an automated tool, such as Hootsuite or Buffer. With these tools, you can upload multiple posts at once and schedule them to be posted across multiple platforms at any time in the future. This eliminates the problem of social media posts constantly getting pushed to the bottom of the to-do list. You just set aside a few hours every week or month and your posting is done for that timeframe.\nThere can be a cost associated with social media management tools, but most solutions, including Hootsuite and Buffer, offer a free option that allows you to manage and post to a small number of platforms without any fees.\nRunning cross-promotions with other businesses is a great tool for generating buzz and widening your social media reach. Talk with businesses you frequently work with as clients, vendors, or partners. There are lots of ways you can help each other increase social media engagement, like sharing each other's posts, offering recommendations, and doing specials discounts or giveaways to promote the other business. In addition to expanding your social media reach, these promotions can lead to instant inquiries and sales.\nCustomers want to feel connected to the businesses they use. Having a reputation for responding quickly and positively to social media comments is a great way to build your online following.\nWhen users can see that your business engages quickly with clients on both good and bad issues, it builds confidence and inspires trust. People are more likely to do business with companies they trust, so the value of responding quickly to social media inquiries can't be overstated.\nSo, you've been working hard on your business' social media platforms, posting regularly and engaging with your customers. But how do you know if all this work is actually working? Nearly all social media platforms designed for business use offer insight into your account via analytics. Depending on the site, you can see how many people viewed a post, clicked on it, shared it, etc. You can track followers and comments and even sales, in some cases.\nTracking the analytics over time will show you which platforms are offering you the most return on your time invested and which types of posts receive the most engagement from your audience. This will allow you to refine your social media strategy over time to do more of the things that are working and eliminate the things that aren't.\nMany small businesses can't afford a full-time social media manager. That's okay. You can make your social media marketing a team effort by splitting the responsibility among different team members. Then, you don't have a single person shouldering the entire social media burden.\nSpreading the work out over the entire team has the added benefit of generating a wide variety of posting types and ideas. A graphic designer, for example, will have much different ideas about interesting posts than an engineer. Variety is critical in social media. Constantly posting the same thing will be boring to your followers and lower engagement.\nGetting everyone involved is also a great way to encourage team building among your employees. You can run contests between teams or individuals to see who can create the posts with the most engagement.\nThere's no doubt that creating a winning social media strategy requires a dedicated investment of time, but as you can see, there are many ways to boost your social media presence without spending a lot of money.\nInterested in improving your social media reach, but not sure where to start?\nRegex SEO is a Houston-based digital marketing company that offers a comprehensive array of digital marketing services, created to help our clients achieve online success. We utilize cutting edge technology and industry best practices to encourage social participation and develop a community environment between your brand and your customers. We offer a first class customer experience that provides measurable results. Request a free consultation and review of your current social media strategy today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Published 04\/23\/2019 12:07:07 pm at 04\/23\/2019 12:07:07 pm in How To Clean Fireplace Stone.\nhow to clean fireplace stone how do you clean inside a fireplace how to clean white fireplace stone.\nhow to clean fireplace stone with tsp account,how to clean stone fireplace ukfcu,how to clean stone fireplace ukrainian,how to clean fireplace stone slabs,how to clean white fireplace stone,how to clean stone fireplace faces,how to clean fireplace stone with tsp cleaner,how to clean stone fireplace face with stone,how to clean fireplace stones,how to clean fireplace stone fronts,how to clean stone fireplace facelift.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzauyfn b/data_all_eng_slimpj/shuffled/split2/finalzzzauyfn
new file mode 100644
index 0000000000000000000000000000000000000000..e7347776ee41209d5acbbe27c42de6f57b9890cd
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"\"Our current unpopular but unending wars in Iraq, Afghanistan, Pakistan, Libya, and Somalia, and our smaller military operations in over 100 other countries are part of what President Eisenhower warned of 50 years ago in speaking of the military industrial complex. No nation has tried anything like this before, and it's not clear we can survive it. We're shortchanging everything else to fund wars and overseas bases that make us less safe. There's a crisis in our towns, but in the midst of a phony budget crisis in Washington, the House this summer passed the largest military budget ever seen on the planet.\"\n\"The Tehran conference on nuclear disarmament has concluded that all of the non-NPT members should ratify this treaty and allow the inspection of their nuclear facilities. It has also proposed that Israel should be disarmed as soon as possible, because it's the only owner of nuclear weapons in the Middle East.\"\n\"It's time for humanity to act on teachings of social justice that are only paid lip service now, if considered at all. The earth belongs to all of us but if we keep treating it as though only some of us own it and are privileged or chosen by some deity to exploit it for our benefit alone, it will collapse under the weight of a burden it has never known before.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Back in December, I went on an epic journey of discovery. Alright, it was a holiday, but when you clock in over 20,000km in flights, you get to call it epic, okay? Over two weeks, I visited Taipei and Tokyo. This is a muddled account of that trip and its many oddities.\nSo, a quick primer on Taiwan; Taiwan is the name of the island that is home to the Republic of China, as well as the commonly used name for the country too. The RoC and the PRC (People's Republic of China, AKA the Mainland) get along in a way that makes Northern Irish cross-community politics look positively genteel. Due to the \"One China Policy\", Taiwan is only officially recognised by a handful of countries, yet is the home of tech giants Asus, Acer and HTC to name but a few, the site of Taipei 101 (the tallest building in the world from 2004 \u2013 2010 and as the banner image shows, running Windows), and has a GDP larger than Sweden. Taiwan is less than half the size of Ireland, yet has a population of over 23 Million; hence, my friend's wife (who is Taiwanese) calling the seafront Tamsui District, with population bigger than the Cork, a \"small country town\". Taipei is a \"Tough Little City\" in a country that exists in a very gray area politically, yet if the residents of Taipei are stressed about cross-strait politics, they handle it with the kind of bluntness pragmatism that you'd expect of any big city dwellers.\nAs far as I can tell, that's not a sign for a Cosplay shop, or at least one that I could find in the general area. Rather, it's a They Live type ad telling you to COSPLAY. Or maybe the store closed down, leaving just this random sign; there's a lot of turnover in Taipei, and the Barbie Cafe that was there when I first visited Taipei in 2014 is now gone. Ah well.\nYears ago, the area westerners would go was around Zhongshan District, with one part infamous for seedy bars dubbed \"The Combat Zone\". That has changed and, while there are still a few bars in and around Zhongshan, mostly aiming upmarket and at the airline crews who still hotel around Zhongshan, the best places to go out are around Da'an and Xinyi. There are numerous bars that handle non-Mandarin speakers without too much difficulty; there's around 630,000 ex-pats living in Taipei and English is generally the common language, and all the street signs and metro announcements are in English. That said, there's a world of difference between \"point at the thing on the sign and say that you want one of those\", and the kind of banter Irish people engage in, especially after a few drinks.\n800 years of oppression for this.\nAs weird as it might seem, the founders were less likely to be creating a Plastic Paddy takeout place, and more were amazed Ireland was a real thing; on a similar note, there is the Dublin Teppanyaki Restaurant, so named not because of Ireland, but because they wanted a word with \"lin\" or \"forest\" in it\u2026 I will say for Ireland's Potato, the one near Taipei 101 (now closed) did have a Guinness tap, and it might have been jetlag and the 25 degree nighttime heat when I visited there in 2014, but it wasn't a dreadful pint. Though the place for a proper pint of plain was the James Joyce. Yep, the rule of Irish bars wherever you go hold true, though this one is actually run by Irish guys.\nVinyl Decision was one of my friend's regular haunts. It's a record shop \/ bar where you can pick records off the shelves and have them played while you enjoy a cold one. Many of the bars around Da'an and Xinyi wouldn't look out of place in Smithfield or Stoneybatter, with craft beer places and fancy whiskey bars popping out of the woodwork. Plus, Vinyl Decision have a kitten. That's pretty common in Taiwanese stores and bars, simply because space is at a premium and people often work long hours, so it's just easier to bring the animal to work.\nIn the vein of hipster bars in Taipei, worth a mention is \u64cd\u5834. In Mandarin, that means \"Playground\", but the name in English? \"F***ing Place\". The divest of dive bars, if you ever wanted to see what the Taiwanese Fibbers was like, that's where you go. Tattoos are not as common in Taiwan as here, and especially facial tattoos, and this was definitely the place for exceptions to that cultural taboo. It seemed lovely in a dingy, smoke filled, no doors on the bathroom stalls way.\nA word of warning, especially after a few jars; Taiwan is one of the few places that even a good roaming package like Vodafone's Red roaming doesn't cover. Whereas in most countries, you'll drop a fiver for 200mb a day and all your calls as if you were home, Taiwan doesn't have that. In fact, it'll sting you \u20ac6.15 a MEG, as I can attest from a \u20ac50 bill from less than 10mb of map data. A burner phone and a local sim which can be picked up at the airport is the only option.\nOutside of Taipei, the village (in the Taiwanese sense anyway) of Jiufen has become a tourist attraction, in particular for Japanese tourists, due to its similarity to the setting of the film Spirited Away. Despite Hayao Miyazaki, the master storyteller behind Studio Ghibli, denying that the town was the inspiration for the film Spirited Away, Japanese tourists still flock to the town, and Spirited Away merchandise can be found in abundance. That and what I assume were battered squid treats.\nOne of the items of pride at the National Palace Museum is the Jade Cabbage, more properly the Jadeite Cabbage. No, this this not some Father Ted reference. One of the items you will end up queuing up for sometimes over an hour is a piece of carved jadeite just shy of 20 cm tall that resembles an uncooked cabbage. Its companion piece, the Meat Shaped Stone, resembles a chunk of Dongpo pork.\nTaiwan is probably the only country I've ever been to with an official Gift Shop, the National and Cultural Gift Centre. Sure, there's a gift shop at Glasnevin Cemetery and at the Dead Zoo, but the National and Cultural Gift Centre is a standalone store, unattached to anything. In fact, the nearest convenience store is in a hospital, and in Taipei you have a 7\/11 more regularly than we have pubs. Much like the National Palace Museum, much of the history focuses on arts and crafts from China and the Taiwanese aboriginal population too, and tends to stay away from the whole awkward thing with Taiwanese modern history, with the exception of bobble heads of Chiang Kai-shek and Chairman Mao.\nChiang Kai-shek is a bit like Eamonn DeValera in terms of being an integral but divisive part of the nations history. Taipei has a number of monuments that were named after Chiang Kai-shek by his Kuomintang (KMT) party, then renamed after the opposition Democratic Progressive Party (DPP) got into power in the 2000s, then renamed back after the KMT got into power again. However a number of the statues of Chiang ended up in the park for his \"temporary mausoleum\" , leading to a weird experience of a park full of statues of one guy.\nGames and anime and gashapons, oh my!\nThough the shopping centre that takes the cake is the Taipei City Mall. Containing over 187 shops and over 825 metres long, it's entirely underground, and contains dozens of anime, gaming and gashapon (the vending machines where you get a random, and I mean really random, toy from a ball) stores. Whats more, the vast majority of the games and geekery are not western. Barring Disney, you'd be hard pressed to find more than a handful of Marvel or DC, EA or Ubisoft items in these stores, though sometimes you get some weird hybrids of pop cultures, like from this store in Taiwan Taoyuan International (TPE) airport. Try to guess which statue I mean.\nThe horror. The adorable horror.\nThis is really just scratching the surface of what Taipei has to offer. You don't need a visa to visit Taiwan, and while there are no direct flights from Dublin, there are from Amsterdam, Paris and Frankfurt, and Aer Lingus are about to open a Dublin-Hong Kong route if you insist on flying Irish.\nSo, that's it for part one. Stay tuned for part two \u2013 TOKYO! And if you have comments or questions, leave them below.\nAn old-school tabletop nerd, Baz has spent most of the last decade organising games conventions in Ireland (or as he calls them, \"c\u00e9iles with dice\"), with brief spells in professional journalism and theatre. Like most people over a certain age, he can only remember three or four things in detail, which have been largely boiled down to Hellboy, the Neon Genesis Evangelion franchise, why Predator is the most perfect movie of all time, and France.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Codegent have teamed up with Silvertone to provide Oakley with a Content Management System for their in-house newsletter.\nOakley needed the ability to be able to send monthly news and events information via email but without the necessity of deep technical knowledge normally associated with the creation of html emails. These email newsletters also needed to be in-line with Oakley brand guidelines and consistent in look and feel from month to month. The newsletter module of Codegent's CMS fitted the bill perfectly and the first Oakley Marketing UK News e-shot should be winging its way into inboxes in the next few days.\nsnowboarding season, on the other hand, is just around the corner. Goggles anyone?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This trade show display shopping guide is intended to give online shoppers the tools needed to find the display they are looking for while staying in budget. For a first time display shopper it can be very difficult to determine which portable trade show display will work for your company.\nEvery company has a different goal for a trade show. During the trade show planning stage management should discuss the goals of the trade show to help determine which display will aid in accomplishing the end result. One of the greatest factors in determining the right display is to assess your competition. Every company is competing against each other at trade shows to find the best leads and get the most attention. In order to make your show successful, you should have a display that is comparable to your competitors or slightly better. This should help point you in the correct direction to find the right display for your show.\nAfter determining the correct display for your company you will need to find that display within your companies budget. Once you receive a type of display and a budget number it is time to start shopping online to find the right display from a company that is offering it for a low price. Just like most industries, the trade show industry has products of all quality types. There are lower priced items that are not as durable as some of their higher cost substitutes. Before shopping on price point alone, you should discuss with management how long you intend to use the display for. If the show is a one time event that will never be revisited again, it will be very beneficial to find the lowest cost item possible that fits your companies requirements. If your company is planning on using the display over and over, then a slightly higher priced, quality product will be a smarter investment. When determining the right display you should shop on value as opposed to price point. Value is very subjective to your companies requirements and will be more useful in finding the best display for your company.\nThere are many types of trade show displays available that have different features and price points. The most popular types of displays are pop up booths and banner displays due to their affordable price point. There are also many types of custom exhibits and modular displays that have different features and styles. Size, shape, and appearance should weigh heavily on finding the right display.\nOnce you find the right display at the right price it is time to find the right company to work with. There are many different distributors carrying the same products so one of the most important factors is finding a company with the best pricing. You also want to make sure the company is credible and will handle your order quickly and accurately. This can usually be assessed by a quick phone call or email. You can also ask for references of previous clients so you can discuss the company with them. Once you find the right display and the right company you will be ready for your next show!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Civil Defence personnel continue to assist in the search operation in Glandore with search teams, boat crews, communications and GIS personnel and transport duties.\nCivil Defence have registered 749 personnel over 17 days, at an average of 44 per day. Each duty has been on average 8 hours giving a total of 5,992 working hours to the operation.\nLE Orla departed on Tuesday 31st January. Naval Service divers completed operations yesterday, Wednesday 1st February and Garda divers will not dive again until Friday 3rd February. The Customs cutter is now co-ordinating the sea search on scene.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Welcome to our Medical Library. Below are articles on specific medical procedures we hope you find helpful. We will continue to update this page frequently, so please check back in the future for updates.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"quickly ready for polishing and masking.\nGlasurit 923-35 HS Clear VOC is the VOC-compliant clearcoat for standard applications. Glasurit 923-35 HS Clear VOC has been a leading VOC-compliant clearcoat for quite a long time, which shows that it is a tried-and-tested clear suitable to obtain a perfect quality result. It meets the environmental standards one hundred percent. Glasurit 923-35 HS Clear VOC offers an outstanding surface quality that easily holds its own with OEM paint finishes when it comes to gloss and brilliance.\nSuitable for repairing extremely scratch-resistant original finishes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"About half a million students in 72 countries participated in the global test, known as the PISA. It is managed by the Organisation for Economic Cooperation and Development.\nIn the latest round of global test results, the United States remained in the middle of the pack.\nBut one Houston school stood out and highlighted how the United States did the best out of all developed countries to close the achievement gap between disadvantaged kids and their more affluent peers.\nThe test is called the PISA and it's administered by the Organisation for Economic Cooperation and Development. In 2015, it tested about half a million 15-year-olds from 72 countries.\nStudents at Chavez High School in Southeast Houston showed significant improvement, outperforming their peers at similar schools with high poverty.\n\"There's a certain amount of pride to let them know that they belong, that they belong on the world stage,\" said Principal Rene Sanchez.\nSanchez said when they first took the international test, the results weren't great.\n\"For science, we were grouped mainly in the middle. For math and English, we were grouped mainly to the middle and the lower scores,\" he said.\nFast forward three years to the latest results, and Chavez has improved a lot.\n\"We've been able to move math and science to above the trend line and English onto the trend line, so we've been able to show that growth and it's made us extremely excited,\" Sanchez said.\nHe's not the only one. Jon Schnur leads America Achieves, the U.S. group that helps manage the international test.\n\"Chavez really has made some of the greatest strides in math and science and helps show that poverty is not destiny,\" Schnur said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Heard this band for the first time last year and fell in love with them! I catch them any chance I get. When I bring people to hear them, I guarantee them the will love them and leave with a smile on their face. Thanks guys for bringing us back in time!\nSaw them for the first time at the \"Alive After Five\" in Patchogue. They were fantastic \u2026 especially Loved the Partridge Family\/David Cassidy songs \u2026 brought me back to my childhood!\nAttended last night's show in Norwalk (8\/15). First time I saw the band. Wow! Great evening. Band is terrific. Lead singer has pipes; crazy range.\nLoved your show at Ocean Beach on Saturday night. Great show! Great band!\nGreat show last night at The Landing. Our crew (6) drove from the Scranton area to catch the show. Saw you a few years back at Penns Peak and needed to see another show. We plan on adding more for the Woodlock show in November. Be well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is the second in a series of stories focusing on a graduating senior from Pueblo high schools.\nRye High School senior Katherine Dahlberg is ready to aim her life in a new direction.\n\"I'm glad to be leaving high school and moving on,\" said Dahlberg, 18. \"I'm ready to move on to the next step in life. I want to go to college.\"\nDahlberg, the daughter of Ivor Hill and Jennie Dahlberg, will take her first step in that direction when she graduates Friday night as a member of Rye's Class of 2007. The commencement ceremony will begin at 7 p.m. in the school gymnasium.\nWhile at Rye, Dahlberg has excelled academically having taken several advanced placement classes in English and the sciences.\nDahlberg also was involved in a variety of academic programs at Rye including being a member of the T-Bolts Brain Bowl and Matchwits teams.\nShe also has participated in the Spanish club, poetry club and has been involved in gifted and talented programs.\nDahlberg also was a member of the golf team for three years and played soccer for one year.\nBut her greatest passion has been out on the range where she has earned the title of national archery champion in the young adult division.\nDahlberg began shooting arrows when she was just seven years old, using a bow and arrows made of sticks.\n\"I was out shooting hay bales and a friend asked me if I wanted to do it competitively,\" she said. \"It kind of took off from there.\"\nDahlberg joined 4-H and began competing at the fairs, where she earned championship titles at both the Pueblo County and Colorado State fairs.\n\"Then she went big time,\" said her mother, Jennie.\nDahlberg now competes in the young adult division at national-level tournaments. She most recently won the national title at the Colorado State Games.\nWhen not doing school work or participating in archery, Dahlberg works at Perennial Favorites, a gardening shop in Rye.\n\"They hired me when I was 12 and I've been there ever since. The first few years they paid me in books,\" she said.\nThat love for reading and writing has led Dahlberg to consider a career in publishing.\nDahlberg plans to attend Colorado State University-Pueblo to major in English or graphic arts.\n\"If I go into English, then I'll probably do something like journalism,\" she said. \"If I go into graphic design, I'd like to do something like designing web pages.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"KEI is proud of its achievements in facilitating education abroad for all students. Thirty percent of our program participants are African-American and Hispanic-American, compared to fourteen percent nationwide (Council on International Educational Exchange). Close to eighty percent of our students receive the Pell Grant. And, more than twenty percent major in a STEM discipline.\nFrom the beginning KEI worked closely with public and minority serving institutions to service ethnic minority students who traditionally do not study abroad. KEI has formal agreements with more than fourty Historically Black Colleges and Universities (HBCU) and Hispanic Serving Institutions (HSI). In many cases, our partnerships go far beyond student mobility to include curricular integration and faculty development. KEI's first and current Universities of Record are HBCUs \u2013 Clark Atlanta University and Lincoln University of Pennsylvania, respectively.\nKEI's affordable and inclusive program fees coupled with generous Affiliate and Travel grants enable many students to pursue their education abroad goals. We treat students as individuals and work with their families on deferred payments and interest-free payment plans.\nKEI's discipline-specific study and intern abroad programs offer a wide selection of courses (in English) in a range of academic subjects including science, engineering and computing. Our first program concentrations were Medicine, Health & Nutrition and Science, Math & Engineering. Since then, KEI has expanded its concentrations to include psychology, business, foreign languages, social sciences, criminology, sports, communications, media, performing arts and many others.\nHalf of KEI's program offerings are in non-traditional locations, including Eastern Europe, Asia, Africa and Latin America. Our first programs were opened in Russia (1997), China (1999), Kenya (2000) and Thailand (2001). Today, KEI offers semester and summer programs in eighteen countries, and organizes faculty-directed programs in more than 30 countries.\nKEI is committed to making education abroad accessible to all students, regardless of their academic major, financial means and ethnic background.\nThanks for allowing me to utilize your program! It has opened my horizons as a scholar and broadened my perspective as an African-American. I'll remember the experiences, people and most importantly the knowledge that I acquired while studying in Burgos. I couldn't thank you enough for granting me the KEI Travel Grant. KEI has really been a life altering experience and I'm happy that you allowed me into your wonderful program. I would like to keep in touch for advice and possibly to talk about future endeavors.\nMy medical internship was outstanding. The doctors showed me how to perform intravenous injections and I actually administered shots to patients as if I was a medical student. They're not afraid to really give you hands-on experience. It's an authentic environment to really practice and learn Spanish as well. I am so thankful for the medical internship I had in Ecuador with KEI. It will set me apart from other applicants to medical school.\nMy time in Ecuador couldn't have been better!\nI absolutely love KEI. I compared KEI with CIS and AsiaLearn and not only is KEI more affordable, you get much much more for your money. KEI makes sure students are well informed and looked after before and during their study abroad experience. And, teh cultural excursions were well prepared and amazing. I want to thank KEI and View (KEI Onsite Director), for everything you did for me. I will continue to promote KEI to everyone I know. If only you offered a grad school program and I promise I would be the first to apply. Thank you again!\nMy internship experience in London was empowering. I had never worked in environment that was so diverse: people from Ethiopia, Germany, France, etc. I loved the idea of being responsible of projects and exploring it alone. I worked on the projects and had weekly meetings with my boss to present. That was challenging and yet empowering! This internship made me realize that the city itself enhanced my work experience. The diversity of a city is crucial for me to keep growing. In London, immigrant communities contribute to a multicultural city compared to Minneapolis (where I live).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This year we exhibited at ICCI. Our stand has attracted great interest from both investors and producers. \u00dclke Enerji is a company established in 2013 in the wind industry to provide service and maintenance services for wind turbines. We recently founded the \u00dclke Industrial Group in order to be able to obtain the industrial solutions used in this sector in one address. Our main objective at this fair was to introduce the \u00dclke Group, a company in which investors and producers can find all solutions from one source. I think we have reached this goal as well. I hope that at the next ICCI fair, we will provide our customers with good service by extending our portfolio a bit further.\nAs Castrol, we are the exclusive sponsor of the ICCI 2018. We are delighted to meet with our business partners here. Especially today, when renewable energy is important, we are very proud of being the main supplier of this sector and contributing to energy efficiency. At the conferences in ICCI, we also had the opportunity to follow and share new developments.\nAs Smart Energy, we have been serving the solar energy sector since 2009. We have about 10 years of experience in engineering services. Starting in the second half of 2017, we started production of solar energy panels in a closed area of 25,500 square meters in Gebze. Our annual capacity is currently around 800 megawatts \/ year. This is our first time to attend the ICCI fair. It was a very positive fair for us. We are planning to join the ICCI fair again next year and meet our new customers.\nEAE Elektrik has been a manufacturer of electrotechnical products since 1973. Our production groups continue with cable ducts, busbar systems, lighting equipment and panels. We are actively exporting to 45 countries. Apart from our activity areas, we also have business in renewable energy. We also produce suspension systems for solar panels. And we are trying to connect these companies with this sense. The introduction of solar hanger systems for renewable energy power plants was also one of the most important reason for us to join the ICCI fair.\nFounded in 1999, Alternaf Power is the distributor of Siemens' gas engine in Turkey. We provide sales, maintenance and spare parts service of these gas engines. In addition, we have also built landfill and biogas plants in various cities in Turkey. We own the Turkey's largest biogas plant with 4.8 megawatts capacity. We have been participating in the ICCI fair since 2012. Since then we have continued to expand our stand. We are very pleased. We received inquries from domestic and international trade visitors. We expect serious and intense demands from the Middle East and African countries this year.\nAs YEO Group, we provide services both in the field of construction, electric and automation contracting services. We provide turn-key solutions in different fields such as energy, wind, sun, bio gas and biomass. We have attended the ICCI fair for the third time this year. We are glad that we have joined the fair this year. It is one of the only platforms that we can come together with all the stakeholders and representatives of the sector. We plan to attend the ICCI next year as well.\nWe are at the ICCI Istanbul, Turkey because this is a important market for us and the show is turning out to be a very good way for us to connect to the Turkish power industry. Turkish power industry need a grow and we have a technology that is very suited to be able to utilize Turkey's low quality lignites and convert them to economical clean power. So we've gotten our first launch project in Turkey at the Soma Kolin Plant. It's a 510 megawatt powerplant which uses affordable Turkish lignite. We're very happy to be participating in the Turkish market and we'll be coming back again next year for more connection to he market.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Portugal first took my husband, Keith, and I into her warm embrace when we visited in 2002. On that vacation overseas, we fell in love with the country's people, architecture, topography, and relaxed way of life.\nYet when down the line we began to think about leaving the U.S., a different country was first on our list. Keith had been an International Living Magazine subscriber for years, and his curiosity was piqued by Ecuador. He spent about a week there, and while he found the country had much to offer, he also felt we were better suited for a European lifestyle.\nWe first considered France. We'd visited frequently over the years, I speak the language, and other attractions include the daily availability of ripe brie, warm baguettes, and fine bottles of Pinot Noir.\nWe also thought of sunny Spain, land of Antoni Gaud\u00ed, flamenco dancing, and paella. The climate is wonderful, and the cost of living quite reasonable.\nBut then we hit on Portugal, the country of my husband's heritage. The process of getting both a long-term visa and subsequent residence application was simple enough, and not expensive.\nWe remembered the folks we'd rented the place from back in 2002. Their quinta (country estate), situated in the north of the country close to Porto and Braga, offered a number of different houses at varying prices. We had been in touch with the family periodically over the years, and when we contacted them to say we were planning to relocate, everything began to fall into place.\nIL wisely recommends renting before buying in a foreign country. We took this advice. We rented close to the northern border with Spain. Then, moving toward the center, we tasted truly rural Portugal, surrounded by olive trees, vineyards, blackberry bushes, and flocks of sheep.\nNext, we made a transition to the Lisbon district, close to surfing meccas. It was at this point\u2014after about three years of living in Portugal\u2014that we arranged to have our personal belongings, which were still in storage in the States, shipped to us.\nThat dramatic step made us feel surer than ever that we were here to stay. A year later, we began to seriously search for property. Today we own our own home and a little land where we have fruit trees and a small vineyard, in the central region of the country, because we found our money stretched much further here than in trendy beach communities, the capital of Lisbon, or the \"second city\" of Porto.\nWe're now able to apply for citizenship. The good news? We found a great lawyer to help us. The not so good news? Now that so many others are also discovering the charms of Portugal, the expected wait time has gone from about six months to a year-and-a-half. But there's more good news: There's no better place to spend that time than Portugal.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This coming Sunday our wonderful Treasurer Sarah Leah will be running the 2018 Virgin London Marathon. This isn't a casual stroll around the capital, this is 26.2 grueling miles that starts in Blackheath and heads east through Woolwich for three miles before turning west and passing the Cutty Sark in Greenwich between miles six and seven. It crosses the river at Tower Bridge and then snakes around the Isle of Dogs, past Canary Wharf, before heading west again. The home stretch from mile 23 takes runners from London Bridge, along the Embankment to Parliament Square, Birdcage Walk and finishes on the Mall in front of Buckingham Palace. The finish area is all around St James' Park and Horse Guards Parade.\nSarah is taking on this challenge with her sister Emma to raise money for the amazing NSPCC.\n\u2190 Senior Training begins Tonight!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u0645\u0633\u0627\u0641\u0631\u0627\u0646 \u062f\u06cc\u062f\u06af\u0627\u0647\u200e\u0647\u0627\u06cc \u0645\u062a\u0641\u0627\u0648\u062a\u06cc \u0646\u0633\u0628\u062a \u0628\u0647 \u0627\u0647\u0645\u06cc\u062a \u0645\u0639\u06cc\u0627\u0631\u0647\u0627\u06cc \u06a9\u06cc\u0641\u06cc\u062a \u062e\u062f\u0645\u0627\u062a \u062f\u0627\u0631\u0646\u062f. \u0648\u0632\u0646 \u0645\u0639\u06cc\u0627\u0631\u0647\u0627\u06cc \u06a9\u06cc\u0641\u06cc\u062a \u062e\u062f\u0645\u0627\u062a \u0646\u0634\u0627\u0646\u200e\u062f\u0647\u0646\u062f\u06c0 \u0627\u0647\u0645\u06cc\u062a \u0622\u0646\u0647\u0627\u0633\u062a. \u062f\u0631 \u0627\u063a\u0644\u0628 \u067e\u0698\u0648\u0647\u0634\u200e\u0647\u0627 \u0645\u06cc\u0632\u0627\u0646 \u0627\u0647\u0645\u06cc\u062a \u0645\u0639\u06cc\u0627\u0631\u200e\u0647\u0627 \u0648\u0627\u0628\u0633\u062a\u0647 \u0628\u0647 \u0646\u0638\u0631 \u062e\u0628\u0631\u0647 \u0627\u0633\u062a \u0648 \u062f\u06cc\u062f\u06af\u0627\u0647\u200e \u0645\u0633\u0627\u0641\u0631\u0627\u0646 \u062f\u0631 \u0646\u0638\u0631 \u06af\u0631\u0641\u062a\u0647 \u0646\u0645\u06cc\u200e\u0634\u0648\u062f. \u0647\u062f\u0641 \u067e\u0698\u0648\u0647\u0634 \u062d\u0627\u0636\u0631\u060c \u0627\u0631\u0627\u0626\u06c0 \u0631\u0648\u0634 \u067e\u06cc\u0634\u0631\u0641\u062a\u0647\u200e\u0627\u06cc \u0628\u0631\u0627\u06cc \u0631\u062a\u0628\u0647\u200e\u0628\u0646\u062f\u06cc \u0634\u0631\u06a9\u062a\u200e\u0647\u0627\u06cc \u0635\u0646\u0639\u062a \u0647\u0648\u0627\u067e\u06cc\u0645\u0627\u06cc\u06cc \u0627\u06cc\u0631\u0627\u0646 \u0627\u0632 \u062f\u06cc\u062f\u06af\u0627\u0647 \u0645\u0633\u0627\u0641\u0631\u0627\u0646 \u0627\u0633\u062a. \u0628\u0631\u0627\u06cc \u062f\u0633\u062a\u06cc\u0627\u0628\u06cc \u0628\u0647 \u0627\u06cc\u0646 \u0645\u0646\u0638\u0648\u0631 \u062f\u0631 \u0645\u0631\u062d\u0644\u06c0 \u0627\u0648\u0644\u060c \u0641\u0627\u0635\u0644\u06c0 \u0627\u0642\u0644\u06cc\u062f\u0648\u0633\u06cc \u0628\u06cc\u0646 \u0631\u0636\u0627\u06cc\u062a \u0627\u0632 \u062a\u06a9\u200e\u062a\u06a9 \u0645\u0639\u06cc\u0627\u0631\u0647\u0627 \u0648 \u0631\u0636\u0627\u06cc\u062a \u06a9\u0644\u06cc \u0628\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 \u0627\u0631\u0632\u06cc\u0627\u0628\u06cc \u0641\u0627\u0632\u06cc \u062a\u0631\u06a9\u06cc\u0628\u06cc \u0648 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0698\u0646\u062a\u06cc\u06a9 \u0645\u062d\u0627\u0633\u0628\u0647 \u0634\u062f \u0648 \u0648\u0632\u0646 \u0645\u0639\u06cc\u0627\u0631\u0647\u0627 \u0628\u0647\u200e\u062f\u0633\u062a \u0622\u0645\u062f. \u062f\u0631 \u0645\u0631\u062d\u0644\u06c0 \u062f\u0648\u0645 \u0628\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 \u0627\u0631\u0632\u06cc\u0627\u0628\u06cc \u0641\u0627\u0632\u06cc \u062a\u0631\u06a9\u06cc\u0628\u06cc\u060c \u0634\u0631\u06a9\u062a\u200e\u0647\u0627\u06cc \u0635\u0646\u0639\u062a \u0647\u0648\u0627\u067e\u06cc\u0645\u0627\u06cc\u06cc \u0627\u06cc\u0631\u0627\u0646 \u0631\u062a\u0628\u0647\u200e\u0628\u0646\u062f\u06cc \u0634\u062f\u0646\u062f. \u0646\u0645\u0648\u0646\u06c0 \u0622\u0645\u0627\u0631\u06cc \u0627\u06cc\u0646 \u067e\u0698\u0648\u0647\u0634 \u0645\u062a\u0634\u06a9\u0644 \u0627\u0632 \u0645\u0633\u0627\u0641\u0631\u0627\u0646 \u0686\u0646\u062f \u0634\u0631\u06a9\u062a \u0647\u0648\u0627\u067e\u06cc\u0645\u0627\u06cc\u06cc \u0628\u0647\u200e \u0646\u0627\u0645\u200e\u0647\u0627\u06cc \u0622\u0633\u0645\u0627\u0646\u060c \u0645\u0627\u0647\u0627\u0646\u060c \u062a\u0627\u0628\u0627\u0646\u060c \u0632\u0627\u06af\u0631\u0633 \u0648 \u0647\u0645\u0627 \u0627\u0633\u062a. \u0646\u062a\u0627\u06cc\u062c \u0646\u0634\u0627\u0646 \u0645\u06cc\u200e\u062f\u0647\u062f \u0645\u06cc\u0632\u0627\u0646 \u0627\u0647\u0645\u06cc\u062a \u0645\u0639\u06cc\u0627\u0631\u0647\u0627 \u062f\u0631 \u06af\u0631\u0648\u0647\u200e\u0647\u0627\u06cc \u0645\u062e\u062a\u0644\u0641 \u0645\u0633\u0627\u0641\u0631\u0627\u0646 \u0628\u0627 \u0647\u0645 \u0645\u062a\u0641\u0627\u0648\u062a \u0627\u0633\u062a \u0648 \u062f\u0631 \u0646\u062a\u06cc\u062c\u0647 \u0631\u062a\u0628\u0647\u200e\u0628\u0646\u062f\u06cc \u0634\u0631\u06a9\u062a\u200e\u0647\u0627\u06cc \u0635\u0646\u0639\u062a \u0647\u0648\u0627\u067e\u06cc\u0645\u0627\u06cc\u06cc \u0627\u0632 \u062f\u06cc\u062f\u06af\u0627\u0647 \u0645\u0633\u0627\u0641\u0631\u0627\u0646 \u0645\u062a\u0641\u0627\u0648\u062a \u0627\u0633\u062a.\nDifferent segments of passengers have different opinions on the importance of criteria based on service quality. The weight values based on service quality express the relative importance of them. In most of researches, weight values depended heavily on expert knowledge and passenger's perspective haven't been considered. The goal of this research, presenting an advanced method for prioritizing Iranian airlines of passengers' opinions. First Euclidean distance between satisfaction from each criteria and overall satisfaction with fuzzy synthetic evaluation, genetic algorithm and, weight of criteria is determined. Second, with fuzzy synthetic evaluation, Iran airlines have prioritized. Statistical society in this study consists of Homa, Aseman, Taban, Mahan and Zagros passengers. The results show that different segments of passengers have different opinions on the importance of criteria, so prioritizing Iranian airlines have different in from od passengers' opinions.\n\u0622\u0630\u0631\u060c \u0639.\u061b \u0645\u0624\u0645\u0646\u06cc\u060c \u0645. (1395). \u0622\u0645\u0627\u0631 \u0648 \u06a9\u0627\u0631\u0628\u0631\u062f\u200e\u0647\u0627\u06cc \u0622\u0646 \u062f\u0631 \u0645\u062f\u06cc\u0631\u06cc\u062a. \u062a\u0647\u0631\u0627\u0646\u060c \u0627\u0646\u062a\u0634\u0627\u0631\u0627\u062a \u0633\u0645\u062a.\n\u0628\u0627\u0632\u0627\u06cc\u06cc\u060c \u0642.\u061b \u062c\u0639\u0641\u0631\u06cc\u060c \u0645. (1395). \u0627\u0631\u0632\u06cc\u0627\u0628\u06cc \u062a\u0623\u062b\u06cc\u0631 \u06a9\u06cc\u0641\u06cc\u062a \u062e\u062f\u0645\u0627\u062a \u0628\u0631 \u0631\u0636\u0627\u06cc\u062a \u0645\u0634\u062a\u0631\u06cc\u0627\u0646 \u062f\u0631 \u062e\u0637\u0648\u0637 \u0647\u0648\u0627\u06cc\u06cc \u0628\u06cc\u0646\u200e\u0627\u0644\u0645\u0644\u0644\u06cc (\u0645\u0648\u0631\u062f \u0645\u0637\u0627\u0644\u0639\u0647: \u0645\u0633\u0627\u0641\u0631\u0627\u0646 \u067e\u0631\u0648\u0627\u0632\u0647\u0627\u06cc \u0628\u06cc\u0646\u200e\u0627\u0644\u0645\u0644\u0644\u06cc \u0634\u0631\u06a9\u062a \u0647\u0648\u0627\u067e\u06cc\u0645\u0627\u06cc\u06cc \u0645\u0627\u0647\u0627\u0646). \u067e\u0627\u06cc\u0627\u0646\u200e\u0646\u0627\u0645\u06c0 \u06a9\u0627\u0631\u0634\u0646\u0627\u0633\u06cc \u0627\u0631\u0634\u062f\u060c \u062f\u0627\u0646\u0634\u06af\u0627\u0647 \u0622\u0632\u0627\u062f\u060c \u062f\u0627\u0646\u0634\u06a9\u062f\u06c0 \u0645\u062f\u06cc\u0631\u06cc\u062a \u0648 \u062d\u0633\u0627\u0628\u062f\u0627\u0631\u06cc\u060c \u062a\u0647\u0631\u0627\u0646.\n\u062f\u0644\u0627\u0648\u0631\u06cc\u060c \u0639. (1380). \u0645\u0628\u0627\u0646\u06cc \u0646\u0638\u0631\u06cc \u0648 \u0639\u0644\u0645\u06cc \u067e\u0698\u0648\u0647\u0634 \u062f\u0631 \u0639\u0644\u0648\u0645 \u0627\u0646\u0633\u0627\u0646\u06cc \u0648 \u0627\u062c\u062a\u0645\u0627\u0639\u06cc. \u062a\u0647\u0631\u0627\u0646\u060c \u0627\u0646\u062a\u0634\u0627\u0631\u0627\u062a \u0631\u0634\u062f.\n\u062f\u0648\u0633\u062a\u0627\u0631\u060c \u0645.\u061b \u0631\u0645\u0636\u0627\u0646\u200e\u067e\u0648\u0631\u060c \u0627. (1392). \u0627\u0631\u0632\u06cc\u0627\u0628\u06cc \u0631\u0641\u062a\u0627\u0631 \u0645\u0633\u0627\u0641\u0631\u0627\u0646 \u062f\u0631 \u0627\u0646\u062a\u062e\u0627\u0628 \u0634\u0631\u06a9\u062a\u200e\u0647\u0627\u06cc \u0647\u0648\u0627\u067e\u06cc\u0645\u0627\u06cc\u06cc \u062f\u0631 \u067e\u0631\u0648\u0627\u0632\u0647\u0627\u06cc \u062f\u0627\u062e\u0644\u06cc. \u067e\u0698\u0648\u0647\u0634\u0646\u0627\u0645\u06c0 \u062d\u0645\u0644 \u0648 \u0646\u0642\u0644\u060c 11 (1)\u060c 421-411.\n\u0633\u0631\u0648\u0633\u062a\u0627\u0646\u06cc\u060c \u0645. (1392). \u062c\u0627\u06cc\u06af\u0627\u0647\u200e\u06cc\u0627\u0628\u06cc \u062e\u0637\u0648\u0637 \u0647\u0648\u0627\u06cc\u06cc \u062f\u0627\u062e\u0644\u06cc \u0627\u06cc\u0631\u0627\u0646 \u0628\u0631 \u0627\u0633\u0627\u0633 \u0645\u0634\u062e\u0635\u0647\u200e\u0647\u0627\u06cc \u06a9\u06cc\u0641\u06cc\u062a \u062e\u062f\u0645\u0627\u062a \u0627\u0632 \u0646\u0642\u0637\u0647 \u0646\u0638\u0631 \u06af\u0631\u062f\u0634\u06af\u0631\u0627\u0646 \u062f\u0627\u062e\u0644\u06cc. \u067e\u0627\u06cc\u0627\u0646\u200e\u0646\u0627\u0645\u06c0 \u06a9\u0627\u0631\u0634\u0646\u0627\u0633\u06cc \u0627\u0631\u0634\u062f\u060c \u062a\u0647\u0631\u0627\u0646: \u062f\u0627\u0646\u0634\u06af\u0627\u0647 \u0639\u0644\u0627\u0645\u0647 \u0637\u0628\u0627\u0637\u0628\u0627\u0626\u06cc\u060c \u062f\u0627\u0646\u0634\u06a9\u062f\u06c0 \u0645\u062f\u06cc\u0631\u06cc\u062a \u0648 \u062d\u0633\u0627\u0628\u062f\u0627\u0631\u06cc.\n\u0639\u0628\u0627\u0633\u06cc\u060c \u0639.\u061b \u0645\u0646\u0635\u0648\u0631\u0622\u0628\u0627\u062f\u06cc\u060c \u0632.\u061b \u0645\u062d\u0645\u0648\u062f\u0634\u0627\u0647\u06cc\u060c \u0634. (1395). \u062a\u0623\u062b\u06cc\u0631 \u0645\u0633\u0626\u0648\u0644\u06cc\u062a \u0627\u062c\u062a\u0645\u0627\u0639\u06cc \u0634\u0631\u06a9\u062a \u0628\u0631 \u0627\u0639\u062a\u0645\u0627\u062f \u0645\u0634\u062a\u0631\u06cc\u0627\u0646 \u0648 \u0642\u0635\u062f \u062e\u0631\u06cc\u062f \u0628\u0627 \u062a\u0623\u06a9\u06cc\u062f \u0628\u0631 \u0646\u0642\u0634 \u0645\u06cc\u0627\u0646\u062c\u06cc \u0631\u0636\u0627\u06cc\u062a \u0645\u0634\u062a\u0631\u06cc\u0627\u0646. \u0641\u0635\u0644\u0646\u0627\u0645\u06c0 \u0639\u0644\u0645\u06cc \u0640 \u067e\u0698\u0648\u0647\u0634\u06cc \u0645\u062f\u06cc\u0631\u06cc\u062a \u0628\u0627\u0632\u0631\u06af\u0627\u0646\u06cc\u060c 8(4)\u060c 902- 855.\n\u063a\u0627\u0632\u0627\u0646\u06cc\u060c \u062f. (1390). \u0645\u0637\u0627\u0644\u0639\u06c0 \u062a\u0623\u062b\u06cc\u0631 \u06a9\u06cc\u0641\u06cc\u062a \u062e\u062f\u0645\u0627\u062a \u062f\u0631 \u0635\u0646\u0639\u062a \u062d\u0645\u0644 \u0648 \u0646\u0642\u0644 \u0647\u0648\u0627\u06cc\u06cc \u0628\u0631 \u0631\u0636\u0627\u06cc\u062a \u0648 \u0648\u0641\u0627\u062f\u0627\u0631\u06cc \u0645\u0633\u0627\u0641\u0631\u0627\u0646 (\u0645\u0637\u0627\u0644\u0639\u06c0 \u0645\u0648\u0631\u062f\u06cc: \u0634\u0631\u06a9\u062a \u0647\u0648\u0627\u067e\u06cc\u0645\u0627\u06cc\u06cc \u0622\u062a\u0627). \u067e\u0627\u06cc\u0627\u0646\u200e\u0646\u0627\u0645\u06c0 \u06a9\u0627\u0631\u0634\u0646\u0627\u0633\u06cc \u0627\u0631\u0634\u062f\u060c \u0633\u0645\u0646\u0627\u0646: \u062f\u0627\u0646\u0634\u06af\u0627\u0647 \u0633\u0645\u0646\u0627\u0646\u060c \u062f\u0627\u0646\u0634\u06a9\u062f\u06c0 \u0639\u0644\u0648\u0645 \u0627\u062c\u062a\u0645\u0627\u0639\u06cc.\nAli, F., Dey, B. L. & Filieri, R. (2015). An assessment of service quality and resulting customer satisfaction in Pakistan International Airlines: Findings from foreigners and overseas Pakistani customers. International Journal of Quality & Reliability Management, 32(5), 486-502.\nAnselmsson, J. (2006). Sources of customer satisfaction with shopping malls: A comparative study of different customer segments. The International Review of Retail, Distribution and Consumer Research, 16(1), 115-138.\nBuhalis, D. & Law, R. (2008). Progress in information technology and tourism management: 20 years on and 10 years after the Internet\u2014The state of eTourism research. Tourism Management, 29(4), 609-623.\nChang, H. H., Wang, Y. H. & Yang, W. Y. (2009). The impact of e-service quality, customer satisfaction and loyalty on e-marketing: Moderating effect of perceived value. Total Quality Management, 20(4), 423-443.\nFeng, S. & Xu, L. D. (1999). Decision support for fuzzy comprehensive evaluation of urban development. Fuzzy Sets and Systems, 105(1), 1-12.\nGounaris, S., Dimitriadis, S. & Stathakopoulos, V. (2010). An examination of the effects of service quality and satisfaction on customers' behavioral intentions in e\u2010shopping. Journal of Services Marketing, 24(2), 142-156.\nHao, J.-X., Yu, Y., Law, R. & Fong, D. K. C. (2015). A genetic algorithm-based learning approach to understand customer satisfaction with OTA websites. Tourism Management, 48, 231-241.\nHapsari, R. & Clemes, M. D. (2017). The impact of service quality, customer engagement and selected marketing constructs on airline passenger loyalty. International Journal of Quality and Service Sciences, 9(1), 21-40.\nHuang, Y. K. (2010). The Effect of Airline Service Quality on Passengers' Behavioural Intentions Using Servqual Scores:A Taiwan Case Study. Journal of the Eastern Asia Society for Transportation Studies, 8, 2330-2343.\nHussain, R. (2016). The mediating role of customer satisfaction: evidence from the airline industry. Asia Pacific Journal of Marketing and Logistics, 28(2), 234-255.\nIp, C., Law, R. & Lee, H.A. (2011). A review of website evaluation studies in the tourism and hospitality fields from 1996 to 2009. International Journal of Tourism Research, 13(3), 234-265.\nIp, C., Law, R. & Lee, H. A. (2012). The evaluation of hotel website functionality by fuzzy analytic hierarchy process. Journal of Travel & Tourism Marketing, 29(3), 263-278.\nJanahi, M. A. & Almubarak, M. (2017). The impact of customer service quality on customer satisfaction in Islamic banking. Journal of Islamic Marketing, 8(4), 595-604.\nJiang, H. & Zhang, Y. (2016). An investigation of service quality, customer satisfaction and loyalty in China's airline market. Journal of Air Transport Management, 57(Supplement C), 80-88.\nKoklic, M. K., Kukar-Kinney, M. & Vegelj, S. (2017). An investigation of customer satisfaction with low-cost and full-service airline companies. Journal of Business Research. Journal of Business Research, 80(Supplement C), 188-196.\nKuo, Y.-F. & Chen, P.-C. (2006). Selection of mobile value-added services for system operators using fuzzy synthetic evaluation. Expert Systems with Applications, 30(4), 612-620.\nLin, C. C., Chen, S. C. & Chu, Y. M. (2011). Automatic price negotiation on the web: An agent-based web application using fuzzy expert system. Expert Systems with Applications, 38(5), 5090-5100.\nNayebpur, H., & Nazem Bokaei, M. (2017). Portfolio selection with fuzzy synthetic evaluation and genetic algorithm. Engineering Computations, 34(7), 2422-2434.\nNumally, J. C. (1978). Psychometric theory (2th ed.). New York, NY.: McGraw Hill.\nOliver, R. L. (2010). Satisfaction: A behavioral perspective on the consumer. ME Sharpe. Inc., New York.\nPakdil, F. & Ayd\u0131n, O. Z. (2007). Expectations and perceptions in airline services: An analysis using weighted SERVQUAL scores. Journal of Air Transport Management, 13(4), 229\u2013237.\nProenca, T., Torres, A., Sampaio, A. S. (2017). Frontline employee empowerment and perceived customer satisfaction. Management Research: Journal of the Iberoamerican Academy of Management, 15(2), 187-206.\nSaha, G. C. & Theingi (2009). Service quality, satisfaction, and behavioural intentions: A study of low-cost airline carriers in Thailand. Managing Service Quality: An International Journal, 19(3), 350-372.\nSaleem, M. A., Zahra, S. & Yaseen, A. (2017). Impact of service quality and trust on repurchase intentions-the case of Pakistan airline industry. Asia Pacific Journal of Marketing and Logistics, 29(5), 1136-1159.\nSingh, A. K. (2016). Competitive service quality benchmarking in airline industry using AHP. Benchmarking: An International Journal, 23(4), 768-791.\nSuki, N. M. (2014). Passenger satisfaction with airline service quality in Malaysia: A structural equation modeling approach. Research in transportation business & management, 10, 26-32.\nTandon, U., Kiran, R., & Sah, A. (2017). Analyzing customer satisfaction: user's perspective towards online shopping. Nankai Business Review International, 8(3), 266-288.\nTing, P.-H., Wang, S.-T., Bau, D.-Y. & Chiang, M.-L. (2013). Website evaluation of the top 100 hotels using advanced content analysis and eMICA model. Cornell Hospitality Quarterly, 54(3), 284-293.\nYuksel, A. & Yuksel, F. (2003). Measurement of tourist satisfaction with restaurant services: A segment-based approach. Journal of vacation marketing, 9(1), 52-68.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Radiant your Skin Care with the [THE FACE SHOP] Mango Seed Silk Moisturizing Lotion 150ml! we aim to provide you best korean products to take good care of your skin, including Cosmetics, Skincare, Moisturizer. Hydra Moisturizer online.\nNon-drying hydration! Silk moisturizing lotion makes your skin shine.\nApply proper amount of the product on your skin.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You'll appreciate its safety and technology features! With just over 20,000 miles on the odometer, this compact vehicle proves that good things DO come in small packages! Top features include front fog lights, a tachometer, variably intermittent wipers, and much more. It features a standard transmission, 4-wheel drive, and a refined 6 cylinder engine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Max enjoyed another fabulous summer day. We started by sleeping in a little this morning. Once we got up we had breakfast and then headed out. We met a bunch of fellow Ivy Creekers and their kids at Sky Zone, an indoor trampoline facility. The kids had so much fun, and we all got to catch up. Max really enjoyed playing with Aiden. Aiden is going to middle school next school year and is quite a bit older than Max, so it was super-sweet for him to take up time with lil' fellow. Max tried the foam pit at Sky Zone for the first time today and loved it. I got a video and we enjoyed showing it to Daddy.\nEveryone was going to lunch, but Max and I weren't able to join them because we forgot Max's shoes. We stopped by our house and got Max's shoes and then met Mimi, Renae, and Renae's mother at The Olive Garden. Max was good for most of the time but got tired at the end. It was Max's naptime when we got home.\nDaddy worked on Max's fourth-year photobook for most of the day.\nMax enjoyed playing with trains when he got up. He built the coolest track in our living room. He spent a lot of time this evening playing with his wooden railroad on his newly built track.\nWe're all about to turn in for the evening. It's storming, so it's great weather for snuggling with lil' fellow.\nWe've enjoyed a pretty fun day together. We got up a little before 9:00 this morning. Once we got up Max played trains and we got ready to go to Sky Zone. We were a little smarter this time by reserving a jump time before leaving home. When we got to Sky Zone there were a lot of people, but the crowd thinned out after a little while. That's when we decided to purchase thirty more minutes of jump time. This was great! Max jumped constantly for one hour, so he was hungry when we left. And after that he was super-tired. He napped for four hours this afternoon.\nI attended a funeral this evening and when I returned home we three went out to eat. Mimi and Grandpa were also at the restaurant. Max went from table to table. He really thought this was neat. When we finished our dinner we went to visit Granny. She was glad to see us. We showed these pictures to Granny and had some ice cream treats before coming home to start our bedtime routine.\nWe told Max that tomorrow was a holiday, so that means we have to sleep in. Let's hope he remembers that in the morning.\nI had a hard time sleeping this morning. I got up around 5:00 am. I went out and did a few errands and then came home and went back to sleep with Max. We didn't get up until after 10:00. Max and I have our clocks set to summertime.\nWhen we got up we went downstairs and played with Max's trains. (I know, that's hard to believe.) Daddy worked on the images below. Once Daddy finished we got ready to head out for some fun. We had lunch and headed to Sky Zone in Suwanee. When we got to Sky Zone we found that they didn't have any available jump times until later, so we left and went to Catch Air. Max had a ball at there. We played for an hour-and-a-half before leaving.\nIt was way past naptime when we got back home, so we headed upstairs. Before Max lay down we watched the Berenstain Bears' Christmas Tree. Mimi even stopped by for a minute to see us and drop off a present for me for my half birthday.\nWe tried to get Max to stay with Mimi or Granny tonight since we have to get up early for my LASIK surgery in the morning, but no dice. He did stay with Granny while Jaddie and I went out to dinner.\nMax has been very hard to handle since we got home. He's served multiple timeouts and lost the media player for the night. We hope he's better tomorrow for Mimi.\nBaybay had another fun-filled day. We started our day around 8:00 am and watched Caillou until about 10:00 am. Then Baybay played with his trains while I got a few things done, and though I'd planned to carry him to Sky Zone, an indoor trampoline park, this morning, it didn't open until 3:00 pm. So we went to Granny's and waited for Sky Zone to open. Max loves Granny. She was waiting for us on her porch when we arrived, and Max said, \"I can't wait to see Granny,\" as I unbuckled him from his car seat.\nI ran an errand for Granny and was a little late in getting back, and by the time Max and I were ready to go, Mama was leaving school and opted to go with us. It was fun, but Max is about the shortest height (forty inches) they'll let on the regular trampolines. He had to learn not to jump in others' squares, and it took him most of his playtime to learn. We opted to let him jump for just a half-hour because it was already far past his naptime, and that was a good decision. He melted down as we left, but he recovered within about five minutes. He didn't want to leave but was deliriously tired.\nOn our way home we got dinner to go for Mama. Our little buddy said he was hungry Mama was about to have dinner, so she shared with him. Right after dinner they got their baths and took a nap. Mama thought she'd slip out of our bedroom without the little fellow in tow, but she was wrong. Max was on her heels down the steps and has been enjoying Cheez-Its, PediaSure, Thomas & Friends, and Caillou for the last couple of hours.\nNow we're about to lie down together and call it a day.\nI do whatever I want as long as it's okay with my wife.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Length 26 3\/8\" 27\" 27 5\/8\" 28 1\/4\" 28 7\/8\"\nWidth 16 1\/4\" 17 1\/4\" 18 1\/4\" 19 3\/4\" 21 1\/4\"\n\"True self is without form.\" A wandering omnic monk that helps those that cross his path, Zenyatta fights to protect the innocent, be they omnic or human. Pass into the iris with his official shirt!\nAvailable in women's sizes S - 2XL, made of 60% cotton and 40% polyester.\nOnly registered users who bought the product can write reviews. Please log in or register and buy the product to write a review!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jeffrey E. Thomas, M.D., F.A.A.N.S., F.A.C.S. is a Cerebrovascular and Neurointerventional Surgeon. Click here for more information about Dr. Thomas.\nJack Rose, M.D. is an expert in neurointensive care and cerebrovascular disease and Board Certified in Neurocritical Care. To learn more about Dr. Rose, click here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzayypf b/data_all_eng_slimpj/shuffled/split2/finalzzzayypf
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@@ -0,0 +1,5 @@
+{"text":"Jim, Cindy and Jerry, I'm so sorry to hear about your mom. She was such a great and fun lady. I always enjoyed running into her at random places. She always had a smile and her laugh was the best. I know you all will miss her but you have some wonderful memories to look back on. My thoughts and prayers are with you and your entire family.\nJohn and I send our deepest sympathy to you all. What a true lady who always had a smile. She certainly was a huge part of Clear Creek. May her new journey be in peace.\nBarb was such a vivacious lady. She always me people feel special. She & Bill had many fun times with my Mom & Dad....Venita & Jim Gallagher. May she rest in peace in the arms of the lord.\nBarbie was one special lady. Between her & Bill you couldn't ask for any better people or good friends. Their was never a dull moment when the 2 of them were together. Always had a great time with them on the golf course. A lot of great memories. Will miss you Barbie. You lived life to the fullest. Not many people would do the zip line at Idaho Springs. But you did, oxygen & all. Miss you & cherish our friendship forever.\nSorry to hear of Barbara's passing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the suburbs of Maryland, the firm of Hendley and Associates meanders its way through financial transactions\u2014but its true purpose is to identify and locate terrorist threats, and then deal with them in whatever manner necessary. \"The Campus\" is always on the lookout for new talent, and its eye happens to light on President John Patrick Ryan's son: Jack Ryan, Jr. While his father moved through the ranks of the CIA and into the Presidency, Jack Jr. received the benefit of years of life experience in special ops, intelligence analysis, and the way the world really works. Now, the Arab terrorist world and the Colombian drug cartels are about to unite forces, and Jack Jr. must put all he has been taught to good use\u2014or else.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Advantages of hammer crusher are as . The Handling Materials and Application Field of Hammer Mill: Handling Materials: pebble . Get Latest Price of Product .\nTechno wings hammer mill offers three series of hammer mills ideally suited for . economical price and day-in . Technical Specifications of Hammer Mill.\nHammer Mill Prices, Hammer Mill Prices Suppliers and .\nHammer Mill Prices, Wholesale Various High Quality Hammer Mill Prices Products from Global Hammer Mill Prices Suppliers and Hammer Mill Prices Factory,Importer,Exporter at Alibaba.\n13 Series Hammer Mill Technical Specifications. Model: Housing; Shaft Diameter Rotor Width: Rotor Diameter Bearings: Integral Fan System Screen Area: Shaft Speed Tip .\nHammermills,automatic hammer mills,hammer mill .\nHammer mills, automatic hammer mills, hammer mill grinder, corn hammermills,Spice and herbs grinding mills, spice grinding equipment, chilli Grinding Machine, hammer crusher, Auto feed Hammer Mills .\nhammer mill specification stone crusher machine. . They are high capacity hammer mills with . Get Price >> Hammer Crusher Operation And Maintenance .\nSpecification For Hammer Mill Crusher . Request a quotation Simply complete the form below click submit you will get the price list and a .\nHammer Clinker Crusher Spesification - . hammer clinker crusher spesification grinding mill hammer clinker crusher spesification [crusher and mill] price suppliers and hammer quality first cement clinker crusher hammer manufacturer model \/ specifications hammer clinker crusher spesification crusherasia electric power plant .\n. engineer, and manufacture hammermills for your exact specifications. . mix mill hammer mills, or grain hammer mills. .\nindustrial hammer mill specification; industrial hammer mill specification. . E-mail or call for specs, availability and price on all the products listed below. .\nHammer Mill With Accessories And Their Specification .\nHammer Mill With Accessories And Their Specification. . in India along with their contact details traders and exporters Browse hammer mill price, specification .\nPulverizer Hammer Mill Specification | Crusher Mills, .\nMeadows\u00ae Hammer Mill Specs & Pricing | Crusher and Mill . Wheat Hammer Mill-Wheat Hammer Mill specifications,price . New Meadows model 5 hammer mill only with materials-handling fan and extended shaft to accept drive.\nhammer mill specifications . Crusher Hammer Mill Specifications - fishfarmme. chicken feed crushers hammer mill price fish mini feed crusher machine,US , .\nCircular saw - Wikipedia. A circular saw is a power-saw using a toothed or abrasive disc or blade to cut different materials using a rotary motion spinning around an arbor.\nWheat Hammer Mill-Wheat Hammer Mill specifications,price . New Meadows model 5 hammer mill only with materials-handling fan and extended shaft to accept drive.\nMeadows\u00ae Hammer Mill Specs & Pricing | Crusher and Mill, Wheat Hammer Mill-Wheat Hammer Mill specifications,price, New Meadows model 5 hammer mill, .\ntypes and specifications of hammer mill . The best technical specification of hammer mill . Hammer mill for sale,suppliers,plant,price,exporter,types .\nStedman hammer mill machines are utilized by a variety of industries including aggregates and mining. Our hammer mills are available in various sizes with feed, .\nMotor Trend reviews the 2017 Kia Sorento where consumers can find detailed information on specs, . of power for the size of the mill. get price ; M36 Hammer mill .\nhammer mill grinder specification \u00ab gravel crusher sale. . 20 May 2014 Sizes And Capacity Of A Hammer Mill superior quality Low price Hammer Rule 2: .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Which Articles were deliberately read, and confirmed again by the subscription of the hands of the Archbishop and Bishops of the Upper-house, and by the subscription of the whole Clergy of the Nether-house in their Convocation, in the Year of our Lord 1571.\n[S]so that two whole and perfect natures, that is to say, the Godhead and manhood, were joined together in one person, never to be divided, whereof is one Christ, very God and very man, who truly suffered, was crucified, dead, and buried, to reconcile His Father to us, and to be a sacrifice, not only for original guilt, but also for all actual sins of men.\nTHE Old Testament is not contrary to the New; for both in the Old and New Testament everlasting life is offered to mankind by Christ, who is the only Mediator between God and man, being both God and man. Wherefore there are not to be heard which feign that the old fathers did look only for transitory promises.\nAlthough the law given from God by Moses, as touching ceremonies and rites, do not bind Christian men, nor the civil precepts thereof ought of necessity to be received in any commonwealth; yet, notwithstanding, no Christian man whatsoever is free from the obedience of the commandments which are called moral.\nand therefore in every person born into this world, it deserveth God's wrath and damnation. And this infection of nature doth remain, yea, in them that are regenerated, whereby the lust of the flesh, called in Greek phronema sarkos (which some do expound the wisdom, some sensuality, some the affection, some the desire of the flesh), is not subject to the law of God.\nAnd although there is no condemnation for them that believe and are baptized, yet the Apostle doth confess that concupiscence and lust hath itself the nature of sin.\nTHE condition of man after the fall of Adam is such, that he cannot turn and prepare himself, by his own natural strength and good works, to faith and calling upon God.\n[W]we have no power to do good works pleasant and acceptable to God, without the grace of God by Christ preventing ('Preventing' meant 'preceding' in Shakespeare's days) us that we may have a good will, and working with us when we have that good will.\nand not for our own works or deservings.\nWherefore that we are justified by faith only is a most wholesome doctrine, and very full of comfort; as more largely is expressed in the Homily of Justification.\nVOLUNTARY works besides, over and above, God's commandments which they call Works of Supererogation, cannot be taught without arrogancy and impiety.\nFor by them men do declare that they do not only render unto God as much as they are bound to do, but that they do more for His sake than of bounden duty is required: Whereas Christ saith plainly, When ye have done all that are commanded to do, say, We be unprofitable servants.\nCHRIST in the truth of our nature was made like unto us in all things, sin only except, from which He was clearly void, both in His flesh and in His spirit.\nand if we say we have no sin, we deceive ourselves, and the truth is not in us.\nNOT every deadly sin willingly committed after Baptism is sin against the Holy Ghost, and unpardonable. Wherefore the grant of repentance is not to be denied to such as fall into sin after Baptism.\nAfter we have received the Holy Ghost, we may depart from grace given and fall into sin, and by the grace of God we may arise again and amend our lives. And therefore they are to be condemned, which say they can no more sin as long as they live here, or deny the place of forgiveness to such as truly repent.\nThis is what Founding fathers called PROVIDENCE!\nand to bring them by Christ to everlasting salvation as vessels made to honour.\n[T]and at length by God's mercy they attain to everlasting felicity.\nno less perilous than desperation.\n[A]and in our doings that will of God is to be followed which we have expressly declared unto us in the word of God.\nTHEY also are to be had accursed that presume to say that every man shall be saved by the law or sect which he professeth, so that he be diligent to frame his life according to that law and the light of nature.\nFor Holy Scripture doth set out to us only the name of Jesus Christ, whereby men must be saved.\nTHE visible Church of Christ is a congregation of faithful men, in the which the pure word of God is preached and the sacraments be duly ministered according to Christ's ordinance in all those things that of necessity are requisite to the same.\n* neither may it so expound one place of Scripture, that it be repugnant to another.\n[Y]yet, as it ought not to decree anything against the same, so besides the same ought it not to enforce anything to be believed for necessity of salvation.\nGENERAL Councils may not be gathered together without the commandment and will of princes (Rulers: Kings and Presidents).\nAnd when they be gathered together, forasmuch as they be an assembly of men, whereof all be not governed with the Spirit and word of God, they may err and sometime have erred, even in things pertaining to God.\nWherefore things ordained by them as necessary to salvation have neither strength nor authority, unless it may be declared that they be taken out of Holy Scripture.\n[B]but rather repugnant to the word of God.\nIT is not lawful for any man to take upon him the office of public preaching or ministering the sacraments in the congregation, before he be lawfully called and sent to execute the same.\nAnd those we ought to judge lawfully called and sent, which be chosen and called to this work by men who have public authority given unto them in the congregation to call and send ministers into the Lord's vineyard.\n[B]but yet have not the like nature of Sacraments with Baptism and the Lord's Supper, for that they have not any visible sign or ceremony ordained of God.\nThe Sacraments were not ordained of Christ to be gazed upon or to be carried about, but that we should duly use them.\nAnd in such only as worthily receive the same, have they a wholesome effect or operation: but they that receive them unworthily, purchase to themselves damnation, as Saint Paul saith.\nyet forasmuch as they do not the same in their own name, but in Christ's, and do minister by His commission and authority, we may use their ministry both in hearing the word of God and in the receiving of the sacraments.\nNeither is the effect of Christ's ordinance taken away by their wickedness, nor the grace of God's gifts diminished from such as by faith and rightly do receive the sacraments ministered unto them, which be effectual because of Christ's institution and promise, although they be ministered by evil men.\nand finally, being found guilty by just judgement, be deposed.\nand grace increased by virtue of prayer unto God.\nThe baptism of young children is in any wise to be retained in the Church as most agreeable with the institution of Christ.\ninsomuch that to such as rightly, worthily, and with faith receive the same, the bread which we break is a partaking of the body of Christ, and likewise the cup of blessing is a partaking of the blood of Christ.\nTransubstantiation (or the change of the substance of bread and wine) in the Supper of the Lord, cannot be proved by Holy Writ, but is repugnant to the plain words of Scripture, over-throweth the nature of a Sacrament, and hath given occasion to many superstitions.\nfor both parts of the Lord's sacrament, by Christ's ordinance and commandment, ought to be ministered to all Christian men alike.\nTHE offering of Christ once made is the perfect redemption, propitiation, and satisfaction for all the sins of the whole world, both original and actual, and there is none other satisfaction for sin but that alone.\nWherefore the sacrifices of Masses, in the which it was commonly said that the priests did offer Christ for the quick and the dead to have remission of pain or guilt, were blasphemous fables and dangerous deceits.\nBISHOPS, Priests, and Deacons are not commanded by God's laws either to vow the estate of single life or to abstain from marriage.\nTherefore it is lawful also for them, as for all other Christian men, to marry at their own discretion, as they shall judge the same to serve better to godliness.\nuntil he be openly reconciled by penance and received into the Church by a judge that hath authority thereto.\nfor at all times they have been diverse, and may be changed according to the diversity of countries, times, and men's manners, so that nothing be ordained against God's word.\nand woundeth the conscience of the weak brethren.\nand therefore we judge them to be read in Churches by the ministers diligently and distinctly, that they may be understanded of the people.\nneither hath it anything that of itself is superstitious or ungodly.\nAnd therefore whosoever are consecrate or ordered according to the rites of that book, since the second year of King Edward unto this time, or hereafter shall be consecrated or ordered according to the same rites, we decree all such to be rightly, orderly, and lawfully consecrated or ordered.\nthat is, that they should rule all estates and degrees committed to their charge by God, whether they be ecclesiastical or temporal, and restrain with the civil sword the stubborn and evil-doers.\nnotwithstanding every man ought of such things as he possesseth liberally to give alms to the poor, according to his ability.\nso it be done according to the Prophet's teaching in justice, judgement, and truth.\nTHIS Book of Articles before rehearsed, is again approved, and allowed to be holden and executed within the Realm, by the assent and consent of our Sovereign Lady ELIZABETH, by the grace of God, of England, France, and Ireland, Queen, Defender of the Faith, &c.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u012e Northrop JB-1 \"Bat\" was a United States surface-to-surface cruise missile that was a prototype jet-powered flying wing. The United States Army Air Forces MX-543 program was initiated in September 1942 to use license-built versions of Frank Whittle's jet engine (General Electric J31). The Northrop Corporation was contracted in late 1943, and only 10 JB-1 airframes were built. A manned version was towed for the 1st flight on \"August 27, 1943\", from Rogers Dry Lake and a glider version was launched from a rocket-propelled sled and crashed in December 1944. An unmanned JB-1 powered by an improvised General Electric B-1 turbojet with a wing span of 28 feet 4 inches (8.64 m) made its 1st flight from Eglin Field's Santa Rosa Island, Florida, on December 7, 1944, and crashed 400 yards from the rail launcher.\nThe Northrop JB-1 \"Bat\" was a United States surface-to-surface cruise missile that was a prototype jet-powered flying wing skaityti daugiau \"\n\u012e \" Northrop F-5A\/B Laisv\u0117s Kovotojas ir F-5E\/F Tiger II dalis vir\u0161garsinio \u0161viesos kovotojas \u0161eimos skaityti daugiau \"\nThe Radioplane BTT, known as RP-71 by the company, as WS-426\/2 by the United States Navy, and as WS-462\/2 by the US Air skaityti daugiau \"\n\u012e \" yakovlev Yak-11 yra treneris, naudojam\u0173 orlaivi\u0173 Soviet\u0173 Oro Paj\u0117g\u0173 ir kit\u0173 Soviet\u0173 \u012ftakos oro paj\u0117g\u0173 nuo 1947 m. skaityti daugiau \"\n\"Lockheed F-94 Starfire buvo pirmos kartos reaktyvini\u0173 orlaivi\u0173 Jungtini\u0173 amerikos valstij\u0173 karini\u0173 Oro Paj\u0117g\u0173 skaityti daugiau \"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzazdda b/data_all_eng_slimpj/shuffled/split2/finalzzzazdda
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index 0000000000000000000000000000000000000000..686c5062a94aa481c73ad2b8b325e0c16767f4a1
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@@ -0,0 +1,5 @@
+{"text":"Aldi says it will become the first supermarket to sell a 3D printer when it launches a device that it claims is \u00a3600 cheaper than its nearest rival.\nAhead of Christmas, the German retailer is bidding to bring sophisticated gadgets to customers at a fraction of the cost of branded alternatives.\nIt promises its Balco 3D printer, available for \u00a3299, requires minimal assembly and setup.\nAldi says: \"This quality device will ensure that prints are clean and sharp. Plus, shoppers can even print directly from an SD card.\nAldi also points out that shoppers will save \"a whopping \u00a3600\" against an \"almost identical model from Curry's PC World\" \u2013 the UP! Plus 2 3D Printer, which retails for \u00a3899.\n3D printers create three dimensional objects using computer aided design (CAD), building up the object by adding successive layers of molten plastic.\nThey have not been without controversy. In October 2013, police in Manchester seized what they suspected to be a 3D plastic magazine and trigger, which they believed could have been fitted together to make a viable 3D gun.\nAldi's 3D printer will be available exclusively from Aldi online from November 23.\nThe supermarket will also be offering a 55 inch Ultra HD LED Smart TV for \u00a3429 and a Bluetooth soundbar for \u00a3129.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What are homogeneity and generalisability?\nFor the drug to have a general applicability for all those suffering the disease, the widest possible range of cases need to be included (the group needs to be general), but this increases variability in the responses and reduces the scientific value of the results.\nThere can be no such thing as a perfect trial, since trials inevitably involve a compromise between homogeneity and generalisability.\nThe individuality of response in patients means that drugs produce side effects, and these may not be identified during the trial.\nAs a result orthodox medicine recognises that controlled trial evidence alone cannot be relied on, but must be compared with evidence from clinical practice, a approach which forms the basis of evidence based medicine (EBM).\nEvidence for the unreliability of medical RCTs was published in 2007. The table below is based on data obtained from an analysis of 1,016 systematic reviews of RCTs of conventional medicine (and therefore of many more than that number of RCTs in total). For comparison, information about trials of homeopathic treatment is included and there is a striking similarity of results. This suggests that the pattern of results relates to the method rather than to the type of medical approach used.\nThe only real scientific solution to this problem is to test potential treatments on healthy people in order to identify all their actual effects on individuals (which is what homeopaths do in provings), and then have a method of systematically relating this information about effects to the individual needs of patients which is what homeopaths do when selecting a remedy).\n2. Harris L Coulter, Homoeopathic Science and Modern Medicine (Berkeley: North Atlantic Books, 1981), pp. 163-167.\n3. R.P. El Dib, A.N. Atallah, R.B. Andriolo, 'Mapping the Cochrane evidence for decision making in health care', Journal of Evaluation in Clinical Practice, 13 (2007), 689\u2013692.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Where To Buy CBD Oil In Tippecanoe, Marshall, Indiana?\nYes, you can still buy CBD Oil in Tippecanoe, Indiana today. This is comprised of low-THC hemp, high-CBD, whereas medical marijuana products are usually crafted from plants which have high psychoactive tetrahydrocannabinol (THC) concentrations. Since hemp only contains trace quantities of THC, these are non-psychoactive hemp oil products in Tippecanoe, Marshall, IN.\nCannabidiol (CBD) is just one of over 85 cannabinoids that have been found to become contained inside of the cannabis plant and it has the next highest quantity of cannabinoid in marijuana following THC. However, in hemp, THC is there only in trace amounts, with CBD dominating the makeup from the plant. CBD is non-psychotropic and doesn't cause a high. It interacts with all the body's naturally occurring systems. Which makes it a less controversial plus a safer option, while still providing significant benefits to folks In Tippecanoe, Marshall.\nWhy Should Cannabidiol CBD Oil Supplements Be Used In Tippecanoe Marshall, IN?\nLately CBD has emerged as a compound that his highly sought after. After receiving significant amounts of media exposure, such as the CNN special called \"Weed\" by Sanjay Gupta, it offers bought out the natural products industry. Since that time, CBD oil products have demostrated up in supermarkets in Tippecanoe, Indiana, medical marijuana dispensaries and doctor's offices. No medical card is necessary to buy it.\nThe way that CBD interacts with our body is by the endocannabinoid system or endogenous cannabinoid system (ECS). The endocannabinoid system was discovered through the late 1980's regulates the homeostasis in the body, or general balance state, which impacts functions such as immune response, pain, hormone regulation, appetite, sleep and mood. The same as an acrobat through to a high wire, our surrounding environment impacts our regular balance. This really is corrected from the endocannabinoid system by mediating the body's reaction in order to keep things level.\nThe endocannabinoid system can be found in all mammals and is made up of numerous various cannabinoid receptor sites that are primarily found throughout the defense mechanisms (the CB2 receptors), nervous system (the CB1 receptors) and brain that act throughout the neural communication.\nThe endocannabinoid system, however, doesn't rely only on phytocannabinoids or plant-based cannabinoids. The body can make its own cannabinoids. 2-AG and Anandamide, which function in the similar method to CBD, will be the endocannabinoids in the body, which are created in order to control communication in the middle cells, to mediate the functions of our body.\nThe endocannabinoid system from Tippecanoe, Indiana clearly is probably the human body's most significant regulatory systems. However, a majority of individual forget to do much to back up the healthy of your system due to cannabinoids not included in the average diet.\nThe non-psychoactive of cannabis and naturally low-THC hemp plant, grows in a number of countries worldwide then gets imported into the U.S. to use in foods, natural supplements, building materials and textiles.\nThere are many different unique hemp varieties in Tippecanoe, Indiana. The cultivars are used for CBD oil contain higher concentrations of CBD compared to others. By utilizing these unique and potent plants, this makes extracting cannabis oil likely that contains significant cannabidiol levels, as well as other non-psychoactive cannabinoids, flavonoids, terpenes, essential fatty acids, minerals and essential vitamins.\nWithin the last decade or more, there have been over 23,000 studies published by peer-reviewed medical journals that detail the impact of cannabinoids, cannabis oil and cannabis on the body.\nWith scientists, legislators, families are getting to be more aware about potential of hemp CBD oil and cannabidiol, delivers the supplement marketplace using a game-changing solution, which offers all the benefits that CBD has, minus the high that marijuana offers, that happen to be counted on by a large number of individuals each and every day.\nHowever, due to United states regulatory limitations that relate to processing and growing hemp, the method for delivering CBD hemp oil products into a customer's doorstep in Tippecanoe, Indiana is becoming more difficult.\nIn the Usa it is actually prohibited to grow hemp for commercial purposes, despite the 2014 Farm Bill was passed. Approved reasons to growing hemp has limitations to pilot programs just for research needs. Despite the fact that hemp corps have recently been planted in these states as Minnesota, West Virginia and Kentucky for the very first time within the last half century. So Usa hemp cultivation carries a extended way to go before it is able to meet current domestic demand.\nFor this reason those finished hemp goods the truth is available for sale in the Usa, from material materials, to clothing and food products, which can be belong to the imported hemp industry has surpassed $500 million per year. The actual size of the key industry for this particular is amongst the main catalysts to the legalization for hemp in america. Hemp can be a renewable source for a range of products, and supplies a thrilling new phase for American agriculture.\nAnd so the hemp which is used for producing the high CBD products that are available to residents in Tippecanoe, Indiana, especially in Indiana within the United states in cultivated in different countries worldwide. For this reason the hemp that is utilized in your CBD oil products comes from farms in Europe. And just any hemp crop is utilized \u2013 specific hemp plant cultivars from the hemp plant are essential for yielding another that is sufficient to use for CBD Oil.\nWhere To Buy CBD Oil In Florence, Switzerland, Indiana?\nWhere To Buy CBD Oil In Montezuma, Parke, Indiana?\nWhere To Buy CBD Oil In Colfax, Clinton, Indiana?\nWhere To Buy CBD Oil In Saint Bernice, Vermillion, Indiana?\nWhere To Buy CBD Oil In Taswell, Crawford, Indiana?\nWhere To Buy CBD Oil In Holland, Dubois, Indiana?\nWhere To Buy CBD Oil In Galveston, Cass, Indiana?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"3DK Nylon is a material with highest potential in many industries. It is widely used in automotive industry, machine and tool making. This filament is chemically resistant to alcohol, oils, acetone, resin and most akali and much more durable than ABS or PLA.\nThe advantages of 3DK Nylon filament are good layer bonding and printout smoothnes, but they require high extruder temperature (about 240-260 Celsius), low printing speeds and heated print surface.\nThis filament is suitable for printing fully functional prototypes and spare parts.\nThermal shrinkage may result in printout being constantly peeled from the printing surface. To eliminate this problem you may use special adhesive mat or apply PVP\/PVA glue on the surface.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please welcome Heather Blake to The Qwillery. The Goodbye Witch, the 4th Wishcraft Mystery, will be published tomorrow by NAL.\nHonestly, some of the time I have no idea where they come from. The ideas are just there. Most of the time, however, an idea comes from a situation I see or hear. It sets off a spark that I then flame into a full-blown plot by playing the \"what if\" game. What if that happened to one of my characters? Then what if my character did this, did that?\nFor The Goodbye Witch, it was a magazine article I read that created the spark. A human-interest piece about a young woman who'd been murdered. Nineteen years old. A high-achieving college student. The whole world ahead of her. There was a preponderance of evidence against the woman's boyfriend that he'd committed the crime. DNA. Witnesses. A confession (though later retracted). Yet\u2026his parents steadfastly refused to believe their son, their pride and joy who was also a college student on the fast-track to greatness, could have had anything to do with this heinous crime. No way. Not their boy. They proclaimed his innocence to any and all who would listen (and still do, despite his conviction).\nThe behavior, the blind faith, of those parents struck me hard. What if that was one of my sons? What would I do in that situation? How would I react? Better yet\u2026what if it happened to one of my characters?\nIn The Goodbye Witch, Wishcrafter Darcy Merriweather's best friend, Starla, is living her worst nightmare. Her fugitive ex-husband, Kyle, has returned to the Enchanted Village. Years ago, he escaped jail after being arrested for Starla's attempted murder. His family \u2013 his parents and two brothers \u2013 and most of the village took Kyle's side after the arrest. As the village's golden child, Kyle could do no wrong in their eyes.\nDarcy, of course, is livid at the absolute allegiance to Kyle. How could everyone simply ignore all the evidence against him?\nIt's not until Kyle's found dead, and Starla becomes the number one suspect in his death, that Darcy learns how easily someone can put blind faith behind a loved one accused of a horrible crime. She'll do anything and everything to clear her friend's name.\nAs Enchanted Village's resident Wishcrafter, Darcy Merriweather has the power to make other people's wishes come true, but what she really wishes is that she had the power to uncloak the invisible man who's stalking her best friend\u2026.\nDarcy's closest friend and fellow witch, Starla Sullivan, hoped she'd never see her ex-husband, Kyle, again. Two years ago he tried to kill her, and he has been a fugitive ever since. Now Starla claims to have seen him back in Enchanted Village, but it seems she's the only one who can see him. To everyone else, her ex is invisible.\nDarcy only wishes his motives were as transparent as the rest of him. Since the police can't arrest someone they can't see, it's up to Darcy to find the secret behind Kyle's latest disappearing act\u2014before he does something they can't see coming\u2026.\nWhen magical florist Harriette Harkette decides to throw a lavish eightieth birthday party for herself, she hires Darcy's Aunt Ve's personal concierge service, As You Wish, to plan the soiree. But turning eighty isn't all Harriette is celebrating\u2014the Floracrafter has recently created the midnight black Witching Hour rose, the first all-natural rose of that color.\nDarcy works hard on planning an extravagant celebration that will make Harriette feel like the belle of the ball. But when cake delivery boy Michael Healey\u2014a former employee at Harriette's greenhouse\u2014is found dead, the celebration takes a turn. Now Michael's ghost has imprinted on Darcy, meaning that they're bonded until she can untangle the thicket surrounding his murder\u2014and what exactly it has to do with the Witching Hour rose\u2026.\nHeather Blake grew up in a suburb of Boston, and enjoys going \"home\" in her novels. As a young mother, she tried her hand at writing novels and hasn't looked back. Currently, Heather lives in the Cincinnati area with her family, and is currently hard at work on her next novel.\nHeather also writes under the name Heather Webber. So far, she has 11 published novels, one ebook, and several short stories in print under her alter ego.\nWho and When: The contest is open to all humans on the planet earth with a US mailing address. Contest ends at 11:59PM US Eastern Time on May 12, 2014. Void where prohibited by law. No purchase necessary. You must be 18 years old or older to enter.\nAlways love to add a new book to my TBR pile!!\nI love the Wishcrafter series!!!\nSo excited about the new book.\nit does pose an internal debate. nothing as serious as murder but thievery has happened with someone i know and most accused her, just based on her past. turned out the missing money fell out by the car the other chick drove but the point was made.\nif it was my child i would have to say unless it felt he was lying then I would believe my baby. cause if your parents won't believe you then it doesn't matter b\/c your heart is broken anyway.\nThis book sounds really good. Congratulations on your new release!\nSounds like a great new series that I must read!!\nAbsolutely a cozy series to read!!\nI enjoy this series, but I believe it's the one where there's a surprise at the end of book 2 which wasn't addressed in book 3. If my memory is correct, I hope we get answers in the Goodbye Witch.\nEnchanted Village sounds like a place that I'd like to visit.\nThe cover really draw me. I hope to read these.\nwhat an interesting take on real life events! Congrats on the new release and thanks for sharing!\nI have read all the books in this series and can't wait till read the next!!\nI am so excited about this latest release!\nI've enjoyed the earlier Wishcraft installments and would love to add THE GOODBYE WITCH to my collection ... thanks for the chance!\nGreat series! Have a knick knack for witches.\nI have one of her Heather Webber books but I never got around to reading it. She's in Cincinnati and I'm also in Ohio -- in Toledo. I like the paranormal element.\nThis post loaded very slowly and on one of the other computers here at the library the giveaway part wouldn't load!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Mate (61,9%), green Pu Erh tea, pineapple, cola nuts, flavouring, Muira Puama, ginseng, cornflowers.\nHealth use: Mate - delivers energy and nutrition, high in antioxidants, vitamins, encourages digestion and metabolism, termogenic properties (increases fat burning). Green Pu-Erh - detoxifies the body, strengthens the immune system, protection against infectious disease. Pineapple - rich in vitamins, minerals, digestive aid, treats rheumatoid arthritis, strengthens bones, reduces blood clotting and helps remove plaque from arterial walls. Cola Nuts - stimulant effect on the central nervous system and heart, diuretic. Muira Puama - herbal treatment for sexual dysfunction, increases sexual desire, libido and performance. Ginseng - stimulant, enhances body system, boosts immunity, anti-aging, helps with circulatory and digestive system. Cornflowers - treats the inflammation of the eye, irritation around the eye area. Various problems in the mouth (gingivitis), rheumatism, gout and kidney problems as well as menstrual disorders.\nRecommended For: aphrodisiac and energy boost, good for digestion and weight loss.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Life Coaching: What do YOU want?\nLife coaching gives you the opportunity to have, as one of my clients put it, \"some quality time with yourself.\" During our time together you can examine your life as it is now and your life as you want it to be in the future, and work on a way to bridge the gap. I can help you stay http:\/\/www.honeytraveler.com\/pharmacy\/ focused, keep you motivated and cheer you on from the sidelines.\nTo read more about some of the things I can help with, click on the pictures below. Or, if you want to dive straight in, contact me now.\nSign up for news, blogs, offers and general loveliness. I promise not to inundate your inbox, nor to spam you or sell your address on to other people.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New releases, news, and anything else important.\nAnything not about Context Free but that might be of interest to Context Free users.\nIf you have a design you're proud of, share the cfdg file here. It's also a good place to ask for feedback and collaborate.\nHere you can discuss and share functionality improvements and helper programs to make Context Free better.\nLet the developers know what you think of the software and what can be done to either improve the CFDG language or the Context Free program.\nQuestions, Discussions, Aides, Conseils, pour les Francophones.\nFragen und Diskussionen auf Deutsch.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lee Smith Electrical carry out industrial installation and maintenance.\nWe follow strict guidelines and comply with the latest 17th Edition IEE Wiring Regulations (BS7671). All our work is tested and certified on completion.\nWe provide 24 hours emergency callout service by agreement.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"From unbelievable adventures that will have you wondering how he is still alive, to the untold stories of American History and Scripture. Tony's performances will take you on an imaginative journey that you will never want to end.\nOffering programs in literacy and creative writing to meet the needs of preschool thru college, in the classroom or assembly. Tony brings stories of history and culture to life in a way that will engage your students minds and hearts.\nHaving served as a pastor for nearly two decades, Tony's narrative communication of scripture will give you a deeper understanding of God, His word, and your place in His story. Tony is available for both story performances and preaching.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..f7724b57c85c00830d7ead8d8c97f4f3176a22c8
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+{"text":"You should be aware that it is illegal to reproduce or distribute copyrighted material without the permission of the copyright owner, Diarmuid MacShane.\nAccessing images or text provided on Web sites does not give you any rights to use them as you wish. Only the copyright owner, or the owner's legal agent, can give you permission to copy, distribute or publicly display protected material. The copyright owner in most cases is the creator. Images are generally owned by the designer or photographer; text is owned by the author.\nCopyright Ownership \u2013 The copyright of images and other material on this web site are owned by Diarmuid MacShane or the individual copyright owners who have authorised their use on the Value Ireland web site, or are specifically credited.\nUsing Copyrighted Items \u2013 You may download and reprint images and other information for non-commercial, private and personal purposes only. You may not manipulate or alter in any way images or information on this web site without specific permission from Diarmuid MacShane or the copyright owner.\nComputer Security and Safety \u2013 I cannot guarantee that files available for downloading, specifically or otherwise, will be free of viruses. You are responsible for implementing sufficient computer safety procedures on your own computer when downloading items.\nPermission to Use Copyrighted Items \u2013 Should you wish to use any of the articles etc on this web site, you should contact me stating your name, company name, address and contact details, the items you wish to use, and the manner in which you wish to use them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One North Shore, located in the vibrant North Shore Community of Chattanooga, is a mixed-use development with over 200 condominium units, and approximately 25,000 square feet of retail. TSW was retained to provide landscape architecture services for site and streetscape master plans, site construction documents, and streetscape construction documents. Site planning and design included existing conditions analysis, site demolition review, and hardscape and landscape for a pool (located on top of parking structure), fountain, and courtyard areas. Streetscape planning and design included preliminary grading and coordination of grading, drainage, and site utilities with civil engineer, coordination of built site elements with structural, mechanical, and electrical engineers, pavement materials selection and layout, street and site furnishings, site lighting, irrigation design coordination, cost analysis, specifications, and construction observation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Planning to spend 2 days in Portland and wondering what you should do, see, and eat? Keep reading to find out all the cool things to do in Portland, Oregon!\nAh, Portland...it is every bit as weird (their slogan is \"keep Portland weird, after all), wacky, and hipster as the show Portlandia portrays it to be, but it is also such a beautiful, vibrant city with a laid back PNW vibe. The city is known for coffee, its eccentric yet hip locals, beer, donuts, the greenery, and views of the Willamette River and surrounding mountains.\nBecause Portland is only a one hour flight away from San Francisco (aka home), it has become one of my favorite places for a weekend getaway. Basically, I decided to write this post because I am itching to get back ASAP (currently stalking some cheap flights as we speak).\nIt also has a strong cafe\/coffee culture, and is a foodie paradise - one trip to Portland, and you may just find yourself transformed into a food and coffee snob, going on about single origin beans and farm to table and all that good stuff. Anyway, food and coffee are two things I love, especially when I travel, so of course Portland has become one of my favorite places to go!\nPortland has your classic Pacific Northwest weather - lots of rain 9 months out of the year, with perfectly pleasant summers. The good news is that when it rains, it never rains very hard - most of the time it is more like a misty drizzle.\nJune-August are definitely the best times to go, with warm, sunny, and long days. Portland is a perfect city for being outdoors, and the summer is definitely the best time to take advantage of this. This is also when all the roses at the International Rose Test Garden are at their peak, and when the roses are in full bloom, it is a beautiful sight (scroll down to read a bit more about this). There are also many outdoor festivals, concerts, and events during the summer months.\nThe downside of being there in the summer is that it is Portland's high season, and airfare and hotel rates tend to be at their highest.\nGoing during shoulder season is also a great option - the city is beautiful when all the flowers are in full bloom in the springtime, and it becomes transformed with all the foliage in the autumn.\nIt may be gloomy and dreary if you go in the winter, but even then, temperatures are mild compared to many other cities in the winter - average highs are in the 40s-50s. It does occasionally snow in the winter, but significant snow is a rare event.\nPortland is served by Portland International Airport (PDX), which sits about 10 miles outside the city and has flights to many destinations all over the United States, along with several international cities. PDX was actually named the best domestic airport by Travel and Leisure magazine, praised for its easy navigation, efficient security lines, and notable dining options. Did you also know that in Oregon, it is apparently illegal to jack up prices just because you're at an airport, so there are no $4 bottles of water here? Yeah, I can definitely get behind that.\nIt is super easy to get into city center from PDX - just follow the signs out of the terminal to get on the MAX Red Line light rail, and you will be dropped off in downtown Portland in 40 minutes. A one-way fare is $2.50, which also makes it a cost-effective option for getting to city center.\nPortland is also served by Amtrak, which provides train service to Seattle and Vancouver to the north via its Cascades route, and as far as Los Angeles to the south via its Coast Starlight line. It is about 3 1\/2 hours to Seattle and 8 hours to Vancouver via train, and 31 hours to go all the way to LA. I hear the scenery along the train route is pretty spectacular, and it is something that is on my bucket list.\nPortland is also connected to Seattle via Bolt Bus - this takes about 4 hours, and can cost as little as $1 each way, if you book as soon as fares go on sale.\nYou can also plan a Portland road trip, either driving up from Los Angeles in the south, or from Seattle or Vancouver in the north!\nPortland is divided up into four quadrants, so it is super easy to navigate just by looking at the address. The city is split into north and south by Burnside Street, and east and west by the Willamette River.\nPortland's downtown is relatively small and easily walkable. The city is also full of green spaces, which makes walking around super pleasant and enjoyable. Strolling through some of the quirky and unique neighborhoods in the city is also fun (i.e. Alberta, Pearl District, SE Division St., SE Hawthorne, etc.), and there are many shops and cafes you can wander through.\nBiking is also a popular way to get around in Portland, and the city has over 350 miles of bike lanes and paths that are designated as \"bicycle boulevards.\" There is even a free bike tour of the city offered by Portland by Cycle. Many of the boutique hotels in the city offer complimentary bike rentals for their guests. There is also a citywide bike share program called Biketown, which offers over 1000 bicycles for rent at 20 kiosks throughout the city. Rates are about $2.50 for 30 minutes of use, and you can use a kiosk, the Biketown website, or smartphone app to reserve a bike.\nPortland has a relatively easy-to-use public transportation system, and the city highly encourages the use of it. The MAX light rail system is clean and efficient, and can connect you to many of the city's points of interest. The MAX runs every 15 minutes at most hours of the day. Fares are $2.50, and you must purchase a ticket before you get on board. The ticket is good for 2 hours, and are also valid for transfer onto the Portland Streetcar and buses. You can also purchase a day pass for $5 (also can be used on buses and streetcars).\nYou can also take the Portland Streetcar for locations in the Pearl District and near PSU. For other locations, there are numerous bus lines that connect you to many points in the city.\nPortland is also served by Uber and Lyft. Cars are readily available, and this is also an easy way to get around the city.\nIf you want to hit up locations outside of the city (Columbia Gorge, mountains, waterfalls, etc.), it is a good idea to rent a car.\nOregon has no state sales tax, so take advantage of it and shop away!\nAccommodation options in Portland include everything from chain hotels, funky boutique hotels, hostels, and Airbnbs. Portland is the home of the popular Ace Hotel, which is known for their signature hipster chic digs. I have yet to stay at this specific Ace location (but am a fan of the Seattle and Palm Springs properties), but I always pop into here on every trip to grab a coffee at Stumptown next door, check out the cool lobby and people watch, and hop into the photo booth (yes, it's one of the old school ones!), and would love to stay here on my next trip to Portland.\nA boutique hotel that I really enjoyed staying at is Hotel Rose, which is conveniently located downtown right next to the waterfront. The property was remodeled a few years ago, and has reinvented itself into a stylish, modern boutique hotel. Hotel Rose has bikes available for rent, which is a great option for getting around and seeing the city. They also have an afternoon reception, coffee, tea, and snacks, featuring their signature pineapple cupcakes - these are the best (and I definitely had multiples)!\nThe vast majority of the hotels in Portland are clustered in the downtown area, along the waterfront, and near Portland State University. Portland is known for their unique, vibrant neighborhoods, so to take advantage of this and to get a more local experience, I would recommend staying in an Airbnb, which are also super affordable. I found that many of the restaurants, cafes, and shops on my list are in these neighborhoods, so it was great to be able to immerse myself into them, and there is always easy access to downtown to check out the main attractions there.\nPortland is known as the \"City of Roses,\" and it is home to the International Rose Test Garden, which boasts over 7000 rose bushes. Located in Washington Park, is free, spans 4.5 acres, and is the oldest continually running rose garden in the United States (it opened in 1917). The roses are here are in bloom from April through October, and are at their best in the summer months. Wandering through the gardens and taking in that luscious smell of the roses is definitely a must-do in Portland if you are in the city during these months. It really puts other rose gardens to shame - I have never seen so many roses in my life!\nAfter visiting the International Rose Test Garden in Washington Park, head across the street to the Portland Japanese Garden. The 5.5 acre space is known as one of the most authentic Japanese gardens outside of Japan - having been to Japan many times over the years, I can say that wandering through the grounds here really reminded me of wandering through the gardens in Kyoto. You will definitely get a sense of tranquility and zen while wandering through the stone walkways here. The garden is covered in leafy trees and greenery, with a gorgeous waterfall, pond, bridge, and pavilion. The landscape here is meant to highlight the changing of the seasons. The Stone Garden is also perfect for a moment of meditation and self-reflection. There is also an authentic tea house, and on a clear day, you can see Mt. Hood in the distance. Definitely a great spot to stop by for some peace and quiet in between exploring the city.\nOkay, sooooo I actually haven't actually been inside the house, but Pittock Mansion is supposed to be an architectural wonder and tells the story of one of Portland's oldest and most influential families. What I came for are the views - the mansion sits over 1000 feet above the city, and there are spectacular views of downtown and beyond On a clear day, you can spot Mt. Hood - you can see it right under the cloud on the left in my photo above. Definitely worth stopping by to check it out, especially if you're a sucker for a good view, like I am.\nIf you're in Portland over a weekend, you should definitely stop by the Portland Saturday Market, which has been in operation since 1974 and is the largest open-air arts and crafts market in the country. The Market is located in Waterfront Park and features over 250 artisans and small businesses selling various handicrafts locally made in the Pacific Northwest region. It is a great spot for some souvenir shopping. The Saturday Market is open every Saturday AND Sunday from March through Christmas Eve.\nMy first time in Portland, I allotted myself 30 minutes to check out Powell's. Two hours later, I walked out after purchasing 5 books. Taking over an entire city block, Powell's is a must for any book lover. Even if you aren't much of a book person, it is worth checking out to see just how massive it is - so much so that they have store maps. Seriously, there's nothing like it. Plan to spend at least an hour here, but don't be surprised if you end up staying longer - I could seriously spend all day here!\nPortland is home to the most breweries than any other city in the world, with over 75. It is also credited as being a pioneer of the craft beer movement since the 1980s. So, drinking beer here is a must - you can learn more about Portland's most notable breweries here. If you want to take it a step further, and learn more about the craft beer movement and brewing process (and you definitely do), you should also take a brewery tour. I did the Widmer tour - the tour is free, and you get an insight into the brewing process, the ingredients they use (one of the most memorable parts of the tour was learning more about IPAs and seeing and smelling hops), the history of Widmer, and best of all - they give you free beer at the end. The brewery is currently under construction at the time of writing this post in March 2018, but is scheduled to reopen later this spring - check here for updates. Also check out this 5 hour microbrewery tour.\nThe Portland Aerial Tram connects the south waterfront to the Oregon Health Sciences University campus. It climbs 3300 linear feet at a speed of 22 miles per hour - the trip takes about 4 minutes. Over 20,000 people ride the tram every day to visit the OHSU campus. From the top, you can take in a spectacular view of downtown Portland, and the Willamette River (and all the bridges that cross it). On a clear day, you can also see Mt. Hood, Mt. Adams, and Mt. St. Helens. The tram runs Monday-Saturday (except holidays). A round-trip fare is $4.70. For more information about hours and fares, see here.\nYes, Pok Pok is hyped, and yes, there's a long wait - and yes, you should definitely go there. This is one of those places I have to go to every time I'm in Portland - those fish sauce wings are the best, perfectly crispy, with the optimal balance of sweet, savory, and with the slightest bit of sprice. Pok Pok's specialty is Thai street and drinking food, and the stuffed hen, papaya salad, and pork belly are also some of my favorites on the menu. Be sure to get an order of sticky rice on the side.\nThe cocktails here are also legit - the Mango Alexander is my favorite. It was described to me once as \"a vacation in a cocktail,\" and that's kinda the perfect way to describe it. Also, Pok Pok owns Whiskey Soda across the street - you can go here to kill time during the wait (they will call over once your table is ready), or if all you want are the famous fish sauce wings, you can grab them here instead.\nMy first time in Portland, I squeezed in Pine State Biscuits on the way to the airport after eating 2 other breakfasts because I head it was THAT good and I really wanted to try it - and it really was THAT good (I am shocked that I didn't explore afterwards, though). The biscuit sandwiches here are on point - try the Reggie, which has fried chicken, bacon, cheese and gravy. And yes, you should definitely get ready for the food coma afterwards, but trust me, it's worth it!\nMy old boss, who was a pizza snob, once declared this the best \"pie\" (he was also a native New Yorker) in America. Yes, they definitely have one of the best pizzas I've ever had in my life. Apizza Scholls serves up Neopolitan style pizzas, but what really makes them is the crust. They make the dough in limited quantities, and once they're out they're out, so be sure to come early. There is also usually a wait here - if you need a place to kill time while you wait for a table, check out the Sapphire Hotel down the street for a cocktail.\nHaving lived in California all my life, I am basically a Mexican food snob, so when someone recommended Por Que No to me, I immediately scoffed. I finally gave it a chance though, and I was pleasantly surprised - the tacos here are actually pretty good! (although not as good as the tacos in California, obvi. Because I'm a snob). There is a good variety of tacos here, and they all use fresh, sustainable, and locally sourced ingredients. Their sangrias and margaritas are tasty as well, and this is a perfect spot for lunch in the summer. The food is reasonably priced, and the vibe here is laid-back and fun, and I love the eclectic decor as well.\nStumptown may have expanded all over the country, but it all started in Portland - and it still makes a damn good latte. Stumptown's focus is on flavor profiles and direct trade. The baristas here also aren't pretentious assholes like they sometimes are at hipster coffee shops - one of them even gave me recommendations for other great coffee spots in the city. My first stop in Portland usually ends up being here, to fuel me for a full day of exploring the city. There are locations all over the city, but the one inside the Ace Hotel is my favorite - I like to take my latte into the lobby to sit and people watch.\nThis is one of Portland's most popular brunch spots. Tasty n Sons is known for its inventive \"new American diner\" cuisine, and puts a Pacific Northwest spin on classic breakfast fare. Be prepared to wait - but it's worth it (and compared to the 2 hours brunch waits in San Francisco, the wait here is nothing). They have a mix of small and large plates, but it is tapas inspired (and you will want to eat ALL THE THINGS on the menu), so sharing is more fun! You can also check out their sister restaurant, Tasty n Alder, which has a similar menu - however, Tasty n Alder is more known for their dinner menu.\nThis was one of the places suggested to me by the Stumptown barista mentioned above, and it was a good one. Heart Coffee's focus is on light roast beans from Africa and Central America. Light roast doesn't mean boring, though - the lattes are still full of flavor and perfectly roasty. If you're into AeroPress coffee, apparently the owner of Heart won the international AeroPress competition in 2013 so you're probably in for some good stuff. There are 3 locations in the city, but the one on Burnside has a cool, sleek, minimalist vibe, which I dig.\nDid you know that there are over 600 food carts in Portland? Most of them are grouped into pods all over the city - the one on 9th & Alder is one of the largest (and conveniently located in downtown and near Powell's). This pod has carts that serve almost anything and everything that you can think of, but my absolute favorite and the reason I keep going to this pod is Nong's Khao Man Gai, which serves up a classic Thai chicken and rice dish that has the most amazingly addictive soy and ginger sauce. They've opened up a bunch of khao man gai places at home in San Francisco to try to replicate this, but I still haven't had one that's as good as Nong's. (PS: if you want to sit down and eat your khao man gai, they've also opened an actual storefront on Burnside). Other popular pods in the city include Cartopia (this one is open late so it's great for a late-night snack - I've stopped by here after a show), PSU pod, Cartlandia, and pods on Third Avenue and Fifth Avenue. Check out this list for more. There is also a street food tour available.\nAs soon as I saw photos of Cheese and Crack's super cute cheese platters all over the interwebs, I knew I would have to make a trip there. Oh my goodness y'all, aren't these platters just the cutest? Like, it's begging to be Instagrammed. Luckily, the items on them are as good as they look. You choose which two cheeses you want on your plate - I highly recommend the cheddar and brie brulee (with the perfectly caramelized outside - yummm) - and each platter comes with crackers, bread, a cookie, cornichons, olives, dijon mustard, honey, and chocolate ganache. You can add more cheeses, as well as meat and\/or pate, to your plate if you wish. Cheese and crack also serves beer and wine, so it's a perfect spot to hang out for a pre-dinner snack and beverage. Yes, yes, and more yes!\nBeast has had a reputation for being one of the best restaurants in Portland since its opening in 2007, and James Beard award winning chef Naomi Pomeroy is a local legend. Beast has an intimate dining room and only seats 24, with 2 dinner seatings Wednesday-Saturday at 6pm and 8:45pm, along with 2 brunch seatings every Sunday at 10am and 12pm. Seating is communal here, so get ready to get to know your neighbors (it's a bit awkward if you go solo, as I did). Beast has a menu that changes weekly based on what Pomeroy finds in the local markets, and has a meat focus (hence, the name \"Beast\") - if you are vegetarian or have any dietary restrictions, you must let them know ahead of time. Dinner is expensive, but brunch is a manageable $40 (including gratuity). Each of the 3 courses was absolutely delightful, and it was such a uniquely Portland dining experience. One of the best meals in Portland!\nOkay, so Voodoo may be the most famous of Portland's donut spots, but it is touristy AF and well, they just aren't very good. Blue Star, on the other hand, is VERY good. The donuts here come in some really fun and unique flavors, such at PB&J, blueberry bourbon basil, passion fruit cocoa nibs, along with other seasonal flavors. These are made using local ingredients, and are always super fresh. Sooooo good. Seriously. Have one, or maybe like 3 during your 2 days in Portland. There are several locations throughout the city, including one at PDX in case you wanted to grab one on the way out.\nSalt & Straw may have expanded outside of Portland these days (hell, we have two of them here in San Francisco now), but it is a Portland favorite, and the ice cream is soooo good. Salt & Straw's flavor list boats some delicious creations such as pear & blue cheese, Arbequina olive oil, and cinnamon snickerdoodle, along with some creative seasonal selections - at the time of writing, they currently listed Pots of Gold & Rainbows, chocolate caramel potato chip cupcake, and sweet corn & waffle. The sea salt with caramel ribbons is always a classic. I highly recommend ordering a flight, even if you think it's waaaay too much ice cream, because I guarantee that you won't be able to narrow it down to 1 or 2 flavors. So much yum here.\nLocated inside the Ace Hotel, Clyde Common describes itself as serving \"domestic and foreign\" cooking, and at its core, is a gastropub serving local fare. Clyde Common is cocktail-focused with a hipster vibe, and offers a quintessential Portland dining experience. The dining room is communal style, like at many restaurants in the city. The food is tasty, and so are the cocktails - the negroni is their specialty. If you don't come for dinner, it's worth making a stop for happy hour.\nLe Pigeon is a bit spendy, but it is worth EVERY. SINGLE. PENNY. You should definitely consider getting the tasting menu here - every course is perfection, and it is sooooo worth it. A five-course dinner is $85 here, and considering that the entrees all average about $30, it is definitely a good deal, especially when you compare it to a tasting menu of a similar caliber in another major city (i.e. San Francisco or NYC). Keep in mind that Le Pigeon is one of the restaurants joining in on the trend of bundling gratuity into the menu price, AND there's no sales tax in Oregon, so what you see is what you pay. Le Pigeon puts a modern, seasonally-focused, and creative PNW spin on classic French cuisine, and it is upscale without being pretentious. Chef Gaberiel Rucker is a two-time James Beard award winner, and one taste of the food here and it's easy to see why. Trust me, you should definitely treat yo' self here. You'll thank me later.\nPortland is known for its live music scene, and the Doug Fir is one of my favorite venues, maybe ever. It's got a cool futuristic log cabin vibe, and plays host to a lot of emerging and indie musical acts. Check the calendar to see if there are any shows while you are in town - and if there are, you should definitely go (even if you've never heard of the band - I've done this)! It's a fun spot to hang out after dinner, grab some drinks, and take in some live music.\nYou are definitely coming to Cacao for their rich, decadent drinking chocolate, and yes, you should get a flight. Cacao also sells a selection of gourmet artisanal chocolates, and there is a great selection of single-origin bars, and truffles from a selection of chocolatiers from the Pacific Northwest region. Definitely stop by for a treat!\nDisclosure: This post also contains affiliate links. This means that if you make a purchase or booking, I may receive a small commission at no additional cost to you.\nHave you ever been to Portland? What are some of your favorite things to do in Portland?\nYou make me wanan visit so badly now! Love all the food pictures.\nThe Japanese Garden is one of my faves!!! They have the best view of Mt. Hood and I remember my friends and I were joking about how it low key looks like Mt. Fuji hahahaa. I'm so sad I missed out on Powell's Books and the sandwiches at Pine State Biscuits. Salt and Lick is amazingggg too, great recommendation! Did you tried the pear and blue cheese? I think that one was my favorite.\nSo much to do in Portland! And I love the look of all the sumptuous food & drinnks. Definitely worh a visit!\nThat food looks amaaazing, as does the Japanese gardens.\nI didn't make it to the States for yeeears, but we now live in Vancouver so it doesn't sound too hard to make it down to Portland from here! I'll try in the summer.\nI still can't believe I've never been to Portland officially. I think I may have breezed through as a kid...but I clearly need to go back and tackle all these amazing recommendations! Incredible post!\nWhat a fun post!! I feel a special connection eith Portland f because it is weird and I'm weird too! And it looks super fun! I would love to check out that bookstore or the mansion or the roes garden. Okay so basically I wanna see all of Oregon like you said, in June. And thank God it rains but not super hard!!\nI had so many things going through my head while reading your post...wow, yum, cool, YUM, beautiful! The Japanese Garden is gorgeous and Portland definitely seems like a good foodie city!\nI have literally spent about an hour in Portland when the Coast starlight stopped there on our way to SF! I so want to see more! Will pin so when I do go I have a great guide!\nOhh I need to visit Portland! Great post and amazing photos - as always! Love, love, LOVE the way you edit them!\nWe want to live here someday :) Excited about the coffee scene there!\nWow, such a detailed and informative post - thank you! I want to go to Portland again just to eat! All these places you described and all the food photos - omg, yum!\nAhh, I'm drooling over your restaurant recommendations! Unfortunately, I've not made it to Portland but hope to.\nPortland seems like such an amazing city! With all those food recommendations I don't think I just want to visit...I want to live there! Thanks for this comprehensive guide.\nPortland sounds like my kind of place and I love the cafes and boutique hotels you've included. Will def bookmark this for future trips.\nThis is such an awesome guide to Portland! We will be visiting for a couple of days next year and will definitely be taking lots of your recommendations - especially when it comes to breweries!\nWhat a great, detailed itinerary! Saving this for my next trip to the US :) Thanks for sharing!\nMan, everything I see about Portland makes me want to visit! I'm hoping to visit there this year as I'm planning to spend a few months in Canada and hop over to the States from Vancouver. :) This is a great guide, thank you!\nHeading out that way this year so this post will come in handy. I've heard about the amazing food but one meal at Le Pigeon followed by the flight at Cacao and I'd call any trip a success :-) They look and sound unreal!\nWhat a great post, all of those amazing food recommendations have my mouth watering and definitely dreaming of a Portland trip!\nPortland looks like such a fun city to visit. Any city known for its coffee and donuts is a winner in my book! :) Excellent guide - will definitely be saving for our own future visit!\nIve never been to Portland (or the pac NW) but i want to go for the food alone!\nGreat post! I have a friend who lives in Portland! Now I know exactly what to do when I visit!!\nYou have the best guides to cities! I've been dying to visit Portland. I feel like a crazy person who's still never been! This just made me want to go even more!\nDid you say food tour? I'm in! And brewery visits? I need to go! I only visited California when I visited the US. PS. Love the vibe of your photo editing!\nPortland sounds like an amazing city to visit! I would definitely prefer to visit during the summer even though it's high season. It's hard to enjoy a place if it's raining the whole time. The food alone looks like it's worth a visit!!! Por Que No!!!! YUM!\nActually, it's on top of my list for my next trip to the States, so these tips are really very well welcomed. Thank you!\nOh I love Portland! Haven't been in years though. I've never been in the summer, which I'm sure would make me love it even more. Thanks for this great guide!\n50 Licks ice cream was awesome, and so was Coava roasters!\nOH WOW. This is a great list and you hit all my hometown favorites! LOVE Pok Pok and Por Que No! I would only swap out Barista in the Pearl for Stumptown - I just like it better! You have seriously made me homesick - and hungry! Cheers from this Oregon Girl Around the World in Copenhagen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the HOUSE! Oh yeah!\nWhen I was a little girl growing up in NYC, we didn't have AC. I shared a room with my cousin and we had an oscillating fan which offered cool air to me, then the room, then my cousin, then the room and back to me, droning all through the night as we thrashed in our beds. Screens in the windows did little to keep out the gargantuan mosquitoes living in the armpit that is NYC in the summer and the humidity? Nothing could really be done about it. It was a fun, hot, itchy time. Here in California we get a dry heat the grapes and produce don't seem to mind but NYC harbors a humidity begetting thunderstorms and sticky, dirty skin at the end of the day.\nThe world has hundreds of climates and humans inhabit most of them. And some of these places get very hot. Heat, while being crucial to our survival can be too much. At the very least it can leave you feeling drained and tired, leaving you to your desires for ice cream and a pool. At the worst it can cause grave injury, raising your body temperature, causing dehydration, sunstroke and even death. Heat and humidity can cause diseases to spread more readily. I am not going to die today in my house because of the heat but in some parts of the world, this is an issue people must take precaution against. If you're not from the country? You should probably bring a fan.\nPeople have been trying to cool down for a long time. Fans, specialized plumbing and even the utilization of ice have all been used to create cooling systems. The more money or time you have at your disposal, the more effective your cooling system can be. The Ancient Egyptians had slaves fan jars of water, causing the water to evaporate more quickly and carry the heat away. People would hang wet sheets on their porches or in their houses and kings and emperors supposedly had snow and ice brought from mountains to make things more bearable in the summer months. A fan in front of a bucket of ice might be the only thing available to a hospital with sick patients while a government official has a refrigeration unit in their home. Hell, some people might decide to skip the heat altogether and relocate to a cooler climate while those unfortunate enough to have to stay put have to literally sweat it out.\nHot weather can put a strain on any adventuring party. It can disrupt much needed sleep at night and drain the party members during the day. Heat will affect not only the adventurers but their supplies. Drastic changes in the temperatures between the day and evening should be accounted for, with many creatures, human and otherwise, preferring to stir in the cooler part of the evening. Tempers can rise with the temperature. Hopefully everyone will be too beat to take that first swing.\nHow hot does it get where the PCs are from? Where they are travelling? Are the temperatures the same for the entire region or are there differences, with cooler locales existing, hotter ones, etc.? Is the year especially hot\/cold for these various areas?\nWhat kind of heat is it? Is it a dry heat? Is it humid?\nHow is temperature measured? What tools do people use to measure how hot it is?\nIs there a huge discrepancy between morning and evening temperatures? How does the local culture incorporate this (such as a long break in the afternoons\/hottest part of the day? Stores keeping later hours)?\nHow do common people beat the heat? Do they just deal with it? How do they dress? How are their houses built?\nAre there any foods or drinks which are supposed to cool people off? Very spicy\/hot foods? Special chilled foods?\nHow do the various social classes cool off? Are there public pools and fountains? Beaches? Rivers? Aqueducts that provide plumbing only the rich can afford? People hired to fan?\nAre there popular 'vacation' spots people frequent in the hot months to get away from the heat? What are those places like? Who lives there year round? Who shows up in the warm months?\nAre there any diseases\/plagues\/swarms more likely to pick up in the warm months? What factors contribute to their spread\/destruction?\nThe local government official has recently thrown a large party where snow from a nearby sacred mountain was collected and consumed by the guests. Several days after the party those who attended are stricken by a strange illness. The PCs must venture back to where the ice was retrieved and try to sort out what has happened. Is the illness from the ice itself? What are the symptoms? What is the cure?\nThe recent heatwave in the city has inspired an important university to sponsor a contest. Whoever can create the object that best cools down a large number of people will win scholarships to the school. The PCs are students at the school and hope to win the contest in order to secure their education. What is the exact criteria for the contest? Do the objects have price caps? How is their efficiency measured? How long do the PCs have? What other teams are they up against?\nWhen an sickness overcomes their village, the PCs are asked to bring back enough ice and snow to cool down the sick rooms. Where is the closest source of ice? How do they expect to get to the ice and then bring it back? How much ice will they need to bring back to make the venture successful?\nThe last two summers have been hotter than usual and appear to have brought strange, terrible creatures upon the land. The beasts seem to show up when the daytime temperature hits its peak and ravage the countryside at night. This summer looks to be warmer than the last and the local towns wants the PCs to go try and find the source of these strange beasts before they can rak their havoc upon the land. Where are the beasts coming from? Why do they only show up in the hottest parts of the summer? Is this the first time such a thing has happened here? Happened anywhere?\nThe heat drives the officials and politicians to cooler climates, leaving everyone else to stew in their own grumpiness. Dissenters take the opportunity to stir up those left behind and riots are starting to break out. The PCs notice with each riot, the temperature seems to rise a little more. Is it a coincidence? Who is stirring up trouble? Which came first, the hot tempers or the heat? Is anyone behind the correlation or is it bigger than one person?\nWhat is a comfortable temperature for you?\nDo you have any personal tricks to stay cool?\nHow does the climate of where you are now compare to where you were raised?\nDo you dress differently in the summer than the winter? How do your wardrobes compare?\nWhat say you? How do you beat the heat? Also, check out this sweet slideshow from the History Channel about how people stayed cool, well\u2026throughout history.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Discussion in 'Electric Vehicles' started by KingsDude, Oct 1, 2012.\nTesla has no equal. Maybe in a decade or so other makers will catch up. The only thing that will be better than a Tesla next year will be next year's Tesla.\nDisagree. I think there are some better options.\nFor example: WHy would you NOT want an electric car with a gas backup? THis way, you can go as far as you want and not have to worry.\nWell... added cost, weight and complexity of the ICE if you don't need it. If your goal is to drive all electric and you accomplish that goal 95% of the time with a limited 40 mile range, wouldn't a 250 mile range take care of 99+%? No one is saying EVs are for everyone. The Volt is a fine car and especially useful if it is your only car. For families with 2 cars and one person with a defined commute, one can easily be an EV.\nDozens of reasons covered here (a forum dedicated to a fully electric car company).\nFor me it's not wanting have all the disadvantages of a gasoline engine being dragged around till I need it. A larger range battery is a more simple solution.\nAfter 50,000 miles with a 240 mile range battery I have \"needed\" to go further exactly four times. And those were in the days before there was any real J1772 options.\nWHy would you NOT want an electric car with a gas backup? THis way, you can go as far as you want and not have to worry.\nGood point. However, I'd rather give up the ICE (and its maintenance), and gain cargo space. I see little point in carrying around an ICE everywhere you go, draining battery, adding weight, spoiling the handling? We can both be right. Depends on our driving needs. For me, the extra ICE is of little use.\nBecause you're hauling around two powertrains, doubling the complexity and ways that things can go wrong.\nBecause neither the BMW nor the Cadillac are actually in production, so it's difficult to know what will actually reach the market, when.\nWhich isn't to say there aren't reasons on the other side: the Volt is an interesting balance, allowing nearly all driving for most people to be pure electric. But like any EREV, it adds range by adding substantially to the drivetrain complexity and greatly reducing internal storage space and passenger cabin space.\nUnless you're an extreme road trip warrior, the question to me is why on earth would anyone want a gas backup?\nThe reason I'd want a PHEV over an EV is just to have the safety of knowing that I will NEVER be stranded in ANY scenario.\nI love the handling of the tesla and i love everything about it, but having the anxiety of the battery may ruin it for me.\nWhen the 40 kw battery can only go 100 miles and I drive 50 miles to school, it's goin gto be TOUGH.\nFurthermore, what happens in 5 years when these batteries are MUCH MORE EFFICIENT - you're then stuck with it.\nI would probably like to lease the car to avoid this issue, but so far, that hasn't come out yet.\nPeople getting stranded in gas cars happens often enough. As gas prices go up AAA reports getting more calls from people stuck on the side of the road due to running out of gas.\nThen there's also this: Bad Karma - it left me stranded!\nSo I'd be careful when stating absolutes, but at any rate PHEVs are a good transition technology.\nSpeaking as someone who has been stranded in a gas car, a PHEV isn't going to solve the \"ANY\" scenario requirement.\nIn that case you need to either (a) find a way to charge at the other end, even if only 110V, or (b) get the next pack size up.\nTesla is already talking about releasing an upgraded pack for the Roadster. Most likely the same thing will happen with the Model S.\nIncidentally the pack is unlikely to get much more efficient, as it is already quite amazingly efficient and it's hard to improve much on near-perfection. What will happen is higher capacity at a lower cost.\nYou can still get stranded when a coolant hose springs a leak, your water pump fails, any one of the belts snap, the starter gets wet, your 12V battery gets drained, or a whole bunch of other complications involved with an ICE in the critical path.\nSame can be said for a PHEV. If in 5 years batteries are much more efficient, then you are still stuck with a 5 year old PHEV (with an outdated ICE and an inefficient battery).\nI have had my share of events where I was stranded using an ICE (gas) vehicle. I am sure we all have. I expect that an BEV will actually last longer and have less mechanical issues. Nowadays with mobile phones in every pocket having a mechanical or fuel releated issue is much less scary.\nPlease post questions in appropriate threads.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Consider window film as an alternative to net curtains!\nOur window films allow you to get rid of those old net curtains!\nReplace your old net curtains with our window film and immediately notice the difference! As a fantastic alternative to net curtains, our window film maintains the look of standard glass, both from the inside and outside while also giving you full privacy without net curtains.\nWe work across Kent, Essex, London, Sussex, Surrey, Middlesex, Hertfordshire, Oxfordshire, Bedfordshire, Hampshire and Berkshire, so call our office today on 01689 854577 to arrange your FREE no obligation quotation and find out how window film is an incredibly effective alternative to net curtains.\nWant to improve your privacy without nets?\nOur window film will do just that by offering a clear view from the inside, but will stop others on the outside from being able to see in; ideal for homes and all residential properties in busy inner city areas or towns with lots of passing traffic and pedestrians.\nWith many other benefits including heat reduction and glare reduction; window film is a perfect choice when looking at replacing old net curtains.\nNot only does window film have heat reduction benefits, it also stops potential thieves from being able to view inside your property. Instead of closing your blinds or shutting your curtains while you're away, our alternative to net curtains window film provides extra coverage by stopping people from being able to peek inside.\nOrnaments on window sills, curtain backs and open blinds are the only visible items when looking into your property from the outside as the window film works from a half a metre distance from the glass so the glass still gives you a natural look from the outside. Window film offers the highest level of privacy from passers-by and protection against would-be thieves.\nImage to the right - All other windows have film applied, apart from Window I. The view from the other windows is completely obscured but the effect is totally neutral.\nWhen the sun comes out, we know how frustrating it is when there is too much glare on your television when you're trying to watch something. Our window film reduces the amount of glare that is able to enter your property so you are able to see the television screen more clearly. Not only that, window film can also prevent the fading of furniture, paintings etc.\nExcessive heat build-up can sometimes render rooms unusable, but with the use of our window film, it creates a barrier to keep the heat outside and aid in keeping your property cooler. The window film even lets more light through to help light up your property.\nBased in Bromley, Kent, we provide residential properties and commercial businesses with the perfect alternative to net curtains - window film! Just some of the locations we cover include Kent, Essex, London, Surrey, Berkshire, Bedfordshire, Hampshire, Sussex and Hertfordshire.\nMany previous installation projects have taken us to Dartford, Gravesend, Bexleyheath, Southend, Brentwood, Ware, Enfield, Reading, Woking, Reigate, Cambridge, Hastings, Tonbridge, Sittingbourne, Canterbury, Orpington, Sevenoaks, Chatham, Maidstone, Folkestone, Gillingham, West Kingsdown, Bromley, London, Colchester, Tunbridge Wells, Brighton, Portsmouth, Ramsgate, Clacton on Sea, Whitstable, Seaford, St Albans, Bishops Stortford, Stevenage, Crawley, Maidenhead, Reading, Bracknell, Guildford, Croydon, Luton and Oxford.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looks good Sawyer! Darn I feel like such a weirdo with a complete lack of a sweet tooth though..\nThanks for the mention, Mr. MacConkey! Geez one day you're Sawyer and the next, you're Mr. MacConkey. Was that the day we went together for the sukiyaki and ice cream? Duh..I think it was.\nI love floats too. They're so very good with soda. Have you tried sherbet floats, a.k.a. ice cream punch? You pour 7up or any lemon-lime soda over scoops of raspberry, orange or lime sherbet (not rainbow because the colors will mix and become ugly) in a large bowl. Don't mix. It's so wonderful and looks beautiful.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This emo beauty queen sure loves to stay in tiptop shape and this is why she decided to turn one of the room in her house into her private\u2026 gym. How cool is that! Now, do you think you could help her decorate it, getting it a cool emo-ish look? Go for different types of fitness equipment looking for those printed with all kinds of emo punk symbols, such as pink skulls or cute little broken hearts, then go for an emo specific color for the walls, too, and keep in mind to select the lucky owner of his funky emo gym as well!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Discussion in 'Plug-ins : Requests' started by jaejgo, Sep 24, 2018.\nHello, first thanks for your answers but I think we are confusing two different things one is the Render Mental Ray engine and another is Nvidia Iray which are two different engines see: https: \/\/la.nvidia.com\/object\/nvidia-iray -the.html Nvidia Mental Ray is free for 3ds max 2018 to rebind a single frame if you want to render an animation asks for license see: https:\/\/la.nvidia.com\/object\/nvidia-mental-ray-products-la. html, in this case it is for Maya but for 3ds max it is the same. For Nvidia Iray for 3ds max x version and following from the previous page to buy it you get to this link: https:\/\/www.irayplugins.com\/iray-for- 3ds-max \/ buy.php.\nAnana I do not think it's a good idea to delete all the folders you mention because in this folder is not only Mental Ray or Iray but also this information for other Nvidia plugins like PhysX, 3DVision, DriverAnalyzer, HDAudio ...... etc. But possibly you know that they can be deleted, if so, I would appreciate it if you would contact me.\nI hope this post serves to clarify and again thanks for your help and interest.\nYou can move that folders to recycle bin, then install mentalray. Then, restore the folders again.\nHi, anana thank you very much your method has worked perfectly. I wonder if you have another method to remove the trial license for Iray.\nSubtance painter to Iray introduced to be able to perform the renderings within the program itself which I find great because of all the render engines that I tested Iray is the one I like the most easy to use, fast and with very excellent quality render without as many complications as others and perfectly compatible with subtance.And if we add that most people who are dedicated to 3D have Nvidia cards and this engine is ideal for these cards I do not understand how in the forum nobody treats this engine and much less than anyone else. I can break.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Anna: [after saving The Man from getting killed] You said one man can't make a difference. How about a woman?\nPersonally i am not a fan of these type of movies usually, the \"desperado\" style. But as a Scott Adkins movie, i had to see it. It was released under Bullet here in quebec, don't know why, as its so generic of a title... Anyway took me a couple weeks after the DVD release to finally watch it, the reviews had me nervous, but I'm glad i decided to watch it anyway.\nScott Adkins can't be Boyka in every movie, and as much as i love that character, it would be bad for him to just play it again and again. Here he is a more ordinary police officer on the run and pretty much fight like your usual big guy hitting hard. A lot of gun fights also but there is enough hand to hand to keep the fans of it satisfy.\nThe movie has a couple jokes and funny lines, i really laugh hard at one scene in particular, and the \"cheap filming\" style didn't bothered me too much, even tough its made like that for artistic purpose.\nIt will not be remembered as Scott Adkins best movie for sure, but its a decent enough action movie in the no brainer category. It serve its purpose well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Samsung hung on to the top spot as the biggest smartphone maker in the world in the third quarter, but new data from Gartner shows a continuation of what has been a story of decline when it comes to the South Korea-based tech giant's handset business.\nSamsung, according to the Gartner data, held on to almost 19% of the global smartphone market in the third quarter of this year, topping Huawei in second place with a 13.4% share and Apple with an 11.8% share. Samsung's share, meanwhile, was down from 22.3% in the third quarter \u2014 and it was also the largest year-over-year drop since Gartner began following global smartphone shipments.\nThis is yet another reason why 2019 will be an all-important year for Samsung's mobile business, since it will see not only the launch of the Galaxy S10, but also the company's much-touted foldable phone, which will bring a new form factor to the marketplace. Among other competitive worries for Samsung, though, Huawei's executives have made it clear they want to dethrone Samsung from the top spot and are working feverishly to accomplish that goal.\nAmong other takeaways from the new Gartner data is the fact that smartphone sales grew 1.4% overall in the third quarter, to hit 389 million units. That's a small increase, but if you take out the performance of Huawei and Xiaomi, you actually get an overall decline for the market.\nAccording to Gartner research director Anshul Gupta: \"If we had taken both Huawei and Xiaomi out of the global smartphone vendors list, smartphone sales to end users would have declined by 5.2 percent.\" The table above shows why that's the case. Huawei, for example, saw a big jump year-over-year, growing demand for its handsets by a whopping 43% in the third quarter.\nIn material accompanying the release of the new data today, Gartner also addresses future technologies, given that global smartphone sales continue to decelerate. That includes technologies like 5G and the foldable form factor that manufacturers hope will gin up new demand. But though major vendors like Samsung, Huawei and LG are moving ahead with foldable phones in 2019, Gartner cautions that it will take some time for interest in those to become mainstream.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Craftastical!: My child has been taken over by aliens bent on the destruction of my house.\nMy child has been taken over by aliens bent on the destruction of my house.\nI am at the end of my rope.\nClimbed the gate into my room, where he got out Daddy's tools and sprayed his sister with throat spray. Also used laundry as a \"slide.\"\nBathroom, where he opened the child-protective door knob with ease. He used the liquid soap and unwound a whole thing of dental floss.\nIn the kitchen, he dumped on the floor a nearly new package of baby carrots that Maxton had been snacking on.\nThe living room, where he destroyed two scrapbook pages. Moving the scrapbooks was LITERALLY ON MY TO DO LIST. Luckily, they weren't great pages, but my goodness!\nThe child locks on the under-sink cupboard, where he got my Costco-sized, nearly new Cascade and dumped the entire thing out in the living room, which he proceeded to get in his eyes--he was ok after a good rinsing. He also dumped the basket of shoes out and took down all the hanging shirts in his closet (I have no idea how he gets that high, I think he's standing on the diaper pail) and broke 4-5 plastic hangers.\nYesterday, he broke three bowls by pushing his high chair over to the upper cabinets in the pantry where we keep the dishes.\nHe's napping now. I'm sure this day will be really funny in a few years, but right now, I just want to lock him in his room! I don't know what to do, because he's not old enough for punishment, he'll be two in October, and he breezes through the child proofing and it's not like I'm somewhere with the door closed or not checking on them every few minutes. I was organizing my scrap room while most of this happened and I can usually hear whatever they are getting into, but I guess he's learned to be silent.\nI so feel your pain! Maddy can be like that on some days, too. Thankfully, she doesn't dare climb over the baby gate in the kitchen (but she can open the fridge lock). Her vice seems to be the bathroom which we can't really try to lock her out since Hyrum needs to be able to get in there with ease. Hopefully this is a short lived stage for The Griffinator!\nPoor Mommy, don't you hate those days? Darcy was such an easy kid to childproof against but now she is much wiser and Parker is so big and strong that as a team they're a real terror.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I know often it's challenging to consider your imaginative side and mix it with business. Truly, what I'm attempting to help you do is to do it by yourself alternatively of relying on these different men and women to industry you. It's even now excellent to have a manager to take care of specific factors when you get huge sufficient. The objective overall though is to keep as individual and linked with all your fans as a lot as feasible.\nWhat is the difficulty that causes noise in most of the goods out there nowadays? In most cases, the vibration inside the item is going to result in the noise and when you have bad high quality rubber that doesn't fit as correctly as it should the noise is going to be greater. The vibrations that you note in the item may well seem only small, but even little vibrations can cause very a bit of noise.\nWe used to wear these all the time in the 80s. It was absolute treasure to locate a 44 gallon drum that nonetheless had 1. I don't forget my step mum telling me that 1 day I would look back and laugh at how we used to put on them. Of course, she is correct!\nInexpensive Twilight Present Tips \u2013 #13: Crew Edward and Crew Jacob custom silicone wristbands. Reminiscent of the Lance Armstrong-fashion Livestrong bracelets, these custom silicone wristbands enable any Twilight fan to express her allegiance to either Edward Cullen or Jacob Black. Four bracelets are incorporated: Two bejeweled ones that go through 'Twilight', a single that reads 'Team Edward', and one particular that reads 'Team Jacob'. Who does your tween fan favor? Locate out now with this clever present. $18.99, and made by NECA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An au pair differs from a nanny in that they are usually young people from another country without formal childcare qualifications who provide live-in care for children in return for board and spending money and are considered like a member of the family. In addition to childcare, they may do light housework.\nThe main purpose is a cultural exchange for the au pair, allowing them to improve their English.\nHost families may be asked to cover travel and insurance costs as well as a language course.\nAmong the pros of hiring an au pair is flexibility. Because they live in they can generally adapt to your schedule, for example, helping out in the morning or evening if you are running late. They can develop a one to one relationship with your child, like an older sibling.\nThey are usually cheaper than alternative childcare, particularly if you have a larger family. The average pocket money they get is around \u00a385 per week. Au pairs may have to pay Income Tax and National Insurance, depending on how much pocket money they get. They are usually not considered employees for tax purposes, however.\nAlso au pairs may speak a different language and teach your children their language and new cultural experiences.\nPotential drawbacks include the fact that they are not qualified and may be young and inexperienced. You may find having someone in your home infringes your privacy and may feel concerned about having a complete stranger not only living with them but also taking responsibility for their children.\nYou can find an au pair through a reputable agency who will help with the matching process. The British Au Pair Agencies Association website was created to help families find reputable and registered au pair agencies throughout the UK.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For the 73rd episode of Podcast at Ground Zero the umm \"topic\" will be Off topic. I've previously mentioned that we will start doing a series of episodes called Off Topic. What I mean by Off Topic is all the stuff that we usually talk about at the start of an episode but not related to the topic of that episode. This way we can talk about and share with you all kinds of apocalyptic stuff without any worries of being off topic because that is the topic for these shows.\nThank you so much for answering me! A few points: 1) I am always grateful; i.e., I am not a millenial gamer\u2014 I'm 56. 2) I am board certified in Psychiatry, and trained at UCLA San Fernando Valley in same. Med school, UTMB, license number in Alabama 16947. So, you can look me up to check. I practice in Texas, seeing badly regressed psychotic patients; I have worked additionally with the Maori in New Zealand and the Ojibwe in Northern Minnesota. I have worked in multiple sites in multiple states and two countries. I also worked last before this job with ICE seeing \"undocumented aliens,\" working with people who spoke about 30 different languages. Lots of PTSD.\n3) If you have psych questions about the apocalypse and treatment of PTSD, I can do literature searches. As someone with a bit of experience in this field, I can also tell you what might work.\nGreat show, even off topic discussion is fun.\nGreat books, 3 so far, about a 50ish guy surviving in a back yard bunk with shroom zombies outside. Get them on Amazon Kindle and audio. Keep up the good work.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"We would like to thank Blauer Tactical Systems\u2122 for use of images and videos throughout the site.\nThoughts Become Actions...what are you thinking?\nThreat \u2013 Menace; Indication or warning of probable trouble; Give an ominous indication of; impeding evil or mischief.\nWhat I once loved now caused me anxiety!\nWhat I once loved now caused me anxiety.\nWhere are your thoughts leading you?\nI got this email newsletter from Brian Willis from Winning Mind Training, and it lines up with what we teach in the PDR sessions and the cycle of behavior\u2122. It's worth a read and further discussion.\n\"Talk to your kids about the 'what-ifs\"\nJust last night, 3.23.15, there was an attempted abduction of a 9 year old from a sports complex in Northern, KY around 5 pm. See story here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Electronic cigarettes have come flooding on to the popular scene in the last decade, although they have actually existed for much longer. They are a controversial new pasttime due to the limited availability of research into their long term health effects. The problem isn't a lack of interest but a lack of time. E-cigarettes have simply not been around long enough for scientists to adequately assess their health risks. Because of the before-mentioned limitations there is a lot of contradictory information out there about e-cigarettes.\nMy grandad is on so many medication he finds it really hard to keep on top of what medications he needs to take and which medications he has already taken. He doesn't want to move into a nursing home, and we can't always be there to keep an eye on him. Luckily, our pharmacist is a great help and has been working with us on some systems to remind him to take medications like alarms on his phone and a schedule of check-in phone calls. Pharmacists want to make sure their patients stay healthy too! This blog is about working with your pharmacist to make sure your elderly relatives are taking their medication.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Join us for a sip & sup spring fundraiser! Beth Shalom is excited to host Manolo Hernandez Martinez and his collection of Spanish, South American and even KOSHER wines. Tastings will be paired with gourmet cheeses and refreshments painstakingly selected to enhance flavor. Anticipate live music, complimentary childcare, and fellowship!\n$35.00 per person. Sponsorships are WELCOME! Bronze ($150), Silver ($250), Gold ($500), and Platinum ($1,000). Each sponsorship includes tickets to the fundraiser.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I have a love-hate relationship with Grandeur Palace: they're one of better dim sum restaurants, in terms of taste for value, but the sheer amount of \"friends and family\" they let go to the front of the queue grates on the nerves while you're waiting. Luckily, they're quiet during dinner and with a small group we've just walked-in. It's so much better when there's no one jumping the line!\nUnlike other Chinese restaurants, Grandeur doesn't serve complimentary soup. So, if you're used to wetting the whistle before the food arrives, plan accordingly. Their soup of the day ($9.98) is reasonably priced and sufficient for about 8-10 people. While it changes, one evening brought us a bone broth made of pork, carrots, apples, and white fungus, arriving piping hot and flavourful (not having been diluted).\nAlmost every table orders the roasted Peking duck special ($19.80), a steal for two courses (duck with wraps and chopped carcass). While it's not the most stellar version of the dish, it's still satisfying. Their biggest flaw being the consistency of the bird \u2013 larger tables are given larger birds.\nMoreover, their wrappers are a little thick and left in the steamer too long causing the top ones to dry out and crack. Nonetheless, they're a good three-bite size ideal for making a duck taco.\nYou will spend more on other dishes \u2013 even a simple vegetable dish is above $15. However, Grandeur doesn't skimp on quality or portion sizes. The sweet and sour pork ($16.80) is made with pork tenderloin, so even older family members could bite through the meat, and there was enough sauce for flavour. Sadly, the large chip is prettier to look at than eat: thinking it was a gigantic shrimp chip, I was disappointed to be greeted with the taste of Styrofoam.\nThe salt and pepper pork chop ($16.80) is a substantial dish. Again, the kitchen ensures it remains tender while creating a crispy crust, it just needs to be spicier.\nWhile it's common in Chinese restaurants, the actual seafood in the tofu, vegetable, and seafood in hot pot ($18.80) has little flavour given it's quickly blanched before cooking. Most tables order the hot pot for the sauce over the natural shrimp, scallop, and squid flavours.\nIf you've never had bamboo fungus, it has an interesting crunchy spongy texture that I love. At Grandeur, you'll find the ingredient in the stir-fried vegetables with bamboo fungus ($15.80), topping broccoli and mixed with black and white fungus.\nThe baby bok choy with salted and preserved egg ($13.80) isn't my favourite dish as the grainy texture of the salted egg yolk is strange against the vegetable. However, it's simple and relatively healthy feeling for those wanting a lighter option.\nAt $19.98 a pound, it doesn't sound expensive for a large lobster, but when you're greeted with a behemoth 6-pound dish ($119.88), it adds up. Best for big tables, large lobsters aren't always as sweet as their younger counterparts but there's more meat, especially in the claws and legs. The traditional stir-fried lobster with green onions and ginger was done well, chopped into large enough pieces so it didn't become overcooked.\nThe braised grouper ($48) is another dish for larger tables \u2013 a platter with a big slab of meaty fish topped with tons of tofu and surrounded with vegetables and mushrooms (these sides alone enough to count as its own dish). The thick grouper was just cooked through and there was sufficient oyster sauce to keep everything flavoured.\nWhoever chopped the deep fried whole chicken ($33.60) did so in a haphazard manner, it arrived disheveled looking. Nonetheless, it was well flavoured, the skin crispy, and the meat cooked through but not tough. Perhaps it was cut while it was still hot - if it's between presentation or temperature, I choose temperature any day.\nAlthough the restaurant doesn't provide soup, customers do leave with a sweet ending. The customary green bean soup with tapioca was hot and sweet and on a weeknight dinner there were also bite-sized mango pudding and cookies.\nWhile dining at Grandeur Palace doesn't make you feel like royalty (even on quiet nights it's hard to get a staff member's attention), their dishes are decent interpretations of Cantonese cuisine. Just go in a table of six or more; they don't skimp on portion size.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tirana has been the capital of Albania since 1920. It is relatively new city was established in the year 1614 from Sulejman Bargjini of Mulleti, feudal of the area at the time who built a settlement in the area around the modern intersection of Rruga Barrikadave and Rruga Luigj Gurakuqi. The statue of Sulejman Bargjini of Mulleti stands today in the little square close to that crossroad. Until Tirana was designated as the capital of Albania in the year 1920, it was just a small, unimportant town whose main virtue for Albania's political leaders was it's geographical position, more or less in the center of the recent independent country of Albania, on the fault line dividing the northern Ghegs from the southern Tosks. Tirana remained a bit of a back water town for several years afterwards, as Albania struggled to stabilize itself in the face of internal lawlessness and invasions by its hostile neighbors. The city began to grow at the beginning of the 18th century. \"Tirana \", it is thought that comes from the word \" Theranda\", mentioned in the ancient Greek and Latin sources, that aborigines called Te Ranat, because the field was formed as a result of stiff materials that the waters from the surrounding mountains brought. Today Tirana is not only the most populated city in Albania, but also the biggest political and economic center in the country. The Adriatic Sea and Dajti mountains are near to the city. It takes you less than one hour drive to reach the sea. A Great Park with an artificial lake is located immediately at the southern part of the city.\nIt was not until Italian influence became pervasive in the late 1920s that the center of Tirana took on the appearance of a capital city. Italian planners created the huge new square, which was named after the National hero of Albania Skanderbeg, and the wide typically Fasist boulevard; Italian architect designed the ministry buildings, the National Bank and the town hall at the southern end of Skanderberg square, as well as the Dajti hotel, the Royal Palce on Rruga Elbasanit, and some of the embassies. During the communist era, the few old buildings still standing on Skanderbeg Square were demolished and the opera house, the National Historical Museum and the hotel Tirana were added as were the public buildings further down the boulevard. All of these 20th century accretions, plus the destruction caused by earthquakes and by the battle for the liberation of Tirana in 1944 means that there is not very much left of the Ottoman Tirana. A good deal of what survived into the 1990s has subsequently disappeared under glittering high rise apartment blocks and shopping centers.\nTirana is well known for its brightly painted apartment blocks. This idea began in the beginning of the elections of the year 2000, which saw an artist and former minister of culture becoming the mayor of the city of Tirana. The new mayor, Edi Rama (the country's prime minister) began by restoring the ministries on and around the famous Skanderberg Square the okra color which they had when they were first built in the 1930s; He went on to also paint the old apartment buildings on the nearby streets, choosing bright colors which, although they did not convince the population of Tirana, at least had the merit of brightening up the city. The colors and patterns became livelier and livelier until even the most progressive citizens of Tirana began to complain that their city was starting to look like a circus. Happily for them, the harsh summer sun bleaches out the brightest colors after a year or two. As well as its strange mix of architectural styles, Tirana has very special museums, lots of green areas and a ranch of cultural activities. There are hundreds of cafes, modern bars that are very popular to the younger people, numerous clubs, some with live music, especially on the weekends and many restaurants. New bars and restaurants open all of the time in the capital city of Albania. The more ephemeral lists all of the latest fashions in eating, drinking and dancing.\nTirana also has many infuriating aspects, mainly the very bad traffic and the Balkan noise level. In general, however, the city center makes an attractive area to stay and its excellent public transport links make it the best base for exploring the rest of Albania.\nBY AIR: The only international airport in Albania is located in Tirana, the airport is seventeen kilometers, an 30 minute drive from the main city. Approved airport taxis can be booked at a kiosk in the area of reclaiming your baggage or hired as you exit into the arrivals hall. You should agree a fare with the taxi driver before accepting his services. The going rate for going into the center of Tirana is 20 Euros. Alternatively, unofficial taxis wait across the road from the perimeter fence of the airport and they charge slightly less. From 7.00 \u2013 19.00 an hourly bus service runs from outside the perimeter fence, into the main city of Tirana. The one way fare is 250 lek. In the center of the city, the bus leaves from the junction of Rruga e Durresit with Rruga Mine Peza, where there is a clearly marked bus stop with a timetable. The last bus to the airport leaves at 18.00.\nThere are several hotels located near the airport, which are useful for early morning departures or late night departures. (See the official website of Tirana airport).\nBY ROAD: Buses and mini buses run to Tirana from all of the other areas of Albania. In the other direction, Tirana does not have a central bus station and sometimes it is difficult to establish where the bus that you want leaves from. As a general rule, buses and mini buses to Durresi and destinations in the north leave from near the Zogu i Zi roundabout, buses to the south west leave from a bus station just off the Rruga e Kavajes and buses to Elbasani and the south east leave from the Northern side of the Qemal Stafa Stadium. The most simple way to locate the vehicle that you need is to hire a taxi and say something like \"autobuset per en Durres (the bus for Durres) to the driver. The receptionist at your hotel should also be able to help you.\nThere are international buses to and from the main cities in all of the neighboring countries. The majority of the bus companies have their office where information is available and tickets can be bought on Bulevardi Zogu i Pare which runs between Skanderbeg Square and the railway station.\nThere are no trains to or from Tirana. The old train station has been demolished and the railway lines have been asphalted over in order to build yet another highway.\nGETTING AROUND TIRANA: Local buses are run by private companies and licensed by the municipality of Tirana. The buses can get very crowed at certain times but they are also incredible cheap with a flat fare of 30 lek which is collected by a ticket collector on the bus itself. The two routes which are most useful for visitors to the city are the Unaza, which goes around the north of the city in a big loop and then along the Lana river and the Tirana e Re\/Stacioni i Trinit which begins near the train station and goes right down the boulevard, before turning off along Rruga Abdyl Frasheri and then making an even wider loop around the western part of the city of Tirana. Other buses depart from the various bus stops around Skanderbeg Square. The buses are different colors depending on the destination and can be boarded or alighted at any of the bus stops along their route. A tram route which will cross the whole of the south east of Albania to the North West axis of the city is planned and will soon be ready.\nIn the center of Tirana, within the ring road, taxis charge 400-500 lek. For longer journies, crossing the ring road, please agree on the fare before setting off. There are taxi ranks at several place in the center of the city, along the side of \"THE BLOCK\" at various points along boulevard Zogu i, convenient for the railway station and buses to and from the north, and beside the Academy of Arts on Mother Teresa Square. Tirana has a few radio taxi companies, the easiest numbers to remember are +355 2244444, +355 2355555 and +355 2377777 all of which are reliable firms with reasonably new vehicles.\nCars can be hired in the city center and at the airport of Albania. Tirana's cycle hire scheme has bicycles in racks at various points around the center, the back packer hostels also have bikes available to rent. It should be noted that driving and especially cycling in the center of Tirana is not for the faint hearted.\nTirana has a wide range of hotels that suit most kinds of people. The very top of the market will charge around 120 Euros a night and these hotels are the Cheraton (down by the university of Mother Teresa Square), the Rogner (on the boulevard opposite the presidency building), and the Grand, (on Rruga Ismail Qemail, opposite Hoxha's villa). All of the aforementioned hotels offer comfortable rooms, good restaurants and swimming pools.\nAre you looking for trustworthy, comfortable and clean accommodation in Tirana?\nWhen it comes to choosing a place to stay in Tirana, this comfortable hotel offers affordable prices, personalized attention, tranquility and a high standard of service, food and hospitality.\nThere are available one day tours of Tirana that you can ask for via this website or at the tourist information office in Tirana. The favorite destinations are Mount Dajti and the Pellumbasi Cave (See sightseeing in Tirana, below).\nSkanderbeg Square: Although the square is no longer the hub of commercial or social life, the square is still referred to as the center of Tirana and it makes a very convenient starting point for a cultural tour of the city of Tirana. On the northern side of the square , you will find the National Historic Museum of Tirana and the Tirana International Hotel, to the west side is the National Bank and the eastern side is entirely taken over with what was once known as the Place of Culture, which still houses the opera house and the National Library. The tables that are outside the Opera House Cafe are wonderfully cool in high summer, thanks to all the marble surrounding them. bisecting the square is the wide boulevard designed by Mossolini's emissaries and is perfect for Fascist parades. To the south of the square down until the university building, it is called Bulevardi Deshmoret i Kombit (the martyrs of the nation boulevard). The northern section is Boulevardi Zogu i Pare, a statue of King Zog now stands at its junction with the ring road. An equestrian statue of Skanderbeg stands at the head of Deshmoret i Kombat. The sculptor was Odhise Paskali, 1903 \u2013 1985, a native of the town of Permet which is located in the far south of Albania. Across the road from the statue are the Mosque of Et'hem Bey and behind it is the clock tower. The 18th century mosque is one of the few really old buildings left in Tirana, and is perhaps also one of the most beautiful buildings. Its minaret was shattered in the battle for the liberation of Tirana but it was subsequently repaired and the status of the mosque as a cultural monument kept it from being damage or destroyed during the atheism campaigns of the late 1960s. There are frescoes on its external walls and more paintings inside the mosques, slim visitors can climb up the narrow, spiral staircase to the gallery from where there is a better view of these. Excellent views of the city can be enjoyed from the top of the 35 meter high clock tower (built in the 1820s). The opening hours are 9.00 \u2013 13.00 on Mondays and 9.00 \u2013 13.00 and 16.00 \u2013 18.00 on Thursdays. Visits are other times may be requested by telephoning +355 2243292. There is also a small exhibition of models of clock towers in Albania which can be seen on request.\nTirana is the capital and the largest city of the country of Albania. Although a new and modern capital city the origins of Tirana as an inhabited center are quite old with several theories and myths associating its current name with ancient versions. One version is that it's name derives from the word 'Theranda' that Greek and Latin sources employ to refer to the area, after the term 'te ranat' used by the inhabitants, meaning 'fallen material', in reference to the composition of the terrain out of hard earth swept down by water from the nearby mountains. Another theory is that it comes from the word 'Tirkan', the name used by the sixth century Byzantine historian Prokop to refer to a castle, first built in the first century BC, on Mount Dajti, and the ruins of which are extant. Some say it comes from 'tyros', the old Greek word for 'dairy', on the hypothesis that it was in the field there that the shepherds of surrounding areas gathered to trade dairy products.\nWell known architects of the Mussolini period in Italy Florestano de Fausto and Armando Brasini, where the masterminds which build the main square, which today bears the name of Albanian National Hero Scanderbeg, the huge boulevard, ministry buildings, national bank, the town hall and the Palace of Brigades (former royal Palace, today Presidential Palace). Today Tirana is the center of the political, economical, and cultural life of the country with over 700 000 inhabitants. In the last few years Tirana has seen substantial changes in its appearance. The dull communist-style apartment blocks have been painted over in bright colors and abstract patterns by an artist turned Mayer. This is not only a quick fix but also an uplifting experience for inhabitants and visitors alike. Furthermore it has seen an increased development in modern infrastructure contributing to the city's metropolitan look.\nThe tourist information office in Tirana is located on Rruga Ded Gjo Luli, behind the National Historical Museum. Unfortunately, it is only open from 1..00 until 16.00 on week days and does not open at all on weekends, this is the reason that we want to give you as much information as possible about the capital of Albania, Tirana. The tourist information office offers you a range of free leaflets and brochures about Tirana and other towns in Albania. The staff speak English.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgfcu b/data_all_eng_slimpj/shuffled/split2/finalzzzbgfcu
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+{"text":"I picked up the Robot Trio last week when Michaels had a sale and I wanted to use one. So I made this birthday card. No challenge or contest, just a card I made for fun. I'll probably give it to my friends son for his birthday in a few months. He loves robots. Thanks for taking a look!\nPerfect background for the adorable robot, Donna! Fun birthday card!\nThis is super cute Donna! These colours are perfect for a boy, love the balloon and cupcake!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Southwest Telehealth Resource Center (SWTRC) was created to advance the effective use of telemedicine services throughout the Southwest. Telemedicine has been shown empirically to improve access to quality medical care for many patients living in rural and medically underserved areas.\nThe SWTRC assists start-up telehealth programs in their development and serves as a resource for existing programs regarding changes in technology and other issues affecting telehealth in the Southwest region.\nThe SWTRC provides services in each of the following modes: one-to-one, peer-to-peer, and one-to-many. The SWTRC serves hospitals, clinics, public health offices, and private-practice healthcare providers in the broader Southwest region. Those seeking advice come from a variety of backgrounds, including clinicians (MD, DO, DDS), healthcare providers (RN, LPN, PhD), hospital administrators (CEO, CFO, COO), technologists (CIO, network manager, network engineer), telemedicine site coordinators, medical students, medical residents, researchers, and community representatives interested in telemedicine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Acenes are compounds constructed as an array of linearly fused benzene rings and of some importance to the field of organic semiconductors. Synthesising them is tricky (the heptacene is currently the longest one) and forcing an acene in a cyclacene (folding a ribbon into a bracelet \/ belt \/ tube ) must be even more challenging. Cycloacenes are of potential interest due to their no doubt interesting electronic properties but also because they and related compounds represent a class of nanotube like building blocks. A cyclo(10)phenacene system (with alternating benzene units as in phenanthrene) was described in 2004, obtained by reducing the north- and south pole of fullerene (DOI).\nIn a recent contribution from the laboratory of Rolf Gleiter, a cycloacene is prepared in a one-pot reaction from a cyclooctadiyne and the cobaltocene CpCo(CO)2 as a trimer (DOI). In one of the isolated intermediates (a trimer prior to cyclisation), the cyclooctatetraene rings, surprisingly, are almost planar!.\nAnother recently described molecule that is guaranteed to bring down your molecular graphics editor is a pyrenophane (pyrene + cyclophane) that is exactly half a nanotube loop (DOI). Its synthesis features a McMurry reaction and a valence isomerization\/dehydrogenation (VID) reaction using DDQ. The authors , Bodwell et al., are as confident today a full aromatic belt is within their reach as they were back in 2001 (DOI).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A couple of ways to really increase the performance potential of a 4-stroke engine is to install a higher performance camshaft(s) & to port your cylinder heads.\nThe FT500 Ascot is much easier to find a performance camshaft due to its close relation to the Honda XR\/XL engines.\nNow technically just about any factory OEM camshaft can be reground to a higher performance profile. This is done in several different ways. Probably about the simplest method is to reduce the diameter of the base circle of the camshaft. This in effect makes for a higher lift lobe profile once the material that was removed, is made up for by adjusting the length of the pushrods in that style of engine, or adjusting the extra lash out of the rocker arms. Another method is to weld additional material to the peak & surrounding area of the camshaft lobe, then grind the additional material to the proper new profile. After that is done, the camshaft is usually re-hardened or heat treated.\nAnother real trick modification to look into, is to install needle bearings at each end of the camshaft vs. a solid bushing & a thin film of oil. The needle bearings reduce some of the internal friction of the engine thereby releasing some additional power & throttle response.\nNow for some type of modification that the average mechanically inclined person can perform at home. Some port & polish work. Now there are professional people out there that are highly talented at portwork & have a flow-bench to test their work with. If you can afford to have a \"master\" port your heads, you most likely will not regret it. But the average guy with a small hand grinder can still accomplish measureable performance gains with some study & some modest grinding\/smoothing in the appropriate areas. The concept that we are after, is to try to \"straighten\" the port & to make any sharp radius less pronounced. Without the benefit of years of experience working with a flow-bench, you do not necessarily want to enlarge the port. For a street bike, we want to maintain a smaller port profile to enhance velocity of the charge. What the average Joe wants to look for, is casting flash that hangs down, or is rough. Something that obviously needs to be removed. Then try to imagine the flow of air, gas, or even the flow of water going through the port. We know that a charge that is moving doesn't like to change direction & will slow down. Now look down into your ports & try to see areas of the port that would resist smooth flow or transitions. Any sharp edges, machining marks, sharp radius with a peak to it, etc. We are not looking to redesign the port, simply make the best of what is already there. If there is a large protrusion around the valve guides, that is usually reduced to a minimum & tear drop shaped. Keep in mind while you are working that if there is more than one intake & exhaust port, you will need to match any changes that you make to one port, to the corresponding port. All intake ports should have as close to the same profile as possible & the same for the exhaust. I recommend that you start with the exhaust port as it will be less critical if you make a mistake. You want to pay special attention to the port \"bowls\" or \"valve pockets\" which is the area of the port closest to the backside of the valve. Most of your performance gains take place in the \"bowl\" & the valve seat area. A three-angle to a multi-angle valve seat really helps to improve the flow numbers. When doing your grinding\/smoothing near the valve seats, you must excercise great care not to damage the valve seats. Often the valve seats are \"high-lighted\" with machinists bluing, or even a large red marker to make them easier to see.\nThese are some of my thoughts on performing some simple port work that can benefit your bike. The real beauty of port-work is that when it is performed properly, you can release large numbers of \"free\" power. Just think of the engine as an \"air-pump\". Anything that you can do to increase the amounts of air that is passed through the engine, or to reduce any frictional or pumping losses, will build power.\nBelow are several articles on performing port work.\nI will add more to this topic as I run across more information.\nThey are in with the 650 Hawk and Transalp which suggests that they could be interchangeable.\nIt looks like they do some excellent work on the Hawk engine & it appears that there may be some interchange possibilities. I noticed that they offer three stages of port work & install 2mm larger intake valves. They also offer big bore kits, highly worked over crankshafts, heavy duty connecting rods, etc. How much transfers over to the VT Ascot, I don't know. I will try to call them in the following week & see what they have to say.\nEDIT: I contacted Hordpower today & was able to speak to J.D. Hord. Unfortunately they do not provide any services for the Ascot.\nOne of the difficulties is getting above 600cc. In the USA it should be possible to fit a Transalp 600 engine in and fit the shaft drive output mechanism from the original engine in place of the chain drive.\n650 head is quite different as the front exhaust port fouls the frame downtube.\nI have often wondered if there were some engine swap candidates. The VT Ascot is not bad stock & is a fun, well balanced bike. I always want a bit more after a while though.\nAny idea how much larger the exterior package of the Transalp 600cc engine is than the VT500 engine?\nThey look exactly the same even down to the air cooled look.\nCylinders could be a little bit taller as the stoke is longer.\nIf you have a VT Ascot, you will need to replace both camshafts at the same time.\nIt is difficult to remove the air-box on a VT Ascot. You might have to drop out the swingarm & rear wheel, fender assembly to slide it out the back. I'm not sure if it will make it out one of the sides. The air-box has a crankcase vent oil seperator attached to the bottom of it. Plus the coolant recovery tank is located below the air-box. It is a very tight fit.\nYou probably need to drop the engine to remove the rear cylinder head. If you are replacing the front head gasket, isn't it a good idea to replace the rear also? The rear cylinder usually runs a bit hotter than the front cylinder does. I'm just wondering if the the front gasket failed, what are the chances that the rear gasket is bad also?\nProbably not what you wanted to hear.\nThe rear valve cover will come off with the air box in place. The snorkel must be removed. Not sure about the head. But the valve cover will get you access to the cam.\nThe air box will come out of the VT Ascot and Shadow without pulling the swing arm. Remove the rear fender and battery box first. Wiggle the air box around and twist the crankcase vent catch chamber 1\/4 turn to get it off the bottom. Removing the wheel might get it out the rear but I haven't tried that. The box will come out the side. The right side I think. Getting it back in will be a challenge. I haven't tried that yet either. But it's coming.\nThanks for that info Buddy. I recently removed a VT air-box, but I was primitive in removing it. The air-box had a couple of cracks in it & looked to be allowing dirty air into the clean side. I wasn't able to get it out of the frame, so I broke out the sawz-all & sliced the air-box. It came out real easy then.\nI don't recommend taking the sawz-all to perfectly good parts. If my air-box hadn't been cracked I wouldn't have cut it.\nMine are cracked also. I'll bet it would be hard to find one that's not. Everyone should check theirs. Cracks in the box lets dirty air into the engine as you mentioned.\nI'm gonna repair\/modify mine and put them back. If it's made out of ABS I'll use acetone and Legos. If not I'll try some caulking I saw at the hardware store.\nBuddy, great info, thanks. Regarding replacing the rear head gasket--I know the compression is good, so I'm just going to roll with it.\nBut this does open up the camshaft upgrade as an easy possibility. Will have to give that place a call once the ascot is running and I'm searching for that extra tad of HP!\nAn issue may be the worm drive for the rev counter.\nOn the later engines (Hawk, Deauville etc) the rev counter is electronic and the cam covers have no outlet.\nMaybe just convert to electronic from a Hawk?\nThere are also a few \"How-To\" articles on porting.\nAir box install update, they go back in real easy with the rear wheel and fender removed. Swing arm is not in the way at all.\nI was once a member of that Cadillac forum. Like the Ascots, the big Cadillac engines don't get the appreciation they deserve. I once had a 500cid Caddy in a 91 Dodge Ramcharger. Plenty of power and 13 mpg.\nI managed to re-install my airbox without removing the rear tire. I did have to move the inner fender back & up against the tire. I don't recall how I managed to finally get the box through the side, but it wasn't easy. My bike does not have the coolant recovery tank in the original location & I'm not running the oil seperater either as I placed a filtered breather on the rear cylinder vent port. It is all a very tight fit & removing the rear tire was next on my agenda if I couldn't get it from the side.\nSo you were a member of Caddy500.com? I started that board originally. It was an active forum at first, then there was a falling out & several members left. Now the Cadillac popularity seems to be on the waning side as more folks look to the LS series of engines for their projects. Please send me a PM with the details on your Ramcharger. I'd like to hear all about it. Do you have a build post somewhere?\nPS. Back to the Ascot airbox. I remember that I had to remove the upper airbox mounting bracket towards the rear, to have enough room to wiggle that box in there.\nIt's not because it's not written, that it is suggested!\n650 Hawk and Transalp 600 could be interchageable, but not with de VT500.\nCause of the gearing of the rev-counter.\nAccording to the company's tech rep the camshafts work in the VT Ascot.\nNow it's possible that you may lose the tach drive on some grinds. Allthough if you are sending in your original camshaft cores to be re-ground, you are pretty much guaranteed that you will still have the tach drive.\nThe question that I forgot to ask, is if there is a needle bearing option for the ends of the VT camshafts like the FT cams have.\nI have both camshafts (hawk and VT5), they are really different in their length.\nAt your home, there is a mystery!\nOr then: they re-cut everything?\nInteresting, are the cams longer, more lift or the overall length of the whole shaft?\nNo possible adaptation in my opinion!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"RetinIn is the cosmeceutical Vitamin A Serum. Vitamin A and its derivatives are the most powerful natural agents, able to modulate the renewal of skin old cells. The exfoliating action of this serum is due to Vitamin A and glycolic acid. Furthermore it contains hydrolyzed low molecular weight DNA and RNA fragments, able to increase the skin tone.\nRetinIn is the serum helping skin regeneration, while helping it keeping its elasticity and compactness.\nVEGETABLE SQUALENE Its action is to prevent loss of hydration and to help keeping skin elasticity and flexibility. Its incredible hydrating power and its penetration ability makes i tone of the favorite ingredients in cosmetic formulations.\nRETINOIC ACID it is considered as one of the best ingredients to treat aging signs due to photo-aging, such as thin lines, wrinkles and hyper-pigmentation.\nGLYCOLIC ACID thanks to its keratolytic properties, glycolic acid is used in cosmetic formulations with exfoliating, anti-age and whitening action.\n\u2013 More stability, because the active is protected from the external environment.\n\u2013 More bioavailability and a longer action span of time, thanks to the slow and controlled in time release.\n\u2013 Possibility to convey lipophilic actives in water.\nHYDROLIZED DNA AND RNA natural hydrating agent, containing DNA and RNA fragments, which are fundamental to keep the skin elastic and flexible. Odorless and colorless active, suitable even for the most sensitive skin types.\nVITAMIN E antioxidant activity. It is able to stop free radicals and oxidative stress, responsible for cellular damages.\nApply in the evenings on cleaned skin. Avoid contact with eyes and mucosa.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"On February 25th, 1999, The Roman Catholic Church excommunicated Father James Callan and the parishioners of Corpus Christi Church in Rochester, NY. Covered by Media nationwide, the excommunication was portrayed as the latest struggle of progressive Catholics against the more conservative Church authorities. Growing from 200 to more than 3,000 parishioners in less than a decade, Corpus Christi became home to an impressive range of social justice programs, including a health care center, a halfway house for former prisoners, a home for recovering addicts, a hospice, and a daycare center. What got Callan and the church into trouble were the practices that went far beyond the norm in Roman Catholicism; blessing homosexual unions, allowing non Catholics to receive communion during mass and allowing his pastoral associate, Mary Ramerman, to wear priestlike vestments and help officiate in the celebration of the Eucharist.\nThis paper discusses the making of the documentary, Spiritus, by Matthew Spaull to fulfill his thesis requirement in order to receive his MFA in film & animation from The Rochester Institute of Technology. This paper takes us from how he learned about this story to the making of the film 15 years later. Matthew navigates not only why he did this, how he did this, but also the challenges that he faced along the way. The paper concludes with the reactions and feedback during the various screenings this film was put through and the awards it has won up until this point.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The NHL and Great Clips are beginning a new multi-year partnership by exploring player hairstyles during the Stanley Cup Playoffs.\nBrands spend more than $65.8 billion globally on sports sponsorships annually. How do marketers properly assess the performance of these partnerships?\nThe agreement between MLB and Little League International provides programming opportunities for sponsors to 2.4 million players and more than 80 nations.\nThe Chief Business Officer of the Symetra Tour sits down with FOS CEO Adam White to talk sponsorship and expanding the LPGA's developmental tour.\nPowered by SAP, fans with NBA front office ambitions used real statistics while competing on the reality show \"GM School.\"\nSponsorshipX is not your typical conference with boring speakers and sessions. With an annual multi-day event, it throws its attendees into the action.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Robert Pattinson will jump right into two new movies after wrapping up promotion of \"Breaking Dawn, Part 2.\" One could put his life in real danger. Yet, he can't wait to get started filming.\nHis upcoming film \"Mission: Blacklist,\" will be shot in Iraq, where conditions are tense and sectarian killings by suicide bombers are still almost an everyday occurrence.\nThe U.S. military presence in Iraq ended in December a year ago, although the country still maintains an embassy in Bagdad and consulates in various cities. Britain's presence in the war-torn country ended seven months earlier.\nBecause the 26-year-old star a British citizen, the British Embassy will be responsible for looking after him while he's in country and will take the lead should anything happen. The U.S. Embassy would likely assist as well.\nHis movie is based on the true story of Army interrogator Eric Maddox, who was involved in the capture of deposed Iraqi president Saddam Hussein. Pattinson's family was opposed to him filming in the country, at first.\nClick here to see all of our Spring Breakers stories!\nHe's also already signed on to shoot \"The Rover,\" a western set in futuristic post-apocalyptic world in Australia. Pattinson plays Rey, a robber who is wounded and abandoned by his gang during a hold up. His victim, played by Guy Pearce, forces him to track down the gang to recover his stolen stuff.\nBut it probably doesn't come close to venturing to Iraq, where he will face real-life danger.\nIs Gwyenth Paltrow Rocking Johnny Depp, Amber Heard Relationship?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Beyonc\u00e9 is constantly killing the style game, especially when it comes to her current pregnancy. Her Madonna-esque announcement (which practically broke Instagram) is iconic. The picture has 11 million likes to date.\nHer mom Tina wanted to try the pose for herself this weekend at the \"Carter Push Party,\" and unsurprisingly, nailed it just like her daughter did.\nBeyonc\u00e9's stylist even jokingly included her own bloated-stomach version of a \"baby bump\" when she took her turn in front of the makeshift photo booth.\nEven Beyonc\u00e9's 73-year-old aunt Flo got in on the fun, recreating the original image complete with the veil.\nBeing Beyonc\u00e9 means hosting a normal baby shower isn't an option, and the \"Carter Push Party\" was anything but. She and husband Jay-Z, already parents to Blue Ivy (age five) wore African-inspired clothing and celebrated with celebrity guests like La La Anthony and Serena Williams.\nBeing pregnant with twins has never looked cooler.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Passionate and versatile film production crew and office team member with a strong work ethic. Looking for a script supervising position in the film and television market where I can demonstrate my skills in a fast-paced production environment. Also interested in pre-production, developement and directing.\nLA based. Dual citizenship USA\/Canada.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..50bc8cf4f3241effccb36d506829be1a37a61102
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+{"text":"think BB was going for a scan today to see the extent.. can't see him being fit for tomorrow. Reach in the middle with Hutch.. Matias & Boyd out wide.\nExpectations Too High This Season.\nI think the good run at the end of last season (even with Jos) and the hope a good pre-season to get players back fit give us the higher expectations. The fact we blooded a few decent youngsters, which to be honest we all liked really, and the fact FF was looking fit and scoring raised our hopes. The rest is history. The league is very open tbh and expect it to be like it next year, I can't see any major spending by any clubs just shuffling players around on loans \/ out of contract. It will be upto Bruce and Co to get the team firing from day one to give us a good start, then we can see how we progress.\nHow many did Roland Nilsson score ?\nUtd may get auto. They will defo be in the play-offs. Utd have handled it well all season. Can't see why they will particularly fail now. Auto is a massive bonus for them, I reckon Play-offs is where they aimed for at the start of the season. We may get in the playoffs if we are lucky and results go for us. Very very long shot.\nBuild from the back.. keep it tight and don't concede gives us a good platform. We all know Palmer and Fox have limitations but currently they are part of the unit that have keep a lot of clean sheets, to drop them would upset that, and if we lose or concede a few by changing it, I'm sure the management would be slated. Hutch is the perfect for the position and role he plays. Bannan needs a rest, Joey isn't upto scratch - we miss a Lee type \/ creative box to box player so much. Fletcher is frustrating, yes tries extremely hard and puts effort in, but lacks the incisive final touch \/ shot. Joao is only good as an impact sub. Reach flatters to deceive and Boyd again huffs and puffs with little end effect. These are the areas that need addressing.\nWe need some pace .. Middlesboro looking at Bolasie and Adama Troure.. surely they should be on our radar ?\nThat flipping belly button makes me gip.. Slightly annoying portrayal character, who really doesn't show Wednesday fans in the best light (stereotypical fat thick Northeners). But big credit to him for the milage he does and probs a decent chap to have a pint with.\nAnything \u00a310m and above we gotta take it. Good player in his time, but not for last 6 months added nothing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home brewers range from casual to obsessive. David Shreffler, of Team Science Brewing Club, demonstrates how an afternoon of brewing can be as casual as making homemade bread. Instead of a mixer, he stirs grain and water together in a 10-gallon Rubbermaid cooler with a three-foot-long glorified spoon called a \"mash paddle.\" Instead of an oven, he uses an outdoor burner commonly used for frying turkeys. Utensils lining the counter attempt to turn this residential kitchen into a commercial lab. Aromas of malted grain mix in the air.\nDavid is barefoot as he stirs the grist, checks temperatures, heats water, measures the amount of sugar in the unfermented beer and sips a brew. Today, he is working without the help of the Team Science Brewing Club, which, David explains, is small for a home brewing group. It consists of about five friends who share ideas before, during and after the brewing session. While not every member may be present for all steps along the way, they will certainly be present at the end to taste and critique each batch. David encourages others to give home fermentation a try \"because you get to make something with your hands.\" His hands remain busy for the next few hours, but his vision for future brewing includes devising a way to use \"small computers and pumps\" to make the brewing process easier, saving the heavy lifting of large kettles of liquid for his son.\nDavid Shreffler uses an outdoor propane burner to boil a batch of unfermented beer or wort during a recent afternoon brewing session.\nTeam Science and David himself represent a band of DIY'ers that would rather make their own version of an IPA than select it from the beer cooler. Their self-reliant and self-sustainable outlook joins the ranks of those growing food, fermenting, preserving, cheese-making, roasting coffee beans and distilling. Part of the allure of home brewing, as well as other DIY crafts, is the challenge to create something from scratch. Sharing the end product creates an excitement for the brewer akin to a chef plating up food for his fellow chefs.\nBrewers of all types circle around boiling kettles for plain fun. Their obsession for unique beer echoes the diverse community they gather in. Gary Grobeck, founder of the Benson Homebrewer Fest, and 20-year veteran of home brewing, speaks of the desolate beer scene before he started making his own, \"Beer wasn't nearly as good then.\" The best Gary could recall was a \"dark beer called Augsburger from Wisconsin.\" The lack of American variety led famous home brewers such as Charlie Papazian (author of the bible to the home brewing movement, The Complete Joy of Homebrewing) to create their own renditions of staples found across the globe but not necessarily in their local grocery store aisle.\nA barefoot David Shreffler uses a mash paddle to evenly disperse the malted grains throughout the water.\nThroughout the last few decades, Gary and other home brewers have fueled the interest for much more exciting varieties of beer that microbreweries now share plentifully. Gary himself has created unique brews such as a basil beer, with herbs snipped from his garden, and a chili beer that uses homegrown peppers and spices. These new makers have shown the country that beer isn't just a post-lawn-mowing refreshment but a potentially complex libation akin to wine.\nA good example of a tinkerer, a home brewer who purposefully messes with a good thing in order to blow the taste buds off his fellow competitors, is DeJuan Cribbs. A member of the Local 402 Home Brewers, he was one of the competitors recently vying behind Krug Park's beautifully hand-built bar for first place at the Benson Homebrewer Fest. On an early November day in preparation for the holidays, DeJuan was serving his Chocolate Candy Cane Porter. The velvety-smooth porter was perfumed with the beautiful menthol smell reminiscent of the white- and red-striped canes and offered notes of dark cocoa and coffee. I imagined sipping it in front of a burning Yule log as I cast my vote for DeJuan.\nTaking up such a hobby should not be left to those who avoid all modern convenience and choose to make all food and drink from scratch. Extract kits, where brewers simply boil syrups in water before adding hops and yeast, are designed to show the masses that crafty beer can be made easily on the kitchen stove. Whether or not you decide to invite a club over to help you in the process or enter your end product in a competition, you will be pursuing, as David puts it, \"an outlet for construction and invention.\" Get in touch with one of the many local brewing groups on Facebook to see the brewing process in action, or visit the new Patriot Homebrew Supply in Elkhorn, Nebraska, to start your own revolution.\nDarian Stout pours coffee slow in the morning, cooks lunch with students, makes baby food for dinner and feeds his chickens the scraps. He dreams nightly of dark chocolate, doughnuts and gelato.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With running, I'm always amazed how a weekend long run can feel so great and then a super short weekday run can be such torture.\nMy 17 miles on Saturday were awesome. It was smooth sailing for most of the run.\nLast night's five miler was terrible. So terrible, in fact, that I cut it down to a 3 miler because my legs were killing me. I thought a nice slow run would help shake things out. I couldn't have been more wrong.\nWhen I felt my left calf seizing up and my right IT band starting to pull on my knee, I knew it was time to call it quits.\nRunning can be humbling like that. Just when you think you're on top of the world and are totally ready to tackle whatever challenge you throw at yourself, a short crappy run will knock you down a few pegs and keep your ego in check.\nI don't think that's a bad thing. We all need to be reminded every once in a while that we're only human. We're not Superman.\nTo get back in the game, I'll be spending lots of time this week stretching and foam rolling to get my legs loosened up. I'll probably try something short again tonight. Yesterday was just a building block in the whole marathon training process.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is Forex Trading Legit or It Is Scam?\nForex trading per se is not a scam.\nBut there are many scams associated with forex.\nAs the forex market is opening up gradually, scams are also increasing.\nPerhaps that is the reason why you hear many questions how legitimate this is?\nIf you ask me, is Forex trading legit, my answer is yes.\nBut the question is how legit are the practitioners.\nThere are many brokers who try to capitalize on the opportunity.\nThe legal formalities are not very strict in this case.\nAs a result, the chance of a scam is fairly easy.\nIn fact, currently, scams are one of the biggest concerns in this space.\nAlmost everyone associated with forex trading will express their concern about it.\nThe primary trigger is the relative novelty of this type of trading.\nSo you have many forex brokers trying to take advantage.\nForex trading can be really profitable if you follow the legitimate way.\nIt functions pretty much like a stock market.\nYou need to have a clear forex strategy, goal and choose relevant counters.\nYou will never gain overnight.\nThe returns in forex trading are exactly like stock markets.\nA slow and steady approach will help you garner maximum profit.\nIt is important to understand some striking forex realities.\nIt is important because in anyways it shows why forex trading is susceptible to scams.\nToday the average daily turnover in forex trading is close to $1 trillion.\nCurrency futures and options have further increased the size of the market.\nIt has also made the forex market extremely lucrative.\nOn any day, you might have another 1 trillion dollar changing hands through these contracts.\nAccording to a 2009 report, international settlements totaled over $3 trillion.\nThe number can, therefore, be even higher today almost a decade later.\nSo, you can well estimate the kind of money that is floating around.\nNeedless to mention, this is what makes forex trading susceptible to scams.\nTo add to the party, the forex spot market is primarily unregulated.\nSo they can only add more fuel to anyone's interest in making some quick bucks.\nHowever, the good news is the Commodity Futures Trading Commission has played a constructive role.\nAnother self-regulatory authority, National Futures Association came into being in 1982.\nTogether they have constantly been working towards weeding out these irritants.\nThat said forex market scams are still a striking reality.\nWhat Is a Scam in Forex Trading?\nSo the basic question that will bother you is what is the scam in it?\nForex trading is legal, the brokers are legit so how do scammers operate?\nFor that, let us understand what exactly comprises a scam.\nA scam is effectively any type of deception to fleece you of money.\nWhen a person is duped, and money is taken away without their knowledge, it is a scam.\nBut this is not the same as a poorly managed firm.\nA firm can be poorly managed, but it may not be a scam.\nBut you can also have an extremely well-managed scam.\nThe problems is non-regulated brokers.\nThough forex trading has been popular since 1999, the relative regulation is poor.\nInitially, you had a huge number of brokers who opened up overnight shops.\nEspecially in countries where it is non-regulated, this created a major problem.\nHowever, the environment is sanitized a lot more now.\nThe instances of these scams are reducing with the passage of time.\nYou will notice a majority of forex brokers are now acquired by financial institutions.\nGlobal currencies are also better synced at the moment.\nWith the backing of reputed banks and institution, forex trading is a lot safer.\nSo, the chances of hoodwinking are relatively much lesser in the current situation.\nWhen you are exploring scams in forex trading, there are many popular variants.\nManipulation of bid\/ask rates on the computer has to be one of the oldest.\nNormally the spread is the extent of profit a broker earns.\nSo they are different for every currency pair.\nThe factors governing the currency rate also impacts the spread.\nBut brokers often use this spread for their convenience.\nLet's say the spread in Dollar-Euro pair is 7 pips.\nBut the broker offers only 3 pips to the customers.\nSo they stand to gain the additional 4 pips.\nWhile just on its own, it may not be huge.\nThink of the trade that is undertaken every day.\nOn a base close to $1 million, this amount can be rather huge.\nSo your profit is taken up as a commission by the forex broker.\nBut a wider internet access and stringent measures in past 10 years impacted these favorably.\nIt has reduced to a large extent.\nHowever, it will be wrong to say that they have disappeared completely.\nThe signal seller is a relatively more popular modern scam.\nThey use only one weapon to get to their target, capitalizing on human greed.\nImagine when a forex trader hears about instant gains or huge gains.\nAlmost inevitably most will fall into this trap.\nMost times, retail investors are pretty na\u00efve in this matter.\nIf someone approaches them with recommendations to become wealthy, they fall for it.\nTo make matters worse, most of the touts involved in the scam have years of experience.\nIt is needless to mention that it has just worsened the situation.\nThey have sufficient resources even to get people to testify on their behalf.\nThey will convince the unsuspecting trader about their prowess and command on the market.\nThe result, the unsuspecting trader hands over a certain sum of money.\nMost simply disappear after they collect the money.\nSome throw in a few recommendations too.\nBut there is no guarantee about who and how they were created.\nNeither is there any guarantee on how effective they will be.\nBut then, not all signal sellers are involved in scams.\nThere are many who offer genuine and worthwhile advice.\nThe trick is identifying the right candidate.\nThis is perhaps one of the latest avatars of forex trading scams: Do Automated Forex Trading Software or Robots Work?\nThe principle essentially is same.\nIt is all about working towards extracting undue benefit from the system.\nNow, most systems can generate automated trade today.\nThis is also known as robot trade.\nThey help you earn a lot of money while you are asleep.\nThis is because the forex markets are working 24\u00d77.\nBut the issue is these systems need to be verified and tested.\nIn the absence of proper verification, they can generate random numbers.\nSo effectively, you are then gambling in unknown territory.\nSo before you integrate such a system, it is important to verify.\nAlso, make sure you verify with third-party voices.\nThey will tend to churn a more realistic result.\nOtherwise, you are again saddled with an untested system backed by touts trying to sell it.\nYou have to make sure that the system you choose is legitimate.\nMake sure if forex trading is legit using these pre-programmed systems.\nAlso, the way you place orders with your forex broker needs scrutiny.\nMany times, many silent scams origin from these too.\nWe've already talked about the scam Forex brokers a lot on this site.\nSo there is no point to focus on them here.\nWhat you have to keep in mind is that retail Forex brokers are not your friends and they don't want you to make money consistently.\nIt is quite interesting to note that most forex trading scams originate from ignorance.\nAlmost inevitably, they are banking heavily on how little the traders know.\nFor example, the trading systems have traditionally been considered expensive.\nTill a few years ago, no one will bat an eyelid if you quoted $5000 for one.\nNow, this is a big scam in itself.\nIdeally, traders should not be paying anything more than some hundred dollars.\nA proper forex trading system costs a few hundred dollars on an average.\nSo if you get an offer for a really pricey system, think twice.\nChances are that it is a major scam in the making.\nPerhaps you should listen to the commentary carefully.\nDo they promise unprecedented gains?\nDo they promise instant results?\nThe chances then are you are being prepped to be duped.\nThe reality is that the forex market is not a magical paradise.\nNeither are you George Soros to land with a mega deal.\nSlow and steady gains are at best the most practical expectation.\nBut that said, there are many genuine trading system sellers.\nYou can use these for reasonable income over a point if time.\nSo the only way you can avoid these scams is by staying alert.\nIt is never wise to fall for promises of instant gains.\nThat is by far the easiest trap.\nAlso keep in mind that trading systems are not magic bullets.\nThey don't work under any conditions.\nIf you ask me is forex trading legit, I will agree.\nBut the point is that does not mean that you trust your broker blindly.\nIt is always wise to question every additional penny that you part with.\nThat brings to limelight the other common type of forex trading scams.\nIn technical terms, it is referred to as the commingling of funds.\nIndividual traders cannot track their exact investment without segregated accounts.\nAs a result, retail firms often misuse this opportunity.\nThey use the funds to pay huge salaries to their employees.\nThis includes large bonus, cars, houses and what not.\nSome are even less sophisticated.\nThey conveniently disappear with the additional customer money.\nHowever, the Commodity Futures Modernization Act of 2000 played a crucial role.\nThey created some stringent measures to segregate brokerage accounts.\nThis empowers clients to opt out of strategies where they are not comfortable.\nHowever, this Act does not have a global bearing.\nSo, if you plan to undertake forex trading elsewhere, read the regulation.\nBe very clear about what or how you are getting into before committing cash.\nThat will make sure that you are making an informed choice.\nAdditionally, you can easily identify doubtful transactions instantly.\nIf you want to capitalize on legit opportunities in forex trading, be alert.\nYou must watch out for warning signs for any potential scams in the making.\nIf you see brokers not allowing withdrawals from investor accounts, it is a sure sign of trouble.\nAre you constantly facing problems with your trading station?\nAre there too many trade glitches lately?\nCheck if you can access your accounts before or during an economic announcement.\nRemember these are the times when the forex market is susceptible.\nThere can be massive pip movement in a short time.\nInevitably the volatility increases too.\nForex trading is often about capitalizing on these opportunities.\nIf you are not able to capitalize on these, there is surely trouble in heaven.\nDon't be buoyed by promises about great returns.\nYou are the best judge of your interest.\nDoes the brokerage satisfy your liquidity or trading expectations?\nI will always be wary of too many promotional literature promising huge gains.\nIt is interesting to note that 193 cases were registered by the NFA in 2008.\nMost of these were pertaining to this type of trading irregularities.\nWhile 166 of these cases were resolved in less than 9 months, only 23% received lost funds.\nSo preventive actions are very important for capital preservation.\nYou can never be sure of receiving your lost funds on lodging a complaint.\nSo, it is best to be careful about the ground you step on to.\nTherefore, a background check is a necessity in every forex trading arrangement.\nYou must be very sure about how your broker operates and what the firm is about.\nIn this context, the Background Affiliation Status Information Center system plays a key role.\nThe best part is the introduction of these measures has deterred many crooks.\nBut the system is not sanitized entirely.\nHowever, the temptation of huge profit is very lucrative.\nIt is hard to assess who or how one falls for it.\nTherefore you need to be careful about the forex trade brokerages.\nThere is never any guarantee of what new type of method they introduce.\nDon't wait to get duped.\nIt is always important to take a strategic viewpoint on the promises that are being made.\nLet's face it; whether you are into forex trading or stocks, overnight gains are mythical.\nIt is always better to look for steady gains.\nSo always be careful about potential forex broker scams.\nThe question is how you can avoid a scam in forex trading?\nThe first step is checking the authenticity of the broker.\nAlways go for a broker who is a tried and tested name in the investment world.\nThe advantage then is you have enough number of people verifying the authenticity.\nCheck where this brokerage is based from and the response rate.\nLook for potential complaints against the broker.\nHas there been any complaint about withdrawing firms?\nRead the fine prints of the agreement carefully.\nIt is very important to be aware of all the contingencies planned.\nPay special attention to the incentives and withdrawals.\nYou must make sure that they are in your favor.\nDon't always rely on the size of the brokerage.\nIt may not be such a sure indicator of the broker being genuine.\nAlways maintain steady communication with the broker.\nIf you fail to do that, raise the warning bells.\nBe stern with your broker and don't lose focus on your trade.\nThat is the best way to make sure you gain from forex trading.\nSo, Is Forex Trading Legit or It Is Scam?\nFortunately, regulations have increased significantly in the past decade.\nAs a result, the business environment has also improved.\nAnother development is the cost ratio for starting a forex trading firm.\nAs a result of these new and enhanced rules, it has become relatively expensive.\nThis is a big dampener for many fly by night operators keen to scam people.\nBut the ground rule is quite simple.\nWhenever you are choosing a broker, you need to check their details.\nLook at how long the firm is doing business?\nThe promoters and the key operators' list is also very important.\nAlso, pay attention to the average number of subscribers.\nIn case you feel you have been scammed despite this, contact the authorities.\nEvery country has a designated regulating authority.\nThey look after all forex trading policies and potential breach.\nIn fact, the interesting fact is most of the forex market regulation were formed that way.\nForex trading has never been structured.\nBased on the user complaints, authorities have been continuously altering it.\nSo the fundamental fact is forex trading is indeed legitimate.\nBut you have to be cautious about scams.\nSo don't question is forex trading legit, instead check if the broker is legit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Reservation was easy and straightforward. I called the Reykjavik office (Kristin? KS)when our flight to Iceland was delayed by one day, and was assure that our car would be held for us. Kristin followed up with a reconfirmation e-mail for the revised pick-up date minutes after the call ended. Check-out was simple. We were given a 2018 Nissan Quashqai 4x4 with less than 3.000 Kilometers in virtually showroom condition. Returning the car 2 weeks later was equally simple. Anita checked the car back in, signed off on the rental form, and we were on our way. I could not have asked for a better experience! Thank you! If we return to Iceland, Reykjavik Rent A Car will definitely be our choice again.\n- Renter from United States. Rented Nissan Qashqai | Manual.\nThe company has an office right outside Keflavik airport. There is a free shuttle bus fm the airport to the office. The stuff is friendly. I had no problems w\/the car whatsoever (considering the fact she was brand new and only had 15 k km run). The technical support and gas discounts work perfectly well. There is a possibility to pay by debit card (that is IF you purchase platinum insurance which, by the way is a wise idea, considering the road conditions). One word of advice to future travellers - small car may not be sufficient for gravel roads regardless if they are F marked or not. Nevertheless we used a Suzuki Swift and she had managed just fine.\n- Renter from Ukraine. Rented Suzuki Swift.\nEverything was possible from the hiring to the delivery of the car. We are very satisfied. If you want to ask Iceland correctly, we recommend taking the highest insured amount. It's a bit more expensive since your on the safe side Kind regards Kiryl & Patrick.\n- Renter from Germany. Rented Suzuki Vitara.\n- Renter from Germany. Rented Suzuki Vitara | Manual.\nGood level of service. Nice contact. Very professional. We only regret the age of the GPS delivered (needs an update). We will come back.\n- Renter from France. Rented Land Rover Discovery Sport | Auto.\nGreat place, polite staff, reliable cars. On Iceland it's better to rent from local rent-a-car agency such as this one than from more expensive franchise.\n- Rented Suzuki Jimny | Manual.\n- Renter from UK. Rented Hyundai i10.\n- Renter from Italy. Rented Suzuki Vitara | Auto.\nEverything went perfect. Our Jimmy (Yes Jimny was our friend so that's why we gave him better name - Jimmy) works nice. 2.200 km without any problems. Thx.\n- Renter from Poland. Rented Suzuki Jimny | Manual.\nGood service, good location , good price.\n- Renter from Ukraine. Rented Dacia Duster | Manual.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Arin Crumley is the director of Four-Eyed Monster and now a co-founder of a movement (OpenIndie) where filmmakers can figure out how to get their films screened in a ton of cities. Arin premiered his film at the same SXSW that we premiered our film Straight Line. It's been really fun to see Arin become such a big indie figure since then. He is building a community of filmmakers to help get indie films some screenings. Check it out. I'll be joining. Where's my credit card? Click here to pledge and become a member of the site. You'll be in good hands!\nWe've past the half way mark towards our funding and I've been reading the amazing discussion taking place on our campaign comments and saw that already we are getting great input for how the site should function from our backers.\nIn addition to Kieran and I's direct responses I wanted to make a post to say that the point is this, the possibilities are endless. Kieran is a computer programmer with over 10 years of experience building similar infrastructures. So we can all bring our ideas to the table and make this thing do anything we want it too. Also I've been planning how this could all function for over 4 years by constantly envisioning this future system every time I made a new distribution move with Four Eyed Monsters. There are pages and pages of notes and ideas and sketches and drawings and concepts for how I see a new film industry functioning and OpenIndie.com is designed to be a huge part of the missing link.\nBut even with all of these ideas we have, it's the backers who will aid more then anybody the actual shape of this new service. First of all you guys bring money to the project which is huge. And secondly you are our base of filmmakers and users that will inform us how the system should be developed. The plan is to regularly update the backers with what the sites components are looking like as the get built. Comments about each update will play into the development and inform the direction we take phase 1. And then of course there will be phase 2 where we all get to try out the new system and learn what can improve.\nSo we are building the future together. What your donation means is that we are all going to make this happen together.\nOf course this only happens if we fund. So please, spread the word. Reach out to a few fans of your work, see if they'll back even just 1 dollar. And if you know feature or short filmmakers, reach out to them as well. Share with them your reasons for backing this project and invite them to do the same.\nAlso, I've noticed many filmmakers pledging 20 dollars, I'd strongly encourage you to increase your pledge to 100 so that your film can actually be one of the 100 films that gets the free 1 hour of consulting and gets to use the service of the site. Kickstarter makes it very easy to increase your donation. Just hit the button and you can type in any number during the time frame that the campaign is live.\nIt's great to see filmmakers alerting other filmmakers to this new method we can make possible with OpenIndie.com.\nAnd it's great to hear you'll be joining. We've all come a long way from straight line, puffy chair, kissing on the mouth, mutual appreciation and all the others films that played at SXSW 2005.\nYes, we have come a long way! Nice to hear from you Arin. SXSW 2005 was a great year. So glad to see you doing so well. Sean says hi too!\nI do plan to join this week! My other film premieres tomorrow. Cheers!\nI pledged $100, and am spreading the word! Thanks for this gem!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a neat item - it's a genuine fossil tooth from an extinct mackerel shark, Otodus obliquus, which lived up to 60 Million years ago! Otodus had a pointed snout, nearly symmetrical tail, and a reputation for aggressiveness; it may have grown up to 30 feet long!\nTHE TOOTH PICTURED IS NOT THE ACTUAL ONE YOU WILL RECEIVE IF YOU PURCHASE THIS ITEM, BUT IS AN EXAMPLE OF THE SIZE AND QUALITY OF THESE FOSSILS. These are our \"B\" grade teeth so they may have repairs, missing side cusps or tips, or other imperfections, but these are genuine fossils at a reasonable price! They are great for kids and for educational purposes!\nThe tooth you receive will measure approximately 1 3\/4 to 2 1\/4 inches in length. Included with the fossil is a nice color information label.\nCHECK OUT OUR OTHER AMAZING FOSSIL TEETH!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lukas Watznauer has been modelling since the end of 2013 and one look at his Instagram page tells you that he is loving every minute of it!\nThe classically handsome Spaniard recently spent time in Turkey during which he shot editorials for magazines including Turkish GQ, Esquire and Vogue.\nFitness Regime: Day 1 \u2013 Chest & Back. Day 2 \u2013 Arms, Shoulders. Day 3 \u2013 Legs & Abs.\nDo you have any Guilty Pleasures?: When I have a free day I sleep until noon.\nWhat's the funniest thing that has happened to you on a shoot?: Once I did a shoot with a girl and a parrot. I got the parrot on my shoulder and suddenly it started to poo on my shoulder. I think he felt a bit uncomfortable but we laughed so hard!\nWhat was the last dream you remember?: I was riding my motorcycle when a police officer stopped me and stole my bike!\nDid you like Lukas Watznauer?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"None of the opinion polls had predicted the historic regional election result in Andaluc\u00eda last Sunday night. A sharp decline in support for the ruling PSOE Socialist party and a drop-off in votes for the left in general gave parties to the right a majority and the chance to wrest control of the Junta de Andaluc\u00eda regional government from the PSOE for the first time in 36 years.\nThe new, joint electoral platform of left-wing parties including Podemos and Izquierda Unida failed to improve on both their individual performances in the last regional elections in March 2015.\nWhile encouraging street protests against the rise of Vox this week, Podemos's national leader, Pablo Iglesias, urged Adelante Andaluc\u00eda leaders, Teresa Rodr\u00edguez and Antonio Ma\u00edllo, to woo Ciudadanos towards supporting the left in an investiture.\nWith almost 28 per cent of the vote, Susana D\u00edaz's Socialists are still the largest party but, with just 33 seats in the 109 regional parliament, they are a long way from an overall majority in Andaluc\u00eda, seen as a traditional bastion of PSOE power.\nThe other main news of the night was the shock emergence of far-right party, Vox, as a parliamentary force, winning 12 seats, its first in any part of Spain, with 11 per cent of the vote. Observers said that it was the country catching up with an international trend in populism, although large demonstrations have been held against the party's policies in cities across the region this week.\nNot even Adelante Andaluc\u00eda, the left-wing alliance of Izquierda Unida and Podemos, could offer its support to the PSOE and keep Susana D\u00edaz as president of the Junta de Andaluc\u00eda government. Teresa Rodr\u00edguez's group lost three seats, down to 17 with 16% of votes.\nCentre-right party, Ciudadanos (Cs), under Juan Mar\u00edn, more than doubled its seats to 21, with 18 per cent of the vote, and was seen as the success story of the night, close to PP and a rival to lead the Junta.\nAlthough the conservative Partido Popular (PP), under Juanma Moreno, lost ground, finishing on 26 seats and 20.7 per cent, its second place position immediately gave the party hope that it could lead a coalition of the right to take power from the PSOE in the region after four decades.\nSusana D\u00edaz's future was still unclear this week, although after hesitation, her national party backed her. D\u00edaz had called the early election after Ciudadanos withdrew its support for her minority control of the Junta de Andaluc\u00eda government, which controls big-spending services such as health, education and social care. Until Sunday's result, polls had shown that she was on course to probably form a government again.\nThe total of MPs for the PP (26), C's (21) and Vox (12) is 59 and a sufficient majority to keep out the PSOE. But a pact needs to be negotiated ahead of a parliamentary investiture debate due by the end of December.\nThe second most-voted group was the conservative Partido Popular, which has lost seven seats, but, with the support of the other right-wing groups, could take the region away from the Socialists for the first time ever.\nCentre-right Ciudadanos increased its representation by 12 seats to a total of 21.\nFar-right newcomers Vox gain representation in the Andalusian parliament for the first time with a surprise result of 12.\nThe left-wing group Adelante Andaluc\u00eda (uniting Podemos and Izquierda Unida, among others) won 17 seats, which, combined with the PSOE's 33, fail to reach the 55 seats needed for an overall majority.\nThe turnout was 58%, significantly lower than in 2015.\nThe results in Malaga threw up a three-party tie. The province, which includes the western and eastern Costa del Sol, the city and inland areas, has a total of 17 seats in the regional parliament. On Sunday Malaga voters gave the three parties PSOE, PP and C's four seats each, with three going to Adelante Andaluc\u00eda and a surprise two to far-right Vox.\nAs in the region as a whole the Socialists won the most votes (24.2%), followed closely by the PP (22.6%) and C's (19.8%). The left-wing alliance Adelante Andaluc\u00eda gained 15.6% and Vox 11.5%.\nThe PSOE won the most votes in 69 of the 103 municipalities in Malaga province (down from 82 in 2015), although the PP won the election in the main Costa towns, with the exception of Marbella, where the PSOE won 500 votes more than the PP, and Malaga city where the Socialists gained the most votes, followed by Ciudadanos, relegating the PP to third place. Ciudadanos won more votes than any other party in four towns, Rinc\u00f3n, Alhaur\u00edn de la Torre, Manilva and Benalm\u00e1dena.\nFar-right newcomers Vox gained 72,455 votes in the province of Malaga, their best result in percentage terms being in Benahav\u00eds, with nearly 20% of the total number of votes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We're always looking for quality individuals to join our crew! We offer pilot applicants a fast track to valuable Captain experience that leads to your dream aviation career quickly. There is also room to advance your career in management positions within FRE\/ACE\u00ae.\nNot a pilot? We have plenty of opportunities for you to take to the skies as a flight attendant or help us on the ground as a mechanic or flight coordinator. No matter the position we can offer you highly competitive pay, comprehensive training, and a great work environment!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjvsb b/data_all_eng_slimpj/shuffled/split2/finalzzzbjvsb
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+{"text":"Feathers Badminton Club (Banbury, Oxfordshire, OX16 9DQ) -- Good Standard Competitive Club, Meeting Wednesday Nights 7pm to 10pm between September and April, Feather Shuttles used for Club Nights and Home Matches.\nLarkmead Badminton (Abingdon, Oxfordshire) -- Friendly group with varying ability who play doubles games purely for fun and sociability.\nWantage Badminton Club (Wantage, Oxfordshire, OX12 9BY) -- The Club plays at Wantage Leisure Centre on Wednesday evenings 8.30 - 10.30pm and welcomes new members of all ages from 6-60 for both recreational and league badminton play.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"- Your China Suspension Expert, Alibaba Highly recommended supplier.\nOur suspension arms and coilover kits are widely used on Nissan, Honda, Toyota, BMW, Mazda, Subaru, Suzuki, Jeep Rubicon, etc..\nSample can be sent out next day upon request.\nPlease send email for sample requirement.\nFor urgent order, we can make it faster. Please send email to negotiate.\nWe work with the most professional forwarders in the world and we share our large shipping discounts with all our customers.\nWe usually acceptT\/T (wire transfer), PAYPAL, Western Union, Escrow.\nOther terms like MONEY GRAM, L\/C, D\/P, D\/A, O\/A can be negotiated.\n1.As a direct manufacturer and also a trading company, we are able to have the best resources integrated in China and offer our clients qualified parts at the most competitive price.\n2.Quality Assurance: 100% QC inspection before delivery; ISO9001:2008, and SGS certified; Full refund or replacement for quality faulty; 90 days quality warranty for ALL products.\n3.Excellent customer service, responded within 24 hours.\n5.We provide OEM service and can make custom parts according to clients' prints or sample pieces.\n6.With 10 year's exporting experience, we've worked with customers worldwide (NORTH AMERICA 80%, EUROPE\/11%, Oceania\/5%, RUSSIA\/2%, South Korea\/1%, Japan\/1%).\n7.Monthly production capacity over 200,000pcs, one of the three largest in our field in China.\n8.Cheapest Shipping: the price we enjoyed is the cheapest in Shanghai, \"84% off from FedEx\" and \"80% off from DHL\".\n9.One-Stop service: we can take care of everything from production, shipping, import clearance, door delivery until your order is right there in your hands.\nThank you for you valuable time. Sincerely hope to cooperation with you.\nAir suspension compressor,air suspension pump.\nUsed on 2005-2011 Land Rover Range Rover LR3 LR4.\nhigh quality with competitive price .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Build a world where people and nature thrive!\nYou must be the change you wish to see in the world. You have the power today to change tomorrow!\nWho we are? \u00c9cologie Engo Solutions is a leading provider of environmental services to commercial and nonprofits working together for a healthy planet. Together, with you, we are determined to change the world for good, creating a cleaner, healthier, fairer world for everyone, for today and generations to come.\n\u00c9cologie bold planis to curb carbon dioxide emissions around the globe.\nFor the 2018 International Coastal Cleanup in September, we will organize groups of volunteers to document and remove debris at a beach near them, as well as learn about the coastal environment.\nThis project rescues, rehabilitates and release many species of animals including large mammals like deer.\nPeople who need to fish for a living are catching turtles in their nets and either injuring them or killing them.\nA quiet crisis playing out in US forests as huge numbers of trees succumb to drought, disease, insects and wildfire.\nYou have the power to protect our Earth. Your support can help save natural resources today and for generations to come.\nOne pair of hands is all we need to make a difference in the world. Find volunteer opportunities near you! Save The Planet!\nWe're taking our resistance to the next level. This summer, activists across the country are standing up against injustice!\nWe are proud to announce the launch of TREES Texas, a new and exciting way for people to easily and quickly give back to the environment with a tangible local impact!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Preparation is the key to controlling those jitters, and we've got a wealth of good advice for you.\nSo you found the job you want, you've devised a winning CV and you've been called up for an interview! Exciting! But also frightening? If you've been to plenty of interviews or if you've never been to one before, don't worry \u2013 it's ok to be nervous.\nPreparation is the key to controlling those jitters, and we've got a wealth of good advice for you. Start here with the kind of questions you can expect, and then move on to wondering what you should wear and how early you should be. No matter how snazzy you look on the day, the way you answer their questions will be the key to your success.\nWait, are you really ready? Do you know what you're wearing?\nThis should be easy to answer. After all, you wanted it enough to study towards the role for several years, apply, dress up and arrive at the required time, so be honest with the interviewer and tell them what you liked about the job ad, and what you think you will gain from the job. If your impression was horribly wrong and you aren't hired then surely that's only a good thing too?\nExample answer: I was excited by the title because it fits in exactly with my area of study, and I thought that a company of this size would be a great one in which to grow my experience while using my knowledge and skills to their best advantage.\nDescribe a situation where you worked in a team?\nThe ability to work as part of a team is important in almost any job you'll ever have, and employers want to know that you can handle the effort it takes to communicate, delegate and accept the responsibilities of team work.\nExample answer: At university we were often put into groups and tasked with presenting a project, and I learned many valuable skills in those situations. I also learnt that I prefer to avoid conflict, but that I am happy to take on the responsibility of designing and submitting the final presentation.\nYou've probably already been told to give an example of a weakness which is actually a strength, such as being a perfectionist or a workaholic. But employers are on to that one, so instead be honest but also describe how you've worked (and continue to work) to overcome this flaw.\nExample answer: I struggle with confidence sometimes, and have found myself speechless in group situations. I realised that this was not going to aid my career growth so I started to work on this aspect while at university. I took advantage of social clubs and events, and made myself join the debate team so I would learn how to speak in public. It's an ongoing process but I'm proud of my progress.\nWhy did you choose to study that degree and why at that university?\nThe interviewer is listening for your evidence of decision-making, planning and logical reasoning skills. So even if you just choose the closest or cheapest university, talk about what you like about the university, the courses offered and bring up ways in which the university is unique.\nExample answer: I choose to go to this university because it was close to home, meaning I could rely on family support which enabled me to focus my attentions on my studies. I also appreciated that the university is one of the top for offering excellent classes in the qualification I completed.\nThese are only a few of the many questions you will need to be ready to answer, but check back here for our next article in the series, coming soon \u2013 we're here to help.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Racism is one believing their race is superior to another and them using that belief to make other people's lives harder.\nThat Eddie bravo video is in poor taste and highly disrespectful. I don't know what to say about people using cultural stereotypes to make fun of a race as a whole except that it's cringey and I don't like it. I dont want to see a person of a different race dressing up as a \"ching chong Chinaman\" \"lazy Mexican\" \"redneck hillbilly\" \"crazy extremist Muslim\" etc etc I am all about laughing with me not at me and with this sort of humor at what point does it become laughing at me and not with me? It would be nice if we could all laugh at each other without getting offended but I just don't think that's possible in this political climate.\nI will take it one step further. I wonder if we have racism in our dna as a defense mechanism. A sort of Hatfield vs the mcoys hatred that gets ingrained in our dna because of past wars\/ skirmishes between groups. Blacks vs whites Mexicans vs whites Turks vs Greeks Muslims vs whites Jews vs Muslims etc etc etc once we can get past these grudges perhaps we can better move forward as a society.\nBack when there used to be a half-dozen large hominid species roaming around, it was probably important for humans to \"stick with their own kind\" and avoid socializing a lot with these non-homo sapien creatures. Most creatures already have a biological imperative not to mix with other species, and we aren't all that different. So we reinforced that biological idea of not mixing with \"the other\" and added a social layer to it, to make is a sin or illegal or what have you. And that rejection of the other is part of racism.\nThe other part is what sociologists call the \"Matching Hypothesis,\" which indicates that people find other people physically or sexually attractive if they look kind of alike. No so alike that they would be your close relatives, though.\nSo we have a couple of very strong biological drives pushing us away from people who are different from us, and drawing us toward those who are similar. That's going to lead to some racism.\nOf course, we don't have to act on our instincts, and we can certainly get along with other types of people as long as we have a good reason to, like gaining something through trade. But we have disliking other, different people in our blood. It's hard to overcome.\nIn my opinion all it takes to be racist is to accept the concept of race as anything other than bull#. If you believe in race you are a racist because simply by holding that belief you've already divided people along superficial irrelivant lines.\nI will never ever buy into the \"refusing to acknowledge race\" makes one part of the problem bull#. It's the refusal of others to take that very same step that's part of the problem.\nAnyone who refuses to acknowledge race as bull# is either ignorant or has something to gain by it.\nNo law and no policy can be made that acts along racial lines without inherently being racist as the mere fact that it does so it both supports, emboldens and provides legitimacy to the bull#.\nYou raise an excellent point, and likely a valid point, though like so many other characteristics and qualities, it is stronger or weaker in different individuals.\nI think if the genetic tendency is reinforced by racist or hateful upbringing, then it can play out more obviously.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Essay with thesis statements jobs health essay writing words pdf. Writing about nature essay paragraph structure greek art essay definition. Discrimination short essay disability photography photo essay kahirapan brainly.\nProfessional education essay qualification internet essay short youth day creative writing teach it quotation marks writing english research paper formats family pets essay junior essay citation format government publication. Essay writing personal experience leaving cert essay narrative topics zoho. Advantage of mobile phone essay revolution write essay online cheap your admission essay sample report pt3 english essay with thesis statements jobs my mother school essay graduates, essay writing on creativity imagination.\nNeed of english language essay variety professional achievement essay for scholarship block letters essay travel essay ??????? light the disaster essay kernel topics for an essay toefl cite on essay deforestation pdf what is a plant essay brother essay on commercial banks functions perfect essay example rebuttal mexican food essay hookah housing problem essay garbage essay examples for application desktop publishing.\nFrench essay structures competition what is quality essay successful marriage about policeman essay kitchen equipment. Essay on graffiti vandalism car insurance about twins essay nature's beauty essay about contracts online shopping pdf block letters essay my personal computer essay possession essay writing about my home plans culture in my life essay ambition wuthering heights essay film 2009 review. Introverts and extroverts essay listening.\nDream places essay community effect essay samples for 5th grade report research paper example pdf mla? essay social responsibility jobs san francisco. Feedback for dissertation m.ed on the other hand synonym essay (family essay in spanish business) about twins essay nature's beauty essay about programming nelson mandela day essay structure and tips opinion perfect essay example rebuttal.\nResearch paper business topics in hindi research worldwide paper kodungaiyur internet problem essay topics list . Writing bank essay wikipedia euthanasia for and against essay pro introducing persuasive writing ks2.\nTalking in class essay essay research paper management system gps tracking. Make notes essay sentences writing dissertation findings by step-by-step guide. Writing sounds essay uk cheap.\nResearch paper term helpers. Essay social responsibility jobs san francisco write a good college essay quality reference format paper research owl purdue research paper references format nhd video argumentative essay examples college board. Essay social problem greatest essay introduction writing for beginners 1 parts of paragraph essay dbq essay about language barrier linguee topics proposal research paper gun control essay on homemade food for soulmate essay on study deforestation and afforestation. Essay social responsibility jobs san francisco example of critiquing a research paper my motorcycle essay kitesurf (essay about academics books vs movies) writing academic english essay york body essay writing with outline sample writing bank essay wikipedia about twins essay earthquake in english. Essay scholarships tips leaving cert history deceptive appearances essay keeping up. Essay introduction writing for beginners 1 travel advantages and disadvantages essay newspapers.\nEssay writing samples essays ibt essay about college admissions ucf. The examples of academic essay rhetorical classroom experience essay best holiday about law essay equality in india. Essay topics of ielts sample answers essay about family problems day outing review on essay writing services dubai.\nEssay about brands discipline in malayalam research paper contents meaning of life studying at home essay work my favorite gadget essay music essay plans example vce. Essay chinese culture upsrtc ielts essay topic on media education.\nSlavery essays in american best software research paper rubric.\nessay writing about my home plans culture in my life essay ambition wuthering heights essay film 2009 review. Introverts and extroverts essay listening.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Led Panels | XESS Corp.\nMy Twitter friend, Jeremiah Johnson, sent me a couple of LED panels in January. These things retail for about $40 each, so I thought I should at least develop a driver for him.\nFrom the picture, you can see the panels are 32 \u00d7 32 arrays of RGB LEDs. Ribbon cables are used to concatenate panels to form larger displays. There's a total of 2 \u00d7 32 \u00d7 32 \u00d7 3 = 6,144 red, green and blue LEDs on both boards which can supposedly take up to 4 amps of current, but I got away with using a single 1A power adapter.\nThe panels don't contain a frame buffer to store an image: the LEDs have to be continuously refreshed. This is done by shifting-in a sequence of bits to activate selected RGB LEDs in a given row and then latching those bits to drive the LEDs while another set of bits is shifted in for the next row. An entire image is created by moving rapidly from one row to the next. And if you adjust the proportion of how often a particular LED bit is on or off, you can also display colors of varying intensity.\nNaturally this takes a fair amount of bit-twiddling over a short period of time, but that's a perfect job for an FPGA. The block diagram for the FPGA-based LED panel driver that I built is shown below.\nEach word of RAM is divided into three bit-fields which each hold the intensity value for the red, green and blue LEDs of a pixel. The bit field values are compared to a threshold and the associated LED is turned on if the bit field value is greater than the threshold. The intensity of the pixel colors is varied by scanning a line of pixels multiple times while ramping-up the threshold. For example, a red LED will be on twice as much (and shine twice as brightly) if its bit field has the value 42 as compared to when it stores 21.\nThe LED panel is divided into top and bottom halves, so the RAM stores pixels for one half in the locations with even addresses and pixels for the other half at odd addresses. The sequencer outputs odd and even addresses on alternate clock phases. The RGB bits for the pixel in one half of the panel are stored temporarily while the bits for the pixel in the other half are generated. Then both sets of bits are sent simultaneously to the panel on the falling-edge of the clock so they'll be stabilized before the rising edge of the clock shifts them into the panel.\nThe sequencer also generates the latching signal and selects the active row of the display. Because several clock cycles are needed to fetch pixels from the RAM and do the threshold comparisons, the latch and active-row signals have to be delayed a few cycles to get them back in sync with the RGB signals.\nThe sequencer also solves a timing problem that arises when it comes time to change the active row of the panel. If the row is changed before the latch is activated, then the pixels for the previous row will appear faintly in the new row before the new pixel values are established. Likewise, if the row changes after the latch signal, then the new pixels will appear briefly in the previous row before they show up in the desired location. Finally, if a pixel is on but then has its driver turned off, the charge residing in the wiring capacitance can partially illuminate other pixels in the same column. These are examples of ghosting. Since a row of pixels is scanned multiple times while ramping the threshold to create varying colors, ghosting can be eliminated by doing one final scan where all the pixels are off (or \"blanked\") before the row address is changed. While this leads to a slight diminution in the overall pixel brightness, it's much less noticeable than the partially-on pixels that occur with ghosting.\nThe circuitry can resize itself at compile time by setting a few generic parameters that describe the width and height of the LED panel and how many are strung together.\nThe dual-port RAM is done using VHDL inference instead of instantiating a vendor-specific block RAM. That makes it easier to port this module to other FPGA families.\nThe VHDL file also contains a version of the LED panel driver with a Wishbone interface. This makes it easy to attach as a peripheral to the ZPUino softcore processor. The combined ZPUino plus LED panel driver can be loaded into a XuLA2 board that is then inserted into a Logic Pod. The Logic Pod provides a convenient connector for mating with the cable to the LED panel; the voltage translation capabilities of the Logic Pod are not needed since the LED panels accept 3.3V logic levels.\nI built the driver starting from the pinout and my best guess as to how the circuitry expected the signals to toggle. Big mistake. I ended up rediscovering all the ghosting problems that others had already found and solved. Next time: read the blogs!\nI originally used a random number generator (RNG) for the threshold. I figured that creating a uniform distribution between 0 and 255 would give the same effect as a linear ramp and would require less sequencing and circuitry. Unfortunately, I noticed a lot of sparkling and correlation effects in the pixels even when using an eight-bit slice of a 32-bit RNG.\nThe output-enable control input for the LED panels is practically useless. I tried to use it to eliminate ghosting by disabling the outputs when the row was switched, but it had no effect. I ended up just enabling the panel all the time by pulling this input to ground.\nPlease, what application do you use to make the nice drawings.\nHi, William. I use the FOSS vector-drawing program Inkscape.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nose trimmer. 5in1 epilation tools. Wholesale razor gold. Black&golden. Wholesale 110v c2h67a. Remove hair face women. Body part: 11.5x2.5. Wholesale lotion spray set. 5d shaving. Cutting machine stainless. Switch: Charger type: 12*3.5cm. Flawless hair remover.\nSn206. Metal + plastic. Number: Bikini ladys. Abs+titanWholesale gliteres nail art. Micro hair tool. Photon tender skin. 2 usb motherboard. 4in1epilator02. 2*aaa battery(not include). Eyebrow shaper trimmer. 160mins. Electric portable clothes lint pill fluff remover. Cleaner blackhead. Blue \/ pink. Incubator egg controller.\n1x aaa batteries (not included). 600mah ni-cd battery. 3 in 1 shaver. Multifunction ipl laser hair removal epilator. 10.2*10.2*1 inch\/260*260*27mm. 5\u2103~35\u2103. Treatment area: Small. Depilation site: Ship from: Ce approved home use multi-function ipl laser hair removal machine. Product input voltage: Lanthome orgasmic. 13.5*19.5*4.5cm. 15.80 x 6.00 x 23.90 cm. Painless touch laser epilator. 2.5x2.5x13cm\/1 x 1 x 5 inchesKm-1107.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lock in a great price for Lennox Ushuaia \u2013 rated 8.2 by recent guests!\nWe were detained in Ushuaia when we finished our boat tour to Antarctica due to the airline strike. We like everyone else were pressed for accommodations. Through Booking we were able to get a room for 2 days while we waited for the crowds to get moving again. We liked the hotel. The staff was very very helpful and so very polite. We actually enjoyed our couple of extra days. It gave us an opportunity to explore the city. It is very like Sitka Alaska. We are from Fairbanks and enjoyed visiting with the folks on the street and in the businesses.\nWithin close walking distance of the commercial pier and waterfront. Boutique hotel feel. Reasonable breakfast included with our room rate.\nLennox Ushuaia This rating is a reflection of how the property compares to the industry standard when it comes to price, facilities and services available. It's based on a self-evaluation by the property. Use this rating to help choose your stay!\nOne of our top picks in Ushuaia.Located a few steps from the Beagle Channel and on Ushuaia's Main Avenue, this boutique hotel offers elegant accommodations with free internet access, balcony and complimentary breakfast buffet.\nRecently renovated, rooms at Lennox Ushuaia feature warm decor and design, featuring a mini-bar, safe, and a private hot tub.\nWidely considered to be the world's Southernmost city, Ushuaia is an ideal base for exploring the island of Tierra del Fuego and the beautiful region of Patagonia.\nWhen would you like to stay at Lennox Ushuaia?\nAn air-conditioned room styled in elegant earth tones and featuring a spa bath and a dining area. It may accommodate 3 beds or a double bed and a single bed.\nThis room cannot accommodate and additional bed.\nCozy decoration, care and attention to details accompanied by the perfect location, will provide guests an unforgettable stay.\nHouse Rules Lennox Ushuaia takes special requests \u2013 add in the next step!\nLennox Ushuaia accepts these cards and reserves the right to temporarily hold an amount prior to arrival.\nThis hotel needs work and only deserves 3 star rating. The lobby furniture is old and shabby looking. Our first room was on the west side of the hotel which was very hot in summer with no AC. The wall paper was peeling up and in need of replacement. We were kindly moved to a more updated room that was nice.\n-- Because we were in Ushuaia for a family emergency and rooms were scarce during high season, we had to move around and ending up staying at 3 different hotels and sometimes had to change rooms within the same hotel. We were booking last minute, so may not have gotten the best possible prices. The staff at all 3 hotels were incredibly accommodating and kind about our situation. -- All 3 hotels are clean and comfortable and the included breakfasts are about the same (coffee, juice, cold cereal, ham & cheese, and many types of bread, most of them very sweet). All have triples available (one queen bed plus one or two twin beds) that can really bring the cost down, though they will charge more for 3 people in a triple than for 2 people in a triple. Of the 3, I'd choose Hotel Tierra del Fuego first if I could get a room with a view of the water, Hotel Cap Polonio second for the price, and then Hotel Lennox. I'm pretty easy to please; others might prefer a more upscale place like the Lennox. Hotel Lennox - Nice place, perfect location. (It's on San Martin, the busy main street full of restaurants and shops.). The breakfast room has big windows and a beautiful view. Most expensive of the 3 but definitely a step up from Cap Polonio: nicer decor, quieter feel, 2 large elevators vs. 1 small one, and more reliable wifi in the rooms. Hotel Cap Polonio - 2 doors down from the Lennox. Cheapest of the 3, with a slight hostel feel. Wifi in the rooms wasn't great. The big and lively lobby \/ cafe \/ breakfast room has good wifi and plenty of power outlets. Note that rooms at Cap Polonio that face the street can be noisy; I assume that's true at most of the hotels on San Martin (and there are many), but our room at the Lennox was toward the back and was quiet. Hotel Tierra del Fuego - on the next street parallel to San Martin but up a steep hill. The street is more deserted than San Martin.\nGood location, near to pier.\nThe beds, very narrow and unstable very cheap mattress.\nWe could have gotten on the plane a few days earlier but the hotel was great. No negative thoughts at all on our accommodations or the food.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"THE HOME YOU WANT: We want you to find the home that's right for you based on your own individual wants and needs.\nNEW HOME BUYING: We help our clients determine a fair offer to make on new homes.\nCURRENT HOME VALUE: We have licensed appraisers on staff to help determine the value of your home in today's market. We want you to get the most out of your investment.\nMARKET AND FEASIBILITY STUDIES: Help determine how quickly, and for what price, you current home will sell.\nFINANCE & MORTGAGE: We can help you find the best value and rates for your new home loan.\nAPPRAISAL TO NEGOTIATIONS: We are there for our clients for both home sales and home purchasing. From the start of the process to after all contracts are signed and beyond.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Getting Serious about My Goal!!\nI just weighed in and logged it in my tracker and ran a new date for completion. If I lose 1.5 pounds per week I will be at 162 (which is my first goal) by March, 2010. 162 puts me below obese, which I'm really tired of being!! My daughter graduates in April, 2010 and I really want to look nice for her graduation!!\nSo, really time to get moving!! I need to work on a visual this weekend. And put pictures of Tracy's Graduation from high school as well as pictures of my hubby to help keep me on track!\nSounds like you have a plan and motivation! Keep looking at those pictures, but remember, you're losing the weight for you!\nLooks like your on the right track! Good luck!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Famous Grouse 30yo blended malt whisky is a fantastic example of the skill that the Master Blender uses to create this malt whisky. The core malts are The Macallan from Speyside and Highland Park from Orkney.\nAnother one from the Rarities\/oddities tasting. This aged blend from Edrington's Famous Grouse stable is 100% malt whisky (i.e. no grain whisky component) all aged over 30 years.\nA very nice blended malt, plenty of character in here that reminded us of well-aged and heavily-sherried Macallan and Highland Park.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"TemplateHack.com - Download Alpha premium blogger template. Alpha is magazine blogger template with simple and great design. It is green blogger template designed especially for news blog, magazine blog, personal blog, tech blog, and etc.\nIt is premium blogger template, customizable blogger template, and designed with html5 base code and you will get many feature to improve your blog visitor. Because, Alpha is designed with search engine optimized feature, responsive, and simple design so your blog can load fastly with this template.\nDownload this template for free, and you will get all amazing thing from its features. Download this premium blogger tempate for free.\nClean, SEO friendly, Slideshow, 1 sidebar, 2 column, 3 column footer, Post thumbnails, Responsive, Social bookmark ready, Ads ready, Email subscription, Widget, Page navigation menu.\nNews blogger template, Tech blogger template, Magazine blogger template, Personal blogger template.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What role do legal trials have in collective processes of coming to terms with a history of mass violence? How does the theatrical structure of a criminal trial facilitate and limit national processes of healing and learning from the past? This study begins with the widely publicized, historic trials of three Nazi war criminals, Eichmann, Barbie, and Priebke, whose explicit goal was not only to punish, but also to establish an officially sanctioned version of the past. The Truth and Reconciliation commissions in South America and South Africa added a therapeutic goal, acting on the belief that a trial can help bring about a moment of closure.\nHorsman challenges this belief by reading works that reflect on the relations among pedagogy, therapy, and legal trials. Philosopher Hannah Arendt, poet Charlotte Delbo, and dramaturg Bertolt Brecht all produced responses to historic trials that reopened the cases those trials sought to close, bringing to center stage aspects that had escaped the confines of their legal frameworks.\nYasco Horsman is University Lecturer of Film and Literary studies at Leiden University. He holds a Ph.D. in comparative literature from Yale University and is author of Theaters of Justice: Judging Staging and Working Through in Arendt, Brecht and Delbo (Stanford University Press, 2010), on the relations between trials, theatre and psychoanalysis. He co-editor of a special issue of Law and Literature on Legal Bodies: Corpus, Persona, Communitas (2016), and has published essays on literature (Kafka, Beckett, Coetzee), cinema (Resnais), Graphic Novels (Spiegelman, Ware, Clowes), Animation (Mickey), drones and flipbooks. He is currently working on a book on Radio and Modernism and collecting notes for a genealogy of the \"funny animal\" figure in early animation and comic strips.\nTogether with Pepita Hesselberth, he organizes a monthly series of NICA Lectures at Leiden University called Lectures in Media Art Politics. For more info, see here.\nTogether with Pepita Hesselberth, I organize a monthly series of NICA Lectures at Leiden University called Lectures in Media Art Politics.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Use WASD for the movement. Use the mouse to aim, click the left mouse to use items.\nClansmen.io is a tough strategy game online with a real-time element. The game gives you a chance to construct a big base following your own style, but the big mission you must focus on doing is to conquer enemies. You start it off by creating the base, collecting many items on the ground, hiring strong units and sending them out into the fight to vanquish all opponents. Perform your strategic ability, defend yourself, and work with friendly people so as to wipe out the tougher ones. Can you lead your units to the final victory? Join the game now!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..0ac6329c6cbdd45d6fddeeea20a260e0e384d7a4
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+{"text":"Turkey's Directorate of Religious Affairs (Diyanet) is set to launch a big campaign to promote the role and status of the women in society, after controversial comments on violence against women and gender relations from rogue preachers prompted outrage across the country.\n\"The Diyanet will touch on the women's issue in a more visible and emphasized way in the new era,\" Deputy Prime Minister and government spokesman Bekir Bozda\u011f told the Ankara bureau chiefs of various media outlets on March 15.\n\"I would like to announce that a huge campaign to convey the honorable place of the women in Islam, in accordance with the nobleness of Islam, to Turkey and to the world through books, television broadcasts, radio programs, seminars, panels and all other means,\" he said.\nBozda\u011f serves as the deputy prime minister responsible for the activities of the Diyanet, whose head Ali Erba\u015f was also present at the meeting.\nBozda\u011f also noted that the Diyanet's works on women will be carried out under the leadership of Diyanet Deputy Chair Huriye Mart\u0131, who was recently appointed as the first female deputy to the directorate since its foundation in 1924.\nDiscussion was sparked in the country after a number of preachers made headlines through controversial remarks on the role of the women in society.\nAmong them was Social Fabric Foundation head Nureddin Y\u0131ld\u0131z, who appeared to justify violence against women in a video posted online on March 3.\n\"Women should be grateful to God, because God allowed men to beat women and be relaxed,\" Y\u0131ld\u0131z said. Other preachers around the same time had attracted attention for suggesting that women and men should not take the same elevators and intensive care units at the hospital should have separate places for female and male patients.\nA few days after Erdo\u011fan's statement, Mart\u0131 was appointed as the Diyanet's new deputy chair.\nBozda\u011f also said more women would be appointed to the directorate, where currently seven women are serving as department heads.\nAt least one female mufti will serve in every province, while female auditors and inspectors have also begun to be recruited in the Diyanet in order to increase women's participation in its works.\n\"Efforts to tarnish our religion through women have coincided with our works [to increase women's status]. That's why we have been really saddened,\" Bozda\u011f stressed, adding that an event is upcoming at which all these issues will be discussed with the participation of female scholars.\n\"We'll bring women's issues to the agenda of the people in a more efficient and visible way,\" he said.\nAnother area where the Diyanet will intensify its works is in \"conveying the right information to the people on the exploitation of Islam by terror organizations, like [the Fethullahist Terror Organization [FET\u00d6] and [the Islamic State of Iraq and the Levant] ISIL,\" Bozda\u011f vowed.\n\"The Diyanet will not only release reports but also carry out effective works in the field to enlighten our people on the terror organizations that exploit our religion, as well as those that are religiously deviant,\" he said.\nDiyanet officials will visit every village, town and province in the country to brief locals on \"the real faces of such terror organizations,\" Bozda\u011f said, adding that these events will also include religious vocational (imam-hatip) high-schools, regular high-schools, and universities.\n\"Thanks to the common sense of our people, these terror organizations do not have much room for recruitment of the terrorists. But we will continue our works in order not to give them space in the future as well,\" he said.\nMeanwhile, Bozda\u011f also drew attention to \"plots by intelligence organizations of some foreign countries that aim to spark new sectarian clashes in the Middle East and in the Islamic world.\"\n\"The intelligence services of some countries have been exerting special efforts. They have reserved special budgets and are try to provoke something as a result,\" he said.\n\"This is a great threat and danger for our country, our people and for all of us. We should all be cautious against those who want to pour fuel on the sectarian fire in our region,\" Bozda\u011f added.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I have purchased a new (used) life raft after my last one got destroyed in the accident during the move and launch from her transit back from New Zealand. I have mounted it in the same location just in front of the dodger as it is the most accessible and out of the way. It is the closest place to the cockpit I could find and still be unobtrusive. And believe it or not it does not impede the view at all!\nThe first photo is taken from inside the cockpit looking forward.\nCopyright \u00a9 1995-2006 Phillip J. Seaman All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are here to build something unique, fuel recognition and amplify differentiation for your brand!\nI am Aspa Chroneou, an independent Brand Identity Designer, based in Athens, specializing in the fields of branding, packaging design and creative direction.\nSince 2005, I have worked collaboratively with a wide range of clients to grow their businesses through distinctive, meaningful and enduring design, helping them to communicate better their products and adding value of what they do.\nI believe that creative work should be unique, authentic targeted idea, and therefore memorable.\nMy creative team strategically launches your project with consequence, methodology and imagination, starting from conception to completion and always ready to take your brand beyond the ordinary. Whether you run a small startup or a multinational corporation, say hello if you like to discuss about your brand and how to start building its identity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tomorrow morning it goes on for the editorial staff of the JewelBlog to the inhorgenta 2010 in Munich. Previously, Jeannine and I meet more friends and from Friday, we are then represented at the inhorgenta. During this time, we will Tweet at hopefully short intervals, the most recent of the inhorgenta in Munich in 2010. Who would like to remain so up to date I put our Twitter account \" Bridgat \" to the heart. Just follow the Bridgat on Twitter and get the latest information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The ruling Awami League has named the individuals who will represent it in the Jatiya Sangsad as part of the reserved seat structure for women in parliament. The party has exercised its right to choose the persons who it thinks are suitable for the seats and naturally will expect them to play a constructive role in the House.\nIt will certainly be the expectation of the nation that the lawmakers selected for the reserved seats in Parliament, despite the fact that they are not chosen by popular vote, will nevertheless play a role that shows them to be fully engaged in the business of overseeing the diverse affairs of state. In the past, there was much criticism, a good deal of it justified, about the essentially non-participation of MPs in reserved seats in parliamentary debate.\nMoreover, there has been the general perception that their non-elected nature does not give such lawmakers the clout which elected MPs enjoy. That is wrong, for all the women who have occupied and will occupy the reserved seats hold positions ordained by the Constitution. To underrate them or to deny them the importance their position entitles them to would thus be wrong.\nThe new crop of women lawmakers as announced by the Awami League should be a bold new addition to the JS. We as citizens will expect them to keep themselves in touch with national and global issues. In other words, through doing their homework and through putting their understanding of national and international issues to good use in parliamentary deliberations, they will do justice to the positions they now formally hold in national politics.\nJust as we expect all our elected lawmakers to inject vigour and substance and intellectual richness to the Jatiya Sangsad over the next five years, so will we tell ourselves that our women MPs in the reserved seats will bring their knowledge and wisdom to bear on the deliberations of the House.We wish them well as we wait for Parliament to serve as a vibrant and meaningful voice for the nation between now and the next election.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Accentuate your personality with casual look by adoring this high rise ankle sneaker by Belvedere. Made of genuine ostrich, Taylor is design to suit your personality with buttery soft leather. It features leather lining and cushion insole.\nWear this casual sneaker with your favourite jeans and t-shirt for accentuating looks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am seeking an energetic and dynamic Deli Manager on behalf of my client in Roscommon. Our client is a well-established retailer who is part of a well-known symbol brand.\nThe Deli Manager will need to be enthusiastic, have a passion for retail and fresh food and control the daily operations of the deli department. ideally, you will have experience as a deli supervisor or Assistant deli manager and be looking to take the next step in your career. you will work closely with the store manager and owner and be expected to deliver high standards in the department.\nComplete orders and stock control and waste management.\nEnsure pristine displays and push the highest standards in the department.\n2 years or more experience working as an Assistant Deli Manager or Deli Supervisor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Work-based learning experiences like internships offer you the opportunity to learn alongside professionals as you develop valuable skills and connections to prepare you for your transition into the workforce.\nInternships are wonderful opportunities that prove time and time again to be important learning experiences for students and highly impactful investments for employers.\nMany programs at NTC offer work based learning experiences. Depending on your program, a work based learning experience may include an internship to help you connect the classroom to the workplace. These experiences offer you the opportunity to learn alongside professionals as you develop valuable skills and connections to prepare you for your transition into the workforce.\nThese programs require students to complete an internship for credit. For more information or to get started with a work-based learning experience, contact one of your instructors.\nNote: The School of Engineering and Technology and the School of Advanced Manufacturing and Transportation have opportunities to complete an internship as elective credit. For more information, please contact Suzi Mathias at 715.803.1767 or email at mathias@ntc.edu.\nAre you interested in an internship but have questions about how it works for you? You may ask your question through our online form or contact Suzi via phone or email.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Color is an advanced theme for Business, Portfolio or Blog usage. It comes with over 100 theme options, 20+ custom widgets, 3 page templates (Blog, About, Contact, Services & Portfolio), Many slideshow animations, very clean HTML5 & CSS3, jQuery scripts integrated (LightBox, ToolTip & others) and also a flash, image and video support. Full documentation, support forums and tutorials are available online.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Archives|MRS. STENZ VICTOR IN TENNIS BATTLE; Defeats Mrs. Baker by 6-1, 6-4, Reaching Semi-Finals in Women's Tourney.\nMRS. STENZ VICTOR IN TENNIS BATTLE; Defeats Mrs. Baker by 6-1, 6-4, Reaching Semi-Finals in Women's Tourney.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Getting to grips with the very earliest social networks: the challenges of using network methodologies to tackle Palaeolithic datasets.\nA series of recent analyses have demonstrated an expansion in the scale and complexity of social networks as larger and more permanent social aggregations grew up with the adoption of a more sedentary, village-based and ultimately agricultural lifestyle during the late Palaeolithic and early Neolithic. However, these developments may represent only a relatively late stage in a much more long-term and gradual process of scaling-up of social networks which took place over the course of hominin and human evolution, from the small, intimate and highly local communities of other primates, to the globally-connected, city-dwelling Homo urbanus of the contemporary world. In this paper I will argue that increasing engagement with material culture was central to this process, providing a mechanism by which humans were able to expand the scale of social relations well beyond those of their closest relatives. Network methodologies provide the best opportunity to investigate this; however, the nature of Palaeolithic datasets makes such analyses challenging; in this paper I will discuss the problems and also some potential ways forward, through which we may be able to gain vital insights into some of the very earliest social networks and the ways in which they subsequently evolved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hosting your event with us this spring? Choose between two exclusive offers: SAVE 15% on your event (with code SPRING15) or enjoy 4 FREE* MOVIE TICKETS (with code MOVIE19). They're available now\u2014but only for a limited time!\nGet ready to bring those epic party plans to life with a celebration that's strikingly different and always unforgettable. Bowl with your friends. Play the latest arcade games. Hit up some old-school classics like beer pong. And experience all this fun as you feast on outrageous eats that were made to party\u2014from our 6-lb. Behemoth Burger and XXL Pretzel to our shareable 100 oz. cocktails.\nOverwhelmed by all this awesome? Lucky for you, our party planning pros are available to help you design the whole thing.\nDon't see what you're looking for? Fill out this form and one of our Personalized Party Planners will contact you to assist!\n* Tax and event fee not included in pricing. All events are subject to our Online Booking Terms.\nThe event fee covers administrative costs for your event and takes care of overhead associated with producing your party. The event fee is not gratuity and will not be distributed to personnel that provide service at the event. No gratuity to personnel is required or expected.\nWings: For game time or anytime! Try our world-famous wings\u2014available in six delicious varieties.\nClassic Pepperoni Pizza: 16\" Pizza made with 100% whole milk mozzarella and topped with pepperoni.\nMozzarella Sticks: Your favorite finger food made with our delicious Parmesan cheese and herb blend and served with warm marinara.\nFRESH VEGETABLE CRUDIT\u00c9S: A healthy mix of crisp, garden-fresh vegetables served with zesty ranch!\nSPINACH & ARTICHOKE DIP: A savory favorite served with fresh-cut pita and tortilla chips.\nEasy to Plan. Awesome for Everyone.\nThis isn't your typical bowling alley\u2014and this ain't no ordinary party. It's an immersive experience that comes fully loaded with blacklight lanes, life-sized video walls, state-of-the-art arcade games, and old-school classics like billiards and foosball. Our events are so fun they should come with a warning. So be advised: spontaneous high-fives and loud laneside chants may occur while partying.\nOur parties are selfie-ready, foodie-inspired, totally unforgettable, and made to be shared, liked, loved, and \"haha\"-ed all over the interweb. From venues designed to impress to menus that stun and amaze, these parties are built to be the envy of your social circle. Get ready for the kind of good times and outrageous eats that only Instagram can tame.\nDon't worry one iota about your party\u2014we got you. We handle all the deets so that your party shines and everyone has a great time. Because every celebration should be a ton of fun (not a ton of work), we feature specially trained event planners, custom food & beverage packages, premium bar options, and party packages that are easy to design and modify.\nNeed a little inspo' for your next event? Check out our pro tips and see what it takes to throw one seriously chill laneside party!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Camden Council our vision is to make Camden a place where everyone has a chance to succeed, and nobody gets left behind. To achieve this we need to think differently about everything that we do, design flexible services that focus on what matters to citizens and continuously learn and improve. As a Systems Thinking Senior Manager, you will join our dedicated systems thinking team which is responsible for building capabilities in the organisation to think and act differently to enable us to work towards our vision.\nThe post holder will build positive relationships with a range of stakeholders to contribute to decision-making and have a relentless focus on our citizens' experience. You will help Camden reimagine its services taking into account the system as a whole, shifting the thinking and behaviour as well as the strategy to challenge current assumptions and drive change. You will play a key leadership role in the organisation.\nAn expert practitioner in systems thinking \/ quality improvement approaches within complex organisations with a focus on innovation and improving services for the end-user. Ideally you will have knowledge and experience of applying Vanguard method, however this is not essential.\nTo apply for this job please follow the \"Apply\" link. In the 'Why you?' section of the application form you will be expected to explain how you meet the key requirements for the role listed in this advert. When explaining how you meet each of the requirements, please give examples that clearly demonstrate your skills, knowledge and experience. When writing your examples give a brief description of the situation or task but focus on the actions you took and the result of your actions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hidden Secrets Tours is taking helpful steps towards Homelessness Solutions.\nIn recent years, we decided we wanted a clearer focus for our community and charitable efforts.\nHomelessness in Melbourne is apparently a hidden secret. But we know it is not.\nThough we do not have skills or resources to directly change this, we know many who do and have chosen to engage with them to bring this issue to light.\nSince we began working with Street Smart, we have moved through a huge economic change in our city, and with global turmoil, huge amounts of infrastructure works, and general higher costs of living, we have increased our fundraising, and now put $2 per booking aside for Street Smart.\nWe have also chosen to give 30% of our donations specifically towards the running of this innovative national enterprise (which is based in Melbourne). Its important that foundations such as this are supported to be sustainable themselves. We love that we can see where our money is making change, as we have nominate that we want to support grassroots programs here in Melbourne.\nAs of April 2018, we have donated over $6700 to Street smart in cash, and further supported them and others (Sacred Heart Dinners, the Big Issue, Books on the Rails and Urban Seed) with vouchers or services in kind.\nIn February 2016 we are moving forward to create a transparent fundraising space for Hidden Secrets Tours \u2013 we have joined up with Street Smart #homelessnessisnotahiddensecret \u2013 we will share more on what we do with that as we progress.\nIn February we committed to donating $1.00 for every person booked directly with us to go to Street Smart \u2013 they will allocate this money towards projects that will help our local Melbourne Community \u2013 we will update you here and on our NEWS pages on our progress.\nIn 2015 we joined with social enterprises, charitable groups, and fundraising events that make positive changes here in the Melbourne community of people identifying as homeless.\nJust a few of our efforts are listed as below.\n* The Big Issue \u2013 we are a sponsor of another one of Melbourne's great community projects, the Big Issue. in January 2015 we joined the Womens Subscription Enterprise and taken a digital subscription rather then paper.\n* We have donated tour product as prized for Kids Under Cover.\nIn the coming months we will have clear programs where our clients and colleagues will be able to add their efforts or finances to this cause!\nEven if you are couch surfing.. everyone wants a place, a bed, to call their own. Safe and Secure.\nKeep your eye on our blog and website to see how we develop this project further.\nYou can also have a look at our Sustainability, Social & Environmental Pride page to see what we've been involved with so far.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A luxury and strong inflatable kayak. 3 air compartments with a quick-fill\/deflate valve. Comes with welded paddle supports, an extra grip en and a grappling rope. Including a handy repair set, 2 aluminum paddles, an inflatable pump. Certifications: T\u00dcV and GS certified. 274 X 76 X 33 cm. 274.3 x 76.2 x 33 cm ; 11.3 Kg. We go above and beyond to ensure you receive the best level of service.\nThe item \"Intex Inflatable Kayak Canoe Challenger K1 Person Man Woman\" is in sale since Wednesday, January 16, 2019. This item is in the category \"Sporting Goods\\Canoeing & Kayaking\\Kayaks\".\nThe seller is \"sugarbabe28cars\" and is located in Luton. This item can be shipped to United Kingdom, Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Australia, United States, Canada, Brazil, Japan, New Zealand, China, Israel, Hong Kong, Norway, Indonesia, Malaysia, Mexico, Singapore, South Korea, Switzerland, Thailand, Saudi arabia, South africa, United arab emirates, Ukraine, Chile.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbnjdf b/data_all_eng_slimpj/shuffled/split2/finalzzzbnjdf
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@@ -0,0 +1,5 @@
+{"text":"Air and hydro excavation \u2014 also known as hydrovac or hydro trenching \u2014 is the state-of-the-art solution for exposing underground utility lines, pipelines and railroad switches and crossings. The process involves using precisely focused flows of air or water to dislodge the earth, while simultaneously vacuuming the loosened debris into a vacuum truck for reuse or disposal.\nAir and hydro excavation services are performed using a highly mobile, all-in-one system that serves as an excavator, loader and transporter. Because just one machine is necessary, it is possible to complete hydro excavations in remote or rough terrain areas that are inaccessible to larger fleets of equipment.\nHydro excavation is a non-destructive technique that virtually eliminates the potential for damage to existing underground infrastructure. As a result, it is suitable for use in environmentally sensitive areas and locations where power lines, water lines or other buried cables are present.\nMost importantly, the main advantage of air and hydro excavation is safety. By protecting sensitive infrastructure, the chances of an on-the-job accident are substantially lowered. This, in turn, helps you meet your timelines and stick to your budget with a minimal risk to your crew.\nThe diversity of potential applications for air and hydro excavation make it an excellent choice for a range of different industries. Power and utility contractors, pipeline and refinery development crews, and airport and railway engineering companies, among others, all frequently use air and hydro excavation services for technically demanding jobs.\nSafety: It virtually eliminates the risk of personal injury or death associated with accidentally striking pressurized gas, water or electrical lines.\nSpeed: It's up to 75% faster than traditional digging methods, allowing your crew to quickly turn around complex projects with a minimal amount of work.\nReduced downtime: Because it is so much faster than other methods, it also decreases the risk of property damage, lost work opportunities, community disruption, ecological damage and insurance liability issues that occur on longer jobs.\nEfficiency: A single hydro excavation truck does the work of an excavator, loader and transporter. As a result, labor and fleet maintenance costs go down, and environmental disruption is minimized.\nVersatility: All-in-one hydro excavation trucks are uniquely capable of working in remote and inaccessible areas where large fleets can't go.\nLower impacts: Similarly, a smaller fleet means your operation has a reduced footprint overall. This minimizes disruptions to roads and surrounding work areas, while keeping your fuel use and carbon emissions lower.\nThese are just a few of reasons why demanding businesses request our air and hydro excavation services. To learn if the process is right for your upcoming project, contact AIMS Companies directly.\nAIMS Companies offers a full range of air and hydro excavation services for customers in Alabama, Georgia and throughout the Southeast. We are a veteran-owned business with a reputation for quality work and honest communication in everything we do. We are available for emergency service as well as scheduled excavation and cleanup work.\nTo request a consultation or an estimate, call 334-855-4775 or use the form on this page to contact us directly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please contact us if you have questions or special requests.\nWe are available by phone Mon-Sat, 10am-5pm Eastern Time.\nRequired fields are in bold. There is a short math question at the end to help prevent \"robots\" from using the form.\nIncoming e-mail messages are monitored on a regular basis to help ensure a quick reply to your question. Thanks again for shopping with us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Responsive web design (RWD) is an approach to web design aimed at crafting sites to provide an optimal viewing and interaction experience\u2014easy reading and navigation with a minimum of resizing, panning, and scrolling. these modern designs work across a wide range of devices (from desktop computer monitors, laptops, tablets, including mobile phones). A site designed with RWD adapts the layout to the viewing environment by using fluid, proportion-based grids, flexible images and CSS3 media queries.\nBelow is a responsive design that is built around the powerful WordPress blogging platform, which exploits the dynamic powerful CMS features of this free blogging software.\nKindred is a tasteful food and lifestyle blog with great styling and a layout to die for. Whether you run an established blog or just published your first post, Kindred will highlight your content beautifully and demand the attention it deserves.\nMailing list integration for MailChimp, Aweber, Constant Contact etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ever wanted to explore the Brecon Beacons national park and learn along the way? If the answer is yes then this workshop is for you. Mark Upfield and Ben Holland have teamed up with Capture Tours to run unique and exciting one day adventures. They will take you on a hike through a popular trail to some of Wales most spectacular waterfalls, Sgwd Ddwli Isaf and Sgwd Ddwli Uchaf to just name a couple.\nWe will be meeting at Pontneddfechan for 9am and starting the hike by 9:30. We will be photographing through till sunset and will return back to the cars no later than 6:30pm.\n\u2022 Transport to meeting point at the start of the day.\nThe book now please e-mail us via the contact form.\nEach workshop will hold a minimum 4 people and maximum 6. If for any reason we do not fill 4 spaces we will reschedule you or offer a refund (but please note this is extremely rare).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Modeling the Best ARIMA Modeling Approach for Forecasting Market Indices in Colombo Stock Exchange, Sri Lanka.\nMadushani, M. L. P.; Erandi, M. W. A.; Madurangi, L. H. L. S.; Sivaraj, L. B. M.; Weerasinghe, W. D. D.; Jayasundara, D. D. M.; Rathnayaka, R. M. K. T.\nCitation: Madushani, M. L. P., Erandi, M. W. A., Madurangi, L. H. L. S., Sivaraj, L. B. M., Weerasinghe, W. D. D., Jayasundara, D. D. M., and Rathnayaka, R. M. K. T. (2017). Modeling the Best ARIMA Modeling Approach for Forecasting Market Indices in Colombo Stock Exchange, Sri Lanka. 8th International Conference on Business & Information ICBI \u2013 2017, Faculty of Commerce and Management Studies, University of Kelaniya, Sri Lanka. p.07.\nGenerally, the movements of the stock prices are highly volatile and make much more dynamics. As a result day by day the large number of companies has been listed on stock exchanges across the world. Under this scenario, examine a suitable model for forecasting stock prices is a biggest challenge in the modern world. The propose of this study is to examine a suitable model for forecasting stock prices in the Colombo Stock Exchange (CSE), Sri Lanka. Since the data has a non-seasonal linear trend, an autoregressive integrated moving average model was used for modeling and forecasting. The empirical results suggested that ARIMA model is more accurate for forecasting ASPI index than other traditional regression methods.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbnqmk b/data_all_eng_slimpj/shuffled/split2/finalzzzbnqmk
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index 0000000000000000000000000000000000000000..20b176a406e4e8f1499fd7cc0c3c28a375492473
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@@ -0,0 +1,5 @@
+{"text":"The latest volume of the prestigious anthology series, published annually across six decades! The Nebula Awards Showcase volumes have been published annually since 1966, reprinting the winning and nominated stories of the Nebula Awards, voted on by the members of the Science Fiction and Fantasy Writers of America (SFWA). The editor, selected by SFWA's anthology Committee (chaired by Mike Resnick), is Jane Yolen, an author of children's books, fantasy, and science fiction. This year's Nebula Award winners are Charlie Jane Anders, Seanan McGuire, William Ledbetter, Amal El-Mohtar, and Eric Heisserer, with David D. Levine winning the Andre Norton Award for Young Adult Science Fiction and Fantasy Book.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Big Freedia: Queen of Bounce season 3 premiere is coming up fast. All-new episodes of the hit series kick off Wednesday, Feb. 25 at 11\/10c, and it's going to be here before you know it.\nDon't end up night of the season premiere suddenly realizing \"Wait, I forgot what happened last season on Big Freedia!\" Check out our 2-minute refresher above now for a quick recap of (almost) everything that went down in seasons 1 and 2.\nRemember when Freedia and her Twerk Team shut down New Orleans traffic in season 1 to twerk on cars, trucks and the busy street for a music video shoot? Or when Team Freedia played South by Southwest, right after Lady Gaga's set? Or last season when new Creative Director Wilberto showed up and caused some major shake ups in the team? Relive it all now so you're ready for season 3 in February.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today, so many exciting things are going on in the cloud software world that my bet is we will see very large improvements in the way things are done driven by higher productivity and innovation. The difficulty in perceiving this now is that there is no thing to \"see\" with this software.\nCompanies with B2B order flows have been slower to understand the important role a distributed order management system can play in increasing customer satisfaction, sales, and helping to insure orders flow to customers in the most profitable way possible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I ' m a 2 3 y e a r s o l d F re e l a n c e D e s i g n e r f ro m V i e t n a m . I l o v e e v e r y t h i n g t h a t h a s t o d o w i t h We b d e s i g n , G r a p h i c d e s i g n , P a c k a g i n g , 3 D d e s i g n , I n d u s t r i a l d e s i g n , a n d m o t i o n a n d I f e e l a t r u e d e v o t i o n f o r t y p o g r a p h y.\nI h a v e 3 y e a r s o f e x p e r i e n c e w o r k i n g a s a f re e l a n c e a n d s t u d y i n g a t s c h o o l , a n d I ' m re a d y f o r v i s i t i n g a w o r k t o i n t e r n ! I l o v e w o r k i n g w i t h p e o p l e , s p e c i a l l y i f t h e y h a v e the same passion I have for what they do.\nOverview This is a subject that helps us have knowledge of the basic principles for the g r a p h i c s i n d u s t r y. a s w e l l a s h e l p i n g s t u d e n t s h a v e m o re d e t a i l e d e x e rc i s e s to practice those basic principles.\nUNITY & HARMONY NOTE This is a stylized drawing from a chase between wolves and deer.\nSAUBASS PRACTICE NOTE This is posterfilms of Brand of brother getting ideas from Saulbass's posters.\nTEXTURE PRACTICE NOTE This is a picture made from beans, ropes and paper. It was inspired by the portrait of a lecturer on the school. It means a great life.\nSAUBASS PRACTICE NOTE This is a sarcastic picture. it talks about some of the corruption problems that have worsened the face of the ruling state.\nWATER COLOR NOTE This is a watercolor painting.\nSELF PORTRAIT NOTE NOTE This is a watercolor painting.\nLOGO NOTE This is one of the logos I've ever designed.\nSTORY BOARD NOTE Here are some artboards I drew for an advertising customer.\n3D MODEL NOTE This is a character created to make a short clip of StarWar.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"See the world with PlanetWare and create extraordinary adventures.\nKey West's Best Beaches Enjoy the island's best strips of sand, from Smathers Beach to the Dry Tortugas.\nTop Cheap Ski Holidays in North America Discover the most affordable places to hit the slopes on your next winter vacation.\nBest Beaches in the Honolulu Area Surf, swim, sunbathe, and snorkel along the city's popular palm-lined coast.\nPlan your travels with PlanetWare and our dedicated group of travel writers who offer their ideas on where to go, what to see, and tips for making the most of your trip. Our experienced writers are based all over the globe and provide their insight to help you create your own extraordinary adventures. Discover the top attractions in destinations around the world, along with maps, and beautiful photos to inspire great ideas.\nPlanetWare has also been creating and licensing attraction POI data (including geographic coordinates) since 1999.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"We live in Florida and saw a need to start-up this new OES rescue group and plan to serve the abandoned, mistreated, or unwanted OES in the \"Land of Cotton\". We have just received our non-profit status and have applied to be 501 (3) (c) corporation. Rosie, our first Dixie rescue dog, was placed in her forever home this week. Michelina has painted us our very own puppy, \"Dixie\", the elite pup with a ball. He is adorable!\nPlease visit us on-line at www.dixieoesrescue.org Linda Allen, our webmaster, has done a fantastic job putting all of our wants and needs on a website. We owe her quite a bit of hair dye for all the gray hair we have given her!\nWe would like to have your permission to have your contact information on our OES Rescue contact list on our website. Please check our website and let me know if your contact information is correct and\/or if any changes need to be made.\nCourtesy postings of Available Dogs from your rescue are also an option to consider on our website. Interested? Please contact us.\nWe would love to hear from you. Pllease consider becoming a member of Dixie -- you can even join on-line! Whistlin'Dixie, our newsletter, will be published quarterly. Our on-line store will have some very unique items available soon.\nWe look forward to seeing y'all at the Rescue Parade at the OESCA meeting in Williamsburg this fall.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The United Kingdom has been at the forefront of Medical Research and Innovation throughout the modern era and British Doctors have been responsible for numerous ground-breaking discoveries. Innovation has the potential to transform the care of patients, the NHS and the UK life sciences sector.\nThis session will explore options available to Clinicians and how they may progress their medical innovations from concept to marketing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"a still from the Alef Beta teaching on acquiring a wife.\nI have heard some strange things said about the position of women in Bible times; the latest being that it was permissible to divorce ones wife if she was childless.\nSome views about the low status of Jewish women make me wonder if people are attributing Islamic practices to the Jews, and how many Christian commentaries are coloured by anti-Jewish stereotypes?\nEarly books of the Bible do tend to show women being treated as possessions, but some factors need to be borne in mind.\nIn Genesis, Woman was created to be a suitable companion for man, and marriages appear to have been undertaken from this point of view.\nSome deplorable things happened to women in the book of Judges, but we must remember that the overview of those times was that people \"did what was right in their own eyes.\" The Bible records both the good and the bad and expects us to learn from both - what to emulate and what to avoid. This is specially true of the treatment of women in some stories.\nWhat happens when we look at times long ago through twenty first century spectacles?\nArranged marriages seem so archaic and unromantic, seen from our time. But what is so good about our present day western culture of falling in love (or lust) and moving in together; with or without a big showy wedding?\nAnswer: a fifty percent divorcee rate and a high proportion of children growing up without a father. An arranged marriage, demands that the couple be committed to making the marriage work, as opposed to what appears to be happening in our day; people walking away from their marriage when the warm and fuzzy feelings wear off.\nSee Weddings for more on Jewish practice.\nWhat are the effects of viewing marriage in the Hebrew scriptures through spectacles of a century of feminism?\nThe Jewish scriptures and traditional practices were certainly not feminist friendly, but that does not make them bad. Right from the beginning we see woman created from man to be a partner, and to bear children, and not to have the same role as man.\nGenesis 2:20 So the person gave names to all the livestock, to the birds in the air and to every wild animal. But for Adam there was not found a companion suitable for helping him. Then God caused a deep sleep to fall upon the person; and while he was sleeping, he took one of his ribs and closed up the place from which he took it with flesh. The rib which Adonai, God, had taken from the person, he made a woman-person; and he brought her to the man-person.\nThe man-person said, \"At last! This is bone from my bones and flesh from my flesh. She is to be called Woman [ Hebrew: ishah ], because she was taken out of Man [ Hebrew: ish ].\" This is why a man is to leave his father and mother and stick with his wife, and they are to be one flesh.\nAfter the fall (Genesis 3) man is cursed to labour to provide for himself and his family, while the woman is cursed to bear children in pain and be subject to her husband.\nGenesis 3:16 To the woman he said, \"I will greatly increase your pain in childbirth. You will bring forth children in pain. Your desire will be toward your husband, but he will rule over you.\" To Adam he said, \"Because you listened to what your wife said and ate from the tree about which I gave you the order, 'You are not to eat from it,' the ground cursed on your account; you will work hard to eat from it as long as you live. It will produce thorns and thistles for you, and you will eat field plants. You will eat bread by the sweat of your forehead till you return to the ground-for you were taken out of it: you are dust, and you will return to dust.\" The man called his wife Havah [ life ], because she was the mother of all living.\nWomen proved, in two world wars, that they were able to take up many of the roles that their menfolk had left to go and fight. But that does not make man and woman the same, and that they should have the same roles in God's plans. There seems to be an implication that men and women being of equal worth is the same as being the same. It should be recognised that there is a cost in pursuing sameness in the world of work, and that it is not possible for a woman to \"have it all.\" Poor rearing of children can all too easily be the cost, with children being raised by professionals.\nOne feature of Jewish wedding practice is the Ketubah, that is a marriage contract.\nThe Get or Gett is the corresponding docket of termination of marriage. Some things said about the Get appears to have more to do with Islamic divorce than Jewish practice.\nPeople study this passage concerning remarriage; but it is about divorce and how God hates it.\nMatthew 19:7 They said to him, \"Then why did Moshe give the commandment that a man should hand his wife a get and divorce her?\"\nYeshua said, Matthew 5:31 \"It was said, 'Whoever divorces his wife must give her a get.' But I tell you that anyone who divorces his wife, except on the ground of fornication, makes her an adulteress; and that anyone who marries a divorcee commits adultery.\nAccording to Wikipedia, A get or gett is a divorce document in Jewish religious law, which must be presented by a husband to his wife to effectuate their divorce. The essential part of the get is very short: the text states \"You are hereby permitted to all men\", which means that the woman is no longer married and that the laws of adultery no longer apply. The get also returns to the wife the legal rights that a husband holds in regard to her in a Jewish marriage.\nThis statement \"You are hereby permitted to all men\" would appear to be an example of the religious leaders devising laws that actually go against God's intentions. As Yeshua quoted Isaiah on one occasion when he clashed with the religious leaders.\nIf women are supposed to be virgins when they marry, how can a document restore that state of virginity?\nYeshua's teaching appears to be improving upon tradition, in the same way as he widened out murder to including holding hatred.\nMalachi 2:16 says \"For I hate divorce,\" says Adonai the God of Isra'el, \"and him who covers his clothing with violence,\" says Adonai-Tzva'ot. Therefore take heed to your spirit, and don't break faith.\nBut, remember the word translated as \"hate\" comes up in several places and is always problematical.\nWe also read that Jacob \"hated\" his first wife Leah. Upon closer reading, however, it becomes clear that Jacob loved Rachel more than Leah (Gen.29:31).\nYeshua's spoke of hating one's father and mother - a violation of the fifth commandment if taken too literally.\nThe key to resolving this difficulty is hidden in the ancient meaning of the Hebrew word soneh - being inaccurately translated as \"hate\". It appears that, in Biblical Hebrew, soneh meant \"loving someone\/something less\".\nchabad.org\/library\/article_cdo\/aid\/560110\/jewish\/The-Husbands-Grounds-for-Divorce.htm is an interesting and balanced outline of Jewish perspective.\n\"The very notion of a get procedure is to fore-stall the instantaneous termination of a marriage. The Rabbinical Court will deliberate, and the first point of deliberation will be to see if outstanding issues between the couple can or cannot be corrected.\"\nDeuteronomy 24 \"Suppose a man marries a woman and consummates the marriage but later finds her displeasing, because he has found her offensive in some respect. He writes her a divorce document, gives it to her and sends her away from his house. She leaves his house, goes and becomes another man's wife; but the second husband dislikes her and writes her a get, gives it to her and sends her away from his house; or the second husband whom she married dies. In such a case her first husband, who sent her away, may not take her again as his wife, because she is now defiled. It would be detestable to Adonai, and you are not to bring about sin in the land Adonai your God is giving you as your inheritance.\nThe acceptable grounds appear to be finding her offensive in some way; presumably concerning her behaviour.\nThere may well have been men who divorced their wives on account of childlessness, but there appears to be no grounds for saying that Jewish law sanctioned this.\nYet there were qualifications, important because the rabbis who interpreted biblical law for later generations built the legal structure on them.\n1 The husband had to write a bill of divorce and present it to his wife before sending her away ( Deuteronomy 24: 1-3; Isaiah 50:1; Jeremiah 3:8). This served as protection for her, as a delaying mechanism so that in a fit of anger a husband could not simply pronounce a declaration of divorce and be done with her.\nThe Chabad article quoted above does say that a husband may ask for a divorce on account of childlessness; not that he may just issue a get and divorce her. Also, remember that the article does not inform us about Torah, but about present day religious law in one branch of Judaism.\nChildlessness features in some scriptural stories and involved consider pain, but not divorce.\nRabbi Dr Eli Lizorkin-Eyzenberg, quoted below, commented in his essay on the Samaritan woman Jesus met at the well, that one of the reasons for having had four husbands could have been that she had been divorced for childlessness. But he does not cite any sources for this opinion.\nPerhaps our society's struggle with marriage stems from not comprehending God's design for man and woman.\nProperly understood, many things in the Torah that appear harsh and unacceptable, within today's frame of reference, were actually big steps forward from the prevailing practices of the time. The instructions about slavery provided a safety net for the poorest in society and prevented the ownership and abuse of fellow human beings.\nVideo studies by Rabbi David Fohrman of AlefBeta include some that throw light on the place of women in the Jewish scriptures.\nThese videos seek to help people get to the heart of what Torah is all about. Through a willingness to question and take a deeper look at our sources, Aleph Beta helps people find their place in stories that have been told for thousands of years.\nIt seems troubling that in a Jewish marriage, a man 'acquires' a woman - is Judaism really so archaic? In this lecture, Rabbi David Fohrman explores this question in the context of God's creation of Eve, the first woman, and shows that true acquisition is not about control, it's about completeness of the self.\nCan a man sell his daughter?\nThis study of Female Servitude in the Torah by Rabbi David Fohrman is very helpful.\nWhen a man was forced by poverty to sell his daughter into servitude, it was not into slavery as we understand slavery, but giving her the chance to move upwards socially through marriage. The daughter was protected by the terms of the Torah instructions.\nRabbi Fohrman also throws light on other issues concerning women, including childlessness, by looking at the story of Hannah.\nFrom Hannah's prayers to God, we may draw challenging lessons of closeness to a distant God.\nScandal in the story of Ruth?\nAnother helpful teaching looks at the stranger aspects of the story of Ruth, by examining the scandalous backstory.\nOne of the areas where the Bible finds itself apparently at odds with the twenty first century is Paul's perceived down on women.\nI would not wish to step in to this minefield, just to urge people to see if they have an issue with Paul or with God.\nWas Paul urging practices that would enable the church to function well in the culture in which it was found, or was there an eternal principle at stake? And are we properly understanding what Paul was saying?\nRabbi DR. ELI LIZORKIN-EYZENBERG wrote this essay. It caused some dismay when I shared it with friends, but I believe we should weigh these teachings and check that we are not adhering to the traditions of men when we put forth interpretations of what scripture says.\nIn 1 Corinthians 14:34 the Apostle Paul's letter states: \" . . . the women should keep silent in the assemblies. For they are not permitted to speak, but should be in submission, as the Law also says.\" There are several major problems with this statement.\nFirst, nowhere does the Jewish Law forbid women to speak in public gatherings. Paul, being a well-educated Jew, certainly would have known this. In fact, there was a law on the books that did forbid women to speak, vote and exercise authority over men by holding public office. It was not a Jewish, but a Roman law. These words would sound far more credible if someone else, other than the Jewish Apostle Paul, had written them.\nIn 1 Corinthians 11:5, he wrote that a woman's head must be covered while she is engaged in speaking in tongues or prophesying in a public assembly. The question was not, therefore, if a woman could speak and teach, but how it should be done in a way that would be right before God, angels and the people of Corinth.\nWhen we read Paul's letters we need to keep in mind that 1 Corinthians was not the beginning of this correspondence. Paul wrote at least one letter to the Corinthians prior to this (1 Cor. 5:9) and the Corinthian leadership had also written to him (1 Cor. 7:1). It is therefore highly probable that the statement in 1 Cor. 14:34-35 is a quotation from a letter that the Corinthian male leadership had addressed to Paul. It was their proposal on how to bring order into the disruptive practice of some women in the congregation as they spoke in tongues and prophesied. Paul, however, disagreed.\n\"Was it from you (masculine) that the word of God first went forth?! Or has it come to you (masculine) only?!\"\nPaul's solution, therefore, was not to exclude half of the congregation from exercising the gifts of the Spirit, but rather to make sure that it was done in a respectful, proper and orderly fashion.\nWas Paul right about women? Absolutely! His Corinthian opponents were not.\nSee also his essay \"What did Jesus teach about divorce?\"\nA man having more than one wife is obviously not what God intended at creation (the two become one flesh), yet we see it several times among the patriarchs. As with much in the Hebrew scriptures, its inclusion does not condone the practice, but honestly records what happened.\nPolygamy obviously took place, and one might argue that it was preferable at that time for a woman to share a husband than to be alone, without support or offspring to support her in her old age. But scripture honestly records the problems with polygamy.\nThe king was told not to take many wives, as was presumably the general practice at that time. (Think of how Esther came to displace Vashti as queen) Collecting many wives and concubines was Solomon's downfall, as they took his heart away to their foreign gods.\nThere appears to be a link between extra wives and childlessness. Remember the wrangling that ensued when Abraham and Sarah attempted to solve the problem of Sarah's childlessness. Check out the stress between Rachel and Leah, when the loved wife was childless. Consider Hannah and her pain over the taunting from her rival, Pinina. Had Elkannah, maybe, taken a second wife because Hannah could not give him an heir?\nLevirate marriage is the obligation of a surviving brother to marry the widow of his brother if he died without having sired children ( Deuteronomy 25:5-6).\nThe closest relative, referred to as the kinsman redeemer in the story of Ruth, had the first obligation to perform this commandment.\nThe explicit purpose of this commandment was for the surviving brother enable the widow to produce an heir to perpetuate the name of his dead brother, so that it would not \"be blotted out of Israel.\" It also served to keep property within the family line.\nThe institution of levirate marriage also served to protect the wife. In numerous verses, the Torah lumps widows with orphans and strangers as the disenfranchised members of society to whom special kindness must be shown. The situation of a widow without children was especially dire, for she had no one to care for her and provide material support. The levirate law guaranteed her a new family, enhanced status, and financial resources.\nThe most famous stories about levirate marriage in the Bible are those of Tamar and Ruth. See the AlefBeta videos mentioned above.\nconcubine A sexual partner, especially a woman, to whom one is not or cannot be married. A woman who lives with a man, but who is not a wife.\nThe practice of taking a concubine goes back thousands of years to the civilizations of ancient Mesopotamia and Babylonia where the elite members of society took concubines , many of whom were slaves, however, the first wife always retained a place of primacy in the family.\nQuestion: \"What is a concubine? Why did God allow men to have concubines in the Bible?\"\nConcubines in the patriarchal age and beyond did not have equal status with a wife. A concubine could not marry her master because of her slave status, although, for her, the relationship was exclusive and ongoing. Sometimes concubines were used to bear children for men whose wives were barren. Concubines in Israel possessed many of the same rights as legitimate wives, without the same respect.\nAlthough it's true the Bible nowhere explicitly condemns concubinage, a condemnation can be found implicitly from the beginning of time.\nAs a matter of fact, a study of the lives of men like King David and King Solomon (who had 300 concubines; 1 Kings 11:3) reveals that many of their problems stemmed from polygamous relationships ( 2 Samuel 11:2-4).\nThe Bible never explains why God allowed men to have concubines. He allowed divorce and polygamy, too, although neither was part of His original plan for marriage. Jesus said God allowed divorce because of the hardness of men's hearts ( Matthew 19:8). We can assume the same hardness of heart led to polygamy and concubinage.\nWe can also surmise a reason based on the culture of the day. Unmarried women in ancient times were completely dependent on their family members, such as their fathers, brothers, etc. If for some reason a woman had no family members or her husband had died or divorced her, she would be left with few options for survival.\nPerhaps, if we understood what God is saying to us in his Torah, instead of judging it by our limited, world influenced, traditional perspective, our society would have better marriages with relationships as God intended them to be.\nRemember what Paul and Peter taught about men and women.\nGalatians 3:28 there is neither Jew nor Gentile, neither slave nor freeman, neither male nor female; for in union with the Messiah Yeshua, you are all one.\nPaul was not saying that men and women are the same, but that we have equal standing in Christ as God's loved children. We retain our identity, as do Jews and Gentiles.\nNotice too that the above verse starts part way through a sentence and should be read in context.\nEphesians 5:28 This is how husbands ought to love their wives like their own bodies; for the man who loves his wife is loving himself.\nColossians 3:18 Wives, subject yourselves to your husbands, as is appropriate in the Lord.\nColossians 3:19 Husbands, love your wives and don't treat them harshly.\n1Peter 3:7 You husbands, likewise, conduct your married lives with understanding. Although your wife may be weaker physically, you should respect her as a fellow-heir of the gift of Life. If you don't, your prayers will be blocked.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On one level, my 2017 Year of Making Things Nice failed dismally.\nOne of the aims was to generate content for this blog. \"I will produce a suite of before\/after posts, generously imbued with beautiful photos of process and procedure. Blog traffic will steadily flow in, comments will be positive, questions will be asked, and the weekly stats will hit a record high for my little corner of the e-world.\" That, I believe, was January's internal monologue.\nAlas, there were only 11 posts on this topic \u2013 12 if you count this one. It was not my best year. The record belongs to the 28 posts of the Year of Efficient Eating, my first year of blogging back in 2012. Novelty is a strong motivator.\nUnfortunately, these 11 posts were not particularly inspirational either. I can't argue for quality over quantity. The stated aim for this year defined 'nice' in terms of potential \u2013 taking something that was not living up to it and making sure it did. It seems I didn't apply that to my blog.\nDid I make anything nice?\nYes, as it turned out, lack of blog posts aside, Making Things Nice was a useful mantra for 2017 when I moved again into a home of my own.\nIt's an old house, with a lot of its old charm intact. The floor layout is of its time. The kitchen has to be seen to be believed. The west-facing enclosed rear verandah is not insulated. The shower-over-bath arrangement is a little dangerous. There is a list.\nIn settling in and working out how to make these spaces work, I took the opportunity to think about how I get things done. A little light reflection on Shuffle Days and Whirlwinds was supplemented with a new strategy I heard on the radio \u2013 Kanban or, as I prefer to call it, Times Threes.\nUnderstanding and documenting overarching process was the blogging outcome rather than detailed step-by-step instructions of specific projects.\nIs the format for Year Of changing?\nThere are some rules for my Year Of format, and they deliberately exclude detailed plans, timelines and ticking off milestones towards a goal. The format is supposed to be exploratory and reactive. Change will not always run smoothly. Direction will twist and turn.\nHowever, if the aim is to also generate interesting and informative blog posts, perhaps I should take note of the numerous how-to-blog articles that recommend a bit of planning and structure.\nPerhaps the centre path will win the day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Muncy Historical Society \u2013 Spend a Weekend in Historic Muncy!\nMUNCY \u2013 Time to shop for the holidays and unique art is always a great choice! Muncy Historical Society's annual juried art exhibition features some of the most talented artists in Pennsylvania. The show will be a rich and varied collection of original work along with signed and numbered limited-edition reproductions, as well as photography, oil painting, jewelry, fashion, pottery, metal- and wood-working, quilting, mosaics, watercolors, sculpture, stained glass and home d\u00e9cor.\nThe 12th annual invitational Artists Exhibition and Sale will be held from 5:30 to 9 p.m. Friday, Nov. 9, and from 9 a.m. until 3 p.m. Saturday, Nov. 10. This year's show will be held at the First United Methodist Church, 206 S. Market Street, Muncy, its new location.\nFriday night's opening includes live music by members of the Williamsport Symphony Youth Orchestra, and light refreshments will be available both Friday night and Saturday. Admission is $10 per person. The artists will be available during both Friday and Saturday hours to meet and greet show attendees.\nRon Cimini, chairman of the board of the First Community Foundation Partnership of Pennsylvania, congratulates William Poulton, president and executive director of the Muncy Historical Society, who accepted a First Community Foundation award on behalf of the society.\nMuncy Historical Society received a grant from the Waldron Memorial Fund, of the First Community Foundation Partnership of Pennsylvania. William Poulton, president and executive director of the Historical Society, represented the community-focused organization at the First Community Foundation award ceremony.\nThis year's opening tournament of the Muncy Cup at the White Deer Golf Complex will take place at 8:30 a.m. Saturday, April 28, 2018.\nThe event is a 4-person scramble team, with prizes paid in two (2) flights based upon 15 teams or more. Player Registration Fee is $80 per golfer ($320 per team), and includes Green fees, Cart rental, Coffee & Donuts in the morning, Lunch in the afternoon, and a \"Goodie\" bag.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"In conjunction with National Diabetes Month and World Diabetes Day on 14th November, we thought it would be informative and interesting to shed some light on diabetes. So here's Part 1, of our diabetes awareness series.\nWe've all heard of it, and most of us already know it relates to having too much sugar in our blood. But how many of us really know what it is, why it happens and what puts us at risk? Did you know, people can often have type 2 diabetes for years without ever realising it? In fact, it's a lot more common that most people know.\nDiabetes is a lifelong condition that disrupts the natural production and use of insulin in our body, a type of hormone that converts sugar, starches and certain food products into the energy source we need in our daily lives. Due to this disruption of insulin efficiency, the sugar continues to remain in a patient's blood stream where it over accumulates and disrupts the body in various ways.\nIf you or a friend suspects you might have diabetes, do get yourselves checked as soon as possible! Guardian Malaysia offers blood glucose checks done by trained pharmacists for only RM5. All you have to do is walk into one of our pharmacy outlets. Locate one here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A Realtor in Pembroke Pines that WOWs you.\nI am not going to tell you what to look into a Realtor in Pembroke Pines, I am just going to tell you a little of my story.\nMy days started in a small town close to Buenos Aires, Argentina called Zarate.\nTwo years after my birth I did my first international move.\nSet course to S\u00e3o Paulo Brazil. Enjoyed the warmth of Brazilian culture until it was time to go back to Argentina.\nIt was not easy to adapt but after a few years i stopped crying every night.\nI met my wonderful wife and the journey started.\nSince then we had 3 beautiful children and lived in Mexico before relocating to become a Realtor in Pembroke Pines.\nIn terms of business I sell homes for the best price.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Immediate payment of AU $109.95 is required.\nSmaller, thinner and lighter than most 120W folding panels on the market, which was important to me. Quality is better than I expected for the price - good quality cable and bag: so-so latches hinges and stand. I bypassed the built-in regulator (using the included anderson style plugs) because I wanted to feed the output directly into my caravan's solar charger, so I can't comment on how well the regulator works. Although, it is worth noting that the regulator is not glued directly on to the back of the panels. It is mounted so that there is a small air gap between it and the panels to assist cooling. However, I can't say whether it would be enough to stop the regulator from frying in hot weather. In my configuration, the panels provide about 8 amps @ 13.8 volts in full sun, which appears to be about right for 120 watt panels.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gilcrest Place is one of the Paternoster guest houses and provides accommodation in 3 guest rooms for tourists visiting the Cape West Coast region.\nThis charming guesthouse is situated 200m from the beach in the quaint West Coast fishing village of Paternoster, an easy 90 minute drive from Cape Town.\nUninterrupted views of the ocean can be enjoyed from all guest suites, as well as from the breakfast room and lounge area. The accent is on 'relax and unwind' and a welcoming and laid back atmosphere is created by the light, spacious, airy rooms offering guests all modern amenities.\nUpstairs spacious room with own entrance, private balcony and magnificent sea views. Queen size bed, DStv, mini-bar fridge, tea and coffee station, hairdryer, walk in cupboard, large bathroom with bath, shower and complimentary toiletries and slippers.\nDownstairs room with own entrance and sea views. Queen size bed, charming Victorian bathroom with bath, shower and complimentary toiletries, DStv, mini bar fridge, hairdryer and slippers.\nUpstairs room with own entrance, private balcony and sea views. Queen size bed, DStv, mini bar fridge, tea and coffee station, hairdryer, slippers, bathroom with shower and complimentary toiletries.\nResponse timeGilcrest Place typically responds to requests in 5-10 minutes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Probashbangla24 online Desk : A mass grave containing 75 bodies has been found near the former jihadist bastion of Sirte in western Libya, an official said Wednesday.\nMohamad al-Amial, the coastal city's municipal council spokesman, told AFP the mass grave was uncovered in an agricultural area of Al-Daheir, west of Sirte.\n\"Seventy-five decomposed bodies were recovered\" in the presence of the public prosecutor, Sirte's security force and the Red Crescent, he said.\nAmial said the find was made \"a few days ago\" and that the bodies were believed to be of Islamic State (IS) group members, although there was no confirmation.\nIS jihadists overran Sirte, 600 kilometres (375 miles) east of Tripoli, in June 2015.\nForces of Libya's UN-backed unity government expelled IS from the coastal city in December 2016 after eight months of deadly combat. The group has since pulled back to desert outposts.\nIt claimed responsibility for a suicide attack last month on the Tripoli headquarters of Libya's National Oil Corporation that killed two staffers as well as an assault in May on the country's electoral commission in which 14 people died.\nIS took advantage of the anarchy in Libya after the NATO-backed overthrow and killing of longtime dictator Moamer al-Kadhafi in 2011 to establish footholds in several parts of the country.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbohyg b/data_all_eng_slimpj/shuffled/split2/finalzzzbohyg
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index 0000000000000000000000000000000000000000..e78afd4f99b859f069ec07cf07c7bcbf9f1dc30c
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+{"text":"Sheep in the road near Musbury prompted police to issue a warning to motorists this morning (Monday, March 18).\nAccording to officers, there were reports of lambs on the A358 Seaton road between Axminster and Colyton.\nAt around 7.10am, Police advised motorists in the area to 'approach with care'.\nOfficers said at 7.50am that the sheep were no longer in the road.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Correction: the minutes of the 10-1-18 meeting should have stated that Kate Cole would ask Chris Frye about a prize.\nEfforts will be made toward adding information about this task force to the Flint Neighborhoods United (FNU) website. We discussed prizes for kids' classrooms, in regard to the \"Name the Campaign\" effort. Theresa suggested we try to come up with a name for the campaign ourselves.\nCade distributed a handout and told us about Grand Rapids' efforts to improve traffic safety, in conjunction with the Michigan Department of Transportation. Grand Rapids currently has a project, including signs, for bicycle safety education and will be adding signs and etc. for pedestrian safety. It was suggested that we use their signs and no one disagreed. Ideas for posters and murals were also brought up; possible participants could be Southwestern Academy students and\/or south side mural artists.\nRaul Vargas said he is caught up with data entry for the surveys; 88 have been turned in. A few surveys mentioned street light outages and\/or potholes and others did not name a particular trouble spot. The bright red spots on the map are where fatalities took place. There was a brief discussion regarding whether others can or should be able to access the map of survey results.\nCade stated that the third training session, \"Planning and Designing Safe Streets,\" of the \"Road to Zero\" project from the organization America Walks will be tomorrow, 11-6-18, and the last two sessions are 11-27-18 and 12-11-18. We are able to participate in this because of a grant.\nTheresa submitted a micro-grant application for $1500 to America Walks for this task force. If we receive it, we will have $3000 and permanent installations, such as a crosswalk sensor setup, would be possible. Per Theresa, the Crim Foundation has obtained a Ruth Mott Foundation grant to work with three north Flint neighborhood groups; data collection will take place. However, some areas are not represented. Cade said that he learned from the training sessions that we have more citizen support than other participating cities but less support from our city government.\nKathy asked about speed limits on streets where no signs are posted. Chris replied that a city ordinance requires that residential streets almost always be 25 mph. Cade stated that adding speed limit signs might be part of the intervention for that area. Nancy shared that the speed limit sign [25 mph] in front of her house doesn't appear to do any good, since many drivers speed by.\nThe next meeting will be Monday, December 3rd at the Neighborhood Engagement Hub. Today's meeting ended at 5:00 p.m.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sole Boys believes that healthy, happy guys make great, thoughtful leaders and stronger communities!\nGuys who live healthy, active, thoughtful lives who respect, encourage, include and support each other to accomplish goals, understand their feelings and embrace their inner awesomeness!\nSupport each other and girls in our communities in an inclusive group. Help each other build confidence.\nWe love changing lives. We do that by listening, and building a community of self-love. When we love ourselves we can be love others more thoughtfully.\nEncourage healthy lifestyles, habits and goal setting. Have A LOT of fun!!\nSole Boys participants are boys ages 5-11 of all running abilities.\nThey are encouraged to run, walk, skip or hop (whatever they feel comfortable doing).\nThrough the 8 week running training, they are motivated, and supported to build their confidence, and self esteem. They will learn goal setting and achievement, healthy living habits, and how to deal with peer challenges, media, self-doubt and lots more!\nBuild self confidence, realize a goal, and be mentally and physically prepared to participate in a 5km fun run at the culmination of the 8 weeks.\nIntroduction to the \"Topic of the day\"\nActivity related to the \"Topic\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This edition of \"Fantasy Review\" covers California Bones by Greg van Eekhout. It is the first in the series, followed by Pacific Fire, and Dragon Coast due out September 15, 2015. Slight spoilers will follow.\nIn an alternate universe Los Angeles, Daniel Blackland is the son of a powerful magic user, Sebastian. However, to cut down a rival, the Kingdom's ruler, the Hierarch, kills Sebastian and cannibalizes him right before his son's eyes, absorbing the magic embedded inside Sebastian's bones. Daniel escapes, and years later, is brought in to steal his father's most precious artifact, a sword made out of Daniel's own magical essence. Daniel puts together a crew of his friends, and someone on the inside of the Hierarch's organization. But Daniel has a hard time trusting someone he doesn't know. Could this insider ruin the heist of the century and send Daniel into the clutches of the Hierarch?\nThis story is about so much more than a heist. While Daniel\u2014growing up on the streets, orphaned\u2014needs to steal for a living, that is only one thread of the plot. More important is Daniel himself, his innate magic, what he gathers for the robbery, and his relations to the people around him. Daniel is a very deep, well-round character with clear motivations, strengths, and weaknesses. He's never been obsessed with vengeance for his father's murderer, showing that he's smart enough to realize the rules of the world. If he sought vengeance, he'd be dead by now. Instead, living is how he gets back at the Hierarch.\nThe other point of view character is Gabriel Argent, the Heirarch's great-nephew, and a bureaucratic worker in the government. His family connections are more a hindrance than a help\u2014Gabriel's own mother was eaten by the Heirarch during the Third Correction, the same event that lead to Sebastian Blackland's death. Gabriel is clever, keeping quiet and unnoticeable within the government, until certain circumstances lead him to the knowledge that Daniel is alive. He goes to higher-ups with his discovery and suddenly becomes a target himself.\nGabriel is fascinating. The story could have easily been told with him as the hero. His view of the world\u2014his morals of working within the system\u2014differ greatly from Daniel, but the two have very similar motivations: only wanting to live and work in peace.\nMr. van Eekhout's writing style is quick and a tad gritty. It needs to be when it details a society that is ruled by a cannibal. But it's expertly done, adding to the overall atmosphere of the Los Angeles setting. The magic system, osteomancy, was refreshing; a new way for power to be used, stolen, and processed. This book barely scratched the surface of what osteomancy can do, and I hope it's explored more as the series progresses.\nDaniel's crew\u2014Moth, a healer, Cassandra, a good shot and lock pick, and Jo, a shapeshifter\u2014are very important to the plot line of stealing the sword. Yet, they could have been more developed as characters. Part of what makes them this way is Daniel's own magic; he makes them loyal to him. So, it could be seen that they only act the way he wants them to. Still, I would have liked to know more about them and their backstories\u2014especially more about Moth's abilities, and how Jo got hers.\nThis story kept me eagerly reading; it was difficult to put down. I'm really excited to start the sequel and find out how much more the magic system will be explored.\nCalifornia Bones can be purchased from the Dragonmount eBook store, and so can the sequel, Pacific Fire.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"outdoor shower kit surprising doors enclosures standard cedar enclosure portable showers for camping.\nbest shower cam portable outdoor showers camping gear handheld adjustable head with cigarette lighter car for enclosure showe.\noutdoor water heater enclosure portable shower showers deck encl.\noutdoor shower screen designs long island ideas portable s.\nportable outside shower outdoor tent changing steel pole dressing toilet stall ou.\nportable outdoor shower how to build enclosure camping.\nportable outdoor shower walmart best camping.\nportable outdoor shower stall solid eucalyptus wood.\nportable water heater here we go for an outdoor shower deck.\nportable outside shower best outdoor ideas images on showers must haves for the backyard stall s.\nportable outdoor shower tent changing warm thickening dressing bath cover simple toilet second open room blue outside showe.\nportable outdoor shower screen pros and cons of an photo by mark ideas out.\nportable outdoor shower south ideas designs and decors with hot water.\nhow portable outdoor shower deck to make an kit lovely best showers images on of.\noutdoor shower ideas enclosure best on portable pictures deck id.\nshowers portable outdoor shower home depot head designs tripod stand enclosures bathroom s pool tower poolside v.\noutdoor shower camping portable travel deck.\nportable outdoor shower outside enclosure stall portab.\ntraditional portable outdoor shower walmart.\nportable outdoor shower tent toilet bath changing fitting room privacy camping one person stall.\npowered portable outdoor shower outside stall.\ncamping shower fabulous portable full size of outdoor free standing seats outside stall fu.\npop up toilet tents silver coated portable outdoor shower tent photography with function in from sports entertainment on sho.\nportable outdoor shower enclosure heaters camping heater a bag outside desig.\npool bathroom ideas outdoor shower and toilet large size of bathrooms for pools superior portable outside stall t.\noutdoor shower ures portable ure camping cape cod ideas for kits vinyl enclosure.\nportable outdoor shower w foot washer ideas.\nportable outdoor shower enclosure table for camping showers knotty cedar boards.\nportable outdoor shower bag with hot water.\nportable outdoor shower walmart showers.\nportable outdoor camping shower water heater bag kit showers by young in solar gas heat with hot beach stone showe.\nportable outdoor showers for modern backyard designs shower enclosure camping.\nportable outdoor shower sensational best enclosure ideas on outside enclosures pic outs enclosu.\nportable outdoor shower designs pool and baths showers for camping.\nbeach outdoor shower portable stall.\noutdoor shower panel china portable enclosure camping.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbosvg b/data_all_eng_slimpj/shuffled/split2/finalzzzbosvg
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@@ -0,0 +1,5 @@
+{"text":"After World War II, most sea surface data were collected from ships. Then came buoys and satellites. Now, NOAA scientists want to send in the drones. \"We could be making the next epochal advancement in oceanography,\" says Craig McLean, NOAA's assistant administrator for oceanic and atmospheric research and acting chief scientist. Within the next decade, hundreds or even thousands of solar- and wind-powered drones could roam the world's oceans, using satellites to relay information gathered from the sea surface and the air above.\nThe drones can't come too soon for scientists who study the El Ni\u00f1o\u2013Southern Oscillation, a set of shifting global temperature and rainfall patterns triggered by warm surface waters that slosh back and forth across the equatorial Pacific every few years. Since the 1980s, NOAA has supported a grid of buoys, moored to the Pacific sea floor, called the Tropical Atmosphere Ocean (TAO) array, to study and forecast these swings. Its success led to similar arrays in the Indian and Atlantic oceans.\nBut earlier this decade, the TAO array had a near-death experience. The marine growth on buoys and their moorings are fish magnets, which makes them a magnet for fishers as well, causing the distinctive TAO grid to appear on maps of global fish catches. As fishers dragged the buoys aside for easy pickings, they damaged them, and maintenance work began to pile up. Meanwhile, budget cuts and the soaring expense of operating research ships meant few new buoys were deployed. \"That was a wake-up call for all of us,\" says Christopher Sabine, an oceanographer at the University of Hawaii in Honolulu.\nUltimately, Congress restored funding for the TAO array, which costs about $10 million a year. But now Japan, which maintains a complementary array in the western Pacific called the Triangle Trans-Ocean Buoy Network, has pulled out almost all of its buoys because of limited funds and a lack of ship time to maintain them, degrading El Ni\u00f1o measurements once again. The crises prompted NOAA and others to look for a more sustainable system to deliver El Ni\u00f1o warnings, which help agencies plan for the heavy rains and droughts that follow in its wake. \"It's an opportunity to look at exactly what the array is and its requirements are,\" says Meghan Cronin, an oceanographer at NOAA's Pacific Marine Environmental Laboratory (PMEL) in Seattle, Washington.\nRichard Jenkins, an engineer and Saildrone's founder, smelled an opportunity. He had built a sailboat on wheels called Greenbird that in 2009 broke the land-speed record for a wind-powered vehicle, reaching 202 kilometers per hour on a dry lake bed in Nevada. Afterward, he helped two ocean-minded philanthropists, Eric and Wendy Schmidt, outfit their research vessel, the R\/V Falkor, at a cost of $60 million. He marveled at the expense and wondered whether a smaller, seafaring version of Greenbird could gather ocean data more cheaply. In theory, an ocean-going saildrone would need no fuel for propulsion; solar panels could power instruments and communications. Simply set waypoints and the drone would ride the wind there. The Schmidts gave Jenkins $2.5 million to get started; by 2013, a test craft had completed its first voyage from California to Hawaii, propelled by a 4.6-meter-tall carbon fiber composite sail.\nSince then, Saildrone has worked with PMEL scientists to rig the boats with sensors and test their limits. In 2015, they survived 40-knot winds during a 3-month foray into the Arctic to assess marine life. That success encouraged scientists to mull whether the drones could help anchor observations in the Pacific. Conceivably, the drones could sail in circles around a virtual mooring point, or run other preplanned patterns, before returning to port every year for cleaning\u2014no ships necessary. The drones could be cheaper, too. Saildrone charges $2500 a day per drone to collect data, whereas ship time can cost $30,000 or more per day. Jenkins thinks his drones can profit off that difference in the Pacific. \"We are anticipating a fleet to service that market exclusively,\" he says.\nEventually, the saildrones caught the wind and escaped. And as Cronin looked back at the data from the cold tongues, she discovered a surprise: shifts in water temperature by 1\u00b0 in less than a kilometer. \"Some of these fronts are much sharper than you would ever imagine,\" Cronin says. \"That was shocking.\" It is the type of detail a satellite's lower resolution would smear out, and something a stationary buoy would have missed. Cronin says current climate models don't account for these sharp gradients, which could churn the atmosphere above.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On this website we recommend many pictures abaout short hairstyles for thin hair over 50 that we have collected from various sites ayrshirearts.com, and of course what we recommend is the most excellent of picture for short hairstyles for thin hair over 50. If you like the picture on our website, please do not hesitate to visit again and get inspiration from our website.\nSimilar Keyword short hairstyles for thin hair over 50 : best hairstyle for fine thin hair over 50, hairstyle for fine thin hair over 50, short bob hairstyles for fine hair over 50, short bob hairstyles for thin hair over 50, short curly hairstyles for thin hair over 50, short haircuts for fine thin hair over 50, short haircuts for round faces and thin hair over 50, short haircuts thin hair over 50, short hairstyles for fine curly hair over 50, short hairstyles for fine hair over 50 african american, short hairstyles for fine hair over 50 round face, short hairstyles for fine thin hair over 50, short hairstyles for fine thin straight hair over 50, short hairstyles for thin hair over 50 2017, short hairstyles for thin hair over 50 african american, short hairstyles for thin hair over 50 round face, short hairstyles for thin hair over 50 with glasses, short hairstyles thin hair over 50, short layered haircuts for thin hair over 50, short length hairstyles for fine hair over 50, short length hairstyles for thin hair over 50, and more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Objective: On-road driving before gaining a valid license (prelicense driving) represents a risk for all road users. Prelicense driving among young people who obtained a provisional license within an enhanced graduated driver licensing program in Queensland, Australia, was investigated.Methods: Recently licensed drivers (n = 1032) aged 17 to 19 years (M = 17.54) completed a survey exploring their driving experiences while on their learner's license. Six months later, 355 of these drivers completed the same survey exploring their experiences on their provisional (intermediate) license.Results: Twelve percent of participants reported prelicense driving. Prelicense drivers reported significantly more risky driving as learners and provisional drivers.Conclusions: Prelicense drivers not only place themselves and other road users at risk at the time but also continue to do so through their subsequent risky driving. Prelicense driving should be discouraged, and parents should be encouraged to monitor car use and the driving behavior of their children.\nCopyright \u00a9 2012 Taylor & Francis Inc. Reproduced with permission. This is an electronic version of an article published in Traffic Injury Prevention, 13:3. 213-218, 2012. Traffic Injury Prevention is available online at: www.tandfonline.com\/10.1080\/15389588.2011.638683.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"3. How did you first get into car audio? Got my first truck and had to have a \"system\"\n26. Whats one thing that most ppl on SSA dont know about you? ... a lot?\n3. How did you first get into car audio? Installed a HU, amps, front coaxials, 6x9s & a sub into my first car, before I could drive it \"legally\"\n21. Whats your favorite saying? Can't teach smart.\nya dude those are bad as hell maybe if i ever go to college i will be able to get one so they better keep makin them so i can have a new one in like 2020 lol, but check them out heres the site.\non line \/ Local shop ???\n21. Whats your favorite saying? dont have one?\n33. SQ or SPL? why? sq. i like listening to music...and dont give me that sql\/lsq crap.\n2. When did you get into car audio?------------------------bout 3 years ago?\n3. How did you first get into car audio?--------------------i bought speekas!\n29. Whats is the most recent thing you have purchased?----Fi BTL 15, 60sqf eDead v1SE2, 1gl eDead v3, 2 RE\/USAmps 1\/0 guage fuse holders, 20ft 1\/0 knu Fleks wire, 20ft 8Ga Knu Purple power wire, Tenzo R GT-6 18\" rims w\/ Nitto Neo gens, & box planes.(this has been within the last 2 weeks.\n33. SQ or SPL? why? sq. i like listening to music...and dont give me that LSQ\/lsq crap.\n29. Whats is the most recent thing you have purchased? SSA Icon 10\"\n31. Who do you think is the most knowledgeable person on SSA? I dont correlate knowledge with screen names i dont even really pay much attention to whom says what ijust read the comments and thats about it.\n33. SQ or SPL? why? little of both, i cant stand straight SPL setups as i hate the none existant linearity, but i hate how most people in SQ are quick to dysect what is or isnt SQ.\n15. Who is our hero in the car audio industry? I'd say Jacob at Sundown, and Mark and Aaron here at SSA for getting these great companies going!\n21. Whats your favorite saying? \"F*** it\"\n29. Whats is the most recent thing you have purchased? Car audio related? My ICON!\nlol nah jk dude. i AM asian tho.\nfor a second, i thought... u almost had me. omg i was kinda scared of saying something lol. that is funny tho. and im william hung lol. but yea im asian too.\nBelieve it or not, but I am asian too! Not full though, just barely. I am one quarter Korean, or Quartorean for short!\nbelieve it or not, but i am white too! not full though, just barely, i am 1\/4 white, or i dont even know what to call it. i had a friend thats half and we call him halfling. but yea i look full asian but if you look just right in the light, my beard, theres a spot thats kinda red. i wanna say that the white is irish i really dont know lol just throwing something out there.\n3. How did you first get into car audio? --- bought a crappy 200 watt kicker amp and a CVR kicker 12\"\n11. Were do you live? ---- TEXAS!!!\n21. Whats your favorite saying? --- Go To Church Ho!\n33. SQ or SPL? why?------ SPL all the way, I'm a warrior at it.\n3. How did you first get into car audio? Went to a car show and heard a truck with 4 12\" team RF subs in a blowthrough S-10 Fell in love.\n11. Were do you live? Dallas Texas baby! YEEHAW!\n29. Whats is the most recent thing you have purchased? Computer?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the stills, Dhanush can be seen spending some quality time with his onscreen wife Amala Paul.\nIn a development that is bound to send Dhanush's fans into a state of frenzy, some stills from his eagerly-awaited 'VIP 2' are going viral and that too for an awesome reason. In them, the 34-year-old star can be seen spending some quality time with his onscreen wife Amala Paul. Needless to say, the two look good together and impress us with their crackling chemistry. Based on these stills it can safely be said that their track will be a major highlight of 'VIP 2'.\nIn case you did not know, 'VIP 2' is touted to be an action-comedy and has been directed by Soundarya Rajinikanth. A sequel to the 2014 release 'VIP', it was originally supposed to hit screens last week. However, it got delayed because of some unforeseen circumstances. It will now hit screens in August. Besides Dhanush and Amala Paul, 'VIP 2', also features Bollywood's Kajol in the lead. She will playing the role of a ruthless businesswoman in this one.\nInterestingly, 'VIP 2' marks Kajol's return to Tamil films after a hiatus of 20 years. She was last seen in the musical-drama 'Mansara Kanavu' which hit screens in 1997 and created a buzz amongst fans because of AR Rahman's lifting tunes. Sometime ago, Kajol had said that she was initially reluctant to take up VIP 2 but Dhanush helped her change her mind.\nSo, did you like Dhanush and Amala's chemistry in these stills? Comments, please!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzboull b/data_all_eng_slimpj/shuffled/split2/finalzzzboull
new file mode 100644
index 0000000000000000000000000000000000000000..5b508a1613a0c9878860b145ebf02ce8aded79c2
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Since 1906, New Balance has maintained its low-profile approach to making high-quality, hand-crafted footwear. Its modest objective to maintain quality builds for comfort and arch support created a legacy that's upheld for generations. The Boston-bred sneaker brand takes pride in maintaining six American factories, responsible for constructing its most popular kicks like the 574 and 999. The minimalist silhouettes often allow for unique colorways and creative collaborations.\nBoston-based New Balance has joined forces with KITH's Ronnie Fieg for a limited-edition New Balance..\nNew Balance 574 improves upon an already successful mild stability running shoe. These trainers feat..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I dag vil jeg vise fram ett konfirmasjonskort til en gutt.\nIkke noe problem \u00e5 lage n\u00e5r man bruker disse flotte arkene fra Kort og Godt.\nToday I will show you a confirmationscard for a boy.\nNo problem to made when you have this gorgeous papers from Kort og Godt.\nEasy and simply work !\nHele kortet er laget av ett ark.\nI have used one DP on this card.\nDet eneste jeg har gjort er \u00e5 stemple hj\u00f8rnene p\u00e5 h\u00f8yre side.\nThe only thing I have done, is to stamp the cornes on the right side.\n\u00d8nsker deg ei flott helg !\nWOW! This is fabulous Berit! Thanks so much for joining us at The Sisterhood 'Free For All (anything goes)' challenge and please come back again soon!\nBeautiful work. Thank you for joining us at Eclectic Ellapu this week.\nWhat an outstanding image and card.\nWe love seeing what you have created and posted with us this week. Thank you!\nVERY cool masculine card, Berit. Your creativity is always so impressive, and I thank you for sharing with us again this week.\nWonderful male themed card. Thank you for sharing your project with us at PCJ Challenge.\nWOW this is a fabulous male card. Thank you for joining us over at Challenge 4 Everybody.\nSuper card Berit, thanks for playing along @ MAWTT this week.\nCute men card. I love comboand embelishments. Thanks for playing with us at Fussy & Fancy challenge.\nUtrolig fint kort! Tusen takk for at du deltar hos Kort o Mania :) Hilsen Kirsten.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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